Synthesis of Control and State Observer for Weakly Nonlinear Systems Based on the Pseudo-Linearization Technique

In this paper an approach for the construction of nonlinear output tracking control on a finite time interval for a class of weakly nonlinear systems with state-dependent coefficients is considered. The proposed method of control synthesis consists of two main stages. On the first stage, a nonlinear state feedback regulator is constructed using a previously proposed control algorithm based on the State-Dependent Riccati Equation (SDRE). On the second stage, the problem of full-order observer construction is formulated and then is reduced to the differential game problem. The form of its solution is obtained with the help of the guaranteed (minimax) control principle, which allows to find the best observer coefficients with respect to a given functional considering the worst-case uncertainty realization. The form of the obtained equations made it possible to use the algorithm from the first stage to determine the observer matrix. The proposed approach is characterized by the nonapplicability of the estimation and control separation principle used for linear systems, since the matrix of observer coefficients turned out to be dependent on the feedback coefficients matrix. The use of numerical-analytical procedures for determination of observer and feedback coefficients matrices significantly reduces the computational complexity of the control algorithm.


INTRODUCTION
Currently, many approaches are used to solve nonlinear control problems. One promising approach is a technique based on solving matrix State-Dependent Riccati Equations (SDREs) (see reviews [1,2]). The essence of this approach is to represent the original system in a pseudo-linear form, in which the system and the control matrices depend on the state. The control law has a standard form for a linear-quadratic problem, however, for its implementation during the control process, it is necessary to numerically find a solution to the corresponding SDRE at each time point, which may run into computational limitations. In [3], an SDRE-based approach was proposed for solving the finite-horizon optimal tracking problem for weakly nonlinear systems with fully measurable state vector. Since finding the exact solution in the general case seems to be a computationally time-consuming task, a numerical-analytical algorithm for approximate synthesis has been proposed, which significantly reduces the computational complexity compared to the standard SDRE technique. However, to apply the approach from [3], it is necessary to construct an appropriate observer, which allows one to obtain estimates of the unknown states of the system. In this paper the underlined problem is reduced to the problem of control synthesis under uncertainty, for the solution of which the minimax principle from [4][5][6] is used.
where x, y and u are the state, output, and control vectors respectively, is some given sufficiently small positive constant, A 0 , B 0 and С are known constant matrices, , are known matrices with elements that are sufficiently smooth and bounded by argument x, X and Y are some bounded sets. It's assumed that for any continuous control u(t) the trajectories of the closed loop system (1) exist, are unique and belong to Х on . Let the reference behavior of system (1) be described by solving a differential equation where x r and y r are the desired (reference) trajectory of the system and its output, is the known constant matrix, and (x r ) is the known matrix with elements that are sufficiently smooth and bounded by the argument x r . The initial states x 0 and are generally assumed to be unknown. If is known, then the entire reference trajectory is actually known in advance. For example, the use of a special master device, the initial state of which can be selected, can correspond to this case. Let us define the following cost function (2) where , , , are given symmetric matrices for , . Hereinafter, the signs 0 ( 0) denote the positive definiteness (semi-definiteness) of the corresponding matrix. It is necessary to find such a continuous output control u(y, , t) that provides an approximate solution to problem (1), (2).
It is known that the original problem (1), (2) can be represented [7] in the form where , are extended state and output vectors, zero blocks in the matrices and are the zero matrices of the corresponding dimensions. Let's define the following equations = ,μ , = , = , ,μ = + , ∈ , ∈ , ∈ , , 0 0 Let's define also the following conditions (i) The trajectories of the closed loop system (1) exist, are unique and belong to Х on for any continuous control u(t), where X is some bounded set of state space, elements of are bounded, continuous, and sufficiently smooth for ; .
(ii) The triple , where , is stabilizable and observable.
(iii) The system matrices the cost function symmetric matrices , , , , and are such that for In [3] under conditions (i)-(iii) for a fully measurable system (the vector x is exactly measured) the following control was proposed that approximately solves problem (3)   (4) where , is the linear part and forms a nonlinear correction. According to [3] under conditions (i)-(iii), we have the following numerical-analytical algorithm for constructing approximate control in the finite-horizon optimal nonlinear tracking problem.
(1) Find as , where is a positive definite solution of the next matrix Riccati equation .
(2) Find as (5) where , and the matrix M P is calculated by means of expression (3) Finally, define the control (4). Let's us make a few comments. Since the future trajectory of the system is unknown, in the proposed algorithm the matrix is calculated under the assumption that and slightly differ near t , i.e. is used instead of . If the reference trajectory is known in advance, then, assuming that x(t 1 ) is close to x r (t 1 ), we can calculate , using x(t 1 ) = x r (t 1 ). The "rigidity" of these two assumptions is mitigated by the fact that the first term in (5) is significant only in the neighborhood of t 1 due to . Note that the algorithm offers an analytical formula for an unknown state-dependent matrix , which significantly reduces the computational complexity of the control algorithm. In addition, the following is true Thus, since for , then the condition (iii) can always be fulfilled for sufficiently small . In addition, the condition (iii) can be met, if for The proposed algorithm can be applied to the output control problem by constructing an appropriate observer that determines estimates of the unknown state coordinates at each time moment.  Let's pass form this system for error dynamics to another one, which has the same eigenvalues for each , , . The following new systems is equivalent to the original one in terms of stability properties in the linear approximation Thus, we get the system (7) where , and , are new control and unknown disturbance accordingly.
Let's consider uncertainty as the control of the opposing player. Such an interpretation leads to a differential game. To solve it, we apply the principle of guaranteed control [4][5][6], which ensures defining the optimal control in the worst implementation from the point of view of the next cost function (8) where , , , , are given weight matrices. Let's define the conditions (iv) A pair is controllable for all , , .
= χ, μ, , = − Γ + μ χ, μ, + = + μ χ, μ, − Γ + μ . (v) By analogy with [6], where the control interval is assumed to be sufficiently large compared to the transition process, when (iv), (v) are fulfilled, the control and disturbance laws are defined as , , where is a positive definite solution of the next SDRE (9) From the obtained relations it follows that (10) The main difficulty here is the solution of differential SDRE (9). However, it can be noted that problem (3) formulated above and problem (7), (8) have a similar structure. Therefore, for an approximate solution of (9) in the problem of observer construction one can apply the above algorithm from [3], which was used to solve a similar SDRE for the control synthesis. To do this, we introduce the conditions (vi) The trajectories of the closed loop system (7) on exist, are unique and for any continuous v(t), (t), matrix elements of are bounded, continuous, and sufficiently smooth for .
(vii) The triple , where , are stabilizable and observable.
(viii) The system matrices , the cost function matrices , , , , and are such that for , . Thus, in accordance with Section 1, the state control is defined firstly. Then, under conditions (iv)-(viii) the observer is constructed by means of the following numerical-analytical procedure.
(1) Find as a positive definite solution to the next equation (2) Find as where (3) Find by means of (10), where .
(4) Finally, find the observer (6), by setting the initial state in an arbitrary way. Now we can apply the control (4), using estimate vector instead of the unknown state vector . Comment 1. In contrast to linear systems [9], in this case, the principle of separation of control synthesis and observer synthesis problems is not fully satisfied, since the observer gain matrix Г depends on the feedback gain matrix K. Comment 2. If the initial state or is known, then for t t 0 using the observer, one can obtain absolutely accurate estimates of the vector x or x respectively, by setting χ 0 = x 0 or = . If the reference path x is known in advance (which means according to the remark above), then, assuming the proximity of x(t 1 ) to x r (t 1 ) and the proximity of (t 1 ) to (t 1 ), one may calculate , using (t 1 ) = (t 1 ) = (t 1 ).