2015-09-15
# Synthesis of Time Quasi-Optimal Asymptotically Stable Control Laws 2015-01-2481

The solution of the both synthesis and implementation problems of high-rapid rates control laws is extremely important for the development of automatic control systems of the aircraft. This is due to the high speed of such vehicles. Along with this, it is imperative that control laws provide that system is asymptotically stable, as the basis for the reliability of their controlled motion. Another important objective of the method of synthesis of control laws for aircraft is compulsory compliance with strict limitations on the values of control inputs at the actuation devices. It is equally important that the control laws provides limitations on the state variables of aircraft, such as velocity, acceleration, etc. Pontryagin's maximum principle is aimed at solving such a time-optimal problem with the limited command variable. However, both the mathematical formalism of this principle, and the mathematical formalism of the methods based on this principle don't provide a solution of the class described tasks. The problem is that, despite the fundamental theoretical validity of these methods, they don't provide robustness synthesized control laws. Robustness is understood here as the insensitivity of the properties of the control system to small variations in the properties of the control object. The reasons for this phenomenon are known. They are as follows: the asymptotic properties absent in positional control, maximum speed law is defined only within a given time interval, the stabilizing feedback is absent. As a result the inadequacy of the dynamic properties of a managed object and calculated data can lead to a complete loss of the control system quality.

The essence of the proposed approach consists in influencing the change of the derivatives of the state variables, which form a system of phase coordinates. Each derivative of the state variable is formed by a special non-linear law. These laws impose a number of requirements: the derivative of the phase coordinates is limited, its rate of change is close to optimal for the designated limit, the law of its change is asymptotically stable. In other words, the state-space of the control object form specific invariant manifolds. These manifolds are formed by specially synthesized control law so that their properties are determined by the above requirements. Mathematically varieties are generated by nonlinear functions of the right sides of the system of differential equations of state of the control object. Structures and function parameters form the mathematical model of the control law. The resulting control action (signal input) sequentially generates the processes of change of phase coordinates in such a way that directs the state variables of the system on the given invariant manifolds. Formed diversity determine the dynamics of the controlled system. They in aggregate form its attractor, which meets the following properties: near-optimal performance, bounded of phase coordinates and asymptotic movement. The procedure of successive synthesis of invariant manifolds is given can be implemented for objects that have a mathematical model in the form of the Cauchy problem with the so-called “triangular” structure. For such a model is a characteristic that every i-th state variable of the system, which has the order of “n” and “r” of the control inputs, depends only on the state variables with index not greater than i + 1. In this case, any derivative of the state variable may depend on only one control input, and the n-th derivative is necessarily dependent on one of these inputs. This result is very important for the practical problems of management of working bodies of aircraft. For them, the control speed is a critical factor in the quality of flight and stability and robustness is a critical factor in the reliability of automatic control. Application of the proposed method is illustrated by the example of the construction and study of the automatic control system of the device controlling airship.

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