Stabilization of discrete-time positive switched T-S fuzzy systems subject to actuator saturation

1.   Introduction

Positive switched systems consist of a series of subsystems and a switching signal that determines the switching behavior of the subsystems ^[1]. In recent years, positive switched systems have received extensive attention ^[2,3,4,5] due to their wide application in various fields, including formation flying ^[6], communication ^[7], water pollution control ^[8].

The nonlinear characteristic is ubiquitous in practical positive switched systems. T-S fuzzy model, as a powerful tool to deal with nonlinearity^[9,10], is used to approximate positive switched nonlinear systems. Such systems are called PSTSFSs in this paper.

The existing research results on PSTSFSs mainly focus on the stabilization problem under the case that switched subsystems are all stabilizable ^[11,12,13]. However, switched subsystems may be unstabilizable in practice ^[14,15,16]. Recently, the stabilization has been investigated for discrete-time PSTSFSs with partially stabilizable switched subsystems in ^[17].

However, little attention is paid to the problem of actuator saturation in ^[17]. In fact, actuator saturation is an unavoidable phenomenon in the practical system due to the inherent constraints on physical actuators. Actuator saturation often leads to system performance degradation or even instability. Hence, the actuator saturation has attracted extensive attention in the past few years^{[18,19,20,21,22,23]}. For continuous-time PSTSFSs with partially stabilizable switched subsystems, the actuator saturation was discussed and sufficient conditions were proposed to ensure the positivity and stability of closed-loop systems in ^[24]. Nevertheless, the control scheme proposed in ^[24] is invalid for discrete-time saturated PSTSFSs. In fact, the controller of practical systems implements in a digital manner. Thus, it is an important and meaningful issue that the stabilization of discrete-time saturated PSTSFSs with actuator saturation. Moreover, Fornasini Ettore also pointed out that the stability condition in the discrete-time case is much more difficult than the stability condition in the continuous-time case in ^[25]. Hence, it is natural to ask the following two questions: how to design the controller to ensure the positivity and stability of closed-loop PSTSFSs subject to actuator saturation in discrete-time domain? Whether the largest ADE of the system can be estimated in discrete-time domain? These two issues motivate us to carry out this work.

The stabilization of discrete-time PSTSFSs with actuator saturation is investigated in this paper. The main innovations include three aspects:

$(1)$ The saturated state feedback fuzzy controller as well as the slow-fast combined MDADT switching signal are co-designed to ensure the positivity and stability of the closed-loop PSTSFSs.

$(2)$ The sufficient conditions for the positivity and stability of discrete-time saturated PSTSFSs with partially stabilizable subsystems are proposed, which can be reduced to the ones applied to the case that switched subsystems are all stabilizable.

$(3)$ The largest ADE for discrete-time saturated PSTSFSs is given by formulating an optimization problem.

The remainder of this paper is arranged as follows. The problem formulation and preliminaries of discrete-time saturated PSTSFSs are presented in Section 2. A saturated fuzzy controller is designed and sufficient conditions for the positivity and stability of discrete-time saturated PSTSFSs are developed in Section 3. The largest ADE of saturated PSTSFSs is given in Section 4. Section 5 provides two simulation examples to verify the availability and superiority of the proposed control scheme. Section 6 summarizes the work of this paper and points out the research in the future.

Notation: $I$ stands for $n$-order identity matrix. $A^{T}$ represents the transpose of matrix $A$. $A \succeq 0 (A\prec0)$ denotes all elements of matrix $A$ are nonnegative (negative). $\left \| \cdot \right \|$ represents Euclidean norm. $\mathbb{R}^n$ and ${\mathbb{R}^{m \times n}}$ represent the set of all $n$-dimensional vectors and the set of all ${m \times n}$ matrices over the real number field, respectively. $\mathbb{R}_+^n$ and ${\mathbb{R}_+^{m \times n}}$ represent the set of all $n$-dimensional vectors and the set of all ${m \times n}$ matrices over the positive real number field, respectively. $L({H})$ stands for the set $\left\{ {\left. {x \in {\mathbb{R}^n}} \right|\left| {{h_{i}}x} \right| \le 1}, {H} \in {\mathbb{R}^{m \times n}} \right\}$, where ${{h_{i}}}$ is the $i$th row of matrix ${H}$. For any constant $c > 0$, $\varepsilon ({\xi_p}, c)$ represents the set $\left\{ {\left. {x(k) \in {\mathbb{R}^n}} \right|{x^T}(k){\xi_p} \le c} \right\}$.

2.   Problem formulation and preliminaries

Consider the following discrete-time PSTSFS with actuator saturation, where each switched subsystem is described by T-S fuzzy model:

$R_p ^i$: IF ${z_{p 1}}(k)$ is $M_{p 1}^i$ and ${z_{p 2}}(k)$ is $M_{p 2}^i$ and $\cdots$ and ${z_{p l}}(k)$ is $M_{p l}^i$, THEN

where $R_p ^i, i \in\mathcal{R}, p\in\mathcal{N}$ is the $i$-th fuzzy rule of the $p$-th positive switched subsystems. $\mathcal{R}\buildrel \Delta \over = \{ 1, 2, \cdots, r\}$ is a set of fuzzy rule numbers with total number $r$. $\mathcal{N}\buildrel \Delta \over = \{ 1, 2, \cdots, N \}$ is a set of positive switched subsystem numbers with total number $N$, and ${\cal N} = {\cal N}_c \cup {\cal N}_{uc}$, where ${\cal N}_c = \{1, 2, \dots, n\}$ and ${\cal N}_{uc} = \{n+1, n+2, \dots, N\}$ represent the set of stabilizable subsystems and the set of unstabilizable subsystems, respectively. ${z_p}(k) = [{{z_{p1}}(k)}\; {{z_{p2}}(k)}\; \cdots \; {{z_{pn}}(k)}]^T$ is a vector composed of premise variables ${z_{pi}}(k)$. $M_{pj}^i, i \in\mathcal{R}, p\in\mathcal{N}, j\in\mathcal{L}$ are fuzzy sets, where $\mathcal{L}\buildrel \Delta \over = \{1, 2, \cdots, l\}$ is a set of fuzzy set numbers with total number $l$. $x(k)\in\mathbb{R}_+^n$ and $u(k)\in\mathbb{R}^m$ are the system state and control input. ${A_{pi}}$ and ${{B}_{pi}}$ are constant matrices of the $p$th subsystems. Define the saturation function ${\rm sat}(u(k)) = {\left[{{\rm sat}({u_1}(k)), {\rm sat}({u_2}(k)), \cdots, {\rm sat}({u_m}(k))} \right]^T}$, where ${\rm{sat}}(u_j(k)) = {\rm{sign}}({u_j}(k)){\rm{min}}\left\{ {\left| {{u_j}(k)} \right|, 1}\right\}$, $j = 1, 2, \cdots, m$.

Applying the fuzzy blending method, the final discrete-time PSTSFS is

where the normalized membership functions ${\theta_{pi}}({z_p}(k))$ satisfy

Some definitions and lemmas related to this paper will be introduced in the follows.

Definition 1 (^[26]). The system is called stabilizable if one can find an unconstrained control input $u(k)$ such that for any finite time $[k_0, k_f]$, the system can always arrive at any terminal state $x(k_f)$.

Definition 2 (^[27]). System (2.1) is controlled positive if under a certain control input ${u}(k)$, the system state ${x}(k)$ is non-negative for any non-negative initial state ${ x}(k_0)$.

Lemma 1 (^[27]). For any $0 < {\theta_{pi}}({z_p}(k)) < 1$, the positivity of system (2.2) with ${\rm sat}(u(k)) = 0$ is equivalent to ${A_{pi}} \succeq 0$, $p\in\mathcal{N}$, $i\in\mathcal{R}$.

Definition 3 (^[28]). System (2.2) is exponentially stable if for any switching signal $\sigma (k)$, we can find two constants $\alpha > 0$ and $0 < \beta < 1$ such that $\left\| x(k) \right\|\le \alpha \beta^{(k-{{k}_{0}})}\left\| x({{k}_{0}}) \right\|$ holds for any $k\ge {{k}_{0}}$.

Definition 4 (^[29]). For given switching signal $\sigma(k) = p$, if one can find two positive constants ${\kappa _{ap}}$ and ${N_{0p}}$ such that

holds for any $k_2 \ge k_1 \ge 0$, then ${\kappa _{ap}}$ is called a MDADT, where $N_{\sigma p}(k_2, k_1)$ and ${T_p}(k_2, k_1)$ denote the total switching numbers and the total time of the $p$th subsystem on $[k_1, k_2)$, respectively.

Definition 5 (^[30]). For given switching signal $\sigma(k) = p$, if one can find two positive constants ${\kappa _{ap}}$ and ${N_{0p}}$ such that

holds for any $k_2 \ge k_1 \ge 0$, then ${\kappa _{ap}}$ is called a fast MDADT, where $N_{\sigma p}(k_2, k_1)$ and ${T_p}(k_2, k_1)$ are the same as Definition 4.

Lemma 2 (^[31]). Given matrices $K, H \in {\mathbb{R}^{m \times n}}$,

holds for any $x(k) \in L(H)$, where ${E_s}\in {\mathbb{R}^{m \times m}}$ are diagonal matrices whose diagonal elements are 0 or 1, and $E_s^ - = I - {E_s}$. Since $s \in Q$, the numbers of matrices ${E_s}$ and ${E_s^ -}$ are both ${2^m}$, thus, ${\rm{sat}}(Kx(k))$ can be further expressed as follows:

where ${\eta _s}(k)$ are nonnegative scalar functions with $0 \le {\eta _s}(k) \le 1$ and $\sum\limits_{s = 1}^{{2^m}} {{\eta _s}} (k) = 1$.

3.   Saturated controller design and stabilization

In this section, our goal is to ensure system (3.4) is controlled positive and exponentially stable by designing a saturated fuzzy controller. And the sufficient conditions for the positivity and stability of closed-loop system with partially stabilizable subsystems are given in Theorem 1. These conditions can be reduced to the ones for the positivity and stability of closed-loop system with all stabilizable subsystems, which is given in Corollary 1.

3.1.   Saturated controller design

If $\sigma(k) = q \in {\cal N}_{uc}$, it is impossible to find a suitable controller to stabilize the subsystem. Therefore, we can ignore the design of the controller, that is to say, ${\rm sat}(u(k)) = 0$. If $\sigma(k) = p \in {\cal N}_{c}$, a suitable controller can be found to stabilize the subsystem, which means ${\rm sat}(u(k))\ne 0$. For any $x(k) \in L(H)$, by Lemma 2, a PDC-based fuzzy controller is designed as follows:

Rule $R_p^i$: IF ${z_{p1}}(k)$ is $M_{p1}^i$ and $\cdots$ and ${z_{pn}}(k)$ is $M_{pn}^i$, THEN

The final fuzzy controller is

with $K_{pj}$, $H_{pj}$ are matrices to be determined.

(3.2) together with (2.2) implies that

where ${\bar A_{pij}} = {A_{pi}} + {B_{pi}}({E_{sj}}{K_{pj}} + E_{sj}^ - {H_{pj}})$.

The final closed-loop PSTSFS with partially stabilizable subsystems in discrete-time domain is

3.2.   Stabilization

Theorem 1. For $p \in {{\cal{N}}_c}$, $q \in {{\cal{N}}_{uc}}$, given constants ${\mu _p} > 1$, $0 < {\lambda _p} < 1$, $0 < {\mu _q} < 1$, ${\lambda _q} > 1$. If we can find a group of vectors ${c_p} \in {\mathbb{R}_+^m}$, $\xi_{p} = [\xi_{p1}, \xi_{p2}, \cdots, \xi_{pn}]\in\mathbb{R}_+^n$, $\xi_{q} = [\xi_{q1}, \xi_{q2}, \cdots, \xi_{qn}]\in\mathbb{R}_+^n$, ${g_{pij}} \in {\mathbb{R}^n}$, ${w_{pij}} \in {\mathbb{R}^n}$, ${z_{pij}}\in {\mathbb{R}}$, such that

then system (3.4) is controlled positive and exponentially stable under the saturated fuzzy controller (3.2) and the slow-fast combined MDADT with

where

and the controller gains are

Proof. It is assumed that the switching instants on $[0, K)$ are $k_i, i\in \{1, 2, \dots, N_\sigma \}$ and when $k \in [{k_i}, {k_{i + 1}})$, the $\sigma(k)$th subsystem is activated.

Case 1: $\sigma(k) = p \in {\cal N}_c$.

Since ${c_p} \in \mathbb{R}_ + ^m, {B_{pi}} \in \mathbb{R}_ + ^{n \times m}, {\xi_p} \in \mathbb{R}_ + ^n$, one can find ${E_{sj}} \in {\mathbb{R}_+^{m \times m}}$ such that ${c_p^TE_{sj}^TB_{pi}^T{\xi_p}} \in \mathbb{R}_+$. It is obtained from (3.5) that

Combining (3.15) and (3.16) yields that

Thus, the positivity of system (3.4) is proved by Lemma 1.

Case 2: $\sigma(k) = q \in {\cal N}_{uc}$.

From (3.6), we can also get the positivity of system (3.4).

In what follows, we will prove that system (3.4) is exponentially stable.

Case 1: $\sigma(k) = p \in {\cal N}_c$.

Constructing the following multiple linear co-positive Lyapunov function candidate:

Let ${d_1} = {\min _{(p, n) \in \mathcal{N} \times \mathcal{N_*}}}\{ {\xi_{pn}}\}$, ${d_2} = {\max _{(p, n) \in\mathcal{N} \times \mathcal{N_*}}}\{ {\xi_{pn}}\}$, where $\mathcal{N_*}$ belongs to the set $\{1, 2, \cdots, n\}$.

From (3.19), one can get that

From (3.4) and (3.20), one has

It follows from (3.7) and (3.21) that

Case 2: $\sigma(k) = q \in {\cal N}_{uc}$.

From (3.8), one can also get

Combining (3.22) and (3.23) gives that

When $k = {k_i}$, the subsystem is assumed that switched from the $\sigma(k_i-1)$th to the $\sigma(k_i)$th.

Case 1: $\sigma(k_i-1) = q \in {\cal N}_{uc}$, $\sigma(k_i) = p \in {\cal N}_c$.

It is obtained from (3.10) that

Case 2: $\sigma(k_i-1) = p' \in {\cal N}_{c}$, $\sigma(k_i) = p \in {\cal N}_c$.

We can also derive from (3.9) that (3.25) holds.

Case 3: $\sigma(k_i-1) = p \in {\cal N}_{c}$, $\sigma(k_i) = q \in {\cal N}_{uc}$.

We can also get from (3.11) that (3.25) holds.

From (3.24) and (3.25), it follows that

Due to ${\mu _p} > 1$, $0 < {\lambda _p} < 1$ and $0 < {\mu _q} < 1$, ${\lambda _q} > 0$, according to Definitions 4 and 5, one can get

and

From (3.26)–(3.28), we have

where

(3.20) together with (3.29) gives that

where $\eta = \frac{{{d _{\rm{2}}}}}{{{d _1}}}\gamma$.

From ${\mu _p} > 1$, $p \in {{\cal N}_c}$, $0 < {\mu _p} < 1$, $p \in {{\cal N}_{uc}}$ and ${d _1} > 0$, ${d _2} > 0$, it follows that $\eta > 0$. From (3.13) and (3.14), one can obtain that $0 < \zeta < 1$. By Definition 3, system (3.4) is exponentially stable.

Then, it will be proved that an ADE of system (3.4) is the set $\bigcap\limits_{p = 1}^N {\varepsilon ({\xi_p}, {\gamma ^{ - 1}})}$. For any initial state $x({k_0}) \in \bigcap\limits_{p = 1}^N {\varepsilon ({\xi_p}, {\gamma ^{ - 1}})} \subset \varepsilon ({\xi _p}, {\gamma^{ - 1}})$, one has $x({k_0}) \in \varepsilon ({\xi_p}, {\gamma ^{ - 1}})$, which implies that ${x^T}({k_0}){\xi_p} \le {\gamma^{ - 1}}$. It is known from (3.30) that $\gamma > 0$. We can get that $\gamma {V_{\sigma ({t_0})}}(x({t_0})) \le 1$ through multiplying both sides of ${x^T}({k_0}){\xi_p} \le {\gamma^{ - 1}}$ by $\gamma$. It yields from $0 < \zeta < 1$ that $0 < {\zeta ^{k - {k_0}}} < 1$. Thereby, it is obtained that ${V_{\sigma (t)}}(x(k)) \le \gamma {\zeta ^{k - {k_0}}}{V_{\sigma ({t_0})}}(x({k_0})) \le 1$, that is to say, $x(k) \in \varepsilon ({\xi_p}, 1) \subset L({H_{pj}})$. Thus, the set $\bigcap\limits_{p = 1}^N {\varepsilon ({\xi_p}, {\gamma ^{ - 1}})}$ is an ADE of system (3.4).

Remark 1. Note that the designed switching signal is invalid for the switching between unstabilizable subsystems. Suppose that the $q$th subsystems is switched to the $r$th one, and then is switched to the $q$th one again, where $p, q, r\in {\cal N}_{uc}$, one has $0 < {\mu _{r}} < 1, 0 < {\mu _q} < 1$. But ${\xi_{r}} - {\mu _{r}}{\xi_q} \prec 0$ and ${\xi_q} - {\mu _q}{\xi_{r}} \prec 0$ imply that ${\mu _{r}}{\mu _q} > 1$. This is in contradiction with $0 < {\mu _{r}} < 1, 0 < {\mu _q} < 1$.

Remark 2. In ^[32], the stabilization for positive switched linear systems with partially stabilizable subsystems and actuator saturation was investigated, which is not applicable to nonlinear systems. Different from ^[32], this paper studies the actuator saturation control of discrete-time PSTSFSs, the obtained result is applicable to nonlinear systems. In addition, the solution to controller gains $K_{pj}$ requires a given matrix $H_{pj}$ in advance in^[32], while this paper can solve $K_{pj}$ and $H_{pj}$ at the same time, which removes the requirement for given $H_{pj}$.

When $N = n$, sufficient conditions in Theorem 1 will reduce to the conditions where subsystems are all stabilizable.

Corollary 1. For any $p, q \in {\cal N}_c$, $p \ne q$ and $i, j\in\mathcal{R}$, given a group of constants $0 < {\lambda _p} < 1$, ${\mu _p} > 1$, and vectors ${c_p} \in {\mathbb{R}_+^m}$. If there exist a set of vectors $\xi_{p}\in\mathbb{R}_+^n$, ${g_{pij}} \in {\mathbb{R}^n}$, ${w_{pij}} \in {\mathbb{R}^n}$, ${z_{pij}} \in {\mathbb{R}}$ such that

and $\varepsilon ({\xi_p}, 1) \subset L({H_{pj}})$, then system (3.4) is controlled positive and exponentially stable under the saturated fuzzy controller (3.2) as well as the switching signal satisfying (3.13), where ${H_{pj}}$, $K_{pj}$ are the same as (3.15) and (3.16).

Note that it is impossible to solve $H_{pj}$ and $K_{pj}$ from the conditions of Theorem 1 due to the existence of (3.12). The conditions for directly solving $H_{pj}$ and $K_{pj}$ will be given in Theorem 2.

Theorem 2. For $p \in {{\cal{N}}_c}$ and $q \in {{\cal{N}}_{uc}}$, given a group of constants ${\mu _p} > 1$, $0 < {\lambda _p} < 1$, $0 < {\mu _q} < 1$, ${\lambda _q} > 1$, and vectors ${c_p} \in {\mathbb{R}_+^m}$. If one can find a family of vectors $\xi_{p}\in\mathbb{R}_+^n$, $\xi_{q}\in\mathbb{R}_+^n$, ${g_{pij}} \in {\mathbb{R}^n}$, ${w_{pij}} \in {\mathbb{R}^n}$, ${z_{pij}}\in {\mathbb{R}}$ such that

then system (3.4) is controlled positive and exponentially stable under the saturated fuzzy controller (3.2) as well as the slow-fast combined MDADT satisfying (3.13) and (3.14).

Proof. To prove Theorem 2, we only need to prove that $(3.12) \Leftrightarrow (3.43)$.

1) ${{(3.12) \Rightarrow (3.43)}}$.

Suppose (3.43) does not hold. It means that

Due to the positivity of $x(k)$, multiplying both sides of (3.44) by $x^T(k)$, we can obtain

For any $x(k)\in L(H_{pj})$, one has $\left| {{h_{pji}}x(k)} \right| \le 1$, namely, $\left| {x^T(k){h^T_{pji}}} \right| \le 1$. Since $x(k)$ is positive, we can get from (3.45) that

It is obtained from (3.46) that $x(k)\in\varepsilon ({\xi_p}, 1)$. Thus, $L({H_{pj}}) \subset \varepsilon ({v_p}, 1)$, which is in contradiction with (3.12), Therefore, $(3.12) \Rightarrow (3.43)$.

2) ${{(3.43) \Rightarrow (3.12)}}$.

For any $x(k) \in \varepsilon ({\xi _p}, 1)$, one has

Multiplying both sides of (3.43) by $x^T(k)$ yields that

Combining (3.47) and (3.48) gives that

Thus, $x(k)\in L(H_{pj})$, which implies that $\varepsilon ({\xi_p}, 1) \subset L({H_{pj}})$. Therefore, $(3.43) \Rightarrow (3.12)$. The proof is completed.

4.   Attraction domain estimation

In this section, we expect to seek a larger attraction domain to reduce the conservatism of the ADE in Theorem 1. The largest ADE is proposed in Theorem 3.

Theorem 3. For $p \in {{\cal{N}}_c}$ and $q \in {{\cal{N}}_{uc}}$, given constants ${\mu _p} > 1$, $0 < {\lambda _p} < 1$, $0 < {\mu _q} < 1$, ${\lambda _q} > 1$, $\gamma > 0$ and vectors ${\xi^*}, {c_p} \in {\mathbb{R}_+^m}$. If there exist a family of vectors $\xi_{p}, \xi_{q} \in\mathbb{R}_+^n$, ${g_{pij}} \in {\mathbb{R}^n}$, ${w_{pij}} \in {\mathbb{R}^n}$, ${z_{pij}}\in {\mathbb{R}}$ such that the following optimization problem

is solvable, then $\varepsilon ({\xi_p}, {\gamma ^{ - 1}})$ is the largest ADE for system (3.4).

Proof. Given a shape reference set ${X_\xi} \subset {\mathbb{R}^n}$, which is a bounded convex set. For the set $S \subset {\mathbb{R}^n}$ and the parameter $\rho > 0$, define ${\rho _\xi}(S)$ as follows:

It is obvious that ${\rho_\xi}(S) \ge 1$, which implies ${X_\xi} \subset S$. When ${\rho _\xi}(S) < 1$, the definition of ${X_\xi}$ is given as follows:

In order to obtain the largest ADE, the maximum of $\rho$ is expected to be found such that $\rho {X_\xi} \subset \varepsilon ({\xi_p}, {\gamma ^{ - 1}})$. Thus, we can transform the ADE problem into the following optimization problem:

However, the above optimization problem is unsolvable due to the existence of $(a')$. In what follows, it is proved that $(a') \Leftrightarrow (a)$.

Necessity(${{(a') \Leftarrow (a)}}$). Suppose ${\xi^*} \prec \gamma \rho {\xi_p}$, then

For any $x(k) \in \varepsilon ({\xi_p}, {\gamma ^{ - 1}})$, one can obtain that

Combining (4.4) and (4.5) gives that

which means that $\frac{1}{\rho }x(k) \in {X_\xi}$, namely, $x(k) \in \rho {X_\xi}$. It follows that $\varepsilon ({\xi_p}, {\gamma ^{ - 1}}) \subset \rho {X_\xi}$. Since it is unconformity to ($a$), the necessity is proved.

Sufficiency (${{(a) \Rightarrow (a')}}$). Multiplying both sides of ($a$) by $x^T(k)$, one has

For any $x(k) \in \rho {X_\xi}$, it can be found that $\frac{1}{\rho }x(k) \in {X_\xi}$. From (4.3), one can get

Combining (4.7) and (4.8) yields that

It follows from (4.9) that $x(k) \in \varepsilon ({\xi_p}, {\gamma ^{ - 1}})$, which means that ($a$) holds. The sufficiency is proved. Hence, ($a'$) are equivalent to ($a$). The proof is completed.

Remark 3. By substituting (3.32)–(3.34) for (3.35)–(3.42) in the condition ($b$) of Theorem 3, it is obtained that the largest ADE of discrete-time PSTSFSs with actuator saturation when switched subsystems are all stabilizable.

5.   Illustrative examples

Two illustrative examples will be provided to verify the availability and advantages of the proposed control scheme.

Example 1. The positive switched nonlinear system with partially stabilizable subsystems

Considering the following positive switched nonlinear system with actuator saturation in discrete-time domain:

Define ${z_1}(k) = {\sin ^2}({x_1}(k))$, ${z_2}(k) = {\sin ^2}({x_2}(k))$, ${z_3}(k) = {\sin ^2}({x_2}(k))$, by sector nonlinearity ^[33], we construct the following T–S fuzzy model:

Rule $R_p ^i$: IF ${z_{p 1}}(k)$ is $M_{p 1}^i$, THEN

where

The final discrete-time PSTSFS with actuator saturation is:

with the following normalized membership functions:

Let ${A_p} = \sum\limits_{i = 1}^2 {{\theta _{pi}}} (z_p(k)){A_{pi}}, p\in\{1, 2, 3\}$, ${B_p} = \sum\limits_{i = 1}^2 {{\theta _{pi}}} (z_p(k)){B_{pi}}$, $p\in \{1, 2, 3\}$, ${K_p} = \sum\limits_{j = 1}^2 {{\theta _{pj}}} (z_p(k)){K_{pj}}, p\in\{2\}.$

For ${\rm sat}(u(k)) = 0$, by Lemma 1, ${A_{p1}}\succeq 0$, ${A_{p2}}\succeq 0$, ${A_{p3}}\succeq 0$ imply the positivity of subsystems ${\Xi _{\rm{1}}}$, ${\Xi _{\rm{2}}}$ and ${\Xi _{\rm{3}}}$. By calculating the eigenvalues of matrices ${A_{p1}}$, ${A_{p2}}$, ${A_{p3}}$, we can get that subsystems ${\Xi _{\rm{1}}}$, ${\Xi _{\rm{2}}}$ and ${\Xi _{\rm{3}}}$ are all unstable.

For ${\rm sat}(u(k))\ne0$, given $\lambda_1 = 2$, $\lambda_2 = 0.8$, $\lambda_3 = 1.8$, $\mu_1 = 0.45$, $\mu_2 = 3$, $\mu_3 = 0.5$, $c_1 = c_2 = c_3 = 1$, $D_{11} = D_{12} = 1$, $D_{11}^- = D_{12}^- = 0$, $D_{21} = D_{22} = 0$, $D_{21}^- = D_{22}^- = 1$, $\xi^* = [3\; 6]$. By Theorem 3, the designed switching signal satisfies $\kappa_{a1} < -\frac{\ln \mu_1 }{\lambda_1 } = 1.1520$, $\tau_{a2} > -\frac{\ln \mu_2 }{ \lambda_2} = 4.9233$, $\tau_{a3} < -\frac{\ln \mu_3}{\lambda_3 } = 1.1792$, the feasible solutions can be obtained and the saturated controller gains are $K_{21} = [-0.4111-0.3618]$, $K_{22} = [-0.4111-0.3618]$.

We can check that ${A_2} + {B_2}{K_2} \succeq 0$. By Lemma $1$, one can get the positivity of closed-loop subsystem ${\Xi _{\rm{2}}}$. Given the initial condition $x(k_0) = {[1,1]^T}$, Figure 1 depicts the state trajectories of closed-loop subsystem ${\Xi _{\rm{2}}}$. From Figure 1, we can observe that the closed-loop subsystem ${\Xi _{\rm{2}}}$ is stable.

According to (3.15) and (3.16), we design the switching signal as follows: $\kappa_{a1} = 1.1 < 1.1520$, $\kappa_{a2} = 5 > 4.9233$ and $\kappa_{a3} = 1.1 < 1.1792$. Hence, it is possible to generate a switching sequence $(1, 2, 3, 2, 1, 3, 2, 1, 2, 3, 2, 1, 2)$. Given $x(k_0) = {[0.7, 0.8]^T}$, the state trajectories of the system are depicted in Figure 2, which shows that the closed-loop system satisfies the positivity and stability. Figures 3 and 4 depict the saturated control input and the largest attraction domain, respectively.

Example 2. The positive switched nonlinear system with all stabilizable subsystems

The water-quality model, borrowed from ^[34,35], can be modeled as the following positive switched nonlinear system:

Similar to Example 1, the final discrete-time PSTSFS with actuator saturation is as follows:

where

Let ${A_p} = \sum\limits_{i = 1}^2 {{\theta _{pi}}} (z_p(k)){A_{pi}}$, ${B_p} = \sum\limits_{i = 1}^2 {{\theta _{pi}}} (z_p(k)){B_{pi}}$, ${K_p} = \sum\limits_{j = 1}^2 {{\theta _{pj}}} (z_p(k)){K_{pj}}$, where $p\in\{1, 2\}$.

When ${\rm sat}(u(k)) = 0$, according to Lemma 1, it follows from ${A_{pi}}\succeq 0, p, j \in\{1, 2\}$ that subsystems ${\Xi _{\rm{1}}}$ and ${\Xi _{\rm{2}}}$ are both positive. It is obtained from the eigenvalues of matrices ${A_{pi}}$, $p, j\in \{1, 2\}$ that subsystems ${\Xi _{\rm{1}}}$ and ${\Xi _{\rm{2}}}$ are both unstable.

When ${\rm sat}(u(k))\ne0$, both subsystems ${\Xi _{\rm{1}}}$ and ${\Xi _{\rm{2}}}$ are checked to be stabilizable. Given $\lambda_1 = 0.5$, $\mu_1 = 3.1$, $\lambda_2 = 0.28$, $\mu_2 = 4$, $c_1 = c_2 = 1$, $D_{11} = D_{12} = 1$, $D_{11}^- = D_{12}^- = 0$, $D_{21} = D_{22} = 0$, $D_{21}^- = D_{22}^- = 1$, $v^* = [3\; 6]$, according to Corollary 1, the corresponding feasible solutions can be obtained and the switching signals satisfy ${\tau_{a1}}\ge-\frac{\ln \mu_1 }{ \lambda_1 } = 1.6323$, $\tau_{2a}\ge-\frac{\ln \mu_2 }{ \lambda_2} = 1.0890$, the saturated controller gains are $K_{11} = [-0.1278 \; 0.0192]$, $K_{12} = [-0.1278 \; 0.0192]$, $K_{21} = [-0.3217 \; -0.2029]$ and $K_{22} = [-0.3217 \; -0.2029]$.

We can check that ${A_1} + {B_1}{K_1}\succeq 0$ and ${A_2} + {B_2}{K_2}\succeq 0$, which ensure that the positivity of the saturated closed-loop subsystems ${\Xi _{\rm{1}}}$ and ${\Xi _{\rm{2}}}$. And Figures 5 and 6 depict the state trajectories of the system with the initial condition $x(t_0) = {[0.7, 0.8]^T}$. From Figures 5 and 6, we can observe that closed-loop subsystems ${\Xi _{\rm{1}}}$ and ${\Xi _{\rm{2}}}$ are both exponentially stable.

By Corollary 1, the switching signals of the system are designed as $\tau_{a1} = 2 > 1.6323$ and $\tau_{a2} = 2 > 1.0890$. Based on the designed swiching signal, we can generate a possible switching sequence. Choose the initial condition $x(t_0) = {[3 \; 5]^T}$, the system state trajectories, together with the switching signal are depicted in Figure 7. From Figure 7, it is observed that the closed-loop system satisfies both positivity and stability. Figure 8 depicts the saturated control input.

Moreover, under the same initial conditions, the comparison results of state trajectories under the controller designed for the system without actuator saturation in ^[17] and the controller proposed in this paper are shown in Figure 9. From Figure 9, it can be verified that the proposed controller in this paper makes the system converge faster than the one in ^[17]. Thus, for the PSTSFS in discrete-time domain, the designed control scheme is superior to the one in ^[17].

6.   Conclusions

The stabilization of discrete-time PSTSFSs with partially stabilizable subsystems has been investigated in the presence of actuator saturation. By the CHT, a PDC-based saturated fuzzy controller is designed and sufficient conditions for the positivity and stability of the closed-loop PSTSFSs are developed by the discrete multiple linear co-positive Lyapunov function and slow-fast combined MDADT approach. The obtained conditions are also applicable to the PSTSFSs with all stabilizable switched subsystems. Furthermore, the largest ADE is presented by an optimization problem. Finally, two illustrative examples are provided to demonstrate the effectiveness and advantages of the proposed control scheme. In practice, due to factors such as measurement costs, the state is often unmeasurable. Thus, the observer-based saturated control of discrete-time PSTSFSs deserves further investigation.

Conflict of interest

The authors declare that there is no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.