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Robustness Bound of LQ Guaranteed Cost Control Systems for Parameter Uncertainty.

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Dev. Chem Eng. Mineral Process., 9(ID),pp.199-204, 2001.
Robustness Bound of LQ Guaranteed Cost
Control Systems for Parameter Uncertainty
Anke Xue", Yao Chen and Youxian Sun
National Laboratory of Industrial Control Technology, Institute of
Industrial Process Control, Zhejiang University, Hangzhou, P R. China
~
~~
The problem of robustness bound for parameter uncertain systems with LQ
guaranteed cost controller is considered in this paper. A parameter dependent
robustness bound for parameter uncertainties is derived. Then, an optimization
problem of improving the robustness of parameter uncertain systems is proposed. And
also, the responding Optimization technique is given in the paper.
Introduction
During the past few years, many results for robust stability bounds of systems with
parameter uncertainties have been presented using a variety of methods [l-31. Most of
those proposed robustness bounds only guarantee the uncertain systems remaining
stable, i.e. robust stability bounds on uncertain systems. Although some results are
derived based upon the robust LQ problem, conservatism may obviously be
introduced in order to minimize the value of performance index by using a quadratic
Lyapunov function. In fact, those robustness bounds, which try to ensure fvred cost
function is minimal, will certainly make the stable domain of the closed-loop system
very small. On the other hand, these results may present high norm gains, since they
consider the maximization of the stability robustness of the closed-loop system
without considering the trade-off between performance and control effort [4].
Therefore, robustness bounds considering the trade-off between stability and
performance would be quite useful in the robust control system design.
A possible way to deal with above problems is to adopt an idea of minimizing a
certain upper bound (not minimal value) for the quadratic criterion of the LQ control
problem for the uncertain closed-loop system. Such idea results from the guaranteed
cost control problem (GCCP) [ 5 ] . The GCCP has been one of the most active areas of
research because the guaranteed cost controllers guarantee the closed-loop system
designed quadratic stability and a certain level of LQ performance for all admissible
values of the uncertainty. Many results for GCCP have been presented [6-91.
However, no general procedure for buildmg a guaranteed cost controller for
parametric uncertainties is available in the literature because of the linear or norm
bound restriction for the uncertainty [7,8], and the difficulty of solving the three or
more coupled design equations with some parameters. Practically, solving these
nonlinear equations is not an easy task. The numerical solution still remains open [9,
* Authorfor correspondence (akxue@iipc..zju.edu.cn).
199
A. Xue, Y. Chen and Y.Sun
101. A difficult thing here is how to determine the robustness bounds of GCCP. We
shall refer to this as robustness bounds of guaranteed cost control (RBGCC).
Problem Formulations
We consider an uncertain continuous-time system of the form
i ( t )= ( A , +AA)x(t)
+ Bu(t),
x(0) = X,
(1)
where x ( t ) E R" is the state. u(t) E R m is the control input, A, and B are real constant
matrices of appropriate dimensions that describe the nominal system, and AA is a
real valued matrix representing the plant uncertainty as follows:
AA
I
= CkiAi
i=l
where ki are real uncertain parameters and A, are given real constant matrices.
Associated with this system is the cost function.
J = 10"(x * (t)Qx(t) + u (t)Ru(t))dt
(3)
where, Q 2 0, R > 0.
Definition 1: The uncertain system (1) is said to be quadratically stable if there
exists a positive definite symmetric ma& such that
(A,+AA)*~+~(A,+AA)cO
(4)
Furthermore, the uncertain system (1) is said to be quadratically stabilizable via linear
state feedback if there exists a h e a r state feedback law of the form
u(t) = f i ( t ) ,
K €Rmxn
such that #he resulting closed-loop system
i ( t ) = ( A + A&(t)
(6)
is quadratically stable, where A = A, + BK .
Definition 2: A linear state feedback controller ( 5 ) is said to be a quadratic
guaranteed cost controller with associated cost matrix for the uncertain system (1) and
cost function (3) if
( A + AA)* P + P ( A + AA) + Q c 0
(7)
for all existing uncertainty AA . The positive-definite matrix P satisfjmg (7) is said to
be a quadratic guaranteed cost matrix, and the resulting uncertain closed-loop system
(6) is then said to be a quadratic guaranteed cost control (QGCC) system.
Lemma 1: If there exists a matrix P > 0 satisfjmg (7) for the uncertain system
(1) and cost function (3), then the system is quadratically stabilizable and the cost
function satisfies the bound
200
Robustness Bound of LQ Guaranteed Cost Control Systems
J IxoT Px,
(8)
Proof: In fact, the Lemma 1 is an immediate consequence of the Definition 1,2.
Remark 1: Lemma 1 shows that a QGCC system is a quadratic stable system.
Robustness Bound of LQ Guaranteed Cost Control
Leta,,(X)
(d,(X))
and a-(X)
( d ~ n ( X ) denote
)
the maximum and
minimum singular values (eigenvalues) of matrix, respectively, and d( X) denote any
eigenvalue of matrix X. Define Hi as
i=1,2, ...1
H~ = A'P+ PA,
(9)
Note that Hi 's are real and symmetric (Hermitian) matrices.
Theorem 1 The uncertain closed-loop system (6) is a QGCC system. if
I
c Jki10-
( H i ) < 1-
i=l
(-( A P + PA + Q))
(10)
for any parameter uncertainty A4 satisfying (2).
Proof: From (2) and (9), we have
s = ( A + AA)'
I
P + P ( A + AA)+ Q = z k , ~+ ,A' P + P A + Q
(11)
i=l
For the system (6) to be a QGCC system, S must be negative, or equivalently, the all
eigenvalues of S are negative, i.e.
1
n(s) = n(x k i ~+iA
~ P P+A + Q ) < o
i=l
(12)
Note that for any Hermitian matrices A and B
A ( A + B ) IA(,
A ) + d,(B)
(13)
then
I
k i d , ( H i )- d ~( -"( A TP + PA + Q))
d(S) I
14)
i=l
Now, since A ,
(Hi ) = CT,
(Hi ) for any Hermitian matrices Hi ,it follows
1
A(S)I c I ~ ~ J ~ , ( H ~ ) - ~ . , ~ ( PA
- ( +Q))
A~P+
i=l
In view of (12), the proof of this theorem is completed.
Remark 2: The significance of the Theorem 1 is that the theorem relates the
weighting and feedback matrices to the parameter uncertainties of the system. Thus,
the robustness of QGCC m a y be improved by choosing the weighting matrices and
201
A. Xue,
Y. Chen and Y. Sun
solving state feedback law.
Now, we give a design approach for QGCC system with parmeter uncertainty.
Using Cauchy- Schwarz inequality, we obtain
I
I
lZlkilcma.x(Hi)/
i= 1
2
1
5 ClkiI *C&x(Hi)
i=l
i=1
(16)
From the Definition 2, we have
M ~ P +
PU C - ( A ' P + P A + Q )
(17)
Note that identity
X T Y + Y T X IX T X + Y T Y
(18)
is hold for any real matrices X and Y. Now using (10) into (9) we obtain
AA~AAI-(A~P+PA+Q+P~)
(19)
From (ll), it is easy to know that A r P + PA + Q c 0 , hence
1- ( - ( A T P + PA + Q)) = 0- ( - ( A T P+ PA + Q))
Clearly, (10) is satisfied if
I
c IkiI2 c
i-1
I
(
0
:
" ( - ( ~ f pPA
+ +
QN~/C
015,(H,)
i=l
Thus, the robust stability of QGCC system can be improved by optimizing the
condition (21). The corresponding optimization problem is shown as follows
Obviously, solving this optimization problem is not an easy task in general. One
possible approach is to first choose a proper feedback gain matrix K , which makes
the closed-loop system quadratically stable and minimizes the cost
function J ( M ,K) , and then to apply optimization technique to (22). However, no
general procedure for choosing such a matrix K is available in the literature. As
stated so far this is still an analytically untractable problem. Many methods for
finding such a matrix K for some class of uncertainties have been presented [6-91.
Note that those methods involve solving several coupled nonlinear equations.
Therefore some numerical difficulties arise certainly in using these methods. In fact,
even for the case where the system is free of uncertain parameters, the numerical
solution for the resulting equations is still an open problem. And also, the question of
when these equations have a solution remains open [9].In this paper, we propose a
most practical algorithm for this problem for the structured parameter uncertainty
with the form of (2).
To remove the dependence of the upper cost bound (8) on the initial condition xo ,
202
Robustness Bound of LQ Guaranteed Cost Control Systems
we consider the initial condition to be a zero-mean random variable with covariance
matrix E { xox: } = I where E (.} denotes the expectation operator.
Considering the cost function
Thus a bound on the closed-loop cost function is found to be
Now, with above assumptions, the problem of finding a feedback matrix K so as
to minimize J(AA,K) is equivalent to finding a feedback matrix K that minimizes
the bound given by (24) on the cost function (23) for the uncertain linear system (1).
To make the problem tractable and numerically solvable we introduce the following
assumption for the uncertainty of the system (1).
Assumption 1 Assume that there exists a linear upper bound p(P) for the
uncertainty (2) such that
p(P)2AATP+PAA
A I
(25)
,
1
where p( P) = C (1 / y P + y AT PA ) = 2 d + C y AT PA, , where
j=1
1 '
a =2
l/y,
,y
> 0 , ( j = 1J)
i
are arbitrary real positive numbers.
j=l
Thus, with the Assumption 1, it is easy to venfy that the positive definte matrix
P of (24) is the solution of the parameter dependent algebraic Riccati equation
( A , + B K ) ~ P + P ( A+, B K ) + Q + K ~ R K + ~ ( P ) = o
(26)
Furthermore, using the results given in [8], we can find a desired feedback matrix
K such that the bound f'{P} of the cost function (23) is minimized.
Theorem 2 [8] If there exists a ma& KO and P > 0 ,such that
1
( A , - B K , ) P + P ( A , - B K , , ) ~ + ~ ~ ~ A ~< Po A ;
j-1
where A, is defined by A, = A, + aI then the desired K = R-' BT , where
the unique positive definte solution of the generalized algebraic Riccati equation
is
I
A , P + PA: +Q- P B R - ' B ~ P +C Y ~ A ; P A=~o
j=I
Using the resulting feedback gain matrix K , the robustness bounds of QGCC
system can be obtained by solving the optimizationproblem (22).
A possible design procedure for QGCC system with structured parameter
203
A. Xue. Y. Chen and Y.Sun
uncertainties may be introduced as follows
Step 1 Calculate the Hermitian matrices H i as defined in (9).
Step 2 Choose a proper matrix KO satisfjmg (27).
Step 3 Find a feedback matrix K by solving Riccati-like equation (28).
Step 4 Solve the optimization problem (22) to obtain weighting matrices.
Step 5 Apply (10) of Theorem 1 to obtain the robustness bounds of QGCC
system.
It is clear that the feedback gain K obtained above is optimal. In t h s case, the
linear state-feedback controller ( 5 ) is a robust optimal guaranteed cost controller.
Conclusions
In this paper, a robustness bound of optimal LQ guaranteed cost control systems with
parameter uncertainties is proposed. As the resulting robustness bounds are parameter
and variable dependent, therefore the robustness of controlled systems can be
improved by solving optimization problems. On the other hand, the proposed results
may reduce the conservatism of the robustness bounds because of no norm bounded
restriction on uncertainties. A design procedure of optimal LQ guaranteed cost
controller for parameter uncertain systems is presented in the paper. This procedure
consists of finding an optimal feedback matrix K and solving an optimization
problem.
Acknowledgments
The work was supported by the National Natural Science Foundation of China under
grants No. 69874036,the Chinese Postdoctoral Science Foundation, and Zhejiang
Provincial Natural Science Key Foundation of China under grants No.ZD9905.
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Petersen, 1. R. 1995. Guaranteed cost LQG control of uncertain linear-system. IEE Proc., Part D,
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Received: 28 October 1999;Accepted afier revision: 12 May 2000.
204
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