Industrial Applications of a Feedback Controller Performance

Ind. Eng. Chem. Res. 2004, 43, 597-607

597

Industrial Applications of a Feedback Controller Performance Assessment of Time-Variant Processes Folake Olaleye,† Biao Huang,*,† and Edgar Tamayo‡ Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6, and Syncrude Canada Ltd., Fort McMurray, Alberta, Canada

This paper is concerned with the performance assessment problem for linear time-variant (LTV) processes. It is a continuation of the work by Huang (J. Process Control 2002, 12, 707-719) with further theoretical development and industrial applications. Systematic algorithms and procedures for the performance assessment of LTV feedback control loops are proposed. The developed algorithms are illustrated through a simulated stirred-tank heater example and, as an industrial case study, applied to a sulfur recovery unit that is under adaptive control. 1. Introduction Automatic process control has been widely used in process industries to achieve objectives that vary from maintainance of safe process operations to process optimization. The extensive control application has inspired a growing interest in the monitoring and assessment of the control-loop performance. Assessment of control loops should preferably not disturb routine operations of the processes. That is, performance monitoring should be noninvasive or at least should be done under closed-loop conditions.2 It is also required that the performance assessment algorithm should be simple and noncomplex and should require minimal process knowledge.3 The performance of control loops is often measured with respect to the response of a process to a step change in the set point (servo performance) or to a load disturbance variable (regulatory performance). Performance characteristics such as the integral of the absolute value of the error (IAE), settling time, overshoot, damping ratio, etc., are calculated and often used for monitoring purposes. This is a simple and useful method when experiments or set-point changes can be made periodically on each control loop.4 However, continuously operating processes are subject to numerous disturbances that make the controlled variable behave as a random time series. The mean square error (MSE) or variance of the process variable is commonly used as the measure for the control-loop performance. The variance (or standard deviation) is used for monitoring because of its direct relationship to process performance and profit.4-6 Significant progress has been made in the assessment of time-invariant processes or to time series that can be made stationary by some simple transformation; see, for example, work by Harris,7 Desborough and Harris,8 Kozub,9 Kendra and Cinar,10 Thornhill et al.,11 and Huang and Shah.12 The list is certainly not complete, and readers are referred to the review papers by Harris et al.2 and Qin13 for details. However, nonstationary * To whom correspondence should be addressed. Tel.: (780) 492-2971 or 9016. Fax: (780) 492-2881. E-mail: Biao.Huang@ ualberta.ca. † University of Alberta. E-mail: [email protected]. ‡ Syncrude Canada Ltd. E-mail: [email protected].

time series are often observed in the performance assessment of control loops because of varying process dynamics, change of disturbance models, nonlinearity of actuators and sensors, etc. Although there exist various performance assessment methods for timeinvariant processes, there are few results available for time-variant processes. In practice, most processes have a certain degree of time-varying behavior, and this has brought about a need to develop performance assessment methods for time-varying processes. The most intuitive extension of the performance assessment technique from linear time-invariant (LTI) processes to linear time-variant (LTV) processes is through the recursive estimation technique, also referred to as the “sequential parameter estimation” or “adaptive algorithm”.14 The recursive identification algorithm (for LTV assessment) uses information from past observations recursively by focusing on the most recent data and discounting remote past measurements exponentially. Several recursive algorithms have been proposed to estimate the control-loop performance in the presence of nonstationary characteristics in the data (for example, work by Desborough and Harris8 and Huang and Shah12). It is found that any recursive time series algorithm can be used to estimate the LTV ARMA model for the performance assessment of LTV processes. However, the potential difficulty in handling LTV operators is seen in the noncommutativity of the multiplication and/or division of these transfer functions.15 A theoretical framework for the feedback controller performance assessment of LTV processes was built by Huang.1 However, the work of Huang1 focused on the derivation of the LTV minimum-variance control law and theoretical feasibility of the LTV control performance assessment from routine operating data with illustrative application examples. This paper extends the work of Huang1 through (1) generalization of the LTV control-loop performance assessment technique by deriving expressions of both the LTV minimum-variance term and the actual variance term and subsequent calculation of the LTV performance index, which has not been achieved in the previous work and (2) detailed simulation and industrial case studies to illustrate the applicability of the LTV control performance assessment techniques in practice. The remainder of this paper is

10.1021/ie020788p CCC: $27.50 © 2004 American Chemical Society Published on Web 12/19/2003

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Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004

organized as follows: some necessary background of the LTV process is reviewed in section 2; our main results and algorithms are developed in section 3; a stirredtank heater simulation example is presented in section 4; the proposed algorithm is applied to an industrial process in section 5; concluding remarks are given in section 6. 2. Revisit of the Control-Loop Performance Assessment of LTV Processes

Figure 1. Schematic of a time-variant SISO process under feedback control.

As has been discussed by Li and Evans15 and Huang,1 the potential difficulty in handling of LTV operators is seen in the noncommutativity of the multiplication and/ or division of these transfer functions. This is illustrated in the multiplication of two LTV polynomials, u(q-1,t) and v(q-1,t) in the backshift operator q-1:

The multiplication of u′(q-1,t) and v′(q-1,t) is given by -1

-1

u′(q ,t) v′(q ,t) ) )

u(q-1,t) ) u0(t) + u1(t) q-1 + ... + un(t) q-n v(q-1,t) ) v0(t) + v1(t) q-1 + ... + vm(t) q-m

(1)

The multiplication of u(q-1,t) and v(q-1,t) is given by

u(q-1,t) v(q-1,t) )

n

∑ ∑ i)0 j)0 n

)

m

ui(t) q-ivj(t) q-j

-1

v(q ,t) u(q ,t) ) )

u(q-1,t)

v(q-1,t)

m

n

m

ui′(t) q-ivj′(t) q-j ∑ ∑ i)0 j)0 ui′(t) vj′(t) q-(i+j) ∑ ∑ i)0 j)0

(5)

The multiplication of v′(q-1,t) and u′(q-1,t) is given by

v′(q-1,t) u′(q-1,t) ) )

m

n

m

n

vj′(t) q-jui′(t) q-i ∑ ∑ j)0 i)0 vj′(t) ui′(t) q-(i+j) ∑ ∑ j)0 i ) 0

(6)

m

ui(t) vj(t-i) q-(i+j) ∑ ∑ i)0 j)0

(2)

The multiplication of v(q-1,t) and u(q-1,t) is given by -1

n

m

n

m

n

vj(t) q-jui(t) q-i ∑ ∑ j)0 i)0 vj(t) ui(t-j) q-(i+j) ∑ ∑ j)0 i)0 v(q-1,t)

(3)

u(q-1,t).

Hence, * The multiplication of u(q-1,t) and v(q-1,t) in eqs 2 and 3 is referred to as normal multiplication of the LTV polynomials. This type of multiplication causes a time delay in the LTV operators. Normal multiplication is, therefore, said to be noncommutative, and this result shows that care has to be taken in the noncommutativity involved in manipulation of LTV transfer functions. This property of LTV polynomials is important and has to be taken into consideration in calculating the minimum-variance term when the LTV ARMA model, for example, is transferred to an LTV MA model. However, unlike normal multiplication, pointwise multiplication does not cause any time delay in the multiplication and/or division of the LTV polynomials, as is illustrated in the following. For two LTV polynomials, u′(q-1,t) and v′(q-1,t) in the backshift operator are

u′(q-1,t) ) u0′(t) + u1′(t) q-1 + ... + un′(t) q-n v′(q-1,t) ) v0′(t) + v1′(t) q-1 + ... + vm′(t) q-m (4)

From eqs 5 and 6, it can be seen that, for pointwise multiplication, u′(q-1,t) v′(q-1,t) ) v′(q-1,t) u′(q-1,t). Thus, pointwise multiplication is said to be commutative. It should be noted that pointwise multiplication may yield erroneous results if the plant or disturbance dynamics has a relatively fast parameter change (as would be seen in the examples). Therefore, normal multiplication, which is noncommutative, is recommended when handling LTV operators because the normal multiplication is the only theoretically correct solution to LTV processes.15,1 3. Control-Loop Performance Assessment of LTV Processes Consider the LTV single-input single-output (SISO) process shown in Figure 1:

˜ (q-1,t) ut + N(q-1,t) at yt ) q-dT

(7)

where d is the process time delay, T ˜ (q-1,t) is the delayfree LTV plant transfer function, N(q-1,t) is the LTV disturbance transfer function, and at is a white-noise sequence with zero mean and variance, σa2(t). By application of time-series analysis to the routine operating data, the LTV closed-loop SISO response can be expressed as an ARMA model:1

Acl(q-1,t) yt ) Ccl(q-1,t) at

(8)

The LTV ARMA model can be transferred to the LTV MA model and obtained as

yt ) [f0(t) + f1(t) q-1 + f2(t) q-2 + ... + fd-1(t) q-(d-1) + fd(t) q-d + ...]at (9)

Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 599

It has been shown by Huang1 that the closed-loop response under time-variant minimum-variance control consists of the first d terms of the MA model

yt|mv ) [f0(t) + f1(t) q-1 + f2(t) q-2 + ... + fd-1(t) q-(d-1)]at (10) The LTV minimum variance can be calculated as 2

2

2

2

2

σmv (t) ) [f0 (t) + f1 (t) + ... + fd-1 (t)]σa (t) (11) Note that in the calculation of the minimum-variance term for LTV processes the noncommutativity associated with LTV operators should be taken into account in transferring the LTV ARMA model to the LTV MA model by using normal multiplication rather than pointwise multiplication or division. The above procedure can be used to calculate the time-variant minimum-variance term for any model structure with any order. The complexity is, however, increased with an increase of the model order. In the following sections, we shall focus on the derivation of the minimum-variance terms for two special, yet most frequently used, model structures, namely, AR(4) and ARMA(2,2). 3.1. Calculation of the LTV Minimum Variance for the AR Model. Let us consider that the process output, yt, is represented by an LTV AR model of order 4, which is a default choice in most applications16 and in the MATLAB System Identification toolbox for example: -1

Acl(q ,t) yt ) at

(12)

where

yt ) [f0(t) + f1(t) q-1 + f2(t) q-2 + f3(t) q-3 + ...]at (13) Acl(q-1,t) ) 1 + R1(t) q-1 + R2(t) q-2 + R3(t) q-3 + R4(t) q-1 (14) Substituting eq 13 into eq 12 yields

Acl(q-1,t) [f0(t) + f1(t) q-1 + f2(t) q-2 + f3(t) q-3 + ...]at ) at (15)

{

It follows from eq 16 that

f0(t) ) 1 f1(t) ) -R1(t) f2(t) ) R1(t) R1(t-1) - R2(t) f3(t) ) R1(t) R2(t-1) - R1(t) R1(t-1) R1(t-2) + R1(t-2) R2(t) - R3(t) l (17)

That is,

yt ) {1 - R1(t) q-1 + [R1(t) R1(t-1) - R2(t)]q-2 + [R1(t) R2(t-1) - R1(t) R1(t-1) R1(t-2) + R1(t-2) R2(t) - R3(t)]q-3 + ...}at (18) However, direct long division of eq 12 gives

yt ) {1 - R1(t) q-1 + [R12(t) - R2(t)]q-2 + [2R1(t) R2(t) - R13(t) - R3(t)]q-3 + ...}at (19) It can be seen that, unlike the result obtained with normal multiplication or division of LTV polynomials, pointwise multiplication does not cause any time delay in the coefficients of the moving-average model as seen in eq 19 and is incorrect. As an example, from eq 17, the appropriate minimum variance for the LTV process with a time delay of 3 can be calculated as

σmv2(t) ) [1 + f12(t) + f22(t)]σa2(t) ) {1 + R12(t) + [R1(t) R1(t-1) - R2(t)]2}σa2(t) (20) 3.2. Calculation of the Actual Time-Variant Variance for the AR Model. To calculate a time-variant performance index, the actual time-variant variance has to be calculated. An algorithm based on the LTV AR(4) model is discussed in this section. The LTV AR model in eq 12 can be expressed as

yt + R1(t) yt-1 + R2(t) yt-2 + R3(t) yt-3 + R4(t) yt-4 ) at (21) The Ri(t)’s represent the parameters, while at is the white-noise sequence. Both sides of eq 21 are multiplied by yt, yt-1, yt-2, yt-3, and yt-4, respectively, and the expectations are taken to obtain

γ0 + R1(t) γ1 + R2(t) γ2 + R3(t) γ3 + R4(t) γ4 ) σa2(t) γ1 + R1(t) γ0 + R2(t) γ1 + R3(t) γ2 + R4(t) γ3 ) 0

The LTV impulse response coefficients are obtained by equating coefficients on the right- and left-hand sides of eq 15:

{

γ2 + R1(t) γ1 + R2(t) γ0 + R3(t) γ1 + R4(t) γ2 ) 0 γ3 + R1(t) γ2 + R2(t) γ1 + R3(t) γ0 + R4(t) γ1 ) 0

f0(t) ) 1 f1(t) ) -R1(t) f0(t-1) f2(t) ) -R1(t) f1(t-1) - R2(t) f0(t-2) f3(t) ) -R1(t) f2(t-1) - R2(t) f1(t-2) - R3(t) f0(t-3) fk(t) ) -R1(t) fk-1(t-1) - R2(t) fk-2(t-2) R3(t) fk-3(t-3) - R4(t) fk-4(t-4) k > 3 (16)

γ4 + R1(t) γ3 + R2(t) γ2 + R3(t) γ1 + R4(t) γ0 ) 0 (22) The γi’s in eq 22 represent the time-variant autocovariance of the process variable, y, with lag i, and σa2(t) is the variance of the white noise (or shock). The equations are reorganized in matrix format and solved to obtain the required time-variant process variance, σy2, which is denoted by γ0.

600


That is,

yt ) {1 + [c1(t) - R1(t)]q-1 + [c2(t) - R1(t) c1(t-1) + R1(t) R1(t-1) R2(t)]q-2 + ...}at (31) From eq 23, the γi’s are calculated as

B ) A-1Cσa2(t)

However, direct long division gives a different result:

(24)

The first element of the vector B represents the process variance, σy2. The control-loop performance can be determined by comparing the minimum variance with the process variance. Note that σy2 is also time-variant and eq 24 calculates the variance at time instant t. 3.3. Calculation of the Time-Variant Minimum Variance for the ARMA Model. A more general representation of a time series is the ARMA model. It is known that ARMA(2,2) is a typical representation of the time-series process and most physical processes can be well fitted by this model. The calculation of the timevariant minimum variance and actual variance for an ARMA model is more involved than that for the AR model. The calculation of the time-variant minimum variance for this model is illustrated next. Consider

Acl(q-1,t) yt ) Ccl(q-1,t) at

(25)

where

yt ) [f0(t) + f1(t) q-1 + f2(t) q-2 + f3(t) q-3 + ...]at (26) Acl(q-1,t) ) 1 + R1(t) q-1 + R2(t) q-2 Ccl(q-1,t) ) 1 + c1(t) q-1 + c2(t) q-2

(27)

R2(t) c1(t)]q-3 + ...}at (32) As an example to calculate the minimum-variance term, from eq 29 the appropriate minimum-variance term for a time delay of 3 can be calculated as

σmv2(t) ) {1 + [c1(t) - R1(t)]2 + [c2(t) - R1(t) c1(t-1) + R1(t) R1(t-1) - R2(t)]2}σa2(t) (33) 3.4. Calculation of the Actual Time-Variant Output Variance for the ARMA Model. The LTV ARMA(2,2) model can further be expressed as

yt + R1(t) yt-1 + R2(t) yt-2 ) at + c1(t) at-1 + c2(t) at-2 (34) Both sides of eq 34 are multiplied by yt, yt-1, and yt-2, respectively, and the expectations are taken to obtain

γ0 + R1(t) γ1 + R2(t) γ2 ) R1(t) c1(t) c2(t) + R12(t) c2(t)]σa2(t)

[c1(t) + c1(t) c2(t) - R1(t) c2(t)]σa2(t)

f3(t) q-3 + ...]at ) Ccl(q-1,t) at (28) The LTV impulse response coefficients are obtained by equating coefficients on the right- and left-hand sides of eq 28 after substituting eq 27 into eq 28:

{

[-R1(t) c2(t) + 2R1(t) R2(t) + R1(t)2 c1(t) - R1(t)3 -

γ1 + R1(t) γ0 + R2(t) γ1 )

Acl(q-1,t) [f0(t) + f1(t) q-1 + f2(t) q-2 +

{

[c2(t) - R1(t) c1(t) + R1(t)2 - R2(t)]q-2 +

[1 - R1(t) c1(t) + c12(t) + c22(t) - R2(t) c2(t) -

Substituting eq 26 into eq 25 yields

f0(t) ) 1 f1(t) ) c1(t) - R1(t) f0(t-1) f2(t) ) c2(t) - R1(t) f1(t-1) - R2(t) f0(t-2) fk(t) ) -R1(t) fk-1(t-1) - R2(t) fk-2(t-2)

yt ) {1 + [c1(t) - R1(t)]q-1 +

γ2 + R1(t) γ1 + R2(t) γ0 ) c2(t) σa2(t)

(35)

In a format similar to that with the LTV AR model, the equations are reorganized and solved to obtain the required time-variant process variance, σy2, which is denoted by γ0.

k>2 (29)

It follows from eq 29 that f0(t) ) 1 f1(t) ) c1(t) - R1(t) f2(t) ) c2(t) - R1(t) c1(t-1) + R1(t) R1(t-1) - R2(t) f3(t) ) -R1(t) c2(t-1) + R1(t) R2(t-1) + R1(t) R1(t-1) c1(t-2) R1(t) R1(t-1) R1(t-2) - R2(t) c1(t-2) + R2(t) R1(t-2) l

(30)

The actual variance γ0 is solved from eq 36, and the control-loop performance is determined by comparing the minimum variance with the process variance. So far, we have been able to show how to estimate the control-loop performance for LTV AR and LTV ARMA models. From the analysis, it can be seen that the proposed methodology for the performance assessment of LTV processes can be obtained from routine


and some parameters for the process are given by Bequette.17

Fs ) 1.0 ft3/min, FCp ) 61.3 Btu/°F‚ft3, V ) 10 ft3, Tis ) 50 °F, Ts ) 125 °F Fjs ) 1.5 ft3/min, FjCpj ) 61.3 Btu/°F‚ft3, Vj ) 1 ft3, Tjis ) 200 °F, Tjs ) 150 °F Figure 2. Jacketted stirred-tank heater.

Linearizing and applying Laplace transform yield

operating data and the results can be extended to higher-order models following the same procedure if necessary.

T ˜ (s) )

4. Evaluation via Simulation In this section, we consider a stirred-tank heater shown in Figure 2. The stirred-tank heater is an example of a mixing vessel that is heated by a jacket surrounding the vessel. A mixing vessel may serve as a chemical reactor, where two or more components are reacted under certain conditions to produce one or more products. The reaction often occurs at a certain temperature to achieve the desired yield. In this example, saturated steam is the heat-transfer medium that is circulated through the jacket to heat the fluid in the tank. The assumptions made in writing the dynamic modeling equations to find the tank temperature include the following: (1) The volume and liquids have constant density and heat capacity. (2) Perfect mixing is assumed in both the tank and jacket. (3) The temperature of the saturated steam is constant throughout the jacket. (4) The flow rate of the saturated steam is timevarying, and this causes the heat-transfer coefficient U to be time-varying. (5) The tank inlet flow rate Fi, tank outlet flow rate F, jacket flow rate Fj, tank inlet temperature Ti, and jacket inlet temperature Tji vary with time. When the work done by the impeller is neglected, energy balance around the tank is used to obtain the modeling equation given by

Q dT F ) (Ti - T) + dt V VFCp

where the rate of heat transfer from the jacket to the tank, Q, is given by

Q ) UA(Tj - T)

Assuming that the temperature measurement has a time delay of 4, eq 39 yields

T ˜ (s) )

(38)

T is the tank temperature, F is the volumetric flow rate, F is the density, Cp is the heat capacity, U is the overall heat-transfer coefficient, and A is the area for heat transfer. The subscripts i, j, and ji denote inlet, jacket, and jacket inlet, respectively. In linearization of the nonlinear model in eq 37, it is assumed that the tank outlet flow rate, F, and the tank temperature, T, are the manipulated and controlled variables, respectively. The overall heat-transfer coefficient U is time-varying, and Ti is considered as the disturbance affecting the system. The steady state is obtained by solving the dynamic equation for dT/dt ) 0. The steady-state values of the system variables

-7.5e-4s F ˜ (s) + s + (0.1 + 0.00163u j) 0.1 T ˜ (s) (40) s + (0.1 + 0.00163u j) i

where u j ) UA. Equation 40 can be written in the general form

y(s) )

K ke-ds u(s) + D (s) τs + 1 τs + 1 0

(41)

A continuous-time transfer function with the form

y(s) )

k x(s) τs + 1

(42)

may be discretized as

yn ) e-Ts/τyn-1 + (1 - e-Ts/τ)kxn-1

(43)

Thus, eq 40 can be expressed as

T ˜ (s) ) (37)

1/V(Tis - Ts) F ˜ (s) + s + Fs/V + UA/FVCp Fs/V T ˜ (s) (39) s + Fs/V + UA/FVCp i

j) -7.5e-4s/(0.1 + 0.00163u F ˜ (s) + 1 s+1 0.1 + 0.00163u j 0.1/(0.1 + 0.00163u j) T ˜ i(s) (44) 1 s+1 0.1 + 0.00163u j

(

)

(

)

and with a sampling time of 1 unit, eq 44 can be discretized as

T(t) ) q

1 7.5(e-(0.1+0.00163uj ) - 1) 0.1 + 0.00163u j -4 1 - e-(0.1+0.00163uj )q-1 0.1 (1 - e-(0.1+0.00163uj )) 0.1 + 0.00163u j 1 - e-(0.1+0.00163uj )q-1

F(t) +

Ti(t) (45)

Let us consider that UA is time-varying and is given by UA ) 183.9[1 + 0.5 sin(t/x)] for illustration purposes, where x/2π is the oscillation period. Thus, eq 45 can be written as

602


T(t) ) (5.03e-0.15 sin(t/x) - 7.5)/[0.4 + 0.15 sin(t/x)] F(t) + q-4 1 - 0.67e-[0.15 sin(t/x)]q-1 (0.1 - 0.067e-0.15 sin(t/x))/[0.4 + 0.15 sin(t/x)] Ti(t) (46) 1 - 0.67e-[0.15 sin(t/x)]q-1 From eq 46, it can be seen that both the process model and the disturbance model are time-variant. In this example, it is assumed that the disturbance has three different time-variant dynamics from relatively slow parameter change to relatively fast parameter change. This time-varying nature is induced by the time-varying steam flow rate for example. This illustration is chosen to compare the performance monitoring methodology (which takes noncommutativity associated with LTV transfer functions into account) with the conventional performance assessment algorithm. That is, the minimum-variance term is calculated using normal multiplication (noncommutative) and pointwise multiplication (commutative), and the differences between the two methods are compared for each of the disturbance dynamics. Although this example is only for illustration purposes as an LTV process, with a small perturbation to the process in the simulation, the linearized timevariant model should be fairly close to the actual process. Assuming that Ti(t) is a random white-noise disturbance representing the driving force of the unmeasured disturbances, then the process model in eq 46 can further be expressed as

φ(t) υ(t) u + at yt ) q-4 -1 t 1 - δ(t) q 1 - δ(t) q-1

(47)

where yt is the process variable and ut is the manipulated variable. The time-variant process and disturbance dynamics are given by

{

φ(t) ) (5.03e-0.15 sin(t/x) - 7.5)/[0.4 + 0.15 sin(t/x)] υ(t) ) (0.1 - 0.067e-0.15 sin(t/x))/[0.4 + 0.15 sin(t/x)] δ(t) ) 0.67e-0.15 sin(t/x) (48)

A proportional-integral (PI) controller is used to control the process and is given by

Q(q-1) )

-0.05 + 0.045q-1 1 - q-1

(49)

Three cases of time-varying dynamics are being considered in ascending order of increasing parameter-varying rate:

case 1:

x ) 10

case 2:

x)1

case 3:

x ) 0.5

(50)

The simulation results in Figure 3 show a comparison of the difference between normal multiplication (solid line) and pointwise multiplication (dotted line). It can be observed that the difference between the minimumvariance terms calculated using the normal multiplication and pointwise multiplication increases as the

Figure 3. Comparison of the time-variant minimum-variance term using normal multiplication and pointwise multiplication.

parameter-varying rate increases from the top to the bottom subplot. This result shows that it is important to use normal multiplication rather than pointwise multiplication in the estimation of the minimum-variance term for time-varying processes. 5. Case Study on the Adaptive Control of a Sulfur Recovery Unit (SRU) 5.1. Process Description. The proposed performance assessment methodology is applied to monitor the control-loop performance of an adaptive controller in a SRU in Syncrude Canada Ltd. The purpose of the SRU is to extract elemental sulfur from the hydrogen sulfide (H2S) component of the acid gas stream obtained as byproducts of plant operations. A simplified schematic of the SRU is shown in Figure 4. 5.2. Adaptive Controller. The adaptive controller is used to control the tail gas ratio effectively by maintaining the appropriate set point for the trim air controller in order to achieve optimal performance. The adaptive controller updates its process model automatically and continuously as required to maintain optimal control of the process. This controller increases the unit efficiency because it is able to handle the long time delay inherent in the process, and it can handle the small variations in the acid gas flow/composition (i.e., feed changes). The adaptive nature of the controller makes the control loop time-variant. The effect of this time-variant nature will appear as the nonstationarity of the closedloop data. Therefore, the proposed time-variant performance assessment algorithm is used to estimate the control-loop performance in the presence of nonstationary characteristics in the data. It involves a systematic solution where the noncommutativity problem associated with time-variant processes is taken into consideration. 5.3. Data Analysis. The time delay of the process including a zero-order hold from a priori analysis is known to be no less or approximately two samples. The sampling interval is 1 min, and a sample size of 2076 data points collected over a 3-day period is used in this analysis. The data are assumed to contain a representative sample of normal process operations. The plot of


Figure 4. Schematic of the SRU.

Figure 5. Plot of the operating data for the adaptive controller in the SRU.

the operating data for the adaptive controller is shown in Figure 5. In addition to the minimum-variance control benchmark used for the performance assessment, other indicators of the control-loop performance are also considered, and these include the closed-loop impulse response and autocorrelation function (ACF) of the output error. An impulse response function curve represents the closed-loop impulse response between the whitened disturbance sequence and the process output, and it is a direct measure of how well the controller is performing in rejecting disturbances or tracking set-point changes.8,12 The ACF of the output error is an approximate measure of how close the existing controller is to the minimumvariance condition. The minimum-variance control performance has been achieved if the ACF decays to zero beyond d - 1 lags, where d is the time delay of the process. The rate at which the autocorrelation tends to zero beyond d - 1 lags indicates the closeness of the existing controller to the minimum-variance condition. It is worthwhile to point out that the impulse response curve or ACF only applies to time-invariant processes. However, both control-loop performance measures are

straightforward to calculate using process data and are therefore used as an initial, approximate estimation of the performance by considering the control loop as timeinvariant. After the preliminary tests, the proposed time-variant performance assessment methodology is used to estimate the performance index of the timevariant control loop. The time-variant impulse response coefficients for the AR model (in eqs 18 and 19) and ARMA model (in eqs 31 and 32) show that the first two terms of the movingaverage processes using pointwise and normal multiplication are the same. It has been discussed that the minimum-variance term consists of the first d terms of the moving-average model. Hence, there will be no difference in the calculation of the minimum-variance term for a time delay of 2 or less between the pointwise multiplication method and the normal multiplication method. However, the exact time delay is not known, and time delay may also be time-varying. The more effective method to account for such time delay uncertainty is to use the extended horizon prediction method;2 that is, we need to calculate the performance index as a function of time delay. With this consideration, the difference between normal multiplication and pointwise

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in assessing the performance of the control loop, and the results are compared with the LTV performance assessment algorithm. Figure 7 shows the performance measure of the control loop using a minimum-variance control benchmark. By visualization of this plot, it can be seen that the controller performance is good and exhibits a performance close to minimum-variance control most of the time. The controller has a performance index that is greater than 0.7, approximately 60% of the sampled data. Figure 8 shows the relative difference (RD) between the estimated performance indices using the LTV performance assessment method and the moving-windowbased method (a conventional method to handle timevarying control loops; no forgetting factor is used in the estimation of the parameters). The formula for calculating the RD is given as follows:

Figure 6. ACF.

multiplication will show up. In this case study, we shall also compare the difference between the LTV assessment methodology (normal multiplication) and the traditional assessment method (pointwise multiplication) over a range of time delays. Figure 6 shows the results for the autocorrelation test where a rather smooth autocorrelation plot is observed. Although it has not achieved minimum-variance control performance, the response is fairly fast (almost settles down in about four samples). It is also observed that the closed-loop impulse response shows a similar response with smooth decay to zero. This result confirms a good performance of the adaptive controller. Further analysis is done on the LTV process by calculating the time-variant performance index using the proposed performance assessment algorithm. The plot of the performance index estimated over the 3-day time period is given in Figure 7. Also, a moving-window size of 50 (with an overlap of 49 data points) using the conventional pointwise multiplication approach is used

RD )

|NM result - PM result| |PM result|

(51)

where NM represents normal multiplication and PM represents pointwise multiplication with a data moving window. Figure 8 clearly reveals that the RD between these two methods could be up to 100%. Hence, the windowbased method is unlikely to produce correct estimates of the performance index of a control loop with timevariant disturbance and/or process dynamics. This result shows that it is important to use the proposed LTV performance assessment algorithm in assessing the performance measure for time-varying processes. Because it is known that there is a fundamental incorrectness in pointwise multiplication for the LTV process, it is always better and safe to use normal multiplication in the analysis of variance for timevariant processes. However, it can be seen that if the parameter-varying rate is slow, the difference between normal multiplication and pointwise multiplication with the forgetting factor based recursive algorithm may not be significant, as will be seen in the following.

Figure 7. Estimation of the performance index for the adaptive controller in the SRU.


Figure 8. RD between performance indices using LTV assessment methodology and a window-based method.

Figure 9. RD between the performance indices using normal multiplication and pointwise multiplication.

The RD between the performance indices using the LTV assessment methodology (recursive parameter estimation with a forgetting factor plus normal multiplication) and the traditional recursive assessment method (recursive parameter estimation with a forgetting factor plus pointwise multiplication) for time delays of 4 is shown in Figure 9. Figure 9 shows the RD between the LTV assessment technique and the traditional recursive method. The difference has not been very significant because of a relatively stationary response of the process under the adaptive control. The difference becomes more significant for a set of more unstationary data, as will be seen next. 5.4. Analysis of the Proportional-IntegralDerivative (PID) Controller of the LTV Process. The proposed LTV performance assessment technique is also applied to analyze the control-loop performance of the PID controller that was used before the installation of the adaptive controller in the SRU (described in section 5.1). This analysis is used to show the

difference between the proposed LTV performance assessment technique and the traditional assessment methodology when the process data are more unstationary. A total of 740 data points are used for this case study. The sampling interval for the data is 1 min. Figure 10 shows the plot of the operating data, which can roughly be divided into three sections according to the observed data trend. The plot reveals that an abnormal disturbance was affecting the process for a time period in the second section and the process and/or disturbance dynamics is clearly time-varying. The first section represents the data sampled from time 0 to 179 min, the second section is between 180 and 400 min, while the third data set is sampled from 401 to 740 min. Figure 11 shows the plots of the RD between the performance indices using the LTV assessment methodology (normal multiplication) and the traditional recursive assessment method (pointwise multiplication) for time delays of 4 and 6. Figure 11a shows that the

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Figure 10. Operating data for the PID controller.

Figure 11. RD between the performance indices using normal multiplication and pointwise multiplication for the PID controller.

RD between the LTV assessment technique and the traditional method can be up to 50%, and the difference increases up to 75% with a larger time delay of 6 as shown in Figure 11b. It is interesting to note that the

difference between the two methods is clearly revealed in the 2nd section of data where the abnormal disturbance is seen to be affecting the process. This result shows that it is important to take noncommutativity


into account in the manipulation of LTV operators especially when the parameter-varying rate is relatively fast. 6. Conclusion The technique for evaluating the benchmark for the control-loop performance assessment of time-variant processes under time-variant or adaptive control has been discussed in this paper. The proposed method provides a way to monitor the control-loop performance of time-variant processes by taking the noncommutativity associated with LTV systems into account in calculating both the minimum-variance term and the process variance. Algorithms have been developed in this paper for the monitoring of LTV processes and are being tested in the plant online. The proposed performance monitoring method has been illustrated through a simulated example and demonstrated by an industrial application. As has been discussed by many researcher and practicing engineers, routine control-loop monitoring is gaining increasing attention. New control-loop performance assessment technologies will permit automated and repeated monitoring of the design, tuning, and upgrading of the control loops. Poor design, tuning, or upgrading of the control loops will be detected, and continuous performance monitoring will indicate which loops should be retuned or which loops have not been effectively upgraded when changes in the disturbances, in the process, or in the controller occur. The proposed LTV performance assessment methodology can be used for timely detection of unwanted control-loop variability. This can give control engineers insight into focusing control retuning and maintenance efforts on such control loops with poor performance. Acknowledgment The financial support from NSERC and Syncrude Canada Ltd. for this work is greatly acknowledged. B.H. also acknowledges the support from Alexander von Humboldt Foundation during the final preparation of this manuscript.

Literature Cited (1) Huang, B. Minimum Variance Control and Performance Assessment of Time-Variant Processes. J. Process Control 2002, 12, 707-719. (2) Harris, T. J.; Seppala, C. T.; Desborough, L. D. Review of Performance Monitoring and Assessment Techniques for Univariate and Multivariate Control Systems. J. Process Control 1999, 9 (1), 1-17. (3) Horch, A. Condition Monitoring of Control Loops. Ph.D. Thesis, Royal Institute of Technology, Sweden, 2000. (4) Stanfelj, N.; Marlin, T.; MacGregor, J. Monitoring and Diagnosing Process Control Performance: The Single Loop Case. Ind. Eng. Chem. Res. 1993, 32 (2), 301-314. (5) Bozenhardt, H.; Dybeck, M. Estimating Savings from Upgrading Process Control. Chem. Eng. Sci. 1986, 93 (3), 99-102. (6) Marlin, T.; Barton, G.; Brisk, M. Advanced Process Control Applications; ISA: Research Triangle Park, NC, 1987. (7) Harris, T. Assessment of Closed Loop Performance. Can. J. Chem. Eng. 1989, 67, 856-861. (8) Desborough, L.; Harris, T. Performance Assessment Measures for Univariate Feedback Control. Can. J. Chem. Eng. 1992, 70 (6), 1186-1197. (9) Kozub, D. Controller Performance Monitoring and Diagnosis: Experience and Challenges; AIChE Symposium Series 316; AIChE: New York, 1993; p 83. (10) Kendra, S. J.; Cinar, A. Controller Performance Assessment by Frequency Domain Techniques. J. Process Control 1997, 7 (3), 181-194. (11) Thornhill, N. F.; Oettinger, M.; Fedenczuk, P. Refinerywide Control Loop Performance Assessment. J. Process Control 1999, 9 (2), 109-124. (12) Huang, B.; Shah, S. L. Control Loop Performance Assessment: Theory and Applications; Springer-Verlag: London, 1999. (13) Qin, S. J. Control Performance MonitoringsA Review and Assessment. Comput. Chem. Eng. 1998, 23 (2), 173-186. (14) Ljung, L.; Soderstrom, T. Theory and Practice of Recursive Identification; MIT Press: London, England, 1983. (15) Li, Z.; Evans, R. J. Minimum-Variance Control of Linear Time-Varying Systems. Automatica 1997, 33 (8), 1531-1537. (16) Ljung, L. System IdentificationsTheory For the User; PTR Prentice Hall: Englewood Cliffs, NJ, 1999. (17) Bequette, B. W. Process Dynamics, Modeling, Analysis and Simulation; Prentice Hall: Englewood Cliffs, NJ, 1998.

Received for review October 4, 2002 Revised manuscript received April 28, 2003 Accepted November 11, 2003 IE020788P

Industrial Applications of a Feedback Controller Performance

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