Closed-Loop Identification of Second-Order Plus Time Delay (SOPTD

Article pubs.acs.org/IECR

Closed-Loop Identification of Second-Order Plus Time Delay (SOPTD) Model of Multivariable Systems by Optimization Method C. Rajapandiyan and M. Chidambaram* Department of Chemical Engineering, Indian Institute of Technology, Madras, Chennai 600036, India ABSTRACT: The closed-loop identification of second-order plus time delay (SOPTD) transfer function models of multivariable systems is presented based on optimization method using the combined step-up and step-down responses. The need for combined step up and step down changes in the set point is brought out for the convergence of the model parameters. A standard nonlinear least-squares optimization method is used to obtain the parameters of the SOPTD model transfer function matrix by minimizing the sum of squared errors between the closed-loop responses of the model and the actual process responses. A simple method is proposed to obtain the initial guess values of SOPTD transfer function model parameters from the main and interaction responses of the actual process. This method was applied to two-input two-output (TITO) second-order plus time delay (SOPTD), higher order and 3 × 3 SOPTD transfer function models of multivariable systems. The proposed method considerably reduces the computational time for the optimization for 2 × 2 SOPTD model systems (11 min and 11 s) when compared with the genetic algorithm (GA) method reported by Viswanathan et al. (Ind. Eng. Chem. Res. 2001, 40, 2818− 2826).

1. INTRODUCTION Most chemical processes are multi-input multi-output systems. The main characteristic of multivariable systems is the presence of interactions among the loops. Due to the presence of interactions between the process inputs and outputs, each loop is affected by other loops. The control of such process becomes more complicated. The multivariable system transfer function matrix models are required to design decentralized/centralized/ decoupled proportional−integral (PI)/ proportional−integral− derivative (PID) controllers.2 Some chemical processes (underdamped behavior or higher-order dynamics) may not be well represented by first-order plus time delay (FOPTD) models. However, second-order plus time delay (SOPTD) model approximation is effectively represented for most chemical processes compared to that of a FOPTD model approximation. Process identification is classified into open-loop identification and closed-loop identification. For the open-loop test, an excitation is introduced in the each input variable of the process, one at a time to get the output responses. Using these output responses, the transfer function matrix is to be identified. Since there is no interaction in this test, the identification of multivariable system is easier. The open-loop identification is simple, but the test is sensitive to the disturbances and it is not applicable for unstable and integrating systems. In the closed-loop identification method, an excitation signal (known magnitude of step/pulse) is introduced in each of the set points of the closed-loop system and the output response of the actual process is obtained. The transfer function matrix model is obtained using the main and interaction responses. The effect of interactions among the loops cannot be neglected in the closedloop identification method. The closed-loop test reduces the effect of disturbances on the estimated model parameters. The model parameters obtained by the closed-loop identification method fit better with the actual process compared to those of open-loop identification.3−5 Further, the controller designed © 2012 American Chemical Society

from the model obtained by the closed-loop identification shows better performance compared to that of the controller designed from the model obtained from open-loop identification.3−5 The closed-loop identification method is classified into three methods:6 direct method, indirect method, and joint inputoutput method. In the direct method, the process input (additional measurement) and output data are used and the identification is carried out similar to that of the open-loop process identification. Sung and Lee7 extended the direct method for a single input-single output (SISO) system to a multiple input-single output (MISO) system. The model parameters for the deterministic part are identified by nonlinear least-squares (Levenberg−Marquardt) method. Sung et al.8 suggested that initial guess values of the parameters can be obtained if possible by the physical insight. Otherwise, the continuous time autoregressive with exogenous input (ARX) model can be identified first, and then, a model reduction method used to obtain the guess values. This method introduces overparameterization. Methods for identifying the lower order model directly are preferable to this approach. In closed-loop identification test, generally two choices are available for introducing the excitation signal in to the system: one is in the process input (manipulated variable) and the second is in the set point (reference). Generally, in multiple inputmultiple output (MIMO) systems, the excitation is introduced into all reference signals simultaneously or sequentially, and then, the obtained data is used for identification test. The analysis of with or without excitation of reference signals on the accuracy of model parameters was conducted by Miskovic et al.,9 who reported that the nonexcitation of one or several references Received: Revised: Accepted: Published: 9620

December 25, 2011 March 7, 2012 June 1, 2012 June 1, 2012 dx.doi.org/10.1021/ie203003p | Ind. Eng. Chem. Res. 2012, 51, 9620−9633

Industrial & Engineering Chemistry Research

Article

least IAE values also, some deviations are found among the time constant and the damping coefficient (similar to what we get in any model reduction technique). They provided success rate, in which how many times they obtained the global minima (exact model parameters) or local accepted minima are not clearly specified. It is to be inferred that attainment of global minimum (in SOPTD model identification test) is difficult to get from the step response. The computational time given is for one trial. The computational time is higher (about 20−50 h) considering together all the 10 trials. They have presented the upper and lower bounds for the parameters in the GA arbitrarily for each of the simulation examples. For the limit values of process gains, Viswanathan et al.1 used a method reported by Papastathopoulou and Luyben,24 and it requires the process input data also for obtaining the guess values of steady state gains. Hence, there is a need to propose a method for obtaining reasonable initial guess values for the model parameters. Rajapandiyan and Chidambaram25 proposed a method for getting the initial guess values of FOPTD model parameters using only the closed-loop step responses of multivariable systems. The computational time for the time domain identification is shown to be reduced significantly (from 1 h 29 min (for GA) to 4 min 17 s for the TITO system). The present work extends their method to identify the SOPTD models and brings the need for step up and step down excitation in the set points to obtain the parameter convergence with significantly reduced computational time. The number of parameters to be estimated increases (from 12 to 16 parameters for TITO systems) as the model order is changed from FOPTD to SOPTD. This poses problems in the convergences of the optimization method. The present method differs from the conventional step test. In the identification method, not only are the dynamics of the step (up) change considered but also the dynamics by removing the step change after the system reaches steady-state conditions (combined step-up and step-down) are considered. For any optimization method, the selection of the initial guess values plays an important role in computational time and convergence. In the present work, a simple and generalized method is proposed for obtaining the reasonable initial guess values for the SOPTD transfer function models parameters. A method to obtain the upper and lower bounds to be used in the optimization routine is also presented. This method is applied to (2 × 2) SOPTD, higher order, and (3 × 3) SOPTD transfer function matrix models of multivariable systems.

almost always impairs the quality of the parameter estimates. By introducing the additional excitations in reference, it improves the accuracy of the model parameters. Ljung10 reported that if persistently exciting the external signal (such as set point changes) applied to the system only the data collected from the closed-loop test have enough information (required for the identification test). Bazanella et al.11 and Bombois et al.12 reported that the identification of multivariable system is possible without any excitation, when the controller is sufficiently complex. In such case, the information for the estimation of parameters is obtained by the excitation due to noise. A large amount of data (for the without excitation case) is required to estimate the parameters to get a prescribed level of accuracy. Yoo and Kim13 reported the control signal of the controller’s set-point change was best for identifying low-frequency information. In the closed-loop identification of integrating and unstable systems, Liu and Gao14 reported that the step change is to be introduced in the set point instead of directly to the process input, in order to avoid the process output from drifting far way from its operational range. Mei and Li15 also highlighted the importance of sequential set point excitation over available discrete time identification methods. Due to its simplicity of step test, in recent years, many works in the literature include closed-loop identification of multivariable processes from step tests.16−19,24−27 McIntosh and Mahalec20 presented a method to obtain the steady state gains of multivariable systems by introducing a step change in one of the set points at a time. From the output data, steady state gains are calculated by matrix operations. The routine operating data of a plant under closed-loop operation can also be used for this method. A frequency response technique for a SISO system21 was extended to a MIMO system by Melo and Friedly.22 They presented a method for the closed-loop identification of multivariable systems in the form of a nonparametric frequency response model. A step change in the set point is introduced in each loop sequentially, and the output transient response is obtained. Fourier transform is used to obtain the frequency response from the measured time response. This method does not give directly the transfer function matrix. Ham and Kim23 applied an optimization method for process identification of FOPTD models of multivariable system from the pulse response; it can also be used for tuning of PID controllers. Mei et al.19 proposed a decentralized closed-loop identification method for a two-input two-output (TITO) process. The coupled closed-loop TITO system was decoupled equivalently in to four individual single open-loop systems. The multivariable identification process is converted into a multisingle-open-loop identification process. This method requires process input and output data. It should be observed that the time domain identification method is easier to formulate, and it is conceptually simple. Viswanathan et al.1 presented a method for the closed-loop identification of FOPTD and SOPTD models by using genetic algorithm (GA) for TITO systems. The computational time for the identification of a SOPTD TITO system using the GA ranges from 1 h 55 min to 5 h 28 min (for different controller settings and test timings).1 Further, the converged values from GA are not reliable. The converged values are refined by using a local optimizer (Broyden−Fletcher−Goldfarb−Shanno method). For the SOPTD model identification, Viswanathan et al.1 reported the success rate for 10 trials ranging from 25% to 100% to get the global minimum. There is no guarantee to get the exact transfer function model (global minimum) in a single trial. If we obtain

2. PROPOSED METHOD Consider a TITO system. G(s) and Gc(s) are process transfer function matrix and decentralized controller matrix with compatible dimensions, respectively, expressed as ⎡ g11 g12 ⎤ ⎥ G (s ) = ⎢ ⎣ g21 g22 ⎦

(1)

⎡g 0 ⎤ c11 ⎥ Gc(s) = ⎢ ⎢⎣ 0 gc22 ⎥⎦

(2)

The controller parameters can be chosen arbitrarily for multivariable systems, such that the closed-loop response of the system is stable with reasonable responses. Consider a decentralized TITO multivariable system, as shown in Figure 1. 9621

dx.doi.org/10.1021/ie203003p | Ind. Eng. Chem. Res. 2012, 51, 9620−9633


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guess values of the time constant of the open-loop system for the optimization problem. Hence, the initial guess value of the time constant26,27 for the model is to be assumed as (ts/8). The initial guess value for the damping coefficient is taken as 1. The initial guess values of the main loop process gains (kp11 and kp22) are obtained by the following method. For lower order model systems, the value28 of kpkc,max is taken as 4 and the design value for kpkc,des = 2. Hence, the guess value for kp is taken as 2/kc. Similarly, for the higher order model systems, the value28 of kpkc,max is 1 and the design value for kpkc,des = 0.5. Hence, the guess value for kp is taken as 0.5/kc. The higher order model and the lower order model are distinguished by checking the initial dynamics of the main responses of the actual process. If the initial slope of the main response is sluggish (zero slope), then we consider the system as higher order model. Similarly, if the initial slope of the main response has the steep response (nonzero slope), then we consider the system as a lower order model. The initial guess values of the cross loop process gains (kp21 and kp12) are obtained by the following method. The Laplace transforms of interaction responses y21(s*) and y12(s*) can be expressed as

The process transfer function models are identified as SOPTD models: gm, ij =

k pm, ije−θijs 2 2 τm, ijs + 2εm, ijτm, ijs + 1

i = 1, 2; j = 1, 2 (3)

Figure 1. Decentralized control system of a TITO process.

In the present work, initially, a unit magnitude of step change is introduced in the set point yr1. After the step response reaches its new steady state, the step excitation is removed from the same set point yr1 with all remaining set points unchanged and all loops kept under closed-loop operation. From the prescribed manner of excitation in set point, yr1, we obtain the main response y11 and the interaction response y21. Similarly, the same magnitude of combined step-up and step-down excitation is introduced in the set point, yr2, and we obtain the main response y22 and interaction response y12. The problem associated with using only the step change excitation is discussed in the Simulation Examples section. The response matrix of the TITO system can be expressed as ⎡ y11 y12 ⎤ ⎥ Y (t ) = ⎢ ⎣ y21 y22 ⎦ (4)

y21(s*) =

∫t

y12 (s*) =

∫t

0

0

y21(t )e−ts *dt

(5)

y12 (t )e−ts *dt

(6)

To evaluate eqs 5 and 6, we need to assume the value of s*. In the present work, the value of s* is considered as 8/ts. Here, ts is the settling time of the closed-loop response. For t ≥ ts, the value of e−8 is close to zero. Then, the integral value does not change further. So, to reduce the computation time of the integral, the value of s* is taken as 8/ts. The upper bound for t is ts. The numerical values for y21(s*) and y12(s*) are obtained by substituting the value of s* in eqs 5 and 6.The closed-loop transfer function for the cross loops are assumed by considering the interaction from the main loop taken as a disturbance to the interacted loop. Some signals are ignored for simplicity, as shown in Figure 2, in order to avoid g12 (see Figure 2a) in deriving eq 7 and to avoid g21 (see Figure 2b) in deriving eq 9. We obtain

The first column in the response matrix (eq 4) is the responses (main and interaction) obtained for the combined step-up and step-down excitations in set point yr1 in the first loop. The second column is the response obtained for the step-up and step-down excitations in set point yr2 in the second loop. From these step responses, the initial guess values of the model parameters are obtained by the method given in the next section. As stated earlier, the selection of initial guess values plays a vital role in any optimization method. The combined dynamics of step-up and step-down responses are used in the optimization routine for the identification of the transfer function model matrix. The initial guess values for the time delay can be taken as the corresponding value of the closed-loop time delay. The initial guess values for the time constant are obtained from the settling time (ts) of the main and interaction responses of the step up response only. From the settling time (to reach 98% of final steady state value and remains within the limit), the closed-loop time constant (τc) can be assumed as (ts/4). The time constant of the open-loop system is assumed to be less than that of the closed-loop time constant. In principle, for SISO systems, the closed-loop response is faster than the open-loop response. In MIMO systems, due to the interaction among the loops, usually the closed-loop response is assumed to slower than the openloop response. This assumption is made only to obtain the initial

y21(s*) yr1(s*)

=

gc11(s*)gm21(s*) (1 + gc11(s*)gm11(s*))(1 + gc22(s*)gm22(s*)) (7)

Substitute the value for yr1(s) as 1/s* and also substitute the controller settings and the guess values of model parameters into eq 7. We obtain ⎧ ⎛ ⎞ ⎛ 1 ⎞ ⎪⎛ 1 ⎞ ⎜ ⎟k ⎜⎜1 + ⎜ ⎟⎟ + τ * y21(s*) = ⎨ s ⎟ c11 d11 ⎪⎝ ⎠ ⎝ ⎝ τI11s* ⎠ ⎠ ⎩ s* ×

⎫ ⎬ + 2εm21τm21s* + 1) ⎪ ⎭

k pm21e−θm21s * (τm221s*2

⎪

⎧⎛ ⎞ ⎛ ⎞ ⎛ 1 ⎞ k mp11e−θm11s * ⎪ ⎟ /⎨⎜⎜1 + kc11⎜⎜1 + ⎜ ⎟ + τd11s*⎟⎟ 2 2 ⎟ ⎪ ⎝ τI11s* ⎠ ⎝ ⎠ (τm11s* + 2εm11τm11s* + 1) ⎠ ⎩⎝ ⎛ ⎞ ⎛ ⎛ 1 ⎞ × ⎜⎜1 + kc22⎜⎜1 + ⎜ ⎟ + τd22s*⎟⎟ ⎝ τI22s* ⎠ ⎠ ⎝ ⎝ ×

9622

⎫ ⎞⎪ ⎟⎬ ⎟ + 2εm22τm22s* + 1) ⎠⎪ ⎭

k mp22e−θm22s * 2 *2 (τm22 s

(8)



Article

Figure 2. Simplified diagram for TITO system (a) for eq 7 and (b) for eq 9.

Similarly, the initial guess values for kp12 is obtained by y12 (s*) yr2 (s*)

=

select the model parameters in order to minimize the sum of squared errors between the model and the actual process responses (obtained from combined step-up and step-down test).

gc22(s*)gm12(s*) (1 + gc22(s*)gm22(s*))(1 + gc11(s*)gm11(s*)) (9)

minimize f =

Substitute the value for yr2(s) as 1/s* and also substitute the controller settings and the guess values of model parameters (time delays, time constants, damping constants and main-loop process gains) in eq 9. We obtain

(kij ,τij , εij , θij)

i = 1, 2

⎫ ⎪ ⎬ 2 *2 ⎪ (τm12s + 2εm12τm12s* + 1) ⎭ k mp12e−θm12s *

⎧ ⎛ ⎞ ⎛ ⎛ 1 ⎞ ⎪ /⎨⎜⎜1 + kc22⎜⎜1 + ⎜ ⎟ + τd22s*⎟⎟ ⎪ ⎝ τI22s* ⎠ ⎠ ⎝ ⎩⎝ ×

⎞ ⎟ 2 *2 s + 2εm22τm22s* + 1) ⎟⎠ (τm22 k mp22e−θm22s *

⎛ ⎞ ⎛ ⎛ 1 ⎞ × ⎜⎜1 + kc11⎜⎜1 + ⎜ ⎟ + τd11s*⎟⎟ ⎝ τI11s* ⎠ ⎠ ⎝ ⎝ ×

⎞⎫ ⎪ ⎟⎬ 2 *2 ⎟ ⎪ (τm11s + 2εm11τm11s* + 1) ⎠⎭

n

2

i=1

j=1

j = 1, 2

[ymij (tn) − yij (tn)]2 Δt

1

(11)

TITO systems have two main and two interaction responses. In the objective function, n is the number of data points in the response of the process and the model. First, we get the error between responses of each transfer function model with the associated process response for various times (to form an error vector for each model). Further, we take the square of the each of the errors and multiply by a fixed Δt (sampling time). The sum of the elements of each vector gives an ISE value. There are four ISE values. The sum of these four ISE values is considered as a scalar objective function (refer to eq 11). The model parameters are selected by minimizing this scalar function. In eq 11, the sampling time Δt need not be included, since it is a constant value. In this present work, the optimization problems are solved by using the routine lsqnonlin in MATLAB and the routine implements the Trust-Region-Reflective algorithm. Although the optimization problem is formulated and solved using the standard routine in MATLAB, the convergence and computational time depends on initial guesses. The main focus of the present work is to propose the simple method to get the initial guess values of the model parameters. In the optimization routine, the parameter values are considered as TolX = 1e−3, TolFun = 1e−6, and the maximum number of iterations = 400. The closed-loop multivariable system is simulated using SIMULINK. The fourth-order Runge− Kutta method with a fixed step size is used for solving the governing initial value ordinary differential equations. All of the computational work for the identification of the multivariable systems was performed in Core 2 Quad (Intel/3.00 GHz) personal computer. To evaluate the proposed method, a known transfer function model matrix is considered with suitable decentralized PI/PID controllers to get a reasonable stable closed-loop response. The closed-loop main and interaction responses of the actual process are obtained by the earlier discussed method. From the closed-loop responses, the model parameters are obtained by the proposed method. The closedloop main and interaction responses of the model with the

⎧ ⎞ ⎛ ⎛ 1 ⎞ ⎪⎛ 1 ⎞ ⎜ ⎟k ⎜⎜1 + ⎜ y12 (s*) = ⎨ ⎟ + τd22s*⎟⎟ c22 ⎪⎝ ⎝ τI22s* ⎠ ⎠ ⎩ s* ⎠ ⎝ ×

2

∑ ∑ ∑

k mp11e−θm11s *

(10)

The guess values for kmp21 and kmp12 are to be obtained by substituting the other known guess values and controller parameters in eqs 8 and 10. After obtaining all the guess values of the model, the same manner of step-up and step-down excitation is introduced to the set point ymr1 of the model with all remaining set points are unchanged and all other loops are kept under closed-loop condition. From the prescribed excitation in set point ymr1, the main response ym11 and interaction response ym21 of the model are obtained. Similarly, the combined step-up and step-down excitation is introduced in the set point ymr2, and the main response ym22 and interaction response ym12 of the model are obtained. These four (main and interaction) responses are compared with the actual process responses. To obtain the model parameters, the optimization problem is formulated as to 9623



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Table 1. Model Identification Results Using PID Controller for Example 1 model (ij) 11

21

12

22

guess convergeda convergedb actual guess convergeda convergedb actual guess convergeda convergedb actual guess convergeda convergedb actual

kpij

τij

θij

εij

0.2500 0.5141 0.5798 0.5800 0.2017 0.3038 0.3498 0.3500 −0.1829 −0.3869 −0.4497 −0.4500 −0.1667 −0.4318 −0.4797 −0.4800

5.0000 2.2175 3.5232 3.5143 6.2500 1.0894 1.8870 1.8300 6.2500 3.6363 4.1289 4.1279 6.2500 1.5547 1.3518 1.3000

1.0000 1.9342 0.9939 1.0000 1.2800 1.7442 1.2439 1.2800 1.0000 1.1944 0.9982 1.0000 1.0000 0.8511 0.9748 1.0000

1.0000 1.5865 1.1709 1.1752 1.0000 2.1350 1.5061 1.5492 1.0000 1.1002 1.0491 1.0514 1.0000 1.4886 1.8758 1.9462

total valuea total valueb a

IAE values

ISE values

0.4499 0.0100

1.78 × 10−02 2.66 × 10−06

0.2370 0.0155

2.80 × 10−03 1.50 × 10−05

0.2424 0.0109

2.10 × 10−03 2.25 × 10−06

0.3396 0.0254

3.60 × 10−03 3.04 × 10−05

1.2689 0.0618

0.0263 5.031 × 10−5

Step (only) response. bCombined step-up and step-down response.

Table 1). The final converged parameters are obtained from the optimization routine. These identified model parameter values deviate from the actual process parameter values. In the estimated SOPTD models, the process gain values nearly matched the actual process gain values. However, the time constant, time delay, and damping coefficient have significant deviations (as we get in any model reduction technique). The converged SOPTD transfer function model parameters obtained from the step response data are reported in Table 1. By using the step response, we obtain the satisfactory closed-loop responses of the model, but the parameters deviate with the true values. Codrons30 reported that for the same controller different models show the same closed-loop behavior even though they have different open-loop behaviors. Two examples are considered to show the validity of this statement. The effect of proportional gain value on the models mismatch is also analyzed. Codrons30 reported that models can only have their quality evaluated for a particular set of experimental conditions. In the present work, a combined step-up and step-down excitation is performed to get the exact model parameters, and it is compared to the model parameters obtained from the conventional step test. The proposed method (combined step up and step down with the guess values) gives the lowest sum of ISE values. The final converged parameters and ISE and IAE values are also listed in Table 1 for the combined step-up and step-down responses. The converged model parameters match well with the actual SOPTD system parameters. The Nyquist plots for the subsystems (g11, g12, g21, and g22) are shown in Figure 3 for the original system, identified model by the step (up) only method and identified model by the combined step-up and step-down method. Figure 3 shows a better fit of the open-loop model (identified by combined step-up and step-down method) with that of the actual system. Figure 4 shows the closed-loop time response matching of the actual system with that of the identified SOPTD system (by combined step-up and step-down method). The computational time for optimization is 11 min 11 s (maximum computational time for 2 × 2 systems) because of the proposed method for obtaining the initial guess values for the model parameters. The computational time reported by Viswanathan et al.1 varies from 1 h 55 min to 5 h 28 min for

decentralized PI/PID controllers are compared with the actual system with the same decentralized PI/PID controller settings.

3. SIMULATION EXAMPLES In this section, three simulation examples are considered to show the effectiveness of the proposed method. 3.1. Example 1. Consider the 2 × 2 (SOPTD) distillation column transfer function model matrix:1,29 ⎡ ⎤ 0.58e−s −0.45e−s ⎢ ⎥ 2 2 ⎢ 12.35s + 8.26s + 1 17.04s + 8.69s + 1 ⎥ G (s ) = ⎢ ⎥ 0.35e−1.28s −0.48e−s ⎢ ⎥ ⎣ 3.35s 2 + 5.67s + 1 1.69s 2 + 5.06s + 1 ⎦ (12)

The decentralized PID controller settings used for the MIMO identification were:1 kc11 = 2, τi11 = 2, τd11 = 0.5; kc22 = −3, τi22 = 2, τd22 = 0.5. For all of the simulation examples, in the optimization routine, we fixed the lower bound value for time constant as 1/20 of the guess value and upper bound value as five times the initial guess value. Similarly, for time delays, the lower bound was specified as 1 /4 of the guess value and the upper bound is specified as four times the initial guess value. Similarly, for the damping coefficient, the lower bound is taken as 0 and the upper bound is taken as 5. In case the initial guess value of process gain was calculated as negative, we then fixed the upper bound as 1/20 of the guess value and lower bound as 20 times the guess value. Similarly, for positive guess values of the process gain, we fixed the lower bound as 1/20 of the guess value and upper bound as 20 times the guess value. These parameter settings in the optimization routine are found to get guarantee in the convergence and also to reduce the computational time. Since the response showed a near zero initial slope, kpkc = 0.5 was used for calculating the guess values of process gains kp11 and kp22. The SOPTD model is assumed, and the initial guess values for the model parameters are obtained by using the step responses. Initially, the step-up (only) signal is used as the set point excitation for the identification of the SOPTD model. The guess values are obtained by the earlier mentioned method (see 9624



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Figure 3. Nyquist plot for the actual system (solid line), identified SOPTD model: combined step-up and step-down method (++), step-up (only) method (•). [Example 1].

Figure 4. Closed-loop response of the process and the identified SOPTD transfer function model (obtained from combined step-up and step-down responses) for example 1 using PID controller (model and process curves overlap).

step-up (only) response poses the same problem in the global convergence, but by using the step-up and step-down responses, the identified model parameters match well with the actual process parameters. The initial guess values, final converged parameters, and ISE and IAE values are listed in Table 2 for both the tests. The effect of initial guess values (for the two controller settings) on the number of iteration, computational time, IAE and ISE values is also studied by changing the guess values +10% and −10% separately. The present method shows a good agreement with the actual process parameters for these perturbed guess values also. The number of iterations, computational time, and ISE* and IAE* values are listed in Table 3 for this perturbed guess values.

one trial, and they have used a minimum of 10 trials, as discussed earlier. Viswanathan et al.1 did not report the values of the converged model parameters for their method and gave only the ISE value as 8.87 × 10−8. The present method (combined step up and step down excitation along with the proposed guess values of the parameters) gives the best fit (that corresponds to the global optimal). The proposed initial guess values are important to achieve faster local convergence to local optima whereas only the combined step up and step down excitation along with the proposed initial guess values lead to a faster global convergence of the optimization problem. The identification method is also carried out by using the closed-loop main and interaction responses obtained by PI controllers (kc11 = 1, τi11 = 3; kc22 = −1, τi22 = 2). Here also, the 9625



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Table 2. Model Identification Results under Using PI Controller (Example 1) kpij

τij

θij

εij

0.5000 0.7802 0.5801 0.5800 0.5151 0.5384 0.3500 0.3500 −0.5607 −0.6467 −0.4501 −0.4500 −0.5000 −0.6593 −0.4800 −0.4800

7.5000 4.2211 3.5138 3.5143 8.7500 2.1056 1.9701 1.8300 10.0000 4.6489 4.1354 4.1279 7.5000 1.6360 1.4619 1.3000

1.0000 0.3029 1.0004 1.0000 1.2800 0.3203 1.1846 1.2800 1.0000 0.4664 0.9924 1.0000 1.0000 8.4989 0.9145 1.0000

1.0000 1.4287 1.1746 1.1752 1.0000 2.3266 1.4582 1.5492 1.0000 1.3975 1.0492 1.0514 1.0000 0.2849 1.7549 1.9462 0.7174

model (ij) 11

21

12

22

guess convergeda convergedb actual guess convergeda convergedb actual guess convergeda convergedb actual guess convergeda convergedb actual


IAE values

ISE values

0.2093 0.0081

1.90 × 10−3 7.25 × 10−7

0.1165 0.0204

4.21 × 10−4 1.94 × 10−5

0.2403 0.0134

1.80 × 10−3 2.07 × 10−6

0.1513 0.0224

1.40 × 10−3 2.14 × 10−5

0.0055 0.0643

4.3895 × 10−5

b

Only step response. Combined step-up and step-down response.

Table 3. Effect of Initial Guess Values of the Model Parameters (IAE and ISE for the sum of main and interaction responses) on the Number of Iterations and Computational Time for Example 1. (Combined Step-up and Step-down Responses) PID Controller

PI controller

perturbation in guess values

−10%

0%

+10%

−10%

0%

+10%

IAE values ISE values no. of iterations computational time (s)

0.0477 3.49 × 10−5 33 323

0.0618 5.03 × 10−5 41 390

0.0618 5.03 × 10−5 40 379

0.0519 2.64 × 10−5 33 509

0.0643 4.35 × 10−5 37 567

0.0643 4.35 × 10−5 44 671

Table 4. Model Identification Results Using PID Controller for Example 1 with Measurement Noise (Combined Step-up and Stepdown Change)a model (ij) 11 21 12 22 a

guess converged guess converged guess converged guess converged

Kpij

τij

θij

εij

0.2500 0.5948 0.2152 0.3673 −0.1824 −0.4653 −0.1667 −0.4988

5.0000 3.8334 5.0000 1.2488 5.0000 4.3886 5.0000 1.7106

1.0000 0.7693 1.3000 1.5696 1.0000 0.7786 1.0000 0.7267

1.0000 1.0948 1.0000 2.2692 1.0000 0.9779 1.0000 1.5988

IAE values

ISE values

0.3386

2.90 × 10−3

0.2057

8.86 × 10−4

0.4083

3.40 × 10−3

0.3889

5.70 × 10−3

−2

Sum of main and interaction values (IAE*) = 1.3415, (ISE*) = 1.29 × 10 .

Golay,31 which filters the noise and gives a smooth curve. From that smooth curve, the initial guess values can be calculated by the proposed method. The proposed method is robust for the controller settings and for the measurement noise. The identified model parameters were closer to those obtained without measurement noise. The initial guess values, final converged values, and IAE and ISE values are also listed in Table 4. The closed-loop main responses and interaction responses using the identified model with the same controller settings are shown in the Figure 5, along with the corrupted process responses 3.2. Example 2. Consider the Niederlinski32 multivariable transfer function model considered by Viswanathan et al.:1

The effect of measurement noise on the guess values and model parameters is also considered by adding a random noise (SD = 0.05) to the actual main and interaction responses, and the sample time of 0.1 s is considered. The noisy output is used for the feedback control action and also for the model identification. To obtain the initial guess values, a smooth curve is first drawn. Because of the integral action in the controller, the main response reaches to one and the interaction reaches to zero in steady state. Hence, a horizontal line is drawn for main response at 1 and a horizontal line is drawn at 0 in the interaction response in the steady state portion only. In the initial dynamics part, a smooth curve is passed through a midpoint of the response. Alternatively, one can use the least-squares method proposed by Savitzky and 9626



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Figure 5. Closed-loop response of the corrupted process with the identified SOPTD model (obtained from combined step-up and step-down response).

Table 5. Model Identification Results under Using PIa Controller (for Example 2) model (ij) 11

21

12

22

guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb

kpij

τij

θij

εij

1.2500 0.4999 0.5005 2.0089 1.0008 1.0002 −2.0089 −1.0015 −0.9992 3.3333 2.3959 2.4015

3.1250 0.3557 0.2818 2.5000 0.2494 0.2061 2.5000 0.1826 0.2267 2.5000 0.4415 0.4110

0.1000 0.0498 0.1049 0.1000 0.0250 0.0859 0.1000 0.0250 0.0667 0.1000 0.2020 0.1905

1.0000 0.7682 0.8734 1.0000 0.9629 1.0254 1.0000 1.2708 0.9727 1.0000 0.9183 0.9901


IAE values

ISE values

0.0132 0.0065

5.67 × 10−5 9.50 × 10−6

0.0145 0.0118

5.85 × 10−5 4.84 × 10−5

0.0112 0.0038

8.79 × 10−5 2.24 × 10−6

0.0165 0.0107 0.0554 0.0328

8.46 × 10−5 2.93 × 10−5 2.87 × 10−4 8.94 × 10−5

Step response. bStep-up and step-down response. ⎡ 0.5 ⎢ 2 (0.1 1) (0.2s + 1)2 s + G(s) = ⎢⎢ 1.0 ⎢ 2 ⎣ (0.1s + 1)(0.2s + 1)

⎤ ⎥ ⎥ ⎥ 2.4 ⎥ (0.1s + 1)(0.2s + 1)2 (0.5s + 1) ⎦

Table 5. In the identification of higher order model in to SOPTD model, also, the step (only) response identification poses the problem of attaining the global minimum (least ISE value). In the higher order transfer function matrix model, we can verify the better fit of the models based on IAE and ISE values. The Nyquist plots for the subsystems (g11, g12, g21, and g22) (Figure 6) shows that the identified model by step (up) only method is not adequate. To obtain the global minima, (least ISE values) the combined step-up and step-down responses are used. The initial guess values, final converged model parameters, ISE and IAE values are listed in Table 5. The Nyquist plot (Figure 6) shows a good match between the estimated SOPTD model by the step-up and step down method with that of the actual process. Figure 7 shows the closed-loop time response of the actual system and the identified model. The computational time is significantly reduced to 3 min 3 s (maximum computational time) when compared to 51 min as reported by Viswanathan et al.1 The SOPTD transfer function model obtained from the combined step-up and step

− 1.0 (0.1s + 1)(0.2s + 1)2

(13)

The higher order TITO process transfer function models are to be approximated as SOPTD transfer function models. The decentralized PI controllers are used for this identification process. The controller parameters for the decentralized PI controllers were1 PIa; kc11 = 0.4, τ i11 = 0.9; kc22 = 0.15, τ i22 = 1.5. The closed-loop step responses of the actual process (with PIa controller) are obtained by using SIMULINK. Since the responses of the actual process show a higher order model behavior (initial zero slope), the kpkc,des value is considered as 0.5 for calculating the guess value of the process gains (main loop). Then, the other initial guess values are obtained by the earlier mentioned methods. The initial guess values, final converged parameters, ISE and IAE values are listed in 9627



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Figure 6. Nyquist plot for the actual system (solid line), identified model: combined step-up step-down method (++) and step (up) only method (•) [example 2].

Figure 7. Closed-loop response of the process and the identified SOPTD transfer function model (obtained from combined step-up and step-down responses) for example 2 using PIa controller (model and process curves overlap).

model parameters are found; it depends on the controller settings. By using the PIb controller also, the combined step-up and step down excitation provides the lower ISE value compared to the values obtained by the model from the step (only) excitation. The initial guess values, final converged parameters, and ISE and IAE values are listed in Table 6 for the combined stepup and step-down responses and for the step (only) response. The effect of initial guess values on the number of iterations, computational time, and IAE* and ISE* values is also studied by changing the guess values by +10% and by −10% separately using

down excitation has the lesser IAE and ISE values as compared to those of the model obtained from step response only. Further, the SOPTD model identification is carried with another set of PIb; kc11 = 0.4, τi11 = 0.4; kc22 = 0.1, τi22 = 0.3. (under damped response). The closed-loop step response adequately matches with the actual process response using the PIb controller, but the final converged model parameters deviate with the parameters obtained by the PIa controller settings. By using the step response, we obtained the satisfactory closed-loop response of the model. In the step (only) response method, the estimated 9628



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Table 6. Model Identification Results Using PIb Controller (for Example 2) kpij

τij

θij

εij

1.2500 0.4974 0.5002 2.4658 1.0009 0.9996 −2.4658 −1.0157 −1.0001 5.0000 2.3896 2.3979

1.2500 0.2861 0.2792 1.2500 0.1981 0.2293 1.2500 0.1313 0.2373 1.2500 0.4785 0.4030

0.1000 0.1155 0.1050 0.1000 0.0999 0.0650 0.1000 0.1003 0.0539 0.1000 0.1200 0.1910

1.0000 0.8401 0.8772 1.0000 1.0298 0.9537 1.0000 1.5929 0.9459 1.0000 0.8915 1.0035

model (ij) 11

21

12

22

guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb


IAE values

ISE values

0.0186 0.0080

5.96 × 10−5 1.04 × 10−5

0.0203 0.0066

9.26 × 10−5 9.52 × 10−6

0.0147 0.0035

4.52 × 10−5 1.74 × 10−6

0.0209 0.0093 0.0745 0.0275

5.80 × 10−5 2.35 × 10−5 2.55 × 10−4 4.52 × 10−5

Step response. bCombined step-up and step-down response.

Table 7. Effect of Initial Guess Values of the Model Parameters on the Number of Iterations and Computational Time for Example 2. (Combined Step-up and Step-down Change) PIb controller

a

PIa controller

perturbation in guess values

−10%

0%

+10%

−10%

0%

+10%

IAEa values ISEa values no. of iterations computational time (s)

0.0463 1.97 × 10−4 37 105

0.0275 4.52 × 10−5 46 140

0.0347 7.67 × 10−5 42 121

0.0302 9.39 × 10−5 30 158

0.0327 8.94 × 10−5 28 151

0.0282 8.25 × 10−5 34 183

Sum of main and interaction response values.

Figure 8. Simplified diagram for 3 × 3 multivariable system for calculating guess values for Kp31 and Kp32.

both the PI controllers. The present method shows a good agreement with the actual process parameters for these perturbed guess values, also. The results obtained for this perturbed guess values are given in Table 7. 3.3. Example 3. Consider the 3 × 3 (SOPTD) transfer function matrix model reported by Garcia et al.33

The PI controller parameters are obtained by using BLT method34 (F = 2) and PI parameters were Kcii = 4.6979, τIii = 2.3612. The guess values for the main-loop (diagonal) process gains Kp11, Kp22, and Kp33 were calculated based on the corresponding controller gains as discussed in example 1. Since the responses of the actual process showed a higher-order model behavior (initial zero slope), the KpKc,des value was considered as 0.5 for calculating the guess value of the process gains (Kp11, Kp22, and Kp33). The nondiagonal process gains (interactions) were calculated using equations similar to eqs 8 and 10. To simplify the calculations, only two loops were considered at a time, and the interaction coming from the remaining loop was considered as negligible. For example, the guess value of the process gain (nondiagonal) Kp21 was estimated by considering the interaction coming from main

⎡ e−0.1s 0.3e−0.01s 0.3e−0.01s ⎤ ⎢ 2 ⎥ 2 2 ⎢ s + 2s + 1 2s + 3s + 1 2s + 3s + 1 ⎥ ⎢ ⎥ 0.3e−0.01s e−0.1s 0.3e−0.01s ⎥ G=⎢ 2 ⎢ 2s + 3s + 1 s 2 + 2s + 1 2s 2 + 3s + 1 ⎥ ⎢ ⎥ 0.3e−0.01s e−0.1s ⎢ 0.3e−0.01s ⎥ ⎢⎣ 2s 2 + 3s + 1 2s 2 + 3s + 1 s 2 + 2s + 1 ⎥⎦ (14) 9629



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Table 8. SOPTD Model Identification Results for Example 3 Using the PI (BLT,34 F = 2) Controller model (ij) 11

21

31

12

22

32

13

23

33

a

guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb guess convergeda convergedb

Kpij

τij

εij

θij

0.1064 0.8337 0.9992 0.0352 0.1452 0.2982 0.0352 0.1452 0.2982 0.0352 0.1452 0.2982 0.1064 0.8337 0.9992 0.0352 0.1452 0.2982 0.0352 0.1452 0.2982 0.0352 0.1452 0.2982 0.1064 0.8337 0.9992

1.0000 0.8593 1.0018 1.5000 0.9993 1.4096 1.5000 0.9993 1.4096 1.5000 0.9993 1.4096 1.0000 0.8593 1.0018 1.5000 0.9993 1.4096 1.5000 0.9993 1.4096 1.5000 0.9993 1.4096 1.0000 0.8593 1.0018

1.0000 1.0810 0.9952 1.0000 0.9509 1.0535 1.0000 0.9509 1.0535 1.0000 0.9509 1.0535 1.0000 1.0810 0.9952 1.0000 0.9509 1.0535 1.0000 0.9509 1.0535 1.0000 0.9509 1.0535 1.0000 1.0810 0.9952

0.1000 0.1749 0.0984 0.01 0.0392 0.0101 0.01 0.0392 0.0101 0.01 0.0392 0.0101 0.1000 0.1749 0.0984 0.01 0.0392 0.0101 0.01 0.0392 0.0101 0.01 0.0392 0.0101 0.1000 0.1749 0.0984

IAE values

ISE values

0.1018 0.0073

3.60 × 10−03 8.65 × 10−06

0.0534 0.0030

4.03 × 10−04 7.20 × 10−07

0.0534 0.0030

4.03 × 10−04 7.20 × 10−07

0.0534 0.0030

4.03 × 10−04 7.20 × 10−07

0.1018 0.0073

3.60 × 10−03 8.65 × 10−06

0.0534 0.0030

4.03 × 10−04 7.20 × 10−07

0.0534 0.0030

4.03 × 10−04 7.20 × 10−07

0.0534 0.0030

4.03 × 10−04 7.20 × 10−07

0.1018 0.0073

3.60 × 10−03 8.65 × 10−06

Step response. bCombined step-up and step-down response.

Figure 9. Nyquist plot for the actual system (solid line), identified model: combined step-up step-down method (++) and step (up) only method (•) [example 3].

was considered. Similarly, the guess values for the other gains Kp13 and Kp23 were calculated. For simulating the model, the sample time was considered as 0.01 s. With these guess values, initially the step (only) change is used as the set point excitation for the identification of the SOPTD model. The identified model parameters (from step

loop only (similar to Figure 2a). The interaction to and from the third loop were assumed to be neglected. This procedure gave explicitly the guess value of the required process gain. Similarly, to obtain the guess values of Kp12, an approach similar to Figure 2b was considered. To obtain the guess values for the gains Kp31 and Kp32, the corresponding Figure 8 9630



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Figure 10. Closed-loop response of the process and the identified SOPTD transfer function model (obtained from combined step-up and stepdown responses) for example 3 using PI (BLT,34 F = 2) controller. (Solid line, model; dashed line, process. Model and process response curves overlap.)

Table 9. Effect of Changing Controller Settings and Effect of Guess Values on the Number of Iterations and Computational Time for Example 3 (Obtained from Combined Step-up and Step-down Responses) PI (BLT,34 F = 3) (Kcii = 3.132, τIii = 3.5417) perturbation in guess values a

IAE values ISEa values no. of iterations computational time (s) a

−10% 0.078 1.14 × 10−04 10 203

0% 0.078 1.14 × 10−04 10 195

Chen and Seborg35 (Kcii = 5.25τIii = 1.08, τDii = 0.271)

+10%

+10% −03

7.84 × 10 9.13 × 10−07 27 554

0.0273 1.79 × 10−05 18 228

0%

+10% −06

8.88 × 10 2.13 × 10−12 13 171

0.0150 4.99 × 10−06 32 393

Sum of main and interaction response values.

numbers of iterations, computational times, and IAE and ISE values are listed in Table 9. The effects of initial guess values on the number of iteration, computational time, and IAE and ISE values were also studied by perturbing the guess values by +10% and separately by −10%. The model parameters obtained by the proposed method showed good agreement with the actual process parameters for these perturbed guess values The numbers of iterations, computational times, and IAE and ISE values are listed in Table 9.

(only) response) deviate from the actual process parameter values, and it is listed in table 8. The Nyquist plots for the subsystems (Figure 9) show that the identified model by this method is not adequate. To converge to the true values, the combined step-up and step-down change in the set point is considered and the output responses are used for the identification test. The proposed method (combined step up and step down with the guess values) gives the lowest sum of ISE value. The final converged parameters, ISE and IAE values are also listed in Table 8 for the combined step-up and step-down responses. The converged model parameters match well with the actual SOPTD system parameters. The Nyquist plot (Figure 9) also shows the better matching of the identified model by the proposed method with that of the actual system. The computational time for the optimization was 3 min 17 s, and the number of iterations was 13. The closed-loop main and interaction responses using the identified model parameters with the same controller settings are also shown in Figure 10. Figure 10 shows a good match between the model and the actual process. The identification method was also carried out using the closed-loop main and interaction responses obtained with different PI/PID settings. The converged model parameters showed good agreement with the actual process parameters. The

4. CONCLUSIONS Closed-loop identification of SOPTD Model Parameters of a TITO Multivariable System based on an optimization method using the combined dynamics of step-up and stepdown response is presented. Compared to that of the conventional step response method, the proposed method (step-up and step-down) shows a good matching between the identified model parameters with that of the actual process. From the three simulation examples, it is shown that the proposed method gives the lower ISE values than the conventional step response. Simple methods for obtaining initial guess values for the model parameters from the main and interaction responses are proposed for getting a quick and a guaranteed convergence. The effects of measurement 9631



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(11) Bazanella, A. S.; Gevers, M.; Miskovic, L. Closed-Loop Identification of MIMO Systems: A New Look at Identifiability and Experiment Design. Proceedings of the European Control Conference 2007, 5694−5699. (12) Bombois, X.; Scorletti, G.; Gevers, M.; Hof, V. C. P. M. J.; Hildebrand, R. Least Costly Identification Experiment for Control. Automatica 2006, 42, 1651−1662. (13) Yoo, C.; Kim, M. H. Industrial Experience of Process Identification and Set-Point Decision Algorithm in a Full-Scale Treatment Plant. J. Environ. Manage. 2009, 90, 2823−2830. (14) Liu, T.; Gao, F. Closed-Loop Step Response Identification of Integrating and Unstable Processes. Chem. Eng. Sci. 2010, 65, 2884− 2895. (15) Mei, H.; Li, S. Decentralized Identification for Multivariable Integrating Processes with Time Delays from Closed-Loop Step Tests. ISA Trans. 2007, 46, 189−198. (16) Jin, Q. B.; Cheng, Z. J.; Dou, L. T.; Wang, K. W. A Novel Closed Loop Identification Method and its Application of Multivariable System. J. Process Control 2012, 22, 132−144. (17) Jeng, J. C. Closed-Loop Identification of Multivariable Systems by Using B-Spline Series Expansions for Step Responses. Ind. Eng. Chem. Res. 2012, 51 (5), 2376−2387. (18) Yagang, W. Closed-Loop Multivariable Process Identification in the Frequency Domain. Proceedings of the 27th Chinese Control Conference; Cheng, D. Z. Li C., Eds; IEEE Press: 2008, 440−445. (19) Mei, H.; Li, S.; Cai, W. J.; Xiong, Q. Decentralized Closed-Loop Parameter Identification for Multivariable Processes from Step Responses. Math. Comput. Simul. 2005, 68, 171−192. (20) McIntosh, A. R.; Mahalec, V. Calculation of Steady-state Gains for Multivariable Systems from Closed-Loop Steady-state Data. J. Process Control 1991, 1, 178−186. (21) Rajakumar, A.; Krishnaswamy, P. R. Time to Frequency Domain Conversion of Step Response Data. Ind. Eng. Chem., Process Des. Dev. 1975, 14 (3), 250−256. (22) Melo, D. L.; Friedly, J. C. On-Line, Closed-Loop Identification of Multivariable Systems. Ind. Eng, Chem. Res. 1992, 31, 274−281. (23) Ham, T. W.; Kim, Y. H. Process Identification and PID Controller Tuning in Multivariable Systems. J. Chem. Eng. Jpn. 1998, 31 (6), 941− 949. (24) Papastathopoulou, H. S.; Luyben, W. L. A New Method for the Derivation of Steady State gains for Multivariable Process. Ind. Eng. Chem. Res. 1990, 29, 366−369. (25) Rajapandiyan, C.; Chidambaram, M. Closed Loop Identification of Multivariable Systems by Optimization Method. Ind. Eng. Chem. Res. 2012, 51, 1324−1336. (26) Pramod, S.; Chidambaram, M. Closed Loop Identification of Transfer Function Model for Unstable Bioreactors for Tuning PID Controllers. Bioprocess Biosyst. Eng. 2000, 22, 185−188. (27) Pramod, S.; Chidambaram, M. Closed Loop Identification by Optimization Method for Tuning PID Controllers. Indian Chem. Eng. A 2001, 43 (2), 90−94. (28) Chidambaram, M. Applied Process Control; Allied Publishers: New Delhi, India, 1998, p 7. (29) Weischedel, K.; McAvoy, T. J. Feasibility of Decoupling in Conventionally Controlled Distillation Columns. Ind. Eng. Chem. Fundam. 1980, 19, 379−384. (30) Codrons, B. Process Modeling for Control: A Unified Framework Using Standard Black-Box Techniques, Springer-Verlag: London, 2005; p 48. (31) Savitzky, A.; Golay, M. J. E. Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Anal. Chem. 1964, 36 (8), 1627−1639. (32) Niderlinski, A. A Heuristic Approach to the Design of Linear Multivariable Interacting Control Systems. Automatica 1971, 7, 691− 701. (33) Garcia, D.; Karimi, A.; Longchamp, R. PID Controller Design for Multivariable Systems Using Gershgorin Bands. IFAC World Congress, Prague, Czech Republic, July 2005.

noise, controller settings, and perturbation in the guess values on the identified model are also studied. The maximum computational time for the optimization method is considerably reduced from 5 h 14 min 1 to 11 min 11 s (max. computational time) for SOPTD (2 × 2) model systems, from 1 h 2 min to 3 min 3 s (max. computational time) for higher order (2 × 2) systems and 9 min 14 s (max. computational time) for SOPTD (3 × 3) model systems.

■

AUTHOR INFORMATION

Corresponding Author

*Email: [email protected]. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE G, Gc = Process and controller transfer function matrices g, gc, gm = process, controller, and model transfer functions km, kp = model and process steady-state gains τm, τ = model and process time constant θm, θ = model and process time delay εm, ε = model and process damping coefficient ymr, yr = model and process set point Y = Closed-loop process response matrix y, ym = closed-loop process and model response s = Laplace transform variable tc = closed time constant of the model t = Time kc = ontroller gain τI = integral time τd = derivative time ts = settling time IAE = integral of absolute error ISE = integral of squared error

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REFERENCES

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Closed-Loop Identification of Second-Order Plus Time Delay (SOPTD

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