Robust Switching Design for Continuous-time Switched LinearSystems

Zhendong Sun

Abstract— In this work, we re-examine the state-feedbackpath-wise switching proposed previously for discrete-timeswitched systems, and extend the switching design approachto continuous-time switched systems. To properly capture thesensitiveness of a switching signal undergoing various switchingperturbations, we define the distance between state-feedbackpath-wise switching laws. Based on the defined distances, weestablish that any asymptotically stabilizable switched linearsystem is robust with respect to switching perturbations.

Index Terms— Switched linear systems, state-feedback path-wise switching, switching distance, robustness

I. Introduction

Switched linear systems have long been investigated in thecontrol literature, and have attracted increasing attention dur-ing the last decade. As a primary issue of concern, stabilizingswitching design has been extensively studied, and variousswitching mechanisms were proposed in the literature. Inparticular, for a system whose state transition matrix is Schurstable along certain switching path, a periodical stabilizingswitching law could be formulated by means of the averageapproach and the Campbell-Baker-Hausdorff formula [2],[15], [13]; for a system whose subsystems admit a stablelinear convex combination, a state-feedback switching lawwas proposed to exponentially stabilize the switched system[1], [16], [17]; when a system admits both time-driven andstate-feedback switching mechanisms, a mixed switching lawwas proposed to reduce the switching frequency and toenhance the robustness [9]; and a state-feedback path-wiseswitching law was presented for stabilization of discrete-timeswitched linear systems [11]. The reader is referred to [8],[5], [20] for more recent developments.

As a further issue, the problem of robust switching designis to seek proper switching mechanism that makes thesystem stable and attenuates possible system disturbances orperturbations. When the subsystems encounter perturbations,robust switching design schemes were developed in manyworks, see, for instance, [6], [7], [9]. When a periodic switch-ing signal encounters perturbations, the robust switchingdesign problem was addressed by several researchers [18],[14], [4], [19]. However, when the switching signal is state-

This work was supported by the National Natural Science Foundation ofChina under grant No. 60925013.

Z. Sun is with College of Automatic Science and Engineering,South China University of Technology, Guangzhou 510460, Chinazdsun@scut.edu.cn

dependent and defined over an infinite time horizon, thereare few works addressing the robust switching problem [12].

In this work, we are to extend the robust switchingschemes established in [11], [12] from discrete-time systemsto continuous-time systems. By adapting the state-feedbackpath-wise switching law to continuous-time systems, weprove that this switching law is universal for stabilization inthe sense that any stabilizable system could be stabilized bysuch a switching law. Furthermore, we introduce the relativedistance between a nominal and perturbed state-feedbackpath-wise switching laws. By establishing a qualitative con-nection between the relative switching distance and the normdeviation of the state transition matrix, we prove that anystabilizable switched linear system is robust with respect tosmall switching perturbations.

Notations. Let be any norm on and the inducedmatrix norm. For a set , denotes the cardinality of theset. For two sets and , set includes each elementof that does not belong to . A real-valued function

is said to be of class if it is continuous,strictly increasing, and . A function

is said to be of class if is ofclass for each fixed and foreach fixed . For a real number , ( ) denotes thesmallest (largest) integer equal or greater (less) than .

II. Preliminaries

In this work, we address the problem of robust switchingdesign for the switched linear system described by

(1)

where is the continuous state,is the discrete state, also known as the switching

signal, and .A switching path is a function defined over a finite time

interval taking values from the index set, . Given a timeinterval with , a switching path

defined over the interval is denoted . The switchingpath is said to be well-defined if there is a finite number ofswitching times within the interval. A switching signal is afunction defined over an infinite time-horizon taking valuesfrom . A switching signal is said to be well-defined ifthere is a finite number of switching times in any interval ofa finite length. Finally, a switching law is a switching rule

258978-1-4577-2074-1/12/$26.00 c©2012 IEEE

that generates a switching path or a switching signal for aset of initial configurations. A switching law is said to bewell-defined if it generates a well-defined switching signalfor any initial state. As a result, a well-defined switching lawcan be represented by the set , where is theswitching signal generated by the switching law with initialstate .

A. Stabilization

For a switched dynamical system, the problem of stabiliza-tion is to find, if possible, a suitable switching law that steersthe switched system asymptotically stable.

Denote by the continuous state of system(1) at time with initial condition and switchinglaw .

Definition 1: Switched linear system (1) is said to beasymptotically stabilizable, if there exists a well-definedswitching law that makes the system asymp-totically stable, i.e.,

where is a class function.A switching law is said to be a stabilizing switching law

if it makes the switched system asymptotically stable.

B. State-feedback Path-wise Switching

Suppose that is a natural number, , areregions in satisfying and forany , and , are well-definedswitching paths. The state-feedback path-wise switching lawvia and , denoted by or in short,is the concatenation of switching paths through

as defined below. For any initial state ,the generated switching signal is recursively defined by

(2)

It is clear that the state-feedback path-wise switchinglaw is always well-defined over for any initial state.For a state-feedback path-wise switching law , theswitching paths are called base paths, and theregions are called the base partitions (of the statespace).

C. Aggregated Systems

Suppose that is a state-feedback path-wise switch-ing law for switched system (1). For each switching path

, there corresponds to a state transition matrixwith the property that for

all . It can be seen that, under the switching law,

switched system (1) can be aggregated into the piecewiselinear system

(3)

which is a discrete-time piecewise linear system. In thesequel, system (3) is said to be the aggregated system ofthe original system (1) with respect to the switching law

.Suppose that is a positive real number. The piecewise

linear system (3) is said to be -contractive with respect toa norm if

(4)

It can be seen that, any state trajectory of the aggregatedsystem is a discrete abstraction of a state trajectory ofthe original system under an appropriate switching signal,and the original system is asymptotically stabilizable if theaggregated system is asymptotically stable.

D. Universality of the Switching Law

It was established that any stabilizable discrete-time switchedlinear system admits a state-feedback path-wise switchinglaw that steers the system stable [11]. That is, the setof state-feedback path-wise switching laws is universal forstabilizibility of discrete-time switched linear systems. Forcontinuous-time switched linear systems, the same conclu-sion holds true, due to the following result.

Lemma 1: For an asymptotically stabilizable continuous-time switched linear system

there is a sampling period , such that the sampledswitched system

(5)

is asymptotically stabilizable.Proof: Note that, for a continuous-time switched linear

system, asymptotic stabilizability implies switched conver-gency [13, Thm. 3.9]. It follows that there exist a naturalnumber , a positive real number , and a set of pairs

, such that

1) is a cone for any , and ;and

2)

Re-write the inequality as

(6)

where is the switching durationsequence of over . Define the functionsfor by

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Fix a . It follows from the continuity of thefunctions that, there exists a positive real number , suchthat

As a result, when the sampling rate is less than or equalto , then there are natural numbers , such that

This, together with relationship (6), yields

This means that, under the sampling rate , the sampled-datasystem (5) is switched convergent and hence is asymptoti-cally stabilizable [13, Thm 3.53]. This completes the proof.

Combining the lemma with [11, Theorem 3.1], we reachthe following conclusion.

Lemma 2: Suppose that switched system (1) is asymptot-ically stabilizable. Then, for any norm and , thereis a state-feedback path-wise switching law that makes theaggregated system -contractive.

A design procedure was presented in [11] for computing aswitching law satisfying Lemma 2. Note that this switchinglaw makes the original system exponentially stable. Fromnow on, we assume that the switched linear system isasymptotically stabilizable. By fixing a norm and a ,we further assume that a -contractive stabilizing switchinglaw has been computed and hence is known.

III. Distance Between Switching Laws

In this section, we present an approach to characterize thedistance between two state-feedback path-wise switchinglaws, which paves the way for the robustness analysisconducted later.

To implement a state-feedback path-wise switching law, it involves a repeated process of first determining

an index according to the online state measurement, andthen coordinating the switching along the switching path

. In this process, perturbations might enter either in indexdetermining or in executing the switching path. The distancebetween the nominal switching and perturbed switchingshould be the summation of the variations of both types insome sense.

Fix a state-feedback path-wise switching law .Given a switched system, it generates a switching signal

for any initial state that is the concatenation of theswitching paths , . This switching signal canbe described in a recursive way as

1) Initiation. From the initial state , choose the activesubsystem index such that ,and let the system evolves along switching path forthe duration of . At the end of the process, the stateis , which is termed as the currentrelay state at .

2) Recursion. From the current relay state at ,choose the active subsystem indexsuch that , and let the system evolves alongswitching path for the duration of . At the end ofthe process, the state is , whichis the current relay state at .

Suppose that we are to implement the (nominal) switchingsignal in the above-mentioned recursive way. Then, pertur-bations might enter in the following possible manners:

1) When implementing a switching path for a durationof , mismanipulations might occur in certain cases.For instance, a time delay in the operation, or com-ponent (subsystem) failures lead to displacement ofswitching paths, etc.

2) At the concatenating time instants , the next pieceof switching path is wrongly selected due to possiblesubsystem disturbances and/or the imprecise measureof the current relay state.

In either case, due to the interaction between the continu-ous state and the switching law, for the same initial state, anysmall perturbation may lead the perturbed switching signalto totally deviated from the nominal switching signal. Thisindicates that the distance between the perturbed switchingand the nominal switching should be formulated based onthe following rules:

1) The distance should be counted locally (piece by piece)rather than globally.

2) Any mismanipulation should be counted only once inthe sense that the future effluence should not be takeninto account to the distance.

3) Once a mismanipulation occurs, the next current relaystate should be taken as the nominal initial state forfuture comparison between the switching laws.

In the light of the above analysis, we re-write the nominalswitching law as

(7)

For comparison, we re-write the perturbed switching law as

(8)

where is the actually selected (perturbed) index of theswitching path for the current initial state , and isthe actual (perturbed) implementation of . It can be seenthat a deviation occurs when .

260 2012 24th Chinese Control and Decision Conference (CCDC)

Definition 2: For a state-feedback path-wise switchinglaw , let and be the nominal switching lawand perturbed switching law, respectively.

1) For a natural number and an initial state , the-distance between and at is defined as

(9)

where is the absolute distance between twoswitching paths as defined in [14];

2) For an initial state , the relative distance betweenand at is defined as

(10)

3) The supremal relative distance between and isdefined as

(11)

IV. Robustness Analysis

Suppose that switched linear system (1) is asymptoticallystabilizable, and the state-feedback path-wise switching law

steers the system asymptotically stable.Let us consider the situation that the system undergoes

switching perturbations. A description of the perturbed sys-tem can be given as

where is the perturbed switching law.The objective of this section is to analyze the robustness

of the switched linear system with respect to the switchingperturbations. For this, we need a technical lemma thatestimates the derivation of the state transition matrix in termsof the switching distance.

Lemma 3: For switched linear system (1), suppose thatand are switching paths defined over intervals and

, respectively, and is an arbitrarily given state. Let. Then, we have

(12)

where .Proof: Let be an arbitrarily given positive real number.

By the definition of the distance between two switchingpaths, there is a common parent path over ofand , such that

Denote , and . Suppose forinstance that

and

Other cases can be treated in exactly the same way. Simplecomputation yields

where the relationships , , ,and have been used. In a similarway, we can prove that

Combining the above facts gives

This completes the proof due to the fact andthe arbitrariness of .

The lemma provides an estimate of deviation of the systemstates with different switching paths by means of the distancebetween switching paths.

As a main result, we establish that, a well-designedstate-feedback path-wise switching law is robust against theperturbations.

Theorem 1: Suppose that the switching lawasymptotically stabilizes the nominal system (1). Then, thereexists a positive real number , such that the switchinglaw asymptotically stabilizes any perturbed switched linearsystem

(13)

with .Proof: Let be any given state, be any given positive

real number, and and are the switching signalsgenerated by the nominal switching law and the perturbedswitching law with respect to initial statefor nominal system and the perturbed system, respectively.Recall that means that

, which further means that the -distance between theperturbed switching and the nominal switching is upperbounded by for sufficiently large .

Re-write the perturbed switching signal as in (8):

We are to prove that the state sequence is expo-nentially convergent, which implies the asymptotic stabilityof the perturbed system.

In view of (4), let , and be a

real number in . Let ,, and . For any non-

negative , let . It is clear that

2012 24th Chinese Control and Decision Conference (CCDC) 261

Applying Lemma 3 yields

ifif

(14)

where , and . Now choose, where is a positive real number to

be determined later. Let be a (sufficiently large) naturalnumber such that, for any , the -distance betweenthe perturbed switching and the nominal switching is upperbounded by . For any , define

By the definition of -distance, we have

Based on the above facts, routine calculation gives

(15)

where , , , and the factthat was used.

Finally, let , which is

clearly independent of . Then, it follows from (15) that

which means that the sequence is exponentiallyconvergent with rate . This completes the proof.

V. Illustrative Example

For the continuous-time switched linear system

(16)

with

and

simple computation exhibits that matrices and donot admit any stable convex combination, and hence thesystem is not quadratically stabilizable [3]. However, it was

shown that the system is asymptotically stabilizable, anda stabilizing state-feedback path-wise switching law waspresented, see [10] for details.

0 10 20 30 40 50 60−2

0

2

4

nominal state trajectory

0 10 20 30 40 50 60−4

−2

0

2

4

sta

te

perturbed state trajectory

d=.1,e=.1

0 10 20 30 40 50 60−4

−2

0

2

4

Time (Sec)

perturbed state trajectory

d=.3,e=.3

Fig. 1. State trajectories of the nominal and perturbed systems

0 10 20 30 40 50 600.5

1

1.5

2

2.5

nominal system

0 10 20 30 40 50 600.5

1

1.5

2

2.5

sw

itchin

g s

ignals

perturbed system

d=.1,e=.1

0 10 20 30 40 50 600.5

1

1.5

2

2.5

Time (Sec)

perturbed system

d=.3,e=.3

Fig. 2. Switching signals of the nominal and perturbed systems

To verify the robustness of the switching law with switch-ing perturbations, we assume that the nominal switching lawis perturbed in the following way: i) a delay of Sec occursat each duration of the second subsystem in a sample period.That is, the sampling period for the second subsystem isin stead of ; and ii) the state measurement is inexact in thatthe measured state is for any state

, where , that is, there is anpercent error in measuring the third state variable. Let the

262 2012 24th Chinese Control and Decision Conference (CCDC)

initial state be

Fig. 1 shows the state trajectories for the nominal systemand the perturbed systems with and

, respectively. It is clear that the state trajectoriesare convergent, but with different transient performance andconvergence rates. The larger switching perturbation, thepoorer performance. The corresponding switching signals aredepicted in Fig. 2.

VI. Conclusion

In this work, we have addressed the problem of robust stabi-lizing switching design for continuous-time switched linearsystems. We proved that, any stabilizable switched systemadmits a state-feedback path-wise switching law, which isa robust switching law. Technically, we characterized thedistance between two state-feedback path-wise switchinglaws, which paves the way for addressing robustness withrespect to switching perturbations. The illustrative exampleverified the effectiveness of the proposed design scheme.

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