An Infinite Model Predictive Controller for Multi Input ...

Proceedings of the ASME 2017 International Mechanical Engineering Congress and Exposition IMECE2017 November 3-9, 2017, Tampa, Florida, USA

IMECE2017-70401

AN INFINITE MODEL PREDICTIVE CONTROLLER FOR MULTI INPUT NONLINEAR PROCESSES Ma’moun Abu-Ayyad Associate Professor, PhD Mechanical Engineering Penn State Harrisburg Middletown, PA, USA

Abdelkader Abdessameud Assistant Professor, PhD Electrical Engineering Penn State Harrisburg Middletown, PA, USA

Issam Abu-Mahfouz Associate Professor, PhD Mechanical Engineering Penn State Harrisburg Middletown, PA, USA

ABSTRACT This paper presents a novel algorithm of an infinite model predictive controller for controlling nonlinear multi-input multioutput (MIMO) processes. The new strategy uses a set of continuous nonlinear functions that captures the nonlinear characteristics of the MIMO plant over a wide operating range resulting in a more accurate prediction of the controlled variables. The method formulates a nonlinear dynamic matrix that is manipulated variable dependent during closed-loop control. The proposed algorithm was implemented on a nonlinear MIMO thermal system comprising of three temperature zones to be controlled with interacting effects. The experimental closed-loop responses of the proposed algorithm were compared to a multi-model dynamic matrix controller (DMC) with improved results for various setpoint trajectories. The MIMO process has nonlinear parameters such as process gain and time constant that are dependent on the size of the control actions. Good disturbance rejection was attained resulting in improved tracking of multi-setpoint profiles in comparison to multi-model DMC.

that are based on linear model predictive control (LMPC) [1]. The performance of LMPC is acceptable when the process operates within a narrow operating range or the rejection of small disturbances. In industry, many processes are nonlinear, multivariable and cannot be modeled and controlled adequately using linear models. The nonlinearities in these applications involve discontinuities (such as stiction), hysteresis, variable deadzone and other plant parameters which can be functions of the manipulated variable. These nonlinear characteristics as well as others make the tracking of setpoint profiles very difficult especially with large operating regimes. Furthermore, if the setpoint or reference trajectory and its changes are non-standard (ramps, parabolic etc.) closed-loop tracking becomes very challenging or almost impossible. To effectively control these MIMO nonlinear processes, a control strategy that uses the nonlinear characteristics of the process should be derived. Over the last decade, the method of predictive control has been shown to provide good control for processes that do not a high degree of nonlinearity. Many researchers have proposed predictive control strategies using nonlinear plant models. AbuAyyad et. al. [2] developed a nonlinear quadratic dynamic matrix control scheme by using a nonlinear model to compute the effect of past manipulated variables on the plant predicted output. In this method, a linear model is obtained by linearizing the nonlinear model every sampling instant. Gattu and Zafiriou [3] incorporated state estimation using a steady-state Kalman filter. This approach was applied to control a 22 continuous

INTRODUCTION Model predictive control (MPC) has been well received by academia and industry because it is intuitive and can explicitly handle multi-input multi-output (MIMO) systems with constraints. Until recently, most of the commercial MPC techniques have relied on linear dynamic models using schemes

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NOMENCLATURE A dynamic matrix of a MIMO system, a normalized response coefficient at time t, e future errors vector, K process gain, M number of manipulated variables, nu control horizon, P prediction horizon, R number of controlled variables, j Si ,k  u j  continuous nonlinear function,

stirred tank reactor and a single-input single-output (SISO) reactor that is unstable in the open-loop mode. Maner et. al. [4] presented a linear MPC algorithm based on the second-order Volterra series model. This method has the ability to describe nonlinear behavior such as symmetric output changes in response to symmetric changes in the input. The main drawback of these methods is that the model calculations can be very time consuming and can be elusive if the nonlinear characteristics of process are not well understood. Shaw and Doyle [5] presented a recurrent dynamic neuron network model to control a 22 nonlinear MIMO distillation column system. Input-Output linearization technique was used with internal model control for the controller design. The complexity of this method can be represented by the huge number of parameters that have to be identified during network training. Foss et. al. [6] developed an approach which can transfer from linear optimizing control to nonlinear optimizing control smoothly by adding constraints to provide an optimal solution for a set of bounding linear models. Therefore, the control actions are computed as a convex combination of control sequence based on the derived linear models. Townsend et. al. [7] used a hybrid learning approach for local models networks constructed from ARX local models and normalized Gaussian basis functions [8]. The local linear models were obtained using singular value decomposition analysis in conjunction with Quasi-Newton optimization for determining the centers and widths of interpolation functions. This method was demonstrated on a nonlinear MIMO system for the control of a pilot pH neutralization plant. Chen et. al. [9] controlled this pH plant under unmeasured disturbances using a neural network model predictive control. In this approach, the Levenberg-Marquardt and least square methods were used synchronously for training the neural network autoregressive exogenous model. A quasi-ARMAX modeling scheme was developed by Waller et al. [10] for the purpose of modeling multivariable nonlinear distillation column at different operating regimes. They showed that this modeling scheme has the ability to capture the nonlinearity and directionality of the process for narrow operating regimes. The major drawback with these algorithms is that model identification becomes difficult and complex when the plant operating regimes is wide ranging and highly nonlinear. This paper focuses on developing a generic control strategy for handling multivariable processes that are nonlinear where the controller parameters are reevaluated continuously. The algorithm is centered on using reformulated MIMO openloop models that are continuous nonlinear functions of the manipulated variables used in controlling the plant every time step. The controller to be formulated is the class of controllers in predictive control. The performance of the IMPC is demonstrated on a highly nonlinear MIMO plant and compared to a multi-model dynamic matrix control (DMC) technique.

Ta uc

Yˆ ysp Q

ᴧ λ τ

ambient temperature, closed-loop control action, process predicted response vector, setpoint vector or reference trajectory, normalized open-loop test coefficients matrix, manipulated variable dependent dynamic matrix, move suppression matrix , process time constant.

Infinite Model Predictive Control Theory In predictive control, the general algorithm uses a fixed model of the plant in order to determine a dynamic matrix. In most cases, this matrix is time invariant or can be of a multimodel form that is reconstructed at setpoint changes. The concept of the proposed algorithm is founded on the fact that the plant dynamic behavior is continuous during control. This leads to the approach that an infinite set of dynamic matrices can be evaluated from the continuous behavior of the plant in the limit as ∆t→0. Therefore, the continuous nature of the proposed controller is termed infinite model predictive control (IMPC). In linear DMC, A matrix is constructed as

AMIMO

1  a1,1  1  a1,2   1  a1, P  1  a2,1  a12,2    a1  2, P   1  aR ,1  a1R ,2    a1  R,P

0 1 a1,1

2 a1,1 2 a1,2

0 2 a1,1

M a1,1 M a1,2

0 M a1,1

a1,1 P 1

a1,2 P

a1,2 P 1

a1,MP

a1,MP 1

0 a12,1

2 a2,1 2 a2,2

0 a12,1

M a2,1 M a2,2

0 M a2,1

a12, P 1

a2,2 P

a2,2 P 1

a2,MP

a2,MP 1

0 a1R ,1

aR2 ,1 aR2 ,2

0 aR2 ,1

aRM,1 aRM,2

0 aRM,1

a1R , P 1

aR2 , P

aR2 , P 1

aRM, P

aRM, P 1

                      R P  M nu 

(1) In AMIMO, the first subscript indicates the sub-process, the second subscript indicates the position on the prediction horizon and the superscript indicates which manipulated variable was used to conduct the open-loop test. The formulation of AMIMO is obtained by conducting open-loop tests

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offline open-loop experiments. From these nonlinear functions, a model matrix of the system is now formulated. This model matrix has the ability to provide an infinite dynamic matrix set thereby capturing the nonlinear plant characteristics at each ∆t. The second component uses the closed-loop control actions and the model matrix to calculate the infinite dynamic matrix every ∆t. In addition, the infinite dynamic matrix is now used to evaluate the plant prediction and the magnitude of the changes in the control actions. The development of the IMPC methodology is based on conducting several open-loop tests (both positive and negative manipulated variable changes) on M as in Figure 1.

on each sub-process that is guided to reach within 95% of its steady-state value. Equation (1) can be expressed as A1M   A11 A12 A A22 A2 M  (2) AMIMO   21     ARM   AR1 AR 2 where Aij is the dynamic matrix of the corresponding controlled variable that is affected by each M. More details on linear conventional MIMO DMC can be found in [11]. IMPC Open-loop Modeling The IMPC structure consists of two main parts, the first being the development of continuous nonlinear functions using

t = Δt

t = kΔt

t = PΔt

u j ,m

y(t)

OL test m

···

·

·

u j ,2 OL test 2 OL test 1

u j ,1

Prediction Horizon Figure 1 Open-loop tests showing time slices The normalized response coefficients

The formulation of Si ,jk  u j  can take other forms and

aij,k of a sub-process

different approaches of analytical development. Also, some values of bnk can be zero. More importantly, Eq. (3) can be used to evaluate the discrete values of the normalized response coefficients k = 1, 2, … P for the controlled variable i excited by any magnitude of the manipulated variable j. Using the scalar values of Eq. (3), the normalized response coefficients

at instant k (circles on vertical slice), extracted from the m open-loop tests at sampling instant k∆t can be analytically expressed as a continuous nonlinear function of the open-loop test signal u j as

Si ,jk  u j ,m    bnk u hj n,m N

(3)

vector

n 1

where N is the order of fitted polynomials, bnk the polynomials coefficients and the exponentials h1 , h2 , are real numbers.

Si j can be expressed as Si j  AijU j

(4)

The sub-process model matrix Aij (not to be confused with Aij)

These exponentials are chosen to provide a best fit between the experimental values aij, k and the scalar outputs of Eq. (3).

contains the coefficients bnk constructed

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b11 b21 Aij  b12 b22  The

vector

Uj

bN 1  bN 2   P N

represents the

a relation between open-loop test signal

values of each open-loop test. This relationship can be expressed as

variable

uj

u j  f  yssj 

(5) and the

T

(6) U j  u hj1 u hj 2 u hj N  The task is to determine the model matrix Aij such that the step response coefficients Si j and the normalized experimental values agree as closely as possible. A simple approach for evaluating Aij is by the least square method [14]. The solution

IMPC Methodology and Structure In the previous section, the important derivation of the MVD dynamic matrix ᴧ was presented with the unique approach that it can be regenerated every sampling instant by exciting Eq. (3) with any set of M. The cornerstone of the IMPC methodology is that closed-loop control actions can be injected into Eq. (3) so as to fictitiously conduct open-loop testing while the system is in closed-loop mode. The significance of this approach is that more accurate plant predictions over P are expected resulting in improved control performance. Based on the above initial derivations and methodology, the general structure of IMPC is shown in Figure 2. First, the open-loop test signal vector u is made equal to uc. This is to facilitate a fictitious online open-loop test every ∆t so as to calculate normalized step responses using the continuous nonlinear functions Si j as in Eq. (3). These normalized M  R

for Aij is





1

Tj Q j

(7)

 j is a matrix of the vectors U j and the corresponding fitted exponentials of all m open-loop tests formulated as  u hj1,1 u hj 2,1 u hj ,1N   h1  u j ,2 u hj 2,2 u hj ,2N  (8)  j     h1  h2 u hj ,Nm  u j ,m u j ,m m N The parameters u j ,1 u j ,m  in Eq. (8) represent the same  

step responses are then used to formulate the nonlinear dynamic matrix ᴧMIMO as in Eq. (9).

manipulated variable j that is used to excite a sub-process at different magnitudes for each of m open-loop tests. The matrix Qj has the same steady state length formulated as T (9) Q j  Q1 Q2 Qm 

In general, the total response of each sub-process comprises of its own nonlinear response and the effect of other manipulated variables due to coupling between them (interactive nonlinear behavior). In this investigation, the total response is assumed to be a summation of M nonlinear responses even though this is not the case in practice since the actual response is due to a nonlinear combination of M which may not be additive. This assumption is aimed at reducing the number of open-loop tests required to formulate a more accurate nonlinear model. This is important since MIMO systems are generally slow reacting and testing of the plant can become complex and lengthy. In this work, the time to reach open-loop steady-state for any of the R processes is in the order of two hours or more.

For MIMO systems, the procedure through Eqs. (3–8) are repeated for each sub-process resulting in M  R matrix models. Based on the above derivations, using Si j , the new nonlinear form of AMIMO in Eq. (1) termed manipulated variable dependent (MVD) dynamic matrix ᴧ can be evaluated as

 MIMO

1  S1,1 (u1 ) 0  1 1 S  1,2 (u1 ) S1,1 (u1 )   1 1  S1, P (u1 ) S1, P 1 (u1 )  1 0  S 2,1 (u1 ) 1 1  S 2,2 (u1 ) S 2,1 (u1 )    S 1 (u ) S 1 (u ) 2, P 1 1  2, P 1   1 0  S R ,1 (u1 )  S R1 ,2 (u1 ) S R1 ,1 (u1 )    S 1 (u ) S 1 (u ) R , P 1 1  R,P 1

2 S1,1 (u2 ) 2 S1,2 (u2 )

0 2 S1,1 (u2 )

S1,2P (u2 ) S1,2P 1 (u2 ) 2 2,1 2 2,2

S (u2 ) S (u2 )

0 S (u2 ) 1 2,1

M S1,1 (uM ) M S1,2 (uM )

0 S1,1M (uM )

S1,MP (uM ) S1,MP 1 (uM ) M S 2,1 (uM ) M S 2,2 (uM )

0 M S 2,1 (uM )

S 2,2 P (u2 ) S 2,2 P 1 (u2 )

S 2,MP (uM ) S 2,MP 1 (uM )

S R2 ,1 (u2 ) S R2 ,2 (u2 )

S RM,1 (uM ) S RM,2 (uM )

0 S R2 ,1 (u2 )

S R2 , P (u2 ) S R2 , P 1 (u2 )

0 S RM,1 (uM )

S RM, P (uM ) S RM, P 1 (uM )

(11)

This use of this relationship will demonstrated in the following IMPC prediction and control section.

corresponding fitted exponentials as

AijT  Tj  j

u j and the steady-state

                    

(10) In ᴧMIMO, the same subscript and superscript notations are used as in Eq. (1). An important feature in IMPC is the derivation of

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uc  t  1

ysp

Future Errors

+

-

+

uc

Optimizer

y

uc

+

Plant

Cost Function

Future Reference

Infinite Model Calculator

u j  uc Si j  u j 

Yˆ Nonlinear Dynamic Matrix Calculator 

Nonlinear Predictor

Figure 2. Structure of IMPC Based on the assumption that the total response of each subprocess is additive, the effect of all M manipulated variable

IMPC Prediction and Control In this section the prediction of the controlled variables and evaluation of the control moves are presented. Using

ᴧ

changes on

(the

yˆi  t  k | t  is evaluated as M k  P 1

yˆi  t  k | t   yˆi  t   

subscript on ᴧMIMO is removed from here on) the objective function for IMPC is T (12) min J  e  uc  e  uc   ucT uc

 S u   S u  u t  k  v | t 

j 1 v  k 1

M



k



j 1 v  k  nu 1

uc

j i ,v

j

In Eq. (14), the first component

yˆ R to form Yˆ as shown in

move at instant t,

M k  P 1

j i ,v

j

j i  k ,v

j

j c

The prediction in Eq. (13) is repeated for each controlled variable to formulate the overall prediction vector Yˆ .

  S u   S u  u t  k  v | t    k 1

 S u   S u  u t  k  v | t  (15)

j 1 v  k 1

ucj both past and future is calculated

k  P 1 v  k 1

yˆi  t  k | t  is calculated as

yˆi  t  k | t   yˆi  t   

using Si j u j as yˆi  t  k | t   yˆi  t  

yˆi  t  is the current prediction

at instant t, the second is the prediction due to past moves and the third is the prediction due to current and future moves. Since the plant will not be affected by the present and future

yˆi  t  k | t  due to any change in any one of the jth

 

j c

(14)

Figure 2. Note that ᴧ in Eq. (12) is recalculated every ∆t. The prediction response vector of each controlled variable

manipulated variable

j

Si ,jv  u j  ucj  t  k  v | t 

e vector is evaluated as the difference between ysp vector and its corresponding predictions yˆ1

j i  k ,v

k

j i ,v

j

j i  k ,v

j

j c

v  k  nu 1

Si ,jv  u j  ucj  t  k  v | t 

P

(13)

5

Initially at t = 0, the nonlinear dynamic matrix ᴧ is calculated as follows  Using the corresponding setpoint values for each controlled variable input these values to solve for an open-loop input u j using Eq. (10).

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

Inject the values of

u j into Eq. (3) to determine normalized

open-loop response coefficients vector Si j for each control variable and each sub-process to formulate an initial nonlinear dynamic matrix ᴧ. These initial steps are unique to IMPC which do not exist in any MPC algorithms. The task of the optimizer in Figure 2 is to minimize the cost function J using a simple least squares method. This minimization results in an unconstrained closed-form solution for the M closed-loop control moves uc as



uc  T    I The

M



1

T e

closed-loop

(16) control

in uc contains

moves

nu  M manipulated variable elements as uc  u 1 1 c

u  nu  1 c

u 1 M c

Figure 3 MIMO temperature control system u

M c

 nu  (17) T

The duty cycle used for the PWM signal has a fixed duration of 5 seconds, i.e the sum of the high and low states. Therefore, in control, the IMPC controller would calculate the durations of the M high logic states. Practical experiments using the proposed algorithm were conducted and compared to the multimodel DMC method with details in the following section.

Since only the first control action of each manipulated variable vector is sent to the plant, the manipulated variable vector uc is obtained as the sum of

uc  uc1 1

uc at t and uc at  t  1 expressed as ucM 1

T

(18)

Open-loop testing and formulation of  Several open-loop tests were conducted on the zones using different input signals to generate the continuous nonlinear functions Si ,jk  u j ,m  for each zone. The open-loop tests are

As shown in Figure 2, the open-loop signals is equated to the control moves in uc as in Eq. (16) given by

u j  uc

conducted by implementing a step function in one of the M manipulated variables while maintaining zero input to the others ~ is a dimensionless value and vice-versa. The input signal u between 0 and 1 representing a portion of the 5 sec. duty cycle that has a high state (heater on). The normalized step response coefficients for each zone is calculated as  T  k t   Ta  (20) i , j   i , j  uj T  T max a   Tmax is the maximum steady state temperature (due to the heater fully on). The open-loop tests were filtered to remove the noise effects. From these tests, K and τ are formulated as functions of the input signal u1 for all zones are shown in Figure 4. It can be

(19)

Therefore using Eq. (19), subsequent evaluations of t  0 can be performed.

ᴧ

for

Real-time application of MIMO IMPC In this study a real-time application of MIMO IMPC was conducted to control the temperature of three steel cylinders which represents a slow reacting nonlinear 3×3 MIMO system. The steel cylinders are in tight contact with each other and are encased circumferentially by individual electrical heater bands (200 Watts) as shown in Figure 3. The dimensions of the cylinders are identical with radius 25 mm and length 150 mm. As a result, the MIMO system has 3 control variables and 3 manipulated variables. The temperature of each cylinder or zone is measured by an ungrounded E-type thermocouple. The energy that heats each zone is manipulated via an electronic solid state relays (one for each control variable) that is pulse width modulated (PWM) using a digital output which can change state (high or low).

seen from these figures that the zones are nonlinear in K and τ over the range in input signal u1 . Other tests were conducted on zone 2 and zone 3 with similar results not shown.

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Closed-loop responses of the IMPC and multi-model DMC schemes are shown in Figure 5. The tuning parameters nu = 2, P=3000, T=5 sec and λ=1.0001 are used to control the temperature of the zones for both control schemes.

Figure 4 Process gains K and time constants using



for all zones

u1

A sixth order fitted polynomial was used for all zones to describe the open-loop tests from which the normalized step response coefficients Si ,jk  u j  are evaluated every time slice

Figure 5 Closed-loop comparisons between IMPC and multimodel DMC

(Figure 1) as

From Figure 5, IMPC performs better than multi-model DMC as it settles faster and has shorter rise time with negligible overshoot in all 3 zones. These results are expected since the IMPC controller exhibits larger control actions uc resulting in

2 3 4 5 6 Si ,jk (u j )  b1k u 0.5 j  b2 k u j  b3k u j  b4 k u j  b5k u j  b6 k u j  b7 k u j (21)

A good agreement (higher that 98% fit) was obtained between the actual and fitted response coefficients Si j for zone 1

u j providing a more aggressive nonlinear dynamic matrix

Figure 6 shows the manipulated variables for both schemes for zone 1 indicating larger values during the transient portion of the close-loop response using IMPC, with other zones having similar results.

as well as for other responses and its corresponding zones. It should be noted that the prediction horizon P is in order of 3000 which allows all zone responses to be within 95% of its steadystate value. As a result, the coefficients in Eq. (18) which are in Aij are not provided. It is now possible using the Eqs. (3–8) to formulate the nonlinear dynamic matrix

ᴧ.

ᴧ including the initial

value of ᴧ. Closed-loop temperature control Closed-loop tests were conducted using setpoints values (test 1) of 240°C, 200°C and 160°C for zones 1, 2 and 3 respectively. The setpoint selection of 200°C and 240°C represent a moderate degree of nonlinearity as indicated in Figure 4, where the range of u j is between 0.4 and 0.6. In addition, the setpoint selection of 160°C represents a higher degree of nonlinearity where the range of u j is less than 0.4. The dynamic matrices of the multi-model DMC were evaluated for 4 equally spaced regions corresponding to a setpoint value for each zone from open-loop tests. As an example for the 240°C setpoint, as the closed-loop response crosses 60°C, the dynamic matrix is switched and so on.

Figure 6 Manipulated variable using IMPC and multi-model DMC for zone 1

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Even though

AMIMO in multi-model DMC is formulated

from a low band or region of the normalized step response coefficients during the transient stage, the algorithm provides large values of uc similar to IMPC during this stage. This is due to the minimization of large error values over P resulting in a high uc. During closed-loop control, the IMPC structure reduces to the ordinary DMC form in the steady state region since uc  0 resulting in negligible changes to ᴧ. Another operating region has been investigated as shown in Figure 7. The new setpoint profiles (test 2) were chosen from a higher nonlinear region where the input signal u j varied between 0.01  0.25 as in Figure 4. The tuning parameters used were the same as the previous tests. It can be seen from the results in Figure 7 that the IMPC controller reaches the setpoints in shorter time in comparison to the multi-model DMC scheme. Figure 8 Closed-loop responses for MIMO temperature control system (test 2) In addition to setpoint tracking, the controllers were tested on rejecting an unexpected disturbance when all 3 zones have reached close to the desired setpoints. The disturbance was applied by generating an airflow over the 3 zones for 65 minutes with an average speed and temperature of 3.1 m/s and 16.20C respectively. The effectiveness of IMPC in rejecting the disturbance was compared to multi-model DMC as shown in Figure 9. It can be seen that the IMPC controller reacts faster to compensate for the unexpected disturbance in comparison to multi-model scheme. The letters S and E represent the starting and ending points of the disturbance.

Figure 7 Closed-loop comparisons between IMPC and multimodel DMC Setpoint changes and disturbance rejecting Figure 8 shows closed-loop responses due to setpoint changes for all zones using the initial setpoint profiles from test 2 with positive changes for each zone. The results show that IMPC controller has the ability to track the setpoint profiles smoothly with minimal overshoot and faster settling time in comparison to multi-model DMC. This improved result of IMPC was achieved because of the change in the nonlinear dynamic matrix

ᴧ

every ∆t capturing the nonlinear characteristics of each controlled variable as they move from one state to another. Figure 9 Disturbance rejection of IMPC and multi-model DMC

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Conclusions A MIMO infinite model predictive control method that is manipulated variable dependent has been developed for processes that have nonlinear parameters such as process gain and time constant that are dependent on the size of the control actions. The basic algorithm of IMPC is founded on the formulation of a manipulated variable dependent system matrix that is reevaluated every sampling instant from continuous nonlinear functions. The major advantage of this approach over other predictive controllers is that it uses a unique mechanism of open-loop testing and modeling during closed-loop control. The proposed algorithm was tested on a challenging MIMO reacting process having a relatively high degree of nonlinearity with improved control performance as compared to a multi-model DMC controller. The method also demonstrated improved tracking of multi-setpoint profiles and good disturbance rejection in comparison to multi-model DMC.

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