Multi-criteria performance assessment based on closed-loop system identiﬁcation

A method to assess the performance of control loops, based on closed-loop system identiﬁcation is proposed. This method allows to take into account the trade-oﬀ between process variable and manipulated variable energy, thus overcoming one of the most important criticisms to Harris’index. To illustrate the proposed approach, a numerical example is given, for which the proposed index is 8% greater than Harris’index.


Introduction
In the process industry sector, companies need to extract actionable informa-2 tion from sensor-based data collected on a daily basis through their integrated IT/OT systems [1]. It is estimated that each operator typically needs to as-4 sess the performance of 90-180 control loops [2], and therefore the interest in developing computer-aided tools to facilitate their work. In this framework, 6 timely and accurate assessment may enable the implementation of appropriate corrective actions as soon as required. 8 A common approach for control loop performance assessment (CPLA) is based on selecting an ideal benchmark to which the actual loop can be compared 10 [3]. As an example, if the Minimum Variance (MV) control loop is selected, it is possible to determine the well-known Harris' index (HI), using only routine data, collected during normal operation, without the need of ad hoc invasive experiments [4]. This index has been extensively used in commercial applications 14 and extended to assess more complex processes [5].
Although numerous of similar techniques are available [6], due to real-world 16 phenomena like non-linearity, unknown disturbances and dynamics, unstable or multivariable loops, etc., this can still be a challenging problem. In this paper 18 we propose a multi-criteria approach based on the following rationales: It may be necessary to consider different aspects related to loop perfor-20 mance, possibly in contradiction, which is not possible using the classic HI.
It is well-known that this index is not realistic because the MV controller 22 is too aggressive. Therefore, in practice it may largely underestimate loop performance.

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Companies are currently implementing, or willing to implement, computing infrastructure able to run more complex analysis algorithms, compared 26 to past situation.
The structure of this paper is as follows. In section 2, problem statement is 28 presented. In section 3, the proposed approach is described and in section 4 a simple case study is presented.

Problem statement
Let S be a commissioned stable single-input single-output closed-loop LTI 32 system, as in Fig. 1, where ε (k) is a disturbance assumed to be unit white noise.
Based on the data collected during the lapse k = 0, 1, 2, ..., N − 1, we define the 34 loop performance P to be a discrete variable Thus, the plant under control is described by the following model where N H + N is the total number of available measurements, with N H N .
As discussed in the previous section, the classical approach to solve this 42 problem is to determine the Harris' index (HI), denoted hereafter η M V . If we denote σ 2 M V the minimum variance achievable by any linear controller, η M V 44 will be given by the ratio: where s 2 y is the measured output variance The estimation of minimum variance σ 2 M V requires the identification of an 48 ARMA model (see for example [3]) from historical data y(−N H )..., y(−2), y(−1), y(0), which allows also to estimate 50 the closed-loop transfer function, given by Knowing σ 2 M V and η M V (see equation 5) the loop performance P is given where η G , η P define performance ranges, usually determined by human experts 54 according to their past experience and a priori information.
The previous approach is well-known and statistical confidence intervals can 56 be calculated for η M V [7]. However, as discussed in the previous section, in practice we may be largely underestimating loop performance.

Proposed approach
We focus on the trade-off between process variable y(k) and manipulated 60 variable u(k) energy. For this, let's denote As in the previous section, we start by estimating A cl (z −1 ), B cl (z −1 ) (see equation 9) from which we obtain the open-loop transfer function Note that in practice, the order of polynomials A(z −1 ), B(z −1 ), namelyn,m, 64 needs to be assumed. Then, letŷ(k+τ ) be the one-step-ahead output prediction: where E(z −1 ) and F (z −1 ) are obtained solving the Diophantine equation For each value α = 1 N P , 2 N P , · · · , 1, with N P 1, we determine the solution u * (k) to the following optimization problem Here, it is possible to show that the solution is u * (k) such that Note that this result (18) was found independently by the author of this paper, however an equivalent result was previously published in [8]. Now, denote 72 S * the set of points s * 2 u,α , s * 2 y,α for α = 1 N P , 2 N P , · · · , 1. Note that S * is a numerical approximation of the Pareto front [9] corresponding to the problem Now, denote p * 2 u,α , p * 2 y,α the point in S * which is nearest to s y,u = s 2 y , s 2 u but dominates s y,u , which means The performance will be assessed based on the following indicator: Note inmediately that

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Consider the first-order system described by with a 1 = −0.9 (also considerd in [3]) and controller As explained in the previous section, first we generate the data y(−N H )..., y(−2), y(−1) and identify the following third-order ARMA model, assumingn = 3,m = 3, and generate S * showed in Fig. 2, with N P = 30, and Note, that, as expected, the proposed multi-criteria index is almost 8% greater than Harris' index. The code to obtain these results is available in the following 94 link: https://github.com/multiopti/research/blob/main/MC_CLPA.ipynb

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A multi-criteria approach for control loop performance assessment was proposed, which allows to take into account the trade-off between process variable 98 and manipulated variable energy, thus overcoming one of the most important criticisms to Harris'index. In future, we plan to extend this method to more 100 complex models, including unstable multivariable processes. URL https://www.automationworld.com/factory/iiot/blog/