Global Practical Stabilization of Discrete-Time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid

I. Introduction

SWITCHED systems are an important subclass of hybrid dynamical systems consisting of a family of continuous-variable subsystems and a switching law that determines which subsystem should be activated at each instant of the time [1]. One of the most important subclasses of switched systems is the switched affine systems that are very common in practice, especially in switching power converters [2]–[8]. In this class of switched systems, there is not any common equilibrium point among all subsystems. Instead, the equilibrium point varies discontinuously during switching among subsystems, and therefore, to achieve asymptotic stability in a desired equilibrium point, the switching frequency should approach infinity, which is not realizable in practice [9]–[11]. To handle chattering problems, one can use space or time regularization techniques. The former enforces a dwell-time using space-based hysteresis logic, and the latter utilizes an explicit timer preventing switching up to a guaranteed dwell-time [11]–[14]. In either case, due to the limitation on the switching frequency and non-existence of a common equilibrium point among subsystems, stability with respect to a set rather than a particular point can be achieved. In this regard, the notion of practical stability has been proposed in the literature to analyze switched systems’ stability without any common equilibria [15]–[17].

It should be noted that the practical stability and stability in the sense of Lyapunov are distinct concepts, and neither implies the other. The states of a system may oscillate around an equilibrium point with sufficiently small deviations, although the system may be mathematically unstable. Such a situation may be quite tolerable in the sense of practical stability, and its performance is acceptable in practice, while the system is still Lyapunov unstable [18], [19]. In this regard, compared to the classical Lyapunov stability analysis of the switched linear and nonlinear systems with a common equilibrium point, there has been little work on the practical stability analysis and stabilization of the discrete-time switched affine systems via state-dependent switching rules. The reader is referred to [20] and [21] for surveys of recent developments on the practical stability and stabilization of switched systems without common equilibria.

Previous works on the global practical stabilization of the switched affine systems via state-dependent switching rules require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type. Among them, the stabilization techniques in [11], [16], [17], [20]–[22] utilize a centralized Lyapunov function and a centralized ellipsoid. In [9], a decentralized Lyapunov function and a centralized ellipsoid are used for controller design and practical stability analysis. The works in [10], [23] employ a centralized Lyapunov function and decentralized ellipsoid. In some cases, such as [24], although shifted Lyapunov functions and ellipsoids are used, the invariant set of attraction is considered, as the level sets of the Lyapunov function and is not parameterized independently.

In this regard, the main objectives and contributions of the paper are summarized as follows. First, new sufficient conditions are provided for the global practical stabilization of discrete-time switched affine systems in a fully decentralized setting by the simultaneous use of a common decentralized quadratic Lyapunov function and a decentralized ellipsoid parameterized independently. Second, a numerical-analytical method is proposed to estimate the invariant set of attraction size. The main purpose in this part is to achieve a set of matrix inequalities with smaller dimensions that may be numerically tractable compared to the existing purely numerical methods. Third, a multiobjective optimization problem is proposed by which the invariant set of attraction size and the steady-state errors of the system trajectories are minimized simultaneously. Finally, the controller design procedure is developed such that the Lyapunov function does not require positive definiteness in the whole state-space but outside of a decentralized ellipsoid. The preceding features imply that the present work's proposed conditions are less restrictive than those in existing works.

The remainder of the paper is organized as follows. Section II provides the practical stabilization of discrete-time switched affine systems via state-dependent switching rules. Some basic definitions of the various types of practical stability are presented in this section. Section III provides the stabilization strategy in a pure numerical setting, including state-dependent switching rules and stability conditions, as a set of matrix inequalities. A numerical-analytical stabilization method with a possibility of a unified switching rule in the whole state space is established in Section IV. Section V formulates the optimization problems corresponding to stability conditions to minimize the size of the invariant set of attraction and steady-state errors. Section VI illustrates the theoretical results via two examples: one academic example and a DC-DC step-down switching converter. Finally, concluding remarks are made in Section VII.

Notations: $ \mathbb{R} $, $ \mathbb{R}_{\geq0} $, $ \mathbb{Z}_{\geq0} $ are used to denote the set of real, nonnegative real, and nonnegative integer numbers, respectively. $ \mathbb{R}^n $ and $ \mathbb{R}^{m\times n} $ denote real-valued n-dimensional column vectors and $ m\times n $ matrices, respectively. We use $ I_{n} $ and $ 0_{n} $ to denote the $ n\times n $ identity matrix and the $ n\times 1 $ zero vector, respectively. For matrix $ M\in\mathbb{R}^{m\times n} $, $ M^T $ denotes its transpose, and for a square matrix $ M\in\mathbb{R}^{n\times n} $, $ M^{-1} $, $ tr(M) $, and $ \lambda_i(M) $ are inverse, trace, and $i{{\rm{th}}}$ eigenvalue of M, respectively. Moreover, $\lambda_{\max}(M)$ and $\lambda_{\min}(M)$ are used to denote the largest and smallest eigenvalue of M, respectively. For $ x\in \mathbb{R}^n $, $ \|x\| $ denotes the Euclidean vector norm, i.e., $ \|x\| = (x^Tx)^{1/2} $. $ M\prec0 $ and $ M\preceq0 $ denote that M is a negative definite and negative semi-definite matrix, respectively. For the set $ {\cal{I}} $, $ |{\cal{I}}| $ denotes its number of elements (cardinality). In symmetric matrices, $ \ast $ denotes each of their symmetric blocks. Assume that the set $ \mathbb{K} = \{1,\ldots,N\} $ is a collection of the N first positive integer numbers, where the convex combination of matrices $ \{M_1,\ldots,M_N\} $ is denoted by $ M_{\lambda} = \sum_{i\in\mathbb{K}}\lambda_iM_i $ with $ \lambda\in \Lambda $ where $ \Lambda: = \{\lambda\in \mathbb{R}^N|\lambda_i\geq0,\sum_{i\in\mathbb{K}}\lambda_i = 1\} $ is the unitary simplex.

II. Problem Statement and Preliminaries

We consider the discrete-time switched affine system as

where $ k\in \mathbb{Z}_{\geq0} $ is the discrete-time instant, $ x(k)\in \mathbb{R}^{n} $ is the state and $ \sigma(x(k)):\mathbb{R}^{n}\rightarrow \mathbb{K} $ is a state-dependent switching function that selects one of the N available subsystems $ (A_i,b_i), i\in \mathbb{K} $ at any instant of time $ k\in \mathbb{Z}_{\geq0} $. It is intended to design the state-dependent switching function $ \sigma(x(k)) $ such that to impose finite time convergence of the state trajectories $ x(k), k\in\mathbb{Z}_{\geq0} $, to a neighborhood of a desired equilibrium point, for all initial conditions $ x_0\in \mathbb{R}^{n} $. In general, such an equilibrium point does not coincide with any other isolated subsystems, namely, $ x_{ei} = (I_n-A_i)^{-1}b_i $. In some literature [2], [9], [11], [25]–[27] based on the common Lyapunov function, it is assumed that this point belongs to a specific set of attainable equilibrium points. For instance, in [9] this set is defined as $ X_e = \{x_e\in \mathbb{R}^n|x_e = (I_n-A_{\lambda})^{-1}b_{\lambda},\lambda\in S\} $ with $ S\subseteq\Lambda $ such that $ A_{\lambda} $ is Schur stable. In the present paper, a desired equilibrium point $ x_e\notin X_e $ may be chosen. This free choice of equilibrium point may extend the application domain of the proposed stabilization technique. More specifically, the situation occurs when the stabilization of the switched affine systems is considered under uncertainty. In this case, the equilibrium point $ x_e $ will be a function of the system matrices’ unknown parameters and may not belong to the set $ X_e $ extracted using the nominal system matrices without uncertainties. Given an equilibrium point $ x_e $, we can always reformulate the stabilization problem around the null equilibrium point with defining the error state vector $ e(k) = x(k)-x_e, \forall k\in \mathbb{Z}_{\geq0} $ that obeys the error dynamics as

with $ \sigma(e(k)) = \sigma(x(k)-x_e) $, $ l_i = (A_i-I_n)x_e+b_i, \forall i\in\mathbb{K} $. We plan to design the state-dependent switching function $ \sigma(e(k)) $ via a general quadratic Lyapunov function candidate $ v(e) $ such that the decentralized ellipsoid $ {\cal{E}}(Q,c,\rho) $ be an invariant set of attraction for the switched affine system (2) based on the following definitions.

We consider the general quadratic Lyapunov function

where $ h\in \mathbb{R}^n $, $ P = P^T\in \mathbb{R}^{n\times n} $, and $ d\in \mathbb{R} $. The decentralized ellipsoid $ {\cal{E}}(Q,c,\rho) $ is given by

where $ Q = Q^T\succ0 $, $ c\in \mathbb{R}^n $ and $ \rho\in \mathbb{R} $ such that $ \rho<c^TQ^{-1}c $. Another representation of the ellipsoid (4) is as

where $ r = c^TQ^{-1}c-\rho $ and $ e_c = -Q^{-1}c $ is the center of the ellipsoid. In this paper, we take $ r = 1 $ yielding $ \rho = c^TQ^{-1}c-1 $. Note that by $ \rho = c^TQ^{-1}c-1 $, the constraint $ \rho<c^TQ^{-1}c $ mentioned above is automatically satisfied. By taking $ e_c = -Q^{-1}c $, the term $ \rho = c^TQ^{-1}c-1 $ can be rewritten as the following matrix equalities without matrix inverse term $ Q^{-1} $:

Definition 1 (Invariant Set): An ellipsoid $ {\cal{E}}(Q,c,\rho) $ in (4) is an invariant set on a given domain $ D\subset\mathbb{R}^n $ for the system (2) by the switching function $ \sigma(e(k)) $ if the following conditions are simultaneously satisfied:

a) $ 0_{n}\in {\cal{E}}(Q,c,\rho) $

b) If $ e(k_0)\in{\cal{E}}(Q,c,\rho) $, $ k_0\in \mathbb{Z}_{\geq0} $, then $ e(k)\in{\cal{E}}(Q,c,\rho) $, $ \forall k\geq k_0 $, where $ k\in \mathbb{Z}_{\geq0} $.

According to (4), to satisfy condition (a) in Definition 1, it is necessary to have

Definition 1 is a modified version of the available definition of the invariant set in the literature [28]–[30] in the sense that the condition a) and a decentralized ellipsoid are considered in our definition. These modifications result in an analogy between the set invariance and local practical stability definitions in the literature [15]–[18]. Indeed, based on condition b), the trajectories starting within $ {\cal{E}}(Q,c,\rho) $, never escape from it. Therefore, according to a), they will remain around the null set point.

Definition 2 (Attractive Set): An ellipsoid $ {\cal{E}}(Q,c,\rho)\subseteq D $ in $ (4) $ is said to be an attractive set on a given domain $ D\subset\mathbb{R}^n $ for the system (2) by the switching function $ \sigma(e(k)) $ if the following conditions are simultaneously satisfied.

a) $ 0_{n}\in {\cal{E}}(Q,c,\rho) $

b) If $ e(0)\in D-{\cal{E}}(Q,c,\rho) $ then there is a $ T = T(e(0))\geq0 $ such that $ e(k)\in {\cal{E}}(Q,c,\rho) $, $ \forall k\geq T $.

According to (8), condition a) in Definition 2 is automatically satisfied.

Definition 3 (Invariant Set of Attraction): An ellipsoid $ {\cal{E}}(Q,c,\rho)\subseteq D $ is an invariant set of attraction on a given domain $ D\subset\mathbb{R}^n $ for the system (2) by the switching function $ \sigma(e(k)) $ if it is both invariant and attractive based on Definitions 1 and 2.

It should be noted that the notion of the set attractiveness in Definition 2 is a bit different from those available in the literature. In [9], [24], [28]–[30], a set M is said to be an attractive set for the system $ (2) $ if any solution $ e(k) $ of $ (2) $ starting from $ e(0)\notin M $ approaches set M when k approaches infinity, i.e.,

where $ dist(e(k),M) = \inf_{\rho\in M}\|\rho-e(k)\| $. According to [15]–[18], the asymptotic practical stability of set M implies that for the state trajectories starting outside M, there is a finite time $ T>0 $ such that the trajectory $ e(k) $ is ultimately inside M. However, the existing definition of set attractiveness in $ (9) $ does not imply such a property. It only guarantees that the distance between the state trajectory $ e(k) $ and set M approaches zero without ensuring that $ e(k) $ enters M eventually. On the other hand, Definition 2 falls into the context of practical stability where the trajectories enter the ellipsoid $ {\cal{E}}(Q,c,\rho) $ in a finite time. Definitions 4–6 clarify these notions.

Definition 4: System (2) is locally practically stable with respect to an ellipsoid $ {\cal{E}}(Q,c,\rho)\subseteq D $ on the domain $ D\subset\mathbb{R}^n $ under switching function $ \sigma(e(k)) $ if the ellipsoid $ {\cal{E}}(Q,c,\rho) $ is an invariant set based on Definition 1.

Definition 5: System (2) is practically asymptotically stable in the large with respect to an ellipsoid $ {\cal{E}}(Q,c,\rho)\subset D $ on the domain $ D\subset\mathbb{R}^n $ under switching function $ \sigma(e(k)) $ if the ellipsoid $ {\cal{E}}(Q,c,\rho) $ is an invariant set of attraction based on Definition 3.

Definition 6: System (2) is practically asymptotically stable in the whole or globally practically asymptotically stable if it is practically asymptotically stable in the large and $ D = \mathbb{R}^n $.

Definition 7: In Definitions 3–6, the set $ D-{\cal{E}}(Q,c,\rho) $ is called the domain of attraction of the system (2) under switching rule $ \sigma(e(k)) $.

The ellipsoid $ {\cal{E}}(Q,c,\rho) $ in Definitions 4–7 is given by (4) and can be replaced by its equivalent form (5). Lemma 1 states the conditions where the autonomous switched nonlinear system (10) is practically asymptotically stable in the large in the sense of Definition 5.

where $ \sigma(e(k))\in\mathbb{K} $ is the switching function. We do not require $ 0_{n} = f_i(0_{n}) $, $ \forall i\in \mathbb{K} $. It is also assumed that the vector fields $ f_i $ satisfy suitable conditions guaranteeing the existence and uniqueness of solutions of (10) starting any $ e(0)\in D $ and $ \forall k\in \mathbb{Z}_{\geq0} $. The switched affine system (2) is a special case of the nonlinear switched system in (10). Moreover, it should be noted that the Definitions 1–6 are valid for the system (10).

Lemma 1: System (10) is practically asymptotically stable in the large in a given domain $ D\subset\mathbb{R}^n $ containing the origin in the sense of Definition 5 if there exists an ellipsoid $ {\cal{E}}(Q,c,\rho)\subset D $ with (8), a scalar function $ v(e(k)):\mathbb{R}^n\rightarrow\mathbb{R} $ and a switching rule $ \sigma(e(k)) $ such that

a) If $ e(k)\in{\cal{E}}(Q,c,\rho) $ then $e(k + 1) = f_{\sigma(e(k))}(e(k)) \in {\cal{E}} (Q,c,\rho)$.

b) If $e(k)\in D\;-\;{\cal{E}}(Q,c,\rho)$ then $v(e(k+1))\;-\;v\;(e(k)) \;= \Delta v(e(k))\leq -\gamma < 0$ where $ \gamma>0 $.

c) $ v(e(k)) $ is positive definite in $ D-{\cal{E}}(Q,c,\rho) $, i.e., $ v(e(k))>0 $ when $ e(k)\in D-{\cal{E}}(Q,c,\rho) $.

Proof: Condition a) is the same as condition b) of Definition 1. Moreover, from (4) and (8), one can conclude $ 0_{n}\in{\cal{E}}(Q,c,\rho) $, and therefore, condition a) of Definitions 1 and 2 is satisfied. Thus, the invariant property of ellipsoid $ {\cal{E}}(Q,c,\rho) $ is concluded. To prove the attractive property of the set $ {\cal{E}}(Q,c,\rho) $, according to condition b) of Definition 2, it is necessary to show that starting any initial state $e(0)\in D- {\cal{E}}(Q,c,\rho)$, $ \exists T = T(e(0))>0 $ such that $ e(k)\in {\cal{E}}(Q,c,\rho) $ for $ k\geq T $. We show this by contradiction as follows. Assume that the state trajectory $ e(k) $ never intersects $ {\cal{E}}(Q,c,\rho) $. Then at any instant k outside $ {\cal{E}}(Q,c,\rho) $ and according to condition b), we have $v(e(k+1))- v(e(k))\leq -\gamma < 0$. Thus, one can write

The right side of (11) will eventually be negative when k takes large values. This leads to a contradiction to the condition c) where it is assumed $ v(e(k)) $ is positive definite on $ D-{\cal{E}}(Q,c,\rho) $. As a result, the state trajectory $ e(k) $ will eventually enter inside the ellipsoid $ {\cal{E}}(Q,c,\rho) $ in finite time T, and due to the condition a), remains within it $ \forall k\geq T $. Therefore, $ {\cal{E}}(Q,c,\rho) $ is an invariant set of attraction according to Definition 3, and as a result, according to Definition 5, system (10) is practically asymptotically stable in the large on the domain D and under the switching function $ \sigma(e(k)) $.■

Using (11), one can obtain an upper bound of the reaching time as $ T = T(e(0))\leq\frac{v(e(0))}{\gamma} $.

Remark 1: In Lemma 1, if $ D = \mathbb{R}^n $, then according to Definition 6, system (2) is globally practically asymptotically stable.

III. Stability Analysis and Controller Synthesis

In this section, the stabilization procedure based on a pure numerical setting is presented. As discussed in Section II, we are planning to design a switching rule $ \sigma(e(k)) $ via a general quadratic Lyapunov function (3) for the system (2) to make it globally practically asymptotically stable. Proposition 1 provides sufficient conditions under which the ellipsoid $ {\cal{E}}(Q,c,\rho) $ is an invariant set for the system (2) under switching rule (12) in the sense of Definition 1.

Proposition 1: If there exist matrix $ Q = Q^T\succ0 $, vectors $ h,c\in \mathbb{R}^n $, scalars $ d \in \mathbb{R} $, $ \rho\leq0 $, nonnegative numbers $ \tau_1\geq0 $, $ \lambda_{1i}\geq0 $, $ i\in \mathbb{K} $, satisfying the system of inequalities

where

then the switching strategy in (12) assures that the ellipsoid $ {\cal{E}}(Q,c,\rho) $ in (4) is an invariant set for discrete-time switched affine system in (2) in the sense of Definition 1.

Proof: Please see the Appendix.■

To obtain the attractiveness property, we introduce the min-type state-feedback switching function as

Proposition 2 provides sufficient conditions that make ellipsoid $ {\cal{E}}(Q,c,\rho) $ in (4) an attractive set for the switched affine system (2) under switching function (18) in the sense of Definition 2.

Proposition 2: If there exist matrices $ Q = Q^T\succ0 $, $ P = P^{T}\in\mathbb{R}^{n\times n} $, vectors $ h,c\in \mathbb{R}^n $, constants $ d \in \mathbb{R} $, $ \rho\leq0 $, and nonnegative numbers $ \lambda_{2i}\geq0 $, $ i\in \mathbb{K} $, $ \tau_2, \tau_3\geq0 $ satisfying the system of inequalities

where

then the switching strategy in (18) assures that the ellipsoid $ {\cal{E}}(Q,c,\rho) $ in (4) is an attractive set for the system (2) on the domain $ D = \mathbb{R}^n $ in the sense of Definition 3.

Proof: Please see the Appendix.■

Condition (19) in Proposition 2 implies the existence of a Schur stable convex combination of subsystems matrices $ A_{\lambda} = \sum_{i\in\mathbb{K}}\frac{\lambda_{2i}}{\sum_{i\in\mathbb{K}}\lambda_{2i}}A_i $. The proof of this statement is similar to the proof of Corollary 1 in [21] and is omitted here. Using Propositions 1 and 2, one can conclude the following result.

Theorem 1: If there exist matrices $ P^T = P $, $ Q^T = Q\succ0 $, vectors $ h,c,l\in \mathbb{R}^n $, scalars $ d \in \mathbb{R} $, $ \rho\leq0 $, and nonnegative numbers $ \tau_1,\tau_2,\tau_3\geq0 $, $ \lambda_{1i},\lambda_{2i}\geq0 $, $ i\in \mathbb{K} $ satisfying the system of inequalities (19)–(21), (13)–(14) and (6)–(7) then, based on Definition 3, the ellipsoid $ {\cal{E}}(Q,c,\rho) $ in (4) is an invariant set of attraction for system (2) under switching rule (12) when $ e(k)\in {\cal{E}}(Q,c,\rho) $ and switching rule (18) when $ e(k)\notin {\cal{E}}(Q,c,\rho) $. As a result, system (2) is globally practically asymptotically stable in the sense of Definition 6.

Theorem 1 is theoretically appealing and provides a single-stage design procedure compared with the double-stage design methods in [9], [10]. Assuming that there is a global optimization tool, the optimal design of the parameters with the proposed conditions in Theorem 1 is performed by one-time implementation of the optimization software. On the other hand, in the double-stage design method, the optimization tool should be implemented many times between two stages to cover the whole search space of the decision variables and obtain the ultimate global optimal solution. Moreover, according to (21), the positiveness of the Lyapunov function in (3) is required only outside ellipsoids (4) or (5), while in [9], [10], it is assumed that the Lyapunov function is positive in the whole state space. Therefore, the proposed conditions in Theorem 1 may be less restrictive than those proposed in [9], [10], and offer some advantages from the optimization of parameters and implementation point of views. However, from a numerical point of view, similar to the proposed conditions in [9], [10], the conditions of Theorem 1 are of BMI type and, therefore, are computationally intractable.

In this regard, in the sequel, we present another method that may be more numerically tractable. In this method, only the attractiveness conditions proposed in Proposition 2 are solved numerically. Next, the ultimate invariant set of attraction is calculated analytically via Proposition 4.

IV. Numerical-Analytical Method and Possibility of the Unified Switching in the Whole State Space

During the numerical experiments in Section VI, we observed that in most cases, the ellipsoid $ {\cal{E}}(Q,e_c,1) $ in (5) is not only attractive by the switching rule (18) but also by the switching law in (12). Such a possibility is of particular importance from several aspects. First, instead of using two different switching rules, i.e., (12) inside of the ellipsoid and (18) outside of it, one can use a unified controller (12) in the whole state space. As a result, the controller logic is simplified, and it can be implemented more easily. Second, in some cases, due to numerical issues, the switching rule (18) does not result in convergence. However, the execution of (12) is successful. Finally, achieving the attractiveness property via switching rule (12) provides a possibility to estimate the invariant set of attraction in a numerical-analytical setting based on only attractiveness conditions in Proposition 2. As a result, without considering the invariant conditions of Proposition 4, the stability conditions of Theorem 2 are developed via matrix inequalities in a smaller dimension, which may be more numerically tractable. This section provides the theoretical basis for the preceding possibilities.

The next lemma provides a sufficient condition for the attractiveness of the ellipsoid $ {\cal{E}}(Q,c,\rho) $ in (4) based on the switching rule (12) rather than (18).

Lemma 2: Ellipsoid $ {\cal{E}}(Q,c,\rho) $ with $ Q = Q^T\succ0 $ and (8) is an attractive set for the system (2) according to Definition 2 and using switching function (12) if $ \forall e(k)\notin{\cal{E}}(Q,c,\rho) $, $ \exists i\in\mathbb{K} $ such that

Proof: Please see the Appendix.■

Based on Lemma 2 and using the equivalency relation between the ellipsoid representations in (4) and (5), one can conclude the following results.

Lemma 3: Let the conditions (6)–(7) be held. The switching rule in (12) is equivalent to the switching rule in

Lemma 4: Let the conditions (6)–(8) be held. Ellipsoid $ {\cal{E}}(Q,e_c,1) $ in (5) with $ Q = Q^T\succ0 $ is an attractive set for the system (2) according to Definition 2 and using switching function (12) if $ \forall e(k)\notin{\cal{E}}(Q,e_c,1) $, $ \exists i\in\mathbb{K} $ such that

Proposition 3 provides sufficient conditions under which the attractiveness conditions of Proposition 2 with the switching rule (18) imply the attractiveness property via switching law (12).

Proposition 3: Consider matrices $ P = P^T $ and $ Q = Q^T\succ0 $, vectors $ h,c\in \mathbb{R}^n $, constants $ d \in \mathbb{R} $, $ \rho\leq0 $ that follow from a feasible solution of Proposition 2. If there exist nonnegative constants $\lambda_{2i}' \geq0$ and $\tau_2' \geq0$ satisfying the system of inequalities

where

then the switching strategy in (12) assures that the ellipsoid $ {\cal{E}}(Q,c,\rho) $ in (4) is an attractive set for discrete-time switched affine system in (2) in the sense of Definition 2.

Proof: We show that the conditions of Proposition 3 imply the condition of Lemma 2. Pre-multiplying (27) by $ [e(k)^T\; 1] $ and post-multiplying it by $ [e(k)^T\; 1]^T $ and using S-procedure, one can obtain

By substituting $ \tilde{M}_1^{'} $, $ \tilde{M}_2^{'} $, and $ \tilde{M}_3^{'} $ from (29)–(31) into (32) and after some algebra, one can reach

Now using (28) and (33), we have

The expression in (34) is identical to the condition of Lemma 2, and as a result, the attractiveness of the ellipsoid $ {\cal{E}}(Q,c,\rho) $ is concluded under switching rule (12).■

It should be noted that given matrices P, Q, vectors $ h, c $, and constants d, $ \rho $ by Proposition 2, the matrix inequalities in Proposition 3 are of the pure LMI type, and therefore, are numerically tractable.

In Theorem 1, the global practical asymptotic stability of the system (2) is achieved via two different switching rules, i.e., (12) inside of the ellipsoid $ {\cal{E}}(Q,c,\rho) $ and (18) outside of it. If the LMI conditions of Proposition 3 are feasible, then one can use switching rule (12) instead of (18) in this theorem. With this approach, the global practical asymptotic stability of the system (2) is achieved via the unified switching rule (12) in the whole state space.

Proposition 4 states that the existence of an attractive ellipsoid implies the existence of an invariant set of attraction.

Proposition 4: Let the conditions (6)–(8) be held, and the ellipsoid $ {\cal{E}}(Q,e_c,1) $ in (5) be an attractive set for the system (2) according to Definition 2. Set R and $ R^{*} $ to be

Then the following statements hold.

a) If $ R\leq1 $, then $ {\cal{E}}(Q,e_c,1) $ is an invariant set of attraction for the system (2) independent of any switching strategy within $ {\cal{E}}(Q,e_c,1) $.

b) If $ R>1 $ and $ R^{*}\leq1 $, then $ {\cal{E}}(Q,e_c,1) $ is an invariant set of attraction for the system (2) under switching rule (12) within $ {\cal{E}}(Q,e_c,1) $.

c) Let the LMI conditions in Proposition 3 be satisfied. If $ R^{*}>1 $, then the ellipsoid $ {\cal{E}}(Q,e_c,R^{*}) $ in (5) is an invariant set of attraction for system (2) under switching strategy (12) within $ {\cal{E}}(Q,e_c,R^{*}) $.

Proof: Please see the Appendix.■

It should be noted that the feasibility requirements of Proposition 3 imposed in item c) of Proposition 4, are not required in items a) and b) where $ R\leq1 $ and $ R^{*}\leq1 $, respectively. On the other hand, the numerical examples show that the case $ R^{*}>1 $ occurs more frequently, and fortunately, the feasibility requirements of Proposition 3 are satisfied in most cases.

Theorem 2 states that one can infer the global practical asymptotic stability of the system (2) whenever an attractive set is provided through conditions of Propositions 2 and 3.

Theorem 2: Assume that there exist matrices $ P^T = P\in \mathbb{R}^{n\times n} $, $ Q^T = Q\succ0 $, $ i\in \mathbb{K} $, vectors $ h,c,l\in \mathbb{R}^n $, scalars $ d\in \mathbb{R} $, $ \rho\leq0 $ nonnegative numbers $ \tau_2,\tau_3\geq0 $, satisfying the system of inequalities (19)–(21) and (6)–(7). Let R and $ R^{*} $ be given by (35) and (36), respectively. Then the following statements hold:

a) The ellipsoid $ {\cal{E}}(Q,e_c,1) $ in (5) with $R\leq1$ is an invariant set of attraction for the system (2) under switching rule (18) when $ e(k)\notin{\cal{E}}(Q,e_c,1) $. Thus, system (2) is globally practically asymptotically stable in the sense of Definition 6.

b) The ellipsoid $ {\cal{E}}(Q,e_c,R^{*}) $ in (5) with $ R^{*}\leq1 $ is an invariant set of attraction for the system (2) under switching rule (12) when $ e(k)\in{\cal{E}}(Q,e_c,1) $ and switching rule (18) when $ e(k)\notin{\cal{E}}(Q,e_c,1) $. Thus, system (2) is globally practically asymptotically stable in the sense of Definition 6.

c) Let the conditions of Proposition 3 be satisfied. Then, the ellipsoid $ {\cal{E}}(Q,e_c,R^{*}) $ in (5) with $ R^{*}>1 $ is an invariant set of attraction for the system (2) under switching rule (12). Thus, the system (2) is globally practically asymptotically stable in the sense of Definition 6.

Proof: The proof follows from the fact that the conditions of Proposition 2 are embedded within the conditions of this theorem. As a result, the ellipsoid $ {\cal{E}}(Q,e_c,1) $ in (5) is an attractive set under switching rule (18) outside of it. Now items a)–c) are inferred from items a)–c) of Proposition 4.■

Thanks to the conditions of Proposition 3 in item c) of Theorem 2, the switching rule in this item is considered identically as (12) in the whole state space. Another switching procedure is to use (12) within $ {\cal{E}}(Q,e_c,R^{*}) $ and (18) outside of it.

Theorem 2 offers the same theoretical advantages of Theorem 1 plus the additional merit of matrix inequalities with smaller dimensions that may be more numerically tractable. This is because the matrix inequalities and the associated decision variables in (13) and (14) related to the invariance condition are eliminated in Theorem 2.

It should be noted that Theorems 1 and 2 cover the following special design cases available in the literature and provide a unified version of the current results.

1) Centralized Lyapunov Function and Centralized Ellipsoid (CLCE): In this case, in (3), $ h = 0_{n},d = 0 $. In addition, $ e_c = 0_{n} $ in (5) or $ c = 0_{n} $, $ \rho = -1 $ in (4). Moreover, one can take $ P = Q $ or $ P\neq Q $ in (3), (4), or (5). The stabilization techniques in [11], [16], [17], [22] fall into this category.

2) Decentralized Lyapunov Function and Centralized Ellipsoid (DLCE): In this case, the vector h and scalar d in (3) are not set to zero necessarily. However, as in the case of the CLCE scenario, $ e_c = 0_{n} $ in (5) or $ c = 0_{n} $, $ \rho = -1 $ in (4). In [9], the authors used these types of Lyapunov functions and ellipsoids in their study.

3) Centralized Lyapunov Function and Decentralized Ellipsoid (CLDE): This case considers the Lyapunov function in (3) with pre-assigned parameters $ h = 0_{n},d = 0 $. However, in contrast to CLCE and DLCE cases, the vectors $ e_c $ and c are not removed in (5) and (4), respectively. Moreover, scalar $ \rho $ is not preset to –1 in (4). The work in [23] and [10] falls into this category.

In what follows, the proposed method in this paper using a decentralized Lyapunov function and decentralized ellipsoid is abbreviated as DLDE. It is noted that the approach in [24] is a particular case of the DLDE method in which the invariant set is considered dependently as the level sets of a shifted quadratic Lyapunov function. While in the general decentralized setting of the DLDE method, the shifted Lyapunov function and the shifted invariant ellipsoid are parameterized independently.

V. Minimization of the Invariant Set of Attraction and Numerical Aspects

To achieve better performance with minimum deviations from the setpoint, we are interested in selecting the invariant set of attraction with a small size as far as possible. In this regard, one technique is to minimize the ellipsoid volume through the minimization of $ det(Q^{-1}) $ [7], [9], [31]. In [28], it is shown that

where $ \rho_i(Q^{-1}) $ is the length of each semiaxis of the ellipsoid $ {\cal{E}}(Q,e_c,1) $ in (5). The product $ \prod_{i = 1}^{n}\rho_i(Q^{-1})\varpropto Vol({\cal{E}}(Q,e_c,1)) $ that is proportional to the ellipsoid volume may admit a substantial value of one semiaxis, while all others may be very small. As a result, the minimum volume criterion through the minimization of $ det(Q^{-1}) $ may not give good performance in all directions. Another approach is to use the trace of the matrix $ Q^{-1} $ that defines the sum of the squares of the ellipsoid $ {\cal{E}}(Q,e_c,1) $ semiaxes [28]. Since the tool YALMIP/MATLAB [32] was not able to handle the objective function $ tr(Q^{-1}) $, in what follows, we have modified it such that the software limitation is relaxed while the main property of the size minimization is preserved. Since we have

the minimization of $\lambda_{\max}(Q^{-1})$, guarantees the minimization of the corresponding maximal semiaxis $\rho_{\max}(Q^{-1}) =$ $\sqrt{\lambda_{\max}(Q^{-1})}$. Moreover, it minimizes an upper-bound for $ tr(Q^{-1}) $. Therefore, the ellipsoid size minimization problem is formulated as

On the other hand, by introducing a slack variable $ t>0 $, we have [33]

Using (38) and Schur complements, the optimization problem in (37) can be equivalently rewritten as

As it will be shown in the next section, the trajectories of the system converge to the center of ellipsoids (4) and (5) at $ e_c = -Q^{-1}c $ which in general does not coincide with the desired equilibrium point $ x_e $. To solve this problem, we add a new term of $ \|e_c\|^2 = e_c^Te_c $ to the optimization problem’s cost function (39) to achieve the following multiobjective nonlinear programming in which w is a weighting factor determined by the designer.

The optimization problems in (37)–(40) are formulated for Theorem 2. They can be easily modified to handle the conditions of Theorem 1 by embedding the additional constraints (13) and (14) and decision variables $ \lambda_{1i} $, $ i\in \mathbb{K} $, $ \tau_1\geq0 $. In this paper, the optimization problems in (39) and (40) are solved via solver PENBMI [34] interfaced by YALMIP [32]. The PENBMI solver is suitable for optimization problems with a quadratic objective function and BMI constraints. It provides a local solution when convergence is achieved [34].

VI. Examples

In this section, various examples are introduced to illustrate the previous results. It should be noted that, in contrast to the pure LMI based optimization problems [35], making a fair comparison among BMI based optimization problems and formulations is a difficult task. The reason is that the BMI based optimization problems are non-convex, and exploring for a global optimum is an NP-hard problem. In most cases, the convergence of the various iterative techniques to find a local optimum is not guaranteed mathematically, and the choice of the initial value is important for convergence to an acceptable solution [36]. Nevertheless, one can always compare various BMI formulations or BMI solution methods on various examples to take some statistical information about various methods [37].

Example 1: This example is borrowed from [10]. Here, to obtain acceptable solutions from PENBMI software, we choose $ T_s = 0.05 $. We consider the discrete-time switched affine system consisting of 2 unstable systems defined by the following state-space matrices:

The state-space matrices $ A_i $ and $ l_i $, $ i\in\{1,2\} $, corresponding to the discrete-time system, can be obtained as

Two different types of equilibrium points are taken for the numerical experiments. First, we adopt an equilibrium point $ x_e = [-1.3415\; 0.5864]^T $ corresponding to $ \lambda = [0.4\; 0.6]^T $ which belongs to $ X_e = \{x_e\in \mathbb{R}^n|x_e = (I_n-A_{\lambda})^{-1}b_{\lambda},\lambda\in\Lambda\} $. $ X_e $ is the set of equilibrium points of the averaged system, which is indeed a convex combination of all subsystems of (1) as $ x(k+1) = \sum_{i\in \mathbb{K}}\lambda_i(A_i x(k)+b_i) $, $ \lambda_i\in\Lambda $. Next, we consider the desired equilibrium point $x_e = [-1\; 0]^T$, which is not the equilibrium point of the averaged system and does not belong to the set $ X_e $. The numerical results are given in Table I. In this table, the symbol – denotes that the optimization problem (40) has not been implemented for the case studies of CLCE and DLCE where a centralized ellipsoid has been used, and, as a result, the center of the ellipsoid is aligned with the chosen equilibrium point. In some cases, the PENBMI software did not provide acceptable solutions with the less conservative positiveness condition in (21). In these cases, this condition is replaced by the conventional and conservative one, namely, $ P\succ0 $ in the case of the centralized Lyapunov function or $ \left[P\; h;h^T\; d\right]\succ0 $ for the decentralized Lyapunov function. The superscript $ ^{*} $ in Tables I and II reflects such situations. Moreover, in this example, the weighting factor w in the optimization problem (40) is set to $ w = 9 $ except in one case for which the optimization problem (40) did not yield a feasible solution, and $ w = 8 $ was selected. According to Table I, the advantages of the proposed DLDE method are highlighted when the equilibrium point is selected out of those that are attainable in the set $ X_e $. In this regard, the state trajectories corresponding to the equilibrium point $ x_e = [-1\; 0]^T\notin X_e $ and respective ellipsoids are shown in Fig. 1. The corresponding switching sequence $ \sigma(k) $ is shown in Fig. 2. In this study, Theorem 1 did not provide any feasible solution for the DLCE method and Theorem 3 is used instead. As can be seen from this figure, with approximately identical steady-state error in all case studies, the proposed DLDE method represents the smallest invariant set of attraction compared to other methods. More specifically, the invariant set’s size is $ 54.03\% $ smaller than that of the best result of other scenarios, i.e., the CLDE method. Fig. 3 shows the state trajectories and corresponding ellipsoids when the multiobjective optimization problem (40) is employed in the CLDE and DLDE methods. As can be seen from this figure, in this case study, the DLDE method's performance is very close to the CLDE and DLCE methods. The time evolutions of the error state vector $ e(kT_s ) $ are illustrated in Fig. 4 where the state trajectory $ x(kT_s) $ starts from $ x_0 = [1\; 1]^T $. It is clear that after the transient the state remains near the equilibrium point when $ x_e\in X_e $. However, there exists a steady state error when $ x_e\notin X_e $.

Table  I.  Ellipsoid Size Comparisons for Numerical Example 1

Case study	Theorem	Optimization problem for $x_e=[-1.3415\;\;0.5864]^T\in X_e$					Optimization problem for $x_e=[-1\;\;0]^T\notin X_e$
		(39)		(40)			(39)		(40)
		t	$\frac{1}{\sqrt{det(Q/r)}}$	t	$\frac{1}{\sqrt{det(Q/r)}}$		t	$\frac{1}{\sqrt{det(Q/r)}}$	t	$\frac{1}{\sqrt{det(Q/r)}}$
CLCE	1	0.3671	0.3213	–	–		0.7851	0.7851	–	–
	2	0.2407	0.3811	–	–		0.3603	0.5656	–	–
DLCE	1	0.3671	0.3213	–	–		Divergence	Divergence	–	–
	2	$0.1909^{*}$	$0.3172^{*}$	–	–		0.3189	0.5131	–	–
CLDE	1	0.2922	0.2730	0.3519	0.3081		0.4290	0.4290	0.5626	0.5626
	2	0.2262	0.7129	0.2380	0.3875		0.3546	0.6459	0.3586	0.5813
DLDE Proposed method	1	0.2261	0.2051	0.3519	0.3081		0.2035	0.1972	$ $$\begin{array}{c}0.5497\\w=8\end{array}$$ $	$ $$\begin{array}{c}0.4982\\w=8\end{array}$$ $
	2	0.1524	1.0809	0.1897	0.3329		0.1477	1.0795	0.2862	0.5218

 | Show Table

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Table  II.  Ellipsoid Size Comparisons for Numerical Example 2

Case study	Theorem	Optimization problem for $x_e=[0.0354\;\;0.3960]^T\in X_e$					Optimization problem for $x_e=[0\;\;0.15]^T\notin X_e$
		(39)		(40)			(39)		(40)
		t	$\frac{1}{\sqrt{det(Q/r)}}$	t	$\frac{1}{\sqrt{det(Q/r)}}$		t	$\frac{1}{\sqrt{det(Q/r)}}$	t	$\frac{1}{\sqrt{det(Q/r)}}$
CLCE	1	0.0240	0.0240	–	–		0.0274	0.0267	–	–
	2	0.0240	0.0267	–	–		0.0266	0.0276	–	–
DLCE	1	Divergence	Divergence	–	–		$0.0275^{*}$	$0.0268^{*}$	–	–
	2	$0.3557$	$0.7819$	–	–		$3.9258\times 10^{-5*}$	$5.2457\times 10^{-4*}$	–	–
CLDE	1	0.0240	0.0240	$ $$\begin{array}{c}0.0240\\w=1\end{array}$$ $	$ $$\begin{array}{c}0.0240\\w=1\end{array}$$ $		Divergence	Divergence	Divergence	Divergence
	2	$0.0240^{*}$	$0.0267^{*}$	$ $$\begin{array}{c}0.0240\\w=5\end{array}$$ $	$ $$\begin{array}{c}0.0267\\w=5\end{array}$$ $		$0.0163^{*}$	$0.0613^{*}$	$ $$\begin{array}{c}0.0251^{*}\\w=17\end{array}$$ $	$ $$\begin{array}{c}0.0295^{*}\\w=17\end{array}$$ $
DLDE Proposed method	1	0.0215	0.0214	$ $$\begin{array}{c}0.0240\\w=4\end{array}$$ $	$ $$\begin{array}{c}0.0239\\w=4\end{array}$$ $		0.0065	0.0065	$ $$\begin{array}{c}0.0112\\w=8\end{array}$$ $	$ $$\begin{array}{c}0.0112\\w=8\end{array}$$ $
	2	Divergence	Divergence	Divergence	Divergence		$9.9635\cdot10^{-5*}$	$0.0012^{*}$	Divergence	Divergence

 | Show Table

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Figure  1.  State trajectories of example 1 corresponding to conditions of Theorem 1 for CLCE, CLDE and DLDE and Theorem 2 for DLCE using optimization (39) when $ x_e = [-1\; 0]^T\notin X_e $.

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Figure  2.  Switching function profile corresponding to Fig. 1, (a) CLCE; (b) DLCE; (c) CLDE; and (d) DLDE.

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Figure  3.  State trajectories of Example 1 corresponding to conditions of Theorem 1 for CLCE, CLDE and DLDE and Theorem 2 for DLCE using optimization (40) for CLDE and DLDE methods when $ x_e = [-1\; 0]^T\notin X_e $.

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Figure  4.  Time evolutions of trajectories $ e(kT_s ) = [e_1 (kT_s )\; e_2 (kT_s )]^T $ in Example 1 by the DLDE method, Theorem 1 and optimization (40). (a) Time evolution of $ e_1(kT_s ) $; (b) Time evolution of $ e_2(kT_s ) $.

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Example 2: Fig. 5 shows the circuit diagram of a DC-DC buck converter. The continuous states of the system are defined as $ x(t) = [i_L(t),v_o(t)]^T $, which are the inductor current and converter output voltage, respectively. The state-space equations of this converter are as follows [38]:

Figure  5.  DC-DC buck Converter

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where

The discrete-time system matrices $ A_i $ and $ l_i $, $ i\in\{1,2\} $, can be obtained by (41). The parameters used for the numerical experiments are given in Table III. To avoid ill-conditioned matrix inequalities and to make the problem more amenable for numerical purposes, we use per unit parameters in the numerical experiments [22], [26], [38]. The per-unit value of a parameter is calculated by dividing its real value into the corresponding base value. The base parameters are selected as $v_{{\rm{base}}} = 50\; {\rm{ V}}$, $i_{{\rm{base}}} = 2.5\sqrt{20}\; {\rm{A}}$ and $T_{{\rm{base}}} = 10\; {\text{μs}}$. The sampling time is set to $T_s = 10\; {\text{μs}}$, and, as a result, the maximum value of switching frequency is limited to ${1}/{2T_s} = 50\; {\rm{kHz }}= f_s$. Similar to Example 1, two different equilibrium points are chosen in this example. One is selected as the equilibrium point of the averaged system as $x_e = (I_n-A_{\lambda})^{-1}b_{\lambda} = [0.0354,0.3960]^T\; {\rm{p.u}}.\in X_e$ corresponding to $ \lambda = [0.4\; 0.6]^T $. The other is chosen arbitrary as $ x_e = [0\; 0.15]^T\notin X_e $. The numerical results are given in Table II. As can be seen from this table, the advantages of the proposed DLDE method are highlighted when the equilibrium point $ x_e $ is outside of the set $ X_e $. Figs. 6 and 7 show the state trajectories and the corresponding switching sequences for the equilibrium point $x_e = [0\; 0.15]^T\notin X_e$ using conditions of Theorem 1 and optimization problem in (39). In this case study, the conditions of Theorem 1 did not provide an acceptable solution for the CLDE method, and therefore, the conditions of Theorem 2 are used. As can be seen from Fig. 6, compared to other scenarios, the proposed DLDE method yields the best solution in terms of the smallest invariant set of attraction and closeness of the state trajectories to the selected equilibrium point. The resulting ellipsoid is $ 75.66\% $ smaller than that of the best result of the other methods in CLCE and DLCE. In the CLDE method, after a certain time, the switching function is fixed on $ \sigma(k) = 2 $, and the state trajectories converge to zero eventually, which is not acceptable in practice for switching converters. This issue highlights the significance of multiobjective optimization in (40). In this regard, Fig. 8 illustrates the state trajectories and the corresponding ellipsoids when the optimization (40) is employed for the CLDE and DLDE methods. The switching signal profiles corresponding to the CLDE and DLDE methods in Fig. 8 are shown in Fig. 9. As can be seen from these figures, utilizing multiobjective optimization (40), the problem associated with the CLDE method discussed in Figs. 6 and 7(c) is fixed. Moreover, in contrast to the similar case study in Example 1 (Fig. 3), the smallest steady-state error and invariant set size are obtained for the proposed DLDE method. The invariant set size is $ 58.05\% $ smaller than that of the best result of the other scenarios, i.e., the CLCE method. Fig. 10 shows the time evolutions of the error state vector $ e(k) $ when the state trajectory $ x(k) $ starts from $ x_0 = [1\; 1]^T $. Similar to Fig. 4, there exists a steady state error when $ x_e\notin X_e $, while it disappears when $ x_e\in X_e $. The preceding examples illustrate the effectiveness and validity of the proposed DLDE stabilization method. It should be noted that the numerical results presented in this paper are extracted via the PENBMI/YALMIP tool that is a local optimizer. As a result, a degree of conservativeness is inherent in the presented results. Although the BMI formulation is by itself reliable, flexible, and less-conservative, however, the optimization problems associated with the BMI problems are non-convex and achieving a global optimal solution is an NP-hard problem. In this respect, developing suitable global optimization tools to achieve better numerical results from the BMI problems will play a key role in this field of research.

Table  III.  The Specifications of the DC-DC Buck Converter

Input voltage $v_s$	50 V	1 p.u.
Converter inductor L	2 mH	$10\sqrt{20}$ p.u.
Output capacitor C	100 μF	$10\sqrt{20}$ p.u.
Load resistance R	50 $\Omega$	$50/\sqrt{20}$ p.u.
Sampling period $T_s$	10 μs	1 p.u.
Inductor series resistance $r_L$	0.5 $\Omega$	$0.5/\sqrt{20}$ p.u.
Capacitor series resistance $r_C$	0.1 $\Omega$	$0.1/\sqrt{20}$ p.u.

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Figure  6.  State trajectories of Example 2 corresponding to conditions of Theorem 1 for CLCE, DLCE and DLDE and Theorem 2 for CLDE using optimization (39) when $ x_e = [0\; 0.15]^T\notin X_e $.

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Figure  7.  Switching function profile corresponding to Fig. 6, (a) CLCE; (b) DLCE; (c) CLDE; and (d) DLDE.

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Figure  8.  State trajectories of Example 2 corresponding to conditions of Theorem 1 for CLCE, DLCE and DLDE and Theorem 2 for CLDE using optimization (40) in CLDE and DLDE when $ x_e = [0\; 0.15]^T\notin X_e $.

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Figure  9.  Switching function profile corresponding to Fig. 8 for CLDE and DLDE methods, (a) CLDE; (b) CLDE enlarged view; (c) DLDE.

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Figure  10.  Time evolutions of trajectories $ e(k) = [e_1(k )\; e_2(k)]^T $ in the Example 2 by the DLDE method, Theorem 1 and optimization (40). (a) Time evolution of $ e_1(k) $; (b) Time evolution of $ e_2(k) $.

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VII. Conclusion

In this paper, the global practical stabilization of the discrete-time switched affine systems via state-dependent switching rules has been investigated. The main contribution relies on extending the existing results by the simultaneous use of a decentralized quadratic Lyapunov function and a decentralized ellipsoid. Two different stability conditions are provided: a purely numerical method and a numerical-analytical one. A multiobjective optimization problem was formulated such that its solution makes it possible to minimize the size of the ultimate invariant set of attraction and the steady-state errors of the system trajectories. The proposed stabilization method was compared with other existing methods in which it is necessary for the Lyapunov function or the employed ellipsoid or both be of the centralized type. The numerical illustrations on an academic example and a DC-DC buck converter show that the proposed stabilization strategy provides less conservative results and yields a smaller invariant set of attraction and less steady-state errors of the system trajectories.

APPENDIX

Proof of Proposition 1: It is shown that (13) and (14) imply condition a) of Lemma 1. Pre-multiplying (13) by $ [e(k)^T\; 1] $ and post-multiplying it by $ [e(k)^T\; 1]^T $ one can reach

$ \forall e(k)\in \mathbb{R}^n $. Inequality (42) can be rewritten as

Using S-procedure [31], (43) implies

By substituting $ M_1 $, $ M_2 $ and, $ M_3 $ from (15)–(17) into (44) and using (4), one can write

According to (14), since $ \sum_{i\in \mathbb{K}}\lambda_{1i}>0 $, $ \lambda_{1i}\geq0, i\in \mathbb{K} $, from (45), one can conclude

Based on the switching rule in (12), one can write

According to (2), $ e(k+1) = A_{\sigma(e(k))}e(k)+l_{\sigma(e(k))} $, thus from (46) and (47), one can conclude

which states that whenever $ e(k)\in{\cal{E}}(Q,c,\rho) $ then $e(k+1)\in {\cal{E}}(Q,c,\rho)$. Thus, according to condition (a) of Lemma 1, (48) implies that the ellipsoid $ {\cal{E}}(Q,c,\rho) $ is an invariant set for the switched affine system (2) under switching rule (12).■

Proof of Proposition 2: It is shown that conditions (19) and (20) imply item b) of Lemma 1, and condition (21) satisfies item c) of this lemma. From (19), one can conclude $ \exists \gamma>0 $ such that

This comes from the fact that for a symmetric matrix $M = M^T\in \mathbb{R}^{(n+1)\times (n+1)}$, we have $\lambda_{\min}(M)I_{n+1}\preceq M\preceq\lambda_{\max}(M)I_{n+1}$. Pre-multiplying (49) by $ [e(k)^T\; 1] $ and post-multiplying it by $ [e(k)^T\; 1]^T $ one can obtain

The second inequality in (50) comes from the fact that $ -\gamma e(k)^Te(k)\leq0 $. (50) can be rewritten as

Using the S-procedure, (51) implies

By substituting $ \tilde{M}_1 $, $ \tilde{M}_2 $, and $ \tilde{M}_3 $ from (22)–(24) into (52) and after doing some algebra, one can reach

According to (20) and (3), from (53) one can conclude

Based on the switching rule (18), we have

From (54) and (55) one can reach

which implies condition (b) in Lemma 1. According to item c) of Lemma 1 we still need to show that the function $ v(e) $ in (3) is positive definite when $ e(k)\notin {\cal{E}}(Q,c,\rho) $. Pre-multiplying (21) by $ [e(k)^T\; 1] $ and post-multiplying it by $ [e(k)^T\; 1]^T $, one can obtain

Using the S-procedure, and according to (3) and (4), (56) implies

which verifies the condition c) of Lemma 1. As a result, conditions b) and c) are satisfied, and the attractiveness of the ellipsoid $ {\cal{E}}(Q,c,\rho) $ under switching rule (18) is concluded. ■

Proof of Lemma 2: We consider the scalar function $ v(e(k)) $ as

which is positive definite $ \forall e(k)\notin {\cal{E}}(Q,c,\rho) $. Thus, condition c) of Lemma 1 is satisfied. In the sequel, it is shown that the condition b) of Lemma 1 is implied by (25). Using (57), we construct the following switching law and show that it makes the ellipsoid $ {\cal{E}}(Q,c,\rho) $ in (4) an attractive set for the system (2):

Now, according to (58), one can write

Using (25) and (59), one can reach

Now, we define function $ \phi_{i}(e(k)) $ as

According to (60) and (61), $ |S|\geq1 $ and $ \phi_{i}(e(k))>0 $ when $ e(k)\notin {\cal{E}}(Q,c,\rho) $. Now, we define the function $ \phi(s) $ as

Function $ \phi(\|e(k\|) $ is nondecreasing and positive definite when $ e(k)\notin {\cal{E}}(Q,c,\rho) $. Moreover, according to (61) and (62), one can write

Since $ e(k)\notin {\cal{E}}(Q,c,\rho) $, using (4), we have

Because $ Q = Q^T\succ0 $, one can write $e(k)^TQe(k)\leq \lambda_{\max} (Q) \|e(k)\|^2$. Moreover, $ c^Te(k)\leq \|c\|\|e(k)\| $. By substituting these inequalities into $ (64) $, one can obtain

According to (65), one can conclude that

Using (66) and the nondecreasing nature of function $ \phi(s) $ in (62), one can get

which together with (63) yields

Therefore, conditions b) and c) in Lemma 1 are fulfilled, and the attractive property of the ellipsoid $ {\cal{E}}(Q,c,\rho) $ is concluded.■

Proof of Proposition 4: i) Let us assume that $e(k)\in {\cal{E}} (Q, e_c,1)$. Then, according to (5), $ (e(k)-e_c)^TQ(e(k)-e_c)\leq1 $ and since $\lambda_{\min}(Q)\|e(k)-e_c\|^2\leq (e(k)-e_c)^TQ(e(k)-e_c)$, then,

Since, $ |\|e(k)\|-\|e_c\||\leq\|e(k)-e_c\| $, from (67), one can conclude

Using (2), one can write

According to (68) and (69), one can conclude

Now, using $(e(k+1)-e_c)^TQ\;(e(k+1)-e_c)\;\leq \;\lambda_{\max}(Q)\;\times \|e(k+1)- e_c\|^2$ and (70), one can write

The term at the right-hand side of (71) is identical to the parameter R in (35). If $ R\leq1 $, from (71) one can conclude that $(e(k+1)-e_c)^T Q(e(k+1)-e_c)\leq1$. Thus, whenever $e(k)\in {\cal{E}}(Q, e_c,1)$ then $ e(k+1)\in {\cal{E}}(Q,e_c,1) $ as well. This means that ellipsoid $ {\cal{E}}(Q,e_c,1) $ is an invariant set independent of any switching strategy within $ {\cal{E}}(Q,e_c,1) $. This result together with the attractiveness property of $ {\cal{E}}(Q,e_c,1) $ implies that the ellipsoid $ {\cal{E}}(Q,e_c,1) $ is an invariant set of attraction independent of any switching rule inside of it.

ii) Let us assume $ e(k)\in {\cal{E}}(Q,e_c,1) $. Under switching rule $ (12) $ that according to Lemma 3 is equivalent to the switching law in (26), one can write

The last inequality in (72) comes from our arguments in the proof of the item i) where the constraint $(e(k)-e_c)^TQ(e(k)- e_c)\leq1$ implies (67). The term at the right-hand side of the last inequality in (72) is equal to the parameter $ R^{*} $ in (36). If $ R^{*}\leq1 $, one can conclude that, under switching rule (12), $ e(k)\in {\cal{E}}(Q,e_c,1) $ implies $ e(k+1)\in {\cal{E}}(Q,e_c,1) $. This means that $ {\cal{E}}(Q,e_c,1) $ is an invariant set via the switching rule (12) within it. This result, together with the attractiveness property of $ {\cal{E}}(Q,e_c,1) $, implies that the ellipsoid $ {\cal{E}}(Q,e_c,1) $ is an invariant set of attraction when the switching rule (12) is implemented inside of it.

iii) If $ R^{*}>1 $, then $ {\cal{E}}(Q,e_c,1)\subset{\cal{E}}(Q,e_c,R^{*}) $. The rest of the proof is provided in two parts:

Case 1 ($e(k)\in{\cal{E}}(Q,e_c,$1)): In this case, following our arguments in the proof of the item ii), the implementation of the switching rule (12) or its equivalent (26), within $ {\cal{E}}(Q,e_c,1) $ implies whenever $ e(k)\in{\cal{E}}(Q,e_c,1) $, then according to (72), $ (e(k+1)-e_c)^TQ(e(k+1)-e_c)\leq R^{*} $ or analogously $e(k+1)\in {\cal{E}} (Q,e_c,R^{*})$.

Case 2 ($ e(k)\in {\cal{E}}(Q,e_c,R^{*})-{\cal{E}}(Q,e_c, $1)): In this case, since conditions of Proposition 3 are valid, so are the equivalent conditions of Lemmas 2 and 4 (please see the proof of Proposition 3). According to Lemma 4, one can conclude $ \exists i\in \mathbb{K} $ such that

$ \forall e(k)\notin{\cal{E}}(Q,e_c,1) $. Using (73), the implementation of the switching rule (12) or its equivalent (26) within $ {\cal{E}}(Q,e_c,R^{*}) $ yields

According to (74), one can conclude $(e(k+1)-e_c)^TQ(e(k+ 1)-e_c) < (e(k)-e_c)^TQ(e(k)-e_c)$. Since $ e(k)\in {\cal{E}}(Q,e_c,R^{*}) $, thus $ (e(k)-e_c)^TQ(e(k)-e_c)\leq R^{*} $ that together with $ (e(k+1)-e_c)^TQ(e(k+1)-e_c)<(e(k)-e_c)^TQ(e(k)-e_c) $ yields $ (e(k+1)-e_c)^TQ(e(k+1)-e_c)<R^{*} $. Therefore in all cases $ e(k)\in {\cal{E}}(Q,e_c,R^{*}) $ implies $ e(k+1)\in {\cal{E}}(Q,e_c,R^{*}) $. This means that $ {\cal{E}}(Q,e_c,R^{*}) $ is an invariant set by the implementation of the switching function (12) inside of it. Since $ R^{*}>1 $ and as a result, $ {\cal{E}}(Q,e_c,1)\subset{\cal{E}}(Q,e_c,R^{*}) $, the attractiveness of $ {\cal{E}}(Q,e_c,1) $ implies the attractiveness of the $ {\cal{E}}(Q,e_c,R^{*}) $. This result together with the invariance property of the $ {\cal{E}}(Q,e_c,R^{*}) $ makes the ellipsoid $ {\cal{E}}(Q,e_c,R^{*}) $ an invariant set of attraction when the switching function (12) is implemented within it.■