Model validation for control and controller validation in a prediction error identification framework—Part II: illustrations

In this paper, we illustrate our new results on model validation for control and controller validation in a prediction error identification framework, developed in a companion paper (Gevers et al., Automatica (2003) 39(3) pii: S005-1098(02)00234-0), through two realistic simulation examples, covering widely different control design applications. The first is the control of a flexible mechanical system (the Landau benchmark example) with a tracking objective, the second is the control of a ferrosilicon production process with a disturbance rejection objective.     

Identification for control of multivariable systems: Controller validation and experiment design via LMIs

This paper presents a new controller validation method for linear multivariable time-invariant models. Classical prediction error system identification methods deliver uncertainty regions which are nonstandard in the robust control literature. Our controller validation criterion computes an upper bound for the worst case performance, measured in terms of the -norm of a weighted closed loop transfer matrix, achieved by a given controller over all plants in such uncertainty sets. This upper bound on the worst case performance is computed via an LMI-based optimization problem and is deduced via the separation of graph framework. Our main technical contribution is to derive, within that framework, a very general parametrization for the set of multipliers corresponding to the nonstandard uncertainty regions resulting from PE identification of MIMO systems. The proposed approach also allows for iterative experiment design. The results of this paper are asymptotic in the data length and it is assumed that the model structure is flexible enough to capture the true system.     

Relations between uncertainty structures in identification for robust control

Various techniques of system identification exist that provide a nominal model and an uncertainty bound. An important question is what the implications are for the particular choice of the structure in which the uncertainty is described when dealing with robust stability/performance analysis of a given controller and when dealing with robust synthesis. It is shown that an amplitude-bounded (circular) uncertainty set can equivalently be described in terms of an additive, Youla parameter and ν-gap uncertainty. As a result, the choice of structure does not matter provided that the identification methods deliver optimal uncertainty sets rather than an uncertainty bound around a prefixed nominal model. Frequency-dependent closed-loop performance functions based on the uncertainty sets are again bounded by circles in the frequency domain, allowing for analytical expressions for worst-case performance and for the evaluation of the consequences of uncertainty for robust design. The results can be used to tune optimal experimental conditions in view of robust control design and in the further development of experiment-based robust control design methods.     

Quantification of frequency domain error bounds with guaranteed confidence level in prediction error identification

This paper considers prediction error identification of linearly parametrized models in the situation where the system is in the model set. For such situation it is easy to construct a confidence ellipsoid in parameter space in which the true parameter lies with an a priori fixed probability level, α. Surprisingly perhaps, the construction of a corresponding uncertainty set in the frequency domain, to which the true system belongs with probability α, is still an open problem. We show in this paper how to construct such frequency domain uncertainty set with a probability level of at least α.     

Control relevant identification for robust optimal control

This paper presents an approach to designing the input signal for an identification experiment, in which the process model estimate is to be used to formulate and solve for a robust (in a worst case sense) optimal controller. The input signal is designed to contain the information that is relevant for the end use of the model, that is for control purposes. The proposed approach uses sensitivity analysis to determine the input signal frequencies that are important with respect to a certain measure of achievable controller performance in conjunction with a frequency sampling filter model of the process. Based on the sensitivity analysis, an iterative experimental design methodology is suggested.     

An alternative structure for next generation regulatory controllers: Part I: Basic theory for design, development and implementation

Even though employed widely in industrial practice, the popular PID controller has weaknesses that limit its achievable performance, and an intrinsic structure that makes tuning not only more complex than necessary, but also less transparent with respect to the key attributes of the overall controller performance, namely: robustness, set-point tracking, and disturbance rejection. In this paper, we propose an alternative control scheme that combines the simplicity of the PID controller with the versatility of model predictive control (MPC) while avoiding the tuning problems associated with both. The tuning parameters of the proposed control scheme are related directly to the controller performance attributes; they are normalized to lie between 0 and 1; and they arise naturally from the formulation in a manner that makes it possible to tune the controller directly for each performance attribute independently. The result is a controller that can be designed and implemented much more directly and transparently, and one that outperforms the classical PID controller both in set-point tracking and disturbance rejection while using precisely the same process reaction curve information required to tune PID controllers. The design, implementation and performance of the controller are demonstrated via simulation on a nonlinear polymerization process.     

Integrated identification and robust control

A framework for integrated identification and control is presented. As part of this framework, frequency domain uncertainty bounds are derived for robust stability tests, a robust stability test for elliptical bounds is developed for SISO systems, a methodology for estimating controller performance is derived, and an optimal experiment design methodology for control-relevant identification is outlined. An example is presented to illustrate how the tools of the framework fit together.     

Experimental evaluation of linear time-invariant models for feedback performance control in real-time systems

In recent years a new class of soft real-time applications operating in unpredictable environments has emerged. Typical for these applications is that neither the resource requirements nor the arrival rates of service requests are known or available a priori. It has been shown that feedback control is very effective to support the specified performance of dynamic systems that are both resource insufficient and exhibit unpredictable workloads. To efficiently use feedback control scheduling it is necessary to have a model that adequately describes the behavior of the system. In this paper we experimentally evaluate the accuracy of four linear time-invariant models used in the design of feedback controllers. We introduce a model (DYN) that captures additional system dynamics, which a previously published model (STA) fails to include. The accuracy of the models are evaluated by validating the models with regard to measured data from the controlled system and through a set of experiments where we evaluate the performance of a set of feedback control schedulers tuned using these models. From our evaluations we conclude that second order models (e.g., DYN) are more accurate than first order models (e.g. STA). Further we show that controllers tuned using second order models perform better than controllers tuned using first order models.     

Improving performance and stability of MPC relevant identification methods

Model Predictive Control (MPC) Relevant Identification (MRI) methods are a good option for identification, if there is model structure mismatch. Herein a new MRI method, named Enhanced Multistep Prediction Error Method (EMPEM), is proposed. EMPEM combines the best characteristics of others MRI methods in a single algorithm. It was developed to identify either closed-loop or open-loop systems; its convergence and stability make it perform better than the other presented methods. To show the advantages of EMPEM, a comparison is made against two other methods (one MRI and one PEM). The statistical analysis indicates that in the cases studied, the performance and the robustness of the new method is equal or better than the other ones.     

Validity of the standard cross-correlation test for model structure validation

In the standard prediction error framework of system identification, statistical properties of estimated models are typically derived under the assumption that the true system is in the model class. The standard model structure validation test for plant models is the sample cross-correlation test between the residuals of the model and the input. It turns out that the standard test itself is valid only under exactly those assumptions it is meant to verify, i.e. the system is in the model class. It is shown that for reliable results of the validation test a vector-valued test is required and that accurate noise modelling is indispensable for reliable model structure validation. This shows the limitation of separate validation of plant and noise model structures. Improvements of the test are presented, and it is motivated by the fact that reserving data only to be used for model validation is not efficient.     

From experiment design to closed-loop control

The links between identification and control are examined. The main trends in this research area are summarized, with particular focus on the design of low complexity controllers from a statistical perspective. It is argued that a guiding principle should be to model as well as possible before any model or controller simplifications are made as this ensures the best statistical accuracy. This does not necessarily mean that a full-order model always is necessary as well designed experiments allow for restricted complexity models to be near-optimal. Experiment design can therefore be seen as the key to successful applications. For this reason, particular attention is given to the interaction between experimental constraints and performance specifications.     

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.     

Identification and control — Closed-loop issues

An overview is given of some current research activities on the design of high-performance controllers for plants with uncertain dynamics, based on approximate identification and model-based control design. In dealing with the interplay between system identification and robust control design, some recently developed iterative schemes are reviewed and special attention is given to aspects of approximate identification under closed-loop experimental conditions.     

Bridging the gap between the linear and nonlinear predictive control: Adaptations for efficient building climate control

The linear model predictive control which is frequently used for building climate control benefits from the fact that the resulting optimization task is convex (thus easily and quickly solvable). On the other hand, the nonlinear model predictive control enables the use of a more detailed nonlinear model and it takes advantage of the fact that it addresses the optimization task more directly, however, it requires a more computationally complex algorithm for solving the non-convex optimization problem. In this paper, the gap between the linear and the nonlinear one is bridged by introducing a predictive controller with linear time-dependent model. Making use of linear time-dependent model of the building, the newly proposed controller obtains predictions which are closer to reality than those of linear time invariant model, however, the computational complexity is still kept low since the optimization task remains convex. The concept of linear time-dependent predictive controller is verified on a set of numerical experiments performed using a high fidelity model created in a building simulation environment and compared to the previously mentioned alternatives. Furthermore, the model for the nonlinear variant is identified using an adaptation of the existing model predictive control relevant identification method and the optimization algorithm for the nonlinear predictive controller is adapted such that it can handle also restrictions on discrete-valued nature of the manipulated variables. The presented comparisons show that the current adaptations lead to more efficient building climate control.     

Box-Jenkins identification revisited—Part I: Theory

In classical time domain Box-Jenkins identification discrete-time plant and noise models are estimated using sampled input/output signals. The frequency content of the input/output samples covers uniformly the whole unit circle in a natural way, even in case of prefiltering. Recently, the classical time domain Box-Jenkins framework has been extended to frequency domain data captured in open loop. The proposed frequency domain maximum likelihood (ML) solution can handle (i) discrete-time models using data that only covers a part of the unit circle, and (ii) continuous-time models. Part I of this series of two papers (i) generalizes the frequency domain ML solution to the closed loop case, and (ii) proves the properties of the ML estimator under non-standard conditions. Contrary to the classical time domain case it is shown that the controller should be either known or estimated. The proposed ML estimators are applicable to frequency domain data as well as time domain data.     

A new error handling algorithm for controller area network in networked control system

An effective error handling mechanism plays an important role to ensure the reliability and robustness of the application of controller area network (CAN) in controlling dynamic systems. This paper addresses a new online error handling approach or named per-sample-error-counting (PSeC) technique that tends to replace native error handling protocol in controller area network (CAN). The mechanism is designed to manage transmission errors of both sensor and control data in networked control system (NCS) used in controlling dynamic system such that the stability of the feedback system is preserved. A new parameter denoted as maximum allowable number of error burst (MAEB) is introduced in which MAEB is selected based on available bandwidth of the CAN network. MAEB serves as the maximum number of attempt of re-transmission of erroneous data per sample which allows the maximum transmission period to be known and guaranteed for time-critical control system. The efficacy of the proposed method is verified by applying the algorithm on the fourth order inverted pendulum system simulated on Matlab/Truetime simulator and the performance is benchmarked with the existing CAN error management protocol. The simulation run under various systems conditions demonstrate that the proposed method results in superior system performance in handling data transmission error as well as meeting control system requirement.     

Identification and robust control for systems with ellipsoidal parametric uncertainty by convex optimization

In this paper, sufficient conditions for robust output feedback controller design for systems with ellipsoidal parametric uncertainty are given in terms of solutions to a set of linear matrix inequalities (LMIs) Performance specifications are in terms of combined pole placement with sensitivity function shaping in the H₂ or H_∞ norm. Furthermore, an optimal input design technique for parameter estimation that is integrated into the robust control design is employed in this paper. This means that performance specifications on the closed‐loop transfer functions are translated into the requirements on the input signal spectrum. The simulation results show the effectiveness of the proposed method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Development of omni-directional correlation functions for nonlinear model validation

In the present study a set of first order correlation functions are proposed to examine the quality of a wide class of identified nonlinear models. The first order correlation functions, defined as omni-directional correlation functions, are integrated into two concise tests to provide more effective auto and cross model error correlation diagnosis than the other approaches proposed from higher order correlation functions. The mechanisms of the novel validity tests are proved in theory and demonstrated with numerical analyses. Two simulated case studies, in the situation of incorrectly detected model structure and estimated parameters, are presented to illustrate the diagnostic power of the new methodology.     

A learning theory approach to system identification and stochastic adaptive control

In this paper, we present an approach to system identification based on viewing identification as a problem in statistical learning theory. Apparently, this approach was first mooted in [E. Weyer, R.C. Williamson, I. Mareels, Sample complexity of least squares identification of FIR models, in: Proceedings of the 13th World Congress of IFAC, San Francisco, CA, July 1996, pp. 239–244]. The main motivation for initiating such a program is that traditionally system identification theory provide asymptotic results. In contrast, statistical learning theory is devoted to the derivation of finite-time estimates. If system identification is to be combined with robust control theory to develop a sound theory of indirect adaptive control, it is essential to have finite-time estimates of the sort provided by statistical learning theory. As an illustration of the approach, a result is derived showing that in the case of systems with fading memory, it is possible to combine standard results in statistical learning theory (suitably modified to the present situation) with some fading memory arguments to obtain finite-time estimates of the desired kind. It is also shown that the time series generated by a large class of BIBO stable nonlinear systems has a property known as β-mixing. As a result, earlier results of [E. Weyer, Finite sample properties of system identification of ARX models under mixing conditions, Automatica, 36 (9) (2000) 1291–1299] can be applied to many more situations than shown in that paper.