Discrete Time Systems Part 7 ppt

Discrete Time Mixed LQR/H ∞ Control Problems 169 1 1 In this section, we will consider the non-fragile discrete-time state feedback mixed LQR/ H∞ control problem with controller uncerta

Discrete Time Mixed LQR/H ∞ Control Problems 169

1 1

In this section, we will consider the non-fragile discrete-time state feedback mixed

LQR/ H∞ control problem with controller uncertainty This problem is defined as follows:

Consider the system (2) (4) satisfying Assumption 1-3 with w ∈ L2[0, )∞ and x(0)=x0, for a

given number γ> and any admissible controller uncertainty, determine an admissible 0

non-fragile controller F∞ such that

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170

2

ˆsup { }

w L

J

+

∈ subject to T zw( )z ∞< γwhere, the controller uncertainty ΔF k( ) considered here is assumed to be of the following

with the elements of ( )F k being Lebesgue measurable

If this controller exists, it is said to be a non-fragile discrete time state feedback mixed

LQR/ H∞ controller

In order to solve the problem defined in the above, we first connect the its design criteria

with the inequality (11)

Lemma 4.1 Suppose that γ> , then there exists an admissible non-fragile controller F0 ∞

that achieves

2

ˆsup { } T

if for any admissible uncertainty ΔF k( ), there exists a stabilizing solution X∞≥ to the 0

inequality (11) such that 2

that A ˆF

∞ is stable and J can be rewritten as follows:

2 1

1 1

∞ is stable It follows

Discrete Time Mixed LQR/H ∞ Control Problems 171

from (19b) that

ˆsup{ } T

w L

J ∈ + =x X x∞ Thus, we conclude that there exists an admissible non-fragile controller such that

2 0 0ˆ

2

T w

w =γ− x X x , z22=x X x0T z 0 Then it follows from (19a) that

∞ depends on the controller uncertainty ΔF k( ), thus it is difficult to find an

upper bound of either of X w and X z This implies that the existence of controller

uncertainty ΔF k( ) makes it difficult to find supw L∈2+{ }J by using (20) Thus, it is clear that

the existence of the controller uncertainty makes the performance of the designed system

become bad

In order to give necessary and sufficient conditions for the existence of an admissible

non-fragile controller for solving the non-non-fragile discrete-time state feedback mixed LQR/ H∞

control problem, we define the following parameter-dependent discrete time Riccati

= + If A is invertible, the

parameter-dependent discrete time Riccati equation (21) can be solved by using the following

symplectic matrix

1 1 2

ρρ

Theorem 4.1 There exists a non-fragile discrete time state feedback mixed LQR/ H∞

controller iff for a given number ρ and a sufficiently small constant δ> , there exists a 0

stabilizing solution X∞≥ to the parameter-dependent discrete time Riccati equation (21) 0

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and achieves sup{ }ˆJ w L∈2+=x X x0T ∞ 0 subject to T zw ∞< γ

Proof: Sufficiency: Suppose that for a given number ρ and a sufficiently small constant 0

δ> , there exists a stabilizing solution X∞≥ to the parameter-dependent Riccati 0equation (21) such that U1 I γ−2B X B1T 1 0

I H U H

ρ − > , we have

2( )( ) ( ) ( ) ( )

Discrete Time Mixed LQR/H ∞ Control Problems 173 Note that A B B X B R B X A Aˆ ˆ( T ˆ ˆ)−1ˆT F∞ γ−2B U B X A1 1−1 1T F∞

− + = + is stable and ΔF k( ) is an admissible uncertainty, we get that 2 1

exists a non- fragile discrete time state feedback mixed LQR/ H∞ controller

Necessity: Suppose that there exists a non-fragile discrete time state feedback mixed

LQR/ H∞ controller By Lemma 4.1, there exists a stabilizing solution X∞≥ to the 0inequality (11) such that 2

= − > , i.e., there exists a symmetric

non-negative-definite solution X∞to the inequality (11) such that 2 1

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174

by using the similar standard matrix manipulations as in the proof of Theorem 3.1 in Souza

& Xie (1992) Note that A B B X B R B X A−ˆ ˆ( T ∞ˆ+ ˆ)−1ˆT ∞ = 2 1

1 1 1T

A∞ γ−B U B X A− ∞

∞+ and ΔF k( ) is

an admissible uncertainty, the assumption that 2 1

a sufficiently small number δ> 0 , the parameter-dependent discrete time Riccati equation

(30) has a stabilizing solution X∞ and U1 I γ−2B X B1T 1 0

In this example, we will design the above system under the influence of state feedback of the

form (3) by using the discrete-times state feedback mixed LQR/ H∞ control method displayed in Theorem 3.1 All results will be computed by using MATLAB The above system is stabilizable and observable, and satisfies Assumption 3, and the eigenvalues of

matrix A are p =1 1.6133, p 0.38272 = , p = −3 0.4919;thus it is open-loop unstable

Discrete Time Mixed LQR/H ∞ Control Problems 175

2

0.6446 0.1352 0.01630.1352 0.2698 0.2882 00.0163 0.2882 0.4626

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176

Based on this, the non-fragile discrete-time state feedback mixed LQR/ H∞ controller is

0.4453 0.1789 0.06820.1613 1.1458 1.0756

we derive the simple approach to discrete time state feedback mixed LQR/ H∞ control problem by combining the Lyapunov method for proving the discrete time optimal LQR control problem with the above extension of the discrete time bounded real lemma, the argument of completion of squares of Furuta & Phoojaruenchanachi (1990) and standard

inverse matrix manipulation of Souza & Xie (1992).A related problem is the standard H∞

control problem (Doyle et al., 1989a; Iglesias & Glover, 1991; Furuta & Phoojaruenchanachai,

1990; Souza & Xie, 1992; Zhou et al 1996), another related problem is the H∞ optimal control problem arisen from Basar & Bernhard (1991) The relations among the two related

problem and mixed LQR/ H∞ control problem can be clearly explained by based on the

discrete time reference system (9)(3) The standard H∞ control problem is to find an

admissible controller K such that the H∞-norm of closed-loop transfer matrix from

disturbance input w to the controlled output z is less than a given number γ> while the 0

H∞ optimal control roblem arisen from Basar & Bernhard (1991) is to find an admissible

controller such that the H∞-norm of closed-loop transfer matrix from disturbance input w

to the controlled output z0 is less than a given number γ> for the discre time reference 0system (9)(3) Since the latter is equivalent to the problem that is to find an admissible

controller K such that supw L∈2+inf { }K ˆJ , we may recognize that the mixed LQR/ H∞ control

problem is a combination of the standard H∞ control problem and H∞ optimal control problem arisen from Basar & Bernhard (1991) The second problem considered by this

chapter is the non-fragile discrete-time state feedback mixed LQR/ H∞ control problem with controller uncertainty This problem is to extend the results of discrete-time state

feedback mixed LQR/ H∞ control problem to the system (2)(4) with controller uncertainty

In terms of the stabilizing solution to a parameter-dependent discrete time Riccati equation,

we give a design method of non-fragile discrete-time state feedback mixed LQR/ H∞

controller, and derive necessary and sufficient conditions for the existence of this non- fragile controller

7 References

T Basar, and Bernhard P (1991) H∞-optimal control and related minmax design problems:

a dynamic game approach Boston, MA: Birkhauser

D S Bernstein, and Haddad W M (1989) LQG control with an H∞ performance bound: A

Riccati equation approach, IEEE Trans Aut Control 34(3), pp 293- 305

J C Doyle, Glover K., Khargonekar P P and Francis B A (1989a) State-space solutions to

standard H2 and H∞ control problems IEEE Trans Aut Control, 34(8), pp

831-847

Discrete Time Mixed LQR/H ∞ Control Problems 177

J C Doyle, Zhou K., and Bodenheimer B (1989b) Optimal control with mixed H2 and H∞

performance objective Proceedings of 1989 American Control Conference, Pittsburh,

PA, pp 2065- 2070, 1989

J C Doyle, Zhou K., Glover K and Bodenheimer B (1994) Mixed H2 and H∞

perfor-mance objectives II: optimal control, IEEE Trans Aut Control, 39(8), pp.1575- 1587

D Famularo, Dorato P., Abdallah C T., Haddad W M and Jadbabaie A (2000) Robust

non-fragile LQ controllers: the static state case, INT J Control, 73 (2),pp.159-165

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control problems Proceedings of 1990 American Control Conference, San Diego, pp

3067-3072, 1990

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with parametric uncertainty and controller gain variations, INT J Control, 73(15),

P P Khargonekar, and Rotea M A.(1991) Mixed H2 / H∞ control: A convex optimization

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theoretic approach to H∞ control for time-varying systems SIAM J Control and Optimization, 30(2), pp.262-283

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mixed H2/ H∞ control IEEE Trans Aut Control, 39(1), pp 69-82

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time algebraic Riccati equation IEEE Trans Aut Control, 25(4), pp 631-641

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robustness Automatica, Vol 29, No 1, pp 225-228

J E Potter (1966) Matrix quadratic solution J SIAM App Math., 14, pp 496-501

P M Makila (1998) Comments ″Robust, Fragile, or Optimal ?″ IEEE Trans Aut Control.,

43(9), pp 1265-1267

M A Rotea, and Khargonekar P P (1991) H2-optimal control with an H∞-constraint: the

state-feedback case Automatica, 27(2), pp 307-316

H Rotstein, and Sznaier M (1998) An exact solution to general four-block discrete-time

mixedH2/ H∞ problems via convex optimization, IEEE Trans Aut Control, 43(10),

pp 1475-1480

C E de Souza and Xie L (1992) On the discrete-time bounded real lemma with application

in the characterization of static state feedback H∞ controllers, Systems & Control

Letters, 18, pp 61-71

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M Sznaier (1994) An exact solution to general SISO mixed H2/ H∞ problems via convex

optimization, IEEE Trans Aut Control, 39(12), pp 2511-2517

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mixedH2/ H∞ control problems, IEEE Trans Aut Control, 45(11), pp.2095-2101

X Xu (1996) A study on robust control for discrete-time systems with uncertainty, A Master

Thesis of 1995, Kobe university, Kobe, Japan, January,1996

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with controller uncertainty Proceedings of the 26th Chinese Control Conference

Zhangjiajie, Hunan, China, pp 635-639, July 26-31, 2007

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linear continuous-time systems Proceedings of the 27th Chinese Control Conference

Kunming, Yunnan, China, pp 678-682, July 16-18, 2008

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controller gain variations, INT J Control, 73(16), pp 1500-1506

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controller gain variations, Automatica, 37, pp 727-737

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and H∞ optimal control, IEEE Trans Aut Control, 37 (3), PP 355-358

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objectives I: robust performance analysis, IEEE Trans Aut Control, 39 (8), PP

1564-1574

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1996

Jun Yoneyama, Yuzu Uchida and Shusaku Nishikawa

Aoyama Gakuin University

Japan

1 Introduction

When we consider control problems of physical systems, we often see time-delay in theprocess of control algorithms and the transmission of information Time-delay often appear inmany practical systems and mathematical formulations such as electrical system, mechanicalsystem, biological system, and transportation system Hence, a system with time-delay is anatural representation for them, and its analysis and synthesis are of theoretical and practicalimportance In the past decades, research on continuous-time delay systems has been active.Difficulty that arises in continuous time-delay system is that it is infinite dimensional and acorresponding controller can be a memory feedback This class of controllers may minimize

a certain performance index, but it is difficult to implement it to practical systems due to

a memory feedback To overcome such a difficulty, a memoryless controller is used fortime-delay systems In the last decade, sufficient stability conditions have been given vialinear matrix inequalities (LMIs), and stabilization methods by memoryless controllers havebeen investigated by many researchers Since Li and de Souza considered robust stabilityand stabilization problems in (8), less conservative robust stability conditions for continuoustime-delay systems have been obtained ((7), (11)) Recently, H∞ disturbance attenuationconditions have also been given ((10), (15), (16))

On the other hand, research on discrete-time delay systems has not attracted as much attention

as that of continuous-time delay systems In addition, most results have focused on statefeedback stabilization of discrete-time systems with time-varying delays Only a few results

on observer design of discrete-time systems with time-varying delays have appeared in theliterature(for example, (9)) The results in (3), (12), (14), (18) considered discrete-time systemswith time-invariant delays Gao and Chen (4), Hara and Yoneyama (5), (6) gave robuststability conditions Fridman and Shaked (1) solved a guaranteed cost control problem.Fridman and Shaked (2), Yoneyama (17), Zhang and Han (19) considered the H∞disturbanceattenuation They have given sufficient conditions via LMIs for corresponding controlproblems Nonetheless, their conditions still show the conservatism Hara and Yoneyama(5) and Yoneyama (17) gave least conservative conditions but their conditions require manyLMI slack variables, which in turn require a large amount of computations Furthermore,

to authors’ best knowledge, few results on robust observer design problem for uncertaindiscrete-time systems with time-varying delays have given in the literature

In this paper, we consider the stabilization for a nominal discrete-time system withtime-varying delay and robust stabilization for uncertain system counterpart The systemunder consideration has time-varying delays in state, control input and output measurement.First, we obtain a stability condition for a nominal time-delay system To this end, we define

Robust Control Design of Uncertain Discrete-Time Systems with Delays

11

a Lyapunov function and use Leibniz-Newton formula and free weighting matrix method.These methods are known to reduce the conservatism in our stability condition, which areexpressed as linear matrix inequality Based on such a stability condition, a state feedbackdesign method is proposed Then, we extend our stabilization result to robust stabilization foruncertain discrete-time systems with time-varying delay Next, we consider observer designand robust observer design Similar to a stability condition, we obtain a condition such thatthe error system, which comes from the original system and its observer, is asymptoticallystable Using a stability condition of the error system, we proposed an observer designmethod Furthermore, we give a robust observer design method for an uncertain time-delaysystem Finally, we give some numerical examples to illustrate our results and to comparewith existing results.

where x(k) ∈ n is the state and u(k) ∈ m is the control A, A d , B and B dare system matrices

with appropriate dimensions d kis a time-varying delay and satisfies 0≤d m≤d k≤d Mand

d k+1≤d k where d m and d Mare known constants Uncertain matrices are of the form



ΔA ΔA d ΔB ΔB d=HF(k)E E d E1 E b

(2)

where F(k) ∈ l ×j is an unknown time-varying matrix satisfying F T(k)F(k) ≤I and H, E, E d,

E1and E bare constant matrices of appropriate dimensions

Definition 2.1. The system (1) is said to be robustly stable if it is asymptotically stable for all admissible uncertainties (2).

When we discuss a nominal system, we consider the following system

x(k+1) =Ax(k) +A d x(k−d k) +Bu(k) +B d u(k−d k) (3)Our problem is to find a control law which makes the system (1) or (3) robustly stable Let usnow consider the following memoryless feedback:

where K is a control gain to be determined Applying the control (4) to the system (1), we have

the closed-loop system

x(k+1) = ((A+ΔA) + (B+ΔB)K)x(k) + ((A d+ΔA d) + (B d+ΔB d)K)x(k−d k) (5)For the nominal case, we have

x(k+1) = (A+BK)x(k) + (A d+B d K)x(k−d k) (6)

In the following section, we consider the robust stability of the closed-loop system (5) and thestability of the closed-loop system (6)

The following lemma is useful to prove our results

Lemma 2.2. ((13)) Given matrices Q=Q T , H, E and R=R T>0 with appropriate dimensions.

3.1 Stability for nominal systems

Stability conditions for discrete-time delay system (6) are given in the following theorem

asymptotically stable if there exist matrices P1>0, P2>0, Q1>0, Q2>0, S>0, M>0,

Proof:First, we note from Leibniz-Newton formula that

and P1, P2, Q1, Q2, S and M are positive definite matrices to be determined Then, we calculate

the differenceΔV=V(k+1) −V(k)and add the left-hand-side of equations (8)-(10)

SinceΔV i(k), i=1,· · ·, 4 are calculated as follows;

3.2 Robust stability for uncertain systems

By extending Theorem 3.1, we obtain a condition for robust stability of uncertain system (5)

robustly stable if there exist matrices P1>0, P2>0, Q1>0, Q2>0, S>0, M>0,