Discrete Time Systems Part 11 potx

Stability Criterion and Stabilization of Linear Discrete-time System with Multiple Time Varying Delay 293 -1 -0.5 0 0.5 1 time-7.. H infinity control of discrete-time linear systems wit

Stability Criterion and Stabilization of

Linear Discrete-time System with Multiple Time Varying Delay 289

0

T

T d T d

0

T

T d T d

Theorem 2: For a given set of upper and lower bounds ,d d for corresponding time-varying i i

delays d ki, if there exist symmetric and positive-definite matrices X1∈ℜn n× ,S i∈ℜn n× and

n n

i

T∈ℜ × , i= …1, ,N and general matrices X and 2 X such that LMIs below hold, the 3

memoryless state-feedback gain is given by 1

1

K FX= − Proof: Now we consider substituting system matrices of (12) into LMIs conditions (13), the LMIs-based conditions of the memoryless state-feedback problem can be obtained directly

Robust state feedback synthesis can be formulated as:

For a given set of upper and lower bounds ,d d for corresponding time varying delays i i d ki,

if there exist symmetric and positive-definite matrices X1∈ℜn n× , n n

Stability Criterion and Stabilization of

Linear Discrete-time System with Multiple Time Varying Delay 291

e S

e S

e T

e

Therefore, a memoryless state-feedback gain is given by K FX= 1−1=[2.0 3.0]

The closed-loop discrete-time system with multiple time-varying time delay is simulated in case of d1=1,d2= , 2 d1=1,d2= , 3 d1=2,d2= , and 2 d1=2,d2= , respectively And 3these results are illustrated in Figure 1, Figure 2, Figure 3 and Figure 4 These figures show that this system is stabilized by the state feedback

-1-0.500.51

Stability Criterion and Stabilization of

Linear Discrete-time System with Multiple Time Varying Delay 293

-1 -0.5 0 0.5 1

time-7 Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant no 6070422), the Fundamental Research Funds for the Central Universities, SCUT 2009ZZ0051, and NCET-08-0206

8 References

Altman, E., Basar, T., & Srikant, R (1999) Congestion control as a stochastic control problem

with action delays In Proceedings of the 34th IEEE conference on decision and control,

pp 1389–1394, New Orleans

Sichitiu, M L., Bauer, P H., & Premaratne, K (2003) The effect of uncertain time-variant

delays in ATM networks with explicit rate feedback: A control theoretic approach

IEEE/ACM Transactions on Networking, 11(4) 628–637

Boukas, E K and Liu, Z K (2001) Robust H infinity control of discrete-time Markovian

jump linear systems with mode-dependent time-delays, IEEE Transactions on

Automatic Control 46, no 12, 1918-1924

Kim, J H and Park, H B, (1999) H infinity state feedback control for generalized

continuous/discrete time-delay system, Automatica 35, no 8, 1443-1451

Song, S H and Kim, J K, (1998) H infinity control of discrete-time linear systems with

norm-bounded uncertainty and time delay in state, Automatica 34, no 1, 137-139

Mukaidani, H., Sakaguchi, S., and Tsuji, T (2005) LMI-based neurocontroller for guaranteed

cost control of uncertain time-delay systems, Proceedings of IEEE International

Symposium on Circuits and Systems, vol 4, Kobe, pp 3407-3050

Chang, Y C., Su, S F., and Chen, S S (2004) LMI approach to static output feedback

simultaneous stabilization of discrete-time interval systems with time delay,

Proceedings of International Conference on Machine Learning and Cybernetics, Vol 7,

Shanghai, pp 4144-4149

Gao, H., Lam, J., Wang, C., and Wang, Y (2004) Delay-dependent output-feedback

stabilisation of discrete-time systems with time-varying state delay, IEE

Proceedings-Control Theory and Applications 151, no 6, 691-698

Guan, X, Lin, Z and Duan, G (1999) Robust guaranteed cost control for discrete-time

uncertain systems with delay, IEE Proc Control Theory, 146, (6), pp 598-602

Chen, W, Guan, Z and Lu, X (2004) Delay-dependent guaranteed cost control for uncertain

discrete-time systems with both state and input delays, Journal of the Franklin

Institute, vol 341, pp 419-430

Chen, W, Guan, Z and Lu, X (2004) Delay-dependent output feedback guaranteed cost

control for uncertain time-delay systems, Automatica, vol 40, pp 1263-1268

Boukas, E K., Discrete-time systems with time-varying time delay: Stability and

Stabilizability, Mathematical Problems in Engineering, Vol 2006, ID 42489, 1-10

Xie, W., Xie, L H., Xu, B G., stability analysis for linear discrete-time systems with multiple

time-varying delays, 2007 IEEE International Conference on Control and Automation, pp 3078-3080

Xie, W., Multi-objective H infinity/alpha-stability controller synthesis of LTI systems, IET

Control Theory and Applications, Vol 2, no 1, pp.51-55, 2008

Xu, B G (1997) Stability robustness bounds for linear systems with multiply time-varying

delayed perturbations, International Journal of Systems Science, Vol.28, No.12, pp 1311-1317

Shi, J X., Ma, Y C, Yang, B and Zhang, X F (2009) Quadratic stabilization for

multi-time-delay uncertain discrete systems, Chinese Control and Decision Conference, pp 4064-4068

Gahinet, P., Nemirovskii, A , Laub, A J and Chilali, M (1995) LMI Control Toolbox

Natick, MA: Mathworks

Valter J S Leite1, Michelle F F Castro2, André F Caldeira3, Márcio F.

1,2,3CEFET–MG / campus Divinópolis

Brazil

1 Introduction

This chapter is about techniques for robust stability analysis and robust stabilization ofdiscrete-time systems with delay in the state vector The relevance of this study is mainlydue to the unavoidable presence of delays in dynamic systems Even small time-delays canreduce the performance of systems and, in some cases, lead them to instability Examples ofsuch systems are robotics, networks, metal cutting, transmission lines, chemical and thermalprocesses among others as can be found in the books from Gu et al (2003), Richard (2003),Niculescu (2001) and Kolmanovskii & Myshkis (1999)

Studies and techniques for dealing with such systems are not new Since the beginning ofcontrol theory, researchers has been concerned with this issue, either in the input-outputapproach or in state-space approach For the input-output approach, techniques such as Padéapproximation and the Smith predictor are widely used, mainly for process control The use

of state space approach allows to treat both cases For both approaches delays can be constant

or time-varying Besides, both the delay and the systems can be precisely known or affected

by uncertainties

In this chapter the class of uncertain discrete-time systems with state delay is studied Forthese systems, the techniques for analysis and design could be delay dependent or delayindependent, can lead with precisely known or uncertainty systems (in a polytopic or in anorm-bonded representation, for instance), and can consider constant or time-varying delays.For discrete-time systems with constant and known delay in the state it is always possible

to study an augmented delay-free system Kapila & Haddad (1998), Leite & Miranda (2008a).However, this solution does not seem to be suitable to several cases such as time-varying delay

or uncertain systems

For these systems, most of the applied techniques for robust stability analysis an robust controldesign are based on Lyapunov-Krasovskii (L-K) approach, which can be used to obtain convexformulation problems in terms of linear matrix inequalities (LMIs)

In the literature it is possible to find approaches based on LMIs for stability analysis, most ofthem based on the quadratic stability (QS), i.e., with the matrices of the Lyapunov-Krasovskiifunction being constant and independent of the uncertain parameters

In the context of QS, non-convex formulations of delay-independent type have been proposed,for example, in Shi et al (2003) where the delay is considered time-invariant In Fridman &

Uncertain Discrete-Time Systems with Delayed State: Robust Stabilization with Performance

Specification via LMI Formulations

17

Shaked (2005a) and Fridman & Shaked (2005b), delay dependent conditions, convex to theanalysis of stability and non-convex for the synthesis, are formulated using the approach ofdescriptor systems These works consider systems with both polytopic uncertainties — seeFridman & Shaked (2005a) — and with norm-bounded uncertainties as done by Fridman &Shaked (2005b).

Some other different aspects of discrete-time systems with delayed state have been studied.Kandanvli & Kar (2009) present a proposal with LMI conditions for robust stability analysis ofdiscrete time delayed systems with saturation In the work of Xu & Yu (2009), bi-dimensional(2D) discrete-time systems with delayed state are investigated, and delay-independentconditions for norm-bounded uncertainties and constant delay are given by means ofnonconvex formulations In the paper from Ma et al (2008), convex conditions have beenproposed for discrete-time singular systems with time-invariant delay Discrete-time switchedsystems with delayed state have been studied by Hetel et al (2008) and Ibrir (2008) Theformer establishes the equivalence between the approach used here (Lyapunov-Krasovskiifunctions) and the one used, in general, for the stability of switched systems with time-varyingdelay The latter gives nonconvex conditions for switched systems where each operation mode

is subject to a norm-bounded uncertainty and constant delay

The problem of robust filtering for discrete-time uncertain systems with delayed state isconsidered in some papers Delayed state systems with norm-bounded uncertainties arestudied by Yu & Gao (2001), Chen et al (2004) and Xu et al (2007) and with polytopicuncertainties by Du et al (2007) The results of Gao et al (2004) were improved by Liu et al.(2006), but the approach is based on QS and the design conditions are nonconvex dependingdirectly on the Lyapunov-Krasovskii matrices

The problem of output feedback has attracted attention for discrete-time systems with delay inthe state and the works of Gao et al (2004), He et al (2008) and Liu et al (2006) can be cited asexamples of on going research In special, He et al (2008) present results for precisely knownsystems with time-varying delay including both static output feedback (SOF) and dynamicoutput feedback (DOF) However, the conditions are presented as an interactive method thatrelax some matrix inequalities

The main objective of this chapter is to study the robust analysis and synthesis of discrete-timesystems with state delay This chapter is organized as follows In Section 2 some notationsand statements are presented, together the problems that are studied and solved in the nextsections In sections 3 and 4 solutions are presented for, respectively, robust stability analysisand robust design, based in a L-K function presented in section 2 In Section 5 some additionalresults are given by the application of the techniques developed in previous sections arepresented, such as: extensions for switched systems, to treat actuator failure and to makedesign with pole location In the last section it is presented the final comments

2 Preliminaries and problem statement

In this chapter the uncertain discrete time system with time-varying delay in the state vector

where k is the k-th sample-time, matrices A(α), A d( α), B(α), B w , C(α), C d( α), D(α)and D w( α)

are time-invariant, uncertain and with adequate dimensions defined in function of the signals

x k =x(k) ∈Rn , the state vector at sample-time k, u k=u(k) ∈Rm, representing the control

vector with m control signals, w k = w(k) ∈ R, the exogenous input vector with input

signals, and z k=z(k) ∈Rp , the output vector with p weight output signals These matrices

can be described by a polytopePwith known vertices

case of d k is not known, it is sufficient to assume K d=0.

297

Uncertain Discrete-Time Systems with Delayed State:

Robust Stabilization with Performance Specification via LMI Formulations

2.1 Stability conditions

Since the stability of system ˜Ω(α)given in (7) plays a central rule in this work, it is addressed

in the sequence Note that, without loss of generality, it is possible to consider the stability of

the system (7) with w k=0,∀k∈N.

Consider the sequence composed by ¯d+1 null vectors

Definition 1(Uniform asymptotic stability) For a given α ∈ Υ, the trivial solution of (7) with

w k =0,∀k∈ N is said uniformly asymptotically stable if for any κ ∈R+ such that for all initial conditions x k∈φ d¯

0,k∈Φκ d¯, k∈ I[−d, 0¯ ], it is verified

lim

t→∞φ d¯t,j,k=0, ∀j∈ I[1, ¯d+1]

This allows the following definition:

Definition 2(Robust stability) System (7) subject to (3), (5) and (8) is said robustly stable if its

respective trivial solution is uniformly asymptotically stable∀α∈Υ.

The main objective in this work is to formulate convex optimization problems, expressed asLMIs, allowing an efficient numerical solution to a set of stability and performance problems

2.2 Problems

Two sets of problems are investigated in this chapter The first set concerns stability issuesrelated to uncertain discrete time with time varying delay in the state vector as presented inthe sequence

Problem 1(Robust stability analysis) Determine if system (7) subject to (3), (5) and (8) is robustly

as stated in the following problems:

Problem 3(H∞guaranteed cost) Given the uncertain system ˜Ω(α) ∈P˜, determine an estimation for γ>0 such that for all w k∈ 2there exist z k∈ 2satisfying

for all α∈Υ In this case, γ is called anH∞guaranteed cost for (7).

Problem 4(RobustH∞control design) Given the uncertain systemΩ(α) ∈P˜, (1), and a scalar

γ >0, determine robust state feedback gains K and K d , such that the uncertain closed-loop system

˜

Ω(α) ∈P˜, (7), is robustly stable and, additionally, satisfies (13) for all w k and z k belonging to2.

It is worth to say that, in cases where time-delay depends on a physical parameter (such asvelocity of a transport belt, the position of a steam valve, etc.) it may be possible to determinethe delay value at each sample-time As a special case, consider the regenerative chatter inmetal cutting In this process a cylindrical workpiece has an angular velocity while a machinetool (lathe) translates along the axis of this workpiece For details, see (Gu et al., 2003, pp 2) Inthis case the delay depends on the angular velocity and can be recovered at each sample-time

k However, the study of a physical application is not the objective in this chapter.

The following parameter dependent L-K function is used in this paper to investigate problems1-4:

The dependency of matrices P(α)and Q(α)on the uncertain parameterα is a key issue on

reducing the conservatism of the resulting conditions Here, a linear relation onα is assumed.

Thus, consider the following structure for these matrices:

conditions, but at the expense of a higher numerical complexity of the resulting conditions

To be a L-K function, the candidate (14) must be positive definite and satisfy

ΔV(α, k) =V(α, k+1) −V(α, k) <0 (19)for all x T k x k T −d

k

T

=0andα∈Υ

The following result is used in this work to obtain less conservative results and to decouple

the matrices of the system from the L-K matrices P(α)and Q(α)

Lemma 1(Finsler’s Lemma) Let ϕ∈ Rn ,M(α) = M(α)T ∈ Rn ×n andG(α) ∈ Rm ×n such

that rank(G(α)) <n Then, the following statements are equivalents:

i) ϕ TM(α)ϕ<0, ∀ϕ : G(α)ϕ=0, ϕ =0

ii) G(α)⊥T

M(α)G(α)⊥<0,

299

Uncertain Discrete-Time Systems with Delayed State:

Robust Stabilization with Performance Specification via LMI Formulations

iii) ∃μ(α) ∈R+:M(α) −μ(α)G(α)TG(α) <0

iv) ∃ X (α) ∈Rn ×m:M(α) + X (α)G(α) + G(α)TX (α)T < 0

In the case of parameter independent matrices, the proof of this theorem can be found in

de Oliveira & Skelton (2001) The proof for the case depending onα follows similar steps.

3 Robust stability analysis andH∞guaranteed cost

In this section it is presented the conditions for stability analysis and calculation of H∞guaranteed cost for system (7) The objective here is to present sufficient convex conditionsfor solving problems 1 and 3

3.1 Robust stability analysis

Theorem 1. If there exist symmetric matrices 0 < P i ∈ Rn ×n , 0 < Q i ∈ Rn ×n , a matrixX ∈

Proof The positivity of the function (14) is assured with the hypothesis of P i = P i T > 0,

Q i=Q T i >0 For the equation (14) be a Lyapunov-Krasovskii function, besides its positivity,

it is necessary to verify (19)∀α ∈ Ω From hereafter, the α dependency is omitted in the expressions V v( k), v=1, , 3, To calculate (19), consider

Using (27) in (25), one gets

M(α) = ∑N

i=1α iQi; G(α) = ∑N

α∈Υ,QiandBigiven in (21) and (23), respectively, completing the proof

An important issue in Theorem 1 is that there is no product between the matrices of the systemand the matrices of the Lyapunov-Krasovskii proposed function, (14) This can be exploited

to reduce conservatism in both analysis and synthesis methods

Example 1(Stability Analysis) In this example the stability analysis condition given in Theorem 1

is used to investigate system (7), with D w=0, where

˜

A1=

0.6 00.35 0.7



and A˜d1=

0.1 00.2 0.1



This system has been investigated by Liu et al (2006), Boukas (2006) and Leite & Miranda (2008a) The objective here is to establish the larger delay interval such that this system remains stable The results are summarized in Table 1.

Although Theorem 1 and the condition from Liu et al (2006) achieve the same upper bound for d k , the L-K function employed by Liu et al (2006) has 5 parts while Theorem 1 uses a function with only 3 parts, as given by (14)-(17).

Consider that (35) is affected by an uncertain parameter being described by a polytope (8) with ˜ A1and

˜

A d1 given by (35) and ˜ A2 =1.1 ˜A1and ˜ A d2=1.1 ˜A d1 In this case the conditions of Boukas (2006) are no longer applicable and those from Liu et al (2006) are not directly applied, because of type of the system uncertainty Using Theorem 1 it is possible to assure the robust stability of this system for

|d k+1−d k| ≤3.

301

Uncertain Discrete-Time Systems with Delayed State:

Robust Stabilization with Performance Specification via LMI Formulations

Condition d ¯ d

Boukas (2006)[Theorem 3.1] 2 10Liu et al (2006) 2 13

Table 1 Maximum delay intervals such that (7) with (35) is stable

3.2 Estimation ofH∞guaranteed cost

Theorem 2 presented in the sequel states a convex condition for checking if a givenγ is anH∞

guaranteed cost for system (7)

Theorem 2. If there exist symmetric matrices 0< P i ∈ Rn ×n , 0 <Q i ∈ Rn ×n , a matrixXH ∈

Proof Following the proof given for Theorem 1, it is possible to conclude that the positivity of

(14) is assured with the hypothesis of P(α) =P(α)T>0, Q(α) =Q(α)T >0and, by (29) that

ΔV(k) ≤xk+1P(α)x k+1+xk[βQ(α) −P(α)]x k−xk −d(k) Q(α)x k −d(k)<0 (39)Consider system (7) as robustly stable with null initial conditions given by (12), assumeμ=γ2

and signals w k and z kbelonging to2 In this case, it is possible to verify that V(α, 0) =0 and

V(α, ∞)approaches zero, whenever w k goes to zero as k increases, or to a constant ˜ φ < ∞,

whenever w kapproachesφ < ∞ as k increases Also, consider the H∞performance indexgiven by

which can be rewritten as

ζ T

kQH(α)ζ k<0 subject to (7), (42)withM(α) = QH(α),ϕ=ζ k,

i.e., eliminating the dependency on the uncertain parameterα — and noting that G(α) =

∑N

i=1α iB Hi, convexity is achieved, and (36) can be used to recover (44) by ΨH(α) =

∑N

i=1α iΨHi,α∈Υ Thus, this assures the negativity of J(α, k)for all w k∈ 2implying that (7)

is robustly stable withH∞guaranteed cost given byγ= √μ.

In case of time-varying uncertainties, i.e α =α k =α(k), the conditions formulated in bothTheorem 1 and Theorem 2 can be adapted to match the quadratic stability approach In this

case, it is enough to use P i = P, Q i = Q, i ∈ I[1, N] This yields conditions similar to (20)and (36), respectively, with constant L-K matrices See Subsection (5.1) for a more detaileddiscussion on this issue

Note that, it is possible to use the conditions established by Theorem 2 to formulate thefollowing optimization problem that allows to minimize the value ofμ=γ2:

4 RobustH∞feedback design

The stability analysis conditions can be used to obtain convex synthesis counterpart

formulations for designing robust state feedback gains K and K d, such that control law (6)applied in (1) yields a robustly stable closed-loop system, and, therefore, provides a solution

to problems 2 and 4 In this section, such conditions for synthesis are presented for both robuststabilization and robustH∞control design

4.1 Robust stabilization

The following Theorem provides some LMI conditions depending on the difference ¯d−d to

design robust state feedback gains K and K dthat assure the robust stability of the closed-loopsystem

303

Uncertain Discrete-Time Systems with Delayed State:

Robust Stabilization with Performance Specification via LMI Formulations