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Linear adaptive robust controllers for a class of uncertain dynamical systems with unknown bounds of uncertainties
LINEAR ADAPTIVE ROBUST CONTROLLERS FOR A CLASS OF ...
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G-2e-17-1
LINEAR ADAPTIVE ROBUST CONTROLLERS FOR A CLASS OF UNCERTAIN DYNAMICAL SYSTEMS WITH UNKNOWN BOUNDS OF UNCERTAINTIES Hansheng Wu and Shinji Shigemaru Department of Information Science Hiroshima Prefect.ural IT niversity Shobara-shi) Hiroshima 727-0023, Japan E-mail: [email protected]
Abstract. The problem of robust stabilization of a class of linear time-varying systems with disturbance and uncertain parameters is considered. ffhe bounds of the disturbance and uncertain parameters are assumed to be unknO\VIl. FOT such uncertain dynamical systems, a class of linear continuous adaptive robust state feedback controllers are proposed. It is shown that the resulting closed-loop systenls are stable in the sense of uniform ultimate boundedness. _t\ numerical example is given to demonstrate the synthesis procedure for the proposed adaptive stabilizing state feedback c.ontrollel'. COP:ljT-ight @1999 IFA(}'f Keywords. Robust stabilizabilit.Yl adaptive control, robust control, uncert.ainty, state feedback, ultimate boundedness.
1,
I~TRODUCTION
Robust stabilizat.ion of systems \vith significant uncertainties has been widely investigated over the last decades. For a class of uncertain systelllS, i.e. the systenl state vector is available, the so-called matching conditions are satisfied, and the upper bound of the uncertainty vector norm is supposed to be known, several types of state feedback controllers have been proposed by mak~ ing use of the techniques based on Lyapunov's direct method (see e.g. Gutman: 1979; Wu and IVlizukami, 1993; Carless and Leitmann, 1981~ and the references therein). In general l the upper bound of the uncertainty vector norm is employed in such state feedback controllers to lead to some desired performance, e.g. asymptotic stabilitYI ultimate boundedness~ and so on t of the resulting closed-loop systerns ,vith uncertainties. In the practical control problems, the bounds of the uncertainties might not be exactly knov'lll. Therefore, for t.he uneertain systems whose uncertainty bounds are partially kno\vn, adaptive
control schenles have been introduced to update these unknovvn bound~. In Corless and LeitInann (1983), for example, three types of systems containing uncertain elements are c.onsidered due to imperfect knowledge about the model and the inPllt~ and the corresponding adaptive robust controllers are proposed. In C~boi a.nd Kim (1993)1 a saturation-type adaptive robust controller is proposed for a class of uncertain system..~, where the uncertainty bounds are partially knoVr"'n. In Brogliato and Trofino Ncto (1995) for a class of nonlinear systems with partially known llneertainties~ a adaptive robust controller is proposed by using a dead~zonc and a hystel'sis function. In this paper the problem of robust stabilization of a class of linear tilue-·varying systems with disturbance and uncertain parameters is considered. Irere the bounds of the disturbance and the uncertain parameters arc assulued to be UIlknO\VIl. For such uncertain ~ystems, a class of linea.r continuous adapt.ive robust state feedback controllers are proposed \vhich can stabilize the resulting closed-loop dynaruical systcnls in the I
l
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LINEAR ADAPTIVE ROBUST CONTROLLERS FOR A CLASS OF ...
sense of uniform ultimate boundedness. In particular, by employing the proposed adaptive robust controller~ it is not required that the bound p(i) of the uncert.ain ~BC) is known. Furthermore! it is not assumed that f.i(t) > -2 for any t E R+ r v~rhich is a necessary condition for robust stability of uncertain systems in the control literatnre the the problem of robust stabilization. A numerical example is given to demonstrate the svnthesis procedure for the proposed adaptive st"abilizing state feedback controller. By making use of the conventional robust state feedback controllers reported in the contl'o11iterature. one cannot stabilize the system given in this eX~IIlple. However] by employing the proposed adaptive robust control scheme, it becorne possible to stabilize this
0;1
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Assumptioll 2.1. The pair {A(-), B(·)} given in system (1) is uniformly completely controllable. Assunlption 2.2. For all (Xi t) E Rn X R there exist continuous lnatrix functions H ( .)) E (-), D(·) of appropriate dimensions such that ~A(V1t)
==
B(t)H(v t) 1
!J..B(t t) ~ H(t)E(v, t) G(t) B(t)D(t) l ;
=
For convenience) the folIolving nota.tions are introduced to represent the bounds of the uncer-
tainties.
system.
p(t)
maxUH(v,t)J1
T'he organization of tbis paper is as follo~vs. In Section 2, the problem to be tackled is stated and sonle standard assurnptions are introduced. In Section 3, a class of adaptive robust c.ontrol schemes are proposed and the corresponding stability analysis is nlade. In Section 4, a nUlnericaI example is given to illustrate t.he use of the results. Finally~ the paper V\7 ill be concluded in Section 5 with a brief discussion of the results.
p(t) :=
~n [~ }.min (E(v, t)+E T (v, t))]
2. PROBLEl\1 FORMITLATION AND .A.SSlJ~1PTIONS
Consider a class of uncertain systems described by the follo,\\7ing differential equations:
d:c( t) dt
::::
[ A(t)
+ ~A(u, t) ] xCt)
+ [ B(t) + ~B( v, t) ] u(t) +C(t}w(t)
(1)
\vhere x(t) E R.n is the state vector, u(t) E R.m is the control vector, wet) E RS is t.he external disturbance vector bounded by II'Ul (t)1l :; Bw(t) where 11 ·11 is the E·uclidean norm of a vector ':." 1 (v, v) E W is the uncertain vector, W C R L is a compact set, A(t), lJ(t)~ G(t) are continuous lIlatrices of appropriate dimensions, and the matrices ~A(·) l LlB(·) represent the system uncertainties and are continuous in all their argurncllts.
v
~~here 11 . 11 is the spectral norm of a matrix a." and ..\min (.) and Amax (.) denote the minimum and maximum eigellvalues of the matrix a." l respectively. ~1oreov€r1 pet), IJ,(t), (Jw (t) are assumed to be continuous and bounded for any t E R+.
1
Rell1.ark 2.1. It is well kno\vn that Assumption 2.1 is standard and denoies the internally stabilizability of the nOIDina] system~ i.e.~ the syst.em in the ahsence of the uncertain ~A(.), ~B(.)) wC')· Assumption 2~2 defines the matching condition about the unccrtainties r and is a rather standard assumption for robust control problem (see, e.g. Gutman, 1979~ Barmish et al.~ 198:3: Wu and :Nlizukami, 199.3; Carless and Leitmann: 1981 ). For dynamical system with matched uncertainties, one can ahvays design some types of stabilizing state feedback controllers. In Carless and Leitmann (1981), for exarrlple, a class of continuous state feedback controllers are proposed such that ultimate uniform boundcdncss for the system can be guaranteed. In Barmish et al. (1983) another class of continuous state feedback controllers '\vith simpler structure are introduced t.o guara.ntee the practical stability (or ultimate uniform boundedncss) of the system. rrhe controllers guaranteeing asyrnptotic and exponential stability of uncertain systems are present.ed in Qu (1992) and u and 1Iizukami (1993) ~ respectively. HOV{eVel· this assertion is not valid for systenlS lvith unmatched un certainties_ FOT 81] ch uncertain systenlS one lllust find some conditions such that some types of stability can be guaranteed (see, e.g. Barmish and Leitmann, 1982; Chen and Lcitmanu l 1987 ).
"r l
Provided that all states aTe available, the state feedback controller can be represented by a function:
u (t) ::::
pC X (t ), t)
(2)
No\v the question is to how to ~ynthesizc a state feedback controller H(t) that ean guarantee the sta.bility of (1) in t.he presence of the uncertain t
~A(-)r ~B(')J u'C)·
l
RClDurk 2.2. The stabilizing state feedback controllers proposed in the control literature for svstern (1) are based on the fact that the bOll~ds
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LINEAR ADAPTIVE ROBUST CONTROLLERS FOR A CLASS OF ...
14th World Congress of IFAC
of the uncertainties arc kno\vn. That is, p(t), p.(t), (}'w(t) are assumed to be the known continuous and bounded functions, and proposed control laws include such bounds p(t), p,(t), 8w (t). However, in a number of practical control problems) such hounds may be unknown. Therefore, some updating la"\\rs to such unknown bounds must be introduced to construct adaptive robust controllers (see, e.g. Carless and LcitUlann 1 1983;
following adaptive laws:
djJ(t) dt
dfJ.(t) dt
Since the bounds pet), J1(t)~ Bw(t) have been assumed to be continuous a.nd bounded 1 it can be supposed that there exist some constants p* , p*, 8* such that for any t E R+,
(3)
Here, it is worth pointing ont that the constants p* , It·, B* are unknown.
It follows from Assumption 2.1 that for any symmetric positive definH,e Inatrix Q E Rnxn, the matrix Ricc.ati equation of the form dF(t)
~
(7a)
-0"2'~Y2e(t)
(7b)
1
-2"/3 k (t)
"B T (t)P(t)x(t) ,,:2 (7e)
""vhere '!i 1 '('i, i ;=: 1,2,3, arc any positive constants, p(t D ), O(t o), jl(t o) are finite, and specially, A(t D) ~ o and iT;!, /3 are selected so that P(t) > -1 for any t E R+.
CONTROLLER
J1(t) 2: 11*
~(Tl1~lP(t)
-U3,'3fi( 1.)
3. LINE,AR ADAPTl\'E ROBllSl'
8u r(t)::; 8*,
flx( t)l!
dt
robust state feedback controllers are proposed for uncertain system (I).
1
(t) P(t)x(t)ll
dO(t)
C·hen, 1992; Choi and Kim, 1993 ). In this paper: another class of linear continuous adaptive
pet) :::; p*
/1 11 B T
Relnark 3.1. It is 1,vorth pointing out that for uncertain systerll (1) it is not assumed that J.L(t) > -1 for any t E R+. It. is "ven knov..'ll that p(t) > -1 for any t E R+ is a necessary condition for robust stability of uncertain system (1) (sec l e.g., Gutman, 1979; Barnlish et al.) 1983 1 and the references relative to robust stabilization of uncertain systenls) ~ Relllark 3 . 2. It is obvious that the solution to (7 c) can be represented as
+ AT (t) pet) + P(t)A(t)
-P(t)B(t)B T (t)P(t) == -Q
(4)
has a solution ,vhich satisfies
for all t, where
0'1
and
(Y2
are positive numbers .
Thus: for system (1) the foHawing adaptive robust state feedback controller is proposed. n( t)
P(X(t)lt) 1
- - k(t)BT(t)P(t)x(t) 2
(6a)
where the control gain function k(t) is given by
k(t) 1
bi
1 + 6i /52 Ct) + r5~ IID(t)1I2g2 (t) 1 + {L(t)
< Alnin(Q)
(6b)
(ne)
It follows frolll (8) that if letting jl(t o ) > 0 and if syst.em (1) can be stabilized r one can selec t al""vays some paralneters 0"3 and 1'3 such that the solutions to (7c) are larger than minus one, i.e. P(t) > -1 for any t E R.+ - F'or instance, for a sufficient small and a sufficient large P(to), frorn (8) one can guarantee that {i.(t) > -1 for any t E R+. However, though the existence of such fl(t o), 13, 0"3 can be guaranteed theoreticanYl ho"\v to select thelll, i.e. a systelnatic selection method, rnay be necessary to further investigation, In the illustrative exarnple given in Section 4 a trial and error method ha.s been employed to select such P,Ci o ), '13, (1'3 t.hat ha.ve guaranteed li,(t) > -1 for any t E R+.
,3
Applying (6) t.o (1) yields a cJosed--loop system of the forul:
and where pet) E Rnxn is the solution of the Riccati equation described by (4) r 01, 62 are t.WO posit.ive constants, and pC), 0(-), jl(.) are respectively the estimates of the unknown parameters p* , ()*: fJ.*, which arc updated respectively by the
dx(t) dt
=
[A -
~kBBT 2
p] xCi)
+ [~A- ~ k~BBTp]X(t)+cW(t)
(9)
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LINEAR ADAPTIVE ROBUST CONTROLLERS FOR A CLASS OF ...
On the other
hand~
14th World Congress of IFAC
letting
+2'l! T (t)r- 1 dw(t) dt
== p(t) -
1fJ(t)
p*
< xT(t)[d~;t) +AT(t)P(t)
;o(t) == 6(t) - 8* qJ ( t) == j1 ( t) - JL >It
+P(t)A(t) - k(l)PBB T p]x(t)
(7) can rewritten as the foJlo,",ring error system
d~~t)
= 1'1
+2p* IIBT(t)P(t)x(t)11 [1x(t)\1
IIB T (t)P(t)x(t)lIl1 x (t)11
-0"11'11f: (t)
-
T
-p*k(t) llB (t)P(t)x(t)11 +28* I1 B
(lOa)
(fl '"'jlP*
d~~t)
_
~ 'Yak(t)
(j21~2():t<
(t)P(t)x(t) 1IIID(t)!1
d,d,(t)
d~~t) = 1'211 BT (t)P(t)x(t) 1IIID(t)11 -U2··y2f.P(t) -
T
2
dw(t)
+21111J)(t)~ + 2r21~(t)-T
+21'3"1.;l(t) d:~t)
(lOb)
(12)
fIBT(t)P(t)x(t)lf2 It can be obtained froln (IQc.) that
(lOc)
-0"3r3tP(t) - tT3i3P.*
~j.l·k(t)
r:l~hen,
the following theorem can be obtained which shows the uniforrn ultirnate boundedness of uncertain e.1osed--loop system (9) and error system (10).
ddJ(t) IIH (t)P(t)x(t)1I 2 +273 1 1J(t)~ ;T
~ jJ,(t )k( t) ~I B T (t )P{t)x( t) ll2 - 2C73q;"2(t) - 2CT3rP(t)P*
= -jj(t)k(t)x T (t)P(t)B(t)B T (t)P(t)x(l)
Theoreul. 3.1. Consider closed-loop system (9) and error system (10) satisfying A ssu'mption 2.1 and Assnmption 2_2. Then, the solutions (x, 1)1, 'Pr rjJ){t; t0 x(io)" 1);(t o ), 'P(to), q'>(t o)) of closed-loop system (9) and error system (la) arc uniformly ultimately bounded in the presence of the uncertainties.
(13)
-2u34'2(t) - 2G'3tf;(t )J-L*
1
Then, substitution of (13) into (12) and using (6b) yields
dV~, '1!) ::;
Proof: First, define for systerns (9) and (10) a. Lya,pnnov function candidate as foJ]o"vs.
- [6lp2(t) + oiIID(t)1I2e2(t)] x I; B T (t)P(t)x(t) I~ 2
~There
P(t) is the solution of Riccati differential equation (4), r- 1 is a positive definite matrix, and r- 1 , w(-) are defined as
wC) == r-1
[~b(-) cli ag {
'PC')
dt
-
+2/111,b(t) d~~t)
}
dV~, w) ~
(14)
0"
+ (lj6rJllx(t)W
2y 2(t) - 0:34>2(t)
-G"1 1i;2(t) - 2rTla/J(t)P*
Il (t)li x
-(T2
IIB (t)P(t)x(t) 2 +20 w (t) ilBT (t)P(t)x(t)l! I!D(t)l~ -J-l(t)k(t)
-x T (t)Qx(t)
-0-1 ~;2(t) -
+P(t).4(t) - k(t)PBB T p]x(t)
T
+ 2/2"lcp(t) d~t)
I\'loreover, substituting (10) into (14) yields
dt
+2p(t) 11 B (t)P(t)x(t) 'j
Ilx(t)lI
-cr3rjJ2(i) - 2cr3cjJ(t)/-l*
() t [dP - -(t)+ AT () tPt
T
j! HT (t)P(t)x(t)lf
+20* 11 ET (t)P(t)x(t)IJ ~~D(t)1l
Let (x(t)r '1t{t) be the solution of el()sed-~loop system (9) and error system (10) for t ~ to. Then by t.aking the derivative of ~f(-) along the trajectories of (9) and (10) it is obtained t.hat dV (x, 'lI) T ( ) ~~--.:...
+2p*
4J(-)) T
r 1 1 , 12 1, 1:3 1
-x T (t)Qx(t) _ tTsq,2(t)
11
~U3q;2 (t.) - 2u 3 6(t)J1*
+ (1/ b~)
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:::; -ryll x (t)I\2 - f7
1lj;2(t)
14th World Congress of IFAC
the paralueters O"i i .; == 1,2,3, and 02 which arc introduced in the control and ada.ptation laV\r.
- \lI T (t)Ew(t)
+ 20"1 hb(t)llp· I
+ 20"2 ftp(t) lle~ r -U3qJ2 (l) + 2U31-P( t) I~j.t* I + (1/ o~)
-0"2 !p2( t)
s
-'r]tlx(t)112-wT(t)~\[I(t)+E
4. l\.N ILLUS'TRXI'lv'"E EXA11PLE Consider an uncertain time-varying linear dynarnicaI systenl described by
(15)
\vhere 7}:=::
A1ll1n (Q) - (1/6;) > 0
diag{-l+~,
A(t)
::=
€:== crllp*12+0'2IBoI
B(t)
==
0
1 JT
(22b)
(16c)
G"f(t) ==-
0
1] T
(22c)
0.5+ 0.5 } (22a) l+t
l+t
(16a)
w(t) :::: 0.05 sin (t)
~ . 4(-)
On the other hand; since P(t) is symmetric positive definite~ it is obtained from the Rayleigh principLe (Franklin, 196B) that for any t E R+,
Am in(P(t))lIx(t)1I 2
<
::;
Amax (P(t»)jJx(t)11
2
=
Then, in terms of (5) it can be known that there exists a. c.onstant O'max such that for any t E R+ ,
(18)
( 19) v...:rhere
From (19)) it is obvious that \l( x, \(1) decreases monotonicalIy along any solution of (9) and (10) until the solution reaches the compact set { (x, w):
Vex, w) $ Vi }
(20)
where
(21) Therefore, it can be concluded that the solution (x, 1/),10, r/;)(t; trJ) .x(t(J)~ ~b(tn), ~(t())) 1J{t o )) of closed-loop system (9) and error systelu (10) are unifofrnly ultirnately bounded ~vith respect to the bound Vj given by (21), • Rell1ark 3.3. It can be knov,rn from (16b) that by decreasing (J'i ~ i::= 1 2,3 and by increasing 62 sufficiently, one can obtain the upper bound on the steady-state x(t) and error iI!(t) as small as desired. That is, t.he system designeT can tune the size of the residual set by adjusting properly 1
1
0.5 C'.os(3t)
1-2.5sin(t)
JT
(22e)
(22f)
It ean be seen that for this system the matching conditions are satisfied with
(17)
Thus, fronI (18) and (15) it is obtained that
0]
: : [0 0.5 cos(3t)
~B(-) == [0
x T (t)P(t)x(t)
(22d)
[0.5 cos(3t)
0.5 cos(3t) ] T
(23a)
1 - 2.5 sin(t)
(23b)
1
(23c)
I,ettlng Q =::: J. Since the p3.lr {A()r BC)} given in (22a) and (22b) is uniformly c.ompletely controllable, from Cl) it can be obtained a solution of P(i.) which is sYlwnetric posit.ive definite. ReIIlark 4.1. It is obvious from (23b) that for any t E R+ ~ J-L(t) = 1 - 2.5 sin(t.) . 'That is~ one cannot guarantee that for any t E R+ ~ fll(t) > -1 For instance, at t :=: t, {let) == -1.5 < -]. Therefore, it is concluded that by making use of conventional robust state feedback controllers, onc cannot stabilize this systelu (see Remark 3.1). Ho\vever, it c.an be seen belo\\'- that by enlploying the adaptive robust control scheme proposed in this paper, thi~ systern can be stabilized. l'vloreover~ for the controller given in (6) select the parameters 01 and 02 as
and for the adaptive lav.."s given in (7) choose the following parameters:
E == diag{ 2.5 2.5 t 1.S} 1
r ==
diag{ 2.0, 2.0, 0.2 } For silnulation~ the initial conditions of the syst.em and the adaptive laws are given by
x(to) == (2.5
2.5 JT
p(to) == 6.0, OCto) == 8.0~
ji.(to) == 10.0
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LINEAR ADAPTIVE ROBUST CONTROLLERS FOR A CLASS OF ...
14th World Congress of IFAC
Finally, a numerical example is given to dernonstrate the synthesis procedure for adaptive stabilizing linear controller proposed in this paper. It is shown from the example and the results of its simulation that the results obtained in this paper are effective and feasjbl~. Therefore, the results may be expected to have S0111e applications to SOlue practical control problcIIlS of uncertain systems 'lllith disturbance and uncertain paraIIleters,
R.EFER.EK(jES Barlnish B.R.. , ]vI,J. Carless and G. Leitmann (1983). A ne\lll"t class of stabilizing controllers for uncerta,jn dynanJical systems. SIA1J,1 J. Control Opti'1n. 21: 246-2.55. Barmish , B.R.. and G. Leitrnann (1982). On ultiITIate boundedness cont.rol of uncertain systems in the absence of matching conditions. IEEE Trans. £4utorn. Cfontr'ol, AC-27, 153l
Q
10
20
30
40
l
Tjme
Fig. 1. Response of state variable x(t) for the closed-loop system. n-------r------.,..-----...,~--__,
12.0
158. Brogliat.o, B. and A. Trofino Neto (1995). f)ractieal sta.bilization of a class of nonlinear systems v.,.-ith partially kno\vn uncertainties . .A.utomatl~ca, 31, 145--150. Chen, Y.H. (1992). A.daptive robust control of uncertain systems ",,.ith measurement noise . .!1-uto1natica, 28, '715-728.
10.0 8.0 6.0
4.0
Chen: Y.H. and G. Lejtmann (1987). Robustness of unc.ertain syst.ems in the absence of matching conditions. Int. J. Control, 45, 1527--
2.0
0.0
153~5.
o
Choi C.H. and H.S. Kiln (1993). Adaptive regulation for a class of uncertain RysterIls "vith partial knowledge of uncertainty bounds. IEEE Trans. Autom. Control, AC-38, 1246~1250. Carless] I\1.J. and G. Leitmann (1981). Continu1
10
20
30
40
Time
Fig. 2. IIistory of the updating parameter fi(l). \OVith the chosen parameter settings, t.he results of a simulation are shown in Figs. 1 and 2. It can be observed from Fig~l that the closed-loop system resulting from the design in this paper is indeed stable in the sense of uniform ultirnate boundedness. It. c.an be seen from Fig.2 that under the chosen parameters 0'"3: /3 y and initial condition jl(t o ), it has been guaranteed that. for any t E R+ pet) > -1. I
,5. CONCLlJSION 'I"he problem of robust. stabilization of tiulevarying systems with disturba.nce and uncertain parameters has been considered. 'rhe bounds of the disturbanc.e and uncertain paranletel's are assUlned to be unknown. For such systems, a cla.ss of linear continuous adaptive robust state fced~ back controllers have been proposed v'lhich can guarant.ee the uniform ultimat.e boundedness of the resulting closed-loop systems.
ous state feedback guaranteeing uniform ultirnate boundedness for uncertain dynamic systelTIS_ IEEE Trans_ Autom. Control1 AC-
26, 1139-1144. :i\l.J ~ and G. Leitmann (l983). Adaptive
C~orlcss,
control of syst.ems c.ontaining uncertain functions and unkno'~ln functions with uncertain bounds. J. Optim~ Theory Applic~1 41, 155--
168. Franklin J.~- (1968). J.~latrix Theory. Prentice Hall, Englewood Cliffs, NJ. Gutma11, S. (1979). lTncertain dynamic.al systems - A Lyapunov min-max approach. IEEE Trans ..A uto1n~ C/onirol, AC-24, 437---443. Qu, Z. (1992). Global stabilization of nonlinear systems \vith a class of unmatched uncertainties. Systems & Control Letters r 18) 301-307, '·Vu, H. and K. Mizukalni (1993). Exponential stability of a class of non linear nyna.mical systems with uncertainties. Systems & Control Letters, 21, 307-313. l
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Copyright
G-2e-17-1
LINEAR ADAPTIVE ROBUST CONTROLLERS FOR A CLASS OF UNCERTAIN DYNAMICAL SYSTEMS WITH UNKNOWN BOUNDS OF UNCERTAINTIES Hansheng Wu and Shinji Shigemaru Department of Information Science Hiroshima Prefect.ural IT niversity Shobara-shi) Hiroshima 727-0023, Japan E-mail: [email protected]
Abstract. The problem of robust stabilization of a class of linear time-varying systems with disturbance and uncertain parameters is considered. ffhe bounds of the disturbance and uncertain parameters are assumed to be unknO\VIl. FOT such uncertain dynamical systems, a class of linear continuous adaptive robust state feedback controllers are proposed. It is shown that the resulting closed-loop systenls are stable in the sense of uniform ultimate boundedness. _t\ numerical example is given to demonstrate the synthesis procedure for the proposed adaptive stabilizing state feedback c.ontrollel'. COP:ljT-ight @1999 IFA(}'f Keywords. Robust stabilizabilit.Yl adaptive control, robust control, uncert.ainty, state feedback, ultimate boundedness.
1,
I~TRODUCTION
Robust stabilizat.ion of systems \vith significant uncertainties has been widely investigated over the last decades. For a class of uncertain systelllS, i.e. the systenl state vector is available, the so-called matching conditions are satisfied, and the upper bound of the uncertainty vector norm is supposed to be known, several types of state feedback controllers have been proposed by mak~ ing use of the techniques based on Lyapunov's direct method (see e.g. Gutman: 1979; Wu and IVlizukami, 1993; Carless and Leitmann, 1981~ and the references therein). In general l the upper bound of the uncertainty vector norm is employed in such state feedback controllers to lead to some desired performance, e.g. asymptotic stabilitYI ultimate boundedness~ and so on t of the resulting closed-loop systerns ,vith uncertainties. In the practical control problems, the bounds of the uncertainties might not be exactly knov'lll. Therefore, for t.he uneertain systems whose uncertainty bounds are partially kno\vn, adaptive
control schenles have been introduced to update these unknovvn bound~. In Corless and LeitInann (1983), for example, three types of systems containing uncertain elements are c.onsidered due to imperfect knowledge about the model and the inPllt~ and the corresponding adaptive robust controllers are proposed. In C~boi a.nd Kim (1993)1 a saturation-type adaptive robust controller is proposed for a class of uncertain system..~, where the uncertainty bounds are partially knoVr"'n. In Brogliato and Trofino Ncto (1995) for a class of nonlinear systems with partially known llneertainties~ a adaptive robust controller is proposed by using a dead~zonc and a hystel'sis function. In this paper the problem of robust stabilization of a class of linear tilue-·varying systems with disturbance and uncertain parameters is considered. Irere the bounds of the disturbance and the uncertain parameters arc assulued to be UIlknO\VIl. For such uncertain ~ystems, a class of linea.r continuous adapt.ive robust state feedback controllers are proposed \vhich can stabilize the resulting closed-loop dynaruical systcnls in the I
l
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sense of uniform ultimate boundedness. In particular, by employing the proposed adaptive robust controller~ it is not required that the bound p(i) of the uncert.ain ~BC) is known. Furthermore! it is not assumed that f.i(t) > -2 for any t E R+ r v~rhich is a necessary condition for robust stability of uncertain systems in the control literatnre the the problem of robust stabilization. A numerical example is given to demonstrate the svnthesis procedure for the proposed adaptive st"abilizing state feedback controller. By making use of the conventional robust state feedback controllers reported in the contl'o11iterature. one cannot stabilize the system given in this eX~IIlple. However] by employing the proposed adaptive robust control scheme, it becorne possible to stabilize this
0;1
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Assumptioll 2.1. The pair {A(-), B(·)} given in system (1) is uniformly completely controllable. Assunlption 2.2. For all (Xi t) E Rn X R there exist continuous lnatrix functions H ( .)) E (-), D(·) of appropriate dimensions such that ~A(V1t)
==
B(t)H(v t) 1
!J..B(t t) ~ H(t)E(v, t) G(t) B(t)D(t) l ;
=
For convenience) the folIolving nota.tions are introduced to represent the bounds of the uncer-
tainties.
system.
p(t)
maxUH(v,t)J1
T'he organization of tbis paper is as follo~vs. In Section 2, the problem to be tackled is stated and sonle standard assurnptions are introduced. In Section 3, a class of adaptive robust c.ontrol schemes are proposed and the corresponding stability analysis is nlade. In Section 4, a nUlnericaI example is given to illustrate t.he use of the results. Finally~ the paper V\7 ill be concluded in Section 5 with a brief discussion of the results.
p(t) :=
~n [~ }.min (E(v, t)+E T (v, t))]
2. PROBLEl\1 FORMITLATION AND .A.SSlJ~1PTIONS
Consider a class of uncertain systems described by the follo,\\7ing differential equations:
d:c( t) dt
::::
[ A(t)
+ ~A(u, t) ] xCt)
+ [ B(t) + ~B( v, t) ] u(t) +C(t}w(t)
(1)
\vhere x(t) E R.n is the state vector, u(t) E R.m is the control vector, wet) E RS is t.he external disturbance vector bounded by II'Ul (t)1l :; Bw(t) where 11 ·11 is the E·uclidean norm of a vector ':." 1 (v, v) E W is the uncertain vector, W C R L is a compact set, A(t), lJ(t)~ G(t) are continuous lIlatrices of appropriate dimensions, and the matrices ~A(·) l LlB(·) represent the system uncertainties and are continuous in all their argurncllts.
v
~~here 11 . 11 is the spectral norm of a matrix a." and ..\min (.) and Amax (.) denote the minimum and maximum eigellvalues of the matrix a." l respectively. ~1oreov€r1 pet), IJ,(t), (Jw (t) are assumed to be continuous and bounded for any t E R+.
1
Rell1.ark 2.1. It is well kno\vn that Assumption 2.1 is standard and denoies the internally stabilizability of the nOIDina] system~ i.e.~ the syst.em in the ahsence of the uncertain ~A(.), ~B(.)) wC')· Assumption 2~2 defines the matching condition about the unccrtainties r and is a rather standard assumption for robust control problem (see, e.g. Gutman, 1979~ Barmish et al.~ 198:3: Wu and :Nlizukami, 199.3; Carless and Leitmann: 1981 ). For dynamical system with matched uncertainties, one can ahvays design some types of stabilizing state feedback controllers. In Carless and Leitmann (1981), for exarrlple, a class of continuous state feedback controllers are proposed such that ultimate uniform boundcdncss for the system can be guaranteed. In Barmish et al. (1983) another class of continuous state feedback controllers '\vith simpler structure are introduced t.o guara.ntee the practical stability (or ultimate uniform boundedncss) of the system. rrhe controllers guaranteeing asyrnptotic and exponential stability of uncertain systems are present.ed in Qu (1992) and u and 1Iizukami (1993) ~ respectively. HOV{eVel· this assertion is not valid for systenlS lvith unmatched un certainties_ FOT 81] ch uncertain systenlS one lllust find some conditions such that some types of stability can be guaranteed (see, e.g. Barmish and Leitmann, 1982; Chen and Lcitmanu l 1987 ).
"r l
Provided that all states aTe available, the state feedback controller can be represented by a function:
u (t) ::::
pC X (t ), t)
(2)
No\v the question is to how to ~ynthesizc a state feedback controller H(t) that ean guarantee the sta.bility of (1) in t.he presence of the uncertain t
~A(-)r ~B(')J u'C)·
l
RClDurk 2.2. The stabilizing state feedback controllers proposed in the control literature for svstern (1) are based on the fact that the bOll~ds
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of the uncertainties arc kno\vn. That is, p(t), p.(t), (}'w(t) are assumed to be the known continuous and bounded functions, and proposed control laws include such bounds p(t), p,(t), 8w (t). However, in a number of practical control problems) such hounds may be unknown. Therefore, some updating la"\\rs to such unknown bounds must be introduced to construct adaptive robust controllers (see, e.g. Carless and LcitUlann 1 1983;
following adaptive laws:
djJ(t) dt
dfJ.(t) dt
Since the bounds pet), J1(t)~ Bw(t) have been assumed to be continuous a.nd bounded 1 it can be supposed that there exist some constants p* , p*, 8* such that for any t E R+,
(3)
Here, it is worth pointing ont that the constants p* , It·, B* are unknown.
It follows from Assumption 2.1 that for any symmetric positive definH,e Inatrix Q E Rnxn, the matrix Ricc.ati equation of the form dF(t)
~
(7a)
-0"2'~Y2e(t)
(7b)
1
-2"/3 k (t)
"B T (t)P(t)x(t) ,,:2 (7e)
""vhere '!i 1 '('i, i ;=: 1,2,3, arc any positive constants, p(t D ), O(t o), jl(t o) are finite, and specially, A(t D) ~ o and iT;!, /3 are selected so that P(t) > -1 for any t E R+.
CONTROLLER
J1(t) 2: 11*
~(Tl1~lP(t)
-U3,'3fi( 1.)
3. LINE,AR ADAPTl\'E ROBllSl'
8u r(t)::; 8*,
flx( t)l!
dt
robust state feedback controllers are proposed for uncertain system (I).
1
(t) P(t)x(t)ll
dO(t)
C·hen, 1992; Choi and Kim, 1993 ). In this paper: another class of linear continuous adaptive
pet) :::; p*
/1 11 B T
Relnark 3.1. It is 1,vorth pointing out that for uncertain systerll (1) it is not assumed that J.L(t) > -1 for any t E R+. It. is "ven knov..'ll that p(t) > -1 for any t E R+ is a necessary condition for robust stability of uncertain system (1) (sec l e.g., Gutman, 1979; Barnlish et al.) 1983 1 and the references relative to robust stabilization of uncertain systenls) ~ Relllark 3 . 2. It is obvious that the solution to (7 c) can be represented as
+ AT (t) pet) + P(t)A(t)
-P(t)B(t)B T (t)P(t) == -Q
(4)
has a solution ,vhich satisfies
for all t, where
0'1
and
(Y2
are positive numbers .
Thus: for system (1) the foHawing adaptive robust state feedback controller is proposed. n( t)
P(X(t)lt) 1
- - k(t)BT(t)P(t)x(t) 2
(6a)
where the control gain function k(t) is given by
k(t) 1
bi
1 + 6i /52 Ct) + r5~ IID(t)1I2g2 (t) 1 + {L(t)
< Alnin(Q)
(6b)
(ne)
It follows frolll (8) that if letting jl(t o ) > 0 and if syst.em (1) can be stabilized r one can selec t al""vays some paralneters 0"3 and 1'3 such that the solutions to (7c) are larger than minus one, i.e. P(t) > -1 for any t E R.+ - F'or instance, for a sufficient small and a sufficient large P(to), frorn (8) one can guarantee that {i.(t) > -1 for any t E R+. However, though the existence of such fl(t o), 13, 0"3 can be guaranteed theoreticanYl ho"\v to select thelll, i.e. a systelnatic selection method, rnay be necessary to further investigation, In the illustrative exarnple given in Section 4 a trial and error method ha.s been employed to select such P,Ci o ), '13, (1'3 t.hat ha.ve guaranteed li,(t) > -1 for any t E R+.
,3
Applying (6) t.o (1) yields a cJosed--loop system of the forul:
and where pet) E Rnxn is the solution of the Riccati equation described by (4) r 01, 62 are t.WO posit.ive constants, and pC), 0(-), jl(.) are respectively the estimates of the unknown parameters p* , ()*: fJ.*, which arc updated respectively by the
dx(t) dt
=
[A -
~kBBT 2
p] xCi)
+ [~A- ~ k~BBTp]X(t)+cW(t)
(9)
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On the other
hand~
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letting
+2'l! T (t)r- 1 dw(t) dt
== p(t) -
1fJ(t)
p*
< xT(t)[d~;t) +AT(t)P(t)
;o(t) == 6(t) - 8* qJ ( t) == j1 ( t) - JL >It
+P(t)A(t) - k(l)PBB T p]x(t)
(7) can rewritten as the foJlo,",ring error system
d~~t)
= 1'1
+2p* IIBT(t)P(t)x(t)11 [1x(t)\1
IIB T (t)P(t)x(t)lIl1 x (t)11
-0"11'11f: (t)
-
T
-p*k(t) llB (t)P(t)x(t)11 +28* I1 B
(lOa)
(fl '"'jlP*
d~~t)
_
~ 'Yak(t)
(j21~2():t<
(t)P(t)x(t) 1IIID(t)!1
d,d,(t)
d~~t) = 1'211 BT (t)P(t)x(t) 1IIID(t)11 -U2··y2f.P(t) -
T
2
dw(t)
+21111J)(t)~ + 2r21~(t)-T
+21'3"1.;l(t) d:~t)
(lOb)
(12)
fIBT(t)P(t)x(t)lf2 It can be obtained froln (IQc.) that
(lOc)
-0"3r3tP(t) - tT3i3P.*
~j.l·k(t)
r:l~hen,
the following theorem can be obtained which shows the uniforrn ultirnate boundedness of uncertain e.1osed--loop system (9) and error system (10).
ddJ(t) IIH (t)P(t)x(t)1I 2 +273 1 1J(t)~ ;T
~ jJ,(t )k( t) ~I B T (t )P{t)x( t) ll2 - 2C73q;"2(t) - 2CT3rP(t)P*
= -jj(t)k(t)x T (t)P(t)B(t)B T (t)P(t)x(l)
Theoreul. 3.1. Consider closed-loop system (9) and error system (10) satisfying A ssu'mption 2.1 and Assnmption 2_2. Then, the solutions (x, 1)1, 'Pr rjJ){t; t0 x(io)" 1);(t o ), 'P(to), q'>(t o)) of closed-loop system (9) and error system (la) arc uniformly ultimately bounded in the presence of the uncertainties.
(13)
-2u34'2(t) - 2G'3tf;(t )J-L*
1
Then, substitution of (13) into (12) and using (6b) yields
dV~, '1!) ::;
Proof: First, define for systerns (9) and (10) a. Lya,pnnov function candidate as foJ]o"vs.
- [6lp2(t) + oiIID(t)1I2e2(t)] x I; B T (t)P(t)x(t) I~ 2
~There
P(t) is the solution of Riccati differential equation (4), r- 1 is a positive definite matrix, and r- 1 , w(-) are defined as
wC) == r-1
[~b(-) cli ag {
'PC')
dt
-
+2/111,b(t) d~~t)
}
dV~, w) ~
(14)
0"
+ (lj6rJllx(t)W
2y 2(t) - 0:34>2(t)
-G"1 1i;2(t) - 2rTla/J(t)P*
Il (t)li x
-(T2
IIB (t)P(t)x(t) 2 +20 w (t) ilBT (t)P(t)x(t)l! I!D(t)l~ -J-l(t)k(t)
-x T (t)Qx(t)
-0-1 ~;2(t) -
+P(t).4(t) - k(t)PBB T p]x(t)
T
+ 2/2"lcp(t) d~t)
I\'loreover, substituting (10) into (14) yields
dt
+2p(t) 11 B (t)P(t)x(t) 'j
Ilx(t)lI
-cr3rjJ2(i) - 2cr3cjJ(t)/-l*
() t [dP - -(t)+ AT () tPt
T
j! HT (t)P(t)x(t)lf
+20* 11 ET (t)P(t)x(t)IJ ~~D(t)1l
Let (x(t)r '1t{t) be the solution of el()sed-~loop system (9) and error system (10) for t ~ to. Then by t.aking the derivative of ~f(-) along the trajectories of (9) and (10) it is obtained t.hat dV (x, 'lI) T ( ) ~~--.:...
+2p*
4J(-)) T
r 1 1 , 12 1, 1:3 1
-x T (t)Qx(t) _ tTsq,2(t)
11
~U3q;2 (t.) - 2u 3 6(t)J1*
+ (1/ b~)
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:::; -ryll x (t)I\2 - f7
1lj;2(t)
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the paralueters O"i i .; == 1,2,3, and 02 which arc introduced in the control and ada.ptation laV\r.
- \lI T (t)Ew(t)
+ 20"1 hb(t)llp· I
+ 20"2 ftp(t) lle~ r -U3qJ2 (l) + 2U31-P( t) I~j.t* I + (1/ o~)
-0"2 !p2( t)
s
-'r]tlx(t)112-wT(t)~\[I(t)+E
4. l\.N ILLUS'TRXI'lv'"E EXA11PLE Consider an uncertain time-varying linear dynarnicaI systenl described by
(15)
\vhere 7}:=::
A1ll1n (Q) - (1/6;) > 0
diag{-l+~,
A(t)
::=
€:== crllp*12+0'2IBoI
B(t)
==
0
1 JT
(22b)
(16c)
G"f(t) ==-
0
1] T
(22c)
0.5+ 0.5 } (22a) l+t
l+t
(16a)
w(t) :::: 0.05 sin (t)
~ . 4(-)
On the other hand; since P(t) is symmetric positive definite~ it is obtained from the Rayleigh principLe (Franklin, 196B) that for any t E R+,
Am in(P(t))lIx(t)1I 2
<
::;
Amax (P(t»)jJx(t)11
2
=
Then, in terms of (5) it can be known that there exists a. c.onstant O'max such that for any t E R+ ,
(18)
( 19) v...:rhere
From (19)) it is obvious that \l( x, \(1) decreases monotonicalIy along any solution of (9) and (10) until the solution reaches the compact set { (x, w):
Vex, w) $ Vi }
(20)
where
(21) Therefore, it can be concluded that the solution (x, 1/),10, r/;)(t; trJ) .x(t(J)~ ~b(tn), ~(t())) 1J{t o )) of closed-loop system (9) and error systelu (10) are unifofrnly ultirnately bounded ~vith respect to the bound Vj given by (21), • Rell1ark 3.3. It can be knov,rn from (16b) that by decreasing (J'i ~ i::= 1 2,3 and by increasing 62 sufficiently, one can obtain the upper bound on the steady-state x(t) and error iI!(t) as small as desired. That is, t.he system designeT can tune the size of the residual set by adjusting properly 1
1
0.5 C'.os(3t)
1-2.5sin(t)
JT
(22e)
(22f)
It ean be seen that for this system the matching conditions are satisfied with
(17)
Thus, fronI (18) and (15) it is obtained that
0]
: : [0 0.5 cos(3t)
~B(-) == [0
x T (t)P(t)x(t)
(22d)
[0.5 cos(3t)
0.5 cos(3t) ] T
(23a)
1 - 2.5 sin(t)
(23b)
1
(23c)
I,ettlng Q =::: J. Since the p3.lr {A()r BC)} given in (22a) and (22b) is uniformly c.ompletely controllable, from Cl) it can be obtained a solution of P(i.) which is sYlwnetric posit.ive definite. ReIIlark 4.1. It is obvious from (23b) that for any t E R+ ~ J-L(t) = 1 - 2.5 sin(t.) . 'That is~ one cannot guarantee that for any t E R+ ~ fll(t) > -1 For instance, at t :=: t, {let) == -1.5 < -]. Therefore, it is concluded that by making use of conventional robust state feedback controllers, onc cannot stabilize this systelu (see Remark 3.1). Ho\vever, it c.an be seen belo\\'- that by enlploying the adaptive robust control scheme proposed in this paper, thi~ systern can be stabilized. l'vloreover~ for the controller given in (6) select the parameters 01 and 02 as
and for the adaptive lav.."s given in (7) choose the following parameters:
E == diag{ 2.5 2.5 t 1.S} 1
r ==
diag{ 2.0, 2.0, 0.2 } For silnulation~ the initial conditions of the syst.em and the adaptive laws are given by
x(to) == (2.5
2.5 JT
p(to) == 6.0, OCto) == 8.0~
ji.(to) == 10.0
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Finally, a numerical example is given to dernonstrate the synthesis procedure for adaptive stabilizing linear controller proposed in this paper. It is shown from the example and the results of its simulation that the results obtained in this paper are effective and feasjbl~. Therefore, the results may be expected to have S0111e applications to SOlue practical control problcIIlS of uncertain systems 'lllith disturbance and uncertain paraIIleters,
R.EFER.EK(jES Barlnish B.R.. , ]vI,J. Carless and G. Leitmann (1983). A ne\lll"t class of stabilizing controllers for uncerta,jn dynanJical systems. SIA1J,1 J. Control Opti'1n. 21: 246-2.55. Barmish , B.R.. and G. Leitrnann (1982). On ultiITIate boundedness cont.rol of uncertain systems in the absence of matching conditions. IEEE Trans. £4utorn. Cfontr'ol, AC-27, 153l
Q
10
20
30
40
l
Tjme
Fig. 1. Response of state variable x(t) for the closed-loop system. n-------r------.,..-----...,~--__,
12.0
158. Brogliat.o, B. and A. Trofino Neto (1995). f)ractieal sta.bilization of a class of nonlinear systems v.,.-ith partially kno\vn uncertainties . .A.utomatl~ca, 31, 145--150. Chen, Y.H. (1992). A.daptive robust control of uncertain systems ",,.ith measurement noise . .!1-uto1natica, 28, '715-728.
10.0 8.0 6.0
4.0
Chen: Y.H. and G. Lejtmann (1987). Robustness of unc.ertain syst.ems in the absence of matching conditions. Int. J. Control, 45, 1527--
2.0
0.0
153~5.
o
Choi C.H. and H.S. Kiln (1993). Adaptive regulation for a class of uncertain RysterIls "vith partial knowledge of uncertainty bounds. IEEE Trans. Autom. Control, AC-38, 1246~1250. Carless] I\1.J. and G. Leitmann (1981). Continu1
10
20
30
40
Time
Fig. 2. IIistory of the updating parameter fi(l). \OVith the chosen parameter settings, t.he results of a simulation are shown in Figs. 1 and 2. It can be observed from Fig~l that the closed-loop system resulting from the design in this paper is indeed stable in the sense of uniform ultirnate boundedness. It. c.an be seen from Fig.2 that under the chosen parameters 0'"3: /3 y and initial condition jl(t o ), it has been guaranteed that. for any t E R+ pet) > -1. I
,5. CONCLlJSION 'I"he problem of robust. stabilization of tiulevarying systems with disturba.nce and uncertain parameters has been considered. 'rhe bounds of the disturbanc.e and uncertain paranletel's are assUlned to be unknown. For such systems, a cla.ss of linear continuous adaptive robust state fced~ back controllers have been proposed v'lhich can guarant.ee the uniform ultimat.e boundedness of the resulting closed-loop systems.
ous state feedback guaranteeing uniform ultirnate boundedness for uncertain dynamic systelTIS_ IEEE Trans_ Autom. Control1 AC-
26, 1139-1144. :i\l.J ~ and G. Leitmann (l983). Adaptive
C~orlcss,
control of syst.ems c.ontaining uncertain functions and unkno'~ln functions with uncertain bounds. J. Optim~ Theory Applic~1 41, 155--
168. Franklin J.~- (1968). J.~latrix Theory. Prentice Hall, Englewood Cliffs, NJ. Gutma11, S. (1979). lTncertain dynamic.al systems - A Lyapunov min-max approach. IEEE Trans ..A uto1n~ C/onirol, AC-24, 437---443. Qu, Z. (1992). Global stabilization of nonlinear systems \vith a class of unmatched uncertainties. Systems & Control Letters r 18) 301-307, '·Vu, H. and K. Mizukalni (1993). Exponential stability of a class of non linear nyna.mical systems with uncertainties. Systems & Control Letters, 21, 307-313. l
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