360 ISSN 1064-2307, Journal of Computer and Systems Sciences International, 2019, Vol. 58, No. 3, pp. 360–373. © Pleiades Publishing, Ltd., 2019. Russian Text © The Author(s), 2019, published in Izvestiya Akademii Nauk, Teoriya i Sistemy Upravleniya, 2019, No. 3, pp. 34–47. Method of Parametric Optimization of Nonlinear Continuous Systems of Joint Estimation and Control L. G. Davtyan a and A. V. Panteleev a, * a Moscow Aviation Institute, Moscow, 125993 Russia *e-mail: avpanteleev@inbox.ru Received November 26, 2018; revised January 25, 2019; accepted January 28, 2019 Abstract—A synthesis problem for the optimal control on average of a bunch of trajectories of a non- linear deterministic dynamic system that originates from the given set of initial states is considered, and the control is performed using the results of estimation of the state vector generated by a nonlinear state observer using the incoming measurements. The control laws for the model of the control plant and for the observer are found simultaneously by solving the parametric optimization problem by one of the metaheuristic algorithms. The proposed algorithm for solving the problem is illustrated by two examples of control of the three-dimensional chaotic nonlinear dynamic systems. DOI: 10.1134/S1064230719030079 INTRODUCTION Applied synthesis problems for nonlinear deterministic systems of joint estimation and control are gen- erally solved by linearization in the neighborhood of a reference trajectory and applying the separation principle for linear systems with a quadratic quality criterion that implies an independent search for the optimal linear controller and asymptotic full- or low-order state observer [1, 2]. Another way to solve the problem is to search for the bounded class of nonlinear control systems such that the separation principle remains valid for it and we can also construct asymptotic observers and the respective stabilizing controls [3–5]. The gain coefficients of the observers are given by either constant functions or functions of time. For the general case of nonlinear systems, the issue of how to find state observers is still open. The struc- ture of a nonlinear observer with the matrix of gain coefficients that depend on both time and generated estimates of the state vector is proposed. As a rule, synthesis problems for the optimal control and for the observer are solved independently. It seems logical to combine the solving process for these problems within a single optimization and simulation procedure without linearization of a nonlinear system. The control laws for the plant and the observation process are sought in the form of functions of the same vari- ables, i.e., time and estimates of the coordinates of the state vector. We consider the behavior of a nonlinear continuous deterministic model of the control plant; its behavior is described by a system of ordinary differential equations. The control is subject to parallelepiped constraints. The initial conditions are given by the compact set of the positive measure, i.e., there is a pri- ori information that characterizes the set of possible initial states. We propose to use the information received from the nonlinear model of the measuring system and the nonlinear observer to restore the full state vector. We assume that the plant’s control uses the information received at the output of the observer (the estimate of the state vector) with the available a priori information and the output of the measuring system taken into account. To minimize the estimation errors, we choose the control of the observer that depends on time and estimates obtained. The control quality is estimated by the value of the Bolza func- tional for the individual trajectory and by its average value over the set of initial states for the bunch of tra- jectories. We state the problem to find the controls for the plant and the observer that minimize the control quality functional of the bunch. Various problems for searching for the optimal programmed control and feedback control for bunches of trajectories were considered in [6–8], including the ones using various metaheuristic global optimiza- tion algorithms in [9, 10], where the approach to find the control that depends on a measurable part of coordinates of the state vector is proposed. In this work, we propose to use incomplete information com- ing from the model of the measuring system to restore the values of all coordinates of the state vector (obtain their estimates) and construct the control that depends on these estimates. Metaheuristic algo- OPTIMAL CONTROL