Design of observers for nonlinear systems with H ∞ performance analysis

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time‐delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H _∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.     

一类非线性系统的自适应反步控制

研究一类带有未知常数参量的非线性系统的镇定及自适应控制器设计问题,提出了一类非线性系统参数估计器设计及自适应反步控制器设计的新方法.构造出Lyapunov函数, 并给出闭环系统全局渐近稳定的新的充分条件.例子表明了所获方法的有效性.     

Reduced order observer design for nonlinear systems

This work is a geometric study of reduced order observer design for nonlinear systems. Our reduced order observer design is applicable for Lyapunov stable nonlinear systems with a linear output equation and is a generalization of Luenberger’s reduced order observer design for linear systems. We establish the error convergence for the reduced order estimator for nonlinear systems using the center manifold theory for flows. We illustrate our reduced order observer construction for nonlinear systems with a physical example, namely a nonlinear pendulum without friction.     

Reduced order observer design for discrete-time nonlinear systems

This work is a geometric study of reduced order observer design for discrete-time nonlinear systems. Our reduced order observer design is applicable for Lyapunov stable discrete-time nonlinear systems with a linear output equation and is a generalization of Luenberger’s reduced order observer design for linear systems. We establish the error convergence for the reduced order estimator for discrete-time nonlinear systems using the center manifold theory for maps. We illustrate our reduced order observer construction for discrete-time nonlinear systems with an example.     

A new time-discretization for delay multiple-input nonlinear systems using the Taylor method and first order hold

A new discretization method is proposed for multi-input-driven nonlinear continuous systems with time-delays, based on a combination of the Taylor series expansion and the first-order hold (FOH) assumption. The mathematical structure of the new discretization scheme is explored. On the basis of this structure, the sampled-data representation of the time-delayed multi-input nonlinear system is derived. First the new approach is applied to nonlinear systems with two inputs, and then the delayed multi-input general equation is derived. The resulting time discretization method provides a finite-dimensional representation for multi-input nonlinear systems with time-delays, thereby enabling the application of existing controller design techniques to such systems. The performance of the proposed method is evaluated using a nonlinear system with time-delays (maneuvering an automobile). Various sampling rates, time-delay values and control inputs are considered to evaluate the proposed method. The results demonstrate that the proposed discretization scheme can meet the system requirements even when using a large sampling period with precision limitations. The discretization results of the FOH method are also compared with those of the zero order hold (ZOH) method. The precision of the FOH method in the discretization procedure combined with the Taylor series expansion is much higher than that of the ZOH method except in the case of constant inputs.     

Gain-Scheduled Worst-Case Control on Nonlinear Stochastic Systems Subject to Actuator Saturation and Unknown Information

In this paper, we propose a method for designing continuous gain-scheduled worst-case controller for a class of stochastic nonlinear systems under actuator saturation and unknown information. The stochastic nonlinear system under study is governed by a finite-state Markov process, but with partially known jump rate from one mode to another. Initially, a gradient linearization procedure is applied to describe such nonlinear systems by several model-based linear systems. Next, by investigating a convex hull set, the actuator saturation is transferred into several linear controllers. Moreover, worst-case controllers are established for each linear model in terms of linear matrix inequalities. Finally, a continuous gain-scheduled approach is employed to design continuous nonlinear controllers for the whole nonlinear jump system. A numerical example is given to illustrate the effectiveness of the developed techniques.     

On the Optimal Stabilisation for a set of Programmed Motions of the Bilinear Control System

During recent years there has been considerable interest in using bilinear systems [1, 2] as mathematical models to represent the dynamic behavior of a wide class of engineering, biological and economic systems. Moreover, there are some methods [3] which may approximate nonlinear control systems by bilinear systems. For the first time Zubov has proposed a method of stabilization control synthesis for a set of programmed motions in linear systems [4]. In papers [5, 6] this method has been developed and used to solve the same problem for bilinear systems. In the present paper the following problems are considered. First, synthesis of nonlinear control as feedback under which the bilinear control system has a given set of programmed and asymptotic stable motions. Because this control is not unique, the second problem concerns optimal stabilization. In this paper a method for the design of nonlinear optimal control is suggested. This control is constructed in the form of a convergent series. The theorem on the sufficient conditions to solve this problem is represented.     

Time-discretization of nonlinear control systems with state-delay via Taylor–Lie series

The state-delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. In this paper, we propose a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption. This scheme is applied to the sample-data representation of a nonlinear system with constant state time-delay. The mathematical expressions of the discretization scheme are presented and the effect of the time-discretization method on equilibrium properties of nonlinear control system with state time-delay is examined. The proposed scheme provides a finite-dimensional representation for nonlinear systems with state time-delay enabling existing controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated using a nonlinear system. For this nonlinear system, various sampling rates and time-delay values are considered.     

Synchronizing chaotic systems using control based on tridiagonal structure

The direct design approach based on tridiagonal structure combines the structure analysis with the design of stabilizing controller and the original nonlinear affine systems is transformed into a stable system with special tridiagonal structure using the method. In this study, the direct method is proposed for synchronizing chaotic systems. There are several advantages in this method for synchronizing chaotic systems: (a) it presents an easy procedure for selecting proper controllers in chaos synchronization; (b) it constructs simple controllers easy to implement. Examples of Lorenz system, Chua’s circuit and Duffing system are presented.     

Fuzzy Partial State Feedback Control of Discrete Nonlinear Systems with Unknown Time-Delay

A new discrete-time fuzzy partial state feedback control method for the nonlinear systems with unknown time-delay is proposed. Ma et al. proposed the design method of the fuzzy controller based on the fuzzy observer and Cao and Frank extend this result to be applicable to the case of the nonlinear systems with the time-delay. However, the time-delay is likely to be unknown in practical. In this paper, the sufficient condition for the asymptotic stability is derived with the assumption that the time-delay is unknown by applying Lyapunov–Krasovskii theorem and this condition is converted into the LMI problem.     

Matching conditions for stability analysis of nonlinear feedback control systems

The differential geometry approach for exact input-output linearizable nonlinear systems usually results in a complicated nonlinear controller that is difficult to implement. To overcome this difficulty, we propose to approximate the nonlinear controller by a canonical piecewise linear expression within an allowable error bound. This design procedure can reduce a tremendous amount of computation in the design and the synthesis. The resulting controller turns out to be fairly simple in general and can achieve many performance specifications. Some sufficient conditions for guaranteeing the closed-loop system stability using this controller design method is derived in the present paper. An application to a chemical reactor system is also briefly discussed.     

Optimal input system identification for nonlinear dynamic systems

The estimation accuracy for nonlinear dynamic system identification is known to be maximized by the use of optimal inputs. Few examples of the design of optimal inputs for nonlinear dynamic systems are given in the literature, however. The performance criterion is selected such that the sensitivity of the measured state variables to the unknown parameters is maximized. The application of Pontryagin's maximum principle yields a nonlinear two-point boundary-value problem. In this paper, the boundary-value problem for a simple nonlinear example is solved using two different methods, the method of quasilinearization and the Newton-Raphson method. The estimation accuracy is discussed in terms of the Cramer-Rao lower bound.     

Exponential stabilization of a class of nonlinear systems: A generalized Gronwall–Bellman lemma approach

In this paper, stabilizing control design for a class of nonlinear affine systems is presented by using a new generalized Gronwall–Bellman lemma approach. The nonlinear systems under consideration can be non Lipschitz. Two cases are treated for the exponential stabilization: the static state feedback and the static output feedback. The robustness of the proposed control laws with regards to parameter uncertainties is also studied. A numerical example is given to show the effectiveness of the proposed method.     

A relation between the output regulation and the observer design for nonlinear systems

Output regulation and observer design are two important problems for nonlinear systems, and there is a vast literature addressing each problem separately in the control literature. Isidori and Byrnes [1] have solved the output regulation problem for nonlinear systems with a Poisson stable exosystem, and Sundarapandian [2] has solved the exponential observer design problem for Lyapunov stable nonlinear systems. In this paper, we demonstrate that for a special class of Lyapunov stable nonlinear systems, namely neutrally stable systems, the exponential observer design problem can be solved by converting it into an output regulation problem and then solving the new problem using the output regulation techniques of Isidori and Byrnes [1]. Finally, we present the corresponding results for the discrete-time case.     

Controlling chaotic systems via nonlinear feedback control

In this article, a new method to control chaotic systems is proposed. Using Lyapunov method, we design a nonlinear feedback controller to make the controlled system be stabilized. A numerical example is given to illuminate the design procedure and advantage of the result derived.     

Local observer design for nonlinear systems

This paper is a geometric study of the local observer design for nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable nonlinear systems. As an application of our local observer design, we consider a class of nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs.     

Synchronization for the coupled stochastic strict-feedback nonlinear systems with delays under pinning control

In this paper, a class of new coupled stochastic strict-feedback nonlinear systems with delays (CSFND) on networks without strong connectedness (NWSC) is considered, and the issue pertaining to the synchronization of the systems is discussed by pinning control. Towards CSFND, the controllers are approached by combining the back-stepping method and the design of virtual controllers. A key novel design ingredient is that the global Lyapunov function is obtained based on each Lyapunov function of stochastic strict-feedback nonlinear systems with delays (SFND). Moreover, a sufficient criterion is presented to realize the exponential synchronization by employing the graph theory and Lyapunov method. As a subsequent result, we apply the obtained theoretical results to the second-order oscillator systems and robotic arm systems. Meanwhile, numerical simulations are provided to demonstrate the validity and feasibility of our theoretical results.     

Observer design for discrete-time nonlinear systems

This paper is a geometric study of the observer design for discrete-time nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable discrete-time nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable discrete-time nonlinear systems. As an application of our local observer design, we consider a class of discrete-time nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs.     

Robust Observer Design for a Class of Nonlinear Systems Using the System Internal Dynamics Structure

In this paper, an observer design strategy is presented for a class of nonlinear systems with structural uncertainty. The modern geometric approach is exploited to simplify the system structure. Then, based on the Lyapunov direct method, a robust observer is proposed using the system internal dynamics structure and the distribution of the uncertainty structure. The bound on the uncertainty, which is employed in the observer design, is allowed to be nonlinear and have a more general form. Simulation shows that the proposed approach is effective.     

Robust digital controllers for uncertain chaotic systems: A digital redesign approach

In this paper, a new and systematic method for designing robust digital controllers for uncertain nonlinear systems with structured uncertainties is presented. In the proposed method, a controller is designed in terms of the optimal linear model representation of the nominal system around each operating point of the trajectory, while the uncertainties are decomposed such that the uncertain nonlinear system can be rewritten as a set of local linear models with disturbed inputs. Applying conventional robust control techniques, continuous-time robust controllers are first designed to eliminate the effects of the uncertainties on the underlying system. Then, a robust digital controller is obtained as the result of a digital redesign of the designed continuous-time robust controller using the state-matching technique. The effectiveness of the proposed controller design method is illustrated through some numerical examples on complex nonlinear systems––chaotic systems.