Recursive exponentially weighted PLS and its applications to adaptive control and prediction

A new and fast recursive, exponentially weighted PLS algorithm which provides greatly improved parameter estimates in most process situations is presented. The potential of this algorithm is illustrated with two process examples: (i) adaptive control of a two by two simulated multivariable continuous stirred tank reactor; and (ii) updating of a prediction model for an industrial flotation circuit. The performance of the recursive PLS algorithm is shown to be much better than that of the recursive least squares algorithm. The main advantage of the recursive PLS algorithm is that it does not suffer from the problems associated with correlated variables and short data windows. During adaptive control, it provided satisfactory control when the recursive least squares algorithm experienced difficulties (i.e., ‘blew’ up) due to the ill-conditioned covariance matrix, (X^TX)_t. For the industrial soft sensor application, the new algorithm provided much improved estimates of all ten response variables.     

Adaptive control of a simple nonlinear system without a prioriinformation on the plant parameters

The authors present an adaptive control scheme for nonlinear systems of the form x=c*^Tf(x)+b*u, where f(x) is Lipschitz, c* is a constant vector, and b* is a constant scalar. The control scheme achieves asymptotical model matching without a priori knowledge of the sign of the b* gain. The adaptive scheme is free from singularities in the sense that the estimate of b*, entering in the denominator of the control law, is bounded away from zero. The singularity has been overcome through a suitable modification of the parameter estimates which is based on standard least squares covariance matrix properties     

The zero divisor problem of multivariable stochastic adaptive control

Stochastic adaptive minimum variance control algorithms require a division by a function of a recursively computed parameter estimate at each instant of time. In order that the analysis of these algorithms is valid, zero divisions must be events of probability zero. This property is established for the stochastic gradient adaptive control algorithm under the condition that the initial state of the system and all finite segments of its random disturbance process have a joint distribution which is absolutely continuous with respect to Lebesgue measure. This result is deduced from the following general result established in this paper: a non-constant rational function of a finite set of random variables {x₁},x_n} is absolutely continuous with respect to Lebesgue measure if the joint distribution function of {x₁,…,x_n} has this property.     

INDIRECT ADAPTIVE CONTROL FOR NONLINEAR SYSTEMS WITH SYSTEM PERTURBATION

This paper presents an indirect adaptive control scheme for a class of input-output linearizable nonlinear systems subjected to system perturbations. System parameters are unknown and estimated recursively by a parameter estimator to obtain approximate system output and output derivatives, and then to derive an adaptive control law. In the parameter estimator, a dead-zone approach is used to avoid the parameter drift problem. A positive switching gain is also set to decrease the dead-zone value to obtain better output tracking performance. Under some assumptions, the indirect adaptive control scheme is proved to be stable.     

基于并行处理的混合自适应控制器

本文研究采用并行处理技术产现对象具有未建模动态时的间接式混合自适应控制算法并重新设计了周期协方差重置序列最小二乘和补偿器增益计算方法，给出便于超大规模集成电路脉动阵列实现的结构，加快了高速高性能自适应控制器的参数综合。     

On parameter convergence in least squares identification and adaptive control

Least squares estimation is appealing in performance and robustness improvements of adaptive control. A strict condition termed persistent excitation (PE) needs to be satisfied to achieve parameter convergence in least squares estimation. This paper proposes a least squares identification and adaptive control strategy to achieve parameter convergence without the PE condition. A modified modeling error that utilizes online historical data together with instant data is constructed as additional feedback to update parameter estimates, and an integral transformation is introduced to avoid the time derivation of plant states in the modified modeling error. On the basis of these results, a regressor filtering–free least squares estimation law is proposed to guarantee exponential parameter convergence by an interval excitation condition, which is much weaker than the PE condition. And then, an identification‐based indirect adaptive control law is proposed to establish exponential stability of the closed‐loop system under the interval excitation condition. Illustrative results considering both identification and control problems have verified the effectiveness and superiority of the proposed approach.     

基于最小二乘算法的最优适应控制器

采用"输入匹配"的方法,建立了"一步超前"最小二乘算法,得以参数估计的收敛速度. 证明了闭环适应系统是全局稳定的,且适应控制收敛于"一步超前"最优控制.     

Robust adaptive neural control for a class of uncertain MIMO nonlinear systems

In this paper, a novel robust adaptive neural control scheme is proposed for a class of uncertain multi-input multi-output nonlinear systems. The proposed scheme has the following main features: (1) a kind of Hurwitz condition is introduced to handle the state-dependent control gain matrix and some assumptions in existing schemes are relaxed; (2) by introducing a novel matrix normalisation technique, it is shown that all bound restrictions imposed on the control gain matrix in existing schemes can be removed; (3) the singularity problem is avoided without any extra effort, which makes the control law quite simple. Besides, with the aid of the minimal learning parameter technique, only one parameter needs to be updated online regardless of the system input–output dimension and the number of neural network nodes. Simulation results are presented to illustrate the effectiveness of the proposed scheme.     

Adaptive H∞ control with application to systemswith structural flexibility

We develop a robust adaptive control algorithm using a combination of H_∞ design and system identification. We derive frequency dependent bounds for the tolerated unmodeled dynamics and show that the approach gives a closed-loop system with bounded l_∞ , and l₂ gain when the model mismatch is small in the frequency range where the control gain is large. Our application focus is systems with structural flexibility. We present a parameter estimation algorithm that uses constrained least squares with prefiltering to overcome the problem of identifying lightly damped antiresonances (a common problem in identification of flexible systems). The estimation and control design are executed at a low frequency and only when parameter updating is needed. This allows us to apply computationally expensive control design and signal processing algorithms. It also eliminates many of the problems of earlier adaptive controllers (such as bursting, parameter drift, etc.) by turning off the estimator. We show results from the application of adaptive H_∞ control to a high-fidelity model of the Martin Marietta flexible beam testbed     

∞ guaranteed cost control for uncertain continuous-time linear systems

This paper deals with the _∞ guaranteed cost control problems for continuous-time uncertain systems. It consist of the determination of a stabilizing state feedback gain which imposes on all possible closed-loop models an _∞ -norm upper bound γ > 0. Assuming that the uncertain domain is convex-bounded and the uncertain system is quadratic-stabilizable with γ disturbance attenuation, it is shown how to determine, by means of a convex programming problem, the global minimum of γ. As a particular and important case, for precisely known linear systems, the last problem reduces to the classical _∞ optimal control problem. The results follow from the definition of a special parameter space on which the above-mentioned problems are convex.     

Robust estimation without positive real condition

The strictly positive real (SPR) condition on the noise model is necessary for a discrete-time linear stochastic control system with unmodeled dynamics, even so for a time-invariant ARMAX system, in the past robust analysis of parameter estimation. However, this condition is hardly satisfied for a high-order and/or multidimensional system with correlated noise. The main work in this paper is to show that for robust parameter estimation and adaptive tracking, as well as closed-loop system stabilization, the SPR condition is replaced by a stable matrix polynomial. The main method is to design a “two-step” recursive least squares algorithm with or without a weighted factor and with a fixed lag regressive vector and to define an adaptive control with bounded external excitation and with randomly varying truncation     

Output‐feedback robust adaptive backstepping control for a class of multivariable nonlinear systems with guaranteed tracking performance

This paper is devoted to output‐feedback adaptive control for a class of multivariable nonlinear systems with both unknown parameters and unknown nonlinear functions. Under the Hurwitz condition for the high‐frequency gain matrix, a robust adaptive backstepping control scheme is proposed, which is able to guarantee the tracking performance and needs only one parameter to be updated online regardless of the system order and input–output dimension. To cope with the unknown nonlinear functions and improve the tracking performance, a kind of high‐gain K‐filters is introduced. It is proved that all signals of the closed‐loop system are globally uniformly bounded. Simulation results on coupled inverted double pendulums are presented to illustrate the effectiveness of the proposed scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

一种不完全信息下递推辨识方法及收敛性分析

针对信号在网络环境下传输带来不完全信息使得在线参数辨识算法和收敛性困难的问题, 不同于传统递推最小二乘方法, 本文提出了一种不完全信息下递推辨识方法并分析其收敛性. 首先运用伯努利分布刻画引起不完全信息的数据丢包特性, 然后基于辅助模型方法补偿不完全信息并构造了新的数据信息矩阵, 并运用矩阵正交变换性质对数据信息矩阵进行QR分解, 推导了融合网络参数的递推辨识新算法, 理论证明了在不完全信息下递推参数辨识算法的收敛性. 最后仿真结果验证了所提方法的可行性和有效性.     

Stable adaptive fuzzy control for MIMO nonlinear systems

In this paper, an indirect adaptive fuzzy control scheme is presented for a class of multi-input and multi-output (MIMO) nonlinear systems whose dynamics are poorly understood. Within this scheme, fuzzy systems are employed to approximate the plant’s unknown dynamics. In order to overcome the controller singularity problem, the estimated gain matrix is decomposed into the product of one diagonal matrix and two orthogonal matrices, a robustifying control term is used to compensate for the lumped errors, and all parameter adaptive laws and robustifying control term are derived based on Lyapunov stability analysis. The proposed scheme guarantees that all the signals in the resulting closed-loop system are uniformly ultimately bounded (UUB). Moreover, the tracking errors can be made small enough if the designed parameter is chosen to be sufficiently large. A simulation example is used to demonstrate the effectiveness of the proposed control scheme.     

A learning theory approach to system identification and stochastic adaptive control

In this paper, we present an approach to system identification based on viewing identification as a problem in statistical learning theory. Apparently, this approach was first mooted in [E. Weyer, R.C. Williamson, I. Mareels, Sample complexity of least squares identification of FIR models, in: Proceedings of the 13th World Congress of IFAC, San Francisco, CA, July 1996, pp. 239–244]. The main motivation for initiating such a program is that traditionally system identification theory provide asymptotic results. In contrast, statistical learning theory is devoted to the derivation of finite-time estimates. If system identification is to be combined with robust control theory to develop a sound theory of indirect adaptive control, it is essential to have finite-time estimates of the sort provided by statistical learning theory. As an illustration of the approach, a result is derived showing that in the case of systems with fading memory, it is possible to combine standard results in statistical learning theory (suitably modified to the present situation) with some fading memory arguments to obtain finite-time estimates of the desired kind. It is also shown that the time series generated by a large class of BIBO stable nonlinear systems has a property known as β-mixing. As a result, earlier results of [E. Weyer, Finite sample properties of system identification of ARX models under mixing conditions, Automatica, 36 (9) (2000) 1291–1299] can be applied to many more situations than shown in that paper.     

随机系统最小二乘辨识的稳健性收敛性及适应控制

基于估计随机系统已建模部分未知参数的推广最小二乘算法,递推地定义了跟踪随机参考信号的适应控制器。我们证明了闭环系统稳定;当未建模动态特性在平均意义下有界时,估计误差随未建模 动态特性的衰减而减少且跟踪误差以一个微小的量偏离其最小值;当未建模动态特性在平均意义下趋于零时,可同时获得估计的强一致性及适应跟踪的渐近最优性。     

Adaptive discrete-time sliding-mode control of nonlinear systems described by Wiener models

In this paper, we propose an adaptive control scheme that can be applied to nonlinear systems with unknown parameters. The considered class of nonlinear systems is described by the block-oriented models, specifically, the Wiener models. These models consist of dynamic linear blocks in series with static nonlinear blocks. The proposed adaptive control method is based on the inverse of the nonlinear function block and on the discrete-time sliding-mode controller. The parameters adaptation are performed using a new recursive parametric estimation algorithm. This algorithm is developed using the adjustable model method and the least squares technique. A recursive least squares (RLS) algorithm is used to estimate the inverse nonlinear function. A time-varying gain is proposed, in the discrete-time sliding mode controller, to reduce the chattering problem. The stability of the closed-loop nonlinear system, with the proposed adaptive control scheme, has been proved. An application to a pH neutralisation process has been carried out and the simulation results clearly show the effectiveness of the proposed adaptive control scheme.     

Singularity-free multivariable model reference adaptive control

In this paper we propose a way to solve the problem of singularities in model reference adaptive control of linear multi-input-multi-output (MIMO) systems using a parameter modification procedure based on the least squares covariance matrix inverse. The scheme does not require any explicit prior knowledge about the leading coefficient matrix associated with the control input and secures a uniform lower bound for the determinant of the estimate of this matrix. A global convergence analysis is presented     

一种基于参数辨识的迭代学习控制算法及应用

针对间歇过程中模型参数变化的问题,提出了一种基于遗忘因子最小二乘法辨识的迭代学习控制算法。迭代学习律的参数随模型参数变化而更新,利用遗忘因子大大减小参数变化时"错误"数据对算法的影响,使算法具有更强的自适应性。把这一算法应用于黄酒发酵过程,提高了发酵过程的优化控制效果。仿真结果表明当模型参数随着批次变化时,系统的跟踪性能得到了改进。     

Overlapping-multi-layer deadzone for alleviating over conservativeness in robot adaptive tracking

We present an overlapping-multi-layer deadzone approach to the adaptive tracking control of robotic manipulators in the presence of an unknown mass matrix and completely unmodeled disturbance torques. Even in the unrealistic case of a known mass matrix, over conservative bounds on the disturbance size lead to an unnecessarily large single-level deadzone and hence an unnecessarily large asymptotic tracking error. Additionally, a single-level deadzone is not assured to prevent the “bursting” phenomenon if the mass matrix is not known. The approach we propose herein is proved to cause convergence of the generalized tracking error to within a user-specified tolerance above the level of the smallest “valid” deadzone size (times a known bound on the square root of the ratio of the largest-possible eigenvalue of the mass matrix to the smallest-possible eigenvalue of the mass matrix), even though this size is not known a priori. The result is that we obtain provable convergence to a tracking error size that is much smaller than we would have with a single-level conservative deadzone. The reason deadzones of sufficient size are needed in the first place is as follows (and this applies even if sigma-modification or e1-modification are used). Even if the mass matrix were unrealistically known, without any deadzone, the following can happen, to the best of our knowledge. We can incur indefinite cycling between (i) and (ii): (i) time periods of Lyapunov function increase, due to a parameter error increase, while the generalized tracking error remains small, (ii) time periods of Lyapunov function decrease due to large reductions in the parameter error while the generalized tracking error simultaneously increases to large peak values before returning back to a small value. With indefinite cycling between (i) and (ii), there is no convergence of the generalized tracking error to small values. This phenomenon has been termed “bursting” by authors such as [P. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall, Englewood Cliffs, NJ, 1996.] and is described as “one of the most annoying phenomena in adaptive control.” To the best of our knowledge, this paper, once published, will represent the first presentation of an overlapping-multi-layer deadzone method for adaptive tracking control of robotic manipulators with unknown mass matrix and disturbance torque.