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A new approach to quantized stabilization of a stochastic system with multiplicative noise
Advances in Difference Equations volume 2013, Article number: 20 (2013)
Abstract
A new quantizationdependent Lyapunov function is proposed to analyze the quantized feedback stabilization problem of systems with multiplicative noise. For convenience of the proof, only a singleinput case is considered (which can be generalized to a multiinput channel). Conditions for the systems to be quantized meansquare polyquadratically stabilized are derived, and the analysis of ${H}_{\mathrm{\infty}}$ performance and controller design is conducted for a given logarithmic quantizer. The most significant feature is the utilization of a quantizationdependent Lyapunov function, leading to less conservative results, which is shown both theoretically and through numerical examples.
1 Introduction
Rapid advancement of digital networks has witnessed a growing interest in investigating efforts of signal quantization on feedback control systems. The emerging networkbased control system where information exchange between the controller and the plant is through a digital channel with limited capacities has further strengthened the importance of the study on quantized feedback control. Different from the classical control theory where data transmission is assumed to have an infinite precision, transmission subject to quantization or limited data capacity in digital networks, the tools in classical control theory may be invalid, so new tools need to be developed for the analysis and design of quantized feedback systems.
The study of quantized feedback control can be traced back to [1]. Most of the early research focuses on the understanding and mitigation of the quantization effects, while the quantization error is considered to impair the performance [2]. In modern control theory where the quantizer is always considered as an information encoder and decoder, one main problem is how much information has to be transmitted in order to make the system achieve a certain objective for the closedloop system. For a discretetime system with a singleinput channel, when the static quantizer is considered, [3] shows the minimum data rate for the system to be stabilized is proved to be characterized by the unstable roots of the system matrix, and the coarsest quantizer is logarithmic. [4] considers the case when the input channel subject to Bernoulli packets dropouts, the minimum data rate is related not only to the unstable roots of the system matrix, but also with the packets dropout probability. As for a discretetime system with single input subject to multiplicative noises in [5], the coarsest static quantizer for the system to be quadratically meansquare stabilized is proved to be logarithmic with infinite levels, and the quantization density can be approximated by solving a Riccati equation; comprehensive study on feedback control systems with logarithmic quantizers is not given. A sector bound approach is proposed in [6] to characterize the quantization error caused by a logarithmic quantizer, by which many quantized problem can be solved by the robust tools. The results are also extended to adaptive control in [7, 8] and the LQRtype problem in [6]. Based on the characterization of the quantized error, [9] gives less conservative conditions of the quantization density to achieve stability by studying the properties of the logarithmic quantizer further; [10] use a method based on Tsypkintype Lyapunov functions to study the absolute stability analysis of quantized feedback control of a discretetime linear system, less conservative conditions than those in the quadratic framework are derived. [11] showed that a finitelevel logarithmic quantizer suffices to approach the wellknown minimum average data rate for stabilizing an unstable linear discretetime system under two basic network configurations, and explicit finitelevel logarithmic quantizers and the corresponding controllers to approach the minimum average data rate are derived. For networked systems, [12] gives the quantized outputfeedback controller for the control with data packets dropout.
In this paper, a new approach to the analysis and synthesis of quantized feedback control for stochastic systems with multiplicative noise is proposed. Using logarithmic quantized statefeedback control, results for meansquare stabilization and ${H}_{\mathrm{\infty}}$ performance analysis as well as the controller synthesis are given. Less conservative results are derived by the utilization of a quantizationdependent Lyapunov function, which is shown both theoretically and through a numerical example.
Notations: $P>0$ ($P\ge 0$) means P is a symmetric positive (semipositive) matrix. ${P}^{T}$ stands for the transposition of matrix P. The space of a square summable infinite sequence is denoted by ${l}_{2}[0,\mathrm{\infty})$, and for $w=\{w(t)\}\in {l}_{2}[0,\mathrm{\infty})$, its norm is given by ${\parallel w\parallel}_{2}=\sqrt{{\sum}_{0}^{\mathrm{\infty}}{w(t)}^{2}}$.
2 Stability and stabilization
2.1 Problem formulation
Consider the following linear discretetime systems with multiplicative noise:
where $x(t)\in {\mathcal{R}}^{n}$ is the system state vector with known initial state ${x}_{0}$; $u(t)\in {\mathcal{R}}^{m}$ is the control input; $\xi (t)\in \mathcal{R}$ is the process noise with $E\xi (t)=0$, $E\xi (t)\xi (j)={\sigma}^{2}{\delta}_{tj}$, and is uncorrelated with initial state ${x}_{0}$. As proved in [5], the coarsest static quantizer for the system (1) to be quadratically meansquare stabilized via quantized statefeedback is proved to be logarithmic. Suppose u is a scalar that has to be quantized, the logarithmic quantizer is in the following form:
with quantization levels as
where ρ is the quantized density of the logarithmic quantizer q, which can be computed using the approach given in [5], with
For the multiinput case with different quantizers, the statefeedback control without quantization is in the form of
which has to be transmitted through a digital network subject to logarithmic quantizers as given in (2), and denote the quantized control as
where ${q}_{i}$, $i=1,\dots ,m$ are quantizers with different quantization density.
Without loss of generality, in this paper only a singleinput case with $m=1$ is considered for simplicity, which can be generalized to a multiinput case. For a quantizer as given in the form of (2), as illustrated in [6], using the sector bound approach, the quantization error $e(t)$ can be characterized as
where $\mathrm{\Delta}(t)\in [\delta ,\delta ]$ with δ given by (4), so the closedloop system with quantized feedback is given by
We mainly focus on the derivation of less conservative sufficient conditions for the system to achieve certain performance. To make the paper selfcontained, the definitions for the system (8) to be meansquare stable and meansquare polyquadratical stable are introduced.
Definition 1 The closed system (8) is called meansquare stable with quantized feedback control in the form of (6) if there exists a control Lyapunov function ${V}_{P}(x)={x}^{T}(t)Px(t)$ satisfying
for all $x(t)\ne 0$ and the given quantization.
Definition 2 The closed system (8) is called meansquare polyquadratically stable with quantized control in the form of (6) if there exists a Lyapunov function
where ${Q}_{1}$ and ${Q}_{2}$ are symmetric positive matrices with proper dimensions satisfying
for all $x(t)\ne 0$ and the given quantization.
Remark 1 When setting ${Q}_{1}={Q}_{2}=P$, the control Lyapunov function proposed in Definition 2 reduces to the one given in Definition 1. We will show that the control Lyapunov function (10) can lead to less conservative conditions for the system (8) to be meansquare polyquadratical stabilized than those deduced by the control Lyapunov function (10).
Problem formulation For the control Lyapunov function (10), deduce the conditions for the system (1) to be meansquare polyquadratical stabilized via quantized feedback control in the form of (6).
2.2 Stability analysis
In this part, we give the conditions for the system (8) to achieve quantized meansquare polyquadratical stability. First, a necessary and sufficient condition is deduced.
Theorem 1 For the discretetime stochastic system (1) and the quantized state feedback control law in the form of (6), given a logarithmic quantizer as in (2), the closedloop system (8) is meansquare polyquadratically stable if and only if there exist matrices ${Q}_{1}>0$, ${Q}_{2}>0$, ${V}_{1}$ and ${V}_{2}$ satisfying
Proof According to Definition 2, for $EV(x(t))$ defined as in (10), the closedloop system is meansquare polyquadratically stable if
for all the $x(t)\ne 0$ and $\mathrm{\Delta}(t)\in [\delta ,\delta ]$. Plugging $EV(x(t))$ into (14), and by considering (6), we have
which is equivalent to
In the next part, we will show that the expressions in (10) and (16) hold if and only if (12) and (13) hold.

(16)
⇒ (12) and (13): By the Schur complement, (16) is equivalent to
$$\left[\begin{array}{ccc}Q(t)& {[A+(1+\mathrm{\Delta}(t))BK]}^{T}Q(t+1)& \sigma {[{A}_{0}+(1+\mathrm{\Delta}(t)){B}_{0}K]}^{T}Q(t+1)\\ \ast & Q(t+1)& 0\\ \ast & 0& Q(t+1)\end{array}\right]<0.$$(17)
Consider the following four cases:
For cases (a) and (b), from (17) we have
For cases (c) and (d), from (17) we have
By selecting ${V}_{i}={V}_{i}^{T}={Q}_{i}$, we can obtain (12) and (13). Therefore, it can be concluded that if (16) holds, there must exist matrices ${Q}_{1}>0$, ${Q}_{2}>0$, ${V}_{1}$ and ${V}_{2}$ satisfying (12) and (13).

(12)
and (13) ⇒ (16): Suppose there exist matrices ${Q}_{1}>0$, ${Q}_{2}>0$, ${V}_{1}$ and ${V}_{2}$ satisfying (12) and (13). First, as ${Q}_{i}>0$, we have ${({V}_{i}{Q}_{i})}^{T}{Q}_{i}^{1}({V}_{i}{Q}_{i})\ge 0$, which implies
$${V}_{i}^{T}{Q}_{i}^{1}{V}_{i}\le {Q}_{i}{V}_{i}^{T}{V}_{i}.$$(21)
From (12) and (21) we have
By multiplying $diag\{I,{Q}_{i}{V}_{i}^{1},{Q}_{i}{V}_{i}^{1}\}$ and $diag\{I,{V}_{i}^{1}{Q}_{i},{V}_{i}^{1}{Q}_{i}\}$ to the left and righthand side of (22) and (23), respectively, we get
$\text{(24)}\times \frac{\delta \mathrm{\Delta}(t+1)}{2\delta}$ and $\text{(25)}\times \frac{\delta +\mathrm{\Delta}(t+1)}{2\delta}$ we get
$\text{(26)}\times \frac{\delta \mathrm{\Delta}(t+1)}{2\delta}$ and $\text{(27)}\times \frac{\delta +\mathrm{\Delta}(t+1)}{2\delta}$ we get
$\text{(28)}\times \frac{\delta \mathrm{\Delta}(t)}{2\delta}$ and $\text{(29)}\times \frac{\delta +\mathrm{\Delta}(t)}{2\delta}$ we can deduce that
The proof is completed. □
2.3 Controller design
In the above section, the controller is assumed to be known for the stability analysis. In practical situations, however, the controller has to be designed to guarantee the closedloop system to achieve stability. The following theorem provides a controller design method based on Theorem 1.
Theorem 2 Consider the system (1) and the state feedback control law in (5). Given a logarithmic quantizer as in (2), the closedloop system (6) is meansquare polyquadratically stabilized if there exist matrices ${\overline{Q}}_{1}>0$, ${\overline{Q}}_{2}>0$, V and K satisfying
In this situation, the controller can be designed as
Proof Suppose that there exist matrices ${\overline{Q}}_{1}>0$ and ${\overline{Q}}_{2}>0$, V and $\overline{K}$ satisfying (31) and (32). From the $(2,2)$ block, we know that ${\overline{Q}}_{i}V{V}^{T}<0$, which means $V+{V}^{T}>{\overline{Q}}_{i}>0$, so V is nonsingular. Performing $diag\{{V}^{T},{V}^{T},{V}^{T}\}$ and $diag\{{V}^{1},{V}^{1},{V}^{1}\}$ to (31) and (32), respectively, yields
By defining the following matrix variables: ${Q}_{i}={V}^{T}{\overline{Q}}_{i}{V}^{1}$, ${V}_{i}={V}^{1}$, $K=\overline{K}{V}^{1}$, if there exist matrices ${Q}_{1}>0$, ${Q}_{2}>0$, ${V}_{1}$ and ${V}_{2}$ satisfying (12) and (13), and using the controller gain given in (33), the system (8) can achieve meansquare polyquadratically stability. □
Theorem 2 is based on Theorem 1 by setting ${V}_{1}={V}_{2}=V$, which increases the conservativeness; the following theorem gives a less conservative condition.
Theorem 3 Consider the system in (1) and the state feedback control law in (5). Given a logarithmic quantizer as in (2), the closedloop system in (8) is meansquare polyquadratically stable if there exist matrices ${Q}_{i}>0$, ${X}_{i}>0$, ${\overline{V}}_{i}$ and K satisfying
Proof First, from the $(2,2)$ block of (36), we can know that ${\overline{V}}_{i}$ is nonsingular. By multiplying $diag\{I,{\overline{V}}_{i}^{T},I,{\overline{V}}_{i}^{T},I\}$ and $diag\{I,{\overline{V}}_{i}^{1},I,{\overline{V}}_{i}^{1},I\}$ to the left and righthand side of (36) and (37), with the Schur complement and the constraint (38), and defining ${\overline{V}}_{i}^{1}={V}_{i}$, we can get the theorem. □
Remark 2 It is worth noting that when ${\delta}_{\mathrm{max}}$ is known, the conditions in Theorem 1 are linear matrix inequalities over the matrix variables ${Q}_{1}>0$, ${Q}_{2}>0$, ${V}_{1}$ and ${V}_{2}$. When Theorem 1 is used to compute the coarsest quantization density ${\delta}_{\mathrm{max}}$ such that the closedloop quantized system is meansquare polyquadratically stable, that is, (12) and (13) are bilinear matrix inequalities. In this case, a line search (such as the bisection method) has to be performed to the variables δ in (12) and (13), and find ${\delta}_{\mathrm{max}}$ iteratively, which can be referred to [13–16].
2.4 Illustrative example
In this part, an example is given to show that the new proposed Lyapunov function can lead to less conservative conditions of the quantization density for the system to achieve stability.
Example 1 For the stochastic discretetime system (1), consider the scalar case of the following form:
It can be proved that the system without control part is unstable in the meansquare sense. Suppose that the statefeedback in (5) is given by $K=[0.8\phantom{\rule{0.25em}{0ex}}0.5\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}1]$, and the quantizer we use is logarithmic in the form of (2). We want to determine the maximum sector bound ${\delta}_{\mathrm{max}}$ below which the stochastic system with quantized state feedback is meansquare asymptotically stable. Table 1 gives the maximum bound of ${\delta}_{\mathrm{max}}$ using the Lyapunov function related to the quantization density proposed in this paper and the general control Lyapunov function.
3 Extension to ${H}_{\mathrm{\infty}}$ performance analysis
For the system
where the state $x(t)$, the input $u(t)$ and the system noise $\xi (t)$ are defined as those of the system (1), $z(t)\in {R}^{n}$ is the control output. A, ${A}_{0}$, B, ${B}_{0}$, C, D, G, F are system matrices with proper dimensions. Suppose the quantizer is given to be logarithmic in the form of (24) and the quantization density is known, so the closedloop system with the quantized state feedback control is given as follows:
where $\mathrm{\Delta}(t)\in [\delta ,\delta ]$. Defining $W=\{w(t)\}\in {l}_{2}[0,\mathrm{\infty})$, the objective of this part is to derive the conditions for the system (42) and (43) to be meansquare asymptotically stable with an ${H}_{\mathrm{\infty}}$ disturbance attention level γ, that is, ${\parallel z(t)\parallel}_{2}<\gamma {\parallel w(t)\parallel}_{2}$ for all the nonzero $w(t)\in {l}_{2}[0,\mathrm{\infty})$ and for all the $\mathrm{\Delta}(t)\in [\delta ,\delta ]$ under zero conditions.
Theorem 4 For the system (40) and (41), considering the control law as given in (5), given a logarithmic quantizer as in (2), the closedloop system in (42) and (43) is meansquare stable with an ${H}_{\mathrm{\infty}}$ disturbance attention level γ if there exist matrices ${Q}_{1}={Q}_{1}^{T}>0$, ${Q}_{2}={Q}_{2}^{T}>0$, ${V}_{1}$ and ${V}_{2}$ satisfying
Proof The theorem is proven based on the Lyapunov function defined in (10). First, (44) and (45) imply (12) and (13), which guarantees the closedloop system in (42) and (43) to be meansquare stable by Theorem 1. To prove the ${H}_{\mathrm{\infty}}$ performance, assume zero initial conditions and consider the following index:
where
Then, along the solutions of (42) and (43), we have
with $\eta (t)=\left[\begin{array}{c}x(t)\\ w(t)\end{array}\right]$, $\mathrm{\Pi}=\left(\begin{array}{cc}{\mathrm{\Pi}}_{11}& {\mathrm{\Pi}}_{12}\\ \ast & {\mathrm{\Pi}}_{22}\end{array}\right)$, where
On the other hand, by similar reasoning as in the proof of Theorem 1, we can conclude from (42) and (43) that $\mathrm{\Pi}<0$. Then from (48) we know that $\mathrm{\aleph}<0$ for all nonzero $w(t)\in {l}_{2}[0,\mathrm{\infty})$. The proof is completed. □
4 Conclusion
The problem of quantized statefeedback control for a stochastic system with multiplicative noises has been investigated through a quantizationdependent approach. Conditions for meansquare polyquadratical stability are obtained by introducing a new quantizationdependent Lyapunov function approach for linear state feedback with a logarithmic quantizer, which are shown to be less conservative than those derived by a common Lyapunov function. Moreover, ${H}_{\mathrm{\infty}}$ performance analysis has also been proposed in the quantizationdependent framework. However, it is worth pointing out that though less conservative conditions are obtained, different from the derivation of the coarsest quantizer, the explicit relation of the system matrices and quantization density is not given. The analysis of relation between the quantization density and the system matrices and the statistical properties of noises in the proposed quantizationdependent framework is a subject worth further researching.
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Acknowledgements
We would like to thank the editorinchief, the associate editor and the reviewers for their valuable comments on the paper which have led to significant improvement on the presentation and quality of the paper. This work is supported by the Taishan Scholar Construction Engineering by Shandong Government, the National Natural Science Foundation (No. 61174141), and the Major State Basic Research Development Program of China (973 Program) (No. 2009cb320600), Yangtse Rive Scholar Bonus Schemes (No. 31400080963017), National Natural Science Foundation (No. 61034007).
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LW carried out the proof of the main part of this article, YY corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.
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Wei, L., Yang, Y. A new approach to quantized stabilization of a stochastic system with multiplicative noise. Adv Differ Equ 2013, 20 (2013). https://doi.org/10.1186/16871847201320
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Keywords
 multiplicative noise
 discretetime systems
 meansquare stability
 logarithmic quantizer
 Lyapunov function