Identification and the Information Matrix: How to Get Just Sufficiently Rich?

In prediction error identification, the information matrix plays a central role. Specifically, when the system is in the model set, the covariance matrix of the parameter estimates converges asymptotically, up to a scaling factor, to the inverse of the information matrix. The existence of a finite covariance matrix thus depends on the positive definiteness of the information matrix, and the rate of convergence of the parameter estimate depends on its “size”. The information matrix is also the key tool in the solution of optimal experiment design procedures, which have become a focus of recent attention. Introducing a geometric framework, we provide a complete analysis, for arbitrary model structures, of the minimum degree of richness required to guarantee the nonsingularity of the information matrix. We then particularize these results to all commonly used model structures, both in open loop and in closed loop. In a closed-loop setup, our results provide an unexpected and precisely quantifiable trade-off between controller degree and required degree of external excitation.      

Application of covariance matrix adaptation–evolution strategy to optimal control of hepatitis B infection

To avoid the cirrhosis and liver cancer, antiviral treatment for chronic hepatitis is necessary. In the literature, several mathematical models have been used to describe the dynamics of viral infections. In addition, several control strategies have been reported in the literature to deal with optimal antiviral therapy problem of infectious diseases. In this paper, three controller structures with optimized parameters using covariance matrix adaptation–evolution strategy algorithm are proposed for optimal control of basic hepatitis B virus (HBV) infection dynamical system. The first structure is an optimized neural-type sigmoid-based closed-loop controller, which is a nonlinear feedback controller. The second structure is an optimized open-loop time-based fuzzy controller in which the control input is approximated using the mixture of Gaussian membership functions. Finally, an optimized closed-loop fuzzy controller is used as the third control structure. After designing the controllers, some parameters of the HBV infection model are considered to be unknown and the robustness of the controllers is studied. Experimental results show that the optimized neural-type sigmoid-based closed-loop controller has the best performance in terms of healthy hepatocytes and free HBVs concentration among the investigated controllers and the optimized closed-loop fuzzy controller is the best in terms of minimum mean control input signal that is the drug usage. Concerning the robustness, the optimized neural-type sigmoid-based closed-loop controller has the best performance.     

Closed loop experiment design for linear time invariant dynamical systems via LMIs

All stationary experimental conditions corresponding to a discrete-time linear time-invariant causal internally stable closed loop with real rational system and feedback controller are characterized using the Youla-Kucera parametrization. Finite dimensional parametrizations of the input spectrum and the Youla-Kucera parameter allow a wide range of closed loop experiment design problems, based on the asymptotic (in the sample size) covariance matrix for the estimated parameters, to be recast as computationally tractable convex optimization problems such as semi-definite programs. In particular, for Box-Jenkins models, a finite dimensional parametrization is provided which is able to generate all possible asymptotic covariance matrices. As a special case, the very common situation of a fixed controller during the identification experiment can be handled and optimal reference signal spectra can be computed subject to closed loop signal constraints. Finally, a brief numerical comparison with closed loop experiment design based on a high model order variance expression is presented.     

Least costly identification experiment for control

All approaches to optimal experiment design for control have so far focused on deriving an input signal (or input signal spectrum) that minimizes some control-oriented measure of plant/model mismatch between the nominal closed-loop system and the actual closed-loop system, typically under a constraint on the total input power. In practical terms, this amounts to finding the (constrained) input signal that minimizes a measure of a control-oriented model uncertainty set. Here we address the experiment design problem from a “dual” point of view and in a closed-loop setting: given a maximum allowable control-oriented model uncertainty measure compatible with our robust control specifications, what is the cheapest identification experiment that will give us an uncertainty set that is within the required bounds? The identification cost can be measured by either the experiment time, the performance degradation during experimentation due to the added excitation signal, or a combination of both. Our results are presented for the situation where the control objective is disturbance rejection only.     

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.     

基于参数辨识的自适应解耦控制算法的研究

多变量模型的复杂结构、强耦合性、被控对象参数的未知、慢时变等问题要求控制器必须具有良好的自适应性,针对以上问题提出了一种基于改进的广义最小方差闭环自适应解耦控制器实现更好的自适应,其由参数可调的控制器和自适应控制律组成,此控制器通过将闭环系统方程的传递函数矩阵等于期望的对角矩阵来实现解耦,同时改进的辨识算法可进行在线辨识控制器的参数实现同步自适应解耦。通过以CARMA为多变量控制模型,采用该方法进行仿真有效的解决了多变量之间的耦合性。结果表明该方法能够适应相应的变化,跟踪性能较好,且具备良好的解耦能力,进而保证了闭环系统的稳定性,从而验证了此方法能够效提高控制系统的稳定性和鲁棒性。     

数据驱动闭环子空间预测控制方法研究与应用

提出一种完全数据驱动的闭环子空间辨识及预测控制器设计方法. 该方法完全由闭环系统的输入输出数据辨识子空间矩阵, 通过子空间矩阵的拆分, 排除了与扰动相关的模型输入, 进而获取子空间矩阵参数的无偏估计; 将辨识得到的闭环系统子空间矩阵描述直接作为预测模型, 设计预测控制器; 将其应用于某钢铁集团焦炉炭化室压力控制系统, 取得了良好的控制效果.

     

随机系统的多模型直接自适应解耦控制器

针对多变量离散时间随机系统, 提出了一种采用广义最小方差性能指标的多模型直接自适应解耦控制器. 该多模型控制器由多个固定控制器和两个自适应控制器构成. 固定控制器用以覆盖系统参数的可能变化范围, 自适应控制器用以保证系统的稳定性和提高暂态性能. 该多模型控制器利用矩阵的伪交换性和拟Diophantine方程性质, 基于广义最小方差性能指标, 将随机系统辨识算法和最优控制器设计相结合, 直接辨识出控制器的参数, 通过广义最小方差性能指标中加权多项式的选取,不但实现了多变量系统的动态解耦控制, 而且消除了稳态误差、配置了闭环极点. 文末给出了全局收敛性分析. 仿真结果表明该方法明显优于常规自适应控制器.     

For model-based control design, closed-loop identification gives better performance

We compare open loop versus closed loop identification when the identified model is used for control design, and when the system itself belongs to the model class, so that only variance errors are relevant. Our measure of controller performance (which is used as our design criterion for identification) is the variance of the error between the output of the ideal closed loop system (with the ideal controller) and that of the actual closed loop system (with the controller computed from the identified model). Under those conditions, we show that, when the controller is a smooth function of the input-output dynamics and the disturbance spectrum, the best controller performance is achieved by performing the identification in closed loop with an operating controller that we characterize. For minimum variance and model reference control design criteria, we show that this ‘optimal operating controller for identification’ is the ideal controller. This then leads to a suboptimal but feasible iterative scheme.     

闭环变量带误差系统的控制器设计

针对目前大多数系统的研究中,只考虑输出被误差干扰的情况,提出了一种闭环变量带误差系统模型(输入和输出信号均被噪声干扰),并对该系统的控制器进行设计。采用最小方差控制来对控制器进行设计,设计完成最小方差控制器以及自校正最小方差控制器。然后应用最小方差性能指标来评估控制器性能,并通过仿真验证控制器性能。     

飞机颤振模态参数辨识试验的快速滤波算法

研究飞机颤振随机模型中实际输入-输出信号序列的最优滤波估计问题,利用矩阵论中的矩阵因式分解和统计信号处理中的条件期望公式,将由新息过程构成的块Toeplitz矩阵进行三角分解,得到一种有效的递推滤波算法。对于滤波输入-输出信号的估计值,推导该算法下的估计误差和方差表达式。最后用仿真算例验证采用滤波后得到的输入-输出信号估计值作为飞机颤振模态参数辨识试验的观测信号可得到较为准确的传递函数,进而使得模态参数的辨识也更精确。     

Optimal experiment design under process noise using Riccati differential equations

In this paper, we present a numerical method for optimal experiment design of nonlinear dynamic processes. Here, we suggest to optimize an approximation of the predicted variance–covariance matrix of the parameter estimates, which can be computed as the solution of a Riccati differential equation. In contrast to existing approaches, the proposed method allows us to take process noise into account and requires less derivative states to be computed compared to the traditional Fisher information matrix based approach. This process noise is assumed to be a time-varying random disturbance which is not known at the time when the experiment is designed. We illustrate the technique by solving an optimal experiment design problem for a fed-batch bioreactor benchmark case study. Here, we concentrate on how the optimal input design and associated accuracy of the parameter identification is influenced when process noise is present.     

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.     

有限字长数字控制器的实稳定半径最优实现

主要讨论了有限字长(FWL)数字控制器的一种最优实现问题,考察了一个典型的采样反馈系统,将实有理稳定半径测度应用到有限字长数字控制器的实现问题中,对实稳定半径测度进行优化,并由此得到控制器的最优状态变换矩阵和最优结构及最小字长.数值算例验证了优化的结果是有效的,优化后较小字长的控制器就可以使系统取得较大的稳定半径.     

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.     

Steady-state optimal controller with actuator noise variancelinearly related to actuator signal variance

The linear-quadratic-Gaussian (LQG) method assumes that the covariance matrix of the actuator noise is given before the design of the feedback matrix. However, in practice, the actuator-noise covariance matrix may depend on the actuator signal energy, which depends on the feedback. Consequently, the feedback from LQG theory degrades the system performance. The authors investigate the steady-state optimal controller when the noise variance of an actuator is linearly related to the variance of an actuator signal. This control system could be much more precise and/or spend much less control energy than the one obtained through the use of the ordinary LQG method     

Conditions when minimum variance control is the optimal experiment for identifying a minimum variance controller

It is well known that if we intend to use a minimum variance control strategy, which is designed based on a model obtained from an identification experiment, the best experiment which can be performed on the system to determine such a model (subject to output power constraints, or for some specific model structures) is to use the true minimum variance controller. This result has been derived under several circumstances, first using asymptotic (in model order) variance expressions but also more recently for ARMAX models of finite order. In this paper we re-approach this problem using a recently developed expression for the variance of parametric frequency function estimates. This allows a geometric analysis of the problem and the generalization of the aforementioned finite model order ARMAX results to general linear model structures.     

Existence of (cheap) minimal norm controllers

It is shown that the spectral norm of the closed-loop system matrix is minimized if a special type of minimum variance control is applied. Furthermore, a sufficient condition is derived for the existence of a controller which, with less control effort than this minimum variance controller, obtains the same minimal closed-loop norm.     

Global convergence for adaptive one-step-ahead optimal controllers based on input matching

This paper establishes global convergence for adaptive one-step-ahead optimal controllers applied to a class of linear discrete time single-input single-output systems. The class of systems includes all stable systems whether they are minimum phase or not, all minimum phase systems whether they are stable or not, and some unstable nonminimum phase systems. The key substantive assumption is that the one-step-ahead optimal controller designed using the true system parameters leads to a stable closed-loop system. Subject to this natural restriction, it is shown that a simple adaptive control algorithm based on input matching is globally convergent in the sense that the system inputs and outputs remain bounded for all time and the input converges to the one-step-ahead optimal input. Both deterministic and stochastic cases are treated.     

Stable and optimal fuzzy control of linear systems

A number of stable and optimal fuzzy controllers are developed for linear systems. Based on some classical results in control theory, we design the structure and parameters of fuzzy controllers such that the closed-loop fuzzy control systems are stable, provided that the process under control is linear and satisfies certain conditions. It turns out that if stability is the only requirement, there is much freedom in choosing the fuzzy controller parameters. Therefore, a performance criterion is set to optimalize the parameters. Using the Pontryagin minimum principle, we design an optimal fuzzy controller for linear systems with quadratic cost function. Finally, the optimal fuzzy controller is applied to a ball-and-beam system