Robust estimation of state vector coordinates in the controlled helicopter motion problem

. The problem of finding a H   observer of the state vector of a linear continuous non-stationary dynamical system with a semi-infinite time of functioning is considered. It is assumed that a mathematical model of a closed-loop linear continuous deterministic dynamical system with an optimal linear regulator, found as a result of minimization of the quadratic quality criterion, is known. For solving the state observer synthesis problem the reduction of the problem to a min-max optimal control problem is used. In this problem, the minimum of the quality criterion is sought by the observer's gain matrix, and the maximum – by the external influence, measurement noise, and initial conditions. To solve this problem, the extension principle is applied and sufficient optimality conditions are obtained that requires the choice of auxiliary functions of the Krotov–Bellman type. As a result of the implementation of the procedure for choosing an auxiliary function and using the rules of matrix differentiation, relations for the synthesis of the observer and formulas for finding the best matrix of observer gains, as well as the laws for choosing the worst external influences and noise, were obtained. We find a solution to the problem of state vector coordinates estimation in the presence of limited external influences and disturbances in a linear model of the measuring system. As an example, the equations of motion of the Raptor-type helicopter are used.


Introduction
The task of designing control systems for aircraft of various types under limited initial conditions, external influences on the system, and measurement errors arises in many applied problems. Due to computational difficulties, when finding the feedback control laws for nonlinear dynamical systems, the procedure of linearization of the mathematical model of motion in the vicinity of the reference trajectory is usually performed. Then the classical problem of designing an optimal linear controller is posed and solved, which ensures that the deviations from the reference trajectory tend to zero and the asymptotic stability of the closed system. The synthesized system must function in the presence of external influences and deviations of the initial state from the reference one. Information about the behaviour of the coordinates of the state vector of the resulting closed-loop system can be obtained using a measuring system that operates under conditions of noise and measurement errors. As a result, there is a problem of operational estimation of the coordinates of the state vector from the accumulated information about the measurements. If accurate statistical information about external influences and interference of the measuring system is not available, the concept of their limitations is used. The most convenient mathematical tool for solving such problems is the H  -control and estimation approach [1][2][3][4][5]. Within the framework of this approach, the apparatus of frequency characteristics [6][7][8], the concept of describing dynamical systems in the state space [9][10][11][12] , the technique of linear matrix inequalities [1,6,[9][10][11] can be used. One of the possible ways to solve the problem is the transition to the min-max problem statement, in which the best control of the object or state observer is sought under the worst external factors [9,10,13]. In this case, various necessary and sufficient conditions [9][10][11]14] are used to find both open-loop control and control with full feedback. In this article, to solve the problem of estimating the coordinates of the state vector [15][16][17], the sufficient conditions obtained in [18,19] are used. The algorithm developed on their basis is used in the applied problem of controlling the Raptor helicopter [20].

Problem statement
Let the plant equation describing the behaviour of linear control system have the form where n x R  is a state vector, is a control vector, The cost functional is quadratic: where is a non-negative definite symmetric matrix of sizes ( S n n  ), and is a positive definite symmetric matrix ( Q  q q  ). In the problem of finding an optimal full feedback control ( ) x  u , it is required to find a control law providing minimization of the functional (2) value for any initial conditions Solving the algebraic Riccati equation it is possible to find the matrix and the explicit form of the optimal full feedback control in the form of the optimal linear regulator Then for any initial conditions the control (4) provides the corresponding control signal .
The equation of the closed-loop system of "control object +optimal controller" type has the form Mathematical model (1) corresponds to the case when it is assumed that there are no external influences. Let us further consider a more general problem, which takes into account the influence of limited external disturbances and initial conditions on the behaviour of the object model, as well as the presence of incomplete information about the state vector coming from the model of the measuring system in the presence of limited disturbances and measurement errors.
Consider the mathematical model of the control object and the model of measuring system  , respectively, are given.
It is assumed that а) , is a non-singular matrix (there are measurement errors in each output equation).

( ) D t
The task is to find the estimation ˆ( ) according to the results of the information obtained from the measuring system, i.e. 0 ),0 } . It is required to minimize the value of the estimation error under the conditions of uncertainty of the information about the vector of initial states 0 x , the laws of variation of disturbance vectors and measurement errors.
Suppose that the structure of the state observer is described by the equation where ˆn x R  is a vector of state vector coordinate estimates, K is an unknown matrix of sizes ( ) n m , 0 x is a vector of initial values of state vector coordinate estimates, set on the basis of available a priori information about possible initial states of the control object model (6). Matrix K performs the function of controlling the observation process.
We obtain an equation describing the change in the estimation error by subtracting equation (8) from equation (6) taking into account (7) and notation ( ) We will use the notation of weighted norm where is a given positive semi-definite symmetric matrix.

Q
It is required to ensure (if it is possible) that the inequality where are symmetric positive definite matrices of corresponding sizes, 0 , , , Q P W V 0   is a given non-negative value. It is desirable to find the minimum value *  , at which the above properties are still valid, which can be achieved by minimizing the value of the numerator of the fraction while maximizing the denominator. In other words, the quality functional has to satisfy the condition which will be performed while minimizing the cost of controlling the estimation process under the worst-case influence of perturbations, initial state, and measurement errors. The multiplier ½ is added to reduce the unwieldiness of the record after the possible application of the differentiation operation.

Synthesis of H  observers
Let us formulate the problem as a game problem, where the first player -matrix is chosen at each from the condition of minimizing the value of the functional, and the second player -composite vector The solution of the problem, obtained using sufficient optimality conditions [18,19] and applying the rules of matrix differentiation to find the following relations, has the form Algorithm 1. Set the matrices included in the mathematical models (1),(2),(6),(7), (9). 2. Solving the algebraic Riccati equation find the matrix and matrix .

Find the matrix , solving the algebraic Riccati equation
Investigate the effect of the parameter  , minimizing its value.
4. Find the control matrix of the observation process Analyze the results of the closed-loop system simulation Matrices defining the functional (2) were selected from the condition of satisfaction of transient processes with the given technical requirements: Matrices defining model (6),(7) and criterion (9) were selected from the condition of ensuring that the norm of the estimation error tends to zero and taking into account the balanced influence of limited initial states, external disturbances and errors of the measuring system: Vector of initial states of the observer was chosen taking into account the availability of a priori information about the possible initial states of the helicopter: For the stability of the control system, the gain matrix of the observer must ensure the negativity of the real parts of the characteristic equation roots corresponding to the differential equation of a closed-loop system. This condition is checked directly or with the help of the Routh-Hurwitz stability criterion.
The   The errors of state vector coordinate estimation tend to zero. The behaviour of the transient process under other limited initial conditions, external influences and measurement errors is similar to that shown in Fig.1.

Conclusions
The problem of synthesis of an observer for estimating the coordinates of the state vector of the helicopter motion model has been solved. It is shown that under limited initial conditions, external influences, and measurement noise, the errors of state vector coordinate estimation tend to zero which demonstrates the robustness property of the state observer.