$$H_{\infty}$$ based anti-windup controller for two-dimensional discrete delayed systems in presence of actuator saturation

$$H_{\infty}$$ based anti-windup controller for two-dimensional discrete delayed systems in... This article addresses the problem of stability analysis for a class of two-dimensional discrete systems described by Roesser model in the presence of actuator saturation and interval-like time varying state delay. The dynamic output feedback $$H_{\infty}$$ stabilization controller appended with an anti-windup compensator is used. The saturation nonlinearity is tackled using sector conditions. Sufficient conditions for asymptotic stability in terms of linear matrix inequality (LMI) are derived using Lyapunov–Krasovskii functional and the domain of attraction is also estimated for the system under consideration. Some examples are provided to illustrate the effectiveness of the proposed approach. 1. Introduction The study of two-dimensional systems is gaining great momentum due to real time applications in many areas such as digital filtering, image processing, signal processing (Bose, 1979; Benzaouia et al., 2016; Chen, 2015; Dey et al., 2012b; Fornasini, 1991; Kaczorek, 1985; Ooba, 2013; Roesser, 1975), industrial process control (paper manufacturing industry, plastic film extrusion, steel sheet production and others) (Zarrop & Wellstead, 2002), iterative learning control, repetitive process control (Liu & Gao, 2010), electricity transmission (Tong et al., 2014), energy exchange process, river pollution and dynamic flood modelling (Costabile & Macchione, 2015), grid based wireless sensor networks (Stefanski, 2014; Sumanasena & Bauer, 2011), medical treatment (Cifor et al., 2013), geophysics (Jeshvaghani & Darijani, 2014), robotics (Kim et al., 2014), heat and mass transfer (Masmoudi et al., 2014), optical fibre network (Han et al., 2015), magnetics (Ueberschar et al., 2015), photovoltaic applications (Koduvelikulathu et al., 2015) and many more. In view of multidimensional modelling, several case studies for practical systems like sensor networks, robot manipulator in batch processing etc. have been addressed in (Rogers et al., 2015). It is known that modelling of two-dimensional systems is more difficult than one-dimensional but two-dimensional systems contain more realistic information about the practical systems. The control problems like that of batch process, thermal systems can easily be described by two-dimensional state space models which inherently contain the transportation and computational delays (Benhayoun et al., 2010; Benzaouia et al., 2011; Chen & Yu , 2013; Dey & Kar, 2014). Such type of delays create instability in the systems and are responsible for poor performance (Chen, 2009; Fridman et al., 2003; Gu et al., 2013; Gorecki et al., 1989; Haidar et al., 2009; Kandanvli & Kar, 2013; Niculescu, 2001; Song et al., 2014; Wang et al., 2013; Zhou et al., 2013) The stability analysis of two-dimensional delayed systems becomes more complicated due to addition of these delay terms in the model of two-dimensional system (Paszke et al., 2003, 2004; Malek-Zavarei et al., 1987). The stabilization of two-dimensional systems subjected to time varying state delays and input nonlinearities is a very challenging work. According to dependence of delay, the available stability criteria for delayed systems can be broadly classified into two categories: delay-dependent and delay-independent. The sufficient conditions for asymptotic stability of two-dimensional systems with state saturation and delay have been derived in Chen (2010a,b); Chen & Yu (2013); Dey & Kar (2012a) using delay dependent approach while delay independent stability criteria are applied in Chen & Fong (2006); Paszke et al. (2003, 2004); Peng & Guan (2009b). Stability conditions derived by delay dependent approach are less conservative because they utilize the information about the size of delay. State feedback controller has been used for stabilization of two-dimensional continuous systems with multi-delays for saturated control in Benzaouia et al. (2011); Benhayoun et al. (2010). Further improvements and modifications in the controller design for two-dimensional systems when subjected to interval like time varying delays and input nonlinearities are reported in Ghous & Xiang (2015); Ghous et al. (2015); Huang et al. (2013); Tadepalli et al. (2015). Saturation is a very common nonlinearity present in control systems due to physical limitation of actuator. The presence of actuator saturation tends the system to the verge of instability and degrades the performance of closed loop system. An anti-windup technique has been widely studied for continuous and discrete systems in presence of actuator saturation nonlinearity and is found to be practical and effective (Bender et al., 2011; Gomes da Silva Jr. et al., 2013; Mesquine et al., 2010; Negi et al., 2012a,b). The problem of actuator saturation has not been studied much for two-dimensional systems and recently it has been addressed in few publications (Benhayoun et al., 2013; Gao & Wang, 2014; Ghous & Xiang, 2015; Ghous et al., 2015; Hmamed et al., 2010; Huang et al., 2013). A state feedback controller has been designed in (Hmamed et al., 2010) for stabilization of two-dimensional saturated continuous systems by using quadratic Lyapunov function. In Gao & Wang (2014), Takagi and Sugeno fuzzy model is employed for two-dimensional nonlinear dynamic system with actuator saturation. $$H_{\infty }$$ controller is used quite often to stabilize two-dimensional discrete systems having nonlinearities and state delays (Huang & Xiang, 2014; Liang et al., 2015; Peng & Guan, 2009a; Xu et al., 2005; Xu & Yu, 2006, 2009). The stability conditions for nonlinear two-dimensional stochastic systems represented by Roesser model with time varying state delay and actuator saturation via state feedback controller are reported in Ghous & Xiang (2015), Huang et al. (2013) and the same case for Fornasini and Marchesini state-space model has been reported in Ghous et al. (2015). The $$H_{\infty}$$ controller is designed for two-dimensional uncertain model for batch process with interval like time varying delay in Wang et al. (2013a,b). The stabilization of two-dimensional discrete systems using $$H_{\infty }$$ based state feedback controller has been reported in Gao & Wang (2014); Ghous & Xiang (2015); Ghous et al. (2015); Huang et al. (2013) in presence of actuator saturation and time varying delay where saturation nonlinearity is described using convex hull. The objective of this article is to obtain stability criterion for two-dimensional discrete system described by Roesser model in presence of input saturation, external disturbances and time varying state delays. Among the available techniques anti-windup gives better results to counter the effect of actuator saturation nonlinearity in comparison to other methods (Gomes da Silva Jr. & Tarbouriech 2006; Mulder et al. 2001, Syaichu-Rohman & Middleton 2004). In the present work, using sector based description of saturated nonlinearity, a technique to compute static anti-windup gain for two-dimensional dynamic compensator for two-dimensional discrete systems described by Roesser model with time varying state delay and disturbances is given. A Lyapunov–Krasovskii functional is used to get the stability condition in terms of LMI and the anti-windup gains are determined such that closed loop system is asymptotically stable. To the best of authors’ knowledge, anti-windup approach has not been reported so far to tackle the problem of actuator saturation in two-dimensional discrete systems with interval like time varying state delays and disturbances. The article is organized as follows. In Section 2, problem is formulated and some necessary definition and lemmas are recalled. The main result is presented in Section 3. Section 4 presents the effectiveness of proposed approach with numerical examples. Notation: throughout this article, $$\boldsymbol{\Re }^{m\times n}$$ represents set of $$m\times n$$ real matrices; $$\boldsymbol{\Re }^{m}$$ represents set of $$m\times 1$$ real matrices; $$\boldsymbol{I}$$ is identity matrix of appropriate dimension; $$\mathbf{0}$$ denotes null matrix or null vector; $$\boldsymbol{{\it\Omega}}^{T}$$ represents transpose matrix of $$\boldsymbol{{\it\Omega} }$$; $$\lambda_{max} ({\it\Omega} )$$ stands for maximum eigenvalue of any given matrix $$\boldsymbol{{\it\Omega} }$$; $$diag\left\{ {a_{1}, a_{2}, \ldots,a_{n} } \right\}$$ is diagonal matrix with diagonal elements $$a_{1}, a_{2}, \ldots,a_{n} $$; $$\boldsymbol{E} = {\bf E}_{1} \oplus \boldsymbol{E}_{2} $$ denotes direct sum i.e. ${\bf E} = \left[\begin{array}{@{}cc@{}} \boldsymbol{E}_{1} & \mathbf{0}\\ \mathbf{0} & \boldsymbol{E}_{2} \end{array} \right]$; $$\left\| \boldsymbol{W} \right\|_{2} =\sqrt {\sum_{i=0}^\infty {\sum_{j=0}^\infty {\left\| \boldsymbol{W}(i,j) \right\|^{2}} } }$$ is $$l_{2} $$ norm of two-dimensional signal $$\boldsymbol{W}(i,j)\in l_{2} \left\{ {\left[{0,\infty } \right),\left[ {0,\infty } \right)} \right\}$$ if $$\left\| \boldsymbol{W} \right\|_{2} <\infty $$; symbol $$\ast $$ is used to represents the symmetric terms in symmetric matrix; $$\left\|. \right\|$$ stands for Euclidean norm. Matrices, if not explicitly mentioned, are assumed to have compatible dimensions. 2. Problem formulation and preliminaries Consider the following two-dimensional discrete system described by Roesser model in presence of actuator saturation, time varying state delay and disturbance (Benhayoun et al., 2013; Chen, 2010b; Huang et al., 2013; Roesser, 1975)   [xh(i+1,j)xv(i,j+1)] =Ap[xh(i,j)xv(i,j)]+Adp[xh(i−dh(i),j)xv(i,j−dv(j))]+B¯w[wh(i,j)wv(i,j)]+Bpu(i,j), 1a  y(i,j) =Cp[xh(i,j)xv(i,j)], 1b  z(i,j) =C¯z[xh(i,j)xv(i,j)]+D¯z[wh(i,j)wv(i,j)], 1c where $$i\in z_{+} $$, $$j\in z_{+} $$ and $$z_{+} $$ denotes the set of nonnegative integers. The $$\boldsymbol{x}^{h}(i,j)\;\in \boldsymbol{\Re}^{n}$$ and $$\boldsymbol{x}^{v}(i,j)\;\in \boldsymbol{\Re}^{m}$$ are the horizontal and the vertical states, respectively. The $$\boldsymbol{u}(i,j)\;\in \boldsymbol{\Re }^{p}$$ is input vector while $$\boldsymbol{y}(i,j)\;\in \boldsymbol{\Re }^{q}$$ and $$\boldsymbol{z}(i,j)\;\in \boldsymbol{\Re }^{t}$$ are measured output and controlled output vectors, respectively. The disturbance input in horizontal and vertical directions are $$\boldsymbol{w}^{h}(i,j)$$ and $$\boldsymbol{w}^{v}(i,j)$$. Matrices $$\boldsymbol{A}_{p} \in \boldsymbol{\Re}^{(n+m)\times (n+m)}$$, $$\boldsymbol{A}_{dp} \in \boldsymbol{\Re}^{(n+m)\times (n+m)}$$, $$\bar{\boldsymbol{B}}_{w} \in \boldsymbol{\Re}^{(n+m)\times (n+m)}$$, $$\boldsymbol{B}_{p} \;\in \;\boldsymbol{\Re}^{(n+m)\times p}$$, $$\boldsymbol{C}_{p} \;\in \;\boldsymbol{\Re}^{q\times (n+m)}$$, $$\bar{{\boldsymbol{C}}}_{z} \;\in \;\boldsymbol{\Re}^{t\times (n+m)}$$ and $$\bar{{\boldsymbol{D}}}_{z} \;\in \;\boldsymbol{\Re}^{t\times (n+m)}$$ are known constant matrices representing the nominal plant. In system (1a), $$ d_{h} (i) $$ and $$ d_{v} (j) $$ represent delays along horizontal direction and vertical directions, respectively. We assume that   dhL⩽dh(i)⩽dhH,dvL⩽dv(j)⩽dvH, 1d where $$ d_{hL} $$, $$ d_{hH} $$, $$ d_{vL} $$ and $$ d_{vH} $$ are constant nonnegative integers representing the lower and upper delay bounds along horizontal and vertical directions, respectively. It is assumed that these delays satisfy (1a). From (1a), it is clear that $$ d_{h} (i) $$ can assume any positive integer value in the interval $$ [d_{hL}, d_{hH} ] $$. Similarly, $$ d_{v} (j) $$ takes any positive integer value in the interval $$ [d_{vL} , d_{vH} ] $$. Therefore, the delays involved in system (1) are considered to be interval-like time varying state delays. It may be mentioned that such type of modelling of time varying delays has been widely used in the literature (Dey & Kar, 2014; Ghous & Xiang, 2015; Ghous et al., 2015; Luo et al., 2016; Tadepalli et al., 2015; Wang et al., 2013a). The initial conditions are defined as follows (Chen, 2010b; Tadepalli et al., 2015; Xu et al., 2013)   X(0) =[xhT(−dhH,0),xhT(−dhH,1),xhT(−dhH,2),…, xhT(1−dhH,0),xhT(1−dhH,1),xhT(1−dhH,2),…, xhT(0,0),xhT(0,1),xhT(0,2),…, xvT(0,−dvH),xvT(1,−dvH,),xvT(2,−dvH),…, xvT(0,1−dvH,),xvT(1,1−dvH),xvT(2,1−dvH),…, xvT(0,0),xvT(1,0),xvT(2,0),…]. 1e Let a linear two-dimensional dynamic compensator stabilizing the system (1) and meeting the desired performance specifications in absence of actuator saturation be given as   [xch(i+1,j)xcv(i,j+1)]=Ac[xch(i,j)xcv(i,j)]+Bcy(i,j), 2a  Vc(i,j)=Cc[xch(i,j)xcv(i,j)]+DcCp[xh(i,j)xv(i,j)], 2b where $$ \boldsymbol{x}_{c}^{h}(i,j)\;\in \boldsymbol{\Re}^{n_{c} } $$ and $$ \boldsymbol{x}_{c}^{v}(i,j)\;\in \boldsymbol{\Re}^{m_{c} } $$ are horizontal and vertical states of the controller, respectively. Vector $$ \boldsymbol{u}_{c} (i,j)=\boldsymbol{y}(i,j)\in \boldsymbol{\Re}^{q} $$ is a controller input vector and $$ \boldsymbol{V}_{c} (i,j)\in \boldsymbol{\Re}^{p} $$ is a controller output vector. The matrices $$\boldsymbol{A}_{c} \in \boldsymbol{\Re }^{(n_{c} +m_{c} )\times (n_{c} +m_{c} )},\boldsymbol{B}_{c} \in \boldsymbol{\Re }^{(n_{c} +m_{c} )\times q} $$, $$ \boldsymbol{C}_{c} \in \boldsymbol{\Re }^{p\times (n_{c} +m_{c} )} $$ and $$ \boldsymbol{D}_{c} \in \boldsymbol{\Re }^{p\times q} $$ are constant matrices of the desired controller. The input vector $$ \boldsymbol{u}(i,j)$$ is subjected to amplitude constrained defined as   −u¯(l)⩽u(l)(i,j)⩽u¯(l), 3 where $$ \overline u_{(l)} >0,l=1,2,\,{\ldots}\,,p$$ denote the control amplitude bounds. Therefore, the actual control signal injected to the system (1) can be written as   u(i,j) =sat(Vc(i,j)) =sat(Cc[xch(i,j)xcv(i,j)]+DcCp[xh(i,j)xv(i,j)]). 4 The saturation nonlinearities are characterized by   sat(Vc(i,j))(l)={−u¯(l)ifVc(l)<−u¯(l)Vc(l)if−u¯(l)⩽Vc(l)⩽u¯(l)u¯(l)ifVc(l)>u¯(l) ,l=1,2,…,p. 5 The actuator saturation causes windup of the controller and to mitigate its undesirable effect an anti-windup term $$\boldsymbol{E}_{c} (sat(\boldsymbol{V}_{c} (i,j))-\boldsymbol{V}_{c} (i,j)) $$ can be added to the controller (2) where $$ \boldsymbol{E}_{c} $$ is static anti-windup gain. Thus, modified system (1) and dynamic output controller (2) are represented by   [xh(i+1,j)xv(i,j+1)] =Ap[xh(i,j)xv(i,j)]+Adp[xh(i−dh(i),j)xv(i,j−dv(j))]+B¯w[wh(i,j)wv(i,j)] +Bp(sat(Cc[xch(i,j)xcv(i,j)]+DcCp[xh(i,j)xv(i,j)])), 6a  z(i,j) =C¯z[xh(i,j)xv(i,j)]+D¯z[wh(i,j)wv(i,j)], 6b  [xch(i+1,j)xcv(i,j+1)] =Ac[xch(i,j)xcv(i,j)]+BcCp[xh(i,j)xv(i,j)]+Ec(sat(Vc(i,j))−Vc(i,j)), 7a  Vc(i,j) =Cc[xch(i,j)xcv(i,j)]+DcCp[xh(i,j)xv(i,j)]. 7b It is noted here that $$ \boldsymbol{A}_{c} $$, $$ \boldsymbol{B}_{c} $$, $$ \boldsymbol{C}_{c}$$ and $$\boldsymbol{D}_{c} $$ are controller design parameters and are selected to meet the desired performance. Let us introduce the elementary matrix $$ \boldsymbol{{\it\Pi}} $$ as   Π=[In00000Inc00Im00000Imc], 8a where $$\boldsymbol{I}_{n}$$ is the identity matrix of order $$n$$ and $$\boldsymbol{{\it\Pi}}^{\boldsymbol{-1}}=\boldsymbol{{\it\Pi}}^{T}$$. Define an extended state vector which comprises of nominal plant and controller states   ξ(i,j)=[xh(i,j)xch(i,j)xv(i,j)xcv(i,j)]∈ℜn^+m^with n^=n+nc,  m^=m+mc. 8b Using (6), (7) and (8) the closed loop system is written as   [ξh(i+1,j)ξv(i,j+1)] =A[ξh(i,j)ξv(i,j)]+Ad[ξh(i−dh(i),j)ξv(i,j−dv(j))]+Bww(i,j)−(B+REc)ψ(K[ξh(i,j)ξv(i,j)]), 9a  z(i,j) =Cz[ξh(i,j)ξv(i,j)]+Dzw(i,j), 9b where   ξh(i+1,j) =[xhT(i+1,j)xchT(i+1,j)]T,ξv(i,j+1)=[xvT(i,j+1)xcvT(i,j+1)]T, 10  A =Π[Ap+BpDcCpBpCcBcCpAc]ΠT,B=Π[Bp0],R=Π[0Inc+mc],K=[DcCpCc]ΠT,Ad =Π[Adp000]ΠT,Bw=Π[B¯w000]ΠT,Cz=Π[C¯z000]ΠT,Dz=Π[D¯z000]ΠT,w(i,j) =[wh(i,j)0wv(i,j)0]∈ℜn^+m^,with the function ψ(Vc(i,j))=Vc(i,j)−sat(Vc(i,j)). 11 The boundary conditions for closed loop system (9) are given by   {ξh(i,j)=hij,∀ 0⩽j⩽r1,−dhH⩽i⩽0ξh(i,j)=0,∀ j>r1,−dhH⩽i⩽0ξv(i,j)=vij,∀ 0⩽i⩽r2,−dvH⩽j⩽0ξv(i,j)=0,∀ i>r2,−dvH⩽j⩽0 , 12 where $$ r_{1} $$ and $$ r_{2} $$ are finite positive integers, $$ \boldsymbol{h}_{ij} $$ and $$ \boldsymbol{v}_{ij} $$ are given vectors. It must be noted that $$ \left\| {\boldsymbol{w}(i,j)} \right\|_2 \leqslant \alpha ^{2} $$ with $$ \alpha $$ is positive constant. Remark 1 We assumed that the closed loop system (9) has a finite set of boundary conditions given by (12). These boundary conditions play a key role for deriving the asymptotic stability conditions for two-dimensional systems. With the appropriate choice of $$ r_{ 1} $$ and $$ r_{ 2} $$, it is possible to define the boundary conditions of dynamic compensator such that (12) holds. Remark 2 The two-dimensional dynamic compensator (2) is used to stabilize the system (1) in absence of saturation which, in turn, requires that the two-dimensional characteristic polynomial of the system (9) should not have any pole on or inside the unit bidisc. In other words, in absence of control bound, the closed loop system would be asymptotically stable for the boundary conditions (12). Generally, state feedback controller is best way to stabilize the control system. In some cases, all states of the system are not available for feedback due to complexity of the system, so output feedback method is preferred. In comparison to static, dynamic method of feedback is more applicable to practical systems in industries (Dong & Yang, 2009; Negi et al., 2012b; Nguyen et al., 2015; Tang et al., 2016; Wei-Wei & Guang-Hong, 2008; Zhang et al., 2014). The problem of designing dynamic compensator with a view to achieve input-to-state and input-to-output stability has been considered in Chai (2015); Gomes da Silva Jr. et al. (2013); Lam et al. (2004); Tang et al. (2016); Tarbouriech et al. (2011). Based on the methodology adopted in Gomes da Silva Jr. & Tarbouriech (2005, 2006); Negi et al. (2012b), it is assumed that the such type of dynamic compensator given in (2) exist. The main aim of this proposed work is to determine the anti-windup gains to guarantee the stability of closed loop system (9) and also to meet the performance requirement. The following definition and lemmas are needed in the proof of main result. Definition 1 (Kaczorek, 1985) The system (9) is asymptotically stable if $$ \lim_{q\to \infty } \chi_{q} =0 $$ for all given boundary conditions (12) where   χq=sup{‖ξ(i,j)‖:i+j=q,i,j⩾1},ξ(i,j)=[ξhT(i,j)ξvT(i,j)]T. 13 Lemma 1 (Qiu et al., 2008) For any constant matrix $$ \boldsymbol{W}\in \;\boldsymbol{\Re}^{m\times m} $$ with $$ \boldsymbol{W}=\boldsymbol{W}^{T}\geqslant \mathbf{0} $$, integers $$ l_{1} <l_{2} $$, vector function $$ \boldsymbol{\omega }:\left\{ {l_{1}, l_{1} +1,\ldots,l_{2} } \right\}\to \boldsymbol{\Re}^{m} $$ such that the sums concerned are well defined, then   (l2−l1+1)∑i=l1l2ωT(i)Wω(i)⩾(∑i=l1l2ω(i))TW(∑i=l1l2ω(i)). 14 Lemma 2 (Boyd et al., 1994) If there exist symmetric matrix $\boldsymbol{T}=\left[ \begin{array}{@{}cc@{}} {\boldsymbol{T}_{11} } & {\boldsymbol{T}_{12} } \\ {\boldsymbol{T}_{12}^{T} } & {\boldsymbol{T}_{22} } \\ \end{array} \right] $ with $$ \boldsymbol{T}_{11} $$ and $$ \boldsymbol{T}_{22} $$ are square matrices then the following statements are equivalent:   T<0T11<0,T22−T12TT11−1T12<0T22<0,T11−T12T22−1T12T<0. 15 Consider a matrix $$ \boldsymbol{G}\in \boldsymbol{\Re}^{p\times (\hat{{n}}+\hat{{m}})} $$ and define a polyhedral set   ℓ≜{ξ∈ℜ(n^+m^);−u¯(l)⩽(K(l)−G(l))ξ(i,j)⩽u¯(l),l=1,2,…,p}. 16 Lemma 3 If $$\boldsymbol{\xi}\in \ell$$ then   δℓ=2ψT(Kξ(i,j))D[ψ(Kξ(i,j))−Gξ(i,j)]⩽0, 17 where $$ \boldsymbol{D} $$ is positive definite diagonal matrix. Proof. The proof of Lemma 3 is very similar to Lemma 1 as given in Negi et al. (2012b); Gomes da Silva Jr. & Tarbouriech (2006). □ It is noted that the Lemma 3 has been used for the stability analysis of two-dimensional discrete system with actuator saturation in Negi et al. (2012b). 3. Main results The main results of the article are stated as follows. 3.1. Stability Analysis without disturbance Theorem 1 Consider two-dimensional system (9) with $$ \boldsymbol{w}(i,j)= $$ 0, for given positive scalars $$ d_{ha} $$ and $$ d_{va} $$ satisfying $$ d_{hL} \leqslant d_{ha} \leqslant d_{hH} $$ and $$ d_{vL} \leqslant d_{va} \leqslant d_{vH} $$, if there exist a diagonal positive definite matrix $$ \boldsymbol{L}\in \boldsymbol{\Re}^{p\times p} $$, symmetric matrices $$ \mathbf{0}<\boldsymbol{P}=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{Q}=\boldsymbol{Q}^{h}\oplus \boldsymbol{Q}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{W}_{k} =\boldsymbol{W}_{k}^{h} \oplus \boldsymbol{W}_{k}^{v} (k=1,2,3)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{R}_{k} =\boldsymbol{R}_{k}^{h} \oplus \boldsymbol{R}_{k}^{v} (k=1,2,3)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{X}_{k} (k=1,\ldots,5)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$ and matrices $$ \boldsymbol{H}\in \boldsymbol{\Re}^{(n_{c} +m_{c} )\times p} $$, $$ \boldsymbol{G}\in \boldsymbol{\Re}^{p\times (\hat{{n}}+\hat{{m}})} $$ satisfying the following set of LMIs   [Γ~11∗∗∗∗0−Q∗∗∗G0−2L∗∗R100W3−W1−R1−R2∗000R2W2−W3−R2−R30000R3AAd(−BL−RH)00AAd(−BL−RH)00d2(A−I)d2Add2(−BL−RH)00d3(A−I)d3Add3(−BL−RH)00d4(A−I)d4Add4(−BL−RH)00  ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−W2−R3∗∗∗∗∗0−2X1+X1PX1∗∗∗∗00−2X2+X2QX2∗∗∗000−2X3+X3R1X3∗∗0000−2X4+X4R2X4∗00000−2X5+X5R3X5]<0, 18  [PK(l)T−G(l)TK(l)−G(l)u0(l)2]>0,l=1,2,…,p, 19 where $$ \boldsymbol{\tilde{{\it\Gamma}}}_{11} =-\boldsymbol{P}+\boldsymbol{W}_{1} +\boldsymbol{d}_{1} \boldsymbol{Q}-\boldsymbol{R}_{1} $$, $ \boldsymbol{d}_{1} =\left[ \begin{array}{@{}cc@{}} {(d_{hH} -d_{hL} )\boldsymbol{I}_{h} } & {\mathbf{0}} \\ {\mathbf{0}} & {(d_{vH} -d_{vL} )\boldsymbol{I}_{v} } \\ \end{array} \right] $, $ \boldsymbol{d}_{2} =\left[ \begin{array}{@{}cc@{}} {d_{hH} \boldsymbol{I}_{h} } & {\mathbf{0}} \\ {\mathbf{0}} & {d_{vL} \boldsymbol{I}_{v} } \\ \end{array} \right] $,   d3=[(dha−dhL)Ih00(dva−dvL)Iv],d4=[(dhH−dha)Ih00(dvH−dva)Iv], 20 then, for the gain matrix $$ \boldsymbol{E}_{c} \;=\boldsymbol{HL}^{-\mbox{1}} $$ the ellipsoid $$ \varepsilon \;(\boldsymbol{P})=\left\{ {\boldsymbol{\xi}\in \Re ^{\hat{{n}}+\hat{{m}}};\boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant 1} \right\} $$, with $$ \boldsymbol{P}=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v} $$, is a region of asymptotic stability for the system (9). Further, an estimate of the domain of attraction for system (9) is given by   Γ(βrh,βrv) =[βrh2(λmax(Ph)+(dhH+1)λmax(Qh)+dhLλmax(W1h) +(dhH−dha)λmax(W2h)+(dha−dhL)λmax(W3h) +0.5(dhH−dhL)(dhH+dhL−1)λmax(Qh)+0.5(dhL)2(1+dhL)λmax(R1h) +0.5(dha−dhL)2(dhL+1+dha)λmax(R2h)+0.5(dhH−dha)2(dhH+dha+1)λmax(R3h)) +βrv2(λmax(Pv)+(dvH+1)λmax(Qv)+dvLλmax(W1v)+(dvH−dva)λmax(W2v) +(dva−dvL)λmax(W3v)+0.5(dvH−dvL)(dvH−1+dvL)λmax(Qv)+0.5(dvL)2(1+dvL)λmax(R1v)+0.5(dva−dvL)2(dvL+1+dva)λmax(R2v) +0.5(dvH−dva)2(dvH+dva+1)λmax(R3v))]⩽1, 21 where $$ \beta_{rh} =\max \left( {\sum\limits_{j=0}^{r_{ 1} } {\left\| {\boldsymbol{\xi}(-\sigma^{h},j)} \right\|} } \right)_{-d_{hH} \leqslant \sigma^{h}\leqslant 0} $$, $$ \beta_{rv} =\max \left( {\sum\limits_{i=0}^{r_{ 2} } {\left\| {\boldsymbol{\xi}(i,-\sigma^{v})} \right\|} } \right)_{-d_{vH} \leqslant \sigma ^{v}\leqslant 0} $$, $$ r_{1} $$ and $$ r_{2} $$ are finite positive integers. Proof. The proof of Theorem 1 is given in Appendix. □ Remark 3 For the given delay bounds $$ d_{hL} $$, $$ d_{vL} $$ and $$d_{ha}$$, $$ d_{va} $$ the bounds on delay $$ d_{hH} $$ and $$ d_{vH} $$ can be obtained by iteratively solving the LMIs of Theorem 1. 3.2. $$H_{\infty}$$ performance analysis with disturbance The performance of system (9) with sufficient condition for $$ H_{\infty } $$ disturbance attenuation is given in this subsection. Theorem 2 Consider two-dimensional system (9), for given positive scalars $$ \alpha $$, $$ \gamma $$, $$ d_{ha} $$ and $$ d_{va} $$ satisfying $$ d_{hL} \leqslant d_{ha} \leqslant d_{hH} $$ and $$ d_{vL} \leqslant d_{va} \leqslant d_{vH} $$, if there exist a diagonal positive definite matrix $$ \boldsymbol{L}\in \boldsymbol{\Re}^{p\times p} $$, symmetric matrices $$ \mathbf{0}<\boldsymbol{P}=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{Q}=\boldsymbol{Q}^{h}\oplus \boldsymbol{Q}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})}$$, $$\mathbf{0}<\boldsymbol{W}_{k} =\boldsymbol{W}_{k}^{h} \oplus \boldsymbol{W}_{k}^{v} ,(k=1,2,3)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{R}_{k} =\boldsymbol{R}_{k}^{h} \oplus \boldsymbol{R}_{k}^{v} (k=1,2,3)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})},\mathbf{0}<\boldsymbol{X}_{k} (k=1,\ldots,5)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$ and matrices $$ \boldsymbol{H}\in \boldsymbol{\Re}^{(n_{c} +m_{c} )\times p} $$, $$ \boldsymbol{G}\in \boldsymbol{\Re}^{p\times (\hat{{n}}+\hat{{m}})} $$ satisfying the following set of LMIs   [Γ~11∗∗∗∗∗∗0−Q∗∗∗∗∗G0−2L∗∗∗∗R100W3−W1−R1−R2∗∗∗000R2W2−W3−R2−R3∗∗0000R3−W2−R3∗000000−γ2IAAd(−BL−RH)000BwAAd(−BL−RH)000Bwd2(A−I)d2Add2(−BL−RH)000d2Bwd3(A−I)d3Add3(−BL−RH)000d3Bwd4(A−I)d4Add4(−BL−RH)000d4BwCz00000Dz  ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−2X1+X1PX1∗∗∗∗∗0−2X2+X2QX2∗∗∗∗00−2X3+X3R1X3∗∗∗000−2X4+X4R2X4∗∗0000−2X5+X5R3X5∗00000−I]<0, 22  [PK(l)T−G(l)TK(l)−G(l)τu0(l)2]⩾0,l=1,2,…,p, 23 where   τ=1/(1+γ2α2), 24 then, for the gain matrix $$ \boldsymbol{E}_{c} \;=\boldsymbol{HL}^{-\mbox{1}} $$ the ellipsoid $$ \varepsilon \;(\boldsymbol{P},1+\gamma^{2}\alpha^{2})=\left\{ {\boldsymbol{\xi}\in \Re^{\hat{{n}}+\hat{{m}}};\;\boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant \;1+\gamma^{2}\alpha^{2}} \right\} $$, with $$ \boldsymbol{P}\;=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v} $$, is a region of asymptotic stability for the system (9). Proof. Consider   β(i,j)=Δv(ξ(i,j))+zT(i,j)z(i,j)−γ2wT(i,j)w(i,j). 25 Adopting the procedure of proof of Theorem 1, we get   β(i,j)⩽ϑ¯T(i,j)I−bϑ¯(i,j), 26 where   ϑ¯(i,j)=[ϑT(i,j)wT(i,j)]T, 27  I−b =[ℏ11∗∗∗∗∗∗ℏ21ℏ22∗∗∗∗∗ℏ31ℏ32ℏ33∗∗∗∗R100−W1+W3−R1−R2∗∗∗000R2W2−W3−R2−R3∗∗0000R3−W2−R3∗ℏ71ℏ72ℏ73000ℏ77], 28  ℏ11 =ATPA−P+ATQA+W1+d1Q+d22(A−I)TR1(A−I)+d32(A−I)TR2(A−I) +d42(A−I)TR3(A−I)−R1+CzTCz, 29  ℏ21 =AdTPA+AdTQA+d22AdTR1(A−I)+d32AdTR2(A−I)+d42AdTR3(A−I), 30a  ℏ22 =AdTPAd+AdTQAd+d22AdTR1Ad+d32AdTR2Ad+d42AdTR3Ad−Q, 30b  ℏ31 =(−B−REc)TPA+(−B−REc)TQA+d22(−B−REc)TR1(A−I) +d32(−B−REc)TR2(A−I)+DG+d42(−B−REc)TR3(A−I), 31a  ℏ32 =(−B−REc)TPAd+(−B−REc)TQAd+d22(−B−REc)TR1Ad+d32(−B−REc)TR2Ad +d42(−B−REc)TR3Ad, 31b  ℏ33 =(−B−REc)TP(−B−REc)+(−B−REc)TQ(−B−REc)+d22(−B−REc)TR1(−B−REc) +d32(−B−REc)TR2(−B−REc)+d42(−B−REc)TR3(−B−REc)−2D, 32a  ℏ71 =BwTPA+BwTQA+d22BwTR1(A−I)+d32BwTR2(A−I)+d42BwTR3(A−I)+DzTCz, 32b  ℏ72 =BwTPAd+BwTQAd+d22BwTR1Ad+d32BwTR2Ad+d42BwTR3Ad, 33a  ℏ73 =BwTP(−B−REc)+BwTQ(−B−REc)+d22BwTR1(−B−REc) +d32BwTR2(−B−REc)+d42BwTR3(−B−REc), 33b  ℏ77 =BwTPBw+BwTQBw+d22BwTR1Bw+d32BwTR2Bw+d42BwTR3Bw+DzTDz−γ2I. 34 In the light of Lemma 2, (22) is equivalent to $$ {\boldsymbol{I}{\kern-3pt-\kern-3pt}\boldsymbol{b}}<\mathbf{0} $$, which implies   β(i,j)=Δv(ξ(i,j))+zT(i,j)z(i,j)−γ2wT(i,j)w(i,j)<0, 35 i.e.   −Δv(ξ(i,j))>zT(i,j)z(i,j)−γ2wT(i,j)w(i,j). 36 If $$ \left\| {\boldsymbol{w}} \right\|_{2} \leqslant \alpha^{2} $$ and $$ \boldsymbol{z}^{T}(i,j)\boldsymbol{z}(i,j)>0 $$ then (36) gives   ∑i+j=rv(i,j)<∑i+j=zv(i,j)+γ2α2. 37 It is seen that (23) corresponds to condition (19) and could be obtained from following   P−τ−1(K(l)−G(l))T(K(l)−G(l))u0(l)−2⩾0,l=1,2,…,p. 38 The satisfaction of (23) follows that the set $$ \varepsilon \;(\boldsymbol{P},1+\gamma^{2}\alpha^{2})=\left\{ {\boldsymbol{\xi}\;\in \Re^{n+n_{c} +m+m_{c} };\boldsymbol{\xi}^{T}\boldsymbol{P}\boldsymbol{\xi}\leqslant 1+\gamma^{2}\alpha^{2}} \right\} $$ is included in polyhedral set $$\ell$$ defined as in (16). Further, applying zero boundary condition on (36), we get   ∑i+j=0∞(zT(i,j)z(i,j))<γ2∑i+j=0∞((wT(i,j)w(i,j)). 39 Therefore, (39) implies   ‖z‖22<γ2‖w‖22. 40 Hence, system (9) achieves $$H_{\infty}$$ disturbance attenuation level $$\gamma$$ in presence of input saturation and interval like time varying state delay. Further, it is shown that $$\boldsymbol{\xi}(i,j)\to \mathbf{0} $$ as $$ i\to \infty$$ and/or $$j\to \infty$$ for boundary conditions given by (12). It follows from $ {\it\Delta} v\left( {\left[ \begin{array}{@{}c@{}} {\boldsymbol{\xi}^{h}(i,j)} \\ {\boldsymbol{\xi}^{v}(i,j)} \\ \end{array} \right]\;} \right)\leqslant 0 $ that   vh(ξh(i+1,j))+vv(ξv(i,j+1))⩽vh(ξh(i,j))+vv(ξv(i,j)) =v([ξh(i,j)ξv(i,j)])∀ξ(i,j)∈ε(P,1). 41 For any nonnegative integer $$ \kappa $$, summing up both side of (41) from $$ 0 $$ to $$ \kappa $$ with respect to $$ i $$ and $$ \kappa $$ to $$ 0 $$ with respect to $$ j $$, we get   vh(ξh(1,κ))+vv(ξv(0,κ+1))+vh(ξh(2,κ−1))+vv(ξv(1,κ)) +…+vh(ξh(κ+1,0))+vv(ξv(κ,1)) ⩽v([ξh(0,κ)ξv(0,κ)])+…+v([ξh(κ,0)ξv(κ,0)]), 42   ∑i+j=κ+1vh(ξh(i,j))+∑i+j=κ+1vv(ξv(i,j))⩽∑i+j=κv([ξh(i,j)ξv(i,j)]), 43   ∑i+j=κ+1v([ξh(i,j)ξv(i,j)])⩽∑i+j=κv([ξh(i,j)ξv(i,j)]). 44 From Definition 1, it can be concluded that   lim i→∞ and/or j→∞ξ(i,j)=limi+j→∞ξ(i,j)→0. 45 Therefore, equation (45) intends that system (9) is asymptotically stable. Similarly, it can also be proved $$\forall i+j\in r\in z_{+}$$ that   ξT(i,j)Pξ(i,j)⩽∑i+j=rv(ξ(i,j))⩽∑j=0r1vh(ξh(0,j))+∑i=0r2vv(ξv(i,0)),∀ξ(i,j)∈ε(P,τ−1) 46   ⩽βrh2(λmax(Ph)+(dhH+1)λmax(Qh)+dhLλmax(W1h)+(dhH−dha)λmax(W2h)+(dha−dhL)λmax(W3h) +0.5(dhH−dhL)(dhH+dhL−1)λmax(Qh)+0.5(dhL)2(1+dhL)λmax(R1h) +0.5(dha−dhL)2(dhL+1+dha)λmax(R2h) +0.5(dhH−dha)2(dhH+dha+1)λmax(R3h))+βrv2(λmax(Pv) +(dvH+1)λmax(Qv)+dvLλmax(W1v)+(dvH−dva)λmax(W2v)+(dva−dvL)λmax(W3v) +0.5(dvH−dvL)(dvH−1+dvL)λmax(Qv)+0.5(dvL)2(1+dvL)λmax(R1v) +0.5(dva−dvL)2(dvL+1+dva)λmax(R2v)+0.5(dvH−dva)2(dvH+dva+1)λmax(R3v)) 47   =Γ(βrh,βrv)⩽1 48 The set $$ {\it\Gamma}_{(\beta_{rh}, \beta_{rv} )} \leqslant 1 $$, and $$ \boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant \;1+\gamma^{2}\alpha ^{2} $$ is satisfied. Thus, all the trajectories of $$ \boldsymbol{\xi}(i,j) $$ starting from $$ {\it\Gamma}_{(\beta_{rh}, \beta_{rv} )} \leqslant 1 $$ remain within $$ \varepsilon \;(\boldsymbol{P},1+\gamma^{2}\alpha^{2}) $$. □ As a direct consequence of Theorem 2, Corollary 1 states the sufficient global asymptotic stability condition for system (9). Corollary 1 For given positive scalars $$\alpha $$, $$ \gamma $$, $$ d_{ha} $$, $$ d_{va} $$ satisfying $$ d_{hL} \leqslant d_{ha} \leqslant d_{hH} $$ and $$ d_{vL} \leqslant d_{va} \leqslant d_{vH} $$, if there exist a diagonal positive definite matrix $$ \boldsymbol{L}\in \boldsymbol{\Re}^{p\times p} $$, symmetric matrices $$ \mathbf{0}<\boldsymbol{P}=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v} \quad \in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{Q}=\boldsymbol{Q}^{h}\oplus \boldsymbol{Q}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{W}_{k} =\boldsymbol{W}_{k}^{h} \oplus \boldsymbol{W}_{k}^{v} \quad (k=1,2,3) \in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{R}_{k} =\boldsymbol{R}_{k}^{h} \oplus \boldsymbol{R}_{k}^{v} (k=1,2,3) \quad \in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{X}_{k} (k=1,\ldots,5)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$ and matrix $$ \boldsymbol{H}\in \boldsymbol{\Re}^{(n_{c} +m_{c} )\times p} $$ satisfying following LMI   [Γ~11∗∗∗∗∗∗0−Q∗∗∗∗∗K0−2L∗∗∗∗R100W3−W1−R1−R2∗∗∗000R2W2−W3−R2−R3∗∗0000R3−W2−R3∗000000−γ2IAAd(−BL−RH)000BwAAd(−BL−RH)000Bwd2(A−I)d2Add2(−BL−RH)000d2Bwd3(A−I)d3Add3(−BL−RH)000d3Bwd4(A−I)d4Add4(−BL−RH)000d4BwCz00000Dz  ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−2X1+X1PX1∗∗∗∗∗0−2X2+X2QX2∗∗∗∗00−2X3+X3R1X3∗∗∗000−2X4+X4R2X4∗∗0000−2X5+X5R3X5∗00000−I]<0, 49 then, for the gain matrix $$ \boldsymbol{E}_{c} \;=\boldsymbol{HL}^{-\mbox{1}} $$ the origin of system (9) is globally asymptotically stable with prescribed $$H_{\infty}$$ disturbance attenuation level $$\gamma$$. Proof. Choosing,   G=K. 50 One can see that (16) is automatically met for all $$\boldsymbol{\xi}\in \boldsymbol{\Re}^{\hat{{n}}+\hat{{m}}}$$. Now, substituting (50) into (22) we obtain the global asymptotic stability condition (49) for system (9). This completes the proof. The condition for global stability is valid only when the open loop system is asymptotically stable (Gomes da Silva Jr. & Tarbouriech, 2006). □ 4. Numerical examples Example 1 Consider the parameters of two-dimensional system represented by (1)   Ap =[0.2150.08⋮0.00.2−0.04⋮0.01⋯⋯⋯⋯⋯⋯⋯⋯0.010.01⋮−0.1],Adp=[0.100⋮0.050.100.02⋮0⋯⋯⋯⋯⋯⋯⋯⋯0.030.12⋮0.03],Bp =[0.80−0.010.01…………0.0010.002]Cp=[10000.10.01]C¯z =[0.010000.060.002],D¯z=[0.0060.0050.0020.006],B¯w=[0.010⋮0.0020.0070.008⋮0.002⋯⋯⋯⋯⋯⋯⋯⋯0.0020.006⋮0.008] and stabilizing dynamic output feedback controller is given by   Ac =[−0.051⋮0………⋮………0⋮−0.501],Bc=[0.5000.6],Cc =[−0.495800−0.51],Dc=[−0.01100−0.527]. The saturation of control signal is characterized by $$ -0.6\leqslant \overline {\boldsymbol{u}} \leqslant 0.6 $$. By selecting $$ d_{hL} = d_{vL} =1, d_{ha} =9 $$, $$ d_{va} =25 $$ and iteratively solving the LMIs (22)–(23) with respect to $$ d_{hH} $$ and $$ d_{vH} $$, it is seen that the system (9) is asymptotically stable for $$ 1\leqslant d_{h} (i)\leqslant 11 $$ and $$ 1\leqslant d_{v} (j)\leqslant 29 $$. For different values of delay ranges, $$ d_{hL}, d_{hH} $$, $$ d_{vL}, d_{vH} $$, $$ d_{ha} $$ and $$ d_{va} $$ the anti-windup controller gains are computed as shown in Table 1. Table 1 Computational results Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 5, \enspace d_{ha} =3$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =5$$  $\left[ \begin{array}{@{}cc@{}} {0.1772} & {-0.0022} \\ {0.0035} & {-0.0001} \\ \end{array} \right]$  $$ 1\leqslant d_{h} (i)\leqslant 8, \enspace d_{ha} =6$$  $$ 1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =11$$  $ \left[ \begin{array}{@{}cc@{}} {0.1565} & {-0.0029} \\ {0.0022} & {0.0001} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 10, \enspace d_{ha} =8$$  $$ 1\leqslant d_{v} (j)\leqslant 22, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.1389} & {-0.0044} \\ {-0.0019} & {0.0022} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 11, \enspace d_{ha} =9$$  $$ 1\leqslant d_{v} (j)\leqslant 29, \enspace d_{va} =25$$  $ \left[ \begin{array}{@{}cc@{}} {0.1196} & {-0.0005} \\ {-0.0043} & {0.0026} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 12 $$  $$ 1\leqslant d_{v} (j)\leqslant 30 $$  Infeasible  Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 5, \enspace d_{ha} =3$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =5$$  $\left[ \begin{array}{@{}cc@{}} {0.1772} & {-0.0022} \\ {0.0035} & {-0.0001} \\ \end{array} \right]$  $$ 1\leqslant d_{h} (i)\leqslant 8, \enspace d_{ha} =6$$  $$ 1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =11$$  $ \left[ \begin{array}{@{}cc@{}} {0.1565} & {-0.0029} \\ {0.0022} & {0.0001} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 10, \enspace d_{ha} =8$$  $$ 1\leqslant d_{v} (j)\leqslant 22, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.1389} & {-0.0044} \\ {-0.0019} & {0.0022} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 11, \enspace d_{ha} =9$$  $$ 1\leqslant d_{v} (j)\leqslant 29, \enspace d_{va} =25$$  $ \left[ \begin{array}{@{}cc@{}} {0.1196} & {-0.0005} \\ {-0.0043} & {0.0026} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 12 $$  $$ 1\leqslant d_{v} (j)\leqslant 30 $$  Infeasible  Table 1 Computational results Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 5, \enspace d_{ha} =3$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =5$$  $\left[ \begin{array}{@{}cc@{}} {0.1772} & {-0.0022} \\ {0.0035} & {-0.0001} \\ \end{array} \right]$  $$ 1\leqslant d_{h} (i)\leqslant 8, \enspace d_{ha} =6$$  $$ 1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =11$$  $ \left[ \begin{array}{@{}cc@{}} {0.1565} & {-0.0029} \\ {0.0022} & {0.0001} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 10, \enspace d_{ha} =8$$  $$ 1\leqslant d_{v} (j)\leqslant 22, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.1389} & {-0.0044} \\ {-0.0019} & {0.0022} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 11, \enspace d_{ha} =9$$  $$ 1\leqslant d_{v} (j)\leqslant 29, \enspace d_{va} =25$$  $ \left[ \begin{array}{@{}cc@{}} {0.1196} & {-0.0005} \\ {-0.0043} & {0.0026} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 12 $$  $$ 1\leqslant d_{v} (j)\leqslant 30 $$  Infeasible  Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 5, \enspace d_{ha} =3$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =5$$  $\left[ \begin{array}{@{}cc@{}} {0.1772} & {-0.0022} \\ {0.0035} & {-0.0001} \\ \end{array} \right]$  $$ 1\leqslant d_{h} (i)\leqslant 8, \enspace d_{ha} =6$$  $$ 1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =11$$  $ \left[ \begin{array}{@{}cc@{}} {0.1565} & {-0.0029} \\ {0.0022} & {0.0001} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 10, \enspace d_{ha} =8$$  $$ 1\leqslant d_{v} (j)\leqslant 22, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.1389} & {-0.0044} \\ {-0.0019} & {0.0022} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 11, \enspace d_{ha} =9$$  $$ 1\leqslant d_{v} (j)\leqslant 29, \enspace d_{va} =25$$  $ \left[ \begin{array}{@{}cc@{}} {0.1196} & {-0.0005} \\ {-0.0043} & {0.0026} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 12 $$  $$ 1\leqslant d_{v} (j)\leqslant 30 $$  Infeasible  The state trajectories of two-dimensional system are depicted in Fig. 1. It can be seen that all the three states are converging to zero with $$\gamma = 0.4$$, $$\alpha = 0.01$$ and boundary conditions are given as   ξh(i,j) ={[−0.030.05−0.6] 0⩽j⩽120j>12 ,ξv(i,j)={[−0.040.51] 0⩽i⩽120i>12 ,{xh(i,j)=[−0.030.05], ∀ 0⩽j⩽12xv(i,j)=−0.04, ∀ 0⩽i⩽12 . Fig. 1. View largeDownload slide Horizontal and vertical state trajectories. Fig. 1. View largeDownload slide Horizontal and vertical state trajectories. Time varying state delays are $$ d_{h} (i)=6+5sin\left( {\frac{\pi i}{2}} \right) $$, $$ d_{v} (j)=15+14sin\left( {\frac{\pi j}{2}} \right) $$ and disturbances are $ \left[\begin{array}{@{}c@{}} {\boldsymbol{w}^{h}(i,j)} \\ \cdots \\ {\boldsymbol{w}^{v}(i,j)} \\ \end{array} \right]=\left[ \begin{array}{@{}c@{}} {0.8e^{-0.05(i+j)}} \\ {0.7e^{-0.05(i+j)}} \\ {\ldots \ldots \ldots \ldots } \\ {0.4e^{-0.05(i+j)}} \\ \end{array} \right] $. Fig. 2. View largeDownload slide Control effort without anti-windup. Fig. 2. View largeDownload slide Control effort without anti-windup. Fig. 3. View largeDownload slide Control effort with anti-windup for $$ \overline u _{(l)} \,=\pm 0.6 $$. Fig. 3. View largeDownload slide Control effort with anti-windup for $$ \overline u _{(l)} \,=\pm 0.6 $$. Figure 2 depicts the control effort without anti-windup. Figure 3 represents the control effort with anti-wind up controller. From Fig. 2, it can be seen that the values of control effort are beyond the limit given by $$ \overline u _{(l)} \,=\pm 0.6 $$. By using anti-windup controller, it is clear from Fig. 3 that control efforts are with in specified control limit i.e. $$ \overline u_{(l)} =\pm 0.6 $$. Remark 4 It is noticed from above example that upper bound on delays found for horizontal and vertical states are 11 and 29, respectively whereas in Ghous & Xiang (2015), the upper bounds on delays in both states are found to be 3. It can be seen that there is remarkable increase in delay range by using anti-windup compensator with sector bounded conditions for saturation nonlinearity as compared to state feedback controller given in Ghous & Xiang (2015) with convex hull technique to represent saturation nonlinearity. It can also be observed that the value of $$\gamma$$ attenuation constant has reduced to 0.4 in the proposed work in comparison to work reported by Ghous & Xiang (2015). Example 2 In this example, the applicability of Theorem 2 to control of dynamical processes in gas absorption, water steam heating and air drying, which are represented by Darboux equation (Du et al., 2001; Ghous & Xiang, 2015; Marszalek, 1984; Negi et al., 2012b) is demonstrated. Consider the Darboux equation given by   ∂2s(x,t)∂x∂t=a1∂s(x,t)∂t+a2∂s(x,t)∂x+a0s(x,t)+ads(x,t−d)+bww(x,t)+bf(x,t), 51 with the initial conditions:   s(x,0)=p(x),s(0,t)=q(t), 52 where $$ s(x,t) $$ is an unknown function at space $$ x\;\in \;[0,x_{f} ] $$ and time $$ t\;\in \;[0,\infty ] $$ ; $$ f(x,t) $$ is the input function subjected to saturation; $$ w(x,t) $$ is disturbance; $$ a_{1}, \;a_{2}, \;a_{0} $$, $$ a_{d} $$, $$ b_{w} $$ and $$ b $$ are real constants. The two-dimensional model of Darboux equation with actuator saturation and delay can be represented in the following form   [xh(i+1,j)xv(i,j+1)]=[(1+a1Δx)(a1a2+a0)ΔxΔt(1+a2Δt)][xh(i,j)xv(i,j)]+[bwΔx0]w(x,t)+[0adΔx00][xh(i−dh(i),j)xv(i,j−dv(j))]+[bΔx0]u(i,j), 53 with the initial conditions   xh(0,j)=z(jΔt),xv(i,0)=p(iΔx). 54 The parameters of two-dimensional system with time varying delay and saturation are   Ap=[0.110.0030.10.24],Adp=[0.0002000],Bp=[0.080.09−0.010.01],Cp=[0.210.0030.10.24],B¯w=[0.010.050.050.001],C¯z=[0.0300.50.02],D¯z=[0.06000.082]. The dynamic output feedback controller which stabilizes the above plant is given by   Ac=[0.0210.0930.160.084],Bc=[0.050.050.00260.10],Cc=[0.010.0360.0020],Dc=[−0.0071−0.0090.01−0.80]. Using MATLAB LMI toolbox (Gahinet et al., 1995), it is verified that (22) and (23) are found feasible for control bound $$ -0.5\leqslant \overline {\boldsymbol{u}} \leqslant 0.5 $$, $$ \gamma = 0.5$$, $$ \alpha = 0.01$$. By selecting $$ d_{hL} = d_{vL} =1, \quad d_{ha} =18 $$, $$ d_{va} =15 $$ and iteratively solving the LMIs (22)–(23) with respect to $$ d_{hH} $$ and $$ d_{vH} $$, it is seen that the system (9) is asymptotically stable for $$ 1\leqslant d_{h} (i)\leqslant 23 $$ and $$ 1\leqslant d_{v} (j)\leqslant 21 $$. Table 2 shows the anti-windup gains for different delay ranges. Table 2 Computational results Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 9, \enspace d_{ha} =5$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =7$$  $\left[ \begin{array}{@{}cc@{}} {0.0060} & {-0.0020} \\ {0.0018} & {-0.0010} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 15, \enspace d_{ha} =11$$  $$1\leqslant d_{v} (j)\leqslant 14, \enspace d_{va} =9$$  $\left[ \begin{array}{@{}cc@{}} {0.0086} & {-0.0021} \\ {0.0055} & {-0.0016} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 20, \enspace d_{ha} =13$$  $$1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =12$$  $\left[ \begin{array}{@{}cc@{}} {0.0117} & {-0.0027} \\ {0.0096} & {-0.0025} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 23, \enspace d_{ha} =18$$  $$ 1\leqslant d_{v} (j)\leqslant 21, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.0135} & {-0.0030} \\ {0.0209} & {-0.0032} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 24$$  $$ 1\leqslant d_{v} (j)\leqslant 22$$  Infeasible  Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 9, \enspace d_{ha} =5$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =7$$  $\left[ \begin{array}{@{}cc@{}} {0.0060} & {-0.0020} \\ {0.0018} & {-0.0010} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 15, \enspace d_{ha} =11$$  $$1\leqslant d_{v} (j)\leqslant 14, \enspace d_{va} =9$$  $\left[ \begin{array}{@{}cc@{}} {0.0086} & {-0.0021} \\ {0.0055} & {-0.0016} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 20, \enspace d_{ha} =13$$  $$1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =12$$  $\left[ \begin{array}{@{}cc@{}} {0.0117} & {-0.0027} \\ {0.0096} & {-0.0025} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 23, \enspace d_{ha} =18$$  $$ 1\leqslant d_{v} (j)\leqslant 21, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.0135} & {-0.0030} \\ {0.0209} & {-0.0032} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 24$$  $$ 1\leqslant d_{v} (j)\leqslant 22$$  Infeasible  Table 2 Computational results Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 9, \enspace d_{ha} =5$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =7$$  $\left[ \begin{array}{@{}cc@{}} {0.0060} & {-0.0020} \\ {0.0018} & {-0.0010} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 15, \enspace d_{ha} =11$$  $$1\leqslant d_{v} (j)\leqslant 14, \enspace d_{va} =9$$  $\left[ \begin{array}{@{}cc@{}} {0.0086} & {-0.0021} \\ {0.0055} & {-0.0016} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 20, \enspace d_{ha} =13$$  $$1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =12$$  $\left[ \begin{array}{@{}cc@{}} {0.0117} & {-0.0027} \\ {0.0096} & {-0.0025} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 23, \enspace d_{ha} =18$$  $$ 1\leqslant d_{v} (j)\leqslant 21, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.0135} & {-0.0030} \\ {0.0209} & {-0.0032} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 24$$  $$ 1\leqslant d_{v} (j)\leqslant 22$$  Infeasible  Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 9, \enspace d_{ha} =5$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =7$$  $\left[ \begin{array}{@{}cc@{}} {0.0060} & {-0.0020} \\ {0.0018} & {-0.0010} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 15, \enspace d_{ha} =11$$  $$1\leqslant d_{v} (j)\leqslant 14, \enspace d_{va} =9$$  $\left[ \begin{array}{@{}cc@{}} {0.0086} & {-0.0021} \\ {0.0055} & {-0.0016} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 20, \enspace d_{ha} =13$$  $$1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =12$$  $\left[ \begin{array}{@{}cc@{}} {0.0117} & {-0.0027} \\ {0.0096} & {-0.0025} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 23, \enspace d_{ha} =18$$  $$ 1\leqslant d_{v} (j)\leqslant 21, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.0135} & {-0.0030} \\ {0.0209} & {-0.0032} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 24$$  $$ 1\leqslant d_{v} (j)\leqslant 22$$  Infeasible  The horizontal and vertical state trajectories are plotted in Fig. 4. The control effort is shown in Fig. 5 without considering the effect of anti-windup compensator. It can be seen from Fig. 6 that the control bounds are within specified limit $$ \overline u_{(l)} \,=\pm 0.5 $$ by using anti-windup compensator. Fig. 4. View largeDownload slide Horizontal and vertical state trajectories. Fig. 4. View largeDownload slide Horizontal and vertical state trajectories. Fig. 5. View largeDownload slide Control effort without anti-windup. Fig. 5. View largeDownload slide Control effort without anti-windup. Fig. 6. View largeDownload slide Control effort with anti-windup for $$\overline u_{(l)} \,=\pm 0.5 $$. Fig. 6. View largeDownload slide Control effort with anti-windup for $$\overline u_{(l)} \,=\pm 0.5 $$. In this example, the boundary conditions are   ξh(i,j) ={[0.040.05] 0⩽j⩽100j>10 ,ξv(i,j)={[−0.060.5] 0⩽i⩽100i>10 ,{xh(i,j)=0.04, ∀ 0⩽j⩽10xv(i,j)=−0.06, ∀ 0⩽i⩽10 . Time varying delays in horizontal and vertical directions are $$d_{h} (i)=12+11sin\left( {\frac{\pi i}{2}} \right) $$ and $$ d_{v} (j)=11+10sin\left( {\frac{\pi j}{2}} \right) $$, respectively. The disturbances are $ \left[ \begin{array}{@{}c@{}} {\boldsymbol{w}^{h}(i,j)} \\ \cdots \\ {\boldsymbol{w}^{v}(i,j)} \\ \end{array} \right]=\left[ \begin{array}{@{}c@{}} {0.5e^{-0.05(i+j)}} \\ \ldots \\ {0.3e^{-0.05(i+j)}} \\ \end{array} \right] $. Remark 5 As quoted in Remark 1 for Example 1, the same observation is make for Example 2. Example 3 Consider the two-dimensional discrete system in Roesser Model setting (1) and the stabilizing controller (2) with   Ap =[0.150.01⋮0.00.2−0.04⋮0.01⋯⋯⋯⋯⋯⋯⋯⋯0.010.01⋮−0.1],Adp=[0.100⋮0.040.100.02⋮0⋯⋯⋯⋯⋯⋯⋯⋯0.030.12⋮0.03],Bp =[0.80−0.010.01…………0.0010.002],Cp=[10000.10.01],B¯w =[0.0080⋮0.0020.0070.008⋮0.002⋯⋯⋯⋯⋯⋯⋯⋯0.0020.006⋮0.008],C¯z=[0.010.500.040.060.008],D¯z=[0.007−0.0050.0020.006],Ac =[−0.097⋮0.02………⋮………0⋮−0.2],Bc=[0.40.060.030.7],Cc =[−0.6−0.80−0.51],Dc=[−0.0400−0.7]. By selecting $$ d_{hL} = d_{vL} =1, \quad d_{ha} =6 $$, $$ d_{va} =10 $$ and iteratively solving the LMI (49) of Corollary 1 with respect to $$d_{hH}$$ and $$d_{vH}$$, it is seen that the system (9) is asymptotically stable for $$ 1\leqslant d_{h} (i)\leqslant 9 $$ and $$1\leqslant d_{v} (j)\leqslant 31$$. For above delay ranges, the control bound is given by $$-1\leqslant \overline {\boldsymbol{u}} \leqslant 1$$ and the values of unknown parameters are calculated as follows-   Ph =[96.45255.2149−8.3618 5.2149111.6712−23.1212−8.3618−23.1212136.2397],Pv=[105.7727−0.5450−0.5450227.6591],H=[−0.00110.0001−0.00050.0027],L =[0.009000.0605]. The anti-windup gain of stabilizing compensator is obtained as   Ec=HL−1=[−0.12180.0010−0.05670.0444]. Remark 6 In the present work, an anti-windup compensator is designed and static anti-windup gains are determined to stabilize the system (1). A sector condition (see (16–17)) is used to characterize the actuator saturation nonlinearity. By contrast, (Benhayoun et al., 2013; Ghous & Xiang, 2015; Hu et al., 2002) deal with the design of state feedback controller where convex hull approach is adopted for characterization of saturation nonlinearity. In Benhayoun et al. (2013); Ghous & Xiang (2015); Hu et al. (2002), the stability conditions are expressed as a convex combination of $$ 2^{p} $$ (where input $$\boldsymbol{u}(i,j)\;\in \boldsymbol{\Re}^{p}) $$ LMIs. Therefore, as compared with Benhayoun et al. (2013); Ghous & Xiang (2015); Hu et al. (2002), the present approach is beneficial in terms of computational complexity. 5. Conclusion In this article, Lyapunov based stability analysis of two-dimensional systems in presence of input saturation, time varying state delay and disturbances has been carried out for Roesser model. The problem is dealt with anti-windup paradigm using generalized sector condition which directly formulates the LMI conditions. The large number of iterations and complexity are reduced in comparison to polytopic differential inclusion (Gomes da Silva Jr. & Tarbouriech, 2005; Tarbouriech et al., 2007). $$H_{\infty}$$ disturbance attenuation performance analysis has been carried out and sufficient conditions are stated in context of local and global stability of closed loop system. Application of the proposed anti-windup controller is demonstrated through processes described by a Darboux equation (Du et al. 2001; Ghous & Xiang 2015; Marszalek 1984; Negi et al. 2012b). A domain of attraction is also estimated. From the numerical examples given in Section 4, it is clear that the anti-windup strategy used along with dynamic output feedback controller for two-dimensional discrete systems gives better results in the sense that the delay range has been increased and also disturbance attenuation level is improved in comparison with the result reported in the previous works (e.g. Ghous & Xiang 2015). Further, following the idea of Chai (2015); Lam et al. (2004); Tang et al. (2016); Tadepalli et al. (2015) this problem can be extended to stabilize the two-dimensional delayed systems with actuator saturation and uncertainties in concern to practical systems in industries, where the parameters of plant are frequently perturbed. Acknowledgements The authors would like to thank the editors, anonymous reviewers for their constructive comments and suggestions to improve the manuscript. The special thanks to Prof. Haranath Kar, for his support, encouragement and fruitful discussion on this topic. Appendix Proof of Theorem 1. Consider a two-dimensional quadratic Lyapunov function Ghous & Xiang (2015)   v([ξh(i,j)ξv(i,j)])=v(ξ(i,j))=vh(ξh(i,j))+vv(ξv(i,j)), A.1 where   vh(ξh(i,j)) =∑k=15vkh(ξh(i,j)), A.2  v1h(ξh(i,j)) =ξhT(i,j)Phξh(i,j), A.3  v2h(ξh(i,j)) =∑r=i−dh(i)iξhT(r,j)Qhξh(r,j), A.4  v3h(ξh(i,j)) =∑r=i−dhLi−1ξhT(r,j)W1hξh(r,j)+∑r=i−dhHi−dha−1ξhT(r,j)W2hξh(r,j)+∑r=i−dhai−dhL−1ξhT(r,j)W3hξh(r,j), A.5  v4h(ξh(i,j)) =∑s=−dhH+1−dhL∑r=i+si−1ξhT(r,j)Qhξh(r,j), A.6  v5h(ξh(i,j)) =dhL∑s=−dhL−1∑r=i+si−1ηhT(r,j)R1hηh(r,j)+(dha−dhL)∑s=−dha−dhL−1∑r=i+si−1ηhT(r,j)R2hηh(r,j) +(dhH−dha)∑s=−dhH−dha−1∑r=i+si−1ηhT(r,j)R3hηh(r,j), A.7  vv(ξv(i,j)) =∑k=15vkv(ξv(i,j)), A.8  v1v(ξv(i,j)) =ξvT(i,j)Pvξv(i,j), A.9  v2v(ξv(i,j)) =∑b=j−dv(j)jξvT(i,b)Qvξv(i,b), A.10  v3v(ξv(i,j)) =∑b=j−dvLj−1ξvT(i,b)W1vξv(i,b)+∑b=j−dvHj−dva−1ξvT(i,b)W2vξv(i,b)+∑b=j−dvaj−dvL−1ξvT(i,b)W3vξv(i,b), A.11  v4v(ξv(i,j)) =∑s=−dvH+1−dvL∑b=j+sj−1ξvT(i,b)Qvξv(i,b), A.12  v5v(ξv(i,j)) =dvL∑s=−dvL−1∑b=j+sj−1ηvT(i,b)R1vηv(i,b)+(dva−dvL)∑s=−dva−dvL−1∑b=j+sj−1ηvT(i,b)R2vηv(i,b) +(dvH−dva)∑s=−dvH−dva−1∑b=j+sj−1ηvT(i,b)R3vηv(i,b). A.13 Define   {ηh(r,j)=ξh(r+1,j)−ξh(r,j)ηv(i,b)=ξv(i,b+1)−ξv(i,b) . A.14 Taking forward difference of Lyapunov functional along trajectories of system (9)   Δv([ξh(i,j)ξv(i,j)]) =∑k=15vkh(ξh(i+1,j))−∑k=15vkh(ξh(i,j))+∑k=15vkv(ξv(i,j+1))−∑k=15vkv(ξv(i,j)), A.15   =Δv1(ξ(i,j))+Δv2(ξ(i,j))+Δv3(ξ(i,j))+Δv4(ξ(i,j))+Δv5(ξ(i,j)), A.16 where   Δv1(ξ(i,j))=Δv1h(ξh(i,j))+Δv1v(ξv(i,j)), A.17  Δv1(ξ(i,j))=[ξh(i+1,j)ξv(i,j+1)]TP[ξh(i+1,j)ξv(i,j+1)]−[ξh(i,j)ξv(i,j)]TP[ξh(i,j)ξv(i,j)], A.18    =ξT(i,j)ATPAξ(i,j)+ξT(i,j)ATP(−B−REc)ψ(Kξ(i,j))  +ξT(i,j)ATPAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]+[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTPAξ(i,j)  +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTPAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTP(−B−REc)ψ(Kξ(i,j))  +ψT(Kξ(i,j))(−B−REc)TPAξ(i,j)+ψT(Kξ(i,j))(−B−REc)TPAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +ψT(Kξ(i,j))(−B−REc)TP(−B−REc)ψ(Kξ(i,j))−ξT(i,j)Pξ(i,j), A.19  Δv2(ξ(i,j))=Δv2h(ξh(i,j))+Δv2v(ξv(i,j)), A.20  Δv2(ξ(i,j))=[ξh(i+1,j)ξv(i,j+1)]TQ[ξh(i+1,j)ξv(i,j+1)]−[ξh(i−dh(i),j)ξv(i,j−dv(j)]TQ[ξh(i−dh(i),j)ξv(i,j−dv(j)], A.21    =ξT(i,j)ATQAξ(i,j)+ξT(i,j)ATQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +ξT(i,j)ATQ(−B−REc)ψ(Kξ(i,j))+[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQAξ(i,j)  +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQ(−B−REc)ψ(Kξ(i,j))  +ψT(Kξ(i,j))(−B−REc)TQAξ(i,j)+ψT(Kξ(i,j))(−B−REc)TQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +ψT(Kξ(i,j))(−B−REc)TQ(−B−REc)ψ(Kξ(i,j))−[ξh(i−dh(i),j)ξv(i,j−dv(j)]TQ[ξh(i−dh(i),j)ξv(i,j−dv(j)]. A.22 Adding a term in (A.22), we get   Δv2(ξ(i,j))⩽ξT(i,j)ATQAξ(i,j)+ξT(i,j)ATQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))] +ξT(i,j)ATQ(−B−REc)ψ(Kξ(i,j))+[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQAξ(i,j) +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))] +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQ(−B−REc)ψ(Kξ(i,j)) +ψT(Kξ(i,j))(−B−REc)TQAξ(i,j)+ψT(Kξ(i,j))(−B−REc)TQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))] +ψT(Kξ(i,j))(−B−REc)TQ(−B−REc)ψ(Kξ(i,j))−[ξh(i−dh(i),j)ξv(i,j−dv(j)]TQ[ξh(i−dh(i),j)ξv(i,j−dv(j)] +[∑r=i+1−dhHi−dhLξh(r,j)∑b=j+1−dvHj−dvLξv(i,b)]TQ[∑r=i+1−dhHi−dhLξh(r,j)∑b=j+1−dvHj−dvLξv(i,b)]. A.23  Δv3(ξ(i,j))=Δv3h(ξh(i,j))+Δv3v(ξv(i,j)). A.24  Δv3(ξ(i,j))=[ξh(i,j)ξv(i,j)]TW1[ξh(i,j)ξv(i,j)]−[ξh(i−dhL,j)ξv(i,j−dvL)]TW1[ξh(i−dhL,j)ξv(i,j−dvL)]  +[ξh(i−dha,j)ξv(i,j−dva)]TW2[ξh(i−dha,j)ξv(i,j−dva)]−[ξh(i−dhH,j)ξv(i,j−dvH)]TW2[ξh(i−dhH,j)ξv(i,j−dvH)]  +[ξh(i−dhL,j)ξv(i,j−dvL)]TW3[ξh(i−dhL,j)ξv(i,j−dvL)]−[ξh(i−dha,j)ξv(i,j−dva)]TW3[ξh(i−dha,j)ξv(i,j−dva)]. A.25  Δv4(ξ(i,j))=Δv4h(ξh(i,j))+Δv4v(ξv(i,j)). A.26  Δv4(ξ(i,j))=[ξh(i,j)ξv(i,j)]T[(dhH−dhL)I00(dvH−dvL)I]Q[ξh(i,j)ξv(i,j)] −[∑r=i−dhH+1i−dhLξh(r,j)∑b=j−dvH+1j−dvLξv(i,b)]TQ[∑r=i−dhH+1i−dhLξh(r,j)∑b=j−dvH+1j−dvLξv(i,b)]. A.27  Δv5(ξ(i,j))=Δv5h(ξh(i,j))+Δv5v(ξv(i,j)). A.28  Δv5(ξ(i,j))=[ηh(i,j)ηv(i,j)]T[dhL2I00dvL2I]R1[ηh(i,j)ηv(i,j)]−[∑r=i−dhLi−1ηh(r,j)∑b=j−dvLj−1ηv(i,b)]T[dhLI00dvLI]R1[ηh(r,j)ηv(i,b)] +[ηh(i,j)ηv(i,j)]T[(dha−dhL)2I00(dva−dvL)2I]R2[ηh(i,j)ηv(i,j)] −[∑r=i−dhai−dhL−1ηh(r,j)∑b=j−dvaj−dvL−1ηv(i,b)]T[(dha−dhL)I00(dva−dvL)I]R2[ηh(r,j)ηv(i,b)] +[ηh(i,j)ηv(i,j)]T[(dhH−dha)2I00(dvH−dva)2I]R3[ηh(i,j)ηv(i,j)] −[∑r=i−dhHi−dha−1ηh(r,j)∑b=j−dvHj−dva−1ηv(i,b)]T[(dhH−dha)I00(dvH−dva)I]R3[ηh(r,j)ηv(i,b)]. A.29 From Lemma 1, (A.29) is rewritten as   (A.30) where   {∑r=i−dhLi−1ξh(r+1,j)−∑r=i−dhLi−1ξh(r,j)=ξh(i,j)−ξh(i−dhL,j)∑r=i−dhai−dhL−1ξh(r+1,j)−∑r=i−dhai−dhL−1ξh(r,j)=ξh(i−dhL,j)−ξh(i−dha,j)∑r=i−dhHi−dha−1ξh(r+1,j)−∑r=i−dhHi−dha−1ξh(r,j)=ξh(i−dha,j)−ξh(i−dhH,j) , A.31  {∑b=j−dvLj−1ξv(i,b+1)−∑b=j−dvLj−1ξv(i,b)=ξv(i,j)−ξv(i,j−dvL)∑b=j−dvaj−dvL−1ξv(i,b+1)−∑b=j−dvaj−dvL−1ξv(i,b)=ξv(i,j−dvL)−ξv(i,j−dva)∑b=j−dvHj−dva−1ξv(i,b+1)−∑b=j−dvHj−dva−1ξv(i,b)=ξv(i,j−dva)−ξv(i,j−dvH) . A.32 Employing Lemma 3 with (A.15)–(A.30), following inequality is stated   Δv(ξ(i,j))⩽ϑT(i,j)dpϑ(i,j)−δℓ, A.33 where   dp =[dp11∗∗∗∗∗dp21dp22∗∗∗∗dp31dp32dp33∗∗∗R100−W1+W3−R1−R2∗∗000R2W2−W3−R2−R3∗0000R3−W2−R3], A.34  dp11 =ATPA−P+ATQA+W1+d1Q+d22(A−I)TR1(A−I)+d32(A−I)TR2(A−I)  +d42(A−I)TR3(A−I)−R1, A.35  dp21 =AdTPA+AdTQA+d22AdTR1(A−I)+d32AdTR2(A−I)+d42AdTR3(A−I), A.36  dp22 =AdTPAd+AdTQAd+d22AdTR1Ad+d32AdTR2Ad+d42AdTR3Ad−Q, A.37  dp31 =(−B−REc)TPA+(−B−REc)TQA+d22(−B−REc)TR1(A−I)  +d32(−B−REc)TR2(A−I)+d42(−B−REc)TR3(A−I), A.38  dp32 =(−B−REc)TPAd+(−B−REc)TQAd+d22(−B−REc)TR1Ad +d32(−B−REc)TR2Ad+d42(−B−REc)TR3Ad, A.39  dp33 =(−B−REc)TP(−B−REc)+(−B−REc)TQ(−B−REc)+d22(−B−REc)TR1(−B−REc)  +d32(−B−REc)TR2(−B−REc)+d42(−B−REc)TR3(−B−REc). A.40 Using inequality (17) of Lemma 3, $$\delta_{\ell }$$ is defined as   δℓ=2ψT(Kξ(i,j))D[ψT(Kξ(i,j))−Gξ(i,j)], where Dis positive definite diagonal matrix A.41 and   ϑ(i,j) =[ξT(i,j)ξdhT(i,j)ψT(Kξ(i,j))ξd_T(i,j)ξaT(i,j)ξd¯T(i,j)]T,ξdh(i,j) =[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]T,ξd_(i,j)=[ξhT(i−dhL,j)ξvT(i,j−dvL]T,ξa(i,j) =[ξhT(i−dha,j)ξvT(i,j−dva)]T,ξd¯(i,j)=[ξhT(i−dhH,j)ξvT(i,j−dvH)]T. A.42 To ensure the local asymptotic stability of closed loop system (9) ${\it\Delta} v\left( {\left[ \begin{array}{@{}c@{}} {\boldsymbol{\xi}^{h}(i,j)} \\ {\boldsymbol{\xi}^{v}(i,j)} \\ \end{array} \right]\;} \right)<0 $. Therefore, it must satisfy that $${\boldsymbol{d}{\kern-3.8pt}\boldsymbol{p}} {\boldsymbol{<0}}$$ with $$\boldsymbol{\vartheta }(i,j)\ne \mathbf{0} $$. Further, applying Schur’s complement (Boyd et al., 1994), (A.33) is equivalent to   [Γ~11∗∗∗∗∗∗∗∗∗∗0−Q∗∗∗∗∗∗∗∗∗DG0−2D∗∗∗∗∗∗∗∗R100W3−W1−R1−R2∗∗∗∗∗∗∗000R2W2−W3−R2−R3∗∗∗∗∗∗0000R3−W2−R3∗∗∗∗∗AAd(−B−REc)000−P−1∗∗∗∗AAd(−B−REc)0000−Q−1∗∗∗d2(A−I)d2Add2(−B−REc)00000−R1−1∗∗d3(A−I)d3Add3(−B−REc)000000−R2−1∗d4(A−I)d4Add4(−B−REc)0000000−R3−1]<0, A.43 Pre and post multiplying (A.43) by $$ diag(\boldsymbol{I},\boldsymbol{I},\boldsymbol{D}^{-1},\boldsymbol{I},\boldsymbol{I},\boldsymbol{I},\boldsymbol{I},\boldsymbol{I},\boldsymbol{I,I,I}) $$ with $$ \boldsymbol{D}^{-1}=\boldsymbol{L} $$ and $$ \boldsymbol{E}_{c} =\boldsymbol{HL}^{-1} $$ along with the following assumptions (Chen & Fong, 2010; Negi et al., 2012a)   {(X1−P−1)(−P)(X1−P−1)⩽0⇒−P−1⩽−2X1+X1PX1(X2−Q−1)(−Q)(X2−Q−1)⩽0⇒−Q−1⩽−2X2+X2QX2(X3−R1−1)(−R1)(X3−R1−1)⩽0⇒−R1−1⩽−2X3+X3R1X3(X4−R2−1)(−R2)(X4−R2−1)⩽0⇒−R2−1⩽−2X4+X4R2X4(X5−R3−1)(−R3)(X5−R3−1)⩽0⇒−R3−1⩽−2X5+X5R3X5 , A.44 (18) is obtained. The satisfaction of condition stated in (19) signifies that the set $$\varepsilon \;(\boldsymbol{P})=\left\{ {\boldsymbol{\xi}\;\in \Re ^{\hat{{n}}+\hat{{m}}};\;\boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant \;1} \right\} $$ is included in polyhedral set $$ \ell $$ as defined in (16). It can be proven that $$ \varepsilon \;(\boldsymbol{P})=\left\{ {\boldsymbol{\xi}\;\in \Re^{\hat{{n}}+\hat{{m}}};\;\boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant \;1} \right\} $$ is equivalent to (Boyd et al., 1994)   P−(K(l)−G(l))T(K(l)−G(l))u0(l)−2⩾0,l=1,2,…,p. A.45 Pre and post multiplication of (A.45) by $$ \boldsymbol{\xi}^{T} $$ and $$ \boldsymbol{\xi} $$ respectively, it follows that $$ \boldsymbol{\xi}\;\in \ell $$ for all $$ \boldsymbol{\xi}\in \varepsilon \;(\boldsymbol{P}) $$. The relation (19) is obtained using Schur’s complement on (A.45). 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$$H_{\infty}$$ based anti-windup controller for two-dimensional discrete delayed systems in presence of actuator saturation

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Abstract

This article addresses the problem of stability analysis for a class of two-dimensional discrete systems described by Roesser model in the presence of actuator saturation and interval-like time varying state delay. The dynamic output feedback $$H_{\infty}$$ stabilization controller appended with an anti-windup compensator is used. The saturation nonlinearity is tackled using sector conditions. Sufficient conditions for asymptotic stability in terms of linear matrix inequality (LMI) are derived using Lyapunov–Krasovskii functional and the domain of attraction is also estimated for the system under consideration. Some examples are provided to illustrate the effectiveness of the proposed approach. 1. Introduction The study of two-dimensional systems is gaining great momentum due to real time applications in many areas such as digital filtering, image processing, signal processing (Bose, 1979; Benzaouia et al., 2016; Chen, 2015; Dey et al., 2012b; Fornasini, 1991; Kaczorek, 1985; Ooba, 2013; Roesser, 1975), industrial process control (paper manufacturing industry, plastic film extrusion, steel sheet production and others) (Zarrop & Wellstead, 2002), iterative learning control, repetitive process control (Liu & Gao, 2010), electricity transmission (Tong et al., 2014), energy exchange process, river pollution and dynamic flood modelling (Costabile & Macchione, 2015), grid based wireless sensor networks (Stefanski, 2014; Sumanasena & Bauer, 2011), medical treatment (Cifor et al., 2013), geophysics (Jeshvaghani & Darijani, 2014), robotics (Kim et al., 2014), heat and mass transfer (Masmoudi et al., 2014), optical fibre network (Han et al., 2015), magnetics (Ueberschar et al., 2015), photovoltaic applications (Koduvelikulathu et al., 2015) and many more. In view of multidimensional modelling, several case studies for practical systems like sensor networks, robot manipulator in batch processing etc. have been addressed in (Rogers et al., 2015). It is known that modelling of two-dimensional systems is more difficult than one-dimensional but two-dimensional systems contain more realistic information about the practical systems. The control problems like that of batch process, thermal systems can easily be described by two-dimensional state space models which inherently contain the transportation and computational delays (Benhayoun et al., 2010; Benzaouia et al., 2011; Chen & Yu , 2013; Dey & Kar, 2014). Such type of delays create instability in the systems and are responsible for poor performance (Chen, 2009; Fridman et al., 2003; Gu et al., 2013; Gorecki et al., 1989; Haidar et al., 2009; Kandanvli & Kar, 2013; Niculescu, 2001; Song et al., 2014; Wang et al., 2013; Zhou et al., 2013) The stability analysis of two-dimensional delayed systems becomes more complicated due to addition of these delay terms in the model of two-dimensional system (Paszke et al., 2003, 2004; Malek-Zavarei et al., 1987). The stabilization of two-dimensional systems subjected to time varying state delays and input nonlinearities is a very challenging work. According to dependence of delay, the available stability criteria for delayed systems can be broadly classified into two categories: delay-dependent and delay-independent. The sufficient conditions for asymptotic stability of two-dimensional systems with state saturation and delay have been derived in Chen (2010a,b); Chen & Yu (2013); Dey & Kar (2012a) using delay dependent approach while delay independent stability criteria are applied in Chen & Fong (2006); Paszke et al. (2003, 2004); Peng & Guan (2009b). Stability conditions derived by delay dependent approach are less conservative because they utilize the information about the size of delay. State feedback controller has been used for stabilization of two-dimensional continuous systems with multi-delays for saturated control in Benzaouia et al. (2011); Benhayoun et al. (2010). Further improvements and modifications in the controller design for two-dimensional systems when subjected to interval like time varying delays and input nonlinearities are reported in Ghous & Xiang (2015); Ghous et al. (2015); Huang et al. (2013); Tadepalli et al. (2015). Saturation is a very common nonlinearity present in control systems due to physical limitation of actuator. The presence of actuator saturation tends the system to the verge of instability and degrades the performance of closed loop system. An anti-windup technique has been widely studied for continuous and discrete systems in presence of actuator saturation nonlinearity and is found to be practical and effective (Bender et al., 2011; Gomes da Silva Jr. et al., 2013; Mesquine et al., 2010; Negi et al., 2012a,b). The problem of actuator saturation has not been studied much for two-dimensional systems and recently it has been addressed in few publications (Benhayoun et al., 2013; Gao & Wang, 2014; Ghous & Xiang, 2015; Ghous et al., 2015; Hmamed et al., 2010; Huang et al., 2013). A state feedback controller has been designed in (Hmamed et al., 2010) for stabilization of two-dimensional saturated continuous systems by using quadratic Lyapunov function. In Gao & Wang (2014), Takagi and Sugeno fuzzy model is employed for two-dimensional nonlinear dynamic system with actuator saturation. $$H_{\infty }$$ controller is used quite often to stabilize two-dimensional discrete systems having nonlinearities and state delays (Huang & Xiang, 2014; Liang et al., 2015; Peng & Guan, 2009a; Xu et al., 2005; Xu & Yu, 2006, 2009). The stability conditions for nonlinear two-dimensional stochastic systems represented by Roesser model with time varying state delay and actuator saturation via state feedback controller are reported in Ghous & Xiang (2015), Huang et al. (2013) and the same case for Fornasini and Marchesini state-space model has been reported in Ghous et al. (2015). The $$H_{\infty}$$ controller is designed for two-dimensional uncertain model for batch process with interval like time varying delay in Wang et al. (2013a,b). The stabilization of two-dimensional discrete systems using $$H_{\infty }$$ based state feedback controller has been reported in Gao & Wang (2014); Ghous & Xiang (2015); Ghous et al. (2015); Huang et al. (2013) in presence of actuator saturation and time varying delay where saturation nonlinearity is described using convex hull. The objective of this article is to obtain stability criterion for two-dimensional discrete system described by Roesser model in presence of input saturation, external disturbances and time varying state delays. Among the available techniques anti-windup gives better results to counter the effect of actuator saturation nonlinearity in comparison to other methods (Gomes da Silva Jr. & Tarbouriech 2006; Mulder et al. 2001, Syaichu-Rohman & Middleton 2004). In the present work, using sector based description of saturated nonlinearity, a technique to compute static anti-windup gain for two-dimensional dynamic compensator for two-dimensional discrete systems described by Roesser model with time varying state delay and disturbances is given. A Lyapunov–Krasovskii functional is used to get the stability condition in terms of LMI and the anti-windup gains are determined such that closed loop system is asymptotically stable. To the best of authors’ knowledge, anti-windup approach has not been reported so far to tackle the problem of actuator saturation in two-dimensional discrete systems with interval like time varying state delays and disturbances. The article is organized as follows. In Section 2, problem is formulated and some necessary definition and lemmas are recalled. The main result is presented in Section 3. Section 4 presents the effectiveness of proposed approach with numerical examples. Notation: throughout this article, $$\boldsymbol{\Re }^{m\times n}$$ represents set of $$m\times n$$ real matrices; $$\boldsymbol{\Re }^{m}$$ represents set of $$m\times 1$$ real matrices; $$\boldsymbol{I}$$ is identity matrix of appropriate dimension; $$\mathbf{0}$$ denotes null matrix or null vector; $$\boldsymbol{{\it\Omega}}^{T}$$ represents transpose matrix of $$\boldsymbol{{\it\Omega} }$$; $$\lambda_{max} ({\it\Omega} )$$ stands for maximum eigenvalue of any given matrix $$\boldsymbol{{\it\Omega} }$$; $$diag\left\{ {a_{1}, a_{2}, \ldots,a_{n} } \right\}$$ is diagonal matrix with diagonal elements $$a_{1}, a_{2}, \ldots,a_{n} $$; $$\boldsymbol{E} = {\bf E}_{1} \oplus \boldsymbol{E}_{2} $$ denotes direct sum i.e. ${\bf E} = \left[\begin{array}{@{}cc@{}} \boldsymbol{E}_{1} & \mathbf{0}\\ \mathbf{0} & \boldsymbol{E}_{2} \end{array} \right]$; $$\left\| \boldsymbol{W} \right\|_{2} =\sqrt {\sum_{i=0}^\infty {\sum_{j=0}^\infty {\left\| \boldsymbol{W}(i,j) \right\|^{2}} } }$$ is $$l_{2} $$ norm of two-dimensional signal $$\boldsymbol{W}(i,j)\in l_{2} \left\{ {\left[{0,\infty } \right),\left[ {0,\infty } \right)} \right\}$$ if $$\left\| \boldsymbol{W} \right\|_{2} <\infty $$; symbol $$\ast $$ is used to represents the symmetric terms in symmetric matrix; $$\left\|. \right\|$$ stands for Euclidean norm. Matrices, if not explicitly mentioned, are assumed to have compatible dimensions. 2. Problem formulation and preliminaries Consider the following two-dimensional discrete system described by Roesser model in presence of actuator saturation, time varying state delay and disturbance (Benhayoun et al., 2013; Chen, 2010b; Huang et al., 2013; Roesser, 1975)   [xh(i+1,j)xv(i,j+1)] =Ap[xh(i,j)xv(i,j)]+Adp[xh(i−dh(i),j)xv(i,j−dv(j))]+B¯w[wh(i,j)wv(i,j)]+Bpu(i,j), 1a  y(i,j) =Cp[xh(i,j)xv(i,j)], 1b  z(i,j) =C¯z[xh(i,j)xv(i,j)]+D¯z[wh(i,j)wv(i,j)], 1c where $$i\in z_{+} $$, $$j\in z_{+} $$ and $$z_{+} $$ denotes the set of nonnegative integers. The $$\boldsymbol{x}^{h}(i,j)\;\in \boldsymbol{\Re}^{n}$$ and $$\boldsymbol{x}^{v}(i,j)\;\in \boldsymbol{\Re}^{m}$$ are the horizontal and the vertical states, respectively. The $$\boldsymbol{u}(i,j)\;\in \boldsymbol{\Re }^{p}$$ is input vector while $$\boldsymbol{y}(i,j)\;\in \boldsymbol{\Re }^{q}$$ and $$\boldsymbol{z}(i,j)\;\in \boldsymbol{\Re }^{t}$$ are measured output and controlled output vectors, respectively. The disturbance input in horizontal and vertical directions are $$\boldsymbol{w}^{h}(i,j)$$ and $$\boldsymbol{w}^{v}(i,j)$$. Matrices $$\boldsymbol{A}_{p} \in \boldsymbol{\Re}^{(n+m)\times (n+m)}$$, $$\boldsymbol{A}_{dp} \in \boldsymbol{\Re}^{(n+m)\times (n+m)}$$, $$\bar{\boldsymbol{B}}_{w} \in \boldsymbol{\Re}^{(n+m)\times (n+m)}$$, $$\boldsymbol{B}_{p} \;\in \;\boldsymbol{\Re}^{(n+m)\times p}$$, $$\boldsymbol{C}_{p} \;\in \;\boldsymbol{\Re}^{q\times (n+m)}$$, $$\bar{{\boldsymbol{C}}}_{z} \;\in \;\boldsymbol{\Re}^{t\times (n+m)}$$ and $$\bar{{\boldsymbol{D}}}_{z} \;\in \;\boldsymbol{\Re}^{t\times (n+m)}$$ are known constant matrices representing the nominal plant. In system (1a), $$ d_{h} (i) $$ and $$ d_{v} (j) $$ represent delays along horizontal direction and vertical directions, respectively. We assume that   dhL⩽dh(i)⩽dhH,dvL⩽dv(j)⩽dvH, 1d where $$ d_{hL} $$, $$ d_{hH} $$, $$ d_{vL} $$ and $$ d_{vH} $$ are constant nonnegative integers representing the lower and upper delay bounds along horizontal and vertical directions, respectively. It is assumed that these delays satisfy (1a). From (1a), it is clear that $$ d_{h} (i) $$ can assume any positive integer value in the interval $$ [d_{hL}, d_{hH} ] $$. Similarly, $$ d_{v} (j) $$ takes any positive integer value in the interval $$ [d_{vL} , d_{vH} ] $$. Therefore, the delays involved in system (1) are considered to be interval-like time varying state delays. It may be mentioned that such type of modelling of time varying delays has been widely used in the literature (Dey & Kar, 2014; Ghous & Xiang, 2015; Ghous et al., 2015; Luo et al., 2016; Tadepalli et al., 2015; Wang et al., 2013a). The initial conditions are defined as follows (Chen, 2010b; Tadepalli et al., 2015; Xu et al., 2013)   X(0) =[xhT(−dhH,0),xhT(−dhH,1),xhT(−dhH,2),…, xhT(1−dhH,0),xhT(1−dhH,1),xhT(1−dhH,2),…, xhT(0,0),xhT(0,1),xhT(0,2),…, xvT(0,−dvH),xvT(1,−dvH,),xvT(2,−dvH),…, xvT(0,1−dvH,),xvT(1,1−dvH),xvT(2,1−dvH),…, xvT(0,0),xvT(1,0),xvT(2,0),…]. 1e Let a linear two-dimensional dynamic compensator stabilizing the system (1) and meeting the desired performance specifications in absence of actuator saturation be given as   [xch(i+1,j)xcv(i,j+1)]=Ac[xch(i,j)xcv(i,j)]+Bcy(i,j), 2a  Vc(i,j)=Cc[xch(i,j)xcv(i,j)]+DcCp[xh(i,j)xv(i,j)], 2b where $$ \boldsymbol{x}_{c}^{h}(i,j)\;\in \boldsymbol{\Re}^{n_{c} } $$ and $$ \boldsymbol{x}_{c}^{v}(i,j)\;\in \boldsymbol{\Re}^{m_{c} } $$ are horizontal and vertical states of the controller, respectively. Vector $$ \boldsymbol{u}_{c} (i,j)=\boldsymbol{y}(i,j)\in \boldsymbol{\Re}^{q} $$ is a controller input vector and $$ \boldsymbol{V}_{c} (i,j)\in \boldsymbol{\Re}^{p} $$ is a controller output vector. The matrices $$\boldsymbol{A}_{c} \in \boldsymbol{\Re }^{(n_{c} +m_{c} )\times (n_{c} +m_{c} )},\boldsymbol{B}_{c} \in \boldsymbol{\Re }^{(n_{c} +m_{c} )\times q} $$, $$ \boldsymbol{C}_{c} \in \boldsymbol{\Re }^{p\times (n_{c} +m_{c} )} $$ and $$ \boldsymbol{D}_{c} \in \boldsymbol{\Re }^{p\times q} $$ are constant matrices of the desired controller. The input vector $$ \boldsymbol{u}(i,j)$$ is subjected to amplitude constrained defined as   −u¯(l)⩽u(l)(i,j)⩽u¯(l), 3 where $$ \overline u_{(l)} >0,l=1,2,\,{\ldots}\,,p$$ denote the control amplitude bounds. Therefore, the actual control signal injected to the system (1) can be written as   u(i,j) =sat(Vc(i,j)) =sat(Cc[xch(i,j)xcv(i,j)]+DcCp[xh(i,j)xv(i,j)]). 4 The saturation nonlinearities are characterized by   sat(Vc(i,j))(l)={−u¯(l)ifVc(l)<−u¯(l)Vc(l)if−u¯(l)⩽Vc(l)⩽u¯(l)u¯(l)ifVc(l)>u¯(l) ,l=1,2,…,p. 5 The actuator saturation causes windup of the controller and to mitigate its undesirable effect an anti-windup term $$\boldsymbol{E}_{c} (sat(\boldsymbol{V}_{c} (i,j))-\boldsymbol{V}_{c} (i,j)) $$ can be added to the controller (2) where $$ \boldsymbol{E}_{c} $$ is static anti-windup gain. Thus, modified system (1) and dynamic output controller (2) are represented by   [xh(i+1,j)xv(i,j+1)] =Ap[xh(i,j)xv(i,j)]+Adp[xh(i−dh(i),j)xv(i,j−dv(j))]+B¯w[wh(i,j)wv(i,j)] +Bp(sat(Cc[xch(i,j)xcv(i,j)]+DcCp[xh(i,j)xv(i,j)])), 6a  z(i,j) =C¯z[xh(i,j)xv(i,j)]+D¯z[wh(i,j)wv(i,j)], 6b  [xch(i+1,j)xcv(i,j+1)] =Ac[xch(i,j)xcv(i,j)]+BcCp[xh(i,j)xv(i,j)]+Ec(sat(Vc(i,j))−Vc(i,j)), 7a  Vc(i,j) =Cc[xch(i,j)xcv(i,j)]+DcCp[xh(i,j)xv(i,j)]. 7b It is noted here that $$ \boldsymbol{A}_{c} $$, $$ \boldsymbol{B}_{c} $$, $$ \boldsymbol{C}_{c}$$ and $$\boldsymbol{D}_{c} $$ are controller design parameters and are selected to meet the desired performance. Let us introduce the elementary matrix $$ \boldsymbol{{\it\Pi}} $$ as   Π=[In00000Inc00Im00000Imc], 8a where $$\boldsymbol{I}_{n}$$ is the identity matrix of order $$n$$ and $$\boldsymbol{{\it\Pi}}^{\boldsymbol{-1}}=\boldsymbol{{\it\Pi}}^{T}$$. Define an extended state vector which comprises of nominal plant and controller states   ξ(i,j)=[xh(i,j)xch(i,j)xv(i,j)xcv(i,j)]∈ℜn^+m^with n^=n+nc,  m^=m+mc. 8b Using (6), (7) and (8) the closed loop system is written as   [ξh(i+1,j)ξv(i,j+1)] =A[ξh(i,j)ξv(i,j)]+Ad[ξh(i−dh(i),j)ξv(i,j−dv(j))]+Bww(i,j)−(B+REc)ψ(K[ξh(i,j)ξv(i,j)]), 9a  z(i,j) =Cz[ξh(i,j)ξv(i,j)]+Dzw(i,j), 9b where   ξh(i+1,j) =[xhT(i+1,j)xchT(i+1,j)]T,ξv(i,j+1)=[xvT(i,j+1)xcvT(i,j+1)]T, 10  A =Π[Ap+BpDcCpBpCcBcCpAc]ΠT,B=Π[Bp0],R=Π[0Inc+mc],K=[DcCpCc]ΠT,Ad =Π[Adp000]ΠT,Bw=Π[B¯w000]ΠT,Cz=Π[C¯z000]ΠT,Dz=Π[D¯z000]ΠT,w(i,j) =[wh(i,j)0wv(i,j)0]∈ℜn^+m^,with the function ψ(Vc(i,j))=Vc(i,j)−sat(Vc(i,j)). 11 The boundary conditions for closed loop system (9) are given by   {ξh(i,j)=hij,∀ 0⩽j⩽r1,−dhH⩽i⩽0ξh(i,j)=0,∀ j>r1,−dhH⩽i⩽0ξv(i,j)=vij,∀ 0⩽i⩽r2,−dvH⩽j⩽0ξv(i,j)=0,∀ i>r2,−dvH⩽j⩽0 , 12 where $$ r_{1} $$ and $$ r_{2} $$ are finite positive integers, $$ \boldsymbol{h}_{ij} $$ and $$ \boldsymbol{v}_{ij} $$ are given vectors. It must be noted that $$ \left\| {\boldsymbol{w}(i,j)} \right\|_2 \leqslant \alpha ^{2} $$ with $$ \alpha $$ is positive constant. Remark 1 We assumed that the closed loop system (9) has a finite set of boundary conditions given by (12). These boundary conditions play a key role for deriving the asymptotic stability conditions for two-dimensional systems. With the appropriate choice of $$ r_{ 1} $$ and $$ r_{ 2} $$, it is possible to define the boundary conditions of dynamic compensator such that (12) holds. Remark 2 The two-dimensional dynamic compensator (2) is used to stabilize the system (1) in absence of saturation which, in turn, requires that the two-dimensional characteristic polynomial of the system (9) should not have any pole on or inside the unit bidisc. In other words, in absence of control bound, the closed loop system would be asymptotically stable for the boundary conditions (12). Generally, state feedback controller is best way to stabilize the control system. In some cases, all states of the system are not available for feedback due to complexity of the system, so output feedback method is preferred. In comparison to static, dynamic method of feedback is more applicable to practical systems in industries (Dong & Yang, 2009; Negi et al., 2012b; Nguyen et al., 2015; Tang et al., 2016; Wei-Wei & Guang-Hong, 2008; Zhang et al., 2014). The problem of designing dynamic compensator with a view to achieve input-to-state and input-to-output stability has been considered in Chai (2015); Gomes da Silva Jr. et al. (2013); Lam et al. (2004); Tang et al. (2016); Tarbouriech et al. (2011). Based on the methodology adopted in Gomes da Silva Jr. & Tarbouriech (2005, 2006); Negi et al. (2012b), it is assumed that the such type of dynamic compensator given in (2) exist. The main aim of this proposed work is to determine the anti-windup gains to guarantee the stability of closed loop system (9) and also to meet the performance requirement. The following definition and lemmas are needed in the proof of main result. Definition 1 (Kaczorek, 1985) The system (9) is asymptotically stable if $$ \lim_{q\to \infty } \chi_{q} =0 $$ for all given boundary conditions (12) where   χq=sup{‖ξ(i,j)‖:i+j=q,i,j⩾1},ξ(i,j)=[ξhT(i,j)ξvT(i,j)]T. 13 Lemma 1 (Qiu et al., 2008) For any constant matrix $$ \boldsymbol{W}\in \;\boldsymbol{\Re}^{m\times m} $$ with $$ \boldsymbol{W}=\boldsymbol{W}^{T}\geqslant \mathbf{0} $$, integers $$ l_{1} <l_{2} $$, vector function $$ \boldsymbol{\omega }:\left\{ {l_{1}, l_{1} +1,\ldots,l_{2} } \right\}\to \boldsymbol{\Re}^{m} $$ such that the sums concerned are well defined, then   (l2−l1+1)∑i=l1l2ωT(i)Wω(i)⩾(∑i=l1l2ω(i))TW(∑i=l1l2ω(i)). 14 Lemma 2 (Boyd et al., 1994) If there exist symmetric matrix $\boldsymbol{T}=\left[ \begin{array}{@{}cc@{}} {\boldsymbol{T}_{11} } & {\boldsymbol{T}_{12} } \\ {\boldsymbol{T}_{12}^{T} } & {\boldsymbol{T}_{22} } \\ \end{array} \right] $ with $$ \boldsymbol{T}_{11} $$ and $$ \boldsymbol{T}_{22} $$ are square matrices then the following statements are equivalent:   T<0T11<0,T22−T12TT11−1T12<0T22<0,T11−T12T22−1T12T<0. 15 Consider a matrix $$ \boldsymbol{G}\in \boldsymbol{\Re}^{p\times (\hat{{n}}+\hat{{m}})} $$ and define a polyhedral set   ℓ≜{ξ∈ℜ(n^+m^);−u¯(l)⩽(K(l)−G(l))ξ(i,j)⩽u¯(l),l=1,2,…,p}. 16 Lemma 3 If $$\boldsymbol{\xi}\in \ell$$ then   δℓ=2ψT(Kξ(i,j))D[ψ(Kξ(i,j))−Gξ(i,j)]⩽0, 17 where $$ \boldsymbol{D} $$ is positive definite diagonal matrix. Proof. The proof of Lemma 3 is very similar to Lemma 1 as given in Negi et al. (2012b); Gomes da Silva Jr. & Tarbouriech (2006). □ It is noted that the Lemma 3 has been used for the stability analysis of two-dimensional discrete system with actuator saturation in Negi et al. (2012b). 3. Main results The main results of the article are stated as follows. 3.1. Stability Analysis without disturbance Theorem 1 Consider two-dimensional system (9) with $$ \boldsymbol{w}(i,j)= $$ 0, for given positive scalars $$ d_{ha} $$ and $$ d_{va} $$ satisfying $$ d_{hL} \leqslant d_{ha} \leqslant d_{hH} $$ and $$ d_{vL} \leqslant d_{va} \leqslant d_{vH} $$, if there exist a diagonal positive definite matrix $$ \boldsymbol{L}\in \boldsymbol{\Re}^{p\times p} $$, symmetric matrices $$ \mathbf{0}<\boldsymbol{P}=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{Q}=\boldsymbol{Q}^{h}\oplus \boldsymbol{Q}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{W}_{k} =\boldsymbol{W}_{k}^{h} \oplus \boldsymbol{W}_{k}^{v} (k=1,2,3)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{R}_{k} =\boldsymbol{R}_{k}^{h} \oplus \boldsymbol{R}_{k}^{v} (k=1,2,3)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{X}_{k} (k=1,\ldots,5)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$ and matrices $$ \boldsymbol{H}\in \boldsymbol{\Re}^{(n_{c} +m_{c} )\times p} $$, $$ \boldsymbol{G}\in \boldsymbol{\Re}^{p\times (\hat{{n}}+\hat{{m}})} $$ satisfying the following set of LMIs   [Γ~11∗∗∗∗0−Q∗∗∗G0−2L∗∗R100W3−W1−R1−R2∗000R2W2−W3−R2−R30000R3AAd(−BL−RH)00AAd(−BL−RH)00d2(A−I)d2Add2(−BL−RH)00d3(A−I)d3Add3(−BL−RH)00d4(A−I)d4Add4(−BL−RH)00  ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−W2−R3∗∗∗∗∗0−2X1+X1PX1∗∗∗∗00−2X2+X2QX2∗∗∗000−2X3+X3R1X3∗∗0000−2X4+X4R2X4∗00000−2X5+X5R3X5]<0, 18  [PK(l)T−G(l)TK(l)−G(l)u0(l)2]>0,l=1,2,…,p, 19 where $$ \boldsymbol{\tilde{{\it\Gamma}}}_{11} =-\boldsymbol{P}+\boldsymbol{W}_{1} +\boldsymbol{d}_{1} \boldsymbol{Q}-\boldsymbol{R}_{1} $$, $ \boldsymbol{d}_{1} =\left[ \begin{array}{@{}cc@{}} {(d_{hH} -d_{hL} )\boldsymbol{I}_{h} } & {\mathbf{0}} \\ {\mathbf{0}} & {(d_{vH} -d_{vL} )\boldsymbol{I}_{v} } \\ \end{array} \right] $, $ \boldsymbol{d}_{2} =\left[ \begin{array}{@{}cc@{}} {d_{hH} \boldsymbol{I}_{h} } & {\mathbf{0}} \\ {\mathbf{0}} & {d_{vL} \boldsymbol{I}_{v} } \\ \end{array} \right] $,   d3=[(dha−dhL)Ih00(dva−dvL)Iv],d4=[(dhH−dha)Ih00(dvH−dva)Iv], 20 then, for the gain matrix $$ \boldsymbol{E}_{c} \;=\boldsymbol{HL}^{-\mbox{1}} $$ the ellipsoid $$ \varepsilon \;(\boldsymbol{P})=\left\{ {\boldsymbol{\xi}\in \Re ^{\hat{{n}}+\hat{{m}}};\boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant 1} \right\} $$, with $$ \boldsymbol{P}=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v} $$, is a region of asymptotic stability for the system (9). Further, an estimate of the domain of attraction for system (9) is given by   Γ(βrh,βrv) =[βrh2(λmax(Ph)+(dhH+1)λmax(Qh)+dhLλmax(W1h) +(dhH−dha)λmax(W2h)+(dha−dhL)λmax(W3h) +0.5(dhH−dhL)(dhH+dhL−1)λmax(Qh)+0.5(dhL)2(1+dhL)λmax(R1h) +0.5(dha−dhL)2(dhL+1+dha)λmax(R2h)+0.5(dhH−dha)2(dhH+dha+1)λmax(R3h)) +βrv2(λmax(Pv)+(dvH+1)λmax(Qv)+dvLλmax(W1v)+(dvH−dva)λmax(W2v) +(dva−dvL)λmax(W3v)+0.5(dvH−dvL)(dvH−1+dvL)λmax(Qv)+0.5(dvL)2(1+dvL)λmax(R1v)+0.5(dva−dvL)2(dvL+1+dva)λmax(R2v) +0.5(dvH−dva)2(dvH+dva+1)λmax(R3v))]⩽1, 21 where $$ \beta_{rh} =\max \left( {\sum\limits_{j=0}^{r_{ 1} } {\left\| {\boldsymbol{\xi}(-\sigma^{h},j)} \right\|} } \right)_{-d_{hH} \leqslant \sigma^{h}\leqslant 0} $$, $$ \beta_{rv} =\max \left( {\sum\limits_{i=0}^{r_{ 2} } {\left\| {\boldsymbol{\xi}(i,-\sigma^{v})} \right\|} } \right)_{-d_{vH} \leqslant \sigma ^{v}\leqslant 0} $$, $$ r_{1} $$ and $$ r_{2} $$ are finite positive integers. Proof. The proof of Theorem 1 is given in Appendix. □ Remark 3 For the given delay bounds $$ d_{hL} $$, $$ d_{vL} $$ and $$d_{ha}$$, $$ d_{va} $$ the bounds on delay $$ d_{hH} $$ and $$ d_{vH} $$ can be obtained by iteratively solving the LMIs of Theorem 1. 3.2. $$H_{\infty}$$ performance analysis with disturbance The performance of system (9) with sufficient condition for $$ H_{\infty } $$ disturbance attenuation is given in this subsection. Theorem 2 Consider two-dimensional system (9), for given positive scalars $$ \alpha $$, $$ \gamma $$, $$ d_{ha} $$ and $$ d_{va} $$ satisfying $$ d_{hL} \leqslant d_{ha} \leqslant d_{hH} $$ and $$ d_{vL} \leqslant d_{va} \leqslant d_{vH} $$, if there exist a diagonal positive definite matrix $$ \boldsymbol{L}\in \boldsymbol{\Re}^{p\times p} $$, symmetric matrices $$ \mathbf{0}<\boldsymbol{P}=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{Q}=\boldsymbol{Q}^{h}\oplus \boldsymbol{Q}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})}$$, $$\mathbf{0}<\boldsymbol{W}_{k} =\boldsymbol{W}_{k}^{h} \oplus \boldsymbol{W}_{k}^{v} ,(k=1,2,3)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{R}_{k} =\boldsymbol{R}_{k}^{h} \oplus \boldsymbol{R}_{k}^{v} (k=1,2,3)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})},\mathbf{0}<\boldsymbol{X}_{k} (k=1,\ldots,5)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$ and matrices $$ \boldsymbol{H}\in \boldsymbol{\Re}^{(n_{c} +m_{c} )\times p} $$, $$ \boldsymbol{G}\in \boldsymbol{\Re}^{p\times (\hat{{n}}+\hat{{m}})} $$ satisfying the following set of LMIs   [Γ~11∗∗∗∗∗∗0−Q∗∗∗∗∗G0−2L∗∗∗∗R100W3−W1−R1−R2∗∗∗000R2W2−W3−R2−R3∗∗0000R3−W2−R3∗000000−γ2IAAd(−BL−RH)000BwAAd(−BL−RH)000Bwd2(A−I)d2Add2(−BL−RH)000d2Bwd3(A−I)d3Add3(−BL−RH)000d3Bwd4(A−I)d4Add4(−BL−RH)000d4BwCz00000Dz  ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−2X1+X1PX1∗∗∗∗∗0−2X2+X2QX2∗∗∗∗00−2X3+X3R1X3∗∗∗000−2X4+X4R2X4∗∗0000−2X5+X5R3X5∗00000−I]<0, 22  [PK(l)T−G(l)TK(l)−G(l)τu0(l)2]⩾0,l=1,2,…,p, 23 where   τ=1/(1+γ2α2), 24 then, for the gain matrix $$ \boldsymbol{E}_{c} \;=\boldsymbol{HL}^{-\mbox{1}} $$ the ellipsoid $$ \varepsilon \;(\boldsymbol{P},1+\gamma^{2}\alpha^{2})=\left\{ {\boldsymbol{\xi}\in \Re^{\hat{{n}}+\hat{{m}}};\;\boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant \;1+\gamma^{2}\alpha^{2}} \right\} $$, with $$ \boldsymbol{P}\;=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v} $$, is a region of asymptotic stability for the system (9). Proof. Consider   β(i,j)=Δv(ξ(i,j))+zT(i,j)z(i,j)−γ2wT(i,j)w(i,j). 25 Adopting the procedure of proof of Theorem 1, we get   β(i,j)⩽ϑ¯T(i,j)I−bϑ¯(i,j), 26 where   ϑ¯(i,j)=[ϑT(i,j)wT(i,j)]T, 27  I−b =[ℏ11∗∗∗∗∗∗ℏ21ℏ22∗∗∗∗∗ℏ31ℏ32ℏ33∗∗∗∗R100−W1+W3−R1−R2∗∗∗000R2W2−W3−R2−R3∗∗0000R3−W2−R3∗ℏ71ℏ72ℏ73000ℏ77], 28  ℏ11 =ATPA−P+ATQA+W1+d1Q+d22(A−I)TR1(A−I)+d32(A−I)TR2(A−I) +d42(A−I)TR3(A−I)−R1+CzTCz, 29  ℏ21 =AdTPA+AdTQA+d22AdTR1(A−I)+d32AdTR2(A−I)+d42AdTR3(A−I), 30a  ℏ22 =AdTPAd+AdTQAd+d22AdTR1Ad+d32AdTR2Ad+d42AdTR3Ad−Q, 30b  ℏ31 =(−B−REc)TPA+(−B−REc)TQA+d22(−B−REc)TR1(A−I) +d32(−B−REc)TR2(A−I)+DG+d42(−B−REc)TR3(A−I), 31a  ℏ32 =(−B−REc)TPAd+(−B−REc)TQAd+d22(−B−REc)TR1Ad+d32(−B−REc)TR2Ad +d42(−B−REc)TR3Ad, 31b  ℏ33 =(−B−REc)TP(−B−REc)+(−B−REc)TQ(−B−REc)+d22(−B−REc)TR1(−B−REc) +d32(−B−REc)TR2(−B−REc)+d42(−B−REc)TR3(−B−REc)−2D, 32a  ℏ71 =BwTPA+BwTQA+d22BwTR1(A−I)+d32BwTR2(A−I)+d42BwTR3(A−I)+DzTCz, 32b  ℏ72 =BwTPAd+BwTQAd+d22BwTR1Ad+d32BwTR2Ad+d42BwTR3Ad, 33a  ℏ73 =BwTP(−B−REc)+BwTQ(−B−REc)+d22BwTR1(−B−REc) +d32BwTR2(−B−REc)+d42BwTR3(−B−REc), 33b  ℏ77 =BwTPBw+BwTQBw+d22BwTR1Bw+d32BwTR2Bw+d42BwTR3Bw+DzTDz−γ2I. 34 In the light of Lemma 2, (22) is equivalent to $$ {\boldsymbol{I}{\kern-3pt-\kern-3pt}\boldsymbol{b}}<\mathbf{0} $$, which implies   β(i,j)=Δv(ξ(i,j))+zT(i,j)z(i,j)−γ2wT(i,j)w(i,j)<0, 35 i.e.   −Δv(ξ(i,j))>zT(i,j)z(i,j)−γ2wT(i,j)w(i,j). 36 If $$ \left\| {\boldsymbol{w}} \right\|_{2} \leqslant \alpha^{2} $$ and $$ \boldsymbol{z}^{T}(i,j)\boldsymbol{z}(i,j)>0 $$ then (36) gives   ∑i+j=rv(i,j)<∑i+j=zv(i,j)+γ2α2. 37 It is seen that (23) corresponds to condition (19) and could be obtained from following   P−τ−1(K(l)−G(l))T(K(l)−G(l))u0(l)−2⩾0,l=1,2,…,p. 38 The satisfaction of (23) follows that the set $$ \varepsilon \;(\boldsymbol{P},1+\gamma^{2}\alpha^{2})=\left\{ {\boldsymbol{\xi}\;\in \Re^{n+n_{c} +m+m_{c} };\boldsymbol{\xi}^{T}\boldsymbol{P}\boldsymbol{\xi}\leqslant 1+\gamma^{2}\alpha^{2}} \right\} $$ is included in polyhedral set $$\ell$$ defined as in (16). Further, applying zero boundary condition on (36), we get   ∑i+j=0∞(zT(i,j)z(i,j))<γ2∑i+j=0∞((wT(i,j)w(i,j)). 39 Therefore, (39) implies   ‖z‖22<γ2‖w‖22. 40 Hence, system (9) achieves $$H_{\infty}$$ disturbance attenuation level $$\gamma$$ in presence of input saturation and interval like time varying state delay. Further, it is shown that $$\boldsymbol{\xi}(i,j)\to \mathbf{0} $$ as $$ i\to \infty$$ and/or $$j\to \infty$$ for boundary conditions given by (12). It follows from $ {\it\Delta} v\left( {\left[ \begin{array}{@{}c@{}} {\boldsymbol{\xi}^{h}(i,j)} \\ {\boldsymbol{\xi}^{v}(i,j)} \\ \end{array} \right]\;} \right)\leqslant 0 $ that   vh(ξh(i+1,j))+vv(ξv(i,j+1))⩽vh(ξh(i,j))+vv(ξv(i,j)) =v([ξh(i,j)ξv(i,j)])∀ξ(i,j)∈ε(P,1). 41 For any nonnegative integer $$ \kappa $$, summing up both side of (41) from $$ 0 $$ to $$ \kappa $$ with respect to $$ i $$ and $$ \kappa $$ to $$ 0 $$ with respect to $$ j $$, we get   vh(ξh(1,κ))+vv(ξv(0,κ+1))+vh(ξh(2,κ−1))+vv(ξv(1,κ)) +…+vh(ξh(κ+1,0))+vv(ξv(κ,1)) ⩽v([ξh(0,κ)ξv(0,κ)])+…+v([ξh(κ,0)ξv(κ,0)]), 42   ∑i+j=κ+1vh(ξh(i,j))+∑i+j=κ+1vv(ξv(i,j))⩽∑i+j=κv([ξh(i,j)ξv(i,j)]), 43   ∑i+j=κ+1v([ξh(i,j)ξv(i,j)])⩽∑i+j=κv([ξh(i,j)ξv(i,j)]). 44 From Definition 1, it can be concluded that   lim i→∞ and/or j→∞ξ(i,j)=limi+j→∞ξ(i,j)→0. 45 Therefore, equation (45) intends that system (9) is asymptotically stable. Similarly, it can also be proved $$\forall i+j\in r\in z_{+}$$ that   ξT(i,j)Pξ(i,j)⩽∑i+j=rv(ξ(i,j))⩽∑j=0r1vh(ξh(0,j))+∑i=0r2vv(ξv(i,0)),∀ξ(i,j)∈ε(P,τ−1) 46   ⩽βrh2(λmax(Ph)+(dhH+1)λmax(Qh)+dhLλmax(W1h)+(dhH−dha)λmax(W2h)+(dha−dhL)λmax(W3h) +0.5(dhH−dhL)(dhH+dhL−1)λmax(Qh)+0.5(dhL)2(1+dhL)λmax(R1h) +0.5(dha−dhL)2(dhL+1+dha)λmax(R2h) +0.5(dhH−dha)2(dhH+dha+1)λmax(R3h))+βrv2(λmax(Pv) +(dvH+1)λmax(Qv)+dvLλmax(W1v)+(dvH−dva)λmax(W2v)+(dva−dvL)λmax(W3v) +0.5(dvH−dvL)(dvH−1+dvL)λmax(Qv)+0.5(dvL)2(1+dvL)λmax(R1v) +0.5(dva−dvL)2(dvL+1+dva)λmax(R2v)+0.5(dvH−dva)2(dvH+dva+1)λmax(R3v)) 47   =Γ(βrh,βrv)⩽1 48 The set $$ {\it\Gamma}_{(\beta_{rh}, \beta_{rv} )} \leqslant 1 $$, and $$ \boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant \;1+\gamma^{2}\alpha ^{2} $$ is satisfied. Thus, all the trajectories of $$ \boldsymbol{\xi}(i,j) $$ starting from $$ {\it\Gamma}_{(\beta_{rh}, \beta_{rv} )} \leqslant 1 $$ remain within $$ \varepsilon \;(\boldsymbol{P},1+\gamma^{2}\alpha^{2}) $$. □ As a direct consequence of Theorem 2, Corollary 1 states the sufficient global asymptotic stability condition for system (9). Corollary 1 For given positive scalars $$\alpha $$, $$ \gamma $$, $$ d_{ha} $$, $$ d_{va} $$ satisfying $$ d_{hL} \leqslant d_{ha} \leqslant d_{hH} $$ and $$ d_{vL} \leqslant d_{va} \leqslant d_{vH} $$, if there exist a diagonal positive definite matrix $$ \boldsymbol{L}\in \boldsymbol{\Re}^{p\times p} $$, symmetric matrices $$ \mathbf{0}<\boldsymbol{P}=\boldsymbol{P}^{h}\oplus \boldsymbol{P}^{v} \quad \in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{Q}=\boldsymbol{Q}^{h}\oplus \boldsymbol{Q}^{v}\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{W}_{k} =\boldsymbol{W}_{k}^{h} \oplus \boldsymbol{W}_{k}^{v} \quad (k=1,2,3) \in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{R}_{k} =\boldsymbol{R}_{k}^{h} \oplus \boldsymbol{R}_{k}^{v} (k=1,2,3) \quad \in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$, $$ \mathbf{0}<\boldsymbol{X}_{k} (k=1,\ldots,5)\in \boldsymbol{\Re}^{(\hat{{n}}+\hat{{m}})\times (\hat{{n}}+\hat{{m}})} $$ and matrix $$ \boldsymbol{H}\in \boldsymbol{\Re}^{(n_{c} +m_{c} )\times p} $$ satisfying following LMI   [Γ~11∗∗∗∗∗∗0−Q∗∗∗∗∗K0−2L∗∗∗∗R100W3−W1−R1−R2∗∗∗000R2W2−W3−R2−R3∗∗0000R3−W2−R3∗000000−γ2IAAd(−BL−RH)000BwAAd(−BL−RH)000Bwd2(A−I)d2Add2(−BL−RH)000d2Bwd3(A−I)d3Add3(−BL−RH)000d3Bwd4(A−I)d4Add4(−BL−RH)000d4BwCz00000Dz  ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−2X1+X1PX1∗∗∗∗∗0−2X2+X2QX2∗∗∗∗00−2X3+X3R1X3∗∗∗000−2X4+X4R2X4∗∗0000−2X5+X5R3X5∗00000−I]<0, 49 then, for the gain matrix $$ \boldsymbol{E}_{c} \;=\boldsymbol{HL}^{-\mbox{1}} $$ the origin of system (9) is globally asymptotically stable with prescribed $$H_{\infty}$$ disturbance attenuation level $$\gamma$$. Proof. Choosing,   G=K. 50 One can see that (16) is automatically met for all $$\boldsymbol{\xi}\in \boldsymbol{\Re}^{\hat{{n}}+\hat{{m}}}$$. Now, substituting (50) into (22) we obtain the global asymptotic stability condition (49) for system (9). This completes the proof. The condition for global stability is valid only when the open loop system is asymptotically stable (Gomes da Silva Jr. & Tarbouriech, 2006). □ 4. Numerical examples Example 1 Consider the parameters of two-dimensional system represented by (1)   Ap =[0.2150.08⋮0.00.2−0.04⋮0.01⋯⋯⋯⋯⋯⋯⋯⋯0.010.01⋮−0.1],Adp=[0.100⋮0.050.100.02⋮0⋯⋯⋯⋯⋯⋯⋯⋯0.030.12⋮0.03],Bp =[0.80−0.010.01…………0.0010.002]Cp=[10000.10.01]C¯z =[0.010000.060.002],D¯z=[0.0060.0050.0020.006],B¯w=[0.010⋮0.0020.0070.008⋮0.002⋯⋯⋯⋯⋯⋯⋯⋯0.0020.006⋮0.008] and stabilizing dynamic output feedback controller is given by   Ac =[−0.051⋮0………⋮………0⋮−0.501],Bc=[0.5000.6],Cc =[−0.495800−0.51],Dc=[−0.01100−0.527]. The saturation of control signal is characterized by $$ -0.6\leqslant \overline {\boldsymbol{u}} \leqslant 0.6 $$. By selecting $$ d_{hL} = d_{vL} =1, d_{ha} =9 $$, $$ d_{va} =25 $$ and iteratively solving the LMIs (22)–(23) with respect to $$ d_{hH} $$ and $$ d_{vH} $$, it is seen that the system (9) is asymptotically stable for $$ 1\leqslant d_{h} (i)\leqslant 11 $$ and $$ 1\leqslant d_{v} (j)\leqslant 29 $$. For different values of delay ranges, $$ d_{hL}, d_{hH} $$, $$ d_{vL}, d_{vH} $$, $$ d_{ha} $$ and $$ d_{va} $$ the anti-windup controller gains are computed as shown in Table 1. Table 1 Computational results Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 5, \enspace d_{ha} =3$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =5$$  $\left[ \begin{array}{@{}cc@{}} {0.1772} & {-0.0022} \\ {0.0035} & {-0.0001} \\ \end{array} \right]$  $$ 1\leqslant d_{h} (i)\leqslant 8, \enspace d_{ha} =6$$  $$ 1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =11$$  $ \left[ \begin{array}{@{}cc@{}} {0.1565} & {-0.0029} \\ {0.0022} & {0.0001} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 10, \enspace d_{ha} =8$$  $$ 1\leqslant d_{v} (j)\leqslant 22, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.1389} & {-0.0044} \\ {-0.0019} & {0.0022} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 11, \enspace d_{ha} =9$$  $$ 1\leqslant d_{v} (j)\leqslant 29, \enspace d_{va} =25$$  $ \left[ \begin{array}{@{}cc@{}} {0.1196} & {-0.0005} \\ {-0.0043} & {0.0026} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 12 $$  $$ 1\leqslant d_{v} (j)\leqslant 30 $$  Infeasible  Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 5, \enspace d_{ha} =3$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =5$$  $\left[ \begin{array}{@{}cc@{}} {0.1772} & {-0.0022} \\ {0.0035} & {-0.0001} \\ \end{array} \right]$  $$ 1\leqslant d_{h} (i)\leqslant 8, \enspace d_{ha} =6$$  $$ 1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =11$$  $ \left[ \begin{array}{@{}cc@{}} {0.1565} & {-0.0029} \\ {0.0022} & {0.0001} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 10, \enspace d_{ha} =8$$  $$ 1\leqslant d_{v} (j)\leqslant 22, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.1389} & {-0.0044} \\ {-0.0019} & {0.0022} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 11, \enspace d_{ha} =9$$  $$ 1\leqslant d_{v} (j)\leqslant 29, \enspace d_{va} =25$$  $ \left[ \begin{array}{@{}cc@{}} {0.1196} & {-0.0005} \\ {-0.0043} & {0.0026} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 12 $$  $$ 1\leqslant d_{v} (j)\leqslant 30 $$  Infeasible  Table 1 Computational results Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 5, \enspace d_{ha} =3$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =5$$  $\left[ \begin{array}{@{}cc@{}} {0.1772} & {-0.0022} \\ {0.0035} & {-0.0001} \\ \end{array} \right]$  $$ 1\leqslant d_{h} (i)\leqslant 8, \enspace d_{ha} =6$$  $$ 1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =11$$  $ \left[ \begin{array}{@{}cc@{}} {0.1565} & {-0.0029} \\ {0.0022} & {0.0001} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 10, \enspace d_{ha} =8$$  $$ 1\leqslant d_{v} (j)\leqslant 22, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.1389} & {-0.0044} \\ {-0.0019} & {0.0022} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 11, \enspace d_{ha} =9$$  $$ 1\leqslant d_{v} (j)\leqslant 29, \enspace d_{va} =25$$  $ \left[ \begin{array}{@{}cc@{}} {0.1196} & {-0.0005} \\ {-0.0043} & {0.0026} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 12 $$  $$ 1\leqslant d_{v} (j)\leqslant 30 $$  Infeasible  Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 5, \enspace d_{ha} =3$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =5$$  $\left[ \begin{array}{@{}cc@{}} {0.1772} & {-0.0022} \\ {0.0035} & {-0.0001} \\ \end{array} \right]$  $$ 1\leqslant d_{h} (i)\leqslant 8, \enspace d_{ha} =6$$  $$ 1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =11$$  $ \left[ \begin{array}{@{}cc@{}} {0.1565} & {-0.0029} \\ {0.0022} & {0.0001} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 10, \enspace d_{ha} =8$$  $$ 1\leqslant d_{v} (j)\leqslant 22, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.1389} & {-0.0044} \\ {-0.0019} & {0.0022} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 11, \enspace d_{ha} =9$$  $$ 1\leqslant d_{v} (j)\leqslant 29, \enspace d_{va} =25$$  $ \left[ \begin{array}{@{}cc@{}} {0.1196} & {-0.0005} \\ {-0.0043} & {0.0026} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 12 $$  $$ 1\leqslant d_{v} (j)\leqslant 30 $$  Infeasible  The state trajectories of two-dimensional system are depicted in Fig. 1. It can be seen that all the three states are converging to zero with $$\gamma = 0.4$$, $$\alpha = 0.01$$ and boundary conditions are given as   ξh(i,j) ={[−0.030.05−0.6] 0⩽j⩽120j>12 ,ξv(i,j)={[−0.040.51] 0⩽i⩽120i>12 ,{xh(i,j)=[−0.030.05], ∀ 0⩽j⩽12xv(i,j)=−0.04, ∀ 0⩽i⩽12 . Fig. 1. View largeDownload slide Horizontal and vertical state trajectories. Fig. 1. View largeDownload slide Horizontal and vertical state trajectories. Time varying state delays are $$ d_{h} (i)=6+5sin\left( {\frac{\pi i}{2}} \right) $$, $$ d_{v} (j)=15+14sin\left( {\frac{\pi j}{2}} \right) $$ and disturbances are $ \left[\begin{array}{@{}c@{}} {\boldsymbol{w}^{h}(i,j)} \\ \cdots \\ {\boldsymbol{w}^{v}(i,j)} \\ \end{array} \right]=\left[ \begin{array}{@{}c@{}} {0.8e^{-0.05(i+j)}} \\ {0.7e^{-0.05(i+j)}} \\ {\ldots \ldots \ldots \ldots } \\ {0.4e^{-0.05(i+j)}} \\ \end{array} \right] $. Fig. 2. View largeDownload slide Control effort without anti-windup. Fig. 2. View largeDownload slide Control effort without anti-windup. Fig. 3. View largeDownload slide Control effort with anti-windup for $$ \overline u _{(l)} \,=\pm 0.6 $$. Fig. 3. View largeDownload slide Control effort with anti-windup for $$ \overline u _{(l)} \,=\pm 0.6 $$. Figure 2 depicts the control effort without anti-windup. Figure 3 represents the control effort with anti-wind up controller. From Fig. 2, it can be seen that the values of control effort are beyond the limit given by $$ \overline u _{(l)} \,=\pm 0.6 $$. By using anti-windup controller, it is clear from Fig. 3 that control efforts are with in specified control limit i.e. $$ \overline u_{(l)} =\pm 0.6 $$. Remark 4 It is noticed from above example that upper bound on delays found for horizontal and vertical states are 11 and 29, respectively whereas in Ghous & Xiang (2015), the upper bounds on delays in both states are found to be 3. It can be seen that there is remarkable increase in delay range by using anti-windup compensator with sector bounded conditions for saturation nonlinearity as compared to state feedback controller given in Ghous & Xiang (2015) with convex hull technique to represent saturation nonlinearity. It can also be observed that the value of $$\gamma$$ attenuation constant has reduced to 0.4 in the proposed work in comparison to work reported by Ghous & Xiang (2015). Example 2 In this example, the applicability of Theorem 2 to control of dynamical processes in gas absorption, water steam heating and air drying, which are represented by Darboux equation (Du et al., 2001; Ghous & Xiang, 2015; Marszalek, 1984; Negi et al., 2012b) is demonstrated. Consider the Darboux equation given by   ∂2s(x,t)∂x∂t=a1∂s(x,t)∂t+a2∂s(x,t)∂x+a0s(x,t)+ads(x,t−d)+bww(x,t)+bf(x,t), 51 with the initial conditions:   s(x,0)=p(x),s(0,t)=q(t), 52 where $$ s(x,t) $$ is an unknown function at space $$ x\;\in \;[0,x_{f} ] $$ and time $$ t\;\in \;[0,\infty ] $$ ; $$ f(x,t) $$ is the input function subjected to saturation; $$ w(x,t) $$ is disturbance; $$ a_{1}, \;a_{2}, \;a_{0} $$, $$ a_{d} $$, $$ b_{w} $$ and $$ b $$ are real constants. The two-dimensional model of Darboux equation with actuator saturation and delay can be represented in the following form   [xh(i+1,j)xv(i,j+1)]=[(1+a1Δx)(a1a2+a0)ΔxΔt(1+a2Δt)][xh(i,j)xv(i,j)]+[bwΔx0]w(x,t)+[0adΔx00][xh(i−dh(i),j)xv(i,j−dv(j))]+[bΔx0]u(i,j), 53 with the initial conditions   xh(0,j)=z(jΔt),xv(i,0)=p(iΔx). 54 The parameters of two-dimensional system with time varying delay and saturation are   Ap=[0.110.0030.10.24],Adp=[0.0002000],Bp=[0.080.09−0.010.01],Cp=[0.210.0030.10.24],B¯w=[0.010.050.050.001],C¯z=[0.0300.50.02],D¯z=[0.06000.082]. The dynamic output feedback controller which stabilizes the above plant is given by   Ac=[0.0210.0930.160.084],Bc=[0.050.050.00260.10],Cc=[0.010.0360.0020],Dc=[−0.0071−0.0090.01−0.80]. Using MATLAB LMI toolbox (Gahinet et al., 1995), it is verified that (22) and (23) are found feasible for control bound $$ -0.5\leqslant \overline {\boldsymbol{u}} \leqslant 0.5 $$, $$ \gamma = 0.5$$, $$ \alpha = 0.01$$. By selecting $$ d_{hL} = d_{vL} =1, \quad d_{ha} =18 $$, $$ d_{va} =15 $$ and iteratively solving the LMIs (22)–(23) with respect to $$ d_{hH} $$ and $$ d_{vH} $$, it is seen that the system (9) is asymptotically stable for $$ 1\leqslant d_{h} (i)\leqslant 23 $$ and $$ 1\leqslant d_{v} (j)\leqslant 21 $$. Table 2 shows the anti-windup gains for different delay ranges. Table 2 Computational results Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 9, \enspace d_{ha} =5$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =7$$  $\left[ \begin{array}{@{}cc@{}} {0.0060} & {-0.0020} \\ {0.0018} & {-0.0010} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 15, \enspace d_{ha} =11$$  $$1\leqslant d_{v} (j)\leqslant 14, \enspace d_{va} =9$$  $\left[ \begin{array}{@{}cc@{}} {0.0086} & {-0.0021} \\ {0.0055} & {-0.0016} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 20, \enspace d_{ha} =13$$  $$1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =12$$  $\left[ \begin{array}{@{}cc@{}} {0.0117} & {-0.0027} \\ {0.0096} & {-0.0025} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 23, \enspace d_{ha} =18$$  $$ 1\leqslant d_{v} (j)\leqslant 21, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.0135} & {-0.0030} \\ {0.0209} & {-0.0032} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 24$$  $$ 1\leqslant d_{v} (j)\leqslant 22$$  Infeasible  Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 9, \enspace d_{ha} =5$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =7$$  $\left[ \begin{array}{@{}cc@{}} {0.0060} & {-0.0020} \\ {0.0018} & {-0.0010} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 15, \enspace d_{ha} =11$$  $$1\leqslant d_{v} (j)\leqslant 14, \enspace d_{va} =9$$  $\left[ \begin{array}{@{}cc@{}} {0.0086} & {-0.0021} \\ {0.0055} & {-0.0016} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 20, \enspace d_{ha} =13$$  $$1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =12$$  $\left[ \begin{array}{@{}cc@{}} {0.0117} & {-0.0027} \\ {0.0096} & {-0.0025} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 23, \enspace d_{ha} =18$$  $$ 1\leqslant d_{v} (j)\leqslant 21, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.0135} & {-0.0030} \\ {0.0209} & {-0.0032} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 24$$  $$ 1\leqslant d_{v} (j)\leqslant 22$$  Infeasible  Table 2 Computational results Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 9, \enspace d_{ha} =5$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =7$$  $\left[ \begin{array}{@{}cc@{}} {0.0060} & {-0.0020} \\ {0.0018} & {-0.0010} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 15, \enspace d_{ha} =11$$  $$1\leqslant d_{v} (j)\leqslant 14, \enspace d_{va} =9$$  $\left[ \begin{array}{@{}cc@{}} {0.0086} & {-0.0021} \\ {0.0055} & {-0.0016} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 20, \enspace d_{ha} =13$$  $$1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =12$$  $\left[ \begin{array}{@{}cc@{}} {0.0117} & {-0.0027} \\ {0.0096} & {-0.0025} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 23, \enspace d_{ha} =18$$  $$ 1\leqslant d_{v} (j)\leqslant 21, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.0135} & {-0.0030} \\ {0.0209} & {-0.0032} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 24$$  $$ 1\leqslant d_{v} (j)\leqslant 22$$  Infeasible  Delay Range     $$d_{hL} \leqslant d_{h} (i)\leqslant d_{hH}$$  $$d_{vL} \leqslant d_{v} (j)\leqslant d_{vH}$$  $$\boldsymbol{E}_{c} =\boldsymbol{HL}^{-1}$$  $$1\leqslant d_{h} (i)\leqslant 9, \enspace d_{ha} =5$$  $$1\leqslant d_{v} (j)\leqslant 10, \enspace d_{va} =7$$  $\left[ \begin{array}{@{}cc@{}} {0.0060} & {-0.0020} \\ {0.0018} & {-0.0010} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 15, \enspace d_{ha} =11$$  $$1\leqslant d_{v} (j)\leqslant 14, \enspace d_{va} =9$$  $\left[ \begin{array}{@{}cc@{}} {0.0086} & {-0.0021} \\ {0.0055} & {-0.0016} \\ \end{array} \right] $  $$1\leqslant d_{h} (i)\leqslant 20, \enspace d_{ha} =13$$  $$1\leqslant d_{v} (j)\leqslant 18, \enspace d_{va} =12$$  $\left[ \begin{array}{@{}cc@{}} {0.0117} & {-0.0027} \\ {0.0096} & {-0.0025} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 23, \enspace d_{ha} =18$$  $$ 1\leqslant d_{v} (j)\leqslant 21, \enspace d_{va} =15$$  $ \left[ \begin{array}{@{}cc@{}} {0.0135} & {-0.0030} \\ {0.0209} & {-0.0032} \\ \end{array} \right] $  $$ 1\leqslant d_{h} (i)\leqslant 24$$  $$ 1\leqslant d_{v} (j)\leqslant 22$$  Infeasible  The horizontal and vertical state trajectories are plotted in Fig. 4. The control effort is shown in Fig. 5 without considering the effect of anti-windup compensator. It can be seen from Fig. 6 that the control bounds are within specified limit $$ \overline u_{(l)} \,=\pm 0.5 $$ by using anti-windup compensator. Fig. 4. View largeDownload slide Horizontal and vertical state trajectories. Fig. 4. View largeDownload slide Horizontal and vertical state trajectories. Fig. 5. View largeDownload slide Control effort without anti-windup. Fig. 5. View largeDownload slide Control effort without anti-windup. Fig. 6. View largeDownload slide Control effort with anti-windup for $$\overline u_{(l)} \,=\pm 0.5 $$. Fig. 6. View largeDownload slide Control effort with anti-windup for $$\overline u_{(l)} \,=\pm 0.5 $$. In this example, the boundary conditions are   ξh(i,j) ={[0.040.05] 0⩽j⩽100j>10 ,ξv(i,j)={[−0.060.5] 0⩽i⩽100i>10 ,{xh(i,j)=0.04, ∀ 0⩽j⩽10xv(i,j)=−0.06, ∀ 0⩽i⩽10 . Time varying delays in horizontal and vertical directions are $$d_{h} (i)=12+11sin\left( {\frac{\pi i}{2}} \right) $$ and $$ d_{v} (j)=11+10sin\left( {\frac{\pi j}{2}} \right) $$, respectively. The disturbances are $ \left[ \begin{array}{@{}c@{}} {\boldsymbol{w}^{h}(i,j)} \\ \cdots \\ {\boldsymbol{w}^{v}(i,j)} \\ \end{array} \right]=\left[ \begin{array}{@{}c@{}} {0.5e^{-0.05(i+j)}} \\ \ldots \\ {0.3e^{-0.05(i+j)}} \\ \end{array} \right] $. Remark 5 As quoted in Remark 1 for Example 1, the same observation is make for Example 2. Example 3 Consider the two-dimensional discrete system in Roesser Model setting (1) and the stabilizing controller (2) with   Ap =[0.150.01⋮0.00.2−0.04⋮0.01⋯⋯⋯⋯⋯⋯⋯⋯0.010.01⋮−0.1],Adp=[0.100⋮0.040.100.02⋮0⋯⋯⋯⋯⋯⋯⋯⋯0.030.12⋮0.03],Bp =[0.80−0.010.01…………0.0010.002],Cp=[10000.10.01],B¯w =[0.0080⋮0.0020.0070.008⋮0.002⋯⋯⋯⋯⋯⋯⋯⋯0.0020.006⋮0.008],C¯z=[0.010.500.040.060.008],D¯z=[0.007−0.0050.0020.006],Ac =[−0.097⋮0.02………⋮………0⋮−0.2],Bc=[0.40.060.030.7],Cc =[−0.6−0.80−0.51],Dc=[−0.0400−0.7]. By selecting $$ d_{hL} = d_{vL} =1, \quad d_{ha} =6 $$, $$ d_{va} =10 $$ and iteratively solving the LMI (49) of Corollary 1 with respect to $$d_{hH}$$ and $$d_{vH}$$, it is seen that the system (9) is asymptotically stable for $$ 1\leqslant d_{h} (i)\leqslant 9 $$ and $$1\leqslant d_{v} (j)\leqslant 31$$. For above delay ranges, the control bound is given by $$-1\leqslant \overline {\boldsymbol{u}} \leqslant 1$$ and the values of unknown parameters are calculated as follows-   Ph =[96.45255.2149−8.3618 5.2149111.6712−23.1212−8.3618−23.1212136.2397],Pv=[105.7727−0.5450−0.5450227.6591],H=[−0.00110.0001−0.00050.0027],L =[0.009000.0605]. The anti-windup gain of stabilizing compensator is obtained as   Ec=HL−1=[−0.12180.0010−0.05670.0444]. Remark 6 In the present work, an anti-windup compensator is designed and static anti-windup gains are determined to stabilize the system (1). A sector condition (see (16–17)) is used to characterize the actuator saturation nonlinearity. By contrast, (Benhayoun et al., 2013; Ghous & Xiang, 2015; Hu et al., 2002) deal with the design of state feedback controller where convex hull approach is adopted for characterization of saturation nonlinearity. In Benhayoun et al. (2013); Ghous & Xiang (2015); Hu et al. (2002), the stability conditions are expressed as a convex combination of $$ 2^{p} $$ (where input $$\boldsymbol{u}(i,j)\;\in \boldsymbol{\Re}^{p}) $$ LMIs. Therefore, as compared with Benhayoun et al. (2013); Ghous & Xiang (2015); Hu et al. (2002), the present approach is beneficial in terms of computational complexity. 5. Conclusion In this article, Lyapunov based stability analysis of two-dimensional systems in presence of input saturation, time varying state delay and disturbances has been carried out for Roesser model. The problem is dealt with anti-windup paradigm using generalized sector condition which directly formulates the LMI conditions. The large number of iterations and complexity are reduced in comparison to polytopic differential inclusion (Gomes da Silva Jr. & Tarbouriech, 2005; Tarbouriech et al., 2007). $$H_{\infty}$$ disturbance attenuation performance analysis has been carried out and sufficient conditions are stated in context of local and global stability of closed loop system. Application of the proposed anti-windup controller is demonstrated through processes described by a Darboux equation (Du et al. 2001; Ghous & Xiang 2015; Marszalek 1984; Negi et al. 2012b). A domain of attraction is also estimated. From the numerical examples given in Section 4, it is clear that the anti-windup strategy used along with dynamic output feedback controller for two-dimensional discrete systems gives better results in the sense that the delay range has been increased and also disturbance attenuation level is improved in comparison with the result reported in the previous works (e.g. Ghous & Xiang 2015). Further, following the idea of Chai (2015); Lam et al. (2004); Tang et al. (2016); Tadepalli et al. (2015) this problem can be extended to stabilize the two-dimensional delayed systems with actuator saturation and uncertainties in concern to practical systems in industries, where the parameters of plant are frequently perturbed. Acknowledgements The authors would like to thank the editors, anonymous reviewers for their constructive comments and suggestions to improve the manuscript. The special thanks to Prof. Haranath Kar, for his support, encouragement and fruitful discussion on this topic. Appendix Proof of Theorem 1. Consider a two-dimensional quadratic Lyapunov function Ghous & Xiang (2015)   v([ξh(i,j)ξv(i,j)])=v(ξ(i,j))=vh(ξh(i,j))+vv(ξv(i,j)), A.1 where   vh(ξh(i,j)) =∑k=15vkh(ξh(i,j)), A.2  v1h(ξh(i,j)) =ξhT(i,j)Phξh(i,j), A.3  v2h(ξh(i,j)) =∑r=i−dh(i)iξhT(r,j)Qhξh(r,j), A.4  v3h(ξh(i,j)) =∑r=i−dhLi−1ξhT(r,j)W1hξh(r,j)+∑r=i−dhHi−dha−1ξhT(r,j)W2hξh(r,j)+∑r=i−dhai−dhL−1ξhT(r,j)W3hξh(r,j), A.5  v4h(ξh(i,j)) =∑s=−dhH+1−dhL∑r=i+si−1ξhT(r,j)Qhξh(r,j), A.6  v5h(ξh(i,j)) =dhL∑s=−dhL−1∑r=i+si−1ηhT(r,j)R1hηh(r,j)+(dha−dhL)∑s=−dha−dhL−1∑r=i+si−1ηhT(r,j)R2hηh(r,j) +(dhH−dha)∑s=−dhH−dha−1∑r=i+si−1ηhT(r,j)R3hηh(r,j), A.7  vv(ξv(i,j)) =∑k=15vkv(ξv(i,j)), A.8  v1v(ξv(i,j)) =ξvT(i,j)Pvξv(i,j), A.9  v2v(ξv(i,j)) =∑b=j−dv(j)jξvT(i,b)Qvξv(i,b), A.10  v3v(ξv(i,j)) =∑b=j−dvLj−1ξvT(i,b)W1vξv(i,b)+∑b=j−dvHj−dva−1ξvT(i,b)W2vξv(i,b)+∑b=j−dvaj−dvL−1ξvT(i,b)W3vξv(i,b), A.11  v4v(ξv(i,j)) =∑s=−dvH+1−dvL∑b=j+sj−1ξvT(i,b)Qvξv(i,b), A.12  v5v(ξv(i,j)) =dvL∑s=−dvL−1∑b=j+sj−1ηvT(i,b)R1vηv(i,b)+(dva−dvL)∑s=−dva−dvL−1∑b=j+sj−1ηvT(i,b)R2vηv(i,b) +(dvH−dva)∑s=−dvH−dva−1∑b=j+sj−1ηvT(i,b)R3vηv(i,b). A.13 Define   {ηh(r,j)=ξh(r+1,j)−ξh(r,j)ηv(i,b)=ξv(i,b+1)−ξv(i,b) . A.14 Taking forward difference of Lyapunov functional along trajectories of system (9)   Δv([ξh(i,j)ξv(i,j)]) =∑k=15vkh(ξh(i+1,j))−∑k=15vkh(ξh(i,j))+∑k=15vkv(ξv(i,j+1))−∑k=15vkv(ξv(i,j)), A.15   =Δv1(ξ(i,j))+Δv2(ξ(i,j))+Δv3(ξ(i,j))+Δv4(ξ(i,j))+Δv5(ξ(i,j)), A.16 where   Δv1(ξ(i,j))=Δv1h(ξh(i,j))+Δv1v(ξv(i,j)), A.17  Δv1(ξ(i,j))=[ξh(i+1,j)ξv(i,j+1)]TP[ξh(i+1,j)ξv(i,j+1)]−[ξh(i,j)ξv(i,j)]TP[ξh(i,j)ξv(i,j)], A.18    =ξT(i,j)ATPAξ(i,j)+ξT(i,j)ATP(−B−REc)ψ(Kξ(i,j))  +ξT(i,j)ATPAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]+[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTPAξ(i,j)  +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTPAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTP(−B−REc)ψ(Kξ(i,j))  +ψT(Kξ(i,j))(−B−REc)TPAξ(i,j)+ψT(Kξ(i,j))(−B−REc)TPAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +ψT(Kξ(i,j))(−B−REc)TP(−B−REc)ψ(Kξ(i,j))−ξT(i,j)Pξ(i,j), A.19  Δv2(ξ(i,j))=Δv2h(ξh(i,j))+Δv2v(ξv(i,j)), A.20  Δv2(ξ(i,j))=[ξh(i+1,j)ξv(i,j+1)]TQ[ξh(i+1,j)ξv(i,j+1)]−[ξh(i−dh(i),j)ξv(i,j−dv(j)]TQ[ξh(i−dh(i),j)ξv(i,j−dv(j)], A.21    =ξT(i,j)ATQAξ(i,j)+ξT(i,j)ATQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +ξT(i,j)ATQ(−B−REc)ψ(Kξ(i,j))+[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQAξ(i,j)  +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQ(−B−REc)ψ(Kξ(i,j))  +ψT(Kξ(i,j))(−B−REc)TQAξ(i,j)+ψT(Kξ(i,j))(−B−REc)TQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))]  +ψT(Kξ(i,j))(−B−REc)TQ(−B−REc)ψ(Kξ(i,j))−[ξh(i−dh(i),j)ξv(i,j−dv(j)]TQ[ξh(i−dh(i),j)ξv(i,j−dv(j)]. A.22 Adding a term in (A.22), we get   Δv2(ξ(i,j))⩽ξT(i,j)ATQAξ(i,j)+ξT(i,j)ATQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))] +ξT(i,j)ATQ(−B−REc)ψ(Kξ(i,j))+[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQAξ(i,j) +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))] +[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]AdTQ(−B−REc)ψ(Kξ(i,j)) +ψT(Kξ(i,j))(−B−REc)TQAξ(i,j)+ψT(Kξ(i,j))(−B−REc)TQAd[ξh(i−dh(i),j)ξv(i,j−dv(j))] +ψT(Kξ(i,j))(−B−REc)TQ(−B−REc)ψ(Kξ(i,j))−[ξh(i−dh(i),j)ξv(i,j−dv(j)]TQ[ξh(i−dh(i),j)ξv(i,j−dv(j)] +[∑r=i+1−dhHi−dhLξh(r,j)∑b=j+1−dvHj−dvLξv(i,b)]TQ[∑r=i+1−dhHi−dhLξh(r,j)∑b=j+1−dvHj−dvLξv(i,b)]. A.23  Δv3(ξ(i,j))=Δv3h(ξh(i,j))+Δv3v(ξv(i,j)). A.24  Δv3(ξ(i,j))=[ξh(i,j)ξv(i,j)]TW1[ξh(i,j)ξv(i,j)]−[ξh(i−dhL,j)ξv(i,j−dvL)]TW1[ξh(i−dhL,j)ξv(i,j−dvL)]  +[ξh(i−dha,j)ξv(i,j−dva)]TW2[ξh(i−dha,j)ξv(i,j−dva)]−[ξh(i−dhH,j)ξv(i,j−dvH)]TW2[ξh(i−dhH,j)ξv(i,j−dvH)]  +[ξh(i−dhL,j)ξv(i,j−dvL)]TW3[ξh(i−dhL,j)ξv(i,j−dvL)]−[ξh(i−dha,j)ξv(i,j−dva)]TW3[ξh(i−dha,j)ξv(i,j−dva)]. A.25  Δv4(ξ(i,j))=Δv4h(ξh(i,j))+Δv4v(ξv(i,j)). A.26  Δv4(ξ(i,j))=[ξh(i,j)ξv(i,j)]T[(dhH−dhL)I00(dvH−dvL)I]Q[ξh(i,j)ξv(i,j)] −[∑r=i−dhH+1i−dhLξh(r,j)∑b=j−dvH+1j−dvLξv(i,b)]TQ[∑r=i−dhH+1i−dhLξh(r,j)∑b=j−dvH+1j−dvLξv(i,b)]. A.27  Δv5(ξ(i,j))=Δv5h(ξh(i,j))+Δv5v(ξv(i,j)). A.28  Δv5(ξ(i,j))=[ηh(i,j)ηv(i,j)]T[dhL2I00dvL2I]R1[ηh(i,j)ηv(i,j)]−[∑r=i−dhLi−1ηh(r,j)∑b=j−dvLj−1ηv(i,b)]T[dhLI00dvLI]R1[ηh(r,j)ηv(i,b)] +[ηh(i,j)ηv(i,j)]T[(dha−dhL)2I00(dva−dvL)2I]R2[ηh(i,j)ηv(i,j)] −[∑r=i−dhai−dhL−1ηh(r,j)∑b=j−dvaj−dvL−1ηv(i,b)]T[(dha−dhL)I00(dva−dvL)I]R2[ηh(r,j)ηv(i,b)] +[ηh(i,j)ηv(i,j)]T[(dhH−dha)2I00(dvH−dva)2I]R3[ηh(i,j)ηv(i,j)] −[∑r=i−dhHi−dha−1ηh(r,j)∑b=j−dvHj−dva−1ηv(i,b)]T[(dhH−dha)I00(dvH−dva)I]R3[ηh(r,j)ηv(i,b)]. A.29 From Lemma 1, (A.29) is rewritten as   (A.30) where   {∑r=i−dhLi−1ξh(r+1,j)−∑r=i−dhLi−1ξh(r,j)=ξh(i,j)−ξh(i−dhL,j)∑r=i−dhai−dhL−1ξh(r+1,j)−∑r=i−dhai−dhL−1ξh(r,j)=ξh(i−dhL,j)−ξh(i−dha,j)∑r=i−dhHi−dha−1ξh(r+1,j)−∑r=i−dhHi−dha−1ξh(r,j)=ξh(i−dha,j)−ξh(i−dhH,j) , A.31  {∑b=j−dvLj−1ξv(i,b+1)−∑b=j−dvLj−1ξv(i,b)=ξv(i,j)−ξv(i,j−dvL)∑b=j−dvaj−dvL−1ξv(i,b+1)−∑b=j−dvaj−dvL−1ξv(i,b)=ξv(i,j−dvL)−ξv(i,j−dva)∑b=j−dvHj−dva−1ξv(i,b+1)−∑b=j−dvHj−dva−1ξv(i,b)=ξv(i,j−dva)−ξv(i,j−dvH) . A.32 Employing Lemma 3 with (A.15)–(A.30), following inequality is stated   Δv(ξ(i,j))⩽ϑT(i,j)dpϑ(i,j)−δℓ, A.33 where   dp =[dp11∗∗∗∗∗dp21dp22∗∗∗∗dp31dp32dp33∗∗∗R100−W1+W3−R1−R2∗∗000R2W2−W3−R2−R3∗0000R3−W2−R3], A.34  dp11 =ATPA−P+ATQA+W1+d1Q+d22(A−I)TR1(A−I)+d32(A−I)TR2(A−I)  +d42(A−I)TR3(A−I)−R1, A.35  dp21 =AdTPA+AdTQA+d22AdTR1(A−I)+d32AdTR2(A−I)+d42AdTR3(A−I), A.36  dp22 =AdTPAd+AdTQAd+d22AdTR1Ad+d32AdTR2Ad+d42AdTR3Ad−Q, A.37  dp31 =(−B−REc)TPA+(−B−REc)TQA+d22(−B−REc)TR1(A−I)  +d32(−B−REc)TR2(A−I)+d42(−B−REc)TR3(A−I), A.38  dp32 =(−B−REc)TPAd+(−B−REc)TQAd+d22(−B−REc)TR1Ad +d32(−B−REc)TR2Ad+d42(−B−REc)TR3Ad, A.39  dp33 =(−B−REc)TP(−B−REc)+(−B−REc)TQ(−B−REc)+d22(−B−REc)TR1(−B−REc)  +d32(−B−REc)TR2(−B−REc)+d42(−B−REc)TR3(−B−REc). A.40 Using inequality (17) of Lemma 3, $$\delta_{\ell }$$ is defined as   δℓ=2ψT(Kξ(i,j))D[ψT(Kξ(i,j))−Gξ(i,j)], where Dis positive definite diagonal matrix A.41 and   ϑ(i,j) =[ξT(i,j)ξdhT(i,j)ψT(Kξ(i,j))ξd_T(i,j)ξaT(i,j)ξd¯T(i,j)]T,ξdh(i,j) =[ξhT(i−dh(i),j)ξvT(i,j−dv(j))]T,ξd_(i,j)=[ξhT(i−dhL,j)ξvT(i,j−dvL]T,ξa(i,j) =[ξhT(i−dha,j)ξvT(i,j−dva)]T,ξd¯(i,j)=[ξhT(i−dhH,j)ξvT(i,j−dvH)]T. A.42 To ensure the local asymptotic stability of closed loop system (9) ${\it\Delta} v\left( {\left[ \begin{array}{@{}c@{}} {\boldsymbol{\xi}^{h}(i,j)} \\ {\boldsymbol{\xi}^{v}(i,j)} \\ \end{array} \right]\;} \right)<0 $. Therefore, it must satisfy that $${\boldsymbol{d}{\kern-3.8pt}\boldsymbol{p}} {\boldsymbol{<0}}$$ with $$\boldsymbol{\vartheta }(i,j)\ne \mathbf{0} $$. Further, applying Schur’s complement (Boyd et al., 1994), (A.33) is equivalent to   [Γ~11∗∗∗∗∗∗∗∗∗∗0−Q∗∗∗∗∗∗∗∗∗DG0−2D∗∗∗∗∗∗∗∗R100W3−W1−R1−R2∗∗∗∗∗∗∗000R2W2−W3−R2−R3∗∗∗∗∗∗0000R3−W2−R3∗∗∗∗∗AAd(−B−REc)000−P−1∗∗∗∗AAd(−B−REc)0000−Q−1∗∗∗d2(A−I)d2Add2(−B−REc)00000−R1−1∗∗d3(A−I)d3Add3(−B−REc)000000−R2−1∗d4(A−I)d4Add4(−B−REc)0000000−R3−1]<0, A.43 Pre and post multiplying (A.43) by $$ diag(\boldsymbol{I},\boldsymbol{I},\boldsymbol{D}^{-1},\boldsymbol{I},\boldsymbol{I},\boldsymbol{I},\boldsymbol{I},\boldsymbol{I},\boldsymbol{I,I,I}) $$ with $$ \boldsymbol{D}^{-1}=\boldsymbol{L} $$ and $$ \boldsymbol{E}_{c} =\boldsymbol{HL}^{-1} $$ along with the following assumptions (Chen & Fong, 2010; Negi et al., 2012a)   {(X1−P−1)(−P)(X1−P−1)⩽0⇒−P−1⩽−2X1+X1PX1(X2−Q−1)(−Q)(X2−Q−1)⩽0⇒−Q−1⩽−2X2+X2QX2(X3−R1−1)(−R1)(X3−R1−1)⩽0⇒−R1−1⩽−2X3+X3R1X3(X4−R2−1)(−R2)(X4−R2−1)⩽0⇒−R2−1⩽−2X4+X4R2X4(X5−R3−1)(−R3)(X5−R3−1)⩽0⇒−R3−1⩽−2X5+X5R3X5 , A.44 (18) is obtained. The satisfaction of condition stated in (19) signifies that the set $$\varepsilon \;(\boldsymbol{P})=\left\{ {\boldsymbol{\xi}\;\in \Re ^{\hat{{n}}+\hat{{m}}};\;\boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant \;1} \right\} $$ is included in polyhedral set $$ \ell $$ as defined in (16). It can be proven that $$ \varepsilon \;(\boldsymbol{P})=\left\{ {\boldsymbol{\xi}\;\in \Re^{\hat{{n}}+\hat{{m}}};\;\boldsymbol{\xi}^{T}\boldsymbol{P}\;\boldsymbol{\xi}\leqslant \;1} \right\} $$ is equivalent to (Boyd et al., 1994)   P−(K(l)−G(l))T(K(l)−G(l))u0(l)−2⩾0,l=1,2,…,p. A.45 Pre and post multiplication of (A.45) by $$ \boldsymbol{\xi}^{T} $$ and $$ \boldsymbol{\xi} $$ respectively, it follows that $$ \boldsymbol{\xi}\;\in \ell $$ for all $$ \boldsymbol{\xi}\in \varepsilon \;(\boldsymbol{P}) $$. The relation (19) is obtained using Schur’s complement on (A.45). 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Published: Jan 11, 2017

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