Soft Constrained MPC Applied to an Industrial Cement Mill Grinding Circuit

Guru Prasatha,b,c, M. Chidambaramc, Bodil Reckeb, John Bagterp Jørgensena,∗

aDepartment of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet, Building 303B, DK-2800 Kgs Lyngby, DenmarkbFLSmidth Automation A/S, Høffdingsvej 34, DK-2500 Valby, Denmark

cDepartment of Chemical Engineering, National Institute of Technology, Tiruchirappalli-620015, TamilNadu, India

Abstract

Cement mill grinding circuits using ball mills are used for grinding cement clinker into cement powder. They use about 40%of the power consumed in a cement plant. In this paper, we introduce a new Model Predictive Controller (MPC) for cement millgrinding circuits that improves operation and therefore has the potential to decrease the specific energy consumption for productionof cement. The MPC is based on linear models that are identified using step tests. The key novelty in the MPC is that it usessoft constraints to form a piecewise quadratic penalty function with a dead zone. The MPC with this penalty function mitigates theeffect of large inevitable uncertainties in the identified models for cement mill grinding circuits. The new MPC accommodates plant-model mismatch and provides better control than conventional MPC and fuzzy controllers. This is demonstrated by simulationsusing linear systems, simulations using a detailed cement grinding circuit simulator, and by tests in an industrial cement millgrinding circuit.

Keywords: Model Predictive Control, Cement Mill Grinding Circuit, Ball Mill, Industrial Process Control, Uncertain Systems

1. Introduction

The annual world consumption of cement is around 1.7 bil-lion tonnes and is increasing at about 1% a year. The elec-trical energy consumed in the cement production is approxi-mately 110 kWh/tonne. 30% of the electrical energy is usedfor raw material crushing and grinding while around 40% ofthis energy is consumed for grinding clinker to cement powder(Fujimoto, 1993; Jankovic et al., 2004; Madlool et al., 2011).Hence, global cement production uses 18.7 TWh which is ap-proximately 2% of the worlds primary energy consumption and5% of the total industrial energy consumption.

Fig. 1 illustrates the cement manufacturing process. Ce-ment manufacturing consists of raw meal grinding, blending,pre-calcining, clinker burning and cement grinding. In thecrusher, mixing bed, and raw mill, limestone and other materi-als containing calcium, silicon, aluminium and iron oxides arecrushed, blended and milled into a raw meal of a certain chemi-cal composition and size distribution. This raw meal is blendedand heated in the pre-heating system (cyclones) to start the dis-sociation of calcium carbonate into calcium oxide. In the kiln,the material is heated, kept at a temperature of 1200 − 1450◦C,and reacts to form calcium silicates and calcium aluminates.The reaction products leave the kiln as a nodular material calledclinker. The clinker is cooled in the clinker coolers before beingstored in the clinker silos. The clinker is ground with gypsumand other materials such as fly ash in a cement mill to formPortland cement.

The final step in manufacture of cement consists of grindingcement clinker into cement powder in a cement mill grinding

∗Corresponding Author. E-mail: jbjo@dtu.dk Tel.: +45 45253088

circuit. Typically, ball mills are used for grinding the cementclinkers because of their mechanical robustness. Fig. 2 illus-trates a ball mill. The cement mill grinding circuit consists of aball mill and a separator as illustrated in Fig. 3. Fresh cementclinker and other materials such as gypsum and fly ash are fedto the ball mill along with recycle material from the separator.The ball mill crushes these materials into cement powder. Thiscrushed material is transported to a separator that separates thefine particles from the coarse particles. The fine particles con-stitute the final cement product, while the coarse particles arerecycled to the ball mill. Ball mills for cement grinding con-sume approximately 40% of the electricity used in a cementplant. Loading the ball mill too little results in early wear ofthe steel balls and a very high specific energy consumption.Conversely, loading the ball mill too much results in inefficientgrinding such that the product quality cannot be met. Loadingthe ball mill too much, may even result in a phenomena calledplugging such that the plant must be stopped and plugged ma-terial must be removed from the mill. Consequently, optimiza-tion and control of the cement mill grinding circuit operationis very important for running the cement plant efficiently, i.e.minimizing the specific electricity consumption and deliveringa consistent product that meets the specifications.

In this paper, we present a new control technology based onModel Predictive Control with soft output constraints used ina novel way for robust operation of cement mill grinding cir-cuits (Prasath and Jørgensen, 2009b; Prasath et al., 2010, 2013).The MPC algorithm uses soft constraints to create a piecewisequadratic penalty function in such a way that the closed loopsystem is less sensitive to model uncertainties than MPC witha quadratic penalty function. Predictive models for cement

Preprint submitted to Control Engineering Practice October 18, 2014

Quary Crusher Transport Mixingbed

RawMill

Dustfilter

Preheater/ calciner

Kiln Clinkercooler

Clinkersilo

Cement mill

Cement silos /Packaging

Figure 1: Cement plant.

Mill inlet

Mill discharge

Ball filling 2. compartment

Ball filling 1. compartment

Drive

Intermediate diaphragm

Discharge diaphragm

Figure 2: Ball mill.

mill grinding circuits are very uncertain. Variations and het-erogeneities of the cement clinker feed affects the gains, timeconstants, and time delays of the cement mill grinding circuit.To avoid the MPC being turned off shortly after commissioningdue to bad closed-loop behavior, it is important that these un-measurable uncertainties are accounted for. Accordingly, com-pared to traditional MPC algorithms, the MPC algorithm de-scribed in this paper gives robust performance to such uncer-tainties and provides easier tuning and maintenance. This inturns leads to longer life time of the control system and theimproved operation resulting from these controllers can poten-tially lead to very large energy savings.

Model predictive control technology is by far the most suc-cessful advanced control technology applied by the process in-dustries (Qin and Badgwell, 2003; Bauer and Craig, 2008).Model predictive control technologies are increasingly beingadopted in the cement industry for control of raw mills, kilns,and cement mill grinding circuits (Sahasrabudhe et al., 2006;Stadler et al., 2011; Samad and Annaswamy, 2011). In the fol-lowing, we review the approaches to control cement mill grind-

Figure 3: Cement mill grinding circuit.

ing circuits. van Breusegem et al. (1994, 1996); Van Breusegemet al. (1996) and de Haas et al. (1995) developed an LQG con-troller for the cement mill circuit. This controller was based ona first order 2 × 2 transfer function model identified from stepresponse experiments. Magni and Wertz (1997), Magni et al.(1999), Wertz et al. (2000), and Grognard et al. (2001) devel-oped a Nonlinear Model Predictive Control algorithm based ona lumped nonlinear model of the cement mill circuit. All thesecontrollers controlled the product and recycle flow rate by ma-nipulating the fresh feed flow rate and the separator speed. Thesame lumped nonlinear model was used by Efe and Kaynak(2002) for nonlinear model reference control and by Topalovand Kaynak (2004) for neural network based adaptive control.Lepore et al. (2002, 2003, 2004, 2007) as well as Boulvin et al.(1998, 1999, 2003) applied a distributed reduced order modelfor Nonlinear Model Predictive Control of a cement mill cir-cuit. They controlled the particle size distribution of the cementproduct by manipulating the fresh feed flow rate and the sepa-rator speed. Martin and McGarel (2001) used a neural networkmodel for Nonlinear Model Predictive Control of the cement

2

mill circuit. Sanchez and Rodellar (1996) control the cementmill circuit using adaptive predictive control.

The cement mill grinding circuit resembles to some extentgrinding circuits used in the mineral industry. Model basedcontrol technologies for such grinding circuits in the mineralprocessing industries have been surveyed by Pomerleau et al.(2000), Hodouin et al. (2001), and Wei and Craig (2009). Raja-mani and Herbst (1991a,b) and Herbst et al. (1992) developednonlinear dynamic models for SAG mills and applied optimalcontrol technology using these models. Craig and MacLeod(1995, 1996) developed robust controllers based on µ-synthesis,while Coetzee et al. (2010) used robust nonlinear model predic-tive control in their control systems design. Najim et al. (1995)handled systems with varying parameters using adaptive con-trol. Chen et al. (2007, 2008, 2009a,b) applied constrainedDMC based on a second order time delay models identifiedfrom step response experiments to ball mill grinding. Lestageet al. (2002) discussed the combination of real-time optimiza-tion and model predictive control for an ore grinding circuit.

This paper is organized as follows. Section 2 reviews the softMPC algorithm with a simple estimator. Section 3 illustratesthe robustness properties of soft MPC by simulating the perfor-mance of the soft MPC for uncertain linear systems. Section4 provides an overview of the cement manufacturing process,while Section 5 discusses the cement mill grinding circuit oper-ating strategy. Section 6 compares the soft MPC to the normalMPC using a commercial rigorous cement mill grinding circuitsimulator. Section 7 describes the implementation of the softMPC for an industrial cement mill grinding circuit and com-pares the performance of the soft MPC to the existing fuzzylogic controller. Conclusions are provided in Section 8.

2. Soft MPC Algorithm

In this section, we describe the Soft MPC algorithm used tocontrol the cement mill grinding circuit (Prasath and Jørgensen,2009b). The controller consists of an estimator and a regulatoras illustrated in Fig. 4. The input to the controller is the setpoints, r, and the measurements, y. The output from the con-troller is the manipulated variables, u. The set points, r, are thetarget values for the controlled variables, z.

2.1. Regulator

Stable processes can be represented by the finite impulse re-sponse (FIR) model

zk = bk +

n∑i=1

Hiuk−i (1)

in which {Hi}ni=1 are the impulse response coefficients (Markov

parameters). bk is a bias term generated by the estimator. bk

accounts for discrepancies between the predicted output and theactual output. In this paper, the output predictions used by theregulator are based on the FIR model (1). Using the FIR model(1), the regularized `2 output tracking problem with hard input

MPC

z

y

ur

b

Regulator

Estimator

Plant

Sensors,Lab analysis

Figure 4: Generic model predictive control system.

constraints and soft `2/`1 output constraints may be formulatedas (Maciejowski, 2002; Prasath and Jørgensen, 2008)

min{z,u,η}

φ =12

N−1∑k=0

‖zk+1 − rk+1‖2Qz

+ ‖∆uk‖2S

+

N∑k=1

12‖ηk‖

2S η

+ s′ηηk (2a)

subject to the constraints

zk = bk +

n∑i=1

Hiuk−i k = 1, . . .N (2b)

umin ≤ uk ≤ umax k = 0, . . .N − 1 (2c)∆umin ≤ ∆uk ≤ ∆umax k = 0, . . .N − 1 (2d)zk ≤ rmax,k + ηk k = 1, . . .N (2e)zk ≥ rmin,k − ηk k = 1, . . .N (2f)ηk ≥ 0 k = 1, . . .N (2g)

in which ∆uk = uk − uk−1. Equation (2) can be converted to adense convex quadratic program and solved efficiently (Prasathand Jørgensen, 2008, 2009b).

Model predictive control is based on the moving horizonimplementation of open-loop optimal control solutions. Ateach sample when new measurements arrive, the bias term iscomputed and the corresponding open-loop optimal control se-quence, {uk}

N−1k=0 , is computed by solution of (2). Only the first

element, u0, of this sequence is implemented on the process.The entire procedure is repeated at the next sample with a newmeasurement. Fig. 5 illustrates the moving horizon implemen-tation principle.

2.2. Soft Constraints and Penalty Functions with a Dead ZoneThe key feature of the convex quadratic program represent-

ing the regulator is its use of soft constraints. Soft constraintsin MPC are not new and has previously been discussed by e.g.Scokaert and Rawlings (1999), Havlena and Lu (2005) andHavlena and Findejs (2005). In this paper, we do not use thesoft constraints to guarantee feasibility of output constraints but

3

Process time

u �Estimation Regulation

��Estimation Regulation

�Estimation Regulation

����Data handling /

computations

Horizonindex

k-2

k-1

k

injected u

Figure 5: The principle of moving horizon estimation and control.

−5 −4 −3 −2 −1 0 1 2 3 4 50

5

10

15

20

25

Error, e = z−r

Pena

lty F

unct

ion

Normal MPCSoft MPC

Figure 6: The principle of soft MPC and normal MPC. The normal MPC usesa quadratic penalty function, while the soft MPC uses a piecewise quadraticpenalty function that is designed to have a dead zone.

to shape the objective function such that the resulting controlbecomes more robust against stochastic noise and model-plantmismatch (Prasath and Jørgensen, 2009b; Huusom et al., 2011).

We use the soft constraints to create a dead zone wherethe penalty function is almost zero (Boyd and Vandenberghe,2004). The penalty function within the dead zone is strictlyconvex and small but not exactly zero. This requirement en-sures that the quadratic program (2) has a unique minimizer.The requirement is fulfilled by selecting a small but positivedefinite penalty matrix, Qz, in (2a). Outside the dead zone, thepenalty function is designed such that it rapidly becomes muchlarger than the penalty function within the dead zone. This isachieved by selecting the soft constraint weights, S η and sη in(2a), such that the associated terms, ‖ηk‖

2S η

and s′ηηk, becomessignificantly larger than the error term, ‖zk − rk‖

2Qz

. The size ofthe dead zone is specified by rmin,k and rmax,k which are boundsaround the set point, rk. Within these bounds, the penalty func-

tion should be small; outside these bounds, the penalty functionshould be large. The size of the bounds may be determinedbased on a covariance analysis of the nominal system. Thered curve in Fig. 6 illustrates a piecewise quadratic penaltyfunction that has been designed according to these principles.For short this penalty function is called soft MPC as this is thepenalty function in MPC with soft constraints. For comparison,we have also shown a quadratic penalty function that is used ina classic MPC. For short we call the penalty function normalMPC.

The effect of the penalty function in the soft MPC is that littlecontrol action is taken as long as the process is within the deadzone. This implies that the controller does not amplify smallerrors by reacting and introducing a real perturbation due to thesignificant plant-model mismatch existing in uncertain systems.Using simulation examples, Prasath and Jørgensen (2009b) andHuusom et al. (2011) illustrate how penalty functions with deadzones may be used to create controllers that are robust (in apractical sense) despite significant plant-model mismatch.

2.3. EstimatorTo have offset free steady state control when unknown step

disturbances occur, we use a simple estimator with integratorsas described in Prasath and Jørgensen (2009b). Integrators inthe feedback loop are necessary to have offset free control de-spite unknown step disturbances. Integrators in the feedbackloop for the FIR based MPC may be obtained using a FIR modelin difference variables. Assume that the relation between the in-puts and outputs may be represented as

∆yk = ∆zk = ek +

n∑i=1

Hi∆uk−i (3)

in which ∆ is the backward difference operator, i.e. ∆yk = yk −

yk−1, ∆zk = zk − zk−1, and ∆uk = uk − uk−1. This representationis identical with the FIR model (1)

yk = zk = bk +

n∑i=1

Hiuk−i (4)

if bk is computed by

ek = ∆yk −

n∑i=1

Hi∆uk−i (5a)

bk = bk−1 + ek (5b)

Note that in the regulator optimization problem b1 = b2 = . . . =

bN = bk at each time instant. This is based on the assumptionthat the disturbances enter the process as constant output distur-bances. Of course this may not be how the disturbances enterthe process in practice, and significant performance deteriora-tion may result as a consequence of this representation. Theperformance of the controller can be improved by using moresophisticated estimators that use a filter to avoid the dead-beatestimate of the simple estimator used in this paper (Jørgensenet al., 2011; Huusom et al., 2012). Alternatively, a moving hori-zon estimation approach can be adopted. Prasath and Jørgensen

4

(2009a) illustrates by simulation the performance of the mov-ing horizon estimator in a closed loop system with an MPC.However, in the trade-off between sophisticated estimators withmany tuning parameters and better estimates versus simple esti-mators with few tuning parameters but slightly worse estimates,we have chosen the latter for the industrial implementation.

3. Soft MPC Simulation for Uncertain Linear Systems

In this section, we demonstrate the performance of the softMPC by simulation of linear systems. Even with significantmodel-plant mismatch, the soft MPC provides good perfor-mance. The performance of the soft MPC is evaluated for bothSISO and MIMO systems with delays and it is compared to theperformance of a normal MPC.

3.1. Linear System Description

The system is simulated by the discrete-time stochastic linearstate space model

xk+1 = Axk + Buk + Bddk + Gwk (6a)zk = Cxk (6b)

x is the state vector, u is the vector of manipulated variables(MVs), d is the vector of unmeasured disturbances, and w isstochastic process noise. z denotes the controlled variables(CVs). The measured outputs, y, are the controlled outputs,z, corrupted by measurement noise, v. Consequently

yk = zk + vk (7)

The initial state, the process noise, and the measurement noiseare assumed to be normally distributed stochastic vectors: x0 ∼

N(x0, P0), wk ∼ Niid(0,Rww), and vk ∼ N(0,Rvv).While (6) is useful for simulation, the more compact

continuous-time transfer function representation

Z(s) = G(s)U(s) + Gd(D(s) + W(s)) (8a)y(tk) = z(tk) + v(tk) (8b)

is used for the system description. In (8) we assume that the in-put signals of (8a) are piecewise constant and that the measure-ments are obtained at discrete times. Using these assumptions,(8) may be realized as the discrete-time state space system (6)-(7). Consequently, we specify our system as a continuous-timetransfer function model (8) but realize and simulate it using thediscrete-time state space model (6)-(7).

3.2. SISO System

Consider a SISO system represented by (8) and the transferfunctions

G(s) =K(βs + 1)

(τ1s + 1)(τ2s + 1)e−τs (9a)

Gd(s) =Kd(βd s + 1)

(τd1s + 1)(τd2s + 1)e−τd s (9b)

The parameters for the nominal system are: Kd = K = 1, τd1 =

τd2 = τ1 = τ2 = 5, βd = β = 2, and τd = τ = 5. The system isconverted to the discrete time state space model (6)-(7) using asample time of Ts = 1 and a zero-order-hold assumption on theinputs. The initial state used for all simulation is x0 = x0 = 0.The variances of the stochastic process noise and measurementnoise are: Rww = 0.01 and Rvv = 0.01.

Both the normal MPC and the soft MPC have been designedbased on the nominal transfer function G(s) and using n = 40impulse response coefficients and a prediction and control hori-zon of N = 3n = 120. In the normal MPC, the tuning pa-rameters used are Qz = 1, S = 10−3, while the soft MPCuses Qz = 0.001, S = 10−3, S η = 1, and sη = 0. The in-puts are constrained such that |uk | ≤ 1 and |∆uk | ≤ 0.2. Thesoft constraint limits for specification of the dead zone usedare rmin,k = −0.2 and rmax,k = 0.2. These values for the softconstraints are obtained based on the noise of the outputs forthe closed loop system. Both the normal MPC and the softMPC provide good control performance for the nominal sys-tem (Prasath and Jørgensen, 2009b).

Fig. 7 illustrates the performance of the soft MPC comparedto the normal MPC for a situation with a model-plant mismatch.The controller uses the nominal gain, K = 1.0, while the planthas a gain of K = 2.0. Fig. 7(a) illustrates the external sig-nals used for the simulation. These signals are unknown tothe controller. The soft MPC has been designed such that con-troller hardly reacts to signals due to measurement noise andstochastic process noise, when the output is within the soft lim-its. However, when the output is outside the soft limits, thesoft MPC is penalized in almost the same way as the normalMPC is penalized. Fig. 7(b) is a closed-loop simulation forthe deterministic case (wk = 0 and vk = 0). It is evident byinspection that the soft MPC handles the disturbance and theplant-model mismatch in an acceptable manner. In contrast, thenormal MPC is very oscillatory, unable to stabilize the system,and useless from a practical point of view. Fig. 7(c) illustratesthe closed-loop performance with stochastic process noise andmeasurements corrupted by stochastic noise. The closed-loopsimulation for the stochastic system confirms that soft MPC isable to stabilize the system in an acceptable manner, while thenormal MPC with the chosen tuning is useless for this situationwith unknown disturbances and a large model-plant mismatch.Prasath and Jørgensen (2009b) provide similar closed-loop sim-ulations for model-plant mismatches in the gain, time delay,zero, and time-constant. The soft MPC is able to handle theseuncertainties while the normal MPC with the chosen tuning isnot.

3.3. MIMO System

Consider a MIMO (2 × 2) system represented by (8) and thetransfer functions

G(s) =

0.62(45s+1)(8s+1) e

−5s 0.29(8s+1)(2s+1)(38s+1) e

−1.5s

−15(60s+1) e

−5s 5(14s+1)(s+1) e

−0.1s

(10a)

Gd(s) =

[ −1(32s+1)(21s+1) e

−3s

60(30s+1)(20s+1)

](10b)

5

0 50 100 150 200 250 3000

0.5

D

0 50 100 150 200 250 300−0.5

0

0.5

v

0 50 100 150 200 250 300−0.5

0

0.5

w

time

(a) External signals.

0 50 100 150 200 250 300−1

−0.5

0

0.5

1

Y

0 50 100 150 200 250 300

−1

−0.5

0

0.5

1

U

time

Soft MPCNormal MPC

(b) Deterministic simulation with gain mismatch.

0 50 100 150 200 250 300−1

0

1

2

Y

0 50 100 150 200 250 300

−1

−0.5

0

0.5

1

U

time

(c) Stochastic simulation with gain mismatch.

Figure 7: Closed-loop simulation with the soft MPC and the normal MPC forthe SISO system (9) when the controllers have been designed based on thenominal gain, K = 1, and the actual plant gain is K = 2.

These transfer functions are identified for a simulated cementmill grinding circuit using ECS/CEMulator (FLSmidth A/S,2014). ECS/CEMulator is a simulator for complete cementplants including the cement mill grinding circuit. Typically,ECS/CEMulator is used for operator training. The model inECS/CEMulator is a detailed nonlinear partial differential equa-tion model for the cement grinding circuit. The controlled vari-ables (CVs) are the elevator load, Y1(s), and the fineness, Y2(s).The manipulated variables (MVs) are the feed rate, U1(s), andthe separator speed, U2(s). The unmeasured disturbance, D(s),to the system is the hardness of the feed material, in particularthe hardness of the clinker. This linear system is simulated toinvestigate the resilience of the normal MPC and the soft MPCto model-plant mismatch for a system resembling the cementmill grinding circuit.

The MPCs for the linear system (8) and (10) are designed us-ing the nominal transfer function (10a) as well as the constraintsumin = [−5;−3], umax = [5; 3], ∆umin = [−1.5;−0.3], and∆umax = [1.5; 0.3]. The horizon of the impulse response modelis n = 75. The control and prediction horizon is N = 3n = 225.The normal MPC is based on the weights Qz = diag([8; 1]) andS = diag([0.001; 0.1]), while the soft MPC is designed withthe weights Qz = diag([0.005; 0.0001]), S = diag[0.001; 0.1]),S η = diag([8; 1]), and sη = [0; 0]. The soft output constraintsare rmin,k = [−0.8; −3.5] and rmax,k = [0.8; 3.5].

Fig. 8 illustrates the closed-loop controlled variables (CVs)and manipulated variables (MVs) for the normal MPC and thesoft MPC in the case with a gain mismatch. The controllers aredesigned for the nominal gain K = [0.62, 0.29; −15.00, 5.00],while the plant gain is K = [1.69, 1.42; −14.98, 5.02]. As forthe SISO system, this case illustrates that the soft MPC is moreresilient to model-plant mismatch than the normal MPC. Usingsimulations, we observed the same resilience properties of softMPC for mismatches in the zeros, the time constants, and thetime delays (Prasath and Jørgensen, 2009b).

4. The Cement Manufacturing Process

As illustrated in Fig. 1, the preparation of cement involvesmining, crushing and grinding of raw materials (principallylimestone and clay), calcining and sintering the materials in arotary kiln to form clinker, cooling the resulting clinker, mixingthe clinker with gypsum, grinding the clinker-gypsum mixtureto cement powder, and storing and bagging the finished cementpowder. The three main steps are 1) preparation of the raw mix-tures, 2) production of the clinker, and 3) grinding the clinkerinto cement powder.

The raw materials used in cement production are limestone,sand, shale and iron ore. The main material, limestone, is nor-mally mined on site while other materials may either be minedon site or in nearby quarries. These materials are crushed andscreened to a size less than 100 mm. The crushed materialis transported to the cement plant at which they are roughlyblended in a pre-homogenization pile (mixing bed). The nextstep in the cement production depends on whether the wet orthe dry process is used. In the wet process each material is pro-portioned to meet a desired chemical composition and fed to a

6

0 100 200 300 400 500−1

0

1

2

Y1

0 100 200 300 400 500−5

0

5

Y2

(a) Controlled variables (CVs).

0 100 200 300 400 500−5

0

5

U1

0 100 200 300 400 500

−2

0

2

U2

time

(b) Manipulated variables (MVs).

Figure 8: Deterministic closed-loop simulation of the linear cement grindingmodel (8) and (10). The plots illustrates the CVs and MVs using the normalMPC (blue) and the soft MPC (red) for the case with a gain mismatch betweenthe plant and the controller model. An unknown disturbance of size 2.5 entersat time 20.

rotating ball mill with water. The raw material is ground to asize where the majority of the material is less than 75 micron.The slurry is pumped to the blending tanks and homogenizedto ensure that the chemical composition is correct. In the dryprocess, each raw material is proportioned to meet a desiredchemical composition and fed to either a rotating ball mill or avertical roller mill. The raw materials are dried with waste pro-cess gases and ground to a size where the majority of the ma-terial is less than 75 micron. The material from either type ofmill is blended to ensure that the chemical composition is wellhomogenized. This so-called raw mix is stored in silos until re-quired. This raw mix is a mixture of calcium carbonate, silicon-, alumina- and iron-oxides. Calcium and silicon are present toform strength producing calcium silicates. Aluminium and ironare present to produce liquid in the kiln burning zone. The liq-

uid act as a solvent for the silicate forming reactions.In the wet process, the slurry is fed to a rotary kiln. In the dry

process the raw mix is fed to the preheater/calciner tower. Thesame basic physical and chemical process takes place in the kilnfor the wet and the dry process. The raw mix is gradually heatedby contact with the hot gases generated by combustion of thekiln feed. A number of chemical reactions take place as thetemperature rises. At 70-110◦C the water is evaporated. At 400-600◦C clay like materials decompose into principally SiO2 andAl2O3. Dolomite, CaMg(CO3)2, decomposes to CaCO3, MgOand CO2. At 650-900◦C CaCO3 reacts with SiO2 to form belite,2CaO·SiO2. At 900-1050◦C the remaining CaCO3 decomposesto CaO and CO2. At 1300-1450◦C the material partial melts(20-30%), and belite reacts with CaO to form alite, 3CaO·SiO2.Alite is the characteristic constituent of Portland cement. Apeak temperature of 1400-1450◦C is required to complete thereaction. The partial melting causes the material to aggregateinto lumps or nodules with a typical diameter of 1-10 mm. Thisis called clinker. As the hot clinker leaves the kiln, they fall intoa cooler that recovers most of the heat and cools the clinker toaround 100◦C. The clinker is stored in clinker silos.

In the final stage in the manufacture of cement, clinker ismixed with approximately 5% gypsum (calcium sulphate) andpossible other materials such as fly ash and grinned to cementpowder. Either a rotating ball mill or a vertical mill is used forgrinding. In this paper we consider a rotating ball mill (Fig. 2).As the mill rotates, the steel balls inside the mill collide withclinker and raw material to form a fine gray powder. Typically,by mass 15% of the particles in this cement powder have a di-ameter less than 5 µm and 5% of the particles have a diameterlarger than 45 µm. Fineness is a measure of the specific surfacearea of the cement powder and it is directly related to the prop-erties of cement. General purpose cement has a specific surfacearea of 3200-3800 cm2/g (320-380 m2/kg), while rapid hard-ening cement has a specific surface area of 4500-6500 cm2/g(450-650 m2/kg). The various cement qualities are stored in ce-ment silos before being packed in bags or shipped by truck, railor boat.

4.1. The Cement Mill Grinding Circuit

To obtain a cement powder with the desired specific surfacearea, the ball mill is operated in conjunction with a separatorthat returns coarse material to the ball mill for further grind-ing. This system is called a cement mill grinding circuit andis illustrated in Fig. 3. As illustrated in Fig. 2, the ball millsused for grinding have two chambers separated by a metallic di-aphragm. The first chamber is filled with large steel balls and issupposed to do the coarse grinding. The second chamber is thefine grinding chamber and is equipped with small steel balls.Both chambers are equipped with classifying liners to ensurethat the ball charge segregate with large balls accumulating atthe inlet of the chamber and small balls accumulating at theoutlet of the chamber.

The fine ground particles leaving the ball mill are lifted bya bucket elevator and sent to an air classifier. As the particlesfrom the bucket elevator enter the classifier, they are suspended

7

in an air stream. The air is sucked from the bottom of the clas-sifier and transports the particles into the rotating equipment.In this rotational gravity field, large and heavy particles impactthe wall and are collected in cyclones as they drop down. Smalland fine particles are transported away towards the center of theclassifier. The coarse particles are recycled to the mill for fur-ther grinding. The fine particles are collected in cement silosand constitute the final cement. Consequently, the overall func-tion of the classifier is to separate coarse particles from fineparticles.

5. Control Strategy for the Cement Mill Grinding Circuit

The objective in controlling the cement mill grinding circuitis to produce cement powder with a desired specific surfacearea while minimizing the energy consumption (cost) in doingso. The energy consumption is largely connected to rotatingthe ball mill and its steel balls. It depends only weakly on thecement material loading in the mill. Therefore, the most prof-itable mode of operation consists of maximizing the productionrate while meeting the specific surface area. In this way, leastenergy is used for cement grinding per produced tonnes of ce-ment.

The specific surface area of the material in the product streamis the primary controlled variable in cement mill grinding cir-cuits. In many cement plants, the specific surface area can onlybe measured by sampling the product stream and analyzing thesample in the lab. In such plants, the specific surface area of theproduct is only measured infrequently and the result is availableto the process control systems with a variable delay of about30 min (the approximate duration of the laboratory procedure).Modern plants are equipped with an on-line particle size ana-lyzer and the specific surface area is directly available to theprocess control system. The particle size distribution and thusthe specific surface area of the product stream depends on therotational speed of the classifier, the air speed, the feed rateto the classifier, and the particle size distribution of the classi-fier feed stream. The air speed is often kept constant and therotational speed of the classifier is manipulated to control thespecific surface area of the product stream.

The grinding efficiency in the ball mill depends on its filling.Too little cement material in the mill causes steel balls to hitsteel balls without using this energy for crushing material. Inaddition, the temperature in the mill becomes easily too high inthis case. Conversely, as the mill gets loaded with too much ce-ment material, the particles leaving the ball mill are too coarse.They are rejected in the separator and recycled back to the ballmill. The consequence is a very low production rate and toohigh specific energy consumption. In the extreme case, mate-rial builds up in the ball mill such that the diaphragm separatingthe first and second chambers plugs and material stops flowing.This situation should be avoided. Consequently, one objectivein most cement mill grinding circuit control strategies is to con-trol the mass of cement material in the ball mill. The mill shouldbe filled as much as possible without plugging as this leads tothe largest production rate. However, the cement mass hold up

cannot be measured directly. A so-called pholaphone (micro-phone) situated outside the cement mill can be used to infer thefilling as an empty mill with steel balls hitting steel balls makesmore noise than a mill filled with cement material. Althoughvery nonlinearly, the material flowing out of the mill is relatedto the filling of the mill. This implies that the power used by thebucket elevator is related to the filling of the mill as the poweris related to the flow. This elevator power is often used in di-rect control of the mill filling while the pholaphone is used foroverride control to avoid plugging.

Consequently, one may control the cement mill grinding cir-cuit by controlling the specific surface area of the cement prod-uct stream and the elevator load. This is done by manipulationof the fresh feed flow rate and the rotational speed of the clas-sifier. The main disturbances affecting the cement mill circuitare the hardness of the clinker as well as their size distribution.These disturbances are unmeasurable and affect the gains andthe time constants of the cement mill circuit. They also affectthe relationship between the output flow rate and the ball millcement hold up. This implies that the elevator load set pointmay be adjusted as consequence of large variations in the hard-ness.

5.1. Predictive Control of Cement Mill Grinding CircuitsIn the cement industry, control of the cement mill grinding

circuit is considered to be a very difficult control problem. Themain reason for this is the presence of significant non-linearitiesand uncertainties. These nonlinearities and uncertainties arisedue to variations in material properties that cannot be measuredonline. Consequently, the gains, the time constants, the zeros,and the time delays in predictive models (10a) for the cementmill grinding circuit are very uncertain and they are dependenton operating conditions. As is evident for linear systems, thenormal MPC is not as resilient to plant-model mismatch as thesoft MPC. Therefore, the soft MPC is expected to be more ap-propriate than the normal MPC for controlling the cement millgrinding circuit.

To demonstrate that the soft MPC can be used to successfullycontrol cement mill grinding circuits in the face of inevitablemodel-plant mismatch, we test the soft MPC for a simulated ce-ment mill grinding circuit using the ECS/CEMulator softwareand in an industrial cement mill grinding circuit. For the simu-lated case study, we compare the soft MPC to a normal MPC.In the industrial cement mill grinding circuit, we were only al-lowed to compare the soft MPC to the existing fuzzy based con-trol system.

For both the simulated and the industrial cement mill grind-ing circuit, the procedure for designing the soft MPC (and thenormal MPC) is to obtain a process model, Y(s) = G(s)U(s),using step tests. As discussed by Prasath et al. (2013), the stepresponses contain large variations and give rise to models withuncertain parameters. We obtain parsimonious well regularizedmodels by fitting low order transfer functions of the type (9a) toeach input-output pair. (10a) is an example of such a model fora cement grinding circuit. This identified model is then usedto design and tune the soft MPC by simulation, i.e. by usingthe linear model, G(s), as well as perturbations of this model to

8

simulate the resulting performance of a specifically tuned softMPC. When an acceptable controller is obtained, it is uploadedto the ECS/SCADA system and used to control the cement millgrinding circuit. The same ECS/SCADA system is used forboth the simulated and the industrial cement mill grinding cir-cuit. Fig. 9 illustrates this ECS/SCADA system.

6. MPC for a Simulated Cement Mill Grinding Circuit

The performance of the soft MPC is tested and com-pared to the performance of the normal MPC by simula-tion using the rigorous cement mill grinding circuit simula-tor, ECS/CEMulator (FLSmidth A/S, 2014). ECS/CEMulatoris based on a nonlinear distributed model and is normally usedfor operator training. In this paper, we use it as a realistic sur-rogate for an industrial cement mill grinding circuit to test ourcontrollers.

Using step response tests, the transfer function in (10a) hasbeen identified. Both the soft MPC and the normal MPCare designed based on this model. The normal MPC and thesoft MPC use the tuning described in Section 3.3. The sam-pling rate is Ts = 1 min. The hard input constraints areumin = [110; 65], umax = [150; 85], ∆umin = [−1.5; −0.3], and∆umax = [1.5; 0.3]. The set points are r = [30; 3100] and thesoft output limits are rmin = [28; 3000] and rmax = [31; 3200].Asymmetric soft output limits for the elevator power, y1, havebeen chosen as it is more important that the controller lowersthe mill filling quickly, if the mill is near the plugging limit,than the controller fills the mill quickly, if its filling is belowthe target. It should be noted that these limits and variables arelisted as physical variables here, but they are converted to devi-ation variables in the MPC software. Both the normal MPC andthe soft MPC performs well close to the nominal conditions.

To illustrate the performance of the soft MPC in the real-istic case with significant uncertainty, we simulated a changeof material properties by changing the grindability of the feed.While the MPCs are designed for the nominal transfer function(10a), this change of grindability corresponds to changing thedynamics to something that can be approximated by the transferfunction

G(s) =

0.2(45s+1)(8s+1) e

−5s 0.12(8s+1)(2s+1)(38s+1) e

−1.5s

−8(60s+1) e

−5s 2(14s+1)(s+1) e

−0.1s

(11)

Comparison of (10a) and (12) demonstrates that this materialproperty change, changes the gain of the linear approximationsignificantly. The time constants, the zeros, and the time delaysare unchanged.

Fig. 10 illustrates a simulation of this situation for the normalMPC and the soft MPC. The uncontrolled ECS/CEMulator isstarted at a steady state different from the target steady state. Attime 1.35 hr, a significant change in hardness of the material isintroduced and the controller is switched on at time 2.0 hr atwhich the cement mill grinding circuit is in a transient phase.We perform the simulation this way to mimic that the controllerin an industrial cement mill grinding circuit is often in suchtransient conditions due to frequent starts and stops. Both the

normal MPC and the soft MPC are able to bring the systemto the target steady state and stabilize it there. However, theinputs for the soft MPC are much more stable than the inputsfor the normal MPC. This indicates that the soft MPC is moreplant friendly and robust towards disturbances and plant-modelmismatch. The plant friendliness and small variations of themanipulated variables of the soft MPC can be translated intoless plant wear and longer equipment life-time.

7. MPC for an Industrial Cement Mill Grinding Circuit

In this section, we describe the implementation and test ofthe soft MPC in an industrial cement grinding circuit. At theplant we were not allowed to implement a normal MPC withthe purpose of comparing it to the soft MPC and expectedlydemonstrate that the normal MPC would not perform as well asthe soft MPC. Instead we compare the performance of the softMPC to the existing fuzzy logic based control system at thatplant.

7.1. An Independent Finish Grinding Plant

The implementation of the soft MPC controller was tested inan industrial cement grinding unit in southern India. Cementgrinding units are independent finish product grinding plantswhere the raw materials are obtained from different locationsand ground to the final cement product. Cement grinding unitsconsist of silos for storage of cement clinker and other raw ma-terials, the cement mill grinding circuit (a ball mill and a sepa-rator), and equipment for cement packing and dispatch (see Fig.1).

The cement mill grinding circuit, where we tested the softMPC, consists of a two-chamber ball mill (see Fig. 2) and aSepax separator in a closed circuit (see Fig. 3). This cementmill grinding circuit has a design capacity of 150 tonnes/hour.Due to the large demand for cement, the cement mill grind-ing circuit is operated at maximum design capacity and be-yond. The recirculation ratio of the circuit is 1.5%. The powerfrequency varies continuously depending on the electricity de-mand in the municipality where the cement mill grinding circuitis located. At periods with low frequency, the rotation speed aswell as the air flows are reduced. This adversely affects theproduction rate and the product quality. The cement mill grind-ing circuit control system must be able to reject the effect ofsuch disturbances. Furthermore, the cement mill grinding cir-cuit runs only at night time and is shut-down in day time wherethe electricity price is high. This implies that the control systemmust be able to handle the effect of transients due to frequentstart-ups and shut-downs. Since the plant, where we implementthe controller, is an independent grinding unit, the clinker pro-duced from multiple plants are transported to the cement grind-ing unit by railway wagons. Due to the multiple sources ofclinker, the hardness and quality variations in the clinker arevery large. This results in continuous and significant variationsin the operating conditions for the cement mill grinding circuit.The final product recipes are Ordinary Portland Cement (OPC)and Puzzalona Portland Cement (PPC). The production of both

9

Figure 9: The ECS/SCADA system for high level control of a cement mill grinding circuit.

OPC and PPC in the cement grinding unit introduces frequentproduct shifts. The control system must be able to facilitatefrequent set-point changes as the result of product demand anddispatch requirements.

All the signals coming from the sensors of the grinding pro-cess are collected in an ECS/SCADA system (see Fig. 9).ECS/SCADA is the Supervisory Control And Data Acquisitionof FLSmidth. The measurement data is obtained from the PLCsystem and logged every 10 seconds in the ECS/SCADA. Thefineness measurement of the product is obtained using an auto-sampling system that collects samples every hour. The time toanalyze a sample is 20-30 min. This introduces a measurementdelay that must be accommodated by the control system.

FLSmidth has developed a high level control system toolcalled PXP that can be used to configure advanced controllers.PXP is used to configure the soft MPC. The MPC toolbox from2-control ApS is used to solve the convex quadratic program as-sociated with the constrained optimal control problem (2). Thissystem is then interfaced to the ECS/SCADA. The executioninterval of the soft MPC is 1 min and the data update intervalin the expert tool is 30 seconds. The measurement data ob-tained from the PLC systems are filtered, scaled and validatedusing the expert system tool before entering the controller. Theoutput from the controller is also scaled and configured forbumpless transfer. The bumpless transfer is important to ob-tain smooth transitions of manipulated variables when the con-troller is shifted from manual to automatic. Interlocks are alsoincluded to accommodate emergency shut down during abnor-mal conditions, e.g. power failure.

7.2. Model Identification

The model used by the soft MPC is obtained from step re-sponse experiments in a single operating point (Prasath et al.,2013). The step responses are obtained by stabilizing the plantat the operating point and individually perturbing the feed flowrate and the separator speed. The elevator power is measuredcontinuously. The fineness measurement for modeling is ob-tained by sampling the product every 15 min and subsequentlaboratory analysis of these samples. Conducting these stepresponse experiments is the most time consuming part in thecommissioning of the soft MPC. The resulting identified modelis

G(s) =

0.47(2s+1)(17s+1)(15s+1) e

−4s 112s+1 e−3s

−0.9(10s+1)(12s+1) e

−5s 2.54s+1

(12)

with Y(s) = G(s)U(s), Y(s) = [Elevator Load; Fineness], andU(s) = [Feed; Separator Speed]. Internally in the MPC, thismodel is converted to a FIR model. Due to the inherent varia-tions in the cement grinding circuit process, the predictions ofthis models are uncertain (Prasath et al., 2013). This is the mo-tivation for the development and use of the soft MPC describedin this paper.

7.3. Controller Performance Comparison

Since the source of raw material for grinding is obtained fromdifferent regions, the physical and chemical properties of theclinker are different for each batch of the clinker used. Thisvaries the grinding pattern for each of the clinker types and im-pacts the grinding efficiency of the cement mill. Therefore, theparameters characterizing mill operation vary significantly and

10

0 2 4 6 8 10 12 14 1625

30

35

time (hours)

Ele

vato

r lo

ad

0 2 4 6 8 10 12 14 162900

3000

3100

3200

3300

time (hours)

Fine

ness

(a) Normal MPC. Controlled variables (CVs).

0 2 4 6 8 10 12 14 1625

30

35

time (hours)

Ele

vato

r lo

ad

0 2 4 6 8 10 12 14 162900

3000

3100

3200

3300

time (hours)

Fine

ness

(b) Soft MPC. Controlled Variables (CVs).

0 2 4 6 8 10 12 14 16110

120

130

140

150

time (hours)

Feed

(TPH

)

0 2 4 6 8 10 12 14 1665

70

75

80

85

time (hours)

Sepa

rato

r Sp

eed(

%)

(c) Normal MPC. Manipulated variables (MVs).

0 2 4 6 8 10 12 14 16110

120

130

140

150

time (hours)

Feed

(TPH

)

0 2 4 6 8 10 12 14 1665

70

75

80

85

time (hours)

Sepa

rato

r Sp

eed(

%)

(d) Soft MPC. Manipulated variables (MVs).

Figure 10: Normal MPC (left) and Soft MPC (right) applied to a rigorous nonlinear cement mill simulator, ECS/CEMulator. The disturbances (change in hardnessof the cement clinker) are introduced at time 1.35 hour (green line) and the controllers are switched on at time 2 hour (purple line). The soft constraints are indicatedby the dashed lines.

continuously. To compensate for the quality variations in thefeed material, we include target adaptation using a real timeoptimizer in the control hierarchy level above the soft MPCand the fuzzy logic controller. This helps in deciding the op-timum operating range of the mill and improves the grindingefficiency. The elevator load is the controlled variable indirectlymeasuring the filling of mill. Since the filling is dependent onthe raw material properties, the target of the elevator load mustbe changed periodically. This periodic adjustment of the eleva-tor target helps to maximize the production while still obtainingthe desired fineness. The controller varies the feed and separa-tor speed to maintain the elevator load and the fineness at thetargets.

The frequent power restrictions limit the operation of the ce-ment mill to maximum 16 hours in a day. Normally, the plant

runs during the night, when the external electricity demand islow, and is stopped during the day. Most of the time, the cementmill produces PPC. Based on the demand, the cement plant mayproduce the OPC product for 3−4 hours. Consequently, we nor-mally get only 12 − 15 hours of continuous mill run to test ourcontrollers. These frequent starts and stops are typical for ce-ment mill grinding circuits all over the world and not only inregions with power limitations.

To have a fair comparison between the fuzzy logic controllerand the soft MPC, both controllers are tested under similar op-erating conditions. We do this by running the controllers forthe same source of clinker. This implies that the controllersare tested for similar feed material properties and the resultingperformance is an indication of their ability to suppress distur-bances and process variations. Target adaptation and optimiza-

11

0 2 4 6 8 10

25

30

35 Soft MPC

Ele

vato

r

0 2 4 6 8 10

340

345

350

Fine

ness

0 2 4 6 8 1065

70

75

time (hours)

Sep

Spee

d (%

)

0 2 4 6 8 10140

145

150

155

160

Feed

(t/h

)

0 5 10 15

25

30

35Fuzzy Controller

0 5 10 15

340

345

350

0 5 10 1565

70

75

time (hours)

0 5 10 15140

145

150

155

160

Figure 11: Comparison of the soft MPC with the fuzzy logic controller for operation of an industrial cement mill grinding circuit. The controlled variables (CVs)are elevator (denotes the elevator load in kW) and fineness (m2/kg). The manipulated variables (MVs) are the feed rate (tph) and the separator speed (%). The redline indicates the targets for the controlled variables and the blue lines indicate the measured values. The dotted lines are used to denote soft constraint limits for theCVs and hard constraint limits for the MVs.

tion to maximize the production are employed for both con-trollers.

In the first test day, the fuzzy logic controller is switched onand controls the cement grinding circuit for 15 hours. In thesecond test day, the soft MPC is switched on and controls thecement grinding circuit for 12 hours. Both controllers are testedwith the cement mill running continuously in a single recipeproducing PPC. It is verified that the fuzzy logic controller iswell tuned such that the soft MPC is tested against the best con-troller available in the plant. As an indication that the fuzzylogic controller can be considered well-tuned, we mention thatbefore the tests with the soft MPC, the plant ran the fuzzy logiccontroller continuously whenever the mill was started and theplant personnel were quite satisfied with the performance of thefuzzy logic controller.

The following configuration are used for the soft MPC ap-plied to the industrial cement mill grinding circuit: Prediction-and control horizon N = 300; number of impulse responseparameters n = 100; the tuning weights on the errors areQz = [5·10−2 0; 0 2.5·10−5]; the tuning weights on the manip-ulated variables are S = [5·102 0; 0 2.5·105]; and the quadraticsoft constraint tuning weights are S η = [9 ·103 0; 0 5 ·102]; thesampling time is Ts = 1 min; the linear soft constraint tuningweight is set to zero; the constraint limits used by the soft MPC

are umin = [135; 64], umax = [152; 74], ∆umin = [−0.5; −0.1],∆umax = [0.5; 0.1], rmin,k = rk − [1; 1], and rmax,k = rk + [1; 1].The tuning weight for the fineness is chosen small comparedto the tuning weight for the elevator load. The reason for thischoice is that the fineness is an hourly sampled measurementand less accurate in comparison to the elevator load. The weightfor the separator speed is set to a large value to smooth separatorspeed movements and avoid that it moves aggressively. Thesechoices of the tuning weights ensure relatively stable operationof the cement mill grinding circuit.

Fig. 11 indicates the performance of the soft MPC and theexisting fuzzy logic control system in the industrial cement millgrinding circuit producing PPC. Fig. 12 reports the perfor-mance of the soft MPC for production of OPC in the cementmill grinding circuit.

Fig. 11 illustrates that the soft MPC is able to control andstabilize the cement mill grinding circuit. The manipulatedvariable trajectories are smooth and the controlled variablesare well controlled. However, the controller runs at its upperlimits and hence the actuator movements are to a large extentrestricted by these limits. This observation indicates that theMPC’s ability to handle input constraints is important as the ce-ment mill grinding circuit often operates at constraints. Com-pared to the fuzzy logic controller, the soft MPC yields smaller

12

0 0.5 1 1.5 2 2.5 3

20

25

30

Ele

vato

r

0 0.5 1 1.5 2 2.5 3270

280

290

300

time (hours)

Fine

ness

(a) Controlled variables (CVs).

0 0.5 1 1.5 2 2.5 358

60

62

64

time (hours)

Sep

Spee

d (%

)

0 0.5 1 1.5 2 2.5 3135

140

145

Feed

(t/h

)

(b) Manipulated variables (MVs).

Figure 12: Production of OPC in an industrial cement mill grinding circuitusing the soft MPC. The plots show the controlled variables (CVs) and themanipulated variables (MVs). The pink line in the plot indicates the recipechangeover from PPC to OPC and is also the time when the soft MPC is takenonline.

variations and better tracking of the elevator load as well as thefineness of the cement product. The reduction of the variationsin the fineness is significant and important for the economic op-timization of the cement mill grinding circuit. When the varia-tions in fineness are reduced, the fineness target can be adjustedsuch that it is closer to the PPC specification constraint. Thisadjustment of the target implies that we can reduce the over-grinding and thereby increase the production capacity of thecement mill grinding circuit. Furthermore, it should be notedthat the soft MPC enables operation along the upper limit of thefeed production rate.

Fig. 12 shows the performance of the soft MPC when theproduction in the cement mill grinding circuit is switched toOPC. In this case, there is a large margin for the controllerto adjust its manipulated variables as the soft MPC is not op-erating at constraints. The soft MPC is designed and tuned

such that it is resilient to disturbances and model-plant mis-match. This feature of the controller manifests itself by the verysmooth trajectories of the manipulated variables. The soft MPCalgorithm uses soft constraints to create a piecewise quadraticpenalty function in such a way that the soft MPC moves themanipulated variables very little when the controlled variablesare within the soft constraints but takes more aggressive actionswhen the controlled variables are outside the soft limit band. Asan example in the period between 1.5 hours and 2 hours wherethe elevator load as well as the fineness are outside the soft con-straint band, the soft MPC penalty function manifests itself bynoticeable adjustments of the manipulated variables. Duringthe tests, it has been observed that the soft MPC handles oper-ating point transitions better than the fuzzy logic controller. Inaddition, the soft MPC reduces the product quality variations(variations in the cement fineness) in comparison to the fuzzylogic controller.

8. Conclusion

We have developed an `2 regularized predictive controllerwith hard input constraints and `2/`1 soft output constraints.The estimator is based on an integrated output disturbancemodel such that the controller provides offset-free steady statecontrol for type 1 disturbances (steps). The numerical imple-mentation of the MPC is based on a finite impulse response(FIR) parametrization. In this paper, the FIR parametrizationis obtained from low order transfer functions with delays thatare obtained from step response experiments. The key noveltyin the model predictive controller formulation is that we usethe soft `2/`1 output constraints to shape the penalty functionsuch that it contains a deadzone within the soft constraint lim-its. This penalty function is convex, almost zero within the softconstraint limits, and much larger outside the soft constraintlimits. This formulation enables design of a socalled soft MPCthat is more resilient towards model-plant mismatch than thenormal MPC. The normal MPC is based on an `2 error penaltyfunction but does not contain the soft `2/`1 output constraintsused to shape the penalty function in the soft MPC.

We demonstrate the resiliency of the soft MPC to model-plant mismatch using closed-loop simulations of linear SISOand MIMO systems with delays. The resiliency of the soft MPCis also confirmed using a commercial cement mill grinding cir-cuit simulator. As the models for cement mill griding circuitsare inherently very uncertain, we use the soft MPC to control anindustrial cement mill grinding circuit and report that it reducesthe variance of the controlled variables compared to the existingfuzzy logic control system. The soft MPC enables operation atthe feed constraint and also reduces the product quality varia-tions. Consequently, over-grinding can be reduced and the pro-duction capacity of the mill increased. The resiliency of the softMPC to model uncertainties eases the tuning and commission-ing and potentially extends the life-time of the MPC system.

13

Acknowledgement

This work was funded in part by EUDP 64013-0558, byFLSmidth A/S, and by 2-control ApS.

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