On the design,robustness, implementation and use of quasi-linear feedback compensators

A quasi-linear feedback compensator is one in which its poles depend in an appropriate way on its gain. The reason for introducing this new concept was the desire to remove the limitation to performance imposed by a plant with more than one pole in excess of its zeros. In this article it is shown that this objective is realized for plants with zeros in the left half of the complex plane. The consequences are surprising. In time domain it is possible to track arbitrarily fast a class of reference inputs despite a large class of disturbances and uncertainty in plant parameters. The response is non-oscillatory for high enough compensator gains, which is explained by the automatic adaptation of the closed loop poles to stability and stability margins for such gains. And in frequency domain the phase margin tends to 90° while the gain margin and crossover frequency become unlimited.

Technically the design procedure of quasi-linear compensators presented here is based on our theoretical result concerning the asymptotic behaviour of the roots of certain polynomials in a complex variable which depend also on a large positive parameter.

We also show how to implement such quasi-linear compensators in practical feedback control schemes, and their use at lower gains which is the case of most industrial applications.     

Limitations on maximal tracking accuracy

This paper studies optimal tracking performance issues pertaining to finite-dimensional, linear, time-invariant feedback control systems. The problem under consideration amounts to determining the minimal tracking error between the output and reference signals of a feedback system, attainable by all possible stabilizing compensators. An integral square error criterion is used as a measure for the tracking error, and explicit expressions are derived for this minimal tracking error with respect to step reference signals. It is shown that plant nonminimum phase zeros have a negative effect on a feedback system's ability to reduce the tracking error, and that in a multivariable system this effect results in a way depending on not only the zero locations, but also the zero directions. It is also shown that if unity feedback structure is used for tracking purposes, plant nonminimum phase zeros and unstable poles can together play a particularly detrimental role in the achievable tracking performance, especially when the zeros and poles are nearby and their directions are closely aligned. On the other hand, if a two parameter controller structure is used, the achievable tracking performance depends only on the plant nonminimum phase zeros     

Improved disturbance rejection with online adaptive pole-zero compensation on a Φ-shaped PZT active suspension

In feedback control systems, the anti-resonant zeros cannot be arbitrarily placed, hence degrading tracking performance as well as input disturbance and noise rejection capabilities due to reduced gain at the frequencies of the zeros. In this paper, an online adaptive inverse control with saturation (OAICS) algorithm is proposed for compensating the minimum phase resonant poles and anti-resonant zeros of a PZT active suspension using measured position error signal. Experimental results on a Φ-shaped PZT active suspension using laser Doppler vibrometer (LDV) shows the proposed OAICS is effective in cancelling the first two dominant minimum phase pole-zero pairs to achieve high servo bandwidth and low sensitivity servo system with small overshoot during set-point tracking.     

Non-interacting control design for multivariable industrial processes

In this paper, a systematic method is proposed for the design of general multivariable controller for complex processes to achieve the goal of fast loop responses with acceptable overshoots and minimum loop interaction while maintaining low complexity of the feedback controller. The design of general transfer function type controller is based on the fundamental relations under decoupling of a multivariable process, and the characterization of the unavoidable time delays and non-minimum phase zeros that are inherent in the decoupled loops. The objective loop transfer functions are then suitably specified to achieve fast loop response taking into account the performance limitation imposed by those non-minimum phase zeros and time delays. The ideal controller is then obtained which is in general a complicated irrational transfer matrix, for which model reduction with recursive least squares is applied in the frequency domain to obtain a much simpler transfer matrix with its elements in the form of rational transfer function plus delay. Simulations show that very satisfactory control performance is achieved.     

Pole and zero assignment by state vector feedback

A procedure is described which enables specified poles and zeros of a scalar transfer function of a controllable, observable, linear system to be obtained by using state vector feedback to two inputs. The number of zeros is equal to the number of zeros in the transfer function, before feedback is applied, from one input or the other, whichever is the greater. Those zeros which can be changed, and those which cannot, are identified. The former can be made equal to, or arbitrarily close to, any assigned values, and the poles can be assigned arbitrarily.     

Frequency domain design trade-offs for linear periodic feedback systems*

The purpose of this paper is to analyse inherent design limitations associated with systems that are linear and periodically time-varying. The contributions of the paper are (i) to relate frequency domain raising methods from signal processing literature to time-domain lifting used in control literature, and (ii) to develop extensions of the Poisson sensitivity and complementary sensitivity integral constraints. In particular, it is shown that there is generally an additional cost associated with having a time invariant target closed loop for a periodic open loop plant. It is also shown that design limitations due to unstable poles and/or non-minimum phase zeros of a discrete linear time-invariant plant remain even if a periodic time-varying controller is used. As a consequence, the utility of periodic control in circumventing design limitations is questioned.     

Signal-to-Noise Ratio Fundamental Limitations in Continuous-Time Linear Output Feedback Control

In the present technical note we study the fundamental limitation on stability that arise when an additive coloured Gaussian noise (ACGN) channel is explicitly considered over either the control or measurement paths of a linear time invariant (LTI) feedback loop. By considering a linear setting we can naturally express the fundamental limitation as a lower bound on the channel signal-to-noise ratio (SNR) required for stabilisability. We start by first obtaining a closed-form expression for the squared L ₂ norm of a partial fraction expansion with repeated poles in the Laplace domain. We then use the squared L ₂ norm result to obtain the closed-form expression for the infimal SNR required for stabilisability. The proposed closed-form includes the case of repeated unstable plant poles and non minimum phase (NMP) zeros.     

多通道约束下的网络控制系统的最优跟踪性能

本文针对双通道约束下的线性时不变网络控制系统的随机信号跟踪性能极限问题进行了研究.网络通信包含通信噪声和通信带宽两种信道因素.被控系统考虑是非最小相位和不稳定系统,并且系统包含多个不同的非最小相位零点和多个不同的不稳定极点.对上行通道和下行通道都存在通信带宽约束及高斯白噪声影响的情形,从频域角度,通过采用双自由度控制器和尤拉参数化方法,获得了此类网络控制系统的最优可达的跟踪性能.研究结果表明网络控制系统的跟踪性能极限完全由被控对象的结构特征(非最小相位零点、不稳定极点以及被控对象的系统增益),参考输入信号和网络特性(高斯白噪声的统计特征、通信信道带宽)所决定.最后,仿真结果检证了所得结果的正确性.     

Semi-global stabilization of linear systems subject to output saturation

It is established that a SISO linear stabilizable and detectable system subject to output saturation can be semi-globally stabilized by linear output feedback if all its invariant zeros are in the closed left-half plane, no matter where the open loop poles are. This result complements a recent result that such systems can always be globally stabilized by discontinuous nonlinear feedback laws, and can be viewed as dual to a well-known result: a linear stabilizable and detectable system subject to input saturation can be semi-globally stabilized by linear output feedback if all its poles are in the open left-half plane, no matter where the invariant zeros are.     

Limitations in tracking systems

In this paper, we obtain information theoretical conditions for tracking in linear time-invariant control systems. We consider the particular case where the closed loop contains a channel in the feedback loop. The mutual information rate between the feedback signal and the reference input signal is used to quantify information about the reference signal that is available for feedback. This mutual information rate must be maximized in order to improve the tracking performance. The mutual information is shown to be upper bounded by a quantity that depends on the unstable eigenvalues of the plant and on the channel capacity. If the channel capacity reaches a lower limit, the feedback signal becomes completely uncorrelated with the reference signal, rendering feedback useless. We also find a lower bound on the expected squared tracking error in terms of the entropy of a random reference signal.We show a misleading case where the mutual information rate does not predict the expected effect of nonminimum phase zeros. However, mutual information rate helps generalize the concept that there is a tradeoff when tracking and disturbance rejection are simultaneous goals, and a constraint communication channel is present in the feedback loop. Examples and simulations are provided to demonstrate some of the results.     

Optimal tracking performance of MIMO discrete‐time systems with communication constraints

The optimal tracking problem for multiple‐input multiple‐output linear‐time‐invariant discrete‐time systems with communication constraints in the feedback path is studied in this paper. The tracking performance is measured by the energy of the error signal between the output of the plant and the reference signal. The objective is to obtain an optimal tracking performance, attainable by all possible stabilizing compensators. It is shown that the optimal tracking performance consists of two parts, one depends on the nonminimum phase zeros and zero direction of the given plant, as well as the reference input signal direction, and the other depends on the nonminimum phase zeros, unstable poles, and pole direction of the given plant, as well as the bandwidth and additive white Gaussian noise of the communication channel. It is also shown that, if the constraint of the communication channel does not exist, the optimal tracking performance reduces to the existing tracking performance of the control system without communication constraints. A typical example is given to illustrate the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

The internal model principle of control theory

The classical regulator problem is posed in the context of linear, time-invariant, finite-dimensional systems with deterministic disturbance and reference signals. Control action is generated by a compensator which is required to provide closed loop stability and output regulation in the face of small variations in certain system parameters. It is shown, using the geometric approach, that such a structurally stable synthesis must utilize feedback of the regulated variable, and incorporate in the feedback path a suitably reduplicated model of the dynamic structure of the disturbance and reference signals. The necessity of this control structure constitutes the Internal Model Principle. It is shown that, in the frequency domain, the purpose of the internal model is to supply closed loop transmission zeros which cancel the unstable poles of the disturbance and reference signals. Finally, the Internal Model Principle is extended to weakly nonlinear systems subjected to step disturbances and reference signals.     

Stabilization of linear systems with input delay and saturation—A parametric Lyapunov equation approach

This paper studies the problem of stabilizing a linear system with delayed and saturating feedback. It is known that the eigenstructure assignment‐based low‐gain feedback law (globally) stabilizes a linear system in the presence of arbitrarily large delay in its input, and semi‐globally stabilizes it when the input is also subject to saturation, as long as all its open‐loop poles are located in the closed left‐half plane. Based on a recently developed parametric Lyapunov equation‐based low‐gain feedback design method, this paper presents alternative, but simpler and more elegant, feedback laws that solve these problems. The advantages of this new approach include its simplicity, the capability of giving explicit conditions to guarantee the stability of the closed‐loop system, and the ease in scheduling the low‐gain parameter on line to achieve global stabilization in the presence of actuator saturation. Copyright © 2009 John Wiley & Sons, Ltd.     

Stabilization of linear systems with time‐varying input delay by event‐triggered delay independent truncated predictor feedback

This article deals with the problem of stabilization of linear systems with time‐varying input delay by an event‐triggered delay independent truncated predictor feedback law, either of the state feedback type or the output feedback type. Only the information of a delay bound rather than the delay itself is required in the design of both control laws and event‐triggering strategies. For both the state feedback case and the output feedback case, an admissible delay bound that guarantees the stabilizability of a general linear system is established, and the Zeno behavior is shown to be excluded. For linear systems with all open‐loop poles at the origin or in the open left‐half plane, stabilization can be achieved for a delay under an arbitrarily large bound.     

Logarithmic integrals, interpolation bounds, and performancelimitations in MIMO feedback systems

We study performance limitation issues found in linear multivariable feedback systems. Our main contributions include Bode and Poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions. These results characterize and quantify explicitly how open-loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance. The role of time delay is also studied in this context. Most notably, we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros, and in particular, on how pole and zero directions are aligned. The latter is characterized by angles measuring the mutual orientation between zero and pole directions, and it is shown to play a crucial role in multivariable system design     

Feedback limitations of linear sampled‐data periodic digital control

We present a set of feedback limitations for linear time‐invariant systems controlled by periodic digital controllers based upon an analysis of the inter‐sample response of the closed‐loop system to sinusoidal inputs. Fundamental sensitivity and complementary sensitivity functions govern the fundamental and harmonic components of the continuous closed‐loop response. The continuous and discrete response of the system is sensitive to variations in the analog plant at frequencies integer multiples of ω_s/N away from the excitation frequency, where ω_s is the sampling frequency and N is the period of the controller. These functions satisfy interpolation and integral constraints due to open‐loop non‐minimum phase zeros and unstable poles. In addition, the use of periodic digital control may result in a reduction in closed‐loop bandwidth. Copyright © 2000 John Wiley & Sons, Ltd.     

基于载波频率辅助相位的GPS信号跟踪算法

针对传统全球定位系统(GPS)接收机在高动态环境下跟踪性能不理想,提出一种基于载波频率辅助相位的GPS信号跟踪算法。利用锁频环(FLL)辅助锁相环(PLL)的方式代替传统单一跟踪环路,通过卡尔曼(Kalman)滤波器对接收机各跟踪通道中频信号进行综合处理。根据多条跟踪通道的伪距和伪距率残差对系统状态参量进行综合估计,并搭建Kalman滤波器的状态方程和量测方程,给出了跟踪环路反馈量,与传统标量跟踪模式下的跟踪性能进行了对比。仿真结果表明,基于载波频率辅助相位的GPS信号跟踪算法进入稳态时间减小了100 ms,位置误差精度提高了5 m,速度误差精度提高了近3 m/s,在接收机用户快速运动的环境下,能够很好地处理高动态信号。     

Autotune identification for systems with right-half-plane poles and zeros

Autotuning based on relay feedback tests have received a great deal of attention recently. The relay feedback tests fail to generate sustained oscillation in some cases. For systems with RHP poles and/or zeros, incorrect autotune identification procedure may give erroneous information and unstable limit cycle. Frequency domain analyses provide a criteria for prediction of the existence of stable limit cycle. Furthermore, a procedure is given to ensure the success of a relay feedback test. The analyses and procedure are illustrated through open loop stable and unstable systems. Results show that, except for some rare occasion, successful relay feedback experiments can be obtained following the proposed procedure.     

Best tracking and regulation performance under control energy constraint

This paper studies optimal tracking and regulation control problems, in which objective functions of tracking error and regulated response, defined by integral square measures, are to be minimized jointly with the control effort, where the latter is measured by the plant input energy. Our primary objective in this work is to search for fundamental design limitations beyond those known to be imposed by nonminimum phase zeros, unstable poles, and time delays. For this purpose, we solve the problems explicitly by deriving analytical expressions for the best achievable performance. It is found that this performance limit depends not only on the plant nonminimum phase zeros, time delays, and unstable poles-a fact known previously-but also on the plant gain in the entire frequency range. The results thus reveal and quantify another source of fundamental performance limitations beyond those already known, which are nonexistent when only conventional performance objectives such as tracking and regulation are addressed without taking into account the control energy constraint. Among other things, they exhibit how the lightly damped poles, the anti-resonant zeros, as well as the bandwidth of the plant may affect the performance.