Contributions to non-identifier based adaptive control in mechatronics

Funnel-Control (FC)–an adaptive (time-varying) MIMO/SISO control strategy–is re-introduced and its applicability introductory for position control of nonlinear, coupled (rigid) robotic systems and for speed control of nonlinear two-mass flexible servo systems is shown. Additionally Error Reference Control (ERC)–a direct derivative of FC–is established. ERC is specially designed with asymmetric boundaries and auxiliary reference, ensuring that the control error evolves within a prespecified tube (a shrinked funnel region, achieving even better tracking performance).FC and hence ERC are based on the high-gain stabilizability of minimum-phase systems with relative degree one and known sign of the high-frequency gain. Although the plant only needs to be known in structure, both concepts assure prescribed transient behavior without identification and/or parameter estimation.As most industrial applications, also the considered robotic and two-mass flexible servo systems, exhibit relative degrees greater than one, FC and ERC cannot directly be applied. Therefore a special state feedback is introduced, reducing the relative degree and retaining the minimum-phase property. The additional implementation of a nominal PI-like extension guarantees good disturbance rejection and asymptotic tracking of (constant) velocity and position reference trajectories.Simulation results for a 6-DOF Manutec r3 robot underpin and compare the achievable position control performance of one overall MIMO Funnel Controller for all joints and one SISO Funnel Controller for each joint (6 controllers). For nonlinear two-mass flexible servo systems, measurement results demonstrate the achievable load speed control performance of FC and ERC in comparison to optimal LQ state feedback.     

Some remarks on adaptive stabilization of infinite-dimensional systems

It is the purpose of this note to show that a first-order adaptive controller stabilizes a large class of infinite-dimensional systems described by strongly continous semigroups. It is assumed that the plant is minimum-phase and has invertible high-frequency gain. Knowledge of the sign of the high-frequency gain is not required.     

一类带不确定输入动态非线性系统的输出反馈鲁棒镇定

研究一类带不确定输入动态非线性系统的输出反馈鲁棒镇定问题.通过在高增益观测器引入新的设计参数,改进了通常的高增益反馈控制的设计方法.在输入动态满足零相对阶最小相位的假设下,基于非分离设计原则给出了动态输出反馈控制器的设计方法,所设计的控制器实现了对任意可允许不确定输入动态的全局鲁棒镇定.     

A new class of nonlinear PID controllers with robotic applications

This paper introduces a new class of simple nonlinear PID controllers and provides a formal treatment of their stability analysis. These controllers are comprised of a sector-bounded nonlinear gain in cascade with a linear fixed-gain P, PD, PI, or PID controller. Three simple nonlinear gains are proposed: the sigmoidal function, the hyperbolic function, and the piecewise–linear function. The systems to be controlled are assumed to be modeled or approximated by second-order transfer functions, which can represent many robotic applications. The stability of the closed-loop systems incorporating nonlinear P, PD, PI, and PID controllers are investigated using the Popov stability criterion. It is shown that for P and PD controllers, the nonlinear gain is unbounded for closed-loop stability. For PI and PID controllers, simple expressions are derived that relate the controller gains and system parameters to the maximum allowable nonlinear gain for stability. A numerical example is given for illustration. The stability of partially-nonlinear PID controllers is also discussed. Finally, the nonlinear PI controller is implemented as a force controller on a robotic arm and experimental results are presented. These results demonstrate the superior performance of the nonlinear PI controller relative to a fixed-gain PI controller. © 1998 John Wiley & Sons, Inc. 15: 161–181, 1998     

Error convergence in an adaptive iterative learning controller

Stability theorems for an adaptive iterative learning control (ILC) system are motivated and described in compact terms that relate the properties of the schemes to systems structure and other important aspects of systems dynamics. The use of high gain feedback is reviewed and a full proof of the convergence of a 'Universal' adaptive scheme based on adaptive gain concepts is given. The results indicate clearly that successful ILC can be achieved in the presence of substantial uncertainty in the detailed knowledge of plant parameters and order. They also suggest that the form and success of the controller will depend crucially on the plant's relative degree and also on its minimum-phase properties.     

Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics

This paper examines the robustness properties of existing adaptive control algorithms to unmodeled plant high-frequency dynamics and unmeasurable output disturbances. It is demonstrated thai there exist two infinite-gain operators in the nonlinear dynamic system which determines the time-evolution of output and parameter errors. The pragmatic implication of the existence of such infinite-gain operators is that 1) sinusoidal reference inputs at specific frequencies and/or 2) sinusoidal output disturbances at any frequency (including dc), can cause the loop gain to increase without bound, thereby exciting the unmodeled high-frequency dynamics, and yielding an unstable control system. Hence, it is concluded that existing adaptive control algorithms as they are presented in the literature referenced in this paper, cannot be used with confidence in practical designs where the plant contains unmodeled dynamics because instability is likely to result. Further understanding is required to ascertain how the currently implemented adaptive systems differ from the theoretical systems studied here and how further theoretical development can improve the robustness of adaptive controllers.     

A framework for stability analysis of high-order nonlinear systems based on the CMAC method

A framework for analyzing the stability of a class of high-order minimum-phase nonlinear systems of relative degree two based on the characteristic model-based adaptive control (CMAC) method is presented. In particular, concerning the tracking problem for such high-order nonlinear systems, by introducing a consistency condition for quantitatively describing modeling errors corresponding to a group of characteristic models together with a certain kind of CMAC laws, we prove closed-loop stability and show that such controllers can make output tracking error arbitrarily small. Furthermore, following this framework, with a specific characteristic model and a golden-section adaptive controller, detailed sufficient conditions to stabilize such groups of highorder nonlinear systems are presented and, at the same time, tracking performance is analyzed. Our results provide a new perspective for exploring the stability of some high-order nonlinear plants under CMAC, and lay certain theoretical foundations for practical applications of the CMAC method.     

Robust output tracking for non-linear systems

The problem of output tracking for a single-input single-output non-linear system in the presence of uncertainties is studied. The notions relative degree and minimum-phase for non-linear systems are reviewed. Given a bounded desired tracking signal with bounded derivatives, a control law is designed for minimum-phase non-linear systems which results in tracking of this signal by the output. This control law is modified in the presence of uncertainties associated with the model vector fields to reduce the effects of these uncertainties on the tracking errors. Two types of uncertainties are considered: those satisfying a generalized matching condition but otherwise unstructured, and linear parametric uncertainties. It is shown that for systems with the first type of uncertainty, high-gain control laws can result in small tracking errors of O(?), where e is a small design parameter. An alternative scheme based on variable structure control strategy is shown to yield zero tracking errors. Adaptive control techniques are used for systems with linear parametric uncertainties. For systems with relative degree larger than one, a new adaptive control scheme is presented which is considerably simpler than the augmented error scheme suggested previously by Narendra et al. (1978) for linear systems and by Sastry and Isidori (1987) for non-linear systems. Contrary to the augmented error scheme, however, this scheme results in small rather than zero tracking errors.     

Universal disturbance rejection for nonlinear systems in output feedback form

This note deals with global disturbance rejection via output feedback of a class of uncertain nonlinear systems subject to a class of unknown disturbances. Both the uncertainty in the system model and the uncertainty in the exosystem are tackled concurrently. The disturbances generated from an unknown linear exosystem are completely rejected. The order of the exosystem is assumed known, and the eigenvalues are distinct. The system is assumed in the format of the minimum-phase output feedback form, with no knowledge of the values of any system parameters, including the high-frequency gain. No other assumptions are needed in the control design. A new set of filters are introduced for state estimation. The stability of the internal model is exploited to design a new auxiliary error, involving both the unknown parameters of the reformatted exosystem and those of the system, which makes it possible to group all the unknown parameters in a format suitable to adaptive control design. A Nussbaum gain is introduced in adaptive control design to tackle the unknown high-frequency gain and a number of control coefficients are also made adaptive so that the disturbance rejection is global with respect to unknown frequencies in the disturbances.     

Output feedback stabilization of nonlinear MIMO systems having uncertain high-frequency gain matrix

The purpose of this paper is to provide a method for (semi-global) asymptotic stabilization of a nonlinear minimum-phase MIMO system, under a mild hypothesis of the so-called “high-frequency gain” matrix. This result is based on a non-trivial extension, to the MIMO setting, of the approach based on the use of extended observers. As a byproduct, a dynamic output feedback control is obtained, that asymptotically stabilizes the equilibrium of the closed-loop system, in spite of uncertainties in the high-frequency gain matrix.     

Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation

In this paper, the problem of robust fault tolerant control for a class of singular systems subject to both time-varying state-dependent nonlinear perturbation and actuator saturation is investigated. A sufficient condition for the existence of a fixed-gain controller is first proposed which guarantees the regularity, impulse-free and stability of the closed-loop system under all possible faults. An optimization problem with LMI constraints is formulated to determine the largest contractively invariant ellipsoid. An adaptive fault tolerant controller is then developed to compensate for the failure effects on the system by estimating the fault and updating the design parameter matrices online. Both of these two controllers are in the form of a saturation avoidance feedback with the advantage of relatively small actuator capacities compared with the high gain counterpart. An example is included to illustrate the proposed procedures and their effectiveness.     

Adaptive backstepping control design for systems with unknownhigh-frequency gain

In this correspondence, Nussbaum gains (1983) are introduced in the backstepping design to obtain adaptive controllers for systems with unknown high-frequency gain. Two kinds of modified backstepping control design schemes are developed. It is shown that both schemes can give asymptotic tracking     

Dynamic control of systems with variable state-delay

The problem of designing dynamic controllers for a class of linear dynamical systems with time-varying delays subject to input disturbance as well as output measurement error and in which some parameters are unknown-but-bounded is considered. It is establishing that when certain structural constraints are met, then two stabilizing controllers can be constructed to render the closed-loop system ‘globally practically stable’. The controllers have three terms: a growth term, an output-driven term and a nonlinear term. Robustness of the developed dynamic controllers against mismatched uncertainties is also proved. Simulation studies are carried out on a water-quality model to illustrate the potential of the control methodology.     

Parameter convergence of a new class of adaptive controllers

A new class of adaptive control schemes for minimum-phase linear time invariant (LTI) systems has recently been developed using nonlinear design techniques which guarantee improved transient performance in addition to closed-loop stability and asymptotic tracking. In this paper we establish the parameter convergence properties of this new class of schemes in the presence of persistently exciting signals and compare them with the properties of the traditional adaptive controllers. We show that the new class of adaptive controllers has stronger parameter convergence properties in the presence of overparameterization     

Multivariable adaptive control using high-frequency gain matrix factorization

In this note, we extend the application of a less restrictive condition about the high-frequency gain matrix to design stable direct model reference adaptive control for a class of multivariable plants with relative degree greater than one. The new approach is based on a control parametrization derived from a factorization of the high-frequency gain matrix K/sub p/ in the form of a product of three matrices, one of them being diagonal. Three possible factorizations are presented. Only the signs of the diagonal factor or, equivalently, the signs of the leading principal minors of K/sub p/, are assumed known.     

动态不确定非线性系统直接自适应模糊backstepping控制

对一类单输入单输出动态不确定非线性系统,提出一种直接自适应模糊backstepping和小增益相结合的控制方法.设计中,首先用模糊逻辑系统逼近虚拟控制器:其次把自适应模糊控制和backstepping控制设计技术相结合.给出了直接自适应模糊控制设计方法.最后基于Lyapunov函数和小增益方法证明了整个闭环系统的稳定性.仿真实例进一步验证了所提方法的有效性.     

Estimation of the Limit Accuracy of Discrete Systems for a Class of Dynamic Controllers Relative to the Output

A problem is considered for the estimation of the limit accuracy of multidimensional discrete systems for a definite class of dynamic controllers relative to the output, which differ in the fact that certain of the poles of a closed system prove to be zero ones, while the remaining poles are either the poles of a linear-quadratic controller or the poles of the Kalman filter that is dual with respect to this controller. Asymptotic properties of such controllers are investigated and for a minimum-phase object, a limit (minimum possible) control error is defined in the explicit form, for which the Euclidean norm of the vector of steady-state values of controllable variables is taken. An example of synthesis of a multidimensional discrete systems by the prescribed requirements for a static accuracy, which illustrates the obtained results, is given.     

Immersion and invariance adaptive control of linear multivariable systems

We show in this paper that it is possible to globally adaptively stabilize linear multivariable systems with reduced prior knowledge of the high-frequency gain. In particular, we relax the restrictive (nongeneric) symmetry condition usually required to solve this problem. Instrumental for the establishment of our result is the use of the new immersion and invariance approach to adaptive control recently proposed in the literature. The controllers obtained with this technique are not certainty equivalent—though smooth and without projections or overparameterizations—and the resulting Lyapunov functions contain cross-terms between the plant states and the parameter errors.     

不确定非线性多智能体系统的分布式模糊自适应镇定控制

针对一类具有未知非线性和未知参数摄动的非线性多智能体系统, 提出一种分布式模糊自适应镇定控制方法. 基于邻接智能体信息和部分智能体的自身信息, 分别设计静态耦合和动态耦合的分布式模糊自适应控制律. 基于Lyapunov 稳定性理论, 证明了所提出的控制器能使得系统状态最终稳定于原点的邻域内. 仿真实例验证了所提出方法的有效性.

     

多变量系统的基于高频增益矩阵分解的直接型模型参考自适应控制

针对相对阶为2的多变量系统,利用高频增益矩阵分解,在适当的假设下,建立了新的参数模型,设计出具有未规范化自适应律的直接型模型参考自适应控制器,保证了闭环系统所有信号的有界性和跟踪误差的收敛性.仿真实例验证了控制方案的有效性.