Stabilization Of Uncertain Nonlinear Systems Research Articles

On the Finite-Time Stabilization of Uncertain Nonlinear Systems With Relative Degree Three

This technical note discusses a finite-time stabilization problem in a nonlinear, uncertain, environment. The main result of the present note is presented making reference to a triple integrator affected by bounded uncertainties and subject to ldquohardrdquo measurement nonlinearities such that the sign of the state variables is the only reliable information available for feedback. We propose a discontinuous control scheme guaranteeing the practical finite-time annihilation of the three state variables. Constructive proof and computer simulations, as well as guidelines for practical implementations, are provided throughout this technical note.

Input-to-state stability of switched systems and switching adaptive control

In this paper we prove that a switched nonlinear system has several useful input-to-state stable (ISS)-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included.

Stabilization of Uncertain Nonlinear Systems via Immersion and Invariance

This paper presents a method for designing asymptotically stabilising and adaptive control laws for uncertain nonlinear systems. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilising control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output-zeroing manifold and its friend, and is well suited, for instance, in situations where we know a stabilising controller of a nominal reduced-order model, which we would like to robustify with respect to higher-order dynamics. In adaptive control problems the method yields stabilising schemes that counter the effect of the uncertain parameters adopting a robustness perspective. This is in contrast with most existing adaptive designs that (relying on certain matching conditions) treat these terms as disturbances to be rejected. The construction does not invoke certainty equivalence, nor requires a linear parameterisation. Furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. To illustrate the concepts presented in this paper we introduce several academic and physical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics, and an adaptive, nonlinearly parameterised, visual servoing problem.

Adaptive H-infinity control of a class of uncertain nonlinear systems

It is concerned with the problem of disturbance attenuation with stability for uncertain nonlinear systems by adaptive output feedback. By a partial-state observer and Backstepping technique, an adaptive output feedback controller was constructed, which can solve the standard gain disturbance attenuation problem with internal stability.

Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization

This paper investigates the problem of robust output feedback stabilization for a family of uncertain nonlinear systems with uncontrollable/unobservable linearization. To achieve global robust stabilization via smooth output feedback, we introduce a rescaling transformation with an appropriate dilation, which turns out to be very effective in dealing with uncertainty of the system. Using this rescaling technique combined with the nonseparation principle based design method, we develop a robust output feedback control scheme for uncertain nonlinear systems in the p-normal form, under a homogeneous growth condition. The construction of smooth state feedback controllers and homogeneous observers uses only the knowledge of the bounding homogeneous system rather than the uncertain system itself. The robust output feedback design approach is then extended to a class of uncertain cascade systems beyond a strict-triangular structure. Examples are provided to illustrate the results of the paper.

Further results on global stabilization of uncertain nonlinear systems by output feedback

AbstractFor a family of uncertain nonlinear systems dominated by a triangular system that satisfies linear growth condition with an output dependent growth rate, we prove that global robust stabilization can be achieved by smooth output feedback. This conclusion has incorporated and generalized the recent output feedback stabilization results, for instance, the work (IEEE Trans. Automat. Control 2002; 47:2068–2073) where the same conclusion was already shown to be true for planar systems, and the work (Proceedings of the 42nd IEEE, CDC, 2003; 1544–1549) where the growth rate is required to be a polynomial function of the system output. There are two key ingredients in the present contribution. One of them is the introduction of a rescaling transformation with a dynamic factor that is tuned on‐line through a Riccati‐like differential equation, which turns out to be extremely effective in dealing with the system uncertainty. The other one is the development of a recursive observer design algorithm making it possible to assign the robust observer gains in a step‐by‐step fashion. Both a smooth state feedback controller and a robust observer are explicitly constructed for the rescaled system using only the knowledge of the bounding system. Copyright © 2005 John Wiley & Sons, Ltd.

An Adaptive Type-2 Fuzzy Neural Controller for Nonlinear Uncertain Systems

This article proposes a new control scheme using type-2 fuzzy neural network (type-2 FNN) and adaptive filter for controlling nonlinear uncertain systems. This type-2 FNN model combines the advantages of type-2 fuzzy logic systems and neural networks. The type-2 FNN system has the ability of universal approximation, that is, identification of nonlinear dynamic systems. We herein adopt it to develop a novel control scheme for nonlinear uncertain systems. The proposed control scheme consists of a PD-type adaptive FNN controller and a pre-filter. The adaptive filter is used to provide better performance under transient response and to treat the problem of disturbance attenuation. The tuning parameters for the filter and the type-2 FNN controller will change according to the learning algorithm. By the Lyapunov stability theorem, the convergence of parameters is given in order to guarantee the stability of nonlinear uncertain systems. The effectiveness of the proposed controller is demonstrated by simulated results.

Robust Stabilization of Uncertain Nonlinear Systems with Multiple Delay

Robust Stabilization of Uncertain Nonlinear Systems with Multiple Delay

A further result on global stabilization of uncertain nonlinear systems by output feedback

For a family of uncertain nonlinear systems dominated by a triangular system that satisfies linear growth condition with an output dependent growth rate, we prove that global robust stabilization can be achieved by smooth output feedback. This conclusion has incorporated and generalized the recent output feedback stabilization results, for instance, the work (Qian and Lin, 2002b) where the same conclusion was already shown to be true for planar systems, and the work (Praly and Jiang, 2003) where the growth rate is required to be a polynomial function of the system output. There are two key ingredients in the present contribution. One of them is the introduction of a rescaling transformation with a dynamic factor that is tuned on-line through a Riccati-like differential equation, which turns out to be extremely effective in dealing with the system uncertainty. The other one is the development of a recursive observer design algorithm making it possible to assign the robust observer gains in a step-by-step fashion. Both a smooth state feedback controller and a robust observer are explicitly constructed for the rescaled system using only the knowledge of the bounding system.

Absolute stability of nonlinear systems with disc and norm‐bounded perturbations

AbstractAbsolute stability of uncertain nonlinear systems is studied using the celebrated Popov and circle criteria. Uncertainty is assumed to exist in terms of disc and norm‐bounded perturbations in the linear plant. The use of circular arithmetic is proposed as an accurate but computationally more demanding alternative to the already existing approach based on strict positive realness conditions which is easier and faster to implement but gives, in general, conservative results. Numerical examples are given in order to illustrate the salient features of the mathematical developments. Copyright © 2004 John Wiley & Sons, Ltd.

Delay-dependent robust stability of uncertain nonlinear systems with time delay

In this paper, robust stability for uncertain systems with time delay is investigated. A new delay-dependent stability criterion is given for the uncertain systems with time delay by using the Lyapunov method and quadratic stability theory. A new upper bound on time delay based on the Hamiltonian matrix and a solvable algebraic Riccati equation is presented. Two illustrative examples are given to show the superiority of the present result to those in the literature.

A unifying framework for the semiglobal stabilization of nonlinear uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing an uncertain nonlinear system consisting of a linear nominal system perturbed by either nonlinearities or model uncertainties. Our approach relies on well-known H/sub /spl infin// linear control tools and allows one to recover and improve, in the unifying framework of a semiglobal separation result, existing results on the semiglobal stabilization via output feedback. In particular, we discuss the case of uncorrupted outputs, input and output nonlinearities, or model uncertainties, which may include, for example, practical situations such as backlash, hysteresis, and saturations. The key feature of our design procedure is given by the choice of two continuous functions: the first one is instrumental in constructing a stabilizing controller; the second one arises in the candidate Lyapunov function for the closed-loop system. Relying on our main theorem, we give general tools for achieving large regions of attraction via bounded measurement feedback for a wide class of nonlinear uncertain interconnected systems.

A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback

It is known that if a system can be (robustly) globally asymptotically stabilized by means of a feedback that is driven by functions that are uniformly completely observable (UCO), then this system can be practically semiglobally stabilized by means of (possibly dynamic) output feedback. This papers discusses a significant structural hypothesis under which the existence of a dynamic feedback driven by UCO functions is guaranteed. The class of systems which satisfy this hypothesis includes any stabilizable and detectable linear system and any relative degree one nonlinear system which is stabilizable by dynamic output feedback. In particular, the hypothesis does not require the system to be minimum phase.

On the Robust Stabilization of Nonlinear Uncertain Systems With Incomplete State Availability

The control problem of a class of nonlinear uncertain processes with incomplete state availability is dealt with by means of second-order sliding mode control technique. With the aim of using a continuous control, a combined scheme is proposed in which a 2-sliding mode differentiator and a 2-sliding mode controller are coupled. The stability of the whole observer-controller system is proved for a class of nonlinear plants, and some simulation results are finally provided. [S0022-0434(00)02704-0]

Recursive scaling design for robust global nonlinear stabilization via output feedback

This paper proposes a novel approach to robust backstepping for global stabilization of uncertain nonlinear systems via output feedback. The design procedure developed in this paper is based on the concept of state-dependent scaling, which handles output-feedback stabilization problems of strict-feedback systems with various structures of uncertainties in a unified way. The proposed method is suitable for numerical computation. The theory of the method employs the Schur complements formula instead of Young's inequality and completing the squares. This paper shows a condition of allowable uncertainty size under which an uncertain system is globally stabilized by output feedback. A class of systems is shown to be always globally stabilizable for arbitrarily large nonlinear size of uncertainties. A recursive procedure of robust observer design for such a class of uncertain systems is presented. Copyright © 2000 John Wiley & Sons, Ltd.

Exponential Stabilization of Uncertain Nonlinear Systems in Chained Form

This paper considers systems in chained form with structured uncertainty. We show that such systems can be robustly and exponentially stabilized.

Robust stabilization of nonlinear uncertain systems with multiple delays

In this paper, a method for robust stabilization is firstly presented for the uncertain linear systems with multiple-state delay. An extension to nonlinear delay systems subject to unmatched and matched uncertainties is extensively investigated. The state feedback controller is constructed via the solution of Riccati matrix equation and known bounding functions of the uncertainties. The proposed schemes are simple since only algebraic Riccati equations need to be solved. Finally, a numerical example is given to illustrate the results.

A General Theorem on the Semiglobal Stabilization of Uncertain Nonlinear Systems via Measurement Feedback

In this paper we study the problem of semiglobally stabilizing uncertain nonlinear systems via measurement feedback. We give a general theorem for the regional stabilization of a class of uncertain nonlinear systems via measurement feedback. These systems are characterized by a nominal linear part and (measurable) uncertain nonlinear terms, which are known only up to some nonlinear bounds. The combination of our theorems gives sufficient conditions for semiglobal stabilization via high and low-gain control laws, recovering well-known results for the case of uncorrupted outputs and saturated inputs.

Closed-Form Solutions of Stochastic and Deterministic Nonlinear Control Problems

We review our recent results on stabilization and optimal disturbance attenuation for nonlinear systems with uncertainties. The systems are affine in the control and the disturbance but otherwise general. We treat both deterministic and stochastic disturbances. In both cases the disturbance attenuation problem is posed as a differential game (in the stochastic case the player opposed to control is not the noise itself but its covariance). The results are based on control Lyapunov function techniques. The full account of the results is given in our new monograph(Stabilization of Nonlinear Uncertain Systems, Springer, 1998).

Robust Hybrid Control of Constrained Robot Manipulators via Decomposed Equations

Basing on a constraint Jacobian induced orthogonal decomposition of the task space and by requiring the force controller to be orthogonal to the constraint manifold, the dynamics of the constrained robots under hybrid control is decomposed into a set of two equations. One describes the motion of robots moving on the constraint manifold, while the other relates the constraint force with the hybrid controller. This decomposition does not require the solution of the constraint equation in partition form. In this setting, the hybrid control of constrained robots can be essentially reduced to robust stabilization of uncertain nonlinear systems whose uncertainties do not satisfy the matching condition. A continuous version of the sliding-mode controller (from Khalil [12]) is employed to design a position controller. The force controller is designed as a proportional force error feedback of high gain type. The coordination of the position controller and the force controller is shown to achieve ultimately bounded position and force tracking with tunable accuracy. Moreover, an estimate of the domain of attraction is provided for the motion on the constraint manifold. Simulation for a planar two-link robot constraining on an ellipse is given to show the effectiveness of a hybrid controller. In addition, the friction effect, viewed as external disturbance to the system, is also examined through simulations.