fast inputs in the problem of control synthesis under uncertainty

ISSN 0012-2661, Differential Equations, 2011, Vol. 47, No. 7, pp. 972–981. c© Pleiades Publishing, Ltd., 2011.Original Russian Text c© A.N. Dar’in, A.B. Kurzhanskii, 2011, published in Differentsial’nye Uravneniya, 2011, Vol. 47, No. 7, pp. 963–971.

CONTROL THEORY

Fast Inputs in the Problem of Control Synthesisunder Uncertainty

A. N. Dar’in and A. B. KurzhanskiiMoscow State University, Moscow, Russia

Received March 16, 2011

Abstract— We present a class of bounded control inputs that permits one to solve targetcontrol synthesis problems for linear systems with geometric (“instantaneous”) constraints onthe perturbations by reduction to simpler programmed control problems.

DOI: 10.1134/S0012266111070068

It is known that the construction of synthesizing control strategies implementing the corre-sponding inputs in the form of feedback is a key point for applications of the mathematical theoryof control under unknown perturbations. In this case, the desired controls depend on time (likeprogrammed controls) as well as on the current state of the system. Such controls are stronglyneeded but are difficult to compute. Below, we present a class of bounded control inputs (so-called“fast inputs”), which permits one to solve target control synthesis problems for linear systems withgeometric (“instantaneous”) constraints on the perturbations by reduction to simpler programmedcontrol problems. It is constructed by approximation of impulse controls, which admit higherderivatives of delta functions [1–4].

1. PROBLEM

We consider a linear system with a control u and an uncertain perturbation (“noise”) v of theform

x(t) = A(t)x(t) + B(t)u(t) + C(t)v(t), t ∈ [t0, t1], (1)

where x ∈ Rn, u ∈ R

m, and v ∈ Rk and the dimensions satisfy m,k ≤ n. The time interval [t0, t1]

is given in advance. The matrix functions A(t) ∈ Rn×n, B(t) ∈ R

n×m, and C(t) ∈ Rn×k are known

and have smoothness sufficient for the subsequent constructions.The noise v(t) is a piecewise continuous function satisfying the constraint

v(t) ∈ Q(t), t ∈ [t0, t1],

where Q(t) is a multifunction ranging in the set conv Rk of convex compact sets in R

k.The function Q(t), which is continuous in the Hausdorff metric, is given, for example, by theinequalities |vi(t)| ≤ νi, i = 1, . . . , k.

The control target is to bring the system to a given target set M ∈ conv Rn at time t1 despite

possible actions of the noise.Let us proceed to the description of control classes. It is known [5, 6] that if B(t) ≡ C(t) and

the control u(t) is a function of the same class as v(t) and satisfies the condition u(t) ∈ P(t) with theconstraint P(t) = α(t)Q(t), |α(t)| ≥ 1, then the solutions of minimax problems (for example,the minimax with respect to u and v of the deviation of the process at the terminal time froma given target set M ∈ conv R

n) obtained in the class of programmed controls and in the class ofsynthesized controls coincide. The situation is different in the case without the above-mentionedrequirement of matching conditions for constraints. Then the solution to the control synthesisproblem is much more complicated than that of the programmed control problem and requiresmuch additional computational work.

972

FAST INPUTS IN THE PROBLEM OF CONTROL SYNTHESIS UNDER UNCERTAINTY 973

Problem 1. Find a class of controls that permits reducing the solution of the synthesis problemto programmed constructions.

In the present paper, we suggest a different class of controls, which provides an effect similar tothe matching property for a broader class of systems than in problems with matching geometricconstraints for the controls and perturbations. It is the class of piecewise constant functions withvariable amplitudes generated by approximations to “ideal controls,” that is, linear combinationsof delta functions and their higher derivatives. It will be described in detail below. The use of suchclass of controls also permits one to reduce the search for a solution of the synthesis problem tothe simpler problem of computing programmed controls.

2. GENERALIZED CONTROLS

Suppose that generalized functions (distributions) of order s are fed as a control input to theinput of the system. According to [7, 8], these functions can be represented as a sum of higher-ordergeneralized derivatives of functions of bounded variation,

u(t) =s∑

j=0

dj+1Uj(t)dtj+1

, Uj(·) ∈ BV ([t0, t1]; Rm). (2)

In particular, as shown in [9], the optimal generalized controls in the problem of bringing a linearsystem to a given state in the noise-free case have the form

u(t) =n∑

i=1

s∑

j=0

hi,jδ(j)(t − τj), (3)

where δ(t) = χ′(t) is the delta function, that is, the generalized derivative of the Heaviside functionχ(t) ∈ BV [t0, t1], the vectors hi,j ∈ R

m determine the direction and amplitude of generalizedimpulses, and the τi are the times of application of these impulses.

After the substitution of a control of the form (2) into the original differential equation (1), weobtain the following system with impulse control (see [9]) :

dx(t) = A(t)x(t) dx(t) + B(t) dU(t) + C(t)v(t) dt, t ∈ [t0, t1], (4)

where B(t) = [L0(t), . . . , Ls(t)], U(t) = [U0(t), . . . , Us(t)] ∈ BV ([t0, t1]; Rm(s+1)) is an impulse con-trol, x(t1+0) ∈ M is the control target, and the Lj(t) are matrix functions found from the recursionrelations

L0(t) = B(t), Lj(t) = A(t)Lj−1(t) −dLj−1(t)

dt, j = 1, . . . , s. (5)

Thus, the use of higher-order impulses can increase the control possibilities in the sense thatRangeB(t) ⊇ RangeB(t). (Here and throughout the following, Range stands for the linear span ofcolumns of a matrix.)

Assumption 1. There exists an s ≤ n−1 such that RangeB(t) ⊇ Range C(t) for all t ∈ [t0, t1].

The assumption holds if A(t) ≡ A, B(t) ≡ B, and [A,B] is a controllable pair. In this case, theminimum value of s coincides with the controllability index of the system.

Let us replace the “ideal” impulse controls in system (4) by physically realizable bounded func-tions; to this end, we subject the control input u(t) = dU/dt to the geometric constraint u(t) ∈ P(t).Then system (4) acquires the form

x(t) = A(t)x(t) + B(t)u(t) + C(t)v(t), t ∈ [t0, t1]. (6)

Here u(t) = [u0(t), . . . , us(t)] ∈ Rm(s+1), and the control target is again x(t1) ∈ M.

DIFFERENTIAL EQUATIONS Vol. 47 No. 7 2011

974 DAR’IN, KURZHANSKII

It is known that, under the “total sweeping out” (generalized matching) condition1

(B(t)P(t) − C(t)Q(t)) + C(t)Q(t) = B(t)P(t), (7)

the solution of the problem on the construction of a synthesizing strategy [5, 6] is simplified dra-matically. This condition is equivalent to the convexity of the difference of support functions of thesets B(t)P(t) and C(t)Q(t).

Our aim is to coordinate the constraints on the control and the noise so as to satisfy condition (7).In particular, there are the following two approaches to such a coordination for a given set Q(t).

1. Choose the corresponding P(t).2. Choose some P(t) such that B(t)P(t) − C(t)Q(t) �= ∅. Next, choose a set Q(t) ⊇ Q(t) for

which the “total sweeping out” condition is satisfied.

3. CHOICE OF THE CONSTRAINT FOR THE CONTROL

3.1. General Case

Lemma 1. For each set N (t) ∈ conv RangeB(t), there exists a set P(t) ∈ convRm(s+1) suchthat B(t)P(t) = N (t).

Assumption 2. The matrices A(t) and B(t) are such that rankB(t) ≡ const and the indicesof columns in B(t) forming a basis in RangeB(t) can be chosen in the form of piecewise constantfunctions of t with finite number of switches.

Lemma 2. Under Assumption 2, the function P(t) can be chosen to be piecewise continuousand upper semicontinuous with respect to inclusion.

Then, to satisfy the total sweep condition, it suffices to set

N (t) = αC(t)Q(t) + N0(t), α ≥ 1,

where N0(t) ∈ conv RangeB(t) is some set that can be chosen in accordance with specific featuresof the physical implementation of the system.

3.2. System of Special Form

Consider a system in which A(t) ≡ A, B(t) ≡ B, and [A,B] is a controllable pair. The noiseparameters Q(t) ≡ Q and C(t) ≡ C have the form described below. We have B(t) ≡ B =[B,AB, . . . , AsB] in this case.

Let v(t) = [v1(t), . . . , vr(t)] be a noise, and let the matrix C have the form

C = [k1Aj1B, . . . , krA

jrB], 0 ≤ j1 < j2 < · · · < jr ≤ s; ki ∈ R.

In addition, suppose that the constraint Q on the noise is given in the form

‖v1(t)‖ ≤ 1, . . . , ‖vr(t)‖ ≤ 1.

Then the constraint for the control can be given in the form

‖u0(t)‖ ≤ μ0, . . . , ‖us(t)‖ ≤ μs;

moreover, to satisfy the total sweeping out condition, it suffices to require that

μj1 ≥ k1, . . . , μjr≥ kr.

1 The symbol − stands for the geometric difference of sets, A − B = {x| x + B ⊆ A}.



3.3. Example

Consider the three-chain vibration system [10]

mw1 = k(w2 − 2w1) + mv1(t),mw2 = k(w3 − 2w2 + w1) + mv2(t),mw3 = k(w2 − w3) + mu(t) + mv3(t),

(8)

which consists of successively joined loads of mass m and springs of rigidity k. The variables wj

describe the displacement of loads from their equilibria. The control u and the noises vj physicallymean the accelerations. We assume that noises satisfy the constraints |vj(t)| ≤ νj, j = 1, 2, 3.

The “total sweeping out” condition fails for system (8), since the control u appears only in thelast equation, while the noise is present in each equation.

Let us rewrite system (8) in normal form (by setting ω = k/m) :

xj = x3+j, j = 1, 2, 3;x4 = ω(x2 − 2x1) + v1(t),x5 = ω(x3 − 2x2 + x1) + v2(t),x6 = ω(x2 − x3) + u(t) + v3(t).

(9)

To satisfy the condition RangeB(t) ⊇ Range C(t), one should use distributions of order s ≥ 4.For the computations in the example, we take s = 5. The matrix B(t) is equal to

B =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 ω2

0 0 0 ω 0 −3ω2

0 1 0 −ω 0 2ω2

0 0 0 0 ω2 0

0 0 ω 0 −3ω2 0

1 0 −ω 0 2ω2 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

The system with this matrix is not covered by the case considered in the preceding section. In thecontrol, we make the change of coordinates

u1 = u1 − ωu3 + 2ω2u5, u3 = ωu3 − 3ω2u5, u5 = ω2u5,

u2 = u2 − ωu4 + 2ω2u6, u4 = ωu4 − 3ω2u6, u6 = ω2u6.

Then

B =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 1

0 0 0 1 0 0

0 1 0 0 0 0

0 0 0 0 1 0

0 0 1 0 0 0

1 0 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, C =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0

0 0 0

0 0 0

0 0 1

0 1 0

1 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

and system (9) acquires the form

x1 = x4 + u6(t), x4 = ω(x2 − 2x1) + u5(t) + v1(t),

x2 = x5 + u4(t), x5 = ω(x3 − 2x2 + x1) + u3(t) + v2(t),

x3 = x6 + u2(t), x6 = ω(x2 − x3) + u1(t) + v3(t).



Now we take the constraints for the control in the form

|u1(t)| ≤ α3ν3, |u3(t)| ≤ α2ν2, |u5(t)| ≤ α1ν1, αj ≥ 1.

The controls u2(t), u4(t), and u6(t) can be constrained by an arbitrary convex set. In particular,one can set u2(t) = u4(t) = u6(t) = 0 to preserve the physical meaning. (The control exerts a directinfluence only on the velocities but not on the displacements of the loads.)

4. CHOICE OF THE CONSTRAINT FOR THE NOISE

Let P(t) be a given set satisfying the following conditions.1. B(t)P(t) − C(t)Q(t) �= ∅.2. B(t)P(t) is a generating set [11, p. 324] in the space RangeB(t).Recall that X ∈ conv R

n is a generating set if, for an arbitrary set Y ⊆ Rn such that X − Y �= ∅,

there exists a set Z ∈ conv Rn such that X − Y + Z = X . Since X − Z + Z = X in this case,

it follows that the “total sweeping out” condition is satisfied for the sets X and Z.Now, by using Lemma 1, one can replace the set Q(t) by Q(t) ⊇ Q(t) so as to satisfy condi-

tion (7).Let us show how to choose a generating set for the constraint on the control.If n = 2, then any convex closed set is generating [11, Th. 4.2.6].Let n ≥ 3, and let Assumption 1 hold. In B(t), we choose basis columns. We assume that these

are the first q columns. (Otherwise one can renumber them.) Then B(t) = [B1(t),B2(t)], whereB1(t) ∈ R

n×q is a matrix of full rank. Accordingly, u(t) = [u1(t),u2(t)], u1(t) ∈ Rq. We subject

the control u(t) to a constraint of the following form.1. A constraint for u1 of one of the following types:(a) |(u1)j| ≤ μj ;(b) ‖u1‖ ≤ μ.2. u2 = 0 (temporary value).Then, by [11, Ths. 4.2.3, 4.2.4, and 4.2.7], B(t)P(t) is a generating set. One should choose the

numbers μj (or the number μ) so as to ensure that B(t)P(t) − C(t)Q(t) �= ∅.After choosing the corresponding extended set Q(t), one can remove the zero value for u2(t) and

keep an arbitrary constraint u2(t) ∈ P2(t).

5. THE FORM OF CONTROLS IN THE ORIGINAL SYSTEM

The above-described approach permits one to obtain a control in the form of synthesis forsystem (6) and, for some realization v(t) of the noise, compute the control input u(t). Next, oneshould indicate what corresponds to this control in the original system (1).

It is impossible to use formula (2) directly since the differentiability (and even the continuity)of the function u(t) is not guaranteed. To eliminate this difficulty, we suggest to approximategeneralized controls by bounded functions in accordance with one of the following two schemes.

1. Replace the derivatives of the delta function in the representation (3) by their approximationsin the class of ordinary functions. In this case, we obtain a system different from (6), for which oneshould use the theory described in the preceding sections.

2. Solve the problem for system (6) and then approximate the resulting control u(t) by sufficientlysmooth functions so as to use formula (2).

5.1. The First Scheme

Following [12, 13], in (3), we replace the derivatives of the delta function by piecewise constantapproximations,

u(t) =n∑

i=1

s∑

j=0

hi,jΔ(j)h (t − τj), (10)



where

Δ(0)h (t) = h−11[0,h](t), Δ(j)

h (t) = h−1(Δ(j−1)h (t) − Δ(j−1)

h (t − h)), j = 1, . . . , s. (11)

Note the required properties of these approximations.1. The weak limit of Δ(j)

h (t) in the space of generalized functions of order j as h → 0 is thederivative δ(j)(t).

2. The recursion relation (5) implies the following closed form of these functions:

Δ(j)h (t) = h−(j+1)

j∑

i=0

(−1)i

(j

i

)1[ih,(i+1)h](t).

Let us write down the Cauchy formula for system (6) :

x(ϑ) = X(ϑ, t0)x0 +s∑

j=0

ϑ∫

t0

X(ϑ, t)Lj(t)uj(t) dt +

ϑ∫

t0

X(ϑ, t)C(t)v(t) dt. (12)

Note that the functions Lj(t) that appear in (5) are given by the relations

Lj(t) = (−1)jX(t, t0)[X(t0, t)B(t)](j).

In the last expression, we represent the derivative as a convolution with the derivative of the deltafunction,

Lj(t) = X(t, t0)∫

R

X(t0, τ)B(τ)δ(j)(τ − t) dτ,

after which we pass to the approximations (11),

M(j)h (t) =

t+(j+1)h∫

t

X(t, τ)B(τ)Δ(j)h (τ − t) dτ. (13)

Theorem 1. The matrix functions M(j)h (t) satisfy the recursion relations

M(j)h (t) = h−1(M (j−1)

h (t) − X(t, t + h)M (j−1)h (t + h)), M

(0)h (t) = h−1

t+h∫

t

X(t, τ)B(τ) dτ.

In particular ,

M(j)h = h−j(I − e−Ah)jM

(0)h , M

(0)h =

1h

[ h∫

0

eAt dt

]B

for A(t) ≡ A and B(t) ≡ B.

Theorem 2. Let the matrix function A(t) be continuous, and let B(t) be s+1 times continuouslydifferentiable. Then the functions M

(j)h (t) converge to Lj(t) uniformly on [t0, t1], j = 0, . . . , s,

as h → 0.

Proof. The assertion of the theorem can be proved by application of the following lemma to allentries of the matrix M

(j)h (t).



Lemma 3. Let a function f(t) be j + 1 times continuously differentiable. Then one has theuniform convergence

t+(j+1)h∫

t

f(τ)Δ(j)h (τ − t) dτ ⇒ (−1)jf (j)(t).

Proof. We partition the interval of integration into the segments [ih, (i + 1)h] and make thechange of variables τ ′ = τ − ih on each of these intervals,

I(h) =

t+(j+1)h∫

t

f(τ)Δ(j)h (τ − t) dτ =

t+h∫

t

f(τ + ih)Δ(j)h (τ + ih − t) dτ

=1h

(−1)j

t+h∫

t

j∑

i=0

(−1)i+jh−jCijf(τ + ih) dτ.

The integrand is a finite-difference approximation to the jth derivative, which is equal to f (j)(t) +O(h)g1(t) under the above-stipulated assumptions, where g1(t) is a bounded function of t. There-fore, I(h) = (−1)jf (j)(t) + O(h)g2(t), where g2(t) is a bounded function as well. The proof of thelemma is complete.

Corollary 1. Under the assumptions of the theorem, the matrix function Mh(t) = (M (0)h (t),

. . . , M(s)h (t)) converges to B(t) uniformly on [t0, t1] as h → 0.

Corollary 2. If rankB(t) ≡ n, then the relation rankMh(t) ≡ n is true for sufficiently smallh > 0 as well.

In (12), we replace the functions Lj(t) by M(j)h (t),

xh(ϑ) = X(ϑ, t0)x0 +s∑

j=0

ϑ∫

t0

X(ϑ, t)M (j)h (t)uj(t) dt +

ϑ∫

t0

X(ϑ, t)C(t)v(t) dt. (14)

This relation is the Cauchy formula for the system

xh(t) = A(t)xh(t) + Mh(t)u(t) + C(t)v(t), t ∈ [t0, t1]. (15)

Theorem 3. The trajectories xh(t) of system (15) uniformly converge to the trajectory x(t) ofsystem (6) on the interval [t0, t1] as h → 0.

Proof. This follows from Theorem 2 and formula (14).We assume that uj(t) = 0 for t < t0. After the substitution of the expression for M

(j)h (t) into (14)

and the change of the order of integration, we obtain the representation

xh(ϑ) = X(ϑ, t0)x0 +s∑

j=0

ϑ+(j+1)h∫

t0

X(ϑ, t)B(t)

t∧ϑ∫

t0

Δ(j)h (t − τ)uj(τ) dτ dt +

ϑ∫

t0

X(ϑ, t)C(t)v(t) dt.

Hence we have the following assertion.

Theorem 4. Let u(t) ≡ 0 and v(t) ≡ 0 for t ∈ (ϑ, ϑ + (s + 1)h]. Then xh(ϑ + (s + 1)h) =x(ϑ + (s + 1)h), where x(t) is the trajectory of the original system (1) for the control

uh(t) =s∑

j=0

t∫

t0

Δ(j)h (t − τ)uj(τ) dτ. (16)



Note that uh(t) depends only on the values of u(τ) for τ ≤ t, i.e., can be computed on the basisof information known by the time t.

The above-proved theorems permit one to indicate a scheme for finding the control inputs forthe original system.

1. Fix h > 0 and pass to system (15).2. Apply the above scheme of choosing the constraints on the control and/or the noise to Mh(t)

as indicated for B(t). [It follows from Corollary 2 that if RangeB(t) = Rn, then RangeMh(t) =

RangeB(t).]3. Construct a control synthesis U(t, x) for system (15) under the chosen constraints.4. Determine the implemented trajectory of the control u(t).5. By formula (16), find the control input for the original system (1). [Since the control uh(t)

depends only on previous value of u(t), we observe that it can be computed on-line.]

5.2. Second Scheme

Let us briefly describe the second scheme for constructing a control input for the original system.Let

u(t) = [u0(t), . . . , us(t)]

be the realized control for system (6). We approximate its components by smooth functions,

u(t) = [u0(t), . . . , us(t)], uj(t) =1h

t1∫

t0

Kj

(t − τ

h

)uj(τ) dτ.

The convolution kernels Kj(t) are subject to the following conditions: Kj(t) = 0 for t < 0,Kj(t) ≥ 0 for t ≥ 0, Kj(t) is j times continuously differentiable, and the normalization condition∫ ∞0

Kj(t) dt = 1 is satisfied.For Kj(t), one can take, for example, the power-law functions

Kj(t) = Cj(t(1 − t))j+1, Cj =(2j + 3)!

((j + 1)!)2.

The following control input for the original system (1) corresponds to the control u(t) :

u(t) =s∑

j=0

u(j)j (t) =

s∑

j=0

h−(j+1)

t1∫

t0

K(j)j

(t − τ

h

)uj(τ) dτ.

This approximation has the following properties.1. uj(t) → uj(t) almost everywhere as h → 0.2. The trajectories x(t) of system (1) for the control u(t) coincide with the trajectories of

system (6) for the control u(t). The latter converge pointwise to the trajectories x(t) of system (6)for the control u(t).

3. u(t) depends only on the values of u(τ) for τ ≤ t, i.e., can be computed on the basis ofinformation known by time t.

5.3. Example

Consider systemx1(t) = x2(t) + v1(t), x2(t) = u(t) + v2(t)

with constraints |v1| ≤ μ1 and |v2| ≤ μ2 for the noise.



Fig. 1. Fig. 2.

For this system, we have

B(t) =

[0 1

1 0

], Mh(t) =

[h/2 1

1 0

].

To use the first scheme for the control, we apply the linear transformation of variables

u1(t) = hu1(t)/2 + u2(t), u2(t) = u1(t)

in the control; then as system (15) we obtain the system

xh1(t) = xh2(t) + u1(t) + v1(t), xh2(t) = u2(t) + v2(t).

For the constraints on the control, one can take the conditions |u1| ≤ ν1 and |u2| ≤ ν2, whereνj ≥ μj.

The use of the second scheme readily produces system (6) of the form

x1(t) = x2(t) + u1(t) + v1(t), x2(t) = u2(t) + v2(t),

where the constraints on the control can be chosen in the form |u1| ≤ ν1 and |u2| ≤ ν2 with νj ≥ μj.Let the control represented in Fig. 1 be realized in system (6). Then Fig. 2 represents the control

inputs for system (1) obtained by the first scheme (a) and the second scheme (b). Here t0 = 0,t1 = 5, and h = 0.5.

ACKNOWLEDGMENTS

The research was supported by the Russian Foundation for Basic Research (project no. 09-01-00589-a), the Federal Task Program “Scientific and Pedagogical Staff of Innovational Russiafor 2009–2013” (contract no. 16.740.11.0426, November 26, 2010), and project no. MK-1111.2011.1.

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fast inputs in the problem of control synthesis under uncertainty

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