Linear Robust Control Theory Research Articles

Multiobjective Robust Controller Synthesis for Nonlinear Mechanical System

In the paper multiobjective robust controller synthesis problem for nonlinear mechanical system described by Lagrange’s equations of the second kind is considered. Such tasks have numerous practical applications, for example in controller design of robotic systems and gyro-stabilized platforms. In practice, we often have to use uncertain mathematical plant models in controller design. Therefore, ensuring robustness in presence of parameters perturbations and unknown external disturbances is an important requirement for designed systems. Much of modern robust control theory is linear. When the actual system exhibits nonlinear behavior, nonlinearities are usually included in the uncertainty set of the plant. A disadvantage of this approach is that resulting controllers may be too conservative especially when nonlinearities are significant. The nonlinear H∞ optimal control theory developed on the basis of differential game theory is a natural extension of the linear robust control theory. Nonlinear theory methods ensure robust stability of designed control systems. However, to determine nonlinear H∞-control law, the partial differential equation have to be solved which is a rather complicated task. In addition, it is difficult to ensure robust performance of controlled processes when using this method. In this paper, methods of linear parameter-varying (LPV) systems are used to synthesize robust control law. It is shown, that Lagrange system may be adequately represented in the form of quasi-LPV model. From the computational point of view, the synthesis procedure is reduced to convex optimization techniques under constraints expressed in the form of linear matrix inequalities (LMIs). Measured parameters are incorporated in the control law, thus ensuring continuous adjustment of the controller parameters to the current plant dynamics and better performance of control processes in comparison with H∞-regulators. Furthermore, the use of the LMIs allows to take into account the transient performance requirements in the controller synthesis. Since the quasi-LPV system depends continuously on the parameter vector, the LMI system is infinite-dimensional. This infinitedimensional system is reduced to a finite set of LMIs by introducing a polytopic LPV representation. The example of multiobjective robust control synthesis for electro-optical device’s line of sight pointing and stabilization system suspended in two-axes inertially stabilized platform is given.

ROBUST STABILITY, STABILISATION AND H-INFINITY CONTROL FOR PREMIUM-RESERVE MODELS IN A MARKOVIAN REGIME SWITCHING DISCRETE-TIME FRAMEWORK

AbstractThe premium pricing process and the medium- and long-term stability of the reserve policy under conditions of uncertainty present very challenging issues in relation to the insurance world. Over the last two decades, applications of Markovian regime switching models to finance and macroeconomics have received strong attention from researchers, and particularly market practitioners. However, relatively little research has so far been carried out in relation to insurance. This paper attempts to consider how a linear Markovian regime switching system in discrete-time could be applied to model the medium- and long-term reserves and the premiums (abbreviated here as the P-R process) for an insurer. Some recently developed techniques from linear robust control theory are applied to explore the stability, stabilisation and robustH∞-control of a P-R system, and the potential effects of abrupt structural changes in the economic fundamentals, as well as the insurer's strategy over a finite time period. Sufficient linear matrix inequality conditions are derived for solving the proposed sub-problems. Finally, a numerical example is presented to illustrate the applicability of the theoretical results.

On System Gains, Nonlinearity Measures, and Linear Models for Nonlinear Systems

This article is concerned with the assessment and derivation of (best) linear models for nonlinear systems. We introduce a general framework reminiscent of uncertainty descriptions in linear robust control theory in order to define a family of six model quality indices for linear models of nonlinear systems. Each quality index corresponds to the gain of an error system, where the nonlinear system can be represented as the interconnection of a nominal linear model and this error system. For an analysis in a certain region of operation, these gains are considered over a specified signal set. A best linear model is a linear model that minimizes the quality index and the minimal quality index serves at the same time as a measure of nonlinearity of the nonlinear system. We show that the model quality indices and nonlinearity measures can be defined under very weak assumptions, and we give conditions that guarantee the existence of optimal linear models. We discuss the structure of the problem of computing the nonlinearity measures. We show that the local linearization of a state space system is a best linear model for the limiting case of a vanishing operating range for one of the measures. We show the relation between the steady-state behavior of a system and its gain and nonlinearity measures. Furthermore we give linear models and upper bounds for nonlinearity measures for systems composed of a linear dynamic and a nonlinear static part. In the case of scalar (SISO) systems, these bounds are given by the sector bounds of the nonlinearity. Finally, it is shown that a lower bound on the nonlinearity measures can be derived using harmonic analysis. Several small examples serve to illustrate the results and the underlying ideas.

Incorporating Robustness Requirements Into Antiwindup Design

This paper treats the problem of synthesizing antiwindup compensators that are able to handle plant uncertainty in addition to controller saturation. The uncertainty considered is of the frequency-weighted additive type, often encountered in linear robust control theory, and representative of a wide variety of uncertainty encountered in practice. The main results show how existing linear matrix inequality based antiwindup synthesis algorithms can be modified to produce compensators that accommodate uncertainty better. Embedded within these results is the ever-present performance - robustness tradeoff. A remarkable feature is that the often criticized internal model control antiwindup solution emerges as an ldquooptimally robustrdquo solution. A simple example demonstrates the effectiveness of the modified algorithms.

Delayed Standard Neural Network Models for Control Systems

In order to conveniently analyze the stability of recurrent neural networks (RNNs) and successfully synthesize the controllers for nonlinear systems, similar to the nominal model in linear robust control theory, the novel neural network model, named delayed standard neural network model (DSNNM) is presented, which is the interconnection of a linear dynamic system and a bounded static delayed (or nondelayed) nonlinear operator. By combining a number of different Lyapunov functionals with S-procedure, some useful criteria of global asymptotic stability and global exponential stability for the continuous-time DSNNMs (CDSNNMs) and discrete-time DSNNMs (DDSNNMs) are derived, whose conditions are formulated as linear matrix inequalities (LMIs). Based on the stability analysis, some state-feedback control laws for the DSNNM with input and output are designed to stabilize the closed-loop systems. Most RNNs and neurocontrol nonlinear systems with (or without) time delays can be transformed into the DSNNMs to be stability-analyzed or stabilization-synthesized in a unified way. In this paper, the DSNNMs are applied to analyzing the stability of the continuous-time and discrete-time RNNs with or without time delays, and synthesizing the state-feedback controllers for the chaotic neural-network-system and discrete-time nonlinear system. It turns out that the DSNNM makes the stability conditions of the RNNs easily verified, and provides a new idea for the synthesis of the controllers for the nonlinear systems.

Discrete-time delayed standard neural network model and its application

A novel neural network model, termed the discrete-time delayed standard neural network model (DDSNNM), and similar to the nominal model in linear robust control theory, is suggested to facilitate the stability analysis of discrete-time recurrent neural networks (RNNs) and to ease the synthesis of controllers for discrete-time nonlinear systems. The model is composed of a discrete-time linear dynamic system and a bounded static delayed (or non-delayed) nonlinear operator. By combining various Lyapunov functionals with the S-procedure, sufficient conditions for the global asymptotic stability and global exponential stability of the DDSNNM are derived, which are formulated as linear or nonlinear matrix inequalities. Most discrete-time delayed or non-delayed RNNs, or discrete-time neural-network-based nonlinear control systems can be transformed into the DDSNNMs for stability analysis and controller synthesis in a unified way. Two application examples are given where the DDSNNMs are employed to analyze the stability of the discrete-time cellular neural networks (CNNs) and to synthesize the neuro-controllers for the discrete-time nonlinear systems, respectively. Through these examples, it is demonstrated that the DDSNNM not only makes the stability analysis of the RNNs much easier, but also provides a new approach to the synthesis of the controllers for the nonlinear systems.

Robust controller design for multivariable nonlinear systems via multi-model H2/ H∞ synthesis

Input/output (I/O) linearization via state feedback provides a convenient framework for designing controllers for multivariable nonlinear systems. However this approach does not account for parametric uncertainty which may lead to loss of stability and performance degradation. In this paper, a multi-model H 2/ H ∞ approach is utilized to design a robust controller for minimum phase multivariable nonlinear systems that are subject to parametric uncertainty. It is first shown that a state feedback (inner loop) based on nominal parameters introduces nonlinear perturbations to the linear sub-system and results in loss of decoupling. The uncertain system is characterized in a form that provides a framework for robust controller design. Recent results from linear robust control theory are utilized to design an outer-loop controller to account for the parametric uncertainty. This methodology is illustrated via simulation of a regulation problem in a continuous stirred tank reactor.