A polynomial approach to nonlinear system controllability

Uses a polynomial approach to present a necessary and sufficient condition for local controllability of single-input-single-output (SISO) nonlinear systems. The condition is presented in terms of common factors of a noncommutative polynomial expression. This result exposes controllability properties of a nonlinear system in the input-output framework, and gives a computable procedure for examining nonlinear system controllability using computer algebra     

Exact and approximate state estimation for nonlinear dynamic systems

This paper presents a modified approach to solve state estimation problems of nonlinear dynamic systems involving noise free, uncorrelated and correlated state and measurement noise processes. The basic approach makes use of the matrix minimum principle together with the Kolmogorov and Kushner's equations to minimize the error-variance, taken to be the estimation criterion. The filtering equations obtained for nonlinear systems with white noise process are exact, but for non-white noise processes the results obtained are approximate.

For systems with polynomial or product types non-linearities, the proposed algorithms can be evaluated without the need of approximation under the assumption that the estimator errors are Gaussian. Such an assumption is significantly different from the most commonly used assumption that the state is Gaussian. Simulation results obtained from the proposed filtering algorithms are compared to various other approximate nonlinear filters. The results indicate the superiority of the proposed filter over those of other filters investigated.     

Global robust output regulation for output feedback systems

The global robust output regulation problem for the class of nonlinear systems in output feedback form has been studied under the assumption that the solution of the regulator equations is polynomial. This assumption essentially requires these systems contain only polynomial nonlinearity and is due to the failure of finding a nonlinear internal model to account for more complex nonlinearities than polynomials. Recently, it was found that a nonlinear internal model can be constructed under some assumption much milder than the polynomial assumption. In this note, we will apply this type of internal model to solve the global robust output regulation problem for the class of nonlinear systems in output feedback form.     

Fundamental limitations of discrete-time adaptive nonlinear control

A particular polynomial is introduced which can be used to determine under what conditions a typical class of discrete-time nonlinear systems with uncertainties in both parameters and noises is not stabilizable by feedback, thus demonstrating the fundamental limitations of discrete-time adaptive nonlinear control. As a consequence, it is shown that for nonlinear systems with unknown parameters and noises, the systems may indeed be nonstabilizable, in general, whenever the usual linear growth condition is relaxed and the number of unknown parameters is large, even though the corresponding noise-free systems are globally stabilizable     

Stabilizing nonlinear Markov jump systems with orthogonality between control and nonlinear terms

This article presents conditions to assure the second‐moment stability for a class of nonlinear Markov jump systems. The main assumption requires that the control and nonlinear terms form orthogonal vectors, a novel condition in the context of Markov jump systems. The second‐moment stability is verified through linear matrix inequalities—the conditions account the case in which the Markovian mode is either accessible or inaccessible to the controller. A real‐time automotive application illustrates the potential benefits of our approach.     

On global controllability for a class of polynomial affine nonlinear systems

In this paper,a necessary and sufficient condition of the global controllability for a class of low dimensional polynomial affine nonlinear systems with special structure is obtained.The condition is imposed on the coefficients of the system only and the methods are based on Green’s formula and the trajectory analysis of planar linear system.Furthermore, I point out that the global controllability does not hold for the corresponding high dimensional polynomial system.     

On strong controllability for planar affine nonlinear systems

Different to linear systems, a controllable nonlinear system does not generally imply that it is strongly controllable. This paper will investigate the strong controllability of planar affine nonlinear systems and obtain its necessary and sufficient condition by introducing the variation function of the control curve. These conditions are imposed on the system structure only. In addition, we also point out that, for a class of polynomial systems, their strong controllability is equivalent to their controllability. Finally, some examples are given to show the application of our results.     

Global stabilization of nonlinear cascaded systems with a Lyapunov function in superposition form

The global stabilization problem of nonlinear cascaded systems has been well studied in literature. In particular, a Lyapunov function in superposition form has been explicitly constructed for the closed-loop system in a recent paper provided the nonlinearities are polynomial. This paper removes this polynomial assumption and gives a more general result. For this purpose, a special version of changing supply function technique is utilized which preserves the superposition form of supply functions during the “changing” procedure.     

Semi-global robust output regulation of minimum-phase nonlinear systems based on high-gain nonlinear internal model

We consider the assumption of existence of the general nonlinear internal model that is introduced in the design of robust output regulators for a class of minimum-phase nonlinear systems with rth degree (r ≥ 2). The robust output regulation problem can be converted into a robust stabilisation problem of an augmented system consisting of the given plant and a high-gain nonlinear internal model, perfectly reproducing the bounded including not only periodic but also nonperiodic exogenous signal from a nonlinear system, which satisfies some general immersion assumption. The state feedback controller is designed to guarantee the asymptotic convergence of system errors to zero manifold. Furthermore, the proposed scheme makes use of output feedback dynamic controller that only processes information from the regulated output error by using high-gain observer to robustly estimate the derivatives of the regulated output error. The stabilisation analysis of the resulting closed-loop systems leads to regional as well as semi-global robust output regulation achieved for some appointed initial condition in the state space, for all possible values of the uncertain parameter vector and the exogenous signal, ranging over an arbitrary compact set.     

Direct adaptive control for nonlinear matrix second-order dynamical systems with state-dependent uncertainty

A direct adaptive control framework for a class of nonlinear matrix second-order dynamical systems with state-dependent uncertainty is developed. The proposed framework guarantees global asymptotic stability of the closed-loop system states associated with the plant dynamics without requiring any knowledge of the system nonlinearities other than the assumption that they are continuous and lower bounded. Generalizations to the case where the system nonlinearities are unbounded are also considered. In the special case of matrix second-order systems with polynomial nonlinearities with unknown coefficients and unknown order, we provide a universal adaptive controller that guarantees closed-loop stability of the plant states.     

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.     

Discrete-time optimal control for stochastic nonlinear polynomial systems

This paper presents a solution to the discrete-time optimal control problem for stochastic nonlinear polynomial systems over linear observations and a quadratic criterion. The solution is obtained in two steps: the optimal control algorithm is developed for nonlinear polynomial systems by considering complete information when generating a control law. Then, the state estimate equations for discrete-time stochastic nonlinear polynomial system over linear observations are employed. The closed-form solution is finally obtained substituting the state estimates into the obtained control law. The designed optimal control algorithm can be applied to both distributed and lumped systems. To show effectiveness of the proposed controller, an illustrative example is presented for a second degree polynomial system. The obtained results are compared to the optimal control for the linearized system.     

Feedback stabilization for high order feedforward nonlinear time-delay systems

This paper investigates the problem of global strong stabilization by state feedback, for a family of high order feedforward nonlinear time-delay systems. The uncertain nonlinearities are assumed to satisfy a polynomial growth assumption with an input or delayed input dependent rate. With the help of the appropriate Lyapunov–Krasovskii functionals, and a rescaling transformation with a gain to be tuned online by a dynamic equation, we propose a dynamic low gain state feedback control scheme. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.     

Discrete-time controller for stochastic nonlinear polynomial systems with Poisson noises

This paper presents a solution to the optimal control problem for discrete-time stochastic nonlinear polynomial systems confused with white Poisson noises subject to a quadratic criterion. The solution is obtained in the following way: a nonlinear optimal controller is first developed for polynomial systems, considering the state vector completely available for control design. Then, based on the solution of the state estimation problem for polynomial systems with white Poisson noises, the state estimate vector is used in the control law to obtain a closed-form solution. Performance of this controller is compared to that of the controller employing the extended Kalman filter and the linear-quadratic regulator and the controller designed for polynomial systems confused with white Gaussian noises.     

Robust output tracking of nonlinear systems with mismatched uncertainties

The problem of robust output tracking for a class of uncertain nonlinear systems which do not satisfy the conventional matching condition is considered. The main assumption on the uncertainty is that the triangularity condition is satisfied. Based on backstepping method and input/output linearization approach, we propose a class of non-adaptive state feedback controllers which can guarantee exponential stability of the tracking error for the uncertain nonlinear systems first. Next, adaptive control laws are developed so that no prior knowledge of the bounds on the uncertainties is required. By updating these upper bounds, we design a class of adaptive robust controllers. It is shown that under the proposed adaptive robust control the tracking error of the controlled system converges to zero as time approaches infinity.     

A stability criterion for a class of discrete nonlinear systems with random parameters

In this paper, discrete nonlinear models with random parameters are studied. A new frame to classify and analyze discrete stochastic nonlinear systems has been developed from deterministic nonlinear systems to stochastic nonlinear systems. This frame is broad and includes a large class of stochastic nonlinear systems. A stability criterion developed for this frame is a non-Lyapunov method of stability analysis and is easily applied. In addition, this derived sufficient condition of stability is obtained without the assumption of stationarity for the random noise as frequently assumed in the literature.     

Robust output regulation for nonlinear systems with nonpolynomial nonlinearity under nonlinear exosystems

This paper studies a robust output regulation problem with nonlinear exosystems without the assumption that the solution to the regulator equations along the trajectory of the exosystems should be polynomial. Based on the extended concepts of steady-state generator and internal model, a set of sufficient conditions for the solvability of the problem are given. Thus, the result developed in this paper can allow the robust output regulation problem with nonlinear exosystems to accommodate much more general nonlinear given plants including those which contain nonpolynomial nonlinearity.     

A general boundedness result for interconnected nonlinear systems

A general boundedness result is given for the feedback interconnection of two nonlinear stable systems. Using the same input-output approach as in the standard small-gain theorem, the sufficient condition given here relaxes the finite-gain stability assumption and does not require a boundedness result for all possible bounded exogenous inputs. This condition has a simple graphical interpretation that utilizes graphs of bounding functions