Monitoring control performance via structured closed-loop response subject to output variance/covariance upper bound

Owing to the process time delays, the closed-loop response can be divided into feedback control invariant part and feedback controller dependent part. If the latter part is replaced by a user specified response trajectory, we refer to the resultant closed-loop response as structured closed-loop response. The user specified structured closed-loop response has been used as an achievable control against which one can assess performance of control loops. In the control performance monitoring literature, the user specified response is often given as a first-order transfer function with some specified performance requirement, such as time constant. In this paper, we solve this problem from a systematic approach, i.e., in viewpoint of a variance/covariance upper bound on the outputs. With available closed-loop routine operating output data and process time delay/interactor matrix, the desired structured closed-loop response can be obtained directly via estimated closed-loop time series model. A significant feature is that the output variance/covariance upper bound constraint can be explicitly specified according to the product specifications and is always satisfied when the problem is feasible. This desired structured closed-loop response can thus be served as a benchmark against which the existing controller performance can be compared. We also show that two approaches, linearizing change of variables and Frank and Wolfe algorithm, are suitable for solving this problem, which result in a full order and a reduced order structured closed-loop response, respectively. Both approaches are illustrated by two case studies.     

Projection based MIMO control performance monitoring: I—covariance monitoring in state space

In this paper we propose a new control performance monitoring method based on subspace projections. We begin with a state space model of a generally non-square process and derive the minimum variance control (MVC) law and minimum achievable variance in a state feedback form. We derive a multivariate time delay (MTD) matrix for use with our extended state space formulation, which implicitly is equivalent to the interactor matrix. We show how the minimum variance output space can be considered an optimal subspace of the general closed-loop output space and propose a simple control performance calculation which uses orthogonal projection of filtered output data onto past closed-loop data. Finally, we propose a control performance monitoring technique based on the output covariance and diagnose the cause of suboptimal control performance using generalized eigenvector analysis. The proposed methods are demonstrated on a few simulated examples and an industrial wood waste burning power boiler.     

Practical solutions to multivariate feedback control performance assessment problem: reduced a priori knowledge of interactor matrices

The research on control loop performance monitoring and diagnostics has been and remains to be one of the most active research areas in process control community. Despite of numerous developments, it remains as a considerably challenging problem to obtain a minimum variance control benchmark from routine operating data for multivariable process since the solution relies on the interactor matrix (or inverse time delay matrix). Knowing the interactor matrix is tantamount to knowing a complete knowledge of process models that are either not available or not accurate enough for a meaningful calculation of the benchmark. However, the order of an interactor matrix (OIM) for a multivariable process, a scalar measure of multivariate time delay, is a relatively simple parameter to know or estimate a priori. This paper investigates the possibility to estimate a suboptimal multivariate control benchmark from routine operating data if the OIM is available. The relation between this suboptimal benchmark and the true multivariate minimum variance control benchmark is investigated. Analytical expressions for the lower and upper bounds of the true multivariate minimum variance are derived. Although not minimum variance control, this benchmark answers important practical questions like “at least how much potential of the improvement does the control have by tuning or redesigning?”. It is further shown that the proposed suboptimal benchmark is achievable by a practical control provided that the system of interest is minimum phase. Simulation examples illustrate the feasibility of the proposed approach.     

前馈反馈线性时变多变量控制系统性能评价

本文研究一类具有可确定“时滞”的线性时变多变量(linear time-varying multiinput-multioutput, LTV MIMO)控制系统的性能评价方法。定义了该类系统中相当于单回路“过程滞后”的矩阵因子——关联矩阵，并利用该关联矩阵和正常乘法提出一套建立线性时变多变量控制回路前馈反馈最小方差(minimum variance, MV)基准的系统化算法和过程．需要考虑到LTV系统特有的计算特性，利用实际最小方差输出而不是关联滤波最小方差输出获得MV基准。理论和仿真研究表明，所提方法能准确和有效地评价线性时变多变量过程．     

Performance assessment of MIMO systems based on I/O delay information

The minimum variance (MV) control is one of the most popular benchmarks in control performance assessment. For a SISO process, the minimum variance can easily be estimated given the information of the process time delay. However, it is more difficult to obtain the MV benchmark for a multivariable system since the solution relies on the process interactor matrix. The computation of the interactor matrix requires knowledge of Markov parameter matrices of the plant, which is tantamount to complete knowledge of the process model. This requirement is usually unrealistic, since the model is either not available or not accurate enough for a meaningful calculation. However, the time delays between the inputs and outputs are relatively easy to obtain and can be used to construct an I/O delay matrix. This paper shows how to estimate upper and lower bounds of the MIMO MV performance from routine operating data with the I/O delay matrix known. In order to estimate the upper bound, the introduction of additional time delays into the controller is normally needed. However, should this be considered restrictive, then another upper bound which has recently been proposed can be used instead. On the other hand, the lower bound can readily be estimated from routine data. The results are illustrated by a simulation example.     

On the assessment of multivariable controllers using closed loop data. Part I: Identification of system models

Controller performance assessment of SISO and MIMO systems requires effective and systematic identification of the associated system models based on closed-loop data. In this work, a new methodology for the identification of the process, controller and disturbance models is presented for the purpose of enabling the evaluation of the performance of MIMO control systems. The methodology is based on subspace identification algorithms for the identification of the controller, process and disturbance models from closed-loop data. However, identification of the process model is enhanced by the estimation of the associated interactor matrix via the Variable Regression Estimation technique, the existence of which is mathematically proved. The proposed identification methodology is applied to two 2 × 2 systems utilizing both step-response and PRBS closed-loop data.     

Self-tuning controller design for systems with arbitrary time delays Part 1. Theoretical development

Self-tuning control of multivariable systems with arbitrary time delays has been an active area of research in recent years. The use of an interactor matrix in deriving controllers for such systems has been suggested by various authors. Factorization of the input polynomial matrix has also been proposed for the controller design stage. It is shown in this paper that these two approaches are not, in general, equivalent. A simple and straightforward procedure for the extraction of the input and output time delay matrices is proposed for the input polynomial matrix factorization approach. Conditions for the time delay matrices to exist are also derived for a class of multivariable systems. These conditions provide a theoretical justification for the use of time delay matrices in self-tuning control of such systems.     

随机时延网络控制系统的H∞输出反馈控制器设计

研究了一类具有随机通讯时延的网络控制系统的控制器设计问题.同时考虑了控制器和执行器之间,控制器和传感器之间这两类由于有限网络带宽而出现的随机传输时延.采用满足Bernoulli分布的二进制序列来描述数据传输的随机时延.利用矩阵不等式方法给出了动态输出反馈控制器存在的充分条件,所设计的控制器使得闭环系统是均方意义下指数稳定并具有给定的H∞性能.采用SLPMM(sequentially linearprogramming matrix method)给出了控制器的求解算法.仿真算例说明该方法的有效性.     

基于用户自定义指标的多变量控制系统性能评估技术

大量的计算以及对一般关联矩阵先验知识的需求导致用户自定义指标在实际工业应用中存在诸多不便,鉴于此,为了减少对过程模型信息的需求,避免估计一般关联矩阵,研究了如何简化用户自定义指标的计算,并提出了两种改进的用户自定义指标的计算方法.该方法仅需过程常规数据和对过程输入/输出时延或关联矩阵阶次的估计便可完成指标的计算.最后通过仿真验证了两种方法的优越性和有效性.     

Robust H/sub /spl infin// control for uncertain discrete-time-delay fuzzy systems via output feedback controllers

This work investigates the problem of robust output feedback H/sub /spl infin// control for a class of uncertain discrete-time fuzzy systems with time delays. The state-space Takagi-Sugeno fuzzy model with time delays and norm-bounded parameter uncertainties is adopted. The purpose is the design of a full-order fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed-loop system and guarantees an H/sub /spl infin// norm bound constraint on disturbance attenuation for all admissible uncertainties. In terms of linear matrix inequalities (LMIs), a sufficient condition for the solvability of this problem is presented. Explicit expressions of a desired output feedback controller are proposed when the given LMIs are feasible. The effectiveness and the applicability of the proposed design approach are demonstrated by applying this to the problem of robust H/sub /spl infin// control for a class of uncertain nonlinear discrete delay systems.     

Performance assessment and robustness analysis using an ARMarkov approach

Application of the ARMarkov model-based formulation offers significant advantages for assessment/monitoring and robustness analysis of process systems. The ARMarkov method does not require a priori specification of the system time delay/interactor matrix, needs only an approximate estimate of model order and can be done using open or closed-loop process data. By appropriate use of standard, linear model estimation techniques, it directly produces statistically consistent estimates of the first few, user-specified number of Markov parameters even in the presence of colored noise. It is shown in this paper that the Markov parameters and the ARMarkov model can be used to calculate the interactor matrix and several process performance metrics including sensitivity/complementary-sensitivity functions and time-domain criteria such as speed of response, minimum variance values etc. In addition it is shown that model-based predictive control (MPC) systems formulated using ARMarkov models have a special state space structure that leads to less conservative robustness bounds for specific types of uncertainties (such as gain mismatch, uncertainty in the fast or slow dynamics, etc.) than applying the Small Gain Theorem directly to the conventional state space model structure.     

一种基于初始闭环系统的性能评估方法

基于初始闭环系统的输出方差和最小方差指标, 提出了一种新的性能评估方法. 在过程时滞变化的情况下, 基于最小方差指标的评估可能会得到错误的结论, 而新的方法可以避免这一缺点. 扩展的性能指标以控制器投运后的初始状态作为零基准, 能够更准确地反映操作工系统性能的变化, 能够很好地替代最小方差指标. 利用交互矩阵可将扩展指标推广到多变量系统的评估中, 本文将这一算法应用于精馏塔过程的评估. 精馏塔过程的仿真示例验证了方法的有效性, 表明过程时滞变化时用扩展指标来进行评估更能反映系统性能的变化.     

Gain-scheduled L-one control for linear parameter-varying systems with parameter-dependent delays

This paper deals with the problem of gain-scheduled L-one control for linear parameter-varying (LPV) systems with parameter-dependent delays. The attention is focused on the design of a gain-scheduled L-one controller that guarantees being an asymptotically stable closed-loop system and satisfying peak-to-peak performance constraints for LPV systems with respect to all amplitude-bounded input signals. In particular, concentrating on the delay-dependent case, we utilize parameter-dependent Lyapunov functions (PDLF) to establish peak-to-peak performance criteria for the first time where there exists a coupling between a Lyapunov function matrix and system matrices. By introducing a slack matrix, the decoupling for the parameter-dependent time-delay LPV system is realized. In this way, the sufficient conditions for the existence of a gain-scheduled L-one controller are proposed in terms of the Lyapunov stability theory and the linear matrix inequality (LMI) method. Based on approximate basis function and the gridding technique, the corresponding controller design is cast into a feasible solution problem of the finite parameter linear matrix inequalities. A numerical example is given to show the effectiveness of the proposed approach.     

控制性能评价中单位关联矩阵的求解

多回路控制性能评价中单位关联矩阵的求解很重要,会影响到时延项的确定与控制性能指标的计算,因此探讨关联矩阵的计算问题是个有意义的话题。关联矩阵求解一般需要知道一些过程模型的信息,这个条件对于实际应用比较苛刻。为避开先验知识的约束,提出了基于闭环数据的单位关联矩阵求解方法。通过对辨识所得各类传函模型的预处理,能从多项式矩阵中直接求解出对角单位关联矩阵或迭代求解出非对角单位关联矩阵。应用结果证明了该方法的有效性。     

Projection based MIMO control performance monitoring: II––measured disturbances and setpoint changes

In this paper the performance monitoring method based on subspace projections from Part I [J. Proc. Cont. 13 (2003) 739] is extended to include measured disturbances and setpoint changes. It was shown in [J. Proc. Cont. 13 (2003) 739] that the minimum variance output space is an optimal subspace of the general closed-loop output space and that orthogonal projections of filtered output data onto past closed-loop output data can be used to assess the performance of feedback controllers. This paper demonstrates that the same framework is directly applicable to systems with measured disturbances by augmenting the data matrix with those measured disturbances. Furthermore, it provides a means of separating suboptimal control performance between that arising from unmeasured disturbances and that due to measured disturbances. The effect of setpoint changes on control performance can be calculated as special feedforward variables. The controller is generally time-varying to include the case of model predictive control. A simulation example and an industrial boiler process are used to demonstrate the effectiveness of the proposed method.     

The unitary interactor matrix and its estimation using closed-loop data

This paper is concerned with the factorization and estimation of the unitary interactor matrix or the time-delay matrix of multivariable systems. The important properties of the unitary interactor matrix for minimum variance control are discussed. An algorithm for factorization of the unitary interactor matrix from the Markov parameters is introduced. A method for direct estimation of the interactor matrix from closed-loop data is proposed. The proposed algorithm is evaluated by application to a simulated example, pilot-scale experiment and actual industrial data.     

Consistency of noise covariance estimation in joint input–output closed-loop subspace identification with application in LQG benchmarking

The LQG trade-off curve has been used as a benchmark for control loop performance assessment. The subspace approach to estimating the LQG benchmark has been proposed in the literature which requires certain intermediate matrices in subspace identification as well as the covariance matrix of the noise. It is shown in this paper that many existing closed-loop identification methods do not give a consistent estimate of the noise covariance matrix. As a result, we propose an alternative subspace formulation for the joint input–output closed-loop identification for which the consistency of the required subspace matrices and noise covariance is guaranteed. Simulation studies and experimental results are provided to demonstrate the utility of the proposed method.     

基于历史性能基准的模型预测控制性能监控

针对实际工业过程中希望只利用过程的日常运行数据对控制系统进行监控的情况,提出了1种用历史性能基准对预测控制系统进行性能评估、监控以及分析的方法.利用运行状态良好的1段历史数据计算出历史性能基准,并将历史性能基准与实际性能的比值用于控制器性能的实时监控.根据历史性能指标的残差监控检测出性能变化时,进一步通过历史数据协方差监控及广义特征值分析,区分出性能显著变差或变好的方向和子空间.为现场工程师提供性能变化的一些原因,用于系统维护.最后通过Shell重油分馏塔仿真证明该方法可以获得很好的评估与监控结果.     

A modified index for control performance assessment

A control performance index which relates actual output variance to minimal achievable variance was first presented by Harris [T.J. Harris, Assessment of control loop performance, The Canadian Journal of Chemical Engineering 67 (1989) 856–861]. In the present paper, a modification of that index is applied. Instead of comparing to minimum variance control — which corresponds to placing all closed-loop poles in the origin — one pole is placed using either control design guidelines or additionally available process knowledge. Some possible choices of the free closed-loop pole are discussed. Based on this modified closed-loop system, a performance index can be calculated which relates actual output variance to the minimum achievable variance using the modified closed-loop system. It is shown that for the calculation of this index no other information apart from the measured process output and the time-delay is necessary. Furthermore, a method is presented which allows the assessment of the accuracy of the calculated index. The original Harris index and the proposed modified index are computed using data from industrial processes.     

Output Feedback Stabilization of Networked Control Systems With Packet Dropouts

In this paper, we discuss the modeling and control of a class of networked control systems (NCSs) with packet dropouts. For the cases that there may be packet dropouts in both the backward and the forward channels in the communication network, and that the network-induced delays are shorter than one sampling period, the closed-loop NCS is modeled as a discrete-time switched system with four subsystems. By using the asynchronous dynamical systems approach and the average dwell-time method, sufficient conditions for the exponential stability of the closed-loop NCS are presented in terms of nonlinear matrix inequalities, and the relation between the packet dropout rate and the stability of the closed-loop NCS is explicitly established. A procedure involving an iterative algorithm is proposed to design the observer-based output feedback controllers. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results.