Impact of model plant mismatch on performance of control systems: An application to paper machine control

Control Engineering Practice 43 (2015) 59–68

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Impact of model plant mismatch on performance of control systems: An application to paper machine control M. Yousefi a,n, R.B. Gopaluni b, P.D. Loewen c, M.G. Forbes d, G.A. Dumont a, J. Backstrom d a

Department of Electrical and Computer Engineering, University of British Columbia, Vancouver BC Department of Chemical and Biological Engineering, University of British Columbia, Vancouver BC c Department of Mathematics, University of British Columbia, Vancouver BC d Honeywell Process Solutions, North Vancouver BC b

art ic l e i nf o

a b s t r a c t

Article history: Received 22 November 2014 Received in revised form 9 July 2015 Accepted 9 July 2015

Model-based controllers based on incorrect estimates of the true plant behaviour can be expected to perform poorly. This work studies the effect of model plant mismatch on the closed loop behaviour and system performance for a certain class of MIMO systems. Performance is measured using a minimum variance index and a closely related user-specified criterion. We study the effect of model plant mismatch on the output variance and performance indices. Under mild assumptions, the performance of each output in a MIMO system can be analysed independently. Moreover, we propose an approach to distinguish the effect of model–plant mismatch from the effect of changes in disturbance characteristics on closed-loop performance. We define a sensitivity measure that relates system performance to model– plant mismatch, and use it to explore this sensitivity for three realistic types of parametric modelling errors. Next, we suggest a quantitative method that compares a system's actual output to its desired response in a transient setting. The performance of the transient response is demonstrably more sensitive to the model–plant mismatch than the steady state performance. The results are illustrated on industrial paper machine data. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Performance monitoring MIMO systems Minimum variance control Sensitivity Paper machines

1. Introduction Performance of industrial control systems must be monitored continuously to maintain product quality. In this work, the term performance refers to the magnitude of perturbations in process outputs. In process industry, these perturbations are mainly caused by stochastic unmeasured disturbances and the control objective is to minimize the effect of the disturbances on process outputs. Harris, Seppala, and Desborough (1999), Huang and Shah (1999) and Joe Qin (1998) present comprehensive surveys of performance assessment techniques for both univariate and multivariable systems. The most popular benchmarks are based on (a) minimum variance control (MVC) and (b) user-specified control benchmarking. These techniques are widely used in industry. Applications of performance assessment techniques to pulp and paper processes are given by Lynch and Dumont (1996), n

Corresponding author. E-mail addresses: [email protected] (M. Yousefi), [email protected] (R.B. Gopaluni), [email protected] (P.D. Loewen), [email protected] (M.G. Forbes), [email protected] (G.A. Dumont), [email protected] (J. Backstrom). http://dx.doi.org/10.1016/j.conengprac.2015.07.005 0967-0661/& 2015 Elsevier Ltd. All rights reserved.

Desborough and Harris (1994), Jofriet and Bialkowski (1996) and Owen, Read, Blekkenhorst, and Roche (1996). These approaches are used to detect poor performance of control systems by comparing the actual output variance with a specific benchmark defined based on the type of performance monitoring algorithm. Model–plant mismatch (MPM) is one of the causes of deterioration in performance of control systems; especially, model based control systems, e.g., model predictive control (MPC). Therefore, the sensitivity of the performance indices to MPM, which shows the ability of the indices to reveal MPM, is of utmost importance. It is easy to observe that the typical performance indices depend on the model used to design a controller. However, there is scant literature on understanding the effect of MPM on minimum variance and user-specified performance indices. For instance, Yousefi et al. (2014) analyse the sensitivity of the various performance indices to different types of parametric MPM. They show that mismatch in different parameters influences the indices differently. However, there is a lack of explanation for their observations. There are also few articles on MPM detection in control systems using other approaches. For instance, in Wang, Hagglund, and Song (2012), where Wang et al. analyse the influence of model–

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M. Yousefi et al. / Control Engineering Practice 43 (2015) 59–68

plant mismatch on control loop behaviour. They introduce Integral Absolute Error (IAE) index to measure performance of control systems to detect MPM. They claim that the smaller the index the better the performance. Indeed, they show that MPM increases IAE index. However, they do not define any benchmark to compare the index with. In fact, it is not clear what the index should be under normal operating conditions. Badwe, Patwardhan, Shah, Patwardhan, and Gudi (2010) define a Relative Sensitivity Index (RSI) by comparing an actual sensitivity function with a designed sensitivity function to quantify the impact of model plant mismatch. This index is defined as follows:

RSI =

Sa Sd

effect of MPM from the effect of disturbance on performance. In Section 6, we define a sensitivity function to quantify the sensitivity of the performance indices to model–plant mismatch. Also, we suggest a technique that measures the performance of control systems in servo control mode. Finally, in Section 7, we present the results of simulations and compare them with our theoretical formulations.

2. Assessment of multi-input/multi-output systems 2.1. Minimum variance benchmarking

∞

(1)

where Sa is the actual sensitivity function and Sd is the designed sensitivity function. To calculate the above index, we only need to estimate Sa. They (Badwe et al., 2010) show that in the presence of MPM in a closed loop system, the RSI is greater than 0 db. They use closed loop information to estimate the actual sensitivity function. To do so, there must be set-point changes in the closed loop system. However, in industry, set-point changes do not happen often and most control systems work in regulatory mode. So, RSI cannot be reliably estimated online to monitor the closed loop performance. The main objective of this work is to quantify the sensitivity of the performance indices to mismatch between a multivariable plant and the MIMO model used in its controller. Yousefi et al. (2014) performed similar analysis for SISO systems. In this paper, we show that MPM affects the closed-loop sensitivity function, and consequently, the output variance as well. We address the question of why the performance indices are more sensitive to certain types of MPM than others. In principle, the performance of each output in a MIMO system can be affected by modelling errors in every single element of the transfer function matrix representing the process model. In this paper, we identify conditions under which MPM in an element of the transfer matrix only deteriorates the performance of the associated output but does not effect the performance of other outputs. In such cases, the performance of each output can be analysed independently. For a given controller, the sensitivity analysis lets us assess the effectiveness of the performance indices in detecting model–plant mismatch of various types. Since the indices are calculated using the steady-state response (regulatory mode) of control systems, they turn out to be insensitive to certain types of parametric mismatch, such as mismatch in time constants. However, we observe that model–plant mismatch has a stronger effect on a system's transient response (servo control mode). We propose a technique that uses transient response to measure controller performance. This analysis provides diagnostic information that helps identify the type of mismatch between the plant and the model. Such diagnostic information on the cause of poor performance can provide useful guidance for intervention, perhaps by focussing the goals of a re-identification experiment. The sensitivity of the performance assessment techniques to model–plant mismatch is analyzed through simulations on a model predictive controller operating on a paper machine. This paper is organized as follows. In Section 2, we briefly review two widely used performance assessment techniques for MIMO systems, namely, minimum variance benchmarking and user specified benchmarking. In Section 3, we describe how MPM changes the sensitivity function and affects the output variance. In Section 4, we describe the possible decoupling between both performance criteria for different output components. In Section 5, we discuss the effect of changes in disturbance characteristics on a system's performance and propose a technique to distinguish the

Huang, Shah, and Kwok (1996) and Harris, Boudreau, and MacGregor (1996) use the minimum variance approach to measure performance of MIMO systems. In Harris (2009), Harris provides some interpretations of the performance bounds for such systems. Huang and Shah (1999) use a statistical signal processing approach, called the filtering and correlation (FCOR) algorithm, to estimate the minimum variance index (MVI) from raw data. In this approach, the only information needed to calculate the MVI is the measured output and the time delay of the system. As shown in Fig. 1, for a multivariable process the output vector Yt, of dimension n, satisfies

Yt = P (q−1) Ut + N (q−1) et .

(2)

Here Ut is the input vector in Rm, et is a noise vector in Rn with zero mean and Var (et ) = Σ e , P (q−1) is a n
m transfer matrix representing the process model, N (q−1) is a disturbance transfer matrix (assumed diagonal, n
n), and q  1 is the back shift operator:

q−1Ut = Ut − 1.

(3) 1

For the sake of simplicity, the dependence on q is not shown explicitly for most transfer functions in this paper. We decompose the matrix P in (2) as

˜ P = D−1P,

(4) 1

where the “interactor matrix” D is the diagonal transfer matrix consisting of the time delays in the diagonal terms of P, and P˜ is the resulting delay-free transfer matrix. The interactor matrix was introduced by Wolovich and Elliott (1983), Wolovich and Falb (1976) and Goodwin and Sin (2013) for MVC and other purposes. Huang and Shah (1999) present a comprehensive survey on the characteristics of the interactor matrix. The minimum variance control law is obtained by choosing the transfer matrix C so that the controller Ut = − CYt minimizes the objective function

J = E ⎡⎣ (Yt − E [Yt ])T (Yt − E [Yt ]) ⎤⎦.

(5)

Proposition. For any linear time invariant system with the transfer function shown in (2), the minimum value of J is

Jmin = tr (Var (Fet )),

Fig. 1. The block diagram of a feedback control system.

(6)

M. Yousefi et al. / Control Engineering Practice 43 (2015) 59–68

where F is a polynomial matrix such that

N=F+

(7)

D−1R

and R is a proper rational transfer function matrix. Since the matrix N is diagonal, R and F are diagonal matrices as well. Proof. With a set-point of zero, the closed loop system obeys

(

˜ Yt = I + D−1PC

−1

)

Net .

(8)

Substituting (7) into (8) gives

(

˜ Yt = I + D−1PC

−1 ⎛ ⎜F

)

⎝

⎞ + D−1R⎟ et ⎠

⎛ = ⎜I − D−1P˜ I + CD−1P˜ ⎝

(

⎞⎛ ⎞ C ⎟⎜F + D−1R⎟ et ⎠⎝ ⎠

−1

)

⎛ = Fet + D−1 ⎜R − P˜ I + CD−1P˜ ⎝

(

−1

)

(

(9)

)

(18)

where GR is a proper transfer function matrix which is assumed to be diagonal in this work. A control law which results in the above output is −1 Cusr = P˜ (R − G R )(F + D−1G R )−1.

where

L = R − P˜ (I + CD−1P˜ )−1 CN.

(10)

In (9), the term Fet is control-invariant. Hence the minimum variance control law is obtained by setting L ¼0, i.e., by defining

˜ −1

Cmv = P RF −1.

Ymv = Fet ,

and using this in (5) reveals the minimum value shown in (6). Accordingly, a performance index for each output in a MIMO system can be defined by

{

},

(20)

n

∑ ηusryi ,

(21)

i=1

(

Σusr, (18) can be written as a

)

Yusr = F0 + F1q−1 + ⋯ + Fd − 1q−d + 1 et D− 1

(G

R,0

+ G R,1

q −1

)

+ ⋯ et .

(22)

Then, we have d−1 T Σ usr = E [Yusr Y usr ]=

∞

∑ Fi Σ et FiT + ∑ GR, i Σ et GRT, i. i=0

i=0

(23)

Thus, the variance of the ith output under user specified control is

n yi . ∑ ηmv

(14)

i=1

d−1

σ y2i , usr =

To calculate the above indices, the minimum variance of outputs must be calculated. The system output under minimum variance control, defined in (12), can be written in the following form:

(

)

Ymv = F0 + F1q−1 + ⋯ + Fd − 1q−d + 1 et

(15)

where d is the maximum time delay in the interactor matrix, D  1. Accordingly, the covariance matrix of the output signal under minimum variance control is d−1

∑ Fi Σ et FiT i=0

(16)

d−1

∑ (Fνii )2Σ eiit. □ ν=0

(17)

∞

∑ (Fνii )2Σ eiit + ∑ (GRii, ν )2Σ eiit. ν=0

ν=0

(24)

3. The impact of MPM on sensitivity functions in SISO systems In this section, it is shown that MPM has a significant effect on the closed loop sensitivity function. A complementary sensitivity function for a SISO system, T, is defined as follows:

T=

where Σ et is a noise covariance matrix. Due to the fact that the noise sequence is statistically independent, the minimum variance of the ith output can be calculated as

σ y2i , mv =

−1

where Σ usr = Var (Yusr ). To calculate moving average process:

(13)

where Σmv is the covariance matrix of the system output under minimum variance control, Σ˜Y = diag (ΣY ) and ΣY = Var (Yt ). Furthermore, a single index can be defined as follows to show the performance of the MIMO system at once:

T Σ mv = E [Ymv Y mv ]=

1 n

+

}

−1 y1 yn [ηmv , …, ηmv ] = diag Σ mv Σ˜Y ,

1 n

{

y1 yn [ηusr , …, ηusr ] = diag Σ usr Σ˜Y

ηusr = (12)

(19)

Accordingly, the user-specified performance index is defined as follows:

(11)

The system output under minimum variance control is

ηmv =

to establish an extremely stringent requirement on the performance of a controller. Moreover, a minimum variance controller is rarely used in practice due to physical limitations. For instance, non-minimum-phase zeros cannot be cancelled by stable controllers and this affects the closed-loop dynamics. Moreover, a minimum variance controller provides very aggressive control moves which are not suitable for industrial implementation. Therefore, Huang and Shah (1999) suggest replacing the minimum variance benchmark with a user-specified benchmark. In this approach, the output covariance matrix is compared with a covariance matrix specified by a user instead of the traditional minimum variance. To do so, an extra term is added to the output under user specified control shown in (12)

Yusr = F + D−1G R et

⎞ CN ⎟ et ⎠

≜ Fet + D−1Let

61

CP , 1 + CP

(25)

where C and P denote the controller and the process model, respectively. The uncertain process model can be written as

P = P ′ + Δ,

(26)

where P′ is a nominal plant model, which is used in the controller design, and Δ is the difference from the nominal model. 3.1. Stability margin

2.2. User-specified benchmarking In practice, the minimum variance benchmark has been found

For a closed-loop system, the stability margin Sm is defined as a shortest distance from the Nyquist curve to the critical point on

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M. Yousefi et al. / Control Engineering Practice 43 (2015) 59–68

the complex plane, which is −1 + j0 (Dorf & Bishop, 2011). The stability margin is a function of the open-loop transfer function and it changes as the open loop transfer function changes. It is straightforward to show

Ms =

1 , Sm

(27)

where Ms stands for the sensitivity function peak value. Accordingly, if the stability margin decreases, i.e., if the open loop gain increases, the sensitivity function peak grows. Among different types of MPM, namely, gain mismatch, time constant mismatch, and delay mismatch, positive gain mismatch is most sensitive to sensitivity peak. The existence of a peak in the Bode diagram of the sensitivity function implies that the closed-loop system not only fails to reject disturbances around that frequency region, but also amplifies them. In fact, a sensitivity peak increase due to MPM leads to amplification of certain frequencies and hence to an increase in output variance. Figs. 2–4 compare the peak sensitivity with different magnitudes of MPM for different types of MPM. These figures show the Bode diagrams of the closed-loop sensitivity function of the dry weight control loop in paper machines. The details of the control system are provided in Section 7. The sensitivity functions plotted in these figures are calculated using the model– based controller and the process model affected by different types of MPM. Comparing these figures shows that positive gain and delay mismatch increase the sensitivity function peak more than time constant mismatch. This is due to the fact that these types of mismatch change the open loop transfer function and decrease the stability margin. The peak of the sensitivity function is less sensitive to time constant as it does not affect the output variance. This implies that the performance indices are not sensitive to time constant mismatch. In addition, negative gain mismatch makes the open loop gain smaller and stability margin larger and consequently, decreases the sensitivity peak. Thus, these types of mismatch sometimes decrease the output variance and improve the performance of the system. In these cases, even though the sensitivity peak does not change, the closed-loop bandwidth changes. This change in the closed-loop system affects the performance of the system in the servo control mode. To increase the sensitivity of the performance indices to MPM, we can take transient responses into account by measuring the performance of the following signal

Fig. 3. Delay mismatch and the sensitivity function.

Fig. 4. Time constant mismatch and the sensitivity function.

instead of the actual output:

Et = Yt − Yt, des ⎡ ˜ = ⎢ I + D−1PC ⎣

(

˜ − Tdes ⎤⎥ rt + I + D−1PC ˜ D−1PC ⎦

−1

)

(

−1

)

Net .

(28)

Yt , des and Tdes are a desired output and a closed-loop transfer (complementary sensitivity function) matrix, respectively, and rt is a vector of set-points. The above definition makes it possible to measure the performance of a system in servo control mode. Accordingly, if there is no model–plant mismatch, and

(I + D

Fig. 2. Gain mismatch and the sensitivity function.

−1PC ˜

−1

)

˜ = Tdes, D−1PC

the performance of Et will be the same as the performance of Yt. On the other hand, if mismatch is present, the difference between the desired response and the system's actual response will help us to detect the mismatch by increasing the variance of Et. This approach has several advantages: (a) Mismatch with no effect on the system's steady state performance can be detected. (b) Performance can be assessed continuously. (Note that current assessment techniques cannot be applied during transients or setpoint changes.) (c) A comparison of the system performance in

M. Yousefi et al. / Control Engineering Practice 43 (2015) 59–68

transients with the performance in the steady state will provide extra information about the type of mismatch. These advantages will be discussed further in the next section.

63

(

)

Yt = F0 + F1q−1 + ⋯ + Fd − 1q−d + 1 et D− 1

+

(L

+ L1

0

q −1

)

+ ⋯ et ,

(37)

the coefficient matrices Lν come from (35): 4. The effect of MPM in MIMO systems

n

L νij =

We now investigate how a MIMO system's output variance is affected by mismatch between the true plant, P, and the model used in its controller, P′. The mismatch is the transfer matrix:

Δ = D−1 (P − P ′).

∑ δ ipΛνipj .

Here Λνipj is the νth infinite impulse response coefficient of Finally, the variance of output component i is given by

(29)

We will show that for output components whose transfer functions are modelled exactly, mismatch in other components has no effect on the variance. This allows for a modest decoupling of the full sensitivity problem. Our analysis relies on the assumptions that both the disturbance transfer matrix N and the filter GR in (18) are diagonal. Similar methods treat both minimum-variance and user-specified control strategies.

4.1. Minimum variance control Using the plant model P′ in (11) produces the mismatched controller:

Cmv = (P˜ − Δ)−1RF −1.

(30)

(38)

p=1

Λipj.

d−1

∑ (Fνii )2Σ eiit

σ y2i =

ν=0 ∞

+

⎡

n

⎡

n

⎡⎛

n

⎞

⎛

n

⎞⎤⎤⎤

⎠

⎝ p=1

⎠⎦⎦⎦

∑ ⎢⎢∑ ⎢⎢ ∑ ⎢⎢ ⎜⎜ ∑ δ ipΛνipq ⎟⎟ Σ eqlt ⎜⎜ ∑ δ ipΛνipl ⎟⎟ ⎥⎥ ⎥⎥ ⎥⎥.

ν=0

⎣ l= 1 ⎣q= 1 ⎣ ⎝ p = 1

(39)

The first term on the right side in (39) is the minimum variance of the ith output. Model–plant mismatch influences the second term only, entering through δip, the transfer-function discrepancy from the pth input to the ith output. If δ ip = 0 for each p, the second term vanishes. That is, if there is no mismatch in the transfer functions associated with the ith output, then the predicted variance of the ith output will be identical to the value predicted using an exact model. Mismatch in transfer functions related to other outputs cannot affect the calculated performance index for component i.

Using this in (10) gives, after simplification,

(

L = R − P˜ I + Cmv D−1P˜

4.2. User-specified control

−1

)

Cmv N

(

−1

)

−1 = R − R I − N −1FR−1ΔP˜ R

.

(31)

Thus, by the matrix inversion lemma,

⎛ −1 L = R − R ⎜I + N −1FR−1Δ I − P˜ RN −1FR−1Δ ⎝

(

(

Cusr = (P˜ − Δ)−1(R − G R )(F + D−1G R )−1.

P R (32)

≜ K ΔM,

K = RN −1FR−1,

(

(33) P R.

Cusr N

n

(35)

p=1

Δip and write the above equation

n

−1 + D−1G R )(R − G R )−1ΔP˜ (R − G R )

−1

)

.

(41)

L = G R + (R − G R ) N −1 (F + D−1G R )(R − G R )−1Δ

(I − P˜

−1

−1

)

(R − G R ) N −1 (F + D−1G R )(R − G R )−1Δ

≜ G R + K ΔM,

(42)

where

K = (R − G R ) N −1 (F + D−1G R )(R − G R )−1,

∑ δ ipΔ˜ip K iiM pj p=1 n

(

(43)

(36)

p=1

where δ is the static gain of Δ , and = Δ˜ ip K iiM pj . = When the output equation (9) is expressed as a moving average process, ip

Δip

δ ipΔ˜ ip

−1 P˜ (R − G R ).

−1

)

−1 M = I − P˜ (R − G R ) N −1 (F + D−1G R )(R − G R )−1Δ

∑ δ ipΛipj ip

−1 (F

−1 P˜ (R − G R )

∑ Δip K iiM pj.

We can extract the static gain of as follows:

(I − N

(34)

The matrix K is diagonal, being a product of diagonal matrices. Thus the elements of matrix L are simply

=

−1

)

Using the matrix inverse lemma, we have

−1 −1 ˜

)

−1 M = I − P˜ RN −1FR−1Δ

Lij =

(

L = R − P˜ I + Cusr D−1P˜ = R − (R − G R )

where

Lij =

(40)

The substitution of the above equation into (10) results in

−1 −1 ˜

)

−1 = RN −1FR−1Δ I − P˜ RN −1FR−1Δ

⎞ P R⎟ ⎠

−1 −1 ˜

)

The same analysis shown above can be carried out for userspecified control. The user-specified control law under mismatch conditions can be written as

(44)

Λipj

Since K is diagonal, the elements of matrix L can be calculated as follows:

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M. Yousefi et al. / Control Engineering Practice 43 (2015) 59–68

⎧ n ⎪ ∑ Δip K iiM pj, i≠j ⎪ p=1 ij L =⎨ n ⎪ ii G R + ∑ Δip K iiM pi, i = j ⎪ ⎪ ⎩ p=1

⎧ n ⎪ ∑ δ ipΛipj , i≠j ⎪ p=1 =⎨ n ⎪ ii G R + ∑ δ ipΛipi , i = j ⎪ ⎪ ⎩ p=1

follows:

S= (45)

where δip is the static gain of Δip and Λipj = Δ˜ ip K iiM pj . Based on the above equation, the elements of the coefficient matrix Lν in (37) are calculated using the following equation:

⎧ n ⎪ ∑ δ ipΛνipj , i≠j ⎪ p=1 ij Lν = ⎨ n ⎪ ii G R , ν + ∑ δ ipΛνipi , i = j. ⎪ ⎪ ⎩ p=1

(46)

In the above equation, is the νth infinite impulse response coefficient of the related transfer function, Λipj. Hence, the variance of the ith output under user-specified control, when there is mismatch between the model used in the controller and the plant, can be calculated as follows:

Λνipj

d−1

σ y2i =

∑ (Fνii )2Σ eiit ν=0 ∞

+

∑ ν=0

⎡ ⎢⎛ + ∑ ⎢ ⎜⎜G Rii, ν + l= 1 ⎢ ⎝ l≠ i ⎢ ⎣

⎤ ⎞⎥ ⎛ n ⎞ ∑ δ ipΛνipi ⎟⎟ Σ eqlt ⎜⎜ ∑ δ ipΛνipl ⎟⎟ ⎥ ⎠⎥ ⎝ p=1 ⎠ p=1 ⎥⎦ n

⎡ ⎢⎛ n ⎛ ⎞ + ∑ ⎢ ⎜⎜ ∑ δ ipΛνipq ⎟⎟ Σ eqlt ⎜⎜G Rii, ν + ⎢ q=1 ⎝ p=1 ⎝ ⎠ q≠i ⎢ ⎣ n

⎛ + ⎜⎜G Rii, ν + ⎝

n

⎞

⎛

p=1

⎠

⎝

∑ δ ipΛνipi ⎟⎟ Σ eiit ⎜⎜GRii, ν

n

∑ p=1

⎤ ⎞⎥

δ ipΛνipi ⎟⎟ ⎥ ⎠⎥ ⎥⎦

⎤ n ⎞⎥ + ∑ δ ipΛνipi ⎟⎟ ⎥. ⎠⎥ p=1 ⎥⎦

(49)

If the bandwidth of N increases, more variation appears in the output which increases output variance. Therefore, when the performance indices drop, we cannot guarantee that it is due to MPM. It is very important to distinguish the effect of MPM from the effect of the disturbance on the system's performance, because this information is useful in determining if an expensive identification exercise is necessary. As we showed in the previous sections, the influence of gain mismatch on the closed loop sensitivity function is more significant than the effect of other types of mismatch. Moreover, gain mismatch happens more often than other types. Hence, in this section we focus on gain mismatch. To distinguish performance drop due to gain mismatch from that due to disturbances, we propose to use the ratio between achieved and designed sensitivity functions at high (jω → ∞) and low frequencies (jω → 0). The results are shown in Table 1, using the notations

Sae → y = Sa Na,

(50)

Sde → y = Sd Nd.

(51)

In the above equations, Sa and Sd denote the actual and designed closed-loop sensitivity functions, respectively. Na is the actual disturbance transfer function. Nd is the disturbance transfer function used in control design. To calculate the limit, Sae → y can be easily estimated by whitening the actual process output. Eq. (48) shows that the source of variations in the output is et. Therefore, when there is an increase in output perturbations, it means either S or N are letting more of et perturb the output. If the problem is with S, it means there exists MPM. In the other case, the increase in the output variance is the result of changes in disturbance characteristics or changes in N. Accordingly, if there is no MPM, Sa ¼Sd. On the other hand, if disturbance characteristics do not change, Na ¼ Nd. Based on (50) and (51), we have

⎡ ⎡ ⎤⎤ ⎡ ⎢ n ⎢ n ⎢⎛ n ⎞ ⎛ n ⎞⎥⎥ ⎢∑ ⎢ ∑ ⎢ ⎜ ∑ δ ipΛνipq ⎟ Σ eql ⎜ ∑ δ ipΛνipl ⎟ ⎥ ⎥ ⎟ t⎜ ⎟⎥⎥ ⎢ l = 1 ⎢ q = 1 ⎢ ⎜⎝ p = 1 ⎠ ⎝ p=1 ⎠ ⎥⎦ ⎥⎦ ⎢⎣ l ≠ i ⎢⎣ q ≠ i ⎢⎣

n

1 . 1 + PC

Sae → y (jω) Na (jω) Sd (jω) = . Sde → y (jω) Nd (jω) Sa (jω) (47)

In the above equation, δip is associated with mismatch in the transfer function from the pth input to the ith output. Accordingly, if δip is zero, the variance of the ith output is same as the userspecified variance (σ y2i , usr ), as shown in (24). In other words, no matter how much mismatch there is in the transfer functions related to other outputs, the performance of the ith output is going to be high if there is no mismatch in transfer functions associated with this output. Based on the above analyses, the performance of each output in an MIMO system is independent of performance of other outputs.

(52)

The limiting values of the above expression for jω → ∞ and jω → 0 are summarized in Table 1. Based on this table, by checking the indicated limit, one can distinguish the effect of MPM from the effect of changes in disturbance characteristics on system's performance. To understand this analysis better, Fig. 5 illustrates Bode diagram of Sae → y/Sde → y in the following cases:

 Case 1: No MPM (Sa ¼Sd) with no changes in N's bandwidth (Na ¼Nd).

 Case 2: þ50% gain mismatch (Sa ≠ Sd ) with no changes in N's bandwidth (Na ¼Nd).

 Case 3: No MPM (Sa ¼Sd) with 50% increase in N's bandwidth (Na ≠ Nd ).

 Case 4: þ50% gain mismatch (Sa ≠ Sd ) with 50% increase in N's bandwidth (Na ≠ Nd ).

5. The effect of changes in disturbance characteristics One of the situations which may result in false positive in MPM detection is when disturbance characteristics changes, or in other words, when the bandwidth of N changes. In this case, the performance indices drop, even if there is no MPM. To understand this effect better and for the sake of simplicity in explanation, we focus on a SISO system. Consider the closed-loop output equation:

yt = SNet

(48)

where S is the closed-closed loop sensitivity function defined as

Fig. 5shows that by the frequency behaviour of the ratio in (52), we can verify the existence of MPM in poor performance situations. The data used to plot the above figure is obtained from a paper machine simulator. The details of the simulator such as the system model and the controller are provided in Section 7. 6. Sensitivity of the performance indices to MPM The process model is a key ingredient in defining performance

M. Yousefi et al. / Control Engineering Practice 43 (2015) 59–68

Table 1 The limit values of the sensitivity function ratio in the cases of gain mismatch and changes in disturbance characteristics. Type of mismatch

S e → y (jω) lim ae → y (jω) jω→ 0 Sd

S e → y (jω) lim ae → y (jω) jω→∞ Sd

Sa ¼Sd, Na ¼ Nd Sa ≠ Sd , Na ¼Nd Sa ¼Sd, Na ≠ Nd

0 dB ≠0 dB 0 dB

0 dB 0 dB

≠0 dB

Sa ≠ Sd , Na ≠ Nd

≠0 dB

≠0 dB

criteria for both minimum-variance and user-specified benchmarking. Lines (39) and (47) show how model–plant discrepancy degrades performance. Sensitivity analysis will reveal how model– plant mismatch affects the performance indices, and help focus reidentification efforts aimed at restoring plant performance. In the most general view, a performance index is a function that maps a model, plant, and controller to a scalar value. Expressing the plant in terms of deviations Δ from a nominal plant P′ suggests defining the general performance function

σ y2i (t , Δip ): R+ × : → R+,

(53)

where : is a set of transfer functions capturing all possible uncertainties Δip. (The controllers considered here are implicitly determined by the plant model, so there is no need to show explicit controller-dependence in the functions above.) For both the minimum variance and user-specified performance indices, the performance criterion is the variance. Thus we take η yi = σ y2i above, and express the relative sensitivity of the performance index to model uncertainty as

S (t , Δip ) =

∂σ y2i (t , Δip ) ∂Δip

Δip σ y2i (t ,

Δip )

, (54)

where Δip is the uncertainty in the transfer function from the pth input to the ith output. In the next section, we use (54) to quantify the sensitivity of the performance indices to different types of parametric MPM.

65

7. Case study In this section we consider a paper machine's control loop which controls their machine directional properties of paper. For this system, it is demonstrated that a poor performance output cannot influence the performance of other outputs as proved in Section 3. We also calculate the sensitivity of a particular to three types of parametric mismatch (gain, time constant and delay) in two different control modes (regulatory and servo control) to see how sensitive the performance indices are to model mismatch. The aim of comparing the sensitivity of the performance indices in two modes is to check if such mismatch affects the performance . Since the related analyses for minimum variance and user specified benchmarks are similar, only the sensitivity of the user-specified control performance index is discussed. Three important product characteristics in paper machines are dry weight (y1), size-press moisture (y2) and reel moisture (y3). The dominant manipulated variables which influence these properties are stock flow (u1) and two different dryer pressures (u2 and u3). The following equation shows the relation between the mentioned variables. This model is widely used in pulp and paper industry (Astrom, 1967; Backström & Baker, 2008; Chu, Gheorghe, Backstrom, Forbes, & Chu, 2011; Shi, Wang, Forbes, Backstrom, & Chen, 2015):

⎡ N11 0 ⎡ P11 0 ⎡ y1 ⎤ 0 ⎤ ⎡ e1 ⎤ 0 ⎤ ⎡ u1 ⎤ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ ⎥ y u 0 N 0 ⎥ ⎢e2 ⎥ = + ⎢ ⎢ P21 P22 0 ⎥ ⎢ 2 ⎥ 22 ⎢ 2⎥ ⎢ 0 ⎥ ⎢P ⎢⎣ y3 ⎥⎦ 0 N33 ⎥⎦ ⎣e3 ⎦ ⎣ ⎣ 31 P32 P33 ⎦ ⎣u3 ⎦  

⏟ Yt

P

⏟ Ut

N

⏟ et

(55)

For such a system, each transfer function Pij is specified as a first order transfer function with a dead time (Table 2). These transfer functions are used as the plant model to design a controller. An MPC controller is implemented to control the above properties. In MacArthur (1996), it is asserted that the controller is very robust to model uncertainties. In fact, the performance indices are not expected to be very sensitive to model mismatch. Our simulations shown in this section will quantify how sensitive the controller and performance indices are to the model mismatches in two different operating modes, the regulatory and servo control modes. For the given control system, the desired transfer function matrix, Tdes, is specified as a diagonal matrix shown as follows:

Tdes

⎡ e−72s ⎢ 0 0 ⎢ 99.9s + 1 ⎢ e−114s =⎢ 0 0 135 s+1 ⎢ ⎢ e−90s ⎢ 0 0 ⎣ 37.26s +

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ 1⎦

(56)

Assuming that G R = Tdes , all of the conditions based on which the analyses in this paper were carried out are satisfied. Accordingly, in the next step, an effect of a poor performance output on other outputs is illustrated based on the simulation results. Fig. 6 shows the performance of 3 outputs of the system in 3 different cases:

 Case 1: No mismatch.  Case 2: 100% gain mismatch in P11.  Case 3: 100% gain mismatch in P22.

Fig. 5. Bode diagram of Sa/Sd .

In the first case, there is no mismatch between the plant and the model used in the controller. Thus, we expect to get good performance for all of the outputs. In the second case, 100% mismatch is added to the gain of P11. In the last case, 100% gain mismatch is added to P22. As we expected, in case 1, all of the outputs

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M. Yousefi et al. / Control Engineering Practice 43 (2015) 59–68

Table 2 Plant model. Output Input

Stock flow, u1

1st dryer pressure, u2

2nd dryer pressure, u3

Dry weight, y1

1.33 e−72s 66.6s + 1 0.296 −90s e 46s + 1 0.553 e−42s 154.8s + 1

–

–

−0.14 −114s e 90s + 1 −0.238 −30s e 211.2s + 1

–

Size-press moisture, y2 Reel moisture, y3

0.0555 e−90s 24.84s + 1

show good performance since the model used in the controller truly represents the plant's dynamics. But, when there is mismatch in the transfer function from the first input to the first output, the performance of the first output deteriorates. However, as proven in the previous sections, this mismatch does not affect the performance of two other outputs. Similarly, as a result of gain mismatch in P22, the performance of the size-press moisture drops, but there is no change in the performance of other outputs. Thus, the sensitivity of the performance indices of each output can be analysed individually. The size-press moisture is selected to analyse the sensitivity of its performance to model–plant mismatch. In the next step, the sensitivity of the user-specified control performance index to three different types of parametric mismatch, namely time constant, time delay and gain mismatch, are calculated, and illustrated. Fig. 7 shows that even a large mismatch in the parameters of P21 does not have a significant effect on the performance of the size-press moisture. This is because of the fact that the dominant control variable which affects the size-press moisture is u2 which is the result of our assumption that the desired transfer function is diagonal. Therefore, in the rest of this section, we analyse sensitivity of the performance index to the model mismatch in P22, assuming that the mismatch in P21 does not change the performance of the size-press moisture. Thus, the sensitivity of the performance index to the different types of mismatch only in P22 is analysed. Furthermore, the performance of the size-press moisture in two different modes, servo control and regulatory modes, is calculated and compared. 7.1. Gain mismatch The gain of P22 is changed. The model used in the controller is kept untouched, but the gain of the transfer function in the plant is changed. Then, the performance index of the size-press moisture is calculated for each gain sample and accordingly the sensitivity of the performance indices to the gain mismatch is calculated based on (54). In this case, the transfer function P22 is specified as follows:

P22 =

k nom (1 + Δk ) −τ dnom s e τnom s + 1

Fig. 6. The effect of a poor performance output on other outputs.

small mismatch in the gain of the transfer function causes a significant drop in the performance index. Fig. 9 shows the sensitivity of the index to the gain mismatch. It illustrates that the index is very sensitive to the mismatch in the gain and the sensitivity increases as the mismatch grows. Furthermore, the sensitivity of the index is almost the same for the both regulatory and servo control modes. 7.2. Delay mismatch Next, the time delay in P22 is changed and the performance index and the sensitivity of the performance to the delay mismatch Δτ d are calculated and plotted. In this case, P22 can be represented as

P22 =

k nom e−τ dnom (1 + Δτ d ) s . τnom s + 1

(58)

Figs. 10 and 11 depict the index and the sensitivity of the index to the delay mismatch for the different values of the delay mismatch, respectively. As illustrated in Fig. 10, increasing the delay mismatch degrades the performance of the system. In addition, the performance of the transient response of the system is affected more in comparison with the performance of the steady state response. This difference is clear in Fig. 11 as well. This figure shows that the index in the servo control mode is more sensitive when there is small mismatch in the time delay. In fact the small delay mismatch affects the performance of the transient response more rather than the performance of the steady state response. 7.3. Time constant mismatch

(57)

where knom, τnom and τdnom are nominal gain, time constant and time delay used in the controller, respectively and Δk is the mismatch between the gain of G22 in the plant and the model used in the controller. The following figures show the user-specified performance index and the sensitivity of the index for different values of gain mismatch. The performance index and the sensitivity of the index are calculated for the response of the system in two different modes, regulatory and servo control modes, and subsequently compared. Fig. 8 shows that the performance index falls dramatically as the mismatch increases in both modes. Indeed, the

In this part, we study the effect of the time constant mismatch on the performance of the system. In the case of time constant mismatch, P22 can be represented as follows:

P22 =

k nom e−τ dnom s τnom (1 + Δτ ) s + 1

(59)

where Δτ is the amount of mismatch in the time constant. Figs. 12 and 13 show the effect of the time constant mismatch on the userspecified performance index and the sensitivity of the index to the mismatch, respectively. According to Fig. 12, the mismatch in the time constant has no significant influence on the performance of the steady state

M. Yousefi et al. / Control Engineering Practice 43 (2015) 59–68

67

Fig. 7. The effect of different types of mismatch in P21 on the performance index.

response of the system. Nonetheless, it affects performance of the system in the servo control mode more. This is confirmed in Fig. 13 as well. This figure shows that the transient response performance of the system is more sensitive to time constant mismatch in comparison with the performance of the system in the regulatory mode. According to the figures shown in this section, for this type of system, we can conclude that if the system shows good performance in the regulatory mode and poor performance in the servo control mode, the problem would be likely due to mismatch in the time constant. This information can be used in experiment design for system identification which is the next step to improve the performance.

8. Conclusion

Fig. 8. The user-specified performance index versus the gain mismatch.

Motivated by applications to paper machines, we studied the effects of various types of model–plant mismatch on two widelyused performance indices for MIMO systems. We showed that MPM changes the closed loop behaviour and leads to performance

Fig. 9. The sensitivity of the user-specified performance index to the gain mismatch.

Fig. 10. The user-specified performance index versus the time delay mismatch.

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deterioration. In addition, we explained why different types of parametric MPM affect the output variance differently. We showed that, in general, under some mild assumptions, the performance of a selected system output will not be affected if the model mismatch is restricted to transfer functions of other outputs. We described an approach to measure the performance of systems in servo control mode instead of regulatory mode. This permits the detection of some types of mismatch, such as time constant mismatch, that do not affect the performance of a system's steady state response. Moreover, we discussed the effect of unmeasured disturbances on the performance indices and proposed a technique to distinguish the effect of disturbance on system's performance from the effect of MPM.

References

Fig. 11. The sensitivity of the user-specified performance index to the time delay mismatch.

Fig. 12. The user-specified performance index versus the time constant mismatch.

Fig. 13. The sensitivity of the user-specified performance index to the time constant mismatch.

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