Stabilization of Continuous-time Switched Linear Systems with Quantized Output Feedback This technical note was partially presented at the 21st international symposium on mathematical theory of networks and systems, July 7-11, 2014, Netherlands.

# Stabilization of Continuous-time Switched Linear Systems with Quantized Output Feedback ††thanks: This technical note was partially presented at the 21st international symposium on mathematical theory of networks and systems, July 7-11, 2014, Netherlands.

Masashi Wakaiki,  and Yutaka Yamamoto,  M. Wakaiki is with the Center for Control, Dynamical-systems and Computation (CCDC), University of California, Santa Barbara, CA 93106-9560, USA (e-mail: masashiwakaiki@ece.ucsb.edu) Y. Yamamoto is with Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan. . M. Wakaiki acknowledges Murata Overseas Scholarship Foundation for the support of this work.
###### Abstract

In this paper, we study the problem of stabilizing continuous-time switched linear systems with quantized output feedback. We assume that the observer and the control gain are given for each mode. Also, the plant mode is known to the controller and the quantizer. Extending the result in the non-switched case, we develop an update rule of the quantizer to achieve asymptotic stability of the closed-loop system under the average dwell-time assumption. To avoid quantizer saturation, we adjust the quantizer at every switching time.

Switched systems, Quantized control, Output feedback stabilization.

## I Introduction

Quantized control problems have been an active research topic in the past two decades. Discrete-level actuators/sensors and digital communication channels are typical in practical control systems, and they yield quantized signals in feedback loops. Quantization errors lead to poor system performance and even loss of stability. Therefore, various control techniques to explicitly take quantization into account have been proposed, as surveyed in [1, 2].

On the other hand, switched system models are widely used as a mathematical framework to represent both continuous and discrete dynamics. For example, such models are applied to DC-DC converters [3] and to car engines [4]. Stability and stabilization of switched systems have also been extensively studied; see, e.g., the survey [5, 6], the book [7], and many references therein.

In view of the practical importance of both research areas and common technical tools to study them, the extension of quantized control to switched systems has recently received increasing attention. There is by now a stream of papers on control with limited information for discrete-time Markovian jump systems [8, 9, 10]. Moreover, our previous work [11] has analyzed the stability of sampled-data switched systems with static quantizers.

In this paper, we study the stabilization of continuous-time switched linear systems with quantized output feedback. Our objective is to solve the following problem: Given a switched system and a controller, design a quantizer to achieve asymptotic stability of the closed-loop system. We assume that the information of the currently active plant mode is available to the controller and the quantizer. Extending the quantizer in [12, 13] for the non-switched case to the switched case, we propose a Lyapunov-based update rule of the quantizer under a slow-switching assumption of average dwell-time type [14].

The difficulty of quantized control for switched systems is that a mode switch changes the state trajectories and saturates the quantizer. In the non-switched case [12, 13], in order to avoid quantizer saturation, the quantizer is updated so that the state trajectories always belong to certain invariant regions defined by level sets of a Lyapunov function. However, for switched systems, these invariant regions are dependent on the modes. Hence the state may not belong to such regions after a switch. To keep the state in the invariant regions, we here adjust the quantizer at every switching time, which prevent quantizer saturation.

The same philosophy of emphasizing the importance of quantizer updates after switching has been proposed in [15] for sampled-data switched systems with quantized state feedback. Subsequently, related works were presented for the output feedback case [16] and for the case with bounded disturbances [17]. The crucial difference lies in the fact that these works use the quantizer based on [18] and investigates propagation of reachable sets for capturing the measurement. This approach also aims to avoid quantizer saturation, but it is fundamentally disparate from our Lyapunov-based approach.

This paper is organized as follows. In Section II, we present the main result, Theorem II.4, after explaining the components of the closed-loop system. Section III gives the update rule of the quantizer and is devoted to the proof of the convergence of the state to the origin. In Section IV, we discuss Lyapunov stability. We present a numerical example in Section V and finally conclude this paper in Section VI.

The present paper is based on the conference paper [19]. Here we extend the conference version by addressing state jumps at switching times. We also made structural improvements in this version.

Notation:  Let and denote the smallest and the largest eigenvalue of . Let denote the transpose of .

The Euclidean norm of is denoted by . The Euclidean induced norm of is defined by .

For a piecewise continuous function , its left-sided limit at is denoted by .

## Ii Quantized output feedback stabilization of switched systems

### Ii-a Switched linear systems

For a finite index set , let be a right-continuous and piecewise constant function. We call a switching signal and the discontinuities of switching times. Let us denote by the number of discontinuities of on the interval . Let be switching times, and consider a switched linear system

 ˙x(t)=Aσ(t)x(t)+Bσ(t)u(t),y(t)=Cσ(t)x(t) (1)

with the jump

 x(tk)=Rσ(tk),σ(t−k)x(t−k) (2)

where is the state, is the control input, and is the output.

Assumptions on the switched system (1) are as follows.

###### Assumption II.1

For every , is stabilizable and is observable. We choose and so that and are Hurwitz.

Furthermore, the switching signal has an average dwell time [14], i.e., there exist and such that

 Nσ(t,s)≤N0+t−sτa(t>s≥0). (3)

We need observability rather than detectability, because we reconstruct the state by using the observability Gramian.

### Ii-B Quantizer

In this paper, we use the following class of quantizers proposed in [13].

Let be a finite subset of . A quantizer is a piecewise constant function . This implies geometrically that is divided into a finite number of the quantization regions . For the quantizer , there exist positive numbers and with such that

 |y|≤M ⇒|q(y)−y|≤Δ (4) |y|>M ⇒|q(y)|>M−Δ. (5)

The former condition (4) gives an upper bound of the quantization error when the quantizer does not saturate. The latter (5) is used for the detection of quantizer saturation.

We place the following assumption on the behavior of the quantizer near the origin. This assumption is used for Lyapunov stability of the closed-loop system.

###### Assumption II.2 ([13, 20])

There exists such that for every with .

We use quantizers with the following adjustable parameter :

 qμ(y)=μq(yμ). (6)

In (6), is regarded as a “zoom” variable, and is the data on transmitted to the controller at time . We need to change to obtain accurate information of . The reader can refer to [13, 20, 7] for further discussions.

###### Remark II.3

The quantized output may chatter on boundaries among quantization regions. Hence if we generate the input by , the solutions of (1) must be interpreted in the sense of Filippov [21]. However, this generalization does not affect our Lyapunov-based analysis as in [12, 13], because we will use a single quadratic Lyapunov function between switching times.

### Ii-C Controller

Similarly to [12, 13], we construct the following dynamic output feedback law based on the standard Luenberger observers:

 ˙ξ(t) =(Aσ(t)+Lσ(t)Cσ(t))ξ(t)+Bσ(t)u(t)−Lσ(t)qμ(t)(y(t)) u(t) =Kσ(t)ξ(t), (7)

where is the state estimate. The estimate also jumps at each switching times :

 ξ(tk)=Rσ(tk),σ(t−k)ξ(t−k).

Then the closed-loop system is given by

 ˙x=Aσx+BσKσξ˙ξ=(Aσ+LσCσ)ξ+BσKσξ−Lσqμ(y). (8)

If we define and by

 z:=[xx−ξ],Fσ:=[Aσ+BσKσ−BσKσ0Aσ+LσCσ], (9)

then we rewrite (8) in the form

 ˙z=Fσz+[0Lσ](qμ(y)−y). (10)

The state of the closed-loop system (8) jumps at each switching time :

 z(tk)=Jσ(tk),σ(t−k)z(t−k),

where

 Jσ(tk),σ(t−k):=[Rσ(tk),σ(t−k)00Rσ(tk),σ(t−k)].

We see from Assumption II.1 that is Hurwitz for each . For every positive-definite matrix , there exist a positive-definite matrix such that

 F⊤pPp+PpFp=−Qp(p∈P). (11)

We define , , , and by

 ¯¯¯λP:=maxp∈Pλmax(Pp),λ––P:=minp∈Pλmin(Pp)λ––Q:=minp∈Pλmin(Qp),Cmax:=maxp∈P∥Cp∥. (12)

Fig. 1 shows the closed-loop system we consider.

### Ii-D Main result

By adjusting the “zoom” parameter , we can achieve global asymptotic stability of the closed-loop system (10). This result is a natural extension of Theorem 5 in [13] to switched systems.

###### Theorem II.4

Define by

 Θ:=2maxp∈P∥Pp^Lp∥λ––Q,~{}~{}where~{}~{}^Lp:=[0Lp]. (13)

and let be large enough to satisfy

 M>max⎧⎪⎨⎪⎩2Δ, ⎷¯¯¯λPλ––PΘΔCmax⎫⎪⎬⎪⎭. (14)

If the average dwell time in (3) is larger than a certain value, then there exists a right-continuous and piecewise-constant function such that the closed-loop system (10) has the following two properties for every and every :

(i) Convergence to the origin:  .

(ii) Lyapunov stability:  To every , there corresponds such that

 |x(0)|<δ⇒|z(t)|<ε   (t≥0).

We shall prove convergence to the origin and Lyapunov stability in Sections III and IV, respectively. We also present an update rule of the “zoom” parameter in Section 3. The sufficient condition on is given by (38) in Theorem III.6 below.

## Iii The proof of convergence to the origin

Define and by

 Γ:=maxp∈P∥Ap∥,Λ:=max{1, maxp,q∈P,p≠q∥Rp,q∥}.

We split the proof into two stages: the “zooming-out” and “zooming-in” stages.

### Iii-a Capturing the state of the closed-loop system by “zooming out”

Since the initial state is unknown to the quantizer, we have to capture the state of the closed-loop system by “zooming out”, i.e., increasing the “zoom” parameter . We first see that can be captured if we have a time-interval with a given length that has no switches.

###### Theorem III.1

Consider the closed-loop system (10). Set the control input . Choose , and define and the observability Gramian

 Wp(τ):=∫τ0eA⊤ptC⊤pCpeAptdt.

Assume that there exists such that we can observe

 |qμ(t)(y(t))|≤Mμ(t)−Δμ(t) (15) σ(t)=σ(s0)=:p (16)

for all . Let the “zoom” parameter be piecewise continuous and monotone increasing in . If we set the state estimate at by

 ξ(s0+τ):=eApτ(Wp(τ)−1∫τ0eA⊤ptC⊤pqμ(s0+t)(y(s0+t))dt) (17)

and if we choose so that

 μ(s0+τ)≥ ⎷¯¯¯λp–λpCmaxM(|ξ(s0+τ)|+2∥Wp(τ)−1∥τΥp(τ)∥∥eApτ∥∥Δμ((s0+τ)−)), (18)

then .

Proof: Since no switch occurs by (16), we can easily obtain this result by extending Theorem 5 in [13] for the non-switched case. We therefore omit the proof; see also the conference version [19].

It follows from Theorem III.1 that in order to capture the state , it is enough to show the existence of satisfying (15) and (16) for all . To this end, we use the following lemma on average dwell time :

###### Lemma III.2

Fix an initial time . Suppose that satisfies the average dwell-time assumption (3). Let . If we choose so that

 N>τaτa−τ(N0−ττa), (19)

then there exists such that .

Proof: Let us denote the switching times by , and fix . Suppose that

 Nσ(t0+υ+τ,t0+υ)>0 (20)

for all . Then we have

 tk−tk−1≤τ(k=1,…,N). (21)

Indeed, if for some and if we let be the smallest such integer, then we obtain

 t¯k−1−t0≤(¯k−1)τ≤(N−1)τ

and . This contradicts (20) with . Thus we have (21).

From (21), we see that for ,

 tN−(t1−ϵ)=N∑k=2(tk−tk−1)+ϵ≤(N−1)τ+ϵ

It follows from (3) that

 N=Nσ(tN,t1−ϵ) ≤N0+(N−1)τ+ϵτa.

Therefore satisfies the following inequality:

 N≤τaτa−τ(N0−τ−ϵτa). (22)

Since was arbitrary, (22) is equivalent to

 N≤τaτa−τ(N0−ττa). (23)

Thus we have shown that if (20) holds for all , then satisfies (23). The contraposition of this statement gives a desired result.

###### Theorem III.3

Consider the closed-loop system (10) with average dwell-time property (3). Set the control input . Fix , , and . Increase in the following way: for ,

 μ(t)=ΛN0⋅(Λ1/τaeΓ)(1+χ)k¯τ (24)

for and . Then there exists such that (15) and (16) hold for all .

Proof: If switches occur in the interval , then we have

 |x(t)|≤(n∏k=1Λ)⋅eΓt⋅|x(0)|.

Since , it follows from (3) that

 |x(t)|≤Λ(N0+tτa)⋅eΓt⋅|x(0)|. (25)

Clearly, this inequality holds in the case when no switches occur. Since (14) shows that and since the growth rate of is larger than that of , there exists such that

 |y(t)|≤Mμ(t)−2Δμ(t)(t≥s′0). (26)

In conjunction with (4), this implies that (15) holds for every . Let be an integer satisfying (19). Then Lemma III.2 guarantees the existence of such that (16) holds for every . This completes the proof.

It follows from Theorems III.1 and III.3 that if we update the “zoom” parameter as in (24) and if we set the state estimate by (17), then the state of the closed-loop system can be captured.

###### Remark III.4

If the initial state is sufficiently small, then in (26) is zero. In this situation, we can capture by for all switching signal with average dwell-time property (3). We use this fact for the proof of Lyapunov stability; see Section 4.

### Iii-B Measuring the output by “zooming in”

Next we drive the state of the closed-loop system to the origin by “zooming-in”, i.e., decreasing the “zoom” parameter . Since increases at each switching time during this stage, the term “zooming-in stage” may be misleading. However, decreases overall under a certain average dwell-time assumption (3), so we use the term “zooming-in” as in [12, 13].

Let us first consider a fixed “zoom” parameter . The following lemma shows that if no switches occur, then the state trajectories move from a large level set to a small level set of the Lyapunov function in a finite time that is independent of the mode :

###### Lemma III.5

Define and as in (9) and (13), respectively. Fix , and consider the non-switched system

 ˙z=Fpz+^Lp(qμ(y)−y). (27)

Choose . If satisfies

 √λ––PM>√¯¯¯λPΘΔ(1+κ)Cmax, (28)

where , , and are defined by (12) and (13), then the following two level sets of the Lyapunov function are invariant regions for every trajectory of (27):

 R1(μ,p) :={z∈Rn:Vp(z)≤λ––PM2μ2C2max} (29) R2(μ,p) :={z∈Rn:Vp(z)≤¯¯¯λP(ΘΔ(1+κ))2μ2}. (30)

Furthermore, if for all , then

 Vp(z(t2))≤Vp(z(t1))−(t2−t1)λ––Qκ(1+κ)(ΘΔμ)2 (31)

for every . Hence if satisfies

 T>λ––PM2−¯¯¯λP(ΘΔ(1+κ)Cmax)2λ––Qκ(1+κ)(ΘΔCmax)2, (32)

then every trajectory of (27) with an initial state satisfies ï¼

Proof: Since the mode is fixed, this lemma is a trivial extension of Lemma 5 in [13] for single-modal systems. We therefore omit its proof; see also the conference version [19].

Using Lemma III.5, we obtain an update rule of the “zoom” parameter to drive the state to the origin.

###### Theorem III.6

Consider the system (27) under the same assumptions as in Lemma III.5. Assume that . For each with , the positive definite matrices and in the Lyapunov equation (11) satisfy

 z⊤J⊤p2,p1Pp2Jp2,p1z≤cp2,p1⋅z⊤Pp1z(z∈R2n) (33)

for some . Define and by

 c:=max{1, maxp1,p2∈P,p1≠p2cp2,p1} (34)
 Ω:= ⎷¯¯¯λPλ––PΘΔ(1+κ)CmaxM<1. (35)

Fix so that (32) is satisfied, and set the “zoom” parameter for all and in the following way: If no switches occur in the interval , then

 μ(t0+kT+t)={μ(t0+kT)(0

otherwise,

 μ(t0+kT+t)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩μ(t0+kT)(0

where are the switching times in the interval . Then for all . Furthermore, if satisfies

 τa>log(c)2log(1/Ω)T, (38)

then .

Proof: To prove that for all , it is enough to show that if , then

 z(t)∈R1(μ(t),σ(t))(t0≤t≤t0+T) (39)

Let us first investigate the case without switching on the interval . We see from Lemma III.5 that for all and that . Since , a routine calculation shows that .

We now study the switched case. Let be the switching times in the interval . Let us define for simplicity of notation. Lemma III.5 implies that () are invariant sets for all , . Moreover, by (33), if , then () for all . Hence leads to

 z(t)∈R1(μ(t),σ(t))(t0≤t

To obtain

 z(tn+1)∈R1(μ(tn+1),σ(tn+1)), (41)

we show that . Assume, to reach a contradiction, that

 z(t−n+1)∉R2(μ(t−n+1),σ(t−n+1)). (42)

Since is an invariant region for all , we also have

 z(t)∉R2(μ(t),σ(t))(t0≤t

Define a Lyapunov function for each . Since a Filippov solution is (absolutely) continuous, exists for each . From (42), we obtain

 limt↗tn+1Vσ(t)(z(t))≥¯¯¯λP(ΘΔ(1+κ))2μ(tn)2. (43)

On the other hand, since for all , (31) gives

 limt↗t1Vσ(t)(z(t))≤(λ––PM2C2max−(t1−t0)λ––Qκ(1+κ)(ΘΔ)2)μ(t0)2,

and hence we have from that

 Vσ(t1)(z(t1)) =z(t−1)⊤J⊤σ(t1),σ(t−1)Pσ(t1)Jσ(t1),σ(t−1)z(t−1) ≤cσ(t1),σ(t0)⋅(limt↗t1Vσ(t)(z(t))) =(λ––PM2C2max−(t1−t0)λ––Qκ(1+κ)(ΘΔ)2)μ(t1)2.

If we repeat this process and use (32), then

 limt↗tn+1Vσ(t)(z(t)) ≤(λ––PM2C2max−Tλ––Qκ(1+κ)(ΘΔ)2)μ(tn)2 <¯¯¯λP(ΘΔ(1+κ))2μ(tn)2, (44)

which contradicts (43). Thus we obtain

 z(t−n+1)∈R2(μ(t−n+1),σ(t−n+1)),

and hence (41) holds.

From (40) and (41), we derive the desired result (39), because .

Finally, since , (3) gives

 μ(t0+mT+t)≤Ωm√cNσ(t0+mT+t,t0)μ(t0)≤√cN0+T/τa⋅(Ω√cT/τa)mμ(t0) (45)

for every and . If , that is, if the average dwell time satisfies (38), then . Since for all , we obtain .

###### Remark III.7

(a)  We can compute by linear matrix inequalities. Moreover, if the jump matrix in (2) is invertible, then Lemma 13 of [22] gives an explicit formula for .

(b)  The proposed method is sensitive to the time-delay of the switching signal at the “zooming-in” stage. If the switching signal is delayed, a mode mismatch occurs between the plant and the controller. Here we do not proceed along this line to avoid technical issues. See also [23] for the stabilization of asynchronous switched systems with time-delays.

(c)  We have updated the “zoom” parameter at each switching time in the “zooming-in” stage. If we would not, switching could lead to instability of the closed-loop system. In fact, since the state may not belong to the invariant region without adjusting , the quantizer may saturate.

(d)  Similarly, “pre-emptively” multiplying at time by does not work, either. This is because such an adjustment does not make invariant for the state trajectories. For example, consider the situation where the state belongs to at due to this pre-emptively adjustment. Then does not converge to the origin. Let be a switching time. Since may not be a subset of , it follows that does not belong to the invariant region at in general.

## Iv The proof of Lyapunov stability

Let us denote by the open ball with center at the origin and radius in . In what follows, we use the same letters as in the previous section and assume that the average dwell time satisfies (38).

The proof consists of three steps:

1. Obtain an upper bound of the time at which the quantization process transitions from the “zoom-out” stage to the “zoom-in” stage.

2. Show that there exists a time such that the state satisfies for all .

3. Set so that if , then for all .

We break the proof of Lyapunov stability into the above three steps.

1) Let satisfy (20) and let be small enough to satisfy

 Cmax⋅ΛN0(Λ1/τaeΓ)Nτδ<Δ0. (46)

We see from the state bound (25) that for from Assumption II.2. As we mentioned in Remark III.4 briefly, Lemma III.2 implies that the time , at which the stage changes from “zooming-out” to “zooming-in”, satisfies for every switching signal with the average dwell-time assumption (3).

2) Fix . By (17), , and hence we see from (18) that achieving can be chosen so that

 α≤μ(t0)≤¯μ, (47)

where is defined by

 ¯μ:=max{α,  2 ⎷¯¯¯λPλ––PΔτCmaxΛN0(Λ1/τaeΓ)(1+χ)NτM⋅maxp∈P(∥Wp(τ)−1∥Υp(τ)∥∥eApτ∥∥)}.

Note that is independent of switching signals.

Let be the smallest integer satisfying

 ¯m>log(¯μM√cN0+T/τa/(εCmax))log(1/(Ω√cT/τa)). (48)

Define . Since and , (36) and (37) give

 μ(tε+kT+t) =μ(t0+(¯m+k)T+t)) ≤√cN0+T/τa⋅(Ω√cT/τa)¯m+kμ(t0) ≤√cN0+T/τa⋅(Ω√cT/τ