**1. Introduction**

Mixed *H*<sup>2</sup> / *H∞* control 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 *H*<sup>2</sup> / *H∞* control problem involves the following linear continuous-time systems

$$\dot{x}(t) = Ax(t) + B\_0 w\_0(t) + B\_1 w(t) + B\_2 u(t), \ x(0) = x\_0$$

$$z(t) = C\_1 x(t) + D\_{12} u(t) \tag{1}$$

$$y(t) = C\_2 x(t) + D\_{20} w\_0(t) + D\_{21} w(t)$$

where,*x*(*t*)∈*<sup>R</sup> <sup>n</sup>* is the state, *u*(*t*)∈*<sup>R</sup> <sup>m</sup>*is the control input, *w*0(*t*)∈*<sup>R</sup> <sup>q</sup>*<sup>1</sup> is one disturbance in‐ put, *w*(*t*)∈*R <sup>q</sup>*<sup>2</sup> is another disturbance input that belongs to*<sup>L</sup>* <sup>2</sup> 0,*∞*), *y*(*t*)∈*<sup>R</sup> <sup>r</sup>* is the measured output.

Bernstein & Haddad (1989) presented a combined LQG/*H<sup>∞</sup>* control problem. This problem is defined as follows: Given the stabilizable and detectable plant (1) with *w*0(*t*)=0 and the expected cost function

© 2012 Xu; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Xu; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

$$J(A\_{\circ}, B\_{\circ}, C\_c) = \lim\_{t \to \ast} E\left[\mathbf{x}^T(t)\mathbf{Q}\mathbf{x}(t) + \mathbf{u}^T(t)\mathbf{R}\mathbf{u}(t)\right] \tag{2}$$

determine an *n*th order dynamic compensator

$$\begin{aligned} \dot{\mathbf{x}}\_c(t) &= A\_c \mathbf{x}(t) + B\_c \mathbf{y}(t) \\ \boldsymbol{\mu}(t) &= \mathbb{C}\_c \mathbf{x}\_c(t) \end{aligned} \tag{3}$$

tively a state space approach and an algebraic Riccati equation approach to discrete-time state feedback mixed LQR/*H∞* control problem, and gave a sufficient condition for the exis‐

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

http://dx.doi.org/10.5772/51019

5

On the other hand, Geromel & Peres (1985) showed a new stabilizability property of the Ric‐ cati equation solution, and proposed, based on this new property, a numerical procedure to design static output feedback suboptimal LQR controllers for linear continuous-time sys‐ tems. 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, Gadewadi‐ kar 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 var‐ ious control problems, such as, static output feedback stabilizability (Kucera & de Souza 1995), *H∞*control problem for linear discrete-time systems (de Souza & Xie 1992), *H∞*control problem for linear continuous-time systems (Gadewadikar et al. 2007), mixed LQR/*H<sup>∞</sup>* con‐

The objective of this chapter is to solve discrete-time 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 discrete-time state feedback mixed LQR/ *H∞* control problem in terms of an algebraic Riccati equation. However, no stochastic inter‐ pretation was provided. This paper thus presents a central solution to the discrete-time state feedback stochastic mixed LQR/*H∞* control problem. This result may be recognied to be a stochastic interpretation of the discrete-time state feedback mixed LQR/*H∞* control problem considered by Xu (2011). The second motivation for our paper is to present a characteriza‐ tion of all admissible state feedback controllers for solving discrete-time stochastic mixed LQR/*H∞* control 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 as‐ sumption that the free parameter matrix is a free admissible controller error. The third moti‐ vation for our paper is to use the above results to solve the discrete-time static output

This chapter is organized as follows: Section 2 introduces several preliminary results. In Sec‐ tion 3, first,we define the state feedback stochastic mixed LQR/*H∞* control problem for linear discrete-time systems. Secondly, we give sufficient conditions for the existence of all admis‐ sible state feedback controllers solving the discrete-time stochastic mixed LQR/*H<sup>∞</sup>* control problem. In the rest of this section, first, we parametrize a central discrete-time state feed‐ back stochastic mixed LQR/*H∞* controller, and show that this result may be recognied to be a stochastic interpretation of discrete-time state feedback mixed LQR/*H<sup>∞</sup>* control problem considered by Xu (2011). Secondly, we propose a numerical algorithm for calclulating a kind

tence of an admissible state feedback controller solving this problem.

trol problem for linear continuous-time systems (Xu 2008).

feedback stochastic mixed LQR/*H∞* control problem.

which satisfies the following design criteria: (i) the closed-loop system (1) (3) is stable; (ii) the closed-loop transfer matrix *Tzw* from the disturbance input *w*to the controlled output *z* satisfies *Tzw <sup>∞</sup>* <*γ*; (iii) the expected cost function *J*(*Ac*, *Bc*, *Cc*)is minimized; where, the dis‐ turbance input *w* is assumed to be a Gaussian white noise. Bernstein & Haddad (1989) con‐ sidered merely the combined LQG/*H∞* control problem in the special case of *Q* =*C*<sup>1</sup> *TC*<sup>1</sup> and *R* =*D*<sup>12</sup> *<sup>T</sup> <sup>D</sup>*<sup>12</sup> and*C*<sup>1</sup> *<sup>T</sup> <sup>D</sup>*<sup>12</sup> =0. Since the expected cost function *J*(*Ac*, *Bc*, *Cc*) equals the square of the *H*2-norm of the closed-loop transfer matrix *Tzw* in this case, the combined LQG/*H∞* prob‐ lem by Bernstein & Haddad (1989) has been recognized to be a mixed *H*<sup>2</sup> / *H∞* problem. In Bernstein & Haddad (1989), they considered the minimization of an "upper bound" of *Tzw* <sup>2</sup> 2 subject to *Tzw <sup>∞</sup>* <sup>&</sup>lt;*γ*, and solved this problem by using Lagrange multiplier techni‐ ques. Doyle et al. (1989b) considered a related output feedback mixed *H*<sup>2</sup> / *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 discrete-time static output feedback mixed *H*<sup>2</sup> / *H∞*controllers in terms of coupled Riccati equations. In Khargonekar & Rotea (1991), they presented a convex optimisation approach to solve output feedback mixed *H*<sup>2</sup> / *H∞* problem. In Limebeer et al. (1994), they proposed a Nash game approach to the state feedback mixed *H*<sup>2</sup> / *H∞* problem, and gave necessary and sufficient conditions for the existence of a solution of this problem. Chen & Zhou (2001) gen‐ eralized the method of Limebeer et al. (1994) to output feedback multiobjective *H*<sup>2</sup> / *H<sup>∞</sup>* problem. However, up till now, no approach has involved the combined LQG/*H<sup>∞</sup>* control problem (so called stochastic mixed LQR/*H<sup>∞</sup>* control problem) for linear continuous-time systems (1) with the expected cost function (2), where, *Q* ≥0and *R* >0are the weighting matri‐ ces, *w*0(*t*)is a Gaussian white noise, and *w*(*t*)is a disturbance input that belongs to*L* <sup>2</sup> 0,*∞*).

In this chapter, we consider state feedback stochastic mixed LQR/*H∞* control problem for linear discrete-time 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 discrete-time systems with uncertainty was presented. When the parameter uncertainty equals zero, the discrete-time state feedback mixed QGC/*H<sup>∞</sup>* control problem reduces to the discrete-time state feedback mixed LQR/*H∞* control problem. Xu (2011) presented respec‐ tively a state space approach and an algebraic Riccati equation approach to discrete-time state feedback mixed LQR/*H∞* control problem, and gave a sufficient condition for the exis‐ tence of an admissible state feedback controller solving this problem.

*J*(*Ac*, *Bc*, *Cc*)=lim

determine an *n*th order dynamic compensator

*R* =*D*<sup>12</sup>

*Tzw* <sup>2</sup>

*<sup>T</sup> <sup>D</sup>*<sup>12</sup> and*C*<sup>1</sup>

4 Advances in Discrete Time Systems

*t*→*∞*

*x*˙ *<sup>c</sup>*(*t*)= *Acx*(*t*) + *Bcy*(*t*)

sidered merely the combined LQG/*H∞* control problem in the special case of *Q* =*C*<sup>1</sup>

which satisfies the following design criteria: (i) the closed-loop system (1) (3) is stable; (ii) the closed-loop transfer matrix *Tzw* from the disturbance input *w*to the controlled output *z* satisfies *Tzw <sup>∞</sup>* <*γ*; (iii) the expected cost function *J*(*Ac*, *Bc*, *Cc*)is minimized; where, the dis‐ turbance input *w* is assumed to be a Gaussian white noise. Bernstein & Haddad (1989) con‐

the *H*2-norm of the closed-loop transfer matrix *Tzw* in this case, the combined LQG/*H∞* prob‐ lem by Bernstein & Haddad (1989) has been recognized to be a mixed *H*<sup>2</sup> / *H∞* problem. In Bernstein & Haddad (1989), they considered the minimization of an "upper bound" of

ques. Doyle et al. (1989b) considered a related output feedback mixed *H*<sup>2</sup> / *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 discrete-time static output feedback mixed *H*<sup>2</sup> / *H∞*controllers in terms of coupled Riccati equations. In Khargonekar & Rotea (1991), they presented a convex optimisation approach to solve output feedback mixed *H*<sup>2</sup> / *H∞* problem. In Limebeer et al. (1994), they proposed a Nash game approach to the state feedback mixed *H*<sup>2</sup> / *H∞* problem, and gave necessary and sufficient conditions for the existence of a solution of this problem. Chen & Zhou (2001) gen‐ eralized the method of Limebeer et al. (1994) to output feedback multiobjective *H*<sup>2</sup> / *H<sup>∞</sup>* problem. However, up till now, no approach has involved the combined LQG/*H<sup>∞</sup>* control problem (so called stochastic mixed LQR/*H<sup>∞</sup>* control problem) for linear continuous-time systems (1) with the expected cost function (2), where, *Q* ≥0and *R* >0are the weighting matri‐ ces, *w*0(*t*)is a Gaussian white noise, and *w*(*t*)is a disturbance input that belongs to*L* <sup>2</sup> 0,*∞*).

In this chapter, we consider state feedback stochastic mixed LQR/*H∞* control problem for linear discrete-time 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 discrete-time systems with uncertainty was presented. When the parameter uncertainty equals zero, the discrete-time state feedback mixed QGC/*H<sup>∞</sup>* control problem reduces to the discrete-time state feedback mixed LQR/*H∞* control problem. Xu (2011) presented respec‐

2 subject to *Tzw <sup>∞</sup>* <sup>&</sup>lt;*γ*, and solved this problem by using Lagrange multiplier techni‐

*<sup>T</sup> <sup>D</sup>*<sup>12</sup> =0. Since the expected cost function *J*(*Ac*, *Bc*, *Cc*) equals the square of

*E*{*x <sup>T</sup>* (*t*)*Qx*(*t*) + *u <sup>T</sup>* (*t*)*Ru*(*t*)} (2)

*<sup>u</sup>*(*t*)=*Ccxc*(*t*) (3)

*TC*<sup>1</sup> and

On the other hand, Geromel & Peres (1985) showed a new stabilizability property of the Ric‐ cati equation solution, and proposed, based on this new property, a numerical procedure to design static output feedback suboptimal LQR controllers for linear continuous-time sys‐ tems. 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, Gadewadi‐ kar 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 var‐ ious control problems, such as, static output feedback stabilizability (Kucera & de Souza 1995), *H∞*control problem for linear discrete-time systems (de Souza & Xie 1992), *H∞*control problem for linear continuous-time systems (Gadewadikar et al. 2007), mixed LQR/*H<sup>∞</sup>* con‐ trol problem for linear continuous-time systems (Xu 2008).

The objective of this chapter is to solve discrete-time 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 discrete-time state feedback mixed LQR/ *H∞* control problem in terms of an algebraic Riccati equation. However, no stochastic inter‐ pretation was provided. This paper thus presents a central solution to the discrete-time state feedback stochastic mixed LQR/*H∞* control problem. This result may be recognied to be a stochastic interpretation of the discrete-time state feedback mixed LQR/*H∞* control problem considered by Xu (2011). The second motivation for our paper is to present a characteriza‐ tion of all admissible state feedback controllers for solving discrete-time stochastic mixed LQR/*H∞* control 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 as‐ sumption that the free parameter matrix is a free admissible controller error. The third moti‐ vation for our paper is to use the above results to solve the discrete-time static output feedback stochastic mixed LQR/*H∞* control problem.

This chapter is organized as follows: Section 2 introduces several preliminary results. In Sec‐ tion 3, first,we define the state feedback stochastic mixed LQR/*H∞* control problem for linear discrete-time systems. Secondly, we give sufficient conditions for the existence of all admis‐ sible state feedback controllers solving the discrete-time stochastic mixed LQR/*H<sup>∞</sup>* control problem. In the rest of this section, first, we parametrize a central discrete-time state feed‐ back stochastic mixed LQR/*H∞* controller, and show that this result may be recognied to be a stochastic interpretation of discrete-time state feedback mixed LQR/*H<sup>∞</sup>* control problem considered by Xu (2011). Secondly, we propose a numerical algorithm for calclulating a kind of discrete-time 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 discrete-time stochastic mixed LQR/*H∞* control problem, and proposes a numerical algo‐ rithm for calculating a discrete-time static output feedback stochastic mixed LQR/*H<sup>∞</sup>* con‐ troller. 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 stabi‐ lizability 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

$$\begin{aligned} \mathbf{x}(k+1) &= A\mathbf{x}(k) + Bu(k); \mathbf{x}(0) = \mathbf{x}\_0 \\ y(k) &= \mathbf{C}\mathbf{x}(k) \end{aligned} \tag{4}$$

Geromel & Peres (1985) showed a new stabilizability property of the Riccati equation solu‐ tion, and proposed, based on this new property, a numerical procedure to design static out‐ put feedback suboptimal LQR controllers for linear continuous-time systems. Geromel et al. (1989) extended this new stabilizability property displayed in Geromel & Peres (1985) to lin‐

−1

holds, *S* ∈*R <sup>n</sup>*×*n*is a positive definite solution of the modified discrete-time Riccati equation

Second, we introduce the well known discrete-time bounded real lemma (see Zhou et al.,

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

*x*(*k* + 1)= *Ax*(*k*) + *B*1*w*(*k*) + *B*2*u*(*k*)

(*B <sup>T</sup> XA* + *D TC*) + *C TC* =0

*B <sup>T</sup> SA* + *L* (8)

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

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7

*<sup>Π</sup><sup>d</sup>* (*S*)=*<sup>Q</sup>* <sup>+</sup> *<sup>L</sup> <sup>T</sup>* (*<sup>R</sup>* <sup>+</sup> *<sup>B</sup> <sup>T</sup> SB*)*<sup>L</sup>* (9)

*<sup>C</sup> <sup>D</sup>* <sup>∈</sup>*RH∞*, then the following two statements are equivalent:

*<sup>z</sup>*(*k*)=*C*1*x*(*k*) <sup>+</sup> *<sup>D</sup>*12*u*(*k*) (10)

ear discrete-time systems. This resut is given by the following theorem.

*K* = −(*R* + *B <sup>T</sup> SB*)

When these conditions are met, the quadratic cost function *J*2 is given by

*Theorem 2.1* (Geromel et al. 1989) For the matrix *L* ∈*R <sup>m</sup>*×*n* such that

Then the matrix (*A* + *BK*)is stable.

Suppose that*<sup>γ</sup>* >0, *<sup>M</sup>* (*z*)= *<sup>A</sup> <sup>B</sup>*

1996; Iglesias & Glover, 1991; de Souza & Xie, 1992).

(*A<sup>T</sup> XB* <sup>+</sup> *<sup>C</sup> <sup>T</sup> <sup>D</sup>*)*U*<sup>1</sup>

under the influence of state feedback of the form

(*D <sup>T</sup> D* + *B <sup>T</sup> XB*)>0.

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

−1

*Lemma 2.1 (Discrete Time Bounded Real Lemma)*

*<sup>J</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup> <sup>T</sup>* (0)*Sx*(0)

i. *M* (*z*) *<sup>∞</sup>* <*γ*.

Riccati equation

*A<sup>T</sup> XA*− *X* + *γ* <sup>−</sup><sup>2</sup>

such that*U*<sup>1</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup>

with quadratic performance index

$$J\_2 \coloneqq \sum\_{k=0}^{\infty} \left\{ \boldsymbol{\omega}^T(k) \boldsymbol{\mathcal{Q}} \boldsymbol{\omega}(k) + \boldsymbol{\omega}^T(k) \boldsymbol{\mathcal{R}} \boldsymbol{\omega}(k) \right\},$$

under the influence of state feedback of the form

$$
\mu(k) = K \alpha(k) \tag{5}
$$

where,*x*(*k*)∈*R <sup>n</sup>* is the state, *u*(*k*)∈*R <sup>m</sup>*is the control input, *y*(*k*)∈*R <sup>r</sup>* is the measured output, *Q* =*Q <sup>T</sup>* ≥0and*R* =*R <sup>T</sup>* >0. We make the following assumptions

*Assumption 2.1*(*A*, *B*) is controllable.

*Assumption 2.2*(*A*, *Q* 1 2 ) is observable.

Define a discrete-time Riccati equation as follows:

$$A\,^T S A - A - A\,^T S B \left(R + B\,^T S B\right)^{-1} B\,^T S A + Q = 0\tag{6}$$

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

$$
\Pi\_d(\mathbb{S}) = \mathbb{Q} \tag{7}
$$

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

*Theorem 2.1* (Geromel et al. 1989) For the matrix *L* ∈*R <sup>m</sup>*×*n* such that

$$K = -\left(R + B^{\top}SB\right)^{-1}B^{\top}SA + L \tag{8}$$

holds, *S* ∈*R <sup>n</sup>*×*n*is a positive definite solution of the modified discrete-time Riccati equation

$$\Pi\_d(S) = Q + L^{-T}(R + B^T S B)L\tag{9}$$

Then the matrix (*A* + *BK*)is stable.

When these conditions are met, the quadratic cost function *J*2 is given by

$$J\_2 = \mathbf{x}^T \text{ (0)} S \mathbf{x} \text{(0)}$$

of discrete-time 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 discrete-time stochastic mixed LQR/*H∞* control problem, and proposes a numerical algo‐ rithm for calculating a discrete-time static output feedback stochastic mixed LQR/*H<sup>∞</sup>* con‐ troller. In Section 6, we give two examples to illustrate the design procedures and their

In this section, we will review several preliminary results. First, we introduce the new stabi‐ lizability 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

*<sup>y</sup>*(*k*)=*Cx*(*k*) (4)

*u*(*k*)= *Kx*(*k*) (5)

is the measured output,

*B <sup>T</sup> SA* + *Q* =0 (6)

*Π<sup>d</sup>* (*S*)=*Q* (7)

*x*(*k* + 1)= *Ax*(*k*) + *Bu*(*k*); *x*(0)= *x*<sup>0</sup>

effectiveness. Section 7 is conclusion.

**2. Preliminaries**

6 Advances in Discrete Time Systems

*<sup>J</sup>*<sup>2</sup> : <sup>=</sup>∑ *k*=0 *∞*

linear discrete-time systems

with quadratic performance index

*Assumption 2.1*(*A*, *B*) is controllable.

1 2

Define a discrete-time Riccati equation as follows:

*Assumption 2.2*(*A*, *Q*

{*x <sup>T</sup>* (*k*)*Qx*(*k*) + *u <sup>T</sup>* (*k*)*Ru*(*k*)}

under the influence of state feedback of the form

where,*x*(*k*)∈*R <sup>n</sup>* is the state, *u*(*k*)∈*R <sup>m</sup>*is the control input, *y*(*k*)∈*R <sup>r</sup>*

*A<sup>T</sup> SA*− *A*− *A<sup>T</sup> SB*(*R* + *B <sup>T</sup> SB*)

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

−1

*Q* =*Q <sup>T</sup>* ≥0and*R* =*R <sup>T</sup>* >0. We make the following assumptions

) is observable.

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 that*<sup>γ</sup>* >0, *<sup>M</sup>* (*z*)= *<sup>A</sup> <sup>B</sup> <sup>C</sup> <sup>D</sup>* <sup>∈</sup>*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 discrete-time Riccati equation

$$\mathbf{A}^T \mathbf{X} \mathbf{A} - \mathbf{X} + \boldsymbol{\gamma}^{-2} (\mathbf{A}^T \mathbf{X} \mathbf{B} + \mathbf{C}^T \mathbf{D}) \mathbf{U}\_1^{-1} (\mathbf{B}^T \mathbf{X} \mathbf{A} + \mathbf{D}^T \mathbf{C}) + \mathbf{C}^T \mathbf{C} = \mathbf{0}$$

such that*U*<sup>1</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup> (*D <sup>T</sup> D* + *B <sup>T</sup> XB*)>0.

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

$$\begin{aligned} \mathbf{x}(k+1) &= A\mathbf{x}(k) + B\_1\mathbf{w}(k) + B\_2\boldsymbol{\mu}(k) \\ \mathbf{z}(k) &= \mathbf{C}\_1\mathbf{x}(k) + D\_{12}\boldsymbol{\mu}(k) \end{aligned} \tag{10}$$

under the influence of state feedback of the form

$$
\mu(k) = K \alpha(k) \tag{11}
$$

*Assumption 2.4*(*A*, *B*2) is stabilizable.

*<sup>T</sup> C*<sup>1</sup> *D*<sup>12</sup> = 0 *I* .

*<sup>A</sup><sup>T</sup> <sup>X</sup>∞A*<sup>−</sup> *<sup>X</sup><sup>∞</sup>* <sup>−</sup> *<sup>A</sup><sup>T</sup> <sup>X</sup>∞<sup>B</sup>*

^ <sup>=</sup> <sup>−</sup> *<sup>I</sup>* <sup>0</sup> <sup>0</sup> *<sup>R</sup>* <sup>+</sup> *<sup>I</sup>* .

problem, this result is given by the following theorem.

*B*<sup>1</sup> *B*<sup>2</sup> ,*R*

sup *w*∈*L* 2+ inf *<sup>K</sup>* {*J*2} <sup>=</sup> *<sup>x</sup>*<sup>0</sup>

> *BKU*<sup>1</sup> −1 *BK*

*<sup>K</sup>* <sup>=</sup> *AK* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

^ (*B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>B</sup>*

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

^ <sup>+</sup> *<sup>R</sup>* ^ ) −1 *B*

Xu (2011) has provided a solution to discrete-time state feedback mixed LQR/*H∞*control

*Theorem 2.2* There exists a discrete-time state feedback mixed LQR/*H<sup>∞</sup>* controller if the dis‐

−1 *B*1 *<sup>T</sup> <sup>X</sup>∞*.

*<sup>T</sup>* (*X<sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>*Xw* <sup>−</sup> *Xz*)*x*0 subject to *Tzw <sup>∞</sup>* <sup>&</sup>lt;*γ*.

*<sup>T</sup> <sup>X</sup>∞BKU*<sup>1</sup>

−2 *BK*

*<sup>T</sup> <sup>X</sup>∞AK <sup>A</sup>* ^ *K k*

crete-time Riccati equation (14) has a stabilizing solution *X∞* and *U*<sup>1</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup>

*<sup>T</sup> <sup>U</sup>*3*B*2 , and*U*<sup>3</sup> <sup>=</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup>2*X∞B*1*U*<sup>1</sup>

Moreover, this discrete-time state feedback mixed LQR/*H∞*controller is given by

In this case, the discrete-time state feedback mixed LQR/*H∞*controller will achieve

*<sup>T</sup> <sup>X</sup>∞AK* ,*Xw* <sup>=</sup>∑

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

*x*(*k* + 1)= *Ax*(*k*) + *B*0*w*0(*k*) + *B*1*w*(*k*) + *B*2*u*(*k*) *z*(*k*)=*C*1*x*(*k*) + *D*12*u*(*k*) *y*(*k*)=*C*2*x*(*k*)

*k*=0 *∞* {(*A* ^ *K k* ) *<sup>T</sup> AK*

^ *<sup>T</sup> <sup>X</sup>∞<sup>A</sup>* <sup>+</sup> *<sup>C</sup>*<sup>1</sup>

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

*TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>* =0 (14)

http://dx.doi.org/10.5772/51019

9

*B*1

*<sup>T</sup> <sup>X</sup>∞B*<sup>1</sup> >0.

} , and

(15)

*Assumption 2.5D*<sup>12</sup>

^ <sup>=</sup> *<sup>γ</sup>* <sup>−</sup><sup>1</sup>

where, *B*

*K* = −*U*<sup>2</sup> −1 *B*2 *<sup>T</sup> <sup>U</sup>*3*<sup>A</sup>*

where, *A*

*Xz* <sup>=</sup>∑ *k*=0 *∞* {(*A* ^ *K k* ) *TCK TCK <sup>A</sup>* ^ *K k* }.

where,*U*<sup>2</sup> =*R* + *I* + *B*<sup>2</sup>

^

**3. State Feedback**

with state feedback of the form

where,*x*(*k*)∈*R <sup>n</sup>* is the state, *u*(*k*)∈*R <sup>m</sup>*is the control input, *w*(*k*)∈*R <sup>q</sup>* is the disturbance input that belongs to*<sup>L</sup>* <sup>2</sup> 0,*∞*),*z*(*k*)∈*<sup>R</sup> <sup>p</sup>* is the controlled output. Let*x*(0)= *x*0.

The associated with this systems is the quadratic performance index

$$\|f\|\_{2} \coloneqq \sum\_{k=0}^{\bullet} \left\{ \mathbf{x}^{\top}(k) \mathbf{Q} \mathbf{x}(k) + \boldsymbol{\mu}^{\top}(k) \mathbf{R} \boldsymbol{u}(k) \right\} \tag{12}$$

where, *Q* =*Q <sup>T</sup>* ≥0and*R* =*R <sup>T</sup>* >0.

The closed-loop transfer matrix from the disturbance input *w* to the controlled output *z* is

$$\mathbf{T}\_{zw}(z) = \begin{bmatrix} A\_K & B\_K \\ C\_K & 0 \end{bmatrix} := \mathbf{C}\_K (zI - A\_K)^{-1} B\_K$$

where, *AK* : = *A* + *B*2*K*,*BK* : = *B*1 ,*CK* : =*C*<sup>1</sup> + *D*12*K*.

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 that*γ* >0,*Tzw*(*z*)∈*RH∞*; then there exists an admissible controller *K* such that *Tzw*(*z*) *<sup>∞</sup>* <*γ* if there exists a stabilizing solution *X<sup>∞</sup>* ≥0 to the discrete time Riccati equation

$$\mathbf{A}\_{\mathbf{K}}^{T}\mathbf{X}\_{\circ\circ}\mathbf{A}\_{\mathbf{K}} - \mathbf{X}\_{\circ\circ} + \boldsymbol{\gamma}^{-2}\mathbf{A}\_{\mathbf{K}}^{T}\mathbf{X}\_{\circ\circ}\mathbf{B}\_{\mathbf{K}}\mathbf{U}\_{1}^{-1}\mathbf{B}\_{\mathbf{K}}^{T}\mathbf{X}\_{\circ\circ}\mathbf{A}\_{\mathbf{K}} + \mathbf{C}\_{\mathbf{K}}^{T}\mathbf{C}\_{\mathbf{K}} + \mathbf{Q} + \mathbf{K}^{T}\mathbf{R}\mathbf{K} = \mathbf{0} \tag{13}$$

such that*U*<sup>1</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup> *BK <sup>T</sup> <sup>X</sup>∞BK* >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/*H∞* control prob‐ lem. Xu (2011) has defined this problem as follows: Given the linear discrete-time systems (10)(11) with *w* ∈ *L* <sup>2</sup>[0,*∞*)and*x*(0)= *x*0, for a given number *γ* >0, determine an admissible controller that achieves

$$\sup\_{w \in \mathcal{L}\_{z\*}} \inf\_{\mathsf{K}} \{ l\_2 \} \text{ subject to} \\ \left\| \begin{array}{c} T\_{zw}(z) \end{array} \right\|\_{\infty} \leqslant \gamma. $$

If this controller *K* exists, it is said to be a discrete-time state feedback mixed LQR/*H<sup>∞</sup>* con‐ troller.

The following assumptions are imposed on the system

*Assumption 2.3*(*C*1, *A*) is detectable.

*Assumption 2.4*(*A*, *B*2) is stabilizable.

*u*(*k*)= *Kx*(*k*) (11)

{*x <sup>T</sup>* (*k*)*Qx*(*k*) + *u <sup>T</sup>* (*k*)*Ru*(*k*)} (12)

is the controlled output. Let*x*(0)= *x*0.

The closed-loop transfer matrix from the disturbance input *w* to the controlled output *z* is

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

*Lemma 2.2* Given the system (10) under the influence of the state feedback (11), and suppose that*γ* >0,*Tzw*(*z*)∈*RH∞*; then there exists an admissible controller *K* such that *Tzw*(*z*) *<sup>∞</sup>* <*γ*

Finally, we review the result of discrete-time state feedback mixed LQR/*H∞* control prob‐ lem. Xu (2011) has defined this problem as follows: Given the linear discrete-time systems (10)(11) with *w* ∈ *L* 2[0,*∞*)and*x*(0)= *x*0, for a given number *γ* >0, determine an admissible

*<sup>K</sup>* {*J*2} subject to *Tzw*(*z*) *<sup>∞</sup>* <sup>&</sup>lt;*γ*.

If this controller *K* exists, it is said to be a discrete-time state feedback mixed LQR/*H<sup>∞</sup>* con‐

*<sup>T</sup> <sup>X</sup>∞AK* <sup>+</sup> *CK*

if there exists a stabilizing solution *X<sup>∞</sup>* ≥0 to the discrete time Riccati equation

−1 *BK*

*<sup>T</sup> <sup>X</sup>∞BKU*<sup>1</sup>

*AK*

sup *w*∈*L* 2+ inf

The following assumptions are imposed on the system

*<sup>T</sup> <sup>X</sup>∞BK* >0.

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

is the disturbance input

*TCK* <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>K</sup> <sup>T</sup> RK* =0 (13)

where,*x*(*k*)∈*R <sup>n</sup>* is the state, *u*(*k*)∈*R <sup>m</sup>*is the control input, *w*(*k*)∈*R <sup>q</sup>*

The associated with this systems is the quadratic performance index

*<sup>J</sup>*<sup>2</sup> : <sup>=</sup>∑ *k*=0 *∞*

that belongs to*<sup>L</sup>* <sup>2</sup> 0,*∞*),*z*(*k*)∈*<sup>R</sup> <sup>p</sup>*

8 Advances in Discrete Time Systems

where, *Q* =*Q <sup>T</sup>* ≥0and*R* =*R <sup>T</sup>* >0.

*CK* <sup>0</sup> : <sup>=</sup>*CK* (*zI* <sup>−</sup> *AK* )−<sup>1</sup>*BK*

*<sup>T</sup> <sup>X</sup>∞AK* <sup>−</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

*BK*

where, *AK* : = *A* + *B*2*K*,*BK* : = *B*1 ,*CK* : =*C*<sup>1</sup> + *D*12*K*.

*AK BK*

*AK*

such that*U*<sup>1</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup>

controller that achieves

*Assumption 2.3*(*C*1, *A*) is detectable.

troller.

*Tzw*(*z*)=

2011).

*Assumption 2.5D*<sup>12</sup> *<sup>T</sup> C*<sup>1</sup> *D*<sup>12</sup> = 0 *I* .

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

$$\left\{ \mathbf{A}^{T} \mathbf{X}\_{\circ \circ} \mathbf{A} - \mathbf{X}\_{\circ \circ} - \mathbf{A}^{T} \mathbf{X}\_{\circ \circ} \hat{\mathbf{B}} \left( \hat{\mathbf{B}}^{T} \mathbf{X}\_{\circ \circ} \hat{\mathbf{B}} + \hat{\mathbf{R}} \right)^{-1} \hat{\mathbf{B}}^{T} \mathbf{X}\_{\circ \circ} \mathbf{A} + \mathbf{C}\_{1}^{T} \mathbf{C}\_{1} + \mathbf{Q} = \mathbf{0} \tag{14}$$

where, *B* ^ <sup>=</sup> *<sup>γ</sup>* <sup>−</sup><sup>1</sup> *B*<sup>1</sup> *B*<sup>2</sup> ,*R* ^ <sup>=</sup> <sup>−</sup> *<sup>I</sup>* <sup>0</sup> <sup>0</sup> *<sup>R</sup>* <sup>+</sup> *<sup>I</sup>* .

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

*Theorem 2.2* There exists a discrete-time state feedback mixed LQR/*H<sup>∞</sup>* controller if the dis‐ crete-time Riccati equation (14) has a stabilizing solution *X∞* and *U*<sup>1</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup> *B*1 *<sup>T</sup> <sup>X</sup>∞B*<sup>1</sup> >0.

Moreover, this discrete-time state feedback mixed LQR/*H∞*controller is given by

$$\mathbf{K} = -\boldsymbol{\mathsf{U}}\boldsymbol{\mathsf{L}}\_2^{-1}\boldsymbol{\mathsf{B}}\_2^T\boldsymbol{\mathsf{U}}\boldsymbol{\mathsf{U}}\_3\boldsymbol{\mathsf{A}}$$

where,*U*<sup>2</sup> =*R* + *I* + *B*<sup>2</sup> *<sup>T</sup> <sup>U</sup>*3*B*2 , and*U*<sup>3</sup> <sup>=</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup>2*X∞B*1*U*<sup>1</sup> −1 *B*1 *<sup>T</sup> <sup>X</sup>∞*.

In this case, the discrete-time state feedback mixed LQR/*H∞*controller will achieve

$$\begin{aligned} \mathop{\rm sup\,inf}\_{w \in L\_{\infty}^{\Gamma} \times \Gamma} & \Big\| \Big( I\_{2} \Big) = \mathop{\rm x\_{0}}^{T} (X\_{m} + \gamma^{-2} X\_{w} - X\_{z}) \mathbf{x}\_{0} \operatorname{subject } \mathop{\rm t \mathbf{u}} \Big\| \Big( T\_{zw} \Big\| \Big)\_{m} < \gamma. \\\\ \text{where,} \quad \overset{\scriptstyle \hat{\mathbf{A}}}{\quad} & A\_{K} = A\_{K} + \gamma^{-2} B\_{K} \mathbf{U}\_{1}^{-1} B\_{K}^{T} X\_{m} A\_{K} X\_{w} = \sum\_{k=0}^{\omega} \Big[ (\hat{A}\_{K}^{k})^{T} A\_{K}^{T} X\_{m} B\_{K} \mathbf{U}\_{1}^{-2} B\_{K}^{T} X\_{m} A\_{K} \overset{\scriptstyle \hat{\mathbf{A}}}{A\_{K}^{k}} \Big], \qquad \text{and} \\\\ \boldsymbol{X}\_{z} = \sum\_{k=0}^{\omega} \Big[ \big( \hat{A}\_{K}^{k} \big)^{T} \mathbf{C}\_{K}^{T} \mathbf{C}\_{K} \overset{\scriptstyle \hat{\mathbf{A}}}{\quad} . \end{aligned}$$

#### **3. State Feedback**

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

$$\mathbf{x}(k+1) = A\mathbf{x}(k) + B\_0\mathbf{z}v\_0(k) + B\_1\mathbf{z}v(k) + B\_2\boldsymbol{\mu}(k)$$

$$\mathbf{z}(k) = \mathbf{C}\_1\mathbf{x}(k) + D\_{12}\boldsymbol{\mu}(k)\tag{15}$$

$$\mathbf{y}(k) = \mathbf{C}\_2\mathbf{x}(k)$$

with state feedback of the form

$$
\mu(k) = K \alpha(k) \tag{16}
$$

the expected cost function (17) subject to *Tzw <sup>∞</sup>* <*γ* for the systems (15) (16) with *w* ∈ *L* <sup>2</sup> 0,*∞*) because the expected cost function (17) is an uncertain function depending on disturbance input*w*(*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 fol‐

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

http://dx.doi.org/10.5772/51019

11

Based on this, we define the discrete-time state feedback stochastic mixed LQR/*H<sup>∞</sup>* control

Discrete-time state feedback stochastic mixed LQR/*H<sup>∞</sup>* control problem: Given the linear discrete-time systems (15) (16) satisfying Assumption 3.1-3.3 with *w*(*k*)∈ *L* <sup>2</sup> 0,*∞*) and the ex‐ pected cost functions (17), for a given number*γ* >0, find all admissible state feedback con‐

where, *Tzw*(*z*)is the closed loop transfer matrix from the disturbance input *w*to the control‐

If all these admissible controllers exist, then one of them *K* = *K* <sup>∗</sup> will achieve the design cri‐

and it is said to be a central discrete-time state feedback stochastic mixed LQR/*H<sup>∞</sup>* control‐

*Remark 3.1* The discrete-time state feedback stochastic mixed LQR/*H<sup>∞</sup>* control problem de‐ fined in the above is also said to be a discrete-time state feedback combined LQG/*H∞* control problem in general case. When the disturbance input*w*(*k*)=0, this problem reduces to a dis‐ crete-time state feedback combined LQG/*H∞* control problem arisen from Bernstein & Had‐

*Remark 3.2* In the case of*w*(*k*)=0, it is easy to show (see Bernstein & Haddad 1989, Haddad et

al. 1991) that *JE* in (17) is equivalent to the expected cost function

lowing design criteria:

*<sup>K</sup>* {*JE*} subject to *Tzw <sup>∞</sup>* <sup>&</sup>lt;*<sup>γ</sup>*

because for all*w* ∈ *L* <sup>2</sup> 0,*∞*), the following inequality always exists.

sup *w*∈*L* 2+ inf

inf

sup *w*∈*L* 2+

teria

sup *w*∈*L* 2+ inf

ler.

*JE* =lim *k*→*∞*

led output*z*.

*<sup>K</sup>* {*JE*} <sup>≤</sup> sup *w*∈*L* 2+ inf *<sup>K</sup>* {*JE*}

problem as follows:

trollers *K* such that

{*JE*} subject to *Tzw <sup>∞</sup>* <*γ*

*<sup>K</sup>* {*JE*} subject to *Tzw <sup>∞</sup>* <sup>&</sup>lt;*<sup>γ</sup>*

dad (1989) and Haddad et al. (1991).

*E*{*x <sup>T</sup>* (*k*)*Qx*(*k*) + *u <sup>T</sup>* (*k*)*Ru*(*k*)}

where,*x*(*k*)∈*<sup>R</sup> <sup>n</sup>* is the state, *u*(*k*)∈*<sup>R</sup> <sup>m</sup>*is the control input, *w*0(*k*)∈*<sup>R</sup> <sup>q</sup>*<sup>1</sup> is one disturbance in‐ put, *w*(*k*)∈*R <sup>q</sup>*<sup>2</sup> is another disturbance that belongs to*<sup>L</sup>* <sup>2</sup> 0,*∞*),*z*(*k*)∈*<sup>R</sup> <sup>p</sup>* is the controlled out‐ put, *y*(*k*)∈*R <sup>r</sup>* is the measured output.

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

*<sup>E</sup>*{*x*(0)} <sup>=</sup> *<sup>x</sup>*¯ <sup>0</sup> cov{*x*(0), *<sup>x</sup>*(0)}: <sup>=</sup>*E*{(*x*(0)<sup>−</sup> *<sup>x</sup>*¯ 0)(*x*(0)<sup>−</sup> *<sup>x</sup>*¯ 0)*<sup>T</sup>* } <sup>=</sup>*R*<sup>0</sup>

The noise process *w*0(*k*) is a Gaussain white noise signal with properties

$$E\left\{\boldsymbol{\varpi}v\_{0}(k)\right\} = \boldsymbol{0}\_{\prime}E\left\{\boldsymbol{\varpi}v\_{0}(k)\boldsymbol{\varpi}v\_{0}^{T}(\boldsymbol{\tau})\right\} = \boldsymbol{R}\_{1}(k)\boldsymbol{\delta}(k-\boldsymbol{\tau})$$

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

Also, we make the following assumptions:

*Assumption 3.1*(*C*1, *A*) is detectable.

*Assumption 3.2*(*A*, *B*2) is stabilizable.

*Assumption 3.3D*<sup>12</sup> *<sup>T</sup> C*<sup>1</sup> *D*<sup>12</sup> = 0 *I* .

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

$$J\_E := \lim\_{T \to \infty} \frac{1}{T} E\left| \sum\_{k=0}^{T} \left( \mathbf{x}^T(k) \mathbf{Q} \mathbf{x}(k) + \boldsymbol{\mu}^T(k) \mathbf{R} \boldsymbol{\mu}(k) - \boldsymbol{\gamma}^2 \|\, \|\, \mathbf{w} \|\, ^2 \right) \right| \tag{17}$$

where,*Q* =*Q <sup>T</sup>* ≥0 ,*R* =*R <sup>T</sup>* >0 , and *γ* >0 is a given number.

As is well known, a given controller *K* is called admissible (for the plant*G*) if *K* is real-ra‐ tional proper, and the minimal realization of *K* internally stabilizes the state space realiza‐ tion (15) of*G*.

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) with*w*(*k*)=0, while the discrete-time state feedback *H<sup>∞</sup>* control problem is to find an admissible controller such that *Tzw <sup>∞</sup>* <*γ* subject to the systems (15) (16) for a given num‐ ber*γ* >0. While we combine the two problems for the systems (15) (16) with *w* ∈ *L* <sup>2</sup> 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 *w*0(*k*) being a Gaussain white noise with known statistics. Thus it is not possible to pose a discrete-time state feedback stochastic mixed LQR/*H∞* control problem that achieves the minimization of the expected cost function (17) subject to *Tzw <sup>∞</sup>* <*γ* for the systems (15) (16) with *w* ∈ *L* <sup>2</sup> 0,*∞*) because the expected cost function (17) is an uncertain function depending on disturbance input*w*(*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 fol‐ lowing design criteria:

sup *w*∈*L* 2+ inf *<sup>K</sup>* {*JE*} subject to *Tzw <sup>∞</sup>* <sup>&</sup>lt;*<sup>γ</sup>*

because for all*w* ∈ *L* <sup>2</sup> 0,*∞*), the following inequality always exists.

$$\inf\_{\mathcal{K}} \{ J\_E \} \le \sup\_{w \in L} \inf\_{z\_\*} \{ J\_E \}$$

*u*(*k*)= *Kx*(*k*) (16)

is another disturbance that belongs to*<sup>L</sup>* <sup>2</sup> 0,*∞*),*z*(*k*)∈*<sup>R</sup> <sup>p</sup>* is the controlled out‐

is one disturbance in‐

)} (17)

where,*x*(*k*)∈*<sup>R</sup> <sup>n</sup>* is the state, *u*(*k*)∈*<sup>R</sup> <sup>m</sup>*is the control input, *w*0(*k*)∈*<sup>R</sup> <sup>q</sup>*<sup>1</sup>

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

The noise process *w*0(*k*) is a Gaussain white noise signal with properties

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

Furthermore, *x*(0)and *w*0(*k*) are assumed to be independent, *w*0(*k*)and *w*(*k*) are also assumed

(*x <sup>T</sup>* (*k*)*Qx*(*k*) + *u <sup>T</sup>* (*k*)*Ru*(*k*)−*γ* <sup>2</sup> *w* <sup>2</sup>

As is well known, a given controller *K* is called admissible (for the plant*G*) if *K* is real-ra‐ tional proper, and the minimal realization of *K* internally stabilizes the state space realiza‐

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) with*w*(*k*)=0, while the discrete-time state feedback *H<sup>∞</sup>* control problem is to find an admissible controller such that *Tzw <sup>∞</sup>* <*γ* subject to the systems (15) (16) for a given num‐ ber*γ* >0. While we combine the two problems for the systems (15) (16) with *w* ∈ *L* <sup>2</sup> 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 *w*0(*k*) being a Gaussain white noise with known statistics. Thus it is not possible to pose a discrete-time state feedback stochastic mixed LQR/*H∞* control problem that achieves the minimization of

*<sup>T</sup>* (*τ*)} <sup>=</sup>*R*1(*k*)*δ*(*<sup>k</sup>* <sup>−</sup>*τ*)

to be independent, where, *E* • denotes expected value.

*<sup>T</sup> C*<sup>1</sup> *D*<sup>12</sup> = 0 *I* .

where,*Q* =*Q <sup>T</sup>* ≥0 ,*R* =*R <sup>T</sup>* >0 , and *γ* >0 is a given number.

is the measured output.

cov{*x*(0), *<sup>x</sup>*(0)}: <sup>=</sup>*E*{(*x*(0)<sup>−</sup> *<sup>x</sup>*¯ 0)(*x*(0)<sup>−</sup> *<sup>x</sup>*¯ 0)*<sup>T</sup>* } <sup>=</sup>*R*<sup>0</sup>

Also, we make the following assumptions:

*Assumption 3.1*(*C*1, *A*) is detectable.

*Assumption 3.2*(*A*, *B*2) is stabilizable.

*JE* : = lim *T* →*∞* 1 *<sup>T</sup> <sup>E</sup>*{∑ *k*=0 *T*

put, *w*(*k*)∈*R <sup>q</sup>*<sup>2</sup>

10 Advances in Discrete Time Systems

put, *y*(*k*)∈*R <sup>r</sup>*

*<sup>E</sup>*{*x*(0)} <sup>=</sup> *<sup>x</sup>*¯ <sup>0</sup>

*E*{*w*0(*k*)} =0,*E*{*w*0(*k*)*w*<sup>0</sup>

*Assumption 3.3D*<sup>12</sup>

tion (15) of*G*.

Based on this, we define the discrete-time state feedback stochastic mixed LQR/*H<sup>∞</sup>* control problem as follows:

Discrete-time state feedback stochastic mixed LQR/*H<sup>∞</sup>* control problem: Given the linear discrete-time systems (15) (16) satisfying Assumption 3.1-3.3 with *w*(*k*)∈ *L* <sup>2</sup> 0,*∞*) and the ex‐ pected cost functions (17), for a given number*γ* >0, find all admissible state feedback con‐ trollers *K* such that

```
sup
w∈L 2+
    {JE} subject to Tzw ∞ <γ
```
where, *Tzw*(*z*)is the closed loop transfer matrix from the disturbance input *w*to the control‐ led output*z*.

If all these admissible controllers exist, then one of them *K* = *K* <sup>∗</sup> will achieve the design cri‐ teria

sup *w*∈*L* 2+ inf *<sup>K</sup>* {*JE*} subject to *Tzw <sup>∞</sup>* <sup>&</sup>lt;*<sup>γ</sup>*

and it is said to be a central discrete-time state feedback stochastic mixed LQR/*H<sup>∞</sup>* control‐ ler.

*Remark 3.1* The discrete-time state feedback stochastic mixed LQR/*H<sup>∞</sup>* control problem de‐ fined in the above is also said to be a discrete-time state feedback combined LQG/*H∞* control problem in general case. When the disturbance input*w*(*k*)=0, this problem reduces to a dis‐ crete-time state feedback combined LQG/*H∞* control problem arisen from Bernstein & Had‐ dad (1989) and Haddad et al. (1991).

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

$$J\_E = \lim\_{k \to \infty} E\left[\mathbf{x}^T(k)\mathbf{Q}\mathbf{x}(k) + \boldsymbol{\mu}^T(k)\mathbf{R}\boldsymbol{u}(k)\right],$$

$$\begin{aligned} \text{Define } &Q = \mathbf{C}\_1^T \mathbf{C}\_1 \text{ and } R = D\_{12}^T D\_{12} \text{ and suppose that} \mathbf{C}\_1^T D\_{12} = 0, \text{ then } I\_E \text{ may be rewritten as} \\\\ \text{If } I\_E = \lim\_{k \to \infty} \mathbb{E} \left[ \mathbf{x}^T(k) \mathbf{Q} \mathbf{x}(k) + \boldsymbol{\mu}^T(k) \mathbf{R} \boldsymbol{\mu}(k) \right] \\\\ \text{=} &\lim\_{k \to \infty} \mathbb{E} \left[ \mathbf{x}^T(k) \mathbf{C}\_1^T \mathbf{C}\_1 \mathbf{x}(k) + \boldsymbol{\mu}^T(k) D\_{12}^T D\_{12} \boldsymbol{\mu}(k) \right] \\\\ = &\lim\_{k \to \infty} \mathbb{E} \left[ \mathbf{z}^T(k) \mathbf{z}(k) \right] \end{aligned}$$

Also, the controlled output *z* may be expressed as

$$z = T\_{zw\_0}(z)w\_0 \tag{18}$$

where, *B*

ieve

sup *w*∈*L* 2+

{*JE*} = lim *T* →*∞* 1 *<sup>T</sup>* ∑ *k*=0 *T tr*(*B*<sup>0</sup>

*ΔK* = *K* − *K* <sup>∗</sup>

^ <sup>=</sup> *<sup>γ</sup>* <sup>−</sup><sup>1</sup>

*B*<sup>1</sup> *B*<sup>2</sup> , *R*

ii. *ΔK*is an admissible controller error.

*<sup>K</sup>* minus the suboptimal controller*<sup>K</sup>* <sup>∗</sup> <sup>=</sup> <sup>−</sup>*U*<sup>2</sup>

time Riccati equation (20), that is,

controller error if it belongs to the set

This result is given by the following lemma.

equation (20); then we have the following lemma.

*Lemma 3.1* Let *x* be normal with mean *m* and covariance*R*. Then

−1 *B*2

*Ω* : ={*ΔK* : *AK* <sup>∗</sup> + *B*2*ΔK* is stable }

mixed LQR/*H∞* controllers.

*E*{*x <sup>T</sup> Sx*} =*m<sup>T</sup> Sm* + *trSR*

are stable.

& Xie (1992) that

*CK* <sup>∗</sup> <sup>=</sup>*C*<sup>1</sup> <sup>+</sup> *<sup>D</sup>*12*<sup>K</sup>* <sup>∗</sup>, and*<sup>K</sup>* <sup>∗</sup> <sup>=</sup> <sup>−</sup>*U*<sup>2</sup>

^ <sup>=</sup> <sup>−</sup> *<sup>I</sup>* <sup>0</sup>

<sup>0</sup> *<sup>R</sup>* <sup>+</sup> *<sup>I</sup>* , *U*<sup>3</sup> <sup>=</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup>2*X∞B*1*U*<sup>1</sup>

*<sup>T</sup> <sup>X</sup>∞B*0*R*1(*k*)) subject to *Tzw <sup>∞</sup>* <sup>&</sup>lt;*<sup>γ</sup>*

In this case, the discrete-time state feedback stochastic mixed LQR/*H∞* controller will ach‐

*Remark 3.3* In Theorem 3.1, the controller error is defined to be the state feedback controller

−1 *B*2

where, *ΔK*is the controller error, *K*is the state feedback controller and *K* <sup>∗</sup> is the suboptimal controller. Suppose that there exists a suboptimal controller *<sup>K</sup>* <sup>∗</sup> such that *AK* <sup>∗</sup> <sup>=</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*<sup>K</sup>* <sup>∗</sup> is stable, then *K* and *ΔK* is respectively said to be an admissible controller and an admissible

*Remark 3.4* The discrete-time state feedback stochastic mixed LQR/*H∞* controller satisfying the conditions i-ii displayed in Theorem 3.1 is not unique. All admissible state feedback con‐ trollers satisfying these two conditions lead to all discrete-time state feedback stochastic

Astrom (1971) has given the mean value of a quadratic form of normal stochastic variables.

For convenience, let*AK* <sup>=</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*K*, *BK* <sup>=</sup> *<sup>B</sup>*1, *CK* <sup>=</sup>*C*<sup>1</sup> <sup>+</sup> *<sup>D</sup>*12*K*,*AK* <sup>∗</sup> <sup>=</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*<sup>K</sup>* <sup>∗</sup> ,*BK* <sup>∗</sup> <sup>=</sup> *<sup>B</sup>*<sup>1</sup> ,

*Lemma 3.2* Suppose that the conditions i-ii of Theorem 3.1 hold, then the both *AK* <sup>∗</sup> and *AK*

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

*<sup>T</sup> <sup>U</sup>*3*A*, where, *X<sup>∞</sup>* <sup>≥</sup>0satisfies the discrete-time Riccati

−1 *B*1

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

*<sup>T</sup> <sup>X</sup>∞*,*U*<sup>2</sup> <sup>=</sup>*<sup>R</sup>* <sup>+</sup> *<sup>I</sup>* <sup>+</sup> *<sup>B</sup>*<sup>2</sup>

*<sup>T</sup> <sup>U</sup>*3*A*, where, *X<sup>∞</sup>* <sup>≥</sup>0satisfies the discrete-

*<sup>T</sup> <sup>U</sup>*3*B*2.

13

http://dx.doi.org/10.5772/51019

where,*Tzw*<sup>0</sup> (*z*)= *AK B*<sup>0</sup> *CK* 0 . If *w*0 is white noise with indensity matrix *I* and the closed-loop sys‐ tems is stable then

$$J\_E = \lim\_{k \to \infty} E\left\{ z^T(k)z(k) \right\} = \left\| \begin{array}{c} T\_{zw\_0} \end{array} \right\|\_2^2$$

This implies that the discrete-time state feedback combined LQG/*H∞* control problem in the special case of *Q* =*C*<sup>1</sup> *TC*1 and *<sup>R</sup>* <sup>=</sup>*D*<sup>12</sup> *<sup>T</sup> <sup>D</sup>*12 and *C*<sup>1</sup> *<sup>T</sup> <sup>D</sup>*<sup>12</sup> =0 arisen from Bernstein & Haddad (1989) and Haddad et al. (1991) is a mixed *H*2/*H∞* control problem.

Based on the above definition, we give sufficient conditions for the existence of all admissi‐ ble state feedback controllers solving the discrete-time stochastic mixed LQR/*H<sup>∞</sup>* control problem by combining the techniques of Xu (2008 and 2011) with the well known LQG theo‐ ry. This result is given by the following theorem.

*Theorem 3.1* There exists a discrete-time state feedback stochastic mixed LQR/ *H<sup>∞</sup>* controller if the following two conditions hold:

i. There exists a matrix *ΔK* such that

$$
\Delta K = K + \mathcal{U}\_2^{-1} \mathcal{B}\_2^T \mathcal{U}\_3 A \tag{19}
$$

and *X∞* is a symmetric non-negative definite solution of the following discrete-time Riccati equation

$$\begin{aligned} & \mathbf{A}^T \mathbf{X}\_{\circ \circ} \mathbf{A} - \mathbf{X}\_{\circ \circ} - \mathbf{A}^T \mathbf{X}\_{\circ \circ} \overset{\wedge}{\mathbf{B}} \mathbf{B}^T \mathbf{X}\_{\circ \circ} \overset{\wedge}{\mathbf{B}} + \overset{\wedge}{\mathbf{B}} \mathbf{I}^{-1} \overset{\wedge}{\mathbf{B}}^T \mathbf{X}\_{\circ \circ} \mathbf{A} \\ & + \mathbf{C}\_1^T \mathbf{C}\_1 + \mathbf{Q} + \Delta \mathbf{K}^T \mathbf{U}\_2 \Delta \mathbf{K} = \mathbf{0} \end{aligned} \tag{20}$$

and *A* ^ *<sup>c</sup>* = *A*− *B* ^ (*B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>B</sup>* ^ <sup>+</sup> *<sup>R</sup>* ^ ) −1 *B* ^ *<sup>T</sup> <sup>X</sup>∞A* is stable and*U*<sup>1</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup> *B*1 *<sup>T</sup> <sup>X</sup>∞B*<sup>1</sup> >0;

$$\text{where, } \stackrel{\frown}{B} = \begin{bmatrix} \gamma^{-1}B\_1 & B\_2 \end{bmatrix} \stackrel{\frown}{R} = \begin{bmatrix} -I & 0 \\ 0 & R+I \end{bmatrix} \cup \begin{bmatrix} \boldsymbol{U}\_3 = \boldsymbol{X}\_{\boldsymbol{\omega}} + \gamma^{-2} \boldsymbol{X}\_{\boldsymbol{\omega}} B\_1 \boldsymbol{U}\_1^{-1} B\_1^T \boldsymbol{X}\_{\boldsymbol{\omega}} \boldsymbol{U}\_2 = \boldsymbol{R} + \boldsymbol{I} + \boldsymbol{B}\_2^T \boldsymbol{U}\_3 B\_2 \boldsymbol{X}\_{\boldsymbol{\omega}}$$

ii. *ΔK*is an admissible controller error.

In this case, the discrete-time state feedback stochastic mixed LQR/*H∞* controller will ach‐ ieve

$$\sup\_{\omega \in \mathcal{L}\_{2^\*}} \left| J\_E \right| = \lim\_{T \to \omega} \frac{1}{T} \sum\_{k=0}^T \operatorname{tr} \left( B\_0^{\top \cdot} X\_{\leadsto} B\_0 R\_1(k) \right) \text{ subject to } \left\| \right\| \left. T\_{zw} \right\|\_{\leadsto} < \gamma$$

*Remark 3.3* In Theorem 3.1, the controller error is defined to be the state feedback controller *<sup>K</sup>* minus the suboptimal controller*<sup>K</sup>* <sup>∗</sup> <sup>=</sup> <sup>−</sup>*U*<sup>2</sup> −1 *B*2 *<sup>T</sup> <sup>U</sup>*3*A*, where, *X<sup>∞</sup>* <sup>≥</sup>0satisfies the discretetime Riccati equation (20), that is,

$$
\Delta K = K - K^\* \ast
$$

Define *Q* =*C*<sup>1</sup>

*<sup>E</sup>*{*<sup>x</sup> <sup>T</sup>* (*k*)*C*<sup>1</sup>

*E*{*z <sup>T</sup>* (*k*)*z*(*k*)}

12 Advances in Discrete Time Systems

(*z*)=

*AK B*<sup>0</sup> *CK* 0

*<sup>E</sup>*{*<sup>z</sup> <sup>T</sup>* (*k*)*z*(*k*)} <sup>=</sup> *Tzw*<sup>0</sup> <sup>2</sup>

*JE* =lim *k*→*∞*

=lim *k*→*∞*

=lim *k*→*∞*

where,*Tzw*<sup>0</sup>

*JE* =lim *k*→*∞*

equation

and *A* ^

*<sup>c</sup>* = *A*− *B* ^ (*B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>B</sup>*

tems is stable then

special case of *Q* =*C*<sup>1</sup>

*TC*1 and *<sup>R</sup>* <sup>=</sup>*D*<sup>12</sup>

*E*{*x <sup>T</sup>* (*k*)*Qx*(*k*) + *u <sup>T</sup>* (*k*)*Ru*(*k*)}

*TC*1*x*(*k*) <sup>+</sup> *<sup>u</sup> <sup>T</sup>* (*k*)*D*<sup>12</sup>

Also, the controlled output *z* may be expressed as

2

*TC*1 and *<sup>R</sup>* <sup>=</sup>*D*<sup>12</sup>

ry. This result is given by the following theorem.

if the following two conditions hold:

i. There exists a matrix *ΔK* such that

+*C*<sup>1</sup>

^ <sup>+</sup> *<sup>R</sup>* ^ ) −1 *B*

(1989) and Haddad et al. (1991) is a mixed *H*2/*H∞* control problem.

*<sup>T</sup> <sup>D</sup>*12 and suppose that*C*<sup>1</sup>

*<sup>T</sup> <sup>D</sup>*12*u*(*k*)}

*z* =*Tzw*<sup>0</sup>

This implies that the discrete-time state feedback combined LQG/*H∞* control problem in the

Based on the above definition, we give sufficient conditions for the existence of all admissi‐ ble state feedback controllers solving the discrete-time stochastic mixed LQR/*H<sup>∞</sup>* control problem by combining the techniques of Xu (2008 and 2011) with the well known LQG theo‐

*Theorem 3.1* There exists a discrete-time state feedback stochastic mixed LQR/ *H<sup>∞</sup>* controller

−1 *B*2

^(*B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>B</sup>*

^ *<sup>T</sup> <sup>X</sup>∞A* is stable and*U*<sup>1</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup>

and *X∞* is a symmetric non-negative definite solution of the following discrete-time Riccati

^ <sup>+</sup> *<sup>R</sup>* ^ )−<sup>1</sup> *B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>A</sup>*

*B*1

*<sup>T</sup> <sup>X</sup>∞B*<sup>1</sup> >0;

*ΔK* = *K* + *U*<sup>2</sup>

*<sup>A</sup><sup>T</sup> <sup>X</sup>∞A*<sup>−</sup> *<sup>X</sup><sup>∞</sup>* <sup>−</sup> *<sup>A</sup><sup>T</sup> <sup>X</sup>∞<sup>B</sup>*

*TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>Δ</sup><sup>K</sup> <sup>T</sup> <sup>U</sup>*2*Δ<sup>K</sup>* =0

*<sup>T</sup> <sup>D</sup>*12 and *C*<sup>1</sup>

*<sup>T</sup> <sup>D</sup>*<sup>12</sup> =0, then *JE* may be rewritten as

(*z*)*w*<sup>0</sup> (18)

*<sup>T</sup> <sup>D</sup>*<sup>12</sup> =0 arisen from Bernstein & Haddad

*<sup>T</sup> <sup>U</sup>*3*<sup>A</sup>* (19)

(20)

. If *w*0 is white noise with indensity matrix *I* and the closed-loop sys‐

where, *ΔK*is the controller error, *K*is the state feedback controller and *K* <sup>∗</sup> is the suboptimal controller. Suppose that there exists a suboptimal controller *<sup>K</sup>* <sup>∗</sup> such that *AK* <sup>∗</sup> <sup>=</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*<sup>K</sup>* <sup>∗</sup> 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

$$
\Omega := \left\{ \Delta K : A\_{\chi\_K \*} + B\_2 \Delta K \text{ : is stable} \right\},
$$

*Remark 3.4* The discrete-time state feedback stochastic mixed LQR/*H∞* controller satisfying the conditions i-ii displayed in Theorem 3.1 is not unique. All admissible state feedback con‐ trollers satisfying these two conditions lead to all discrete-time 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 covariance*R*. Then

*E*{*x <sup>T</sup> Sx*} =*m<sup>T</sup> Sm* + *trSR*

For convenience, let*AK* <sup>=</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*K*, *BK* <sup>=</sup> *<sup>B</sup>*1, *CK* <sup>=</sup>*C*<sup>1</sup> <sup>+</sup> *<sup>D</sup>*12*K*,*AK* <sup>∗</sup> <sup>=</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*<sup>K</sup>* <sup>∗</sup> ,*BK* <sup>∗</sup> <sup>=</sup> *<sup>B</sup>*<sup>1</sup> , *CK* <sup>∗</sup> <sup>=</sup>*C*<sup>1</sup> <sup>+</sup> *<sup>D</sup>*12*<sup>K</sup>* <sup>∗</sup>, and*<sup>K</sup>* <sup>∗</sup> <sup>=</sup> <sup>−</sup>*U*<sup>2</sup> −1 *B*2 *<sup>T</sup> <sup>U</sup>*3*A*, where, *X<sup>∞</sup>* <sup>≥</sup>0satisfies 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 *AK* <sup>∗</sup> and *AK* are 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

$$\left(\stackrel{\bigwedge}{B}^T X\_{\ll} \stackrel{\bigwedge}{B} + \stackrel{\bigwedge}{R}\right)^{-1} = \begin{bmatrix} -\mathcal{U}\_1^{-1} + \mathcal{U}\_1^{-1} \stackrel{\bigwedge}{B}\_1 \mathcal{U}\_2^{-1} \stackrel{\bigwedge}{B}\_1 \mathcal{U}\_1^{-1} & \mathcal{U}\_1^{-1} \stackrel{\bigwedge}{B}\_1 \mathcal{U}\_2^{-1} \\\ \mathcal{U}\_2^{-1} \stackrel{\bigwedge}{B}\_1^T \mathcal{U}\_1^{-1} & \mathcal{U}\_2^{-1} \end{bmatrix}$$

where,*B* ^ <sup>1</sup> <sup>=</sup>*<sup>γ</sup>* <sup>−</sup><sup>1</sup> *B*1 *<sup>T</sup> <sup>X</sup>∞B*2.Thus we have

$$A\,^T X \,^T \hat{B} \{\hat{B}^T X \,^T\_{\text{ov}} \hat{B} + \hat{R}\}^{-1} \hat{B}^T X \,\_{\text{ov}} A = -\gamma^2 A \,^T X \,\_{\text{ov}} \hat{B}\_1 \, \mathcal{U}\_1^{-1} \mathcal{B}\_1^T X \,\_{\text{ov}} A + A \,^T \mathcal{U}\_3 \mathcal{B}\_2 \mathcal{U}\_2^{-1} \mathcal{B}\_2^T \mathcal{U}\_3 A$$

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

$$\begin{aligned} & \mathbf{X}\_{\boldsymbol{\omega}} = \mathbf{A}^{T} \mathbf{X}\_{\boldsymbol{\omega}} \mathbf{A} + \boldsymbol{\gamma}^{\cdot \cdot 2} \mathbf{A}^{T} \mathbf{X}\_{\boldsymbol{\omega}} \mathbf{B}\_{1} \mathbf{U}\_{1}^{-1} \mathbf{B}\_{1}^{T} \mathbf{X}\_{\boldsymbol{\omega}} \mathbf{A} - \mathbf{A}^{T} \mathbf{U}\_{3} \mathbf{B}\_{2} \mathbf{U}\_{2}^{-1} \mathbf{B}\_{2}^{T} \mathbf{U}\_{3} \mathbf{A} \\ &+ \mathbf{C}\_{1}^{T} \mathbf{C}\_{1} + \mathbf{Q} + \boldsymbol{\Delta} \mathbf{K}^{T} \mathbf{U}\_{2} \boldsymbol{\Delta} \mathbf{K} \\ = & \mathbf{A}\_{K}^{T} \mathbf{A}\_{\boldsymbol{\omega}} \mathbf{X}\_{K} \mathbf{A}\_{\boldsymbol{\omega}} + \boldsymbol{\gamma}^{\cdot - 2} \mathbf{A}\_{K}^{T} \mathbf{A}\_{\boldsymbol{\omega}} \mathbf{B}\_{K} \mathbf{A}\_{\boldsymbol{\omega}} \mathbf{U}\_{1}^{-1} \mathbf{B}\_{K}^{T} \mathbf{A}\_{\boldsymbol{\omega}} \mathbf{A}\_{K} \mathbf{A}\_{\boldsymbol{\omega}} + \mathbf{C}\_{K}^{T} \mathbf{C}\_{K} \mathbf{A}\_{\boldsymbol{\omega}} + \mathbf{Q} \\ &+ \mathbf{K}^{\ast \cdot \prime} \mathbf{R} \mathbf{K} \mathbf{A}^{\ast} + \boldsymbol{\Delta} \mathbf{K}^{T} \mathbf{U}\_{2} \boldsymbol{\Delta} \mathbf{K} \end{aligned}$$

that is,

$$\begin{aligned} & \mathbf{A}\_{\mathbf{K}}^{T} \mathbf{A}\_{\bullet} \mathbf{A}\_{\mathbf{K}} \mathbf{\*} - \mathbf{X}\_{\bullet} + \mathbf{y}^{-2} \mathbf{A}\_{\mathbf{K}}^{T} \mathbf{X}\_{\bullet} \mathbf{B}\_{\mathbf{K}} \mathbf{\*} \mathbf{U}\_{1}^{-1} \mathbf{B}\_{\mathbf{K}}^{T} \mathbf{X}\_{\bullet} \mathbf{A}\_{\mathbf{K}} \mathbf{\*} + \mathbf{C}\_{\mathbf{K}}^{T} \mathbf{C}\_{\mathbf{K}} \mathbf{\*} + \mathbf{Q} \\ & + \mathbf{K} \, \, \mathbf{^{\*}T} \mathbf{R} \mathbf{K} \, \, ^{\ast} + \boldsymbol{\Delta} \mathbf{K}^{T} \mathbf{U}\_{2} \boldsymbol{\Delta} \mathbf{K} = \mathbf{0} \end{aligned} \tag{21}$$

On the other hand, we can rewrite the discrete-time Riccati equation (20) by using the same

−1 *B*2 *<sup>T</sup> <sup>U</sup>*3*<sup>A</sup>*

*<sup>T</sup> <sup>X</sup>∞AK* <sup>+</sup> *CK*

It follows from Lemma 2.2 that *Tzw <sup>∞</sup>* <*γ*.Completing the squares for (22) and substituting

*U*1 −1 *BK*

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*)

*<sup>T</sup> <sup>X</sup>∞AK* <sup>+</sup> *CK*

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

)}

*U*1 −1 *BK*

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

2

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*)

*TCK* )*x*(*k*)

*<sup>T</sup> <sup>X</sup>∞B*0*w*0(*k*)

*<sup>T</sup> <sup>X</sup>∞B*0*w*0(*k*)

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*)

*<sup>T</sup> <sup>X</sup>∞B*0*w*0(*k*))}

*B*0*w*(*i*)

*<sup>T</sup> <sup>X</sup>∞B*1*w*(*k*)

2

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

− *x <sup>T</sup>* (*k*)(*Q* + *K <sup>T</sup> RK*)*x*(*k*)

2

*TCK* <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>K</sup> <sup>T</sup> RK* =0 (23)

http://dx.doi.org/10.5772/51019

15

*<sup>T</sup> <sup>X</sup>∞A*<sup>−</sup> *<sup>A</sup><sup>T</sup> <sup>U</sup>*3*B*2*U*<sup>2</sup>

−1 *BK*

standard matrix manipulations as in the proof of Lemma 3.2 as follows:

*<sup>T</sup> <sup>X</sup>∞BKU*<sup>1</sup>

1 2 (*w* −*γ* <sup>−</sup><sup>2</sup>

> −1 *BK*

*<sup>T</sup> <sup>X</sup>∞B*1*w*(*k*) <sup>+</sup> *<sup>w</sup>*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞B*1*w*(*k*) <sup>+</sup> *<sup>w</sup>*<sup>0</sup>

1 2 (*w* −*γ* <sup>−</sup><sup>2</sup>

*<sup>T</sup> <sup>X</sup>∞B*1*w*(*k*) <sup>+</sup> *<sup>w</sup>*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*) <sup>+</sup> <sup>2</sup>*w*<sup>0</sup>

*B*1*w*(*i*)

*<sup>T</sup> <sup>X</sup>∞AK*∑ *i*=0 *k*−1 *AK k* −*i*−1

*k x*0

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*U*1 −1 *BK*

*<sup>T</sup> <sup>X</sup>∞BKU*<sup>1</sup>

−1 *B*1

*AK*

*AK*

1 2 (*w* −*γ* <sup>−</sup><sup>2</sup>

(−*Δ<sup>V</sup>* (*x*(*k*))<sup>−</sup> *<sup>z</sup>* <sup>2</sup> <sup>−</sup>*<sup>γ</sup>* <sup>2</sup> *<sup>U</sup>*<sup>1</sup>

*<sup>B</sup>*0*w*0(*i*) <sup>+</sup> ∑

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*i*=0 *k*−1 *AK k* −*i*−1

*B*0*w*0(*i*) + *w*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞AK AK*

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

(*x <sup>T</sup>* (*k*)*Qx*(*k*) + *u <sup>T</sup>* (*k*)*Ru*(*k*)−*γ* <sup>2</sup> *w* <sup>2</sup>

*<sup>A</sup><sup>T</sup> <sup>X</sup>∞B*1*U*<sup>1</sup>

*<sup>A</sup><sup>T</sup> <sup>X</sup>∞A*<sup>−</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

or equivalently

*AK*

(23) in (22), we get

<sup>+</sup>*<sup>x</sup> <sup>T</sup>* (*k*)(*AK*

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

Thus, we have

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

+2*w*<sup>0</sup>

+2*w*<sup>0</sup>

*JE* = lim *T* →*∞* 1 *<sup>T</sup> <sup>E</sup>*{∑ *k*=0 *T*

= lim *T* →*∞* 1 *<sup>T</sup> <sup>E</sup>*{∑ *k*=0 *T*

+2*w*<sup>0</sup>

≤ lim *T* →*∞* 1 *<sup>T</sup> <sup>E</sup>*{∑ *k*=0 *T*

+*w*<sup>0</sup> *<sup>T</sup>* (*k*)*B*<sup>0</sup>

*x*(*k*)= *AK*

*w*0 *<sup>T</sup>* (*k*)*B*<sup>0</sup>

+*w*<sup>0</sup> *<sup>T</sup>* (*k*)*B*<sup>0</sup>

we have

*TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>Δ</sup><sup>K</sup> <sup>T</sup> <sup>U</sup>*2*Δ<sup>K</sup>* =0

*<sup>T</sup> <sup>X</sup>∞AK* <sup>−</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

*<sup>Δ</sup><sup>V</sup>* (*x*(*k*))= <sup>−</sup> *<sup>z</sup>* <sup>2</sup> <sup>+</sup> *<sup>γ</sup>* <sup>2</sup> *<sup>w</sup>* <sup>2</sup> <sup>−</sup>*<sup>γ</sup>* <sup>2</sup> *<sup>U</sup>*<sup>1</sup>

*<sup>T</sup> <sup>X</sup>∞AK* <sup>−</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*) <sup>+</sup> <sup>2</sup>*w*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*) <sup>+</sup> <sup>2</sup>*w*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*) <sup>+</sup> <sup>2</sup>*w*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞B*0*w*0(*k*))}

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*)=*w*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞AK*∑ *i*=0 *k*−1 *AK k* −*i*−1

*<sup>k</sup> <sup>x</sup>*<sup>0</sup> <sup>+</sup> ∑ *i*=0 *k*−1 *AK k* −*i*−1

Note that *x*(*∞*)=lim*<sup>T</sup>* <sup>→</sup>*∞x*(*T* )=0and

(−*ΔV* (*x*(*k*)) + 2*w*<sup>0</sup>

<sup>=</sup> <sup>−</sup> *<sup>z</sup>* <sup>2</sup> <sup>+</sup> *<sup>γ</sup>* <sup>2</sup> *<sup>w</sup>* <sup>2</sup> <sup>−</sup>*<sup>γ</sup>* <sup>2</sup> *<sup>U</sup>*<sup>1</sup>

+*C*<sup>1</sup>

Since the discrete-time Riccati equation (20) has a symmetric non-negative definite solution *X∞* and *A* ^ *<sup>c</sup>* = *A*− *B* ^ (*B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>B</sup>* ^ <sup>+</sup> *<sup>R</sup>* ^ ) −1 *B* ^ *<sup>T</sup> <sup>X</sup>∞A* is stable, and we can show that *<sup>A</sup>* ^ *<sup>c</sup>* = *AK* <sup>∗</sup> + *γ* <sup>−</sup><sup>2</sup> *BK* <sup>∗</sup>*U*<sup>1</sup> −1 *BK* <sup>∗</sup> *<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup>, the discrete-time Riccati equation (21) also has a symmetric nonnegative definite solution *X∞* and *AK* <sup>∗</sup> <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup> *BK* <sup>∗</sup>*U*<sup>1</sup> −1 *BK* <sup>∗</sup> *<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup> also is stable. Hence, (*U*<sup>1</sup> −1 *BK* <sup>∗</sup> *<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup>, *AK* <sup>∗</sup>)is detectable. Based on this, it follows from standard results on Lya‐ punov equations (see Lemma 2.7 a), Iglesias & Glover 1991) that *AK* <sup>∗</sup> is stable. Also, note that *ΔK* is an admissible controller error, so *AK* = *AK* <sup>∗</sup> + *B*2*ΔK* is 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 *AK* <sup>∗</sup> and *AK* are stable. This implies that*Tzw*(*z*)∈*RH∞*.

Define*<sup>V</sup>* (*x*(*k*))= *<sup>x</sup> <sup>T</sup>* (*k*)*X∞x*(*k*), where, *X∞*is the solution to the discrete-time Riccati equation (20), then taking the difference*ΔV* (*x*(*k*)), we get

$$\begin{aligned} \Delta V(\mathbf{x}(k)) &= \mathbf{x}^T(k+1)\mathbf{X}\_{\circ \circ}\mathbf{x}(k+1) - \mathbf{x}^T(k)\mathbf{X}\_{\circ \circ}\mathbf{x}(k) \\ &= \mathbf{x}^T(k)(A\_K^T \mathbf{X}\_{\circ \circ}A\_K - \mathbf{X}\_{\circ \circ})\mathbf{x}(k) + 2\mathbf{w}^T(k)B\_K^T \mathbf{X}\_{\circ \circ}A\_K \mathbf{x}(k) \\ &+ \mathbf{w}^T(k)B\_K^T \mathbf{X}\_{\circ \circ}B\_K\mathbf{w}(k) + 2\mathbf{w}\_0^T(k)B\_0^T \mathbf{X}\_{\circ \circ}A\_K \mathbf{x}(k) \\ &+ 2\mathbf{w}\_0^T(k)B\_0^T \mathbf{X}\_{\circ \circ}B\_1\mathbf{w}(k) + \mathbf{w}\_0^T(k)B\_0^T \mathbf{X}\_{\circ \circ}B\_0\mathbf{w}\_0(k) \end{aligned} \tag{22}$$

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:

$$\begin{aligned} & \mathbf{A}^T \mathbf{X}\_{\circ \circ} \mathbf{A} - \mathbf{X}\_{\circ \circ} + \mathbf{y}^{-2} \mathbf{A}^T \mathbf{X}\_{\circ \circ} \mathbf{B}\_1 \mathbf{U}\_1^{-1} \mathbf{B}\_1^T \mathbf{X}\_{\circ \circ} \mathbf{A} - \mathbf{A}^T \mathbf{U}\_3 \mathbf{B}\_2 \mathbf{U}\_2^{-1} \mathbf{B}\_2^T \mathbf{U}\_3 \mathbf{A} \\ &+ \mathbf{C}\_1^T \mathbf{C}\_1 + \mathbf{Q} + \Delta \mathbf{K}^T \mathbf{U}\_2 \Delta \mathbf{K} = \mathbf{0} \end{aligned}$$

or equivalently

(*B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>B</sup>*

where,*B* ^ <sup>1</sup> <sup>=</sup>*<sup>γ</sup>* <sup>−</sup><sup>1</sup> *B*1

*<sup>A</sup><sup>T</sup> <sup>X</sup>∞<sup>B</sup>* ^ (*B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>B</sup>*

+*C*<sup>1</sup>

= *AK* <sup>∗</sup>

that is,

*X∞* and *A*

*BK* <sup>∗</sup>*U*<sup>1</sup> −1 *BK* <sup>∗</sup>

*γ* <sup>−</sup><sup>2</sup>

(*U*<sup>1</sup> −1 *BK* <sup>∗</sup>

^ <sup>+</sup> *<sup>R</sup>* ^ ) <sup>−</sup><sup>1</sup> =

14 Advances in Discrete Time Systems

*<sup>X</sup><sup>∞</sup>* <sup>=</sup> *<sup>A</sup><sup>T</sup> <sup>X</sup>∞<sup>A</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

*TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>Δ</sup><sup>K</sup> <sup>T</sup> <sup>U</sup>*2*Δ<sup>K</sup>*

*<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup> <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

<sup>+</sup>*<sup>K</sup>* <sup>∗</sup>*<sup>T</sup> RK* <sup>∗</sup> <sup>+</sup> *<sup>Δ</sup><sup>K</sup> <sup>T</sup> <sup>U</sup>*2*Δ<sup>K</sup>*

*AK* <sup>∗</sup>

^

*<sup>c</sup>* = *A*− *B* ^ (*B* ^ *<sup>T</sup> <sup>X</sup>∞<sup>B</sup>*

−*U*<sup>1</sup>

^ <sup>+</sup> *<sup>R</sup>* ^ ) −1 *B*

<sup>−</sup><sup>1</sup> <sup>+</sup> *<sup>U</sup>*<sup>1</sup> −1 *B* ^ <sup>1</sup>*U*<sup>2</sup> −1 *B* ^ 1 *<sup>T</sup> <sup>U</sup>*<sup>1</sup>

*<sup>A</sup><sup>T</sup> <sup>X</sup>∞B*1*U*<sup>1</sup>

*<sup>T</sup> <sup>X</sup>∞BK* <sup>∗</sup>*U*<sup>1</sup>

*AK* <sup>∗</sup>

*<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup> <sup>−</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

<sup>+</sup>*<sup>K</sup>* <sup>∗</sup>*<sup>T</sup> RK* <sup>∗</sup> <sup>+</sup> *<sup>Δ</sup><sup>K</sup> <sup>T</sup> <sup>U</sup>*2*Δ<sup>K</sup>* =0

negative definite solution *X∞* and *AK* <sup>∗</sup> <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

(20), then taking the difference*ΔV* (*x*(*k*)), we get

<sup>=</sup> *<sup>x</sup> <sup>T</sup>* (*k*)(*AK*

<sup>+</sup>*<sup>w</sup> <sup>T</sup>* (*k*)*BK*

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

+2*w*<sup>0</sup>

^ <sup>+</sup> *<sup>R</sup>* ^ ) −1 *B*

*U*2 −1 *B* ^ 1 *<sup>T</sup> <sup>U</sup>*<sup>1</sup>

*<sup>T</sup> <sup>X</sup>∞B*2.Thus we have

^ *<sup>T</sup> <sup>X</sup>∞A*<sup>=</sup> <sup>−</sup>*<sup>γ</sup>* <sup>2</sup>

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

−1 *B*1

> −1 *BK* <sup>∗</sup>

*AK* <sup>∗</sup>

*<sup>T</sup> <sup>X</sup>∞BK* <sup>∗</sup>*U*<sup>1</sup>

<sup>−</sup><sup>1</sup> *<sup>U</sup>*<sup>1</sup> −1 *B* ^ <sup>1</sup>*U*<sup>2</sup> −1

−1

−1 *B*1

> −1 *B*2 *<sup>T</sup> <sup>U</sup>*3*<sup>A</sup>*

*<sup>T</sup> CK* <sup>∗</sup> <sup>+</sup> *<sup>Q</sup>*

*<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup> <sup>+</sup> *CK* <sup>∗</sup>

^ *<sup>T</sup> <sup>X</sup>∞A* is stable, and we can show that *<sup>A</sup>*

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*)

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*)

*<sup>T</sup> <sup>X</sup>∞B*0*w*0(*k*)

*<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup>, the discrete-time Riccati equation (21) also has a symmetric non-

*BK* <sup>∗</sup>*U*<sup>1</sup> −1 *BK* <sup>∗</sup>

*<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup>, *AK* <sup>∗</sup>)is detectable. Based on this, it follows from standard results on Lya‐

punov equations (see Lemma 2.7 a), Iglesias & Glover 1991) that *AK* <sup>∗</sup> is stable. Also, note

Proof of Theorem 3.1: Suppose that the conditions i-ii hold, then it follows from Lemma 3.2

Define*<sup>V</sup>* (*x*(*k*))= *<sup>x</sup> <sup>T</sup>* (*k*)*X∞x*(*k*), where, *X∞*is the solution to the discrete-time Riccati equation

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

*<sup>T</sup>* (*k*)*B*<sup>0</sup>

that *ΔK* is an admissible controller error, so *AK* = *AK* <sup>∗</sup> + *B*2*ΔK* is stable. Q. E. D.

*<sup>Δ</sup><sup>V</sup>* (*x*(*k*))= *<sup>x</sup> <sup>T</sup>* (*<sup>k</sup>* <sup>+</sup> 1)*X∞x*(*<sup>k</sup>* <sup>+</sup> 1)<sup>−</sup> *<sup>x</sup> <sup>T</sup>* (*k*)*X∞x*(*k*)

*<sup>T</sup> <sup>X</sup>∞BKw*(*k*) <sup>+</sup> <sup>2</sup>*w*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞B*1*w*(*k*) <sup>+</sup> *<sup>w</sup>*<sup>0</sup>

*<sup>T</sup> <sup>X</sup>∞AK* <sup>−</sup> *<sup>X</sup>∞*)*x*(*k*) <sup>+</sup> <sup>2</sup>*<sup>w</sup> <sup>T</sup>* (*k*)*BK*

that the both *AK* <sup>∗</sup> and *AK* are stable. This implies that*Tzw*(*z*)∈*RH∞*.

*<sup>T</sup> <sup>X</sup>∞<sup>A</sup>* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>U</sup>*3*B*2*U*<sup>2</sup>

−1 *B*2 *<sup>T</sup> <sup>U</sup>*3*<sup>A</sup>*

*<sup>T</sup> CK* <sup>∗</sup> <sup>+</sup> *<sup>Q</sup>*

(21)

*<sup>c</sup>* = *AK* <sup>∗</sup> +

(22)

^

*<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup> also is stable. Hence,

<sup>−</sup><sup>1</sup> *<sup>U</sup>*<sup>2</sup>

*<sup>A</sup><sup>T</sup> <sup>X</sup>∞B*1*U*<sup>1</sup>

*<sup>T</sup> <sup>X</sup>∞A*<sup>−</sup> *<sup>A</sup><sup>T</sup> <sup>U</sup>*3*B*2*U*<sup>2</sup>

*<sup>T</sup> <sup>X</sup>∞AK* <sup>∗</sup> <sup>+</sup> *CK* <sup>∗</sup>

−1 *BK* <sup>∗</sup>

Since the discrete-time Riccati equation (20) has a symmetric non-negative definite solution

$$\mathbf{A}\_{\mathbf{K}}^{T}\mathbf{X}\_{\bullet \bullet}\mathbf{A}\_{\mathbf{K}} - \mathbf{X}\_{\bullet \bullet} + \boldsymbol{\gamma}^{-2}\mathbf{A}\_{\mathbf{K}}^{T}\mathbf{X}\_{\bullet \bullet}\mathbf{B}\_{\mathbf{K}}\mathbf{U}\_{1}^{-1}\mathbf{B}\_{\mathbf{K}}^{T}\mathbf{X}\_{\bullet \bullet}\mathbf{A}\_{\mathbf{K}} + \mathbf{C}\_{\mathbf{K}}^{T}\mathbf{C}\_{\mathbf{K}} + \mathbf{Q} + \mathbf{K}^{T}\mathbf{R}\mathbf{K} = \mathbf{0} \tag{23}$$

It follows from Lemma 2.2 that *Tzw <sup>∞</sup>* <*γ*.Completing the squares for (22) and substituting (23) in (22), we get

*<sup>Δ</sup><sup>V</sup>* (*x*(*k*))= <sup>−</sup> *<sup>z</sup>* <sup>2</sup> <sup>+</sup> *<sup>γ</sup>* <sup>2</sup> *<sup>w</sup>* <sup>2</sup> <sup>−</sup>*<sup>γ</sup>* <sup>2</sup> *<sup>U</sup>*<sup>1</sup> 1 2 (*w* −*γ* <sup>−</sup><sup>2</sup> *U*1 −1 *BK <sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*) 2 <sup>+</sup>*<sup>x</sup> <sup>T</sup>* (*k*)(*AK <sup>T</sup> <sup>X</sup>∞AK* <sup>−</sup> *<sup>X</sup><sup>∞</sup>* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup> *AK <sup>T</sup> <sup>X</sup>∞BKU*<sup>1</sup> −1 *BK <sup>T</sup> <sup>X</sup>∞AK* <sup>+</sup> *CK TCK* )*x*(*k*) +2*w*<sup>0</sup> *<sup>T</sup>* (*k*)*B*<sup>0</sup> *<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*) <sup>+</sup> <sup>2</sup>*w*<sup>0</sup> *<sup>T</sup>* (*k*)*B*<sup>0</sup> *<sup>T</sup> <sup>X</sup>∞B*1*w*(*k*) <sup>+</sup> *<sup>w</sup>*<sup>0</sup> *<sup>T</sup>* (*k*)*B*<sup>0</sup> *<sup>T</sup> <sup>X</sup>∞B*0*w*0(*k*) <sup>=</sup> <sup>−</sup> *<sup>z</sup>* <sup>2</sup> <sup>+</sup> *<sup>γ</sup>* <sup>2</sup> *<sup>w</sup>* <sup>2</sup> <sup>−</sup>*<sup>γ</sup>* <sup>2</sup> *<sup>U</sup>*<sup>1</sup> 1 2 (*w* −*γ* <sup>−</sup><sup>2</sup> *U*1 −1 *BK <sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*) 2 − *x <sup>T</sup>* (*k*)(*Q* + *K <sup>T</sup> RK*)*x*(*k*) +2*w*<sup>0</sup> *<sup>T</sup>* (*k*)*B*<sup>0</sup> *<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*) <sup>+</sup> <sup>2</sup>*w*<sup>0</sup> *<sup>T</sup>* (*k*)*B*<sup>0</sup> *<sup>T</sup> <sup>X</sup>∞B*1*w*(*k*) <sup>+</sup> *<sup>w</sup>*<sup>0</sup> *<sup>T</sup>* (*k*)*B*<sup>0</sup> *<sup>T</sup> <sup>X</sup>∞B*0*w*0(*k*)

Thus, we have

$$\begin{split} & \| I\_{E} = \lim\_{T \to \infty} \frac{1}{T} E \Big[ \sum\_{k=0}^{T} \left( \mathbf{x}^{\top}(k) \mathbf{Q} \mathbf{x}(k) + \mathbf{u}^{\top}(k) \mathbf{R} \mathbf{u}(k) - \boldsymbol{\gamma}^{\ \ \ \mathbf{z}} \| \mathbf{w} \| \right)^{2} \Big] \\ & \leq \lim\_{T \to \infty} \frac{1}{T} E \Big[ \sum\_{k=0}^{T} \left( -\Delta V \left( \mathbf{x}(k) \right) - \left\| \mathbf{z} \right\|^{2} - \boldsymbol{\gamma}^{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{x} - \boldsymbol{\gamma}^{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$

Note that *x*(*∞*)=lim*<sup>T</sup>* <sup>→</sup>*∞x*(*T* )=0and

$$\begin{aligned} \mathbf{x}(k) &= A\_{\mathbf{K}}^{k} \mathbf{x}\_{0} + \sum\_{i=0}^{k-1} A\_{\mathbf{K}}^{k-i-1} B\_{0} \mathbf{z}\_{0}(i) + \sum\_{i=0}^{k-1} A\_{\mathbf{K}}^{k-i-1} B\_{1} \mathbf{z} v(i) \\ \mathbf{w}\_{0}^{T}(k) B\_{0}^{T} \mathbf{X}\_{\Leftrightarrow} A\_{\mathbf{K}} \mathbf{x}(k) &= \mathbf{w}\_{0}^{T}(k) B\_{0}^{T} \mathbf{X}\_{\Leftrightarrow} A\_{\mathbf{K}} A\_{\mathbf{K}}^{k} \mathbf{x}\_{0} \\ &+ \mathbf{w}\_{0}^{T}(k) B\_{0}^{T} \mathbf{X}\_{\Leftrightarrow} A\_{\mathbf{K}} \sum\_{i=0}^{k-1} A\_{\mathbf{K}}^{k-i-1} B\_{0} \mathbf{w}\_{0}(i) + \mathbf{w}\_{0}^{T}(k) B\_{0}^{T} \mathbf{X}\_{\Leftrightarrow} A\_{\mathbf{K}} \sum\_{i=0}^{k-1} A\_{\mathbf{K}}^{k-i-1} B\_{0} \mathbf{w}(i) \end{aligned}$$

we have

$$\begin{aligned} E\left[\sum\_{k=0}^{T} \boldsymbol{w}\_{0}^{T} \left(\boldsymbol{k}\right) \boldsymbol{\mathcal{B}}\_{0}^{T} \, \boldsymbol{X}\_{\boldsymbol{\omega}} \boldsymbol{A}\_{K} \boldsymbol{x} \langle \boldsymbol{k} \rangle \right] &= \mathbb{E}\left[\sum\_{k=0}^{T} \left\{\boldsymbol{w}\_{0}^{T} \left(\boldsymbol{k}\right) \boldsymbol{\mathcal{B}}\_{0}^{T} \, \boldsymbol{X}\_{\boldsymbol{\omega}} \boldsymbol{A}\_{K} \sum\_{i=0}^{k-1} \boldsymbol{A}\_{K}^{k-i-1} \boldsymbol{\mathcal{B}}\_{0} \boldsymbol{w}\_{0} \langle \boldsymbol{i} \rangle \right\} \right] \\ &= \sum\_{k=0}^{T} \sum\_{i=0}^{k-1} \operatorname{tr}\left\{ \boldsymbol{\mathcal{B}}\_{0}^{T} \, \boldsymbol{X}\_{\boldsymbol{\omega}} \boldsymbol{A}\_{K}^{k-i} \boldsymbol{\mathcal{B}}\_{0} \boldsymbol{E} \left\{ \boldsymbol{w}\_{0} (\boldsymbol{i}) \boldsymbol{w}\_{0}^{T} \left(\boldsymbol{k} \right) \right\} \right\} = 0 \end{aligned}$$

Based on the above, it follows from Lemma 3.1 that

$$\begin{split} &\mathbb{E}\sup\_{\boldsymbol{\omega}\in\boldsymbol{\mathcal{E}}\_{\boldsymbol{\omega}\boldsymbol{\omega}}}\Big[\boldsymbol{I}\_{E}\Big] = \lim\_{T\to\boldsymbol{\infty}}\frac{1}{T}E\Big\Big\{\boldsymbol{\omega}^{\top}\{\boldsymbol{0}\}\boldsymbol{X}\_{\boldsymbol{\omega}\boldsymbol{\omega}}\boldsymbol{x}\big(\boldsymbol{0}\big) + \sum\_{k=0}^{T}\boldsymbol{w}\_{0}^{\top}\{\boldsymbol{k}\}\boldsymbol{B}\_{0}^{\top}\boldsymbol{X}\_{\boldsymbol{\omega}\boldsymbol{\omega}}\boldsymbol{B}\_{0}\boldsymbol{w}\_{0}(\boldsymbol{k}\}\Big\} \\ &= \lim\_{T\to\boldsymbol{\infty}}\frac{1}{T}\Big{\Big{}}\Big{\Big{}}\boldsymbol{\bar{x}}\_{0}^{\top}\boldsymbol{X}\_{\boldsymbol{\omega}\boldsymbol{\omega}}\boldsymbol{\bar{x}}\_{0} + \operatorname{tr}\{\boldsymbol{X}\_{\boldsymbol{\omega}\boldsymbol{\omega}}\boldsymbol{B}\_{0}\} + \sum\_{k=0}^{T}\operatorname{tr}\{\boldsymbol{B}\_{0}^{\top}\boldsymbol{X}\_{\boldsymbol{\omega}\boldsymbol{\omega}}\boldsymbol{B}\_{0}\boldsymbol{B}\_{1}(\boldsymbol{k})\Big\}\Big{\\ &= \lim\_{T\to\boldsymbol{\omega}}\frac{1}{T}\sum\_{k=0}^{T}\operatorname{tr}\{\boldsymbol{B}\_{0}^{\top}\boldsymbol{X}\_{\boldsymbol{\omega}\boldsymbol{\omega}}\boldsymbol{B}\_{0}\boldsymbol{B}\_{1}(\boldsymbol{k})\Big\}\end{split}$$

Thus, we conclude that

$$\sup\_{w \in L\_{z\_{\ast}}} \left| f\_E \right| = \lim\_{T \to \infty} \frac{1}{T} \sum\_{k=0}^{T} tr(B\_0^T X\_{\ast \ast} B\_0 R\_1(k)) \text{ subject to } \left\| \left. T\_{zw} \right\|\_{\\*\*\ast} \leqslant \gamma \text{ Q.E.D.}$$

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