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Dynamic Data Reconciliation for Enhancing Performance of Minimum Variance Control in Univariate and Multivariate Systems Zhengjiang Zhang† and Junghui Chen*,‡ †

College of Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, People’s Republic of China Department of Chemical Engineering, Chung-Yuan Christian University, Chung-Li, Taoyuan 32023, Taiwan, Republic of China

‡

Ind. Eng. Chem. Res. 2016.55:10990-11002. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 09/27/18. For personal use only.

S Supporting Information *

ABSTRACT: Healthy controllers are required in order for the control systems to maintain a high level of performance. In past research, minimum variance control (MVC) played a crucial role as a benchmark in performance monitoring because of the attractive theoretical and computational properties associated with it. Since the influence of measurement errors has not been explicitly considered in the MVC theory in stochastic control systems, this paper first analyzes the influence of measurement errors on the control performance of MVC in both univariate and multivariate systems. And then the dynamic data reconciliation (DDR) methods are proposed and combined in the procedure of MVC/multivariate MVC (MMVC) to decrease the influence of measurement errors and to enhance the control performance. Considering both random measurement errors and gross errors, the effectiveness of MVC/MMVC combined with DDR on the variances of process outputs is illustrated by two simulated cases, including both univariate and multivariate systems. An actual pilot scale experiment with two inputs and two outputs is used to demonstrate the effectiveness of MMVC combined with DDR. Results of simulation and experiment show that MVC/MMVC combined with DDR can efficiently enhance control performance. application of the linear stochastic control theory.5 The basic idea of MVC is to express the minimum variance prediction of the plant output in terms of the control input so that the predicted output can be driven to follow a desired output by solving a simple linear equation for appropriate control action.6 This groundbreaking research has received increasing attention in the process control field. Then several research efforts on the evaluation of the control performance have been made. The use of MVC has been extended to multivariate control systems based on the multivariate interpretation of the delay term, known as the interactor matrix, by Harris et al., Huang et al., and Yan et al.7−9 Ko and Edgar extended MVC to the issue of the cascade control system performance assessment.10 They also investigated the minimum variance performance based PID controller in the single loop case and developed an iterative solution for the calculation of the best achievable (minimum variance) PID control performance later.11 In recent years, MVC has been also developed for time-variant process systems,12,13 model predictive controllers,14,15 PCA-based minimum variance performance,16 decentralized controllers,17 adaptive optics systems,18 stochastic and deterministic control performance of batch processes,19 and nonlinear multivariate

1. INTRODUCTION In order to keep the plant operation stable, safe, and up to environmental standards, robust performance is required in control systems. Process control engineers and instrumentation technicians have the daunting task of maintaining and troubleshooting control systems. However, a typical modern industrial process operation consists of hundreds or even thousands of control loops. The poorer performing control loops may go unnoticed indefinitely or for a long period of time. A survey conducted by Desborough and Miller showed that only 16% of the controllers operate at their best to give the optimal performance and about 36% of the loops operate in a manual mode.1 It is not feasible for the operating personnel to manually monitor all the control loops in a plant. An automatic performance monitoring system should be used to continuously evaluate product quality and the production levels of a plant. The performance monitoring system is built upon measurable plant data that are reworked (usually statistically) to provide a clear and simple picture of the operating states of a plant. Then, one needs a means of comparison to evaluate the current performance. The research on control system performance assessment can be traced back to the 1960s and 1970s. Astrom and DeVries and Wu carried out the fundamental research.2,3 In the literature of control loop performance assessments, minimum variance control (MVC) had been widely used as a reference bound on the achievable performance since Harris proposed it.4 MVC is a simple, practical control strategy resulting from the © 2016 American Chemical Society

Received: Revised: Accepted: Published: 10990

July 2, 2016 August 24, 2016 September 8, 2016 September 8, 2016 DOI: 10.1021/acs.iecr.6b02532 Ind. Eng. Chem. Res. 2016, 55, 10990−11002

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Industrial & Engineering Chemistry Research

Figure 1. Stochastic univariate (or multivariate) feedback control system.

systems.20,21 Shardt et al. gave a good review of the current techniques for performance assessment.22 One issue with using a minimum variance controller as a standard is that there are only a few industrial applications of MVC. Hugo concluded that there are three reasons why MVC is not always applicable in practice:23 (a) MVC contains no move suppression, so the resultant controller may be overly aggressive; (b) MVC assumes that the process is perfectly known and is linear; (c) Minimum variance controllers explicitly employ a full ARIMA model of the disturbance, as opposed to the limited disturbance model of industrial controllers. As described by Hugo,23 the first two points are not significant deterrents to MVC while the last reason is more problematic because the ARIMA model of the disturbance does not explicitly describe the influence of measurement errors in the sensor devices. In most of the research papers, the measurement noise (also called random errors), which is small fluctuations of the measured variable from the true value and can be electrically induced or process-induced,2 has been included in the original MVC through the concept of innovation process. However, the MVC theory for stochastic control systems does not explicitly consider the influence of measurement errors (containing two types of errors, random errors and gross errors) presented in actual sensor devices. When measurements are corrupted by random variations of the environment, they are said to be affected by random measurement errors. Gross errors in measurements usually occur for many different reasons, such as human error, instrumental errors, fraudulent behavior, and faults in systems. Gross errors can be quite large. Both errors have a detrimental effect on process control, which requires accurate estimated states and measurements. In that situation, the MVC benchmark, which is an indication of the inherent optimum, would not be trustworthy. In the actual process system, the output signals are usually measured by sensor devices, and the measured signals are then the feedback of the input signals of the controller for comparison. Since the sensor devices are not ideal and error-free, it is necessary to consider the influence of measurement errors on the control performance and to develop some corresponding dynamic data reconciliation (DDR) methods to enhance the control performance. The DDR methods have not been completely developed in the area of the control performance assessment yet. Shown in this paper are several developments on MVC as follows:

(a) With additional consideration of measurement errors in the sensor devices, the influence of measurement errors on the control performance of MVC in both univariate and multivariate systems is analyzed. (b) In order to decrease the influence of measurement errors on MVC and enhance the control performance, DDR methods are proposed and combined in the procedure of MVC. (c) Considering both random measurement errors and gross errors, the effect of MVC combined with DDR on the variances of the process output is illustrated. The remainder of this paper is organized as follows: MVC in stochastic univariate and multivariate systems is revisited in the next section. The influence of measurement noise on the control performance of MVC in univariate and multivariate systems is then analyzed in section 3. The DDR methods are proposed and combined in the procedure of MVC to enhance the control performance in section 4. In section 5, the proposed MVC with DDR methods are used in three case studies to illustrate its effectiveness. Those case studies contain both univariate and multivariate systems. Finally, conclusions are drawn.

2. REVISIT OF MVC IN STOCHASTIC UNIVARIATE AND MULTIVARIATE SYSTEMS Performance benchmarks are necessary to determine what the optimum operating state of the process is. This optimum operating state contains the inherent variance that cannot be compensated for by controllers. The operating optimum is then used as a benchmark to do comparative evaluation of the current operation. There are various methods, and most of them originate from the minimum variance theory developed by Harris.4 The advantage of MVC is that no extra perturbations are necessary to determine the performance; only routine closed loop operating data are used. In this section, the MVC benchmark concept in univariate and multivariate systems is reviewed. 2.1. MVC in Stochastic Univariate System. Consider the stochastic univariate feedback control system shown in Figure 1. In order to determine what the controller algorithm should be, consider a general controlled autoregressive moving average model of the closed loop response, which can be described by y(t ) = z −k

B(z −1) D(z −1) u ( t ) + d (t ) A(z −1) A(z −1)

(1)

where u(t) is the manipulated input and d(t) is a random external disturbance, which is assumed to be zero mean and Gaussian white noise of variance σ2; y refers to the controlled 10991

DOI: 10.1021/acs.iecr.6b02532 Ind. Eng. Chem. Res. 2016, 55, 10990−11002

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The difficulty in the multivariate controller performance assessment is the factorization of the time delay matrix, which is known as the interactor matrix. Assume that A(z−1) is a full rank transfer function matrix. There is a unique, nonsingular, unitary interactor matrix (D), and

variable or the process output. The scaling should be chosen so that y = 0 when the process is at the desired set-point or steady state. Control will not be completely free from variance seeing that the manipulated variable takes k time instant to affect the process output, y. Thus, MVC is used to minimize the variance of the output at time t + k given all the information up to time t, 2

2

σy = E[y(t + k) ]

lim D(z −1)A(z −1) = −lim Ã (z −1) = K 1

z −1→ 0

(2)

E indicates the statistical expectation. The cost function in eq 2 is the equation that describes the deviation of the output from its desired value, so it contains the variance to be minimized by the controller. Since z−k in eq 1 represents a k-step delay in the control signal, the disturbance term D(z−1) can be decomposed into two parts, one related to the past time and the other related to the future time. D(z−1) = A(z −1)F(z −1) + z −kG(z −1)

ỹ (t ) = z −k Ã (z −1)u(t ) + N(z −1)d(t ) −k

(3)

+ z −k R(z −1)B(z −1)−1y(t )]

−1

(10)

(11)

After the feedback dependent, the square bracket term in eq 11, is reduced to zero, minimum variance control of the closedloop response can be obtained. (σy 2)MMVC = E[ỹ (t )T ỹ (t )] = trace{Cov[Fd(t )]}

(5)

(12)

where trace is the sum of the diagonal elements of a matrix. It is shown that multiplying a unitary interactor matrix to the output variables does not change the quadratic measure or the variance of the corresponding output variables. Thus, the benchmark of the multivariate control system is obtained. These existing results with minimum variance control as the benchmark did not explicitly consider the influence of measurement errors. However, according to practical experiences, there is no sensor measurement without measurement noise. Measurement noise is the variation in the sensor reading that does not correspond to changes in the process and can be caused by background electrical interference, mechanical vibration, and process fluctuations. For cases in which there is a relatively large amount of noise, the impact of the measurement noise on the performance of control systems will be discussed in the next section.

(6)

One can determine what the variance of the uncontrollable part in the output is, and the variance will be the theoretical lower bound performance benchmark. The benchmark is an indication of the inherent optimum set by the process design and equipment. With the inherent optimum capability defined in a control loop, the detection of the poor control performance is a relatively simple task. What one needs to do is just to compare the normal variance of the output signal with the benchmark. 2.2. MVC in Stochastic Multivariate System. The stochastic feedback control system with multiple inputs and multiple outputs (MIMO) is shown in Figure 1. The MIMO linear time-invariant stationary stochastic process model can be described by y(t ) = A(z −1)u(t ) + B(z −1)d(t )

−1

ỹ (t ) = F(z −1)d(t ) + [F(z −1)B(z −1)−1A(z −1)u(t )

(4)

Now the objective function in eq 2 to be minimized for the value of the output at the k time instant in the future can be represented by (σy 2)MVC = E[(Fd(t + k))2 ]

(9) −k

Like the univariate system, the output is simplified as an uncontrollable part of the output signals and a controllable part because there is time delay of the process.

The task is now to minimize the cost function (eq 2) by reducing the feedback dependent term represented by the terms in the square bracket of eq 4. The terms in the square bracket need to be zero by changing the manipulated input (u). B(z −1)F(z −1) G(z −1) + u t y(t ) = 0 ( ) D(z −1) D(z −1)

−1

N(z −1) = F(z −1) + z −k R(z −1)

⎡ B(z −1)F(z −1) y(t + k) = Fd(t + k) + ⎢ u(t ) ⎣ D(z −1) ⎤ G(z −1) ( ) y t ⎥ ⎦ D(z −1)

−1

where y(̃ t) = z D(z )y(t) and N(z ) = z D(z )B(z ). Huang and Shah have found that a unitary interactor matrix is an optimal factorization of time delays for multivariate systems in terms of minimum variance control and controller performance assessment.24 The disturbance term N(z−1) can be decomposed into two parts, one related to the past time and the other related to the future time,

Then the output is simplified as a controllable variance part of the output signal and an uncontrollable part caused by time delay of the process.

+

(8)

z →0

where Ã (z−1) is the delay-free transfer function matrix of A(z−1) and K is a full rank constant matrix. Premultiplying both sides of eq 7 by z−kD(z−1), where k is the maximum delay order of interactor matrix D(z−1), yields

3. INFLUENCE OF MEASUREMENT NOISE ON MVC Measured process data inevitably contain some inaccurate information because measurements are obtained by imperfect instruments, but the bound of the control loop performance assessment discussed before was estimated from the data without considering the influence of measurement noise. With the influence of measurement noise, the feedback signals are no long the actual output signals (y(t) for univariate systems and y(t) for multivariate systems), and the measured signals contain the inherent variability associated with the measurement noise.

(7)

where A(z−1) and B(z−1) are proper, rational transfer function matrices for the process plant and disturbances, respectively; y(t) and u(t) are the output vector and the input vector separately; d(t) represents a white noise vector of disturbances with zero mean and covariance matrix Σd. 10992


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Figure 2. Stochastic univariate (or multivariate) system under the consideration of measurement noise.

3.2. MVC in the Stochastic Multivariate System with Measurement Noise. Like the univariate system, the measured signals ym(t) in the multivariate system shown in Figure 2 can also be expressed as

The influence of measurement errors that degrades the control performance of MVC in univariate systems and of MMVC in multivariate systems are separately discussed in this section. 3.1. MVC in the Stochastic Univariate System with Measurement Noise. Under the consideration of the measurement noise for the univariate system shown in Figure 2, the raw process measurement (ym(t)) can be expressed as

ym (t ) = y(t ) + ε(t )

ym(t ) = y(t ) + ε(t )

where ε(t) represents an uncorrelated random noise vector at the point t with zero mean and the covariance matrix Σε. After eq 17 is substituted into eq 11, feedback-invariant and feedback-dependent terms can be yielded. The multivariate process model can be described as

(13)

where ε(t) is the measurement noise assumed to be normally distributed (ε(t) ∼ N(0,ρ2)) and ρ2 is the variance of measurement noise. After substitution of eq 13 into eq 4 and regrouping of the terms into feedback-invariant and feedback-dependent terms, the process model can be described as

ỹ(t ) = F(z −1)d(t ) − z −d R(z −1)F(z −1)−1z −d D(z −1)ε(t ) + z −d[Ã (z −1)u(t ) + R(z −1)F(z −1)−1z −d D(z −1) ym(t )]

B(z −1)F(z −1) G(z −1) y(t + k) = u(t ) + (y (t ) − ε(t )) −1 D(z ) D(z −1) m

(σy 2)MMVC = trace{Cov[Fd(t )]} + trace{Cov[R(z −1)F(z −1)−1D(z −1)ε(t )]} (19)

Equality in eq 19 is obtained when the minimum variance controller is used. Note that the first term on the right-hand side of eq 19 is generated by the process disturbances, and the second term is caused by the measurement noise. Compared with eq 12, eq 19 indicates that the variances of outputs under MMVC are also increased because of the influence of measurement noise.

(14)

Therefore, the output variance of the minimum variance controller satisfies the following equality ⎡⎛ ⎞2 ⎤ G(z −1) (σy 2)MVC = E[(Fd(t + k))2 ] + E⎢⎜ ε ( t ) ⎟⎥ ⎢⎣⎝ D(z −1) ⎠ ⎥⎦

(15)

4. MVC COMBINED WITH DDR TO ENHANCE CONTROL PERFORMANCE When the measurements containing measurement errors are used by the controller, the performance of MVC would be affected. Bai demonstrated that the DDR was an effective tool to decrease propagation of measurement errors inside control loops.25 DDR can be used to decrease measurement errors inside control loops. With the uncertainty ranges of measurements determined by the measurement errors of measuring instruments as well as instrument degradation,26 the DDR uses dynamic process models to reconcile the process measurements. Therefore, it can be combined with MVC to enhance the control performance. In this section, the effectiveness of the

Equality in eq 15 is obtained when the minimum variance controller is used. The minimum variance controller for this case can be obtained by solving the following equation: B(z −1)F(z −1) G(z −1) u(t ) + y (t ) = 0 −1 D(z ) D(z −1) m

(18)

Therefore, the output variance of the minimum variance controller satisfies the following equality.

+ Fd(t + k) G(z −1) = Fd(t + k) − ε( t ) D(z −1) ⎤ ⎡ B(z −1)F(z −1) G(z −1) +⎢ u ( t ) + y (t )⎥ −1 −1 m ⎦ ⎣ D(z ) D(z )

(17)

(16)

Note that the first term on the right-hand side of eq 15 is generated by the process disturbance and the second term is generated by the measurement noise. Compared with eq 6, eq 15 indicates that the output variance under MVC in the univariate system is increased because of the influence of measurement noise. 10993


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Figure 3. Stochastic univariate (or multivariate) feedback control system with DDR.

DDR based MVC (DDR−MVC) is proposed to overcome the degradation of the control performance in the presence of significant measurement noise. 4.1. DDR−MVC in a Stochastic Univariate System. For univariate systems, at the time point t, the process output in eq 1 can be decomposed into two parts. One is the predicted part (ŷ(t)) from the past input measurements; the other is the unpredicted part (δ(t)), which is the inherent variability associated with the process disturbance, B(z −1) D(z −1) − + y(t ) = u t k ( ) d (t ) −1 A( z −1 )  A(z )    y (̂ t ) δ(t )

p(yr (t )|ym (t ), y (̂ t )) ∝

2 ⎧ ⎡ ⎪ 1 1 ⎢ (ym (t ) − yr (t )) ⎨ − exp ⎪ (2π )2 ρη ρ2 ⎩ 2 ⎢⎣

(y (̂ t ) − yr (t ))2 ⎤⎫ ⎪ ⎥⎬ + 2 ⎪ ⎥⎦⎭ η

The estimation of yr(t) based on a principle similar to the maximum likelihood is the maximum posterior distribution, yr (t ) = arg max p(yr (t )|ym (t ), y (̂ t )) yr (t )

(20)

yr (t ) = y (̂ t ) + (1 + ρ2 η−2)−1(ym (t ) − y (̂ t ))

(26)

The estimation error of DDR can be given as γ(t ) = yr (t ) − y(t )

(27)

Thus, with DDR, the model output is y(t + k) =

p(yr (t )|ym (t ), y (̂ t )) ∝ L(ym (t )|yr (t ))L(y (̂ t )|yr (t ))p(yr (t )) (21)

B(z −1)F(z −1) G(z −1) u t + ( ) (y (t ) − γ(t )) D(z −1) D(z −1) r + Fd(t + k)

where L(ym(t)|yr(t)) and L(ŷ(t)|yr(t)) are the likelihood functions. Based on the assumptions about the measurement noise and the model predictions, the likelihood functions can be shown as

⎛ 1 (y ̂(t ) − y (t ))2 ⎞ 1 r ⎟⎟ L(y ̂(t )|yr (t )) = exp⎜⎜ − 2 2πη 2 η ⎝ ⎠

(25)

Since p(yr(t)|ym(t), ŷ(t)) is proportional to L(ym(t)|yr(t))L(ŷ(t)| yr(t)), yr(t) minimizes the augmented likelihood L(ym(t)|yr(t)) L(ŷ(t)|yr(t)). The reconciled signal yr(t) is

where δ(t) represents the model prediction error with the Gaussian distribution (δ(t) ∼ N(0, η2)) and η2 is the corresponding variance. The stochastic univariate feedback control system with DDR is shown in Figure 3. According to Bayes’ rule, the posterior distribution of the reconciled output (yr(t)) given the measurements (ym(t)) and the predicted output (ŷ(t)) is proportional to the product of two likelihood functions of the measured output and the predicted output,

⎛ 1 (y (t ) − y (t ))2 ⎞ 1 m r ⎟⎟ L(ym (t )|yr (t )) = exp⎜⎜ − 2 2πρ ρ ⎝ 2 ⎠

(24)

= Fd(t + k) −

G(z −1) γ (t ) D(z −1)

⎤ ⎡ B(z −1)F(z −1) G(z −1) +⎢ u(t ) + y (t )⎥ −1 −1 r ⎦ ⎣ D(z ) D(z )

(28)

Like eq 15, the output variance of the minimum variance controller satisfies the following equality

(22)

2 MVC

(σy ) (23)

⎡⎛ ⎞2 ⎤ G(z −1) ⎢ = E[(Fd(t + k)) ] + E ⎜ γ (t ) ⎟ ⎥ ⎢⎣⎝ D(z −1) ⎠ ⎥⎦ 2

(29)

Using eq 26, the variance of γ(t) can be calculated by

where ρ2 and η2 are the variances of the measured error and the predicted error, respectively; p(yr(t)) is a priori probability for the reconciled signal yr(t), and it is a constant. Thus, the posterior distribution becomes

var(γ(t )) = (ρ−2 + η−2)−1

(30)

The detailed derivations of the equations for the SISO system and for the MIMO system are similar; to save space, 10994


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Industrial & Engineering Chemistry Research only equations for the MIMO system are derived and can be found in Supplement A of the Supporting Information. Equation 30 indicates that the variance (γ(t)) of the reconciled error yr(t) is less than those of measured error ε(t) and model prediction error δ(t). Thus, when eq 29 is compared with eq 15, the output variance under DDR−MVC is decreased since the variance of γ(t) is less than that of the measurement noise ε(t). When the variance of the measurement noise (ρ) increases significantly, the variance of the model prediction error (η) will give the main contribution to the variance of γ(t). Thus, the influence of the process disturbance and measurement instrument bias, outliers, and slow drifts can be reduced efficiently by DDR. In the SISO case, since only one signal enters the DDR block, DDR can be considered as a kind of digital filter. 4.2. DDR−MVC in a Stochastic Multivariate System. In a multivariate system, at the time point t the process output in eq 9 can be decomposed into two parts, −1

p(yr(t )|ym(t ), y (̂ t )) 1 ∝ N (2π ) |ΣεΣδ|1/2

{ − 12 [(y (t) − y (t)) Σ

Σδ = Cov(B(z )d(t ))

yr(t )

(37)

yr(t ) = ŷ(t ) + (Σε−1 + Σδ−1)−1Σε−1(ym(t ) − y (̂ t )) = y (̂ t ) + K(ym(t ) − y (̂ t ))

(31)

−1

−1 −1

(38)

−1

where K = (Σε + Σδ ) Σε . The estimation error of DDR can be given by γ(t ) = yr(t ) − y(t ) = δ(t ) + K(ε(t ) − δ(t ))

(39)

Thus, under DDR, the model output is ỹ (t ) = F(z −1)d(t ) − z −d R(z −1)F(z −1)−1z −d D(z −1)γ(t ) + z −d[Ã (z −1)u(t ) + R(z −1)F(z −1)−1z −d D(z −1) yr(t )]

(40)

Like eq 19, the output variance of the minimum variance controller satisfies the following equality

(33)

(σy 2)MVC = trace{Cov[Fd(t )]}

where p(yr(t)|ym(t), ŷ(t)) represents the posterior knowledge based on both the measurement information ym(t) and the process model information ŷ(t). L(ym(t)|yr(t)) and L(ŷ(t)| yr(t)) are the likelihood functions. Based on the assumptions about the measurement noise and the model predictions, the likelihood functions follow that

+ trace{Cov[G(z −1)F(z −1)−1D(z −1)γ(t )]} (41)

Using eq 39, the variance of γ(t) can be calculated by Vr = Cov(γ(t )) = Cov(δ(t ) + K(ε(t ) − δ(t )))

1

1 (2π )N /2 |Σδ|1/2 ⎛ 1 exp⎜ − (yr(t ) − ŷ(t ))T Σδ−1 ⎝ 2 ⎞ (yr(t ) − ŷ(t ))⎟ ⎠

(36)

Since the p(yr(t)|ym(t), ŷ(t)) is proportional to L(ym(t)|yr(t)) L(ŷ(t)|yr(t)), yr(t) minimizes the augmented likelihood L(ym(t)|yr(t))L(ŷ(t)|yr(t)). The reconciled signals yr(t) are

p(yr(t )|ym(t ), ŷ(t )) ∝ L(ym(t )|yr(t ))L(ŷ(t )|yr(t ))p(yr(t ))

L(y ̂(t )|yr(t )) =

(yr(t ) − ym(t ))

yr(t ) = arg max p(yr(t )|ym(t ), ŷ(t ))

(32)

(2π )N /2 |Σε|1/2 ⎛ 1 exp⎜ − (yr(t ) − ym(t ))T Σε−1 ⎝ 2 ⎞ (yr(t ) − ym(t ))⎟ ⎠

ε

The estimation of yr(t), based on a principle similar to maximum likelihood is the maximum posterior distribution,

The stochastic feedback control MIMO system with DDR can be shown in Figure 3. According to Bayes’ rule, DDR uses both the measurement and the process model information to derive the reconciled signals, yr(t),

L(ym(t )|yr(t )) =

m

}

where δ(t) is normally distributed (N(0, Σδ)). Σδ is the corresponding covariance matrix, which can be obtained by −1

r

+ (yr(t ) − ŷ(t ))T Σδ−1(yr(t ) − ŷ(t ))]

−1

y(t ) =  A(z )u( t) +  B(z )d( t) ŷ (t ) δ(t )

−1

T

exp

= (Σε−1 + Σδ−1)−1

(42)

The detailed derivations of eqs 38, 39 and 42 can be found in Supplement A of the Supporting Information. Equation 42 indicates that the covariance matrix of the reconciled errors (γ(t)) is a function of Σε (the covariance matrix of the measured error) and Σδ (the covariance matrix of the model prediction error). Since both Σε and Σδ are positive definite symmetric matrices, the trace of covariance matrix Vr is less than that in Σε and in Σδ, which means that the reconciled signals are more precisely estimated than the measured signals. If MMVC uses the reconciled signals as the feedback signals instead of the measured signals, the variances of outputs will be decreased. Therefore, the DDR based MMVC (DDR− MMVC) can efficiently enhance the control performance. The control structure in Figure 1 is widely used for benchmarking purposes (like MVC). It is a typical feedback control structure used in the actual process. The traditional MVC does not explicitly consider the influence of measurement errors presented in actual sensor devices. The measurement errors would affect the variances of process outputs. In that

(34)

(35)

where Σε and Σδ are the covariances of the measured error and the predicted error, respectively; p(yr(t)) is a priori probability for the reconciled signals yr(t), and it is a constant. Thus, the posterior distribution becomes, 10995


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Industrial & Engineering Chemistry Research situation, the MVC benchmark, which is an indication of the inherent optimum would not be trustable. In this paper, the influence of measurement errors on the control performance of MVC is considered, so the DDR method combined in the procedure of MVC is proposed to reduce the influence of measurement errors. An additional DDR filter shown in Figure 3 is used to improve the control performance, but the feedback control structure is still the same as the original MVC benchmark of the feedback control structure. This means that the tight control performance of the proposed DDR−MVC would be gotten. The control structure that contains an additional DDR filter may be far from the common practice, but modern controllers on microprocessors and digital computers can be implemented easily.

(1 − 0.92z −1)(1 − z −1)y(t ) = 0.08z −3u(t ) + (1 − 0.92z −1)d(t ) y(t ) − 1.92z −1y(t ) + 0.92z −2y(t ) = 0.08 z −3u(t ) + (1 − 0.92z −1)d(t ) y(t ) = 1.92y(t − 1) − 0.92y(t − 2) + 0.08u(t − 3)  y (̂ t ) 1 + (1 − 0.92 z − t) )d(  

δ(t )

(44)

If the standard deviation of the disturbance is given, the standard deviation of the model prediction error (η) can be calculated by

5. CASE STUDIES

η2 = var[(1 − 0.92z −1)d(t )] = (1 + 0.922)σ 2

To illustrate the effectiveness of the proposed methods developed in section 4, three case studies are presented. For easy explanation, two simulation cases that consist of the given dynamic system with the measurement noise are used because they illustrate how to calculate the DDR−MVC and DDR− MMVC performance assessment indices and how to enhance the indices when the controlled system contains the measurement noise. Then in the third case, the data from the real experiment of the feedback control loop are presented to help readers compare the DDR−MMVC based performance assessment method with the conventional MVC based performance assessment method in terms of the performance improvement. 5.1. Case Study 1: MVC in a Stochastic Univariate Feedback Control system. The case study based on Hugo23 is selected to explain the performance assessment procedure developed in section 4.1. The stochastic univariate feedback control system considered is

(45)

Thus, from eq 26, the reconciled output under MVC and the corresponding estimation error of DDR can be calculated (eq 27). When ρ = 0.025, the results of MVC based on measured signal and reconciled signal are shown in Figure 4. The

0.08z −3 1 u(t ) + d (t ) −1 1 − 0.92z 1 − z −1 0.08z −3(1 − z −1) = u(t ) (1 − 0.92z −1)(1 − z −1)

y(t ) =

+

Figure 4. Results of MVC for the univariate system based on the measured signal and the reconciled signal when there is measurement noise with ρ = 0.025 in the univariate system.

1 − 0.92z −1 d (t ) (1 − 0.92z −1)(1 − z −1)

standard deviations of the process outputs with MVC based on the measured signal and the reconciled signals are 0.0252 and 0.0224, respectively. The results demonstrate that the variance of the process outputs decreases using DDR. When ρ = 0.05, the results of MVC based on measured signal and reconciled signal are shown in Figure 5. The standard deviations of the process outputs with MVC based on the measured signal and the reconciled signal are 0.0367 and 0.0277, respectively. The results show that the variance of the process output with MVC based on measured signal is sensitive to the measurement noise. While the standard deviation of measurement noise is increased to be 0.05, the variance of the process output with MVC based on measured signal increases obviously, from 0.0252 to 0.0367. However, the variance of the process output with MVC based on reconciled signal increases a little, from 0.0224 to 0.0277, which demonstrates that the performance of MVC can be enhanced by DDR. 5.1.2. Condition 2: Measurements with Noise and Gross Errors. The outliers are added to the measurements at time step

(43)

where d(t) is the zero mean Gaussian random variable, which has a standard deviation of σ = 0.01. The parameters of the PI controller k2 =

(

−0.92 , 3 × 0.08 23

k1 + k 2z −1 1 − z −1

) are

selected to be k1 =

1 3 × 0.08

and

both of which are derived from the control law

of MVC. Three sets of conditions are used to demonstrate the performance improvement of this SISO control system based on the proposed DDR−MVC: 5.1.1. Condition 1: Measurements with Two Different Sizes of Measurement Noise. In the first condition, the measurement noise (ε(t)) is assumed to be normally distributed (ε(t) ∼ N(0, ρ2)). Two different sizes of measurement noise with ρ = 0.025 and ρ = 0.05 are compared. From eq 43, the predicted part (ŷ(t)) and the unpredicted part (δ(t)) can be derived as follows, 10996


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exponential filter, which is by far the most commonly used first order filter in industrial applications,27 is used for comparisons. The measurement with only random errors (measurement noise) is considered first, and then measurement with both noise and gross errors is considered as the second case of measurement. The simulation is run 500 times repetitively in each case of measurement. The expected standard deviations of measured signal, filtered signal, and reconciled signal are provided in Table 1 to illustrate the effectiveness of DDR. Table 1. Comparison Results for the Univariate System with 500-Repetition Simulations expected standard deviation

Figure 5. Results of MVC for the univariate system based on the measured signal and the reconciled signal when there is measurement noise with ρ = 0.05 in the univariate system.

100 and 300. The magnitude of outliers is equal to 3.0. The bias is introduced into the measurements from time step 100 to 200. The magnitude of bias is equal to 0.05. Slow linear drift happens from time step 500 to 600 with a gradient of 0.0005. Like the previous discussion in condition 1, the results of MVC based on measured signal and reconciled signal are shown in Figure 6. The standard deviations of the process outputs with

conditions of measurement

measured signal

filtered signal

reconciled signal

random errors with at ρ = 0.025 random errors with ρ = 0.025 and gross errors

0.0342 0.0354

0.0301 0.0308

0.0287 0.0291

As can be seen in Table 1, the expected standard deviations of both filtered signal and reconciled signal are decreased compared with the expected standard deviation of measured signal. The exponential filter can decrease the expected standard deviation of measured signal by 12.00% (from 0.0342 to 0.0301) with measurement noise, while the DDR can decrease the expected standard deviation of measured signal by 16.08% (from 0.0342 to 0.0287) with measurement noise. When measurement with both noise and gross errors is considered, the exponential filter can decrease the expected standard deviation by 13.00% (from 0.0354 to 0.0308), while the DDR can decrease the expected standard deviation by 17.80% (from 0.0354 to 0.0291). The results indicate that the influence of measurement noise and gross errors can be reduced much more efficiently by DDR. The effectiveness of DDR is more obvious compared with that of exponential filter. 5.2. Case Study 2: MVC in a Stochastic Multivariate Feedback Control System. The case study for the multivariate system comes from Huang et al.28 The system involves two manipulated variables and two controlled variables, which are widely used in literature.5,29 The process and disturbance transfer function matrices are defined as follows: ⎡ z −1 ⎢ −1 1 ⎢ 1 − 0.4z A(z −1) = ⎢ −1 ⎢ 0.3z ⎢⎣ 1 − 0.1z −1 1 ⎡ 1 ⎢ 1 − 0.5z −1 B(z −1) = ⎢ ⎢ 0.5 ⎢ ⎣ 1 − 0.5z −1

Figure 6. Results of MVC for the univariate system based on the measured signal and the reconciled signal when there is measurement noise with ρ = 0.05 and gross errors in the univariate system.

MVC based on measured signal and reconciled signal are 0.0325 and 0.0266, respectively. With the influence of measurement instrument bias, outliers, and slow drifts, the variance of process output with MVC based on measured signal increases obviously, from 0.0252 (condition 1) to 0.0325 (condition 2). The variance of process output with MVC based on reconciled signal increases a little, from 0.0224 to 0.0266. The results indicate that the influence of measurement noise can be reduced efficiently by DDR. And the performance of MVC can be improved by DDR, and it is robust to the measurement instrument bias, outliers, and slow drifts. 5.1.3. Condition 3: Comparison Results with Monte Carlo Simulations. In the last condition, Monte Carlo simulations are used to demonstrate the reliable comparison results. The

4z −2 ⎤ ⎥ − 0.1z −1 ⎥ ⎥ z −2 ⎥ − 0.8z −1 ⎥⎦ −0.6 ⎤ ⎥ 1 − 0.5z −1 ⎥ ⎥ 1 −1 ⎥ 1 − 0.5z ⎦

(46)

The set points are assumed to be zeros for both outputs. The disturbances are two-dimensional normal-distributed white noise with the covariance matrix Σd = I. The system has a general interactor matrix, which can be determined as ⎡−0.9578z − 0.2873z ⎤ D(z −1) = ⎢ ⎥ ⎣ 0.2873z 2 − 0.9578z 2 ⎦ 10997

(47)


Article

Industrial & Engineering Chemistry Research The MVC controllers are designed according to the two single loops without considering the interactions:5,29

And the MMVC benchmark based on the reconciled signals can be calculated as JDDR−MMVC = trace[ΣDDR−MMVC] = 3.6564. If the measured signals are used as feedback signals, MMVC will result in larger variances of the process outputs. However, since the dynamic data reconciliation reduces the variances of the measured signals obviously, MMVC based on the reconciled signals will result in smaller variance of the process outputs. 5.2.2. Condition 2: Measurements with Measurement Noise and Gross Errors. The outliers are added to the first measurements (process output 1) at time step 100 and 300, and the second measurements (process output 2) at time step 600 and 800. The magnitude of outliers is equal to 3.0. The sign of outliers in the process output 1 is positive, and the sign of outliers in the process output 2 is negative. The biases are introduced into the first measurements from time step 100 to 200 and the second measurements from time step 300 to 400. The magnitude of biases is equal to 2.5. Slow linear drift happens at the same gradient of 0.05 from time step 500 to 600 for the first measurement and from time step 600 to 700 for the second measurement. Based on the proposed DDR−MMVC, the results of the reconciled signals of the process outputs are shown in Figure 8. The variances of measured signals are

⎡ 0.5 − 0.2z −1 ⎤ ⎢ ⎥ 0 −1 ⎢ 1 − 0.5z ⎥ −1 C(z ) = ⎢ ⎥ −1 0.25 − 0.2z ⎢ ⎥ 0 ⎢⎣ (1 − 0.5z −1)(1 + 0.5z −1) ⎥⎦ (48)

The vector of the measurement noise ε(t) is assumed to be normally distributed with covariance matrix Σε = I. MMVC based on measured signals and based on reconciled signals are studied in this example. Three sets of conditions are used to demonstrate the performance improvement of this MIMO control system based on the proposed DDR−MMVC. 5.2.1. Condition 1: Measurements with Measurement Noise. With the given covariance matrix of the disturbance and the model prediction error (Σδ in eq 32), the standard deviation of the model prediction error (Σδ) is given by Σδ = Cov(B(z −1)d(t ))

(49)

Thus, from eq 38, the reconciled output under MVC can be calculated, and the corresponding estimation error of DDR can also be calculated (eq 39). In Figure 7, the variances of the

Figure 8. Measured signals and reconciled signals of the process outputs when there are two measurement noises with Σε = I and gross errors in the multivariate system. Figure 7. Measured signals and reconciled signals of the process outputs when there are two measurement noises with Σε = I in the multivariate system.

obviously greater than the variances of the reconciled signals. From the simulation results, the covariance of the process outputs for MMVC based on measured signals is ⎡ 2.2983 0.0246 ⎤ ΣMMVC = ⎢ ⎥ ⎣ 0.0246 4.0533 ⎦

measured signals are obviously greater than the variances of the reconciled signals. In fact, the covariance of the process outputs for MMVC based on the measured signals is ⎡ 2.1088 0.0705 ⎤ ΣMMVC = ⎢ ⎣ 0.0705 2.0780 ⎥⎦

And the calculated MMVC benchmark based on the measured signals is JMMVC = trace[ΣMMVC] = 6.3515. After the simulation, the covariance of the process outputs for MMVC based on the reconciled signals is

(50)

The MMVC benchmark based on the measured signals can be calculated as JMMVC = trace[ΣMMVC] = 4.1868. The covariance of the process outputs for MMVC based on the reconciled signals after simulation is ⎡ 1.8962 − 0.0332 ⎤ ΣDDR − MMVC = ⎢ ⎥ ⎣−0.0332 1.7602 ⎦

(52)

⎡ 1.9090 − 0.1388 ⎤ ΣDDR − MMVC = ⎢ ⎥ ⎣−0.1388 2.5633 ⎦

(53)

And the calculated MMVC benchmark based on the reconciled signals is JDDR−MMVC = trace[ΣDDR−MMVC] = 4.4723. Comparing the above results with those in the first condition, the variances

(51) 10998


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Industrial & Engineering Chemistry Research Table 2. Comparison Results for the Multivariate System with 500-Repetition Simulations expected standard deviation filtered signals

measured signals

reconciled signals

conditions of measurements

output 1

output 2

output 1

output 2

output 1

output 2

random errors with Σε = I Random errors with Σε = I and gross errors

1.7432 1.9263

1.7652 2.1149

1.2582 1.4750

1.3299 1.7526

1.0931 1.2125

1.0778 1.2922

of the disturbance and the measurement noise. The explicit models are usually unavailable in advance in many real processes. In this experiment, a pilot scale of a quadruple tank apparatus, shown in Figure 9, is used to illustrate the data

of the process outputs with MMVC based on the measured signals increase obviously, from 2.1088 (condition 1) to 2.2983 (condition 2) for the process output 1, and from 2.0780 (condition 1) to 4.0533 (condition 2) for the process output 2. The variance of process output 2 especially increases over 95%. With the proposed DDR−MMVC, the variances of the reconciled signals increase only a little, from 1.8962 (condition 1) to 1.9090 (condition 2) for process output 1 and from 1.7602 (condition 1) to 2.563 (condition 2) for process output 2. Moreover, the MMVC benchmarks based on the measured signals increase 51.70%, while the MMVC benchmarks based on the reconciled signals increase only 22.31%. The results show that the application of DDR can decrease the variances of the measured signals obviously. DDR is robust to the influence of the measurement instrument bias, outliers, and slow drifts. When the MMVC uses the reconciled signals as feedback signals instead of the measured signals, the variances of the outputs are decreased. DDR−MMVC can efficiently enhance the control performance. 5.2.3. Condition 3: Comparison Results with Monte Carlo Simulations. Like case study 1, in this condition, Monte Carlo simulations are also used to demonstrate the effectiveness of DDR in case study 2 with the multivariable system. The exponential filter is also used for comparisons. The measurements containing only random errors (measurement noise) are considered first, and then measurements with both random errors and gross errors are considered as the second case of measurement. The simulation is run 500 times repetitively in each case of measurement. The expected standard deviations of measured signals, filtered signals, and reconciled signals are provided in Table 2. As can be seen in Table 2, the expected standard deviations of the reconciled signals are decreased more significantly compared with the results of the univariate system with Monte Carlo simulations. Considering measurement noise only, the DDR can decrease the expected standard deviation of measured signal by 37.29% (from 1.7432 to 1.0931) in process output 1 and by 38.94% (from 1.7652 to 1.0778) in process output 2, while the DDR can decrease the expected standard deviation only by 16.08% in the univariate system. Considering the system with measurement noise and gross error, the DDR can decrease the expected standard deviation of measured signal by 37.05% (from 1.9263 to 1.2125) in process output 1 and by 38.90% (from 2.1149 to 1.2922) in process output 2, while the DDR can decrease the expected standard deviation only by 17.80% in the univariate system. The results suggest that more efficient error reduction can be derived in the multivariate system because more redundant information on the process model is used by the DDR. The results also indicate that the influence of measurement noise and gross errors can be reduced much more efficiently by DDR compared with the exponential filter. 5.3. Case Study 3: Experiments on Air-Pressure Tank Systems. In the previous cases, the derivations for DDR− MVC or DDR−MMVC are based on the full ARIMA models

Figure 9. Schematic of the four-tank air pressure system in case study 3.

driven based modeling strategy with all information retrieved from routine closed loop measurement data to enhance the current control performance. The MIMO experimental system shown in Figure 9 consists of four interconnected air tanks arranged in two parallel trains of two tanks in series built upon a steel framework. Supply air flows into the system through two air-actuated control valves, which serve as the manipulated variables for the system. The air flows through tanks before exiting to the atmosphere. Specifically, the air flowing through control valve 1 (Cv1) proceeds into tank 1 and subsequently into tank 2 downstream before exiting the system. Additionally, a portion of the flow from the control valve can be routed into the downstream tank of the adjacent train (tank 4). Similarly, control valve 2 (Cv2) affects the pressure in tanks 3 and 4, with cross-flow effect on tank 2. Valves V14 and V23 are directly responsible for the cross-train flow. In the system, the measured outputs are the pressures in tank 2 and tank 4, P2 and P4 (voltages). This arrangement makes both pressures in the two bottom tanks become a function of the two control valves. The tanks are equipped with differential-pressure-to-current (DP/I) transducers to provide a continuous measurement of the pressure. The computer is connected to a AD-622 analog/digital I/O expansion card from Advantech. The expansion board uses a 14-bit converter; therefore, the digital signals are 14-bit. The analog signals from the measured levels are amplified and 10999


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Figure 10. Process outputs with MVC based on the measurement signals and the reconciled signals in tank 2 and tank 4 in the air-pressure tank experiments.

Figure 11. Process outputs with PI controllers based on the measurement signals and the reconciled signals in tank 2 and tank 4 in the air-pressure tank experiments.

The pseudo-random binary sequence (PRBS) test signals are used to derive a total of 1200 sets of data collected from the closed loop operation; then the conventional identification algorithms can be directly applied to identifying the process model and the disturbance model. Now the operating performance of the controlled system using the proposed method is evaluated. Two different kinds of the control strategies (MVC and PI) are separately applied. Figures 10 and 11 show the two outputs of MVC and of PI, respectively. In Figure 10, the variances of the process output with MVC based on the measurement signals and the reconciled signals in tank 2 are 7.62 × 10−5 and 3.86 × 10−5. The variances in tank 4 are 1.961 × 10−4 and 1.210 × 10−4. In Figure 11, the variances of the process output with PI based on the measurement signals and the reconciled signals in tank 2 are 1.769 × 10−4 and 8.37 × 10−5. The variances in tank 4 are 3.071 × 10−4 and 1.799 × 10−4. This result demonstrates that the variances of the process outputs decrease because the

conditioned using EDM35 (4−20 mA/0−5 V) modules. The visual programming control interface and the controller algorithms (MVC and PI) are developed using MATLAB for this apparatus. From the derivations in section 4.2, the reconciled signals (yγ(t)) rely on covariance matrices (Σε and Σδ) of the measurement noise and model prediction error (eq 38). The variance or the covariance matrices needs to be estimated based on routine closed loop process measurement data. The covariance matrix Σε of the vector of measurement noise can be provided by sensor manufactures. The data based estimation for the covariance matrix (Σδ) from the autoregressive model is given in Supplement B of the Supporting Information. To get the reconciled signals (yγ(t)) in eq 38, the prediction outputs (ŷ(t)) should be calculated. In order to estimate the process model (A(z−1)) and the disturbance model (B(z−1)), the persistent excitation condition should be imposed on the test signals to guarantee that the estimation yields unique solutions. 11000


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as well as feedforward and estimation schemes and many others. Besides, it can also be used to formulate optimization problems for controller design and is widely used in the robust control theory.31,32 Considering the disturbance, measurement noise or even model uncertainty and uncertain time delay, the use of the generalized control structure combined with the robust controller can provide more accurate results. The use of generalized control structure would be a good direction for our further research.

application of a DDR to the process measurement would allow the controllers to access more accurate and precise data. The corresponding control performance would be improved.

6. CONCLUSIONS From the research reports, about 66% to 80% of the controllers of industrial processes are not performing as well as they are expected. This detrimental effect on the process profitability causes the product variance. MVC plays a crucial role as a benchmark in performance monitoring. However, the influence of measurement errors is not explicitly considered in the theory for MVC of stochastic control systems. In fact, measurements such as flow rates, pressures, and temperatures from a chemical process are inherently inaccurate. They are contaminated by random errors and possibly gross errors. With additional consideration of measurement errors in the sensor devices, the variance of the output under MVC is obviously increased. Such influence of measurement errors degrades the control performance of MVC in both univariate and multivariate systems. A model-based approach, DDR, is considered to decrease measurement noise inside control loops. The derivations for MVC/MMVC combined with DDR demonstrate that the reconciled signals by DDR are more precisely estimated than the measured signals. The variances of outputs by DDR− MVC/DDR−MMVC will be decreased, and the control performance can then be enhanced. The effectiveness of DDR−MVC or DDR−MMVC is illustrated by simulation cases and an actual pilot scale experiment, including both univariate and multivariate systems. Results show that both MVC and MMVC benchmarks can be improved by reconciled signals as feedback signals. DDR−MVC or DDR−MMVC can efficiently enhance the control performance. The conventional MVC approach would not be applied when the disturbance distribution is non-Gaussian. Entropy is a general measure of uncertainty. It has more physical relevance than the variance of the arbitrary random variables, so a more general approach by incorporating the entropy for the feedback control loop subjected to the unknown disturbance distribution is worth investigation in the future. In some practice applications when MVC is implemented, there are some problems, such as high gain, wide bandwidth, and unrealistically large control signal variation. The actuators would result in saturation and excessive wear and tear. In the future, the extension work will be focused on DDR combined with minimal achievable variance of measured output, such as generalized minimum variance control and linear quadratic Gaussian (LQG) control, which consider both control signals and output signals. Also, since DDR is based on the information on operating data and process model, the process model needs to be identified first based on the previous operating data. DDR combined with a data-driven based subspace model under LQG control to establish the benchmark will be further developed in the future. The aim of using DDR in the procedure of MVC is to decrease the influence of measurement errors and to enhance the control performance. It can derive accurate approximate results of the actual outputs (y). However, the control structure considered in this paper is known as one degree of freedom control structure. This structure has the disadvantage that the feedback loop properties cannot be designed independently of the reference tracking transfer-function.30 The generalized control structure can represent a wide class of controllers, including the one and two degrees of freedom configurations,

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b02532. The detailed derivations of eqs 38, 39, and 42 for DDR− MVC in a stochastic multivariate system and the estimation of the covariance matrix (Σδ for the multivariate system) of the model prediction errors (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*Tel: +886-3-2654107. Fax: +886-3-2654199. E-mail: jason@ wavenet.cycu.edu.tw. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge the Ministry of Science and Technology, Taiwan, R.O.C. (MOST 103-2221-E-033-068MY3), National Natural Science Foundation of China (No.61503271; No.61374167), Natural Science Foundation of Zhejiang Province (No.LQ14F030006), and Science and Technology Planning Project of Zhejiang Province (No.2015C31157; 2014C31074; 2014C31093) for the financial support.

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REFERENCES

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