Stability Analysis for Switched Systems with Sequence-based Average Dwell Time
aa r X i v : . [ c s . S Y ] N ov Stability Analysis for Switched Systems withSequence-based Average Dwell Time
Dianhao Zheng, Hongbin Zhang,
Senior Member, IEEE , J. Andrew Zhang,
Senior Member, IEEE ,and Steven W. Su,
Senior Member, IEEE
Abstract —This note investigates the stability of both linearand nonlinear switched systems with average dwell time. Twonew analysis methods are proposed. Different from existing ap-proaches, the proposed methods take into account the sequence inwhich the subsystems are switched. Depending on the predecessoror successor subsystems to be considered, sequence-based averagepreceding dwell time (SBAPDT) and sequence-based averagesubsequence dwell time (SBASDT) approaches are proposed anddiscussed for both continuous and discrete time systems. Theseproposed methods, when considering the switch sequence, havethe potential to further reduce the conservativeness of the existingapproaches. A comparative numerical example is also given todemonstrate the advantages of the proposed approaches.
Index Terms —Sequence-based average dwell time, sequence-based average subsequence dwell time, sequence-based averagepreceding dwell time, stability, switched systems.
I. I
NTRODUCTION
Switched systems consist of a finite number of subsystemsand a logical rule that orchestrates switching between thesesubsystems [1], and are often used to model many physicalor man-made systems [2], [3]. In recent years, considerableresearch efforts have been devoted to the study of the switchedsystem [4]–[7], for its wide range of applications, such ascooperation of multi-agent systems [8], [9], modeling ofelectronic switching systems.The stability of dynamical systems [10], [11] is a broadarea in system and control theory. In recent years, the stabilityanalysis of switched systems becomes one of the activeresearch topics. In 1996, paper [12] presented a definition ondwell time, being set as a constant, to solve the problem of su-pervisory control of families of linear controllers. Three yearslater, a more general and flexible definition for the switchingsignal, the signal with average dwell time (ADT) switching,was proposed in [13]. This scheme requires that the average
Manuscript received xxxx; revised xxx; accepted xxxx. This work wassupported in part by National Natural Science Foundation of China (Grantno. 61374117). Recommended by xxxx. Corresponding author: xxxxxDianhao Zheng is with the School of Information and Communication Engi-neering, University of Electronic Science and Technology of China, Chengdu611731, China, is also with the Faculty of Engineering and InformationTechnology, University of Technology Sydney, Ultimo, NSW 2007, Australia(e-mail: [email protected]).Hongbin Zhang is with the School of Information and Communication,University of Electronic Science and Technology of China, Chengdu 611731,China (e-mail:[email protected]).J. Andrew Zhang is with the Global Big Data Technologies Cen-tre, University of Technology Sydney, Ultimo, NSW 2007, Australia (e-mail:[email protected]).Steven W. Su is with the Faculty of Engineering and IT, University of Tech-nology Sydney, Ultimo, NSW 2007, Australia (e-mail:[email protected]) interval between consecutive discontinuities is not less thana constant. On the basis of this innovative idea, a numberof useful techniques have been developed [2], [14]–[16]. In2011, paper [3] proposed a new concept for analysis of thestability of switched systems, mode-dependent average dwelltime (MDADT), which could reduce the restrictions on theaverage dwell time (ADT). The key idea is the specification,in which each mode in the underlying system has its ownADT. On the basis of the concept of MDADT, several newresearch results have been reported [4]–[7].However, improving stability analysis for switched systemsis far from completion yet, even in the linear context. Tobe more specific, for existing methods, if the settings ofthe switching scenarios can be further refined, the conserva-tiveness of the switching law, e.g., the average dwell time,could be further reduced. Under this consideration, to relaxthe restrictions of the switching law, motivated by [3], wenot only consider the ADT for each individual mode , butalso the predecessor and successor of each switching mode.That is, rather than just considering the ADT for each mode,we further investigate the ADT for each mode with their switching sequence taking into account.In this study, we propose two new methods for analyzing thestability of linear and nonlinear switched systems and designthe corresponding feedback controllers in both continuous anddiscrete time situations. The major contributions of this notecan be summarized as follows: 1) Two new concepts havebeen originally proposed and defined, namely the sequence-based average subsequence dwell time and the sequence-basedaverage preceding dwell time. 2) We develop two new meth-ods, which distinguish subsequence ADT from preceding ADTand propose parallel stability conditions for switched linearand nonlinear systems. 3) On the basis of the proposed newmethods, we derive two less conservative stability conditions,which could further reduce the mode-dependent average dwelltime compared to the existing approaches. Furthermore, weshow that many existing methods can be generalized as specialand simplified versions of the proposed methods. Finally, anillustrative example is provided, and the effectiveness of theproposed approaches are demonstrated.The rest of this note is organized as follows. In Section II,some basic concepts and lemmas are introduced. In Section IIItwo sequence-based average dwell time methods are proposedand applied to linear systems. Section IV provides an exampleto verify the obtained results in this note.
Notation:
The notations used in this note are very stan-dard. The symbol ” × ” represents the multiplication opera- tion or Cartesian product of sets. The symbol → meansapproaching zero. Throughout this note, for a given matrix P , P > ( < ) 0 implies that the matrix P is symmetricand positive (or negative) definite. A superscript “T” standsfor the matrix transpose, and a function κ is said to be ofclass K ∞ function if the function κ : [0 , ∞ ) → [0 , ∞ ) , κ (0) = 0 , is strictly increasing, continuous, unbounded. Thetime t , t , t , · · · , t l , t l +1 , · · · or k , k , .., k l , k l +1 ,..., are theswitching times of subsystems. The flags t − l and t l stand forthe time just before or after subsystems switching at time t l .The function σ ( t ) : [0 , + ∞ ) → S = { , , · · · , s } is theswitching signal and s is the total amount of subsystems. Themark p | q stands for the case that the p th subsystem is activatedimmediately after the q th subsystem.II. PRELIMINARIESIn this section, we first recall some concepts about switchedsystems and then we review some recent related results.It is important for switched systems to reach a stable status.In order to describe the stability of switched systems, wefirst give the definition of global uniformly exponential stable(GUES). Definition 1: [3] For a given system and a switchingsignal σ ( t ) , an equilibrium x ∗ = 0 is globally uniformlyexponentially stable (GUES) if there exist constants η > , γ > (or < ς < ) for any initial conditions x ( t ) (or x ( k ) ), such that the solution of the system satisfies k x ( t ) k≤ η k x ( t ) k e − γ ( t − t ) , ∀ t ≥ t for the continuous-time situation (or k x ( k ) k≤ η k x ( k ) k ς ( k − k ) , ∀ k ≥ k forthe discrete-time case).In order to analyse the stability of switched systems, [3]and [13] proposed important concepts as follows. Definition 2: [3] For any switching times t ≥ t ≥ witha switching signal σ ( t ) , let N σp ( t , t ) stand for the numberof the p th subsystem that is activated over the time interval [ t , t ] , and T p ( t , t ) denote the total running time of the p th subsystem over the time interval [ t , t ] , p ∈ S . It is said that σ ( t ) has a mode-dependent average dwell time τ ap if thereexist two subsequence numbers N p ( N p is called the mode-dependent chatter bound here) and τ ap such that N σp ( t , t ) ≤ N p + T p ( t , t ) τ ap . Consider a class of continuous-time or discrete-timeswitched linear systems: ˙ x ( t ) = A σ ( t ) x ( t ) + B σ ( t ) u ( t ) , (1) x ( k + 1) = A σ ( k ) x ( k ) + B σ ( k ) u ( k ) (2)where x ( t ) ( x ( k ) ) ∈ R n and u ( t ) ( u ( k ) ) are the state vectorand the control input.For systems (1) and (2), some useful results can be foundin recent literature. Lemma 1: [3](Continuous-time Situation) For the system (1)when u ( t ) ≡ and given constants λ p > , µ p > , p ∈ S .Suppose that there exist matrices P p > , such that ∀ p ∈ S , A Tp P p + P p A p ≤ − λ p P p and ∀ ( p, q ) ∈ S × S , p = q , itholds that P p ≤ µ p P q . Then the system is globally uniformlyasymptotically stable with the MDADT condition τ ap ≥ τ ∗ ap = lnµ p /λ p . (3) Lemma 2: [3](Discrete-time Situation) For the system (2),when u ( k ) ≡ , and the given constants > λ p > , µ p ≥ , p ∈ S , suppose that there exist matrices P p > such that ∀ p ∈ S , A Tp P p A p − P p ≤ − λ p P p and ∀ ( p, q ) ∈ S × S , p = q ,it holds that P p ≤ µ p P q . Then the system is globally uniformlyasymptotically stable with the MDADT condition τ ap ≥ τ ∗ ap = − lnµ p / (1 − λ p ) . (4)In the situation of u ( t ) , the work [3] considered a classof feedback u ( t ) = K σ ( t ) x ( t ) for the continuous-time caseor u ( k ) = K σ ( k ) x ( k ) for the discrete-time case, where K p , p ∈ S , is the controller gain. The closed-loop system is givenby ˙ x ( t ) = ( A σ ( t ) + B σ ( t ) K σ ( t ) ) x ( t ) . (5) x ( k + 1) = ( A σ ( k ) + B σ ( k ) K σ ( k ) ) x ( k ) . (6)For the systems (5) and (6), important results were presentin some recent papers. Lemma 3: [3](Continuous-time Situation) For the system(5) and the given constants λ p > , µ p > , p ∈ S , supposethat there exist matrices U p > , and T p such that ∀ p ∈ S , A p U p + B p T p + U p A Tp + T Tp B Tp ≤ − λ p U p and ∀ ( p, q ) ∈S × S , p = q , it holds that U q ≤ µ p U p . Then there exists a setof stabilizing controllers such that the system (5) is globallyuniformly asymptotically stable with the MDADT condition(3). The controller gain can be given by K p = T p U − p . (7) Lemma 4: [3](Discrete-time Situation) For the system (6)and the given constants > λ p > , µ p ≥ , p ∈ S , supposethat there exist matrices U p > , and T p such that ∀ p ∈ S , (cid:20) − U p A p U p + B p T p ∗ − (1 − λ p ) U p (cid:21) ≤ and ∀ ( p, q ) ∈ S × S , p = q , it holds that U q ≤ µ p U p . Then there exists a set of stabilizing controllers such that thesystem (6) is globally uniformly asymptotically stable with theMDADT condition (4). The controller gain can be given by(7).More detailed contents can be found in recent works [2],[3], [13]. By reviewing the existing results about switchedsystems, we found that the influence of switching sequenceon dwell time has not been explicitly considered so far. Inthe next section of this note, two switching sequence basedmethods will be proposed.III. MAIN RESULTSThis section includes three subsections. In the first subsec-tion, we propose a sequence-based average subsequence dwelltime method. In the second subsection, we propose a sequence-based average preceding dwell time method. In the third part,we use the two proposed methods to analyze the stability oflinear switched systems.
A. Sequence-based Average Subsequence Dwell Time Method
In this subsection, we propose a sequence-based averagesubsequence dwell time method. According to this method,the setting of the parameter µ p or µ is referred to switchingsequences. The relationships among sequences, the stability ofswitched-systems, average dwell times are reconsidered.The definition of the sequence-based average subsequencedwell time is first given as follows. Definition 3:
For any switching times t > t ≥ with aswitching signal σ ( t ) , let N σp | q ( t , t ) stand for the number ofthe sequence that the p th subsystem is activated immediatelyafter the q th subsystem over the time interval [ t , t ) , andlet T p,p | q ( t , t ) denote the total running time of the p th subsystem activated immediately after the q th subsystems overthe time interval [ t , t ) , p ∈ S . It is said that σ ( t ) has asequence-based average subsequence dwell time τ a ( p,p | q ) ifthere exist two subsequence numbers N p,p | q ) ( N p,p | q ) iscalled the sequenced-based subsequence chatter bound here)and τ a ( p,p | q ) such that N σp | q ( t , t ) ≤ N p,p | q ) + T p,p | q ( t , t ) τ a ( p,p | q ) (8) Remark 1:
Definition 3 constructs a novel set of switchingsignals referring to the sequence-based average subsequencedwell time (SBASDT). It considers the switching sequence ofthe switched systems. Obviously, this definition is differentfrom Definition 2 as the successor of each individual modehas been explicitly considered here.On the basis of the new Definition 3, we present thefollowing two theorems.
Theorem 1: (Continuous-time Situation) Consider aswitched system described by ˙ x ( t ) = f σ ( t ) ( x ( t )) . (9)Let λ q > , λ p > , µ p | q > be given constants. Supposethere exist C functions V σ ( t ) : R n → R , and functions k p , k p , k q and k q of class K ∞ , such that ∀ ( t i = p, t − i = q ) ∈S × S and p = q , k q ( k x ( t ) k ) ≤ V q ( x ( t )) ≤ k q ( k x ( t ) k ) , (10) ˙ V q ( x ( t )) ≤ − λ q V q ( x ( t )) , (11) k p ( k x ( t ) k ) ≤ V p ( x ( t )) ≤ k p ( k x ( t ) k ) , (12) ˙ V p ( x ( t )) ≤ − λ p V p ( x ( t )) , (13)and V p ( x ( t i )) ≤ µ p | q V q ( x ( t i )) , (14)then the system is GUAS for any switching signals withSBASDT τ a ( p,p | q ) ≥ τ ∗ a ( p,p | q ) = lnµ p | q /λ p . (15) Proof:
Let t = 0 and use t , t , t , · · · to denote theswitching moments. According to (13) and (14), for any time t > t and t ∈ [ t l , t l +1 ) , l ∈ Z + , we have V σ ( t ) ( x ( t )) ≤ exp (cid:8) − λ σ ( t l ) ( t − t l ) (cid:9) V σ ( t l ) ( x ( t l )) ≤ exp (cid:8) − λ σ ( t l ) ( t − t l ) (cid:9) µ σ ( t l ) | σ ( t − l ) V σ ( t l − ) ( x ( t l − )) ≤ exp (cid:8) − λ σ ( t l ) ( t − t l ) (cid:9) µ σ ( t l ) | σ ( t − l ) × exp (cid:8) − λ σ ( t l − ) ( t l − t l − ) (cid:9) V σ ( t l − ) ( x ( t l − ))= µ σ ( t l ) | σ ( t − l ) exp (cid:8) − λ σ ( t l ) ( t − t l ) (cid:9) × exp (cid:8) − λ σ ( t l − ) ( t l − t l − ) (cid:9) V σ ( t l − ) ( x ( t l − )) (16)Recursively, we have V σ ( t ) ( x ( t )) ≤ (cid:26) l Q k =1 µ σ ( t k ) | σ ( t − k ) (cid:27) exp (cid:8) − λ σ ( t l ) ( t − t l ) · · ·− λ σ ( t ) ( t − t ) (cid:9) × exp (cid:8) − λ σ (0) ( t − (cid:9) V σ (0) ( x (0)) . (17)On the basis of the set S = { , , · · · , s } (where s is the totalamount of subsystems), we create a new set S ′ which is theCartesian product of S and S , i.e., S ′ = S × S = { ( p, q ) : p ∈S , and q ∈ S} . It is obvious that the set S ′ includes all thepossible ordered pairs ( σ ( t i ) , σ ( t − i )) , i ∈ Z + , and the totalnumber of the elements in S ′ is s ′ = s ( s − .Consider the system switching up to l , and define a newrelated set S ′′ = { ( p, q ) : p ∈ σ ( t i ) , q ∈ σ ( t − i ) , i =1 , , , · · · , l } . Obviously, S ′′ ⊆ S ′ .If σ ( t n ) = p and σ ( t − n ) = q , we denote µ σ ( t n ) | σ ( t − n ) as µ p | q , ∀ n = 1 , , · · · , l . The purpose of the formal transformation isto make it more readable under some situations. Furthermore,we list and number all the elements of the set S ′′ , i.e. we use [ p | q ] ( k ) to denote the pair ( p, q ) which is the k th element of theset S ′′ . N σ [ p | q ] ( k ) ( t, and T p, [ p | q ] ( k ) ( t, denote the activatednumbers and total subsequence dwell times of the k th elementin the time interval [0 , t ) respectively. Assuming that thenumber of elements in S ′′ is s ′′ , we have k ∈ { , , · · · , s ′′ } , s ′′ ≤ s ′ , and s ′′ ≤ l .Hence, V σ ( t ) ( x ( t )) ≤ ( s ′′ Q k =1 µ N σ [ p | q ]( k ) ( t, p | q ] ( k ) ) exp ( s ′′ P k =1 − λ p, [ p | q ] ( k ) T p, [ p | q ] ( k ) ( t, ) × exp (cid:8) − λ σ (0) ( t − (cid:9) V σ (0) ( x (0)) . (18)On the basis of (8), we can get V σ ( t ) ( x ( t )) ≤ s ′′ Q k =1 µ N p | q ]( k ) + Tp, [ p | q ]( k ) ( t, τa ( p, [ p | q ]( k )) [ p | q ] ( k ) × exp ( s ′′ P k =1 − λ p, [ p | q ] ( k ) T p, [ p | q ] ( k ) ( t, ) × exp (cid:8) − λ σ (0) ( t − (cid:9) V σ (0) ( x (0)) ≤ exp ( s ′′ P k =1 N p | q ] ( k ) ln µ [ p | q ] ( k ) ) × exp ( s ′′ P k =1 T p, [ p | q ]( k ) ( t, τ a ( p, [ p | q ]( k )) ln µ [ p | q ] ( k ) ) × exp ( s ′′ P k =1 − λ p, [ p | q ] ( k ) T p, [ p | q ] ( k ) ( t, ) × exp (cid:8) − λ σ (0) ( t − (cid:9) V σ (0) ( x (0))= exp ( s ′′ P k =1 N p | q ] ( k ) ln µ [ p | q ] ( k ) ) × exp ( s ′′ P k =1 ( ln µ [ p | q ]( k ) τ a ( p, [ p | q ]( k )) − λ p, [ p | q ] ( k ) ) T p, [ p | q ] ( k ) ( t, ) × exp (cid:8) − λ σ (0) ( t − (cid:9) V σ (0) ( x (0)) . (19)If for all k ∈ { , , · · · , s ′′ } τ a ( p, [ p | q ] ( k ) ) ≥ ln µ [ p | q ] ( k ) /λ p, [ p | q ] ( k ) , (20)then, we have V σ ( t ) ( x ( t )) ≤ exp ( s ′′ P k =1 N p | q ] ( k ) ln µ [ p | q ] ( k ) ) × exp (cid:26) max (cid:26) max k (cid:26) ln µ [ p | q ]( k ) τ a ( p, [ p | q ]( k )) − λ p, [ p | q ] ( k ) (cid:27) , − λ σ (0) (cid:27) t (cid:27) × V σ (0) ( x (0)) . (21)Therefore, we conclude that V σ ( t ) ( x ( t )) convergences tozero as t → + ∞ if the SBASDT satisfies the condition (15).Then, the asymptotic stability can be deduced according toDefinition 1. Theorem 2: (Discrete-time Situation) Consider a discreteswitched system described by x ( k + 1) = f σ ( k ) ( x ( k )) , (22)and let > λ q > , > λ p > , µ p | q > be givenconstants. Assume there exist C functions V σ ( k ) : R n → R and functions k p , k p , k q and k q of class K ∞ , such that ∀ ( k i = p, k i − q ) ∈ S × S , p = q , k q ( k x ( k ) k ) ≤ V q ( x ( k )) ≤ k q ( k x ( k ) k ) , (23) V q ( x ( k + 1)) − V q ( x ( k )) ≤ − λ q V q ( x ( k )) , (24) k p ( k x ( k ) k ) ≤ V p ( x ( k )) ≤ k p ( k x ( k ) k ) , (25) V p ( x ( k + 1)) − V p ( x ( k )) ≤ − λ p V p ( x ( k )) , (26)and V p ( x ( k i )) ≤ µ p | q V q ( x ( k i )) , (27) then the system is GUAS for any switching signals withSBASDT τ a ( p,p | q ) ≥ τ ∗ a ( p,p | q ) = − lnµ p | q /ln (1 − λ p ) . (28) Proof:
This theorem can be proved by using a methodsimilar to the proof of the continue-time situation. Due to thespace limitation, detailed proof is omitted here.
Remark 2:
Actually, the conditions (10)-(13) (or (23)-(26))are equivalent to the corresponding conditions of the MDADTmethod [3]. This remark is also valid for the sequence-basedaverage preceding dwell time method.
Remark 3:
Compared to the existing results, new methodsallow a subsystem to have different switching restrictions µ p | q for different sequences. Remark 4:
MDADT is a popular method for the analysisof the switched systems. For the p th subsystem, if we set thevalues of µ p | q as a fixed value µ p , without considering thepreceding different q th subsystems, the SBASDT retrogressesto MDADT. Therefore, the method in this note has lessconservativeness and restriction than MDADT. B. Sequence-based Average Preceding Dwell Time Method
In this subsection, we propose another sequence-basedmethod, the sequence-based average preceding dwell timemethod.The definition of the sequence-based average precedingdwell time is given as follows.
Definition 4:
For any switching times t > t ≥ with aswitching signal σ ( t ) , let N σp | q ( t , t ) stand for the number ofthe sequence that the p th subsystem is activated immediatelyafter the q th subsystem over the time interval [ t , t ) , andlet T q,p | q ( t , t ) denote the total running time of the q th subsystem when the p th subsystem is activated immediatelyafter the q th subsystems over the time interval [ t , t ) , p ∈ S .It is said that σ ( t ) has a sequence-based average precedingdwell time τ a ( q,p | q ) if there exist two preceding numbers N q,p | q ) ( N q,p | q ) is called the sequenced-based precedingchatter bound here) and τ a ( q,p | q ) such that N σp | q ( t , t ) ≤ N q,p | q ) + T q,p | q ( t , t ) τ a ( q,p | q ) (29) Remark 5:
Definition 4 also constructs a novel set ofswitching signals referring to the sequence-based averagepreceding dwell time method (SBAPDT). Both Definition 3and Definition 4 are based on switching sequence.On the basis of Definition 4, we can get the following twotheorems.
Theorem 3: (Continuous-time Situation) Consider aswitched system depicted by (9), and let λ q > , λ p > , µ p | q > be given constants. Suppose there exist C functions V σ ( t ) : R n → R , and functions k p , k p , k q and k q of class K ∞ , such that ∀ ( t i = p, t − i = q ) ∈ S × S , p = q , k q ( k x ( t ) k ) ≤ V q ( x ( t )) ≤ k q ( k x ( t ) k ) , (30) ˙ V q ( x ( t )) ≤ − λ q V q ( x ( t )) , (31) k p ( k x ( t ) k ) ≤ V p ( x ( t )) ≤ k p ( k x ( t ) k ) , (32) ˙ V p ( x ( t )) ≤ − λ p V p ( x ( t )) , (33)and V p ( x ( t i )) ≤ µ p | q V q ( x ( t i )) , (34)then the system is GUAS for any switching signals withSBAPDT τ a ( q,p | q ) ≥ lnµ p | q /λ q . (35) Proof:
Let t = 0 and use t , t , t , · · · to denote thesubsystems switching times. According to (33) and (34), forany time t > t and t ∈ [ t l , t l +1 ) , l ∈ Z + , we have systems(16)-(17).We also use the definitions of S , S ′ , S ′′ , and [ p | q ] ( k ) fromthe proof of Theorem 1. The symbol T q, [ p | q ] ( k ) ( t, denotesthe total dwell times of the q th subsystem for the k th sequenceunion in the time interval [0 , t ) .Hence, V σ ( t ) ( x ( t )) ≤ ( s ′′ Q k =1 µ N σ [ p | q ]( k ) ( t, p | q ] ( k ) ) × exp ( s ′′ P k =1 − λ q, [ p | q ] ( k ) T q, [ p | q ] ( k ) ( t, ) × exp (cid:8) − λ σ ( t l ) ( t − t l ) (cid:9) V σ (0) ( x (0)) . (36)According to (29), one can get V σ ( t ) ( x ( t )) ≤ s ′′ Q k =1 µ N q, [ p | q ]( k )) + Tq, [ p | q ]( k ) ( t, τa ( q, [ p | q ]( k )) [ p | q ] ( k ) × exp ( s ′′ P k =1 − λ q, [ p | q ] ( k ) T q, [ p | q ] ( k ) ( t, ) × exp (cid:8) − λ σ ( t l ) ( t − t l ) (cid:9) V σ (0) ( x (0)) ≤ exp ( s ′′ P k =1 N q, [ p | q ] ( k ) ) ln µ [ p | q ] ( k ) ) × exp ( s ′′ P k =1 T q, [ p | q ]( k ) ( t, τ a ( q, [ p | q ]( k )) ln µ [ p | q ] ( k ) ) × exp ( s ′′ P k =1 − λ q, [ p | q ] ( k ) T q, [ p | q ] ( k ) ( t, ) × exp (cid:8) − λ σ ( t l ) ( t − t l ) (cid:9) V σ (0) ( x (0))= exp ( s ′′ P k =1 N q, [ p | q ] ( k ) ) ln µ [ p | q ] ( k ) ) × exp ( s ′′ P k =1 ( ln µ [ p | q ]( k ) τ a ( q, [ p | q ]( k )) − λ q, [ p | q ] ( k ) ) T q, [ p | q ] ( k ) ( t, ) × exp (cid:8) − λ σ ( t l ) ( t − t l ) (cid:9) V σ (0) ( x (0)) . (37)If there exist constants τ a ( q, [ p | q ] ( k ) ) ≥ ln µ [ p | q ] ( k ) /λ q, [ p | q ] ( k ) , we have V σ ( t ) ( x ( t )) ≤ exp ( s ′′ P k =1 N q, [ p | q ] ( k ) ) ln µ [ p | q ] ( k ) ) × exp (cid:26) max (cid:26) max k (cid:26) ln µ [ p | q ]( k ) τ a ( q, [ p | q ]( k )) − λ q, [ p | q ] ( k ) (cid:27) , − λ σ ( t l ) (cid:27) t (cid:27) × V σ (0) ( x (0)) (38)Therefore, we conclude that V σ ( t ) ( x ( t )) convergences to zeroas t → + ∞ if the SBAPDT satisfies the condition (35). Then,the asymptotic stability can be deduced according to Definition1. Theorem 4: (Discrete-time Situation) Consider a discreteswitched system described by (22), and let > λ q > , >λ p > , µ p | q > be given constants. Assume there exist C functions V σ ( k ) : R n → R , and functions k p , k p , k q and k q of class K ∞ , such that ∀ ( k i = p, k i − q ) ∈ S × S , p = q , k q ( k x ( k ) k ) ≤ V q ( x ( k )) ≤ k q ( k x ( k ) k ) , (39) V q ( x ( k + 1)) − V q ( x ( k )) ≤ − λ q V q ( x ( k )) , (40) k p ( k x ( k ) k ) ≤ V p ( x ( k )) ≤ k p ( k x ( k ) k ) , (41) V p ( x ( k + 1)) − V p ( x ( k )) ≤ − λ p V p ( x ( k )) , (42)and V p ( x ( k i )) ≤ µ p | q V q ( x ( k i )) , (43)then the system is GUAS for any switching signals withSBAPDT τ a ( q,p | q ) ≥ − lnµ p | q /ln (1 − λ q ) . (44) Proof:
This theorem can be proved by using a methodsimilar to the proof of the continue-time situation. To savespace, the proof is omitted here.
C. Linear Switched Systems with two Sequence-based AverageDwell Time Methods
In this subsection, we study the stability of switched systemsusing SBASDT and SBAPDT.First, we consider the general linear system (1) and (2).When u ( t ) ≡ , we can obtain Corollary 1 and 2 below. Corollary 1: (Continuous-time Situation) Consider aswitched system depicted by (1) with u ( t ) ≡ . Let λ q > , λ p > , µ p | q > be given constants. Suppose that thereexist matrices P q > , P p > , p, q ∈ S , such that ∀ p ∈ S , ∀ ( t i = p, t − i = q ) ∈ S × S , p = q , A Tq P q + P q A q ≤ − λ q P q , (45) A Tp P p + P p A p ≤ − λ p P p , (46)and P p ≤ µ p | q P q , (47)then the system is GUAS for any switching signals withSBASDT τ a ( p,p | q ) ≥ τ ∗ a ( p,p | q ) = lnµ p | q /λ p . (48)or with SBAPDT τ a ( q,p | q ) ≥ τ ∗ a ( q,p | q ) = lnµ p | q /λ q . (49) Proof:
This corollary can be proved according to Theo-rems 1 and 3 and the proof of Lemma 1. Detailed proof isomitted here.
Corollary 2: (Discrete-time Situation) Consider a discreteswitched system described by (2) with u ( t ) ≡ . Let >λ q > , > λ p > , µ p | q > be given constants. Supposethat there exist matrices P q > , P p > , p, q ∈ S , such that ∀ ( k i = p, k i − q ) ∈ S × S , p = q , A Tq P q A q − P q ≤ − λ q P q (50) A Tp P p A p − P p ≤ − λ p P p (51)and P p ≤ µ p | q P q , (52)then the system is GUAS for any switching signals withSBASDT τ a ( p,p | q ) ≥ − lnµ p | q /ln (1 − λ p ) . (53)or with SBAPDT τ a ( q,p | q ) ≥ − lnµ p | q /ln (1 − λ q ) . (54) Proof:
This corollary can be proved by using a methodsimilar to the proof of the continue-time situation. Detailedproof is omitted here.For the case u ( t ) , we have Corollaries 3 and 4.Their proofs are very similar to those of Corollaries 1 and2, respectively, and hence are omitted. Corollary 3: (Continuous-time Situation) Consider aswitched system depicted by (1) with u ( t ) = K σ ( t ) x ( t ) . Let λ q > , λ p > , µ p | q > be given constants. Suppose thatthere exist matrices P q > , P p > , p, q ∈ S , such that ∀ ( t i = p, t − i = q ) ∈ S × S , p = q , ( A q + B q K q ) T P q + P q ( A q + B q K q ) ≤ − λ q P q , (55) ( A p + B p K p ) T P p + P p ( A p + B p K p ) ≤ − λ p P p , (56)and P p ≤ µ p | q P q , (57)then the system is GUAS for any switching signals withthe SBASDT switching condition (48) or with the SBAPDTswitching condition (49). Corollary 4: (Discrete-time Situation) For a discreteswitched system (2) and u ( k ) = K σ ( k ) x ( t ) , let > λ q > , > λ p > , µ p | q > be given constants. Suppose thatthere exist matrices P q > , P p > , p, q ∈ S , such that ∀ ( k i = p, k i − q ) ∈ S × S , p = q ( A q + B q K q ) T P q ( A q + B q K q ) − P q ≤ − λ q P q (58) ( A p + B p K p ) T P p ( A p + B p K p ) − P p ≤ − λ p P p (59)and P p ≤ µ p | q P q , (60)then the system is GUAS for any switching signals withthe SBASDT switching condition (53) or with the SBAPDTswitching condition. (54).If we want to get a solution on controller gain K σ ( t ) and K σ ( k ) , the following two corollaries can provide solutions. Corollary 5: (Continuous-time Situation) Consider aswitched system depicted by (1), and let λ q > , λ p > , µ p | q > be given constants. Suppose that there exist matrices U p > , and T p , p ∈ S , such that ∀ ( t i = p, t − i = q ) ∈ S × S , p = q , A p U p + B p T p + U p A Tp + T Tp B Tp ≤ − λ p U p (61) A q U q + B q T q + U q A Tq + T Tq B Tq ≤ − λ q U q (62)and U q ≤ µ p | q U p . (63)Then there exists a set of stabilizing controllers such that thesystem is globally uniformly asymptotically stable with thewith SBASDT (48) or with SBAPDT condition (49).The controller gain can be calculated by K p = T p U − p . (64) K q = T q U − q . (65) Proof:
This corollary can be established according toCorollary 3 and the proof of lemma 3. Detailed proof isomitted here.
Corollary 6: (Discrete-time Situation) Consider a discreteswitched system described by (2), and let > λ q > , >λ p > , µ p | q > be given constants. Suppose that thereexist matrices U q > , U p > , T p , T q , p, q ∈ S , such that ∀ ( k i = p, k i − q ) ∈ S × S , p = q , (cid:20) − U p A p U p + B p T p ∗ − (1 − λ p ) U p (cid:21) ≤ (66) (cid:20) − U q A q U q + B q T q ∗ − (1 − λ q ) U q (cid:21) ≤ (67)and U q ≤ µ p | q U p , (68)Then there exists a set of stabilizing controllers such that thesystem is globally uniformly asymptotically stable with theSBASDT condition (53) or with the SBAPDT condition (54).The controller gain can be designed through (64)-(65). Proof:
This corollary can be derived according to Corol-lary 4 and the proof of lemma 4. Detailed proof is omittedhere. IV. NUMERICAL EXAMPLEIn this section, we use a numerical example to show theeffectiveness of the proposed methods and the derived results.Because of the space limitation and the similarity betweencontinuous-time and discrete-time systems, we only verifyCorollary 5. Other obtained results can be verified in the sameway.Consider switched linear systems including three subsys-tems described as A = (cid:20) . − . − . − . (cid:21) , B = (cid:2) − . − . (cid:3) , A = (cid:20) . − .
22 10 . (cid:21) , B = (cid:2) − . (cid:3) , A = (cid:20) . − . . (cid:21) , B = (cid:2) . − . (cid:3) , TABLE I: compution for the MDADT and SBASDT switching
Schemes MDADT switching SBASDT switchingCriteria Theorem 1 in [2] Corollary 5 in this noteParameters λ =3; λ = 1 . ; λ = 2 . ; λ =3; λ = 1 . ; λ = 2 . ; µ = 18 ; µ = 2 . ; µ = 41 . µ | = 18 ; µ | = 2 . ; µ | = 41 ; µ | = 13 ; µ | → ; µ | = 17 . K . , − . ; K1 =[371.7662, -100.0154]Controller K . , − . K2 =[2.8330, -0.4917] K − . , − . . K3 =[-7.9922, -2.0744].Average τ ∗ a =0.96; τ ∗ a (1 , | = 0 . ; τ ∗ a (1 , | = 0 . ;Dwell Time τ ∗ a =0.56; τ ∗ a (2 , | = 0 . ; τ ∗ a (2 , | → ;Thresholds τ ∗ a =1.5. τ ∗ a (3 , | = 1 . ; τ ∗ a (3 , | = 1 . . TABLE II: dwell times of the SBASDT and SBAPDT switching
Schemes SBASDT switching SBAPDT switchingAverage τ ∗ a (1 , | = 0 . ; τ ∗ a (1 , | = 0 . ; τ ∗ a (2 , | = 1 . ; τ ∗ a (3 , | = 1 . ;Dwell Time τ ∗ a (2 , | = 0 . ; τ ∗ a (2 , | → ; τ ∗ a (1 , | = 0 . ; τ ∗ a (3 , | → ;Thresholds τ ∗ a (3 , | = 1 . ; τ ∗ a (3 , | = 1 . . τ ∗ a (2 , | = 1 . ; τ ∗ a (1 , | = 1 . . In this example, we want to get a set of proper sequence-based stabilizing controller gains and search the admissibleminimal SBADT to make the closed-loop system stable.In order to show the advantage of the sequence-basedaverage dwell time, we use Table I to show the numericalresults for the mode-dependent average dwell time switching[3] and the sequence-based average subsequence dwell timeswitching.It can be seen that the dwell time thresholds are reducednotably according to Table I. For the first dwell time set, . and 0.86 repalce 0.96. Two values with one being 0.56 and theother approaching 0 replace 0.56 for the second set. The values1.1 and 1.5 replace 1.5 for the third set.The control gains arealso shown in Table I. The dwell times of two sequence-basedmethods are shown in Table II. We take the first sequence, | , in Table II as an example. According to the conditions(48) and (49), we have τ ∗ a (1 , | = 0 . and τ ∗ a (2 , | = 1 . .The value τ ∗ a (1 , | is smaller than τ a (2 , | for λ > λ . Forthe same reason, τ ∗ a (2 , | > τ ∗ a (1 , | .V. CONCLUSIONIn this note, considering the switching sequence of sub-systems, we have proposed two new analysis methods, forstudying the stability of the switched systems. The proposedsequence based average dwell time switching is less conser-vative than the mode-dependent average dwell time switch-ing, as well as the traditional average dwell time switching.Both linear and non-linear systems in continuous-time anddiscrete-time situations were analyzed, and the associated statefeedback controllers were designed. Finally, we presented anillustrative example to show the advantages and effectivenessof the proposed new methods.R EFERENCES[1] H. Lin and P. J. Antsaklis, “Stability and stabilizability of switchedlinear systems: a survey of recent results,”
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