- Email: [email protected]
Robust Control for Linear Parametric Uncertain Systems with Output Feedback
Copyright @ IFAC Control Systems Design, Bratislava, Slovak Republic, 2000
ROBUST CONTROL FOR LINEAR PARAMETRIC UNCERTAIN SYSTEMS WITH OUTPUT FEEDBACK Vojtech Vesely
1
Dept. of Automatic Control Systems, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Bratislava, Slovak Republic. E-mail [email protected]
Abstract: The paper addresses the design of a robust controller with output feedback for uncertain linear systems in the time domain. The necessary and sufficient conditions for static output feedback stabilizability of linear continuous and discrete time systems are the basis for the proposed robust controller design procedure. The proposed approach does not employ matching conditions. Copyright @2000 IFAC Keywords: Robust control, Linear systems, LQR problem
1. INTRODUCTION
Dynamic systems with bounded uncertainties have been widely used to model physical systems. During the last two decades numerous papers dealing with the design of robust control schemes to stabilize such systems have been published (Ljashevskij, 1991; Peterson and Hollot,1986; Sharev-Shapiro, Palmor and Steinberg, 1998, Kozakova, 1997; Gyurkovics and Takacs,1995; Pakshin, 1997; Yang-Yan Gao and You-Xian Sun, 1998) and references therein. In (Ioannis and Panos, 1995) new conditions for robust stability in linear continuous and discrete time systems are derived when all matrices of the state-space model are perturbed by uncertain parameters and static output feedback is applied. In (Halicka and Rosinova, 1994) robustness bound estimates, based on the direct Lyapunov method for discrete time linear systems are analysed and compared. In (Ljashevskij, 1991; Peterson and Hollot, 1986) the authors study the problem of designing robust state controllers using the Riccati function approach for different types of model uncertainties. 1 The work has been supported by the Slovak Scientific Grant Agence.
421
In (Sharev-Shapiro, Palmor and Steinberg, 1998) both Lyapunov and Riccati based min-max output control law is treated and a simple criterion for the existence of a min-max output controller is provided when all uncertainties are lumped in terms of input matrix which is multiplied by the cone bounded uncertainties as a function of state variable. The necessary and sufficient conditions to stabilize the continuous time system via static output feedback can be found in (Kucera and deSouza, 1995). Robust controller design via sensitivity functions shaping could be found in (M'Saad et aI., 1999). In this paper two approaches have been developed. The necessary and sufficient conditions to stabilize continuous and discrete time systems via static output feedback are used to design the robust controller for different type of uncertainty. In the second approach we pursue the idea of the structurally constrained state feedback, proposed in (Geromel and Peres, 1985) for the design of robust controller for different type of uncertainty. The paper is organized as follows. In Section 2 the problem formulation and some preliminary results are given. The main results of the robust controller design for linear continuous and discrete
time systems are given in Section 3. In Section 4 the obtained theoretical results are applied to the design of robust PI controllers for a DC motor and stabilization of the lateral axis dynamics for an L-1011 aircraft.
2.
where U, Al and A 2 are known matrices and W is an unknown matrix satisfying WTW :S I, 1identity matrix of corresponding dimension. Now let us investigate the robustness bounds for linear continuous and discrete time systems. Lemma 1. Let the closed-loop system with the norm bounded uncertainty (3) is described by
PROBLEM FORMULATION
(8)
In the context of robustness analysis and synthesis of robust controller for linear systems the following uncertain model is commonly used :i: = (A
+ JA)x + (B + JB)u,
where A c = A + BKG and JA c = JA + JBKG. Suppose that A c is Hurwitz with p = maxj (Re(Aj (A c ))) :S 0, where Ai (A c ), i = 1,2, ... , n are eigenvalues of the matrix A c . If for the
y = Gx (1)
norm bounded uncertainty the following inequality holds
for continuous time systems and
x(t
+ 1) = (A + JA)x(t) + (B + JB)u(t)
(9)
(2)
and ii~P ~ JA T PJA, then the closed-loop system (8) is asymptotically stable. 0 Lemma 2. Let the closed loop linear discrete time system with uncertainties is described as follows
y(t) = Gx(t) where x, x(t) E Rn, u, u(t) E R m , and y, y(t) E RI are the state, control and output vector, respectively;A, Band G are known matrices of appropriate dimensions; JA, JB are unknown but bounded matrices. The following types of uncertainty descriptions are often used in the robustness investigations.
x(t
iia:S (3)
where I . I represents the modules of corresponding matrix, Am, Bm are matrices with nonnegative entries and corresponding dimensions, respectively. (3) Matrix polytope type uncertain structure: p
:S
L
fjA j ,
j=1
JB<",B - L... J J
-Al(A~ A c) + j>'M(AI A c) + a
(5)
j=1
u = KCx, where A j , B j are known matrices; fj, Ij are unknown parameters. (4) Matrix type bounded uncertain structure:
JB = UWA 2 ,
(11)
u(t) = KGx(t)
(12)
or static structurally constrained state feedback
u = (K
+ L)x,
u(t) = (K
+ L)x(t)
(13)
so that the closed-loop systems (8) and (10) is robustly asymptotically stable and simultaneously guaranteeing the suboptimal control with respect to the following performance index
where la, Ib are nonnegative constants; Qa, Qb are nonnegative definite matrices. (5) The uncertainties satisfying "matching conditions" : JA = UWA 1 ,
(10)
where iia = qqa, ij~P ~ JA T PJA and iiaAM(AI A)P : AI PA c, a is the exponential stability decay rate of the matrix A c or a = 1- p2, P is spectral radius of matrix A c . 0 Equations (9) and (11) with necessary and sufficient conditions for stabilization of the linear systems via static output feedback imply the new design procedure for the design of robust controllers for linear continuous and discrete time systems proposed in this paper. The problem studied in this paper can be formulated as follows. For continuous and discrete time linear systems described by (1) and (2) with uncertainties (3)-(7) design the static output feedback robust controller with the control algorithm
where 11.11 represents any matrix norm, and qa and qb are nonnegative constants. (2) Element bounded uncertainties, structured model:
JA
= (A c + JAc)x(t)
and control law u(t) = KCx(t) with A c being Hurwitz and for the uncertainties JA c the inequality (3) holds. The perturbed closed-loop system is asymptotically stable if
(1) Norm bounded uncertainties, unstructured model:
IIJAII :S qa, IIJBII:S qb
+ 1)
00
Jc = !(xTQx+uTRu)dt,
(7)
o
422
(14)
co
h
=L
x(tf Qx(t)
+ u(tf Ru(t)
(2) Solve the equation (15) AT Pi
t=O
where Q = QT ~ 0, and R = R T matrices of compatible dimensions.
3. DESIGN OF ROBUST CONTROLLERS
Let us consider the system described by (1). In the following we assume that the nominal system given by u=Kx
(16)
is stabilizable via state feedback. In this section, we will present the new procedure to design robust controllers via static output feedback. Lemma 3. The nominal system (16) is static output feedback stabilizable if and only if there exists a symmetric and positive definite matrix P and a matrix K satisfying the following matrix equality (A
+ BKCf P +
P(A
for Pi symmetric and nonnegative definite. G i = R- ~ B T Pi (I - C T (CC T )-IC)(23) increase i by 1 and go to Step 2.
If the sequence Po, PI, ... , converges, say to P, K is given by (20). The convergence remains to be proved, however. To guarantee the exponential stability of the uncertain closed-loop system the following theorem holds. Theorem 1. The uncertain closed-loop system (1) is robustly exponentially stable with a decay rate a if there exist matrices P = pT > 0, Q = QT ~ 0, R = R T > 0 and a matrix K of appropriate dimensions satisfying the following matrix inequality
3.1 Continuous Time Systems. Output Feedback.
y=Cx,
a
<0
where matrix QI estimates the uncertainties as follows
where Qo = Q~ > o. 0 Due to Lemma 1 and Lemma 3 the gain matrix I{ could be chosen by minimization of the following performance index
P(A
a
G T G - P BR- I B T P
+ BKC) + Qo = 0(17)
= minTrace{(A + BKCf P+ K
T
(A+2"I) P+P(A+2"I)+Q+QI+ (24)
QI
La
PiBR-I B T Pi + (22)
> 0 are the (3)
x=Ax+Bu,
+ PiA + Q -
= II(SA + SBKCf P + P(SA + SBKC)III(25)
and matrix G is given by (21). 0 Calculation of the gain matrix K can be realized by the Algorithm A.
(18) 3.2 Continuous Time Systems. Structurally Constrained State Feedback.
+ BKC) + Qo}
In this section the idea of the structurally constrained state optimal control problem (Geromel and Peres, 1985) has been used for design of robust controllers. Assume the uncertain system The obtained solution of (18) is described by (1). The design problem is to find the static structurally constrained state feedback K = -R- I B T PCT(CCT)-I (20) (13) guaranteeing the robustness properties of the closed-loop system. The necessary and sufficient SHbstituting (19) to (17) after some manipulation conditions for the design of robust controllers are gives given in the following theorem. Theorem 2. Let the uncertain system (1) with AT P + PA + Q - P BR- I B T P + G T G = 0(21) a static structurally constrained state feedback (13) be given. Then, the following statements are where equivalent . where
• The system (1) is static structurally constrained state feedback stabilizable. • There exists a positive definite matrix P and a matrices K and L satisfying the following matrix inequality
Eq. (21) inspires the similar algorithm to those given in (Geromel and Peres, 1985) for calculation of the gain matrix K which guarantee the suboptimal solution of performance index (14). Algorithm A (1) Set i
[A
= 1 and Go = O.
+ SA + (B + SB)(K + L)jT P + P[A+(26) SA
423
+ (B + JB)(K + L)] < 0
• There exist matrices P = pT > 0 and R = RT > 0 and matrices f{ and L that the following inequality holds
3.3
A T p+PA-PB(I+R- 1)B T p+(27) C T R- 1C+ (L+BTpf(L+BTp)f{T Rf{ - LT L P(JA
+ (JA + JB(I{ + L)f P+
+ JB(I{ + L)) < 0
= 1,L o = O,f{o = O,Po = 0,
R- I )-I,QIO=O (2) Calculate -
Qoi = Q
RI
Let the discrete time uncertain system be described by (2). In this section we will present a new procedure for the design of robust controllers via static output feedback. The following theorem will be employed in the further development. Theorem 3. The linear discrete time system (2) is static output feedback stabilizable if and only if (1) (A, B) is stabilizable and (A, C) is detectable, and (2) there exist real matrices f{ and C such that
where C = Rf{ + B T P 0 The algorithm to handle the LQR suboptimal problem with uncertain system via structurally constrained state feedback can be stated as follows. Algorithm B.
(1) i
Discrete Time Systems. Output Feedback
where P is the real symmetric nonnegative definite solution of
= (I +
AT PA - P
+ (Li- 1 + B T Pi-d T (L i- 1+
+Q -
AT PB(B T PB
+ R)-1(30
BTpA+CTG = 0
The matrices Q = QT ~ 0, R = R T > 0 can be taken from the performance index (15).
(3) Calculate Pi =
pT > 0 from
+ PiA + Qoi
AT Pi
Proof. Proof of this theorem is given in(Rosinova, Vesely and Kucera, 2000). 0 The gain matrix f{ is
- Pi BR l 1B T Pi = 0
(4) Calculate f{i = - R- 1B T Pi (5) The entries of the matrix Li are
Equations (30), (31) have inspired the following algorithm for the calculation of robust suboptimal static output feedback gain matrix f{ for discrete time systems. Algorithm C.
Li(j, k) = -f{i(j, k).M(j, k)
The matrix M is given below. (6) Calculate Qli (7) Calculate er = IIL i - L i - 1 IL if er ~ error stop, else i = i + 1 and go to step 2.
(1) i = 1, Go = 0, Q10 (2) Qoi = Q + Qli-l (3) Solve Riccati equation
where for the type of uncertainty (5)
Qli
AT PiA - Pi
P
$
1=1
k=l
= I {L IIAT Pi + PiAtlltlm + L
+ Qoi
R)-I B T PiA
Ikm(28)
=0 - AT PiB(BT PiB+
+ CT-I G i - 1 =
0
pT > O.
for Pi = (4) Calculate
=
=
where tim maxltd, Ijm maxbj I. The matrix M E R mxn defines the structural constraints of the state feedback. The entries of M are defined as follows M(j, k) = 0 iffor thejth input the state Xk is feasible, and M (j, k) = 1 if for the jth input the state Xk is unfeasible. Although the convergence of the Algorithm B has not been formally proven, it has converged for all tests performed in connection with this research. Typically, the number of iterations for convergence varies from 20 - 50 depending on the type of uncertainty and values of matrices Q and R.
f{i
= _(B T PiB + R)-1 B T Pi ACT (CC T )-1 Gi
= (B T PiB + R)-~BT PiA+ (B T PiB
+ R) ~ f{iC
(5) Calculate Q1 i· Q 1 estimates the uncertainties. inrease i by one and go to Step 2. If the sequence Po, PI, ... converges, say to P, the gain matrix f{ is given by (31). The convergence remains to be proved, however.
424
(3) Calculate Pi = ARE
3.4 Discrete Time Systems. State Feedback.
The linear discrete time uncertain systems can be modelled by (2). The problem to be solved is to find the static structurally constrained state feedback (13) which guarantees the suboptimal solution of performance index (15) for all admissible uncertainties. The closed-loop system with static structurally constrained state feedback (13) IS given as
x(t
+ 1) = (AL + oA + oBL + (B+
pl
> 0 from the following
ALpiALi - Pi - ALpiB(BT PiB+ R)-I B T PiALi
+ Qoi =
0
(4) The gain matrix f{i
f{i
= _(B T PjB + R)-l B T PjALi
(5) Entries of matrix L i are
(32)
L j U, k) = -I\;(j, k)M U, k)
oB) K)x(t) (6) Calculate Qli (7) Calculate er = IILi - L i -Ill if er else i = i + 1 and go to Step 2.
where AL = A+BL. The necessary and sufficient conditions for the design of a robust discrete time controller with structurally constrained state feedback is given by the following theorem. Theorem 4. Let the uncertain system (2) with the static structurally constrained state feedback (14) be given. Then, the following statements are equivalent.
where
As a real example we have taken the problem of to design a PI robust controller to control a small DC motors' speed. The DC motor model is given by (1) where
+ oA + (B + oB)(K + L)f P[A+(33) oA
A
(3) There exist matrices P = pT > 0, R = R T > oand K and L satisfying the following matrix inequality.
oA
where Qo = Q + Q1'
o For the uncertainties (5) the matrix Q1 is s
I(L>imIIA; PII + I>rjmll(L + Kf (35)
j=1
i=l p
BJ PII)(L timllAill+ i=l
s
+
L
IjmlIBj(L
= [ ~5.0218
.715
.0422 ]
00]
= [ 1.6885 0 0
oB =
[
~.0597
The uncertainties have been modelled by (5) with oA = 2:;=1 tjA i , tj E< -1,1 > oB = L~=lljBj, Ij E< -1,1 > where each matrix Ai and B i has only one nonzero element, the position of which is that of the 8A(i,j) and oB(k,l). We assume that each entry of matrices aA, oB changes independently from others. The ti, Ij can vary in time arbitrarily, provided that each element is within the given boundary. The results of calculation employing the derived algorithms are summarized in the following. Output feedback. The output matrix is
+ K)II + 211AL + Bf{11)
C_[100] - 00 1
j=l An algorithm to handle the LQR suboptimal problem with structured constrained state feedback can be stated as follows Algorithm D.
(1) i = 1, L o = 0, QlO = 0, Ko = 0 (2) Calculate A Li = A + BL i _ I , Qoi Qli-1 - K[ 1 Rf{i-1
-.0721 ]
B
000
+ Qo - AI P B(B T P B+(34)
p
-4.701 1 0] -8.2986 0 0 [1 0 0
and the uncertainties are
R)-I B T PAL = 0
Q1 =
=
+ (B + oB)(K + L))- P < 0
AIpA L - P
is given in (35)
4. DESIGN OF ROBUST CONTROLLERS. EXAMPLES.
(1) The system (2) is static structurally constrained state feedback stabilizable. (2) There exists the positive definite matrix P and a matrices K and L satisfying the following matrix inequality
[A
Qli
< error stop
For
tim
= Ijm = .65 and = 0 the gain matrix is
= [.153
Q'
1.0706) The eigenvalues of the closedloop nominal model are {-4.2138, -.2495 ± j1.898} and PI controller parameters are as follows f{
Q+ FR(s) = .153(1
425
1
+ .1429s)
State structurally constrained feedback. The matrix M is M = [0 1 0] For tim = 'Yjm = .8, Q = .001I, R = 10000 and Q' = 0 the gain matrix is ]{ = [.0138 0 .4344] and the eigenvalues of the closed-loop nominal model are {-2.5407, -1.08074 ± j1.l833}. Note that for Q = .001 and R = 100000 the above algorithm converges for tim = 'Yjm = .95. The PI controller parameters are
FR(S)
Geromel,J.C. and Peres, P.L.D.: Decentralized load-frequency control, (1985), Proc.IEE, PtD 132, 225-230. Gyurkovics,E and Takacs, T. : Exponential stabilization of discrete time uncertain systems under control constraints, (1995), 3rd ECC'95, 3636-3641 Halicka, M. and Rosinova, D.: Stability robustness bound estimates of discrete systems: Analysis and comparison, (1994), IntJ Control, 60, N2,297-314. Konstantopoulos Ioannis, K. and Antsaklis Panos, J. : New bounds for robust stability on continuous and discrete time systems under parametric uncertainty, (1995), ]{ ybernetika, 31, N6, 623-636. Kozakova, A.:Robust decentralized control of MIMO systems in the frequency domain, (1997) 2nd IFAC, NTDCS'97, Slovakia, 415420 . Kucera, V. and De Souza, C.E. : A Necessary and Sufficient Condition for Output Feedback Stabilizability, (1995),A utomatica, Vol 31, N9, 1357-1359. Ljashevskij, S.E. : Stabilization of continuous time systems with parametric uncertainties, (1991), Elektromechanika, N12, 38-44 (In russian). Luo, J.S., Johnson, A. and Van den Bosch, P.P.J. : On Lyapunov stability robustness bounds for linear continuous time systems with structured uncertainty, (1993), ECC'93, 374-379 M'Saad,M., Latifi, M.A. , Corriou, J.P. and Miklovicova, E.: Partial State Reference Model Adaptive Control of Batch and Fedbatch Chemical Reactors, (1999),Computers and Chemical Engineering, Vol 23, 301-304. Pakshin, P.V. : Robust stability and stabilization of the family of jumping stochastic systems, (1997) ,Nonlinear A nalysis, Theory and Applications, Vol 30, N5, 2855-2866. Peterson, I.R. and Hollot, C.V. : A Riccati equation approach to the stabilization of uncertain linear systems, (1986),Automatica, Vol 22, N4, 397-411. Rosinova, D., Vesely, V. and Kucera, V.: A necessary and sufficient condition for output feedback stabilizability of discrete time linear systems, IFA C Conf. CSD 2000, Slovakia, June 18-20, 2000, accepted. Sharev-Schapiro, N. Palmor, Z.J. and Steinberg, A. : Output stabilizing robust control for discrete uncertain systems, (1998) ,A utomatica, Vol 34, N6, 731-739. Yong-Yan Gao and You-Xian Sun: Static output feedback simultaneous stabilization: LMI approach, (1998), IntJControl, Vol 70, N5, 803-814.
1 = .0138(1 + .031768 )
The second example is borrowed from (Benton and Smith, 1999). It concerns the stabilization of the lateral axis dynamics for an L-1011 aircraft. -2.98 -.99
A= [
o .39
qdt) 0 -.034] -.21 .035 -.0011 0 0 1 -5.555 0 -1.89
B T = [-.032
0 0
- 1.6]
C=[~~~n :s
:s
with -.57 ql(t) 2.43 for all time. Output feedback controller gain matrix for t = 1, R = 1, Q = 5 * I is given as follows
]{ = [17.8852 19.7118] The eigenvalues of closed loop nominal model are {-28.6454, -2.52, -.6594, -1.0052}.
5. CONCLUSION The main aim of this paper has been to propose the new methods for solving the problem of a design of robust controllers via static output feedback and structurally constrained state feedback for linear continuous and discrete time systems with parametric uncertainty. A novel approach for the robust stability analysis of continuous and discrete time systems has been presented. The main point of this approach to the stability analyse of uncertain systems is not to maximize the stability region but the above criterions provide an interesting view on the "physical properties" of robust controllers and give a basis for a new way of designing robust controllers.
6. REFERENCES Benton, R.E.,JR. and Smith, D.:A non-iterative LMI-based algorithm for robust static-output feedback stabilization, (1999), IntJ Control, Vol 72,4, 1322-1330.
426
ROBUST CONTROL FOR LINEAR PARAMETRIC UNCERTAIN SYSTEMS WITH OUTPUT FEEDBACK Vojtech Vesely
1
Dept. of Automatic Control Systems, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Bratislava, Slovak Republic. E-mail [email protected]
Abstract: The paper addresses the design of a robust controller with output feedback for uncertain linear systems in the time domain. The necessary and sufficient conditions for static output feedback stabilizability of linear continuous and discrete time systems are the basis for the proposed robust controller design procedure. The proposed approach does not employ matching conditions. Copyright @2000 IFAC Keywords: Robust control, Linear systems, LQR problem
1. INTRODUCTION
Dynamic systems with bounded uncertainties have been widely used to model physical systems. During the last two decades numerous papers dealing with the design of robust control schemes to stabilize such systems have been published (Ljashevskij, 1991; Peterson and Hollot,1986; Sharev-Shapiro, Palmor and Steinberg, 1998, Kozakova, 1997; Gyurkovics and Takacs,1995; Pakshin, 1997; Yang-Yan Gao and You-Xian Sun, 1998) and references therein. In (Ioannis and Panos, 1995) new conditions for robust stability in linear continuous and discrete time systems are derived when all matrices of the state-space model are perturbed by uncertain parameters and static output feedback is applied. In (Halicka and Rosinova, 1994) robustness bound estimates, based on the direct Lyapunov method for discrete time linear systems are analysed and compared. In (Ljashevskij, 1991; Peterson and Hollot, 1986) the authors study the problem of designing robust state controllers using the Riccati function approach for different types of model uncertainties. 1 The work has been supported by the Slovak Scientific Grant Agence.
421
In (Sharev-Shapiro, Palmor and Steinberg, 1998) both Lyapunov and Riccati based min-max output control law is treated and a simple criterion for the existence of a min-max output controller is provided when all uncertainties are lumped in terms of input matrix which is multiplied by the cone bounded uncertainties as a function of state variable. The necessary and sufficient conditions to stabilize the continuous time system via static output feedback can be found in (Kucera and deSouza, 1995). Robust controller design via sensitivity functions shaping could be found in (M'Saad et aI., 1999). In this paper two approaches have been developed. The necessary and sufficient conditions to stabilize continuous and discrete time systems via static output feedback are used to design the robust controller for different type of uncertainty. In the second approach we pursue the idea of the structurally constrained state feedback, proposed in (Geromel and Peres, 1985) for the design of robust controller for different type of uncertainty. The paper is organized as follows. In Section 2 the problem formulation and some preliminary results are given. The main results of the robust controller design for linear continuous and discrete
time systems are given in Section 3. In Section 4 the obtained theoretical results are applied to the design of robust PI controllers for a DC motor and stabilization of the lateral axis dynamics for an L-1011 aircraft.
2.
where U, Al and A 2 are known matrices and W is an unknown matrix satisfying WTW :S I, 1identity matrix of corresponding dimension. Now let us investigate the robustness bounds for linear continuous and discrete time systems. Lemma 1. Let the closed-loop system with the norm bounded uncertainty (3) is described by
PROBLEM FORMULATION
(8)
In the context of robustness analysis and synthesis of robust controller for linear systems the following uncertain model is commonly used :i: = (A
+ JA)x + (B + JB)u,
where A c = A + BKG and JA c = JA + JBKG. Suppose that A c is Hurwitz with p = maxj (Re(Aj (A c ))) :S 0, where Ai (A c ), i = 1,2, ... , n are eigenvalues of the matrix A c . If for the
y = Gx (1)
norm bounded uncertainty the following inequality holds
for continuous time systems and
x(t
+ 1) = (A + JA)x(t) + (B + JB)u(t)
(9)
(2)
and ii~P ~ JA T PJA, then the closed-loop system (8) is asymptotically stable. 0 Lemma 2. Let the closed loop linear discrete time system with uncertainties is described as follows
y(t) = Gx(t) where x, x(t) E Rn, u, u(t) E R m , and y, y(t) E RI are the state, control and output vector, respectively;A, Band G are known matrices of appropriate dimensions; JA, JB are unknown but bounded matrices. The following types of uncertainty descriptions are often used in the robustness investigations.
x(t
iia:S (3)
where I . I represents the modules of corresponding matrix, Am, Bm are matrices with nonnegative entries and corresponding dimensions, respectively. (3) Matrix polytope type uncertain structure: p
:S
L
fjA j ,
j=1
JB<",B - L... J J
-Al(A~ A c) + j>'M(AI A c) + a
(5)
j=1
u = KCx, where A j , B j are known matrices; fj, Ij are unknown parameters. (4) Matrix type bounded uncertain structure:
JB = UWA 2 ,
(11)
u(t) = KGx(t)
(12)
or static structurally constrained state feedback
u = (K
+ L)x,
u(t) = (K
+ L)x(t)
(13)
so that the closed-loop systems (8) and (10) is robustly asymptotically stable and simultaneously guaranteeing the suboptimal control with respect to the following performance index
where la, Ib are nonnegative constants; Qa, Qb are nonnegative definite matrices. (5) The uncertainties satisfying "matching conditions" : JA = UWA 1 ,
(10)
where iia = qqa, ij~P ~ JA T PJA and iiaAM(AI A)P : AI PA c, a is the exponential stability decay rate of the matrix A c or a = 1- p2, P is spectral radius of matrix A c . 0 Equations (9) and (11) with necessary and sufficient conditions for stabilization of the linear systems via static output feedback imply the new design procedure for the design of robust controllers for linear continuous and discrete time systems proposed in this paper. The problem studied in this paper can be formulated as follows. For continuous and discrete time linear systems described by (1) and (2) with uncertainties (3)-(7) design the static output feedback robust controller with the control algorithm
where 11.11 represents any matrix norm, and qa and qb are nonnegative constants. (2) Element bounded uncertainties, structured model:
JA
= (A c + JAc)x(t)
and control law u(t) = KCx(t) with A c being Hurwitz and for the uncertainties JA c the inequality (3) holds. The perturbed closed-loop system is asymptotically stable if
(1) Norm bounded uncertainties, unstructured model:
IIJAII :S qa, IIJBII:S qb
+ 1)
00
Jc = !(xTQx+uTRu)dt,
(7)
o
422
(14)
co
h
=L
x(tf Qx(t)
+ u(tf Ru(t)
(2) Solve the equation (15) AT Pi
t=O
where Q = QT ~ 0, and R = R T matrices of compatible dimensions.
3. DESIGN OF ROBUST CONTROLLERS
Let us consider the system described by (1). In the following we assume that the nominal system given by u=Kx
(16)
is stabilizable via state feedback. In this section, we will present the new procedure to design robust controllers via static output feedback. Lemma 3. The nominal system (16) is static output feedback stabilizable if and only if there exists a symmetric and positive definite matrix P and a matrix K satisfying the following matrix equality (A
+ BKCf P +
P(A
for Pi symmetric and nonnegative definite. G i = R- ~ B T Pi (I - C T (CC T )-IC)(23) increase i by 1 and go to Step 2.
If the sequence Po, PI, ... , converges, say to P, K is given by (20). The convergence remains to be proved, however. To guarantee the exponential stability of the uncertain closed-loop system the following theorem holds. Theorem 1. The uncertain closed-loop system (1) is robustly exponentially stable with a decay rate a if there exist matrices P = pT > 0, Q = QT ~ 0, R = R T > 0 and a matrix K of appropriate dimensions satisfying the following matrix inequality
3.1 Continuous Time Systems. Output Feedback.
y=Cx,
a
<0
where matrix QI estimates the uncertainties as follows
where Qo = Q~ > o. 0 Due to Lemma 1 and Lemma 3 the gain matrix I{ could be chosen by minimization of the following performance index
P(A
a
G T G - P BR- I B T P
+ BKC) + Qo = 0(17)
= minTrace{(A + BKCf P+ K
T
(A+2"I) P+P(A+2"I)+Q+QI+ (24)
QI
La
PiBR-I B T Pi + (22)
> 0 are the (3)
x=Ax+Bu,
+ PiA + Q -
= II(SA + SBKCf P + P(SA + SBKC)III(25)
and matrix G is given by (21). 0 Calculation of the gain matrix K can be realized by the Algorithm A.
(18) 3.2 Continuous Time Systems. Structurally Constrained State Feedback.
+ BKC) + Qo}
In this section the idea of the structurally constrained state optimal control problem (Geromel and Peres, 1985) has been used for design of robust controllers. Assume the uncertain system The obtained solution of (18) is described by (1). The design problem is to find the static structurally constrained state feedback K = -R- I B T PCT(CCT)-I (20) (13) guaranteeing the robustness properties of the closed-loop system. The necessary and sufficient SHbstituting (19) to (17) after some manipulation conditions for the design of robust controllers are gives given in the following theorem. Theorem 2. Let the uncertain system (1) with AT P + PA + Q - P BR- I B T P + G T G = 0(21) a static structurally constrained state feedback (13) be given. Then, the following statements are where equivalent . where
• The system (1) is static structurally constrained state feedback stabilizable. • There exists a positive definite matrix P and a matrices K and L satisfying the following matrix inequality
Eq. (21) inspires the similar algorithm to those given in (Geromel and Peres, 1985) for calculation of the gain matrix K which guarantee the suboptimal solution of performance index (14). Algorithm A (1) Set i
[A
= 1 and Go = O.
+ SA + (B + SB)(K + L)jT P + P[A+(26) SA
423
+ (B + JB)(K + L)] < 0
• There exist matrices P = pT > 0 and R = RT > 0 and matrices f{ and L that the following inequality holds
3.3
A T p+PA-PB(I+R- 1)B T p+(27) C T R- 1C+ (L+BTpf(L+BTp)f{T Rf{ - LT L P(JA
+ (JA + JB(I{ + L)f P+
+ JB(I{ + L)) < 0
= 1,L o = O,f{o = O,Po = 0,
R- I )-I,QIO=O (2) Calculate -
Qoi = Q
RI
Let the discrete time uncertain system be described by (2). In this section we will present a new procedure for the design of robust controllers via static output feedback. The following theorem will be employed in the further development. Theorem 3. The linear discrete time system (2) is static output feedback stabilizable if and only if (1) (A, B) is stabilizable and (A, C) is detectable, and (2) there exist real matrices f{ and C such that
where C = Rf{ + B T P 0 The algorithm to handle the LQR suboptimal problem with uncertain system via structurally constrained state feedback can be stated as follows. Algorithm B.
(1) i
Discrete Time Systems. Output Feedback
where P is the real symmetric nonnegative definite solution of
= (I +
AT PA - P
+ (Li- 1 + B T Pi-d T (L i- 1+
+Q -
AT PB(B T PB
+ R)-1(30
BTpA+CTG = 0
The matrices Q = QT ~ 0, R = R T > 0 can be taken from the performance index (15).
(3) Calculate Pi =
pT > 0 from
+ PiA + Qoi
AT Pi
Proof. Proof of this theorem is given in(Rosinova, Vesely and Kucera, 2000). 0 The gain matrix f{ is
- Pi BR l 1B T Pi = 0
(4) Calculate f{i = - R- 1B T Pi (5) The entries of the matrix Li are
Equations (30), (31) have inspired the following algorithm for the calculation of robust suboptimal static output feedback gain matrix f{ for discrete time systems. Algorithm C.
Li(j, k) = -f{i(j, k).M(j, k)
The matrix M is given below. (6) Calculate Qli (7) Calculate er = IIL i - L i - 1 IL if er ~ error stop, else i = i + 1 and go to step 2.
(1) i = 1, Go = 0, Q10 (2) Qoi = Q + Qli-l (3) Solve Riccati equation
where for the type of uncertainty (5)
Qli
AT PiA - Pi
P
$
1=1
k=l
= I {L IIAT Pi + PiAtlltlm + L
+ Qoi
R)-I B T PiA
Ikm(28)
=0 - AT PiB(BT PiB+
+ CT-I G i - 1 =
0
pT > O.
for Pi = (4) Calculate
=
=
where tim maxltd, Ijm maxbj I. The matrix M E R mxn defines the structural constraints of the state feedback. The entries of M are defined as follows M(j, k) = 0 iffor thejth input the state Xk is feasible, and M (j, k) = 1 if for the jth input the state Xk is unfeasible. Although the convergence of the Algorithm B has not been formally proven, it has converged for all tests performed in connection with this research. Typically, the number of iterations for convergence varies from 20 - 50 depending on the type of uncertainty and values of matrices Q and R.
f{i
= _(B T PiB + R)-1 B T Pi ACT (CC T )-1 Gi
= (B T PiB + R)-~BT PiA+ (B T PiB
+ R) ~ f{iC
(5) Calculate Q1 i· Q 1 estimates the uncertainties. inrease i by one and go to Step 2. If the sequence Po, PI, ... converges, say to P, the gain matrix f{ is given by (31). The convergence remains to be proved, however.
424
(3) Calculate Pi = ARE
3.4 Discrete Time Systems. State Feedback.
The linear discrete time uncertain systems can be modelled by (2). The problem to be solved is to find the static structurally constrained state feedback (13) which guarantees the suboptimal solution of performance index (15) for all admissible uncertainties. The closed-loop system with static structurally constrained state feedback (13) IS given as
x(t
+ 1) = (AL + oA + oBL + (B+
pl
> 0 from the following
ALpiALi - Pi - ALpiB(BT PiB+ R)-I B T PiALi
+ Qoi =
0
(4) The gain matrix f{i
f{i
= _(B T PjB + R)-l B T PjALi
(5) Entries of matrix L i are
(32)
L j U, k) = -I\;(j, k)M U, k)
oB) K)x(t) (6) Calculate Qli (7) Calculate er = IILi - L i -Ill if er else i = i + 1 and go to Step 2.
where AL = A+BL. The necessary and sufficient conditions for the design of a robust discrete time controller with structurally constrained state feedback is given by the following theorem. Theorem 4. Let the uncertain system (2) with the static structurally constrained state feedback (14) be given. Then, the following statements are equivalent.
where
As a real example we have taken the problem of to design a PI robust controller to control a small DC motors' speed. The DC motor model is given by (1) where
+ oA + (B + oB)(K + L)f P[A+(33) oA
A
(3) There exist matrices P = pT > 0, R = R T > oand K and L satisfying the following matrix inequality.
oA
where Qo = Q + Q1'
o For the uncertainties (5) the matrix Q1 is s
I(L>imIIA; PII + I>rjmll(L + Kf (35)
j=1
i=l p
BJ PII)(L timllAill+ i=l
s
+
L
IjmlIBj(L
= [ ~5.0218
.715
.0422 ]
00]
= [ 1.6885 0 0
oB =
[
~.0597
The uncertainties have been modelled by (5) with oA = 2:;=1 tjA i , tj E< -1,1 > oB = L~=lljBj, Ij E< -1,1 > where each matrix Ai and B i has only one nonzero element, the position of which is that of the 8A(i,j) and oB(k,l). We assume that each entry of matrices aA, oB changes independently from others. The ti, Ij can vary in time arbitrarily, provided that each element is within the given boundary. The results of calculation employing the derived algorithms are summarized in the following. Output feedback. The output matrix is
+ K)II + 211AL + Bf{11)
C_[100] - 00 1
j=l An algorithm to handle the LQR suboptimal problem with structured constrained state feedback can be stated as follows Algorithm D.
(1) i = 1, L o = 0, QlO = 0, Ko = 0 (2) Calculate A Li = A + BL i _ I , Qoi Qli-1 - K[ 1 Rf{i-1
-.0721 ]
B
000
+ Qo - AI P B(B T P B+(34)
p
-4.701 1 0] -8.2986 0 0 [1 0 0
and the uncertainties are
R)-I B T PAL = 0
Q1 =
=
+ (B + oB)(K + L))- P < 0
AIpA L - P
is given in (35)
4. DESIGN OF ROBUST CONTROLLERS. EXAMPLES.
(1) The system (2) is static structurally constrained state feedback stabilizable. (2) There exists the positive definite matrix P and a matrices K and L satisfying the following matrix inequality
[A
Qli
< error stop
For
tim
= Ijm = .65 and = 0 the gain matrix is
= [.153
Q'
1.0706) The eigenvalues of the closedloop nominal model are {-4.2138, -.2495 ± j1.898} and PI controller parameters are as follows f{
Q+ FR(s) = .153(1
425
1
+ .1429s)
State structurally constrained feedback. The matrix M is M = [0 1 0] For tim = 'Yjm = .8, Q = .001I, R = 10000 and Q' = 0 the gain matrix is ]{ = [.0138 0 .4344] and the eigenvalues of the closed-loop nominal model are {-2.5407, -1.08074 ± j1.l833}. Note that for Q = .001 and R = 100000 the above algorithm converges for tim = 'Yjm = .95. The PI controller parameters are
FR(S)
Geromel,J.C. and Peres, P.L.D.: Decentralized load-frequency control, (1985), Proc.IEE, PtD 132, 225-230. Gyurkovics,E and Takacs, T. : Exponential stabilization of discrete time uncertain systems under control constraints, (1995), 3rd ECC'95, 3636-3641 Halicka, M. and Rosinova, D.: Stability robustness bound estimates of discrete systems: Analysis and comparison, (1994), IntJ Control, 60, N2,297-314. Konstantopoulos Ioannis, K. and Antsaklis Panos, J. : New bounds for robust stability on continuous and discrete time systems under parametric uncertainty, (1995), ]{ ybernetika, 31, N6, 623-636. Kozakova, A.:Robust decentralized control of MIMO systems in the frequency domain, (1997) 2nd IFAC, NTDCS'97, Slovakia, 415420 . Kucera, V. and De Souza, C.E. : A Necessary and Sufficient Condition for Output Feedback Stabilizability, (1995),A utomatica, Vol 31, N9, 1357-1359. Ljashevskij, S.E. : Stabilization of continuous time systems with parametric uncertainties, (1991), Elektromechanika, N12, 38-44 (In russian). Luo, J.S., Johnson, A. and Van den Bosch, P.P.J. : On Lyapunov stability robustness bounds for linear continuous time systems with structured uncertainty, (1993), ECC'93, 374-379 M'Saad,M., Latifi, M.A. , Corriou, J.P. and Miklovicova, E.: Partial State Reference Model Adaptive Control of Batch and Fedbatch Chemical Reactors, (1999),Computers and Chemical Engineering, Vol 23, 301-304. Pakshin, P.V. : Robust stability and stabilization of the family of jumping stochastic systems, (1997) ,Nonlinear A nalysis, Theory and Applications, Vol 30, N5, 2855-2866. Peterson, I.R. and Hollot, C.V. : A Riccati equation approach to the stabilization of uncertain linear systems, (1986),Automatica, Vol 22, N4, 397-411. Rosinova, D., Vesely, V. and Kucera, V.: A necessary and sufficient condition for output feedback stabilizability of discrete time linear systems, IFA C Conf. CSD 2000, Slovakia, June 18-20, 2000, accepted. Sharev-Schapiro, N. Palmor, Z.J. and Steinberg, A. : Output stabilizing robust control for discrete uncertain systems, (1998) ,A utomatica, Vol 34, N6, 731-739. Yong-Yan Gao and You-Xian Sun: Static output feedback simultaneous stabilization: LMI approach, (1998), IntJControl, Vol 70, N5, 803-814.
1 = .0138(1 + .031768 )
The second example is borrowed from (Benton and Smith, 1999). It concerns the stabilization of the lateral axis dynamics for an L-1011 aircraft. -2.98 -.99
A= [
o .39
qdt) 0 -.034] -.21 .035 -.0011 0 0 1 -5.555 0 -1.89
B T = [-.032
0 0
- 1.6]
C=[~~~n :s
:s
with -.57 ql(t) 2.43 for all time. Output feedback controller gain matrix for t = 1, R = 1, Q = 5 * I is given as follows
]{ = [17.8852 19.7118] The eigenvalues of closed loop nominal model are {-28.6454, -2.52, -.6594, -1.0052}.
5. CONCLUSION The main aim of this paper has been to propose the new methods for solving the problem of a design of robust controllers via static output feedback and structurally constrained state feedback for linear continuous and discrete time systems with parametric uncertainty. A novel approach for the robust stability analysis of continuous and discrete time systems has been presented. The main point of this approach to the stability analyse of uncertain systems is not to maximize the stability region but the above criterions provide an interesting view on the "physical properties" of robust controllers and give a basis for a new way of designing robust controllers.
6. REFERENCES Benton, R.E.,JR. and Smith, D.:A non-iterative LMI-based algorithm for robust static-output feedback stabilization, (1999), IntJ Control, Vol 72,4, 1322-1330.
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