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Process of breakage
Most comminution processes apply compression too reparticles. Elastic bodies distort when compressed by flattening in the direction of compression and bulging at right angles to the compression force. This bulging of the particle induces tensile stresses inside the particle. The tensile stresses are concentrated at the edges of flaws in the particles. The greater the flaw area is, the greater the concentration of force. If the tensile stress at the flaw edges reaches a critical value, the intermolecular bonds break. As the bonds break, the area of the flaw increases and the concentration of force at the edge of the flaw is increased, which causes more molecular bonds to break. The flaw is almost instantaneously converted to a crack. As the crack propagates, other flaws are activated and start to crack. This results in the particle being covered in a network of cracks that divide it into roughly equal fragments (Kelly and Spottiswood, 1990, Stanley, 1987).
The crack directs compressive strain energy in equal amounts to the two parts on either side of it. If the energy is sufficiently large, it can cause the pieces to break further. The two fragments are unlikely to be equally large in terms of mass, and the smaller fragment is therefore more likely to break owing to the greater energy per mass concentration of that fragment. The breakage processis concentrated on smaller and smaller particles. Everytime a fragment splits, the energy is divided between the fragments, with the smaller fragment being more likely to break, until the energy levels fall below the threshold to support further breakage (Kelly and Spottiswood, 1990, Stanley, 1987). The comminution process needs to apply enough energy to reach the critical level to cause the initial crack network to form. Once breakage starts to occur, most of the energy is then converted to heat in the form of vibrations within the particle fragments (Kelly and Spottiswood, 1990, Stanley, 1987).
STABILITYOFMPC
The inability of both GPC and DMC to guarantee stability caused researchers to focus more on modifying PN(x) to ensure stability, owing to increased criticism (Bitmead et al., 1990) of the makeshift approach of using tuning to attain stability. With terminal equality constraints, the system is forced to the origin by the controller that takes the formE(x)=0,as there is no terminal costand the terminal setisXf ={0}. Keerthi and Gilbert, as discussed in Mayne et al. (2000), proposed this stabilising strategy for constrained,non linear,discrete time systems and showed a stability analysis of this version(terminal equality constraints) of discrete-time receding horizon control. MPC, with a terminal equality constraint, can be used to stabilise a system that cannot be stabilised by continuous feedback controllers, according to Meadows et al. (as discussed in Mayne et al. (2000)). Usingaterminalcostfunctionisanalternativeapproachtoensurestability. Heretheterminal cost is E(·), but there is no terminal constraint and the terminal set is thus Xf = Rnx. For unconstrained linear systems the terminal cost of E(x)= 1 2xTPfx is proposed by Bitmead et al. (1990).
Terminal constraint sets differ from terminal equality constraints, in that subsets of Rnx that include a neighbourhood of the origin are used to stabilise the control, not just the origin. The terminal constraint set, like the terminal equality constraint, does not employ a terminal cost, thus E(x)=0. The MPC controller should steer the system to Xf within a finite time, after which a local stabilising controller κf(·) is employed. This methodology is usually referred to as dual mode control and was proposed by Michalska and Mayne (1993) in the context of constrained, nonlinear, continuous systems using a variable horizon N. A terminal cost and constraint set is employed in most modern model predictive controller theory (Mayne et al., 2000). If an infinite horizon objective function can be used, on-line optimisation is not necessary and stability and robustness can be guaranteed. In practical systems, constraints and other nonlinearities make the use of infinite horizons impossible, butitispossibletoapproximateaninfinitehorizonobjectivefunctionifthesystemissuitably close to the origin. By choosing the terminal set Xf as a suitable subset of Rnx, the terminal costE(·)can be chosen to approximate an infinite horizon objective function. Aterminal cost and constraint set controller the refore need saterminal constraint set Xf in which the terminal cost E(·) and infinite horizon feedback controller Kf are employed. To synthesise these, Sznaier and Damborg (as discussed in Mayne et al. (2000)) proposed that the terminal cost E(·) and feedback controller Kf of a standard linear-quadratic (LQ) problem be used, which is an unconstrained infinite horizon problem, when the system is linear (f(x,u)=Ax+Bu) and the state and input constraint sets, X and U, are polytopes. The terminal constraint set Xf is chosen to be the maximal output admissible set (Gilbert and Tan, 1991) of the system f(x,u)=(A+BKf)x.
Most industrial or commercial MPC controllers do not use terminal costs or constraints, because they do not even provide nominal stability that these terminal costs and constraints are designed to provide and can be attributed to their DMC and IDCOM heritage (Qin and Badgwell, 2003). Most industrial MPC controllers, therefore, require brute-force simulation to evaluate the effects of model mismatch on closed – loop stability (Qinand Badgwell,2003). Time spent tuning and testin go find ustrial controller scan,however,besignificantlyreduced if the controllers implement nominal and potentially robust stability measures, even though closed-loop stability of industrial MPC itself is not perceived to be a serious problem by industry practitioners (Qin and Badgwell, 2003).
LINEARISEDMODELSFORSIMCTUNING METHOD
Linearised models are necessary to design the PI controllers using the SIMC method. Linearised models can be created by performing step tests on the nonlinear model and performing system identification (SID) on the step responses. The output-input pairings for single-loop controllers on multivariable systems are very important, because the output should be paired with the input that has the most influence on thatoutputandtheinputshouldhavetheleastinteractionwithotheroutputs. Thetraditional output-input pairings used on milling circuits are LOAD-MFS, PSE-SFW and SLEV-CFF (Chen et al., 2007b, Conradie and Aldrich, 2001, Lynch, 1979, Napier-Munn and Wills, 2006). This pairing is not without its problems when used on industrial plants, as described byChenetal.(2007b): “DecoupledPIDcontrolhadbeenfrequentlyinterruptedbychanges in mineral ore hardness, feed rate, feed particle size, etc., …” This statement was supported by preliminary simulations using the above-mentioned pairing where the sump would either overflow or underflow as soon as ore hardness and composition disturbances were introduced. Craig and MacLeod (1996) also found SLEV control to be the most problematic aspect of controlling the milling circuit.
Analternativeoutput-inputpairingwastheninvestigatedthatpairedLOAD-MFS,PSE-CFF and SLEV-SFW (Smith et al., 2001). The pairing of SLEV-SFW, rather than PSE-SFW and SLEV-CFF,wastraditionallyusedonlywhenafixedspeedsumpdischargepumpwasavailable (Lynch, 1979). The pairing LOAD-MFS, PSE-CFF and SLEV-SFW, however, shows muchbetterrobustnesstothefeeddisturbancessubjecttoactuatorlimitations,asshownlater in Section 5.3.2 and Addendum B. The single-loop PI controllers are designed with the above-mentioned output-input pairings. The interactions between loops are ignored, because they cannot be included in the PI controllerdesignusingtheSIMCtuningmethod,unlikeothermethods(Pomerleauetal.,2000). The PI controllers are SISO controllers and the three controlled variables (PSE, LOAD and SLEV) will be independently controlled by three independent PI loops. Neither a multivariablecompensator(VázquezandMorilla,2002)noracentraliseddesign(Morillaetal.,2008) will be used for the PI controllers, because most plants that use PI controllers employ only single-loop PI controllers (Wei and Craig, 2009). An attempt at decentralised PID tuning that takes interactions into account was made in Addendum C.
1 INTRODUCTION
1.1 MILL CIRCUIT DESCRIPTION
1.2 OBJECTIVES IN MILL CONTROL
1.3 AIMS AND OBJECTIVES
1.4 ORGANISATION
2 MILLING THE ORYAND MODELLING
2.1 INTRODUCTION
2.2 THEORY OF MILLING
2.3 MILLING MODELLING
2.4 CONCLUSION
3 MODELPREDICTIVECONTROL
3.1 INTRODUCTION
3.2 HISTORICAL BACKGROUND
3.3 STABILITY OF MPC
3.4 ROBUST MPC – STABILITY OF UNCERTAIN SYSTEMS
3.5 ROBUST NONLINEAR MPC FORMULATIONS
3.6 NONLINEAR MODEL PREDICTIVE CONTROL
3.7 ROBUST NONLINEAR MODEL PREDICTIVE CONTROL
3.8 STATE OBSERVERS
3.9 CONCLUSION
4 PIDCONTROL
4.1 INTRODUCTION
4.2 PI CONTROL WITH ANTI-WINDUP
4.3 LINEARISED MODELS FOR SIMC TUNING METHOD
4.4 SIMC TUNING METHOD
4.5 IMPLEMENTATION
4.6 SUMMARY
5 MILLINGCIRCUITCONTROLSIMULATIONSTUDY
5.1 INTRODUCTION
5.2 PERFORMANCE METRICS
5.3 SIMULATION RESULTS
5.4 DISCUSSION
5.5 SIMULATION SUMMARY
6 CONCLUSIONSANDFURTHERWORK
6.1 SUMMARY AND EVALUATION
6.2 FURTHER WORK
