1. ● Ore grinding is one of the most power consuming but also
lowly efficient industrial activities, with only 10 % of the energy
being used for actual size reduction of ore, while the rest is
transformed into heat. With thousands of tones of ore being
processed each day, optimization of the comminution process is
highly imperative.
● In a project named Advanced Grinding Control, SIEMENS AG
took the initiative of improving the efficiency of the SAG mill by
using Model Predictive Control.
● DTU IMM approached SIEMENS with the proposal of using a
robust and flexible MPC framework based on ARX modelling of
the plant and QP solving of the cost function.
Motivation
● Semi-Autogenous Grinding (SAG) mills are the largest units in the comminution chain, with a diameter in between 8 and 12 meters and
a power range of 5 to 30 MW.
● Although highly empirical, the grinding process can be roughly modelled based on mass balance and concentration equations.
● Coarse ore (300 mm) and water are fed to the mill, which is rotated with a percent (ψ ≈ 75 %) of its critical speed, that is the speed at
which the centripetal force acting on the material equals its weight.
● The physical output of the mill is fine ore (20 mm) and water, while soft sensors give the value of the fill level (Φ) and drawn power (P).
● The breakage function S, represents the rate at which coarse ore mass (xc) is reduced to fine ore mass (xf).
The SAG mill and the grinding process
˙
(
xc
x f
xw
)=
(
−S(xc , ψ)
S(xc ,ψ)−x f ∗αout
−xw αout
)+
(
uc
0
uw
)
(
y f
ϕ
P )=
(
x f∗αout
ϕ(xc ,mill size)
P(ϕ ,ψ) )
uc
uw
(
xc
x f
xw
)
ψ
y f
ϕ
P
● A preliminary control objective is to maintain the fill level (Φ)
at a reference value, but a key problem for the controller, in
this particular application, is the step-like disturbances
introduced by the unknown hardness of the ore, which affects
the breakage rate.
● The advantage of the MPC based on an ARX model of the
process is that it can be tuned in order to reject exactly this
kind of disturbances, as well as compensate for the model
mismatch introduced by the linearisation.
● The controller, in state space in innovation form, is computed
directly from the observer canonical realization of the MIMO
ARX model of the plant.
ARX based MPC
● By employing a receding horizon strategy, the
MPC can take into account the output reference
change long before it actually takes place and
starts the controlling action for an optimal tracking,
thus being most suited to plants with long delays.
● As designed, the MPC correctly rejects a step
disturbance in the ore hardness, that appears 46
hours in the simulation.
● The versatility of the MPC framework allows for
multiple extensions, such as gain scheduling and
variable input penalty, for faster response and
better noise rejection.
ConclusionsSimulation
Offset free ARX based MPC applied to an Ore Grinding Mill
(Ir+ ∑i=1
n
Ai q
−i
)y(t)=(∑i=1
n
Bi q
−i
)u(t)+ ε(t), ε(t)∈Ν(0,Σr )
Ap=
[
−A1 I r 0 0 ⋯ 0
−A2 0 Ir 0 ⋯ 0
−A3 0 0 Ir ⋯ 0
⋮ ⋮ ⋮ ⋮ ⋱ ⋮
−An−1 0 0 0 ⋯ Ir
−An 0 0 0 ⋯ 0
], Bp=
[
B1
B2
B3
⋮
Bn−1
Bn
], K p=
[
−A1
−A2
−A3
⋮
−An−1
−An
],
C p=[Ir 0 0 0 ⋯ 0]
Valeriu Ohan1
, Florian Steinke2
, Michael Metzger2
, Thomas Runkler2
, John Bagterp Jørgensen1
1. DTU, Department of Informatics and Mathematical
Modelling, 2800 Kgs. Lyngby, Denmark.
2. Siemens AG, Corporate Technology, Intelligent
Systems and Control, Otto-Hahn-Ring 6, 81739 München,
Germany.