Set-valued solutions to impulsive differential inclusions

Abstract

This paper deals with the state estimation problem for impulsive control systems described by differential inclusions with measures. The problem is studied under uncertainty conditions with set-membership description of uncertain variables which are taken to be unknown but bounded with given bounds. Such problems arise from mathematical models of dynamical and physical systems for which we have an incomplete description of their generalized coordinates (e.g. the model may contain unpredictable errors without their statistical description). In this setting instead of an isolated trajectory of the dynamical control system we have a tube of such trajectories and the phase state vector should be replaced by the set of its possible values. The techniques of constructing the trajectory tubes and their cross-sections that may be considered as set-valued state estimates to differential inclusions with impulses are studied.

Keywords:

• Uncertainty
• control
• differential inclusions
• impulsive system

1. Introduction

In this paper the impulsive control problem for a dynamical systems under uncertainty conditions is studied. In many applications related to control problems the evolution of the dynamic control system depends not only on the current system state but also on uncertain disturbances or errors in modelling. There are many publications devoted to different treatments of uncertain dynamical systems, e.g. [1 –  5].

The model of uncertainty considered here is deterministic, with a set-membership description of uncertain items which are taken to be unknown but bounded with given bounds. We consider a dynamic control system described by a differential equation with measure [1,7 –  13]

with unknown but bounded initial condition

Here u(t) is the usual (measurable) control with constraint

and v(t) is an impulsive control function which is continuous from the right, with

It is well-known that this control system can be modelled by a differential inclusion

with unknown but bounded initial condition

and with certain control variables represented by vector measures dv(t) (generalized or impulsive controls). In such problems the trajectories x(t) are discontinuous and belong to a space of functions with bounded variation. Among many problems related to the treatment of dynamical systems of this type let us mention the results devoted to the precise definition of a solution to (3) [13] and publications on optimal control problems [10,12 –  14].

In the estimation problems the so-called measurement equation is also considered

with ξ(t) being the unknown but bounded ‘noise’ or disturbance. The latter equation may be expressed as the state (‘viability’ [2]) constraint:

where G is a given set-valued map.

One of the principal points of interest in the theory of control under uncertainty conditions is to study the set of all solutions x(t) to (3) –  (6). The ‘guaranteed’ estimation problem consists in describing the set X|t| = ∪{x|t|} that is actually the reachable set (the information domain) of the system at instant t. The set X(t) may be treated as the unimprovable set-valued estimate of the unknown state x(t) of the system (3 –  6).

The mathematical background for investigations of set-valued estimates X(t) of the states of ordinary differential inclusions (without impulsive components) may be found in [16,17]. In this paper we discuss the set-membership approach to the description of the information states for a nonlinear system with impulsive disturbances.

2. The estimation problem

In this section we apply the set-membership (bounding) approach to the estimation of unknown states for a system of type (1), (2) but in an autonomous case and without the restriction (6). Consider a dynamic control system

with initial condition

Following the idea of [1] the information sets are treated here as level sets of the generalized solutions V(t,x) to the HJB (Hamilton – Jacobi – Bellman) equation, where V(t,x) is the value function of type

with φ being a given function (e.g. φ(t ₀,x) = d ²(x,X ⁰) with X ⁰ defined in (8) where d(x,M) is the distance function from x to M ⊂ R ⁿ ).

We assume that the Lipschitz condition

is true and

with some constants L, K > 0. Assume also that the sets

are convex.

Let us introduce a control system of type

with state variables x,v ₁ and control functions u(t), w(t).

Definition [10]. A function x(·) with bounded variation and continuous from the right is called a generalized trajectory to (7) – (8) if there exist a function v ₁(·)also continuous from the right, with bounded variation, and a sequence of controls ( u _n(·), w _n (·)) for the system (10) – (11) such that the sequence of respective solutions (x_n (t), v _{1 n} (t)) of (10) – (11) tends to {x(t),v ₁(t)] at every point t of continuity of {x(·), v ₁(·)}.

The set of all such pairs {x(·), v ₁(·)} is a weak *-closure of the set of classical solutions to (10) – (11).

For all s ∈ [0,T + μ], y ∈ R ⁿ , z,η ∈ R ¹ let us introduce the value function

where the minimum is taken over all solutions

to the auxiliary control system ([10]):

with terminal conditions

and with ordinary (measurable) control functions α, n, e such that

The proof of the next theorem follows from the results of [13].

Theorem 1. The cross-section

of the trajectory tube

to the system (7) – (8) is a subset of the following set

where π _yM denotes the projection

Remark 1. It should be mentioned here that the value function

in the optimization problem (12) can be found through the techniques of viscosity ([18,19]) or minimax ([20]) solutions of the corresponding HJB equation

with boundary condition

Theorem 1 gives us the possibility of producing other upper estimates for the information sets X(t) through the comparison principle that allows us to connect the given approach to the techniques of ellipsoidal or box-valued calculus developed for systems with linear structure ([5,21]).

Consider the variational inequality

with boundary condition

Theorem 2. If there exists a continuously differentiable function

such that the inequalities (18) – (19) are satisfied then the inequality

is valid.

The proof of this theorem is based on the verification function techniques applied to the HJB equation (16) [5,18,19].

Theorem 2 produces many estimates for the value function

if a series of different functions ω is taken.

The following result is a direct consequence of Theorems 1 and 2.

Theorem 3. The cross-section X[T] of the trajectory tube X(·) to the system (1) – (2) is a subset of the projection of the level set taken for the value function

:

for any function

that satisfies (18) – (19).

3. The viability and the estimation problems under state constraints

In this section we consider the control system of type (3) – (4)

with state constraints

where x ∈ R ⁿ , B(t) is a continuous matrix function, u(t) is a control function with bounded variation, F, and G(t,x) are continuous multivalued maps

that satisfy the Lipschitz condition with constant L > 0, namely

and also the condition (for all t,x and some constant c > 0)

where S = {x ∈ R ⁿ ||x|| ⩽ 1}.

Definition 2. A function x[t] = x(t,t ₀,x ₀) will be called a solution to (21) if

where a function

is a selector of F

The last integral in (23) is taken as the Riemann – Stieltjes integral.

Following the scheme of the proof of the well-known Caratheodory theorem we can prove the existence of the solution x[t] = x(t,t ₀,x ₀) for all x ₀ ∈ X ₀ ∈ compR_n .

Let P be a convex closed cone in R ^m with a vertex at 0 ∈ R ^m . Denote

Assume that there exists at least one solution

( together with a starting point

and u ∈ U) that satisfies the condition (22).

Let X(·,t ₀,X ₀) be the set of all solutions to the inclusion (21) that emerge from X ₀ (the ‘solution bundle’) with some u∈U. Let

be its cross-section at instant t ₁. It is not difficult to observe that X[t ₁] is actually the attainability domain (or the ‘reachable set’) at instant t ₁ for the differential inclusion (21) with state constraint (22) constructed over all admissible x ₀ and u(·).

Denote the restriction F_G (t,x) of the map F(t,x) to the map G by

Lemma 1 [17]. A function x(t) defined on the interval [t ₀,t ₁] with x ₀ ∈ X ₀ is a solution to (21), (22), (24) if and only if there exists u ∈ U such that x(t) is a solution to

We represent F_G as the intersection of some set-valued functions based on the following auxiliary assertion.

Lemma 2 [22]. Suppose A is a bounded set, B a convex closed set, both in R ⁿ . Then

From the lemmas we obtain the following theorem.

Theorem 4. A function x(·) defined on an interval [t ₀,t ₁] with x(t ₀) ∈ X ₀ is a solution to (21), (22), (24) iff the inclusion

is true for some u ∈ U and all t ∈ [t ₀,t ₁].

We introduce a set of differential inclusions that depend on the matrix function L(t,x). These are given by

Denote by z[·] = z(·,t ₀,z ₀,L) the solution to (25) defined on the interval [t ₀,t ₁] with z[t ₀] = z ₀ ∈ X ₀ and with a function u ∈ U. Also denote

where Z(·,t ₀,z ₀,L) is the bundle of all the trajectories z[·] issued at time t ₀ from point z ₀ with all admissible u(t) and defined on [t ₀,t ₁]. The cross-sections of the set Z(·,t ₀,X ₀,L) at time t ₁ are then denoted as Z(t ₁,t ₀,X ₀,L).

Following the schemes of the proofs of related results in [22,23,17] devoted to the uncertain problems for differential systems with usual control functions we obtain the following characterization of the trajectory tubes.

Theorem 5. The following equality is true

Corollary. The following inclusion is true

Remark 2. For a linear differential impulsive system the relation (27) is actually an equality (the proof of this fact may be seen by using a similar scheme as in [17]).

4. Impulsive systems with ellipsoidal constraints

Let us consider the linear control system

with impulsive control u(·) restricted by a set U that will be defined later; X ₀ is convex and compact in R ⁿ .

Let

be an ellipsoid in R ^m where M is a symmetric positive definite matrix.

Denote

and let us take U = E* where E* is the conjugate ellipsoid to E.

We assume in this section that the admissible controls u satisfy the restriction

In particular it follows from (29) that the jumps Δu(t_i ) = u(t_{i  + 1}) – u(t_i ) of the admissible controls have to belong to an ellipsoid

The following theorem concerns the structure of the cross-section of the trajectory tube and generalizes the results of [1].

Theorem 6. The reachable set X(t,t ₀,X ₀) is convex and compact for all t ∈ [0,T]. Every state vector x ∈ X(t,t ₀,X ₀) may be generated by a solution x(·) (i.e. x(t) = x) to (28) with the piecewise constant control u(·) whose (n + 1) jumps belong to the set

.

5. Properties of set-valued states

Based on the techniques of approximation of the discontinuous generalized trajectory tubes to (1) – (2) by the solutions of the usual differential systems without measure terms [7,16] it is possible to study the dependence of generalized trajectory tubes and their cross-sections (reachable sets) on parameters that define the restrictions on uncertain values (initial data, a variation of impulses, constraints on measurable controls).

Let us mention here that by using this method for the problem without state constraints studied in Section 1 we can prove the parameter continuity of the solution tubes under not very restrictive assumptions on the problem data. But it is not difficult to observe that if the state constraints are assumed to be involved in the problem then the reachable sets for impulsive differential systems may become semicontinuous with respect to Hausdorff metrics.

6. Conclusions

The set-valued estimates for the tubes of solutions of a differential inclusion with impulsive components are given. The techniques of constructing the trajectory tubes and their cross-sections that may be considered as set-valued current ‘state vectors’ (the information states) to impulsive differential inclusions under uncertainty are studied.

We discuss in this paper the set-membership approach to the description of s for a nonlinear differential system with impulsive disturbances or controls. The schemes developed here may be connected to the techniques of set-valued estimating by ellipsoids or polytopes for linear control systems and to the techniques of level sets for the generalized (viscosity) solutions of the Hamilton – Jacobi – Bellman equation.

Acknowledgements

The research was supported by the Russian Foundation for Basic Research (RFBR) under project No. 03-01-00528, by the grant ‘Russian Scientific Schools’, No. 1889.2003.1, and by the Russian Academy of Sciences under project ‘The Program for Basic Researches, No. 19’.