1 Introduction

In recent years, switched systems have received considerable attention and a number of important results have emerged [1–8]. This increasing attention has taken place mainly because of many practical systems such as process control systems [9], networked control systems [10], mobile robotics [11], and aerospace systems [12, 13] and can be modeled by means of switched systems. In the study of switched systems, the main focus has been put to problems of stability and stabilization of such systems [3, 8, 14–16]. Many methods have been developed in the study of switched systems such as common Lyapunov function [17], multiple Lyapunov functions [18, 19], and switched Lyapunov functions [20].

Meanwhile, as a class of important design methods of switching laws, the average dwell time (ADT) method is usually acknowledged to have more flexibility and availability in stability and stabilization for switched systems [21]. The property of ADT switching is that the ADT of each subsystem activated is no smaller than a constant in any finite interval. The arbitrary switching can be regarded as an extreme case of the ADT switching, and the dwell time switching can be seen as a special case of the ADT switching. Moreover, there have appeared some extensions of the ADT method in [22–26].

On the other hand, most of the practical systems have states/output constraints due to some safety consideration or inherent limits of devices. Presently, a number of achievements have been obtained on the stabilization problem for both linear systems, e.g., [27, 28], and nonlinear systems, e.g., [29, 30], with saturation phenomena and state constraints. However, there exist only a few results on stability and stabilization of switched systems with state constraints. [31] investigates the time optimal control problem for a class of integrator switched systems with polyhedral state constraints. [32] studies the stabilization problem for a class of planar switched systems with state constraint. It is worth pointing out, however, that both of the above literatures are aiming at either the switched system in a special structure, or the two-dimensional switched systems. In other words, to the best of authors’ knowledge, the problems of stability and stabilization for more general switched systems with state constraints have not been addressed in the existing literature, which partly motivates our present work.

Another reason which inspires us for this study is as follows. As aforementioned, ADT is an effective approach to study the stabilization problem for switched systems. This design approach is, of course, expected to be useful for the switched systems with state constraints. However, the ADT switching is inappropriate to be directly extended to the switched systems with state constraints. It is well known that the minimum of admissible ADT is computed by two parameters, i.e., the jump rate of the adjacent Lyapunov functions at switching instants and the decay rate of each Lyapunov function during the running time of the subsystem. It is straightforward that such a setup of the common decay rate regardless of the saturations occur or not will bring about a certain conservativeness. Besides, it is worth noting that, for a switched system without state constraints, the state trajectory is always smooth during the running time of each subsystem; the nonsmooth state trajectory only appears at the switching times. However, for a switched system with state constraints, the situation is completely different. In fact, the nonsmooth state not only appears at the switching times, but also appears at the instants when the state becomes saturated from unsaturated, or becomes unsaturated from saturated, during the running time of each subsystem. Therefore, how to improve the standard ADT method to better handle the switched systems with state constraints is worthwhile to proceed, which is another reason to motivate our present study.

This paper investigates the stabilization problem for a class of switched systems with state constraints in both continuous-time and discrete-time settings. The system model is defined on a closed hypercube as all the state variables are constrained to a unit hypercube. An improved ADT method is presented to take into account different decay rates of a Lyapunov function related to an active subsystem according to whether saturation has occurred or not. Sufficient conditions for stability and stabilizability of such switched systems are derived, and moreover the stabilizing state-feedback controllers are designed.

The mains contributions of this paper are twofold. Firstly, a novel ADT method is proposed. The applications of this ADT method produces smaller average dwell time than the standard ADT method does due to the different decay rates of Lyapunov function (related to an active subsystem) adopted according to whether saturation has occurred or not. Secondly, sufficient for stability and stabilizability of a more broad class of general switched systems with state constraints are derived for the first time in this study.

This paper is organized as follows. Section 2 provides the problem statements and preliminaries. Problems of stability and stabilization for the switched system with state constraints are investigated in Sect. 3 and Sect. 4, respectively. In Sect. 5, an iterative algorithm is introduced for verifying the solvability of proposed linear matrix inequalities (LMIs). Two examples are given in Sect. 6. Finally, some conclusions are drawn in the last section.

Notations

Throughout this paper, the following notations are used. ℝn denotes the n-dimensional Euclidean space, and ℤ+ represents the set of nonnegative integers, the notation ∥⋅∥ refers to the Euclidean vector norm. \(\mathcal {C}^{1}\) denotes the space of continuously differentiable functions. A function φ(⋅) is said to be of class \(\mathcal {K}\) if it is continuous, strictly increasing and φ(0)=0. In addition, I denotes the identity matrix with compatible/relevant dimensions; Symbol ∗ indicates the symmetry elements in symmetric matrices. The Hermitian part of a square matrix M is denoted by He(M):=M+M T. Finally, a function ceil (ε) represents rounding real number ε to the nearest integer greater than or equal to ε.

2 Problem statement

Consider a class of state-constrained switched systems described by

$$ \delta x=h(A_{\sigma}x+B_{\sigma}u_{\sigma}), $$
(1)

where \(x=(x_{1},x_{2},\ldots,x_{n})^{T}\in\mathbb{D}^{n}=\{-1\leq x_{i}\leq 1,\forall i\in \mathcal {N}\}\in\mathbb{R}^{n}\), \(\mathcal {N}=\{1,2,\ldots,n\}\) is the constrained state, the symbol δ denotes the derivative operator in the continuous-time context (\(\delta x(t)=\frac{d}{dt}x(t)\)), and the shift forward operator in the discrete-time case (δx(k)=x(k+1)). σ is a piecewise constant function of time, called a switching signal, which takes its values in the finite set \(\mathcal {P}=\{1,2,\ldots,m\}\), m is the number of subsystems. Also, for a switching sequence 0<t 1<⋯<t i <⋯, symbol t i denotes the moment of the ith switching. When σ=p, we say that the pth subsystem is active, and we define the symbol p i denotes that the pth subsystem is active for the ith switching.

The control input u σ(t)(t) (or u σ(k)(k)) in (1) is used to achieve system stability or some certain performances for certain switching signals. In this paper, the state feedback is considered with u σ(t)(t)=K σ(t) x(t) (or u σ(k)(k)=K σ(k) x(k)), where K p are the controller gains to be determined. Then, for example, for the continuous-time case, the resulting closed-loop system is given by

$$ \delta x(t)=h\bigl(\mathcal {A}_px(t) \bigr)=\begin{pmatrix} h_1(\sum_{j=1}^n\hat{a}_{1j}^px_j(t))\\ h_2(\sum_{j=1}^n\hat{a}_{2j}^px_j(t))\\ \vdots\\ h_n(\sum_{j=1}^n\hat{a}_{nj}^px_j(t)) \end{pmatrix}, $$
(2)

where \(\mathcal {A}_{p}:=A_{p}+B_{p}K_{p}=[\hat{a}_{ij}^{p}]\in\mathbb{R}^{n\times n}\) denotes the closed-loop system matrix of the pth subsystem, for each \(i\in\mathcal {N}\), \(\forall p\in\mathcal {P}\)

(3)

Remark 1

The nonswitched version of system (1) is established in [31, 32]. Obviously, for system (1), according to (3), the function h(⋅) is not continuous with respect to time, and the state of system (1) is nonsmooth not only at the switching points, but also at the points of the saturations occurrence and disappearance during the running time of each subsystem. This is a significant distinction between the system (1) and the switched system without state constraints.

Our objective is to find a more general set of admissible switching signals and the corresponding state-feedback controllers, such that the state-constrained switched system (1) is globally uniformly asymptotically stable (GUAS) operating on the unit hypercube \(\mathbb{D}^{n}\) in ℝn.

Remark 2

GUAS refers to the stability of the equilibriums state of systems (1) operating on the unit hypercube \(\mathbb{D}^{n}\) and not on the usually taken ℝn (ℤn, respectively). In other words, by GUAS of the state space origin we mean that the origin is locally uniformly asymptotic stable (LUAS) within the unit hypercube \(\mathbb{D}^{n}\), instead of being attraction domain of the usual ℝn (ℤn, respectively). This expression is widely used in the relevant literature, e.g., see [33–35], for instance, where the stability problem of state-constrained systems is studied.

Before proceeding further, it is necessary to recall the definition of the ADT and the stability results for switched nonlinear systems with an ADT.

Definition 1

([21])

For a switching signal σ(t) and each t 2≥t 1≥0, let N σ(t)(t 1,t 2) denote the number of discontinuities of σ(t) in the open interval (t 1,t 2). We say that σ(t) has an average dwell time τ a if there exist two positive numbers N 0 and τ a such that

$$N_{\sigma(t)}(t_1,t_2)\leq N_0+\frac{t_2-t_1}{\tau_a},\quad \forall t_2\geq t_1\geq0. $$

Remark 3

Definition 1 means that if there exists a positive number τ a such that a switching signal has the ADT property, the ADT between any two consecutive switching is no smaller than a common constant τ a for all subsystems.

Lemma 1

([21]) Consider the continuous-time switched system \(\dot{x}(t)=f_{\sigma(t)}(x(t))\), \(\sigma(t)\in\mathcal {P}\) and let λ>0, μ>1 be given constants. Suppose that there exist positive definite \(\mathcal {C}^{1}\) functions V σ(t):ℝn→ℝ, and two class \(\mathcal {K}_{\infty}\) functions κ 1, κ 2 such that, \(\forall p\in\mathcal {P}\)

and \(\forall(\sigma(t_{i})=p,\sigma(t_{i}^{-})=q)\in\mathcal {P}\times\mathcal {P}\), p≠q,

$$V_p\bigl(x(t_i)\bigr)\leq\mu V_q\bigl(x(t_i)\bigr), $$

then the system is GUAS for any switching signal with ADT

$$ \tau_a\geq\tau_a^*=\frac{\ln\mu}{\lambda} . $$
(4)

Lemma 2

([36]) Consider the discrete-time switched system x(k+1)=f σ(k)(x(k)), \(\sigma(k)\in\mathcal {P}\) and let 0<λ<1, μ>1 be given constants. Suppose that there exist positive definite \(\mathcal {C}^{1}\) functions V σ(k):ℝn→ℝ, and two class \(\mathcal {K}_{\infty}\) functions κ 1, κ 2 such that, \(\forall p\in\mathcal {P}\)

and \(\forall(\sigma(k_{i})=p,\sigma(k_{i}^{-})=q)\in\mathcal {P}\times\mathcal {P}\), p≠q,

$$V_p\bigl(x(k_i)\bigr)\leq\mu V_q\bigl(x(k_i)\bigr), $$

then the system is GUAS for any switching signal with ADT

$$ \tau_a\geq\tau_a^*=-\frac{\ln\mu}{\ln(1-\lambda)} . $$
(5)

3 Stability analysis

Now, the following lemmas present the stability results for the state-constrained switched nonlinear systems with ADT associated with saturations. For concise notation, let t i and t i+1, ∀i∈ℤ+, denote the starting time and ending time of some active subsystem, meanwhile, \(\mathcal {T}_{f}(t_{i},t_{i+1})\) and \(\mathcal {T}_{s}(t_{i},t_{i+1})\), imply the total length of the dispersed intervals during which the state is saturated and unsaturated within the interval [t i ,t i+1), respectively. Then we denote \(\mathcal {T}_{f}(t_{0},t):=\sum_{j=1}^{i}{\mathcal {T}}_{f}(t_{j-1},t_{j})+{\mathcal{T}}_{f}(t_{i},t)\) and \(\mathcal {T}_{s}(t_{0},t):= \sum_{j=1}^{i}\mathcal {T}_{s}(t_{j-1},t_{j})+{\mathcal{T}}_{s}(t_{i},t)\). Moreover, we let α, β denote the different decay rates of a Lyapunov function related to an active subsystem corresponding to the saturations occur or not, and we suppose that α>β.

Lemma 3

(Continuous-time Case)

Consider a continuous-time switched nonlinear system with state constraints

$$ \dot{x}(t)=f_{\sigma(t)}\bigl(x(t)\bigr),\quad x(t)\in\varOmega \subset\mathbb{R}^n,\ \sigma(t)=p\in\mathcal {P}, $$
(6)

with f p (0)=0, \(\forall p\in\mathcal {P}\). Assume that all trajectories remain inside Ω. Let α, β and μ be some given constants satisfying α>β>0, μ≥1. If there exist some piecewise continuous functions V p (x(t)):Ω→ℝ, for all \(p\in\mathcal {P}\), such that

(7)
(8)

and \(\forall (\sigma(t_{i})=p, \sigma(t_{i}^{-})=q)\in\mathcal {P}\times\mathcal {P}\), p≠q

$$ V_{p}\bigl(x(t_{i})\bigr)\leq \mu V_{q}\bigl(x(t_{i})\bigr),\quad \forall x(t)\in\varOmega, $$
(9)

where φ 1, φ 2 are some class \(\mathcal {K}\) functions, then the state-constrained switched system (6) is GUAS for any switching signal with ADT

$$ \tau_{a}\geq \tau_{a}^{*}= \frac{\ln \mu}{\zeta},\qquad \frac{\mathcal {T}_{f}(t_0,t)}{\mathcal {T}_s(t_0,t)}\geq \frac{\zeta-\beta}{\alpha-\zeta}>0. $$
(10)

Proof

When t belongs to the nonsaturated period, i.e., \(t>t_{f}\in {\mathcal {T}}_{f}(t_{i}, t_{i+1})\), i∈ℤ+, it holds from (8) that

$$ V_{p}\bigl(x(t)\bigr)\leq e^{-\alpha(t-t_f)}V_{p} \bigl(x(t_f)\bigr). $$
(11)

When t belongs to the saturated period, i.e., \(t>t_{s}\in{\mathcal {T}}_{s}(t_{i},t_{i+1})\), i∈ℤ+, it holds from (8) that

$$ V_{p}\bigl(x(t)\bigr)\leq e^{-\beta(t-t_s)}V_{p} \bigl(x(t_s)\bigr). $$
(12)

Thus, when t>t i , according to (9), (11), and (12), we have

(13)

By (10), we have

$$ \alpha\mathcal {T}_f(t_0,t)+\beta \mathcal {T}_s(t_0,t)\geq \zeta(t-t_0), $$
(14)

then (13) and (14) imply that

Thus, if the ADT satisfies (10), one can obtain

$$\mu^{\frac{1}{\tau_a}}e^{-\zeta}\leq\mu^{\frac{\zeta}{\ln\mu}}e^{-\zeta} =e^{\zeta}e^{-\zeta}=1. $$

Therefore, we conclude that \(V_{p_{i}}(x(t))\) convergences to zero as t→∞, if the ADT satisfies (10), then the asymptotic stability can be deduced with the aid of (7). □

Remark 4

It is worth pointing out that α>β is a rather common condition which also appears in [33, 34] which studies the stability problem for the non-switched version of the system (1). Readers are recommended to refer to the detailed discussion in [31] and the references therein.

Remark 5

It is noted that the improved ADT (10) associated with two different decay rates is always smaller than the standard ADT (4) associated with only one common decay rate, since 0<β<ζ<α. Note also that if we choose α=β, then (10) will readily reduce to (4), thus, Lemma 1 can be regarded as a special case of Lemma 3.

Remark 6

In some cases, \(\mathcal {T}_{s}^{\max}\), which is defined at the beginning of this section, is known in advance, and that the value of \(\mathcal {T}_{s}^{\max}\) satisfies \(\mathcal {T}_{s}^{\max}\leq\frac{(\alpha-\zeta)\ln\mu}{(\alpha-\beta)\beta}\), then we can establish another improved ADT described as follows:

$$ \tau_{a}\geq \tau_{a}^{*}= \frac{{\mathcal {T}}^{\max}_{s}(\alpha-\beta)+\ln \mu}{\alpha} , $$
(15)

where \({\mathcal {T}}^{\max}_{s}=\sup_{i\in \mathbb{Z}^{+}}\{{\mathcal {T}}_{s}(t_{i},t_{i+1})\}\).

It can be seen clearly that (15) presents a more general ADT than (4) which corresponding to the special case \(\mathcal {T}_{s}^{\max}=0\) (i.e. the saturations never occur), thus, the ADT (4) in Lemma 1 can be also regarded as a special case of (15). Note also that both of the above two improved ADT (10) and (15) are smaller than the standard ADT (4). Furthermore, in this case, it is easy to show that the ADT (15) is smaller than the ADT (10).

Similar to the continuous-time version, for concise notation, we let k i and k i+1, ∀i∈ℤ+, denote the starting time and ending time of some active subsystem, while \(\mathcal {T}_{f}(k_{i},k_{i+1})\) and \(\mathcal {T}_{s}(k_{i},k_{i+1})\), imply the total length of the dispersed intervals during which the state is saturated and unsaturated within the interval [k i ,k i+1), respectively. Then we denote \(\mathcal {T}_{f}(k_{0},k):=\sum_{j=1}^{i}{\mathcal {T}}_{f}(k_{j-1},k_{j})+{\mathcal{T}}_{f}(k_{i},k)\) and \(\mathcal {T}_{s}(k_{0},k):=\sum_{j=1}^{i}{\mathcal {T}}_{s}(k_{j-1},k_{j})+{\mathcal{T}}_{s}(k_{i},k)\).

Lemma 4

(Discrete-time Case)

Consider a discrete-time switched nonlinear system with state constraints

(16)

with f p (0)=0, \(\forall p\in\mathcal {P}\). Assume that all trajectories remain inside Ω. Let α, β, and μ be some constants satisfying 1>α>β>0, μ≥1. If there exist some piecewise continuous functions V p (x(k)):Ω→ℝ, for all \(p\in\mathcal {P}\), such that

(17)
(18)

and \(\forall (\sigma(k_{i})=p, \sigma(k_{i}^{-})=q)\in\mathcal {P}\times\mathcal {P}\), p≠q

$$ V_{p}\bigl(x(k_{i})\bigr)\leq \mu V_{q}\bigl(x(k_{i})\bigr),\quad \forall x(k)\in\varOmega, $$
(19)

then the state-constrained switched system (16) is GUAS for any switching signal with ADT

(20)

Proof

When k belongs to the nonsaturated period, i.e., \(k\in{\mathcal {T}}_{f}(k_{i}, k_{i+1})\), i∈ℤ+, it holds from (18) that

$$ V_{p}\bigl(x(k+1)\bigr)\leq (1- \alpha)V_{p}\bigl(x(k)\bigr). $$
(21)

When k belongs to the saturated period, i.e., \(k\in{\mathcal {T}}_{s}(k_{i}, k_{i+1})\), i∈ℤ+, it holds from (18) that

$$ V_{p}\bigl(x(k+1)\bigr)\leq (1- \beta)V_{p}\bigl(x(k)\bigr). $$
(22)

Thus, when k>k i , according to (19), (21), and (22), we have

(23)

By (20), we have

$$ (1-\alpha)^{\mathcal {T}_f(k_0,\ k)}(1-\beta)^{\mathcal {T}_s(k_0,\ k)}\leq (1- \zeta)^{k-k_0}, $$
(24)

then (23) and (24) imply that

Thus, if the ADT satisfies (20), one can obtain

Therefore, we conclude that V p (x(k)) converges to zero as k→∞, if the ADT satisfies (20), then the asymptotic stability can be deduced with the aid of (17). □

Remark 7

It is noted that the improved ADT (20) associated with two different decay rates is always smaller than standard ADT (5) associated with only one common decay rate, since 0<β<ζ<α<1. Note also that if we choose α=β, then (20) will readily reduce to (5), thus, Lemma 2 can be regarded as a special case of Lemma 4.

Remark 8

In some cases, if \(\mathcal {T}_{s}^{\max}\) is known in advance, and that, the value of \(\mathcal {T}_{s}^{\max}\) satisfies \(\mathcal {T}_{s}^{\max}\leq \frac{(\ln(1-\zeta)-\ln(1-\alpha))\ln\mu}{(\ln(1-\beta)-\ln(1-\zeta))\ln(1-\beta)}\), we can establish another improved ADT described as follows:

(25)

where \({\mathcal {T}}^{\max}_{s}=\sup_{i\in \mathbb{Z}^{+}}\{{\mathcal {T}}_{s}(k_{i},k_{i+1})\}\).

It can be seen clearly that (25) presents a more general ADT than (5) which corresponding to the special case \(\mathcal {T}_{s}^{\max}=0\) (i.e., the saturations never occur), thus the ADT (5) in Lemma 2 can be also regarded as a special case of (25). Note also that both of the above two improved ADT (20) and (25) are smaller than the standard ADT (5). Furthermore, in this case, it is easy to show that ADT (25) is smaller than ADT (20).

As mentioned before, there are several methods to handle saturations. Here, we introduce the convex hull method form [34] that translates the saturation function h(⋅) into the vertex of a convex hull in order to deal with saturations.

Let D n be the set of n×n diagonal matrices whose diagonal elements are either 1 or 0. For example, if n=2, then

$$D_2=\left \{\left [\begin{array}{cc} 1 & \quad 0 \\ 0 & \quad 1 \end{array} \right ],\left [\begin{array}{cc} 1 &\quad 0 \\ 0 &\quad 0 \end{array} \right ],\left [\begin{array}{cc} 0 &\quad 0 \\ 0 &\quad 1 \end{array} \right ],\left [\begin{array}{cc} 0 &\quad 0 \\ 0 &\quad 0 \end{array} \right ] \right \}. $$

Obviously, there are 2n elements in D n . Suppose that each element of D n is labeled as D s , \(s\in\mathcal {L}:=\{1,2,\ldots,2^{n}\}\), and denote \(D_{s}^{-}=I-D_{s}\). Clearly, \(D_{s}^{-}\) is also an element of D n if D s ∈D n . Then we know that

$$ h(Ax)\in\mbox{co}\bigl\{D_s(Ax)+D_s^{-}Gx, \ s\in\mathcal {\mathcal {L}}\bigr\},\quad \forall x\in \mathbb{D}^n, $$
(26)

where G=[g ij ]∈ℝn×n is row diagonally dominant and the diagonal is composed of negative elements, that is, the matrix G satisfies \(|g_{ii}|> \sum_{j=1,j\neq i}^{n}|g_{ij}|\) and g ii <0, for all \(i\in \mathcal {N}\).

Now, with the above discussions, we will give the stability conditions for system (1) with ADT associated with saturations.

Theorem 1

(Continuous-time Case)

Consider the following continuous-time state-constrained switched system:

$$ \dot{x}(t)=h\bigl(A_{\sigma(t)}x(t)\bigr),\quad \sigma(t)=p\in \mathcal {P}, $$
(27)

let α, β, and μ be some given constants satisfying α>β>0, μ≥1. If there exist positive matrices P p and row diagonally dominant matrices \(G_{p},\ p\in \mathcal {P}\) such that, \(\forall (p,\ q)\in \mathcal {P}\times\mathcal {P}\), p≠q, \(s\in\mathcal {S}=\{1,2,\ldots,2^{n}-1\}\), D s ≠I

(28)
(29)
(30)

then the switched system (27) is GUAS for any switching signal with ADT satisfying (10).

Proof

Choose the Lyapunov functions as follows:

$$ V_{p}\bigl(x(t)\bigr)=x^{T}(t)P_{p}x(t), \quad p\in \mathcal {P}. $$
(31)

When t belongs to the non-saturated period, i.e., \(t\in{\mathcal {T}}_{f}(t_{i}\ t_{i+1})\), i∈ℤ+, it holds from (8), (9), and (31) that

When t belongs to the saturated period, i.e., \(t\in{\mathcal {T}}_{s}(t_{i}\ t_{i+1})\), i∈ℤ+, it holds from (8), (9), and (31) that

where \(\tilde{A}_{p}=D_{s}A_{p}+D_{s}^{-}G_{p}\) and D s ≠I.

Thus, if (28), (29), and (30) hold, system (1) is GUAS for any switching signals with ADT satisfying (10) by Lemma 3. □

Remark 9

When σ(t)≡1, Theorem 1 can readily reduce to the result of [35] which considers the nonswitched systems case, that is, the result of [35] can be regarded as a special case of Theorem 1.

Theorem 2

(Discrete-time Case)

Consider the following discrete-time state-constrained switched system:

$$ x(k+1)=h\bigl(A_{\sigma(k)}x(k)\bigr),\quad \sigma(k)=p\in \mathcal {P}, $$
(32)

let α, β, and μ be some given constants satisfying 0<β<α<1, μ>1. If there exist positive matrices P p and row diagonally dominant matrices G p , \(p\in \mathcal {P}\) such that \(\forall (p,\ q)\in \mathcal {P}\times\mathcal {P}\), p≠q, \(s\in\mathcal {S}\), D s ≠I

(33)
(34)
(35)

then the switched system (32) is GUAS for any switching signal with ADT satisfying (20).

Proof

Choose the Lyapunov functions as follows:

$$ V_{p}\bigl(x(k)\bigr)=x^{T}(k)P_{p}x(k), \quad p\in \mathcal {P}. $$
(36)

When k belongs to the nonsaturated period, i.e., \(k\in{\mathcal {T}}_{f}(k_{i}\ k_{i+1})\), i∈ℤ+, it holds from (18), (19), and (36) that

When k belongs to the saturated period, i.e., \(t\in{\mathcal {T}}_{s}(k_{i}\ k_{i+1})\), i∈ℤ+, it holds from (18), (19), and (36) that

where \(\tilde{A}_{p}=D_{s}A_{p}+D_{s}^{-}G_{p}\) and D s ≠I.

By Schur complement [25], \(A_{p}^{T}P_{p}A_{p}-P_{p}+\alpha P_{p}<0\) becomes (33) and \(\tilde{A}_{p}^{T}P_{p}\tilde{A}_{p}-P_{p}+\beta P_{p}<0\) becomes (34). Thus, if (33), (34), and (35) hold, system (1) is GUAS for any switching signal with ADT satisfying (20) by Lemma 4. □

4 Stabilization

Now, we are in a position to give the existence conditions of a set of stabilizing controllers for the state-constrained switched system (1) with the ADT associated with saturations.

Theorem 3

(Continuous-time Case)

Consider the state-constrained switched system (1), let α, β, and μ be some given constants satisfying α>β>0, μ≥1. If there exist positive matrices Q p , row diagonally dominant matrices G p and matrices F p , \(p\in \mathcal {P}\) such that \(\forall (p,\ q)\in \mathcal {P}\times\mathcal {P}\), p≠q, \(s\in\mathcal {S}\), D s ≠I

(37)
(38)
(39)

then the switched system (1) is GUAS for any switching signal with ADT satisfying (10). Moreover, if (37), (38), and (39) are feasible, the controller gains can be given by

$$ K_{p}=F_{p}Q_{p}^{-1}. $$
(40)

Proof

Theorem 1 implies that for system (1),

(41)
(42)
(43)

where \(\tilde{\mathcal {A}}_{p}=D_{s}\mathcal {A}_{p}+D_{s}^{-}G_{p}\), then system (1) is GUAS for any switched signal with ADT satisfying (10). Substitute A p +B p K p for \(\mathcal {A}_{p}\), in (41), and replace \(\tilde{\mathcal {A}}_{p}\) in (42) by \(D_{s}A_{p}+D_{s}B_{p}K_{p}+D_{s}^{-}G_{p}\), D s ≠I. Pre- and post-multiplying \(P_{p}^{-1}\) to (41), (42), and setting \(Q_{p}=P_{p}^{-1}\), \(F_{p}=K_{p}P_{p}^{-1}\), we can obtain that if (37) and (38) hold, then (41) and (42) are satisfied. In addition, P p , P q are all positive definite matrices, then Q p >0, Q q >0. Then we have (39) is equivalent to (43).

Thus, if (37), (38), and (39) are feasible, the admissible controller gains can be given by \(K_{p}=F_{p}Q_{p}^{-1}\), which completes the proof. □

Theorem 4

(Discrete-time Case)

Consider the state-constrained switched system (1), let α, β, μ be some given constants satisfying 1>α>β>0, μ≥1. If there exist positive matrices Q p , row diagonally dominant matrices G p and matrices F p , \(p\in \mathcal {P}\) such that \(\forall (p,\ q)\in \mathcal {P}\times\mathcal {P}\), p≠q, \(s\in\mathcal {S}\), D s ≠I

(44)
(45)
(46)

then the switched system (1) is GUAS for any switching signal with ADT satisfying (20). Moreover, if (44), (45), and (46) are feasible, the controller gains can be given by

$$ K_{p}=F_{p}Q_{p}^{-1}. $$
(47)

Proof

Theorem 2 implies that

(48)
(49)
(50)

where \(\tilde{\mathcal {A}}_{p}=D_{s}\mathcal {A}_{p}+D_{s}^{-}G_{p}\), then system (1) is GUAS for any switched signal with ADT satisfying (20). Substitute A p +B p K p for \(\mathcal {A}_{p}\), in (48), and replace \(\tilde{\mathcal {A}}_{p}\) in (49) by \(D_{s}A_{p}+D_{s}B_{p}K_{p}+D_{s}^{-}G_{p}\), D s ≠I. Pre- and post-multiplying \(\mbox{diag}\{P_{p}^{-1},\ P_{p}^{-1}\}\) to (48), (49) and setting \(Q_{p}=P_{p}^{-1}\), \(F_{p}=K_{p}P_{p}^{-1}\), we can obtain that if (44) and (45) hold, then (48) and (49) are satisfied. In addition, P p , P q are all positive definite matrices, then Q p >0, Q q >0. Then we have (46) is equivalent to (50).

Thus, if (44), (45), and (46) are feasible, the admissible controller gains can be given by \(K_{p}=F_{p}Q_{p}^{-1}\), which completes the proof. □

5 Algorithm

In this section, we will follow the idea of [35] to propose an iterative LMI algorithm for verifying the sufficient conditions of theorems for the continuous-time case. Let \(\mathcal {V}\) be the set of n-dimensional row vectors in which there is only one nonzero element which is 1. Denote υ i , \(i\in\mathcal {N}\), as an element of \(\mathcal {V}\) in which the ith element is 1. Denote \(\mathcal {W}_{i}\) be the set of n-dimensional column vectors in which the ith element is 1 and other elements are either 1 or −1. The elements of \(\mathcal {W}_{i}\) are denoted as ω ij , \(j\in\mathcal {M}:=\{1,\ldots,2^{n-1}\}\). Then the condition that each G p is row diagonally dominant and the diagonal is composed of negative elements can be expressed as the following LMIs:

$$\upsilon_iG_p\omega_{ij}<0,\quad i\in\mathcal {N},\ j\in\mathcal {M},\ \forall p\in\mathcal {P}. $$

Algorithm: Stabilization for Continuous-time Case

Step 1. Select a W p >0, and solve Q p and F p from the following Lyapunov equation:

$$\mathit{He}\bigl(Q_{p}A_{p}^{T}+F_{p}^{T}B_{p}^{T}\bigr)+\alpha Q_{p}=-W_{p}, $$

where \(K_{p}=F_{p}Q_{p}^{-1}\) are chosen such that A p +B p K p are Hurwitz.

Set k=0.

Step 2. Using Q p and F p obtained previously, solve the following LMI optimization problem for G p and γ:

If k=0 and γ≤0, go to Step 4. If k>0, γ≤0, or γ≰γ k , go to Step 4. Otherwise, set k=k+1, γ k =γ, go to the next step.

Step 3. Using G p obtained in the previous step. Solve the following LMI optimization problem for Q p and F p and γ:

If γ≤0 or γ≰γ k , go to Step 4. Otherwise, set k=k+1, γ k =γ, go to Step 2.

Step 4. If γ≤0, the system (2) is GUAS at the origin. And the current K p is the calculated feedback again. Otherwise, no conclusion can be drawn. A different W p may be selected and the algorithm may be repeated from Step 1.

The iterative LMI algorithm for the discrete-time case is similar to the algorithm above, therefore, we omit it due to space limitations.

6 Examples

In this section, two examples are given to illustrate the applicability and the effectiveness of our proposed method.

Example 1

(Stability for Continuous-time Case) We consider the following continuous-time switched systems with sate constraints:

(51)

where

Given α=0.3, β=0.1, μ=1.4, and choose the switching signal satisfying \(\mathcal {T}_{f}(0,t)/\mathcal {T}_{s}(0.t)=0.4\), we set ζ=0.15, and the conditions (28), (29), and (30) in Theorem 1 are feasible with

Thus, by Theorem 1, the switched system (51) is GUAS for any switching signal with ADT satisfying τ a ≥2.2431. Choose the switching signal satisfying τ a =2.5, shown in (a) of Fig. 1, the trajectory of the switched system (51) is shown in (b) of Fig. 1. From these two figures, it can be seen that the result proposed in Theorem 1 is effective.

Fig. 1
figure 1

(a) Switching signal under τ a =2.5, (b) Trajectory of system (51)

Full size image

Example 2

(Stabilization for Continuous-time Case)

Consider a longitudinal motion of highly maneuverable aircraft technology (HiMAT) vehicle model given in [37] is shown to illustrate the applicability and the effectiveness of our design method. For statement convenience, we select two operating points within the flight envelope as depicted in Table 1 and construct a switched system using longitudinal short period linear models as

where x(t)=[x 1(t),x 2(t)]T, x 1(t) and x 2(t) denote the angle of attack and the pitch rate, respectively.

Table 1 Two operation points of HiMAT vehicle
Full size table

Given parameters α=0.3, β=0.2, μ=1.4, and choose the switching signal satisfying \(\mathcal {T}_{f}(0,t)/\allowbreak \mathcal {T}_{s}(0,t)=1\), then we set ζ=0.25, and the minimal ADT can be calculated as \(\tau_{a}^{*}=1.3459\). We can obtain the controller gains by solving the LMIs (37), (38), and (39):

The simulation result of the trajectory of the closed-loop system with the switching signal by choosing ADT satisfying τ a =2 is shown in Fig. 2. It can be seen from the trajectory of the state that although x 1(t) has occurred saturations, the closed-loop system is GUAS with the initial state x 0=[1,1]T, under the switching signal designed by our method, which adequately illustrates the applicability and the effectiveness of our proposed method.

Fig. 2
figure 2

Saturated trajectory of the closed-loop system under switching signal with τ a =2

Full size image

7 Conclusion

The stabilization problem for a class of switched systems with state constraints has been studied in both continuous-time and discrete-time contexts. There are three points that deserve to be summarized again. Firstly, a key point is to realize that the state of the switched system with state constraints is nonsmooth, not only at the switching times, but also at the instants when the state becomes saturated from unsaturated during the running time of each subsystem. This feature partly motivates our present study. Secondly, the improved ADT method is proposed that takes into consideration different decay rates of a Lyapunov function, which is related to an active subsystem, according to whether saturations have occurred or not. By making use of this improved ADT method, a smaller actual ADT can be obtained than the traditional ADT method can give. Thirdly, we have established sufficient conditions for stability and stabilizability of switched systems with state constraints. Furthermore, the stabilizing state feedback controllers are designed via the LMI technique.