Dynamical Systems with Controllable Singularities: Multi-Scale and

Proceedings of the 40th IEEE Conference on Decision and Control Orlando, Florida USA, December 2001

ThP12-1

Dynamical Systems with Controllable Singularities: Multi-Scale and Limit Representations and Optimal Control Joseph Bentsman 1. and Boris M. Miller 2.* *Department of Mechanical and Industrial Engineering, University of Illinois at UrbanaChampaign,1206 West Green Street, Urbana, IL 61801. E-mail: [email protected] **Institute for Information Transmission Problems, Russian Academy of Sciences, B. Karetny 19, 101447, Moscow, Russia, e-mail: [email protected]

Abstract

the relation

A new class of systems: dynamical systems with controllable singularities, is considered. This class refers to systems that admit introduction of the impulsive control actions during singular phases of their motion, such as changes in dimension, discontinuities in the state, and other nonsmooth types of motion. A well-posed representation of the discontinuous in the limit behavior of these systems is given in terms of differential equations with measure and the corresponding generalized (discontinuous) solution of the new type is introduced. This representation, which admits discontinuity of the entire state, is put into correspondence with the detailed multi-scale system description via a space-time transformation followed by a limit procedure. Finally, using the framework developed, an approach to constructive optimal controller synthesis for this class of systems is presented.

G(X(t),t) = 0 .

(3)

In (1) and (3)functions F(x, u, t) and G ( x , t ) a r e continuous in all variables, and F is Lipschitz with respect to x, so that

liE(x1, u,

t) - f ( x 2 ,

u,

t)ll ~

Lllxl

-

x~ll

(4)

for any xl,x2 E R n, u E U, t E [0, T] with some positive constant L < oc. Therefore, equation (1) has the unique solution for any Lebesgue measurable control u(-) defined on the whole interval [0, T] with an arbitrary initial condition x0. Suppose that originally G(X(O), 0) < 0, and define a sequence of the intersection times 0 O. Let (I)(x, s, w(-), ~-i) denote the shift operator along the paths of (7) until the constraint boundary. This implies that q)(x, s, w(-), Ti) is the general solution of (7) with initial condition y(0) = x. Then the discontinuity of the path X(-) at instant t = ri can be viewed as the result of the action of this translation operator and described by

In spite of the regularity assumptions in the problem statement, this problem belongs to the class of extremely irregular ones due to the possibility of solution to be inextensible (if the set of points s in (9) is empty or infimum is equal to infinity). The first case leads to a so-called "sliding mode" along the set G(x, t) = O, as in the systems with discontinuous right-hand side; the second one corresponds to the case of the solution inextensible beyond the point of accumulation of jumps. However, in the case when the jump behavior is described by some shift-type operator, one possibility is the reduction of this problem to the more regular one by using the discontinuous time transformation as in the impulsive control problems.

x ( ~ ) = e ( x ( ~ - ) , ~ * ( x ( - ~ - ) ) , ~,,(.), ~), wh~e

s*(X(ri)) = inf{s > 0: G(¢o(X(Ti--),s, w(-), r i ) = 0}, (9) 2 G e n e r a l i z e d S o l u t i o n s of S y s t e m s w i t h Controlled Singularities and Differential Equations with Measure

G(q~(X(ri-), s, w(.)) > 0 on the interval (0, s* (X(Ti)). Shift-operator jump representation (10)

with

is admissible for any discrete-continuous system that has trajectories robust with respect to variation of the impulsive input.

Suppose for some control u(-) a solution of (1), say

X(t), is defined on the interval [0, T] and has a finite number of jumps at points {Ti, i = 1, ..., N}. This means that for every i = 1, ...,N the set of controls {w~-,(-)} and the set of s~" = s*(X(ri-)) < oo, such that

Introduce the notation

x ( ~ ) - x ( ~ - ) + ~ ( x ( ~ - ) , ~,(.)), ~).

(~0)

x(~) = e(x(~-), Then by combining the equations (1) and (10) yields the general dynamic description in the form of the discrete-continuous system

~ * ( x ( ~ - ) ) , ~(.), ~),

is defined. Consider the time interval [0, T1], where N

T~ - T + Z

x(t) -

F(X(t), u(t), t) + Z

~(x(~-),

(11)

~,

(~4)

i=1

~ , (.), ~)~(t - ~),

and define on [0, 7'1] the function

r.~_ 0} the system of differential equations for Xg(t) be of the form

, ( T ) < o~,

(25)

localized on the set G(X(t), t) = 0;

where B(x, w, t) describes an additional contact force, which arises due to the constraints violation. Function B is the same as in (7), continuous and Lipschitzian with respect to x, in the area G(x, t) > O.

• the set of controls w~(.) defined .for each atom ~- C D~ of measure #(dr);

such

that X(.)

satisfies

the

equation

(24)

(22)

and

C(X(t), t) corresponds to a scalar product of vectors, i.e., the left-hand side is the total derivative of function G along the paths of the system of differential equations ic = F (x, t). T h e following result is the first a p p r o x i m a t i o n theorem for the generalized solution.

>o

This implies t h a t for finite value of # there exists a non-zero time interval of the constraints violation. Introduce the following space-time transformation of the coordinates, for s > 0 :

Ytt(s) -- Xtt(r nt- # - 1 8 ) ' - ~ ( t - ~).

u(T -Jc-# - 1 8 ) , 7" -~- ~ - 1 8 )

a(x,t)

(~. 0 .

s*

=

i n f ( s > 0, G(9(s),s) = O,

-a(v,~) ds B (9(s),~)


#o, 8*(#)

with initial condition y"(0) = X " ( r ) . Due to the continuity properties of functions F and B the terms in the right-hand side of system (28) converge for any (y, u, w, r, s) to the limit

=

inf{s > 0, G(~t~(s),s)

--

O, ~

d ]

a(y, ,)

B-t-P"-iF

< 0} (9"(s),s)

exists as well. Here B + # - i F corresponds to the righthand side of (28). Moreover

lim B(y, w, r + # - 1 8 ) = B(y, w, r), tz---+cx3 lim # - i F ( y , u, 7 -t- ~ - 1 8 )

d ~ ?7

Suppose also that .for the chosen control w°(.) solution of equation (29) with the initial condition y(O) - X"(T) is such that there exists

(27)

~.(~) - B(v"(,), ~0(~), ~ + ~ - ~ )

4 Suppose that (X"(r), T) is such that

a ( x . ( ~ ) , ~) - o,

Then, taking into account the representation (26) of the control w(t, #) yields the following equation for y"(s):

tt---+oo

< VxG(x, t), F(x, t) >

°a(x,t)

Theorem

+#-lf(ytt(s),

-

+ Ot

3.1.3 Space-Time Transformation in t h e Vicinity of the Singularity Point: Let the system motion start from the initial condition X ~ (0) = x0 such t h a t G(xo, 0) _< 0 and r is the first time where the system hits constraint, so t h a t

a ( x " (T), r) -- O,

2

lim s* (#) -- s*.

-- 0,

and for y E Y,s E [0, S ] , S < oo, where Y is any b o u n d e d domain in R ~, this convergence is uniform on Y x U x W x [0, T] x [0, S]. Therefore, one can obtain the following result.

The next result shows t h a t in the vicinity of the point where system (11) hits the constraint, solution X " ( t ) converges to the discontinuous solution of (11). This immediately follows from T h e o r e m 4.

3 On any given.finite interval [0, S] solution of equation (28) y"(s) converges uniformly to the ~(s), where [1(s) is the solution of equation

Corollary 1 Let (X•(r),r) satisfy the conditions of Th~o~,~ 4. If X"(~) - X ( ~ - ) , the,~

Theorem

~(~) - B(9(~), ~o(,), T),

Xtt(T -~- ~ - 1 8 " ( ~ ) )

~

(~(X(T--),8*, wO('), T) -- X ( T ) ,

(29)

where g2 is the general solution of system (7).

with initial condition [1(0)- y " ( O ) - X"(T).

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Next theorem lutions is the in the week-* this closure is topology[2].

Theorem 2 one can obtain the equivalence of the original optimization problem, which contains measures, to the auxiliary problem of nonsmooth optimization.

shows that the set of generalized soclosure of the ordinary solutions set topology. One can demonstrate that equivalent to the closure in the curves

D e f i n i t i o n 3 Consider the optimal control problem.for the system (16) with control signals {c~,Ul, Wl} satis.fying (20), and such that the integral constraints (19) are valid. Assume the non:fixed time interval [0, T1], such that T1 < oo : ~?(T1) = T with the performance criterion

T h e o r e m 5 Let X ( t ) be some generalized solution of (11). Then there exist: • a sequence of #n ~ oo;

J' {y(.), ct(.),Ul(-), Wl(.)} - ¢0(y(T1)) ~ min,

• a sequence of subsets A~ c_ [0, T], with Lebesgue measure A(A,~) --+ 0; •

where ¢0 is the same as in (13).

a sequence of admissible controls {u~(-), w~(-);

T h e o r e m 6 Suppose that the sets F ( X , U , t ) 2 t~"

=

F ( X " ~ ( t ) , u n ( t ) , t ) -+-

F(X. U. t)

. ~ B ( X . ~ ( t ) . w ~ ( t ) , t)IA~ (t), with initial condition X"~(O) = X(O), converges to X ( t ) at all points of continuity. Moreover,

f G ( X ",~ (t), t)dt --+ O. 0

In conclusion, one can mention that sliding mode along the constraint boundary, if it exists, can be described by the relation

dt aF+(1-a)B

B(X. W. t)

R e m a r k 4 The auxiliary problem belongs to the class of nonsmooth optimal control problems due to the nondifferentiability of functions G + and G - . However, by applying the methods recently developed, it becomes possible to derive the necessary optimality conditions in the m a x i m u m principle form and to design the computational algorithms. As an example of such approach one can refer to the paper [1], where the optimality conditions in hybrid system with phase constraints and singular phase, which arise when the system hits the constraint boundary were derived.

which after the inverse time transformation gives the following relation --

i~(t) = -

{F(X. ~. t) ~ e U}. {B(X.w.t)lw ~ W}

~(dt).

G -0,

d G dt r < VG, B >

-

are convex .for any (X, t) E R ~ x [0, T], the set of the admissible ordinary solutions with the .finite number of jumps is non empty, and the set of the appropriate T1 such that ~(T1) - T is uniformly bounded. Then~ the auxiliary problem has the optimal solution and the original optimal control problem has a solution within the class of the generalized ones, and the optimal generalized solution of the original problem satisfies the equation (13) with some control u(.), w(.) and measure

T

d

and

B(x, w, t), i.~.,

such that solution of equation

(30) (X(t),u(t),w(t),t)

for the continuous component of measure #C(dt) = ) ( t ) d t . Thus, the condition /2(t) _> 0 takes place if d I' G > 0, which occurs when the system in its natudt ,F ral phase hits the constraint and also if < VG, B > < 0 when during the singular phase the system is pushed out of the inhibited area.

References

[1] B.M. Miller and J. Bentsman, "A new approach to control of dynamic systems with unilateral constraints," Proc. l~th IFAC World Congress, Beijing, China, v. C, pp. 199-203, 1999.

4 An Approach to the Optimal Control Problems

[2] A. Bressan and F. Rampazzo, "Impulsive control systems without commutativity assumptions," J. Optim. Theory and Appl. vol. 81, pp. 435-457, 1994.

The concept of the generalized solution naturally leads to a search for the solution of the optimal control problem within the class of the generalized solutions. From

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