Abstract. We consider dynamical systems containing uncertain elements due to imperfect knowledge about the model and the input. Since these uncertainties ...
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 41, No. I, SEPTEMBER 1983
Adaptive Control of Systems Containing Uncertain Functions and Unknown Functions with Uncertain Bounds 1 M.
CORLESS
2 AND
G.
LEITMANN
3
Dedicated to L. Cesari
Abstract. We consider dynamical systems containing uncertain elements due to imperfect knowledge about the model and the input. Since these uncertainties may result in unstable behavior, we seek controllers which guarantee that all possible responses of the system are uniformly bounded and approach a desired response. Toward that end, we present a class of adaptive controllers.
Key Words. Uncertain systems, deterministic control, adaptive control, guaranteed stability.
1. Introduction In order to control the behavior of a system in the real world, be it physical, biological, or socioeconomic, the system analyst seeks to capture the system's salient features in a mathematical model. This abstraction of the real system always contains uncertain elements; these may be parameters, constant or varying, which are unknown or imperfectly known, or they may be unknown or imperfectly known inputs into the system. Despite such imperfect knowledge about the chosen mathematical model, one often seeks to devise controllers which will steer the system in some desired fashion, for example so that the system response will approach or track a desired reference response; by suitable definition of the system (state) variables, such problems can always be cast into the form of stability problems. 1This paper is based on research supported by the National Science Foundation, Grant No. ECS-78-13931, and carried out, in part, while G. Leitmann was a recipient of a US Senior Scientist Award of the Alexander von Humboldt Foundation. 2 Research Assistant, Department of Mechanical Engineering, University of California, Berkeley, California. 3 Professor, Department of Mechanical Engineering, University of California, Berkeley, California. 155 0022-3239/83/0900-0155503.00/0 © 1983 Plenum Publishing Corporation
156
JOTA: VOL 41, NO. 1, SEPTEMBER 1983
Two main avenues are open to the analyst seeking to control an uncertain dynamical system. He may choose a stochastic approach in which information about the uncertain elements as well as about the system response is statistical in nature; for example, see Refs. 1 and 2. Loosely speaking, when modelling via random variables, one is content with desirable behavior on the average. The other approach to the control of uncertain systems, and the one for which we shall opt in the present discussion, is deterministic. Available, or assumed, information about uncertain elements is deterministic in nature. Here, one seeks controllers which assure the desired response of the dynamical system. In this paper, the mathematical model is embodied in ordinary differential equations, the state equations of the system. We divide the systems under consideration into three subclasses depending on the type of potentially destabilizing uncertainties present in the system description (model uncertainty) and in the way the control enters into the description (input uncertainty). For each of the systems considered, there exists a state feedback control which assures that the zero state is globally uniformly asymptotically stable. However, these controls depend on constants in the system description which are not known; for example, such constants are the values of unknown constant disturbances or unknown bounds on time-varying parameters or inputs. We propose controllers which may be regarded as adaptive versions of the feedback controls mentioned above; in place of the unknown constants, one employs quantities which change or adapt as the state of the system evolves. Under some circumstances, these adaptive quantities may be considered to be estimates of the unknown constants. The method of devising these adaptive controllers is based on the constructive use of Lyapunov theory as suggested, in a somewhat different context, in Refs. 3-8.
2. Systems under Consideration All of the systems under consideration belong to one main class ($4). However, we introduce first three subclasses (S1, $2, $3) each of which is contained in the main class. System Class Sl. In this class, we consider systems which are subject to unknown constant disturbances. The systems are described by ~(t) =f(t, x(t))+B~l~(t, x(t))[t#(u~l~(t)+d~)+db],
(1)
where t ~ R , x(t)~R n is the state, u ~ ( t ) ~ R ml is the control, d~, db ~R '~1
JOTA: VOL. 41, NO. 1, SEPTEMBER 1983
157
are constant disturbances, f : R × R" ~ R ~, B (1)'.RX ~ . . ~. .R , and 0: R '~ ~ R 'm. Concerning the function f, we introduce the following assumption.
AssumptionA1. (i) f is Caratheodory (see Appendix) andf(t, 0) = 0 for all t ~ R. (ii) There exist a C l - f u n c t i o n V: ~ x R n ~ R + and continuous nondecreasing functions 3'1, 3'2, Y3: ~+ ~ R+ which satisfy T~(0) = 0,
r >0~T~(r)>0,
!irn T~(r) = oo,
i= 1,2,3, i = 1, 2,
(2) (3)
such that, for all (t, x) ~ R x ~ ,
l(llxll)
,=(llx II),
v ( t , x)
a W(t, x)/at + [OV(t, x)/ax ]f(t, x) ~
o , lim w ( k ) = oo. We also make the following additional assumptions.
(8) (9)
158
JOTA: VOL. 41, NO. 1, SEPTEMBER 1983
AssumptionA3.
(i)
The function B (~) is strongly Caratheodory (see
Appendix).
Assumption A4. Condition C1. satisfies
One of the following two conditions is satisfied. There exists a continuous function 3'4: R+-* ~+ which lim 3"4(r) = co,
(10)
r-~Oo
such that, for all (v~, w ) ~ R "~ ×R m', [~/(k~ ) - ~ (w)]T (l~ -- w) >i 3"4([]1,~-- w 11)[11~-- kiz,[[. Condition C2. all (t, x)~l~ xlR n,
(11)
For each d~>O, there exists b~(d)>~O such that, for
IIx II
Ila {')(t, x)ll bl(d),
d
(12)
where a(1)(t, x) = B(1)~ (t, x)[OVT(t, x)/Ox].
(13)
Assumption A5. (i) At each t ~ R, a(1)(4 x (t)) is known. Remark 2.1.
(a)
If f is linear time-invariant, i.e.,
f(t,x)=Ax,
V(t, x) ~ R xl~",
(14)
where A ~ , , × n and A is strictly stable (i.e., all of its eigenvalues have negative real parts), then Assumption Al(i) is satisfied and Assumption Al(ii) may be satisfied by letting V(x)
1 TPx, = ~x
(15)
where, for any given positive-definite symmetric O ~R n×n, the matrix P ~ R n×" is the unique positive-definite symmetric solution of P A + a Tp = _ O ;
(16)
see Refs. 5, 9-11. If, in addition, B (1) is constant, i.e., B(1)(t,x)=B,
V ( t , x ) c ~ xR",
(17)
where B ~ R"×"', then a (1)(t, x ) = B Tpx, and Condition C2, and hence Assumption A4, is satisfied.
(18)
JOTA:
VOL.
41, NO.
1, S E P T E M B E R
1983
159
(b) As a particular example of a function which satisfies Assumption A2 and Condition C1, consider any function 0 given by O(w) =Fw,
(19)
where F ~ R "~I×"~t is symmetric positive definite. The existence of F -1 implies that Assumption A2(i) is satisfied. Assumptions A2(ii) and A2(iii) are shown to hold by letting xt/(w)
= gw 1 T Fw,
and Condition C1 is assured with y4(r)
=
Ami,(F)r,
(20)
where ,~min(F) denotes the smallest eigenvalue of F, and ami.(F)> 0. (c) As a more general example of a function which satisfies Assumption A2, consider any function 0 given by 4t(w) = L rg~(Lw),
(21)
where L ~ ~"1×"1 is nonsingular and ~b: R ml ~ R "1 has the form ~(o-) = (4~1(cr0, ~b2(o'2). . . . . 4~,~l(crml))T,
(22)
.,
(23)
(Ovl, O'Z, • •
Ovml) ~---Or T ,
each 4~i: ~ ~ ~ being continuous, strictly increasing, and satisfying lira ~i(o'i) = ~ ,
o-i -+¢c3
lim ~bi(o-i)= - ~ .
o-~-..~-cx3
(24)
For a proof that g, satisfies Assumption A2, see Ref. 12. (d) Assumption A5(i) is made in order to ensure that there is sufficient information available to implement the proposed controllers for this system class. Note that this assumption is completely independent of ~ and does not require complete knowledge of f and B a). For example, some of the controlled systems presented in Refs. 13-15 contain an uncertain f which satisfies Assumption A1 for a known V. There, B ~ is known, so that the function o~~a~is known. For another example, see Ref. 12. (e) In the previous literature, known linear time-invariant systems subject to unknown constant disturbances [i.e., systems with f linear timeinvariant, B m constant, and if(w)= w] have been treated using different approaches in Refs. 16-21, among others. In these references, the requirement that A be strictly stable is usually relaxed to the requirement that (A, B ~1)) be stabilizable. (f) Lur'e-type systems [i.e., systems with f linear time-invariant, B a~ constant, and 4t in general nonlinear] have been considered in Refs. 22 and 23, subject to fewer restrictions on A but more restrictions on 0.
160
JOTA: VOL. 41, NO. 1, SEPTEMBER 1983
System Class $2. In this class, we consider systems whose descriptions contain unknown constant parameters, namely, ~(t) =f(t, x(t))+B~E)(t, x(t))[Fu(2)(t)+Dh(t, x (t))],
(25)
where t, x (t), f are as defined for $1, u(2)(t) ~ ~"2 is the control, F c R m2×m2 and D ~ R r~×p contain unknown parameters, Bt2): R × R" ~ R"×~2, and h: R × R ~ ~ R p. In addition to assuming that f satisfies Assumption A1, we also make the following assumptions for this class.
Assumption A6. The matrix F is symmetric positive definite. Note that Assumptions A1 and A6 imply that, for each F and D, there exists a state feedback control, given by u ~2~(t)= K h (t, x (t)),
K = -F-1D,
(26)
such that the zero state is a g.u.a.s, equilibrium point of (25). However, the matrices F and D are not assumed to be known.
Assumption A3. (ii) The functions B ¢2~ and h are strongly Caratheodory.
Assumption AS. (ii) At each t ~ R , o~2~(t,x(t)) and h(t,x(t)) are known, where, for all (4 x) ~ R × R ", a ~2)(t,x) = B (2)r (t, x)[0 V r (t, x)/Ox ].
(27)
The following condition, which is not an assumption, will affect the choice of one of the parameters in the proposed controllers for this system class.
Condition C3. all ( t , x ) ~ R x R ",
For each d/> 0, there exists b2(d)>i 0 such that, for
Ilxll ~ d ~ llh (t, x )llllo~2~(t, x )ll ~ bz(d).
(28)
Remark 2.2. (a) Quite frequently, the error equation which arises in the problem of requiring a linear time-invariant system to track a reference model falls into this class of systems (see Refs. 6-8, 24-32). (b) For a particular example of a system in this class, see Ref. 12.
System Class $3. In this class, we consider systems which contain possibly destabilizing uncertainties of a more general nature than those considered in $1 and $2. The systems are described by 2 (t) = [(t, x (t)) + B ~3~(t, x (t))g (t, x (t), u (3)(t)),
(29)
JOTA: VOL. 41, NO. 1, SEPTEMBER 1983
161
where t, x(t), f are as defined for S1, u(3)(t)EN m3 is the control, B(3~: R x R ~ -->R~×r~, and g: R xt~ ~ x I~"~-->I~m~. In addition to assuming that f satisfies Assumption A1, the following assumptions are made for this class.
Assumption AT. (i) There exist a function p : R × R " ~ R + and a constant/30 > 0 such that, for all (t, x, u) ~ R × R n × R m~,
u~g(t, x, u) ~>/3011ullEIlull-p(t, x)].
(30)
(ii) There exist a constant/3 e Rk+ and a known function P: ~ x R" x R k+-->~+ such that, for all ( t , x ) e R xR",
p(t, x) = P(t, x, fl).
(31)
That is, we do not assume that the bound p(t,x) is known; we assume that it depends in a known manner on an unknown constant vector/3. (iii) For each (4 x ) e R x I~", the function P(t, x,. ): R k+~ R+ is C1 and nondecreasing with respect to each component of its argument fl, and -P(t, x, ") is convex.
Assumption A3. (iii) The function B (3) is Caratheodory and g, P and OP/Ofl are strongly Caratheodory. Assumption A5. for all ( t , x ) e R x R " ,
(iii)
At each t c ~ , a(3~(t,x(t)) is known, where,
a~3)(t, x) = BI3)T (t, x )[OVT (t, x )/Ox ].
(32)
Remark 2.3. (a) In the earlier literature (see Refs. 15, 33-40), systems of this class have been considered, where g is of the form
g(t,x, u) =[I +E(t,x)]u +e(t,x),
(33)
and E(t, x), e(t, x) satisfy
IIE(t,x)He~s(t,x,[3, e)=a(a)(t,x)/l[ot(3)(t,x)[],
(53)
tz(t, x, ~) =P(t, x,/3)a (3)(t, x),
(54)
where for all (t, x,/3, ~) E R × ~" x (0, oo) TM.A particular example of such a function s is given by
s(t, x, ¢~, e) = sat[tz (t, x,/~)/e],
(55)
164
JOTA: VOL. 41, NO. 1, SEPTEMBER 1983
where sat(n)
! n, tn/ll,711,
I1~[1~ 1, 11,711>1.
(56)
These controllers may be considered as adaptive versions of those presented in Refs. 34-38 and modified in Ref. 39 for systems of this class where g is of the form considered in Remark 2.3(a). When p is known,
OP/O~ = 0; and these controllers, with e = 0, reduce to those presented in Refs. 34-38 and, with • = const > 0, reduce to those presented in ReI. 39.
Controller Class C4. System Class S4. Roughly speaking, the controllers in this class are combinations of controllers from the preceding three classes. More precisely, they are given by u r ( t ) = (u(alr(t), u~Z~r(t), u(3~r(t)), where u(~)(t), U(2)(t),u(3)(t) C3, respectively.
are
(57)
given by controllers in Classes C1, C2,
4. Properties of Systems with Proposed Controls Before stating a theorem, let us consider any system belonging to Class $4, subject to any corresponding controller in Class C4. By defining the
parameter estimate vector
4 = (e~ ~1, ~2 ..... ~m~, F, ,)~,
(58)
where/~, i = 1, 2 . . . . , m2, are the rows of/~, and by appropriately defining fro: R x R " x ~ ~ R n and f(2): R x R " x ~ -~R r (see Ref. 12), where = R "1 x R "2~ x (0, ee) k÷l,
r = m l + m 2 p + k + l,
(59) (60)
such a controlled system may be described by
~(t) = 7(~(t, x(t), ,~(t)),
(61)
4(t) = f(2)(t, x(t), ~(t)). This is a system whose complete state (x, ~) belongs to R" × ~ . By a solution of (61), we shall mean an absolutely continuous function (x ("), ~ (")) : [to, h) ~ R" x 2, where tl ~ (to, co), which satisfies (61) for all t e [to, h), except on a set of Lebesgue measure zero.
JOTA: VOL. 41, NO, 1, SEPTEMBER 1983
165
Defining the parameter vector
q = (v T, kl, k2 . . . . . k.,2, flY, 0) r
(62)
where ki, i = 1, 2 . . . . . m2, are the rows of K, we are now ready to state a theorem. Theorem 4.1. Consider any system belonging to Class $4 and subject to any corresponding controller in Class C4. The resulting controlled system may be described by (61) and has the following properties.
Property PI. Existence of Solutions. For each (to, x0, C~o)E R × R" × 2, there exists a solution (x (-), 4 (")) : [to, tl)--> R" x ~ of (61), with (x (to), 4(to)) = (xo, 40).
Property P2. Uniform Stability of (0, q). For each 7/> 0, there exists 8 > 0 such that, if (x(. , q(. ))is any solution of (61) with ilX(to)ll, IIq(to) -qlt 6, then A
*
•
IIx (t)ll, IIq(t) -qll < r/,
•
A
for all t e [to, tl).
Property P3. Uniform Boundedness of Solutions. For each rx, r2 > 0, there exist dl(rl, r2), d2(rl, rz)>>-O such that, if (x(.), ~(')) is any solution of (61) with fix(to)If~
