Research Article | Open Access
Valter J. S. Leite, Márcio F. Miranda, "Robust Stabilization of Discrete-Time Systems with Time-Varying Delay: An LMI Approach", Mathematical Problems in Engineering, vol. 2008, Article ID 875609, 15 pages, 2008. https://doi.org/10.1155/2008/875609
Robust Stabilization of Discrete-Time Systems with Time-Varying Delay: An LMI Approach
Sufficient linear matrix inequality (LMI) conditions to verify the robust stability and to design robust state feedback gains for the class of linear discrete-time systems with time-varying delay and polytopic uncertainties are presented. The conditions are obtained through parameter-dependent Lyapunov-Krasovskii functionals and use some extra variables, which yield less conservative LMI conditions. Both problems, robust stability analysis and robust synthesis, are formulated as convex problems where all system matrices can be affected by uncertainty. Some numerical examples are presented to illustrate the advantages of the proposed LMI conditions.
Time-delay systems have received a lot of attention from both academics and industrial engineers in the last decades. This can be verified by the great number of papers published in this area. See, for example, [1–5] and the references therein. It is worthwhile to mention that time delays affect both continuous and discrete-time systems and are presented in several systems like thermal processes, communication systems, internet dataflow, biological systems, regenerative chatter in metal cutting, and so on [2, 4]. The presence of delay yields, in general, performance degradation and, eventually, leads the system to instability.
There are two main classes of robust stability analysis that have been investigated, namely, delay-dependent and delay-independent conditions. For a system whose stability does not depend on the time-delay value, the analysis performed through delay-dependent conditions can be very conservative. Also, delay-independent conditions cannot be obtained as a limit case of delay-dependent ones just by imposing the maximum delay value , leading to a gap between these two types of delay-stability conditions [4, page 146].
An important approach used in the last years to deal with delay in the states of the systems is the use of Lyapunov-Krasovskii functionals. This approach has been largely used to obtain convex conditions mainly for continuous-time systems subject to retarded states and for neutral systems . On the other hand, discrete-time systems with state delay (DTSSD) have received little attention compared with its continuous-time counterpart. The main reason for this is that for precisely known discrete-time systems with constant time-delay, it is always possible to obtain an augmented system without delayed states . This approach, however, does not seem to be suitable for time-varying delay, delay-independent stability characterization, and for robust-system stabilization. Despite the fact that the digital control has increased in importance for practical robust control applications in the last decades, there are few papers dealing with DTSSD. Besides this, the majority of the results available in the literature for DTSSD are formulated for norm-bounded uncertainty employing constant quadratic Lyapunov-Krasovskii functionals, which may lead to conservative results . This approach, called quadratic stability, is the base of several results available in the literature as it can be seen in [9–12].
Recent results on DTSSD can be found in  where the real stability radii of time-invariant state-delayed systems have been considered. In , there has been presented the descriptor approach to obtain stability and design conditions for discrete time-delay systems. See also  where some nonconvex conditions were proposed to the synthesis problem. In the context of systems with Markovian jumps, see [3, 16, 17] and references therein. More recently, in [18, 19], robust stabilizing conditions have been proposed, but most of the results have been obtained through quadratic stability approach and the state feedback gain is designed directly from the Lyapunov-Krasovskii matrices. In , a dynamic controller is designed achieving a guaranteed cost control for norm-bounded DTSSD, but the design conditions are nonlinear. In , precisely known discrete-time systems with time-varying delays in the state are investigated and convex conditions are given to design a memoryless control law. The design conditions presented in  do not seem to be suitable to deal with polytopic systems since the matrices of the system multiply the Lyapunov-Krasovskii functional matrices. Observe that convex extensions of these results to treat polytopic type of uncertainty in the context of robust stability analysis and robust stabilization are not straightforward. Delay-independent conditions for robust stabilization of DTSSD have been considered in , where a parameter-dependent functional has been used, but only for time-invariant delays.
In this paper, new results for robust stability analysis as well as for robust stabilization of uncertain DTSSD with time-varying delay are proposed as convex conditions depending on the maximum and minimum values assumed by the delay, and , respectively. A Lyapunov-Krasovskii functional similar to that employed in  is used. Differently from other results in the literature, both analysis and synthesis conditions are formulated as LMI feasibility problems that can be solved efficiently in polynomial time by specialized numerical algorithms. The main contribution of this paper is the improvement of the recently published results on DTSSD with time-varying delay given in . Furthermore, the scope of the results of  is extended to deal with polytopic-type uncertainty affecting DTSSD and the possibility of design memory state feedback control gain. Parameter-dependent Lyapunov-Krasovskii functional used conjoint some extra matrices to achieve less conservative results. The approach proposed here does not introduce any additional dynamics and leads to a product separation between the matrices of the system and the matrices from the functional, allowing a convex formulation for the synthesis problem.
In Section 2 some definitions and the problem formulation are provided. Then, in Section 3, the main results are presented for both robust-stability analysis and for the synthesis of state feedback gains assuring the robust stability of the closed-loop system. The computational complexity of the proposed conditions and decentralized control design are discussed. In Section 4, some examples are given to illustrate the efficiency of the proposed conditions. In Section 5 some conclusions are presented. Notation. The notation used here is quite standard. is the set of real numbers and is the set of natural numbers. and denote, respectively, the identity matrix and the null matrix of appropriate dimensions. () means that matrix is positive (negative) definite. is the transpose of . The symbol stands for symmetric blocks in the LMIs.
2. Preliminaries and Problem Formulation
Consider the following discrete-time system with a time-varying delay in the state:where is the sampling time, is the state vector, is the input control signal, is the time-varying delay, where its time variation is bounded as with and being the minimum and maximum delay values, respectively. are unknown constant matrices belonging to a polytope ,where the vertices are precisely known. In special, observe that if , then the delay is uncertain, belonging to , but it is time invariant. Note that if , then (2.1) is rewritten as . Also define
In this paper, the following control law is considered: where are the robust state feedback gains that assure the robust stability of the closed-loop system, that is, the stability of (2.1)–(2.4) with (2.6) is assured for all . Therefore, this uncertain closed-loop system is given bywith where withIt is worth to mention that if the delay is not known at each sample time , then it is enough to make in (2.6). If is known at each sample time , then the possibility of using and may improve the performance of the closed-loop system (2.7).
The objective of this paper is to give convex conditions solving the following problems.
Remark 2.1. Generally speaking, in the cases where the time-delay depends on a physical parameter (such as velocity of a transport belt, the stem position of a valve, etc.) it may be possible to determine the delay value at each sample-time. As a special case, consider the regenerative chatter in metal cutting. In this process a cylindrical workpiece has an angular velocity while a machine tool (lathe) translates along the axis of this workpiece. For details, see [2, page 2]. In this case the delay depends on the velocity and thus, if this angular velocity can be measured, the delay could be determined at each instant. Note, however, that a detailed study on physical application is not the focus of the paper.
3. Main Results
First, sufficient LMI conditions to solve Problem 1 are given. The approach used here does not introduce any dynamics and leads to a product separation between the matrices of the system and those from the Lyapunov-Krasovskii functional. Then, these conditions are exploited to provide some convex synthesis results. The following theorems provide some LMI conditions depending on the values and to determine the robust stability of (2.7) or to design robust state feedback gains and that assure the robust closed-loop stability.
3.1. Robust Stability Analysis
Theorem 3.1. System (2.7) subject to (2.2), (2.4),
and (2.9) is robustly stable if there exist symmetric matrices and , such that one of the following equivalent conditions
(b) there exist parameter-dependent matrices , , and , such that where denotes and denotes with given by (2.5). In this case, the functional with is such that and is called a Lyapunov-Krasovskii functional, assuring the robust stability of (2.7).Proof. The positivity of the functional (3.3)) is assured with the hypothesis of , . For (3.3)) is a Lyapunov-Krasovskii functional, besides its positivity, it is necessary to verify (3.5) . From hereafter, the dependency is omitted in the expressions , , for simplicity of the notation. To calculate (3.5), considerObserve that the third term in (3.7) can be rewritten asUsing (3.9) in (3.7), one getsSo, taking into account (3.6), (3.8), and (3.10) the following upper bound for (3.5) can be obtained:Replacing in (3.11) by the right-hand side of (2.7) one gets (3.1). The equivalence between (3.1) and (3.2) can be established as follows. First, note that (3.1) can be rewritten as which by Schur complement is equivalent to Therefore, the equivalence between (a) and (b) is the same as that between (3.2) and (3.13). So, if (3.13) is verified, then (3.2) is true for , . On the other hand, if (3.2) is verified, then with completing the proof.
Note that (3.7) keeps a relation with Finsler's lemma where the slack variables and depend on the uncertain parameter . The conditions presented in Theorem 3.1 are of infinite dimension if belongs to a continuous domain. These conditions can be numerically treated by using different approaches such as those presented in [23, 24], where it is possible to consider the products of matrices depending on by means of LMI relaxations. In case of belonging to a discrete, countable, and finite domain, the conditions of Theorem 3.1 state a finite set of LMIs defined at each value of . In this paper, the structure of matrices and is supposed to be linear in : and . The extra matrices are chosen to be fixed , , and . Although other structures may lead to a less conservative condition for Problem 1, no improvement is expected for Problem 2 as it deals with constant state feedback gains and . Also observe that, since the conditions of Theorem 3.1 depend on the size of delay variation, , and not on the delay value itself, these conditions are delay-independent conditions. Theorem 3.2. System (2.7) subject to (2.2), (2.4), and (2.9) is robustly stable if there exist symmetric matrices and , such that are verified with given by (2.5). Beside this, (3.3)) with (3.4) and (3.15) is a Lyapunov-Krasovskii functional for (2.7).Proof. Observe that can be obtained by multiplying (3.16) by and summing it up, that is, , .
It is worth to mention that the parameter-dependent structure, imposed to and , (3.15), cannot be directly used in the conditions presented in (3.1) and (3.13). This limitation is due to the products between the system matrices and the Lyapunov-Krasovskii candidate matrices.
The approach based on quadratic stability can be recovered from (3.1), (3.2), (3.13) (or (3.16)). In this case, it is sufficient to impose and replace and by and , respectively. Then it is necessary to test those conditions for . Note that all the quadratic stability conditions obtained as described above are equivalent and the difference between them is just the computational burden necessary to solve each one.
3.2. Robust Feedback Design
The stability analysis conditions presented in Theorem 3.2 can be used to obtain a convex condition for robust synthesis of the gains and such that the control law (2.6) applied in (2.1) results in a robust stable closed-loop system, and, therefore, resulting in a solution to Problem 2.
Theorem 3.3. If there exist symmetric matrices , matrices and , such that the following LMIs are verified with given by (2.5), then system given by (2.1)–(2.4) is robustly stabilizable with (2.6), where the static feedback gains are given by yielding a convex solution to Problem 2. Beside this, (3.3)–(3.4) is a Lyapunov-Krasovskii functional that guarantees the robust stability of the resulting closed-loop system with and given in (3.15).Proof. Condition (3.17) is obtained from (3.16) by choosing , replacing and by and , respectively, and making the change of variables and .
Note that conditions presented in Theorem 3.3 encompass quadratic stability, since it is always possible to choose and . Also observe that if is not available at each sample, and therefore cannot be used in the feedback, then it is enough to choose leading to a control law given by . Another relevant note is that Theorem 3.3 presents some LMI conditions for the synthesis of state feedback gains, differently from other approaches found in the literature where nonconvex techniques are employed.
The results of Theorem 3.3 can be used to deal with decentralized control. This can be done by imposing a diagonal structure on , , and , being the number of subsystems. In this case, robust block-diagonal state feedback gains given by and can be obtained. Beside this, no structure is imposed to the matrices of the Lyapunov-Krasovskii functional, and , which may lead to less conservative results.
The computational complexity of the conditions presented in this paper can be determined by the number of scalar variables, , and the number of rows, , involved in the optimization problems. Theorem 3.2 presents scalar variables and LMI rows and in Theorem 3.3 it is found that and . In the case of using LMI control toolbox , the computational complexity is and using the solver SeDuMi  the computational complexity is .
4. Numerical Examples
Example 4.1. This example shows that the
condition presented by Theorem 3.2 can be less conservative than other conditions
available in the literature. Consider the
DTSSD given by (2.7),
whereThis system has been
investigated in 
and its stability has been assured for by using a
Lyapunov-Krasovskii with terms instead
of the terms presented
here (see (3.3))-(3.4)). Using Theorem 3.2 the same delay interval is obtained, but
with a lower-computational complexity. Using [21, Theorem 3.1] that employs a
Lyapunov-Krasovskii functional similar to that used here, it is possible to
verify the stability of this system but with a narrower delay interval: .
Consider now that this system is affected by an uncertain parameter such that it can be described by a polytope (2.3) with and given above and and . In this case, the conditions of [18, 21] are no longer applicable, since the system is uncertain. Using Theorem 3.2, it is possible to assure the robust stability of this system for . Thus, a more flexible analysis condition is provided by Theorem 3.2.Example 4.2. Consider the discrete-time system with delayed states which is described by (2.1), where the system matrices are given byThis system has been investigated for in  where a state feedback gain has been obtained by means of application of its Theorem 4.1 that has  scalar variables and  rows. On the other hand, the conditions of Theorem 3.3 have scalar variables and rows. Thus, by using LMI control toolbox , the condition proposed in  is more complex than that in Theorem 3.3. Conditions of Theorem 3.3 yieldand , which is close to the gain obtained by  with a lower computational cost. Therefore, this example illustrates that the conditions proposed here are numerically more efficient than those proposed in .Example 4.3. Consider the system investigated in Example 4.2 where some uncertainties have been added as follows:with , , and . These parameters lead to a polytope with vertices determined by the combination of the extreme values of , , and . Keeping the same delay variation interval considered in Example 4.2, that is, and , the conditions of Theorem 3.3 are used to stabilize this uncertain system resulting in The behavior of the states of the closed-loop response of this uncertain discrete-time system with time-varying delay is shown in Figure 1. It has been simulated that the time response of this system at each vertex of the polytope was used in the controller design. The respective control signals are shown in Figure 2. The initial conditions have been chosen as , , and the value of the delay, , has been varied randomly as shown in Figure 2(b). In Figure 1, the stability of the uncertain closed-loop system is illustrated, assured by the state feedback calculated by means of Theorem 3.3, for . The control effort is shown in Figure 2(a). It is worth to mention that the results presented in  cannot be directly used with polytopic uncertain systems and cannot be applied in this case. This example shows the efficacy of the conditions proposed here applied to uncertain discrete-time delayed systems.
Using Theorem 3.3 with and , , , it is possible to obtain that assures the quadratic stability of the system for and . Therefore, , , and , . The closed-loop system behavior is illustrated in Figures 4 and 5. The state behavior of the closed-loop system is shown in Figure 4(a) and the control signal is presented in the bottom part. This system has been simulated for and as seen in Figure 5(a), with initial conditions . The time-varying delay has been randomly generated as indicated in Figure 5(b).
Therefore, this example shows that conditions of Theorem 3.3 can be used in the context of time-varying systems encompassing, in this case, quadratic stability conditions.
Some sufficient convex conditions were proposed to solve two problems: the robust stability analysis and the synthesis of robust state feedback gains for the class of polytopic discrete-time systems with time-varying delay. The presented LMI conditions include some extra variables and no additional dynamic in the investigated system, thus yielding less conservative results. Some examples, with numerical simulation, are given to demonstrate some relevant characteristics of the proposed design methodology such as robust stabilization using memory or memoryless state feedback gains in the control law, decentralized control, and design for time-varying discrete-time systems with time-varying delay. Some of these examples have been compared with other results available in the literature.
This paper has been supported by the Brazilian agencies CNPq (485496/2006-2) and FAPEMIG (TEC ). The authors are grateful to the anonymous reviewers for their careful reading and valuable suggestions and comments that have helped to improve the presentation of the paper.
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Copyright © 2008 Valter J. S. Leite and Márcio F. Miranda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.