Stability and constrained control for positive two-dimensional systems with delays in the second FM model

Stability and constrained control for positive two-dimensional systems with delays in the second... Abstract This paper addresses the stability and control problem of linear positive two-dimensional discrete-time systems with multiple delays in the second Fornasini–Marchesini model. The contribution lies in three aspects. First, a novel proof is provided to establish necessary and sufficient conditions of asymptotic stability for positive two-dimensional delayed systems. Then, a state-feedback controller is designed to ensure the non-negativity and stability of the closed-loop systems. Finally, a sufficient condition for the existence of constrained controllers is developed under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Two examples are given to validate the proposed methods. 1. Introduction Two-dimensional systems have attracted considerable attention from researchers in the recent decades due to the wide range of applications in electricity transmission, circuit analysis, digital image processing, process control and modeling of partial differential equations, e.g. Darboux equation used in dynamic processes in gas absorption, water stream heating and air drying (Roesser, 1975; Bracewell, 1995; Kaczorek, 1985; Dymkov & Dymkou, 2012). Two-dimensional systems can be represented by different models such as Roesser model, Fornasini-Marchesini (FM) model and Kurek model (Roesser, 1975; Fornasini & Marchesini, 1976, 1978; Kurek, 1985). Numerous results on stability analysis and stabilization of two-dimensional discrete systems are available (Benton et al., 2010; Fernando, 1988; Hinamoto, 1997; Bouagada & Dooren, 2013; Gao et al., 2004; Paszke et al., 2004; Wu et al., 2009; Xiang & Huang, 2013; Yang et al., 2006). A system is said to be positive if its states and outputs are non-negative whenever the initial conditions and inputs are non-negative. Positive systems model many real world physical systems, which involve non-negative variables, for example, population levels, absolute temperature and concentration of substances (Benvenuti et al., 2003; Berman et al., 1989; Farina & Rinaldi, 2000; Li, 2014). As a special class of two-dimensional systems, positive two-dimensional systems have also been extensively studied due to their significant applications in various fields such as the wave equation in fluid dynamics, river pollution and self-purification process, gas absorption, etc. (Kaczorek, 2001; Fornasini, 1991, 1996). Hence, researchers have paid considerable attention to stability and stabilization of positive two-dimensional systems during the past years (Przyborowski & Kaczorek, 2009; Chu & Liu, 2007; Twardy, 2007; Duan et al., 2013, 2014; Kaczorek, 2009a, 2009b). In addition, time delays widely exist naturally in many practical systems and are frequently a source of instability and poor performance, even make systems out of control. The aforementioned papers did not consider the effects of delays. Kaczorek (2009a) gave one sufficient and necessary condition of asymptotic stability for positive two-dimensional systems with delays by transferring the two-dimensional model with delays into one system without delays but with higher dimension. Then, Kaczorek (2009b) improved the results in Kaczorek (2009a). By co-positive Lypunov function method, Duan et al. (2013, 2014) achieved several results on stability and performance analysis for positive two-dimensional systems with delays but the conditions are usually only sufficient. On the other hand, real plants always involve constrained variables, such as saturated control signal. Bounded control is also an active research issue (Blanchini, 1990; Henrion et al., 2001; Rami & Tadeo, 2007; Liu et al., 2008), where the control inputs and the states of the closed-loop systems are required to be bounded. Rami & Tadeo (2007) provided constrained controller design for positive one-dimensional system and Liu et al. (2008) extended the result to the case with delays. To the best of authors’ knowledge, no one has directly considered constrained control problem for positive two-dimensional systems. With regard to these previous works, this paper addresses the constrained control problem for the positive two-dimensional systems with delays in the second FM model. In this paper, one necessary and sufficient condition is proposed to test the stability of positive two-dimensional systems with multiple time delays described by the second Fornasini–Marchesini model. Then, a necessary and sufficient condition for the existence of state-feedback controllers is established for general delayed two-dimensional systems, which ensures the non-negativity and the stability of the resulting closed-loop systems. Finally, a constrained controller is designed to stabilize the addressing systems such that the control inputs and the states of the corresponding closed-loop systems are non-negative and bounded. Two examples illustrate the feasibility of the proposed approaches. The remainder of this paper is organized as follows: problem formulation and necessary preliminaries are presented in Section 2. Section 3 establishes the necessary and sufficient stability criteria for positive two-dimensional systems with multiple constant delays. The bounded controller for stabilizing delayed two-dimensional systems is designed in Section 4. Section 5 provides two examples and Section 6 concludes this paper. Notations 1 In this paper, the superscript ‘T’ denotes the transpose. $$A\ \underline {\succ }\ 0$$($$\underline {\prec }\ 0$$) means that all entries of matrix A are non-negative (non-positive). A ≻ 0 (≺ 0) means that all entries of matrix A are positive (negative). Rn×m denotes the set of n × m real matrices. The set of real n × m matrices with non-negative entries will be denoted by $$R_{+}^{n\times m} $$ and the set of non-negative integers will be denoted by Z+. The n × n identity matrix will be denoted by In. Vector $${b_{i}^{T}} $$ is the ith row vector of matrix B, and $$A_{1\tau } =[a_{1\tau }^{ij} ]$$. The l1 norm of a two-dimensional signal $$w\left (k,l\right )=\left [w_{1} \left (k,l\right ),w_{2} \left (k,l\right ),\cdot \cdot \cdot ,w_{m} \left (k,l\right )\right ]^{T} $$ is given by $$ \left\| w(k,l)\right\| =\sum_{i=1}^{m}\left|w_{i}(k,l)\right|.$$ 2. Problem formulation and preliminaries Consider following positive two-dimensional discrete system in FM model: $$ x(k,l)=\sum_{\tau =0}^{h}\left[A_{1\tau } x(k-1-\tau,l)+A_{2\tau } x(k,l-1-\tau )\right] $$ (1) where x(k, l) ∈ Rn is the state vector, k and l are two integers in Z+, A1τ, A2τ ∈ Rn×n are parameter matrices with τ = 0, 1, …, h, h ∈ Z+ denote the number of delays, τ are constant delays. The positive boundary conditions of system (1) are defined by $$ x(k,l)=\begin{cases} f_{k,l}, & -h\le k\le 0,\ 0\le l\le z_{2}\\ 0, & -h\le k\le 0,\ l>z_{2}\\ g_{k,l}, & 0\le k\le z_{1},\ -h\le l\le 0\\ 0, & k>z_{1},\ -h\le l\le 0\end{cases} $$ (2) where fk, l and gk, l are given non-negative vectors, $$z_{1} <\infty $$, and $$z_{2} <\infty $$ are positive integers. Definition 1 (Kaczorek, 2009a) System (1) is said to be positive if for any positive boundary conditions (2) satisfying $$f_{k,l}\ \underline {\succ }\ 0,g_{k,l}\ \underline {\succ }\ 0$$, we have $$x(k,l)\ \underline {\succ }\ 0$$ for all k, l ≥ 0. Lemma 1 (Kaczorek, 2009a) System (1) is positive if and only if $$A_{1\tau } \ \underline {\succ }\ 0$$ and $$A_{2\tau } \ \underline {\succ }\ 0$$ with τ = 0, 1, …, h. Definition 2 (Yeganefar et al., 2013) System (1) is said to be asymptotically stable if $$\mathop {\lim }\limits _{i\to \infty } X_{i} =0$$ for all bounded boundary conditions (2), where $$ X_{i} =\sup \left\{\left\| x(k,l)\right\| :k+l=i,k,l\ge 1\right\} .$$ (3) The aim of this paper is to establish the asymptotic stability criterion for system (1) and furthermore based on this condition, to solve the constrained control problem for the general positive two-dimensional systems with multiple delays. 3. Stability analysis Theorem 1 System (1) is asymptotically stable if and only if there exists a strictly positive vector $$\lambda \in R_{+}^{n} $$ such that $$ \left(\sum_{\tau =0}^{h}(A_{1\tau} +A_{2\tau}) -I_{n}\right)\lambda \prec 0 .$$ (4) Proof Necessity: For p = k + l with k ≥ 0 and l ≥ 0, we define $$ X(p)=\sum_{k+l=p}x(k,l) =x(p,0)+x(p-1,1)+x(p-2,2)+\cdots+x(k,l)+\cdots+x(1,p-1)+x(0,p) .$$ (5) When constant time delay τ happens to X(p) in horizontal direction, define the following signal \begin{align} X^{-} (p,\tau )=\sum_{k+l=p}x(k-\tau,l)=&\,x(p-\tau,0)+x(p-1-\tau,1)+\cdots x(k-\tau,l)\cdots\nonumber\\ &+x(1-\tau,p-1)+x(-\tau,p).\end{align} (6) Similarly, define X+(p, τ) when time delay τ happens to X(p) in vertical direction as follows \begin{align} X^{+} (p,\tau )=\sum_{k+l=p}x(k,l-\tau )=&\,x(p,-\tau )+x(p-1,1-\tau )+\cdots x(k,l-\tau )\cdots\nonumber\\ &+x(1,p-1-\tau )+x(0,p-\tau).\end{align} (7) Combining (6)–(8), one obtains $$ X^{-} (p,\tau )=\begin{cases} \sum\limits_{i=\tau -p}^{\tau }x(-i,p-\tau +i), & \forall 0\le p\le \tau\\ X(p-\tau )+\sum\limits_{i=1}^{\tau }x(-i,p-\tau +i), & \forall p>\tau\end{cases} $$ (8) $$ X^{+} (p,\tau )=\begin{cases} \sum\limits_{i=\tau -p}^{\tau }x(p-\tau +i,-i), & \forall 0\le p\le \tau\\ X(p-\tau )+\sum\limits_{i=1}^{\tau }x(p-\tau +i,-i), & \forall p>\tau\end{cases}.$$ (9) Taking p = k + l, from the evolution of system (1) we have \begin{align} x(p-1,1)=&\,A_{10} x(p-2,1)+A_{20} x(p-1,0)+\sum_{\tau =1}^{h}\left[A_{1\tau } x(p-2-\tau,1)+A_{2\tau } x(p-1,-\tau )\right]\nonumber\\ &\qquad\cdots\nonumber\\ x(k,l)=&\,A_{10} x(k-1,l)+A_{20} x(k,l-1)+\sum_{\tau =1}^{h}\left[A_{1\tau } x(k-1-\tau,l)+A_{2\tau } x(k,l-1-\tau )\right]\nonumber\\ &\qquad\cdots\nonumber\\ x(1,p-1)=&\, A_{10} x(0,p-1)+A_{20} x(1,p-2)+\sum_{\tau =1}^{h}\left[A_{1\tau } x(-\tau,p-1)+A_{2\tau } x(1,p-2-\tau )\right] .\end{align} (10) Denote the boundary condition by $$X_{0}^{-} (p)=x(p,0)$$ and $$X_{0}^{+} (p)=x(0,p)$$. Summing the above equalities in (10), one obtains \begin{align} &X(p)-X_{0}^{-} (p)-X_{0}^{+} (p)\nonumber\\ &\quad=A_{10}\left[X(p-1)-X_{0}^{-} (p-1)\right]+A_{20}\left[X(p-1)-X_{0}^{+} (p-1)\right]\nonumber\\ &\qquad+A_{11}\left[X^{-} (p-1,1)-X_{0}^{-} (p-1-1)\right]+A_{21} \left[X^{+} (p-1,1)-X_{0}^{+} (p-1-1)\right]\nonumber\\ &\qquad+\cdots+A_{1h} \left[X^{-} (p-1,h)-X_{0}^{-} (p-1,-1)\right]+A_{2h}\left[ X^{+} (p-1,h)-X_{0}^{+} (p-1-h)\right]\nonumber\\ &\quad=A_{10} \left[X(p-1)-X_{0}^{-} (p-1)\right]+A_{20} \left[X(p-1)-X_{0}^{+} (p-1)\right]\nonumber\\ &\qquad+\sum_{\tau =1}^{h}\left\{A_{1\tau } \left[X^{-} (p-1,\tau )-X_{0}^{-} (p-1-\tau )\right]+A_{2\tau } \left[X^{+} (p-1,\tau )-X_{0}^{+} (p-1-\tau )\right]\right\} .\end{align} (11) Thus, for p = 1, 2, …, q (q is a positive integer), from (12) we have respectively \begin{align} X(1)-X_{0}^{-} (1)-X_{0}^{+} (1)=&\,0\nonumber\\ X(2)-X_{0}^{-} (2)-X_{0}^{+} (2)=&\,A_{10} \left[X(1)-X_{0}^{-} (1)\right]+A_{20} \left[X(1)-X_{0}^{+} (1)\right]\nonumber\\ &+\sum_{\tau =1}^{h}\left\{A_{1\tau }\left[X^{-}(1,\tau)-X_{0}^{-} (1-\tau)\right]+A_{2\tau} \left[X^{+} (1,\tau )-X_{0}^{+} (1-\tau )\right]\right\}\nonumber\\ &\cdots\nonumber\\ X(q)-X_{0}^{-}(q)-X_{0}^{+}(q)=&\,A_{10}\left [X(q-1)-X_{0}^{-} (q-1)\right]+A_{20}\left [X(q-1)-X_{0}^{+}(q-1)\right]\nonumber\\ &+\sum_{\tau =1}^{h}\left\{A_{1\tau }^{T}\left[X^{-} (q-1,\tau )-X_{0}^{-} (q-1-\tau )\right]\right.\nonumber\\ &\quad\qquad\left.+\ A_{2\tau }^{T} \left[X^{+} (q-1,\tau )-X_{0}^{+} (q-1-\tau )\right]\right\} .\end{align} (12) Summing the above equalities in (12), one gets \begin{align} &X(q)-X_{0}^{-} (q)-X_{0}^{+} (q)+\sum_{p=1}^{q-1}\left\{X(p)-X_{0}^{-} (p)-X_{0}^{+} (p)\right\}\nonumber\\ &\quad=\sum_{p=1}^{q-1}\left\{A_{10}\left[X(p)-X_{0}^{-} (p)\right]+A_{20}\left[X(p)-X_{0}^{+} (p)\right]\right\}\nonumber\\ &\quad\quad+\sum_{\tau =1}^{h}\sum_{p=1}^{q-1}\left\{A_{1\tau }\left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]+A_{2\tau } \left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]\right\} .\end{align} (13) Furthermore, the above equality can be rewritten in the following form: \begin{align} &X(q)-\sum_{p=1}^{q}\left[X_{0}^{-} (p)+X_{0}^{+} (p)\right]\nonumber\\ &\quad=\sum_{p=1}^{q-1}(A_{10} +A_{20} -I_{n} )X(p) -\sum_{p=1}^{q-1}\left\{A_{10} X_{0}^{-} (p)+A_{20} X_{0}^{+} (p)\right\}\nonumber\\ &\quad\quad+\sum_{\tau =1}^{h}\sum_{p=1}^{q-1}\left\{A_{1\tau } \left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]+A_{2\tau } \left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]\right\} .\end{align} (14) Due to the positive property, the asymptotic stability of system (1) can be equivalent to $$ \mathop{\lim }\limits_{q\to \infty } X(q)= \mathop{\lim }\limits_{q\to \infty } \sum_{k+l=q}x(k,l) =0 .$$ (15) When q tends to infinity, it follows from (14) and (15) that \begin{align} -\sum_{p=1}^{\infty }\left[X_{0}^{-} (p)+X_{0}^{+} (p)\right]=&\,\sum_{p=1}^{\infty }(A_{10} +A_{20} -I_{n} )X(p) -\sum_{p=1}^{\infty }\left\{A_{10} X_{0}^{-} (p)+A_{20} X_{0}^{+} (p)\right\}\nonumber\\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau }\left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]+A_{2\tau }\left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]\right\} .\end{align} (16) Adding $$\sum _{p=1}^{\infty }[X_{0}^{-} (p)+X_{0}^{+} (p)] -\sum _{p=1}^{\infty }\left \{A_{10} X_{0}^{+} (p)+A_{20} X_{0}^{-} (p)\right \} $$ to the both sides of (18) leads to $$ -\sum_{p=1}^{\infty }\left\{A_{10} X_{0}^{+} (p)+A_{20} X_{0}^{-} (p)\right\} =\sum_{p=1}^{\infty }(A_{10} +A_{20} -I_{n} )\left[X(p)-X_{0}^{-} (p)-X_{0}^{+} (p)\right] +\Pi $$ (17) with \begin{align*} \Pi =&\,\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau } \left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]+A_{2\tau } \left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]\right\}\\ =&\,\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau } X^{-} (p,\tau )+A_{2\tau } X^{+} (p,\tau )\right\} -\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau } X_{0}^{-} (p-\tau )+A_{2\tau } X_{0}^{+} (p-\tau )\right\}\\ =&\,\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } X(p-\tau )+A_{2\tau } X(p-\tau )\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\sum_{i=\tau -p}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\sum_{i=1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\\ & -\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau } X_{0}^{-} (p-\tau )+A_{2\tau } X_{0}^{+} (p-\tau )\right\}\\ =&\,\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } \left[X(p-\tau )-X_{0}^{-} (p-\tau )-X_{0}^{+} (p-\tau )\right]\right.\\[-2pt] &\left.\quad\qquad\qquad+\ A_{2\tau } \left[X(p-\tau )-X_{0}^{-} (p-\tau )-X_{0}^{+} (p-\tau )\right]\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\sum_{i=1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\} \\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\sum_{i=\tau -p}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } X_{0}^{+} (p-\tau )+A_{2\tau } X_{0}^{-} (p-\tau )\right\} -\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\left\{A_{1\tau } X_{0}^{-} (p-\tau )+A_{2\tau } X_{0}^{+} (p-\tau )\right\} \end{align*} \begin{align} =&\,\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\!\!\left\{A_{1\tau } \!\left[X(p-\tau )-X_{0}^{-} (p-\tau )-X_{0}^{+} (p-\tau )\right]+A_{2\tau }\!\left[X(p-\tau )-X_{0}^{-} (p-\tau )-X_{0}^{+} (p-\tau )\right]\right\} \nonumber\\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\sum_{i=\tau -p+1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\nonumber\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\sum_{i=1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\nonumber\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } X_{0}^{+} (p-\tau )+A_{2\tau } X_{0}^{-} (p-\tau )\right\} .\end{align} (18) Letting $$Y(p)=X(p)-X_{0}^{-} (p)-X_{0}^{+} (p)$$, we have $$ -\psi =\sum_{p=1}^{\infty }(A_{10} +A_{20} -I_{n} )Y(p) +\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }(A_{1\tau } +A_{2\tau } )Y(p-\tau )=\left(\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau }\right) -I_{n} )\sum_{p=1}^{\infty }Y(p) $$ (19) with \begin{align*} \psi =&\,\sum_{p=1}^{\infty }\left\{A_{10} X_{0}^{+} (p)+A_{20} X_{0}^{-} (p)\right\} +\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } X_{0}^{+} (p-\tau )+A_{2\tau } X_{0}^{-} (p-\tau )\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\sum_{i=\tau -p+1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\sum_{i=1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}.\end{align*} Without loss of generality, not all fk, l and gk, l in the boundary conditions (2) are zero vectors. It follows that the left hand side of (19) is strictly negative and hence $$ \left(\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau }) -I_{n} \right)\sum_{p=1}^{\infty }Y(p) \prec 0 .$$ (20) The condition (20) is equivalent to (4) with $$\lambda =\sum _{p=1}^{\infty }Y\left (p\right ) $$. Sufficiency: Consider the dual system of system (1): $$ x(k,l)=\sum_{\tau =\mathrm{0}}^{h}\left[A_{1\tau }^{T} x(k-1-\tau,l)+A_{2\tau }^{T} x(k,l-1-\tau )\right] $$ (21) which is positive and stable if and only if the original system (1) is positive and asymptotically stable (Haddad & Chellaboina, 2004). Utilizing the similar line of the deducing process from system (1) to system (21), we have \begin{align} &X(q)-X_{0}^{-} (q)-X_{0}^{+} (q)\nonumber\\ &\quad=A_{10}^{T} \left[X(q-1)-X_{0}^{-} (q-1)\right]+A_{20}^{T} \left[X(q-1)-X_{0}^{+} (q-1)\right]\nonumber\\ &\qquad+\sum_{\tau =1}^{h}\left\{A_{1\tau }^{T} \left[X^{-} (q-1,\tau )-X_{0}^{-} (q-1-\tau )\right]+A_{2\tau }^{T} \left[X^{+} (q-1,\tau )-X_{0}^{-} (q-1-\tau )\right]\right\} .\end{align} (22) According to (15), the stability of system (1) is equivalent to the stability of system (22). Define the following Lyapunov-like function for system (22) $$ V(p)=X^{T}(p)\lambda +\sum_{j=1}^{h}\left[X^{-} (p,j)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda+\sum_{j=1}^{h}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda $$ (23) which is positive for non-zero $$X\left (p\right )$$ and strictly positive vector λ ≻ 0. From (22) and (23), we have \begin{align} \Delta V(p)=&\,V(p+1)-V(p)=\left[X(p)-X_{0}^{-} (p)\right]^{T} A_{10} \lambda +\left[X(p)-X_{0}^{+} (p)\right]^{T} A_{20} \lambda\nonumber\\ &+\sum_{\tau =1}^{h}\left\{\left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]^{T} A_{1\tau } \lambda +\left[X^{+} (p,\tau )\right]\right\}\nonumber\\ &+\left(X_{0}^{-} (p\mathrm{+}1)\right)^{T} \lambda +X_{0}^{+} (p\mathrm{+}1))^{T} \lambda +\sum_{j=1}^{h}[X^{-} \left(p+1,j\right)-X_{0}^{-} (p+1-j)]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda \nonumber\\ &+\sum_{j=1}^{h}\left[X^{+} (p+1,j)-X_{0}^{+} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda -X^{T} \left(p\right)\lambda\nonumber\\ &-\sum_{j=1}^{h}\left[X^{-} \left(p,j\right)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda -\sum_{j=1}^{h}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda .\end{align} (24) Assuming p > h, from (6) and (7), one obtains \begin{align} &\sum_{\tau =1}^{h}\left\{\left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]^{T} A_{1\tau } \lambda +\left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]^{T} A_{2\tau } \lambda \right\}\nonumber\\ &\quad-\sum_{j=1}^{h}\left[X^{-} \left(p,j\right)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda -\sum_{j=1}^{h}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda\nonumber\\ &\qquad=-\sum_{j=1}^{h-1}\left[X^{-} \left(p,j\right)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j+1}^{h}A_{1\tau } \lambda -\sum_{j=1}^{h-1}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j+1}^{h}A_{2\tau } \lambda.\end{align} (25) Applying (8) and (9) to (25) leads to \begin{align*} \Delta V(p)=&\,\left[X(p)-X_{0}^{-} (p)\right]^{T} A_{10} \lambda +\left[X(p)-X_{0}^{+} (p)\right]^{T} A_{20} \lambda+\left(X_{0}^{-} (p+1)\right)^{T} \lambda\nonumber\\ &+\left(X_{0}^{+} (p+1)\right)^{T} \lambda -X^{T}(p)\lambda +\Lambda^{-} (p)+\Lambda^{+} (p) \end{align*} (26) with \begin{align*}\Lambda^{-} (p)=&\,\sum_{j=1}^{h} \left[X^{-} \left(p+1,j\right)-X_{0}^{-} (p+1-j) \right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda -\sum_{j=1}^{h-1} \left[X^{-} \left(p,j\right)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j+1}^{h}A_{1\tau } \lambda\\ \Lambda^{+} (p)=&\,\sum_{j=1}^{h}\left[X^{+} (p+1,j)-X_{0}^{+} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda -\sum_{j=1}^{h-1}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j+1}^{h}A_{2\tau } \lambda.\end{align*} It should be noted that $$ \Lambda^{-} (p)=X(p)\sum_{\tau =1}^{h}A_{1\tau } \lambda +\Theta_{0}^{-} (p) $$ (27) with \begin{align*} \Theta_{0}^{-} (p)=&\,\sum_{j=1}^{h}\left[\sum_{i=1}^{j}x(-i,p+1-j+i) -X_{0}^{-} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda\\ &-\sum_{j=1}^{h-1}\left[\sum_{i=1}^{j}x(-i,p+1-j+i) -X_{0}^{-} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda.\end{align*} Similarly, we have $$ \Lambda^{+} (p)=X(p)\sum_{\tau =1}^{h}A_{2\tau } \lambda +\Theta_{0}^{+} (p) $$ (28) with \begin{align*} \Theta_{0}^{+} (p)=&\,\sum_{j=1}^{h}\left[\sum_{i=1}^{j}x(p+1-j+i,-i) -X_{0}^{-} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda \\ &-\sum_{j=1}^{h-1}\left[\sum_{i=1}^{j}x(p+1-j+i,-i) -X_{0}^{+} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda.\end{align*} Therefore, (26) can be rewritten as: \begin{align*} \Delta V(p)=&\,\left[X(p)-X_{0}^{-} (p)\right]^{T} A_{10} \lambda +\left[X(p)-X_{0}^{+} (p)\right]^{T} A_{20} \lambda +(X_{0}^{-} (p+1))^{T} \lambda +(X_{0}^{+} (p+1))^{T} (p)\lambda\\ &+X^{T} (p)\sum_{\tau =1}^{h}A_{1\tau } \lambda+\Theta_{0}^{-} (p)+X^{T} (p)\sum_{\tau =1}^{h}A_{2\tau } \lambda +\Theta_{0}^{+} (p)\\ &=X(p)\left[\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } ) -I_{n} \right]\lambda +M(p)\end{align*} with $$ M(p)=\Theta_{0}^{-} (p)+\Theta_{0}^{+} (p)-\left(X_{0}^{-} (p)\right)^{T} A_{10} \lambda -\left(X_{0}^{+} (p)\right)^{T} A_{20} \lambda+\left(X_{0}^{-} (p+1)\right)^{T} \lambda +\left(X_{0}^{+} (p+1)\right)^{T} \lambda.$$ Due to the boundness of the boundary condition (2), for $$p>\max \left \{z_{1},z_{2} \right \}$$, we have \begin{align*} X_{0}^{-} \left(p\right)=&\,x(p,0)=0,X_{0}^{+} \left(p\right)=x(0,p)=0,\\ x(p,-\tau )=&\,0,x(-\tau,p)=0,\tau =1,2,\ldots,h. \end{align*} Thus, for any $$p>\max \left \{z_{1},z_{2} \right \}$$, it holds that M(p) = 0 and condition (4) implies $$ \Delta V(p)=X^{T} (p)\left[\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } ) -I_{n} \right]\lambda \prec 0,\quad \forall p>\max \left\{z_{1},z_{2} \right\} .$$ (29) Then, the positive system (22) is asymptotically stable, which implies $$ \mathop{\lim }\limits_{q\to \infty } X\left(q\right)= \mathop{\lim }\limits_{q\to \infty } \sum_{k+l=q}x(k,l) =0\Leftrightarrow \mathop{\lim }\limits_{k+l\to \infty } x(k,l)=0.$$ Thus, system (1) is asymptotically stable. This completes the proof. Remark 1 Kaczorek (2009a) provided a necessary and sufficient condition of asymptotic stability for two-dimensional positive systems with delays using ‘augmented system approach’ (Liu et al., 2008). However, the derived stability condition is delay-dependent and with high computational complexity. Kaczorek (2009b) improved the results in Kaczorek (2009a) and presented a delay independent stability criteria without detailed proof. In this paper, a strict proof of stability criteria is provided via a simple and novel method. Remark 2 It should be noted that the stability in Definition 2 is stricter than ones used in Kaczorek (2009a, 2009b), since $$k+l\to \infty $$ includes the case $$k,l\to \infty $$ used in Kaczorek (2009a, 2009b), and another two cases k = i, $$l\to \infty $$ and $$k\to \infty $$, l = j with $$i,j<\infty $$ being constant integers. However, the sufficient and necessary condition obtained in this paper is the same as that in Kaczorek (2009a, 2009b). Remark 3 Different from the existing results that the stability is usually closely related to the magnitude of the delays for general two-dimensional delayed systems, the magnitude of delays has no any impact on the asymptotic stability for positive two-dimensional delayed systems, which is shown in Theorem 1. It shows that the delays in system (1) are required to be consecutive and the delays along horizontal and vertical orientations take the same value. Actually, system (1) could be used to describe many kinds of actual systems through taking different matrix values. For example, it is readily to obtain that system (1) is equivalent to $$ x(k,l)=A_{10} x(k-1,l)+A_{20} x(k,l-1)+\sum_{\alpha =1}^{\kappa_{1} }A_{1\theta_{\alpha } } x(k-1-\theta_{\alpha },l) +\sum_{\beta =1}^{\kappa_{2}}\left[A_{2\vartheta_{\beta } } x(k,l-1-\vartheta_{\beta } )\right] $$ (30) by defining: \begin{align*}h=&\,\mathop{\max }\limits_{i\in \underline{\kappa_{1} },j\in \underline{\kappa_{2} }} \{ \theta_{i},\vartheta_{j} \}\\ A_{1\tau } =&\,0,\quad \forall \tau \ne \theta_{1},\theta_{2},\cdots,\theta_{\kappa_{1} }\\ A_{2\tau } =&\,0,\quad \forall \tau \ne \vartheta_{1},\vartheta_{2},\cdots,\vartheta_{\kappa_{2}},\end{align*} where κ1, κ2 ∈ Z+ denote the numbers of delays in two directions, θα and ϑβ are distributed constant delays along horizontal and vertical orientations, respectively. In this case, Theorem 1 reduces to the following theorem. Theorem 2 Positive two-dimensional discrete-time system (1) is asymptotically stable if and only if there exists a strictly positive vector $$\lambda \in R_{+}^{n} $$ such that $$ \left(\sum_{\alpha =0}^{\kappa_{1} }A_{1\theta_{\alpha}}+\sum_{\beta =0}^{\kappa_{2}}A_{2\vartheta_{\beta}}-I_{n} \right)\lambda \prec 0 .$$ (31) 4. Control synthesis In this section, our interest is to design state-feedback controllers for general two-dimensional discrete systems with delays, that guarantee the positivity and stability of the resulting closed-loop system. The first subsection is devoted to the control problem without constraints for the two-dimensional discrete systems in FM model, and the second subsection extends the result to the case with constraints. 4.1 Unconstrained control Consider the following two-dimensional discrete systems in FM model $$ x(k,l)=\sum_{\tau =0}^{h}\left[A_{1\tau}x(k-1-\tau,l)+A_{2\tau}x(k,l-1-\tau)\right]+Bu(k,l) $$ (32) where $$ u(k,l)=\sum_{\tau =0}^{h}\left[F_{1\tau}x(k-1-\tau,l)+F_{2\tau}x(k,l-1-\tau)\right] $$ (33) with F1τ, F2τ ∈ Rm×n. The positive boundary conditions are defined by (2). Substituting (33) into (32), the closed-loop system is $$ x(k,l)=\sum_{\tau =0}^{h}\left[(A_{1\tau}+BF_{1\tau})x(k-1-\tau,l)+(A_{2\tau}+BF_{2\tau})x(k,l-1-\tau)\right] .$$ (34) Theorem 3 There exist matrices F1τ and F2τ, such that system (34) is stable and positive, if and only if there exist vectors $$f_{1\tau }^{\,j},f_{2\tau }^{\,j} \in R^{m} $$, and $$\lambda =\left [{\lambda ^{1}} \quad {\lambda ^{2}} \quad {\cdot \cdot \cdot} \quad {\lambda ^{n}}\right ]^{T} \in R_{+}^{n} $$, with τ = 0, 1, …, h, and j = 1, 2, …, n, such that the following conditions hold: $$ a_{1\tau }^{ij} \lambda^{j} +{b_{i}^{T}} f_{1\tau }^{\,j} \ge 0, a_{2\tau }^{ij} \lambda^{j} +{b_{i}^{T}} f_{2\tau }^{\,j} \ge 0, $$ (35) $$ \sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } ) +\sum_{\tau =0}^{h}\sum_{j=1}^{n}\left(Bf_{1\tau }^{\,j} +Bf_{2\tau }^{\,j} \right) -\lambda \prec 0 $$ (36) where $$a_{1\tau }^{ij} $$ and $$a_{2\tau }^{ij} $$ denote the elements located at (i, j) of matrices A1τ and A2τ, respectively, vector $${b_{i}^{T}} $$ is the ith row vector of matrix B, with i, j = 1, 2, …, n. Then, the controller gain matrices can be computed by \begin{align*}F_{1\tau } =&\,\left[\begin{array}{cccc} f_{1\tau }^{1} /\lambda^{1} & f_{1\tau }^{2} /\lambda^{2} & \cdots & f_{1\tau }^{\,n} /\lambda^{n}\end{array}\right],\\ F_{2\tau } =&\,\left[\begin{array}{cccc} f_{2\tau }^{1} /\lambda^{1} & f_{2\tau }^{2} /\lambda^{2} & \cdots & f_{2\tau }^{\,n} /\lambda^{n}\end{array}\right].\end{align*} Proof First, since $$a_{1\tau }^{ij},a_{2\tau }^{ij} \ge 0$$, (35) implies $$a_{1\tau }^{ij} +{b_{i}^{T}} f_{1\tau }^{\,j} /\lambda ^{j} \ge 0$$ and $$a_{2\tau }^{ij} +{b_{i}^{T}} f_{2\tau }^{\,j} /\lambda ^{j} \ge 0$$ with τ = 0, 1, …, h, and i, j = 1, 2, …, n, which in turn is equivalent to the fact that $$A_{1\tau } +BF_{1\tau }\, \underline {\succ }\,0$$ and $$A_{2\tau } +BF_{2\tau }\, \underline {\succ }\,0$$. Therefore, according to Lemma 1, system (34) is positive if and only if (35) holds. Second, since $$F_{1\tau } \lambda =\left [ {f_{1\tau }^{1} /\lambda ^{1} } \quad {f_{1\tau }^{2} /\lambda ^{2} } \quad {\cdot \cdot \cdot } \quad {f_{1\tau }^{\,n} /\lambda ^{n} } \right ]\lambda =\sum _{j=1}^{n}f_{1\tau }^{\,j} $$, $$F_{2\tau } \lambda =\sum _{j=1}^{n}f_{2\tau }^{\,j} $$, (36) is equivalent to $$ \left(\sum_{\tau =0}^{h}\left(A_{1\tau } +BF_{1\tau } +A_{2\tau } +BF_{2\tau }\right) -I_{n} \right)\lambda \prec 0.$$ (37) According to Theorem 1, system (34) is positive and stable if and only if (37) holds. The proof is completed. 4.2 Constrained control For the widely existence of physical bounds on the state and control variables, so a fundamental requirement is that the resulting closed-loop system fulfills these constraints. In this subsection, we focus on solving the constrained control problem for the two-dimensional systems with multiple delays. This subsection deals with the following constrained system: \begin{align} x(k,l)=&\,\sum_{\tau =0}^{h}\left[A_{1\tau } x(k-1-\tau,l)+A_{2\tau } x(k,l-1-\tau )\right] +Bu(k,l)\nonumber\\ 0\underline{\prec}&\,u(k,l)\underline{\prec }\bar{u} \end{align} (38) where $$\bar {u}$$ is a constant vector serving as the upper bound of the input u(k, l) defined by (33). The boundary conditions for system (38) are defined by (2). Substituting (33) into (38), the resulting closed-loop system is \begin{align} x(k,l)=&\,\sum_{\tau =0}^{h}\left[(A_{1\tau } +BF_{1\tau } )x(k-1-\tau,l)+(A_{2\tau } +BF_{2\tau } )x(k,l-1-\tau )\right]\nonumber\\ 0\ \underline{\prec}&\ u(k,l)\ \underline{\prec }\ \bar{u} .\end{align} (39) In this situation, the goal is to find out matrices F1τ and F2τ with τ = 0, 1, …, h, such that there exists a state-feedback control law (33) satisfying $$0\,\underline {\prec }\,u(k,l)\,\underline {\prec }\,\bar {u}$$, under which the following two constraints are satisfied. The closed-loop system is positive and stable. $$0\,\underline {\prec }\ \hat {x}(k,l)\ \underline {\prec }\ \lambda $$ for the boundary condition (2) satisfying \begin{align} 0\,\underline{\prec}\ &\,f_{k,l}\, \underline{\prec }\,\lambda (-h\le k\le 0,\ -h\le l\le z_{2})\nonumber\\ 0\ \underline{\prec}\ &\,g_{k,l} \,\underline{\prec }\,\lambda (-h\le k\le z_{1},\ -h\le l\le 0) .\end{align} (40) To achieve this goal, we need to introduce Lemma 2. Lemma 2 Let the solution to system (1) be $$\hat {x}(k,l)$$. For a given vector $$\lambda \in R_{+}^{n} $$ and any initial condition (2) satisfying (40), it holds that $$0\,\underline {\prec }\ \hat {x}(k,l)\,\underline {\prec }\ \lambda $$ if and only if $$A_{1\tau } \ \underline {\succ }\ 0$$, $$A_{2\tau } \ \underline {\succ }\ 0$$, and $$ \left(\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau}) -I_{n}\right)\lambda \prec 0 $$ (41) with τ = 0, 1, …, h. Proof Necessity: If the system is necessarily positive, then $$A_{1\tau } \ \underline {\succ }\ 0$$ and $$A_{2\tau } \ \underline {\succ }\ 0$$, with τ = 0, 1, …, h. Take the values of fk, l and gk, l in boundary condition (2) as follows: $$ f_{k,l} =\lambda (-h\le k\le 0,\ -h\le l\le z_{2} ), g_{k,l} =\lambda (-h\le k\le z_{1},-h\le l\le 0) .$$ (42) Since the solution to system (1) satisfies $$0\,\underline {\prec }\ \hat {x}(k,l)\,\underline {\prec }\ \lambda $$, then $$\hat {x}(1,1)\,\underline {\prec }\ \lambda $$, that is $$ \hat{x}(1,1)=\sum_{\tau =0}^{h}A_{1\tau } \hat{x}(-\tau,1)+A_{2\tau}\hat{x}(1,-\tau)=\sum_{\tau =0}^{h}(A_{1\tau }+A_{2\tau})\lambda \prec \lambda .$$ (43) It is easy to obtain that $$(\sum _{\tau =0}^{h}(A_{1\tau } +A_{2\tau } ) -I_{n} )\lambda \prec 0$$. Sufficiency: For given bounded boundary conditions, according to inequality (41), we easily have $$ \hat{x}(1,1)=\sum_{\tau =0}^{h}\left[A_{1\tau } \hat{x}(-\tau,1)+A_{2\tau } \hat{x}(1,-\tau )\right]=\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } )\lambda \prec \lambda .$$ (44) For $$\hat {x}(1,j)\,\underline {\prec }\ \lambda , j\ge 1$$, it follows that $$ \hat{x}(1,j+1)=\sum_{\tau =0}^{h}\left[A_{1\tau } \hat{x}(-\tau,j+1)+A_{2\tau } \hat{x}(1,j-\tau )\right]\underline{\prec }\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } )\lambda \underline{\prec }\ \lambda .$$ (45) Using the mathematical induction, we have $$ \hat{x}(1,l)\ \underline{\prec }\ \lambda,\ l\in Z_{+} .$$ (46) By a similar manner, we can show that $$ \hat{x}(k,1)\ \underline{\prec }\ \lambda,\ k\in Z_{+} .$$ (47) Based on the state revolution of system (1), we further have the state value located at the point (2, 2) depends on those at the adjacent points (1, 2) and (2, 1) and their delayed points (1 − τ, 2) and (2, 1 − τ) (boundary points) along the horizontal and vertical directions, respectively. Then we obtain $$ \left.\begin{array}{r} {\hat{x}(1,2)\ \underline{\prec }\ \lambda } \\ {\hat{x}(2,1)\ \underline{\prec }\ \lambda } \\ {\hat{x}(1-\tau,2)\ \underline{\prec }\ \lambda } \\ {\hat{x}(2,1-\tau )\ \underline{\prec }\ \lambda } \end{array}\right\}\Rightarrow \hat{x}(2,2)\ \underline{\prec }\ \lambda .$$ Applying the mathematical induction as in (44)–(47), we get $$ \hat{x}(2,l)\ \underline{\prec }\ \lambda,\ l\in Z_{+} $$ (48) $$ \hat{x}(k,2)\ \underline{\prec }\ \lambda,\ k\in Z_{+} .$$ (49) Repeating the above procedures repetitively, it follows that $$ \hat{x}(k,l)\ \underline{\prec }\ \lambda,\ \ \forall k,\ l\in Z_{+} .$$ (50) The proof is completed. With the help of the derived results, the constrained control problem of two-dimensional systems with delays in the second FM model gets solved in the following theorem. Theorem 4 For arbitrary $$0\le \bar {u}\in R^{m} $$, suppose that there exist vectors $$0\ \,\underline {\prec }\ \ f_{1\tau }^{j},f_{2\tau }^{j} \in R^{m} $$ and $$\lambda =\left [{\lambda ^{1} } \quad {\lambda ^{2} } \quad {...} \quad {\lambda ^{n} } \right ]^{T} \in R_{+}^{n} $$, with τ = 0, 1, …, h and j = 1, 2, …, n, such that the following conditions hold: $$ a_{1\tau }^{ij} \lambda^{j} +{b_{i}^{T}} f_{1\tau }^{j} \ge 0,\ a_{2\tau }^{ij} \lambda^{j} +{b_{i}^{T}} f_{2\tau }^{j} \ge 0 $$ (51) $$ \sum_{\tau =0}^{h}\sum_{j=1}^{n}\left(f_{1\tau }^{j} +f_{2\tau }^{j}\right) \underline{\prec }\ \bar{u} $$ (52) $$ \underline{\prec }\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } )\lambda +\sum_{\tau =0}^{h}\sum_{j=1}^{n}\left(Bf_{1\tau }^{j} +Bf_{2\tau }^{j}\right)-\lambda \prec 0 .$$ (53) Let $$F_{1\tau } =\left [{f_{1\tau }^{1} /\lambda ^{1} } \quad {f_{1\tau }^{2} /\lambda ^{2} } \quad {...} \quad {f_{1\tau }^{n} /\lambda ^{n} } \right ]$$, $$F_{2\tau } =\left [ {f_{2\tau }^{1} /\lambda ^{1} } \quad {f_{2\tau }^{2} /\lambda ^{2} } \quad {...} \quad {f_{2\tau }^{n} /\lambda ^{n} } \right ]$$. Then, the corresponding closed-loop system (39) is stable and positive, and 0 ≤ x(k, l) ≤ λ and $$0\,\underline {\prec }\ u(k,l)\,\underline {\prec }\ \bar {u}$$ for any boundary condition (2) satisfying (40). Proof According to Theorem 1, if (51) and (53) hold, then $$ \left(\sum_{\tau =0}^{h}\left(A_{1\tau } +BF_{1\tau } +A_{2\tau } +BF_{2\tau } \right) -I_{n} \right)\lambda \prec 0$$ and system (39) is stable and positive. Based on Lemma 2, for any boundary condition (2) satisfying (40), we have $$0\,\underline {\prec }\ x(k,l)\,\underline {\prec }\ \lambda $$. Due to $$F_{1\tau },F_{2\tau } \ \underline {\succ }\ 0$$ with τ = 0, 1, …, h. It follows from (52) that $$ u(k,l)=\sum_{\tau =0}^{h}\left[F_{1\tau } x(k-1-\tau,l)+F_{2\tau } x(k,l-1-\tau )\right]\underline{\prec}\sum_{\tau =0}^{h}(F_{1\tau } \lambda +F_{2\tau } \lambda ) =\sum_{\tau =0}^{h}\sum_{j=1}^{n}\left(f_{1\tau }^{\,j} +f_{2\tau }^{\,j} \right) \underline{\prec }\ \bar{u}.$$ That is $$0\,\underline {\prec }\ u(k,l)\,\underline {\prec }\ \bar {u}$$. The proof is completed. 5. Illustrative examples In this section, two examples are provided to illustrate the theoretical results, Example 1 Consider the two-dimensional FM model (32) with h = 3 and the following system matrices. \begin{align} A_{10} =&\,\left[\begin{array}{cc} {-\text{0.2}} & {-0.1} \\ {0.2} & {0.2} \end{array}\right],\ A_{20} =\left[\begin{array}{cc} {0.2} & {0.1} \\ {0.5} & {0.1} \end{array}\right],\ A_{1\mathrm{3}} =\left[\begin{array}{cc} {0.1} & {0.1} \\ {0.2} & {0.2} \end{array}\right],\nonumber\\ A_{2\mathrm{2}}=&\,\left[\begin{array}{cc} {0.1} & {0.1} \\ {0.1} & {0} \end{array}\right],\ B=\left[\begin{array}{c} {-0.3} \\ {0.1} \end{array}\right],\ A_{11} =A_{1\mathrm{2}} \mathrm{=}A_{\mathrm{2}1} \mathrm{=}A_{\text{23}} \text{=0.} \end{align} (54) The boundary conditions are $$ x(k,l)=\begin{cases}\left[\begin{array}{cc} {20} & {15} \end{array}\right]^{T}, & -3\le k\le 0,0\le l\le 10,\text{ or }-3\le k\le 10,-3\le l\le 0 \\ \left[\begin{array}{cc} {0} & {0} \end{array}\right]^{T}, & -3\le k\le 0, l>10,\text{ or }k>10,0\le l\le 0. \end{cases}$$ Applying Theorem 3, one easily gets the gain matrices as follows. \begin{align} F_{10} =&\,\left[\begin{array}{cc} {-0.9231} & {-0.5466} \end{array}\right],\ F_{20} =\left[\begin{array}{cc} {0.1336} & {0.1472} \end{array}\right],\nonumber\\ F_{1\mathrm{3}} =&\,\left[\begin{array}{cc} {-0.0112} & {0.0727} \end{array}\right],\ F_{2\mathrm{2}} =\left[\begin{array}{cc} {0.0769} & {0.2424} \end{array}\right]. \end{align} (55) It should be noted that two-dimensional system (32) with (54) in this example is not positive. Under feedback control law (33) with (55), the closed-loop system is stable and positive, as illustrated in Figure 1–2. Fig. 1. View largeDownload slide The response of state x1(k, l). Fig. 1. View largeDownload slide The response of state x1(k, l). Fig. 2. View largeDownload slide The response of state x2(k, l). Fig. 2. View largeDownload slide The response of state x2(k, l). Example 2 Consider the thermal processes in chemical reactors, which can be expressed in the following partial differential equation with time delays (Kaczorek, 1985) $$ \frac{\partial T\left(x,t\right)}{\partial x} =-\frac{\partial T\left(x,t\right)}{\partial t} -a_{0} T\left(x,t\right)-a_{1} T\left(x,t-\tau \right)+bu\left(x,t\right) $$ (56) where $$T\left (x,t\right )$$ is the temperature at $$x\in \left [0,x_{f} \right ]$$ (space) and $$t\in [0,\infty ]$$ (time), $$u\left (x,t\right )$$ is the input function, τ is the time delay and a0, a1, b are real coefficients. Taking \begin{align*}T\left(k,l\right)=&\,T\left(k\Delta x,l\Delta t\right), u\left(k,l\right)=u\left(k\Delta x,l\Delta t\right), \sigma \left(k,l\right)=\sigma \left(k\Delta x,l\Delta t\right),\\ \frac{\partial T\left(x,t\right)}{\partial x} \approx&\, \frac{T\left(k,l\right)-T\left(k-1,l\right)}{\Delta x}, \frac{\partial T\left(x,t\right)}{\partial t} \approx \frac{T\left(k,l+1\right)-T\left(k,l\right)}{\Delta t},\end{align*} we can rewrite (56) in the discrete form $$ T\left(k,l+1\right)=\left(1-\frac{\Delta t}{\Delta x} -a_{0} \right)T\left(k,l\right)+\frac{\Delta t}{\Delta x} T\left(l-1,l\right)-a_{1} \Delta tT\left(k,l-\tau \right)+bu\left(x,t\right). $$ (57) Let $$x(k,l)=\left [ {T\left (k,l+1\right )} \quad {T\left (k,l+1\right )} \right ]^{T} $$, then the system can be converted into a two-dimensional FM model (38) with h = 4, and the following system matrices \begin{align*}A_{10} =&\,\left[\begin{array}{cc} {\mathrm{0}} & {1} \\ {0} & {0} \end{array}\right], A_{20} =\left[\begin{array}{cc} {0} & {0} \\ {-0.4} & {0.2} \end{array}\right], A_{2\mathrm{4}} =\left[\begin{array}{cc} {0} & {0} \\ {0} & {0.1} \end{array}\right], B=\left[\begin{array}{c} {0} \\ {0.2} \end{array}\right].\\ A_{1j} =&\,0,j=1,2,3,4, A_{2j} =0,j=1,2,3.\end{align*} Choosing $$\bar {u}=100$$ and according to Theorem 3, the feasible gain matrices and positive vector variable λ are obtained: \begin{align*}F_{10} =&\,\left[\begin{array}{cc} {\text{0.4119}} & {\text{0.5580}} \end{array}\right], F_{20} =\left[\begin{array}{cc} {\text{2.3638}} & {\text{0.5580}} \end{array}\right],\\[10pt] F_{2\mathrm{4}} =&\,\left[\begin{array}{cc} {\text{0.4119}} & {\text{0.5580}} \end{array}\right], \lambda =\left[\begin{array}{c} {\text{20.1644}} \\ {\text{14.8843}} \end{array}\right].\end{align*} Then, the following boundary conditions satisfying (40) are given as follows: $$ x(k,l)=\begin{cases}\left[\begin{array}{cc} {20} & {14} \end{array}\right]^{T}, & k\mathrm{=}0,0\le l\le 10,\text{ or }0\le k\le 10,-4\le l\le 0 \\[10pt] \left[\begin{array}{cc} {0} & {0} \end{array}\right]^{T}, & k\mathrm{=}0, l>10,\text{ or }k>10,0\le l\le 10.\end{cases}$$ It can be shown that closed-loop system (39) is positive and all state trajectories converge to the origin and satisfy that 0 ≤ x(k, l) ≤ λ, which can be seen from Figure 3–4. The evolution of control inputs is given in Figure 5, obviously, the constraint condition $$0\,\underline {\prec }\ u(k,l)\,\underline {\prec }\ \bar {u}$$ is satisfied. Fig. 3. View largeDownload slide The response of state x1(k, l). Fig. 3. View largeDownload slide The response of state x1(k, l). Fig. 4. View largeDownload slide The response of state x2(k, l). Fig. 4. View largeDownload slide The response of state x2(k, l). Fig. 5. View largeDownload slide Control input u(k, l). Fig. 5. View largeDownload slide Control input u(k, l). 6. Conclusions Necessary and sufficient stability conditions have been proposed for positive two-dimensional linear systems with multiple time delays described by the second Fornasini–Marchesini model. A state-feedback controller has been designed to ensure the non-negativity and the stability of the unconstrained closed-loop systems. Finally, a sufficient condition has been provided under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Acknowledgements The authors gratefully thank the National Natural Science Foundation of China under Grant No. 61573007 and No. 61603188 for financial support. References Benton , S. E. , Rogers , E. & Owens , D. H. ( 2010 ) Stability tests for a class of 2D continuous-discrete linear systems with dynamic boundary conditions . Int. J. Control , 75 , 55 -- 60 . Benvenuti , L. , Santis , A. & Farina , L. ( 2003 ) Positive Systems, Lecture Notes in Control and Information Sciences . Berlin : Springer . Berman , A. , Neumann , M. & Stern , R. J. ( 1989 ) Nonnegative Matrices in Dynamic Systems . New York : Wiley . Blanchini , F. ( 1990 ) Feedback control for linear time-invariant systems with state and control bounds in the presence of disturbance . IEEE Trans. Automat. Control , 35 , 1231 -- 1234 . Google Scholar CrossRef Search ADS Bouagada , D. & Dooren , P. V. ( 2013 ) On the stability of 2D state-space models . Numeri. Linear Algebra Appl. , 20 , 198 -- 207 . Google Scholar CrossRef Search ADS Bracewell , R. N. ( 1995 ) Two-Dimensional Imaging, Prentice-Hall Signal Processing Series . Englewood Cliffs, NJ : Prentice-Hall . Chu , B. & Liu , Y. ( 2007 ) On the asymptotic stability of positive 2-D systems described by the Roesser model . IEEE Trans. Circuits Syst. II: Express Briefs , 54 , 1102 -- 1104 . Google Scholar CrossRef Search ADS Duan , Z. , Karimi , H. & Xiang , Z. ( 2013 ) Stability and l1 -gain analysis for positive 2D systems with state delays in the Roesser model . Math. Probl. Eng. , 1 -- 11 . doi: 0.1155/2013/169713 . Duan , Z. , Xiang , Z. & Karimi , H. ( 2014 ) Stability and l1 -gain analysis for positive 2D T-S fuzzy state-delayed systems in the second FM mode . Neurocomputing , 142 , 209 -- 215 . Google Scholar CrossRef Search ADS Dymkov , M. & Dymkou , S. ( 2012 ) Repetitive and 2-D systems theory approach for modeling in gas networks . Proceedings of 4th International Conference in Problems of Cybernetics and Informatics , pp. 1 -- 4 . Farina , L. & Rinaldi , S. ( 2000 ) Positive linear systems: theory and applications . New York : Wiley . Google Scholar CrossRef Search ADS Fernando , K. ( 1988 ) Stability of 2D state space . Systems Application of Matrix Theory . Fornasini , E. & Marchesini , G. ( 1976 ) State-space realization theory of two-dimensional filters . IEEE Trans. Automat. Control , 21 , 484 -- 491 . Google Scholar CrossRef Search ADS Fornasini , E. & Marchesini , G. ( 1978 ) Doubly indexed dynamical systems: state-space models and structural properties . Math. Syst. Theory , 12 , 59 -- 72 . Google Scholar CrossRef Search ADS Fornasini , E . ( 1991 ) A 2-D systems approach to river pollution modelling . Multi. Syst. Signal Proc., 2 , 233 -- 265 . Google Scholar CrossRef Search ADS Fornasini , E. & Valcher , M. E. 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Google Scholar CrossRef Search ADS Kaczorek , T. ( 1985 ) Two-Dimensional Linear Systems . Berlin : Springer . Kaczorek , T. ( 2001 ) Positive 1D and 2D Systems . London : Springer-Verlag . Kaczorek , T. ( 2009a ) Asymptotic stability of positive 2D linear systems with delays . Bull. Pol. Academy Sci.: Tech. Sci. , 57 , 133 -- 138 . Kaczorek , T. ( 2009b ) Independence of asymptotic stability of positive 2D linear systems with delays of their delays . Int. Applied Math. Comput. Sci. , 19 , 255 -- 261 . Kurek , J . ( 1985 ) The general state-space model for a two-dimensional linear digital system . IEEE Trans. Automatic Control , 30 , 600 -- 602 . Google Scholar CrossRef Search ADS Li , P . ( 2014 ) Positive filtering for continuous-time positive systems under performance . Int. J. Control , 87 , 1906 -- 1913 . Google Scholar CrossRef Search ADS Liu , X. , Wang , L. & Yu , W. ( 2008 ) Constrained control of positive discrete-time systems with delays . IEEE Trans. Circuits Syst. II: Express Briefs , 55 , 193 -- 197 . Google Scholar CrossRef Search ADS Paszke , W. , Lam , J. , Gałkowski, K., Xu, S. & Lin, Z . ( 2004 ) Robust stability and stabilisation of 2D discrete state-delayed systems. Syst. Control Lett. , 51 , 277 -- 291 . Przyborowski , P. & Kaczorek , T. ( 2009 ) Positive 2D discrete-time linear Lyapunov systems . Int. J. Applied Math. Comput. Sci. , 19 , 95 -- 106 . Google Scholar CrossRef Search ADS Rami , M. A. & Tadeo , F. ( 2007 ) Controller synthesis for positive linear systems with bounded controls . IEEE Trans. Circuits Syst. II: Express Briefs , 54 , 151 -- 155 . Google Scholar CrossRef Search ADS Roesser , R. P. ( 1975 ) A discrete state-space model for linear image processing . IEEE Trans. Automatic Control , 20 , 1 -- 10 . Google Scholar CrossRef Search ADS Twardy , M. ( 2007 ) An LMI approach to checking stability of 2D positive systems . Bull. Pol. Academy Sci. Tech. Sci. , 55 , 385 -- 395 . Wu , L. , Lam , J. & Wang , C. 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Stability and constrained control for positive two-dimensional systems with delays in the second FM model

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Abstract

Abstract This paper addresses the stability and control problem of linear positive two-dimensional discrete-time systems with multiple delays in the second Fornasini–Marchesini model. The contribution lies in three aspects. First, a novel proof is provided to establish necessary and sufficient conditions of asymptotic stability for positive two-dimensional delayed systems. Then, a state-feedback controller is designed to ensure the non-negativity and stability of the closed-loop systems. Finally, a sufficient condition for the existence of constrained controllers is developed under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Two examples are given to validate the proposed methods. 1. Introduction Two-dimensional systems have attracted considerable attention from researchers in the recent decades due to the wide range of applications in electricity transmission, circuit analysis, digital image processing, process control and modeling of partial differential equations, e.g. Darboux equation used in dynamic processes in gas absorption, water stream heating and air drying (Roesser, 1975; Bracewell, 1995; Kaczorek, 1985; Dymkov & Dymkou, 2012). Two-dimensional systems can be represented by different models such as Roesser model, Fornasini-Marchesini (FM) model and Kurek model (Roesser, 1975; Fornasini & Marchesini, 1976, 1978; Kurek, 1985). Numerous results on stability analysis and stabilization of two-dimensional discrete systems are available (Benton et al., 2010; Fernando, 1988; Hinamoto, 1997; Bouagada & Dooren, 2013; Gao et al., 2004; Paszke et al., 2004; Wu et al., 2009; Xiang & Huang, 2013; Yang et al., 2006). A system is said to be positive if its states and outputs are non-negative whenever the initial conditions and inputs are non-negative. Positive systems model many real world physical systems, which involve non-negative variables, for example, population levels, absolute temperature and concentration of substances (Benvenuti et al., 2003; Berman et al., 1989; Farina & Rinaldi, 2000; Li, 2014). As a special class of two-dimensional systems, positive two-dimensional systems have also been extensively studied due to their significant applications in various fields such as the wave equation in fluid dynamics, river pollution and self-purification process, gas absorption, etc. (Kaczorek, 2001; Fornasini, 1991, 1996). Hence, researchers have paid considerable attention to stability and stabilization of positive two-dimensional systems during the past years (Przyborowski & Kaczorek, 2009; Chu & Liu, 2007; Twardy, 2007; Duan et al., 2013, 2014; Kaczorek, 2009a, 2009b). In addition, time delays widely exist naturally in many practical systems and are frequently a source of instability and poor performance, even make systems out of control. The aforementioned papers did not consider the effects of delays. Kaczorek (2009a) gave one sufficient and necessary condition of asymptotic stability for positive two-dimensional systems with delays by transferring the two-dimensional model with delays into one system without delays but with higher dimension. Then, Kaczorek (2009b) improved the results in Kaczorek (2009a). By co-positive Lypunov function method, Duan et al. (2013, 2014) achieved several results on stability and performance analysis for positive two-dimensional systems with delays but the conditions are usually only sufficient. On the other hand, real plants always involve constrained variables, such as saturated control signal. Bounded control is also an active research issue (Blanchini, 1990; Henrion et al., 2001; Rami & Tadeo, 2007; Liu et al., 2008), where the control inputs and the states of the closed-loop systems are required to be bounded. Rami & Tadeo (2007) provided constrained controller design for positive one-dimensional system and Liu et al. (2008) extended the result to the case with delays. To the best of authors’ knowledge, no one has directly considered constrained control problem for positive two-dimensional systems. With regard to these previous works, this paper addresses the constrained control problem for the positive two-dimensional systems with delays in the second FM model. In this paper, one necessary and sufficient condition is proposed to test the stability of positive two-dimensional systems with multiple time delays described by the second Fornasini–Marchesini model. Then, a necessary and sufficient condition for the existence of state-feedback controllers is established for general delayed two-dimensional systems, which ensures the non-negativity and the stability of the resulting closed-loop systems. Finally, a constrained controller is designed to stabilize the addressing systems such that the control inputs and the states of the corresponding closed-loop systems are non-negative and bounded. Two examples illustrate the feasibility of the proposed approaches. The remainder of this paper is organized as follows: problem formulation and necessary preliminaries are presented in Section 2. Section 3 establishes the necessary and sufficient stability criteria for positive two-dimensional systems with multiple constant delays. The bounded controller for stabilizing delayed two-dimensional systems is designed in Section 4. Section 5 provides two examples and Section 6 concludes this paper. Notations 1 In this paper, the superscript ‘T’ denotes the transpose. $$A\ \underline {\succ }\ 0$$($$\underline {\prec }\ 0$$) means that all entries of matrix A are non-negative (non-positive). A ≻ 0 (≺ 0) means that all entries of matrix A are positive (negative). Rn×m denotes the set of n × m real matrices. The set of real n × m matrices with non-negative entries will be denoted by $$R_{+}^{n\times m} $$ and the set of non-negative integers will be denoted by Z+. The n × n identity matrix will be denoted by In. Vector $${b_{i}^{T}} $$ is the ith row vector of matrix B, and $$A_{1\tau } =[a_{1\tau }^{ij} ]$$. The l1 norm of a two-dimensional signal $$w\left (k,l\right )=\left [w_{1} \left (k,l\right ),w_{2} \left (k,l\right ),\cdot \cdot \cdot ,w_{m} \left (k,l\right )\right ]^{T} $$ is given by $$ \left\| w(k,l)\right\| =\sum_{i=1}^{m}\left|w_{i}(k,l)\right|.$$ 2. Problem formulation and preliminaries Consider following positive two-dimensional discrete system in FM model: $$ x(k,l)=\sum_{\tau =0}^{h}\left[A_{1\tau } x(k-1-\tau,l)+A_{2\tau } x(k,l-1-\tau )\right] $$ (1) where x(k, l) ∈ Rn is the state vector, k and l are two integers in Z+, A1τ, A2τ ∈ Rn×n are parameter matrices with τ = 0, 1, …, h, h ∈ Z+ denote the number of delays, τ are constant delays. The positive boundary conditions of system (1) are defined by $$ x(k,l)=\begin{cases} f_{k,l}, & -h\le k\le 0,\ 0\le l\le z_{2}\\ 0, & -h\le k\le 0,\ l>z_{2}\\ g_{k,l}, & 0\le k\le z_{1},\ -h\le l\le 0\\ 0, & k>z_{1},\ -h\le l\le 0\end{cases} $$ (2) where fk, l and gk, l are given non-negative vectors, $$z_{1} <\infty $$, and $$z_{2} <\infty $$ are positive integers. Definition 1 (Kaczorek, 2009a) System (1) is said to be positive if for any positive boundary conditions (2) satisfying $$f_{k,l}\ \underline {\succ }\ 0,g_{k,l}\ \underline {\succ }\ 0$$, we have $$x(k,l)\ \underline {\succ }\ 0$$ for all k, l ≥ 0. Lemma 1 (Kaczorek, 2009a) System (1) is positive if and only if $$A_{1\tau } \ \underline {\succ }\ 0$$ and $$A_{2\tau } \ \underline {\succ }\ 0$$ with τ = 0, 1, …, h. Definition 2 (Yeganefar et al., 2013) System (1) is said to be asymptotically stable if $$\mathop {\lim }\limits _{i\to \infty } X_{i} =0$$ for all bounded boundary conditions (2), where $$ X_{i} =\sup \left\{\left\| x(k,l)\right\| :k+l=i,k,l\ge 1\right\} .$$ (3) The aim of this paper is to establish the asymptotic stability criterion for system (1) and furthermore based on this condition, to solve the constrained control problem for the general positive two-dimensional systems with multiple delays. 3. Stability analysis Theorem 1 System (1) is asymptotically stable if and only if there exists a strictly positive vector $$\lambda \in R_{+}^{n} $$ such that $$ \left(\sum_{\tau =0}^{h}(A_{1\tau} +A_{2\tau}) -I_{n}\right)\lambda \prec 0 .$$ (4) Proof Necessity: For p = k + l with k ≥ 0 and l ≥ 0, we define $$ X(p)=\sum_{k+l=p}x(k,l) =x(p,0)+x(p-1,1)+x(p-2,2)+\cdots+x(k,l)+\cdots+x(1,p-1)+x(0,p) .$$ (5) When constant time delay τ happens to X(p) in horizontal direction, define the following signal \begin{align} X^{-} (p,\tau )=\sum_{k+l=p}x(k-\tau,l)=&\,x(p-\tau,0)+x(p-1-\tau,1)+\cdots x(k-\tau,l)\cdots\nonumber\\ &+x(1-\tau,p-1)+x(-\tau,p).\end{align} (6) Similarly, define X+(p, τ) when time delay τ happens to X(p) in vertical direction as follows \begin{align} X^{+} (p,\tau )=\sum_{k+l=p}x(k,l-\tau )=&\,x(p,-\tau )+x(p-1,1-\tau )+\cdots x(k,l-\tau )\cdots\nonumber\\ &+x(1,p-1-\tau )+x(0,p-\tau).\end{align} (7) Combining (6)–(8), one obtains $$ X^{-} (p,\tau )=\begin{cases} \sum\limits_{i=\tau -p}^{\tau }x(-i,p-\tau +i), & \forall 0\le p\le \tau\\ X(p-\tau )+\sum\limits_{i=1}^{\tau }x(-i,p-\tau +i), & \forall p>\tau\end{cases} $$ (8) $$ X^{+} (p,\tau )=\begin{cases} \sum\limits_{i=\tau -p}^{\tau }x(p-\tau +i,-i), & \forall 0\le p\le \tau\\ X(p-\tau )+\sum\limits_{i=1}^{\tau }x(p-\tau +i,-i), & \forall p>\tau\end{cases}.$$ (9) Taking p = k + l, from the evolution of system (1) we have \begin{align} x(p-1,1)=&\,A_{10} x(p-2,1)+A_{20} x(p-1,0)+\sum_{\tau =1}^{h}\left[A_{1\tau } x(p-2-\tau,1)+A_{2\tau } x(p-1,-\tau )\right]\nonumber\\ &\qquad\cdots\nonumber\\ x(k,l)=&\,A_{10} x(k-1,l)+A_{20} x(k,l-1)+\sum_{\tau =1}^{h}\left[A_{1\tau } x(k-1-\tau,l)+A_{2\tau } x(k,l-1-\tau )\right]\nonumber\\ &\qquad\cdots\nonumber\\ x(1,p-1)=&\, A_{10} x(0,p-1)+A_{20} x(1,p-2)+\sum_{\tau =1}^{h}\left[A_{1\tau } x(-\tau,p-1)+A_{2\tau } x(1,p-2-\tau )\right] .\end{align} (10) Denote the boundary condition by $$X_{0}^{-} (p)=x(p,0)$$ and $$X_{0}^{+} (p)=x(0,p)$$. Summing the above equalities in (10), one obtains \begin{align} &X(p)-X_{0}^{-} (p)-X_{0}^{+} (p)\nonumber\\ &\quad=A_{10}\left[X(p-1)-X_{0}^{-} (p-1)\right]+A_{20}\left[X(p-1)-X_{0}^{+} (p-1)\right]\nonumber\\ &\qquad+A_{11}\left[X^{-} (p-1,1)-X_{0}^{-} (p-1-1)\right]+A_{21} \left[X^{+} (p-1,1)-X_{0}^{+} (p-1-1)\right]\nonumber\\ &\qquad+\cdots+A_{1h} \left[X^{-} (p-1,h)-X_{0}^{-} (p-1,-1)\right]+A_{2h}\left[ X^{+} (p-1,h)-X_{0}^{+} (p-1-h)\right]\nonumber\\ &\quad=A_{10} \left[X(p-1)-X_{0}^{-} (p-1)\right]+A_{20} \left[X(p-1)-X_{0}^{+} (p-1)\right]\nonumber\\ &\qquad+\sum_{\tau =1}^{h}\left\{A_{1\tau } \left[X^{-} (p-1,\tau )-X_{0}^{-} (p-1-\tau )\right]+A_{2\tau } \left[X^{+} (p-1,\tau )-X_{0}^{+} (p-1-\tau )\right]\right\} .\end{align} (11) Thus, for p = 1, 2, …, q (q is a positive integer), from (12) we have respectively \begin{align} X(1)-X_{0}^{-} (1)-X_{0}^{+} (1)=&\,0\nonumber\\ X(2)-X_{0}^{-} (2)-X_{0}^{+} (2)=&\,A_{10} \left[X(1)-X_{0}^{-} (1)\right]+A_{20} \left[X(1)-X_{0}^{+} (1)\right]\nonumber\\ &+\sum_{\tau =1}^{h}\left\{A_{1\tau }\left[X^{-}(1,\tau)-X_{0}^{-} (1-\tau)\right]+A_{2\tau} \left[X^{+} (1,\tau )-X_{0}^{+} (1-\tau )\right]\right\}\nonumber\\ &\cdots\nonumber\\ X(q)-X_{0}^{-}(q)-X_{0}^{+}(q)=&\,A_{10}\left [X(q-1)-X_{0}^{-} (q-1)\right]+A_{20}\left [X(q-1)-X_{0}^{+}(q-1)\right]\nonumber\\ &+\sum_{\tau =1}^{h}\left\{A_{1\tau }^{T}\left[X^{-} (q-1,\tau )-X_{0}^{-} (q-1-\tau )\right]\right.\nonumber\\ &\quad\qquad\left.+\ A_{2\tau }^{T} \left[X^{+} (q-1,\tau )-X_{0}^{+} (q-1-\tau )\right]\right\} .\end{align} (12) Summing the above equalities in (12), one gets \begin{align} &X(q)-X_{0}^{-} (q)-X_{0}^{+} (q)+\sum_{p=1}^{q-1}\left\{X(p)-X_{0}^{-} (p)-X_{0}^{+} (p)\right\}\nonumber\\ &\quad=\sum_{p=1}^{q-1}\left\{A_{10}\left[X(p)-X_{0}^{-} (p)\right]+A_{20}\left[X(p)-X_{0}^{+} (p)\right]\right\}\nonumber\\ &\quad\quad+\sum_{\tau =1}^{h}\sum_{p=1}^{q-1}\left\{A_{1\tau }\left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]+A_{2\tau } \left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]\right\} .\end{align} (13) Furthermore, the above equality can be rewritten in the following form: \begin{align} &X(q)-\sum_{p=1}^{q}\left[X_{0}^{-} (p)+X_{0}^{+} (p)\right]\nonumber\\ &\quad=\sum_{p=1}^{q-1}(A_{10} +A_{20} -I_{n} )X(p) -\sum_{p=1}^{q-1}\left\{A_{10} X_{0}^{-} (p)+A_{20} X_{0}^{+} (p)\right\}\nonumber\\ &\quad\quad+\sum_{\tau =1}^{h}\sum_{p=1}^{q-1}\left\{A_{1\tau } \left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]+A_{2\tau } \left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]\right\} .\end{align} (14) Due to the positive property, the asymptotic stability of system (1) can be equivalent to $$ \mathop{\lim }\limits_{q\to \infty } X(q)= \mathop{\lim }\limits_{q\to \infty } \sum_{k+l=q}x(k,l) =0 .$$ (15) When q tends to infinity, it follows from (14) and (15) that \begin{align} -\sum_{p=1}^{\infty }\left[X_{0}^{-} (p)+X_{0}^{+} (p)\right]=&\,\sum_{p=1}^{\infty }(A_{10} +A_{20} -I_{n} )X(p) -\sum_{p=1}^{\infty }\left\{A_{10} X_{0}^{-} (p)+A_{20} X_{0}^{+} (p)\right\}\nonumber\\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau }\left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]+A_{2\tau }\left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]\right\} .\end{align} (16) Adding $$\sum _{p=1}^{\infty }[X_{0}^{-} (p)+X_{0}^{+} (p)] -\sum _{p=1}^{\infty }\left \{A_{10} X_{0}^{+} (p)+A_{20} X_{0}^{-} (p)\right \} $$ to the both sides of (18) leads to $$ -\sum_{p=1}^{\infty }\left\{A_{10} X_{0}^{+} (p)+A_{20} X_{0}^{-} (p)\right\} =\sum_{p=1}^{\infty }(A_{10} +A_{20} -I_{n} )\left[X(p)-X_{0}^{-} (p)-X_{0}^{+} (p)\right] +\Pi $$ (17) with \begin{align*} \Pi =&\,\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau } \left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]+A_{2\tau } \left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]\right\}\\ =&\,\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau } X^{-} (p,\tau )+A_{2\tau } X^{+} (p,\tau )\right\} -\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau } X_{0}^{-} (p-\tau )+A_{2\tau } X_{0}^{+} (p-\tau )\right\}\\ =&\,\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } X(p-\tau )+A_{2\tau } X(p-\tau )\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\sum_{i=\tau -p}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\sum_{i=1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\\ & -\sum_{\tau =1}^{h}\sum_{p=1}^{\infty }\left\{A_{1\tau } X_{0}^{-} (p-\tau )+A_{2\tau } X_{0}^{+} (p-\tau )\right\}\\ =&\,\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } \left[X(p-\tau )-X_{0}^{-} (p-\tau )-X_{0}^{+} (p-\tau )\right]\right.\\[-2pt] &\left.\quad\qquad\qquad+\ A_{2\tau } \left[X(p-\tau )-X_{0}^{-} (p-\tau )-X_{0}^{+} (p-\tau )\right]\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\sum_{i=1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\} \\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\sum_{i=\tau -p}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } X_{0}^{+} (p-\tau )+A_{2\tau } X_{0}^{-} (p-\tau )\right\} -\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\left\{A_{1\tau } X_{0}^{-} (p-\tau )+A_{2\tau } X_{0}^{+} (p-\tau )\right\} \end{align*} \begin{align} =&\,\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\!\!\left\{A_{1\tau } \!\left[X(p-\tau )-X_{0}^{-} (p-\tau )-X_{0}^{+} (p-\tau )\right]+A_{2\tau }\!\left[X(p-\tau )-X_{0}^{-} (p-\tau )-X_{0}^{+} (p-\tau )\right]\right\} \nonumber\\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\sum_{i=\tau -p+1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\nonumber\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\sum_{i=1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\nonumber\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } X_{0}^{+} (p-\tau )+A_{2\tau } X_{0}^{-} (p-\tau )\right\} .\end{align} (18) Letting $$Y(p)=X(p)-X_{0}^{-} (p)-X_{0}^{+} (p)$$, we have $$ -\psi =\sum_{p=1}^{\infty }(A_{10} +A_{20} -I_{n} )Y(p) +\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }(A_{1\tau } +A_{2\tau } )Y(p-\tau )=\left(\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau }\right) -I_{n} )\sum_{p=1}^{\infty }Y(p) $$ (19) with \begin{align*} \psi =&\,\sum_{p=1}^{\infty }\left\{A_{10} X_{0}^{+} (p)+A_{20} X_{0}^{-} (p)\right\} +\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\left\{A_{1\tau } X_{0}^{+} (p-\tau )+A_{2\tau } X_{0}^{-} (p-\tau )\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=1}^{\tau }\sum_{i=\tau -p+1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}\\ &+\sum_{\tau =1}^{h}\sum_{p=\tau +1}^{\infty }\sum_{i=1}^{\tau }\left\{A_{1\tau } x(-i,p-\tau +i)+A_{2\tau } x(p-\tau +i,-i)\right\}.\end{align*} Without loss of generality, not all fk, l and gk, l in the boundary conditions (2) are zero vectors. It follows that the left hand side of (19) is strictly negative and hence $$ \left(\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau }) -I_{n} \right)\sum_{p=1}^{\infty }Y(p) \prec 0 .$$ (20) The condition (20) is equivalent to (4) with $$\lambda =\sum _{p=1}^{\infty }Y\left (p\right ) $$. Sufficiency: Consider the dual system of system (1): $$ x(k,l)=\sum_{\tau =\mathrm{0}}^{h}\left[A_{1\tau }^{T} x(k-1-\tau,l)+A_{2\tau }^{T} x(k,l-1-\tau )\right] $$ (21) which is positive and stable if and only if the original system (1) is positive and asymptotically stable (Haddad & Chellaboina, 2004). Utilizing the similar line of the deducing process from system (1) to system (21), we have \begin{align} &X(q)-X_{0}^{-} (q)-X_{0}^{+} (q)\nonumber\\ &\quad=A_{10}^{T} \left[X(q-1)-X_{0}^{-} (q-1)\right]+A_{20}^{T} \left[X(q-1)-X_{0}^{+} (q-1)\right]\nonumber\\ &\qquad+\sum_{\tau =1}^{h}\left\{A_{1\tau }^{T} \left[X^{-} (q-1,\tau )-X_{0}^{-} (q-1-\tau )\right]+A_{2\tau }^{T} \left[X^{+} (q-1,\tau )-X_{0}^{-} (q-1-\tau )\right]\right\} .\end{align} (22) According to (15), the stability of system (1) is equivalent to the stability of system (22). Define the following Lyapunov-like function for system (22) $$ V(p)=X^{T}(p)\lambda +\sum_{j=1}^{h}\left[X^{-} (p,j)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda+\sum_{j=1}^{h}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda $$ (23) which is positive for non-zero $$X\left (p\right )$$ and strictly positive vector λ ≻ 0. From (22) and (23), we have \begin{align} \Delta V(p)=&\,V(p+1)-V(p)=\left[X(p)-X_{0}^{-} (p)\right]^{T} A_{10} \lambda +\left[X(p)-X_{0}^{+} (p)\right]^{T} A_{20} \lambda\nonumber\\ &+\sum_{\tau =1}^{h}\left\{\left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]^{T} A_{1\tau } \lambda +\left[X^{+} (p,\tau )\right]\right\}\nonumber\\ &+\left(X_{0}^{-} (p\mathrm{+}1)\right)^{T} \lambda +X_{0}^{+} (p\mathrm{+}1))^{T} \lambda +\sum_{j=1}^{h}[X^{-} \left(p+1,j\right)-X_{0}^{-} (p+1-j)]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda \nonumber\\ &+\sum_{j=1}^{h}\left[X^{+} (p+1,j)-X_{0}^{+} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda -X^{T} \left(p\right)\lambda\nonumber\\ &-\sum_{j=1}^{h}\left[X^{-} \left(p,j\right)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda -\sum_{j=1}^{h}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda .\end{align} (24) Assuming p > h, from (6) and (7), one obtains \begin{align} &\sum_{\tau =1}^{h}\left\{\left[X^{-} (p,\tau )-X_{0}^{-} (p-\tau )\right]^{T} A_{1\tau } \lambda +\left[X^{+} (p,\tau )-X_{0}^{+} (p-\tau )\right]^{T} A_{2\tau } \lambda \right\}\nonumber\\ &\quad-\sum_{j=1}^{h}\left[X^{-} \left(p,j\right)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda -\sum_{j=1}^{h}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda\nonumber\\ &\qquad=-\sum_{j=1}^{h-1}\left[X^{-} \left(p,j\right)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j+1}^{h}A_{1\tau } \lambda -\sum_{j=1}^{h-1}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j+1}^{h}A_{2\tau } \lambda.\end{align} (25) Applying (8) and (9) to (25) leads to \begin{align*} \Delta V(p)=&\,\left[X(p)-X_{0}^{-} (p)\right]^{T} A_{10} \lambda +\left[X(p)-X_{0}^{+} (p)\right]^{T} A_{20} \lambda+\left(X_{0}^{-} (p+1)\right)^{T} \lambda\nonumber\\ &+\left(X_{0}^{+} (p+1)\right)^{T} \lambda -X^{T}(p)\lambda +\Lambda^{-} (p)+\Lambda^{+} (p) \end{align*} (26) with \begin{align*}\Lambda^{-} (p)=&\,\sum_{j=1}^{h} \left[X^{-} \left(p+1,j\right)-X_{0}^{-} (p+1-j) \right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda -\sum_{j=1}^{h-1} \left[X^{-} \left(p,j\right)-X_{0}^{-} (p-j)\right]^{T} \sum_{\tau =j+1}^{h}A_{1\tau } \lambda\\ \Lambda^{+} (p)=&\,\sum_{j=1}^{h}\left[X^{+} (p+1,j)-X_{0}^{+} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda -\sum_{j=1}^{h-1}\left[X^{+} (p,j)-X_{0}^{+} (p-j)\right]^{T} \sum_{\tau =j+1}^{h}A_{2\tau } \lambda.\end{align*} It should be noted that $$ \Lambda^{-} (p)=X(p)\sum_{\tau =1}^{h}A_{1\tau } \lambda +\Theta_{0}^{-} (p) $$ (27) with \begin{align*} \Theta_{0}^{-} (p)=&\,\sum_{j=1}^{h}\left[\sum_{i=1}^{j}x(-i,p+1-j+i) -X_{0}^{-} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda\\ &-\sum_{j=1}^{h-1}\left[\sum_{i=1}^{j}x(-i,p+1-j+i) -X_{0}^{-} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{1\tau } \lambda.\end{align*} Similarly, we have $$ \Lambda^{+} (p)=X(p)\sum_{\tau =1}^{h}A_{2\tau } \lambda +\Theta_{0}^{+} (p) $$ (28) with \begin{align*} \Theta_{0}^{+} (p)=&\,\sum_{j=1}^{h}\left[\sum_{i=1}^{j}x(p+1-j+i,-i) -X_{0}^{-} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda \\ &-\sum_{j=1}^{h-1}\left[\sum_{i=1}^{j}x(p+1-j+i,-i) -X_{0}^{+} (p+1-j)\right]^{T} \sum_{\tau =j}^{h}A_{2\tau } \lambda.\end{align*} Therefore, (26) can be rewritten as: \begin{align*} \Delta V(p)=&\,\left[X(p)-X_{0}^{-} (p)\right]^{T} A_{10} \lambda +\left[X(p)-X_{0}^{+} (p)\right]^{T} A_{20} \lambda +(X_{0}^{-} (p+1))^{T} \lambda +(X_{0}^{+} (p+1))^{T} (p)\lambda\\ &+X^{T} (p)\sum_{\tau =1}^{h}A_{1\tau } \lambda+\Theta_{0}^{-} (p)+X^{T} (p)\sum_{\tau =1}^{h}A_{2\tau } \lambda +\Theta_{0}^{+} (p)\\ &=X(p)\left[\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } ) -I_{n} \right]\lambda +M(p)\end{align*} with $$ M(p)=\Theta_{0}^{-} (p)+\Theta_{0}^{+} (p)-\left(X_{0}^{-} (p)\right)^{T} A_{10} \lambda -\left(X_{0}^{+} (p)\right)^{T} A_{20} \lambda+\left(X_{0}^{-} (p+1)\right)^{T} \lambda +\left(X_{0}^{+} (p+1)\right)^{T} \lambda.$$ Due to the boundness of the boundary condition (2), for $$p>\max \left \{z_{1},z_{2} \right \}$$, we have \begin{align*} X_{0}^{-} \left(p\right)=&\,x(p,0)=0,X_{0}^{+} \left(p\right)=x(0,p)=0,\\ x(p,-\tau )=&\,0,x(-\tau,p)=0,\tau =1,2,\ldots,h. \end{align*} Thus, for any $$p>\max \left \{z_{1},z_{2} \right \}$$, it holds that M(p) = 0 and condition (4) implies $$ \Delta V(p)=X^{T} (p)\left[\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } ) -I_{n} \right]\lambda \prec 0,\quad \forall p>\max \left\{z_{1},z_{2} \right\} .$$ (29) Then, the positive system (22) is asymptotically stable, which implies $$ \mathop{\lim }\limits_{q\to \infty } X\left(q\right)= \mathop{\lim }\limits_{q\to \infty } \sum_{k+l=q}x(k,l) =0\Leftrightarrow \mathop{\lim }\limits_{k+l\to \infty } x(k,l)=0.$$ Thus, system (1) is asymptotically stable. This completes the proof. Remark 1 Kaczorek (2009a) provided a necessary and sufficient condition of asymptotic stability for two-dimensional positive systems with delays using ‘augmented system approach’ (Liu et al., 2008). However, the derived stability condition is delay-dependent and with high computational complexity. Kaczorek (2009b) improved the results in Kaczorek (2009a) and presented a delay independent stability criteria without detailed proof. In this paper, a strict proof of stability criteria is provided via a simple and novel method. Remark 2 It should be noted that the stability in Definition 2 is stricter than ones used in Kaczorek (2009a, 2009b), since $$k+l\to \infty $$ includes the case $$k,l\to \infty $$ used in Kaczorek (2009a, 2009b), and another two cases k = i, $$l\to \infty $$ and $$k\to \infty $$, l = j with $$i,j<\infty $$ being constant integers. However, the sufficient and necessary condition obtained in this paper is the same as that in Kaczorek (2009a, 2009b). Remark 3 Different from the existing results that the stability is usually closely related to the magnitude of the delays for general two-dimensional delayed systems, the magnitude of delays has no any impact on the asymptotic stability for positive two-dimensional delayed systems, which is shown in Theorem 1. It shows that the delays in system (1) are required to be consecutive and the delays along horizontal and vertical orientations take the same value. Actually, system (1) could be used to describe many kinds of actual systems through taking different matrix values. For example, it is readily to obtain that system (1) is equivalent to $$ x(k,l)=A_{10} x(k-1,l)+A_{20} x(k,l-1)+\sum_{\alpha =1}^{\kappa_{1} }A_{1\theta_{\alpha } } x(k-1-\theta_{\alpha },l) +\sum_{\beta =1}^{\kappa_{2}}\left[A_{2\vartheta_{\beta } } x(k,l-1-\vartheta_{\beta } )\right] $$ (30) by defining: \begin{align*}h=&\,\mathop{\max }\limits_{i\in \underline{\kappa_{1} },j\in \underline{\kappa_{2} }} \{ \theta_{i},\vartheta_{j} \}\\ A_{1\tau } =&\,0,\quad \forall \tau \ne \theta_{1},\theta_{2},\cdots,\theta_{\kappa_{1} }\\ A_{2\tau } =&\,0,\quad \forall \tau \ne \vartheta_{1},\vartheta_{2},\cdots,\vartheta_{\kappa_{2}},\end{align*} where κ1, κ2 ∈ Z+ denote the numbers of delays in two directions, θα and ϑβ are distributed constant delays along horizontal and vertical orientations, respectively. In this case, Theorem 1 reduces to the following theorem. Theorem 2 Positive two-dimensional discrete-time system (1) is asymptotically stable if and only if there exists a strictly positive vector $$\lambda \in R_{+}^{n} $$ such that $$ \left(\sum_{\alpha =0}^{\kappa_{1} }A_{1\theta_{\alpha}}+\sum_{\beta =0}^{\kappa_{2}}A_{2\vartheta_{\beta}}-I_{n} \right)\lambda \prec 0 .$$ (31) 4. Control synthesis In this section, our interest is to design state-feedback controllers for general two-dimensional discrete systems with delays, that guarantee the positivity and stability of the resulting closed-loop system. The first subsection is devoted to the control problem without constraints for the two-dimensional discrete systems in FM model, and the second subsection extends the result to the case with constraints. 4.1 Unconstrained control Consider the following two-dimensional discrete systems in FM model $$ x(k,l)=\sum_{\tau =0}^{h}\left[A_{1\tau}x(k-1-\tau,l)+A_{2\tau}x(k,l-1-\tau)\right]+Bu(k,l) $$ (32) where $$ u(k,l)=\sum_{\tau =0}^{h}\left[F_{1\tau}x(k-1-\tau,l)+F_{2\tau}x(k,l-1-\tau)\right] $$ (33) with F1τ, F2τ ∈ Rm×n. The positive boundary conditions are defined by (2). Substituting (33) into (32), the closed-loop system is $$ x(k,l)=\sum_{\tau =0}^{h}\left[(A_{1\tau}+BF_{1\tau})x(k-1-\tau,l)+(A_{2\tau}+BF_{2\tau})x(k,l-1-\tau)\right] .$$ (34) Theorem 3 There exist matrices F1τ and F2τ, such that system (34) is stable and positive, if and only if there exist vectors $$f_{1\tau }^{\,j},f_{2\tau }^{\,j} \in R^{m} $$, and $$\lambda =\left [{\lambda ^{1}} \quad {\lambda ^{2}} \quad {\cdot \cdot \cdot} \quad {\lambda ^{n}}\right ]^{T} \in R_{+}^{n} $$, with τ = 0, 1, …, h, and j = 1, 2, …, n, such that the following conditions hold: $$ a_{1\tau }^{ij} \lambda^{j} +{b_{i}^{T}} f_{1\tau }^{\,j} \ge 0, a_{2\tau }^{ij} \lambda^{j} +{b_{i}^{T}} f_{2\tau }^{\,j} \ge 0, $$ (35) $$ \sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } ) +\sum_{\tau =0}^{h}\sum_{j=1}^{n}\left(Bf_{1\tau }^{\,j} +Bf_{2\tau }^{\,j} \right) -\lambda \prec 0 $$ (36) where $$a_{1\tau }^{ij} $$ and $$a_{2\tau }^{ij} $$ denote the elements located at (i, j) of matrices A1τ and A2τ, respectively, vector $${b_{i}^{T}} $$ is the ith row vector of matrix B, with i, j = 1, 2, …, n. Then, the controller gain matrices can be computed by \begin{align*}F_{1\tau } =&\,\left[\begin{array}{cccc} f_{1\tau }^{1} /\lambda^{1} & f_{1\tau }^{2} /\lambda^{2} & \cdots & f_{1\tau }^{\,n} /\lambda^{n}\end{array}\right],\\ F_{2\tau } =&\,\left[\begin{array}{cccc} f_{2\tau }^{1} /\lambda^{1} & f_{2\tau }^{2} /\lambda^{2} & \cdots & f_{2\tau }^{\,n} /\lambda^{n}\end{array}\right].\end{align*} Proof First, since $$a_{1\tau }^{ij},a_{2\tau }^{ij} \ge 0$$, (35) implies $$a_{1\tau }^{ij} +{b_{i}^{T}} f_{1\tau }^{\,j} /\lambda ^{j} \ge 0$$ and $$a_{2\tau }^{ij} +{b_{i}^{T}} f_{2\tau }^{\,j} /\lambda ^{j} \ge 0$$ with τ = 0, 1, …, h, and i, j = 1, 2, …, n, which in turn is equivalent to the fact that $$A_{1\tau } +BF_{1\tau }\, \underline {\succ }\,0$$ and $$A_{2\tau } +BF_{2\tau }\, \underline {\succ }\,0$$. Therefore, according to Lemma 1, system (34) is positive if and only if (35) holds. Second, since $$F_{1\tau } \lambda =\left [ {f_{1\tau }^{1} /\lambda ^{1} } \quad {f_{1\tau }^{2} /\lambda ^{2} } \quad {\cdot \cdot \cdot } \quad {f_{1\tau }^{\,n} /\lambda ^{n} } \right ]\lambda =\sum _{j=1}^{n}f_{1\tau }^{\,j} $$, $$F_{2\tau } \lambda =\sum _{j=1}^{n}f_{2\tau }^{\,j} $$, (36) is equivalent to $$ \left(\sum_{\tau =0}^{h}\left(A_{1\tau } +BF_{1\tau } +A_{2\tau } +BF_{2\tau }\right) -I_{n} \right)\lambda \prec 0.$$ (37) According to Theorem 1, system (34) is positive and stable if and only if (37) holds. The proof is completed. 4.2 Constrained control For the widely existence of physical bounds on the state and control variables, so a fundamental requirement is that the resulting closed-loop system fulfills these constraints. In this subsection, we focus on solving the constrained control problem for the two-dimensional systems with multiple delays. This subsection deals with the following constrained system: \begin{align} x(k,l)=&\,\sum_{\tau =0}^{h}\left[A_{1\tau } x(k-1-\tau,l)+A_{2\tau } x(k,l-1-\tau )\right] +Bu(k,l)\nonumber\\ 0\underline{\prec}&\,u(k,l)\underline{\prec }\bar{u} \end{align} (38) where $$\bar {u}$$ is a constant vector serving as the upper bound of the input u(k, l) defined by (33). The boundary conditions for system (38) are defined by (2). Substituting (33) into (38), the resulting closed-loop system is \begin{align} x(k,l)=&\,\sum_{\tau =0}^{h}\left[(A_{1\tau } +BF_{1\tau } )x(k-1-\tau,l)+(A_{2\tau } +BF_{2\tau } )x(k,l-1-\tau )\right]\nonumber\\ 0\ \underline{\prec}&\ u(k,l)\ \underline{\prec }\ \bar{u} .\end{align} (39) In this situation, the goal is to find out matrices F1τ and F2τ with τ = 0, 1, …, h, such that there exists a state-feedback control law (33) satisfying $$0\,\underline {\prec }\,u(k,l)\,\underline {\prec }\,\bar {u}$$, under which the following two constraints are satisfied. The closed-loop system is positive and stable. $$0\,\underline {\prec }\ \hat {x}(k,l)\ \underline {\prec }\ \lambda $$ for the boundary condition (2) satisfying \begin{align} 0\,\underline{\prec}\ &\,f_{k,l}\, \underline{\prec }\,\lambda (-h\le k\le 0,\ -h\le l\le z_{2})\nonumber\\ 0\ \underline{\prec}\ &\,g_{k,l} \,\underline{\prec }\,\lambda (-h\le k\le z_{1},\ -h\le l\le 0) .\end{align} (40) To achieve this goal, we need to introduce Lemma 2. Lemma 2 Let the solution to system (1) be $$\hat {x}(k,l)$$. For a given vector $$\lambda \in R_{+}^{n} $$ and any initial condition (2) satisfying (40), it holds that $$0\,\underline {\prec }\ \hat {x}(k,l)\,\underline {\prec }\ \lambda $$ if and only if $$A_{1\tau } \ \underline {\succ }\ 0$$, $$A_{2\tau } \ \underline {\succ }\ 0$$, and $$ \left(\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau}) -I_{n}\right)\lambda \prec 0 $$ (41) with τ = 0, 1, …, h. Proof Necessity: If the system is necessarily positive, then $$A_{1\tau } \ \underline {\succ }\ 0$$ and $$A_{2\tau } \ \underline {\succ }\ 0$$, with τ = 0, 1, …, h. Take the values of fk, l and gk, l in boundary condition (2) as follows: $$ f_{k,l} =\lambda (-h\le k\le 0,\ -h\le l\le z_{2} ), g_{k,l} =\lambda (-h\le k\le z_{1},-h\le l\le 0) .$$ (42) Since the solution to system (1) satisfies $$0\,\underline {\prec }\ \hat {x}(k,l)\,\underline {\prec }\ \lambda $$, then $$\hat {x}(1,1)\,\underline {\prec }\ \lambda $$, that is $$ \hat{x}(1,1)=\sum_{\tau =0}^{h}A_{1\tau } \hat{x}(-\tau,1)+A_{2\tau}\hat{x}(1,-\tau)=\sum_{\tau =0}^{h}(A_{1\tau }+A_{2\tau})\lambda \prec \lambda .$$ (43) It is easy to obtain that $$(\sum _{\tau =0}^{h}(A_{1\tau } +A_{2\tau } ) -I_{n} )\lambda \prec 0$$. Sufficiency: For given bounded boundary conditions, according to inequality (41), we easily have $$ \hat{x}(1,1)=\sum_{\tau =0}^{h}\left[A_{1\tau } \hat{x}(-\tau,1)+A_{2\tau } \hat{x}(1,-\tau )\right]=\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } )\lambda \prec \lambda .$$ (44) For $$\hat {x}(1,j)\,\underline {\prec }\ \lambda , j\ge 1$$, it follows that $$ \hat{x}(1,j+1)=\sum_{\tau =0}^{h}\left[A_{1\tau } \hat{x}(-\tau,j+1)+A_{2\tau } \hat{x}(1,j-\tau )\right]\underline{\prec }\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } )\lambda \underline{\prec }\ \lambda .$$ (45) Using the mathematical induction, we have $$ \hat{x}(1,l)\ \underline{\prec }\ \lambda,\ l\in Z_{+} .$$ (46) By a similar manner, we can show that $$ \hat{x}(k,1)\ \underline{\prec }\ \lambda,\ k\in Z_{+} .$$ (47) Based on the state revolution of system (1), we further have the state value located at the point (2, 2) depends on those at the adjacent points (1, 2) and (2, 1) and their delayed points (1 − τ, 2) and (2, 1 − τ) (boundary points) along the horizontal and vertical directions, respectively. Then we obtain $$ \left.\begin{array}{r} {\hat{x}(1,2)\ \underline{\prec }\ \lambda } \\ {\hat{x}(2,1)\ \underline{\prec }\ \lambda } \\ {\hat{x}(1-\tau,2)\ \underline{\prec }\ \lambda } \\ {\hat{x}(2,1-\tau )\ \underline{\prec }\ \lambda } \end{array}\right\}\Rightarrow \hat{x}(2,2)\ \underline{\prec }\ \lambda .$$ Applying the mathematical induction as in (44)–(47), we get $$ \hat{x}(2,l)\ \underline{\prec }\ \lambda,\ l\in Z_{+} $$ (48) $$ \hat{x}(k,2)\ \underline{\prec }\ \lambda,\ k\in Z_{+} .$$ (49) Repeating the above procedures repetitively, it follows that $$ \hat{x}(k,l)\ \underline{\prec }\ \lambda,\ \ \forall k,\ l\in Z_{+} .$$ (50) The proof is completed. With the help of the derived results, the constrained control problem of two-dimensional systems with delays in the second FM model gets solved in the following theorem. Theorem 4 For arbitrary $$0\le \bar {u}\in R^{m} $$, suppose that there exist vectors $$0\ \,\underline {\prec }\ \ f_{1\tau }^{j},f_{2\tau }^{j} \in R^{m} $$ and $$\lambda =\left [{\lambda ^{1} } \quad {\lambda ^{2} } \quad {...} \quad {\lambda ^{n} } \right ]^{T} \in R_{+}^{n} $$, with τ = 0, 1, …, h and j = 1, 2, …, n, such that the following conditions hold: $$ a_{1\tau }^{ij} \lambda^{j} +{b_{i}^{T}} f_{1\tau }^{j} \ge 0,\ a_{2\tau }^{ij} \lambda^{j} +{b_{i}^{T}} f_{2\tau }^{j} \ge 0 $$ (51) $$ \sum_{\tau =0}^{h}\sum_{j=1}^{n}\left(f_{1\tau }^{j} +f_{2\tau }^{j}\right) \underline{\prec }\ \bar{u} $$ (52) $$ \underline{\prec }\sum_{\tau =0}^{h}(A_{1\tau } +A_{2\tau } )\lambda +\sum_{\tau =0}^{h}\sum_{j=1}^{n}\left(Bf_{1\tau }^{j} +Bf_{2\tau }^{j}\right)-\lambda \prec 0 .$$ (53) Let $$F_{1\tau } =\left [{f_{1\tau }^{1} /\lambda ^{1} } \quad {f_{1\tau }^{2} /\lambda ^{2} } \quad {...} \quad {f_{1\tau }^{n} /\lambda ^{n} } \right ]$$, $$F_{2\tau } =\left [ {f_{2\tau }^{1} /\lambda ^{1} } \quad {f_{2\tau }^{2} /\lambda ^{2} } \quad {...} \quad {f_{2\tau }^{n} /\lambda ^{n} } \right ]$$. Then, the corresponding closed-loop system (39) is stable and positive, and 0 ≤ x(k, l) ≤ λ and $$0\,\underline {\prec }\ u(k,l)\,\underline {\prec }\ \bar {u}$$ for any boundary condition (2) satisfying (40). Proof According to Theorem 1, if (51) and (53) hold, then $$ \left(\sum_{\tau =0}^{h}\left(A_{1\tau } +BF_{1\tau } +A_{2\tau } +BF_{2\tau } \right) -I_{n} \right)\lambda \prec 0$$ and system (39) is stable and positive. Based on Lemma 2, for any boundary condition (2) satisfying (40), we have $$0\,\underline {\prec }\ x(k,l)\,\underline {\prec }\ \lambda $$. Due to $$F_{1\tau },F_{2\tau } \ \underline {\succ }\ 0$$ with τ = 0, 1, …, h. It follows from (52) that $$ u(k,l)=\sum_{\tau =0}^{h}\left[F_{1\tau } x(k-1-\tau,l)+F_{2\tau } x(k,l-1-\tau )\right]\underline{\prec}\sum_{\tau =0}^{h}(F_{1\tau } \lambda +F_{2\tau } \lambda ) =\sum_{\tau =0}^{h}\sum_{j=1}^{n}\left(f_{1\tau }^{\,j} +f_{2\tau }^{\,j} \right) \underline{\prec }\ \bar{u}.$$ That is $$0\,\underline {\prec }\ u(k,l)\,\underline {\prec }\ \bar {u}$$. The proof is completed. 5. Illustrative examples In this section, two examples are provided to illustrate the theoretical results, Example 1 Consider the two-dimensional FM model (32) with h = 3 and the following system matrices. \begin{align} A_{10} =&\,\left[\begin{array}{cc} {-\text{0.2}} & {-0.1} \\ {0.2} & {0.2} \end{array}\right],\ A_{20} =\left[\begin{array}{cc} {0.2} & {0.1} \\ {0.5} & {0.1} \end{array}\right],\ A_{1\mathrm{3}} =\left[\begin{array}{cc} {0.1} & {0.1} \\ {0.2} & {0.2} \end{array}\right],\nonumber\\ A_{2\mathrm{2}}=&\,\left[\begin{array}{cc} {0.1} & {0.1} \\ {0.1} & {0} \end{array}\right],\ B=\left[\begin{array}{c} {-0.3} \\ {0.1} \end{array}\right],\ A_{11} =A_{1\mathrm{2}} \mathrm{=}A_{\mathrm{2}1} \mathrm{=}A_{\text{23}} \text{=0.} \end{align} (54) The boundary conditions are $$ x(k,l)=\begin{cases}\left[\begin{array}{cc} {20} & {15} \end{array}\right]^{T}, & -3\le k\le 0,0\le l\le 10,\text{ or }-3\le k\le 10,-3\le l\le 0 \\ \left[\begin{array}{cc} {0} & {0} \end{array}\right]^{T}, & -3\le k\le 0, l>10,\text{ or }k>10,0\le l\le 0. \end{cases}$$ Applying Theorem 3, one easily gets the gain matrices as follows. \begin{align} F_{10} =&\,\left[\begin{array}{cc} {-0.9231} & {-0.5466} \end{array}\right],\ F_{20} =\left[\begin{array}{cc} {0.1336} & {0.1472} \end{array}\right],\nonumber\\ F_{1\mathrm{3}} =&\,\left[\begin{array}{cc} {-0.0112} & {0.0727} \end{array}\right],\ F_{2\mathrm{2}} =\left[\begin{array}{cc} {0.0769} & {0.2424} \end{array}\right]. \end{align} (55) It should be noted that two-dimensional system (32) with (54) in this example is not positive. Under feedback control law (33) with (55), the closed-loop system is stable and positive, as illustrated in Figure 1–2. Fig. 1. View largeDownload slide The response of state x1(k, l). Fig. 1. View largeDownload slide The response of state x1(k, l). Fig. 2. View largeDownload slide The response of state x2(k, l). Fig. 2. View largeDownload slide The response of state x2(k, l). Example 2 Consider the thermal processes in chemical reactors, which can be expressed in the following partial differential equation with time delays (Kaczorek, 1985) $$ \frac{\partial T\left(x,t\right)}{\partial x} =-\frac{\partial T\left(x,t\right)}{\partial t} -a_{0} T\left(x,t\right)-a_{1} T\left(x,t-\tau \right)+bu\left(x,t\right) $$ (56) where $$T\left (x,t\right )$$ is the temperature at $$x\in \left [0,x_{f} \right ]$$ (space) and $$t\in [0,\infty ]$$ (time), $$u\left (x,t\right )$$ is the input function, τ is the time delay and a0, a1, b are real coefficients. Taking \begin{align*}T\left(k,l\right)=&\,T\left(k\Delta x,l\Delta t\right), u\left(k,l\right)=u\left(k\Delta x,l\Delta t\right), \sigma \left(k,l\right)=\sigma \left(k\Delta x,l\Delta t\right),\\ \frac{\partial T\left(x,t\right)}{\partial x} \approx&\, \frac{T\left(k,l\right)-T\left(k-1,l\right)}{\Delta x}, \frac{\partial T\left(x,t\right)}{\partial t} \approx \frac{T\left(k,l+1\right)-T\left(k,l\right)}{\Delta t},\end{align*} we can rewrite (56) in the discrete form $$ T\left(k,l+1\right)=\left(1-\frac{\Delta t}{\Delta x} -a_{0} \right)T\left(k,l\right)+\frac{\Delta t}{\Delta x} T\left(l-1,l\right)-a_{1} \Delta tT\left(k,l-\tau \right)+bu\left(x,t\right). $$ (57) Let $$x(k,l)=\left [ {T\left (k,l+1\right )} \quad {T\left (k,l+1\right )} \right ]^{T} $$, then the system can be converted into a two-dimensional FM model (38) with h = 4, and the following system matrices \begin{align*}A_{10} =&\,\left[\begin{array}{cc} {\mathrm{0}} & {1} \\ {0} & {0} \end{array}\right], A_{20} =\left[\begin{array}{cc} {0} & {0} \\ {-0.4} & {0.2} \end{array}\right], A_{2\mathrm{4}} =\left[\begin{array}{cc} {0} & {0} \\ {0} & {0.1} \end{array}\right], B=\left[\begin{array}{c} {0} \\ {0.2} \end{array}\right].\\ A_{1j} =&\,0,j=1,2,3,4, A_{2j} =0,j=1,2,3.\end{align*} Choosing $$\bar {u}=100$$ and according to Theorem 3, the feasible gain matrices and positive vector variable λ are obtained: \begin{align*}F_{10} =&\,\left[\begin{array}{cc} {\text{0.4119}} & {\text{0.5580}} \end{array}\right], F_{20} =\left[\begin{array}{cc} {\text{2.3638}} & {\text{0.5580}} \end{array}\right],\\[10pt] F_{2\mathrm{4}} =&\,\left[\begin{array}{cc} {\text{0.4119}} & {\text{0.5580}} \end{array}\right], \lambda =\left[\begin{array}{c} {\text{20.1644}} \\ {\text{14.8843}} \end{array}\right].\end{align*} Then, the following boundary conditions satisfying (40) are given as follows: $$ x(k,l)=\begin{cases}\left[\begin{array}{cc} {20} & {14} \end{array}\right]^{T}, & k\mathrm{=}0,0\le l\le 10,\text{ or }0\le k\le 10,-4\le l\le 0 \\[10pt] \left[\begin{array}{cc} {0} & {0} \end{array}\right]^{T}, & k\mathrm{=}0, l>10,\text{ or }k>10,0\le l\le 10.\end{cases}$$ It can be shown that closed-loop system (39) is positive and all state trajectories converge to the origin and satisfy that 0 ≤ x(k, l) ≤ λ, which can be seen from Figure 3–4. The evolution of control inputs is given in Figure 5, obviously, the constraint condition $$0\,\underline {\prec }\ u(k,l)\,\underline {\prec }\ \bar {u}$$ is satisfied. Fig. 3. View largeDownload slide The response of state x1(k, l). Fig. 3. View largeDownload slide The response of state x1(k, l). Fig. 4. View largeDownload slide The response of state x2(k, l). Fig. 4. View largeDownload slide The response of state x2(k, l). Fig. 5. View largeDownload slide Control input u(k, l). Fig. 5. View largeDownload slide Control input u(k, l). 6. Conclusions Necessary and sufficient stability conditions have been proposed for positive two-dimensional linear systems with multiple time delays described by the second Fornasini–Marchesini model. A state-feedback controller has been designed to ensure the non-negativity and the stability of the unconstrained closed-loop systems. Finally, a sufficient condition has been provided under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Acknowledgements The authors gratefully thank the National Natural Science Foundation of China under Grant No. 61573007 and No. 61603188 for financial support. References Benton , S. E. , Rogers , E. & Owens , D. H. ( 2010 ) Stability tests for a class of 2D continuous-discrete linear systems with dynamic boundary conditions . Int. J. Control , 75 , 55 -- 60 . Benvenuti , L. , Santis , A. & Farina , L. ( 2003 ) Positive Systems, Lecture Notes in Control and Information Sciences . Berlin : Springer . Berman , A. , Neumann , M. & Stern , R. J. ( 1989 ) Nonnegative Matrices in Dynamic Systems . New York : Wiley . Blanchini , F. ( 1990 ) Feedback control for linear time-invariant systems with state and control bounds in the presence of disturbance . IEEE Trans. Automat. Control , 35 , 1231 -- 1234 . 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