Open access peer-reviewed chapter

Stochastic Mixed LQR/H∞ Control for Linear Discrete-Time Systems

By Xiaojie Xu

Submitted: March 4th 2012Reviewed: June 25th 2012Published: December 5th 2012

DOI: 10.5772/51019

Downloaded: 832

1. Introduction

Mixed H2/Hcontrol has received much attention in the past two decades, see Bernstein & Haddad (1989), Doyle et al. (1989b), Haddad et al. (1991), Khargonekar & Rotea (1991), Doyle et al. (1994), Limebeer et al. (1994), Chen & Zhou (2001) and references therein. The mixed H2/Hcontrol problem involves the following linear continuous-time systems

x˙(t)=Ax(t)+B0w0(t)+B1w(t)+B2u(t),x(0)=x0z(t)=C1x(t)+D12u(t)y(t)=C2x(t)+D20w0(t)+D21w(t)E1

where,x(t)Rnis the state, u(t)Rmis the control input, w0(t)Rq1is one disturbance input, w(t)Rq2is another disturbance input that belongs toL2[0,), y(t)Rris the measured output.

Bernstein & Haddad (1989) presented a combined LQG/Hcontrol problem. This problem is defined as follows: Given the stabilizable and detectable plant (1) with w0(t)=0and the expected cost function

J(Ac,Bc,Cc)=limtE{xT(t)Qx(t)+uT(t)Ru(t)}E2

determine an nth order dynamic compensator

x˙c(t)=Acx(t)+Bcy(t)u(t)=Ccxc(t)E3

which satisfies the following design criteria: (i) the closed-loop system (1) (3) is stable; (ii) the closed-loop transfer matrix Tzwfrom the disturbance input wto the controlled output zsatisfiesTzw<γ; (iii) the expected cost function J(Ac,Bc,Cc)is minimized; where, the disturbance input wis assumed to be a Gaussian white noise. Bernstein & Haddad (1989) considered merely the combined LQG/Hcontrol problem in the special case of Q=C1TC1and R=D12TD12andC1TD12=0. Since the expected cost function J(Ac,Bc,Cc)equals the square of the H2-norm of the closed-loop transfer matrix Tzwin this case, the combined LQG/Hproblem by Bernstein & Haddad (1989) has been recognized to be a mixed H2/Hproblem. In Bernstein & Haddad (1989), they considered the minimization of an “upper bound” of Tzw22subject toTzw<γ, and solved this problem by using Lagrange multiplier techniques. Doyle et al. (1989b) considered a related output feedback mixed H2/Hproblem (also see Doyle et al. 1994). The two approaches have been shown in Yeh et al. (1992) to be duals of one another in some sense. Haddad et al. (1991) gave sufficient conditions for the exstence of discrete-time static output feedback mixed H2/Hcontrollers in terms of coupled Riccati equations. In Khargonekar & Rotea (1991), they presented a convex optimisation approach to solve output feedback mixed H2/Hproblem. In Limebeer et al. (1994), they proposed a Nash game approach to the state feedback mixed H2/Hproblem, and gave necessary and sufficient conditions for the existence of a solution of this problem. Chen & Zhou (2001) generalized the method of Limebeer et al. (1994) to output feedback multiobjective H2/Hproblem. However, up till now, no approach has involved the combined LQG/Hcontrol problem (so called stochastic mixed LQR/Hcontrol problem) for linear continuous-time systems (1) with the expected cost function (2), where, Q0and R>0are the weighting matrices, w0(t)is a Gaussian white noise, and w(t)is a disturbance input that belongs toL2[0,).

In this chapter, we consider state feedback stochastic mixed LQR/Hcontrol problem for linear discrete-time systems. The deterministic problem corresponding to this problem (so called mixed LQR/Hcontrol problem) was first considered by Xu (2006). In Xu (2006), an algebraic Riccati equation approach to state feedback mixed quadratic guaranteed cost and Hcontrol problem (so called state feedback mixed QGC/Hcontrol problem) for linear discrete-time systems with uncertainty was presented. When the parameter uncertainty equals zero, the discrete-time state feedback mixed QGC/Hcontrol problem reduces to the discrete-time state feedback mixed LQR/Hcontrol problem. Xu (2011) presented respectively a state space approach and an algebraic Riccati equation approach to discrete-time state feedback mixed LQR/Hcontrol problem, and gave a sufficient condition for the existence of an admissible state feedback controller solving this problem.

On the other hand, Geromel & Peres (1985) showed a new stabilizability property of the Riccati equation solution, and proposed, based on this new property, a numerical procedure to design static output feedback suboptimal LQR controllers for linear continuous-time systems. Geromel et al. (1989) extended the results of Geromel & Peres (1985) to linear discrete-time systems. In the fact, comparing this new stabilizability property of the Riccati equation solution with the existing results (de Souza & Xie 1992, Kucera & de Souza 1995, Gadewadikar et al. 2007, Xu 2008), we can show easily that the former involves sufficient conditions for the existence of all state feedback suboptimal LQR controllers. Untill now, the technique of finding all state feedback controllers by Geromel & Peres (1985) has been extended to various control problems, such as, static output feedback stabilizability (Kucera & de Souza 1995), Hcontrol problem for linear discrete-time systems (de Souza & Xie 1992), Hcontrol problem for linear continuous-time systems (Gadewadikar et al. 2007), mixed LQR/Hcontrol problem for linear continuous-time systems (Xu 2008).

The objective of this chapter is to solve discrete-time state feedback stochastic mixed LQR/Hcontrol problem by combining the techniques of Xu (2008 and 2011) with the well known LQG theory. There are three motivations for developing this problem. First, Xu (2011) parametrized a central controller solving the discrete-time state feedback mixed LQR/Hcontrol problem in terms of an algebraic Riccati equation. However, no stochastic interpretation was provided. This paper thus presents a central solution to the discrete-time state feedback stochastic mixed LQR/Hcontrol problem. This result may be recognied to be a stochastic interpretation of the discrete-time state feedback mixed LQR/Hcontrol problem considered by Xu (2011). The second motivation for our paper is to present a characterization of all admissible state feedback controllers for solving discrete-time stochastic mixed LQR/Hcontrol problem for linear continuous-time systems in terms of a single algebraic Riccati equation with a free parameter matrix, plus two constrained conditions: One is a free parameter matrix constrained condition on the form of the gain matrix, another is an assumption that the free parameter matrix is a free admissible controller error. The third motivation for our paper is to use the above results to solve the discrete-time static output feedback stochastic mixed LQR/Hcontrol problem.

This chapter is organized as follows: Section 2 introduces several preliminary results. In Section 3, first,we define the state feedback stochastic mixed LQR/Hcontrol problem for linear discrete-time systems. Secondly, we give sufficient conditions for the existence of all admissible state feedback controllers solving the discrete-time stochastic mixed LQR/Hcontrol problem. In the rest of this section, first, we parametrize a central discrete-time state feedback stochastic mixed LQR/Hcontroller, and show that this result may be recognied to be a stochastic interpretation of discrete-time state feedback mixed LQR/Hcontrol problem considered by Xu (2011). Secondly, we propose a numerical algorithm for calclulating a kind of discrete-time state feedback stochastic mixed LQR/Hcontrollers. Also, we compare our main result with the related well known results. As a special case, Section 5 gives sufficient conditions for the existence of all admissible static output feedback controllers solving the discrete-time stochastic mixed LQR/Hcontrol problem, and proposes a numerical algorithm for calculating a discrete-time static output feedback stochastic mixed LQR/Hcontroller. In Section 6, we give two examples to illustrate the design procedures and their effectiveness. Section 7 is conclusion.

2. Preliminaries

In this section, we will review several preliminary results. First, we introduce the new stabilizability property of Riccati equation solutions for linear discrete-time systems which was presented by Geromel et al. (1989). This new stabilizability property involves the following linear discrete-time systems

x(k+1)=Ax(k)+Bu(k);x(0)=x0y(k)=Cx(k)E4

with quadratic performance index

J2:=k=0{xT(k)Qx(k)+uT(k)Ru(k)}

under the influence of state feedback of the form

u(k)=Kx(k)E5

where,x(k)Rnis the state, u(k)Rmis the control input, y(k)Rris the measured output, Q=QT0andR=RT>0. We make the following assumptions

Assumption 2.1 (A,B)is controllable.

Assumption 2.2 (A,Q12)is observable.

Define a discrete-time Riccati equation as follows:

ATSAAATSB(R+BTSB)1BTSA+Q=0E6

For simplicity the discrete-time Riccati equation (6) can be rewritten as

Πd(S)=QE7

Geromel & Peres (1985) showed a new stabilizability property of the Riccati equation solution, and proposed, based on this new property, a numerical procedure to design static output feedback suboptimal LQR controllers for linear continuous-time systems. Geromel et al. (1989) extended this new stabilizability property displayed in Geromel & Peres (1985) to linear discrete-time systems. This resut is given by the following theorem.

Theorem 2.1 (Geromel et al. 1989) For the matrix LRm×nsuch that

K=(R+BTSB)1BTSA+LE8

holds, SRn×nis a positive definite solution of the modified discrete-time Riccati equation

Πd(S)=Q+LT(R+BTSB)LE9

Then the matrix (A+BK)is stable.

When these conditions are met, the quadratic cost function J2is given by

J2=xT(0)Sx(0)

Second, we introduce the well known discrete-time bounded real lemma (see Zhou et al., 1996; Iglesias & Glover, 1991; de Souza & Xie, 1992).

Lemma 2.1 (Discrete Time Bounded Real Lemma)

Suppose thatγ>0, M(z)=[ABCD]RH, then the following two statements are equivalent:

i.M(z)<γ.

ii. There exists a stabilizing solution X0(X>0if (C,A)is observable ) to the discrete-time Riccati equation

ATXAX+γ2(ATXB+CTD)U11(BTXA+DTC)+CTC=0

such thatU1=Iγ2(DTD+BTXB)>0.

Next, we will consider the following linear discrete-time systems

x(k+1)=Ax(k)+B1w(k)+B2u(k)z(k)=C1x(k)+D12u(k)E10

under the influence of state feedback of the form

u(k)=Kx(k)E11

where,x(k)Rnis the state, u(k)Rmis the control input, w(k)Rqis the disturbance input that belongs toL2[0,),z(k)Rpis the controlled output. Letx(0)=x0.

The associated with this systems is the quadratic performance index

J2:=k=0{xT(k)Qx(k)+uT(k)Ru(k)}E12

where, Q=QT0andR=RT>0.

The closed-loop transfer matrix from the disturbance input wto the controlled output zis

Tzw(z)=[AKBKCK0]:=CK(zIAK)1BK

where, AK:=A+B2K,BK:=B1,CK:=C1+D12K.

The following lemma is an extension of the discrete-time bounded real lemma ( see Xu 2011).

Lemma 2.2 Given the system (10) under the influence of the state feedback (11), and suppose thatγ>0,Tzw(z)RH; then there exists an admissible controller Ksuch that Tzw(z)<γif there exists a stabilizing solution X0to the discrete time Riccati equation

AKTXAKX+γ2AKTXBKU11BKTXAK+CKTCK+Q+KTRK=0E13

such thatU1=Iγ2BKTXBK>0.

Proof: See the proof of Lemma 2.2 of Xu (2011). Q.E.D.

Finally, we review the result of discrete-time state feedback mixed LQR/Hcontrol problem. Xu (2011) has defined this problem as follows: Given the linear discrete-time systems (10)(11) with wL2[0,)andx(0)=x0, for a given number γ>0,determine an admissible controller that achieves

supwL2+infK{J2}subject toTzw(z)<γ.

If this controller Kexists, it is said to be a discrete-time state feedback mixed LQR/Hcontroller.

The following assumptions are imposed on the system

Assumption 2.3 (C1,A)is detectable.

Assumption 2.4 (A,B2)is stabilizable.

Assumption 2.5 D12T[C1D12]=[0I].

The solution to the problem defined in the above involves the discrete-time Riccati equation

ATXAXATXB^(B^TXB^+R^)1B^TXA+C1TC1+Q=0E14

where, B^=[γ1B1B2],R^=[I00R+I].

Xu (2011) has provided a solution to discrete-time state feedback mixed LQR/Hcontrol problem, this result is given by the following theorem.

Theorem 2.2 There exists a discrete-time state feedback mixed LQR/Hcontroller if the discrete-time Riccati equation (14) has a stabilizing solution Xand U1=Iγ2B1TXB1>0.

Moreover, this discrete-time state feedback mixed LQR/Hcontroller is given by

K=U21B2TU3A

where,U2=R+I+B2TU3B2, andU3=X+γ2XB1U11B1TX.

In this case, the discrete-time state feedback mixed LQR/Hcontroller will achieve

supwL2+infK{J2}=x0T(X+γ2XwXz)x0subject toTzw<γ.

where, A^K=AK+γ2BKU11BKTXAK,Xw=k=0{(A^Kk)TAKTXBKU12BKTXAKA^Kk}, andXz=k=0{(A^Kk)TCKTCKA^Kk}.

3. State Feedback

In this section, we consider the following linear discrete-time systems

x(k+1)=Ax(k)+B0w0(k)+B1w(k)+B2u(k)z(k)=C1x(k)+D12u(k)y(k)=C2x(k)E15

with state feedback of the form

u(k)=Kx(k)E16

where,x(k)Rnis the state, u(k)Rmis the control input, w0(k)Rq1is one disturbance input, w(k)Rq2is another disturbance that belongs toL2[0,),z(k)Rpis the controlled output, y(k)Rris the measured output.

It is assumed that x(0)is Gaussian with mean and covariance given by

E{x(0)}=x¯0cov{x(0),x(0)}:=E{(x(0)x¯0)(x(0)x¯0)T}=R0

The noise process w0(k)is a Gaussain white noise signal with properties

E{w0(k)}=0,E{w0(k)w0T(τ)}=R1(k)δ(kτ)

Furthermore, x(0)and w0(k)are assumed to be independent, w0(k)and w(k)are also assumed to be independent, where, E[]denotes expected value.

Also, we make the following assumptions:

Assumption 3.1 (C1,A)is detectable.

Assumption 3.2 (A,B2)is stabilizable.

Assumption 3.3 D12T[C1D12]=[0I].

The expected cost function corresponding to this problem is defined as follows:

JE:=limT1TE{k=0T(xT(k)Qx(k)+uT(k)Ru(k)γ2w2)}E17

where,Q=QT0,R=RT>0, and γ>0is a given number.

As is well known, a given controller Kis called admissible (for the plantG) if Kis real-rational proper, and the minimal realization of Kinternally stabilizes the state space realization (15) ofG.

Recall that the discrete-time state feedback optimal LQG problem is to find an admissible controller that minimizes the expected quadratic cost function (17) subject to the systems (15) (16) withw(k)=0, while the discrete-time state feedback Hcontrol problem is to find an admissible controller such that Tzw<γsubject to the systems (15) (16) for a given numberγ>0. While we combine the two problems for the systems (15) (16) with wL2[0,), the expected cost function (17) is a function of the control input u(k)and disturbance input w(k)in the case of γbeing fixed and x(0)being Gaussian with known statistics and w0(k)being a Gaussain white noise with known statistics. Thus it is not possible to pose a discrete-time state feedback stochastic mixed LQR/Hcontrol problem that achieves the minimization of the expected cost function (17) subject to Tzw<γfor the systems (15) (16) with wL2[0,)because the expected cost function (17) is an uncertain function depending on disturbance inputw(k). In order to eliminate this difficulty, the design criteria of discrete-time state feedback stochastic mixed LQR/Hcontrol problem should be replaced by the following design criteria:

supwL2+infK{JE}subject to Tzw<γ

because for allwL2[0,), the following inequality always exists.

infK{JE}supwL2+infK{JE}

Based on this, we define the discrete-time state feedback stochastic mixed LQR/Hcontrol problem as follows:

Discrete-time state feedback stochastic mixed LQR/Hcontrol problem: Given the linear discrete-time systems (15) (16) satisfying Assumption 3.1-3.3 with w(k)L2[0,)and the expected cost functions (17), for a given numberγ>0, find all admissible state feedback controllers Ksuch that

supwL2+{JE}subject to Tzw<γ

where, Tzw(z)is the closed loop transfer matrix from the disturbance input wto the controlled outputz.

If all these admissible controllers exist, then one of them K=Kwill achieve the design criteria

supwL2+infK{JE}subject to Tzw<γ

and it is said to be a central discrete-time state feedback stochastic mixed LQR/Hcontroller.

Remark 3.1 The discrete-time state feedback stochastic mixed LQR/Hcontrol problem defined in the above is also said to be a discrete-time state feedback combined LQG/Hcontrol problem in general case. When the disturbance inputw(k)=0, this problem reduces to a discrete-time state feedback combined LQG/Hcontrol problem arisen from Bernstein & Haddad (1989) and Haddad et al. (1991).

Remark 3.2 In the case ofw(k)=0, it is easy to show (see Bernstein & Haddad 1989, Haddad et al. 1991) that JEin (17) is equivalent to the expected cost function

JE=limkE{xT(k)Qx(k)+uT(k)Ru(k)}

Define Q=C1TC1and R=D12TD12and suppose thatC1TD12=0, then JEmay be rewritten as

JE=limkE{xT(k)Qx(k)+uT(k)Ru(k)}=limkE{xT(k)C1TC1x(k)+uT(k)D12TD12u(k)}=limkE{zT(k)z(k)}

Also, the controlled output zmay be expressed as

z=Tzw0(z)w0E18

where,Tzw0(z)=[AKB0CK0]. If w0is white noise with indensity matrix Iand the closed-loop systems is stable then

JE=limkE{zT(k)z(k)}=Tzw022

This implies that the discrete-time state feedback combined LQG/Hcontrol problem in the special case of Q=C1TC1and R=D12TD12and C1TD12=0arisen from Bernstein & Haddad (1989) and Haddad et al. (1991) is a mixed H2/Hcontrol problem.

Based on the above definition, we give sufficient conditions for the existence of all admissible state feedback controllers solving the discrete-time stochastic mixed LQR/Hcontrol problem by combining the techniques of Xu (2008 and 2011) with the well known LQG theory. This result is given by the following theorem.

Theorem 3.1 There exists a discrete-time state feedback stochastic mixed LQR/ Hcontroller if the following two conditions hold:

i. There exists a matrix ΔKsuch that

ΔK=K+U21B2TU3AE19

and Xis a symmetric non-negative definite solution of the following discrete-time Riccati equation

ATXAXATXB^(B^TXB^+R^)1B^TXA+C1TC1+Q+ΔKTU2ΔK=0E20

and A^c=AB^(B^TXB^+R^)1B^TXAis stable andU1=Iγ2B1TXB1>0;

where, B^=[γ1B1B2], R^=[I00R+I], U3=X+γ2XB1U11B1TX,U2=R+I+B2TU3B2.

ii. ΔKis an admissible controller error.

In this case, the discrete-time state feedback stochastic mixed LQR/Hcontroller will achieve

supwL2+{JE}=limT1Tk=0Ttr(B0TXB0R1(k))subject to Tzw<γ

Remark 3.3 In Theorem 3.1, the controller error is defined to be the state feedback controller Kminus the suboptimal controllerK=U21B2TU3A, where, X0satisfies the discrete-time Riccati equation (20), that is,

ΔK=KK

where, ΔKis the controller error, Kis the state feedback controller and Kis the suboptimal controller. Suppose that there exists a suboptimal controller Ksuch that AK=A+B2Kis stable, then Kand ΔKis respectively said to be an admissible controller and an admissible controller error if it belongs to the set

Ω:={ΔK:AK+B2ΔK  is stable }

Remark 3.4 The discrete-time state feedback stochastic mixed LQR/Hcontroller satisfying the conditions i-ii displayed in Theorem 3.1 is not unique. All admissible state feedback controllers satisfying these two conditions lead to all discrete-time state feedback stochastic mixed LQR/Hcontrollers.

Astrom (1971) has given the mean value of a quadratic form of normal stochastic variables. This result is given by the following lemma.

Lemma 3.1 Let xbe normal with mean mand covarianceR. Then

E{xTSx}=mTSm+trSR

For convenience, letAK=A+B2K, BK=B1, CK=C1+D12K,AK=A+B2K,BK=B1, CK=C1+D12K, andK=U21B2TU3A, where, X0satisfies the discrete-time Riccati equation (20); then we have the following lemma.

Lemma 3.2 Suppose that the conditions i-ii of Theorem 3.1 hold, then the both AKand AKare stable.

Proof: Suppose that the conditions i-ii of Theorem 3.1 hold, then it can be easily shown by using the similar standard matrix manipulations as in the proof of Theorem 3.1 in de Souza & Xie (1992) that

(B^TXB^+R^)1=[U11+U11B^1U21B^1TU11U11B^1U21U21B^1TU11U21]

where,B^1=γ1B1TXB2.Thus we have

ATXB^(B^TXB^+R^)1B^TXA=γ2ATXB1U11B1TXA+ATU3B2U21B2TU3A

Rearranging the discrete-time Riccati equation (20), we get

X=ATXA+γ2ATXB1U11B1TXAATU3B2U21B2TU3A+C1TC1+Q+ΔKTU2ΔK=AKTXAK+γ2AKTXBKU11BKTXAK+CKTCK+Q+KTRK+ΔKTU2ΔK

that is,

AKTXAKX+γ2AKTXBKU11BKTXAK+CKTCK+Q+KTRK+ΔKTU2ΔK=0E21

Since the discrete-time Riccati equation (20) has a symmetric non-negative definite solution Xand A^c=AB^(B^TXB^+R^)1B^TXAis stable, and we can show that A^c=AK+γ2BKU11BKTXAK, the discrete-time Riccati equation (21) also has a symmetric non-negative definite solution Xand AK+γ2BKU11BKTXAKalso is stable. Hence, (U11BKTXAK,AK)is detectable. Based on this, it follows from standard results on Lyapunov equations (see Lemma 2.7 a), Iglesias & Glover 1991) that AKis stable. Also, note that ΔKis an admissible controller error, so AK=AK+B2ΔKis stable. Q. E. D.

Proof of Theorem 3.1: Suppose that the conditions i-ii hold, then it follows from Lemma 3.2 that the both AKand AKare stable. This implies thatTzw(z)RH.

DefineV(x(k))=xT(k)Xx(k), where, Xis the solution to the discrete-time Riccati equation (20), then taking the differenceΔV(x(k)), we get

ΔV(x(k))=xT(k+1)Xx(k+1)xT(k)Xx(k)=xT(k)(AKTXAKX)x(k)+2wT(k)BKTXAKx(k)+wT(k)BKTXBKw(k)+2w0T(k)B0TXAKx(k)+2w0T(k)B0TXB1w(k)+w0T(k)B0TXB0w0(k)E22

On the other hand, we can rewrite the discrete-time Riccati equation (20) by using the same standard matrix manipulations as in the proof of Lemma 3.2 as follows:

ATXAX+γ2ATXB1U11B1TXAATU3B2U21B2TU3A+C1TC1+Q+ΔKTU2ΔK=0

or equivalently

AKTXAKX+γ2AKTXBKU11BKTXAK+CKTCK+Q+KTRK=0E23

It follows from Lemma 2.2 thatTzw<γ.Completing the squares for (22) and substituting (23) in (22), we get

ΔV(x(k))=z2+γ2w2γ2U112(wγ2U11BKTXAKx)2+xT(k)(AKTXAKX+γ2AKTXBKU11BKTXAK+CKTCK)x(k)+2w0T(k)B0TXAKx(k)+2w0T(k)B0TXB1w(k)+w0T(k)B0TXB0w0(k)=z2+γ2w2γ2U112(wγ2U11BKTXAKx)2xT(k)(Q+KTRK)x(k)+2w0T(k)B0TXAKx(k)+2w0T(k)B0TXB1w(k)+w0T(k)B0TXB0w0(k)

Thus, we have

JE=limT1TE{k=0T(xT(k)Qx(k)+uT(k)Ru(k)γ2w2)}=limT1TE{k=0T(ΔV(x(k))z2γ2U112(wγ2U11BKTXAKx)2+2w0T(k)B0TXAKx(k)+2w0T(k)B0TXB1w(k)+w0T(k)B0TXB0w0(k))}limT1TE{k=0T(ΔV(x(k))+2w0T(k)B0TXAKx(k)+2w0T(k)B0TXB1w(k)+w0T(k)B0TXB0w0(k))}

Note that x()=limTx(T)=0and

x(k)=AKkx0+i=0k1AKki1B0w0(i)+i=0k1AKki1B1w(i)w0T(k)B0TXAKx(k)=w0T(k)B0TXAKAKkx0+w0T(k)B0TXAKi=0k1AKki1B0w0(i)+w0T(k)B0TXAKi=0k1AKki1B0w(i)

we have

E{k=0Tw0T(k)B0TXAKx(k)}=E{k=0T(w0T(k)B0TXAKi=0k1AKki1B0w0(i))}=k=0Ti=0k1tr{B0TXAKkiB0E(w0(i)w0T(k))}=0

Based on the above, it follows from Lemma 3.1 that

supwL2+{JE}=limT1TE{xT(0)Xx(0)+k=0Tw0T(k)B0TXB0w0(k)}=limT1T{x¯0TXx¯0+tr(XR0)+k=0Ttr(B0TXB0R1(k))}=limT1Tk=0Ttr(B0TXB0R1(k))

Thus, we conclude that

supwL2+{JE}=limT1Tk=0Ttr(B0TXB0R1(k))subject to Tzw<γQ.E.D.

In the rest of this section, we give several discussions.

A. A Central Discrete-Time State Feedback Stochastic Mixed LQR/ HController

We are to find a central solution to the discrete-time state feedback stochastic mixed LQR/Hcontrol problem.This central solution involves the discrete-time Riccati equation

ATXAXATXB^(B^TXB^+R^)1B^TXA+C1TC1+Q=0E24

where, B^=[γ1B1B2],R^=[I00R+I]. Using the similar argument as in the proof of Theorem 3.1 in Xu (2011), the expected cost function JEcan be rewritten as:

JE=limT1TE{k=0T(xT(k)Qx(k)+uT(k)Ru(k)γ2w2)}=limT1TE{k=0T[ΔV(x(k))z2γ2U112(wγ2U11BKTXAKx)2+xT(AKTXAKX+γ2AKTXBKU11BKTXAK+CKTCK+Q+KTRK)x+2w0T(k)B0TXAKx(k)+2w0T(k)B0TXB1w(k)+w0T(k)B0TXB0w0(k)]}E25

Note that

AKTXAKX+γ2AKTXBKU11BKTXAK+CKTCK+Q+KTRK=ATXAXATXB^(B^TXB^+R^)1B^TXA+C1TC1+Q+(K+U21B2TU3A)TU2(K+U21B2TU3A)E26

It follows from (25) and (26) that

JE=limT1TE{k=0T(xT(k)Qx(k)+uT(k)Ru(k))}=limT1TE{k=0T[ΔV(x(k))z2+γ2w2γ2U112(wγ2U11BKTXAKx)2+U212(K+U21B2TU3A)x2+2w0T(k)B0TXAKx(k)+w0T(k)B0TXB0w0(k)]}E27

IfK=U21B2TU3A, then we get that

supwL2+infK{JE}=limT1T{x¯0TXx¯0+tr(XR0)+k=0Ttr(B0TXB0R1(k))}=limT1Tk=0Ttr(B0TXB0R1(k))

by using Lemma 3.1 and the similar argument as in the proof of Theorem 3.1. Thus, we have the following theorem:

Theorem 3.2 There exists a central discrete-time state feedback stochastic mixed LQR/Hcontroller if the discrete-time Riccati equation (24) has a stabilizing solution X0andU1=Iγ2B1TXB1>0.

Moreover, if this condition is met, the central discrete-time state feedback stochastic mixed LQR/ Hcontroller is given by

K=U21B2TU3A

where,U3=X+γ2XB1U11B1TX,U2=R+I+B2TU3B2.

In this case, the central discrete-time state feedback stochastic mixed LQR/Hcontroller will achieve

supwL2+infK{JE}=limT1Tk=0Ttr(B0TXB0R1(k))subject to Tzw<γ

Remark 3.5 WhenΔK=0, Theorem 3.1 reduces to Theorem 3.2.

Remark 3.6 Notice that the condition displayed in Theorem 3.2 is the same as one displayed in Theroem 2.2. This implies that the result given by Theorem 3.2 may be recognied to be a stochastic interpretation of the discrete-time state feedback mixed LQR/Hcontrol problem considered by Xu (2011).

B. Numerical Algorithm

In order to calculate a kind of discrete-time state feedback stochastic mixed LQR/ Hcontrollers, we propose the following numerical algorithm.

Algorithm 3.1

Step 1: Fix the two weighting matrices QandR, seti=0, ΔKi=0, U2(i)=0and a small scalarδ, and a matrix Mwhich is not zero matrix of appropriate dimensions.

Step 2: Solve the discrete-time Riccati equation

ATXiAXiATXiB^(B^TXiB^+R^)1B^TXiA+C1TC1+Q+ΔKiTU2(i)ΔKi=0

for Xisymmetric non-negative definite such that

A^ci=AB^(B^TXiB^+R^)1B^TXiAis stable andU1(i)=Iγ2B1TXiB1>0.

Step 3: CalculateU3(i), U2(i)and Kiby using the following formulas

U3(i)=Xi+γ2XiB1U1(i)1B1TXiU2(i)=R+I+B2TU3(i)B2Ki=U2(i)1B2TU3(i)A+ΔKiE28

Step 4: Let ΔKi+1=ΔKi+δM(orΔKi+1=ΔKiδM) andU2(i+1)=U2(i).

Step 5: If Ai=A+B2Kiis stable, that is, ΔKiis an admissible controller error, then increase iby1, goto Step 2; otherwise stop.

Using the above algorithm, we obtain a kind of discrete-time state feedback stochastic mixed LQR/Hcontrollers as follows:

Ki=U2(i)1B2TU3(i)A±iδM(i=0,1,2,,n,)

C. Comparison with Related Well Known Results

Comparing the result displayed in Theorem 3.1 with the earlier results, such as, Geromel & Peres (1985), Geromel et al. (1989), de Souza & Xie (1992), Kucera & de Souza (1995) and Gadewadikar et al. (2007); we know easily that all these earlier results are given in terms of a single algebraic Riccati equation with a free parameter matrix, plus a free parameter constrained condition on the form of the gain matrix. Although the result displayed in Theorem 3.1 is also given in terms of a single algebraic Riccati equation with a free parameter matrix, plus a free parameter constrained condition on the form of the gain matrix; but the free parameter matrix is also constrained to be an admissible controller error. In order to give some interpretation for this fact, we provided the following result of discrete-time state feedback stochastic mixed LQR/Hcontrol problem by combining directly the proof of Theorem 3.1, and the technique of finding all admissible state feedback controllers by Geromel & Peres (1985) ( also see Geromel et al. 1989, de Souza & Xie 1992, Kucera & de Souza 1995).

Theorem 3.3 There exists a state feedback stochastic mixed LQR/Hcontroller if there exists a matrix Lsuch that

L=K+U21B2TU3AE29

and Xis a symmetric non-negative definite solution of the following discrete-time Riccati equation

ATXAXATXB^(B^TXB^+R^)1B^TXA+C1TC1+Q+LTU2L=0E30

and AK+γ2BKU11BKTXAKis stable andU1=Iγ2B1TXB1>0.

Where, B^=[γ1B1B2], R^=[I00R+I], U3=X+γ2XB1U11B1TX,U2=R+I+B2TU3B2.

Note thatAK=AK+B2ΔK,K=K+ΔKand

A^c=AB^(B^TXB^+R^)1B^TXA=AK+γ2BKU11BKTXAK

This implies that AK+γ2BKU11BKTXAKis stable if A^cis stable and ΔKis an admissible controller error. Thus we show easily that in the case ofΔK=L, there exists a matrix Lsuch that (29) holds, where, Xis a symmetric non-negative definite solution of discrete-time Riccati equation (30) and AK+γ2BKU11BKTXAKis stable if the conditions i-ii of Theorem 3.1 hold.

At the same time, we can show also that if ΔK=Lis an admissble controller error, then the calculation of the algotithm 3.1 will become easilier. For an example, for a given admissible controller errorΔKi, the step 2 of algorithm 3.1 is to solve the discrete-time Riccati equation

ATXiAXiATXiB^(B^TXiB^+R^)1B^TXiA+Q^=0

for Xibeing a stabilizing solution, where,Q^=C1TC1+Q+ΔKiTU2(i)ΔKi. Since A^ci=AB^(B^TXiB^+R^)1B^TXiAis stable and ΔKiis an admissible controller error, so AK+γ2BKU1(i)1BKTXiAKis stable. This implies the condition ii displayed in Theorem 3.1 makes the calculation of the algorithm 3.1 become easier.

4. Static Output Feedback

This section consider discrete-time static output feedback stochastic mixed LQR/Hcontrol problem. This problem is defined as follows:

Discrete-time static ouput feedback stochastic mixed LQR/Hcontrol problem: Consider the system (15) under the influence of static output feedback of the form

u(k)=Fy(k)

withwL2[0,), for a given numberγ>0, determine an admissible static output feedback controller Fsuch that

supwL2+{JE}subject to Tzw<γ

If this admissible controller exists, it is said to be a discrete-time static output feedback stochastic mixed LQR/Hcontroller. As is well known, the discrete-time static output feedback stochastic mixed LQR/Hcontrol problem is equivalent to the discrete-time state feedback stochastic mixed LQR/Hcontrol problem for the systems (15) (16), where, Kis constrained to have the form ofK=FC2. This problem is also said to be a structural constrained state feedback stochastic mixed LQR/Hcontrol problem.Based the above, we can obtain all solution to discrete-time static output feedback stochastic mixed LQR/Hcontrol problem by using the result of Theorem 3.1 as follows:

Theorem 4.1 There exists a discrete-time static output feedback stochastic mixed LQR/ Hcontroller if the following two conditions hold:

i.There exists a matrix ΔKsuch that

ΔK=FC2+U21B2TU3AE31

and Xis a symmetric non-negative definite solution of the following discrete-time Riccati equation

ATXAXATXB^(B^TXB^+R^)1B^TXA+C1TC1+Q+ΔKTU2ΔK=0E32

and A^c=AB^(B^TXB^+R^)1B^TXAis stable andU1=Iγ2B1TXB1>0.

Where, B^=[γ1B1B2], R^=[I00R+I], U3=X+γ2XB1U11B1TX,U2=R+I+B2TU3B2.

ii. ΔKis an admissible controller error.

In this case, the discrete-time static output feedback stochastic mixed LQR/Hcontroller will achieve

supwL2+{JE}=limT1Tk=0Ttr(B0TXB0R1(k))subject to Tzw<γ

Remark 4.1 In Theorem 4.1, define a suboptimal controller asK=U21B2TU3A, thenΔK=FC2K. As is discussed in Remark 3.1, suppose that there exists a suboptimal controller Ksuch that AK=A+B2Kis stable, then ΔKis an admissible controller error if it belongs to the set:

Ω:={ΔK:A+B2FC2is stable}

It should be noted that Theorem 4.1 does not tell us how to calculate a discrete-time static output feedback stochastic mixed LQR/HcontrollerF. In order to do this, we present, based on the algorithms proposed by Geromel & Peres (1985) and Kucera & de Souza (1995), a numerical algorithm for computing a discrete-time static output feedback stochastic mixed LQR/Hcontroller Fand a solution Xto discrete-time Riccati equation (32). This numerical algorithm is given as follows:

Algorithm 4.1

Step 1: Fix the two weighting matrices QandR, seti=0, ΔKi=0, andU2(i)=0.

Step 2: Solve the discrete-time Riccati equation

ATXiAXiATXiB^(B^TXiB^+R^)1B^TXiA+C1TC1+Q+ΔKiTU2(i)ΔKi=0

for Xisymmetric non-negative definite such that

A^ci=AB^(B^TXiB^+R^)1B^TXiAis stable andU1(i)=Iγ2B1TXiB1>0.

Step 3: CalculateU3(i+1), U2(i+1)and ΔKi+1by using the following formulas

U3(i+1)=Xi+γ2XiB1U1(i)1B1TXiU2(i+1)=R+I+B2TU3(i+1)B2ΔKi+1=U2(i+1)1B2TU3(i+1)A(C2T(C2C2T)1C2I)

Step 4: If ΔKi+1is an admissible controller error, then increase iby1, and goto Step 2; otherwise stop.

If the four sequencesX0,X1,,Xi,, U1(1),U1(2),,U1(i),,U2(1),U2(2),,U2(i),, and U3(1),U3(2),,U3(i),converges, say toX,U1,U2andU3, respectively; then the both two conditions displayed in Theorem 4.1 are met. In this case, a discrete-time static output feedback stochastic mixed LQR/Hcontrollers is parameterized as follows:

F=U21B2TU3AC2T(C2C2T)1

In this chapter, we will not prove the convergence of the above algorithm. This will is another subject.

5. Numerical Examples

In this section, we present two examples to illustrate the design methods displayed in Section 3 and 4 respectively.

Example 5.1 Consider the following linear discrete-time system (15) under the influence of state feedback of the formu(k)=Kx(k), its parameter matrices are

A=[0240.2],B0=[0.10.2],B1=[0.50.3],B2=[10]C1=[1000],C2=[1001],D12=[01]

The above system satisfies Assumption 3.1-3.3, and the open-loop poles of this system are p1=2.7302,p2=2.9302; thus it is open-loop unstable.

LetR=1,Q=[1001], γ=9.5, δ=0.01,M=[0.041.2]; by using algorithm 3.1, we solve the discrete-time Riccati equation (20) to getXi,U1(i), Ki(i=0,1,2,,10)and the corresponding closed-loop poles. The calculating results of algorithm 3.1 are listed in Table 1.

It is shown in Table 1 that when the iteration indexi=10,X100and U1(10)=0.29270, thus the discrete-time state feedback stochastic mixed LQR/Hcontroller does not exist in this case. Of course, Table 1 does not list all discrete-time state feedback stochastic mixed LQR/Hcontrollers because we do not calculating all these controllers by using Algorithm in this example. In order to illustrate further the results, we give the trajectories of state of the system (15) with the state feedback of the form u(k)=Kx(k)for the resulting discrete-time state feedback stochastic mixed LQR/Hcontroller K=[0.30712.0901]. The resulting closed-loop system is

x(k+1)=(A+B2K)x(k)+B0w0(k)+B1w(k)z(k)=(C1+D12K)x(k)

where, A+B2K=[0.30710.09014.00000.2000],C1+D12K=[1.000000.30712.0901].

IterationIndexiSolution of DARE X iAdditionalConditionU1(i)State FeedbackControllerKiThe Closed-Loop Poles
0X0>00.5571[0.28861.9992]p1=0.2951,
p2=0.2065
1X1>00.5553[0.28922.0113]p1=0.1653,
p2=0.0761
2X2>00.5496[0.29032.0237]p1,2=0.0451
±j0.1862
3X3>00.5398[0.29182.0363]p1,2=0.0459
±j0.2912
4X4>00.5250[0.29402.0492]p1,2=0.0470
±j0.3686
5X5>00.5040[0.29692.0624]p1,2=0.0485
±j0.4335
6X6>00.4739[0.30102.0760]p1,2=0.0505
±j0.4911
7X7>00.4295[0.30712.0901]p1,2=0.0535
±j0.5440
8X8>00.3578[0.31672.1049]p1,2=0.0584
±j0.5939
9X9>00.2166[0.33602.1212]p1,2=0.0680
±j0.6427
10X10>00.2927

Table 1.

The calculating results of algorithm 3.1.

To determine the mean value function, we take mathematical expectation of the both hand of the above two equations to get

x¯(k+1)=(A+B2K)x¯(k)+B1w(k)z¯(k)=(C1+D12K)x¯(k)

where, E{x(k)}=x¯(k), E{z(k)}=z¯(k),E{x(0)}=x¯0.

Letw(k)=γ2U11B1TX(A+B2K)x¯(k), then the trajectories of mean values of states of resulting closed-loop system with x¯0=[32]Tare given in Fig. 1.

Figure 1.

The trajectories of mean values of states of resulting system in Example 5.1.

Example 5.2 Consider the following linear discrete-time system (15) with static output feedback of the formu(k)=Fy(k), its parameter matrices are as same as Example 5.1.

When C2is quare and invertible, that is, all state variable are measurable, we may assume without loss of generality that C2=I; letγ=6.5, R=1andQ=[1001], by solving the discrete-time Riccati equation (24), we get that the central discrete-time state feedback stochastic mixed LQR/Hcontroller displayed in Theorem 3.2 is

K=[0.37192.0176]

and the poles of resulting closed-loop system arep1=0.1923,p2=0.0205.

WhenC2=[15.4125], letγ=6.5,R=1, Q=[1001], by using Algorithm 4.1, we solve the discrete-time Riccati equation (32) to get

X=[148.90068.83168.83169.5122]>0,U1=0.0360

Thus the discrete-time static output feedback stochastic mixed LQR/Hcontroller displayed in Theorem 4.1 isF=0.3727. The resulting closed-loop system is

x(k+1)=(A+B2FC2)x(k)+B0w0(k)+B1w(k)z(k)=(C1+D12FC2)x(k)

where, A+B2FC2=[0.37270.01744.00000.2000],C1+D12FC2=[1.000000.37272.0174].

Taking mathematical expectation of the both hand of the above two equations to get

x¯(k+1)=(A+B2FC2)x¯(k)+B1w(k)z(k)=(C1+D12FC2)x¯(k)

where, E{x(k)}=x¯(k), E{z(k)}=z¯(k),E{x(0)}=x¯0.

Letw(k)=γ2U11B1TX(A+B2FC2)x¯(k), then the trajectories of mean values of states of resulting closed-loop system with x¯0=[12]Tare given in Fig. 2.

Figure 2.

The trajectories of mean values of states of resulting system in Example 5.2.

6. Conclusion

In this chapter, we provide a characterization of all state feedback controllers for solving the discrete-time stochastic mixed LQR/Hcontrol problem for linear discrete-time systems by the technique of Xu (2008 and 2011) with the well known LQG theory. Sufficient conditions for the existence of all state feedback controllers solving the discrete-time stochastic mixed LQR/Hcontrol problem are given in terms of a single algebraic Riccati equation with a free parameter matrix, plus two constrained conditions: One is a free parameter matrix constrained condition on the form of the gain matrix, another is an assumption that the free parameter matrix is a free admissible controller error. Also, a numerical algorithm for calculating a kind of discrete-time state feedback stochastic mixed LQR/Hcontrollers are proposed. As one special case, the central discrete-time state feedback stochastic mixed LQR./Hcontroller is given in terms of an algebraic Riccati equation. This provides an interpretation of discrete-time state feedback mixed LQR/Hcontrol problem. As another special case, sufficient conditions for the existence of all static output feedback controllers solving the discrete-time stochastic mixed LQR/Hcontrol problem are given. A numerical algorithm for calculating a static output feedback stochastic mixed LQR/Hcontroller is also presented.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Xiaojie Xu (December 5th 2012). Stochastic Mixed LQR/H∞ Control for Linear Discrete-Time Systems, Advances in Discrete Time Systems, Magdi S. Mahmoud, IntechOpen, DOI: 10.5772/51019. Available from:

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