LMI approach for robust stabilization of Takagi–Sugeno descriptor systems with input saturation

LMI approach for robust stabilization of Takagi–Sugeno descriptor systems with input saturation Abstract The aim of this article is to present a systematic framework for the design of a robust controller for uncertain nonlinear systems represented in the Takagi–Sugeno descriptor form. The design takes explicitly into account state constraints and also the saturation of the control input. Based on Lyapunov theory, the tuning of the control parameters can be done by solving an optimization problem with linear matrix inequality constraints. Two numerical examples are included to illustrate the effectiveness of the proposed method. 1. Introduction More and more often, in sectors such as automotive or robotics, the control design of a process needs to take in consideration its nonlinear feature. The presence of non-linearities in the model usually makes the synthesis of the controller more difficult to perform. One way to overcome such difficulties is to use a Takagi–Sugeno (T–S) model of the system (Takagi and Sugeno, 1985). As explained in Tanaka and Wang (2001), if a nonlinear model based, for instance, on physical principles is available for the process to control, an equivalent T–S representation valid on a specified domain may be obtained by using the sector-nonlinearity approach. Based on Lyapunov stability theory and particular choices of control laws such as parallel distributed compensation (PDC), the design of the nonlinear controller can then be realized through a general framework as described in the monograph (Tanaka and Wang 2001) or the survey articles (Sala et al., 2005; Feng, 2006; Guerra et al., 2015b). In this framework, the requirements of closed-loop stability and other performances are translated in terms of linear matrix inequalities (LMI) problems that can be efficiently solved by convex optimization techniques (Boyd et al., 1994). Nevertheless, it is well known that the number of rules in the T–S representation grows exponentially with the number of nonlinearities appearing in the original nonlinear model, leading to possible numerical problems in the resolution of the optimization problems. It is thus of the highest interest to obtain a T–S representation of a given nonlinear system with a reduced number of linear models. For many systems, especially for mechanical devices, a natural model is obtained on descriptor form (see, for instance Luenberger (1977)). Its transformation in a standard T–S representation often leads to a model with numerous rules. A more convenient way to proceed is to preserve the original structure of the nonlinear model by rewriting it on the form of a T–S descriptor model as proposed by (Taniguchi et al., 1999). As emphasized in that article, but also in Guelton et al. (2008); Vermeiren et al. (2012), this may lead to models more adequate for the control design. The design approach developed for standard T–S models has been extended to the descriptor TS models in Taniguchi et al. (1999, 2000); Bouarar et al. (2007, 2008, 2010); Vermeiren et al. (2012); Chadli et al. (2014). Extensions to time-delay systems have been considered in Gassara et al. (2014); Luo et al. (2014). Descriptor representations may also simplify the design of controllers or observers for standard Takagi–Sugeno models as shown in the references Tanaka et al. (2007); Guelton et al. (2009); Guerra et al. (2015a). Due to technological or safety reasons, the actuator saturation is certainly the most frequently encountered nonlinear phenomenon in practice. Ignoring its effects in the design of a controller can lead to a significant degradation of the closed-loop performance and, even to the destabilization of the controlled system. Therefore, the synthesis of stabilizing controllers for dynamical systems under input saturation have attracted the attention of several researchers. For linear systems, there are many works dedicated to this problem; the interested readers may refer to the monographs Hu and Lin (2001), Kapila and Grigoriadis (2002), Hippe (2006) or Tarbouriech et al. (2011). Concerning T–S models, relying on the polytopic representation of the saturation function due to (Hu and Lin, 2001), the authors of (Cao and Lin, 2003) propose a method for the synthesis of a PDC controller taking into account the actuator saturation. In Ariño et al. (2010), another approach for dealing with input constraints is proposed based on an optimal iterative design. Recently, in Bezzaoucha et al. (2013) (see also Bezzaoucha et al. (2015)), the authors used the sector nonlinearity approach to obtain a new polytopic representation of the saturation characteristic. A main limitation of this approach is that it applies only to T–S models involving local linear models that are all open-loop stable. In order to overcome this drawback, the authors in Dang et al. (2015b) used a similar approach involving a smaller number of decision variables. This approach has been extended to the case of descriptor T–S models in Dang et al. (2015a). To our knowledge, only the works (Gassara et al., 2014; Luo et al., 2014) deal with the stabilization of saturated descriptor systems. Note that these two articles consider descriptor models of singular type with a common matrix on the state derivative (i.e. $$E_k=E$$, for $$k=1,\dots,ne$$ in this article notation). We propose in this article a new LMI-based framework which allows the synthesis of stabilizing controllers for uncertain T–S descriptor models under actuator saturation. The design takes also explicitly into account constraints on the state variables that may due to modelling reason or other technical considerations. 2. Notations The following notations will be used in this article: $$\Re^{n\times m}$$ is the set of $$n\times m$$ real matrices; $$\Re^{n}$$ denotes the set of real vectors of size $$n$$; $$\mathbb{N}_{n}$$ denotes the set of natural numbers $$ \{1,...,n\}$$; $$\mathbb{S}_{n}$$ denotes the subset of $$\Re^n$$ defined by: $$\mathcal{S}_{n}=\{x\in\Re^n:(\forall i\in \mathbb{N}_{n},\ x_i\ge0)\text{ and }\sum_{i=1}^{n}x_i=1\}$$; $$A_h$$ will denote the convex combination $$A_h = \sum\limits_{i = 1}^n h_i A_i$$ associated with a family $$(A_1,\dots,\ A_n)$$ of matrices having the same dimensions and a vector $$h\in\mathcal{S}_{n}$$. Similarly, $$A_{hh}$$ will denote a convex combination of the form $$ \sum\limits_{i = 1}^{n} \sum\limits_{j = 1}^{n} h_i\,h_j\,A_{ij}$$,... $$A^T$$ denotes the transpose of the matrix $$A$$; $$A_{(i)}$$ represents the $$i$$-th row of $$A$$; $$I$$, $$0$$ are the identity and zero matrices of appropriate dimensions; $${\rm 1}\kern-0.24em{\rm I}$$ a vector of adequate dimension with all entries equal to 1; $${\text{diag}}\left(A_1,\dots,\ A_n\right)$$ is the block diagonal matrix with square matrices $$A_1,\dots,$$$$\ A_n$$ as diagonal elements; Given two symmetric matrices $$P$$ and $$Q$$, $$P \succ Q$$ means that the matrix $$P-Q$$ is positive definite. Relations ‘$$\prec$$’ and ‘$$\preceq$$’ are defined similarly. Notation $$P\succ0$$ will be a shortcut for meaning that $$P$$ is a symmetric and positive-definite matrix; $${\it{\Omega}}_{P}(K,\ u)$$ denotes the polyhedral region associated with a matrix $$K \in {\Re ^{nu \times nx}}$$ and a vector $$u\in\Re ^{nu}$$ defined by ΩP(K, u)={x∈ℜnx:|K(i)x|≤ui, ∀i∈Nnu}; For $$P\succ0$$, $${\it{\Omega}}_{E}(P) = \left\{x\in \Re^{nx}: x^{T}P^{-1}x\le 1\right\}$$, that is, an ellipsoidal set containing the origin; * denotes elements that can be deduced by symmetry in a symmetric matrix. 3. Problem statement 3.1. System description In this article, we consider a class of uncertain nonlinear systems expressed in the following descriptor form: (E(x)+ΔE(t,x))x˙=(A(x)+ΔA(t,x))x+(B(x)+ΔB(t,x))sat(u) (3.1) where the state variable $$x(t)\in\Re^{nx}$$ and the control input $$u(t)\in \Re^{nu}$$. The vector $${\text{sat}}(u)\in\Re^{nu}$$ has for $$i$$-th entry sat(u)i=sign(ui)min(u0,i, |ui|), (3.2) where $${u_{0,i}} > 0$$ is the saturation bound of the $${i}$$th control input. For technological or safety reasons, we assume that the state $$x$$ remains, or has to remain, in a polyhedral region $${\it{\Omega}}_x$$ characterized by a matrix $$N\in \Re^{nc\times nx}$$ and defined as Ωx=ΩP(N,1I). (3.3) It is assumed that, on the set $${\it{\Omega}}_x$$ and for all $$t\ge 0$$, all entries of the different system matrices $$E(x)$$,..., $${\it{\Delta}} B(t,x)$$ are bounded, and that the matrix $$E(x)+{\it{\Delta}} E(t,x)$$ is regular for any uncertain matrix $${\it{\Delta}} E(t,x)$$ as described below. The descriptor system (3.1) is therefore of non-singular type. Applying the sector nonlinearity approach described in Tanaka and Wang (2001), a Takagi–Sugeno representation of the system (3.1), valid on $${\it{\Omega}}_x$$, may be obtained: (Ew+ΔEw(t))x˙=(Ah+ΔAh(t))x+(Bh+ΔBh(t))sat(u), (3.4) where the scheduling variables $$w(t)$$ and $$h(t)$$ are, respectively, in $$\mathbb{S}_{re}$$, and $$\mathbb{S}_{ra}$$. The uncertain matrices $${\it{\Delta}} A_i$$, ...admit the following decompositions: ΔEk(t)=HkeΔe(t)WkeΔAi(t)=HiaΔa(t)WiaΔBi(t)=HibΔb(t)Wib (3.5) for $$k\in \mathbb{N}_{re}$$ and $$i\in\mathbb{N}_{ra}$$. The matrices $$H_k^e$$, $$H_i^a$$, $$H_i^b$$, $$W_k^e$$, $$W_i^a$$ and $$W_i^b $$ are assumed constant and known, while the time-varying matrices $${\it{\Delta}}_e$$, $${\it{\Delta}}_a$$ and $${\it{\Delta}}_b$$ may be unknown but satisfy: Δℓ(t)TΔℓ(t)⪯I,∀t≥0, with ℓ=e, a, or b (3.6) 3.2. Control problem definition The main goal of this article is to propose a systematic framework for designing a controller that will stabilize an uncertain nonlinear system represented in the T–S descriptor form (3.4) with the state and input constraints (3.3)–(3.2). In order to achieve this goal, the following properties will be considered: Property 1 ($$\alpha $$-stability) There exists an ellipsoidal set $${{\it{\Omega}}_e}\subset \Re^{nx}$$ such that any trajectory of the closed-loop system (CLS) starting from one of its point converge exponentially to the origin; the decay rate being less than a positive real number $$\alpha $$. Property 2 (State constraint satisfaction) Any trajectory of the CLS starting from a point of $${{\it{\Omega}}_e}$$ remains in the polyhedral region $${{\it{\Omega}}_x}$$ described in (3.3). 4. Synthesis of the control law In order to stabilize the uncertain T–S descriptor system (3.4), we consider the following modified PDC law: u =∑j=1ra∑k=1rehj(z)wk(z)KjkP11−1x =KhwP11−1x, (4.1) where the feedback gain matrices $$K_{jk}$$ and the matrix $$P_{11}\succ 0$$ are to be determined such that the properties in Section 3.2 are respected. In order to take into account the saturation of the input, we will consider the polytopic representation proposed in Hu and Lin (2001). Let $$\left\{ V_{p}^{+} : p \in \mathbb{N}_{2^{nu}}\right\}$$ be the set of all diagonal matrices in $$\Re^{nu\times nu}$$ whose diagonal elements take the value 0 or 1. For a matrix $$V_{p}^{+}$$ in this set, the associated matrix $$V_{p}^{-}$$ is defined as $$V_{p}^{-} = I-V_{p}^{+}$$. Then, for any $$u,\ v \in \Re^{nu}$$ with $$v={\text{sat}}(v)$$, there exists a $$\mu\in\mathcal{S}_{ru}$$ (with $$ru=2^{nu}$$) such that sat(u)=∑p=1ruμp(u)(Vp+u+Vp−v)=Vμ+u+Vμ−v (4.2) Let us define the polyhedral region $${{\it{\Omega}} _u}\subset\Re^{nx}$$ as follows: Ωu=⋂j∈Nra,k∈NreΩP(LjkP11−1,u0), (4.3) where $$L_{jk}\in\Re^{nu\times nx}$$ are matrices to be defined later. From (4.1) and (4.2), for any $$x \in {{\it{\Omega}} _u}$$, the saturated control input can be expressed as: sat(u) =∑j=1ra∑k=1re∑p=1ruhj(z)wk(z)μp(u)(Vp+Kjk+Vp−Ljk)P11−1x =(Vμ+Khw+Vμ−Lhw)P11−1x (4.4) By introducing the augmented state variable $\bar{x}=\begin{bmatrix} x^{T} & \dot{x}^{T} \end{bmatrix}^T$, the closed-loop system can be written as: E¯x¯˙=(A¯+ΔA¯)x¯, (4.5) where E¯ =[I000]A¯ =[0IAh+Bh(Vμ+Khw+Vμ−Lhw)P11−1−Ew]ΔA¯ =[00ΔAh+ΔBh(Vμ+Khw+Vμ−Lhw)P11−1−ΔEw] Theorem 4.1 For a given $$\alpha>0$$, the uncertain T–S descriptor system (3.4) is stabilized by a modified PDC control law of the form (4.1) and satisfies properties 1 and 2 if there exist matrices $$P_{11}\succ 0$$, $$P_{ij}^{21}$$, $$P_{ij}^{22} \in \Re^{nx \times nx}$$, $$K_{jk}$$, $$L_{jk} \in \Re^{nu \times nx}$$ and positive real numbers $$\tau_i^a$$, $$\tau_{ijkp}^b$$, $$\tau_{ijk}^e$$ such that the following inequalities are satisfied: [−P11T∗Ljk(l)−u0,l2I] ⪯0 (4.6) [−P11T∗N(q)P11−I] ⪯0 (4.7) Υiikp ≺0 (4.8) 2ra−1Υiikp+Υijkp+Υjikp ≺0(for i≠j) (4.9) for all $$(i,j,k,p,l,q)\in \mathbb{N}_{ra} \times \mathbb{N}_{ra} \times\mathbb{N}_{re} \times \mathbb{N}_{ru}\times\mathbb{N}_{nu}\times\mathbb{N}_{nc}$$, where the symmetric, partitioned matrix $$\Upsilon_{ijkp}=\left\{\Upsilon^{mn}_{ijkp}\right\}_{(m,n)\in \mathbb{N}_{5}\times\mathbb{N}_{5}}$$ is defined by: Υijkp11 =Pij21+(Pij21)T+2αP11 (4.10) Υijkp21 =AiP11+(Pij22)T−EkPij21+Bi(Vp+Kjk+Vp−Ljk) (4.11) Υijkp22 =−EkPij22−(Pij22)TEkT+τiaHia(Hia)T+τijkpbHib(Hib)T+τijkeHke(Hke)T (4.12) Υijkp31 =WiaP11 (4.13) Υijkp33 =−τiaI (4.14) Υijkp41 =Wib(Vp+Kjk+Vp−Ljk) (4.15) Υijkp44 =−τijkpbI (4.16) Υijkp51 =−WkePij21 (4.17) Υijkp52 =−WkePij22 (4.18) Υijkp55 =−τijkeI (4.19) (unspecified blocks that cannot be obtained by symmetry are assumed equal to a zero matrix). Proof. Inequalities (4.6) imply that the ellipsoidal set $${\it{\Omega}}_e={\it{\Omega}}_E(P_{11})$$ contains the polyhedral set $${\it{\Omega}}_u$$. Similarly, inequalities (4.7) imply that $${\it{\Omega}}_e$$ contains the set $${\it{\Omega}}_x$$. We will now prove that the set $${\it{\Omega}}_e$$ is invariant with respect to the closed-loop dynamics and contained in the domain of attraction of $$x=0$$. For that, note first that, according to the relaxation lemma given in Tuan et al. (2001), Conditions 4.8 and 4.9 imply that inequality ∑i=1ra∑j=1ra∑k=1re∑p=1ruhihjwkμpΥijkp≺0 (4.20) holds for all $$(h,w,\mu)\in \mathcal{S}_{ra}\times \mathcal{S}_{re}\times \mathcal{S}_{ru}$$. One may then deduce that $$P_{hh}^{22}$$ is a regular matrix for all $$h\in\mathcal{S}_{ra}$$. Let us consider the Lyapunov candidate function: V(x)=xTP11−1x (4.21) Function $$V$$ can be rewritten as $$V(x)=\bar{x}^{T}\bar{E}P_{hh}^{-1}\bar{x}$$, where the matrix $$P_{hh}$$ is defined as Phh=[P110Phh21Phh22]. (4.22) The time-derivative of the Lyapunov candidate function is V˙(x) =x¯˙TE¯Phh−1x¯+x¯TE¯Phh−1x¯˙+x¯TE¯P˙hh−1x¯ =x¯T(A¯+ΔA¯)TPhh−1+(Phh−1)T(A¯+ΔA¯)x¯ (4.23) The solution $$x=0$$ of the closed-loop system (4.5) is exponentially stable with a decay rate less than $$\alpha $$ if: V˙(x¯)+2αV(x¯)=x¯T[(A¯+ΔA¯)TPhh−1+(Phh−1)T(A¯+ΔA¯)+2αE¯Phh−1x¯<0 (4.24) for all $$x\neq 0$$. This is satisfied if (A¯+ΔA¯)TPhh−1+(Phh−1)T(A¯+ΔA¯)+2αE¯Phh−1≺0 (4.25) Applying a congruence transformation with $$P_{hh}$$, the previous inequality (33) can be shown to be equivalent to PhhTA¯T+A¯Phh+2αPhhTE¯+PhhTΔA¯T+ΔA¯Phh≺0 (4.26) The uncertain terms in (4.26) can be decomposed as PhhTΔA¯T+ΔA¯Phh=HΔW+WTΔTHT, (4.27) where H =[000HhaHhbHwe];Δ=[Δa000Δb000Δe];W =[WhaP110Whb(Vμ+Khw+Vμ−Lhw)0−WwePhh21−WwePhh22] Completing the square, it comes straightforwardly the following inequality PhhTΔA¯T+ΔA¯Phh⪯HSHT+WTS−1W (4.28) with $$S = {\text{diag}}(\tau_{h}^{a}I,\ \tau_{hhw\mu}^{b}I,\ \tau_{hhw}^{e}I)$$. Applying Schur complement, it can be derived that inequality (4.26) is satisfied if 4.20 holds. □ Remark 4.1 We have used in the preceding result the relaxation scheme in Tuan et al. (2001). Weaker conditions can be obtained as, for instance, in Liu and Zhang (2003), or Ariño and Sala (2007). However, these improvements are obtained at the cost of an increased complexity since additional slack variables have to be introduced. Remark 4.2 The sole performance criterion consider in Theorem 4.1 is the exponential rate $$\alpha$$. However, its optimization may lead to a closed-loop system with a very small region of attraction. In order to prevent this phenomenon, a solution consists in imposing that the ellipsoidal set $${{\it{\Omega}} _e}$$ (included in the domain of attraction) contains another reference ellipsoidal set of the form $$\beta^{-1/2}\,{\it{\Omega}}_{E}(R)$$, with $$\beta>0$$ and $$R\succ0$$. This can be imposed by adding to the LMI constraints of Theorem 4.1 the following one: [−βR∗R−P11]⪯0 (4.29) 5. Illustrative examples Example 5.1 Consider the uncertain nonlinear system represented in the T–S descriptor form (3.1) with $$re = ra = 2$$, $$nu = 1$$ and E1 =[1a−2.5+b1];H1e =H2e=[01];u0=2;E2 =[11.5−2+a0.75];W1e =W2e=[0.150]T;A1 =[−1.5+a−4−13+b];H1a =H2a=[01];A2 =[−2.5+a−4−11+b];W1a =W2a=[00.2]T;B1 =B2=[01];N1 =[10];N2=[01] (5.1) The existence of a controller satisfying properties 1 and 2, with $$\alpha = 0.5$$, has been tested for parameter values $$(a,b)\in [-4,\ 4]\times [-8,\ 8]$$ by checking the feasibility of LMIs condition of Theorem 1 in Dang et al. (2015a) and those of Theorem 4.1 in this article. The feasibility sets are represented in Fig. 1. As it can be seen on this figure, the conservatism of our new conditions are reduced with respect to those in Dang et al. (2015a). Fig. 1. View largeDownload slide Feasibility sets of LMI conditions for Example 5.1. Circles correspond to points found feasible with Theorem 4.1 of this article; crosses to point obtained with Theorem 1 of Dang et al. (2015a). Fig. 1. View largeDownload slide Feasibility sets of LMI conditions for Example 5.1. Circles correspond to points found feasible with Theorem 4.1 of this article; crosses to point obtained with Theorem 1 of Dang et al. (2015a). Example 5.2 Consider the following nonlinear system that is unstable in open-loop and described by: E(x)x˙=A(x)x+Bu, (5.2) where E(x) =[100.11cos⁡x11+0.12x2sin⁡x1];B=[01]A(x) =[010.13+0.17sin⁡x1x1−1+0.1x22cos⁡x1] and $$u$$ is the control input under the saturation limit $$\left| u \right| \le 0.5$$. The state variables are assumed to satisfy the following bounds: $$|x_{1}| \le \dfrac{\pi}{2}$$, $$|x_{2}| \le 1.5$$. In order to simplify the model, we will consider the terms weighted by $${\delta _1} = 0.12{x_2}\sin {x_1}$$ and $${\delta _2} = 0.1x_2^2\cos {x_1}$$ as uncertainties. For $$j=1,2$$, denoting $$\delta _j^{\max }$$ the maximal value of $${\delta _j}$$ on the domain described by the previous inequalities leads to: δj=δjmaxfjwith |fj|≤1,∀j=1,2 (5.3) The premise variables are chosen as follows: z1=0.11cos⁡x1;z2=0.13+0.17sin⁡x1x1 (5.4) Therefore, the membership functions are defined as: w1 =z1−z1minz1max−z1min;w2=1−w1h1 =z2−z2minz2max−z2min;h2=1−h1, (5.5) where, for $$j=1$$ or $$2$$, the quantities $$z_{j\max}$$ and $$z_{j\min}$$ are the maximal and minimal values of the premise variable $$z_{j}$$. By using the nonlinear sector approach (Tanaka and Wang 2001), the system (5.2) can be expressed in the uncertain T–S descriptor form (3.1) with $$re = 2, ra = 2, nu = 1$$ and E1 =[10z1max1];E2=[10z1min1];A1 =[01z2max−1];A2=[01z2min−1];H1e =H2e=H1a=H2a=B1=B2=[01]T;W1e =W2e=[0δ1max];W1a=W2a=[0δ2max];N1 =[2π0];N2=[023];u0=0.5 (5.6) In this section, it will be shown that the proposed control law satisfies all predefined properties stated in Section 3.2. First, we solve the LMI conditions in Theorem 4.1 with $$\alpha = 0$$, these conditions are feasible along with: P11=[1.1276−0.1582−0.15820.2918] (5.7) On the other hand, following Remark 4.2, by taking $$R = I$$ and searching the minimal value of $$\beta$$ with the additional constraint 4.29, we obtain: P11∗=[2.4674−1.4824−1.48242.25] (5.8) The boundaries of the two estimated domain of attraction $${\it{\Omega}}_{E}(P_{11}^{*})$$ and $${\it{\Omega}}_{E}(P_{11})$$ are plotted in Fig. 2. One may see that the estimated basin obtained with the optimization design is much larger than the one obtained without optimization. For the two designs, simulation of several trajectories initialized on the boundary of the corresponding set $${\it{\Omega}}_{e}$$ have been done and confirm that these sets are positively invariant with respect to the closed-loop dynamics and contained in the domain of attraction of the equilibrium state $$x=0$$. Figure 3 shows the control input signal and the closed-loop response from an initial state fixed at $$x(0) = [-1.56\ 0.79]^{T}$$; the controller being designed with the optimization approach. It can be seen that the system is stable with the designed controller even in the presence of control input saturation. Fig. 2. View largeDownload slide Sets of constraints, estimated domains of attraction and closed-loop trajectories for Example 5.2. Fig. 2. View largeDownload slide Sets of constraints, estimated domains of attraction and closed-loop trajectories for Example 5.2. Fig. 3. View largeDownload slide Left: control input. Right: closed-loop response. Fig. 3. View largeDownload slide Left: control input. Right: closed-loop response. 6. Conclusion This article presents a LMI-based systematic framework to deal with the problem of actuator saturation of nonlinear systems represented in descriptor form. 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ISA Trans. , 51 , 596 – 608 . Google Scholar CrossRef Search ADS PubMed © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

LMI approach for robust stabilization of Takagi–Sugeno descriptor systems with input saturation

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Abstract

Abstract The aim of this article is to present a systematic framework for the design of a robust controller for uncertain nonlinear systems represented in the Takagi–Sugeno descriptor form. The design takes explicitly into account state constraints and also the saturation of the control input. Based on Lyapunov theory, the tuning of the control parameters can be done by solving an optimization problem with linear matrix inequality constraints. Two numerical examples are included to illustrate the effectiveness of the proposed method. 1. Introduction More and more often, in sectors such as automotive or robotics, the control design of a process needs to take in consideration its nonlinear feature. The presence of non-linearities in the model usually makes the synthesis of the controller more difficult to perform. One way to overcome such difficulties is to use a Takagi–Sugeno (T–S) model of the system (Takagi and Sugeno, 1985). As explained in Tanaka and Wang (2001), if a nonlinear model based, for instance, on physical principles is available for the process to control, an equivalent T–S representation valid on a specified domain may be obtained by using the sector-nonlinearity approach. Based on Lyapunov stability theory and particular choices of control laws such as parallel distributed compensation (PDC), the design of the nonlinear controller can then be realized through a general framework as described in the monograph (Tanaka and Wang 2001) or the survey articles (Sala et al., 2005; Feng, 2006; Guerra et al., 2015b). In this framework, the requirements of closed-loop stability and other performances are translated in terms of linear matrix inequalities (LMI) problems that can be efficiently solved by convex optimization techniques (Boyd et al., 1994). Nevertheless, it is well known that the number of rules in the T–S representation grows exponentially with the number of nonlinearities appearing in the original nonlinear model, leading to possible numerical problems in the resolution of the optimization problems. It is thus of the highest interest to obtain a T–S representation of a given nonlinear system with a reduced number of linear models. For many systems, especially for mechanical devices, a natural model is obtained on descriptor form (see, for instance Luenberger (1977)). Its transformation in a standard T–S representation often leads to a model with numerous rules. A more convenient way to proceed is to preserve the original structure of the nonlinear model by rewriting it on the form of a T–S descriptor model as proposed by (Taniguchi et al., 1999). As emphasized in that article, but also in Guelton et al. (2008); Vermeiren et al. (2012), this may lead to models more adequate for the control design. The design approach developed for standard T–S models has been extended to the descriptor TS models in Taniguchi et al. (1999, 2000); Bouarar et al. (2007, 2008, 2010); Vermeiren et al. (2012); Chadli et al. (2014). Extensions to time-delay systems have been considered in Gassara et al. (2014); Luo et al. (2014). Descriptor representations may also simplify the design of controllers or observers for standard Takagi–Sugeno models as shown in the references Tanaka et al. (2007); Guelton et al. (2009); Guerra et al. (2015a). Due to technological or safety reasons, the actuator saturation is certainly the most frequently encountered nonlinear phenomenon in practice. Ignoring its effects in the design of a controller can lead to a significant degradation of the closed-loop performance and, even to the destabilization of the controlled system. Therefore, the synthesis of stabilizing controllers for dynamical systems under input saturation have attracted the attention of several researchers. For linear systems, there are many works dedicated to this problem; the interested readers may refer to the monographs Hu and Lin (2001), Kapila and Grigoriadis (2002), Hippe (2006) or Tarbouriech et al. (2011). Concerning T–S models, relying on the polytopic representation of the saturation function due to (Hu and Lin, 2001), the authors of (Cao and Lin, 2003) propose a method for the synthesis of a PDC controller taking into account the actuator saturation. In Ariño et al. (2010), another approach for dealing with input constraints is proposed based on an optimal iterative design. Recently, in Bezzaoucha et al. (2013) (see also Bezzaoucha et al. (2015)), the authors used the sector nonlinearity approach to obtain a new polytopic representation of the saturation characteristic. A main limitation of this approach is that it applies only to T–S models involving local linear models that are all open-loop stable. In order to overcome this drawback, the authors in Dang et al. (2015b) used a similar approach involving a smaller number of decision variables. This approach has been extended to the case of descriptor T–S models in Dang et al. (2015a). To our knowledge, only the works (Gassara et al., 2014; Luo et al., 2014) deal with the stabilization of saturated descriptor systems. Note that these two articles consider descriptor models of singular type with a common matrix on the state derivative (i.e. $$E_k=E$$, for $$k=1,\dots,ne$$ in this article notation). We propose in this article a new LMI-based framework which allows the synthesis of stabilizing controllers for uncertain T–S descriptor models under actuator saturation. The design takes also explicitly into account constraints on the state variables that may due to modelling reason or other technical considerations. 2. Notations The following notations will be used in this article: $$\Re^{n\times m}$$ is the set of $$n\times m$$ real matrices; $$\Re^{n}$$ denotes the set of real vectors of size $$n$$; $$\mathbb{N}_{n}$$ denotes the set of natural numbers $$ \{1,...,n\}$$; $$\mathbb{S}_{n}$$ denotes the subset of $$\Re^n$$ defined by: $$\mathcal{S}_{n}=\{x\in\Re^n:(\forall i\in \mathbb{N}_{n},\ x_i\ge0)\text{ and }\sum_{i=1}^{n}x_i=1\}$$; $$A_h$$ will denote the convex combination $$A_h = \sum\limits_{i = 1}^n h_i A_i$$ associated with a family $$(A_1,\dots,\ A_n)$$ of matrices having the same dimensions and a vector $$h\in\mathcal{S}_{n}$$. Similarly, $$A_{hh}$$ will denote a convex combination of the form $$ \sum\limits_{i = 1}^{n} \sum\limits_{j = 1}^{n} h_i\,h_j\,A_{ij}$$,... $$A^T$$ denotes the transpose of the matrix $$A$$; $$A_{(i)}$$ represents the $$i$$-th row of $$A$$; $$I$$, $$0$$ are the identity and zero matrices of appropriate dimensions; $${\rm 1}\kern-0.24em{\rm I}$$ a vector of adequate dimension with all entries equal to 1; $${\text{diag}}\left(A_1,\dots,\ A_n\right)$$ is the block diagonal matrix with square matrices $$A_1,\dots,$$$$\ A_n$$ as diagonal elements; Given two symmetric matrices $$P$$ and $$Q$$, $$P \succ Q$$ means that the matrix $$P-Q$$ is positive definite. Relations ‘$$\prec$$’ and ‘$$\preceq$$’ are defined similarly. Notation $$P\succ0$$ will be a shortcut for meaning that $$P$$ is a symmetric and positive-definite matrix; $${\it{\Omega}}_{P}(K,\ u)$$ denotes the polyhedral region associated with a matrix $$K \in {\Re ^{nu \times nx}}$$ and a vector $$u\in\Re ^{nu}$$ defined by ΩP(K, u)={x∈ℜnx:|K(i)x|≤ui, ∀i∈Nnu}; For $$P\succ0$$, $${\it{\Omega}}_{E}(P) = \left\{x\in \Re^{nx}: x^{T}P^{-1}x\le 1\right\}$$, that is, an ellipsoidal set containing the origin; * denotes elements that can be deduced by symmetry in a symmetric matrix. 3. Problem statement 3.1. System description In this article, we consider a class of uncertain nonlinear systems expressed in the following descriptor form: (E(x)+ΔE(t,x))x˙=(A(x)+ΔA(t,x))x+(B(x)+ΔB(t,x))sat(u) (3.1) where the state variable $$x(t)\in\Re^{nx}$$ and the control input $$u(t)\in \Re^{nu}$$. The vector $${\text{sat}}(u)\in\Re^{nu}$$ has for $$i$$-th entry sat(u)i=sign(ui)min(u0,i, |ui|), (3.2) where $${u_{0,i}} > 0$$ is the saturation bound of the $${i}$$th control input. For technological or safety reasons, we assume that the state $$x$$ remains, or has to remain, in a polyhedral region $${\it{\Omega}}_x$$ characterized by a matrix $$N\in \Re^{nc\times nx}$$ and defined as Ωx=ΩP(N,1I). (3.3) It is assumed that, on the set $${\it{\Omega}}_x$$ and for all $$t\ge 0$$, all entries of the different system matrices $$E(x)$$,..., $${\it{\Delta}} B(t,x)$$ are bounded, and that the matrix $$E(x)+{\it{\Delta}} E(t,x)$$ is regular for any uncertain matrix $${\it{\Delta}} E(t,x)$$ as described below. The descriptor system (3.1) is therefore of non-singular type. Applying the sector nonlinearity approach described in Tanaka and Wang (2001), a Takagi–Sugeno representation of the system (3.1), valid on $${\it{\Omega}}_x$$, may be obtained: (Ew+ΔEw(t))x˙=(Ah+ΔAh(t))x+(Bh+ΔBh(t))sat(u), (3.4) where the scheduling variables $$w(t)$$ and $$h(t)$$ are, respectively, in $$\mathbb{S}_{re}$$, and $$\mathbb{S}_{ra}$$. The uncertain matrices $${\it{\Delta}} A_i$$, ...admit the following decompositions: ΔEk(t)=HkeΔe(t)WkeΔAi(t)=HiaΔa(t)WiaΔBi(t)=HibΔb(t)Wib (3.5) for $$k\in \mathbb{N}_{re}$$ and $$i\in\mathbb{N}_{ra}$$. The matrices $$H_k^e$$, $$H_i^a$$, $$H_i^b$$, $$W_k^e$$, $$W_i^a$$ and $$W_i^b $$ are assumed constant and known, while the time-varying matrices $${\it{\Delta}}_e$$, $${\it{\Delta}}_a$$ and $${\it{\Delta}}_b$$ may be unknown but satisfy: Δℓ(t)TΔℓ(t)⪯I,∀t≥0, with ℓ=e, a, or b (3.6) 3.2. Control problem definition The main goal of this article is to propose a systematic framework for designing a controller that will stabilize an uncertain nonlinear system represented in the T–S descriptor form (3.4) with the state and input constraints (3.3)–(3.2). In order to achieve this goal, the following properties will be considered: Property 1 ($$\alpha $$-stability) There exists an ellipsoidal set $${{\it{\Omega}}_e}\subset \Re^{nx}$$ such that any trajectory of the closed-loop system (CLS) starting from one of its point converge exponentially to the origin; the decay rate being less than a positive real number $$\alpha $$. Property 2 (State constraint satisfaction) Any trajectory of the CLS starting from a point of $${{\it{\Omega}}_e}$$ remains in the polyhedral region $${{\it{\Omega}}_x}$$ described in (3.3). 4. Synthesis of the control law In order to stabilize the uncertain T–S descriptor system (3.4), we consider the following modified PDC law: u =∑j=1ra∑k=1rehj(z)wk(z)KjkP11−1x =KhwP11−1x, (4.1) where the feedback gain matrices $$K_{jk}$$ and the matrix $$P_{11}\succ 0$$ are to be determined such that the properties in Section 3.2 are respected. In order to take into account the saturation of the input, we will consider the polytopic representation proposed in Hu and Lin (2001). Let $$\left\{ V_{p}^{+} : p \in \mathbb{N}_{2^{nu}}\right\}$$ be the set of all diagonal matrices in $$\Re^{nu\times nu}$$ whose diagonal elements take the value 0 or 1. For a matrix $$V_{p}^{+}$$ in this set, the associated matrix $$V_{p}^{-}$$ is defined as $$V_{p}^{-} = I-V_{p}^{+}$$. Then, for any $$u,\ v \in \Re^{nu}$$ with $$v={\text{sat}}(v)$$, there exists a $$\mu\in\mathcal{S}_{ru}$$ (with $$ru=2^{nu}$$) such that sat(u)=∑p=1ruμp(u)(Vp+u+Vp−v)=Vμ+u+Vμ−v (4.2) Let us define the polyhedral region $${{\it{\Omega}} _u}\subset\Re^{nx}$$ as follows: Ωu=⋂j∈Nra,k∈NreΩP(LjkP11−1,u0), (4.3) where $$L_{jk}\in\Re^{nu\times nx}$$ are matrices to be defined later. From (4.1) and (4.2), for any $$x \in {{\it{\Omega}} _u}$$, the saturated control input can be expressed as: sat(u) =∑j=1ra∑k=1re∑p=1ruhj(z)wk(z)μp(u)(Vp+Kjk+Vp−Ljk)P11−1x =(Vμ+Khw+Vμ−Lhw)P11−1x (4.4) By introducing the augmented state variable $\bar{x}=\begin{bmatrix} x^{T} & \dot{x}^{T} \end{bmatrix}^T$, the closed-loop system can be written as: E¯x¯˙=(A¯+ΔA¯)x¯, (4.5) where E¯ =[I000]A¯ =[0IAh+Bh(Vμ+Khw+Vμ−Lhw)P11−1−Ew]ΔA¯ =[00ΔAh+ΔBh(Vμ+Khw+Vμ−Lhw)P11−1−ΔEw] Theorem 4.1 For a given $$\alpha>0$$, the uncertain T–S descriptor system (3.4) is stabilized by a modified PDC control law of the form (4.1) and satisfies properties 1 and 2 if there exist matrices $$P_{11}\succ 0$$, $$P_{ij}^{21}$$, $$P_{ij}^{22} \in \Re^{nx \times nx}$$, $$K_{jk}$$, $$L_{jk} \in \Re^{nu \times nx}$$ and positive real numbers $$\tau_i^a$$, $$\tau_{ijkp}^b$$, $$\tau_{ijk}^e$$ such that the following inequalities are satisfied: [−P11T∗Ljk(l)−u0,l2I] ⪯0 (4.6) [−P11T∗N(q)P11−I] ⪯0 (4.7) Υiikp ≺0 (4.8) 2ra−1Υiikp+Υijkp+Υjikp ≺0(for i≠j) (4.9) for all $$(i,j,k,p,l,q)\in \mathbb{N}_{ra} \times \mathbb{N}_{ra} \times\mathbb{N}_{re} \times \mathbb{N}_{ru}\times\mathbb{N}_{nu}\times\mathbb{N}_{nc}$$, where the symmetric, partitioned matrix $$\Upsilon_{ijkp}=\left\{\Upsilon^{mn}_{ijkp}\right\}_{(m,n)\in \mathbb{N}_{5}\times\mathbb{N}_{5}}$$ is defined by: Υijkp11 =Pij21+(Pij21)T+2αP11 (4.10) Υijkp21 =AiP11+(Pij22)T−EkPij21+Bi(Vp+Kjk+Vp−Ljk) (4.11) Υijkp22 =−EkPij22−(Pij22)TEkT+τiaHia(Hia)T+τijkpbHib(Hib)T+τijkeHke(Hke)T (4.12) Υijkp31 =WiaP11 (4.13) Υijkp33 =−τiaI (4.14) Υijkp41 =Wib(Vp+Kjk+Vp−Ljk) (4.15) Υijkp44 =−τijkpbI (4.16) Υijkp51 =−WkePij21 (4.17) Υijkp52 =−WkePij22 (4.18) Υijkp55 =−τijkeI (4.19) (unspecified blocks that cannot be obtained by symmetry are assumed equal to a zero matrix). Proof. Inequalities (4.6) imply that the ellipsoidal set $${\it{\Omega}}_e={\it{\Omega}}_E(P_{11})$$ contains the polyhedral set $${\it{\Omega}}_u$$. Similarly, inequalities (4.7) imply that $${\it{\Omega}}_e$$ contains the set $${\it{\Omega}}_x$$. We will now prove that the set $${\it{\Omega}}_e$$ is invariant with respect to the closed-loop dynamics and contained in the domain of attraction of $$x=0$$. For that, note first that, according to the relaxation lemma given in Tuan et al. (2001), Conditions 4.8 and 4.9 imply that inequality ∑i=1ra∑j=1ra∑k=1re∑p=1ruhihjwkμpΥijkp≺0 (4.20) holds for all $$(h,w,\mu)\in \mathcal{S}_{ra}\times \mathcal{S}_{re}\times \mathcal{S}_{ru}$$. One may then deduce that $$P_{hh}^{22}$$ is a regular matrix for all $$h\in\mathcal{S}_{ra}$$. Let us consider the Lyapunov candidate function: V(x)=xTP11−1x (4.21) Function $$V$$ can be rewritten as $$V(x)=\bar{x}^{T}\bar{E}P_{hh}^{-1}\bar{x}$$, where the matrix $$P_{hh}$$ is defined as Phh=[P110Phh21Phh22]. (4.22) The time-derivative of the Lyapunov candidate function is V˙(x) =x¯˙TE¯Phh−1x¯+x¯TE¯Phh−1x¯˙+x¯TE¯P˙hh−1x¯ =x¯T(A¯+ΔA¯)TPhh−1+(Phh−1)T(A¯+ΔA¯)x¯ (4.23) The solution $$x=0$$ of the closed-loop system (4.5) is exponentially stable with a decay rate less than $$\alpha $$ if: V˙(x¯)+2αV(x¯)=x¯T[(A¯+ΔA¯)TPhh−1+(Phh−1)T(A¯+ΔA¯)+2αE¯Phh−1x¯<0 (4.24) for all $$x\neq 0$$. This is satisfied if (A¯+ΔA¯)TPhh−1+(Phh−1)T(A¯+ΔA¯)+2αE¯Phh−1≺0 (4.25) Applying a congruence transformation with $$P_{hh}$$, the previous inequality (33) can be shown to be equivalent to PhhTA¯T+A¯Phh+2αPhhTE¯+PhhTΔA¯T+ΔA¯Phh≺0 (4.26) The uncertain terms in (4.26) can be decomposed as PhhTΔA¯T+ΔA¯Phh=HΔW+WTΔTHT, (4.27) where H =[000HhaHhbHwe];Δ=[Δa000Δb000Δe];W =[WhaP110Whb(Vμ+Khw+Vμ−Lhw)0−WwePhh21−WwePhh22] Completing the square, it comes straightforwardly the following inequality PhhTΔA¯T+ΔA¯Phh⪯HSHT+WTS−1W (4.28) with $$S = {\text{diag}}(\tau_{h}^{a}I,\ \tau_{hhw\mu}^{b}I,\ \tau_{hhw}^{e}I)$$. Applying Schur complement, it can be derived that inequality (4.26) is satisfied if 4.20 holds. □ Remark 4.1 We have used in the preceding result the relaxation scheme in Tuan et al. (2001). Weaker conditions can be obtained as, for instance, in Liu and Zhang (2003), or Ariño and Sala (2007). However, these improvements are obtained at the cost of an increased complexity since additional slack variables have to be introduced. Remark 4.2 The sole performance criterion consider in Theorem 4.1 is the exponential rate $$\alpha$$. However, its optimization may lead to a closed-loop system with a very small region of attraction. In order to prevent this phenomenon, a solution consists in imposing that the ellipsoidal set $${{\it{\Omega}} _e}$$ (included in the domain of attraction) contains another reference ellipsoidal set of the form $$\beta^{-1/2}\,{\it{\Omega}}_{E}(R)$$, with $$\beta>0$$ and $$R\succ0$$. This can be imposed by adding to the LMI constraints of Theorem 4.1 the following one: [−βR∗R−P11]⪯0 (4.29) 5. Illustrative examples Example 5.1 Consider the uncertain nonlinear system represented in the T–S descriptor form (3.1) with $$re = ra = 2$$, $$nu = 1$$ and E1 =[1a−2.5+b1];H1e =H2e=[01];u0=2;E2 =[11.5−2+a0.75];W1e =W2e=[0.150]T;A1 =[−1.5+a−4−13+b];H1a =H2a=[01];A2 =[−2.5+a−4−11+b];W1a =W2a=[00.2]T;B1 =B2=[01];N1 =[10];N2=[01] (5.1) The existence of a controller satisfying properties 1 and 2, with $$\alpha = 0.5$$, has been tested for parameter values $$(a,b)\in [-4,\ 4]\times [-8,\ 8]$$ by checking the feasibility of LMIs condition of Theorem 1 in Dang et al. (2015a) and those of Theorem 4.1 in this article. The feasibility sets are represented in Fig. 1. As it can be seen on this figure, the conservatism of our new conditions are reduced with respect to those in Dang et al. (2015a). Fig. 1. View largeDownload slide Feasibility sets of LMI conditions for Example 5.1. Circles correspond to points found feasible with Theorem 4.1 of this article; crosses to point obtained with Theorem 1 of Dang et al. (2015a). Fig. 1. View largeDownload slide Feasibility sets of LMI conditions for Example 5.1. Circles correspond to points found feasible with Theorem 4.1 of this article; crosses to point obtained with Theorem 1 of Dang et al. (2015a). Example 5.2 Consider the following nonlinear system that is unstable in open-loop and described by: E(x)x˙=A(x)x+Bu, (5.2) where E(x) =[100.11cos⁡x11+0.12x2sin⁡x1];B=[01]A(x) =[010.13+0.17sin⁡x1x1−1+0.1x22cos⁡x1] and $$u$$ is the control input under the saturation limit $$\left| u \right| \le 0.5$$. The state variables are assumed to satisfy the following bounds: $$|x_{1}| \le \dfrac{\pi}{2}$$, $$|x_{2}| \le 1.5$$. In order to simplify the model, we will consider the terms weighted by $${\delta _1} = 0.12{x_2}\sin {x_1}$$ and $${\delta _2} = 0.1x_2^2\cos {x_1}$$ as uncertainties. For $$j=1,2$$, denoting $$\delta _j^{\max }$$ the maximal value of $${\delta _j}$$ on the domain described by the previous inequalities leads to: δj=δjmaxfjwith |fj|≤1,∀j=1,2 (5.3) The premise variables are chosen as follows: z1=0.11cos⁡x1;z2=0.13+0.17sin⁡x1x1 (5.4) Therefore, the membership functions are defined as: w1 =z1−z1minz1max−z1min;w2=1−w1h1 =z2−z2minz2max−z2min;h2=1−h1, (5.5) where, for $$j=1$$ or $$2$$, the quantities $$z_{j\max}$$ and $$z_{j\min}$$ are the maximal and minimal values of the premise variable $$z_{j}$$. By using the nonlinear sector approach (Tanaka and Wang 2001), the system (5.2) can be expressed in the uncertain T–S descriptor form (3.1) with $$re = 2, ra = 2, nu = 1$$ and E1 =[10z1max1];E2=[10z1min1];A1 =[01z2max−1];A2=[01z2min−1];H1e =H2e=H1a=H2a=B1=B2=[01]T;W1e =W2e=[0δ1max];W1a=W2a=[0δ2max];N1 =[2π0];N2=[023];u0=0.5 (5.6) In this section, it will be shown that the proposed control law satisfies all predefined properties stated in Section 3.2. First, we solve the LMI conditions in Theorem 4.1 with $$\alpha = 0$$, these conditions are feasible along with: P11=[1.1276−0.1582−0.15820.2918] (5.7) On the other hand, following Remark 4.2, by taking $$R = I$$ and searching the minimal value of $$\beta$$ with the additional constraint 4.29, we obtain: P11∗=[2.4674−1.4824−1.48242.25] (5.8) The boundaries of the two estimated domain of attraction $${\it{\Omega}}_{E}(P_{11}^{*})$$ and $${\it{\Omega}}_{E}(P_{11})$$ are plotted in Fig. 2. One may see that the estimated basin obtained with the optimization design is much larger than the one obtained without optimization. For the two designs, simulation of several trajectories initialized on the boundary of the corresponding set $${\it{\Omega}}_{e}$$ have been done and confirm that these sets are positively invariant with respect to the closed-loop dynamics and contained in the domain of attraction of the equilibrium state $$x=0$$. Figure 3 shows the control input signal and the closed-loop response from an initial state fixed at $$x(0) = [-1.56\ 0.79]^{T}$$; the controller being designed with the optimization approach. It can be seen that the system is stable with the designed controller even in the presence of control input saturation. Fig. 2. View largeDownload slide Sets of constraints, estimated domains of attraction and closed-loop trajectories for Example 5.2. Fig. 2. View largeDownload slide Sets of constraints, estimated domains of attraction and closed-loop trajectories for Example 5.2. Fig. 3. View largeDownload slide Left: control input. Right: closed-loop response. Fig. 3. View largeDownload slide Left: control input. Right: closed-loop response. 6. Conclusion This article presents a LMI-based systematic framework to deal with the problem of actuator saturation of nonlinear systems represented in descriptor form. 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Published: Apr 5, 2017

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