DEVELOPMENT OF STABLE ALGORITHMS FOR ADAPTIVE CONTROL OF NONLINEAR DYNAMIC OBJECTS

ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 304 DEVELOPMENT OF STABLE ALGORITHMS FOR ADAPTIVE CONTROL OF NONLINEAR DYNAMIC OBJECTS Akhmedov D.A Quite often, control for nonlinear objects is usually sought using Lyapunov functions in the form of quadratic forms [5]. However, to obtain constructive results, the Lyapunov function, its derivative, or the desired control must reflect the structure of the given object to some extent. Let us consider an analytical procedure for the synthesis of stabilizing controls based on the polynomial approach [5] and some modification of the Lyapunov function. Consider the control object described by the equation [1,3,4]: , (1) where – state vector, , хi – state variables, ; – control, – - function matrix, – vector function. We will assume that there are continuous derivatives (2) Elements matrices A(х ) and bi(x) vector b(х) at . Assuming some state variables хi, observables, we pose the problem of designing a stabilizing control и, for which the zero solution х=0 equation (1) is globally asymptotically stable. Additional restrictions on the properties of the object (1), related to the solvability of the problem, are given below. Based on [5], we write the equations for the control part of the system in the form ,(3) where – state vector, R(х) – -, L(х) – - function matrix , – vector function., . In this case, the matrix R(x) is an accompanying one [6]. Combining the vectors х and z into one state vector of the closed system (1), (3), we have resulted the equation , (4) Where vector , function matrix has the form . (5) Despite the fact that the coefficients of equations (1) and (3) functions of the state variables of the object (1), to determine the dimension r and matrix elements R(х), L(x) and vector l(х) it is possible to apply practically without changes the polynomial method of control synthesis for linear systems, the main relations of which are given in [5]. According to [5], all components of the matrices R(x), ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 305 L(x) and the vector l(x) are found by solving an algebraic system of linear equations. The conditions for the solvability of the indicated system of algebraic equations are determined only by the properties of the object itself (1) and the set of observed variables xi and shown in [5]. The specificity of the synthesis of nonlinear control (3) affects only the fact that the conditions specified in [5] must be satisfied in the entire state space of the object (1), and the solutions of the algebraic system are scalar functions of the state vector х. We emphasize that if the solvability conditions are satisfied, then the eigenvalues of the matrix H(w) system (4) can be assigned arbitrarily. In what follows, we will assume that object (1) satisfies these conditions and control (3) is found such that all eigenvalues of the matrix H(w) in (4) are different and have negative real parts. Moreover, using the relations from [5], one can show that the existence of derivatives (2) guarantees the existence of continuous derivatives. , (6) elements hij(w) of the matrix H(w) for For brevity, nonlinear systems (4) whose matrix H(w) has different eigenvalues with negative real parts will be called Hurwitz. The negativity of the real parts of the eigenvalues of the matrix H(w) in the general case, of course, is not enough for the stability of system (4). Considering the nonlinear Hurwitz system (4), we assume that for all derivatives (6) are defined and the conditions are satisfied (7) Here are the eigenvalues of the matrix H(w) – are the roots of the equation , where E is the identity matrix [6], . In addition, a vector was found , such that the derivatives are defined , (8) and the conditions (9) (10) at . Here Sp(∙) – matrix trace (∙), (∙)* – matrix complex conjugate [6] to the ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 306 matrix (∙), ρ, K, ε - positive numbers, matrix (11) Where (12) Here – are polynomial coefficients We will assume that all measurable [2-4]. Then, according to [5] , the control action has the form (15) The vector from equation (1) is taken as a vector. In this case, the conditions for the existence of derivatives (6) and (8) is replaced by the requirement for the existence of derivatives (2). Inequalities (9) and (10) become the following: (16) (17) Where . These inequalities follow from expressions (9) and (10) when the vectors and the matrix H(w) are replaced by х, b(х) and А(х), respectively. Conditions (10) or (17) are quite strict, but in many cases they are violated only in the region of large deviations, practically outside the dynamic range of changes in the real variables of the object (1). In such cases, one can consider the possibility of an appropriate modification of the object model or use another method, synthesis [1-4]. We indicate one particular case [ 3]. Let the matrix have non-zero elements only in the nth row, i.e. (18) here is the derivative before time due to the system (1) of the functional matrix , defined by the expressions (11), (12) at vector is a zero matrix. In this case, the control u can be found, following [4], by the formula (19) ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 307 Where is the nth row of the matrix ; are the components of the vector ; - are the coefficients of the desired polynomial (20) In practice, system (1), (19) under conditions (16) and (18) turns out to be reducible since a nonsingular transformation, it is reduced to a linear system with a characteristic polynomial (20). 1. Bohlin T. Practical Grey-box Process Identification. Theory and Applications. - London: Springer, 2006. - 351 pp. 2. Heij C., Ran A., van Schagen F. Introduction to Mathematical Systems Theory. Linear Systems, Identification and Control. - Basel: Birkhauser, 2007. - 166 pp. 3. Landau I.D., Lozano R. Unification of discrete time explicit model reference adaptive control designs // Automatica. 1981. V.17. № 4. P.593-611. 4. Math Bolen, Irene Gu. Signal Processing of Power Quality Disturbances. Wiley, 2006 (IEEE Press Series on Power Engineering). - Wiley-IEEE Press, 2006. - 861 рр.