Model Deficiency Diagnosis and Improvement via Model Residual

Article pubs.acs.org/IECR

Model Deficiency Diagnosis and Improvement via Model Residual Assessment in Model Predictive Control Lijuan Li,* Jianquan Song, Xiaoxiao Zhang, Jing Ye, and Shipin Yang College of Electrical Engineering and Control Science, Nanjing Tech University, Nanjing, 211816, China ABSTRACT: To reduce the effort and cost of model maintenance in model predictive control (MPC) systems, this paper explored a model deficiency diagnosis and improvement method by the assessment of model residual and optimization of disturbance model. A model quality index (MQI) method was first presented to evaluate the model performance with the routine input and output process data. Based on MQI, a leave-one-out method was proposed to further assess the performances of submodels in multi-input−multi-output (MIMO) MPC processes. The root deficient submodel could be diagnosed based on a comparison of the overall MQI with the submodel index by moving it to the disturbance channel. Further, a model performance improvement method through the upgraded optimal disturbance model was proposed to enhance the model performance without altering the control model. The experiment results on the Wood−Berry distillation column process and an industrial process indicated the validity of the proposed model diagnosis and improvement method.

1. INTRODUCTION Control performance monitoring (CPM) is concerned with the methods monitoring the control system performance in real time by using some assessing criteria. The most studied criterion is minimum variance control (MVC), which was first proposed by Harris,1 with many other extensions arising in succession,2−10 including the multi-input−multi-output (MIMO) version,2,3 time variant version,4 and constrained MPC version.5 To get the MVC benchmark of a MIMO system, the interactor matrix needs to be obtained first, which is difficult work because time delay and some leading impulse response coefficients of the process are required. Yu and Qin6 proposed a combined left/right diagonal interactor to ease the computation of a general one. To avoid computing the interactor, some other more practical alternative benchmarks were also proposed, such as the linear quadratic Gaussian (LQG) benchmark,7 user-specified benchmark,8 designed versus achieved controller performance,9,10 and normalized multivariate impulse response (NMIR) curve.7 These measures of multivariate controller performance have been applied in several industrial MPC processes.11,12 Nonlinear control performance assessment methods were studied in refs 13 and 14. From a more global view, economically motivated performance measures were also presented.15,16 Excellent reviews of control-loop performance assessment technology and applications were summarized by Qin,17 Harris et al.,18 Huang and Shah,7 and Jelali.19 CPM technology aims to evaluate the performance of an entire control system. However, poor control performance of an MPC system is usually caused by deficient prediction models, dynamic changes in disturbances, inappropriate controller parameters, actuator nonlinearities, and so on.20 More recent work has focused on evaluating the performances of separate portions of a © XXXX American Chemical Society

control system. Tian et al. proposed a data-driven classifier to diagnose the cause of control performance degradation by giving the performance index of a model, the controller, and the disturbance variety.20 The detection tools for control valve stiction were studied in refs 21−23. The performance criterion of a controller itself was proposed to select suitable controller parameters in the design of an MMPC controller.24 Model prediction accuracy is a key factor of the performance of industrial MPC systems. With the long-term operation of chemical units, it is usually degraded by some problems, such as the drift of object characteristics, catalyst deactivation, and heatexchanger fouling. The maintenance of models is a costly practice and sometimes even requires processes to be halted. If the model performance is known at any time during unit operation and advice regarding the necessity of model maintenance is given, some unnecessary model maintenance work and costs can be avoided. Hence, a systematic model assessing method without interrupting the process is desirable in practice. Model performance evaluation has recently attracted more attention in the field of control performance assessment.25−28 The designed control error benchmark was proposed to assess the impact of model−plant mismatch on control performance.25 A model error detection method was presented by comparing the tested process frequency responses and frequency responses of the current MPC model.26 Badwe et al. proposed a method to detect the precise location of the mismatch for MIMO processes based Received: Revised: Accepted: Published: A

June 24, 2017 September 23, 2017 September 26, 2017 September 26, 2017 DOI: 10.1021/acs.iecr.7b02598 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research on the partial correlations between model residuals and manipulated variables.27 Sun and Qin deduced the theoretical minimum process innovations, on the basis of which a model quality index (MQI) was defined to evaluate the performance of the MPC model.28 The contribution of ref 29 was to identify whether the mismatch lies in the disturbance model or in the process model based on the order of innovations. The MQI method is capable of evaluating the model mismatch of MPC by using the MPC set point and output data.28 The method yields the performance of the overall MPC model. However, most of the industrial processes are typically multiinput−multi-output systems. Poor MQI does not necessarily mean that all the submodels are deficient. Badwe et al.’s methodology can locate the deficient submodels, but it is on the premise of sufficient set point excitation, and the disturbance on the object interferes with the result (it interferes with the result on the premise of sufficient set point excitation and the disturbance on the object).27 In this paper, we explored a tool to diagnose the location of deficient submodel based on the theory of MQI. It also achieves the mismatch location result but overcomes the disadvantages of Badwe et al.’s work. Moreover, the further deficient submodel optimization is conveniently implemented based on the proposed method. Separate MQIs of the overall model of each controlled variable (CV) can be easily obtained by using the CV and the corresponding manipulated variables (MVs) data. Therefore, the performance of each CV model can be inferred based on its own MQI. However, each CV model still consists of several submodels because there are often several MVs that correspond to one CV. The separate MQI of each CV still cannot yield the performance of a one-to-one CV−MV submodel. It is unfortunate that we cannot obtain the decomposed CV data caused by an MV because a CV is the resulting combined response of all MVs and disturbances. Thus, the MQI of each one-to-one CV−MV submodel cannot be computed directly based on the corresponding MV and CV data. To evaluate the performance of an MV−CV submodel, we present a leave-oneout method by removing the submodel from the control channel to the disturbance channel. According to the comparison of the MQIs of a CV before and after removing a submodel, the contribution of the submodel to the model prediction can be inferred, and thus the performance of the removed submodel is assessed. Prediction model deficiency can be caused by a poor control model, a poor disturbance model, or both. An updated disturbance model may compensate for the mismatch of a control model and improve the control performance without altering the control model, thus reducing the cost of model reidentification or model maintenance. In this paper, a model improvement method was deduced by optimizing the model index and finding a more appreciate disturbance model. A higherorder disturbance model was tested to show the effectiveness of this approach. The remainder of this paper is organized as follows. In Section 2, an alternative MQI method using routine closedloop data is introduced. The deficient submodel diagnosis and related theoretical derivation are addressed in Section 3. Improvement of the prediction performance by upgrading the disturbance model is discussed in Section 4. Case studies are given in Section 5, followed by conclusions in Section 6.

Figure 1. Schematic of a model predictive control.

2. MQI CRITERION VIA DISTURBANCE INNOVATION Consider the MPC control system in Figure 1, where y(k) is a Ny × 1-dimensional CV vector, ŷ(k) is a Ny × 1-dimensional prediction CV vector, u(k) is a Nu × 1-dimensional MV vector, r(k) is a Ny × 1-dimensional reference trajectory, G0(q) is the Ny × Nu-dimensional true process model, H0(q) is the Ny × Ny-dimensional diagonal disturbance model, Gm(q) is the control prediction model, H(q) is the disturbance prediction model, Ny × Nu-dimensional Gc(q) is the prediction controller, e0(k) and d0(k) are the Ny × 1-dimensional process innovation and disturbance, respectively, and e(k) and d(k) are the Ny × 1-dimensional prediction error and control model error, respectively. According to the theory of Sun and Qin,28 process innovation is theoretically the minimum prediction error. If there is no mismatch, the prediction error e(k) should equal to e0(k). Otherwise, the innovation e0(k) is the projection component of e(k) projected to the space spanned by the previous output y(k) and set point r(k). Therefore, the MQI is defined as N

η=

∑k = 1 e 0(k)T Q (k)e 0(k) N

∑k = 1 e(k)T Q (k)e(k)

(1)

where Q(k) is the Ny × Ny-dimensional output weight matrix at time k chosen in the MPC design stage and N is the data length. e0(k) = [e01(k) ... e0Ny(k)]T can be obtained based on routine CV data y(k) and set point data r(k), and e0(k) = [e1(k) ... eNy(k)]T can be computed by using the prediction model and process output y(k). MQI η is in (0, 1]. A larger η indicates a better prediction model and disturbance model, or vice versa. The MQI of the lth CV is defined as N

ηl =

∑k = 1 el0(k)T Q l(k)el0(k) N

∑k = 1 el(k)T Q l(k)el(k)

(2)

e0l (k)

where and el(k) are the process innovation and prediction error corresponding to the lth CV, respectively, and Ql(k) is the corresponding weight matrix. 2.1. Estimating Disturbance Innovations from Input− Output Data. The below theorem is used to estimate disturbance innovations e0(k), which is similar to that of Sun and Qin28 except that it uses more general input−output data rather than set point−output data for the set point is usually a constant value in practice. Theorem 1. Consider the multi-input−multi-output (MIMO) process under linear-time-invariant (LTI) control in Figure 2. y(k) is the Ny × 1-dimensional output vector (CV); u(k) is the Nu × 1-dimensional input vector (MV). B

DOI: 10.1021/acs.iecr.7b02598 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Hence, e0(k) can be computed based on the closed-loop data y(k) and u(k). The vector expression of eq 8 is yp(k) = L̅pZp̅ (k) + ep0(k)

(9)

where L̅ p = [H1 ··· HM G1 ··· GN] and the other vectors are as defined in eq 3. 1 Post-multiply eq 9 by p Π⊥Zp̅ (k) as follows 1 1 1 y (k)Π⊥Zp̅ (k)= L̅pZp̅ (k)Π⊥Zp̅ (k)+ ep0(k)Π⊥Zp̅ (k) p p p p

Figure 2. Schematic diagram of a closed-loop controlled process.

Define

For

yp(k) = [ y(k) y(k − 1) ... y(k − p + 1)]

Zp̅ (k)Π⊥Zp̅ (k)

[Zp̅ (k)Zp̅ (k)T ]−1 Zp̅ (k)

up(k) = [ u(k) u(k − 1) ... u(k − p + 1)]

Furthermore, data

YM(k − 1) = [ yp(k − 1) yp(k − 2) ... yp(k − M )]T UN (k − 1) = [ up(k − 1) up(k − 2) ... up(k − N )]T (3)

0 0

y(k) = G u(k) + H e (k)

approaches 0 as p → ∞. Thus, eq 5 is concluded. 2.2. Estimating Prediction Errors. In MPC algorithms, the one-step-ahead predictive output is computed by solving y(k) = Gm(q)u(k) + H(q)e(k)

(4)

y (̂ k|k − 1) = (I − H −1)y(k) + H −1Gmu(k)

(5)

e(k) = y(k) − y (̂ k|k − 1) = H −1(y(k) − ym(k))

y (̂ k|k − 1) = (I − H0−1)y(k) + H0−1G0u(k)

N0

ym(k) = y ̅ (k) +

(7)

∞

∑ Hiq−i

⎡ a11i a12i ⎢ ⎢ a 21i a 22i ai = ⎢ ⋮ ⎢ ⋮ ⎢ a n 1i a n 2i y ⎣ y

i=1 ∞

∑ Giq−i i=1

where Hi and Gi are the coefficients. Thus, for sufficiently large M and N, eq 7 becomes ∞

y(k) =

≈

∞

i=1

M

N

y(k) = ym(k) + H(q)e(k) N0

∑ Hiy(k − i) + ∑ Giu(k − i) + e 0(k) i=1

i=1

⋯ a1nui ⎤ ⎥ ⋯ a 2nui ⎥ ⎥ ⋱ ⋮ ⎥ ⋯ anynui ⎥⎦

Δu(k) = [Δu1(k) Δu2(k) ... ΔuNu(k)]T, Nu is the input number, and N0 is the model length. Hence,

∑ Hiy(k − i) + ∑ Giu(k − i) + e 0(k) i=1

(15)

where y(k) ̅ is the initial state value of y(k), ai(i = 1, ..., N0) is the ith step response coefficient matrix of the process

where H and G are strictly causal. Denote

H0−1G0 =

∑ aiΔu(k − i) i=1

0

I − H0−1 =

(14)

where ym(k) = Gmu(k) is the predictive output of the MPC control model at time k. In Dynamic Matrix Control (DMC), a finite step response model is often used to compute ym(k)

From Ljung,31 the one-step-ahead prediction of the output ŷ(k|k − 1) is

0

(13)

where Gm(q) is the Ny × Nu-dimensional control model and H(q) is the Ny × Ny-dimensional diagonal disturbance model. The prediction and prediction errors are as follows31

(6)

y(k) = y (̂ k|k − 1) + e 0(k)

(12)

1

Proof. From Figure 2, the closed-loop output y(k) is the sum of the input response and the disturbance response 0

∀i≥1

Hence, p ep0(k)Zp̅ (k)T → 0 as p → ∞. The second item of eq 11

For a linear process controlled by an LTI controller, disturbance innovations e0p(k) can be computed by as p → ∞

(11)

is uncorrelated to the past input and output

E[ep0(k)up(k − i)T ] = 0,

where p is the time window, yp(k) and are Ny × p dimensional, up(k) is Nu × p dimensional, YM(k−1) is (Ny × M) × p dimensional, UN(k−1) is (Nu × N) × p dimensional, and Z̅ p(k) is (Ny × M + Nu × N) × p dimensional. Define the projection to the orthogonal complement of the row space of Z̅ p(k)

ep0(k) = yp(k)Π⊥Zp̅ (k)

e0p(k)

E[ep0(k)yp(k − i)T ] = 0

e0p(k)

Π⊥Zp̅ (k)=I − Zp̅ (k)T [Zp̅ (k)Zp̅ (k)T ]−1 Zp̅ (k)

= 0, eq 10 becomes

1 1 1 yp(k)Π⊥Zp̅ (k)= ep0(k) − ep0(k)Zp̅ (k)T p p p

ep0(k) = [ e 0(k) e 0(k − 1) ... e 0(k − p + 1)]

Zp̅ (k) = [ YM(k − 1) UN (k − 1)]T

(10)

= y ̅ (k ) +

(8)

∑ aiΔu(k − i) + H(q)e(k) i=1

C

(16) DOI: 10.1021/acs.iecr.7b02598 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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l The response ∑Ni=10 aj(N Δuj(k + i − N0 − 1) of uj(k) is 0−i+1) ignored. The prediction error el(j)(k) after removing uj(k) is given by

In the practice of industrial MPC, a step disturbance model, which is equivalent to a random walk disturbance model, ⎧ 1 1 ⎫ ⎬ H(q) = diag⎨ , ..., −1 1 − q −1 ⎭ ⎩1 − q

el(j)(k) = (1 − q−1)(yl (k) − ylm(j)(k))

is generally applied. In this case, the prediction error

uj(k). From the proof of Theorem 1, we have

e(k) = (1 − q−1)(y(k) − ym(k)) −1

= (1 − q )d(k)

M

yl(j)(k) =

(17)

+

(20)

N

ηl(j) =

∑k = 1 el0(j)(k)T Q l(j)(k)el0(j)(k) N

∑k = 1 el(j)(k)T Q l(j)(k)el(j)(k)

(21)

and the MQI impact factor due to removing the input uj as ηl(j) κl(j) = ηl (22) Compared to the value of ηl, a larger value of ηl(j) implies that the contribution of the CVl−MVj submodel to the prediction residual is negative. In this case, the removed CVl−MVj submodel is considered deficient. Conversely, it is considered a good submodel if ηl(j) is smaller than ηl. Hence, κl(j) > 1 indicates that the removed submodel is deficient, or vice versa. 3.2. Procedure of Diagnosis of Model Performance. As a systematic model diagnosis tool, we first need to assess the performance of the overall model, and only in cases of the emergence of deficiency is the leave-one-out method required to judge which submodel is the cause of the deficiency. The procedure is summarized in Figure 3. The steps are listed as follows. STEP 1: Collect routine data y(k) and u(k) and compute disturbance innovations e0(k) by using eq 5. STEP 2: Compute model output ym(k) by using eq 15; then, further calculate model error e(k) by using eq 17. STEP 3: Calculate η by solving eq 1, using e0(k) and e(k) obtained in STEP 1 and STEP 2; if η is larger than a threshold, the prediction model is considered to be good and the assessment ends; otherwise, go to STEP 4. STEP 4: Compute ηl one by one by using eq 2 and evaluate the model of each CV; if ηl is larger than a threshold, the prediction model is considered to be good and the procedure ends; otherwise, go to STEP 5. STEP 5: Remove the MVs one by one; compute the corresponding el(j)(k) by solving eq 19. STEP 6: Calculate ηl(j) by using eq 21 and κl(j) by using eq 22 and assess the corresponding submodel performance based on the value of κl(j); if κl(j) > 1, the submodel is considered to be deficient; otherwise, the corresponding submodel is considered to be good; go to STEP 7. STEP 7: Repeat STEP 4 to STEP 6 until all the CVs and related MVs are treated. 3.3. Computational Complexity. In MQI theory, the most time-consuming is the computation of disturbance innovations by eq 5 because the inversion of Z̅ p(k)Z̅ p(k)T is included.

N0

∑ ∑ ahl (N0− i + 1)Δuh(k + i − N0− 1)(k) h=1 i=1

∑ ∑ ahl (N0− i + 1)Δuh(k + i − N0 − 1) h=1

i=1

The response − i) is ignored in eq 20 to compute e0l(j)(k). It can be given by eq 5, but Z̅ p(k) is the matrix after removing all items related to uj. Define the MQI for the lth output after removing the jth input as

Nu(h ≠ j) N0

= yl(̅ j) (k) +

h=1

el0(j)(k)

∑Ni=1Gljiuj(k

3. ASSESSMENT OF DEFICIENT SUBMODELS In Section 2, the overall MQI and the MQI of each CV are defined to assess the overall model performance and the model performance of each CV. However, the separate MQI of each CV still cannot give the performance of the CV−MV submodel. Because we cannot get the decomposed CV data caused by an MV, the MQI of each CV−MV submodel cannot be computed directly. A leave-one-out method that involves removing the MVs one by one is proposed to evaluate the performance of an MV−CV submodel. 3.1. MQI of Submodels Based on the Leave-One-Out Method. Take the jth MV and the lth CV submodel as examples. Assume that e0lj and elj are the disturbance innovation and prediction error corresponding to the CVl−MVj submodel. MQIlj is defined as the MQI of the CVl−MVj submodel. To compute e0lj and elj by eqs 5 and 17, the data of uj(k) and its response ylj(k) are required. Unfortunately, a CV is the combined response of all MVs, and it is difficult to extract the response of the jth MV to the lth CV. Therefore, we cannot obtain MQIlj directly. Our idea is to infer the contribution of a submodel to the overall model prediction by removing it from the control channel to the disturbance channel in the computation of the MQI. That is, the function of the submodel is still included in the output response, but it is considered as a disturbance. Thus, the output remains as before the submodel was removed and the running data can be used to compute the MQI. Based on the comparison of the MQI of a CV before and after removing the submodel, the positive or negative contribution of the removed submodel can be assessed. If the submodel shows negative contribution, it can be considered deficient. The method uses all the submodels except for one CV−MV submodel one time to compute the corresponding MQI. For example, for the lth CV, the jth MV is removed. e0l(j) and el(j) are defined as the disturbance innovation and corresponding prediction error after the removal, respectively. From eq 15, the model output ylm(j) of CVl after removing the jth MV can be computed by solving Nu

Nu(h ≠ j) N

∑ Hilyl(j)(k − i) + ∑ ∑ Gjiluh(k − i) i=1

where d(k) = y(k) − ym(k) is the estimated disturbance. With the estimated disturbance innovations e0(k) by eq 5 and the prediction error e(k) by eq 17), the model quality index can be obtained by eq 1. Only the routine input u(k) and output y(k) are required to compute e0(k). The input u(k), output y(k), and model coefficients ai are required to compute e(k).

ylm̂ (j) (k) = yl(̅ j) (k) +

(19)

Correspondingly, we compute e0l(j)(k) by removing the effect of

i=1

(18) D


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Figure 3. Procedure of the systematic model diagnosis.

As explained in Section 2.1, Z̅ p(k) is (Ny × M + Nu × N) × p dimensional where Ny is the number of CVs, Nu is the number of MVs, p is the sliding time window, and M and N are the sufficiently large order of the ARX (HOARX) model.28 Hence, Z̅ p(k)Z̅ p(k)T is (Ny × M + Nu × N) × (Ny × M + Nu × N) dimensional, and the computational complexity of eq 5 is O(p4 × (Ny × M + Nu × N)3). Those of eqs 17 and 2 are O(Ny4 × Nu) and O(2 × N̂ 2) where N is the number of data length for the evaluation, respectively. The computational complexity for one MQI is O(p4 × (Ny × M + Nu × N)3 + Ny4 × Nu + 2 × N̂ 2). For the leave-one-out method, the total computational complexity of the Ny output Nu input process is O((Ny × Nu + 1) × (p4 × (Ny × M + Nu × N)3 + Ny4 × Nu + 2 × N2)). The computation of disturbance innovation accounts for a large proportion O(p4 × (Ny × M + Nu × N)3) for M and N must be large enough. QR decomposition can be applied to simplify the computation of MQI.28 By the QR decomposition, the computational complexity of disturbance innovation is O(2 × (Ny × M + Nu × N + 1) × p2 + (Ny × M + Nu × N)2 × p2), that is, O(p2 × ((Ny × M + Nu × N)2 + 2 × (Ny × M + Nu × N) + 2). Defining h = Ny × M + Nu × N, it becomes O(p2 × (h2 + 2 × h + 2), which is much smaller than O(p4 × h3) by the direct computation 5. Althogh the computation of the proposed MQI method is huge, it is acceptable in practice because the diagnosis can be implemented for a long period.

From the theory of MQI, the difference between disturbance innovation e0(k) and prediction residual e(k) indicates the prediction accuracy. The smaller the difference, the higher the prediction accuracy. For the MPC system in Figure 1, by keeping the control model Gm(q) fixed, we can construct the optimization problem N

min J = H(q)

∑ (e(k) − e 0(k))2 (23)

k=1

to find the optimal disturbance model H*(q), where e (k) is given by eq 5 and e(k) can be expressed as 0

e(k) = H −1(q)(y(k) − Gm(q)u(k))

(24)

according to eq 16. As shown in eq 23, the optimal solution H*(q) is at e(k) = e0(k). Hence, substituting e(k) with e0(k) in eq 24, we obtain H *(q)e 0(k) = y(k) − Gm(q)u(k)

(25)

0

In eq 25, the data series of e (k) can be computed by solving eq 5, and the data series y(k) − Gm(q)u(k) can be easily obtained by using the process data and the control model Gm(q). Therefore, the optimal disturbance model H*(q) can be fitted by using different kinds of data-based modeling algorithms. The upgraded disturbance models H*(q) may vary in complexity. Consider the lth single output system; the simplest upgrade over the random walk model is the IMA(1, 1) disturbance model, i.e.,

4. IMPROVEMENT OF PREDICTION PERFORMANCE BY UPGRADING A DISTURBANCE MODEL As shown in eq 13, the predictive output y(k) is the combined contribution of control model Gm(q) and disturbance model H(q). If a significant difference between prediction errors e(k) and disturbance innovations e0(k) is assessed, the difference may be caused by a poor control model or a poor disturbance model. The predictive output of a deficient control model Gm(q) can be compensated for by using an appropriate disturbance model output. Hence, it is possible to assess how the prediction errors can be improved by using an upgraded disturbance model without altering the control model. If such an improvement is significant, it would be desirable in practice to improve the model prediction by simply adopting an upgraded disturbance model without reidentifying the control model, thus reducing the cost of model reidentification or model maintenance.

Hl*(q) =

1 + αq − 1 1 − q −1

(26)

In this case, the single output formation of eq 13 becomes yl (k) = Gml(q)u(k) +

1 + αq − 1 1 − q −1

el0(k)

(27)

It is obvious that the second item of eq 27, i.e., the IMA(1, 1) disturbance model, can compensate for the mismatch of the control model Gml(q) by choosing the proper model parameter α. With a higher order H*(q), higher prediction precision can be achieved. With the upgrade of the disturbance model, the predictive error is compensated for and hence the model prediction E


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e0(k) is the independent white noise with covariance diag{1.5, 2.6}. The MPC prediction and control horizons are chosen to be 100 and 10, respectively. The weight matrices are Q = diag{1, 50} and S = diag{1, 2}. The model length is M = N = 20, and the time window is p = 4460 in the experiment. The proposed and Badwe’s methodologies were both experimented. To make Badwe’s method applicable, the set points of the two CVs are assumed varying stochastically in [88, 92] and [4.5, 5.5], respectively. To demonstrate the effectiveness of the proposed algorithm, several model cases were tested in the experiment, including the no-mismatch case and different kinds of mismatch cases. 5.1.1. Case 1: No Mismatch Both in the Control Model and the Disturbance Model. Models 31 and 32 were applied as the predictive control model and disturbance model, respectively, in MPC simulation. The MQI values of y1 and y2 are 0.9460 and 0.9204, respectively. The indices both are close to 1, which implies the good performance of the predictive models. The result is anastomotic with the fact. Badwe’s partial correlation analysis method was also tested in the experiment. The comparison of the two methodologies is listed in Table 1. It can be seen that the assessment results are consistent and they reflect the truth.

performance is greatly enhanced. Theoretically, the prediction error equals the disturbance innovations after H*(q) is upgraded because the error is determined based on the condition e(k) = e0(k). However, the model performance deterioration may be caused by several submodels, while usually only the most severe one is considered as the deficient submodel based on the threshold, with the others being considered as good because their performances degrade slightly. Hence, the improvement of one disturbance model is not enough to compensate for all model deterioration. There is still some difference between the prediction error and the disturbance innovations. Define the prediction error el′(k) when H*(q) is upgraded. The MQI index N

η* = l

∑k = 1 el0(k)T Q (k)el0(k) N

∑k = 1 el′(k)T Q (k)el′(k)

(28)

is still used to evaluate the improvement. Define the improvement index by replacing the random walk disturbance model with the optimal high-order H*(q) ξl =

ηl* − ηl ηl

× 100% (29)

ξl shows the improvement of disturbance model H*(q) over the random walk one. Theoretically, the model performance with disturbance model H*(q) is not inferior to that with the random walk one because H*(q) equals to the random walk model if its order is 0. Hence, the prediction model with the upgraded disturbance model H*(q) has higher prediction ability. Therefore, ξl ≥ 0. A larger ξl indicates better prediction performance by the upgraded disturbance model H*(q).

Table 1. Result of No Mismatch Case Badwe’s method

proposed method

5. SCENARIOS 5.1. Wood−Berry Distillation Column Process. The presented systematic model performance diagnosis tool and improvement method were applied in the Wood−Berry distillation column process.30 The process transfer function is ⎡ 12.8e−s ⎢ ⎢ 16.7s + 1 G(s) = ⎢ − 7s ⎢ 6.6e ⎣ 10.9s + 1

−18.9e−3s ⎤ ⎥ 21.0s + 1 ⎥ ⎥ −19.4e−3s ⎥ 14.4s + 1 ⎦

e1 ↔ u2

e2 ↔ u1

e 2 ↔ u2

0.0044

0.0051

0.0173

0.0067

good

good

good

good

MQI assessment result

y1

y2

0.9460 good

0.9204 good

⎡ 1 ⎤ 0 ⎥ ⎢ −1 ⎢1 − q ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ 1 − q −1 ⎦ ⎣

(30)

(33)

was simulated, and the MQIs of y1 and y2 were 0.6649 and 0.3529, respectively. Compared to Case 1, their MQIs both decreased. Hence, the value of the MQI correctly reflected the model performance. 5.1.3. Location of Deficient Submodel. DMC with the mismatched control model ⎡ −2 ⎤ 0.744 −0.8789 q −4 ⎢q −1 −1 ⎥ 1 − 0.9535q ⎥ ⎢ 1 − 0.9419q Gm(q) = ⎢ 0.5786 −1.302 ⎥ −4 ⎢ q −8 ⎥ 3 q × ⎢⎣ 1 − 0.9123q−1 1 − 0.65q−1 ⎥⎦

(31)

with a sampling time of 1 min. Assume that the actual disturbance model is ⎡ 1 − 0.5q−1 ⎤ ⎢ ⎥ 0 −1 1 − q ⎢ ⎥ H 0(q) = ⎢ ⎥ −1 1 − 0.7q ⎥ ⎢ 0 ⎢ ⎥ 1 − q −1 ⎦ ⎣

e1 ↔ u1

5.1.2. Case 2: Mismatch in Disturbance Model. DMC with accurate control model 31 and random walk disturbance model

The two controlled variables are the compositions of the top product and bottom product of the column (in mol %). The two manipulated variables of the process are the reflux rate and steam flow rate (in lb/min). The discretized process transfer function is ⎡ −2 −0.8789 ⎤ 0.744 q −4 ⎢q ⎥ −1 1 − 0.9535q−1 ⎥ ⎢ 1 − 0.9419q G 0(q) = ⎢ ⎥ −1.302 0.5786 −4 ⎢ q −8 ⎥ q ⎢⎣ 1 − 0.9123q−1 1 − 0.9329q−1 ⎥⎦

partial correlation assessment result

(34)

and the random walk disturbance model 33 was simulated where there was a heavy gain mismatch in the y2 ↔ u2 submodel. It is denoted by Case 3. The corresponding MQIs of y1 and y2 were 0.6533 and 0.2162, respectively. The result indicates that mismatch emerges in y2 related submodels because there is a large decrease of y2’s MQI.

(32) F


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Industrial & Engineering Chemistry Research Table 2. Diagnosis Result of Deficient Submodel Badwe’s method

partial correlation assessment result running time

e1 ↔ u1

e1 ↔ u2

e2 ↔ u1

e2 ↔ u2

−0.0327 good

0.0258 good

0.0183 good

0.1937 deficient

6.3 s removing y2 ↔ u1

y2 proposed method

MQI assessment result running time

0.2162 deficient

0.1326 good 14.5 s

removing y2 ↔ u2 0.4202 deficient

Table 3. Impact of Noises on the Result Badwe’s method Partial correlation

MQI method

Diag{0.35, 0.65}

Diag{0.74, 1.3}

Diag{1.5, 2.6}

−0.0129 0.0161 0.0261 0.2193 0.5696 0.2412

0.0297 0.0334 −0.0174 0.2071 0.6703 0.2450

−0.0327 0.0258 0.0183 0.1937 0.6859 0.2479

e1 ↔ u1 e1 ↔ u2 e2 ↔ u1 e2 ↔ u2 y1 y2

Figure 4. Diagram of noise impact on the model indices (left: Badwe’s method; right: MQI method).

5.1.4. Impact of Noise. To test the impact of noises on the assessment result, three different noises were imposed on the mismatched system, respectively. MQI and Badwe’s methods were both experimented under the three different noises. The predictive control model is eq 34, the same as in Case 3. The result is in Table 3 and visualized in Figure 4. It can be seen from Table 3 and Figure 4 that the partial correlation of mismatched submodel e2 ↔ u2 tends to decrease while others would increase with the increase of noises in Badwe’s method. It indicates that the larger noise interferes with the assessment of mismatch because mismatch in submodel e2 ↔ u2 should exhibit large partial correlation. This is consistent with the theoretical analysis.27 However, in the MQI method, the index of matched submodel y1 becomes larger and the difference between the matched submodel y1 and mismatched submodel y2 becomes distinct with the increase of noises. The reason is that the MQI is calculated based on the estimation of disturbance innovation (i.e., noise), and larger noise helps to obtain better estimation accuracy. 5.1.5. Improvement of Deficient Submodels. DMC with the mismatched control model

To further diagnose the location of deficient submodels, the leave-one-out method was imposed in y2 related submodels. The y2 ↔ u1 submodel diagnosis, denoted by Case 4, was implemented by removing u1 in Case 3. Likely, the y2 ↔ u2 submodel diagnosis, denoted by Case 5, was implemented by removing u2 in Case 3. The result is presented in Table 2 accompanying with Badwe’s result. We set the threshold of Badwe’s index to 0.1. It is obvious that only the partial correlation of submodel y2 ↔ u2 is larger than the threshold while others are much smaller. Hence, the mismatch in the y2 ↔ u2 submodel is detected by Badwe’s method. Also, the submodel y2 ↔ u2 is assessed mismatched by the leave-one-out method because the MQI increases from 0.2162 to 0.4202 after removing the submodel while another MQI is degraded by removing the submodel y2 ↔ u1. This gives the conclusion that the mismatch locates in submodel y2 ↔ u2. The result of the proposed method is consistent with Badwe’s and the actual case. The running times of the two methods are also listed in Table 2. By the same computer, the Badwe’s method spent 6.3 s to get the conclusion, and the leave-one-out method took 14.5 s. Though the running time of the proposed method is about two times that of Badwe’s, it satisfies the industrial requirement for the model characteristic changes rather slowly and the interval of mismatch diagnosis is much longer than the running time.

⎡ 0.744 − 0.8789 ⎤ −2 3 × q −4 ⎢4 × q ⎥ −1 1 − 0.9419 q 1 − 0.9535q−1 ⎥ ⎢ Gm(q) = ⎢ ⎥ 0.5786 − 1.302 ⎢ q −8 ⎥ 6 × q −4 −1 −1 ⎢⎣ 1 − 0.9123q 1 − 0.9329q ⎥⎦ (35) G


Article


Figure 5. Prediction comparison before and after disturbance model upgrade.

and the random walk disturbance model was first simulated, as denoted by Case 6. The MQIs of y1 and y2 are 0.5038 and 0.4513, respectively. The disturbance model was upgraded by ⎡ 1.003 − 0.4026z −1 ⎤ ⎢ ⎥ 0 −1 ⎢ 1 − 0.996z ⎥ ⎢ −1 ⎥ 1.032 − 0.8661z ⎥ ⎢ 0 ⎢⎣ 1 − 0.9943z −1 ⎥⎦

proposed methodology based on MQI is not so sensitive to large noises as Badwe’s in view of the result of noise tests. And furthermore, improvement of the prediction model cannot be developed on the basis of Badwe’s method because the optimal model cannot be identified by the optimization of correlation coefficients. In addition, enough set point excitation was supplied to make Badwe’s method applicative. If it is unsatisfied, Badwe’s method is inoperative while the proposed method works. Although the running time was much longer than Badwe’s, it is totally acceptable because the diagnosis interval is much longer in practice. 5.2. Industrial Process. The presented model monitoring and improvement method was applied to an industrial MPC process. For the purpose of confidentiality, the details of the process are not given, and all the controlled variables and manipulated variables are denoted as CV1, CV2, MV1, MV2, etc. There were 10 controlled variables (CV) and 9 manipulated variables (MV) in the MPC process. The correlations between the 10 most significant CVs and their corresponding MVs in the actual control system are listed in Table 5. It can be seen from Table 5 that CV3, CV4, CV5, CV6, CV9, and CV10 are controlled by single manipulated variables and that CV1, CV2, CV7, and CV8 are controlled by multi-inputs. Hence, there is only one prediction model for CV3, CV4, CV5, CV6, CV9, and CV10 and more than one submodel for CV1, CV2, CV7, and CV8. To test the presented algorithm, 5432 routine closed-loop data points were used. 5.2.1. Model Performance Assessment of Each CV. The MQI of each CV with the random walk disturbance model was calculated by using the routine data and control model coefficients. The results are listed in Table 6, where the MQI threshold with respect to whether a model is considered to be good or deficient is set to 0.6. It can be seen that some MQIs are relatively good, while others are very low and even approach 0. The model performance of each CV is clearly shown by its MQI. Seven CV models (CV1, CV3, CV4, CV5, CV6, CV8, CV10) are considered to be good, while the other 3 CV models (CV2, CV7, CV9) are considered to be deficient. 5.2.2. Diagnosis of Deficient Submodels. Of the three CVs with deficient models in Table 6, CV9 has only one input and there is only one model. Therefore, it can be concluded that the submodel CV9−MV2 is deficient. However, for the rest (CV2, CV7), there are several submodels corresponding to each. Hence, the leave-one-out method was used to diagnose the deficient channel for CV2 and CV7. The results are shown in Table 7, where the bold rows correspond to the deficient submodels.

(36)

This is denoted as Case 7. After the improvement, MQI of y1 was enhanced greatly from 0.5038 to 0.8443, and MQI of y2 was also enhanced from 0.4513 to 0.8276. To visually show the improvement of the upgraded disturbance model, the prediction output and actual output curves before and after the upgrade are given in Figure 5. It can be seen from Figure 5 that the model prediction is more accurate after the disturbance model upgrade. The covariance of the prediction error decreased from 3624 to 3148 for y1 and from 14680 to 9737 for y2. Hence, the adoption of a more appropriate disturbance model is quite effective in the improvement of prediction performance without the need to alter the control model. All of the cases are summarized in Table 4. According to results, the following can be concluded: (1) The value of the MQI correctly reflected the model performance in the Wood−Berry distillation column process. For the matched model in case 1, the MQIs are close to 1. For those cases existing mismatch either in the control model or the disturbance model (Case 2, Case 3, Case 6), the MQIs deviate from 1 markedly. (2) The deficient submodel can be diagnosed by the leaveone-out method. Which output the mismatch locates in can be figured out by the separate output MQIs. The further submodel mismatch location can be determined by the variation of MQI after transferring the input from the control channel to the disturbance channel one by one. In this experiment, mismatch was first located in y2 related submodels and then finally concluded in submodel y2 ↔ u2 by removing the function of u2 from the control channel to the disturbance channel. (3) The upgrade of the disturbance model based on the optimization of MQI is effective in the prediction of the process output. The prediction accuracy in Case 7 was greatly improved after the upgrade. (4) The proposed leave-one-out method and Badwe’s method obtained almost the same factual conclusion. However, the H


Article

Industrial & Engineering Chemistry Research Table 4. MQIs of the Wood−Berry Column in Difference Cases

other MVs. Hence, mismatch in the CV2−MV2 submodel is detected. Badwe’s method was also experimented for every CV−MV pair. The result is listed in Table 8 and visualized in Figure 6 (using the absolute value of partial correlation coefficients). It can be found that the largest three correlations are in CV2, CV7, and CV9, and, hence, mismatches are considered in corresponding CVs. Furthermore, CV2−MV2 and CV7−MV4 have the largest correlation coefficient in CV2 and CV7, respectively. There is only one model in CV9, i.e., CV9−MV2. Hence, submodels CV2−MV2, CV7−MV4, and CV9−MV2 are considered deficient by Baswe’s method, which is consistent with the conclusion of the leave-one-out method. CV8’s submodels can also be considered as deficient if the threshold is set to 0.1 for the corresponding coefficients that are close to 0.1. Likewise, CV8’s MQI in Table 6 also approaches to the MQI threshold 0.6. Hence, the conclusions of mismatch location by the two methods are anastomotic. The running time of the two methods was also recorded. 228 s and 32 s were taken by MQI and Bodwe’s method in the same computer, respectively. This is acceptable in practice. 5.2.3. Upgrade of Disturbance Models. To improve the prediction performance, a high-order disturbance model is applied in the prediction. The estimated disturbance models and MQIs are listed in Table 9. It can be seen from Table 9 that

Table 5. Relationships between CVs and MVs in the Industrial Processa CV1 CV2 CV3 CV4 CV5 CV6 CV7 CV8 CV9 CV10 a

MV1

MV2

MV3

MV4

▲ ▲

▲ ▲

▲ ▲

▲ ▲

MV5

MV6

MV7

MV8

MV9

▲

▲ ▲

▲ ▲

▲ ▲ ▲ ▲ ▲

▲

▲ ▲

▲ denotes correlation.

As shown in Table 7, it is clear that most of the values of κj approach 1, which means that leaving out MVs does not significantly influence the MQI and that the corresponding submodel performance is not poor. For CV2 and CV7, an obvious improvement emerges after removing MV2 and MV4, repectively. The corresponding submodels CV2−MV2 and CV7−MV4 are considered to be deficient. Taking CV2 as an example, the MQI of CV2 increased from 0.0361 to 0.1621 and the corresponding MQI impact factor was much larger than 1 after removing MV2, while it hardly changed after removing I


Article

Industrial & Engineering Chemistry Research Table 6. MQIs of All CVs in the Industrial Process MQIs conclusion

CV1

CV2

CV3

CV4

CV5

CV6

CV7

CV8

CV9

CV10

0.9393 good

0.0361 deficient

0.9088 good

0.9680 good

0.9045 good

0.7428 good

0.3281 deficient

0.5473 good

0.2854 deficient

0.9569 good

Table 7. Model Diagnosis Results for the Industrial Process CV

overall MQI

Leaving out MVs

MQI_j

κj

CV2

0.0361

CV7

0.3281

MV1 MV2 MV3 MV4 MV7 MV4 MV6 MV8 MV9

0.0363 0.1621 0.0365 0.0302 0.0373 0.4758 0.3125 0.3263 0.3255

1.0055 4.4903 1.0111 0.8366 1.0332 1.4502 0.9735 0.9947 0.9921

almost all the MQIs increased, which is identical to the result of the theory analysis that higher-order disturbance model performance is not inferior to the random walk one in Section 4. The MQIs of most CVs increased significantly, and the improvement index became rather larger after using the higher-order disturbance model. Those CVs are promising because their prediction ability can be improved by using an upgraded disturbance model rather than by remodeling the step response coefficients. A comparison between the original predictive output and improved predictive output is shown in Figure 7. The corresponding root-mean-square error (RMSE) is shown in Table 10. It is obvious that the predictive performances of all the three controlled variables, which were diagnosed as deficient, improved greatly. According to the experiment result in the industrial process, the following can be concluded: (1) Model performance assessment base on MQI value is feasible in industrial practice according to the consistent conclusion with Badwe’s method. (2) The location of mismatched submodels could be realized by the leave-one-out method in practice. The location result is also in line with Badwe’s. (3) The running time is short enough to satisfy the industrial requirement although it far exceeds Badwe’s. (4) The improvement of prediction can be realized by upgrading of disturbance model based on the optimization of MQI in practice. The prediction was much more accurate after the upgrade in the experiment. It cannot be done by Badwe’s method.

Figure 6. Diagram of partial correlation by Badwe’s method.

Table 9. MQI Improvement Result after Disturbance Model Upgrade CV

original MQI

upgraded MQI

improvement index ξι

upgraded disturbance model

CV2

0.0361

0.0653

80.9%

0.9427 + 0.2153z−1 1 − 0.8953z−1

CV7

0.3281

0.5327

83.8%

0.8721 + 0.1523z−1 1 − 0.9531z−1

CV9

0.2854

0.5704

99.86%

0.9825 + 0.8371z−1 1 − 0.9214z−1

6. CONCLUSIONS This paper first proposed a deficient submodel diagnosis method. Compared to other model performance assessment methods, ours can not only give the assessment result of the overall model but also further diagnose the location of the submodel that caused the overall model deficiency. It overcomes the disadvantage of Badwe’s method that sufficient set point excitation is required and large noises interfere with the result. By applying our method to the Wood−Berry distillation column and an industrial process, it is verified that all poor models and deficient submodels can be correctly detected by the diagnosis algorithm. The diagnosis results can reveal which models need to be maintained and thus reduce the work and cost needed to maintain all the models.

Table 8. Partial Correlation by Badwe’s Method

CV1 CV2 CV3 CV4 CV5 CV6 CV7 CV8 CV9 CV10

MV1

MV2

MV3

MV4

MV5

MV6

MV7

MV8

MV9

−0.0156 0.0151 0.0001 0.0019 −0.0135 −0.0954 0.0048

0.0107 0.3341 0.1604 -

−0.0196 0.1374 -

0.0183 0.0983 −0.1293 -

0.0073 -

−0.088 -

0.1043 -

0.0248 -

0.0855 0.1038 -

J


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Table 10. RMSE Comparison before and after the Disturbance Model Upgrade before after

CV2

CV7

CV9

0.0073 0.0043

0.0437 0.0101

7.7710 1.0657

Another contribution of the paper is an effective model improvement method that yields improvement by optimizing the model performance index and upgrading the disturbance model. By the optimal disturbance model, the control model mismatch can be compensated for without influencing the normal running of an MPC system. The two cases studied verified the disturbance model compensation method. While the paper provides engineers with a practical approach to diagnose and improve deficient models, there is still some work that requires further consideration. In the submodel diagnosis process, mismatch may exist in several submodels, but the mismatch function of these submodels is offset by that of others. Thus, the mismatch cannot be shown using the input−output data, and the mismatch may not be diagnosed by the presented method.

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AUTHOR INFORMATION

Corresponding Author

*(L.L.) Tel.: +86-25-58139517. E-mail: [email protected]. ORCID

Lijuan Li: 0000-0002-3694-5570 Jianquan Song: 0000-0002-7012-8726 Xiaoxiao Zhang: 0000-0003-2772-3009 Jing Ye: 0000-0003-2975-0457 Shipin Yang: 0000-0003-3725-4293 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research is supported by the National Natural Science Foundation of China (61203072, 61403190) and the Six Talent Peak Project of Jiangsu Province.

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REFERENCES

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Model Deficiency Diagnosis and Improvement via Model Residual

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