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.
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.
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  and to car engines . Stability and stabilization of switched systems have also been extensively studied; see, e.g., the survey [5, 6], the book , 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  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 .
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  for sampled-data switched systems with quantized state feedback. Subsequently, related works were presented for the output feedback case  and for the case with bounded disturbances . The crucial difference lies in the fact that these works use the quantizer based on  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 . 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
with the jump
where is the state, is the control input, and is the output.
Assumptions on the switched system (1) are as follows.
For every , is stabilizable and is observable. We choose and so that and are Hurwitz.
Furthermore, the switching signal has an average dwell time , i.e., there exist and such that
We need observability rather than detectability, because we reconstruct the state by using the observability Gramian.
In this paper, we use the following class of quantizers proposed in .
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
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.
We use quantizers with the following adjustable parameter :
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.
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 . 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.
where is the state estimate. The estimate also jumps at each switching times :
Then the closed-loop system is given by
If we define and by
then we rewrite (8) in the form
The state of the closed-loop system (8) jumps at each switching time :
We see from Assumption II.1 that is Hurwitz for each . For every positive-definite matrix , there exist a positive-definite matrix such that
We define , , , and by
Fig. 1 shows the closed-loop system we consider.
Ii-D Main result
and let be large enough to satisfy
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
Iii The proof of convergence to the origin
Define and by
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.
Consider the closed-loop system (10). Set the control input . Choose , and define and the observability Gramian
Assume that there exists such that we can observe
for all . Let the “zoom” parameter be piecewise continuous and monotone increasing in . If we set the state estimate at by
and if we choose so that
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 :
Fix an initial time . Suppose that satisfies the average dwell-time assumption (3). Let . If we choose so that
then there exists such that .
Proof: Let us denote the switching times by , and fix . Suppose that
for all . Then we have
Indeed, if for some and if we let be the smallest such integer, then we obtain
From (21), we see that for ,
It follows from (3) that
Therefore satisfies the following inequality:
Since was arbitrary, (22) is equivalent to
Proof: If switches occur in the interval , then we have
Since , it follows from (3) that
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
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.
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 :
Choose . If satisfies
Furthermore, if for all , then
for every . Hence if satisfies
then every trajectory of (27) with an initial state satisfies ï¼
Using Lemma III.5, we obtain an update rule of the “zoom” parameter to drive the state to the origin.
for some . Define and by
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
where are the switching times in the interval . Then for all . Furthermore, if satisfies
Proof: To prove that for all , it is enough to show that if , then
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
we show that . Assume, to reach a contradiction, that
Since is an invariant region for all , we also have
Define a Lyapunov function for each . Since a Filippov solution is (absolutely) continuous, exists for each . From (42), we obtain
On the other hand, since for all , (31) gives
and hence we have from that
If we repeat this process and use (32), then
which contradicts (43). Thus we obtain
and hence (41) holds.
Finally, since , (3) gives
for every and . If , that is, if the average dwell time satisfies (38), then . Since for all , we obtain .
(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  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:
Obtain an upper bound of the time at which the quantization process transitions from the “zoom-out” stage to the “zoom-in” stage.
Show that there exists a time such that the state satisfies for all .
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
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).
where is defined by
Note that is independent of switching signals.