Solving output control problems using Lyapunov gradient-velocity vector function

Received Sep 20, 2018 Revised Mar 13, 2019 Accepted Mar 21, 2019 This paper describes a controller and observer parameter definition approach in one input-one output (closed-loop) control systems using Lyapunov gradient-velocity vector function. Construction of the vector function is based on the gradient nature of the control systems and the parity of the vector functions with the potential function from the theory of catastrophe. Investigation of the closed-loop control system’s stability and solution of the problem of controller (determining the coefficient of magnitude matrix) and observer (calculation of the matrix elements of the observing equipment) synthesis is based on the direct methods of Lyapunov. The approach allows to select parameters based on the requested characteristics of the system.


INTRODUCTION
In practice the state vector is available for manipulation less often than the plant output. This leads to the use of the state value in the control law instead of the state variables received by the observers [1][2][3][4]. This, in return, requires that the dynamic properties of the system change accordingly. We aim to observe how the replacement of the state variables by state values affects the properties of the system. In the modal control [1,5] case characteristic polynomials are found through output.
The characteristic polynomial [6] of the closed-loop system with a controller that uses state values with an observer requires that the roots of the polynomial with a modal control be combined with the observer's own number [1,3,6]. This way synthesizing the modal controller with the observer becomes a challenging task.
The known iterative algorithms [1,3] of separate own value control are based on the preliminary matrix triangularization or block-diagonalization. Notice that the control matrix is used as the nonsingular transition matrix in this canonical transformation, and it is defined by its own vectors in complicated unequivocal ways described here [6,7].
The paper considers the control systems with one input and one output. To research the dynamic equalizer we use the Lyapunov gradient-velocity vector function [8][9][10][11]. Construction of the vector function is based on the gradient nature of the control systems and the parity of the vector functions with the potential function from the theory of catastrophe [12,13]. Investigation of the closed-loop control system's stability and solution of the problem of controller (determining the coefficient of magnitude matrix) and observer (calculation of the matrix elements of the observing equipment) synthesis is based on the direct methods of Lyapunov [14][15][16]. The approach offered in this paper can be considered as a way of determining the parameters of the controller and observer for a closed-loop with certain transitional characteristics.

RESEARCH
Assume the control system can be described by this set of equations [1][2][3][4]: Modify the state equation (1)- (3). For this we will use the estimation error ) Then we can write is as: , and Equations (1)-(3) will transform to: For brevity consider the system with one input and one output, hence the system looks like:

 
The set of (4), (5) will transform into: Notice that in the absence of the external impact, the process in the set (4), (5) must asymptotically approach the processes of a system with a controller, as if the closed-loop system according to a state vector, was affected by the impact of the convergent disturbance waves. These disturbances are caused by the ) (t K polynom in the Equation (5). The error must converge and the speed of convergence is defined during the synthesis of the observer. The main property of the set (4) and (5) lies in the asymptomatical stability. This way we found the requirement for the asymptotical stability of the system using the gradient-velocity method of the Lyapunov functions [8][9][10][11].   From (6) we find the decomposition of the velocity vector to the coordinates ). ,..., , ,..., ( To research the stability of the system (6) we use the basics of Lyapunov's direct method [14][15][16].
For the system to achieve the asymptotical equilibrium we need to secure the existence of a positive function ) , (  x V so that its total derivative on the time axis along the state function (6) is a negative function. The total derivative from Lyapunov function with regard to the state Equation (6) is defined as a scalar product of the gradient (7) from Lyapunov and the velocity vector. (8): From (9) we derive that the total time derivative of the vector function will be negative. Lyapunov function from (7) can be represented in the scalar view: The condition for the positive certainty (10) i.e. existence of Lyapunov function will be defined: The quality and the stability of the control system is dictated by the elements of the matrix of the closed-system. That is determining the target values of coefficients in a closed-loop system will prodivide smooth transitional processes in a system and result in higher quality control. The set of inequalities (11) and (12) serve as the necessary condition for the robust dynamic equalizer. The condition (11) allows for the stability in the state vector. Imagine a control system with a set of desired transition processes with one input and one output:  [8]. From (13) we find the components of the gradient-vector function )) ( ),...,   (14) From (13) we determine the decomposition of the velocity vector according to the coordinates: The total time derivative from the vector function is defined as a scalr product of the gradient vector (14) and the velocity vector (15) From (16) follows that the total time derivative of the vector-gradient function is negative. The vector function in the scalar view from (14) can be represented as:  the observer's own number. This process requires substantial calculations and requires preliminary matrix block-diagonalization. This nonsingular matrix in the canonical transformation is defined by its own vectors in. Researching the closed-loop control system using the gradient-velocity method of Lyapunov function gave us the opportunity to develop an approach for controller and observer parametrization that provides us with a system of our desire without extraneous calculations.