CONSTRUCTION OF MINIMAX CONTROL FOR ALMOST CONSERVATIVE CONTROLLED DYNAMIC SYSTEMS WITH THE LIMITED PERTURBATIONS

The problem is considered for constructing a minimax control for a linear stationary controlled dynamical almost conservative system (a conservative system with a weakly perturbed coefficient matrix) on which an unknown perturbation with bounded energy acts. To find the solution of the Riccati equation, an approach is proposed according to which the matrix-solution is represented as a series expansion in a small parameter and the unknown components of this matrix are determined from an infinite system of matrix equations. A necessary condition for the existence of a solution of the Riccati equation is formulated, as well as theorems on additive operations on definite parametric matrices. A condition is derived for estimating the parameter appearing in the Riccati equation. An example of a solution of the minimax control problem for a gyroscopic system is given. The system of differential equations, which describes the motion of a rotor rotating at a constant angular velocity, is chosen as the basis.


Introduction
In solving practical problems of mechanics, gyroscopy and navigation, models of almost conservative systems, in particular controlled ones, are traditionally used, which characteristics can be substantially improved by means of optimal control methods. Using the specificity of the coefficient matrix in the equations of almost conservative systems, namely, the presence of a large non-degenerate skew-symmetric matrix and a small parameter in the perturbation matrix, the process of solving optimal control problems can be greatly simplified.

Computer Sciences and Mathematics
Here S -a positive definite matrix-solution of the matrix Riccati equation of the following form the matrix P is alternating. Thus, in order to find the minimax control, it is required to perform an estimation of the parameter γ and to indicate a method for finding the matrix-solution P of the matrix Riccati equation.

Materials and methods of research
To find the matrix solution P of equation (7), the approach proposed for solving the problem of optimal control of almost conservative systems is applied [6][7][8]. The matrix-solution is represented as a series expansion in powers of a small parameter. As a result of equating the coefficients for the same powers, an infinite system of matrix equations is obtained. A consistent solution of a certain number of equations of this system gives the desired approximation of the solution.
To estimate the parameter γ and to fulfill the condition of non-negative definiteness of the matrix included in the Riccati equation, a number of theorems are formulated, the proofs of which are based on the fundamental concepts of matrix theory [9][10][11]. In particular, the dependencies between the ranks of matrices and the dimensions of their zero spaces are used, as well as the properties of the ranks of matrices under transpose. Some properties of the eigenvalues of the matrix are used, in particular, the property of the continuous dependence of the eigenvalues of the matrix on its elements.

Results of the investigation of the matrix Riccati equation of a special form 1. A necessary condition for the existence of a solution of equation
Let's write equation (7) in a form convenient for further investigation ( ) Based on the minimax control problem formulation, it is evident that the non-zero elements of the matrix Ψ can be placed only in those rows in which there are non-zero elements of the matrix B. It should be a necessary condition for the existence of solutions of the equation (8), namely, It is known [9] that T rang(AA ) rang(A) = . Then condition (9) can be rewritten in the form TT rang(BB ) rang( ). ≥ ΨΨ (10) In this case, the matrix must be non-negative definite.

2. Additive operations on definite parametric matrices.
where min δ -the minimum among the positive eigenvalues of matrix pencil AB −δ .
Proof. Let's carry out the proof for non-negative definite matrices A, B. In the case of their non-positive definiteness, it can consider a non-negative definite matrix of the form ( ) Necessity. If the first condition (12) is not satisfied, i. e. rang(A) rang[A, B] < , then the zero-space of the matrix A will be of greater dimension than the zero-space of the matrix [A, B]. And this means that there is a vector x0 ≠ for which the following is true: Thus, it follows from (13) that if the first condition (12) is not satisfied, the matrix AB −δ is not a non-negative definite matrix for an arbitrary 0 δ> . Let's suppose that equality rang(A) rang[A, B] = holds. Then for an arbitrary vector x of zero-space of the matrix A the following is true: TT xA x xB x 0 == . Let's consider a matrix pencil AB µ− having, for a parameter 1 δ= µ, the range of values that is the same as a pencil AB −δ .

Computer Sciences and Mathematics
Let's propose one more criterion of identical determinacy of matrices A and AB −δ , less convenient for practice, and based on the theorem of simultaneous (by one non-degenerate transformation) reduction to diagonal form of two definite matrices [12].

Finding a solution of the Riccati equation of a special kind for almost conservative systems
To solve equation (8), let's apply the approach presented in [6,7], proposed for solving the problem of optimal control of almost conservative systems with a small parameter. Let's find a matrix-solution in the form of an expansion in the small parameter Substituting (29) and (30) in (8), and equating the coefficients for the same powers ε, let's obtain an infinite system of algebraic equations of Riccati type: To find the desired approximation of the matrix solution P, it is necessary to solve successively the corresponding number of equations of the given system.

An example of solving the minimax control problem for a gyroscopic system
Let's consider a system of differential equations describing the motion of a rotor rotating at a constant angular velocity [13]  +α + +αω =ε +εψ After applying the transformation T diag{k,1, k,1} = and taking into account the replacement 1 xx = , 2 xx = ɺ, 3 xy = , 4 xy = ɺ , the model (33) will fully correspond to the form (1)

Conclusions
The minimax control problem for almost conservative systems with a small parameter is investigated.
A necessary condition for the existence of a solution of the Riccati equation of a special form is formulated.
Theorems on additive operations on definite matrices are formulated.
A condition is obtained for estimating the parameter appearing in the Riccati equation.
A possible approach is proposed for finding a solution of the Riccati equation of a special form for almost conservative systems. An example of the application of the proposed algorithms to the model of a rotor rotating at a constant angular velocity is given.
In the applied plan, the studies presented in this article are effective for the development of gyroscopic and navigation systems that are stable to perturbations.