Performance limitations in the robust servomechanism problem for discrete-time LTI systems

Fundamental limitations for error tracking/regulation are obtained for the robust servomechanism problem (RSP) for a sampled system. In studying this problem, the cheap control problem for a multi-input/multi-output discrete time system is considered, and explicit expressions are obtained for the limiting steady state solution of its associated algebraic Riccati equation (ARE), as the weight on the control energy tends to zero. Application of these results is then made to obtain explicit expressions for the limiting performance costs associated with error tracking/regulation in the RSP. These limitations can be characterized by the system order, the dimension of the outputs, the number of the system's transmission zeros and the location of the system's nonminimum phase transmission zeros.     

Limiting performance of optimal linear discrete filters

The limiting variance of the estimation error is obtained for the optimal H₂ filter for discrete time systems when the intensity of the measurement noise tends to zero. In particular, an explicit expression is obtained for the lowest achievable mean-square error when the measurement noise is slowly varying and has intensity which tends to zero. This limitation can be characterized completely by the system order, the dimension of the process noise, the number of the system's transmission zeros, and the location of the system's unstable transmission zeros.     

Properties and calculation of transmission zeros of linear multivariable systems

A new definition of transmission zeros for a linear, multivariable, time-invariant system is made which is shown to be equivalent to previous definitions. Based on this new definition of transmission zeros, new properties of transmission zeros of a system are then obtained; in particular, it is shown that a system with an unequal number of inputs and outputs almost always has no transmission zeros and that a system with an equal number of inputs and outputs almost always has either n−1 or n transmission zeros, where n is the order of the system; transmission zeros of cascade systems are then studied, and it is shown how the transmission zeros of a system relate to the poles of a closed loop system subject to high gain output feedback. An application of transmission zeros to the servomechanism problem is also included. A fast, efficient, numerically stable algorithm is then obtained which enables the transmission zeros of high order multivariable systems to be readily obtained. Some numerical examples for a 9th order system are given to illustrate the algorithm.     

Perfect control of the robust servomechanism problem

The problem of finding a robust servomechanism controller which achieves exact asymptotic tracking/regulation for a modeled class of reference and disturbance signals, arbitrarily good approximate error regulation (AGAER) for reference/disturbance signals which lie outside of this design class, and arbitrarily good transient response is considered. It is shown that a necessary condition that must be satisfied to accomplish this requirement is that the initial conditions of the servocompensator should be zero. An explicit algorithm for designing such controllers is then given. A simple controller, called the high gain servomechanism controller is also presented, which has the property that it gives AGAER and arbitrarily good transient response, when the robust exact asymptotic tracking/regulation requirement is relaxed. This controller shows that dynamics in a controller is not essential for achieving good error regulation in the presence of unknown unmeasurable disturbances. It is also shown that using the controllers proposed, exact error regulation occurs, for a larger class of design signals, called the extended disturbance/reference input signals, even if the conditions for perfect control are not satisfied. Some examples are included to illustrate the type of results obtained.     

Asymptotic tracking in uncertain Volterra systems

This paper studies the robust tracking problem for an uncertain Volterra system via the frequency-domain approach. Based on the knowledge that the steady-state response of a Volterra system under the sinusoidal input contains possibly infinitely many frequency components, it is shown that (1) for any specified finitely many frequency components possibly contained in the steady-state tracking error, there exists a linear controller that robustly annihilates these components; and (2) robust tracking is possible by a linear controller provided that the Volterra system of the closed-loop system is polynomial. These results can be viewed as a nonlinear analog of the well-known linear robust servomechanism theory developed in the 1970s.     

Robust estimation for discrete time‐varying systems with limited communication capacity

In this paper, we consider the state estimation problem for linear discrete time‐varying systems subject to limited communication capacity which includes measurement quantization, random transmission delay and data‐packet dropouts. Based on transforming the three communication limitations into the system with norm‐bounded uncertainties and stochastic matrices, we design a robust filter such that, for all the communication limitations, the error state of the filtering process is mean square bounded. An upper bound on the variance of the state estimation error is first found, and then, a robust filter is derived by minimizing the prescribed upper bound in the sense of the matrix norm. It is shown that the desired filter can be obtained in terms of the solutions to two Riccati‐like difference equations which also provide a recursive algorithm suitable for online computation. A simulation example is presented to demonstrate the effectiveness and applicability of the proposed algorithm. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

The robust decentralized control of a general servomechanism problem

The decentralized robust control of a completely general servomechanism problem is considered in this paper. Necessary and sufficient conditions, together with a characterization of all decentralized robust controllers which enables asymptotic tracking to occur, independent of disturbances in the plant and perturbations in the plant parameters, and gains of the system, are obtained. A new type of compensator called a decentralized sercocompensator which is quite distinct from an observer is introduced. It is shown that this compensator, which corresponds to an integral controller in classical control theory, must be used in any decentralized servomechanism problem to assure that the controlled system is stabilizable and achieves robust control; in particular, it is shown that a decentralized robust controller of a general servomechanism problem consists of two devices 1) a decentralized servocompensator and 2) a decentralized stabilizing compensator. It is then shown that, under certain mild conditions, there almost always is a solution to the robust decentralized servomechanism problem for any composite system consisting of a number of subsystems interconnected in any arbitrary manner. This last observation has important implications for process control.     

A new tracking controller for discrete‐time SISO non minimum phase systems

A new tracking controller for discrete‐time Single Input Single Output (SISO) non‐minimum phase (NMP) systems is presented. In the proposed method, after cancelation of poles and cancelable zeros of the system, the controller adds some NMP zeros to compensate the effect of NMP zero (zeros) of the system. As a result, the phase of the overall transfer function will be almost linear and its magnitude approaches unity for all frequencies. The method can be applied even to the systems with complex conjugate NMP zeros. As well, it is applicable to the systems for which the conventional methods cannot properly be used. Furthermore, a generalization of method to continuous‐time systems is another given result. Several examples are provided to illustrate the effectiveness of the method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust compensation of time delayed plant by continuous‐time high frequency periodic controller with negligible output ripples

The present work addresses the problem of ensuring robust stability to time delayed plants, compensated with continuous‐time high frequency periodic controller. An efficient design methodology is proposed to synthesize the periodic controller for robust compensation of time delayed linear time‐invariant plants. The periodic controller, by virtue of its loop zero‐placement capability, is shown to achieve superior gain as well as phase/delay margin compensation, especially for non‐minimum phase plants having right half plane poles and zeros in close vicinity to each other. The periodic controller is considered in the observable canonical form which results in realizable bounded control input as well as ensuring insignificant periodic oscillations in the plant output. As a consequence, this paper, furthermore, establishes the fact that the periodic controller designed and synthesized with the proposed methodology can be implemented in real‐time with an assurance of model matching and robust zero‐error tracking. Simulation and experimental results are illustrated to establish the veracity of the claims. The closed‐loop system comprising of time‐delayed linear time‐invariant plant with the periodic controller is analyzed employing the averaging principle and presented here explicitly in a meticulous approach. Copyright © 2015 John Wiley & Sons, Ltd.     

A minimal k-step delay controller for robust tracking of non-minimum phase systems

This paper considers the problem of look-ahead robust tracking for a non-minimum phase system under a sensitivity constraint. Because of non-minimum phase zeros, the exact tracking of arbitrary reference input is impossible. However, with an introduction of some delay, tracking is possible within any given “tolerance” ?₁ > 0. It is shown in the paper that the minimum delay required to satisfy the tracking performance is decoupled from a sensitivity/disturbance rejection constraint. Moreover, it is shown that calculation of the minimum delay reduces to a binary search problem.     

Augmented System Approach to Measurement-Based Robust Tracking

This paper considers the robust tracking problem using an augmented system approach. Our results extend previous full-state results to observer-based feedback. The system matrix is assumed to contain time-invariant uncertainty. Two cases are considered. In the first, the tracked output is available for feedback and a controller can be obtained by considering a robust stabilization problem. The resulting control law can be applied to polynomial reference signals of known order and results in zero steady-state error. In the second case, where the tracked output is not available for feedback, our approach is restricted to step references, and the steady-state error is non-zero, although bounded by a prescribed value. To obtain the controller, a robust H problem must be solved rather than a robust stabilization problem.     

GLOBAL ROBUST SERVOMECHANISM OF A CLASS OF NONLINEAR SYSTEMS USING OUTPUT FEEDBACK

The robust servomechanism problem (alternatively, output regulation problem) of the class of nonlinear systems in lower triangular form has been extensively studied in recent years. The semi‐global solution was first given by either state feedback or output feedback. The global solution by state feedback was given very recently. However, the global solution by output feedback has long been an open problem. In this paper, we present a set of solvability conditions of the global robust servomechanism problem for this class of nonlinear systems by output feedback.     

H∞ output tracking control for uncertain networked control systems via a switched system approach

In this paper, the problem of H_∞ output tracking control for networked control systems with random time delays and system uncertainties is investigated. Effective sampling instant that is tightly related with transmission delay from sensor to actuator is proposed to ensure that the random variable time delay is always shorter than one effective sampling period. By using both active time‐varying sampling period strategy and hybrid node‐driven mechanism, the switching instant is coincided with the effective sampling instant. An augmented time‐varying networked tracking system model is provided by including the output tracking error as an additional state. However, random transmission delay causes indeterminate sampling period, which induces infinite subsystems. Gridding approach is introduced to transform the continuous time axis into discrete‐time sequences, which guarantees the finite number of switching rules. By employing multiple Lyapunov–Krasovskii functions, linear matrix inequality (LMI)‐based output tracking H_∞ performance analysis is presented, and robust switching H_∞ model reference tracking controller for networked control systems with communication constraints and system uncertainties is designed to guarantee asymptotic tracking of prescribed reference outputs while rejecting disturbances. Finally, simulation results illustrate the correctness and effectiveness of the proposed approaches. Copyright © 2015 John Wiley & Sons, Ltd.     

Modelling of non-minimum phase effects in discrete-time norm optimal iterative learning control

The subject of this article is the modelling of the influence of non-minimum phase discrete-time system dynamics on the performance of norm optimal iterative learning control (NOILC) algorithms with the intent of explaining the observed phenomenon and predicting its primary characteristics. It is established that performance in the presence of one or more non-minimum phase plant zeros typically has two phases. These consist of an initial fast monotonic reduction of the L ₂ error norm (mean square error) followed by a very slow asymptotic convergence. Although the norm of the tracking error does eventually converge to zero, the practical implications over a finite number of trials is apparent convergence to a non-zero error. The source of this slow convergence is identified using the singular value distribution of the system's all pass component. A predictive model of the onset of slow convergence behaviour is developed as a set of linear constraints and shown to be valid when the iteration time interval is sufficiently long. The results provide a good prediction of the magnitude of error norm where slow convergence begins. Formulae for this norm and associated error time series are obtained for single-input single-output systems with several non-minimum phase zeros outside the unit circle using Lagrangian techniques. Numerical simulations are given to confirm the validity of the analysis.     

周期时变系统的鲁棒自适应重复控制

针对周期时变系统,提出一种鲁棒自适应重复控制方法.该方法利用周期学习律估计周期时变参数,并结合鲁棒自适应方法处理非周期不确定性.与现有重复控制不同的是,在控制器设计中引入了新变量—周期数,利用周期系统的重复特性,使界的逼近误差随周期数的增加而逐渐减少,保证了系统的全局渐近稳定性.同时将该方法应用于一类非线性参数化系统,使系统在非参数化扰动的情形下,输出误差仍能收敛于0,倒立摆模型的仿真验证了此结果.该设计方法适用于消除神经网络逼近误差对重复控制系统的影响,理论证明了基于神经网络的鲁棒自适应重复控制系统中所有变量的有界性和输出误差的渐近收敛性,关于机械臂模型的仿真结果验证了受控系统具有良好的跟踪性能.     

Adaptive error feedback regulation problem for 1D wave equation

By using the adaptive control approach, we solve the error feedback regulator problem for the one‐dimensional wave equation with a general harmonic disturbance anticollocated with control and with two types of disturbed measurements, ie, one collocated with control and the other anti‐collocated with control. Different from the classical error feedback regulator design, which is based on the internal mode principle, we give the adaptive servomechanism design for the system by making use of the measured tracking error (and its time derivative) and the estimation mechanism for the parameters of the disturbance and of the unknown reference. Constructing auxiliary systems and observer and applying the backstepping method for infinite‐dimensional system play important roles in the design. The control objective, which is to regulate the tracking error to zero and to keep the states bounded, is achieved.     

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。     

The decentralized control expanding system problem

Suppose a controlled system denoted by S_old exists with the property that robust tracking and regulation occurs in each of the system's local control agents. We wish to examine the conditions under which a new subsystem and a new set of local control agents denoted by S_new can be connected to S_old, so that the new outputs and previous outputs are regulated, without the need for returning the original control agents. This problem is a generalization of the results obtained in [4] which deals with the case when the control agents of S_new are centralized.     

Arbitrarily fast and robust tracking by feedback

The output of a singe-input-single-output linear feedback system with more than one pole in excess over the zeros in the loop transmission cannot track arbitrarily fast its input (by the root locus). In this work we extend the linear feedback so that some of the open loop poles may depend on the open loop gain; we call this new class quasi-linear feedback systems. We then derive time domain, pole-zero, and frequency domain conditions which ensure arbitrarily fast and robust tracking by quasi-linear feedback, for an arbitrary number of poles in excess over the zeros. We prove that in a particular case these conditions are equivalent, and that the boundedness in frequency of the closed loop transfer function is no longer necessary for achieving arbitrarily fast tracking. The robustness is to external disturbances and initial conditions, and the open loop has to be minimum phase. Some examples are presented which illustrate these results. They also show that this good performance can be obtained with a reduced control effort, and that quasi-linear feedback can alleviate the limitation on performance of non-minimum phase open loops.