- Email: [email protected]
ROBUST MPC OF A RUN-OF-MINE ORE MILLING CIRCUIT
ROBUST MPC OF A RUN-OF-MINE ORE MILLING CIRCUIT L.C. Coetzee ∗ I.K. Craig ∗∗
∗
Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria, 0002, South Africa. Tel. +27 12 998 9853 Fax.+27 82 131 558 2699 Email: [email protected] ∗∗ Email: [email protected]
Abstract: In this article a robust MPC controller is presented for a run-of-mine milling circuit. The objective of the controller is to reduce the variability in the particle size from the milling circuit which leads to increased recovery of gold from downstream processes. The controller design is evaluated through a simulation c study. Copyright 2007 IFAC Keywords: Robust model predictive control, RMPC, run-of-mine ore milling circuit, ROM milling circuit
1. INTRODUCTION Milling of ore from a mine is an important step in the metallurgical extraction process. The process is complicated by significant input and plant uncertainties, because the feed ore forms part of the grinding medium and the variation in feed ore contributes to the input uncertainty. Proper design of the milling circuit alone can not eliminate the disturbances. Feedback control systems play an important part in reducing the effects of disturbances and increasing efficiency. Reduction of output variations does not only have a positive effect on the economics of the milling circuit itself but is also beneficial in downstream processing. In milling circuits it is difficult to control important variables like the product particle size, because independent control of the amount, size and hardness of the grinding medium in the mill is not possible. This causes significant uncontrollable disturbances and uncertain plant dynamics (Craig and MacLeod, 1995). To address the issues of uncertainty and performance specifications, Craig et al. (Craig and
MacLeod, 1995, 1996) used µ-synthesis (Doyle, 1983, 1987). This allowed model uncertainty and performance specifications to be incorporated explicitly into the controller synthesis. Craig et al. used over one hundred step tests to obtain the model uncertainties, and showed that the ROM milling problem can be formalized in a general synthesis and analysis framework (Craig and MacLeod, 1995). The controller synthesised through µ-synthesis was implemented on a real plant (Craig and MacLeod, 1996). Craig et al. conclude however that the effort needed to design the controller is not worth it, because simpler techniques such as the previously used inverse Nyquist array (INA) method provided adequate performance (Hulbert et al., 1990). The application of robust MPC to this problem was therefore investigated, because of the input and model uncertainties inherent in this problem. Robust MPC guarantees stability for all adequately described model realizations and disturbances. The performance of the closed-loop can be tuned without affecting the stability of the system. A robust model predictive control
1987). The mill discharges pulp through an enddischarge grate into a sump. The pulp is diluted with water in the sump and pumped to the hydro-cyclone. The hydro-cyclone has an internal diameter of 1m. The underflow of the cyclone, water and feed ore constitute the mill feed. The variables of the mill (figure 1) that can be controlled are the level of the slurry in the sump (SLEV), the product particle-size (PSE) and the mass of material in the mill (LOAD). The inputs to the mill that can be manipulated are the feedrate of water to the sump (SFW), the flow rate of slurry to the cyclone (CFF) and the feed-rate of solids to the mill (MFS). There is an additional input, the rate of water fed to the mill inlet (MIW) that was not modelled for control purposes. The solids feed-rate to the mill (MFS) does effect the particle size (PSE), but has an order of magnitude larger time constant and time delay compared to feed-rate of water to the sump (SFW) and flow rate of slurry to the cyclone (CFF). The effect of MFS on PSE can be compensated for by the faster SFW and CFF and is therefore left out of the model.
Figure 1. Milling Circuit Schematic technique by Pannocchia et al. (Pannocchia, 2004; Pannocchia and Kerrigan, 2003) was investigated for this system. This technique constructs a robust invariant set for a dynamic state-feedback controller that can accommodate asymmetric constraints and has a much bigger feasible area than their ellipsoid counterparts. This method provides offset free tracking by using a dynamic statefeedback controller rather than just a static statefeedback gain controller. It uses a simple MPC algorithm similar to nominal MPC that is based on a quadratic program (QP) rather than semidefinite programming (SDP) as used in linear matrix inequality (LMI) based methods (Kothare et al., 1996).
The milling circuit transfer function model (figure 2) without ∆LOAD/∆M F S has the following form: ∆P SE g g ∆SF W = 11 13 ∆SLEV g31 g33 ∆CF F with g11 and g13 of the form gij =
kij e−θij s τij s + 1
(1)
and g31 and g33 of the form kij −θij s e . (2) s Using the standard deviation of each model parameter, the relative uncertainty in each transfer function element is given by gij =
2. ROM MILL CIRCUIT 2.1 ROM Milling Circuit Description An industrial run-of-mine circuit with single stage classification is discussed.
35% 31% − −
19% 18% − − − 27% θij : − −
kij :
The circuit is fed gold-bearing ore at about 100 tons/hour and grinds it to give a product with particle size of 70% to 75% smaller than 75 µm. The ROM mill is operated in closed circuit with a hydro-cyclone that separates the product from the out-of-specification material which is recycled to the mill. The gold is then extracted through a leaching process downstream.
τij :
(3) (4) (5)
where the dashes indicate for that particular parameter, the uncertainty is insignificant or the parameter itself does not exist.
A typical mill has dimensions of 5m in diameter and a length of 9m. The mill is supported by pressurized-oil circumferential bearings. The mill features lifter bars and solid white-iron liners and it is operated at 90% of critical speed (Stanley,
2.2 Objectives in Mill Control The control of the milling circuit has multiple objectives, firstly to stabilize the system and sec2
∆P SE ∆SLEV
0.14 −40s 175s+1 e
−0.0575 −40s 167s+1 e
= −0.00299 s
0.00253 s
∆SF W ∆CF F
Figure 2. Nominal plant transfer function ondly to optimize the economics of the process (Hulbert, 1989). The economic objective is divided into sub-objectives that each contribute to the overall economic objective of the milling process. A set of possible sub-objectives for the milling circuit are to (Craig and MacLeod, 1995):
3. ROBUST MODEL PREDICTIVE CONTROLLER The controller proposed by Pannocchia and Kerrigan (2003, 2005) is a dual-mode controller. It consists of an unconstrained dynamic state-feedback controller. Free perturbations are added to the control action of the dynamic controller to enlarge the feasible region and impose the constraints. For brevity, only the main results are shown and the reader is referred to the references for more detail.
(1) improve product quality (a) by increasing grind fineness, (b) and decreasing the fluctuations in product size, (2) maximize throughput, (3) minimize the amount of steel that is consumed for each ton of fines produced, (4) and to minimize the power consumed for each ton of fines produced, etc.
The discrete-time linear time-invariant plant is given by x+ = Ax + Bu + Ed
The objectives above are interrelated and require trade-offs to be made. There is a trade-off between particle size of the product and the throughput of solids (objectives 1a and 2). More gold can be extracted at a finer product size (objective 1a), but the variation in particle size also influences the recovery (objective 1b).
z = Cz x
(6) (7)
in which x ∈ Rn is the plant state, x+ is the next plant state, u ∈ Rm is the control input (manipulated variable), d ∈ Rr is a persistent, unmeasured disturbance and z ∈ Rp is the controlled variable, the variable to be controlled to some time-varying setpoint s ∈ S ⊂ Rp . The constraints on the input and state is given by affine inequality constraints
It is assumed that the throughput of the mill is maximized when it draws maximum power from the mill motor. The ∆LOAD/∆M F S circuit is therefore often under power peak seeking control Craig et al. (1992). This is contrary to objective 4 which is to minimize electrical power, but given the value of the milling product versus the cost of electricity, objective 4 is not considered important.
x ∈ X ⊂ Rn , u ∈ U ⊂ R m ,
(8)
where X is a polyhedron (i.e. a closed and convex set that can be described by a finite number of affine inequality constraints) and U is a polytope (i.e. a bounded polyhedron) and X × U contains the origin.
Objectives 1 and 3 are interrelated. Steel is added to the mill to stabilize the conditions inside the mill and also increase throughput. A controller that is capable of stabilizing the particle size, will reduce the need for steel and thus objective 3 will be addressed when objective 1b is met.
3.1 Dynamic Controller A dynamic nonlinear time-invariant state feedback controller to be designed for the system given in (6)-(7) is assumed to have the following structure:
A possible control strategy is to maximize throughput at a certain particle size setpoint. This strategy considers both objectives 1 and 2. The particle size setpoint may be determined by throughput targets or if throughput is not a consideration, the particle sized can be optimized. A trade-off exists between throughput and grind, and grind and residue (product that is not recovered) (Craig et al., 1992). The aim of control would be to increase throughput, while keeping grind constant.
σ + = α(x, σ, s),
(9)
u = γ(x, σ, s),
(10)
where σ ∈ Rl is the controller state, σ + is the next controller state, α : Rn ×Rl ×Rp → Rl is controller state dynamics map and γ : Rn × Rl × Rp → Rm is the controller output map. The plant dynamics (6), together with the controller (9)-(10), forms a closed-loop system where the dynamics are given by 3
ζ
+
=
Ax + Bγ(x, σ, s) E + d. α(x, σ.s) 0
O∞ := {ζ∈Θ|ζ(k+1)=AK ζ(k)+Ed(k)+F s(k)∈Θ
(11)
for all s(k)∈S, all d(k)∈D and all k∈N}.
where ζ is the closed-loop state and ζ + is the next closed-loop state. The controller dynamics (9) can be defined by using the following auxiliary system: x ˆ = Ax + Bu + (dˆ + x − x ˆ), + dˆ = dˆ + x − x ˆ. +
(20)
3.3 Receding Horizon Controller The receding horizon controller uses the approach of ”pre-stabilizing” the plant by letting the linear control be modified with a perturbation term as follows: u = Kζ + Ls + v, (21)
(12) (13)
Combining the plant dynamics (6), with the auxiliary system (12)-(13), leads to the following augmented system:
where v ∈ Rm is the perturbation term. This leads to the augmented closed-loop system:
ζ + = Aζ + Bu + Ed,
The objective function for calculating the perturbation term is given by:
z = Cζ,
ζ + = AK ζ + Bv + Ed + Fs.
(14) (15)
JN (v) :=
in which x A 0 0 B ˆ , A := I + A −I I , B := B , ζ := x I −I I 0 dˆ E E := 0 , C := Cz 0 0 . (16) 0
N −1 X
vkT W vk ,
(22)
(23)
k=0
where the (positive definite) matrix W ∈ Rm×m is given by W := R + B T P B, R is the weighting matrix on the control action, P = P T > 0 is the solution to the discrete algebraic Riccati equation P = Q + AT P A −
The controller state σ ∈ Rl , with l := 2n, to be the states of the auxiliary system (12)-(13), i.e. x ˆ σ := ˆ . (17) d
AT P B(R + B T P B)−1 B T P A.
(24)
The objective function (23) must be solved subject to:
If the augmented system (14)-(16) is put under linear control u = Kζ + Ls as
VN (ζ,s) = {v∈RmN |F v≤b+Gd d+Gs s+H ζ ζ+H s s
ζ + = AK ζ + Ed + Fs,
(18)
AK := A + BK,
(19)
where the matrices F ∈ Rq×mN , Gd ∈ Rq×rN , Gs ∈ Rq×p(N −1) , H ζ ∈ Rq×3n , H s ∈ Rq×p and the vector b ∈ Rq depends on the augmented system dynamics (22) and the reader is referred to Pannocchia and Kerrigan (2003) for details on how to calculated them.
for all s∈SN −1 and all d∈DN },
where and the reader is referred to Pannocchia and Kerrigan (2003) for the details in calculating K, L and F.
(25)
4. CONTROLLER SYNTHESIS 3.2 The Maximal Constraint-admissible Robustly Positively Invariant Set
The robust model predictive controller consists of an offline part that is done only when the plant model changes. The online implementation is in the form of a quadratic program (QP) with linear constraints.
The constraint-admissible set Θ is defined as all the augmented states for which the plant state and input constraints are satisfied, for any choice of setpoint s ∈ S, if the control is given by u = Kζ + Ls (Pannocchia and Kerrigan, 2003).
Offline The first step is to discretize the system model in figure 2 with a sampling time of T s = 10s that is typically used for this application, and to convert it to a state-space representation. Next, the dynamic unconstrained state-feedback controller is designed using the weighting matrices defined in table 2. The weighting on the control
The maximal constraint-admissible robustly positively invariant set O∞ for the closed-loop system (18) is defined as all initial states in Θ for which the evolution of the system remain within Θ for all allowable infinite setpoint and disturbance sequences, i.e. 4
Table 1. Constraints for ROM milling circuit
PSE LOAD SLEV
Nominal Value 80% 40% 50%
SFW
300
m3 hour
-100
200
tons hour
-50
850
tons hour
-350
Constraint
MSF CFF
Min Value
Max Value
-10% -10% -30%
+10% +10% +30%
m3 hour tons hour tons hour
+100
m3 hour
+100
tons hour
+100
tons hour
258s) and the time delay of g13 is much longer (θ = 72.4s) in the ”real” plant model compared to the nominal model. The setpoint from nominal at time t = 0s is [5, -10], at time t = 2000s is [5, 10], at time t = 4000s is [-5, 10] and at time t = 6000s is [-5, -10]. These setpoint changes are chosen to stress the controller for the purpose of this simulation study. In practice, the plant would be set to a certain setpoint determined by the production strategy and the primary function of the controller would be to reject disturbances and reduce particle size variations.
Table 2. Weighting matrices on state error and control move size Matrix name
Matrix value I5
Q
R
The robust MPC with a static state-feedback gain controller (figure 3) shows offset in tracking particle size (PSE), but not the sump level (SLEV), because the plant is an integrator with respect to sump level. Integrator action would be required to eliminate the offset. This is addressed by the robust MPC with dynamic state-feedback controller, which tracks the setpoint without offset.
125 0 0 75
The simulations (figure 3) show only the deviation from the operating point (table 1, nominal values).
action is much higher than on the states to limit the severity of the control action. This is necessary to construct a feasible maximal constraintadmissible robustly positively invariant set O∞ . The flow rate of slurry to the cyclone (CFF) is weighted less than the flow rate of water to the sump (SFW), because CFF was employed less than SFW relative to its constraints by the controller.
6. CONCLUSIONS This study shows that robust model predictive control can be applied to a run-of-mine milling circuit. The controller controls particle size and sump water level, by manipulating the inflow of water to the sump and flow rate of slurry that is pumped from the sump to the cyclone.
The state constraint polyhedron X and input constraint polytope U is constructed from the system constraints as shown in table 1. They are employed in constructing the maximal constraintadmissible robustly positively invariant set O∞ . The set O∞ together with the augmented closed loop dynamics (22) are used to construct the set of admissible perturbation VN (ζ, s).
The throughput of the mill is a function of the feed-rate of solids to the mill (MFS). This acts as a disturbance to the controlled system, while the controller aims to maintain the particle size at its setpoint and minimize particle size variations.
On-line
The robust MPC with dynamic state-feedback control tracks the setpoint well compared to the robust MPC with static state-feedback gain control. This is significant, because most robust MPC implementations suffer from tracking errors (Pannocchia, 2004).
The on-line implementation of the robust controller involves solving the QP problem with objective function (23) subject to the constraints (25). The control law at each time-step is given by u = Kζ + Ls + v0∗ , where is v0∗ is the first term of the optimal solution v∗ of the QP problem.
An improvement can be made by reconstructing the state-space system to consist of previous inputs and outputs as described by Maciejowski (2002), which has the property that the states are always known, if it is assumed that the inputs are known and the outputs are measured. This representation has the added advantage that the states correspond to the actual system. This enables the controller to do output feedback, rather than state-feedback.
5. SIMULATION A simulation study is shown where the robust model predictive controller of section 3 is applied to the ROM milling model presented in section 2. The controller is compared to a robust MPC technique that only uses a static state-feedback gain controller (Pluymers et al., 2005a,b).
In practice, the load of the mill is controlled by a power optimiser Craig et al. (1992). It is believed that the mill is most efficient when maximum power is drawn from the mill motor. The load of
In this simulation scenario the actual plant differs from the nominal model. The time constants of g11 and g13 are much larger (τ11 = 275s and τ13 = 5
Plant Outputs
0 −10 0
2000 4000 6000 Time (seconds)
8000
20 0 −20 0
2000 4000 6000 Time (seconds)
8000
CFF (tons / hour) SFW (m3 / hour)
PSE (% < 75 µ m) SLEV (% Full)
10
(a) Plant Outputs
100
Plant Inputs
0 −100 0
2000 4000 6000 Time (seconds)
8000
2000 4000 6000 Time (seconds)
8000
100 0 −100 0
(b) Plant Inputs
Figure 3. Dual-mode Robust MPC with Dynamic state-feedback controller (Solid), Dual-mode Robust MPC with Static state-feedback gain controller (Dashed) and Setpoint (Dotted). the mill acts as a disturbance on the rest of the system variables. The controller can be improved by adding the load of the mill as a measured disturbance into the design.
Hulbert, D. G., I. K. Craig and M. L. Coetzee (1990). Multivariable control of a run-of-mine milling circuit. Journal of the South African Institute of Mining and Metallurgy 90(7), 173– 181. Kothare, M. V., V. Balakrishnan and M. Morari (1996). Robust constrained model predictive control using linear matrix inequalities. Automatica 32, 1361–1379. Maciejowski, J. M. (2002). Predictive Control with constraints. Prentice Hall. Great Britain. Pannocchia, G. (2004). Robust model predictive control with guaranteed setpoint tracking. Journal of Process Control 14, 927–937. Pannocchia, G. and E. C. Kerrigan (2003). Offset-free receding horizon control of constrained linear systems subject to time-varying setpoints and persistent unmeasured disturbances. Technical report. Department of Engineering, University of Cambridge. Cambridge, UK. CUED/F-INFENG/TR.468. Pannocchia, G. and E. C. Kerrigan (2005). Offsetfree receding horizon control of constrained linear systems. AIChE Journal 51(12), 3134–3146. Pluymers, B., J. A. Rossiter, J. A. K. Suykens and B. De Moor (2005a). The efficient computation of polyhedral invariant sets for linear systems with polytopic uncertainty. In: Proceedings of the American Control Conference 2005. Portland, USA. pp. 804–809. Pluymers, B., J. A. Rossiter, J. A. K. Suykens and B. De Moor (2005b). A simple algorithm for robust MPC. In: Procedings of the 16th IFAC World Congress. Prague, Czech Republic. Stanley, G. G. (1987). The Extractive Metallurgy of Gold in South Africa. Technical Report Vol 1. South African Institute of Mining and Metallurgy. Johannesburg.
A further improvement can be made by calculating the explicit solution (Bemporad et al., 2002) for the control law. This will significantly reduce the online computational burden, especially for large complex problems.
REFERENCES Bemporad, A., M. Morari, V. Dua and E. N. Pistikopoulos (2002). The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20. Craig, I. K. and L. M. MacLeod (1995). Specification framework for robust control of a runof-mine ore milling circuit. Control Engineering Practice 3(5), 621–630. Craig, I. K. and L. M. MacLeod (1996). Robust controller design and implimentation for a runof-mine ore millingcircuit. Control Engineering Practice 4(1), 1–12. Craig, I. K., D. G. Hulbert, G. Metzner and S. P. Moult (1992). Optimized multivariable control of an industrial run-of-mine milling circuit. Journal of the South African Institute of Mining and Metallurgy 92(6), 169–176. Doyle, J. C. (1983). Synthesis of robust controllers and filters. In: 22nd IEEE Decision and Control Conference. Doyle, J. C. (1987). A review of µ for case studies in robust control. In: IFAC World Congress (Munich). Hulbert, D. G. (1989). The state of the art in the control of milling circuits.. In: 6th IFAC Symposium on Automation in Mining, Mineral and Metal Processing (Buenos Aires). 6
∗
Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria, 0002, South Africa. Tel. +27 12 998 9853 Fax.+27 82 131 558 2699 Email: [email protected] ∗∗ Email: [email protected]
Abstract: In this article a robust MPC controller is presented for a run-of-mine milling circuit. The objective of the controller is to reduce the variability in the particle size from the milling circuit which leads to increased recovery of gold from downstream processes. The controller design is evaluated through a simulation c study. Copyright 2007 IFAC Keywords: Robust model predictive control, RMPC, run-of-mine ore milling circuit, ROM milling circuit
1. INTRODUCTION Milling of ore from a mine is an important step in the metallurgical extraction process. The process is complicated by significant input and plant uncertainties, because the feed ore forms part of the grinding medium and the variation in feed ore contributes to the input uncertainty. Proper design of the milling circuit alone can not eliminate the disturbances. Feedback control systems play an important part in reducing the effects of disturbances and increasing efficiency. Reduction of output variations does not only have a positive effect on the economics of the milling circuit itself but is also beneficial in downstream processing. In milling circuits it is difficult to control important variables like the product particle size, because independent control of the amount, size and hardness of the grinding medium in the mill is not possible. This causes significant uncontrollable disturbances and uncertain plant dynamics (Craig and MacLeod, 1995). To address the issues of uncertainty and performance specifications, Craig et al. (Craig and
MacLeod, 1995, 1996) used µ-synthesis (Doyle, 1983, 1987). This allowed model uncertainty and performance specifications to be incorporated explicitly into the controller synthesis. Craig et al. used over one hundred step tests to obtain the model uncertainties, and showed that the ROM milling problem can be formalized in a general synthesis and analysis framework (Craig and MacLeod, 1995). The controller synthesised through µ-synthesis was implemented on a real plant (Craig and MacLeod, 1996). Craig et al. conclude however that the effort needed to design the controller is not worth it, because simpler techniques such as the previously used inverse Nyquist array (INA) method provided adequate performance (Hulbert et al., 1990). The application of robust MPC to this problem was therefore investigated, because of the input and model uncertainties inherent in this problem. Robust MPC guarantees stability for all adequately described model realizations and disturbances. The performance of the closed-loop can be tuned without affecting the stability of the system. A robust model predictive control
1987). The mill discharges pulp through an enddischarge grate into a sump. The pulp is diluted with water in the sump and pumped to the hydro-cyclone. The hydro-cyclone has an internal diameter of 1m. The underflow of the cyclone, water and feed ore constitute the mill feed. The variables of the mill (figure 1) that can be controlled are the level of the slurry in the sump (SLEV), the product particle-size (PSE) and the mass of material in the mill (LOAD). The inputs to the mill that can be manipulated are the feedrate of water to the sump (SFW), the flow rate of slurry to the cyclone (CFF) and the feed-rate of solids to the mill (MFS). There is an additional input, the rate of water fed to the mill inlet (MIW) that was not modelled for control purposes. The solids feed-rate to the mill (MFS) does effect the particle size (PSE), but has an order of magnitude larger time constant and time delay compared to feed-rate of water to the sump (SFW) and flow rate of slurry to the cyclone (CFF). The effect of MFS on PSE can be compensated for by the faster SFW and CFF and is therefore left out of the model.
Figure 1. Milling Circuit Schematic technique by Pannocchia et al. (Pannocchia, 2004; Pannocchia and Kerrigan, 2003) was investigated for this system. This technique constructs a robust invariant set for a dynamic state-feedback controller that can accommodate asymmetric constraints and has a much bigger feasible area than their ellipsoid counterparts. This method provides offset free tracking by using a dynamic statefeedback controller rather than just a static statefeedback gain controller. It uses a simple MPC algorithm similar to nominal MPC that is based on a quadratic program (QP) rather than semidefinite programming (SDP) as used in linear matrix inequality (LMI) based methods (Kothare et al., 1996).
The milling circuit transfer function model (figure 2) without ∆LOAD/∆M F S has the following form: ∆P SE g g ∆SF W = 11 13 ∆SLEV g31 g33 ∆CF F with g11 and g13 of the form gij =
kij e−θij s τij s + 1
(1)
and g31 and g33 of the form kij −θij s e . (2) s Using the standard deviation of each model parameter, the relative uncertainty in each transfer function element is given by gij =
2. ROM MILL CIRCUIT 2.1 ROM Milling Circuit Description An industrial run-of-mine circuit with single stage classification is discussed.
35% 31% − −
19% 18% − − − 27% θij : − −
kij :
The circuit is fed gold-bearing ore at about 100 tons/hour and grinds it to give a product with particle size of 70% to 75% smaller than 75 µm. The ROM mill is operated in closed circuit with a hydro-cyclone that separates the product from the out-of-specification material which is recycled to the mill. The gold is then extracted through a leaching process downstream.
τij :
(3) (4) (5)
where the dashes indicate for that particular parameter, the uncertainty is insignificant or the parameter itself does not exist.
A typical mill has dimensions of 5m in diameter and a length of 9m. The mill is supported by pressurized-oil circumferential bearings. The mill features lifter bars and solid white-iron liners and it is operated at 90% of critical speed (Stanley,
2.2 Objectives in Mill Control The control of the milling circuit has multiple objectives, firstly to stabilize the system and sec2
∆P SE ∆SLEV
0.14 −40s 175s+1 e
−0.0575 −40s 167s+1 e
= −0.00299 s
0.00253 s
∆SF W ∆CF F
Figure 2. Nominal plant transfer function ondly to optimize the economics of the process (Hulbert, 1989). The economic objective is divided into sub-objectives that each contribute to the overall economic objective of the milling process. A set of possible sub-objectives for the milling circuit are to (Craig and MacLeod, 1995):
3. ROBUST MODEL PREDICTIVE CONTROLLER The controller proposed by Pannocchia and Kerrigan (2003, 2005) is a dual-mode controller. It consists of an unconstrained dynamic state-feedback controller. Free perturbations are added to the control action of the dynamic controller to enlarge the feasible region and impose the constraints. For brevity, only the main results are shown and the reader is referred to the references for more detail.
(1) improve product quality (a) by increasing grind fineness, (b) and decreasing the fluctuations in product size, (2) maximize throughput, (3) minimize the amount of steel that is consumed for each ton of fines produced, (4) and to minimize the power consumed for each ton of fines produced, etc.
The discrete-time linear time-invariant plant is given by x+ = Ax + Bu + Ed
The objectives above are interrelated and require trade-offs to be made. There is a trade-off between particle size of the product and the throughput of solids (objectives 1a and 2). More gold can be extracted at a finer product size (objective 1a), but the variation in particle size also influences the recovery (objective 1b).
z = Cz x
(6) (7)
in which x ∈ Rn is the plant state, x+ is the next plant state, u ∈ Rm is the control input (manipulated variable), d ∈ Rr is a persistent, unmeasured disturbance and z ∈ Rp is the controlled variable, the variable to be controlled to some time-varying setpoint s ∈ S ⊂ Rp . The constraints on the input and state is given by affine inequality constraints
It is assumed that the throughput of the mill is maximized when it draws maximum power from the mill motor. The ∆LOAD/∆M F S circuit is therefore often under power peak seeking control Craig et al. (1992). This is contrary to objective 4 which is to minimize electrical power, but given the value of the milling product versus the cost of electricity, objective 4 is not considered important.
x ∈ X ⊂ Rn , u ∈ U ⊂ R m ,
(8)
where X is a polyhedron (i.e. a closed and convex set that can be described by a finite number of affine inequality constraints) and U is a polytope (i.e. a bounded polyhedron) and X × U contains the origin.
Objectives 1 and 3 are interrelated. Steel is added to the mill to stabilize the conditions inside the mill and also increase throughput. A controller that is capable of stabilizing the particle size, will reduce the need for steel and thus objective 3 will be addressed when objective 1b is met.
3.1 Dynamic Controller A dynamic nonlinear time-invariant state feedback controller to be designed for the system given in (6)-(7) is assumed to have the following structure:
A possible control strategy is to maximize throughput at a certain particle size setpoint. This strategy considers both objectives 1 and 2. The particle size setpoint may be determined by throughput targets or if throughput is not a consideration, the particle sized can be optimized. A trade-off exists between throughput and grind, and grind and residue (product that is not recovered) (Craig et al., 1992). The aim of control would be to increase throughput, while keeping grind constant.
σ + = α(x, σ, s),
(9)
u = γ(x, σ, s),
(10)
where σ ∈ Rl is the controller state, σ + is the next controller state, α : Rn ×Rl ×Rp → Rl is controller state dynamics map and γ : Rn × Rl × Rp → Rm is the controller output map. The plant dynamics (6), together with the controller (9)-(10), forms a closed-loop system where the dynamics are given by 3
ζ
+
=
Ax + Bγ(x, σ, s) E + d. α(x, σ.s) 0
O∞ := {ζ∈Θ|ζ(k+1)=AK ζ(k)+Ed(k)+F s(k)∈Θ
(11)
for all s(k)∈S, all d(k)∈D and all k∈N}.
where ζ is the closed-loop state and ζ + is the next closed-loop state. The controller dynamics (9) can be defined by using the following auxiliary system: x ˆ = Ax + Bu + (dˆ + x − x ˆ), + dˆ = dˆ + x − x ˆ. +
(20)
3.3 Receding Horizon Controller The receding horizon controller uses the approach of ”pre-stabilizing” the plant by letting the linear control be modified with a perturbation term as follows: u = Kζ + Ls + v, (21)
(12) (13)
Combining the plant dynamics (6), with the auxiliary system (12)-(13), leads to the following augmented system:
where v ∈ Rm is the perturbation term. This leads to the augmented closed-loop system:
ζ + = Aζ + Bu + Ed,
The objective function for calculating the perturbation term is given by:
z = Cζ,
ζ + = AK ζ + Bv + Ed + Fs.
(14) (15)
JN (v) :=
in which x A 0 0 B ˆ , A := I + A −I I , B := B , ζ := x I −I I 0 dˆ E E := 0 , C := Cz 0 0 . (16) 0
N −1 X
vkT W vk ,
(22)
(23)
k=0
where the (positive definite) matrix W ∈ Rm×m is given by W := R + B T P B, R is the weighting matrix on the control action, P = P T > 0 is the solution to the discrete algebraic Riccati equation P = Q + AT P A −
The controller state σ ∈ Rl , with l := 2n, to be the states of the auxiliary system (12)-(13), i.e. x ˆ σ := ˆ . (17) d
AT P B(R + B T P B)−1 B T P A.
(24)
The objective function (23) must be solved subject to:
If the augmented system (14)-(16) is put under linear control u = Kζ + Ls as
VN (ζ,s) = {v∈RmN |F v≤b+Gd d+Gs s+H ζ ζ+H s s
ζ + = AK ζ + Ed + Fs,
(18)
AK := A + BK,
(19)
where the matrices F ∈ Rq×mN , Gd ∈ Rq×rN , Gs ∈ Rq×p(N −1) , H ζ ∈ Rq×3n , H s ∈ Rq×p and the vector b ∈ Rq depends on the augmented system dynamics (22) and the reader is referred to Pannocchia and Kerrigan (2003) for details on how to calculated them.
for all s∈SN −1 and all d∈DN },
where and the reader is referred to Pannocchia and Kerrigan (2003) for the details in calculating K, L and F.
(25)
4. CONTROLLER SYNTHESIS 3.2 The Maximal Constraint-admissible Robustly Positively Invariant Set
The robust model predictive controller consists of an offline part that is done only when the plant model changes. The online implementation is in the form of a quadratic program (QP) with linear constraints.
The constraint-admissible set Θ is defined as all the augmented states for which the plant state and input constraints are satisfied, for any choice of setpoint s ∈ S, if the control is given by u = Kζ + Ls (Pannocchia and Kerrigan, 2003).
Offline The first step is to discretize the system model in figure 2 with a sampling time of T s = 10s that is typically used for this application, and to convert it to a state-space representation. Next, the dynamic unconstrained state-feedback controller is designed using the weighting matrices defined in table 2. The weighting on the control
The maximal constraint-admissible robustly positively invariant set O∞ for the closed-loop system (18) is defined as all initial states in Θ for which the evolution of the system remain within Θ for all allowable infinite setpoint and disturbance sequences, i.e. 4
Table 1. Constraints for ROM milling circuit
PSE LOAD SLEV
Nominal Value 80% 40% 50%
SFW
300
m3 hour
-100
200
tons hour
-50
850
tons hour
-350
Constraint
MSF CFF
Min Value
Max Value
-10% -10% -30%
+10% +10% +30%
m3 hour tons hour tons hour
+100
m3 hour
+100
tons hour
+100
tons hour
258s) and the time delay of g13 is much longer (θ = 72.4s) in the ”real” plant model compared to the nominal model. The setpoint from nominal at time t = 0s is [5, -10], at time t = 2000s is [5, 10], at time t = 4000s is [-5, 10] and at time t = 6000s is [-5, -10]. These setpoint changes are chosen to stress the controller for the purpose of this simulation study. In practice, the plant would be set to a certain setpoint determined by the production strategy and the primary function of the controller would be to reject disturbances and reduce particle size variations.
Table 2. Weighting matrices on state error and control move size Matrix name
Matrix value I5
Q
R
The robust MPC with a static state-feedback gain controller (figure 3) shows offset in tracking particle size (PSE), but not the sump level (SLEV), because the plant is an integrator with respect to sump level. Integrator action would be required to eliminate the offset. This is addressed by the robust MPC with dynamic state-feedback controller, which tracks the setpoint without offset.
125 0 0 75
The simulations (figure 3) show only the deviation from the operating point (table 1, nominal values).
action is much higher than on the states to limit the severity of the control action. This is necessary to construct a feasible maximal constraintadmissible robustly positively invariant set O∞ . The flow rate of slurry to the cyclone (CFF) is weighted less than the flow rate of water to the sump (SFW), because CFF was employed less than SFW relative to its constraints by the controller.
6. CONCLUSIONS This study shows that robust model predictive control can be applied to a run-of-mine milling circuit. The controller controls particle size and sump water level, by manipulating the inflow of water to the sump and flow rate of slurry that is pumped from the sump to the cyclone.
The state constraint polyhedron X and input constraint polytope U is constructed from the system constraints as shown in table 1. They are employed in constructing the maximal constraintadmissible robustly positively invariant set O∞ . The set O∞ together with the augmented closed loop dynamics (22) are used to construct the set of admissible perturbation VN (ζ, s).
The throughput of the mill is a function of the feed-rate of solids to the mill (MFS). This acts as a disturbance to the controlled system, while the controller aims to maintain the particle size at its setpoint and minimize particle size variations.
On-line
The robust MPC with dynamic state-feedback control tracks the setpoint well compared to the robust MPC with static state-feedback gain control. This is significant, because most robust MPC implementations suffer from tracking errors (Pannocchia, 2004).
The on-line implementation of the robust controller involves solving the QP problem with objective function (23) subject to the constraints (25). The control law at each time-step is given by u = Kζ + Ls + v0∗ , where is v0∗ is the first term of the optimal solution v∗ of the QP problem.
An improvement can be made by reconstructing the state-space system to consist of previous inputs and outputs as described by Maciejowski (2002), which has the property that the states are always known, if it is assumed that the inputs are known and the outputs are measured. This representation has the added advantage that the states correspond to the actual system. This enables the controller to do output feedback, rather than state-feedback.
5. SIMULATION A simulation study is shown where the robust model predictive controller of section 3 is applied to the ROM milling model presented in section 2. The controller is compared to a robust MPC technique that only uses a static state-feedback gain controller (Pluymers et al., 2005a,b).
In practice, the load of the mill is controlled by a power optimiser Craig et al. (1992). It is believed that the mill is most efficient when maximum power is drawn from the mill motor. The load of
In this simulation scenario the actual plant differs from the nominal model. The time constants of g11 and g13 are much larger (τ11 = 275s and τ13 = 5
Plant Outputs
0 −10 0
2000 4000 6000 Time (seconds)
8000
20 0 −20 0
2000 4000 6000 Time (seconds)
8000
CFF (tons / hour) SFW (m3 / hour)
PSE (% < 75 µ m) SLEV (% Full)
10
(a) Plant Outputs
100
Plant Inputs
0 −100 0
2000 4000 6000 Time (seconds)
8000
2000 4000 6000 Time (seconds)
8000
100 0 −100 0
(b) Plant Inputs
Figure 3. Dual-mode Robust MPC with Dynamic state-feedback controller (Solid), Dual-mode Robust MPC with Static state-feedback gain controller (Dashed) and Setpoint (Dotted). the mill acts as a disturbance on the rest of the system variables. The controller can be improved by adding the load of the mill as a measured disturbance into the design.
Hulbert, D. G., I. K. Craig and M. L. Coetzee (1990). Multivariable control of a run-of-mine milling circuit. Journal of the South African Institute of Mining and Metallurgy 90(7), 173– 181. Kothare, M. V., V. Balakrishnan and M. Morari (1996). Robust constrained model predictive control using linear matrix inequalities. Automatica 32, 1361–1379. Maciejowski, J. M. (2002). Predictive Control with constraints. Prentice Hall. Great Britain. Pannocchia, G. (2004). Robust model predictive control with guaranteed setpoint tracking. Journal of Process Control 14, 927–937. Pannocchia, G. and E. C. Kerrigan (2003). Offset-free receding horizon control of constrained linear systems subject to time-varying setpoints and persistent unmeasured disturbances. Technical report. Department of Engineering, University of Cambridge. Cambridge, UK. CUED/F-INFENG/TR.468. Pannocchia, G. and E. C. Kerrigan (2005). Offsetfree receding horizon control of constrained linear systems. AIChE Journal 51(12), 3134–3146. Pluymers, B., J. A. Rossiter, J. A. K. Suykens and B. De Moor (2005a). The efficient computation of polyhedral invariant sets for linear systems with polytopic uncertainty. In: Proceedings of the American Control Conference 2005. Portland, USA. pp. 804–809. Pluymers, B., J. A. Rossiter, J. A. K. Suykens and B. De Moor (2005b). A simple algorithm for robust MPC. In: Procedings of the 16th IFAC World Congress. Prague, Czech Republic. Stanley, G. G. (1987). The Extractive Metallurgy of Gold in South Africa. Technical Report Vol 1. South African Institute of Mining and Metallurgy. Johannesburg.
A further improvement can be made by calculating the explicit solution (Bemporad et al., 2002) for the control law. This will significantly reduce the online computational burden, especially for large complex problems.
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