ROBUST H[Infinity]-ALMOST DISTURBANCE DECOUPLING PROBLEM WITH STABILIT...
14th World Congress of IFAC
E-2c-07-2
Copyright © 1999 IFAC 14th Triennial \\7orld Congress, Beijing, P.R. China
ROBUST Hcc-ALMOST DISTURBANCE DECOUPLING PROBLEM "WITH STABILITY FOR A CLASS OF NONLINEAR TIME-DELAY SYSTEMS 1 Guo-ping Lu * Vu-fan Zheng ** Daniel "W. C. Ho *** * Department of Automatic Control, Nantong Institute of Technology, Jiangsu 226007,
P.R~
China
** Institute of Systems Science} East China Normal University, Shanghai 200062, P.R~ China *** Department of Mathematics, City University of Hong Kong, Hong Kong Email:
[email protected]
Abstract: This paper deals with the robust H(X)-almost disturbance decoupling
problem v.rith stability (RADDPS) for a class of nonlinear time-delay systems with uncertainty. The paper presents a sufficient condition for solvability of RADDPS, and a design of state feedback controller if it is solvable. Illustrative example is given to demonstrate the applicability of the proposed approach. Copyright © 1999 IFAC Keywords: Nonlinear system, Time-delay, Robust Haa-control, Almost disturbance decoupling.
1. INTRODUCTION Disturbance decoupling problems with stability
(DDPS) have been discussed for linear and nonlinear systems intensively in literature (~1"onham 1985, Isidori 1995). It is known that almost disturbance decoupling problem ('~leiland, et al. 1989) is also an interesting and practical topic in control theory for many years. The almost disturbance decoupling problem is to find an feedback controllaw such that the closed-loop system achieves an arbitrary small level of disturbance attenuation in the sense that, for ex a.mple, the L 2 -norm ratio between disturbance and output functions
is less than an arbitrarily small number l' > 0 while maintaining internal stability (Weiland, et al. 1989).
1
This work was supported by )Jatural Science Foundation
of China, Doctoral Education Foundation of China, the Australia Research Council, Hong Kong R,esearch Grant
For linear systems the above mentioned problem has been completely solved. For nonlinear systems relevant results are available only for single-input and single-output systems (Isidori 1996). In a different formulation, (Lin, et al. 1996, Lin 1997) discuss the H C/O- ahnost disturbance decQupling
problems with internal stability of linear systems subject to input saturation and input-additive disturbance. In this paper we assume that: (1) state feedback is applied; and (2) the output variables are state variables. Under the above two assumptions, the almost disturbance decoupling problem with sta-
bility (ADDPS) for a class of nonlinear systems is developed~ We shall further extend ADDPS to the robust He
problem with stability (RADDPS) in this
work~
It is known to us that time delay is commonly encountered in various engineering systems, such as chemical processes, hydraulic and rolling mill systems. As time delay usually results in unsat-
Council #9040285.
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isfactory performances and is frequently a source of instability, many researchers have paid serious attention to the problems caused by time delay. Recently several authors have used different approaches such as quadratic Lyapunov function, robust H cc control, linear matrix inequalities to study the linear systems with time delay and uncertainty (Lee, et al. 1994, 'frinh, et al. 1994). In this paper we study the RADDPS for a class of nonlinear systems with time delay and uncertain ties. The class of nonlinear systems with time delays and parameter uncertainties considered in
Assumption 1.2: Assume the parameter matrices D B, Hb and the nonlinear vector f satisfy 1
the following conditions
(i) there exist constant matrices Do) H bo with appropriate dimensions such that
(ii) the.re exist a constant Ci > 0 and constant = 0, 1,2, ... ,r) with appropriate dimensions such that
matrices E i (i
r
f' f ::; r
+ ~Ai(t)]X(t -
d,.)
(4)
D ==BD o ,
this paper are described as follows: x(t) == 2:[A i
14th World Congress of IFAC
2: CiXd~ E~EiXdi
(5)
i=O
+ [A r + 1
where f == f[x(t),x(t - d1 ),··· ,x(t - d r )].
1=0
The precise description of RADDPS for a class of nonlinear time-delay systems will be given in
next section. Throughout this paper, the following +[B
+ 6.B(t)]u(t) + Dw(t)
(1)
where x(t) E Rn, u(t) E R In and w(t) E RP are respectively the state, the control input and the disturbance input; do = 0 and d i ~ 0 (i = 1,2, ... , r) are the time delays; and Ai, B, D are known constant matrices "\vith appropriate dimensions (i == O~ 1, ... ,r + 1); ..6..B(t), .6.Ai (t)
are appropriately dimensional matrices representing time-varying parameter uncertainties, which are sometimes denoted as ~B and ~Ai for i == 0,1,·· *, r + 1; f[x(t), xCt - d 1 ) , · · · , x(t - d r )] is known vector-valued continuous function. It is also a vector-valued function of time t. For convenience, V,te denote Xdi ;:::: x(t-d i ), (i == 0,1,··· ~ r), xdo as x. The following assumption is important for our analysis in our paper:
Assutnption 1.1: The uncertainty pammeters ~Ai(t); i == 0,1,· ~ · , r + 1, and ~B(t) satisfy the
following condition LlA i (t) = HiFi (t)Ni ,
notations will be used.
X' denotes the transpose of matrix X E R nx1n , IIXII denotes Euclidean norm of X. I denotes an identity matrix with appropriate dimensions. Given two matrices X and Y with the same dimensions, the notation X > y (X ~ Y) means that X - Y is positive definite (semidefinite). For x ERn, Ilxll denotes the Euclidean norm of x. L~ denotes the set of all measurable functions oo x(t) E Rn,O ::; t < 00 such that f o Ilx(t)1I 2 dt < 00. The L 2-norm of xCt) E £2 is defined as IlxllL2 == (Jooo IIx(t) 112dt) 1/2. The rest of this paper is organized as follows: In Section 2, we give a precise formulation for RADDPS, which is considered in this paper. Section 3 presents the main results of the paper, i.e., the solvability conditions for the proposed RADDPS. An illustrative example is presented in Section 4 and the conclusion remarks are drawn in Section 5~
flB(t) =HBNFb (t)Nb
2. PROBLEM STATEMENT OF RADDPS
(2) where Hi, Ni, H b , Nb are known constant matrices with appropriate dimensions. The matrix--valued functions Fi(t), Fb(t) (i = 0,1,2,· ~., r + 1) are time-varying Lebesgue integrable functions, which describe system parameter uncertainties, satisfying that
where i
== 0,1,2,··· ,T + 1.
IT the .6.Ai (t) and l:1.B(t) satisfy Assumption 1.1, then they are called allowable uncertainty parameters.
Consider the following general nonlinear system with time delays and parameter uncertainties: xCt) = f[x(t), x(t - d 1 ), ~ ~1 (t),
..
,xCt - d r »)
... , 8 q (t), w(t)]
(6)
where x(t) E Rn and wet) E RP for t E Rare the state and the disturbance input, respectively. Let ~j be a set of piece-wise-continuous matrixvalued functions of time t, j = 1, 2) ... ,q. By ~j(t) E ~j, we describe the time-varying parameter uncertainties, Now we give two definitions for the system (6):
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Definition 2.1: (Robust L2 -Stability) The system (6) is of robust L 2 -stability if the solution x(t); t ~ of system (6) with initial condition x(t) == 0 (t :S 0) satisfies xCi) E £'2 subject that w(t) E L~ and /).i (t) E (j == 1, 2, ... ,q).
o
and it is always possible to find a sufficiently small Ei
> 0 such that EiNINi < I
(t
~
(i = 1,2,·· . ,r + 1).
Now let
Li.;
Definition 2.2: (Robust L2-"Y-Gain Performance) Given "y > 0 system (6) is of robust L2-7-g ain performance if with initial condition x(t) == 0
14th World Congress of IFAC
T i := Ai(l - €iNINi)-l A~
+ E;l Hi HI
(9)
and establish a matrix Riccati equation as follows. r+l
0)
IlxllL2 :S for any w(t) 1,2,··· ,q).
E
L~
P A o + A~P - P(BReB J
,II w llL2
and 6. j (t)
E
-
2:Ti - coHoH~)P i=l
dj (j
r
+L
ciE~Ei +
to
l
N~No + rI + Q
=0
(10)
i=l
Based on the above definitions, we present the
where Q and
definition of the RADDPS for the system (1) as follows.
positive definite matrix and positive constant,
RADDPS (Robust Ho:;. Almost Disturbance Decoupling Problem with Internal Stability) For a
> 0, to find,
priori given J
if possible, a lin-
are properly chosen constant
EO
respectively~
If there exists a positive definite solution P to the algebraic Riccati equation (10), then we can find a state feedback law as
ear state feedback law, under which the resulting closed loop system of (1) has the following prop-
u(t) == -(1
follows~
+ TJ)B Px(t) 1
11 ~ 0
(11)
erties.
(1) In the absence of the disturbance wet), x = 0 is globally asymptotically stable equilibrium.
The following theorem is the main result of this paper.
(2) It is of robust L 2 -stability and robust L2-,gain performance subject to all allowable param-
Theorem 3.1: A.ssume that system (1) satisfies
eter uncertainties
conditions~
the conditions (2)-(5), then its RADDPS is solvable
Our main result is shown in the following
Lemma~
LernIlla 2.1: (Wang, et al. 1992) Assume that
feedback law (11) gives a solution to the RADDPS for system (1). Proof: Choose a Lyapunov functional candidate
A, H, N are real constant matrices with appropriate dimensions and F(t) is a matrix-valued
v == x' (t)Px(t)
junction with appropriate dimension, then the fol-
+?= J x'(s)(I +c.;E~Ei)X(s)ds, r
lowing matrix-inequalities hold. (a) For any HF(t)N
E
>0
+ N' F'(t)H
if there exists a positive definite solution P for
algebraic Riccati equation (10). Furthermore, the
t
(12)
~=lt-di
1
::;
£.-1 HH
1
+ €N1N
(7)
Thus, the derivative of V along any trajectory
x(t); t E R of the closed-loop system (1) under (11) is
if F'(t)F(t) ::; I.
(b) r
V == 2x' P
[A + HF(t)NHA + HF(t)N]' :S A(I - €N(lV)-l A' + E- 1 HHJ with ~I.
€
2: (Ai + ~Ai)Xd, + 2x' P(Ar +
1
i=O
+~Ar+l)f + 2x' P(B
+ ~B)u
(13)
r
> 0 satisfying EN' N < I and F ' (t)F(t)
+2x' P Dw
+ Xl 2:(1 + CiE:Ei)X i=l
r
- 2: X~i (I + CiE;Ei)Xdi
3. MAIN RESULTS In what follows we assume that there exists an €
(14)
i=l
By (4) and Lemma 2.1 we have
> 0 such that
== 2x P BR;/2 R;1/2 Dow 1 ::; TjX' P BRrc.B' Px + 1J- W ' D~R;l Dow 2x l P Dw
(8)
l
(15)
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14th World Congress of IFAC
r+l
Using (5), (11) and (15), for some 71 > 0, we have
M~ -PLTiP
(23)
i==l
r
11 ::; 2x' P LCAi + ~Ai)Xdi + 2x P(Ar +l l
By Lemma 2.1, for any i
== 1, 2, .. ~ ,r + 1, we have
i=O
+AAr+1)f - 2(1
+ 71) x' P(B + llB)B' Px
r
+x
t
P(A i
r
ECl + ciE~Ei)X -
2: Xd. (1 + CiE~Ei)Xdi
i=l
i=l
CiXdiE~EiXdi - j' f
r+l
+11-1W' DoR;l Dow
(16)
Let
(~
M
+ 17 x ' PBRt;B1px
i=l
0:=
(24)
Furthermore, we have
r
+L
+ ~Ai)(Ai + li.A i )' P :5 PTiP
no) -1
(17)
where
+ (A o + LlAo)'P -(1 + 1J)P[(B + ~B)BI + B(B + ~B)']P
+ pEp(Ai + ~Ai)(Ai + ~Ai)'P ~ 0 i=l
By means of Schur Complement Lemma, (20) holds, see e.g., (Iwasaki~ et al" 1994). Therefore, (16) implies
11 ~ == 0,
-x/Qx
+ E(1 + CiE:Ei) + Q + 7]P BR~B' P
(18)
i=l
(25)
If w then for any allowable parameter uncertainties Fi(t) , Fb(t), (i = 0,1,2,···, r + 1)
iT:::;
M:= P(A o + ~Ao) T
+ 1J- 1 W' DhR;l Dow
(26)
-x'Qx
Therefore, we conclude that the resulting closed loop system is asymptotically stable. L·et x(t) be the trajectories of the closed-loop system of (1) under (12) with initial condition x(t) == 0 (t :s; O)~ Integrating both sides of (25) from 0 to t, we have
then t
11 ::;
-x'Qx + TJ-lwID~R;l Dow
+ xnx'
Amin(Q)
JIIxll ds ~ '1-11ID~R;1 DolI!lIwll2ds t
2
o
(19)
0
(27) where
X :== (x' Now
Vi-Te
x'dl
Obviously, it follows from w(t) E L~ that x(t) E and for any wet) E L~ (27) implies
/' )
£2
show that (28)
(20) For any to, f
> 0,
i.e., for any I
by Lemma 2.1, we have
> 0,
we can choose
*( ) _ IID~R;lDoll 1]
If TJ
+ 11)P[(B + ~B)B' + B(E + tlB)']P == -2(1 + 77)PBB P - (1 + 1])P B(Hbo FbNb + N'b F/b HIbo )B p -(1
I
~ 7]* ( -r),
l' -
Amin (Q)
.
then we have IlxlIL~
This,
,2
therefore~
:5 ,llwllL 2
completes the proof of this theo-
rem. 0
1
::; -PBRe-E' p - f]P BREB' P
(21)
By (18) and (21), we obtain
Relllarks:
(1) If (Aa, B) is stabilizable and there exist constant matrices Ho = BHoo , A Oi ' H Oi and Ei > 0 such that Hi = BHoi , Ai = BAoi (i = 1, 2, ... ,r+ 1) and
r
+£'0 1 N61'l o +
E ctE~Ei + r I + Q i=l
(22)
r+l
R(;-
L
[AOi(I - ciNtNi)-l ~i
+
f;l
HOiH~iJ > 0
i=l
It follows from (10) that (22) is equivalent to
then, let
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ROBUST H[Infinity]-ALMOST DISTURBANCE DECOUPLING PROBLEM WITH STABILIT...
If we choose
r+l
R(€o)
:=
R£ - L[Aoi(I - EiNINi)-l ~i
€o
i=l
+f:;1 HOiH~i] - €oHooHbo'
14th World Congress of IFAC
(29)
= 1,
E
=
El
=
1
'4)
Q
= 0.11,
then the algebraic Riccati equation (11) has a unique positive definite solution as follows by
the matrix-valued function ReEo) is continuous
:rvlATLAB.
function of EO. Therefore, there exists a sufficiently small EO > 0 such that R(t:o) > O. It further implies that the algebraic Riccati equation (10) has a unique positive definite matrix solution P, see e.g., (Knobloch, et at 1993). By Theorem 4.1,
p = (7.4010
0.3810
0.3810) 1.0210
Therefore, the state feedback controller
the RAnDPS for the system (1) is solvable. u(t)
(2) The parameters €, €i introduced in the algebraic Riccati equation (10) is helpful to find a solvable solution. This fact can be shown in the illustrative example in Section 4.
== -(1 + 11) (1.5239xl (t) + 4.0839x 2(t)) (31)
solves the RADDPS for the system (30). That is, if we choose
(3) When r == 0, system (1) describes a class of nonlinear systems without time delay. In this
1]* (-y)
case, Theorem 4 also gives a sufficient condition
= 1:1'2'
then the control law (31) solves the RADDPS.
of RADDPS, and a design of state feedback law for a class of nonlinear systems without time delay. 0
The simulation results are given in Figures 1-3, w here the initial condition is taken by Xl (t) == 1, X2(t) == 1 for t S; 0 and the rj == 0.5. In this case
the
4. NUMERICAL EXAMPLE
r < 1.
Consider the second order nonlinear system with time delay described as follows.
d?y(t) _ (4 + sint) dy(t) dt 2 dt
5. CONCLUSION
+ (3 + sint) dyCt -
This short paper highlights some new ideas of
1)
dt
implementing a robust controller for RADDPS for a special class of nonlinear time-delay systems.
-costy(t - 1) + (1 - Isintl)
Under the strong assumptions in (4) and (5), a state feedback controller is found via solving an algebraic Riccati equation. Further analysis are carried out in future study to relax these strong
x
assumptions on RADDPS to a wider class of = (4
+
+ w(t)
lcostl)u(t)
nonlinear systems.
(30)
Let
6. REFERENCES
d~~t) = xCt)
X2(t),
yet)
= (Xl (t)
X2
Isidori~
A., (1995) Nonlinear control systems, 3rd Edition, Spinger- Verlag, Berlin Isidori, A., (1996) A note on almost disturbance decoupling for nonlinear minimum phase sys-
= Xl (t),
Ct) )' ,
JxI(t) + x~(t) + xHt - 1) + xHt - 1),
9=
tem, Systems & Control Letters, VaL 27, pp.
191-194 Isidori, A., (1996) Global almost disturbance de-
then the system (30) can be transformed into
system (1), where r
= 1,
Fo(t)
= sint,
= (~
1)'
Do = HbO ]\'0=(-1
Ho
1
= 4' 1)~
=
== cost,
F2 (t)
==
coupling with stability for non minimumphase single-input single-output nonlinear
Isintl,
systems, Systems &1 Control Letters, Vcl. 27,
AO=(~3 ~), Al=(~ ~1)'
Fb(t)=lcostl,
A2
F 1 (t)
B = Co
(~),
= Cl =
D =
(~) ,
1, Eo = El = I,
N 1 ==(1 -1), N 2 =Nb =1, H 1 = H 2 = H b = (0 1 )' .
pp. 115-122
Iwasaki, T. and R. E. Skelton, (1994) All controllers for the general Hoc control problem: LMI existence condition and state space formulas, Automatica, Val. 30, pp. 1301-1317 Knobloch, H. W., A. Isidori and D. Flock-
erzi, (1993) Topics in Control Theory, Basel, Boston, Berlin, Birkhiiuser, Verlag
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ROBUST H[Infinity]-ALMOST DISTURBANCE DECOUPLING PROBLEM WITH STABILIT...
Lee, J. H., S. W. Kim and ~7. H. Kwon, (1994) Memoryless H oo controllers for state delayed systems, IEEE Trans. Automat. Contr., VoL 39, pp. 784-788 Lin, Z.~ A. Saberi and A. R. TeeI, (1996) The almost disturbance decoupling problem with internal stability for linear systems subject to input saturation-state feedback case, A utomatica, Val. 32, pp. 619-624 Lin, Z., (1997) Hoo-ahnost disturbance decoupling problem with internal stability for linear systems subject to input saturation, IEEE Trans. Automat. Contr., Vo!. 42, pp . 992-995 Saberi, A. and P. Sannuti, (1988) Global stabilization with almost disturbance decQupling of a class of uncertain nonlinear systems, Int. J . Contr., Va!. 41, pp. 1655-1704 Trinh, H and 1\1. Aldeen, (1994) Stabilization of uncertain dynamic delay systems by memoryless feedback controllers, Int. J. Control, VaL 59 1 pp. 1525-1542 Wang, Y., L. Xie and C. E. de Souza, (1992) Robust control of a class of uncertain nonlinear systems, Systems & Control Letters, Va!. 19, pp. 139-149 \Veiland, S. and J. C. Willems, (1989) Almost disturbance decoupling, IEEE Trans. Automat. Cont.,.~, Vol. 34, pp. 277-286 Wonham, W. M., (1985) Linear Multivariable Control: a Geometric Approach, Springer Verlag
Fig..
2.1~
14th World Congress of IFAC
wet)
Fig. 2.2. y(t)
06
,! ~ !
I
I
I
11."
I
i
I
02f 0'-
i
I
~2!--------'------"~---'----'-~"""",------L.._.-L......------L-_""",-------_ ,,, ,~
16
iI
:20
Fig. 3.1. wet)
Fig. 1.1. wet)
os
o·
o
2
Fig. 1.2. yet)
16
Fig. 3.2. yet)
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