Performance bounds for nonlinear systems with a nonlinear ℒ2-gain property

Nonlinear ?₂-gain is a finite gain concept that generalises the notion of conventional (linear) finite ?₂-gain to admit the application of ?₂-gain analysis tools of a broader class of nonlinear systems. The computation of tight comparison function bounds for this nonlinear ?₂-gain property is important in applications such as small gain design. This article presents an approximation framework for these comparison function bounds through the formulation and solution of an optimal control problem. Key to the solution of this problem is the lifting of an ?₂-norm input constraint, which is facilitated via the introduction of an energy saturation operator. This admits the solution of the optimal control problem of interest via dynamic programming and associated numerical methods, leading to the computation of the proposed bounds. Two examples are presented to demonstrate this approach.     

Multiple-model-and-neural-network-based nonlinear multivariable adaptive control

A multivariable adaptive controller feasible for implementation on distributed computer systems (DCS) is presented for a class of uncertain nonlinear multivariable discrete time systems. The adaptive controller is composed of a linear adaptive controller, a neural network nonlinear adaptive controller and a switching mechanism. The linear controller can provide boundedness of the input and output signals, and the nonlinear controller can improve the performance of the system. The purpose of using the switching mechanism is to obtain the improved system performance and stability simultaneously. Theory analysis and simulation results are presented to show the effectiveness of the proposed method.     

3-DOF-PID nonlinear ℋ∞ estimators

In this article, we present the one-degree-of-freedom (1-DOF) or proportional (P), the 2-DOF or proportional-integral (PI) and the 3-DOF or proportional-integral-derivative (PID) ?_∞ nonlinear filters as counterparts of the popular P, PI and PID controllers, respectively. We use the idea of certainty-equivalence principle in which the worst-case disturbance or noise is determined as a function of the estimated state and under the assumption that this converges to the actual value asymptotically. Sufficient conditions for the solvability of the various filter gains are derived in terms of new Hamilton–Jacobi–Isaacs equations, which do not involve the system states. Simulation results are also presented to support the theory.     

Development of nonlinear disturbance observer based control and nonlinear PID: A personal note

This paper gives an overview of early development of nonlinear disturbance observer design technique and the disturbance observer based control (DOBC) design. Some critical points raised in the development of the methods have been reviewed and discussed which are still relevant for many researchers or practitioners who are interested in this method. The review is followed by the development of a new type of nonlinear PID controller for a robotic manipulator and its experimental tests. It is shown that, under a number of assumptions, the DOBC consisting of a predictive control method and a nonlinear disturbance observer could reduce to a nonlinear PID with special features. Experimental results show that, compared with the predictive control method, the developed controller significantly improves performance robustness against uncertainty and friction. This paper may trigger further research and interests in the development of DOBC and related methods, and building up more understanding between this group of control methods with comparable ones (particularly control methods with integral action).     

Dissipative-based adaptive neural control for nonlinear systems

A dissipative-based adaptive neural control scheme was developed for a class of nonlinear uncertain systems with unknown nonlinearities that might not be linearly parameterized. The major advantage of the present work was to relax the requirement of matching condition, i.e., the unknown nonlinearities appear on the same equation as the control input in a state-space representation, which was required in most of the available neural network controllers. By synthesizing a state-feedback neural controller to make the closed-loop system dissipative with respect to a quadratic supply rate, the developed control scheme guarantees that the L2-gain of controlled system was less than or equal to a prescribed level. And then, it is shown that the output tracking error is uniformly ultimate bounded. The design scheme is illustrated using a numerical simulation.     

Coordinated formation control of multiple nonlinear systems

A general method of controller design is developed for the purpose of formation keeping and reconfiguration of nonlinear systems with multiple subsystems, such as the formation of multiple aircraft, ground vehicles, or robot arms. The model consists of multiple nonlinear systems. Controllers are designed to keep the subsystems in a required formation and to coordinate the subsystems in the presence of environmental changes. A step-by-step algorithm of controller design is developed. Sufficient conditions for the stability of formation tracking are proved. Simulations and experiments are conducted to demonstrate some useful coordination strategies such as movement with a leader, simultaneous movement, series connection of formatiom, and human-machine interaction.     

On Hamiltonian realization of time-varying nonlinear systems

This paper investigates Hamiltonian realization of time-varying nonlinear (TVN) systems, and proposes a number of new methods for the problem. It is shown that every smooth TVN system can be expressed as a generalized Hamiltonian system if the origin is the equilibrium of the system. If the Jacopian matrix of a TVN system is nonsingu-lar, the system has a generalized Hamiltonian realization whose structural matrix and Hamiltonian function are given explicitly. For the case that the Jacobian matrix is singular, this paper provides a constructive decomposition method, and then proves that a TVN system has a generalized Hamiltonian realization if its Jacobian matrix has a non-singular main diagonal block. Furthermore, some sufficient (necessary and sufficient) conditions for dissipative Hamiltonian realization of TVN systems are also presented in this paper.     

Parameterization of nonlinear H∞ state-feedback controllers

State-space formulas are derived for a family of controllers solving the nonlinear H^∞ state-feedback control problem. These controllers are obtained by interconnecting the ‘central controller’ with an asymptotically stable, free system that satisfies one additional cascade condition. All proofs given are simple and clear, and also provide a deeper insight into the synthesis of the corresponding linear H^∞ controllers.     

H-infinity control for cascade minimum-phase switched nonlinear systems

1Introduction H_∞control theory has become a powerful tool to solverobust stabilization or disturbance attenuation problems.Many results about linear H∞control have appeared,andlinear H∞theory has been generalized to nonlinear systems[1~5].Two major approaches have been used to providesolutions to nonlinear H∞control problems.One is basedon the dissipativity theory and differential games theory[2,6].The other is based on the nonlinear versionofclassical bounded real lemma[3~5].Both of th…     

Stochastic resonance–a nonlinear control theory interpretation

Stochastic resonance (SR) is an effect that has been known (Benzi, R., Sutera, A., and Vulpiani, A. (1981 Chapeau-Blondeau, F. 1997. Input-Output Gains for Signal in Noise in Stochastic Resonance. Physics Letters A, 232: 41–48.  [Google Scholar]), ‘The Mechanism of Stochastic Resonance’, Journal of Physics, A14, L453–L457) for almost three decades and has been extensively studied in biology, statistics, signal processing and in numerous other eclectic areas (Wiesenfeld, K., and Moss, F. (1995 Xu, B, Jiang, Z-P, Wu, X and Repperger, DW. 2009. Investigation of Two-dimensional Parameter-Induced Stochastic Resonance and Applications in Nonlinear Image Processing. Journal of Physics A–Mathematical and Theoretical, 42: 145207, 1-45207, 9 [Google Scholar]), ‘Stochastic Resonance and the Benefits of Noise: From Ice Ages to Crayfish and Squids’, Nature, 373, 33–36). Herein, a nonlinear control theory analysis is conducted on how to better understand the class of systems that may exhibit the SR effect. Using nonlinear control theory methods, equilibrium points are manipulated to create the SR response (similar to shaping dynamical response in a phase plane). From this approach, a means of synthesising and designing the appropriate class of nonlinear systems is introduced. New types of nonlinear dynamics that demonstrate the SR effects are discovered, which may have utility in control theory as well as in many diverse applications. A numerical simulation illustrates some powerful attributes of these systems.     

Robustness of nonlinear state feedback—A survey

A survey of robustness of nonlinear state feedback is given. For series perturbations there are fairly complete results, showing that under mild restrictions an optimal controller can tolerate an infinite increase in gain. For gain reductions there are some results for systems linear in the control. In particular there is a 50% reduction tolerance if the control penalty is quadratic. Usually the optimal controller cannot be computed exactly. There are some results showing the effects of truncation on the robustness. Essentially robustness is maintained but in a reduced (computable) part of the state space.     

A new nonlinear output tracking controller via output-feedback

1 Introduction Since the introduction of backstepping design [1], exten- sive research has been investigated on control design for the systems in the triangular structure [1～6], merged with non- linear damping [7], tuning functions [4], MT filters [5] and other techniques. Two classes of output-feedback nonlinear control sys- tems have been investigated intensely: the first is that non- linearities only depend on the measurable states. By output injection, a much simple but effectual observe…     

Stabilization of a class of switched nonlinear systems

The stabilization of a class of switched nonlinear systems is investigated in the paper. The systems concerned are of （generalized） switched Byrnes-Isidori canonical form, which has all switched models in （generalized） Byrnes- Isidori canonical form. First, a stability result of switched systems is obtained. Then it is used to solve the stabilization problem of the switched nonlinear control systems. In addition, necessary and sufficient conditions are obtained for a switched affine nonlinear system to be feedback equivalent to （generalized） switched Byrnes-Isidori canonical systems are presented. Finally, as an application the stability of switched lorenz systems is investigated.     

Nonsmooth finite-time control of uncertain second-order nonlinear systems

Nonsmooth finite-time stabilizing control laws have been developed for the double integrator system. The objective of this paper is to further explore the finite-time tracking control problem of a general form of uncertain second-order affine nonlinear system with the new forms of terminal sliding mode (TSM). Discontinuous and continuous finite-time controllers are also developed respectively without the singularity problem. Complete robustness can be acquired with the former, and enhanced robustness compared with the conventional boundary layer method can be expressed as explicit bounded function with the latter. Simulation results on the stabilizing and tracking problems are presented to demonstrate the effectiveness of the control algorithms.     

Adaptive control of Hammerstein–Wiener nonlinear systems

The Hammerstein–Wiener model is a block-oriented model, having a linear dynamic block sandwiched by two static nonlinear blocks. This note develops an adaptive controller for a special form of Hammerstein–Wiener nonlinear systems which are parameterized by the key-term separation principle. The adaptive control law and recursive parameter estimation are updated by the use of internal variable estimations. By modeling the errors due to the estimation of internal variables, we establish convergence and stability properties. Theoretical results show that parameter estimation convergence and closed-loop system stability can be guaranteed under sufficient condition. From a qualitative analysis of the sufficient condition, we introduce an adaptive weighted factor to improve the performance of the adaptive controller. Numerical examples are given to confirm the results in this paper.     

A simplified NARMAX method using nonlinear input-output data

A system identification method for nonlinear systems with unknown structure is presented using short input-output data. The method simplifies the original NARMAX method. It introduces more general model structures for nonlinear systems. The group method of data handling （GMDH） method is employed to obtain the model terms and parameters. Effectiveness of the proposed method is illustrated by a typical nonlinear system with unknown structure and deficient input-output data.