Performance assessment of multivariable feedback control systems

doi:10.1016/S0005-1098(01)00032-2

Automatica

Volume 37, Issue 6, June 2001, Pages 899-905

https://doi.org/10.1016/S0005-1098(01)00032-2 Get rights and content

Abstract

A new procedure is developed for the estimation of minimum-variance performance bounds in multivariable feedback control systems. The algorithm used in this paper does not require any knowledge of the interactor matrix. Instead, it uses the first few Markov parameters of the plant (up to the delay order of the process) and a set of closed-loop operating data to estimate an expression for the outputs under minimum-variance control. The minimum-variance term of a multivariable feedback control system is shown to be identical to the behavior of the outputs under a finite-horizon minimum-variance control with specified horizon length. The utility of the proposed method in the performance assessment of multivariable control systems is demonstrated through a simulated example.

Introduction

In the literature of control loop performance assessment, the minimum-variance (MV) control has been widely used as a reference bound on achievable performance since the work of Harris (1989). Using this MV performance bound as a performance benchmark has some merits in that (1) it is the absolute lower bound on achievable performance for linear systems and (2) it is the only performance measure that can be evaluated without complete knowledge of the process model. Therefore, methodologies for the assessment of a MV benchmark have been reported in a variety of control applications including (1) single-loop feedback control (Harris, 1989; Lynch & Dumont, 1996), (2) feedforward/feedback control (Desborough & Harris, 1993), (3) cascade control (Ko & Edgar, 2000), and (4) multivariable feedback control (Harris, Boudreau, & MacGregor, 1996; Huang, Shah, & Kwok, 1997b). Two comprehensive review papers by Qin (1998) and Harris, Seppala, and Desborough (1999) provide a detailed review of the research on control loop performance assessment.

In the performance assessment of multivariable feedback control, the interactor matrix played an important role in the recent papers by Harris et al. (1996) and Huang et al. (1997b) as a multivariable generalization of the time-delay term encountered in single-loop. Harris et al. (1996) have suggested using multivariate spectral factorization of the interactor matrix to obtain the closed-loop behavior of the outputs under MV control assuming the interactor matrix is known a priori. On the other hand, Huang, Shah, and Kwok (1997b) have introduced a unitary interactor matrix for the separation of feedback controller-invariant term of the outputs and used multivariate filtering and correlation (FCOR) algorithm to calculate multivariate performance indices using routine operating data. Unlike the time-delay term in SISO processes, the interactor matrix, in general, cannot be constructed from the knowledge of time-delays only. Recently, Huang, Shah, and Fujii (1997a) have shown that the interactor matrix can be estimated from the first few Markov parameters of the process using the algorithm given in Rogozinski, Paplinski, and Gibbard (1987). Even though the interactor matrix is meaningful as a multivariable generalization of the time-delay term encountered in single-loop systems, its calculation and the concept itself have been an obstacle for use by practicing control engineers (Kozub, 1999). Hence, the elimination of the requirement to develop the (unitary) interactor matrix would simplify the calculation of multivariable performance index.

In this paper we develop a simple method that is applicable to both square and non-square processes for the estimation of MV performance bounds without any knowledge of the interactor matrix. This method is based on the first few Markov parameters (up to the delay order) of the process and a set of closed-loop operating data. The first few Markov parameters up to the delay order of the process define the delay structure of the multivariable process and thus are needed for the calculation of the MV performance bound. To be precise, the proposed method does not reduce any a priori knowledge for the calculation of performance index compared to the previous approaches, but it simplifies the calculation of performance index and gives an explicit “one-shot” solution for the output expression under MV control. The capability of giving a direct solution is an important feature of our approach. In addition, unlike previously developed multivariable performance assessment methods, the proposed method is robust to changes in the output weighting matrix, and any additional analysis is not required for the estimation of performance bounds.

The organization of this paper is as follows. We first derive a finite-horizon minimum-variance controller for multivariable systems in Section 2. In Section 3, we show that the expression for the outputs under the infinite-horizon minimum-variance controller is in fact identical to that under a certain finite-horizon minimum-variance controller utilizing singular value analysis and the nullity increasing property of the block Toeplitz matrix. Section 4 describes how the outputs under MV control can be estimated from routine closed-loop output data. The application of the proposed method and comparison of the results with those from Huang's method (1997b) are made in Section 5. Conclusions are drawn in Section 6.

Section snippets

Finite-horizon minimum-variance control of MIMO systems

This section describes how the finite-horizon MV controller can be designed using the plant Markov parameters of an MIMO system with a stable inverse. In particular, the expression obtained for the outputs under MV control will be further analyzed in subsequent sections.

Consider the following multivariable system with m inputs and p outputs represented by a linear time-invariant process with additive noise at the output: $y_{t} =H(z^{−1})u_{t} +N(z^{−1})a_{t} = ∑ i=1 ∞ H_{i} z^{−i} u_{t} + ∑ i=0 ∞ N_{i} z^{−i} a_{t},$ where H_i and N_i are Markov

Expression for the outputs under the infinite-horizon MV control

In the previous section, the outputs under a finite-horizon MV control were expressed in terms of the matrix G_n+1, which is called the block Toeplitz matrix. In this section we will show that the behavior of the outputs under the infinite-horizon MV control is identical to that under a finite-horizon MV control with certain horizon length if the rank of the transfer function matrix H(z⁻¹) is equal to the number of outputs for almost all z. To show this we will first simplify the expression for

Performance assessment

In the previous section it was shown that the outputs under the infinite-horizon MV control or the minimum-variance term can be expressed with the first few Markov parameters of the plant and the disturbances. In this section we describe how the MV term can be estimated from a set of closed-loop operating data and utilized in the calculation of performance bounds for linear multivariable feedback control systems. Let the current feedback controller be described by $u_{t} =−K(z^{−1})y_{t},$ where the

An example

In this section we demonstrate that the expressions for the MV term given in (18) and (28) do result in the same outputs as those of Huang et al.’s (1997b) approach that uses the unitary interactor matrix. Although the proposed method is applicable to both square and non-square systems, only a square multivariable system used in Huang et al. (1997b) will be considered here. The process and disturbance transfer functions are $H(z^{−1})= z^{−1} 1−0.4z^{−1} K_{12} z^{−2} 1−0.1z^{−1} 0.3z^{−1} 1−0.1z^{−1} z^{−2} 1−0.8z^{−1},N(z^{−1})= 1 1−0.5z^{}$

Conclusions

A methodology for the estimation of multivariable minimum-variance performance bounds from routine closed-loop output data has been developed that does not require any knowledge of the interactor matrix. The proposed approach simplified the calculation of multivariable performance index over the previous approaches by eliminating the requirement to develop the (unitary) interactor matrix and giving an explicit “one-shot” solution for the output expression under MV control. In this paper, it was

Byung-Su Ko received his B.S. and M.S. degrees in chemical engineering in 1991 and 1994, respectively, from Seoul National University, Korea, and a Ph.D. in chemical engineering from the University of Texas at Austin in May 2000. Since June 2000, he has been working as an Applications Engineer with ExxonMobil Chemical Company in Baton Rouge, LA. His research interest is on the application of statistical methods in control-loop performance assessment.

Thomas F. Edgar–For a photograph and

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Thomas F. Edgar–For a photograph and biographical sketch of Thomas F. Edgar, see p. 1603 of Vol. 36, No. 11 (November 2000).

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. A. Johansen under the direction of Editor Sigurd Skogestad.

¹: Now at ExxonMobil Chemical Company, P.O. Box 241, Baton Rouge, LA 70821, USA.

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Brief PaperPerformance assessment of multivariable feedback control systems☆

Abstract

Introduction

Section snippets

Finite-horizon minimum-variance control of MIMO systems

Expression for the outputs under the infinite-horizon MV control

Performance assessment

An example

Conclusions

Automatica

Journal of Processing Control

Journal of Processing Control

Automatica

Automatica

Performance assessment measures for univariate feedforward/feedback control

Canadian Journal of Chemical Engineering

Brief Paper
Performance assessment of multivariable feedback control systems☆