STATE ESTIMATION FOR BILINEAR IMPULSIVE CONTROL SYSTEMS UNDER UNCERTAINTIES

The state estimation problem for uncertain impulsive control systems with a special structure is considered. The initial states are taken to be unknown but bounded with given bounds. We assume here that the coefficients of the matrix included in the differential equations are not exactly known, but belong to the given compact set in the corresponding space. We present here algorithms that allow to find the external ellipsoidal estimates of reachable sets for such bilinear impulsive uncertain systems.


Introduction
The paper deals with an impulsive control systems with unknown but bounded uncertainties related to the case of a set-membership description of uncertainty [Kurzhanski and Valyi, 1997;Schweppe, 1973;Walter and Pronzato, 1997;Boyd, El Ghaoui, Feron and Balakrishnan, 1994]. Systems with such uncertainties may be found in many applied areas such as engineering problems in physics and cybernetics [Ceccarelli and etc., 2006], economics, biological and ecological modeling when it occurs that a stochastic nature of errors is questionable. A special type of nonlinear system is considered in the paper. The matrix of the system is uncertain and only the bounds of the admissible values of these matrix coefficients are known. For such bilinear systems the reachable sets are star-shaped sets and, in particular, can be non-convex.
Such systems can simulate various types of systems whose parameters are unknown, but can vary within certain limits, when the stochastic nature of errors is questionable due to limited data or because of the complexity of the model [August, Lu and Koeppl, 2012;Boscain, Chambrion and Sigalotti, 2013;Boussaïd, Caponigro and Chambrion, 2013;Ceccarelli and etc., 2006;Gough, 2008;Nihtila, 2010]. For instance one can indicate mechanical systems in which the stiffness or friction coefficients are given inaccurately. Electrical systems where the resistance, capacitance, inductance, or feedback coefficients are known with a certain accuracy can also be described within the framework of this model.
The paper deals with the guaranteed state estimation problem and uses ellipsoidal calculus [Chernousko, 1994;Kurzhanski and Valyi, 1997] to construct external reachable sets estimates for such systems. Here we develop the set-membership approach based on ellipsoidal calculus for the considered system. Also we generalize earlier results [Filippova and Matviychuk, 2015;Filippova, 2016;Matviychuk, 2017a], in particular we consider more complicated model of the control system than in [Matviychuk, 2017b]. In this paper the control function of studied bilinear impulsive control system is a pair of a classical (measurable) control and an impulsive control function. It is assumed that a classical control should belongs to a given finitedimensional ellipsoid and an impulsive control function is the scalar function of bounded variation. The algorithms of constructing external ellipsoidal estimates for studied systems are given.

Basic Notations
Let R n be the n-dimensional vector space, comp R n be the set of all compact subsets of R n , conv R n be the set of all convex and compact subsets of R n , R n×n stands for the set of all real n × n-matrices and x ′ y = (x, y) = ∑ n i=1 x i y i be the usual inner product of x, y ∈ R n with prime as a transpose, ∥x∥ = (x ′ x) 1/2 . Let I ∈ R n×n be the identity matrix, Tr(A) be the trace of n × n-matrix A (the sum of its diagonal ele- For a set A ⊂ R n we denote its closed convex hull as co A. We denote by B(a, r) = {x ∈ R n : ∥x − a∥ ≤ r} the ball in R n with a center a ∈ R n and a radius r > 0 and by E(a, Q) = {x ∈ R n : (Q −1 (x − a), (x − a)) ≤ 1} the ellipsoid in R n with a center a ∈ R n and a symmetric positive definite n × n-matrix Q. Denote by h(A, B) the Hausdorff distance between sets A, B ∈ R n .

Problem Formulation
Consider the following bilinear impulsive control system here x ∈ R n , vector-function B(·) ∈ R n is continuous on [t 0 , T ]. The initial condition x(t 0 − 0) = x 0 to the system (1) is assumed to be unknown but bounded Let us assume that the control function u(t) in (1) is Lebesgue measurable on [t 0 , T ] and satisfies the constraint whereâ ∈ R n ,Q ∈ R n×n . The impulsive control function v(·) ∈ R n is a scalar function of bounded variation, monotonically increasing and right-continuous for t ∈ [t 0 , T ]. Also it is assumed that for some given µ > 0 we have (1) has the special form where A 0 ∈ R n×n is given and the measurable, be a solution of the system (1)-(5) with initial state x 0 ∈ X 0 , with controls u ∈ U, v ∈ V and with a matrix A(·) ∈ A(·).
The trajectory tube X (·) = X (·; X 0 , A, U, V) of the system (1)-(5) is defined as the following set (see also [Filippova and Matviychuk, 2011]) and the reachable set of the system (1) at the time t is the cross-sections The main problem considered in this paper is to find the external ellipsoidal estimates for reachable sets X (t) of the dynamic control systems (1)-(5) with uncertain matrix of the system and uncertain initial state basing on the special structure of the data A, U, V and X 0 .

Preliminary Results
Consider first some auxiliary results.

Bilinear System
Consider first the following bilinear systeṁ where x ∈ R n , the set A is defined in (5). (9) is defined as the following set Note that the reachable sets X (t) for the bilinear system (9) are star-shaped sets.
The set of all star-shaped compact subsets Z ⊆ R n with center c will be denoted as We will assume further that Assumption 1 is satisfied.
Let ρ(l|C) be the support function of a convex compact set C ∈ conv R n , i.e., We will denote the Minkowski function of a set M ∈ St R n by We need the following notation Then the evolution equation known as the integral funnel equation [Kurzhanski and Filippova, 1993;Kurzhanski and Valyi, 1997] that describes the dynamics of star-shaped trajectory tubes is given in the following theorem.
Theorem 2. [Filippova and Lisin, 2000] The trajectory tube X (t) of the bilinear differential system (9) with constraints (2), (5) is the unique solution to the evolution equation From Theorem 2 we have Taking into account (5), we note that where sets A 1 and A 2 are defined in (6) and (7) respectively.
Consider the auxiliary bilinear systeṁ The external ellipsoidal estimate of set (I + σ(A 0 + A 1 )) * X 0 may be found by applying the following theorem.
Theorem 3. [Chernousko, 1996] Let a * (t) and Q * (t) be the solutions of the following system of nonlinear differential equationṡ Here the maximum is taken over all σ ij = ±1, i, j = 1, . . . , n, such that c ij ̸ = 0 and v is a number of such indices i for which we have: c ij = 0 for all j = 1, . . . , n. Then the following external estimate for the reachable set X (t) of the system (13) is true  Figure 1. Trajectory tube X (t) and its ellipsoidal estimating tube E(a * (t), Q * (t)) for the bilinear system with uncertain initial states.
The following example illustrates the Theorem 3.
The trajectory tube X (t) and its external ellipsoidal estimating tube E(a * (t), Q * (t)) calculated by the Theorem 3 are given in the Figure 1.
The following theorem is hold.
Theorem 4. [Filippova and Lisin, 2000] For every z ∈ R n such that z i ̸ = 0 (i = 1, . . . , n) the following formula is true Remark 1. [Filippova and Lisin, 2000] Let the set A 2 is defined in (7) and X 0 = E(0, Q 0 ), then the following formula is true The external ellipsoidal estimate of set σA 2 * X 0 may be found by applying the following theorem.
Theorem 5. [Matviychuk, 2016] For X 0 = E(a 0 , Q 0 ) and all σ > 0 the following external estimate is true where σ −1 o(σ) → 0 for σ → +0, Here p is the unique positive root of the equation ..., n) being the roots of the following equation Then an external ellipsoidal estimate of the trajectory tube X (t) of the system (9) may be found by applying the following new result.
The following algorithm is based on Theorem 6 and may be used to produce the external ellipsoidal estimates for the reachable sets of the system (9).
• The next step repeats the previous iteration beginning with new initial data. . Trajectory tube X (t) and its ellipsoidal estimating tube E(a + (t), Q + (t)) for the bilinear control system with uncertain initial states.
At the end of the process we will get the external estimate of the tube X (·) of the system (9) with accuracy tending to zero when m → ∞. The following example illustrates the Algorithm 1. Example 2. Consider the following system The trajectory tube X (t) and its external ellipsoidal estimate E(a + (t), Q + (t)) calculated by Algorithm 1 are given in the Figure 2.
The following lemma explains the reason of construction of the auxiliary differential inclusion (20).
The next iterative algorithm is based on Theorem 7 and allows to find the external ellipsoidal estimates of the reachable sets of the studied bilinear impulsive control system (1)-(5).
At this step we find the ellipsoidal estimate for the union of a finite family of ellipsoids [Filippova and Matviychuk, 2011;Matviychuk, 2012]. 4. Find the projection of E ε (w 1 (σ), O 1 (σ)) at the subspace of variables z by Lemma 1: E(a 1 , Q 1 ) = π z E ε (w 1 (σ), O 1 (σ)). 5. Consider the system on the next subsegment [t 1 , t 2 ] with E(a 1 , Q 1 ) as the initial ellipsoid at instant t 1 . 6. The next step repeats the previous iteration beginning with new initial data.
At the end of the process we will get the external estimate E(a + (T ), Q + (T )) of the reachable set X (T ) of the impulsive control system (1)-(5) with uncertain matrix of the system and uncertain initial state basing on the special structure of the data A, U and X 0 .

Conclusion
The problem of state estimation of the reachable sets for uncertain impulsive control systems for which we assume that the initial state is unknown but bounded with given constraints and the matrix in the linear part of state velocities is also unknown but bounded was considered in this paper. The modified state estimation method which uses the special constraints on the controls and uncertainty and allows to construct the external ellipsoidal estimates of reachable sets is presented here. This method is based on results of ellipsoidal calculus developed earlier for some classes of uncertain systems.