Discrete Time Systems by Mario A. Jordan and Jorge L. Bustamante - HTML preview

ˆ

sup { J}

T

= x

<

0 X∞ x 0 subject to

zw

T

γ

∞

w∈ 2

L +

if for any admissible uncertainty F

Δ ( k) , there exists a stabilizing solution X∞ ≥ 0 to the

inequality (11) such that

2

−

T

U = − γ

>

1

I

Bˆ X∞ Bˆ

0 .

F∞

F∞

Proof: Suppose that for any admissible uncertainty F

Δ ( k) , there exists a stabilizing solution

X

−

T

∞ ≥ 0 to the inequality (11) such that

2

U = − γ

>

1

I

Bˆ X∞ Bˆ

0 . This implies that the

F∞

F∞

solution X

−

−

T

∞ ≥ 0 is such that

2

1

Aˆ + γ Bˆ U B X∞ A is stable. Then it follows from

F∞

F

1

ˆ

ˆ

∞

F∞

F∞

Lemma 2.3 that

<

zw

T

γ . Using the same argument as in the proof of Lemma 2.3, we get

∞

that Aîs stable and J can be rewritten as follows:

F∞

∞

2

1

2

2

2

2

2

2

−

1

J = ∑{−Δ V( (

x k))

−

T

− z + γ w − γ U

− γ

1 ( w

U 1 Bˆ X∞ Aˆ x)

F∞

F∞

k=0

(18)

T

T

2

−

T

1

−

T

T

ˆ T ˆ

+ x ( Aˆ X∞ Aˆ − X + γ

∞

Aˆ X∞ Bˆ U B X∞ A + C C + Q + F∞ RF∞ x

F∞

F∞

F∞

F

1

ˆ

ˆ

ˆ

ˆ

) }

∞

F∞

F∞

F∞ F∞

Substituting (11) for (18) to get

2

1

T

2

2

2

2

2

2

−

1

−

T

J < x

−

+ γ

− γ

− γ

0 X∞ x 0

z

w

U ( w

U B X∞ A x) (19a)

2

2

1

1

ˆ

ˆ

F∞

F∞

2

or

2

1

T

2

ˆ

2

2

2

−

1

−

T

J < x

−

− γ

− γ

0 X∞ x 0

z

U ( w

U B X∞ A x) (19b)

2

1

1

ˆ

ˆ

F∞

F∞

2

By letting

2

−

1

−

T

w = γ U

ˆ

=

1 Bˆ X∞ Aˆ x for all k ≥ 0 , we get that (

x k)

k

A x with

F

ˆ F 0

∞

F∞

∞

ˆ

2

−

1

−

T

A

ˆ

ˆ = A ˆ + γ

Bˆ U B X∞ A which belongs to L

+∞ since A is stable. It follows

F

2[0,

)

ˆ

∞

F∞

F

1

ˆ

ˆ

∞

F∞

F∞

F∞

Discrete Time Mixed LQR/H∞ Control Problems

171

from (19b) that

ˆ

sup{ J}

T

=

. Thus, we conclude that there exists an admissible

w∈

x X∞ x

2

L +

0

0

non-fragile controller such that

ˆ

sup{ J}

T

=

subject to

< γ . Q. E. D.

w∈

x X∞ x

zw

T

2

L +

0

0

∞

Remark 4.1 In the proof of Lemma 4.1, we let

2

−

1

−

T

w = γ U 1 Bˆ X∞ Aˆ x for all k ≥ 0 to get that

F∞

F∞

ˆ

(

x k)

k

= A

ˆ

−

−

T

ˆ

ˆ x with

2

1

A = A + γ B U B X∞ A which belongs to L

+∞ since A is

F

0

ˆ

ˆ

ˆ

F

2[0,

)

ˆ

∞

F∞

F

1

ˆ

ˆ

∞

∞

F∞

F∞

F∞

stable. Also, we have

2

4 T

w

γ −

=

x X x ,

2

T

z = x X x .

2

0

w 0

2

0

z 0

Then it follows from (19a) that

T

2

J < x

+ γ −

−

0 ( X∞

X

X )

w

z x 0 (20)

∞

∞

where,

ˆ k T T

2

−

T

ˆ

X = ∑{(

k

ˆ k T T

ˆ k

w

Aˆ ) Aˆ X∞ B U B X∞ A A

, and X = ∑{( A ) C C A } .

F

z

ˆ

ˆ

ˆ

ˆ

∞

F

1 1

1

ˆ

ˆ }

∞

F∞ F∞

F∞

F∞ F∞ F∞

k=0

k=0

Note that ˆ Aˆ depends on the controller uncertainty F

Δ ( k) , thus it is difficult to find an

F∞

upper bound of either of Xw and Xz . This implies that the existence of controller

uncertainty Δ F( k) makes it difficult to find sup ∈ { }

w

by using (20). Thus, it is clear that

2

L

J

+

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

become bad.

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

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

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

equation:

T

T

ˆ ˆ T

ˆ

ˆ 1

− ˆ T

2

A X∞ A − X∞ − A X∞ (

B B X∞ B + R)

T

T

B X∞ A + ρ

+

+

=

K

E K

E

C 1 C 1 Qδ 0 (21)

⎡− I

0 ⎤

where, ˆ

1

B = γ −

⎡

⎤ ˆ

⎣

=

= + δ

δ >

1

B

2

B ⎦ , R ⎢

, Qδ Q

I with

0 being a sufficiently small

0

I R⎥

+

⎣

⎦

constant, ρ is a given number satisfying 2

T

ρ I − H

>

T

= − γ −

>

KU 2 H

0

K

,

2

U 1 I

1

B X∞ 1

B

0 ,

T

U =

+ +

T

=

+ γ −

−

2

2

B U 3 2

B

I R and

2

1

U 3 X∞

X∞ 1

B U 1 1

B X∞ . If A is invertible, the parameter-

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

symplectic matrix

⎡

ˆ ˆ 1

− ˆ T − T

2 T

T

ˆ ˆ 1

− ˆ

A + BR B A (

T

− T

ρ

+

+

−

⎤

K

E K

E

C 1 C 1 Qδ )

ˆ

BR B A

S∞ := ⎢

⎥

− T

2

⎢

− A (

T

T

− T

ρ

+

+

K

E K

E

C 1 C 1 Qδ )

A

⎥

⎣

⎦

The following theorem gives the solution to non-fragile discrete time state feedback mixed

LQR/ H∞ control problem.

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

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

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

such that

2 T

U = − γ −

>

T

ρ −

>

1

I

1

B X∞ 1

B

0 and 2 I HKU 2 H

0

K

.

172

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

1

−

T

F∞ = U

− 2 2

B U 3 A

and achieves

ˆ

sup{ J}

T

=

<

w∈

x X∞ x subject to

.

2

L +

0

0

zw

T

γ

∞

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

δ > 0 , there exists a stabilizing solution X∞ ≥ 0 to the parameter-dependent Riccati

equation (21) such that

2 T

U = − γ −

>

T

ρ −

>

1

I

1

B X∞ 1

B

0 and 2 I HKU 2 H

0

K

. This implies that the

solution X

ˆ ˆ T

ˆ

ˆ − ˆ T

∞ ≥ 0 is such that

1

A − (

B B X∞ B + R) B X∞ A is stable. Define respectively the

state matrix and controlled output matrix of closed-loop system

1

−

T

Aˆ = A + B U

−

B U A + H F k E

F

2 (

2

2

3

( ) )

K

K

∞

1

−

T

C ˆ = C + D

U

−

B U A + H F k E

F

1

12 (

2

2

3

( ) )

K

K

∞

and let

1

−

T

= −

−

T

−

+

F

A

A

2

B U 2 2

B U 3 A and

1

F∞ = U B U A H F k E , then it follows from the

∞

2

2

3

( )

K

K

square completion that

T

2

−

T

1

−

T

T

T

Aˆ X∞ Aˆ − X + γ

∞

Aˆ X∞ Bˆ U B X∞ A + C C + Q + F∞ RF

F

∞

∞

F∞

F∞

F

1

ˆ

ˆ

ˆ

ˆ

∞

F∞

F∞

F∞ F∞

T

2

−

T

1

−

T

T