Quadratic embedding into algebras and global stabilization for a class of nonlinear control systems

In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise stabilization. This method requires a sequential investigation of lower-dimensional subsystems. Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing powers can be applied to systems of higher dimensions.

In this study the tracking problem for a class of non-linear uncertain systems is analyzed. The considered class of non-linear systems are restricted by those verifying the global Lipschitz condition on non-linearities making them linear-like. A new sliding mode neurocontroller is suggested to solve this problem. The controller desing includes the on-line state estimates construction and the corresponding tracking control based on sliding mode approach and the reconstructed dynamics generated by a special non-linear observer. A special sliding mode technique during the “off-line training” to estimate the right-hand side of the given dynamics in finite-time was applied. This procedure allows use of these estimates for the best (in LQ-sense) nominal weights selection in the neuro observer designed. A switching (sign) type term is incorporated in the observer structure to correct the current state estimates using just the on-line measurable output. This observer is supplied with a new learning procedure with switching structure. Two illustrative examples are presented: a benchmark problem related with a nominal linear system and a real chemical procedure dealing with the organic compounds elimination by ozone effect.

This paper presents a unified fault isolation scheme for process faults and sensor faults in a class of nonlinear uncertain systems. The proposed fault diagnosis architecture consists of a fault detection estimator and a bank of isolation estimators, each corresponding to a particular fault type. Based on the class of nonlinear systems and fault types under consideration, adaptive thresholds are derived for the isolation estimators, and fault isolability conditions are rigorously investigated, characterizing the class of process faults and sensor faults that are isolable by the proposed scheme.

This paper addresses the design of static anti-windup compensators for a class of nonlinear systems subject to actuator saturation. Considering that a (possibly nonlinear) dynamic output feedback control law has been previously designed disregarding the saturation, a method to compute a static anti-windup gain which ensures the local stability of the closed-loop system is proposed. The results are based on a differential algebraic representation of rational systems and on a modified version of the generalized sector condition to deal with the saturation effects. From these elements, an algorithm based on the solution of LMIs is proposed to calculate an anti-windup gain to enlarge the region of attraction of the closed-loop system. Moreover, extensions of the results are presented to deal with parameter uncertainty and the presence of energy bounded disturbances. A numerical example is presented to illustrate the proposed method.

We address the problem of giving a new canonical form for a class of (time varying) nonlinear noncontrollable systems. We define and solve the partial linearization problem (PLP) by static state feedback. Our notion of linearization is weaker than the one found in the literature, since we want to transform the given system to a controllable linear system that is affected by a (perhaps nonlinear) autonomous system, which plays the role of its noncontrollable part. For the class of nonlinear systems for which the PLP is solvable, we show that the influence of the noncontrollable part in the linear part can be always removed by a convenient static state feedback. In particular, a new canonical form for this class of noncontrollable nonlinear systems is obtained.

In this paper, an optimal control scheme for a class of nonlinear systems with time delays in both state and control variables with respect to a quadratic performance index function is proposed using a new iterative adaptive dynamic programming(ADP) algorithm.By introducing a delay matrix function, the explicit expression of the optimal control is obtained using the dynamic programming theory and the optimal control can iteratively be obtained using the adaptive critic technique.Convergence analysis is presented to prove that the performance index function can reach the optimum by the proposed method.Neural networks are used to approximate the performance index function, compute the optimal control policy, solve delay matrix function, and model the nonlinear system, respectively, for facilitating the implementation of the iterative ADP algorithm.Two examples are given to demonstrate the validity of the proposed optimal control scheme.

State-estimation-based economic model predictive control of nonlinear systems

On Switched Controller Design for Robust Control of Uncertain Polynomial Nonlinear Systems Using Sum of Squares

Adaptive output feedback event-triggered tracking control for nonlinear systems with unknown control coefficient

Diagnosability of fuzzy discrete event systems

A self-organizing fuzzy controller to augment a sliding-mode control (SOFSMC) scheme for a class of nonlinear systems is proposed. The motivation behind this scheme is to combine the best features of self-organizing fuzzy control and sliding-mode control to achieve rapid and accurate tracking control of a class of nonlinear systems. The chatter encountered by most sliding-mode control schemes is greatly alleviated without sacrificing invariant properties. A stability analysis is presented; the design guidelines and the class of applicable systems are clearly identified. To verify the scheme, the authors performed experiments on its implementation in a magnetic levitation system. The results show that both alleviation of chatter and robust performance are achieved; the advantages of the scheme are indicated in comparison with the conventional sliding-mode design. >

For a class of non-linear dynamic systems, a quadratic regulator problem is considered in which the cost functional contains a small parameter multiplied to the control term. In the analysis of the problem, an appropriate transformation of the state variables and an eigenvector technique are utilized, which were devised originally for analysing complex linear cheap control problems and for solving a Lyapunov-type system of linear equations, respectively. In so doing, the class of nonlinear systems is characterized and extended in a mathematically simple way, for which the optimal cost becomes zero as the small parameter goes to zero.

Disturbance rejection and output regulation are big topics in control system design. In this chapter, rejection of deterministic disturbances in a class of nonlinear output feedback systems is discussed. The chapter starts from sinusoidal disturbances with unknown frequencies, then disturbances generated from nonlinear exosystems and then general periodical disturbances, etc. Adaptive control techniques are used to deal with the unknown disturbance frequencies and unknown parameters in the system. The presented design concepts can be applied to other classes of nonlinear systems.

The aim of the paper is the output feedback stabilization of a class of single-input single-output (SISO) nonlinear systems. We deal with a class of systems, so-called triangular form systems, that may have unobservable and unstabilizable approximate linearization about the origin. This kind of systems, referred to as singular triangular forms, may neither be stabilized by a smooth state feedback nor smoothly detectable. However, under certain simple conditions, it is known to be stabilizable by a static state feedback. A solution to continuous dynamic output feedback stabilization is proposed. This is based on the construction of the continuous observer and nonlinear analogue of the well known separation principle. The resulting observer may be computed and tuned in a straightforward constructive way. Numerical simulation of selected examples are performed to underline the applicability of the presented approach and to illustrate good convergence of both observer and controller system.

A robust output feedback control which globally stabilizes a class of SISO minimum-phase nonlinear systems with known and constant relative degree containing a vector of unknown parameters is developed. The constant parameter vector is not restricted to enter linearly in the state equations but it is assumed to belong to a known compact set and an imprecise knowledge of the nonlinearities (e.g., lookup tables) is allowed. The class of nonlinear systems is determined by geometric conditions; an additional assumption, which generalizes the knowledge of the sign of high-frequency gain for linear systems, is also required. The nonlinearities are restricted to depend, in suitable coordinates, on the output only: no growth conditions, such as sector or Lipschitz, are required. The robust output feedback control stabilizes the system for every value of the parameter vector in a known compact set. The order of the compensator is equal to the relative degree minus one. >