Stanislaw H. Żak

Comparative study of non-linear state-observation techniques

ABSTRACT This paper contains a comparative study of four techniques for observing the slates of non-linear systems. The first technique examined is inspired by Bestle and Zeitz, and ICrener and Respondek, In this method a non-linear transformation is found that brings the system into a canonical form, from where observer design is facilitated. The second technique is attributable to Thau. In this method, the error between the systems true state and the output of the observer is shown to be asymptotically convergent to zero provided that an additional assumption is valid. The third technique is due to Baumann and Rugh. In this method, extended or pseudolinearization of the error differential equation about a family of equilibrium points results in an observer design such that the eigenvalues of the linearized error equation are locally invariant. Finally, techniques from variable-structure systems are utilized to design an observer that yields an exponentially decaying error like Thaus observer, but, unl...

Brief paper: Sliding-mode observers for systems with unknown inputs: A high-gain approach

Sliding-mode observers can be constructed for systems with unknown inputs if the so-called observer matching condition is satisfied. However, most systems do not satisfy this condition. To construct sliding-mode observers for systems that do not satisfy the observer matching condition, auxiliary outputs are generated using high-gain approximate differentiators and then employed in the design of sliding-mode observers. The state estimation error of the proposed high-gain approximate differentiator based sliding-mode observer is shown to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters. Finally, the unknown input reconstruction using the proposed observer is analyzed and then illustrated with a numerical example.

On discrete-time variable structure sliding mode control

The purpose of this paper is to show the limitations of discrete-time variable structure sliding mode control and that the equivalent control must be used in order to have sliding in a neighborhood of the switching surface. Conflicting requirements for the sliding mode controller behavior in the continuous and discrete-time domains are revealed and analyzed. A linear control law for an uncertain discrete-time linear plant, with bounded uncertainties, is analyzed and its superiority over nonlinear controllers is demonstrated. The conclusion of the obtained results is that in the discrete-time variable structure sliding mode controller design, unlike in the continuous-time, the designer may have limited flexibility in selecting controller architectures.

State-feedback control of non-linear systems†

A design method for state-feedback controllers for single-input non-linear systems is proposed. The method makes use of the transformations of the non-linear system into ‘controllable-like’ canonical forms. The resulting non-linear state feedback is designed in such a way that the eigenvalues of the linearized closed-loop model are invariant with respect to any constant operating point. The method constitutes an alternative approach to the design methodology recently proposed by Baumann and Rugh. Also a review of different transformation methods for non-linear systems is presented. An example and simulation results of different control strategies are provided to illustrate the design technique.

Robust control synthesis for uncertain/nonlinear dynamical systems

This paper addresses the problem of robust output-feedback controller design for uncertain and nonlinear dynamic systems. First a robust state-feedback nonlinear control law is synthesized. This control strategy practically stabilizes the closed-loop system. Then a state estimator for the nonlinear/uncertain plant is designed and its performance analyzed. Finally the control law and the estimator are combined and the practical stability region is estimated. The results are illustrated by their application to a benchmark problem for robust control design proposed by Wie and Bernstein (1990).

On the brain-state-in-a-convex-domain neural models

We propose and investigate new types of neural network models. They can be viewed as discrete linear systems operating on closed and bounded, that is, compact, convex domains. We first analyze the dynamic behavior of a neural network model on an arbitrary convex domain. Then, we analyze two specific cases: when the convex domain is a ball, and the case when the convex domain is a simplex. The equilibrium points of the proposed neural models are located and their stability is investigated. Copyright 1996 Elsevier Science Ltd

Practical stabilization of nonlinear/uncertain dynamic systems with bounded controllers

We address the problem of practically stabilizing nonlinear and/or uncertain continuous-time dynamic systems when the norm of the control input is subject to a given fixed bound. We consider a class of systems whose nominal part is linear and whose nonlinear/uncertain part does not satisfy the matching condition. In this treatment no statistical information regarding the uncertainties is needed. Only a norm-bound on each nonlinear/uncertain quantity is assumed. We propose a bounded state-feedback controller whose design incorporates elements from both variable structure control theory and the deterministic control methodology. Using the second method of Lyapunov we obtain sliding domains, regions of uniform ultimate boundedness, and estimates of the domain of attraction for systems employing this controller. The closed-loop stability analysis as well as the design of the controller is facilitated by introducing a special coordinate transformation. The approach is illustrated by a numerical example.

Neural network prediction of peptide separation in strong anion exchange chromatography

MOTIVATION The still emerging combination of technologies that enable description and characterization of all expressed proteins in a biological system is known as proteomics. Although many separation and analysis technologies have been employed in proteomics, it remains a challenge to predict peptide behavior during separation processes. New informatics tools are needed to model the experimental analysis method that will allow scientists to predict peptide separation and assist with required data mining steps, such as protein identification. RESULTS We developed a software package to predict the separation of peptides in strong anion exchange (SAX) chromatography using artificial neural network based pattern classification techniques. A multi-layer perceptron is used as a pattern classifier and it is designed with feature vectors extracted from the peptides so that the classification error is minimized. A genetic algorithm is employed to train the neural network. The developed system was tested using 14 protein digests, and the sensitivity analysis was carried out to investigate the significance of each feature. AVAILABILITY The software and testing results can be downloaded from ftp://ftp.bbc.purdue.edu.

On the estimation of sliding domains and stability regions of variable structure control systems with bounded controllers

This article examines the problem of the control of a class of multivariable linear time-invariant uncertain dynamic systems with bounded controllers using the variable structure control (VSC) approach and the second method of Lyapunov. A special coordinate transformation is utilized to facilitate the analysis. Sliding domains and estimates of the domain of attraction along with the regions of asymptotic stability (RAS) are obtained with the aid of Lyapunov-type arguments. Numerical examples are given to illustrate the approach developed in the article.

Low-order state estimators and compensators for dynamical systems with unknown inputs

Abstract We propose a new type of state estimator for a class of dynamical systems with unknown inputs. We synthesize the combined low-order estimator-controller compensator and show the stability of the closed-loop system. The results are illustrated with a numerical example.