Adaptive control of nonlinear systems via approximate linearization

This paper presents an adaptive control scheme for nonlinear systems that violates some of the common regularity and structural conditions of current nonlinear adaptive schemes such as involutivity, existence of a well-defined relative degree, and minimum phase property. While the controller is designed using an approximate model with suitable properties, the parameter update law is derived from an observation error based on the exact model described in suitable coordinates. The authors show that this approach results in a stable, closed-loop system and achieves adaptive tracking with bounds on the tracking error and parameter estimates. The authors also present a constructive procedure for adaptive state regulation which is based on the quadratic linearization technique via dynamic state feedback. This regulation scheme does not impose any restriction on the location of the unknown parameters and is applicable to any linearly controllable nonlinear system     

Composite control of a class of nonlinear singularly perturbed discrete-time systems via D-SDRE

In this paper, the regulation problem of a class of nonlinear singularly perturbed discrete-time systems is investigated. Using the theory of singular perturbations and time scales, the nonlinear system is decoupled into reduced-order slow and fast (boundary layer) subsystems. Then, a composite controller consisting of two sub-controllers for the slow and fast subsystems is developed using the discrete-time state-dependent Riccati equation (D-SDRE). It is proved that the equilibrium point of the original closed-loop system with a composite controller is locally asymptotically stable. Moreover, the region of attraction of the closed-loop system is estimated by using linear matrix inequality. One example is given to illustrate the effectiveness of the results obtained.     

First-order corrections in optimal feedback control of singularly perturbed nonlinear systems

In this paper first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications. This improvement is based on an explicit solution of the integrals arising from the first-order matching conditions and leads to correct the initial values of the slow costate variables in the boundary layer. Consequently, a uniformly valid feedback control law, corrected to the first-order, can be synthesized. The new method is applied to an example of a constant speed minimum-time interception problem. Comparison of the zeroth- and first-order feedback control laws to the exact optimal solution demonstrates that first-order corrections greatly extend the domain of validity of the approximation obtained by singular perturbation methods.     

On approximate linearization of nonlinear control systems

A method for the approximate linearization of nonlinear control systems based on the ‘state-space exact linearization’ method is presented. An explicit procedure, both for the single-input and for the multiple-input case, is given, which is straightforward to implement.     

Global external linearization of nonlinear systems via feedback

This note presents necessary conditions and sufficient conditions for an affine nonlinear system to be globally feedback equivalent to a controllable linear system over an open subsetVof Rⁿ. WhenVequals Rⁿ, necessary and sufficient conditions are obtained.     

Economic model predictive control of nonlinear singularly perturbed systems

We focus on the development of a Lyapunov-based economic model predictive control (LEMPC) method for nonlinear singularly perturbed systems in standard form arising naturally in the modeling of two-time-scale chemical processes. A composite control structure is proposed in which, a “fast” Lyapunov-based model predictive controller (LMPC) using a quadratic cost function which penalizes the deviation of the fast states from their equilibrium slow manifold and the corresponding manipulated inputs, is used to stabilize the fast dynamics while a two-mode “slow” LEMPC design is used on the slow subsystem that addresses economic considerations as well as desired closed-loop stability properties by utilizing an economic (typically non-quadratic) cost function in its formulation and possibly dictating a time-varying process operation. Through a multirate measurement sampling scheme, fast sampling of the fast state variables is used in the fast LMPC while slow-sampling of the slow state variables is used in the slow LEMPC. Appropriate stabilizability assumptions are made and suitable constraints are imposed on the proposed control scheme to guarantee the closed-loop stability and singular perturbation theory is used to analyze the closed-loop system. The proposed control method is demonstrated through a nonlinear chemical process example.     

Constrained feedback control of imperfectly known singularly perturbed non-linear systems

Global attractors are investigated for a class of imperfectly known, singularly perturbed, dynamic control systems. The uncertain systems are modelled as non-linear perturbations to a known non-linear idealized system and are represented by two time-scale subsystems. The two subsystems, which depend on a scalar singular perturbation parameter, represent a singularly perturbed system which has the property that the system reduces to one of lower order when the singular perturbation parameter is set to zero. It is assumed that the full-order system is subject to constraints on the control inputs. A class of constrained feedback controllers is developed which assures global uniform attraction of a compact set, containing the state origin, for all values of the singular perturbation parameter less than some threshold value.     

A state space embedding approach to approximate feedback linearization of single input nonlinear control systems

This contribution presents a numerical approach to approximate feedback linearization which transforms the Taylor expansion of a single input nonlinear system into an approximately linear system by considering the terms of the Taylor expansion step by step. In the linearization procedure, higher degree terms are taken into account by using a state space embedding such that the corresponding system representation has not to be computed in every linearization step. Linear matrix equations are explicitly derived for determining the nonlinear change of coordinates and the nonlinear feedback that approximately linearize the nonlinear system. If these linear matrix equations are not solvable, a least square solution by applying the Moore–Penrose inverse is proposed. The results of the paper are illustrated by the approximate feedback linearization of an inverted pendulum on a cart. Copyright © 2006 John Wiley & Sons, Ltd.     

一类非线性奇异摄动系统的近似最优控制

针对一类非线性奇异摄动系统,基于自适应动态规划算法提出了一种新型的近似最优控制设计方法.该方法基于奇异摄动系统的快、慢Hamilton-Jacobi-Bellman(HJB)方程,从初始性能指标开始,通过神经网络的近似和控制律与性能指标的逐步更新迭代,最终收敛到最优的性能指标,而不用直接求解复杂的HJB方程.同时给出了...     

Contraction-based stabilisation of nonlinear singularly perturbed systems and application to high gain feedback

Recent development of contraction theory-based analysis has opened the door for inspecting differential behaviour of singularly perturbed systems. In this paper, a contraction theory-based framework is proposed for stabilisation of singularly perturbed systems. The primary objective is to design a feedback controller to achieve bounded tracking error for both standard and non-standard singularly perturbed systems. This framework provides relaxation over traditional quadratic Lyapunov-based method as there is no need to satisfy interconnection conditions during controller design algorithm. Moreover, the stability bound does not depend on smallness of singularly perturbed parameter and robust to additive bounded uncertainties. Combined with high gain scaling, the proposed technique is shown to assure contraction of approximate feedback linearisable systems. These findings extend the class of nonlinear systems which can be made contracting.     

On linearization of nonlinear singularity perturbed systems

This note considers external (feedback) linearization of nonlinear singularly perturbed systems. When more accurate representation of some dynamic systems is taken into account. The necessary condition for linearization may be violated. In these cases the slow manifold theory is shown to provide a practical tool for solving this problem. Furthermore, for a class of systems with fast actuators, direct application of present methodology in the literature is shown to be valid provided the linear equivalent system contains fast dynamics as well.     

Robust stabilization of singularly perturbed nonlinear systems

Consider a singularly perturbed nonlinear system endowed with a control input and suppose a static nonlinear feedback law is designed. We show that the operations ‘compute the reduced order system’ (i.e., let the singular perturbation parameter μ = 0) and ‘close the feedback loop’ commute, i.e. the closed loop reduced-order system is unambiguously determined. We then show that, if the reduced order system associated with the original system is stabilizable or has uncertainties matched with the input (a condition frequently used in the design of robust controllers), then the closed loop reduced-order system enjoys the same property. As shown by an example, this result can be used to simplify the structure of a composite controller.     

Composite control of non-linear singularly perturbed systems: a geometric approach

Using a geometric approach, a composite control—the sum of a slow control and a fast control—is derived for a general class of non-linear singularly perturbed systems. A new and simpler method of composite control design is proposed whereby the fast control is completely designed at the outset. The slow control is then free to be chosen such that the slow integral manifold of the original system approximates a desired design manifold to within any specified order of ε accuracy.     

Input linearization of nonlinear systems via pulse-width control

In this note, it is shown that a general nonlinear system can be transformed to an affine nonlinear system by virtue of the use of pulsewidth control. Therefore, the combined system from the pulse width control input to the nonlinear system output behaves as an affine (linear-in-control) system. By this way, a cumbersome nonlinear system model can be transformed to a simpler linear-in-control form without increasing the system dimension; this in turn enables simpler control design. Furthermore, using this methodology, some control design methods developed only for affine systems can be adapted to general nonlinear systems.     

Anti-windup control for nonlinear singularly perturbed switched systems with actuator saturation

This paper considers the problem of anti-windup (AW) control for nonlinear singularly perturbed switched systems with actuator saturation. An AW controller consisting of a dynamic state feedback (DSF) controller and an AW compensator is firstly constructed. Then, two methods are proposed to determine the AW controller gain matrices by a common Lyapunov function. One of which assumes that the singular perturbation parameter ? is available and designs ?-dependent AW controller gains simultaneously. The other one, which considers the case that ? is unknown but sufficiently small, designs the ?-independent DSF controller gain matrices first and then designs the ?-independent AW compensator gain matrices. Both of the methods are reduced to solving convex optimisation problems and can achieve larger stability bound and basin of attraction than the existing results. Finally, examples are used to illustrate the feasibility and advantages of the proposed methods.     

Gain scheduling control of nonlinear systems based on approximate input-output linearization

The gain scheduling control mostly has been developed based on Jacobian linearization around the operating points related with scheduling variables. In this paper, We introduce a gain scheduling control method based on approximate input-output linearization. First, the nonlinear system is approximately input-output linearized via a diffeomorphism. Then, a gain scheduling controller with derivative information is developed. The proposed controller consists of two parts. The outer loop controller is like a feedback linearizing controller and the internal controller is a gain scheduling controller. It is shown that the overall resulting controller has a simple structure and at the same time achieves better tracking performance over the existing Jacobian-based gain scheduling controller.     

Multivariable singularly perturbed feedback systems with time delay

Multivariable singularly perturbed systems with both small and large time delays in the feedback loop are considered. Conditions for stability of the full-order closed-loop system are given in terms of properties of the slow and fast subsystems.