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where,x(t)∈Rn is 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/H∞ control problem. This problem is defined as follows: Given the stabilizable and detectable plant (1) with w0(t)=0 and the expected cost function
J(Ac,Bc,Cc)=limt→∞E{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 closedloop system (1) (3) is stable; (ii) the closedloop transfer matrix Tzw from the disturbance input wto the controlled output zsatisfies‖Tzw‖∞<γ; (iii) the expected cost function J(Ac,Bc,Cc)is minimized; where, the disturbance input w is assumed to be a Gaussian white noise. Bernstein & Haddad (1989) considered merely the combined LQG/H∞ control problem in the special case of Q=C1TC1 and R=D12TD12 andC1TD12=0. Since the expected cost function J(Ac,Bc,Cc) equals the square of the H2norm of the closedloop transfer matrix Tzw in this case, the combined LQG/H∞ problem by Bernstein & Haddad (1989) has been recognized to be a mixed H2/H∞ problem. In Bernstein & Haddad (1989), they considered the minimization of an “upper bound” of ‖Tzw‖22 subject to‖Tzw‖∞<γ, and solved this problem by using Lagrange multiplier techniques. Doyle et al. (1989b) considered a related output feedback mixed H2/H∞ problem (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 discretetime static output feedback mixed H2/H∞controllers in terms of coupled Riccati equations. In Khargonekar & Rotea (1991), they presented a convex optimisation approach to solve output feedback mixed H2/H∞ problem. In Limebeer et al. (1994), they proposed a Nash game approach to the state feedback mixed H2/H∞ problem, 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/H∞ problem. However, up till now, no approach has involved the combined LQG/H∞ control problem (so called stochastic mixed LQR/H∞ control problem) for linear continuoustime systems (1) with the expected cost function (2), where, Q≥0and 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/H∞ control problem for linear discretetime systems. The deterministic problem corresponding to this problem (so called mixed LQR/H∞ control problem) was first considered by Xu (2006). In Xu (2006), an algebraic Riccati equation approach to state feedback mixed quadratic guaranteed cost and H∞ control problem (so called state feedback mixed QGC/H∞ control problem) for linear discretetime systems with uncertainty was presented. When the parameter uncertainty equals zero, the discretetime state feedback mixed QGC/H∞ control problem reduces to the discretetime state feedback mixed LQR/H∞ control problem. Xu (2011) presented respectively a state space approach and an algebraic Riccati equation approach to discretetime state feedback mixed LQR/H∞ control 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 continuoustime systems. Geromel et al. (1989) extended the results of Geromel & Peres (1985) to linear discretetime 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), H∞control problem for linear discretetime systems (de Souza & Xie 1992), H∞control problem for linear continuoustime systems (Gadewadikar et al. 2007), mixed LQR/H∞ control problem for linear continuoustime systems (Xu 2008).
The objective of this chapter is to solve discretetime state feedback stochastic mixed LQR/H∞ control 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 discretetime state feedback mixed LQR/H∞ control problem in terms of an algebraic Riccati equation. However, no stochastic interpretation was provided. This paper thus presents a central solution to the discretetime state feedback stochastic mixed LQR/H∞ control problem. This result may be recognied to be a stochastic interpretation of the discretetime state feedback mixed LQR/H∞ control problem considered by Xu (2011). The second motivation for our paper is to present a characterization of all admissible state feedback controllers for solving discretetime stochastic mixed LQR/H∞ control problem for linear continuoustime 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 discretetime static output feedback stochastic mixed LQR/H∞ control 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/H∞ control problem for linear discretetime systems. Secondly, we give sufficient conditions for the existence of all admissible state feedback controllers solving the discretetime stochastic mixed LQR/H∞ control problem. In the rest of this section, first, we parametrize a central discretetime state feedback stochastic mixed LQR/H∞ controller, and show that this result may be recognied to be a stochastic interpretation of discretetime state feedback mixed LQR/H∞ control problem considered by Xu (2011). Secondly, we propose a numerical algorithm for calclulating a kind of discretetime state feedback stochastic mixed LQR/H∞ controllers. 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 discretetime stochastic mixed LQR/H∞ control problem, and proposes a numerical algorithm for calculating a discretetime static output feedback stochastic mixed LQR/H∞ controller. In Section 6, we give two examples to illustrate the design procedures and their effectiveness. Section 7 is conclusion.
In this section, we will review several preliminary results. First, we introduce the new stabilizability property of Riccati equation solutions for linear discretetime systems which was presented by Geromel et al. (1989). This new stabilizability property involves the following linear discretetime 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)∈Rn is the state, u(k)∈Rmis the control input, y(k)∈Rris the measured output, Q=QT≥0andR=RT>0. We make the following assumptions
Assumption 2.1(A,B) is controllable.
Assumption 2.2(A,Q12) is observable.
Define a discretetime Riccati equation as follows:
ATSA−A−ATSB(R+BTSB)−1BTSA+Q=0E6
For simplicity the discretetime 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 continuoustime systems. Geromel et al. (1989) extended this new stabilizability property displayed in Geromel & Peres (1985) to linear discretetime systems. This resut is given by the following theorem.
Suppose thatγ>0, M(z)=[ABCD]∈RH∞, then the following two statements are equivalent:
i.‖M(z)‖∞<γ.
ii. There exists a stabilizing solution X≥0 (X>0if (C,A)is observable ) to the discretetime Riccati equation
ATXA−X+γ−2(ATXB+CTD)U1−1(BTXA+DTC)+CTC=0
such thatU1=I−γ−2(DTD+BTXB)>0.
Next, we will consider the following linear discretetime 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)∈Rn is the state, u(k)∈Rmis the control input, w(k)∈Rqis the disturbance input that belongs toL2[0,∞),z(k)∈Rp is 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=QT≥0andR=RT>0.
The closedloop transfer matrix from the disturbance input w to the controlled output z is
Tzw(z)=[AKBKCK0]:=CK(zI−AK)−1BK
where, AK:=A+B2K,BK:=B1 ,CK:=C1+D12K.
The following lemma is an extension of the discretetime 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 K such that ‖Tzw(z)‖∞<γ if there exists a stabilizing solution X∞≥0 to the discrete time Riccati equation
Proof: See the proof of Lemma 2.2 of Xu (2011). Q.E.D.
Finally, we review the result of discretetime state feedback mixed LQR/H∞ control problem. Xu (2011) has defined this problem as follows: Given the linear discretetime systems (10)(11) with w∈L2[0,∞)andx(0)=x0, for a given number γ>0, determine an admissible controller that achieves
supw∈L2+infK{J2} subject to‖Tzw(z)‖∞<γ.
If this controller K exists, it is said to be a discretetime state feedback mixed LQR/H∞ controller.
The following assumptions are imposed on the system
Assumption 2.3(C1,A) is detectable.
Assumption 2.4(A,B2) is stabilizable.
Assumption 2.5D12T[C1D12]=[0I].
The solution to the problem defined in the above involves the discretetime Riccati equation
ATX∞A−X∞−ATX∞B^(B^TX∞B^+R^)−1B^TX∞A+C1TC1+Q=0E14
where, B^=[γ−1B1B2],R^=[−I00R+I].
Xu (2011) has provided a solution to discretetime state feedback mixed LQR/H∞control problem, this result is given by the following theorem.
Theorem 2.2 There exists a discretetime state feedback mixed LQR/H∞ controller if the discretetime Riccati equation (14) has a stabilizing solution X∞ and U1=I−γ−2B1TX∞B1>0.
Moreover, this discretetime state feedback mixed LQR/H∞controller is given by
K=−U2−1B2TU3A
where,U2=R+I+B2TU3B2 , andU3=X∞+γ−2X∞B1U1−1B1TX∞.
In this case, the discretetime state feedback mixed LQR/H∞controller will achieve
where,x(k)∈Rn is 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)∈Rp is the controlled output, y(k)∈Rris the measured output.
It is assumed that x(0)is Gaussian with mean and covariance given by
As is well known, a given controller K is called admissible (for the plantG) if K is realrational proper, and the minimal realization of K internally stabilizes the state space realization (15) ofG.
Recall that the discretetime 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 discretetime state feedback H∞ control problem is to find an admissible controller such that ‖Tzw‖∞<γ subject to the systems (15) (16) for a given numberγ>0. While we combine the two problems for the systems (15) (16) with w∈L2[0,∞), the expected cost function (17) is a function of the control input u(k) and disturbance input w(k) in the case of γ being 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 discretetime state feedback stochastic mixed LQR/H∞ control problem that achieves the minimization of the expected cost function (17) subject to ‖Tzw‖∞<γ for the systems (15) (16) with w∈L2[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 discretetime state feedback stochastic mixed LQR/H∞ control problem should be replaced by the following design criteria:
supw∈L2+infK{JE} subject to ‖Tzw‖∞<γ
because for allw∈L2[0,∞), the following inequality always exists.
infK{JE}≤supw∈L2+infK{JE}
Based on this, we define the discretetime state feedback stochastic mixed LQR/H∞ control problem as follows:
Discretetime state feedback stochastic mixed LQR/H∞ control problem: Given the linear discretetime systems (15) (16) satisfying Assumption 3.13.3 with w(k)∈L2[0,∞) and the expected cost functions (17), for a given numberγ>0, find all admissible state feedback controllers K such that
supw∈L2+{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=K∗ will achieve the design criteria
supw∈L2+infK{JE} subject to ‖Tzw‖∞<γ
and it is said to be a central discretetime state feedback stochastic mixed LQR/H∞ controller.
Remark 3.1 The discretetime state feedback stochastic mixed LQR/H∞ control problem defined in the above is also said to be a discretetime state feedback combined LQG/H∞ control problem in general case. When the disturbance inputw(k)=0, this problem reduces to a discretetime state feedback combined LQG/H∞ control problem arisen from Bernstein & Haddad (1989) and Haddad et al. (1991).
where,Tzw0(z)=[AKB0CK0]. If w0 is white noise with indensity matrix I and the closedloop systems is stable then
JE=limk→∞E{zT(k)z(k)}=‖Tzw0‖22
This implies that the discretetime state feedback combined LQG/H∞ control problem in the special case of Q=C1TC1 and R=D12TD12 and C1TD12=0 arisen from Bernstein & Haddad (1989) and Haddad et al. (1991) is a mixed H2/H∞ control problem.
Based on the above definition, we give sufficient conditions for the existence of all admissible state feedback controllers solving the discretetime stochastic mixed LQR/H∞ control 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 discretetime state feedback stochastic mixed LQR/ H∞ controller if the following two conditions hold:
i. There exists a matrix ΔK such that
ΔK=K+U2−1B2TU3AE19
and X∞ is a symmetric nonnegative definite solution of the following discretetime Riccati equation
In this case, the discretetime state feedback stochastic mixed LQR/H∞ controller will achieve
supw∈L2+{JE}=limT→∞1T∑k=0Ttr(B0TX∞B0R1(k)) subject to ‖Tzw‖∞<γ
Remark 3.3 In Theorem 3.1, the controller error is defined to be the state feedback controller K minus the suboptimal controllerK∗=−U2−1B2TU3A, where, X∞≥0satisfies the discretetime Riccati equation (20), that is,
ΔK=K−K∗
where, ΔKis the controller error, Kis the state feedback controller and K∗ is the suboptimal controller. Suppose that there exists a suboptimal controller K∗ such that AK∗=A+B2K∗ is stable, then K and ΔK is 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 discretetime state feedback stochastic mixed LQR/H∞ controller satisfying the conditions iii displayed in Theorem 3.1 is not unique. All admissible state feedback controllers satisfying these two conditions lead to all discretetime state feedback stochastic mixed LQR/H∞ controllers.
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 x be normal with mean m and 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∗=−U2−1B2TU3A, where, X∞≥0satisfies the discretetime Riccati equation (20); then we have the following lemma.
Lemma 3.2 Suppose that the conditions iii of Theorem 3.1 hold, then the both AK∗ and AK are stable.
Proof: Suppose that the conditions iii 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
Since the discretetime Riccati equation (20) has a symmetric nonnegative definite solution X∞ and A^c=A−B^(B^TX∞B^+R^)−1B^TX∞A is stable, and we can show that A^c=AK∗+γ−2BK∗U1−1BK∗TX∞AK∗, the discretetime Riccati equation (21) also has a symmetric nonnegative definite solution X∞ and AK∗+γ−2BK∗U1−1BK∗TX∞AK∗ also is stable. Hence, (U1−1BK∗TX∞AK∗,AK∗)is detectable. Based on this, it follows from standard results on Lyapunov equations (see Lemma 2.7 a), Iglesias & Glover 1991) that AK∗ is stable. Also, note that ΔK is an admissible controller error, so AK=AK∗+B2ΔK is stable. Q. E. D.
Proof of Theorem 3.1: Suppose that the conditions iii hold, then it follows from Lemma 3.2 that the both AK∗ and AK are stable. This implies thatTzw(z)∈RH∞.
DefineV(x(k))=xT(k)X∞x(k), where, X∞is the solution to the discretetime Riccati equation (20), then taking the differenceΔV(x(k)), we get
On the other hand, we can rewrite the discretetime Riccati equation (20) by using the same standard matrix manipulations as in the proof of Lemma 3.2 as follows:
supw∈L2+{JE}=limT→∞1T∑k=0Ttr(B0TX∞B0R1(k)) subject to ‖Tzw‖∞<γ Q.E.D.
In the rest of this section, we give several discussions.
A. A Central DiscreteTime State Feedback Stochastic Mixed LQR/H∞Controller
We are to find a central solution to the discretetime state feedback stochastic mixed LQR/H∞ control problem.This central solution involves the discretetime Riccati equation
ATX∞A−X∞−ATX∞B^(B^TX∞B^+R^)−1B^TX∞A+C1TC1+Q=0E24
where, B^=[γ−1B1B2],R^=[−I00R+I]. Using the similar argument as in the proof of Theorem 3.1 in Xu (2011), the expected cost function JE can be rewritten as:
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 discretetime state feedback stochastic mixed LQR/H∞ controller if the discretetime Riccati equation (24) has a stabilizing solution X∞≥0 andU1=I−γ−2B1TX∞B1>0.
Moreover, if this condition is met, the central discretetime state feedback stochastic mixed LQR/ H∞ controller is given by
K=−U2−1B2TU3A
where,U3=X∞+γ−2X∞B1U1−1B1TX∞ ,U2=R+I+B2TU3B2.
In this case, the central discretetime state feedback stochastic mixed LQR/H∞ controller will achieve
supw∈L2+infK{JE}=limT→∞1T∑k=0Ttr(B0TX∞B0R1(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 discretetime state feedback mixed LQR/H∞ control problem considered by Xu (2011).
B. Numerical Algorithm
In order to calculate a kind of discretetime state feedback stochastic mixed LQR/ H∞ controllers, we propose the following numerical algorithm.
Algorithm 3.1
Step 1: Fix the two weighting matrices Q andR, seti=0, ΔKi=0, U2(i)=0and a small scalarδ, and a matrix Mwhich is not zero matrix of appropriate dimensions.
Step 4: Let ΔKi+1=ΔKi+δM(orΔKi+1=ΔKi−δM) andU2(i+1)=U2(i).
Step 5: If Ai=A+B2Ki is stable, that is, ΔKiis an admissible controller error, then increase i by1, goto Step 2; otherwise stop.
Using the above algorithm, we obtain a kind of discretetime state feedback stochastic mixed LQR/H∞ controllers 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 discretetime state feedback stochastic mixed LQR/H∞ control 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/H∞controller if there exists a matrix L such that
L=K+U2−1B2TU3AE29
and X∞ is a symmetric nonnegative definite solution of the following discretetime Riccati equation
This implies that AK+γ−2BKU1−1BKTX∞AK is stable if A^c is stable and ΔKis an admissible controller error. Thus we show easily that in the case ofΔK=L, there exists a matrix L such that (29) holds, where, X∞is a symmetric nonnegative definite solution of discretetime Riccati equation (30) and AK+γ−2BKU1−1BKTX∞AK is stable if the conditions iii of Theorem 3.1 hold.
At the same time, we can show also that if ΔK=L is 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 discretetime Riccati equation
ATXiA−Xi−ATXiB^(B^TXiB^+R^)−1B^TXiA+Q^=0
for Xi being a stabilizing solution, where,Q^=C1TC1+Q+ΔKiTU2(i)ΔKi. Since A^ci=A−B^(B^TXiB^+R^)−1B^TXiA is stable and ΔKi is an admissible controller error, so AK+γ−2BKU1(i)−1BKTXiAK is stable. This implies the condition ii displayed in Theorem 3.1 makes the calculation of the algorithm 3.1 become easier.
This section consider discretetime static output feedback stochastic mixed LQR/H∞ control problem. This problem is defined as follows:
Discretetime static ouput feedback stochastic mixed LQR/H∞ control problem: Consider the system (15) under the influence of static output feedback of the form
u(k)=F∞y(k)
withw∈L2[0,∞), for a given numberγ>0, determine an admissible static output feedback controller F∞ such that
supw∈L2+{JE} subject to ‖Tzw‖∞<γ
If this admissible controller exists, it is said to be a discretetime static output feedback stochastic mixed LQR/H∞ controller. As is well known, the discretetime static output feedback stochastic mixed LQR/H∞ control problem is equivalent to the discretetime state feedback stochastic mixed LQR/H∞ control problem for the systems (15) (16), where, Kis constrained to have the form ofK=F∞C2. This problem is also said to be a structural constrained state feedback stochastic mixed LQR/H∞ control problem.Based the above, we can obtain all solution to discretetime static output feedback stochastic mixed LQR/H∞ control problem by using the result of Theorem 3.1 as follows:
Theorem 4.1 There exists a discretetime static output feedback stochastic mixed LQR/ H∞controller if the following two conditions hold:
i.There exists a matrix ΔK such that
ΔK=FC2+U2−1B2TU3AE31
and X∞ is a symmetric nonnegative definite solution of the following discretetime Riccati equation
In this case, the discretetime static output feedback stochastic mixed LQR/H∞ controller will achieve
supw∈L2+{JE}=limT→∞1T∑k=0Ttr(B0TX∞B0R1(k)) subject to ‖Tzw‖∞<γ
Remark 4.1 In Theorem 4.1, define a suboptimal controller asK∗=−U2−1B2TU3A, thenΔK=F∞C2−K∗. As is discussed in Remark 3.1, suppose that there exists a suboptimal controller K∗ such that AK∗=A+B2K∗ is stable, then ΔK is an admissible controller error if it belongs to the set:
Ω:={ΔK:A+B2F∞C2 is stable}
It should be noted that Theorem 4.1 does not tell us how to calculate a discretetime static output feedback stochastic mixed LQR/H∞ controllerF∞. 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 discretetime static output feedback stochastic mixed LQR/H∞ controller F∞ and a solution X∞ to discretetime Riccati equation (32). This numerical algorithm is given as follows:
Algorithm 4.1
Step 1: Fix the two weighting matrices Q andR, seti=0, ΔKi=0, andU2(i)=0.
Step 4: If ΔKi+1 is an admissible controller error, then increase i by1, 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 ,U2 andU3, respectively; then the both two conditions displayed in Theorem 4.1 are met. In this case, a discretetime static output feedback stochastic mixed LQR/H∞ controllers is parameterized as follows:
F∞=−U2−1B2TU3AC2T(C2C2T)−1
In this chapter, we will not prove the convergence of the above algorithm. This will is another subject.
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 discretetime system (15) under the influence of state feedback of the formu(k)=Kx(k), its parameter matrices are
The above system satisfies Assumption 3.13.3, and the openloop poles of this system are p1=−2.7302,p2=2.9302; thus it is openloop unstable.
LetR=1,Q=[1001] , γ=9.5, δ=0.01,M=[−0.04−1.2]; by using algorithm 3.1, we solve the discretetime Riccati equation (20) to getXi,U1(i) , Ki(i=0,1,2,⋯,10)and the corresponding closedloop 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,X100 and U1(10)=−0.29270, thus the discretetime state feedback stochastic mixed LQR/H∞ controller does not exist in this case. Of course, Table 1 does not list all discretetime state feedback stochastic mixed LQR/H∞ controllers 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 discretetime state feedback stochastic mixed LQR/H∞ controller K=[−0.3071−2.0901]. The resulting closedloop system is
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)=γ−2U1−1B1TX∞(A+B2K)x¯(k), then the trajectories of mean values of states of resulting closedloop system with x¯0=[32]T are given in Fig. 1.
Example 5.2 Consider the following linear discretetime system (15) with static output feedback of the formu(k)=F∞y(k), its parameter matrices are as same as Example 5.1.
When C2 is 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 discretetime Riccati equation (24), we get that the central discretetime state feedback stochastic mixed LQR/H∞ controller displayed in Theorem 3.2 is
K∗=[−0.3719−2.0176]
and the poles of resulting closedloop 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 discretetime Riccati equation (32) to get
X∞=[148.90068.83168.83169.5122]>0,U1=0.0360
Thus the discretetime static output feedback stochastic mixed LQR/H∞ controller displayed in Theorem 4.1 isF∞=−0.3727. The resulting closedloop system is
Letw(k)=γ−2U1−1B1TX∞(A+B2F∞C2)x¯(k), then the trajectories of mean values of states of resulting closedloop system with x¯0=[12]T are given in Fig. 2.
In this chapter, we provide a characterization of all state feedback controllers for solving the discretetime stochastic mixed LQR/H∞ control problem for linear discretetime 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 discretetime stochastic mixed LQR/H∞ control 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 discretetime state feedback stochastic mixed LQR/H∞ controllers are proposed. As one special case, the central discretetime state feedback stochastic mixed LQR./H∞ controller is given in terms of an algebraic Riccati equation. This provides an interpretation of discretetime state feedback mixed LQR/H∞ control problem. As another special case, sufficient conditions for the existence of all static output feedback controllers solving the discretetime stochastic mixed LQR/H∞ control problem are given. A numerical algorithm for calculating a static output feedback stochastic mixed LQR/H∞ controller is also presented.
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Written By
Xiaojie Xu
Submitted: 04 March 2012Published: 05 December 2012