Approximate methods of H-inﬁnity control of nonlinear dynamic systems output

. The problem of ﬁnding H ∞ -regulators in the output control problem is considered. Two approaches for approximate synthesis of the closed loop systems are proposed. The paper considers the problem of ﬁnding the H-inﬁnity control of a nonlinear continuous dynamical system output. The system is linear in control and perturbation, with a ﬁnite time of the system operation. Meth-ods for ﬁnding H-inﬁnity control are used to solve problems of synthesis of controllers, particularly for solving the problem of synthesis of controllers in conditions of incomplete state information. In such problems, it is di ﬃ cult to ﬁnd the structure of the regulator and its parameters. Su ﬃ cient H ∞ -control conditions are formulated and proved. Two approximate methods for ﬁnding the H-inﬁnity controller of closed loop nonlinear continuous systems were proposed. To illustrate the application of the methods, the problem of stabilization of the ZD559-Lynx helicopter, whose ﬂight takes place during the imposed time period of the system operation, was considered.


Introduction
Problems and methods for finding H ∞ -control laws are considered to be one of the most relevant problems of designing complex aerospace systems.The developing algorithms and software needed to find matrix gains of optimal controllers allow getting the desired quality of control over the object in conditions where the mathematical model and external perturbations contain indeterminacies.It's one of the main reasons why methods of finding H ∞ -control have become so popular [1][2][3][4][5][6].For many applied problems it is required to synthesize a controller in conditions of incomplete information about the state vector of the system.This article proposes applying methods like the ones described in [7,8] for finding optimal regulator of a nonlinear system with a finite time of the system operation [7,9].As a result of the work, sufficient conditions of control of nonlinear dynamical system, which are required to design output feedback controller, will be formulated.To do so, we have to prove the sufficient conditions theorem, find controller in conditions of incomplete information and propose a solving method, which could be universal for various control problems and provide the desired quality of transient processes as well as the asymptotic stability of the system.
The ZD559-Lynx helicopter stabilization problem [10], which illustrates the effectiveness of the proposed approaches for solving applied control problems, is solved.

Statement of the problem
The mathematical model of the plant is and model of a measuring system is where 2) is fully controllable and observable.Let's denote �z(t)� 2 = f 0 (t, y(t)) + u T (t) Q(t) u(t) as performance output, where f 0 (t, y) is a given continuous function, Q(t) is a positive definite symmetric matrix q × q; �F(t 1 )� 2 = F(y(t 1 )) as performance output at the final moment of the control process.
It is required to ensure (if possible) the fulfillment of the inequality where P(t) ∈ R p×p is a positive definite symmetric matrix, γ > 0 is a given non-negative value, as well as the asymptotic stability of a closed loop system.It is preferred to find the minimum value of γ at which these properties are still valid, which can be achieved by minimizing the value of the fraction numerator while maximizing the denominator.In other words, the cost functional must satisfy the condition which will be fulfilled when the output vector tends to zero, minimizing control costs under the worst influence of disturbances possible.

Synthesis of full state vector H ∞ -control
Let there be a function V(t, x) ∈ C 1,1 and the expression where Theorem.If a function V(t, x) ∈ C 1,1 exists, satisfying the condition V(t, 0) = 0 and where and function V(t, x) ∈ C 1,1 satisfies the equation then the inequality (3) is true.
Proof.Let the assertion conditions be satisfied.Let's find min u max w R(t, x, u, w) using the necessary conditions for an extremum: Here u * (t, x), w * (t, x) are structures for controlling the plant and disturbance (external influence).Sufficient conditions for an unconstrained minimum with respect to u are fulfilled because ∂u T ∂u = 2Q(t) > 0, as well as sufficient conditions for maximum with respect to w are fulfilled because i.e. there is a saddle point.
Let the available function V(t, x) ∈ C 1,1 satisfy the condition V(0, 0) = 0 and R t, x, u * (t, x), w * (t, x) = 0. Since the system is completely controllable and observable, for any u(.
Along the trajectories of the dynamical system it is true that Let's consider the left hand side of inequality ( 8), namely R t, x, u We integrate the left and right hand parts in time from 0 to t 1 : Therefore, condition (3) is fulfilled.The proof is over.
Remark.If P(t) = E and the energy of disturbances acting on the system is limited, i.e.

Synthesis of Output H ∞ -control
If m = n and the matrix C(t) is non-degenerate, then the state vector could be found directly from the output vector: If m ≤ n, rg C(t) = m ∀t ∈ T , one can apply two approaches.The first one is related to finding a pseudo-solution x of system y = C(t) x and its application in control law, the second-to the synthesis of the state observer that develops the estimate x of the state vector, and using estimates in control.
First approach.Find system pseudo-solution y = C(t)x using a pseudo-inverse matrix Second approach.Synthesize an asymptotic full order observer where K(t) ∈ R n×m is the gain observer matrix, x * 0 is a column containing the a priori information about the initial state.Matrix K(t) is selected with respect to condition, providing that the error ε(t) = x(t) − x(t) asymptotically tends to zero.Then the control law has the form where y t 0 = {y(τ), 0 ≤ τ ≤ t} is an accumulated information about the output measurement results.

Algorithm of Output H ∞ -control
Step 1. Set parameter γ > 0. Find a solution to equation x at fixed γ, satisfying stability conditions of a closed loop system.Sequentially reducing γ, find the minimum value under which all conditions remain fulfilled.
Step 2. Find the control of a plant of the form ( 9) or (11) and the law of change of the ∂V(t, x) ∂x .
Step 3. Find output control law and closed-loop system trajectories.Case 1.Control system with arbitrary disturbances satisfying the condition Case 2. Control system with the worst perturbations w(t, x) = Verify the asymptotic stability of the closed loop system for each case.

Results and discussion
Based on the outlined algorithms, software was developed to find the parameters of synthesized controllers, to simulate processes in dynamic systems for a given period of time, to obtain information about the quality of transient processes and analyze the asymptotic stability of a closed loop system.We used the MATLAB computing environment.Following is a model example, for which an analytical solution is obtained.Numerical experiment was carried out using the data of the model example.

Example
Let us consider the problem of stabilizing the ZD559-Lynx aircraft model [10,11], where in ( 1)-( 2 [10] are listed for the Lynx at flight speeds from hover to 140 knots in straight and level flight, i.e. trim angular velocities in fuselage axes system are equal to zero.Let us approximate these matrices.The equations of the linearized system are obtained for eight fixed values of the helicopter speed in [10] V i = (i − 1) ∆V, i = 1, . . ., 8, where ∆V = 20 knots: It is proposed to use an extended model where V(t) can be found by solving the equation V(t) = 140/T , V(0) = 0, which is used for mode with linearly increasing speed or another equation V(t) = 280t/T 2 , V(0) = 0, which is used for mode with positive acceleration.As a result, a � t, x(t) ) can be approximated with law: where -finite function with parameter p. Then the other matrix in plant model is described by the following one: The matrix in the equation of the measuring system has the form: The state vector is as follows: x 1 -velocity along aircraft x-axis, x 2 -velocity along aircraft z-axis, x 3 -angular velocity along aircraft y-axis, x 4 -pitch angle, x 5 -velocity along aircraft y-axis, x 6 -angular velocity along aircraft x-axis, x 7 -angle of attack, x 8 -angular velocity along aircraft z-axis.

Simulations
Simulation Settings.For the correct computation of the output feedback control, it is necessary to select Q.During a search for suitable elements, we changed their design several times.After all we selected the design matrices as in table 1, which contains parameters of numerical simulations.
In addition, we choose function given in table 1 to describe disturbance, which affects the state vector.
Numerical Simulations.For one of the described cases, namely the case of control system with arbitrary disturbances, we performed solution of the model example with the parameter γ = 0.05, which satisfies (3) and still guarantees the fulfillment of the asymptotic stability property of a closed loop system.
Following are plots of the changing angle of attack for the first and second approaches (figure 1).
Analysis of plots indicates that problems of synthesis of suboptimal H ∞ -controller can be solved by both approaches, which were proposed in the paper.Results for the both methods on the example presented on the figure 1 for the coordinate x 7 look similar due to the fact that each of the methods results in the optimal trajectory of the coordinate.Simulations give good results.A rather large influence is exerted by the elements of matrix Q in the quality criteria.
The coordinate x 7 and other coordinates of the state vector asymptotically converge to zero, which indicates the robustness of the system and the correct selection of parameters, for which the system retains its property of stability for any given disturbances.
Generally, both of the approaches produce comparable results observed with the processes change over time, which allows to make a conclusion about them both being a viable option in further research.

Summary
The purpose of this study was to design output feedback controller using the H ∞ -control for nonlinear dynamical system.Based on the obtained results we can state that proposed algorithms of synthesis of controllers provide the desired quality of transient processes and asymptotic stability of closed loop systems.Using H ∞ -control allows to effectively neutralize the negative influence of limited disturbances on the system.
H ∞ -control method was successfully applied to solving the considered example.The described method successfully deals with the problem, demonstrating good results.The proposed methods could be useful in autopilot design problems for aircrafts and rotorcrafts.
Nowadays, methods such as LQG-synthesis or frequency response conditions are used to find H ∞ -control.Often finding the control by these methods is reduced to solving linearmatrix inequalities, so the main difference between the methods proposed in the paper and other methods is that the proposed methods allow to avoid the solution of linear matrix inequalities and are distinguished by their simplicity for application.

Table 1 .
Numerical simulation parameters