Event-Based Predictive PI Control for Mobile Crushing Plants

2nd IFAC Workshop on Multivehicle Systems The International Federation of Automatic Control Espoo, Finland. October 3-4, 2012

Event-Based Predictive PI Control for Mobile Crushing Plants Pasi Airikka* *Metso Corporation, Tampere, P.O Box 237, FIN-33101 FINLAND (Tel: +358-40-5872730; e-mail: [email protected]).

Abstract: Mobile aggregate crushers operate with wireless communication for plant-wide operability, controls and monitoring. Mobile crushing plants typically have relatively long material transportation delays making processes rather dead time dominant. Consequently, plant-wide controls rely on single predictive PI controllers designed for compensating process dead times. However, due to wireless communication in harsh outdoor conditions, measurements may not always be available for process controls. To solve this, this paper suggests an event-based control strategy for predictive PI controllers. Keywords: Event-based, Predictive PI control, Mobile, Crushers.

1.

Årzen (1999) was the first to propose an event-based version of the PID controller for asynchronous, that is, eventtriggered control to replace conventional synchronous timetriggered control. Owing to Årzen’s idea, several improvements and modifications have been introduced since.

INTRODUCTION

Mobile crushing plants comprise of mobile crushing units having crushers, screens, conveyors, feeders and feed hoppers. Typically, there are one to three mobile crushing units that may be linked to each other in various order to establish the most appropriate mobile crushing process for particular circumstances and operation needs at hand.

Durand and Marchand (2009a, 2009b) pointed out a few improvements on Årzen’s version for a event-based PID controller. First, they showed that Euler’s forward approximation of a numerical integral cannot be applied due to non-existing a priori information of the next real control period. Second, they introduced a separation method for dividing control error term into two parts for long idle times not having control actions. Third, they wanted to get rid of the safety condition set for limiting the maximum allowable control period. Fourth, they introduced a minimum control period for not allowing too small control periods.

A mobile crushing unit is typically equipped with machine controls allowing an independent operation as a stand-alone unit. The mobile units with cone crushers, for example, have closed-side setting (CSS) and eccentric speed (ES) as manipulated variables. Apart from that, there is not much else to be manipulated in automatic in single crushers. Therefore, to enable plant-wide controls for regulating crushing plant capacity, productivity, quality and yield of different endproduct aggregate fractions, there must be a plant-wide automation system linking all the crushing units together.

Since then, there has been an obvious interest in two-degreeof-freedom PID controllers and removal of sticking and limit cycle issues that may derive from a badly implemented eventbased controller. Nevertheless, this paper proposes a way to implement an event-based PPI controller with integral antiwind-up. Furthermore, it announces a new way to handle the I controller in cases when it has been long since the last control action has been executed.

Mobile crushing units communicate wirelessly with a plantwide automation and information management system. Operation in harsh outdoor conditions obviously brings challenges for estabishing and maintaining reliable and synchronous communication. Therefore, it would be of importance to implement the plant-wide automation system’s PID control loops in such a way that control performance does not get disturbed by abrupt communication breaks.

2.

The main target of a mobile crushing plant is to process the large-sized rocks (feed material) into targeted aggregate fractions having dimensions measured in millimeters through crushing and screening the feed material in different stages with recycling for recrushing oversized material and bypasses for fine materials. To accomplish all this, a mobile crushing plant is typically formed of several crushing units that can be linked to each other. Figure 1 illustrates a fleet of three mobile crushers (jaw crusher + 2 cone crushers) that can be

As crushing plant processes for material flow, crusher level, conveyor level or hopper level mainly contains relatively long dead times and residence times due to material transportation through conveyors, crushing and screening, it is essential to have dead time compensation techniques available in the plant-wide automation system. A well-known way to compensate dead times is a predictive PI (PPI) controller (Hägglund, 1996) and its variants (e.g Airikka, 2011, 2012).

978-3-902823-15-1/12/$20.00 © 2012 IFAC

CRUSHING PROCESSES

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10.3182/20121003-3-SF-4024.00008

2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012

operated in different combinations (e.g 1+2+3 or 1+3) to create a mobile crushing plant.

Predictive PI controller was first introduced by Hägglund (1996). It is an elegant simplification of Smith Predictor (Smith, 1958) with only three tuning parameters: proportional gain k p , integral time t i and delay estimate L. For simplicity, Hägglund proposed to set the prediction gain k pred = 1 / t i for making that parameter trivial for controller design. For a practical reason, however, it may be subject to fine tuning resulting in having four adjustable tuning parameters in total.

Figure 1. Fleet of Metso Lokotrack mobile crushing units (from left to right: LT120, LT300GP, LT300GPB). The crusher units have their purpose-built machine-specific control system to support their independent functions and machine conditioning. However, to automate the whole crushing plant all the crushers need to be linked to a single DCS system for providing smooth and efficient coordination of capacity, quality and energy efficiency. Figure 2 shows a main operator screen for monitoring and controlling a mobile crushing plant.

t

t

∫

u (t ) = k p e(t ) + k i e(τ )dτ − k pred 0

∫ u(τ )dτ

(1)

t−L

where k i = k p / t i . For implementation, the PPI controller can be reformulated to have the same integration time limit for both integral and predictive part: t

t

∫

∫ (u(τ ) − u(τ − L))dτ

0

0

u (t ) = k p e(t ) + k i e(τ ) − k pred

(2)

For applicability, the PPI controller must be equipped with an integrator antiwind-up having an adjustable tracking time constant T as described in Åström (1995) using a dynamic tracking from saturated control signal u sat t

t

∫

∫

0

0

u (t ) = k p e(t ) + k i e(τ )dτ − k pred ∆u (τ )dτ +

1 T

t

∫ v(τ ) 0

t

1   = k p e(t ) +  k i e(τ ) − k pred ∆u (τ ) + v(τ ) dτ 123 T   0 44444424444443 u p (t ) 1

∫

Figure 2. Operator main screen for a mobile crushing plant in Metso DNA automation system.

ui ( t )

with Primary controlled variables in crushing processes are feed flows and material levels in hoppers, feed chutes, conveyors and crushers. These variables typically come with dynamics which mostly involve material transport delays (dead times). Consequently, most of the crushing plant processes are dead time dominant and their minimum-phase dynamics can be characterized by FOPDT (First-Order Plus Dead Time) or SOPDT (Second-Order Plus Dead Time) models. 3.

(3)

∆u (τ ) = u (τ ) − u (τ − L) v (τ ) = u sat (τ ) − u (τ )

4.

(4a) (4b)

EVENT-BASED PPI CONTROLLER

The basic idea of an event-based controller is to call the controller routine only when there is an obvious reason for it. Consequently, the control action is always event-triggered (asynchronous). However, for judging if a control action is required, there must be a time-triggered (synchronous) decision-making procedure. In his novel breakthrough paper, Årzen (1999) named decision-making as an event detection procedure.

PPI CONTROLLER

For widely adopted PID controllers (Proportional-IntegralDerivative), dead times are a real challenge resulting in rather inadequate control performance due to either sluggish (overdamped, slow) or aggressive (under-damped, overshooting, oscillatory) control responses. The main reason for this is that the PID controller has no capability of predicting process behaviour but, instead, it only relies on the past and the present behaviour of the control loop. To overcome this, there are variants of PID controllers having prediction involved such as Smith predictor (Smith, 1950), PPI controller (Hägglund, 1996) and improved PPI variants (Normey-Rico, 2001) and (Airikka, 2011, 2012).

The fundamental difference between conventional periodic time-triggered and non-periodic event-triggered control is the control period that remains constant for time-triggered control but is time-variant for event-based control. This apparently insignificant distiction, however, makes controller implementation rather tedious and cumbersome. It is a known fact that event-based control is an emerging field of control system theory and, in its infancy, it is still lacking a mature theory for stability and robustness analysis and control design.

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2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012

control instant and the whole history since the last control execution for cases when hact > hmax

In event-based control, the actual control period hact at time of control execution t act is calculated as hact = t act − t old

hact e(t act ) ≈ (hact − h)elim + he(t act )

(5)

The upper limit for control period can be set e.g to hmax = 10h . In this paper, it is recommended to set the limit

where t act is current (actual) time instant and t old is time instant for the previous control action. In this paper, it is assumed that the minimum of hact equals to a nominal fixed control period h which is also the execution period of the time-triggered event detection procedure.

for the PPI controller to hmax = L . The limit value elim in (9) may not be needed used for the event detection procedure. Thus, it would be another new tuning parameter for implementation. To avoid this, it is proposed to make implementation more straighforward by applying a trapezoidal approximation for the first term using the control error values at two time instants for hact > hmax

Despite a lacking solid system theory for event-based systems, a number of publications based on simulations have pointed out that there are certain pitfalls to be avoided by a careful controller implementation. The main pitfalls are called sticking and limit cycles. Sticking means a case where the controller freezes, that is, gets stuck with having a control error offset that is not acceptable for that particular control design. Limit cycle is oscillation with a limited amplitude around the given setpoint with amplitude and frequency determined by the controller parameters.

hact e(t act ) ≈ 0.5(hact − h)(e(t act − h) + e(t old ) ) + he(t act )

The proportional P control can be implemented straigthforward as

To solve the issue of control action bumps, it is proposed to ignore the control error history by removing the first term of (9) totally and to approximate the product of control period and control error as follows for hact > hmax

(6)

However, the integral control with prediction and antiwindup obviously requires more attention. The integral term ui(t) in (3) can be discretized as

hact e(t act ) ≈ he(t act )

h u i (t act ) = u i (t old ) + k i hact e(t act ) − k pred hact ∆u (t act ) + act v(t act ) T 1   = u i (t old ) + hact  k i e(t act ) − k pred ∆u (t act ) + v(t act )  (7) T  

with

∆u (t act ) = u (t act ) − u (t act − L)

(8a)

v (t act ) = u sat (t act ) − u (t act )

(8b)

(10)

Yet, neither of implementation variants (9) or (10) totally removes the issue of having large control actions due to long idle times with no control execution. Originally Årzen (1999) proposed to have a safety limit hact ≤ hmax but having such an upper bound severely limits the achievable computational load reduction. And to avoid any performance limitations, the safety limit is omitted here.

4.1 Discretization of PPI controller with antiwind-up

u p (t act ) = k p e(t act )

(9)

Similarly, it is recommended to apply the approximation method for the prediction term in (7)

(11) same

hact ∆u (t act ) ≈ 0.5(hact − h )(∆u (t act − h) + ∆u (t old ) ) + h∆u (t act ) ≈ h∆u (t act )

(12) Finally, the integrator antiwind-up in (7) can be approximated in a same manner

To implement numerical integration without requiring any prediction on the next control period, Euler’s backward approximation has been used in (7) as proposed by Durand and Marchand (2009b).

hact v(t act ) ≈ 0.5(hact − h )(v(t act − h) + v(t old ) ) + hv(t act ) ≈ hv(t act )

(13)

If the condition hact > hmax does not hold, the product of the

4.2 Long idle times with no control actions

actual control period

hact and any of the signal e(t act ), ∆u (t act ) or v(t act ) can be calculated as thus.

In addition to issues of sticking and limit cycles, there is another issue with approximation of numerical integration for controllers having integration part such as PID and PPI controllers. Between two consequtive control actions having a large hact even a small non-zero control error may contribute strongly to an integral control action at the time of the next control execution.

4.3 Event detection There are numerous methods proposed for event detection and triggering control actions. A good reference to these methods with control strategy implementation examples is given by Vilanova and Visioli (2012).

To avoid strong bumps after a long idle time between the consequtive control actions, Durand & Marchand proposed (2009b) a separation method for separating the current

Mobile crushers that communicate wirelessly may not provide sensor or machine condition information at fixed equidistant sampling instants. Therefore, event detection 68

2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012

operating at fixed time instants may not always have information for judging if control actions are required. Hence, for simplicity, it is recommended not to execute any control action if there is no information available to support such an act.

if hact>hmax, he = h * e ; else he = hact * e ; end

However, to make a decision whether a control action is required or not, it is recommended to trigger an event if the following conditions is true

% P control up = kp*e ;

e(t act ) − e(t old ) > ∆elim OR e(t act ) > elim

% I control (forward appr.) ui = ui + ki*he ;

(14)

The first condition is to make sure that there are control error activities going on until the controller is again called to execute its routine. Control error changes smaller than ∆elim do not require any control actions. However, to prevent sticking it is necessary to have a separate condition. The latter condition ensures that the control loop does not get stuck unless the control error is smaller than e lim .

% Total control utotal = up + ui ; % Predictive control du = utotal - u(delayed by L) ; if hact>hmax, hdu = h*du ; else hdu = hact * du ; end ui = ui - kpred*hdu ;

4.4 Implementation of event-based PPI controller The event-based PPI controller with anti-windup is suggested to be implemented as given in (6), (7), (8a) and (8b) using approximations given in (11), (12) and (13). For user-friendly implementation, the PPI controller is given using pseudocode syntax as below.

% Integral anti-windup us = max(umin,min(utotal,umax)) ; v = us – utot ; if hact>hmax, hv = h*v ; else hv = hact*v ; end ui = ui +hv/T ;

% Controller parameters kp = proportional gain ki = integral gain kpred = predictive gain L = process dead time estimate T = tracking time for integral anti-windup

% Reset & storage hact = 0 ; eold = e ; u = us ; uold = u ;

% Limit values delim = control error change limit elim = control error limit h = nominal control period hmax = soft upper limit for control period umin = control minimum limit umax = control maximum limit

else hact = hact + h ; u = uold ;

% Signals e = control error = setpoint – output eold = control error at previous control action instant u = controller output up = proportional control ui = integral control utot = total control

end 5.

SIMULATION RESULTS

Consider a FOPTD crushing process with static gain of 0.5, time constant of 5 sec and dead time of 20 sec controlled by a PPI controller with parameters k p = 2 , t i = 5 , k pred = 0.2 and

% Values hact = actual control period

h = 1 . The event-based PPI controller is set up with a nominal control period hnom = 1 sec, ∆elim = 0.02 and e lim = 0.05.

% Event detector edif = abs(e - eold) ; eabs = abs(e) ;

Figure 3 illustrates setpoint and load disturbance responses and controller outputs for both time-triggered and eventbased PPI controlled system. At time t = 0 there is a unit setpoint change up and at t = 150 sec the unit load disturbance with the same process dynamic than that of the

if (edif > delim) | (eabs > elim), % Product of control period and control error 69

2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012

process strikes the closed loop. The immediate conclusion is that the event-based system’s response is rather similar to that of the time-triggered system althrough it has significantly fewer control actions.

+ e lim (dash). Middle: Absolute control error change with the limit ∆elim (dash). Bottom: Control execution signal. Figure 5 shows breakdown of control periods (top), control period time trend (middle) and development of control actions (bottom). The average control period is 3.0 sec and the maximum control period is as high as 134 sec. The CPU load which is calculated as a ratio between actual control actions (73) and a number of nominal control period (300 x 1) is 24.3%.

1

0.5

0

0

50

100

150

200

250

300

Distrib. of control periods [%]

Output

1.5

4

Control

3 2 1

0

50

100

150 Time [sec]

200

250

300

Control periods [sec]

0

Figure 3. Time-triggered PPI controller (blue dashed) and event-based PPI controller (red, solid). Upper: Process output. Lower: Control signal.

50 40 30 20 10 0

0

2

4

6

8

%

12

14

16

18

20

6 4 2 0 50

Figure 4 shows the control error response for both timetriggered and event-based PPI controlled system with the control error limit e lim . It also shows the control error change

10

8

100

150

200

250

40

100

20

50

∆e with error change limit ∆elim . When either of the given

limits elim or ∆elim is violated, the controller is executed. The execution time instants are also given in the figure 4. As expected, most of the control actions are taken as a result of setpoint and load disturbance change. All in all, there are 73 control actions during 300 seconds (24.3%).

0

Control error

1

-0.5

Control error change

-1 100

150

200

250

300

0

50

100

150

200

250

300

0.2 0.15 0.1 0.05 0

150 Time [sec]

200

250

Control period [sec]

e lim = 0.05

0.5

CPU load [%]

∆elim

Min

Average

Max

0.01

1

2.5

31

36 7

0.05

1

2.9

133

23.7

0.1

1

3.0

139

23.0

0 0

50

100

150 Time

200

250

0 300

Table 6. Performance statistics for an event-based PPI controller with various control error change limit ∆elim .

1 0:off, 1:on

100

Tables 6 and 7 give performance indeces (average control period, maximum control period and CPU load) with respect to various limit values for ∆elim and e lim for the event-based PPI controlled case system. The conclusion is rather expected. The larger the limits are, the less there are control actions resulting in higher control period averages and lower CPU load rates.

0

50

50

Figure 5. Control periods of PPI controller. Top: Distribution of control periods. Middle: Control period time trend. Bottom: Development of control actions (total = green, relatively = blue).

0.5

0

0

300

Figure 4. Time-triggered (blue dashed) and event-based PPI controller (red, solid). Top: Control error with limits of

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2nd IFAC Workshop on Multivehicle Systems Espoo, Finland. October 3-4, 2012

Figure 8 illustrates comparison of process output and control responses for different control error change limits ∆elim as given in table 6. It can be seen that when the limit gets higher, the responses slighty start deteoriate in performance. Hence, setting control error limits requires making a trade-off between CPU load performance and control performance. Figure 9 reveals how CPU load gets higher and average actual control period gets smaller when the normalized dead time τ = L /( L + T ) increases.

6.

This paper introduced a method to implement an event-based PPI controller with an integrator antiwind-up for industrial usage. The proposed method is particularly suitable for dead time dominating processes having limited possibilities for synchronous control execution. The proposed event-based PPI algorithm is particularly suitable for mobile crushing plant processes but it can be equally applied to any other process plant having a need or a possibility for asynchronous event-based control. The proposed method was simulated and tested with various event-triggering parameters. The preliminary results give confidence to carry on the research for practical implementation and usage on a real mobile crushing plant.

Table 7. Performance statistics for an event-based PPI controller with various control error limit e lim . Control period [sec]

∆elim = 0.02

CPU load [%]

e lim

Min

Average

Max

0.01

1

1.8

95

43.3

0.05

1

3.0

134

24.3

0.1

1

3.1

135

23.3

REFERENCES Airikka P. (2011). Extended Predictive Proportional-Integral Controller for Typical Industrial Processes, The 18th IFAC World Congress, Milan, Italy, CD-ROM Proceedings. Airikka P. (2012). Another Novel Modification Of Predictive PI Controller For Processes With Long Dead Times, IFAC workshop on Advances in PID Control, Brescia, Italy, 2830.3.2012.

Output

1

0.5

0

0

50

100

150

200

250

Durand S., N. Marchand (2009a). An Event-based PID Controller with Low Computational Cost, Proceedings of 8th International Conference on Sampling Theory and Applications (SampTA’09), Marseille, France.

300

3 Control

CONLUSION

Durand S., N. Marchand (2009b). Further Results on EventBased PID controller, Proceedings of 10th European Control Conference, Budapest, Hungary.

2 1 0

0

50

100

150 Time [sec]

200

250

Hägglund T. (1996). An industrial dead-time compensating PI controller, Control Eng. Practice, Vol. 4, No. 6, pp. 749756.

300

6

60

4

40

2

20

Ingimundarson A., T. Hägglund (2001). Robust tuning procedures for dead-time compensating controllers, Control Engineering Practice, 9, pp. 1195-1208. Smith O.J.M (1957). Closer control of control loops with dead time, Chem. Eng. Progr., 53, pp. 217-219. Vilanova R., A. Visioli (2012) (editors). PID Control in the Third Millennium – Lessons Learned and New Approaches, pp. 495-526 on Event-Based PID Control, Springer-Verlag, London.

CPU load [%]

Average control period [sec]

Figure 8. Comparison of PPI controller responses for different control error change limits Top: Process outputs. Bottom: Control signals (Red solid = 0.01, Red dash = 0.05, Green dash dot = 0.1).

0

0

0.1

0.2

0.3

0.4 0.5 0.6 Normalized dead time [0-1]

0.7

0.8

0.9

Åström K.J, T. Hägglund (1995), PID Controllers: Theory, Design and Tuning, 2nd ed., Instrument Society of America, USA.

0 1

Årzen K.E (1999). A Simple Event-based PID Controller, Proceedings of IFAC 14th World Congress, Beijing, China.

Figure 9. Average control periods and CPU loads of eventbased PPI controllers for different normalized dead times.

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