*A. A Central Discrete-Time State Feedback Stochastic Mixed LQR/ H<sup>∞</sup> Controller*

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

$$\left\{ \left\{ \boldsymbol{A}^{T} \boldsymbol{X}\_{\boldsymbol{m}} \boldsymbol{A} - \boldsymbol{X}\_{\boldsymbol{m}} - \boldsymbol{A}^{T} \boldsymbol{X}\_{\boldsymbol{m}} \boldsymbol{\widehat{\boldsymbol{B}}} \boldsymbol{\widehat{\boldsymbol{B}}}^{T} \boldsymbol{X}\_{\boldsymbol{m}} \boldsymbol{\widehat{\boldsymbol{B}}} + \boldsymbol{\widehat{\boldsymbol{R}}} \right\}^{-1} \boldsymbol{\widehat{\boldsymbol{B}}}^{T} \boldsymbol{X}\_{\boldsymbol{m}} \boldsymbol{A} + \boldsymbol{C}\_{1}^{T} \boldsymbol{C}\_{1} + \boldsymbol{Q} = \boldsymbol{0} \tag{24}$$

*AK*

*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*

+ *U*<sup>2</sup> 1 2 (*K* + *U*<sup>2</sup> −1 *B*2 *<sup>T</sup> <sup>U</sup>*3*A*)*<sup>x</sup>*

> −1 *B*2

*<sup>K</sup>* {*JE*} <sup>=</sup> lim *T* →*∞* 1 *<sup>T</sup>* {*x*¯ <sup>0</sup>

the following theorem:

*B*1

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

LQR/ *H∞* controller is given by

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

If*K* = −*U*<sup>2</sup>

sup *w*∈*L* 2+ inf

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

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

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

achieve

sup *w*∈*L* 2+ inf +(*K* + *U*<sup>2</sup>

It follows from (25) and (26) that

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

−1 *B*2

<sup>=</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>*

*<sup>T</sup> <sup>U</sup>*3*A*, then we get that

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

*AK*

*<sup>T</sup> <sup>U</sup>*3*A*)*<sup>T</sup> <sup>U</sup>*2(*<sup>K</sup>* <sup>+</sup> *<sup>U</sup>*<sup>2</sup>

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

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

*<sup>T</sup> <sup>X</sup>∞x*¯ <sup>0</sup> <sup>+</sup> *tr*(*X∞R*0) <sup>+</sup> ∑

−1 *B*1

*Remark 3.5* When*ΔK* =0, Theorem 3.1 reduces to Theorem 3.2.

<sup>−</sup>*Δ<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>* (*k*)*B*<sup>0</sup>

*k*=0 *T tr*(*B*<sup>0</sup>

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

*Theorem 3.2* There exists a central discrete-time state feedback stochastic mixed LQR/*H∞* con‐ troller if the discrete-time Riccati equation (24) has a stabilizing solution *X<sup>∞</sup>* ≥0 and

Moreover, if this condition is met, the central discrete-time state feedback stochastic mixed

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

*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

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

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

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

−1 *BK*

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

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

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

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

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

*<sup>T</sup> <sup>X</sup>∞B*0*R*1(*k*))}

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

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

*TCK* <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>K</sup> <sup>T</sup> RK*

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

*TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>*

*U*1 −1 *BK*

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

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

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

2

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

(26)

17

(27)

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>* . Using the similar argument as in the proof of Theo‐ rem 3.1 in Xu (2011), the expected cost function *JE* can be rewritten as:

*JE* = lim *T* →*∞* 1 *<sup>T</sup> <sup>E</sup>*{∑ *k*=0 *T* (*x <sup>T</sup>* (*k*)*Qx*(*k*) + *u <sup>T</sup>* (*k*)*Ru*(*k*)−*γ* <sup>2</sup> *w* <sup>2</sup> )} = lim *T* →*∞* 1 *<sup>T</sup> <sup>E</sup>*{∑ *k*=0 *T* <sup>−</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> 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>* (*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* <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>K</sup> <sup>T</sup> RK*)*<sup>x</sup>* +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*) } (25)

Note that

#### Stochastic Mixed LQR/H∞ Control for Linear Discrete-Time Systems http://dx.doi.org/10.5772/51019 17

$$\begin{aligned} & \mathbf{A}\_{K}^{T} \mathbf{X}\_{\text{ov}} \mathbf{A}\_{K} - \mathbf{X}\_{\text{ov}} + \mathbf{y}^{-2} \mathbf{A}\_{K}^{T} \mathbf{X}\_{\text{ov}} \mathbf{B}\_{K} \mathbf{U}\_{1}^{-1} \mathbf{B}\_{K}^{T} \mathbf{X}\_{\text{ov}} \mathbf{A}\_{K} + \mathbf{C}\_{K}^{T} \mathbf{C}\_{K} + \mathbf{Q} + \mathbf{K}^{T} \mathbf{R} \mathbf{K} \\ &= \mathbf{A}^{T} \mathbf{X}\_{\text{ov}} \mathbf{A} - \mathbf{X}\_{\text{ov}} - \mathbf{A}^{T} \mathbf{X}\_{\text{ov}} \mathbf{B} \mathbf{B}^{T} \mathbf{X}\_{\text{ov}} \mathbf{B} + \mathbf{R} \mathbf{I}^{-1} \mathbf{B}^{T} \mathbf{X}\_{\text{ov}} \mathbf{A} + \mathbf{C}\_{1}^{T} \mathbf{C}\_{1} + \mathbf{Q} \\ &+ \left( \mathbf{K} + \mathbf{U}\_{2}^{-1} \mathbf{B}\_{2}^{T} \mathbf{U}\_{3} \mathbf{A} \right)^{T} \mathbf{U}\_{2} \left( \mathbf{K} + \mathbf{U}\_{2}^{-1} \mathbf{B}\_{2}^{T} \mathbf{U}\_{3} \mathbf{A} \right) \end{aligned} \tag{26}$$

It follows from (25) and (26) that

*E*{∑ *k*=0 *T w*0 *<sup>T</sup>* (*k*)*B*<sup>0</sup>

sup *w*∈*L* 2+

= lim *T* →*∞* 1 *<sup>T</sup>* {*x*¯ <sup>0</sup>

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

sup *w*∈*L* 2+

where, *B*

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

Note that

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

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

{*JE*} = lim *T* →*∞* 1

16 Advances in Discrete Time Systems

Thus, we conclude that

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

*<sup>T</sup> <sup>X</sup>∞AK <sup>x</sup>*(*k*)} <sup>=</sup>*E*{∑

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

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

*k*=0 *T tr*(*B*<sup>0</sup>

=∑ *k*=0 *T* ∑ *i*=0 *k*−1 *tr*{*B*<sup>0</sup>

Based on the above, it follows from Lemma 3.1 that

*<sup>T</sup> <sup>E</sup>*{*<sup>x</sup> <sup>T</sup>* (0)*X∞x*(0) <sup>+</sup> ∑

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

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

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

*<sup>T</sup> <sup>X</sup>∞x*¯ <sup>0</sup> <sup>+</sup> *tr*(*X∞R*0) <sup>+</sup> ∑

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

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

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

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

*<sup>T</sup> <sup>X</sup>∞B*0*R*1(*k*))}

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

We are to find a central solution to the discrete-time state feedback stochastic mixed LQR/

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

> 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>

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

<sup>0</sup> *<sup>R</sup>* <sup>+</sup> *<sup>I</sup>* . Using the similar argument as in the proof of Theo‐

)}

*U*1 −1 *BK*

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

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

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

*TCK* <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>K</sup> <sup>T</sup> RK*)*<sup>x</sup>*

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

2

(25)

*TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>* =0 (24)

*H∞* control problem.This central solution involves the discrete-time Riccati equation

*A. A Central Discrete-Time State Feedback Stochastic Mixed LQR/ H<sup>∞</sup> Controller*

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

rem 3.1 in Xu (2011), the expected cost function *JE* can be rewritten as:

<sup>−</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>

*AK*

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

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

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

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

*<sup>T</sup>* (*k*))} =0

$$\begin{split} \mathcal{I}\_{E} &= \lim\_{T \to \omega} \frac{1}{T} E\left| \sum\_{k=0}^{T} \{\mathbf{x}^{T}(k)Q\mathbf{x}(k) + \mathbf{u}^{T}(k)Ru(k)\} \right| \\ &= \lim\_{T \to \omega} \frac{1}{T} E\left| \sum\_{k=0}^{T} \mathbf{I} - \Delta V \{\mathbf{x}(k)\} - \left\| \left\| \mathbf{z} \right\|^{2} + \gamma^{2} \left\| \left\| \mathbf{w} \right\| \right\|^{2} - \gamma^{2} \left\| \mathbf{U}\_{1} \right\|^{2} \{\mathbf{w} - \gamma^{-2} \mathbf{U}\_{1}^{-1} \mathbf{B}\_{\mathbf{K}}^{T} \mathbf{X}\_{\omega} A\_{\mathbf{K}} \mathbf{x} \} \right\|^{2} \\ &+ \left\| \left\| \mathbf{U}\_{2} \right\|^{2} (\mathbf{K} + \mathbf{U}\_{2}^{-1} \mathbf{B}\_{2}^{T} \mathbf{U}\_{3} A) \mathbf{x} \right\|^{2} + 2w\_{0}^{T}(k) \mathbf{B}\_{0}^{T} \mathbf{X}\_{\omega} A\_{\mathbf{K}} \mathbf{x}(k) + w\_{0}^{T}(k) \mathbf{B}\_{0}^{T} \mathbf{X}\_{\omega} B\_{0} w\_{0}(k) \right\| \end{split} \tag{27}$$

If*K* = −*U*<sup>2</sup> −1 *B*2 *<sup>T</sup> <sup>U</sup>*3*A*, then we get that

$$\begin{aligned} &\underset{w\in\mathbb{L}\_{\mathcal{Z}\_{\omega}}}{\operatorname{sup}}\inf\_{\boldsymbol{X}}\Big[\boldsymbol{J}\_{E}\Big]=\lim\_{T\to\boldsymbol{\omega}}\frac{1}{T}\Big[\bar{\boldsymbol{x}}\_{0}^{T}\boldsymbol{X}\_{\circ\circ}\bar{\boldsymbol{x}}\_{0}+tr\left(\boldsymbol{X}\_{\circ\circ}\boldsymbol{R}\_{0}\right)+\sum\_{k=0}^{T}tr\left(\boldsymbol{B}\_{0}^{T}\boldsymbol{X}\_{\circ\circ}\boldsymbol{B}\_{0}\boldsymbol{R}\_{1}(k)\right)\Big] \\ &=\lim\_{T\to\boldsymbol{\omega}}\frac{1}{T}\sum\_{k=0}^{T}tr\left(\boldsymbol{B}\_{0}^{T}\boldsymbol{X}\_{\circ\circ}\boldsymbol{B}\_{0}\boldsymbol{R}\_{1}(k)\right) \end{aligned}$$

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

*Theorem 3.2* There exists a central discrete-time state feedback stochastic mixed LQR/*H∞* con‐ troller if the discrete-time Riccati equation (24) has a stabilizing solution *X<sup>∞</sup>* ≥0 and *<sup>U</sup>*<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, if this condition is met, the central discrete-time state feedback stochastic mixed LQR/ *H∞* controller is given by

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

where,*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>∞* ,*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*B*2.

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

$$\limsup\_{w \in \mathbb{Z}\_{z\_{\omega}}} \inf\_{K} \left| J\_{E} \right| = \lim\_{T \to \omega} \frac{1}{T} \sum\_{k=0}^{T} tr\left(B\_{0}^{T} X\_{\omega} B\_{0} R\_{1}(k)\right) \text{ subject to } \left\| \left. T\_{zw} \right\|\_{\ast \ast} < \gamma \right\|\_{\ast}$$

*Remark 3.5* When*ΔK* =0, Theorem 3.1 reduces to Theorem 3.2.

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

#### *B. Numerical Algorithm*

In order to calculate a kind of discrete-time state feedback stochastic mixed LQR/ *H<sup>∞</sup>* con‐ trollers, we propose the following numerical algorithm.

#### *Algorithm 3.1*

Step 1: Fix the two weighting matrices *Q* and*R*, set*i* =0, *ΔKi* =0, *U*2(*i*)=0and a small scalar*δ*, and a matrix *M* which is not zero matrix of appropriate dimensions.

Step 2: Solve the discrete-time Riccati equation

$$\Delta \boldsymbol{A}^{T} \mathbf{X}\_{i} \mathbf{A} - \mathbf{X}\_{i} - \mathbf{A}^{T} \mathbf{X}\_{i} \overset{\wedge}{\mathbf{B}} \mathbf{(B}^{T} \mathbf{X}\_{i} \overset{\wedge}{\mathbf{B}} + \overset{\wedge}{\mathbf{R}})^{-1} \overset{\wedge}{\mathbf{B}}^{T} \mathbf{X}\_{i} \mathbf{A} + \mathbf{C}\_{1}^{T} \mathbf{C}\_{1} + \mathbf{Q} + \boldsymbol{\Delta} \mathbf{K}\_{i}^{T} \mathbf{U}\_{2\{i\}} \boldsymbol{\Delta} \mathbf{K}\_{i} = \mathbf{0}$$

for *Xi* symmetric non-negative definite such that

$$\stackrel{\frown}{A}\_{cl} = A - \stackrel{\frown}{B} (\stackrel{\frown}{B}^T X\_i \stackrel{\frown}{B} + \stackrel{\frown}{R})^{-1} \stackrel{\frown}{B}^T X\_i A \text{ is stable and} \\ U\_{1(i)} = I - \gamma^{-2} B\_1^T X\_i B\_1 > 0.$$

Step 3: Calculate*U*3(*i*) , *U*2(*i*) and *Ki* by using the following formulas

$$\begin{aligned} \mathbf{U}\_{3(i)} &= \mathbf{X}\_i + \boldsymbol{\gamma}^{-2} \mathbf{X}\_i \mathbf{B}\_1 \mathbf{U}\_{1(i)}^{-1} \mathbf{B}\_1^T \mathbf{X}\_i \\ \mathbf{U}\_{2(i)} &= \mathbf{R} + \boldsymbol{I} + \mathbf{B}\_2^T \mathbf{U}\_{3(i)} \mathbf{B}\_2 \\ \mathbf{K}\_i &= -\boldsymbol{\Lambda} \mathbf{I}\_{2(i)}^{-1} \mathbf{B}\_2^T \mathbf{U}\_{3(i)} \mathbf{A} + \boldsymbol{\Delta} \mathbf{K}\_i \end{aligned} \tag{28}$$

rameter matrix is also constrained to be an admissible controller error. In order to give some interpretation for this fact, we provided the following result of discrete-time state feedback stochastic mixed LQR/*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

*Theorem 3.3* There exists a state feedback stochastic mixed LQR/*H∞*controller if there exists a

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

^

, the step 2 of algorithm 3.1 is to solve the discrete-time Riccati equation

*<sup>T</sup> XiAK* is stable. This implies the condition ii displayed in Theorem 3.1

*TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>Δ</sup>Ki*

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

−1 *B*1

*<sup>T</sup> <sup>U</sup>*3*<sup>A</sup>* (29)

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

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

*<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>c</sup>* is stable and *ΔK*is an admissible

*<sup>T</sup> <sup>X</sup>∞AK* is stable if the conditions i-ii of Theorem 3.1

*<sup>T</sup> <sup>U</sup>*2(*i*)

is an admissible controller error, so

*ΔKi*

. Since *A* ^ *ci* = *A*−

(30)

19

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

−1 *B*2

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

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

*<sup>T</sup> <sup>X</sup>∞AK* is stable if *<sup>A</sup>*

controller error. Thus we show easily that in the case of*ΔK* = *L* , there exists a matrix *L* such that (29) holds, where, *X∞*is a symmetric non-negative definite solution of discrete-time Ric‐

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

^ =0

^ <sup>=</sup>*C*<sup>1</sup>

^ *<sup>T</sup> XiA* <sup>+</sup> *<sup>Q</sup>*

(1985) ( also see Geromel et al. 1989, de Souza & Xie 1992, Kucera & de Souza 1995).

*L* = *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>*

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

*BKU*<sup>1</sup> −1 *BK*

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

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

*BKU*<sup>1</sup> −1 *BK*

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

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

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

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

Note that*AK* <sup>=</sup> *AK* <sup>∗</sup> <sup>+</sup> *<sup>B</sup>*2*ΔK*,*<sup>K</sup>* <sup>=</sup> *<sup>K</sup>* <sup>∗</sup> <sup>+</sup> *<sup>Δ</sup>K* and

*BKU*<sup>1</sup> −1 *BK*

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

This implies that *AK* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

controller error*ΔKi*

*<sup>A</sup><sup>T</sup> XiA*<sup>−</sup> *Xi* <sup>−</sup> *<sup>A</sup><sup>T</sup> XiB*

*BKU*1(*i*) <sup>−</sup><sup>1</sup> *BK*

cati equation (30) and *AK* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

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

being a stabilizing solution, where,*Q*

makes the calculation of the algorithm 3.1 become easier.

^ *<sup>T</sup> XiA* is stable and *ΔKi*

matrix *L* such that

equation

and *AK* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>

Where, *B*

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

*A* ^

hold.

for *Xi*

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

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

Step 4: Let *ΔKi*+1 =*ΔKi* + *δM* (or*ΔKi*+1 =*ΔKi* −*δM* ) and*U*2(*i*+1)=*U*2(*i*) .

Step 5: If *Ai* = *A* + *B*2*Ki* is stable, that is, *ΔKi* is an admissible controller error, then increase *i* by1, goto Step 2; otherwise stop.

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

$$\mathbf{K}\_{i} = -\boldsymbol{\mathsf{U}}\_{2(i)}^{-1} \boldsymbol{\mathsf{B}}\_{2}^{T} \boldsymbol{\mathsf{U}}\_{3(i)} \boldsymbol{\mathsf{A}} \pm i \boldsymbol{\delta} \boldsymbol{\mathsf{M}}$$
 
$$(i = \boldsymbol{0}, \boldsymbol{1}, \boldsymbol{2}, \cdots, \boldsymbol{n}, \cdots)$$

#### *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 con‐ strained 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 pa‐ rameter matrix is also constrained to be an admissible controller error. In order to give some interpretation for this fact, we provided the following result of discrete-time state feedback stochastic mixed LQR/*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

$$L \quad = K + \mathcal{U}\_2^{-1} \mathcal{B}\_2^T \mathcal{U}\_3 A \tag{29}$$

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

$$\begin{aligned} \mathbf{A}^T \mathbf{X}\_{\circ \circ} \mathbf{A} - \mathbf{X}\_{\circ \circ} - \mathbf{A}^T \mathbf{X}\_{\circ \circ} \overset{\wedge}{\mathbf{B}} & \overset{\wedge}{\mathbf{B}} (\overset{\wedge}{\mathbf{B}}^T \mathbf{X}\_{\circ \circ} \overset{\wedge}{\mathbf{B}} + \overset{\wedge}{\mathbf{R}})^{-1} \overset{\wedge}{\mathbf{B}}^T \mathbf{X}\_{\circ \circ} \mathbf{A} \\ + \mathbf{C}\_1 \overset{\wedge}{\mathbf{C}} \mathbf{C}\_1 + \mathbf{Q} + \mathbf{L} \overset{\wedge}{\mathbf{L}} \mathbf{U}\_2 \mathbf{L} &= \mathbf{0} \end{aligned} \tag{30}$$

and *AK* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup> *BKU*<sup>1</sup> −1 *BK <sup>T</sup> <sup>X</sup>∞AK* 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.

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>* , *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>∞*,*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*B*2.

Note that*AK* <sup>=</sup> *AK* <sup>∗</sup> <sup>+</sup> *<sup>B</sup>*2*ΔK*,*<sup>K</sup>* <sup>=</sup> *<sup>K</sup>* <sup>∗</sup> <sup>+</sup> *<sup>Δ</sup>K* and

stochastic interpretation of the discrete-time state feedback mixed LQR/*H∞* control problem

In order to calculate a kind of discrete-time state feedback stochastic mixed LQR/ *H<sup>∞</sup>* con‐

Step 1: Fix the two weighting matrices *Q* and*R*, set*i* =0, *ΔKi* =0, *U*2(*i*)=0and a small scalar*δ*,

*TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>Δ</sup>Ki*

by using the following formulas

*A* + *ΔKi*

*<sup>T</sup> <sup>U</sup>*3(*i*) *B*2

Using the above algorithm, we obtain a kind of discrete-time state feedback stochastic mixed

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 con‐ strained 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 pa‐

<sup>−</sup><sup>1</sup> *<sup>B</sup>*<sup>1</sup> *<sup>T</sup> Xi*

*B*1

*<sup>T</sup> <sup>U</sup>*2(*i*)

*<sup>T</sup> XiB*<sup>1</sup> >0.

.

is an admissible controller error, then increase *i*

*ΔKi* =0

(28)

^ *<sup>T</sup> XiA* <sup>+</sup> *<sup>C</sup>*<sup>1</sup>

^ *<sup>T</sup> XiA* is stable and*U*1(*i*)<sup>=</sup> *<sup>I</sup>* <sup>−</sup>*<sup>γ</sup>* <sup>−</sup><sup>2</sup>

*<sup>U</sup>*3(*i*)<sup>=</sup> *Xi* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup>*XiB*1*U*1(*i*)

<sup>−</sup><sup>1</sup> *<sup>B</sup>*<sup>2</sup> *<sup>T</sup> <sup>U</sup>*3(*i*)

*U*2(*i*)=*R* + *I* + *B*<sup>2</sup>

*Ki* = −*U*2(*i*)

Step 4: Let *ΔKi*+1 =*ΔKi* + *δM* (or*ΔKi*+1 =*ΔKi* −*δM* ) and*U*2(*i*+1)=*U*2(*i*)

is stable, that is, *ΔKi*

considered by Xu (2011).

18 Advances in Discrete Time Systems

trollers, we propose the following numerical algorithm.

symmetric non-negative definite such that

and *Ki*

Step 2: Solve the discrete-time Riccati equation

, *U*2(*i*)

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

and a matrix *M* which is not zero matrix of appropriate dimensions.

*B. Numerical Algorithm*

*<sup>A</sup><sup>T</sup> XiA*<sup>−</sup> *Xi* <sup>−</sup> *<sup>A</sup><sup>T</sup> XiB*

Step 3: Calculate*U*3(*i*)

Step 5: If *Ai* = *A* + *B*2*Ki*

<sup>−</sup><sup>1</sup> *<sup>B</sup>*<sup>2</sup> *<sup>T</sup> <sup>U</sup>*3(*i*)

(*i* =0,1,2,⋯, *n*, ⋯)

*Ki* = −*U*2(*i*)

by1, goto Step 2; otherwise stop.

LQR/*H∞* controllers as follows:

*A* ± *iδM*

*C. Comparison with Related Well Known Results*

*Algorithm 3.1*

for *Xi*

*ci* = *A*− *B* ^ (*B* ^ *<sup>T</sup> XiB* ^ <sup>+</sup> *<sup>R</sup>* ^ ) −1 *B*

*A* ^

$$\stackrel{\frown}{A}\_{\boldsymbol{c}} = \boldsymbol{A} - \stackrel{\frown}{B} (\stackrel{\frown}{B}^T \boldsymbol{X}\_{\circ \circ} \stackrel{\frown}{B} + \stackrel{\frown}{R})^{-1} \stackrel{\frown}{B}^T \boldsymbol{X}\_{\circ \circ} \boldsymbol{A} = \boldsymbol{A}\_{\boldsymbol{K}^\* \ast} + \stackrel{\frown}{\gamma}^{-2} \boldsymbol{B}\_{\boldsymbol{K}^\*} \, \! \cdot \boldsymbol{U}\_1^{-1} \boldsymbol{B}\_{\boldsymbol{K}^\*}^T \, \! \boldsymbol{X}\_{\circ \circ} \boldsymbol{A}\_{\boldsymbol{K}^\*} \, \! \cdot \boldsymbol{U}\_1$$

This implies that *AK* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup> *BKU*<sup>1</sup> −1 *BK <sup>T</sup> <sup>X</sup>∞AK* is stable if *<sup>A</sup>* ^ *<sup>c</sup>* is stable and *ΔK*is an admissible controller error. Thus we show easily that in the case of*ΔK* = *L* , there exists a matrix *L* such that (29) holds, where, *X∞*is a symmetric non-negative definite solution of discrete-time Ric‐ cati equation (30) and *AK* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup> *BKU*<sup>1</sup> −1 *BK <sup>T</sup> <sup>X</sup>∞AK* is stable if the conditions i-ii 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 error*ΔKi* , the step 2 of algorithm 3.1 is to solve the discrete-time Riccati equation

$$A^T X\_i A - X\_i - A^T X\_i \overset{\wedge}{B} \overset{\wedge}{(B}^T X\_i \overset{\wedge}{B} + \overset{\wedge}{R})^{-1} \overset{\wedge}{B}^T X\_i A + \overset{\wedge}{Q} = 0$$

for *Xi* being a stabilizing solution, where,*Q* ^ <sup>=</sup>*C*<sup>1</sup> *TC*<sup>1</sup> <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>Δ</sup>Ki <sup>T</sup> <sup>U</sup>*2(*i*) *ΔKi* . Since *A* ^ *ci* = *A*− *B* ^ (*B* ^ *<sup>T</sup> XiB* ^ <sup>+</sup> *<sup>R</sup>* ^ ) −1 *B* ^ *<sup>T</sup> XiA* is stable and *ΔKi* is an admissible controller error, so *AK* <sup>+</sup> *<sup>γ</sup>* <sup>−</sup><sup>2</sup> *BKU*1(*i*) <sup>−</sup><sup>1</sup> *BK <sup>T</sup> XiAK* is stable. This implies the condition ii displayed in Theorem 3.1 makes the calculation of the algorithm 3.1 become easier.
