Self-tuning control of nonminimum-phase systems

The instability that arises from the pole-zero cancellation of elementary self-tuners and most MRAC algorithms has given rise to a misapprehension that nonminimum-phase systems can pose insoluble problems with the application of adaptive control. Since most practical processes exhibit some form of nonminimum-phase behaviour (particularly when controlled digitally) this appeared to restrict the usefulness of self-tuning methods. However, during the last few years new algorithms have been suggested which partly overcome these difficulties and have been shown to be effective in industrial trials. The paper describes the practical features that a self-tuning controller must possess and discusses how plant dead-time and excess continuous-time poles can lead to discrete-time nonminimum-phase zeros. The source of instability of elementary self-tuners is analysed and various suggested approaches for overcoming this problem are reviewed. These include the use of a generalized minimum variance cost-function incorporating control weighting, factorization methods which avoid cancellation of the offending zeros, adaptive control based on state-space linear quadratic Gaussian (LQG) theory, and explicit pole-placement via solution of a Diophantine or Bezoutian equation. Gawthrop's hybrid controller, which avoids those nonminimum-phase zeros due to continuous-time pole excess, is briefly discussed. A series of simulations in which a nonminimum-phase plant is subjected to a pattern of set-point changes is presented to show some of the features of the algorithms. In applications the order of the process is unlikely to be known (or indeed be finite); the pole-placement algorithm which is particularly effective in the known-order case is shown to lack robustness to underspecified order. The use of a large control weighting however restores stability to the generalized minimum-variance controller. The conclusions are that with appropriate design and engineering judgement adaptive control is viable even for nonminimum-phase processes.     

A multivariable self-tuning controller

A multivariable self-tuning controller is derived extending the scalar version of Clarke and Gawthrop (1975). Because the control terms are penalized in the cost function, fluctuations and peaking in control signals are reduced compared with the multivariable minimum variance self-tuning controller (Borisson, 1979). Time-varying reference signals as well as certain nonminimum-phase systems can be handled without difficulty with the proposed controller. Several examples illustrate the power of the derived self-tuning controller.     

Fuzzy model based control: stability, robustness, and performanceissues

A nonlinear controller based on a fuzzy model of MIMO dynamical systems is described and analyzed. The fuzzy model is based on a set of ARX models that are combined using a fuzzy inference mechanism. The controller is a discrete-time nonlinear decoupler, which is analyzed both for the adaptive and the fixed parameter cases. A detailed stability analysis is carried out, and the main result is that the closed loop is globally stable and robust with respect to unstructured uncertainty, which may include modeling error and disturbances. In addition, bounds on the asymptotic and transient performance are given. The main assumptions on the system and model are that they must not have strong nonminimum-phase effects, except time-delay, and the unstructured uncertainty must not be too large. A simulation example illustrates some of the properties of the modeling method and model based control structure     

新型多变量广义预测自校正控制器

本文对多变量线性系统提出了一种新的广义预测自校正控制器(MGPC).由于本算法是利用辨识结果直接求解控制器,不需要在线求解Diophantine方程,因而大大减少了计算量,并简化了控制系统的设计过程。仿真结果表明:本文提出的算法能适用于具有有色噪声干扰的不稳定和(或)非最小相位多变量随机系统。     

Observations-weighted controllers for single-input single-output stochastic systems using a perturbed-type dual criterion

An observations-weighted (OW) controller is presented for linear discrete-time stochastic single-input single-output systems. A dual criterion to be minimized includes the usual OW-type cost function, but has in addition the terms of the sensitivity and complementary sensitivity weightings. A stabilizing controller is obtained by using the polynomial equation approach. The dual criterion OW controller proposed here is shown to be a usual OW controller with perturbed weighting functions for the tracking error and control signals. Furthermore, it is asserted that the resulting controller is useful for improving the robustness of the usual OW controller and is very simple to implement. Thus, the controller may be suitable for use in robust self-tuning control systems.     

一种适用于非最小相位系统的模糊控制方法

针对某一类非最小相位系统的特性,提出一种新型模糊控制器的设计方法,改进了ＰＩＤ型模糊控制器的结构,对输入变量设计了一种简单的可调整宽度的三角形隶属度函数;根据偏差和偏差变化率的相平面图,建立了调节系统的动态特性和规则基之间的定性关系,使规则基的设计趋于简单、合理。水轮机调节系统的仿真结果表明了这种方法的有效性。     

Self-tuning leader-follower games

The optimization of a class of linear discrete-time stochastic systems with unknown parameters and multiple decision-makers or controllers each having his own single-stage quadratic objective is considered. Based on a centralized information pattern, a steady-state solution is obtained for the stochastic adaptive leader-follower game problem. It is shown that this class of leader-follower game problems can be transformed to an equivalent class of multiple-input-multiple-output self-tuning control problems with a single decision-maker. The convergence of the multiple-input-multiple-output self-tuning controller is established and applied to the leader-follower game. A numerical simulation illustrates the proposed self-tuning algorithm for an adaptive leader-follower strategy.     

Regarding self-tuning controllers for nonminimum phase plants

This correspondence is in response to the recent excellent survey of Clarke (1984, Automatica, 20, 501–517), which points out that minimum variance self-tuning controllers fail in the control of nonminimum-phase plants. It is also in response to recent algorithms of Ydstie (1984, Preprints 9th IFAC World Congress, Budapest) which have application in the control of such systems. A new control algorithm, designed to control nonminimum-phase plants, is described here. This “Control Advance Moving Average Controller” (CAMAC) determines the input that minimizes the variance between the output and the setpoint at a time advance equal to or greater than the input-output (I/O) transport delay, based upon an assumption made on future inputs. The CAMAC algorithm is tuned by determining the appropriate control time advance. This approach is similar to that of Ydstie (1984, Preprints 9th IFAC World Congress, Budapest), but uses a different assumption on future inputs which yields substantially different controller performance. Its performance (in simulation) compares very favorably with that of self-tuning control algorithms reviewed in Clarke (1984, Automatica20, 501–517) and those described in Ydstie (1984, Preprints 9th IFAC World Congress, Budapest) in the control of nonminimum-phase plants. Since it is not restricted to controlling the plant at the assumed I/O transport delay, it is also suitable for plants with unknown or time-varying dead times. The CAMAC algorithm has been successfully implemented in the control of mean arterial pressure in anesthetized animals.     

基于死区吸引律的离散重复控制

针对周期参考信号下的不确定离散时间系统,提出一种离散重复控制方法,利用死区函数设计新型的吸引律,将干扰补偿、抑制措施“嵌入”吸引律,构造理想误差动态,并基于此导出重复控制器。为了进行具体的控制器参数整定和表征闭环系统的误差动态行为,推导出了稳态误差带、单调减区域和绝对吸引层边界的表达式。所设计的离散重复控制器能够完全抑制周期对称干扰信号,控制器设计方法也适用于常值调节问题的定位控制。数值仿真及在电机伺服系统上的实验结果验证了所提出控制方法的有效性。     

Deterministic convergence of a Clarke-Gawthrop self-tuning controller

A proof of global asymptotic stability of a single input-single output deterministic self-tuning controller is given in this paper. The controller considered minimizes a quadratic cost measure of the Clarke-Gawthrop type which penalizes input and output deviations from the desired values as well as excessive control action. The parameter estimation scheme employs a variable forgetting factor. It is established that the output error converges to zero, even for nonminimum phase systems, for appropriate cost functions.     

Nonlinear model-state feedback control for nonminimum-phase processes

A general nonlinear controller design methodology for continuous-time nonminimum-phase systems is presented, which utilizes synthetic outputs that are statically equivalent to the original process outputs and make the system minimum-phase. A systematic procedure is proposed for the construction of statically equivalent outputs with prescribed transmission zeros. The calculated outputs are used to construct a model-state feedback controller. The proposed method is applied to a nonminimum-phase chemical reactor control problem where a series/parallel reaction is taking place.     

无抖振离散重复控制器的设计与实现

针对周期参考/干扰信号下的不确定离散时间系统,提出一种基于吸引律的重复控制方法,在吸引律中"嵌入"干扰抑制措施,构造理想误差动态,并基于此设计重复控制器.文中推导出单调减区域、绝对吸引层和稳态误差带边界的表达式,用于整定控制器参数和表征闭环系统的跟踪性能,并给出了跟踪误差在无干扰时收敛于原点及在干扰存在时收敛进入稳态误差带内所需最多步数的表达式.设计的重复控制器不仅能够完全抑制周期干扰信号,而且可以消除系统抖振.在电机实验装置上的应用结果表明了所提出控制方法的有效性.     

基于约束输入变速吸引律的离散重复控制器设计

针对离散时间线性系统的周期跟踪问题, 提出一种能够约束控制输入变化速度的变速吸引律, 结合干扰抑制措施构造了理想误差动态, 并由此导出离散重复控制器. 分析表明, 该变速吸引律能使跟踪误差在有限时间内单调收敛至零, 且误差收敛速度可控. 为刻画误差动态行为, 推导了有界扰动下的误差单调收敛域、绝对值收敛域和稳态误差带, 并给出了收敛步数. 针对伺服电机系统的仿真与实验结果验证了所提出控制方案的有效性.

     

多变量自校正解耦PID控制器

将前馈解耦控制与自校正ＰＩＤ控制相结合，提出了多变量自校正解耦ＰＩＤ控制器。该控制器适于具有任意延时结构和非最小相位系统，本文利用根轨变法证明了这一算法的全局收敛性。     

Discrete-time neural net controller for a class of nonlineardynamical systems

A family of two-layer discrete-time neural net (NN) controllers is presented for the control of a class of mnth-order MIMO dynamical system. No initial learning phase is needed so that the control action is immediate; in other words, the neural network (NN) controller exhibits a learning-while-functioning-feature instead of a learning-then-control feature. A two-layer NN is used which is linear in the tunable weights. The structure of the neural net controller is derived using a filtered error approach. It is indicated that delta-rule-based tuning, when employed for closed-loop control, can yield unbounded NN weights if: 1) the net cannot exactly reconstruct a certain required function, or 2) there are bounded unknown disturbances acting on the dynamical system. Certainty equivalence is not used, overcoming a major problem in discrete-time adaptive control. In this paper, new online tuning algorithms for discrete-time systems are derived which are similar to ϵ-modification for the case of continuous-time systems that include a modification to the learning rate parameter and a correction term to the standard delta rule     

基于模糊神经网络的积分自适应控制

将模糊神经网络应用于传统线性积分自适应控制,构造了一类模糊神经自适应方法,用于消除非线性系统响应偏差.模糊神经网构成直接非线性自适应控制器.对线性及非线性对象的仿真控制以及与经典自适应控制的比较,表明了模糊神经自适应控制器的有效性.     

Adaptive control of nonminimum-phase systems using shifted Laurent series

We present a direct discrete-time output-feedback adaptive control algorithm for single-input, single-output systems that are possibly unstable and nonminimum phase. The plant modeling information is given by impulse response components, and the plant is modelled within the algorithm by a truncated shifted Laurent series. A shifted Laurent series is a Laurent series at a point different from the origin in the complex plane and about infinity. The shifted Laurent series is analysed, including its convergence and its relationship to other Laurent series. In particular, we provide a technique for constructing a truncated shifted Laurent series using impulse response components. Numerical examples show that retrospective cost adaptive control can achieve asymptotic command following for a class of exponentially unstable, nonminimum-phase systems.     

不确定离散广义系统的D稳定鲁棒控制

研究了具有圆盘区域极点约束的一类不确定离散广义系统的鲁棒控制问题.首先,研究了控制输入项不含扰动的不确定离散广义系统,提出了广义二次D镇定的概念,基于矩阵不等式和广义Riccati方程,给出了一种广义二次D镇定器的设计方法,所得到的结论能够实现研究目标;然后,讨论了控制输入项含有扰动的不确定离散广义系统,在一定的假设条件下,给出了期望状态反馈增益阵的存在条件及其解析表达式.最后,用数值示例说明所给方法的有效性及可行性.     

A new approach to the zero-dynamics assignment problem for nonlinear discrete-time systems using functional equations

The present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear discrete-time real analytic system by constructing the requisite synthetic output maps. The problem under consideration is motivated by the need to adequately address the control problem of nonminimum-phase nonlinear discrete-time systems, since the latter represent a rather broad class of systems due to the well-known effect of sampling on the stability of zero-dynamics. In the proposed approach, the above control objective can be attained through: (i) a systematic computation of synthetic output maps that induce minimum-phase behavior while being statically equivalent to the original output maps (both vanish on the equilibrium manifold) and (ii) the subsequent integration into the methodological framework of currently available nonminimum-phase compensation schemes for nonlinear discrete-time systems that rely on output redefinition. The mathematical formulation of the zero-dynamics assignment problem is realized via a system of nonlinear functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of functional equations can be proven to be locally analytic, and this enables the development of a solution method that is easily programmable with the aid of a symbolic software package. The synthetic output maps that induce the prescribed zero dynamics for the original nonlinear discrete-time system can be explicitly computed on the basis of the solution to the aforementioned system of functional equations.     

Multilayer discrete-time neural-net controller with guaranteedperformance

A family of novel multilayer discrete-time neural-net (NN) controllers is presented for the control of a class of multi-input multi-output (MIMO) dynamical systems. The neural net controller includes modified delta rule weight tuning and exhibits a learning while-functioning-features. The structure of the NN controller is derived using a filtered error/passivity approach. Linearity in the parameters is not required and certainty equivalence is not used. This overcomes several limitations of standard adaptive control. The notion of persistency of excitation (PE) for multilayer NN is defined and explored. New online improved tuning algorithms for discrete-time systems are derived, which are similar to sigma or epsilon-modification for the case of continuous-time systems, that include a modification to the learning rate parameter plus a correction term. These algorithms guarantee tracking as well as bounded NN weights in nonideal situations so that PE is not needed. An extension of these novel weight tuning updates to NN with an arbitrary number of hidden layers is discussed. The notions of discrete-time passive NN, dissipative NN, and robust NN are introduced. The NN makes the closed-loop system passive.