Qualitative Exponential Stability and Instability of Dynamical Systems and Range Estimation of Parameter Acceptable Changes

The main purpose of the research is the extending of the concept of qualitative exponential stability and instability for a wider class of the dynamical systems and objects as well as developing of analytic and calculating technologies for analyzing the quality of processes and projecting of control devices for control systems. And if the property of asymptotic stability indicates the convergence or divergence of the processes in time, the exponential stability provides information about the speed of convergence or divergence processes, thereby characterizing the rapidness of the system. Meeting the conditions of quality exponential stability evaluates the average rate of convergence or divergence of processes, as well as ongoing processes of deviations of the time-average behavior, the last gives information about the behavior of transients (oscillation, overshoot). Development of analytical and computational techniques for the analysis of stability and instability of comparison systems and, as a result of multiply-connected systems, as well as the processes quality is almost an essential task for the study.


Introduction
Modern hardware of computers and computer technology analysis of the behavior of multiply dynamic systems provides efficient algorithms for comparison of systems based on the Lyapunov functions, which are the result of the synthesis of multiply-connected systems. The modular Lyapunov functions greatly simplify the procedures for the investigation of such system's behavior. The terms of exponential and qualitative exponential stability and instability, obtained on the basis of modular Lyapunov functions, can judge the behavior of processes and their quality of multiply-connected systems., Algorithms and software, developed on the basis of the results, obtained in [1,9], allow to solve the problems of analysis and synthesis of continuous and discrete multiply-connected systems with linear objects. The ideology worked out permit to consider qualitative exponential stability and instability for systems and objects with continuous and discrete time similarly, and determine the sufficient local conditions to ensure these types of stability.

Task Statement
Consider a discrete system, the movement of which is specified by the difference equation (1) Where m -the integer number of discrete intervals, x -n-dimensional state vector, F (q (m, x (m))) -a square matrix of size n x n, the elements of which depend on the changing values of the vector q (m, x (m)), the component values which in turn depend on the number of discrete intervals and values state vector, and q (m, x (m)) -one-dimensional vector-valued function, continuous in each variable.
It is assumed that for arbitrary values of m and arbitrary values of the state vector x (m) ∈ R n vector values varying parameters are limited in the parameter space R l by some simply connected closed domain Dq, ie q(m,x(m))∈Dq при ∀m, ∀x(m)∈R n .
We define the quality estimation processes in the system (1) with the inequalities where ρ ≥ 1, the degree of attenuation 0 < λ < 1, β = λ − r, Qualitative Exponential Stability and Instability of Dynamical Systems and Range Estimation of Parameter Acceptable Changes where 0 < r ≤ λ . Note that if the inequality (2) is valid for (1) then the system is exponentially stable, and this inequality allows to measure system performanceevaluation of the transient time [6,7] over the set of trajectories with initial values of the state vector x (0) for which the value of the norm is constant.
The system is exponentially unstable if parameter λ>1in the inequality (2), while if more, and 1<β+ r<1 in the inequality (3), then the system is qualitatively exponentially unstable. The inequality (3) for the value r = λ (β = 0) gives (2). The inequality (3) for (1) gives a local estimate of the behavior of processes in the system, namely the estimate of the norm deviation from exponential decaying process βmx(0) (|β| < 1), which provides a stabilization system's evaluation of the first release and overshoot over the set of trajectories (||x(0)|| = d, где d > 0), where d > 0). Figure 1 shows a graphical interpretation of (2) and indicate the graphical definition estimates of the transient time tn, estimates of the values of first release σ0 and of deregulation σ, characterizing the behavior of processes in time. In order to estimate the values of tn, σ0 and σ, by (3) we can obtain analytical expressions, depending on the values of λ, β, ρ. For (1) the following the task is set: to find within parameter space Rl such a range of permissible changes of the parameters Dq, that for an arbitrary time changing settings, limited by the values of parameters within this range, quality estimations of (2) and (3) types are guaranteed for the system (1).

Sufficient Conditions for Meeting the Required Quality Estimations
Let's state sufficient conditions for meeting the required quality estimations (2) and (3). In order to satisfy the evaluations of quality processes (2) and (3) (4) is associated with the use of beam properties of quadratic forms [3,7,8]. Consider the characteristic equation (5) and assume that at a fixed value of q the maximum root µ + (q) is found. Then the inequality (6) gives the validity of (4). Note that the maximum root of the characteristic equation (5) coincides with the maximum eigenvalue of the matrix . Thus, the calculation of the maximal root µ + (q), depending on the values of the parameter vector q with the following checking of the inequality (6), can form a base for establishing the boundaries of acceptable change settings. However, this method of determining the boundaries of Dq(β,r) is effective only when q is a scalar. If q is a vector quantity, then to simplify the calculations, we find further ellipsoidal estimation of the parameters acceptable change range.
We formulate such a condition for a continuous system, the movement of which is set by (7) where all variables and matrixes have the same meaning as in equation (1), moreover it is assumed that the matrix is such that the solution of (7) for any initial conditions exists and is unique. In a continuous system (7) the quality estimation like (2), (3) are defined like (8) , if α > 0, then the system is exponentially stable, if α < 0 and β = α + r > 0, ( r > 0), then the system is qualitatively exponentially stable.
( ) ( ) To make the system (7) with the values of the parameter q (t, x (t)) of the limited bounded domain Dq meet the quality estimations (8) and (9)  Checking of (10) is to find the maximum eigenvalue µ + (q) of the matrix and verification of (6) for all q∈Dq. Thus, to verify the conditions of the provisions for discrete and continuous systems we use the same software. Note that the results of works [3,6,9] say that the condition (4) guarantees the position of all the eigenvalues λ i (q) (i = 1, 2, … , n) of the matrix F (q) for all values of q∈Dq within a circle with radius r centered at (β, j 0 ) of the complex plane, and condition (10) -within a circle of radius r centered at (-β, j 0 ) of the complex plane. In other words, changing q∈Dq the trajectory of discrete or continuous system roots are limited by above said areas of the complex plane.

Construction of the Ellipsoidal Estimates
We assume that the matrix F (q) of systems (1) and (7) can be represented as (11) where F0 -square matrix of size n×n with constant elements, Bi -matrix size n×1, qi -n-dimensional vectors, which are functions of time and of current value of the state vector . It is thought out that the set Bi, i = 1, 2, ..., k (1 ≤ k ≤ n) form a linearly independent system of vectors. Introduce Lyapunov function (12) given by the quadratic form with a positive definite symmetric matrix P of n×n size. First, consider the case when the matrix of a closed system is presented in the form (11) with k = 1. Let the matrix F0 be such that the solution of the Lyapunov equation Let the matrix P 0 >0 is a solution of the Lyapunov equation (13), and the system B i , e i = 1, 2, …, k (1 ≤ k ≤ n) forms a system of linearly independent mutually orthogonal vectors in the sense of (20) Then, in order to allow for the system (1) to meet the quality estimations (2), (3) with the parameters β and r ≥ r 0 ,, it is enough that for each m and x (m) values of the vectors qi belong to areas Dq i , limited by the surface of the ellipsoid where Di are positively defined symmetric matrixes, such as For continuous systems with the equation of motion (7) with respect to the quality estimates (8) and (9) we have the same assertion of Theorem 3 and the corollary to it, if in (13) ÷ (16) and (21) ÷ (24) to replace matrix with ( ) ( )  (8) and (9). Note that the values qN (16) (qNi (24)), determine the minimum value of the function (25) i.e. when on the trajectories of the system (1), if the matrix F (q) has the representation (11). In other words, the values qNi ∈ Dqi ask such vectors' changing values of parameters for which there is a minimum sensitivity on the trajectories of the system. Note also that in the received results it is assumed that the functions qi(m,x) are of such type that for all values of x ∈ Rn values of the vector of varying parameters belong to Dqi(qi(m,)) ∈ Dqi area. If these conditions are not valid and qi are functions only of the state vector (qi(m,x)= qi(x)), then equation (21) give the surfaces in the state space, limiting the area of Dix, in which for x ∈ Dix run restrictions for changing parameters qi(x). Let these areas are simply connected and include the origin of coordinates. Then, for any initial condition x (0) from the state space bounded by a surface where for x ∈ Dx trajectories of the system (1) or (7) we have the appropriate evaluation of quality (2), (3) or (8), (9).
The developed theoretical concepts come from the need to solve practical problems of analysis and motion control of moving objects, especially in the automation of the most complicated modes of aircraft landing on mobile and immobile base, spatial tracking system during capturing and auto-tracking mode, the trajectory movements in robotic complexes, the management of chemical and other types of processes subjected to increased risk of accidents.