Vladimir N. Chestnov

Synthesizing H∞-controllers for multidimensional systems with given accuracy and degree of stability

We consider linear multidimensional systems with output controllers subject to external perturbances from the class of polyharmonic functions with unknown amplitudes and bounded powers. We formulate the synthesis problem for continuous and discrete output controllers that guarantee a given accuracy with respect to the object’s controlled variables. We introduce the notion of a stabilized state radius of the closed system with respect to controlled variables and reformulate the problem of guaranteeing a given accuracy as a problem of guaranteeing the necessary or minimal possible stabilized state radius. Controller synthesis reduces to a standard H∞-problem of suppressing external perturbances, and its numerical solution is based on the linear matrix inequalities (LMI) approach implemented in the MATLAB package LMI Control Toolbox. We show a way to take into account a given degree of stability of the closed system which determines control time. We show a sample controller synthesis for an interconnected electric drive.

Design of robust H∞-controllers of multivariable systems based on the given stability degree

For the linear multivariable plants whose physical parameters are subject to deviations from the rated values, the measurable output-based controllers were designed providing the given degree of stability of the closed-loop system which defines the desired time of control. This approach relies on the “technique of opening the plant-controller system” for the investigated physical parameters and comes to a standard problem of H∞-optimization. An example was presented.

Linear-quadratic regulator. I. a new solution

In this paper, the classical linear-quadratic regulator problem is solved via use of the linear matrix inequality technique. This approach is shown to yield the optimal solution obtained by using the matrix Riccati equation. Various undesirable effects are discussed, which may appear when applying other solution methods known from the literature; numerical examples are presented.

H ∞ -approach to controller synthesis under parametric uncertainty and polyharmonic external disturbances

We consider the robust stabilization problem for linear multivariable systems whose physical parameters may deviate from computed (nominal) in some known bounds, and the control object is subject to non-measurable polyharmonic external disturbances (with unknown amplitudes and frequencies) bounded in power. We pose the problem of synthesizing a controller that guarantees robust stability of the closed-loop system and additionally ensures given errors with respect to controlled variables in the established nominal mode. The solution of this problem is based on the technique of opening the object–controller system with respect to varying object parameters and can be reduced to a standard H∞-optimization procedure, while the necessary accuracy is achieved by choosing the weight matrix for controlled object variables. We show the solution for a well-known benchmark problem.

Synthesis of discrete H∞-controllers with given stability margin radius and settling time

For linear multivariable systems, we construct discrete output controllers that guarantee a given stability margin radius on the input or output of a control plant. Besides, given control time is also taken into account. We show that solving such problems reduces to a certain specially constructed standard H∞-optimization problem. A numerical solution has been implemented in MATLAB with the Robust Control Toolbox suite based on the method of linear matrix inequalities (LMI).

Synthesis of digital H∞-controllers for multidimensional systems with given accuracy with the mean squared criterion

We consider linear multidimensional objects subject to power bounded polyharmonic perturbances and measurement noise that contains an arbitrary number of harmonics with unknown amplitudes and frequencies. For such objects, we propose a synthesis method for digital state controllers and controllers with respect to measurable output. The problem of guaranteeing a desired accuracy is formulated as the problem of guaranteeing a given mean squared radius of the stabilized state [1]; our solution of this problem is based on the choice of weight matrices in the minimal quadratic functional of a discrete H∞-optimization problem. We give a synthesis algorithm for a digital controller in the LMI Control Toolbox package and a numerical example for an interconnected electric drive.

Controllers Design via Given Oscillation Index: Parametric Uncertainty and Power-Bounded External Disturbances

The problem is considered of output controllers design for linear multivariable systems with deviating in prescribed bounds physical parameters of the plant, and subjected to the influence of unknown polyharmonic external disturbances, limited only in power. The controller is built so that to provide the prescribed bounds of mean-square values of controlled variables in addition to the robust stability of the closed-loop system. The problem solution is reduced to the H∞-optimization procedure produced in some specific way. The solution of the well-known “benchmark” problem is considered.

Maximum achievable precision of linear systems with discrete controllers

Consideration was given to the problem of maximum achievable precision of linear systems with discrete state and output controllers. The external perturbations affecting the system are the bounded step-type and harmonic (of unknown frequency) vector functions of time which the control theory regards as standard. The condition for asymptotic stability of the closed-loop system is the only requirement on the controllers aside from their physical realizability. Therefore, the conclusions of the present paper apply to the entire set of the discrete stabilizing controllers, no matter what method was used to design them.

Output Feedback H-infinity Control Design for Synchronizing of the Main Drives of the Pipe-Rolling Mill

Abstract We formulate the discrete-time output feedback controller design problem where the desired accuracy of controlled variables in the mean-square sense is guaranteed in the presence of bounded polyharmonic disturbances with a priori unknown number of harmonics, amplitudes, and frequencies. The amplitudes of the harmonics must satisfy a condition that results in the boundedness of the power of each polyharmonic component. The solution is based on the discrete-time H ∞ -optimization procedure by properly choosing the corresponding weighting matrices of the minimax cost criterion. Numerical solution in state-space is based on the method of linear matrix inequalities, using the LMI Control Toolbox (MATLAB application). We give the numerical example for the main drives pipe-rolling mill where the digital controller is synthesized using the LMI Control Toolbox package.

State feedback control design for multivariable discrete time systems with required accuracy via mean-square criterion

Abstract We formulate the discrete-time controller design problem where the desired accuracy of controlled variables in the mean-square sense is guaranteed in the presence of bounded polyharmonic disturbances with a priori unknown number of harmonics, amplitudes, and frequencies. The amplitudes of the harmonics must satisfy a condition that results in the boundedness of the power of each polyharmonic component. The concept of mean-square radius of the steady state of the closed-loop system is introduced which accounts for the bounds on the mean-square values of both the controlled variables and control inputs (for each controlled and control variable). The attainment of the desired accuracy of the control system is formulated as the problem of ensuring the required mean-square radius of the steady state. The solution is based on the discrete-time H ∞ -optimization procedure by properly choosing the corresponding weighting matrices of the minimax cost criterion.