Stabilization of Continuoustime 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 711, 2014, Netherlands.
Abstract
In this paper, we study the problem of stabilizing continuoustime 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 nonswitched case, we develop an update rule of the quantizer to achieve asymptotic stability of the closedloop system under the average dwelltime assumption. To avoid quantizer saturation, we adjust the quantizer at every switching time.
I Introduction
Quantized control problems have been an active research topic in the past two decades. Discretelevel 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 DCDC 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 discretetime Markovian jump systems [8, 9, 10]. Moreover, our previous work [11] has analyzed the stability of sampleddata switched systems with static quantizers.
In this paper, we study the stabilization of continuoustime 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 closedloop 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 nonswitched case to the switched case, we propose a Lyapunovbased update rule of the quantizer under a slowswitching assumption of average dwelltime 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 nonswitched 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 sampleddata 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 Lyapunovbased approach.
This paper is organized as follows. In Section II, we present the main result, Theorem II.4, after explaining the components of the closedloop 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 leftsided limit at is denoted by .
Ii Quantized output feedback stabilization of switched systems
Iia Switched linear systems
For a finite index set , let be a rightcontinuous 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
(1) 
with the jump
(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
(3) 
We need observability rather than detectability, because we reconstruct the state by using the observability Gramian.
IiB 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
(4)  
(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 closedloop system.
We use quantizers with the following adjustable parameter :
(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 Lyapunovbased analysis as in [12, 13], because we will use a single quadratic Lyapunov function between switching times.
IiC Controller
Similarly to [12, 13], we construct the following dynamic output feedback law based on the standard Luenberger observers:
(7) 
where is the state estimate. The estimate also jumps at each switching times :
Then the closedloop system is given by
(8) 
If we define and by
(9) 
then we rewrite (8) in the form
(10) 
The state of the closedloop system (8) jumps at each switching time :
where
We see from Assumption II.1 that is Hurwitz for each . For every positivedefinite matrix , there exist a positivedefinite matrix such that
(11) 
We define , , , and by
(12) 
Fig. 1 shows the closedloop system we consider.
IiD Main result
By adjusting the “zoom” parameter , we can achieve global asymptotic stability of the closedloop system (10). This result is a natural extension of Theorem 5 in [13] to switched systems.
Theorem II.4
Define by
(13) 
and let be large enough to satisfy
(14) 
If the average dwell time in (3) is larger than a certain value, then there exists a rightcontinuous and piecewiseconstant function such that the closedloop 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 “zoomingout” and “zoomingin” stages.
Iiia Capturing the state of the closedloop system by “zooming out”
Since the initial state is unknown to the quantizer, we have to capture the state of the closedloop system by “zooming out”, i.e., increasing the “zoom” parameter . We first see that can be captured if we have a timeinterval with a given length that has no switches.
Theorem III.1
Consider the closedloop system (10). Set the control input . Choose , and define and the observability Gramian
Assume that there exists such that we can observe
(15)  
(16) 
for all . Let the “zoom” parameter be piecewise continuous and monotone increasing in . If we set the state estimate at by
(17) 
and if we choose so that
(18) 
then .
Proof: Since no switch occurs by (16), we can easily obtain this result by extending Theorem 5 in [13] for the nonswitched 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 dwelltime assumption (3). Let . If we choose so that
(19) 
then there exists such that .
Proof: Let us denote the switching times by , and fix . Suppose that
(20) 
for all . Then we have
(21) 
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:
(22) 
Since was arbitrary, (22) is equivalent to
(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
Proof: If switches occur in the interval , then we have
Since , it follows from (3) that
(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
(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.
IiiB Measuring the output by “zooming in”
Next we drive the state of the closedloop system to the origin by “zoomingin”, i.e., decreasing the “zoom” parameter . Since increases at each switching time during this stage, the term “zoomingin stage” may be misleading. However, decreases overall under a certain average dwelltime assumption (3), so we use the term “zoomingin” 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 nonswitched system
(27) 
Choose . If satisfies
(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):
(29)  
(30) 
Furthermore, if for all , then
(31) 
for every . Hence if satisfies
(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 singlemodal 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
(33) 
for some . Define and by
(34) 
(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
(36) 
otherwise,
(37) 
where are the switching times in the interval . Then for all . Furthermore, if satisfies
(38) 
then .
Proof: To prove that for all , it is enough to show that if , then
(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
(40) 
To obtain
(41) 
we show that . Assume, to reach a contradiction, that
(42) 
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
(43) 
If we repeat this process and use (32), then
(44) 
which contradicts (43). Thus we obtain
and hence (41) holds.
Finally, since , (3) gives
(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 timedelay of the switching signal at the “zoomingin” 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 timedelays.
(c) We have updated the “zoom” parameter at each switching time in the “zoomingin” stage. If we would not, switching could lead to instability of the closedloop system. In fact, since the state may not belong to the invariant region without adjusting , the quantizer may saturate.
(d) Similarly, “preemptively” 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 preemptively 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 “zoomout” stage to the “zoomin” 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
(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 “zoomingout” to “zoomingin”, satisfies for every switching signal with the average dwelltime assumption (3).