Nonlinear control performance assessment in the presence of valve stiction

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Abstract

Control performance assessment or CPA is a useful tool to establish the quality of industrial feedback control loops, but in practice its usefulness is hampered by nonlinearities in the feedback loop. The Harris index is a popular and easily automated metric used to quantify performance of the loop. However, computing the Harris index simply ignoring common process nonlinearities such as valve stiction will lead to an over-estimate of the index, and consequently a false sense of security. In this paper, we propose two complimentary strategies for accurately assessing the quality of the control performance for loops suffering from modest valve stiction thus avoiding the bias inherent in the standard CPA calculations.

Introduction

It is perhaps not surprising that instrument and control engineers are overwhelmed by the sheer number of loops that need attention on any typical industrial processing plant. Many loops are mis-tuned, if tuned at all, as noted by audits [1], [2] and many control valves are only maintained when something catastrophic occurs. However, the economic benefits from improving the performance of control loops, even those operating at a cursory glance acceptably, is often grossly under estimated.

Clearly this suboptimal state of affairs persists partly because the sheer number of loops makes it a substantial task to monitor and retune, and the fact that there are a number of possible reasons for a control loop to under-perform. CPA is applied in the refining, petrochemicals, pulp and paper, and the mineral processing industries as noted by [3], [4], [5], while a recent practical overview is given in [6] and an automated system intended for plant wide use is described in [7].

In this paper, we will extend one of the more important techniques used to quantify performance indices to a common practical nonlinear problem—control valve stiction. It is now widely recognised that valve stiction is a common industrial problem [8], prompting [9], [10], [11], [12], [13], [14] to investigate ways to diagnose the issue, while [15], [16] are two of the few attempts to regress parametric stiction models and thereby indirectly quantify the stiction problem.

In this work, we are not overly concerned with excessive stiction as that is relatively easily recognised (perhaps by its tell-tale triangular periodic waveform or one of the many strategies outlined in [14], [9], [11], [12], [17], [18] and compared in [19]), but rather in the cases where the valve stiction is relatively small, and hence easily overlooked, but still insidious. After all, when the output data exhibits excessive and obvious oscillation due to things other than poor tuning, the only sensible option is to first service the valve, and only then perform a CPA.

The danger is that under moderate valve stiction where perhaps the tell-tale limit cycle oscillation is buried under the process noise, and one performs a regular performance index calculation using the linear CPA techniques, one overestimates the quality of the performance. Consequently, the neglect of the nonlinear valve stiction phenomena causes a bias which leads the control engineers into a false sense of security.

In the case of nonlinear systems, [20] superimposes a differentiable nonlinear dynamic model to an additive linear or partially nonlinear disturbance where it is shown that a minimum variance feedback invariant exists and the minimum variance performance can be estimated from routine operating data.

In [20], a general nonlinear model, polynomial AR model, is used to estimate the performance index however this method is only valid for nonlinear systems in which the function of nonlinearity is differentiable. However, the valve stiction is non-differentiable, so to provide more reliable estimates of performance index for the process with the moderate valve stiction, we propose two methods. The first is a semi-parametric method based on spline smoothing [21], and the second exploits the steady-state periods when the valve is stuck [22].

The layout of the paper is as follows. In Section 2, the problem statement and model including valve stiction is introduced. Section 3 describes the existence of a minimum variance lower bound and the biased estimation of a performance index. Section 4 outlines the proposed two methods which can be used to estimate the minimum variance lower bound with a valve stiction problem. In Section 5, several simulations are used to illustrate the proposed methodology. This is followed by a discussion and conclusions highlighting both the limitations and potential of the proposed methods.

Section snippets

Process description

We assume the plant can be adequately modelled byyt=B(q−1)A(q−1)q−but+dtwhereA(q−1) andB(q−1) are polynomials in the backshift operator q−1, and b is the time delay of the system whose upper bound is assumed known. The disturbance dt is modelled as the output of a linear Autoregressive-Integrated-Moving-Average (ARIMA) filter driven by white noise at of zero mean and variance σa2 of the formdt=θ(q−1)ϕ(q−1)∇hatwhere ∇=def(1−q−1) is the difference operator and h is a non-negative integer,

Minimum variance lower bound with valve stiction cases

If a process follows the form of Eq. (3), it was proved in [20] that a feedback invariant exists. We only need to show that the b-step ahead prediction error, et+b|t, is independent of the manipulated variable action. The feedback invariant is given inyt+b=B(q−1)A(q−1)f(ut)+dt+b|t+et+b|tyt+b=yt+b|t+et+b|twhereet+b|t=(1+ϕ1q−1+⋯+ϕb−1q−(b−1))at+band the ϕ weights are the impulse coefficients of the θ(q−1)/ϕ(q−1)∇h transfer function andA(q−1)yt+b|t=B(q−1)f(ut)+Pb(q−1)ϕ(q−1)∇hat=B(q−1)f(ut)+Pb(q−1)ϕ(

Estimating the MVPLB in the case of valve stiction

An obvious CPA strategy is simply to ignore the presence of the nonlinearity, and compute the MVPLB in standard manner assuming a purely linear system. Unfortunately if we do this, we incur a bias. To show this, let β(⋅) be the linear approximator of f(⋅) in Eq. (12)Ã(q−1)yt+b|t=B˜(q−1)β(γ(q−1)(yt−ysp))+C˜(q−1)yt+ϵtwhere ϵ is the bias of the approximator and only involves yt−i,i=0,1,… values.

If we use a linear ARMA model in Eq. (16) to estimate yt+b|t and assume that the parameter estimation is

Simulation experiments

To illustrate the two proposed methods to reliably quantify the Harris index in situations of modest plant stiction, we have chosen a simple single-input, single-output (SISO) plant with time constants 10 and 2, and steady-state gain of 3, sampled at Ts=1,Gp=BA=(0.04338+0.03755q−1)q−41−1.621q−1+0.6483q−2with a PI controllerGc=0.11−0.1q−11−q−1and an additive ARMA disturbancedt=at1−1.6q−2+0.8q−1,at∈N(0,0.005)where at is a sequence of independently and identically distributed (i.i.d.) Gaussian

Discussion

The results from the numerical experiments show that both strategies establish the minimum variance performance lower bound given loops suffering from moderate valve stiction. While both methods on average deliver values for the Harris index closer (and lower) to the true value, by comparing the uncertainty ranges in Fig. 8, Fig. 11, it is evident that the stuck-valve strategy is to be preferred provided the stuck periods are long enough.

While the stuck-valve method is specific to

Conclusions

The strategies proposed in this paper reliably establish the minimum variance performance lower bound in the case of moderate valve stiction using only observable signals and crude estimates of the plant dominant time constants and plant delay. The importance of this work is that one can estimate the achievable controlled performance for an industrial control loop despite the valve suffering from poor maintenance causing valve stiction. Furthermore if one were to simply ignore the valve

Acknowledgments

Financial support for this project from the Industrial Information and Control Centre, Faculty of Engineering, The University of Auckland, New Zealand is gratefully acknowledged.

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