Controller design for nonlinear affine systems by control Lyapunov functions

This paper addresses the stabilization problems for nonlinear affine systems. First of all, the explicit feedback controller is developed for a nonlinear multiple-input affine system by assuming that there exists a control Lyapunov function. Next, based upon the homogeneous property, sufficient conditions for the continuity of the derived controller are developed. And then the developed control design methodology is applied to stabilize a class of nonlinear affine cascaded systems. It is shown that under some homogeneous assumptions on control Lyapunov functions and the interconnection term, the cascaded system can be globally stabilized. Finally, some interesting results of finite-time stabilization for nonlinear affine systems are also obtained.     

一类仿射非线性网络控制系统的稳定性分析

利用采样数字控制系统的方法分析了一类混杂动态系统模型描述的仿射非线性网络控制系统的稳定性问题.针对一类仿射非线性对象和线性数字控制器组成的网络控制系统，考虑了网络诱导延时对系统稳定性的影响，得到了仿射非线性网络控制系统一致渐近稳定的条件．仿真实例验证了理论分析的正确性．     

Singular L2‐gain control for switched nonlinear control systems under arbitrary switching

This article concerns with the synthesis of L₂ ‐gain state feedback controllers, without the standard regular assumption, for multi‐input switched nonlinear control‐affine systems under arbitrary switching. A common control storage function approach is developed for deriving sufficient conditions for the existence of uniform L₂ ‐gain controllers. Moreover, an explicit formula for constructing L₂ ‐gain controllers is presented. A numerical example is given for illustration.     

Finite-time disturbance attenuation of nonlinear systems

This paper is devoted to the finite-time disturbance attenuation problem of affine nonlinear systems. Based on the finite time Lyapunov stability theory, some finite-time H _∞ performance criterions are derived. Then the state-feedback control law is designed and the structure of such a controller is investigated. Furthermore, it is shown that the H _∞ controller can also make the closed-loop system satisfy finite-time H _∞ performance for nonlinear homogeneous systems. An example is provided to demonstrate the effectiveness of the presented results.     

Output feedback design and controllable cascade connections of nonlinear systems

For linear control systems necessary and sufficient conditions have been established for transforming a linear multiinput multioutput controllable system to a single input controllable system by use of linear output feedback. This problem is studied here for nonlinear systems affine in control, extending previous work of the authors dealing with the more restricted case of state feedback. The results obtained are subsequently applied to analyze controllability of cascade connections of nonlinear systems. A notable consequence of the development presented here, is that for linear systems the above questions can be answered under milder conditions than those required by the linear theory, provided that nonlinear output feedback is allowed.     

Stochastic nonlinear stabilization - I: A backstepping design

While the current robust nonlinear control toolbox includes a number of methods for systems affine in deterministic bounded disturbances, the problem when the disturbance is unbounded stochastic noise has hardly been considered. We present a control design which achieves global asymptotic (Lyapunov) stability in probability for a class of strict-feedback nonlinear continuous-time systems driven by white noise. In a companion paper, we develop inverse optimal control laws for general stochastic systems affine in the noise input, and for strict-feedback systems. A reader of this paper needs no prior familiarity with techniques of stochastic control.     

仿射非线性奇异系统的分析

从机器人控制中抽象出一类仿射非线性广义系统,给出此类系统的简化条件及方法,讨 论了它的输出跟踪问题,并推出了可稳的充分条件.     

一类非线性系统镇定问题的分层控制方法

通过利用分层控制设计方法,研究了一类仿射非线性系统的镇定问题.为了镇定这类系统,分层方法构建了一个简单的抽象系统,然后根据抽象系统的控制律求得原始系统的控制律.最后,一个四维仿射非线性系统的仿真例子说明了这种方法的有效性.     

Guaranteed robustness bounds for matched-disturbance nonlinear systems

This paper considers input affine nonlinear systems with matched disturbances and shows how to compute an a priori upper bound of the H_∞ attenuation level achieved by the optimal L₂ controller and the suboptimal H_∞ central controller. The case where the disturbance contains a constant term is also discussed. These bounds are shown to depend only on the function mapping the control input to the performance variable. This result is used to derive a robust control design for a special, but practically important, class of non-input affine nonlinear systems consisting of the series connection of a nonlinear state and input dependent map and of a nonlinear input affine dynamical system. Approximate inversion of the nonlinear static map leads to a robust control problem which fits into the framework. The effectiveness of the theoretical results is shown by its use for the robust control design of a diesel engine test bench.     

State-feedback ℋ∞ control for stochastic time-delay nonlinear systems with state and disturbance-dependent noise

This article focuses on the state-feedback ?_∞ control problem for the stochastic nonlinear systems with state and disturbance-dependent noise and time-varying state delays. Based on the maxmin optimisation approach, both the delay-independent and the delay-dependent Hamilton–Jacobi-inequalities (HJIs) are developed for synthesising the state-feedback ?_∞ controller for a general type of stochastic nonlinear systems. It is shown that the resulting control system achieves stochastic stability in probability and the prescribed disturbance attenuation level. For a class of stochastic affine nonlinear systems, the delay-independent as well as delay-dependent matrix-valued inequalities are proposed; the resulting control system satisfies global asymptotic stability in the mean-square sense and the required disturbance attenuation level. By modelling the nonlinearities as uncertainties in corresponding stochastic time-delay systems, the sufficient conditions in terms of a linear matrix inequality (LMI) and a bilinear matrix inequality (BMI) are derived to facilitate the design of the state-feedback ?_∞ controller. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed methods.     

Determining Dissipativity of Switched Nonlinear Systems Using Linearization

Local strict QSR‐dissipativity of a switched nonlinear system is studied using the linearization technique in this paper. We obtain local strict QSR‐dissipativity of a switched system even if each subsystem is not locally strictly QSR dissipative by designing a switching law. The derived dissipative sufficient condition is characterized by a modified Lyapunov‐Metzler inequality that can be simplified as an LMI by assuming specific forms. Two special forms of local strict QSR‐dissipativity, local input state strict passivity and local L₂‐gain, are considered. When the approximate errors of a switched affine system satisfy certain conditions, local strict passivity can be drawn from its linearization. Finally, a numerical example is given to illustrate how to apply the proposed method to achieve passivity of switched nonlinear systems.     

SENSITIVITY APPROACH TO OPTIMAL CONTROL FOR AFFINE NONLINEAR DISCRETE‐TIME SYSTEMS

This paper deals with the optimal control problem for a class of affine nonlinear discrete‐time systems. By introducing a sensitivity parameter and expanding the system variables into a Maclaurin series around it, we transform the original optimal control problem for affine nonlinear discrete‐time systems into the optimal control problem for a sequence of linear discrete‐time systems. The optimal control law consists of an accurate linear term and a nonlinear compensating term, which is an infinite sequence of adjoint vectors. In the present approach, iteration is required only for the nonlinear compensation series. By intercepting a finite sum of the series, we obtain a suboptimal control law that reduces the complexity of the calculations. A numerical simulation shows that the algorithm can be easily implemented and has a fast convergence rate.     

不确定非线性系统的鲁棒H∞控制分析与综合

讨论一类仿射非线性系统的鲁棒H_∞ 控制问题. 针对自由系统, 外部干扰输入矩阵和控制输入矩阵都具有加法不确定性的系统, 将鲁棒性能的分析和综合归结为求解一组Hamilton_Jacobi不等式, 给出了确保非线性控制系统满足鲁棒干扰抑制性能的充分条件, 获得了非线性鲁棒控制器的一种形式. 一个仿真例子验证了所得结论的正确性.     

Stabilization of a class of switched nonlinear systems

The stabilization of a class of switched nonlinear systems is investigated in the paper. The systems con- cerned 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 prob- lem 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.     

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.     

Learning from neural control of nonlinear systems in normal form

A deterministic learning theory was recently proposed which states that an appropriately designed adaptive neural controller can learn the system internal dynamics while attempting to control a class of simple nonlinear systems. In this paper, we investigate deterministic learning from adaptive neural control (ANC) of a class of nonlinear systems in normal form with unknown affine terms. The existence of the unknown affine terms makes it difficult to achieve learning by using previous methods. To overcome the difficulties, firstly, an extension of a recent result is presented on stability analysis of linear time-varying (LTV) systems. Then, with a state transformation, the closed-loop control system is transformed into a LTV form for which exponential stability can be guaranteed when a partial persistent excitation (PE) condition is satisfied. Accurate approximation of the closed-loop control system dynamics is achieved in a local region along a recurrent orbit of closed-loop signals. Consequently, learning of control system dynamics (i.e. closed-loop identification) from adaptive neural control of nonlinear systems with unknown affine terms is implemented.     

Mixed ℋ︁2/ℋ︁∞ nonlinear filtering

In this paper, we consider the mixed ??₂/??_∞ filtering problem for affine nonlinear systems. Sufficient conditions for the solvability of this problem with a finite‐dimensional filter are given in terms of a pair of coupled Hamilton–Jacobi–Isaacs equations (HJIEs). For linear systems, it is shown that these conditions reduce to a pair of coupled Riccati equations similar to the ones for the control case. Both the finite‐horizon and the infinite‐horizon problems are discussed. Simulation results are presented to show the usefulness of the scheme, and the results are generalized to include other classes of nonlinear systems. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Control of nonstationary LPV systems

This paper considers control of nonstationary linear parameter-varying systems, and is motivated by interest in the control of nonlinear systems along prespecified trajectories. In the paper, synthesis conditions are derived for such systems using an operator theoretical framework with the ?₂ induced norm as the performance measure. These conditions are given in terms of structured operator inequalities. In general, evaluating the validity of these conditions is an infinite dimensional convex optimization problem; however, if the initial system is eventually periodic, they reduce to a finite dimensional semi-definite programming problem. The paper concludes with an in-depth example on the control of a two-thruster hovercraft along an eventually periodic trajectory.