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.


Introduction
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 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.R is the set of real numbers and N is the set of natural numbers.I n and 0 denote, respectively, the n × n identity matrix and the null matrix of appropriate dimensions.M > 0 M < 0 means that matrix M is positive negative definite.M is the transpose of M. The symbol stands for symmetric blocks in the LMIs.

Preliminaries and problem formulation
Consider the following discrete-time system with a time-varying delay in the state: where k is the sampling time, x k ∈ R n is the state vector, u k ∈ R n is the input control signal, d k is the time-varying delay, where its time variation is bounded as with d and d being the minimum and maximum delay values, respectively.A α , A d α , B α ≡ A, A d , B α ∈ R n×2n are unknown constant matrices belonging to a polytope P, where the vertices A i , A di , B i ≡ A, A d , B i are precisely known.In special, observe that if d d, then the delay is uncertain, belonging to 0, d , but it is time invariant.Note that if d k 0, then 2.1 is rewritten as In this paper, the following control law is considered: where K, K d ∈ R ×2n 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 α ∈ Ω.Therefore, this uncertain closed-loop system is given by Mathematical Problems in Engineering with where A, A d α ∈ P with It is worth to mention that if the delay d k is not known at each sample time k, then it is enough to make K d 0 in 2. 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 ω 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 ω 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.

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 d and d to determine the robust stability of 2.7 or to design robust state feedback gains K and K d that assure the robust closed-loop stability.

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 0 < P α ∈ R n×n and 0 < Q α ∈ R n×n , such that one of the following equivalent conditions is verified: with β given by 2.5 .In this case, the functional 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 P α P α > 0, Q α Q α > 0. For 3.3 is a Lyapunov-Krasovskii functional, besides its positivity, it is necessary to verify 3.5 for all α ∈ Ω.From hereafter, the α dependency is omitted in the expressions V v k , v 1, . . ., 3, for simplicity of the notation.To calculate 3.5 , consider Observe that the third term in 3.7 can be rewritten as 3.9 Using 3.9 in 3.7 , one gets So, taking into account 3.6 , 3.8 , and 3.10 the following upper bound for 3.5 can be obtained: Replacing x k 1 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 completing the proof.
Note that 3.7 keeps a relation with Finsler's lemma where the slack variables F α and G α depend on the uncertain parameter α.The conditions presented in Theorem 3.1 are of infinite dimension if α 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 α by means of LMI relaxations.In case of α 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 α.In this paper, the structure of matrices P α and Q α is supposed to be linear in α 25 : and α ∈ Ω.The extra matrices are chosen to be fixed F α F, G α G, and H α H.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 K and K d .Also observe that, since the conditions of Theorem 3.1 depend on the size of delay variation, d − d , 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 0 < P i ∈ R n×n and 0 < Q i ∈ R n×n , i 1, . . ., N, such that It is worth to mention that the parameter-dependent structure, imposed to P α and Q α , 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 P α P, Q α Q P i P, Q i Q, i 1, . . ., N and replace A α and A d α by A i and A di , respectively.Then it is necessary to test those conditions for i 1, . . ., N. 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.
Observe that conditions presented in Theorems 3.1 and 3.2 can be used to test the robust stability of both systems A α , A d α and A α , A d α , since their eigenvalues are the same.

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 K and K d 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 0 are verified with β 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 Note that conditions presented in Theorem 3.3 encompass quadratic stability, since it is always possible to choose P i P and Q i Q, i 1, . . ., N. Also observe that if d k is not available at each sample, and therefore x k − d k cannot be used in the feedback, then it is enough to choose W d 0 leading to a control law given by u k Kx k .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.

Decentralized control
The results of Theorem 3.3 can be used to deal with decentralized control.This can be done by imposing a diagonal structure on F F D block-diag{F 1 , . . ., F κ }, W W D block-diag{W 1 , . . ., W κ }, and W d W dD block-diag{W d1 , . . ., W dκ }, κ being the number of subsystems.In this case, robust block-diagonal state feedback gains given by K D W D F D −1 and K dD W dD F D −1 can be obtained.Beside this, no structure is imposed to the matrices of the Lyapunov-Krasovskii functional, P α and Q α , which may lead to less conservative results.

Computational complexity
The computational complexity of the conditions presented in this paper can be determined by the number of scalar variables, K, and the number of rows, R, involved in the optimization problems.Theorem 3.2 presents K T 2 n 3n N n 1 scalar variables and R T 2 5Nn LMI rows and in Theorem 3.3 it is found that K T 3 n 2 n N n 1 and R T 3 3Nn.In the case of using LMI control toolbox 26 , the computational complexity is O K 3 R and using the solver SeDuMi 27 the computational complexity is O K 2 R 2.5 R 3.5 .

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 , where This system has been investigated in 18 and its stability has been assured for 2 ≤ d k ≤ 13 by using a Lyapunov-Krasovskii with 5 terms instead of the 3 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: 2 ≤ d k ≤ 10.Consider now that this system is affected by an uncertain parameter such that it can be described by a polytope 2.3 with A 1 and A d1 given above and A 2 1.1A 1 and A d2 1.1A d1 .
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 |d k 1 −d k | ≤ 3. Thus, a more flexible analysis condition is provided by Theorem 3.  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 . ., 0, and the value of the delay, d k , 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 1 ≤ d k ≤ 10.The control effort is shown in Figure 2 a .It is worth to mention that the results presented in 21 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.

4.6
By imposing a block diagonal structure on F and W given by {2, 2} and choosing W d 0, it is possible to apply the conditions of Theorem 3.3 getting  described by a polytope of matrices with vertices given by

4.8
The open-loop system is not robustly stable as can be noticed by the eigenvalues of A 1 , given by 0.0233 and 2.7667, and the eigenvalues of A 2 , given by −0.1514 and 1.6614.The unforced system is simulated for an initial state given by x k 1, −1 , k ∈ −2, 0 and a constant α 1 0.4, that is, A α 0.4 A 1 0.6 A 2 , A d α 0.4 A d1 0.6 A d2 , and B α 0.4 B 1 0.6 B 2 .The unstable behavior of the states is shown in Figure 3.
Using Theorem 3.3 with W d 0 and P i P , Q i Q, i 1, 2, it is possible to obtain K − 3.1599, 3.4971 that assures the quadratic stability of the system for d 0 and d 19.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.

Conclusions
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 timevarying discrete-time systems with time-varying delay.Some of these examples have been compared with other results available in the literature.

Figure 1 :Figure 2 :
Figure 1: The behavior of the states x 1 k a and x 2 k , with b 1 ≤ d k ≤ 10 see Figure 2 b , K 109670, 2.7170 .

Figure 3 :Figure 4 :
Figure 3: The behavior of the states x 1 k × and x 2 k • for the unforced system with 1 ≤ d k ≤ 2 and α 1 0.4.

Figure 5 :
Figure 5: α 1 k a and time-varying delay, d k b .
6.If d k is known at each sample time k, then the possibility of using K and K d 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.
N, 3.16are verified with β 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 M α can be obtained by multiplying 3.16 by α i and summing it up, that is, M α Condition 3.17 is obtained from 3.16 by choosing G H 0, replacing A i and A di by A i B i K and A di B i K d , respectively, and making the change of variables FK W and FK d W d .
2. This system has been investigated for d 1 ≤ d k ≤ 10 d in 21 where a state feedback gain K 2.0005, 2.9051 has been obtained by means of application of its Theorem 4.1 that has K 21 19 scalar variables and R 21 16 rows.On the other hand, the conditions of Theorem 3.3 have K T 3 12 scalar variables and R T 3 6 rows.Thus, by using LMI control toolbox 26 , the condition proposed in 21 is more complex than that in Theorem 3.3.Conditions of Theorem 3.3 yield .9929 , which is close to the gain obtained by 21 with a lower computational cost.Therefore, this example illustrates that the conditions proposed here are numerically more efficient than those proposed in 21 .Example 4.3.Consider the system investigated in Example 4.2 where some uncertainties have been added as follows: Keeping the same delay variation interval considered in Example 4.2, that is, d 1 and d 10, the conditions of Theorem 3.3 are used to stabilize this uncertain system resulting in P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 with |ρ| ≤ 0.07, |θ| ≤ 0.1, and |η| ≤ 0.1.These parameters lead to a polytope with 8 vertices determined by the combination of the extreme values of ρ, θ, and η.