Maximal Admissible Nonparametric Uncertainty in a Control Design Problem

Copyright @ IFAC Control Applications of Optimization, St. Petersburg, Russia, 2000

MAXIMAL ADMISSIBLE NONPARAMETRIC UNCERTAINTY IN A CONTROL DESIGN PROBLEM Victor V. Koulaguin Institute of Problems of Mechanical Engineering Russian Academy of Sciences 61 Bolshoi ave. , V.O., Saint Petersburg, Russia E-mail: kulagin @ensure.ipme.ru

Abstract: A problem how to let constraints and uncertainty be compatible when setting up control design conditions is discussed. If constraints are given, an uncertainty set cannot have its "size" more than the admissible "size". In this paper a correct assignment of an uncertainty set is considered. As a result a new kind of a design problem and some method of its solution are obtained. A design problem of that kind for some dynamic system is set forth. Copyright @ 2000 IFAC Keywords: automatic control, control system design, dynamic systems, feedback control, optimal control, systems design , uncertainty, uncertain dynamic systems.

1. INTRODUCTION

Finally, these general considerations are applied to a control design problem. This problem is a new one because an objective function not as a characteristic of trajectories of a dynamic system, but as a characteristic of an admissible uncertainty set is defined. As an example, an optimal control design problem for a single mass point, to be held within an interval of a straight line under disturbance of that kind of uncertainty, is stated. It should be noted that, instead of disturbance function, only some its property is known, so that the large-scale uncertainty set is a domain in a function space, i.e. a non parametric case is considered.

There is a following practical way to make a design decision under uncertainty. Let a design problem include three parts: a set of design decisions, a quality demand (constraints), an uncertainty set. And let an uncertainty set be also uncertain. In that case, the uncertainty set is supposed to be large enough (it is named a large-scale uncertainty set). The fact is that, for each design decision, there is an admissible uncertainty set, i.e. the set of all uncertain parameters values or/and uncertain functions realizations, such that a quality demand is satisfied. Under such a condition, a designer will choose a design decision that would maximize some size of the admissible uncertainty set. In this paper a formalization of the practical method and its specification with respect to some double problem are considered. Then an equivalence between two double problems, as a possible method to solve a starting one, is formulated .

2. SOME PROBLEM OF DESIGN UNDER UNCERTAINTY Let p(x , y) be a system, x E X be a design, y EYe Ey be an uncertainty, p(x, y) E P be a constraint. The problem is to find a design x· E X 193

such that p(x',y) E P, Vy E Y.

is compatible; the design x is a solution of problem (1) for P = P",. To construct unique (if possible) set P"" where x EX, let ~ (P",) be some functional. An optimal constraint design problem is

(1)

If sets X, Y, P are arbitrary, there might be difficulties with the problem (1), for example it might have no solution or each x E X might be a solution. So there is a related problem - to construct sets X, Y, P such that a unique solution of problem (1) exists.

~(P",)

Remark 3. considered.

The construction of a compatible triplet < X, Y, P> might be done in the following way: preassign two of sets X, Y, P and construct the third one such that the triplet < X, Y, P> is compatible.

Let 'l/J(y) be a convex functional;

= {y E Ey I p(x,y) E P},

Y(-y) = {y E Ey

is compatible, moreover this design x is a solution of problem (1) for Y = Y",. It is also clear that there is a nonuniqueness in the related problem. To construct a unique (if possible) set Y"" let
I

'l/J(y)::; " , E Rd

be a Lebesgue set of the functional 'l/J(y). Let us map the set Y(-y) into the set

P",(-y) = {p(x,y) E Ep lyE Y(-y)}. Assume that, as, increases, the set Y(-y) evolves inside the set Y"" the set P", (,) evolves inside the set P. Let p(x, ,) be a function which estimates some distance between sets P and P",(-y). Assume that for each x E X the function p( x, ,) decreases, with, increasing, and there exists " such that

(2)

A solution of problem (2) (if any) is a pair (x', Y",.); it is obvious that the design x' is a solution of problem (1) for Y = Y",.. Thus the problem (2) is an optimal elaboration of problem (1).

= 0, P",(-y') uP = P, p(x, ,')

Remark 1. There is the estimation how the largescale uncertainty set W might be preassigned:

P",(-y'

+c

> 0) UP:/; P, Vc> O.

Hence, the value " is a solution of the equation p(x, ,) = 0 and is denoted by root')' p(x, ,)

W2A, where A

In this paper only problem (2) is

In the section, some technique how to make the estimation
Let X, P be given sets, Y be one to be constructed. It is clear that for some x E X the triplet , where

",EX

(3)

3. HOW TO SPECIFY THE FUNCTIONAL
2.1 An optimal uncertainty design problem.


",EX

A solution of problem (3) is a pair (x', P",.); the design x· is a solution of problem (1) for P = P", •. The problem (3) might have a meaning of a minimal scattering of vector p.

Definition 1. A triplet < X, Y, P> is called compatible, if there exists a solution x'.

Y",

-+ min.

= U"'EX Y",.

"

Remark 2. A set Y", might have (with a view to applications) a meaning of a domain of reliable work for the system p(x, V). An estimation
= rootp(x,,). ')'

The value " represents the fact of a quality demand violation, therefore, the value estimates the set Y"" i.e.


Thus, the specification of problem (2) is:

2.2 An optimal constraint design problem.

root p(x, ,) -+ max,

Let X, Y be given sets, P be one to be constructed. It is clear that, for some x EX, the triplet < X, Y, P", >, where

P",

= {p(x,y)

')'

",EX

or max root p(x, ,). "'EX

lyE Y}, 194

')'

(4)

,z

Definition 2. A loading parameter is called an admissible loading parameter for the control x, if each disturbance y E Y (,z) does not make a single mass point out of the interval.

As it will be shown in the sequel, the double problem (4) might be solved by passing to the equivalent double problem root max p(x, I )' 'Y

z EX

(5)

Definition 3. A maximal admissible loading parameter (denoted by,;) is called a resource of the system under control x .

4. ON SOME EQUIVALENCE BETWEEN DOUBLE PROBLEMS

The problem is to design a feedback control x· such that a resource is maximal:

Theorem (on equivalence) (Koulaguin, 1999). Let X eRn, Y c RI be compact sets, f(x, y) be a continuous function on X x Y . Assume that there exists the unique solution y; of the equation f(x , y)=O ,'v'xEX. If8f~~y; ) < 0, then max rootf(x,y) z EX

yEY

= root yEY

maxf(x,y), z EX

Let us set up the problem more exactly. There is the dynamic system

(6)

z(t)

+ x(t, z(t), z(t), y(t))

t E [0, TJ,

and there exists a set of points {(x,y)}, each of which is a solution of both the right double problem and the left double problem in equality (6).

= z (O) = 0,

z (O)

The general system is

p(x, y) 5. AN OPTIMAL CONTROL DESIGN PROBLEM

= tE[O,Tj max Iz(t; x, y)l .

The set of admissible control functions is X = {x( ·)

First, there are some specifications of the above terms to the field of control:

I Ixl

~

a},

(8)

where a > 0 is a given constant.

p - a vector of all functionals we are interested in

which are defined on trajectories of a dynamic system;

The set of disturbances without a change of sign is denoted by W. The characteristic of each disturbance from W is a total mechanical impuls

x E X - a control;

t

y E Y - a disturbance; 1j;(y)

p(x, y) E P - constraints; -

(7)

where z - a coordinate, x(·) - a control (a design), y(.) - a disturbance. The solution of Cauchy problem (7) is z(t; x(·), y(.)) or z(t; x, y).

From the theorem it follows that the left double problem and the right double problem are equivalent, i.e. their solutions are the same.

Yz

= y(t),

= tE[O max I rY(T)dTI . ,Tj ~

°

a domain of reliable work;

A constraint is given in the following form:

1jJ(y) - a characteristic of a disturbance; , - a loading parameter;

p(x, y)

Y{J) - a loading set;

~

(9)

{J,

where {J > 0 is a given constant.

p(x, ,) - a resource function ;

tp(Yz )

-

The domain of reliable work is

an objective function, a resource of a dynamic system.

Yz

Let us consider the following control design problem. Two forces operate a single mass point which moves along a straight line. The single mass point is supposed to be held within an interval of the line. There is an uncertainty of the following kind: instead of a disturbance function, one, who constructs a control function, knows only that property of the disturbance function: it has no change of sign. Each disturbance of this uncertainty set is characterized by its total mechanical impuls 1jJ(y).

= {y E W I p(x,y) ~ {J} .

The loading parameter , determines the loading set Y(J) = {y E W I 1jJ(y) ~ ,} and the set of states p(x, y) loaded

Pz{J) = [ inf p(x, y), sup p(x, y)) = yEY(-y)

=[ inf

max Iz(t;x,y)I, sup

yEY(-y) tE[O ,Tj

195

yEY(-y)

max Iz(t;x,y)lJ·

yEY(-Y) tE[O ,Tj

is attained, and that the value of the minimum in (14) is equal to -y2/2O'.

A resource function p( x, -y) is given in the following form:

p(x,-y)=/3- sup

max Jz(t;x,y)J,

Thus, the solution of problem (13) is

(10)

y E Y(-y) tE[O ,T]

where

(16)

/3 is given in (9) .

An objective function
An auxiliary optimum problem. Let us consider the problem (4) again

rootp(x,')').

root max p(x, -y).

-y

-y

Then the maximal resource problem is max root p(x, ')'). "' E X

The value (17) can be calculated from (16) by the equation /3 - (-y2/2O') = O. It is obvious that the value (17) is equal to

(11)

-y

In order to obtain a solution of problem (11), hereafter the several preliminary considerations are made.

(18) Hence, the solution of problem (17) is the pair (-y+ , x+), where the optimal control function x+ is defined in (15) (it is the same for each ')'), the value -y+ is defined in (18) .

Some properties of the function p(x, -y). In (Bolotnik, 1976), for dynamic system (7), constraint (8), and disturbance set W, a solution of problem in the right-hand part of the definition (10), i.e. the solution of problem

max max Jz(t; x, y)J,

y E Y(-Y) t E [O ,T]

Remarks to the Theorem (on equivalence) . One can show that the Theorem is true under the following assumptions:

(12)

(a) the set X is such that there is a solution of problem (4);

is obtained. It is shown that the maximum in (12) is attained by 8-impuls y(t) = ')'8(t), moreover the maximum in (12) increases as -y increases. For some admissible control functions the function p(x, -y) ~ 0, V-y > 0 (i.e. a resource is equal to zero), for some ones the function p( x, -y) has the value p(x,O) > 0 which decreases as -y increases and has a unique value of the argument when p(x,-y) = O.

(b) the function

",EX

(d) for some x E X, the equation f(x,y) = 0 might not have a solution;

"'EX yEY(-y) tE[O,T]

(e) instead of the partial derivative with respect to y, it is sufficient to have a change of sign of the function f(x,y) from "+" to "-" in the point y;.

(13) In (Bolotnik, 1976) also the problem in the right part of (13), i.e.

min max max Jz(t; x, y)J,

(14)

These considerations allow to use the Theorem (on equivalence) to make the following statement: problems (11), (17) are equivalent, i.e. the pair (-y;., x*) and the pair (-y+, x+) are the same. Thus, the starting problem (11) has the solution: x· is x+; -y;. = -y+ = ..j2(ifJ.

is solved. It is shown that the minimum in (14), for each -y > 0, by the control function known as "Coulomb friction" i > 0, i < 0, i = 0, where

Y;

y;;

/3 - min max max Jz(t; x, y)J.

",EX yEY(-y) tE[O ,T]

f (x, y) is continuous on

(c) there is a finite number of isolated roots y", of the equation f(x,y) = 0, x E X, and/or there is a finite number of closed intervals each point of which is also a root of the latter equation; in this case it is necessary to take a maximal root, in the capacity of the root

Some optimum problem for the resource function (1 0). Let us consider the problem

maxp(x, -y) =

(17)

"' E X

(15) 6. CONCLUSION In the paper the problem how to make a control design decision, if an uncertainty set is unknown, is discussed. As an answer, an optimal uncertainty design problem (2) is stated and then to the double

y > a, a, sataY = -a, y < -a, { y, JyJ ~ a,

196

problem (4) is specified. The double problem (4) is an optimum condition for problems of control design under uncertainty of that kind. As it is shown in the example, the problem (4) might be solved by means of a passage to the equivalent problem (5). REFERENCES Bolotnik, N.N. (1976). Shock isolation systems for sets of disturbances. Izvestiya of Academy of Sciences of the USSR, Mechanics of Solids, 4, 34-45. Koulaguin, V.V. (1999). On control and design under large-scale uncertainty. In: Proceedings of 6-th Saint Petersburg Symposium on Adaptive Systems Theory, Vol. 1, p. 235, Vol. 2, pp. 203-205, St.Petersburg.

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