Copyright © IFAC 12th TrielUlial World Congress, Sydney, Australia, 1993
ADAPTIVE CONTROL OF UNCERTAIN COUPLED MECHANICAL SYSTEMS. J. Rodellar Benede* and E.P. Ryan** -Department ofApplied MathemaJics 1//, Technical University ofCatalunya. 08034 BarcelollQ. Spain "School ofMatMmaJical SciellCes, University ofBath, Bath BA2 7AY, UK
Abstract. A problem of feedback stabilization is addressed for a class of uncertain nonlinear mechanical systems ~ with n degrees of freedom and ne < n control inputs. Each system has the structure of two coupled subsystems with ne and n r degrees of freedom, respectively: a prototype being an uncertain flexible structure with n degrees of freedom actively controlled via actuators applying forces to specific degrees of freedom, ne < n in number. Two nonlinear adaptive feedback strategies are described, with (i) continuous and (ii) discontinuous state dependence, which guarantee (i) a form of practical stability and (ii) asymptotic stability of the zero state. Key Words: Adaptive control; mechanical systems; nonlinear feedback; robustness; stabilization.
1. INTRODUCTION
systems 2; with n degrees of freedom and ne < n control inputs. Each system has the structure of two coupled subsystems, 2;e and 2;r) with ne and n r degrees of freedom, respectively, n = ne + n r , and described by equations of motion of the following form.
The problem of actively controlling flexible structures has been extensively studied in the literature (see, for example, Meirovitch 1990, Soong 1990). Whilst in many cases controllers have been designed under the assumption of having a perfectly known structural model, there has also been considerable interest in questions of robustness. In some cases, the controller is designed for a nominal system and a sensitivity analysis is carried out to check the performance of the control for different uncertainties (Meirovitch and Baruh 1983, Baruh and Silverberg 1985, Lopez Almansa and Rodellar 1990). Hoc techniques have been used, amongst others, by Blelloch and Mingori (1986) and Joshi (1988). Lyapunov based design of practically stable controllers have been also used for flexible structures by Kelly, Leitmann and Soldatos (1987), Joshi (1988), Rodellar, Leitmann and Ryan (1992). The present paper essentially falls into the latter category and focusses on the construction of adaptive controllers for a class of uncertain mechanical
2;r :
Mr(qr(t))ijr(t) =
+ 9r(qr(t), qr(t))
h(qe(t), qe(t)),
(qr(tO),qr(tO)) = (q~,v~) 2;e: Me(qe(t))ije(t) +ge(t, qr(t), qr(t), qe(t), qe(t)) = u(t), (qe(t O)' qe(tO)) = (q~, v~). qr(t) E
jRn r ,
qe(t) E
jRn c
are vectors of generalized coordinates and u(t) E jRn c is the vector of control forces; the matrix-valued functions M r and Mc represent inertias, and the (nonlinear) functions 9r) h, ge model damping, stiffness, coupling and Coriolis effects, as well as extraneous inputs and disturbances. Assumptions 1 to 6 below complete the description of the system class 2;.
1Based on work supported by the British CouncillSpanish Ministry of Education & Science under the Acciones Integradas Programme
571
AI. The function M r is continuous with uniformly bounded inverse: that is, for some (unknown) positive scalar rn, IIM;l(qr)11 ~ rn for all qr E IRn r
where M r , Mc are inertia matrices, C n Cc are damping matrices, and K n Kc are stiffness matrices. The control forces constitute the vector u(t), while f(t) represents extraneous forces/disturbances. A1-A6 hold under the following conditions: M r invertible, Mc symmetric positive-defi~ite, Cr and K r positive definite, f E LOO(]R., jR.n e ).
•
A2. The function Mc is continuous and such that, for some (unknown) positive scalars rn, rn and known continuous function j.L, the following hold for all qc E IRn e : (i) IIMc-1(qc)1I ~ rnj.L(qc), and (ii) M;l(qc) ;::: rnI (in the sense that, for all v E re, (v, M;l (qc)v) ;::: rnllvll 2 •
The question to be addressed is twofold:(a) Does there exist an adaptive feedback strategy, parameterized by A > 0, which, for every system (unknown to the controller) of class E, every solution of the feedbackcontrolled initial-value problem (1) is asymptotic to a ball centred at zero in jR.n of radius p(A), where p(A) ~ as A ~ ? In Section 2, we answer this question affirmatively by explicit construction of one such continuous feedback strategy. (b) Does there exist an adaptive feedback strategy which achieves the above objective with A = 0, that is, a strategy which renders the zero state globally attractive? In Section 3, we provide a construction which achieves this more stringent objective: however, in this case, we need recourse to discontinuous feedback.
A3. The function gr is continuous.
A4. h is continuous, with h(O,O) = 0. AS. With h == 0, the subsystem Er zs quadratically asymptotically stable, in the sense that there exists an (unknown) positive definite quadratic form v;. on rr such that, for some (unknown) scalar c > 0, :t Vr(qr(t), cir(t))
°
~ -cVr(qr(t), cir(t))
for a. a. t on every solution (qn cir)(- ). A6. The function gc is of Caratheodory class and such that, for some known continuous function" the following holds for some (unknown) scalar a:
IIgc (t, qn Vn qc, vc) 1I ~ a,( qn Vn qc, vc) for a.a. t E IR and all (qn Vn qc, vc) E
2. CONTINUOUS STRATEGY
jR.2n.
Thus, the only a priori system information available to the controller is the pair of continuous functions, and j.L: in particular, we stress that the uncertainty bounding parameters rn, rn, rn and a are unknown.
Throughout this section, we assume A > 0. We first introduce some notation. Let d>. denote the function defined (on jR.n e , jR.n r , jR.2n e or 1R2n as context dictates) by r
d>. : v
Example. By way of example, consider a flexible structure with n degrees of freedom and actively controlled via actuators supplying forces on specific degrees of freedom (ne < n in total) which are also subject to external disturbances. Under an assumption of linearity and separating the coordinates of the directly controlled and uncontrolled degrees of freedom, we may write
Er: Mriir(t)
Ec
:
Mciic(t)
t-t
Ilvll {
0,
A,
Ilvll ;::: A Ilvll < A
Let s>. denote the function defined on
jR.ne
by
The proposed continuous adaptive strategy, parameterized by A > 0, is given by
+ Crcir(t) + Krqr(t) + Crccic(t) +Krcqc(t) =
°
u(t) = -k(t)U>.(qr(t), cir(t) , qc(t),Pc(t))
°
Pc(t)
+ Cccic(t) + Kcqc(t) + Ccrcir(t) +Kcrqr(t) = u(t) + f(t)
= cic(t) + qc(t)
k(t) = K>.(qr(t), cir(t), qc(t),Pc(t)) k(t o) = kO, 572
where the functions U>. and K>. are given by
Theorem l. Let A > 0 and (to,
X O)
E ~
X ]RN.
maximal solution x(·) = (qT> qT> qc, Pc, k)(·) : [to, w) -+ of the initial-value problem (1),
+ 9r(qr(t), qr(t)) Remarks. By the above theorem, we see that the proposed adaptive continuous feedback strategy ensures a form of practical stability for the system class E. In essence, for any prescribed A > 0, the subsystem state (qc(t),Pc(t)) is asymptotic to that ball (centred at zero in ]R2n c ) of radius A, the remaining subsystem state (qr(t), qr(t)) is asymptotic to a ball (centred at zero in ]R2n of radius CA - however, the scale factor c > 0 depends on the unknown function h and so is not computable from a priori system information. This is somewhat unsatisfactory from an applications viewpoint. In the next section, we construct a strategy which ensures asymptotic stability of the zero state of the full system: the price we pay for such precision in guaranteed performance is that of discontinuous feedback.
= h(qc(t),pc(t) - qc(t))
qc(t) = -qc(t)
+ Pc(t)
Pc(t) = P>.(t, qr(t), qr(t), qc(t),Pc(t), k(t)) k(t) = K>.(qr(t)qr(t), qc(t),Pc(t)) (qr (to), qr(to), qc(to), Pc(to), k(to)) =
]RN
(i) w = 00; (ii) limt--too k(t) exists and is finite; (iii) d>.(qc(t)), d>.(Pc(t)) -+ 0 as t -+ 00; (iv) dch(>.)(qr(t),qr(t)) -+ 0 as t -+ 00, for some positive scalar c.
The overall controlled system representation on ]RN, N = 2(nr + ne) + 1 now becomes
Mr(qr(t))ijr(t)
For every
(qO VD qO r' r' c'
poc) kO ) =..
XO
r
E ]RN .
where the function P>. is given by
P>.(t, qr, Vr , qc,Pc, k) := Pc - qc - M c- 1(qc)[9c(t, qTl VT> qc,Pc - Qc) +kU>.(qT> VT> qc,Pc)] Equivalently, writing
)
3. DISCONTINUOUS STRATEGY we have
x(t) = F>.(t,x(t)),
x(to) = X O
The proposed discontinuous adaptive strategy is given by
(1)
where
u(t) E -k(t)U(qr(t), qr(t), qc(t),Pc(t)) Pc(t) = qc(t) (VT> Mr-1[h(qClPc - Qc) - 9r(qT> Vr)], -Qc
+ Pc,
k(t) = K(qr(t), qr(t),Pc(t))
P>.(t, x), K>.(x)) .
k(t o) = kO,
This system satisfies the Caratheodory conditions and so, for every (to, X O) E ]R X ]RN, the above initial-value problem has a solution and every solution can be extended into a maximal solution. On [0,00), define
h:AN
+ qc(t)
where the set-valued map U and function K are defined as follows
sup{llh(qClPc - qc)" I d>.(qc) = 0 = d>.(pc)},
which, by virtue of Assumption A4, is continuous with h(A) -+ 0 as A -+ o. 573
The set-valued map 'l/J is a generalized "bangbang" component defined as
Theorem 2. Let (to, XO) E ~ X ~N. For every maximal solution x(·) = (qn qr, qc,Pc, k)(-) : [to, w) --+ ~N of the initial-value problem (2),
(i) w =
(ii) limt-+CXl k(t) exists and is finite; (iii) (qr(t), qr(t), qc(t),Pc(t)) --+ 0 as t --+
In words, 'l/J(Pc) is a singleton consisting of the unit vector in the direction of Pc whenever Pc i:- 0 while 'l/J(Pc) is the closed unit ball centred at zero in ~nc whenever Pc = O.
Aubin, J-P. and Cellina, A. (1984), Differential Inclusions, Springer-Verlag. Baruh, H. and Silverberg, L.M. (1985), Robust natural control of distributed systems, Journal of Guidance, Control and Dynamics, 8, 717-724.
+ gr(qr(t), qr(t))
= h(qc(t),Pc(t) - qc(t)) qc(t) = -qc(t)
Blelloch, P.A. and Mingori, D.L. (1986), Modified LTR robust control for flexible structures, Proc. AIAA Guidance, Nav. f3 Control Conference, 314-318.
+ Pc(t)
Pc(t) E P(qr(t), qr(t), qc(t),Pc(t), k(t)) k(t)
= K(qr(t)qr(t), qc(t),Pc(t))
Joshi, S.M. (1988), Control of Large Flexible Space Structures, Lect. Notes in Control & Info. Sciences, 131, Springer Verlag.
(qr (to), qr (to), qc (to), Pc (to), k(to)) =
c' poc' k
(qO VO qO
r' r'
O)
=: X O E ~N
,
Kelly, J.M., Leitmann, G. and Soldatos, A. (1987), Robust control of base-isolated structures under earthquake excitation, J. Opt. Theory fj Appls., 53, 159-181.
where the set-valued map P is given by
1
{Pc - qc - M c- (qc)[w
L6pez Almansa, F. and Rodellar, J. (1990), Feasibility and robustness of predictive control of building structures by active cables, Earthquake Eng. fj Strucural Dynamics, 19, 157-171.
+ ku]1
Ilwll ::; O'.,(qr,vn qc,Pc -
qc),
u E U(qn V n qClPC)}
Meirovitch, L. and Baruh, H. (1983), Robustness of the IMSC method, J. Guidance, Control fj Dynamics, 6, 20-25.
Equivalently,
x(t) E F(x(t)),
x(to) =
XO
(2)
Meirovitch, L. (1990), Dynamics and Control of Structures, John Wiley.
where
F: x
00.
4. REFERENCES
The overall controlled system representation on ~N, N = 2(n r +n c )+1 can now be embedded in the following autonomous differential inclusion formulation
Mr(qr(t))ijAt)
00;
= (qnvnqc,Pc,k)
Rodellar, J., Leitmann, G. and Ryan, E.P. (1992), On output feedback control of uncertain coupled systems, Int. J. Control, to appear.
t--+
{v r } x {M;l[h(qc,Pc - qc) - gr(qr,v r )]} x{ -qc
+ Pc}
x P(x) x {K(x)}
Ryan, E.P. (1990), Discontinuous feedback and universal adaptive stabilization, in Control of Uncertain Systems (D. Hinrichsen & B. Martensson, eds), Birkhiiuser.
It is readily verified that P (and hence F) is upper semicontinuous with convex and compact values. Therefore, by Aubin & Cellina (1984: Theorem 2.1.3), for every (to, XO) E ~ X ~N, the above initial-value problem has a solution and every solution can be extended into a maximal solution (Ryan 1990).
Soong, T. (1990), Active Structural Control, Longman Scientific & Technical. 574