A Simulation Study of Multi-Step Predictive Control Applied to a Mill Feeder System

Copyright © !FAC Automation in Mining. Mineral and Metal Processing. Sun City. South Africa. 1995

A SIMULATION STUDY OF MULTI-STEP PREDICTIVE CONTROL APPLIED TO A MILL FEEDER SYSTEM R.L.J. Bolton' and I.M. MacLeod' 'University of the Witwatersrand , Department of Electrical Engineering, Joha.nnesburg , South Africa

:Abstract: A model predictive :on~rol technique known as Multi-Step Predictive Control (MSC) for the control of multIvanable systems that exhibit significant time delays. A software SImulatIOn study of MSC control applied to a grinding mill feeder system is presented in order to evaluate this technique. The abilities of the controller to deal with the practical issues of process control are emphasized. The results show that MSC control provides robust control in the face of modelling errors and sensor noise. The incorporation of disturbance feedforward and the ability of the controller to handle process constraints and dead time are demonstrated. I~ usef~

Keywords: Model predictive control; process control; computer control; mill feeder control

1. INTRODUCTION Model predictive control refers to a family of control techniques which make direct use of an explicit and separately-identifiable plant model for the purpose of feedback control. Model predictive controllers have been successful in industrial applications. An important reason for this is their ability to handle process constraints in both the design and implementation stages . This is a significant improvement over traditional control techniques which tend to disregard process constraints during the design phase and then handle them in an ad hoc manner during implementation (Garcia et al., 1989) . Model predictive techniques lead to savings in effort be'cause it is not necessary to study each control loop extensively to ensure that constraints are not violated. Maintenance is also simpler because the constraint handling ability of model predictive controllers is insensitive to plant changes. Multi-Step Predictive Control (MSC) (Bolton, 1994; Bolton and MacLeod, 1994) is one such technique. It provides a useful framework for controlling multi variable plants that have significant dead times and readily allows measurable disturbance feedforward variables to be incorporated. The application of supplying a continuous and accurately-controlled feed of ore to a grinding mill is used as a case study to asses the MSC technique. In the South African mineral processing industry ore feed is usually controlled by varying the discharge of material from a storage bin or silo onto a moving surface, typically a conveyor

belt, which carries it to the mill. Different devices are used for varying the flow of ore out of the bin-however, they all tend to have highly uncertain behaviour. This, together with the dead time or pure delay due to the transport of material to the point where the mass flow rate can be measured, makes it difficult to achieve accurate control of mass flow rate into the mill. The present investigation concentrates on the design of a controller for a mill feed system based on one particular kind of discharge device, namely, the drum feeder (Stanley, 1987) , although similar considerations apply to other types of feeder . A simulation study of MSC control of grinding mill feeders is discussed , with emphasis placed on the ability of the controller to provide multivariable control with constraints on the feeder inputs. MacLeod (1992) discusses an alternative approach based on a fuzzy logic adaptive mechanism. The remainder of this paper is organised as follows. Section 2 gives a brief overview of the mill feed system and its dynamic characteristics. A brief overview of MSC control is then presented and important aspects of the technique are discussed in section 3. Considerations involved in applying MSC control to the mill feeder system are discussed in section 4. Details of the simulation, the simulation results and the conclusions are then presented.

2. TYPICAL MILL FEED SYSTEMS The aim of the mill feeder system shown in Fig. 1 is to deliver run-of-mine ore to a grinding mill

241

Silo filled with ore

. Mill

Fig. 1. Typical mill feeder system function model of the form

at an accurately-controlled flow rate. In a drum feeder the surface of a rotating cylinder on a horizontal axis forms the bottom plane of the discharge chute of the ore storage bin. The ore is moved out of the chute and onto a conveyor belt by the rotation of the drum. The drum is rotated by a variable-speed drive, delivery rate being varied by drum speed. The conveyor belt travels at constant speed and a transport delay of 50 s before the ore reaches a point where the mass flow rate can be measured is typical.

yes) , e- t4 $ --=!\. - u(s) PTs+1

(1)

where s is the Laplace transform variable, u is the control input to the plant and y is the plant output, i.e., the measured mass flow rate. The time constant T is typically 5 s, the delay time td is 50 s and the gain Kp is highly variable.

3. OVERVIEW OF MULTI-STEP PREDICTIVE CONTROL

The highly uncertain behaviour of an individual drum feeder is largely the result of the "as hoisted" nature of the ore which can vary from wet, slushy material to fine dry powder to large rock pieces, depending on mining conditions. The delivery rate of a drum feeder for a given drum speed is strongly affected by the highly-variable ore flow characteristics. The feed system for one mill would typically consist of four drum feeders operating in parallel and controlled by one mass flow rate controller. Blockages of one or more feeders occur frequently during operation.

Multi-Step Predictive Control (MSC) (Bolton, 1994; Bolton and MacLeod, 1994) is a computationally-intensive control technique that is based on a "receding horizon" control policy (Athans and Falb, 1966). Control actions are continually updated as new measurement information becomes available. Feedback, which is essential to the success of the control strategy, is inherent in the algorithm because each control step is based on the measured current state of the system.

From experience it is known that acceleration of the rock particles is not a significant factor in the overall dynamic behaviour of a drum feeder (approximately constant inertia). _A first-order dynamic response where the time constant is primarily determined by the speed response time of the variable speed drive is therefore a good representation. When the transport delay due to the conveyor belt is included this leads to a plant transfer

This technique relies on a pre-calculated model of the plant known as the predictor model, which describes the dominant dynamics of the plant. This model is used to compute the future control actions based on the projected performance of the system over the prediction horizon, HP steps long. The selection of the optimal control action for each step is based on the solution of an open-loop optimal control problem. This calculation solves the optimisation problem of finding the best fu-

242

Senaor

r

~

y•

Uz

MSC Controller

l'.J

Y Plant

u.

d

Proc:esa Current

H

Current Senaors

FeederMotor Cu rrants

Fig. 2. Signal flow in MSC control

It is important for a process controller to have integral action , which is necessary for nulling constant offset disturbances which may affect the process outputs. This property was incorporated into the MSC algorithm discussed in the following sections. Feedforward control of measured disturbances was also included in the MSC algorithm by incorporating the projected plant outputs due to the measured disturbances into the optimisation problem.

ture control actions that satisfy the control objectives. This optimisation procedure is repeated for each sampling period and is based on the current state of the system, thus yielding a feedback control policy. The optimal control problem is solved using quadratic programming with inequality constraints which describe the constraints on the plant inputs. The criteria for control are described by means of a "cost" function, HP

Cost

L

[A(k)2(Y·(t+k) - Y(t+k))2

4. CONSIDERATIONS IN APPLYING MSC TO A MILL FEEDER SYSTEM

k=l

+ ,8(k)2(~u(t+k-l))21

(2)

The overall block diagram of the MSC control scheme is shown in Fig . 2. The first step in applying MSC control is to identify the dominant dynamics of the plant to be controlled . The resulting model is critical to the success of the controller. The more accurate the predictor model, the higher performance the controller can deliver.

where ~ = 1 - q-l and q-l is the conventional time shift operator. Y· and Y are the reference and predicted output trajectories, respectively. This function gives a quantitative measure of the mismatch between the extrapolated plant outputs and the proposed future outputs. The set of future plant inputs which gives the lowest cost and does not violate any process constraints is used as the set of optimal control actions for that particular step. It is important to note that only the first predicted control action is implemented. The remaining predicted control actions are discarded.

The usual approach to mill feeder control is to lump the feeders together and to use one controller to control all four feeders . MSC control, however, is a true multivariable control technique. For this reason the feeders are controlled separately. According to (1) a suitable predictor model is

The cost function used in MSC control has two com~nents. The first component describes the predicted mismatch between the predicted and proposed plant output trajectories. The second component describes the cost of the rate of change of the plant inputs . These components have weighting factors A and ,8, respectively, which tune the controller.

1 1 1 1] P( s) = [ 5s + 1 5s + 1 5s + 1 5s + 1

(3)

The next step is to consider the sampling period of the controller and the number of samples necessary in the prediction horizon. The sampling period is dependent on the dynamics of the plant and should satisfy the Nyquist sampling criterion .

243

Table 1 Simulation parameters Sampling Period Prediction Horizon Control Horizon Simulation time Weightings Weightings

--

Tsp HP CP Tend

= = = -

).

(3

2 30 6 2500 0.1 [0 .500.550 .650.75)

seconds steps steps seconds

Constraints (T <= 1800 s)

dU/dt U

=

[-0.1 -0.1 -0 .1 -0.1) [ 0.1 0.1 0.1 0.1) [0 .0 0.0 0.0 0.0) [ 0.1 0.5 0 .5 0.5)

-

[-0 .1 -0 .1 -0.1 [ 0.1 0.1 0.1 [0.0 0.0 0.0 [0 .0 0.0 0.0

tlUmin tlUma:r: = Umin Uma:r: =

..

Constraints (T > 1800 s)

dU/dt U

tlUmin tlUma:r: Umin Uma:r:

The prediction horizon should be selected such that the trajectory describing the projected output of the plant is longer than the settling time of the closed loop response. The sampling period , T., is selected as 2 s. The MSC controller needs a prediction horizon of at least 60 s. The first 50 s allows for the dead time of the plant, while the last 10 s is the controllable portion of the horizon . The prediction horizon (HP) is therefore selected as 30 steps.

= = =

-0 .1) 0.1) 0.0) 0.5)

The ability of the MSC controller to handle process constraints allows the designer more freedom to deal with the practical issues of controlling the plant . MSC control allows constraints on the rate of change of the input to the plant ( dU/ dt) and the input to the plant (U). The dU/ dt constraint may be used to protect the motors (assuming direct control) from current surges which might occur if the input signal to the feeder motors drives changes quickly. Constraints on the input to the plant (U) can be changed (on-line) to meet the requirements of the plant. For example, if a feeder needs to run on reduced power or if the feeder motor needs to be shut down then the controller can be changed by redefining the constraints on that particular feeder motor input.

The computational effort and processing speed required of the microprocessor running the MSC algorithm is dependent on the sampling period , number of inputs and outputs, order of the predictor model, prediction horizon (HP steps) and the number of process constraints. The selection of these parameters needs to be considered when selecting the computer needed to run the MSC algorithm.

The MSC controller settings used in the simulation tests are summarised in Table 1. It is possible in the MSC control algorithm to include feedforward control to improve the rejection of measurable disturbances. In the feeder system, for example, it is possible to predict a blocked feeder before the effect on the mass feed rate is measured 50 slater. The current drawn by a feeder motor gives an indication of the load on that feeder . For certain kinds of blockage the load torque will increase, causing the motor current to increase. The measured motor current can therefore be used for feedforward control.

Tuning of the MSC controller is dependent on the dynamics of the plant and the closed-loop performance demanded . The two tuning parameters of the controller are (3 and), in (2) . The). variables are weighting factors describing the cost of tracking error over the prediction horizon. The robustness of the controller is tuned using the (3 parameters. It is found through experience that suitable parameters are (3 = [0.50 0.50 0.50 0.50] and ). 0.1. In order to emphasize the independent control that the feeders have, the (3 parameter is changed to (3 = [0 .500.55 0.65 0.75]. Each (3 index describes the weighting of the rate of change of the input to the plant. In this case feeder 1 has a (31 = 0.50 which is less than the (3 of feeder 4 ((34 = 0.75), which results in feeder 4 having a more sluggish response than feeder 1.

=

5. SIMULATION Consider the following scenario. Of the four feeders in the mill feeder system, feeder 1 cannot operate at full power, for example, due to a damaged gear box. It is decided that in order to prevent

244

Sensor Output 2r-----~--------------------__,

Plant output

2 1.5

>

>

o

0 .5

·1~--------------~----~--~ o 500 tOOO 1500 2000 2500

0.6

500

1000

1500

Time (sec)

Time (sec)

Plant Inputs

Computation time delay

2000

2500

2000

2500

2000

2500

2000

2500

r---~--~--~--~----,

E :: 0 .5 E

;:::

500

1000

1500

2000

2500

500

1000

Tim. (sec)

1500

Time (sec)

Fig. 3. Simulation results (feedback only)

Sensor Output

Plant output

2 1.5 >

>

o

0.5

·1 ~--~--~----------~

o

0.6

500

1000

1500

2000

2500

500

1000

1500

Time (sec)

Time (sec)

Plant Inputs

Feedlorward input

r---~--~--~--~----.

0 .4

..,

0.5 0

0.2

I

~ .S

500

1000

1500

2000

2500

Computation time delay

E :: 0.5

~~

1000

tSOO Time (.ec)

500

1000

1500

Time (sec)

Time (sec)

500

0

2000

2500

Fig. 4. Simulation results (feedback plus feed forward)

245

further damage to the motor the input signal to the feeder drive should not exceed 0.1. The remaining feeder inputs should not exceed 0.50 . The constraints on the inputs are therefore selected as Umin [0000] and Umar [0 .100.500 .500 .50] . The plant is initially at steady state with a measured output Y = O. The set-point of the con1 at time T 0 s. The troller is set to Y set-point is changed to Y 0.6 at time T 480 s. Feeders 1, 2 and 3 become blocked at time T = 800 s. Feeders 1, 2 and 3 are unblocked at time T = 1500 s. The controller is changed at time T = 1800 s. Feeders 1, 2 and 3 are shut down (constraints on U changed) , leaving only feeder 4 to provide ore.

=

=

=

=

=

=

Simulation model. The model used by the controller is based on the dominant or low frequency dynamics of the plant . Little is known about the high frequency dynamics of the mill feeder . The simulator model synthesises the plant dynamics and is based on the predictor model defined in (3) . High frequency dynamics are included in this model by adding to each element in the model a pole which is a decade higher than the dominant pole.

suIted were correct by the integral action of the controller. In any control system the controller attempts t< track the noise just as readily as the comman< signal. The effect of measurement noise on the plant output can be seen in the plot "Plant Out· put", which shows the output of the simulatiOl model as a function oftime . This output oscillate: around the set-point which can be attributed t< sensor noise.

7. CONCLUSIONS

A simulation study of Multi-Step Predictive Con trol applied to a mill feeder system has been de· scribed . This plant is multi variable and exhibit: a long dead time and a highly-variable gain . MSC is a multivariable control technique whid has inherent integral action, the ability to han· dIe process constraints and is well suited to plant: that have significant dead time. In addition t( these properties it allows easy incorporation 0 feedforward control. The simulation results show that the MSC con· t roller provides robust control in the face of mod· elling errors and sensor noise. The ability of thE controller to handle constraints, dead time an( feedforward disturbances is shown in the simula tion results.

Sensor noise. Gaussian noise with a standard deviation of 0.15 units is added to the simulated plant output to give signals typical of those seen in practice (MacLeod, 1992) .

6. SIMULATION RESULTS 8. REFERENCES

The feeder system is simulated using MSC control first with only feedback and then with feedback and feedforward control. Simulation results are shown in Fig. 3 and Fig. 4.

Athans , M and P Falb (1966) . Optimal Control New York Mcgraw-hill. Bolton , R .L.J . (1994). Development and evalua· tion of a new predictive control algorithm fOl the control of multivariable systems. Master ': thesis. University of the Witwatersrand, Johan· nesburg. Bolton , R.L.J . and I.M . MacLeod (1994) . A simulation study of the control of a multivariable system using multi-step predictive control. Sub· mitted to S.A . Institute of Electrical Engineers Garcia, C .E., D.M. Prett and M. Morari (1989) Model predictive control:theory and practice-c survey. Automatica pp. 335-348. MacLeod, I.M . (1992) . A controller incorporatin~ a fuzzy logic adaptive mechanism for process~ with long dead times . Proc. S.A. council fo : A utomation and Computation symposium 01 Fuzzy Logic in control systems pp . 1-7. Stanley, G .G . (Ed .) (1987) . The extrative metal lurgy of gold in south africa. S. A . Institu te o. Mining and Metallurgy, Johannesburg 1 , 84 .

The results illustrate the constraint-handling ability of the MSC controller. At each control step the MSC algorithm solves a quadratic program with inequality constraints. When the plant operates close to a constraint the computational delay in solving the quadratic program increases. Refer to the plot showing the normalised computational time delay at each control step . The improved disturbance rejection of the MSC feedforward controller can be seen in Fig. 4. This simulation relies on the prediction of the disturbance based on the current of the feeder motors. The MSC algorithm has inherent integral action , which results in the controller having high gain at low frequencies. This property is useful for the nulling of constant offset disturbances affecting the plant outputs. The disturbances found at time T=800 sand T=1500 s caused the gain of the plant to change. The offset errors which re-

246

PAPER PRESENTED BY RLJBOLTON

DISCUSSION FOLLOWING THE PRESENTATION:

B D 0 Anderson: What happens if the dead-time varies from that assumed? R L J Bolton: The system is fairly robust, but one may have to increase the prediction time. K Furuta: How did you choose the horizon of the criterion? R L J Bolton: By considering the dead time (50 s) and dynamics (5 s) of the plant. We need about 10 seconds more than the dead time, i.e. about 60 seconds. With a sample time of 2 seconds, this requires about 30 steps. R Perez-Correa: How do you get the predicted output? How does this differ from QDMC? R L J Bolton: The MSC algorithm is based on a transfer function linear modeL unlike DMC which is a step response model. MSC, like QDMC, uses a quadratic program with inequality constraints to solve the optimal control problem for each step. MSC is formulated in a different way to QDMC, but both result in the same quadratic program. A van Cauwenberghe: Did you study the influence of the length of the prediction horizon, as well as of the control horizon on the control performance? Is the prediction horizon long enough to obtain robust control performance and is your control horizon not unnecessarily long? R L J Bolton: Not in great detail. I selected the HP & CP values that gave a suitable response. If more robustness is required then it may be necessary to increase HP and decrease CP. This may be important when the uncertainty of the transport delay is considered.

247