Downloaded from orbit.dtu.dk on: Jan 2, 219 A uning Procedure for ARX-based MPC of Multivariate Processes Olesen, Daniel Haugård; Huusom, Jakob Kjøbsted; Jørgensen, John Bagterp Published in: 213 American Control Conference (ACC) Publication date: 213 Document Version Publisher's PDF, also known as Version of record Link back to DU Orbit Citation (APA): Olesen, D., Huusom, J. K., & Jørgensen, J. B. (213). A uning Procedure for ARX-based MPC of Multivariate Processes. In 213 American Control Conference (ACC) (pp. 1721-1726). IEEE. American Control Conference General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
213 American Control Conference (ACC) Washington, DC, USA, June 17-19, 213 A uning Procedure for ARX-based MPC of Multivariate Processes Daniel Haugaard Olesen, Jakob Kjøbsted Huusom, John Bagterp Jørgensen Abstract We present an optimization based tuning procedure with certain robustness properties for an offset free Model Predictive Controller (MPC). he MPC is designed for multivariate processes that can be represented by an ARX model. he stochastic model of the ARX model identified from inputoutput data is modified with an ARMA model designed as part of the MPC-design procedure to ensure offset-free control. he MPC is designed and implemented based on a state space model in innovation form. Expressions for the closed-loop dynamics of the unconstrained system is used to derive the sensitivity function of this system. he closed-loop expressions are also used to numerically evaluate absolute integral performance measures. Due to the closed-loop expressions these evaluations can be done relative quickly. Consequently, the tuning may be performed by numerical minimization of the integrated absolute error subject to a constraint on the maximum of the sensitivity function. he latter constraint provides a robustness measure that is essential for the procedure. he method is demonstrated for two simulated examples: A Wood-Berry distillation column example and a cement mill example. I. INRODUCION Model Predictive Control (MPC) has evolved to become an industrial standard in advanced process control [1]. Using a model of the system to predict the process output over some future horizon, MPC computes a trajectory of manipulated inputs such that the predicted future output is as desirable as possible. Only the inputs related to the first period in this trajectory are implemented. As new measurements become available, the estimation and regulation windows are shifted and the estimation and optimization procedures are repeated. In this paper we consider MPC based on ARX models. An ARX model representation of the plant may be obtained from input-output data using convex optimization methods [2]. o ensure offset free control, integrators have to be introduced in the plant model in case of persistent unmeasured disturbances and/or plant model mismatch. In such cases, the observer that guarantees offset free control introduces a plant model mismatch. his plant model mismatch complicates the tuning of the controller [3] [6]. Despite the growing popularity of MPC, a systematic tuning practice has not evolved, and only few guidelines exist. he topic has not been short of research, as there are numerous academic publications on the subject. A comprehensive review of proposed tuning methods is presented by [7] and loop transfer recovery procedures have also been investigated [8], [9]. Our study relies on a closed loop description of the D.H. Olesen and J.B. Jørgensen are with DU Compute, echnical University of Denmark, DK-28 Kgs Lyngby, Denmark. Corresponding author: J.B. Jørgensen, jbjo@dtu.dk J.K. Huusom is with DU Chemical Engineering, echnical University of Denmark, DK-28 Kgs Lyngby, Denmark controller and the process model to assess the performance of an MPC with a given tuning. It has previously been proposed to use a closed loop description for synthesis of a MPC by application of robust design techniques [1]. In this paper, we state the tuning problem as an inequality constrained optimization problem. We propose a deterministic tuning objective function related to the integrated absolute error for a number of pre-defined scenarios and use a bound on the maximum sensitivity to ensure robustness. he paper is organized as follows. Section II describes an ARX-based MPC for multivariate processes. Section III derives a state-space model for the closed-loop system and uses this state space model for covariance computation and sensitivity function computation. IAE measures and the sensitivity function are used to formulate an optimization problem for selecting the tuning parameters of the MPC. Section IV demonstrates the procedure for a Wood-Berry binary distillation example, while Section V provides a case study for a simulated cement mill. Conclusions are presented in Section VI. II. ARX-BASED MPC FOR MIMO SYSEMS In this section, we derive a state space representation for an unconstrained MPC based on MISO ARX-models modified with a filtered integrated white noise stochastic model. First, we represent the MISO ARX model as a state space model in innovation form. Subsequently, we use this state space model in innovation form to derive the correct control law for the unconstrained MPC. As the control law is linear the resulting controller may be represented in a state space form. A. State Space Model in Innovation Form he MISO ARX model A i (q 1 )y i,k = B i (q 1 )u k +ε i,k i = 1,...,n y (1) with y i,k R for i = 1,...,n y, u k R nu, and ε i,k R for i = 1,...,n y has been used in a number of MPC applications. he advantage of this model parametrization is that the parameters may be identified using standard system identification techniques based on convex optimization. o have offset-free control from the MPC based on this model, the stochastic part of the model is modified to be a filtered white noise process ε i,k = 1 α iq 1 1 q 1 e i,k i = 1,...,n y (2) where e i,k N iid (,R ee ). he coefficients α i are design parameters of the MPC. 978-1-4799-178-4/$31. 213 AACC 1721
he representation of the MIMO system from these MISO systems is not unique. One straightforward representation leading to a compact notation is A(q 1 )y k = B(q 1 )u k + ( I Iq 1) 1 (I Aq 1 )e k (3) with A(q 1 ) = diag([a 1 (q 1 );...; A ny (q 1 )]), B(q 1 ) = [B 1 (q 1 );...;B ny (q 1 )], and A = diag([α 1 ;...;α ny ]). his model can be represented as an ARMAX model with Ā(q 1 )y k = B(q 1 )u k + C(q 1 )e k (4) Ā(q 1 ) = (I Iq 1 )A(q 1 ) B(q 1 ) = (I Iq 1 )B(q 1 ) C(q 1 ) = I Aq 1 Denote the coefficients of Ā(q 1 ) and B(q 1 ) as Ā(q 1 ) = I +Ā1q 1 +Ā2q 2 +...+Ānq n B(q 1 ) = B 1 q 1 + B 2 q 2 +...+ B n q n (5a) (5b) (5c) (6a) (6b) hen the system (1)-(2) may be represented as a state space model in innovation form x k+1 = Âx k + ˆBu k + ˆKe k y k = Ĉx k +e k (7a) (7b) with the state space matrices (Â, ˆB, ˆK,Ĉ) realized in observer canonical form Ā1 I Ā2 I  =........ ˆB = Ān 1 I Ān Ĉ = [ I ] B 1 B 2. B n 1 B n A Ā1 Ā2 ˆK =. Ān 1 Ān B. Unconstrained MPC for State Space Models in Innovation Form he filtered state estimation and the one-step prediction may for state space models in innovation form (7) be combined to give the following expressions for computation of the innovation, e k [5]: ˆx k k 1 = ˆx k 1 k 2 + ˆBu k 1 + ˆKe k 1 ŷ k k 1 = Ĉˆx k k 1 e k = y k ŷ k k 1 (8a) (8b) (8c) Initially, ˆx 1 is known and the one-step prediction (8a) is not needed. Knowing the innovation, e k, the predictions in the state space model in innovation form may be represented as [5] ˆx k+1 k = ˆx k k 1 + ˆBû k k + ˆKe k (9a) ˆx k+1+j k = ˆx k+j k + ˆBû k+j k, j = 1,...,N 1 (9b) ŷ k+j k = Ĉˆx k+j k, j = 1,...,N (9c) It is important to notice the term ˆKe k in (9a). his term is important for derivation of the correct control law [5]. Let the objective of the MPC be φ = 1 2 N 1 j= min {û k+j j} N 1 j= ŷk+j+1 k r k+j+1 k 2 Q + ûk+j k 2 S (1) in which the second term, ûk+j j 2, is a regularization S { } term. We assume the reference parametrization, N rk+j k = {r j=1 k,...,r k }. he tuning parameters in this objective function are the matrices Q = diag([q 1 ;...;q ny ]) and S = diag([s 1 ;...;s nu ]). As indicated, these matrices are restricted to diagonal matrices. he unconstrained MPC may be represented as the convex quadratic optimization problem { φ = φ( { } } N 1 û k+j j ;ˆx j= k k 1,r k,u k 1,e k ) : (9) which has the solution U k = [û k k,...,û k+n 1 k ] with [5] u k = û k k = L xˆx k k 1 +L w e k +L u u k 1 +L r r k (11) he specific expressions for and derivation of L x, L w, L u and L r are given in [5]. It must be emphasized that most available expressions for linear-quadratic controllers misses the term L w e k that arises due to the term ˆKe k in (9a). Define the controller states as x c k = [ˆx k k 1;u k 1 ] such that the unconstrained MPC consisting of (8) and (11) may be represented in the state space form x c k+1 = A c x c k +B cy y k +B cr r k u k = C c x c k +D cy y k +D cr r k with [ (  A c = ˆKĈ)+ ˆB(L ] x L w Ĉ) ˆBLu L x L w Ĉ L u [ ] ] ˆK + ˆBLw [ˆBLr B cy = B L cr = w L r C c = [ ] L x L w Ĉ L u (12a) (12b) (13a) (13b) (13c) D cy = L w D cr = L r (13d) In addition to the model (1), this controller representation depends on the tuning parameters A = diag([α 1 ;...;α ny ]) Q = diag([q 1 ;...;q ny ]) S = diag([s 1 ;...;s nu ]) III. CLOSED-LOOP SYSEM AND MEASURES Let the system be a LI system in the form x k+1 = Ax k +Bu k +Ed k +Gw k z k = Cx k y k = z k +v k (14a) (14b) (14c) (15a) (15b) (15c) where x k is states, u k is manipulated inputs, d k is unknown disturbances, w k N iid (,R ww ) is process noise, z k is outputs, v k N iid (,R vv ) is measurement noise, and y k is measurements, i.e. the outputs, z k, corrupted by 1722
R(z) Controller C ur(z) C uy(z) D(z) W(z) Y(z) U(z) Process G zd(z) G zw(z) G zu(z) Fig. 1. Closed-loop system. he transfer functions C ur(z) and C uy(z) forms the unconstrained MPC. he controlled outputs of the process is described by G zu(z), G zw(z) and G zd (z). measurement noise, v k. his model (A,B,E,G,C) is not necessarily identical to the model (Â, ˆB,Ĉ, ˆK) used by the MPC. Using the system model (15) and the MPC state space representation (12), the closed-loop system may be represented as x cl k+1 = A cl x cl k +B wcl w k +B vcl v k +B rcl r k +B dcl d k (16a) z k = C cl x cl k (16b) y k = C cl x cl k +v k (16c) u k = C ucl x cl k +D vcl v k +D rcl r k (16d) with x cl k = [x k;x c k ] and [ ] [ A+BDcyC BC A cl = c G, B B cyc A wcl =, c ] [ ] [ ] [ BDcy BDcr E B vcl =, B B rcl =, B cy B dcl =, cr ] C cl = [ C ], C ucl = [ D cyc C c ], D vcl = D cy, D rcl = D cr. Z(z) V(z) (17) his representation depends on the MPC tuning parameters, (A,Q,S), and is used extensively to compute measures for the controller performance. An obvious measure is the eigenvalues of the closed-loop system, λ = eig(a cl ), as acceptable tunings must provide stable closed loop systems. A. Covariance he covariance of the outputs, (z k,y k,u k ), for the closedloop system as response to the exogenous stochastic signals, w k and v k, is one measure for the performance of the MPC. Provided that A cl is stable, the covariance of the states of the closed loop system, R xx, may be computed by solution of the discrete Lyapunov equation R xx = A cl R xx A cl +B wcl R ww B wcl +B vcl R vv B vcl (18) he corresponding output covariances are R zz = C cl R xx C cl R yy = C cl R xx C cl +R vv R uu = C ucl R xx C uc +D vcl R vv D vcl (19a) (19b) (19c) B. Sensitivity Fig. 1 illustrates the transfer functions in the process model and the model predictive controller. he transfer function model of the open-loop system (15) is Y(z) = G zu (z)u(z)+g zd (z)d(z)+g zw W(z)+V(z) (2) and the transfer function model of the MPC control law (12) may be represented as U(z) = C uy (z)y(z)+c ur (z)r(z) (21) G zu (z), G zd (z), G zw (z), C uy (z), and C ur (z) may be computed from the associated state-space representations in the standard way. Combining (2) and (21) yields a transfer function for the closed-loop system (16) Y(z) = S(z) D(z)+(z)R(z) (22) with D(z) = G zd (z)d(z)+g zw (z)w(z)+v(z) and S(z) = C cl (zi A cl ) 1 B vcl +I (z) = C cl (zi A cl ) 1 B rcl (23a) (23b) S(z) is the sensitivity function and (z) is the complementary sensitivity function. he sensitivity function, S(z), is related to the robustness of the system in relation to modelplant mismatch as well as process and measurement noise [11]. In particular the H norm of S(z) M S = S(z) = max ω σ(s(e jωs )) (24) has been used as a measure of robustness. σ denotes the maximum singular value and s denotes the sampling time. C. Integrated Absolute Error he integrated absolute error (IAE) is a classical way to measure control systems performance for certain reference and disturbance scenarios of systems without noise (w k = and v k = ). Consider a scenario starting from steady state and specified by [r(t)] t f t = {r k } n f 1 and [d(t)] t f t = {d k } n f 1 with r(t) = r k and d(t) = d k for t k t < t k+1 using t k = t +k s and t f = t +n f s. he IAE of this scenario is approximated by euler integration to be: J i = n f 1 y i,k r i,k i = 1,...,n y (25) Equation (25) is evaluated by simulation using the deterministic part of (16), the initial steady state, x cl =, and the specified scenario, {r k } n f 1 and {d k } n f 1. he scenarios, j S, for evaluation of the IAE-measures, J ij with i = 1,...,n y can be chosen according to the tasks of a given control system. In this paper we consider two standard type of scenarios. he first type of scenarios are related to individual set-point changes and consist of a set of n y scenarios, S r, with unit step changes in each individual reference, (r j ) k = 1 for k < n f and j S r. We denote the performance matrix associated with these scenarios J r = [J ij ] for i = 1,...,n y and j S r. he second type of scenarios are related to disturbance rejection. 1723
his set of scenarios, S d, consists of n d scenarios with a unit step in each individual disturbance, (d j ) k = 1 for k < n f and j S d. he performance matrix associated with these scenarios is denoted J d = [J ij ] with i = 1,...,n y and j S d. D. uning In the tuning of the MPC, the control and prediction horizon, N, is chosen sufficiently large such that the resulting controller for all practical purposes corresponds to an infinite horizon controller. he remaining tuning parameters, (A,Q,S), are chosen by solution of the constrained optimization problem min A,Q,S J = J r (A,Q,S) 2 + J d (A,Q,S) 2 (26a) s.t. M S (A,Q,S) M S,max (26b) A I (26c) Q Q max (26d) S S max (26e) he objective minimizes some measure related to the IAE of the chosen scenarios. In the cases studied in this paper, we have used the sum of the 2-norms of the matrices associated with the IAE of setpoint changes and disturbance rejections. One could also use the sum of all scenarios, ny i=1 J ij with S = S r S d, and expect J = j S similar results. It is critical for the usefulness of the resulting tuning that the robustness constraint (26b) is included in the optimization problem. Useless results are obtained if the robustness constraint (26b) is discarded by using a large upper bound. In such cases, the resulting controller is far too aggressive and useless in practice. M S,max is a user selected parameter used for deciding how robust the resulting closed loop system should be. Smaller values gives a less aggressive and more robust controller. Equation (26) is a constrained nonlinear optimization problem which is not necessarily convex. Accordingly, we cannot guarantee location of the global optimum of (26) when using solvers such as fmincon, KNIRO, IPOP, NLOP, or SNOP. IV. WOOD-BERRY DISILLAION COLUMN We consider a Wood-Berry binary distillation column that has the input-output description Y(s) = G u (s)u(s)+g d (s)(d(s)+w(s))+v(s) (27) with u(t) = u k, d(t) = d k and w(t) = w k N iid (,R ww ) being piecewise constant in the interval t k t < t k+1 and v(t k ) = v k N iid (,R vv ). he transfer functions are [12] [ 12.8 18.9 ] G u (s) = 16.7s+1e s 21.s+1 e 3s (28a) G d (s) = 6.6 19.4 1.9s+1e 7s 14.4s+1 [ e 3s 3.8 ] 14.9s+1 e 8.1s 4.9 13.2s+1 e 3.4s (28b) he Wood-Berry binary distillation column separates water and methanol. Y 1 is the methanol mole fraction in the ABLE I PERFORMANCE MERICS FOR HE CLOSED-LOOP WOOD-BERRY SYSEM USING HE UNING OBAINED FROM (26). Design Simulation 9.78 1.78 11.14 3.19 J r 2.85 12.8 4.39 13.45 7.22 8.43 J d 2.27 21.54 1.34 1 R 4.19 1 4 1.28 1 4.14 1 4 yy.19 1 4 1.78 1 4.14 1 4 1.52 1 4 R uu 9.26 1 6 1.1 1 6 1.1 1 6 5.31 1 6 8.93 1 6.51 1 6.51 1 6 4.43 1 6 distillate [mol%], Y 2 is the methanol mole fraction in the bottom product [mol%], U 1 is the reflux flow rate [lb/min], U 2 is the steam flow rate [lb/min], and D is the unmeasured feed flow rate [lb/min]. he sampling time of the system is s = 1 [min]. he process and measurement noise covariance for the system are: R ww =.1 and R vv =.1 I. he resulting system is realized as a discrete LI state space system (15). he control and prediction horizon for the MPC is selected to 4 min, i.e. N = 4. A(q 1 ) and B(q 1 ) in (1) are identified such that there is an exact match to G u (s) in (27). Using a robustness bound of M S,max = 1.775, the described tuning procedure, i.e. solution of (26), yields the following tuning parameters A = diag([.963;.933]) Q = diag([87.3; 57.8]) S = diag([4.87 1 4 ; 6.88 1 4 ]) able I shows the metrics, J r and J d, obtained by solution of (26) for the nominal system with the specified scenarios, S. he corresponding theoretical covariances, R yy and R uu, are also illustrated. he simulation-column in able I shows the metrics, J r and J d, for the scenarios simulated with additional process and measurement noise. It is evident that for the chosen tuning, these measures, J r and J d, do not deteriorate significantly. he covariances R yy and R uu in the simulation-column of able I are computed empirically by a Monte Carlo method from a finite sequence of process and measurement noise applied to the closed-loop system without deterministic disturbances and set-point changes. he nice properties of the selected tuning of the controller manifest itself by covariance-matrices having the same size as the covariance matrices for the design case. We have made an additional simulation using an operating scenario from [12]. he reference for top methanol (distillate) is changed from 96.25 [mol%] to 97 [mol%] and at t = [min], a change in the feed occurs. Fig. 2 shows a nominal and a plant-model mismatch simulation for this scenario. For the nominal simulation we assume that G u (s) is exactly known and the model used for the controller design is identical to the model used for simulation. In the mismatch case, the time constants in G u (s) are 75% of the nominal values for the simulation model. he tuning is robust, since the deviations between the nominal and mismatch case is 1724
op Methanol [mol%] Reflux Flowrate [lb/min] 97.5 97 96.5.75 96 2.3 2.2 2.1 2.75 1.9 Bottom Methanol [mol%] Steam Flowrate [lb/min] 1.4 1.2 1.8.6.75.4 1.9 1.85 1.8 1.75.75 1.7 Fig. 2. Nominal and plant-model mismatch simulation of the Wood-Berry distillation column controlled by the ARX based MPC tuned using (26). marginal. Furthermore, it can be concluded, that the MPC with the selected tuning rejects the disturbance nicely and have good tracking properties. V. CEMEN MILL CIRCUI In this section, we illustrate the tuning procedure and the role of M S,max, for the cement mill system described by [13]. he cement mill is modeled as a continuous-time stochastic input-output model (27) with piecewise constant input signals, i.e. u(t) = u k, d(t) = d k and w(t) = w k N iid (,R ww ) in the interval t k t < t k+1 and v(t k ) = v k N iid (,R vv ). he transfer functions are G u (s) = G d (s) =.62e 5s (45s+1)(8s+1) 15e 5s 6s+1 1.e 3s (32s+1)(21s+1) 6 (3s+1)(2s+1).29(8s+1)e 1.5s (2s+1)(38s+1) 5e.1s (14s+1)(s+1) (29a) (29b) he variables in the model are: Y 1 is the elevator load [kw], Y 2 is the cement fineness [cm 2 /g], U 1 is the feed flow rate [PH], U 2 is the separator speed [%], and D is the clinker hardness [HGI]. he continuous-time input-output model (27) is converted to a discrete-time state space model (15) using a sample time of s = 2 [min]. he covariances of the process and measurement noise are: R ww = 1. and R vv = diag([.1; ]). he ARX-based MPC is designed for a sampling time of s = 2 [min] with a prediction and control horizon of 8 [min], i.e. N = 4. o illustrate the role of the robustness bound, the tuning is performed using two different values for M S,max. In the first case, we use M S,max = 1.775. he tuning parameters obtained by solution of (26) are A = diag([.985;.]) Q = diag([14.1; 91.6]) S = diag([9.68 1 5 ; 9.29 1 3 ]) ABLE II PERFORMANCE MERICS FOR A CEMEN MILL CONROLLED BY AN ARX BASED MPC UNED WIH M S,max = 1.775 IN (26). Design 81.63 2.35 J r.85 7.15 42.54 J d 22.7.4.77 R yy.77 154.16 46.1 452.1 R uu 452.1 446. Simulation 212.35 176.35 3769.4 3769. 184.26 3766.8.3.86.86 136.41 416.46 41.12 41.12 45.5 able II lists the associated performance metrics for the cement mill controlled by an ARX based MPC using these parameters. he design-column lists the metrics obtained for the nominal system used in selecting the tuning parameters (A,Q,S). he simulation-column lists the metrics obtained by simulating the closed-loop system with stochastic process and measurement noise using the determined tuning parameters. By inspection of the design-column, J r, J d and R yy look reasonable. he only exception should be the high variance on the cement fineness, Y 2. However, R uu is very large and suggests that the proposed controller is sensitive to process and measurement noise. his suggestion is confirmed by the simulation-column. When the system is simulated for the scenarios S with additional process and measurement noise, the integrated absolute error measures, J r and J d, deteriorates significantly. he empirically obtained covariances obtained by a stochastic simulation do not change significantly. his illustrates the usefulness of these covariances in assessing the sensitivity of the system. In particular, the sensitivity is often revealed through the magnitude of the input covariances, R uu. o illustrate how useless the proposed tuning with M S,max = 1.775 is, the system is simulated for a scenario in which the elevator load is initially changed from 26 [kw] to 3 [kw] and a change in the clinker hardness is introduced at t = 8 [min]. Fig. 3 illustrates closed-loop simulations for the nominal case and a plant-model mismatch case where the dead times of G u (s) is increased by 5%. In both cases, the tuning gives variances of the input signals that are ridiculous large. Accordingly, the resulting controller is useless in practice. o improve the robustness of the system and make it less sensitive to noise, the robustness bound is reduced to M S,max = 1.3 and a new tuning is computed from (26) A = diag([.992;.852]) Q = diag([382; 76]) S = diag([9.91 1 5 ; 4.87 1 5 ]) able III illustrates the performance metrics of the controlled system with this tuning. From a deterministic point of view, the reduced robustness bound, M S,max, results in worse disturbance rejection and reference tracking. However, the associated input covariance, R uu, is significantly lower for the tuning with M S,max = 1.3 compared to the tuning with 1725
35 3 35 3 Elevator Load [kw] 3 25 1.5d d 2 Fineness [cm 2 /g] 315 3 35 3 Elevator Load [kw] 3 25 1.5d d 2 Fineness [cm 2 /g] 315 3 35 3 Feed [PH] 3 Seperator Speed [%] 15 5 5 Fig. 3. Nominal and plant-model mismatch simulations of a cement mill controlled by an ARX-based MPC using a tuning with M S,max = 1.775. Feed [PH] 3 Fig. 4. Nominal and plant-model mismatch simulations of a cement mill controlled by an ARX-based MPC using a tuning with M S,max = 1.3. Seperator Speed [%] 15 5 ABLE III PERFORMANCE MERICS FOR A CEMEN MILL CONROLLED BY AN ARX BASED MPC UNED WIH M S,max = 1.3 IN (26). Design 97.78 2.13 J r 2.52 1.69 J d 155.49 139.75.17.2 R yy.2 121.26 3.45 6.29 R uu 6.29 12.24 Simulation 179.9 128.3 3245.6 3245.5 199.5 3277.4.17.27.27 18.95 3.19 5.77 5.77 11.18 M S,max = 1.775. he typical working scenario is also simulated for a system using an MPC with this tuning. his simulation is illustrated in Fig. 4. he controller with this tuning is less aggressive. he tracking and disturbance rejection properties are slightly affected, but the system is much less sensitive to stochastic process and measurement noise. By inspection of Fig. 3 and Fig. 4, it is obvious that the tuning obtained with M S,max = 1.3 gives the best controller and a controller with acceptable performance. For this controller, the deviations between the nominal and the mismatch case are nearly indistinguishable. his illustrates the nice robustness properties of the controller. VI. CONCLUSION In this paper, we have presented a procedure to tune an ARX-based MPC for multivariate processes. he ARXbased MPC has been designed such that it gives offsetfree control for type I disturbances (steps). he suggested tuning is obtained by minimizing a measure related to the integrated absolute error for a set of pre-determined scenarios. Robustness of the resulting tuning is obtained by restricting the maximum of the sensitivity function by an upper bound. he method has been demonstrated for a binary distillation column and for a cement mill example. he setup in this paper, using disturbance scenarios, has been used for illustrative purposes. In practice the model from the unknown disturbance to the output is not necessarily known. In such cases the disturbance rejection scenarios are replaced by scenarios in which parameters of the ARX model is varied; i.e. the gain, the time delay, or the time constants of the corresponding transfer function model are varied. REFERENCES [1] S. J. Qin and. A. Badgwell, A survey of industrial model predictive control technology, Control Engineering Practice, vol. 11, pp. 733 764, 3. [2] D. A. Finan, J. B. Jørgensen, N. K. Poulsen, and H. Madsen, Robust model identification applied to type 1 diabetes, in Proceedings of the 21 American Control Conference, Baltimore, MD, USA, June 3 - July 2, 21 21, pp. 221 226. [3] J. K. Huusom, N. K. Poulsen, S. B. Jørgensen, and J. B. Jørgensen, uning of methods for offset free MPC based on ARX model representations, in 21 American Control Conference, 21, pp. 2355 236. [4], Noise modelling and MPC tuning for systems with infrequent step disturbances, in Proceedings of the 18. IFAC World Congress, 211, pp. 11 226 11 232. [5] J. B. Jørgensen, J. K. Huusom, and J. B. Rawlings, Finite horizon MPC for systems in innovation form, in 5th IEEE Conference on Decision and Control and European Control Conference, 211, pp. 1896 193. [6] J. K. Huusom, N. K. Poulsen, S. B. Jørgensen, and J. B. Jørgensen, uning SISO offset-free model predictive control based on ARX models, Journal of Process Control, vol. 22, pp. 1997 7, 212. [7] J. L. Garriga and M. Soroush, Model predictive control tuning methods: A review, Ind. Eng. Chem. Process Design and Developement, vol. 49, pp. 355 3515, 21. [8] J. M. Maciejowski, Predictive Control with Constraints. Prentice Hall, 2. [9] R. R. Bitmead and M. Gevers, Adaptive Optimal Control. he hinking Man s GPC. Prentice Hall International, 199. [1] J. H. Lee and Z. H. Yu, uning of model predictive controllers for robust perfomance, Computers in Chemical Engineering, vol. 18, pp. 15 37, 1994. [11] S. Skogestad and I. Postlethwaithe, Multivariable Feedback Control - Analysis and Design. Wiley, 5. [12] C. Zhou, J. R. Whiteley, E. A. Misawa, and K. A. M. Gasem, Application of enhanced LQG/LR for distillation control, IEEE Control Systems Magazine, vol. 15, pp. 56 63, 1995. [13] G. Prasath, B. Recke, M. Chidambaram, and J. B. Jørgensen, Application of soft constrained MPC to a cement mill circuit, in Proceedings of the 9th International Symposium on Dynamics and Control of Process Systems (DYCOPS 21), 21, pp. 288 293. 1726