Elsevier

Automatica

Volume 131, September 2021, 109786
Automatica

Brief paper
Data-driven tuning of model-reference controllers for stable MIMO plants

https://doi.org/10.1016/j.automatica.2021.109786Get rights and content

Abstract

A new data-driven tuning method of a linearly parametrized controller is proposed for stable multi-input multi-output plants. A criterion that evaluates the difference between responses of the reference model and those of the feedback system is employed for tracking control. Because the criterion is nonconvex with respect to controller parameters, its second order Taylor expansion, which is convex, is used for parameter tuning. To represent the approximated criterion using plant responses instead of a plant mathematical model, it is proposed to use the responses of all entries of the plant for a test function, typically a step function. The parameter value that minimizes the approximated data-based criterion is obtained adopting the linear least-squares method. The proposed tuning method is compared with virtual reference feedback tuning and its extended methods for multi-input multi-output plants.

Introduction

In many practical control applications, a mathematical model of the plant is unavailable and a data-driven controller tuning method that does not demand a mathematical model is desired. A design specification of forcing the response of the closed-loop transfer function to follow that of the reference model is useful for the design of tracking control. The design problem is represented as a model reference control problem, where the control objective is the minimization of the criterion that evaluates the difference between the response of the closed-loop system and that of the reference model. Properties of the closed-loop system can be assigned via the reference model, which is preferable especially for multi-input multi-output (MIMO) plants. This paper studies data-driven tuning of model-reference controllers for the tracking control of MIMO plants.

In industrial control systems, fixed structure controllers, typically proportional–integral–derivative (PID) controllers, are usually used, and a data driven method that estimates the controller parameter is useful for such control systems. There are two classes of off-line methods for parameter tuning. One is an iterative method that conducts an experiment to evaluate the criterion at each iteration (Hjalmarsson et al., 1998, Mišković et al., 2007), and the other is a non-iterative method that uses one-shot response (Campi et al., 2002, van Heusden et al., 2011, Saeki, 2014). As for on-line methods, there are a method that estimates the parameter using the recursive least square method (Wakasa et al., 2011) and an adaptive method that removes falsified controllers from a set of candidate controllers at each step (Safonov & Tsao, 1997). These off-line and online methods mostly use model reference criteria. In Hou and Jin (2011) and Xiong and Hou (2020), a virtual equivalent dynamical linearization data model is built at each dynamic operation point of the closed-loop system for nonlinear plants and the control input is calculated using the discrete-time data model. This adaptive method does not use a fixed structure controller and guarantees the stability of the closed-loop system. In this paper, we deal with the off-line parameter tuning problem for a fixed structure controller. In the following, an unresolved issue of the virtual reference feedback tuning (VRFT) for MIMO plants is outlined and the aim of study is explained.

VRFT provides a data-driven method for tracking control (Campi et al., 2002). This method presents a tuning method of a linearly parametrized controller for stable single-input single-output (SISO) plants. We refer to this method as SISO VRFT in this paper. Features of this method are that the one-shot plant response is used for tuning; the tracking performance is assigned via a reference model; and the controller is efficiently obtained using the least-squares method. In the SISO VRFT, the criterion of virtual reference approach is given and the controller parameter is obtained by minimizing the criterion. However, the criterion looks different from the criterion of the model reference control problem, and the following two important properties are shown: Property (a) The virtual reference criterion takes its minimum value zero for the ideal controller that attains the exact model matching. Property (b) The virtual reference criterion becomes equivalent to the second-order Taylor expansion of the model reference criterion around the ideal controller by using a certain filtered plant response.

VRFT has been applied to several practical problems relating to SISO systems (Formentin et al., 2019, Formentin et al., 2013) and MIMO systems (Campestrini et al., 2016, Formentin et al., 2012, Nakamoto, 2004, Passenbrunner et al., 2011, Rojas et al., 2012). For MIMO systems, extension of the SISO VRFT to MIMO systems has been studied. However, the equivalence does not hold in the case of MIMO plants, because the property that scalar transfer functions are commutative is used to show the equivalence in the SISO VRFT and matrix transfer functions are not commutative. VRFT is restrictive when applied to MIMO plants. The criterion given in Nakamoto (2004) does not have property (a) in general, but has the property by restricting the transfer function of the reference model to be an identity matrix multiplied by a scalar transfer function. The same criterion is used in Formentin et al., 2012, Passenbrunner et al., 2011 and Rojas et al. (2012). The criterion given in Campestrini et al. (2016) has property (a) without the restriction on the transfer function of the reference model. A practical filter is derived in Campestrini et al. (2016), but the Taylor expansion is not used in the derivation, which implies that property (b) is not examined in a precise sense.

Property (b) implies that the second order Taylor expansion of the model reference criterion is minimized using the virtual reference criterion with an appropriate filter for SISO plants, but this approach seems to be difficult for MIMO plants from the above review. In this paper, we will propose a new data-driven method that has the following features. We do not use the virtual reference criterion and examine the minimization of the approximated criterion, namely the second order Taylor expansion, directly. To avoid the problem of commutativity of matrix product, we use a certain class of response data that are measured for each entry of the MIMO plant by applying the same time function. Then, the second order expansion can be expressed using the class of response data and the controller parameters, and the solution that minimizes the approximated criterion is obtained adopting the least-squares method. Corresponding to property (a), the approximated criterion has a property of taking its minimum value zero for the ideal controller that attains the exact model matching.

The remainder of the paper is organized as follows. Section 2 presents a preliminary discussion and describes a feedback system. Section 3 defines the performance index for the model matching problem of MIMO plants, represents the index using plant responses and controller parameters, and gives the tuning procedure. Section 4 reviews the VRFT methods and explains the relationship between the VRFT methods and our proposed method. Section 5 numerically demonstrates the usefulness of the proposed method for two-input two-output plants.

Section snippets

Description of the feedback system

Let us consider a feedback system described by y=G(z)ue=ryu=K(z,θ)e, where y,r,eRm and uRn. G is an n-input m-output plant. The condition nm is assumed so that Td may have full rank m. The plant G is assumed to be stable for the safe measurement of the plant responses in open loop operation. Meanwhile, K is an m-input n-output controller, whose entries are described by discrete time transfer functions: Kij=cijT(z)θij,i=1,2,,n,j=1,2,,m.Here, cij(z) is a p-dimensional column vector of

Performance index

The feedback system from r to y is expressed as y=T(z,θ)r, where T(I+GK)1GK. The desirable tracking performance is specified by the reference model: y=Td(z)r.The ideal controller is the one satisfying Td=(I+GKd)1GKd, but there generally exists no ideal controller because, typically, {K(z,θ)}Kd(z). We therefore obtain K by making the difference between T and Td small.

We evaluate the difference using responses for the test inputs rj(t)=eju0(t),t=1,2,,N, j=1,2,,m, where ejRm. That is to

Comparison with VRFT methods

In this section, we review VRFT methods and compare our methods with them.

First, we explain the SISO VRFT. The model reference criterion: JMR(θ)=12πππ|G(z)K(z,θ)1+G(z)K(z,θ)Td(z)W(z)|2dω, where z=ejω, is considered. The parameter θ is determined by minimizing the different criterion: JVRN(θ)=t=1N(L(z)u(t)K(z,θ)L(z)e(t))2,where a virtual reference r̄(t) that satisfies y(t)=Td(z)r̄(t) and the corresponding tracking error e(t)=r̄(t)y(t) are calculated from the measured plant response u(t)

Non-oscillatory plant

Consider a stable, non-oscillatory, and non-minimum phase plant with the following transfer function (Goncalves da Silva et al., 2016): G1(z)=z(z0.9)(z0.8)0.6z0.91z0.90.2z0.9.Here, G1 has the unstable transmission zero z0=1.2. The sampling period Δt is 0.01 s and N=201. u0(t) is a unit step function. Fig. 1 shows the responses y1(t) and y2(t), where the red dotted line shows the signals without noise and black solid line shows the signals with noise. Noises of the responses are given by 1.2

Conclusion

A new data-driven method for tuning a linearly parametrized controller was proposed for stable n-input m-output plants. The responses of all entries of the MIMO plants are used in the tuning. The criterion for tracking control is defined by the sum of squares of the difference between the responses of the reference model and those of the feedback system. The model-reference criterion is approximated by its second order Taylor expansion, and expressed using the response data and the controller

Masami Saeki is a Professor Emeritus at Hiroshima University, Japan. He received the B.E., M.E., and Ph.D. degrees, all in engineering, from Kyoto University in 1976, 1978, and 1982, respectively. In 1981 he became an Instructor in the Department of Electrical Engineering, Kyoto University. From 1982 to 1992, he was an Assistant Professor and then an Associate Professor with the Institute of Information Sciences and Electronics, University of Tsukuba. From 1992 to 2018, he was a Professor in

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Masami Saeki is a Professor Emeritus at Hiroshima University, Japan. He received the B.E., M.E., and Ph.D. degrees, all in engineering, from Kyoto University in 1976, 1978, and 1982, respectively. In 1981 he became an Instructor in the Department of Electrical Engineering, Kyoto University. From 1982 to 1992, he was an Assistant Professor and then an Associate Professor with the Institute of Information Sciences and Electronics, University of Tsukuba. From 1992 to 2018, he was a Professor in the Department of Mechanical Systems Engineering, Hiroshima University. His research interests include robust control and optimization, parameter space design, anti-windup control, unfalsified control, and data-driven control.

This work was supported by JSPS KAKENHI Grant Number JP17K06496. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Juan I. Yuz under the direction of Editor Torsten Söderström. I thank the anonymous reviewers for useful comments and Glenn Pennycook, MSc, from Edanz Group (https://en-author-services.edanzgroup.com/ac) for editing a draft of this manuscript.

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