Control and stabilization of nonholonomic dynamic systems

A class of inherently nonlinear control problems has been identified, the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system. Models are presented for mechanical systems with nonholonomic constraints represented both by differential-algebraic equations and by reduced state equations. Control issues for this class of systems are studied and a number of fundamental results are derived. Although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback, a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested     

Quadratic stability and stabilization of dynamic interval systems

This note presents necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems. The results are obtained in terms of linear matrix inequalities (LMIs) and extended to the quadratic stability and stabilization of linear systems with uncertain parameters. With the powerful LMI toolbox, it is very convenient to solve these problems. The illustrative examples show that this method is effective and less conservative to check the robust stability and to design the stabilizing controller for dynamic interval systems.     

Adaptive stabilization of uncertain dynamic non-holonomic systems

This paper investigates the stabilization problem of uncertain dynamic non-holonomic chained systems with unknown constant inertia parameters. Globally time-varying adaptive stabilizing control laws for such systems are presented. The application of it to a non-holonomic wheeled mobile robot is described. Simulation results show that the proposed approach is effective.     

On stabilization of uncertain dynamic nonholonomic systems

This paper considers the stabilization problem of uncertain dynamic nonholonomic systems. New robust and adaptive robust control laws are presented with an aim to stabilize the system to the origin with a simple design procedure and no extensive online computations. The designed controllers have been implemented in a nonholonomic wheeled mobile robot, and the application are discussed. Simulation study demonstrates the effectiveness of the proposed method.     

Adaptive robust stabilization of dynamic nonholonomic chained systems

In this article, the stabilization problem is investigated for dynamic nonholonomic systems with unknown inertia parameters and disturbances. First, to facilitate control system design, the nonholonomic kinematic subsystem is transformed into a skew‐symmetric form and the properties of the overall systems are discussed. Then, a robust adaptive controller is presented in which adaptive control techniques are used to compensate for the parametric uncertainties and sliding mode control is used to suppress the bounded disturbances. The controller guarantees the outputs of the dynamic subsystem (the inputs to the kinematic subsystem) to track some bounded auxiliary signals which subsequently drive the kinematic subsystem to the origin. In addition, it can also be shown all the signals in the closed loop are bounded. Simulation studies on the control of a unicycle wheeled mobile robot are used to show the effectiveness of the proposed scheme. © 2001 John Wiley & Sons, Inc.     

On stabilization of decentralized dynamic output feedback systems

A dynamic output feedback law is presented for large scale interconnected systems. If the coupling terms between the subsystems are neglected, the control scheme can be interpreted as decentralized reconstruction of the state variables from the output of the subsystems. A class of systems stabilizable by decentralized state feedback is investigated. A sufficient condition is determined for systems to be stabilizable with local state feedback and reconstruction with observers of full order. The resulting closed-loop systems are connectively stable. Moreover, any prescribed degree of stability can be achieved and the closed-loop systems are robust with respect to bounded nonlinearities in the couplings between the subsystems.     

基于分散动态补偿的矩形广义系统的正则化与镇定

采用代数方法研究基于分散动态补偿的矩形广义系统的正则化、无脉冲, 以及镇定问题. 首先给出了补偿后闭环系统正则与无脉冲的充要条件, 进而给出矩形广义系统能通过分散动态补偿镇定的充要条件. 这些条件涉及一系列简单不等式与等式是否存在正整数解问题. 所得结果揭示矩形系统及其动态补偿器的许多新的性质, 而且进一步说明对应方形系统与方形或矩形补偿器的结果仍是矩形系统结果的特例. 因而, 本文结果可以认为是方形系统相应结果的自然推广. 另外, 给出几个数字例子说明所得结果.     

Robust stabilization of non-linear systems by discontinuous dynamic state feedback

In this paper, a discontinuous dynamic state feedback that robustly stabilizes affine in control uncertain non-linear systems is proposed. The formulation is based on Hamilton-Jacobi-Isaacs partial differential equations with two boundary conditions. The resulting dynamic state feedback is then expressed in terms of the solution of the related PDEs. An interesting feature is that the internal state of the resulting dynamic state feedback may have discontinuous behaviour as a function of time. The proposed scheme is illustrated through the example of the stabilization of the angular velocities of a rigid body under two actuators.     

Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control

A general dynamic model is proposed for describing a large class of nonholonomic systems including extended chained systems, extended power systems, underactuated surface vessel systems etc. By introducing an assistant state variable and a time-varying state transformation based on the concept of minimal dilation degree, this class of nonholonomic systems is transformed into linear time-varying control systems, and the asymptotic exponential stability is thus achieved by using a smooth time-varying feedback control law. The existence and uniqueness of the minimal dilation degree for the discussed systems are also proved under certain conditions.     

Comments on "Quadratic stability and stabilization of dynamic interval systems"

In the above paper, it was claimed that the necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems are given in terms of linear matrix inequalities (LMIs). In this note, numerical examples are presented to show that the necessity of the conditions given in does not hold.     

Observer-based feedback stabilization of linear systems with event-triggered sampling and dynamic quantization

We consider the problem of output feedback stabilization in linear systems when the measured outputs and control inputs are subject to event-triggered sampling and dynamic quantization. A new sampling algorithm is proposed for outputs which does not lead to accumulation of sampling times and results in asymptotic stabilization of the system. The approach for output sampling is based on defining an event function that compares the difference between the current output and the most recently transmitted output sample not only with the current value of the output, but also takes into account a certain number of previously transmitted output samples. This allows us to reconstruct the state using an observer with sample-and-hold measurements. The estimated states are used to generate a control input, which is subjected to a different event-triggered sampling routine; hence the sampling times of inputs and outputs are asynchronous. Using Lyapunov-based approach, we prove the asymptotic stabilization of the closed-loop system and show that there exists a minimum inter-sampling time for control inputs and for outputs. To show that these sampling routines are robust with respect to transmission errors, only the quantized (in space) values of outputs and inputs are transmitted to the controller and the plant, respectively. A dynamic quantizer is adopted for this purpose, and an algorithm is proposed to update the range and the centre of the quantizer that results in an asymptotically stable closed-loop system.     

Output feedback stabilization of SISO nonlinear systems via dynamic sliding modes

Previous work on asymptotic stabilization of SISO nonlinear systems using dynamic sliding mode control to produce dynamic state feedback has been developed further to produce dynamic output feedback. All the states in the feedback controller are replaced with estimated states which come from a semi-high gain observer. An upper bound of the semi-high gain parameter is explicitly given. It is proved that if the given differential I-O system is minimum phase and proper, local uniform asymptotic output feedback stabilization can be achieved. Pertinent examples are given to show how the method works for both differential inputoutput models and state space models.     

Robust output feedback stabilization of uncertain dynamic systems with bounded controllers

We propose a new class of infinitely many bounded output feedback controllers for uncertain dynamic systems with bounded uncertainties. No statistical information about the uncertainties is assumed. A variable structure systems approach is employed in the synthesis of the proposed output feedback controllers. The role of the system zeros in the output feedback stabilization using the variable structure approach is discussed. We show that the proposed controllers guarantee the practical stability of the closed-loop system and give estimates of the regions of practical stability.     

Memoryless stabilization of uncertain dynamic delay systems:Riccati equation approach

The authors present a procedure for obtaining the memoryless linear state feedback control of uncertain dynamic delay systems. The uncertainties are time varying and within a given compact set. This method is an extension of the Riccati equation approach proposed by I.R. Petersen and C.V. Hollot (1986). The extension is straightforward. Also the uncertainties do not need to satisfy the matching conditions     

Output feedback stabilization of MIMO non-linear systems via dynamic sliding mode

Previous work on asymptotic stabilization of MIMO non-linear systems using dynamic sliding mode control to produce dynamic state feedback has been generalized to dynamic output feedback. All the states in the feedback controller are replaced with estimated states which come from a semi-high-gain observer. The bound on the observer gain is explicitly given, which depends on some previously chosen design parameters. If the given differential input–output (I–O) system is minimum phase and proper, local uniform asymptotic output feedback stabilization can be achieved. In addition, the restriction on the stability of the zero dynamics has been relaxed to weakly minimum phase and semi-global results are obtained under some mild conditions. The stability of the closed-loop system is based on some stability results of triangular systems when continuous or discontinuous controllers are adopted. Two pertinent examples are given to illustrate the design method. Copyright © 1999 John Wiley & Sons, Ltd.     

Decentralized adaptive stabilization for interconnected systems with dynamic input-output and nonlinear interactions

This paper considers the problem of robust decentralized adaptive output feedback stabilization for a class of interconnected systems with dynamic input and output interactions and nonlinear interactions by using MT-filters and the backstepping design method. It is shown that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the L₂ and L_∞ norms of the system outputs are also be bounded by functions of design parameters. The scheme is demonstrated by a simulation example.     

具有动态和静态关联项大系统的鲁棒分散自适应镇定

对于一类具有动态和静态关联项的大系统，利用MT-滤波器和反推设计方法设计了一种鲁棒分散自适应输出反馈控制器．在各子系统为最小相位、静态关联项满足Lipschitz条件、动态关联项稳定且严格正则的假设下，证明了闭环系统的所有信号全局一致有界，且除了参数估计外的其他信号皆以指数速率收敛到零，该结果好于其他相关文献中的结果，仿真结果进一步验证了该方法的有效性。     

Decentralized adaptive backstepping stabilization of interconnected systems with dynamic input and output interactions

So far there is still no result available for backstepping based decentralized adaptive stabilization of unknown systems with interactions directly depending on subsystem inputs, even though such interactions commonly exist in practice. In this paper, we provide a solution to this problem by considering both input and output dynamic interactions. To clearly illustrate our approaches, we will start with linear systems and then extend the results to nonlinear systems. Each local controller, designed simply based on the model of each subsystem by using the standard adaptive backstepping technique without any modification, only employs local information to generate control signals. It is shown that the designed decentralized adaptive backstepping controllers can globally stabilize the overall interconnected system asymptotically. The L₂ and L_∞ norms of the system outputs are also established as functions of design parameters. This implies that the transient system performance can be adjusted by choosing suitable design parameters.