NONLINEAR PREDICIVE CONROL OF A ROUGHER FLOAION UNI USING LOCAL MODELS A. Desbiens *, D. Hodouin ** and M. Mailloux * GRAIIM (Groupe de Recherche sur les Applications de l Informatique à l Industrie Minérale) * Department of electrical and computer engineering, ** Department of mining and metallurgy, Université Laval, Pavillon Adrien-Pouliot, Sainte-Foy (Québec), Canada GK 7P4 Abstract: Global Predictive Control (GlobPC) is a predictive control algorithm using three different controllers for tracking, disturbance rejection and feedforward, thus leading to perfect decoupling of all modes of control. Since rougher flotation units are nonlinear, the process is more adequately represented by a nonlinear model. he parameters of the nonlinear model are obtained by interpolating the parameters of three local linear models. Application of nonlinear GlobPC to a rougher flotation phenomenological simulator leads to better performances and robustness than linear control. Copyright 998 IFAC Keywords: Process control, Predictive control, Nonlinear control, Nonlinear models, Process simulators.. INRODUCION Flotation processes are solid-solid separation systems commonly used in the mineral industry. Because they are continuously disturbed by the variation of the feed characteristics and by the changing interactions between the solid surfaces and the fluid chemistry, the flotation units require to be carefully controlled to maintain their separation efficiency at a suitable performance level (McKee, 99). Optimizing a flotation process consists of selecting the setpoints according to an economical criteria. However, before implementing the optimizing stage, an efficient stabilizing strategy must be designed. Since flotation units present nonlinear behaviour, adaptive control (Desbiens et al., 994) or nonlinear control must be used. his paper is restricted to nonlinear stabilization of the concentrate grade by manipulating the collector feedrate. Stabilization is performed by a nonlinear predictive control algorithm. 2. GLOBAL PREDICIVE CONROL he Global Predictive Control (GlobPC) algorithm is based on an Internal Model Control (IMC) structure (Garcia and Morari, 982) where the tracking, the regulation and the feedforward dynamics are obtained by minimizing three independent quadratic cost functions (Figures and 2) as suggested by Hodouin et al. (995). he IMC structure restricts GlobPC to stable processes. Figure shows that the control action u(t) is the sum of u (t), the tracking control action, u F (t), the feedforward control action and u R (t), the regulation control action. he process output, the unmeasured
DP//G F (q - ) r F (t) G F (q - ) w F (t) R F (t+h F /t) - C R (t+h /t) DP//G (q - ) r (t) G (q - ) C F u F (t) u (t) + + + u R (t) C R u(t) ξ(t) v(t) Process G P (q - ) y(t) - ξ(t) w R (t) - G P2 (q - ) + - y v (t) w (t) R R (t+h R /t) SP//G R (q - ) G P3 (q - ) G R (q - ) r R (t) Fig.. GlobPC structure R i (t+h i /t) + ε i (t+h i /t) - C i DP//G i (q - ) Fig. 2. Predictive controller i =, F or R Min J i y i (t) G i (q - ) = G P (q - ) F i (q - ) u i (t) disturbance and the measured disturbance are respectively y(t), ξ(t) and v(t). When the process output and the disturbances are not measured, the only possible action is the tracking control action u (t). If the process is not disturbed, this pure tracking controller is sufficient to adequately control the process, provided that there is no modelling error. he tracking control action is calculated by the tracking optimal regulator, C. he corresponding setpoint (the tracking setpoint) is given by the operator and is denoted w (t). he tracking reference trajectory r (t) is calculated using the tracking reference model G (q - ). With the tracking model, deterministic predictions over a future horizon of the tracking reference trajectory can be calculated: R( t [ r( + H / t) = t + / t) r( t + 2 / t) K () r( t + H / t) ] hese predictions are the input to the optimal controller (figure 2). he signal y (t) is the output of G P the process model ( q ) (the model giving the relation between u(t) and y(t)). his output may be filtered by F (q - ). he prediction of the model output are: Y ( t [ y + H / t) = ( t + / t) y ( t + 2 / t) K (2) y ( t + H / t) ] he resulting control action is obtained by minimizing the following quadratic cost function: J = H 2 [ F + + ] ( q ) y ( t j / t) r ( t j / t) j= + K j= λ 2 [ Q ( q ) u ( t + j ) ] (3) subject to u (t + j) = 0 for j = K. he transfer function Q (q - ) allows a frequency weighting of the command. he resulting control action is adequate for the model G P ( q ) and therefore for the process if there are neither disturbance nor modelling error. Suppose now that there are no setpoint changes and that the process output is not measured. he only possible control action is the feedforward rejection of
the measured disturbance. If there are no modelling errors of the effect of u(t) and v(t) on the process output, this feedforward controller is adequate to control the process. he effect of the measured disturbance on the process output is ˆ ( t). It can be G P 2 estimated using the model ( q ). o fight this measured disturbance, the feedforward control action must be such that its effect on the process output is ˆ ( t), which therefore corresponds to the y v feedforward setpoint w F (t). As in the tracking situation, a reference trajectory and its predictions are calculated and an optimal regulator is used. hese elements are similar to those described in the preceding paragraph. If there are no setpoint changes and if the disturbance v(t) is not measured, the only possible control action is given by the regulation algorithm which compensates for the disturbances and the modelling errors. he differences between the model and the process outputs (modelling errors) and the unmeasured disturbances are evaluated using the IMC structure. he resulting signal is ξ ˆ( t ). he control action must be calculated so that its effect on the model output will cancel out ξ ˆ( t ). herefore, its setpoint w R (t) must be ξ ˆ( t ). As before, a reference model can be used to generate the regulation reference trajectory r R (t). A stochastic predictor calculates the predictions of the reference trajectory and an optimal controller, similar to the tracking and feedforward ones, minimizes a quadratic cost function in order to obtain the regulation control action. he stochastic prediction G P 3 is based on ( q ), which models the effect of 3 ξ (t) on y(t). If unknown, G P ( q ) is used as a tuning knob for the regulation dynamics. It can be shown (Plamondon et al., 998) that control algorithms such as Generalized Predictive Control (GPC) (Clarke et al., 987) and Partial State Reference Model (PSRM) Control (M Saad et al., 986; Irving et al., 986) are particular cases of GlobPC for stable processes. o reproduce GPC, all three controllers must have the same settings since GPC strategy makes use of a single objective function. he PSRM has long-range predictive regulation capabilities but is limited to one-step ahead setpoint tracking. y v G P dynamics. Hence, ( q ) must be a linear model. If the process is nonlinear, it can be modelled by a set of local linear models. For instance, in the example of the next section, three local models were identified at three different operating regimes and interpolation functions are used to extrapolate the model parameters between each regime (Foss et al., 995; Johansen and Foss, 993). he operating regime is defined by the value of the manipulated variable. he local models identified around the operating points u(t) = u, u(t) = u 2 and u(t) = u 3 are respectively G MU (s), G MU2 (s) and G MU3 (s), where P3 i s G s) = P ( + P s) e ( + P s)( + P s). he ( ) ( ) ( MUi i 2i 4i 5i Laplace domain was selected because the physical meaning of the parameters is more obvious than with discrete transfer functions. he parameters of the resulting process model are the normalized weighted sum of the parameters of the local models: Gˆ ( s, t) = P wherep ( t) = k P3 ( t) s P( t) ( + P ( t) s) e 2 ( + P ( t) s)( + P ( t) s) 3 4 i= W ( u( t) ) P ki i 5 (4) he normalized weighting factors W i are shown in figure 3. he discrete model G P ( q ) used in the control algorithm is the z-transform of Gˆ P ( s, ). t he model G P ( q ) must be evaluated at each sampling period since the weighting factors depend on the time through the variable u(t). Discrete statespace representation of the model is used to facilitate the predictions calculations. Since the model structure is fixed, the relation between the Laplace domain model and the discrete state-space representation can be calculated once for all. he nonlinear model described above is used to write the prediction equations needed for optimal control. A prediction equation consists in a series of equations depending on the (unknown) future values of the input (smaller than u, between u and u 2, between u 2 and u 3 or larger than u 3 ). herefore, the predictions over the horizon H remain functions of u(t + i), i = to H, as in the case of a linear model, since the model parameters over the prediction horizon are also function of u(t + i), i = to H. 3. NONLINEAR MODEL With GlobPC, linearity is assumed in order to be able to separate tracking, feedforward and regulation 4. FLOAION PROCESS he nonlinear GlobPC has been tested on a phenomenological simulator of a rougher flotation
W i (u(t)) 0.8 0.6 0.4 0.2 0 i = i = 2 i = 3 u u 2 u 3 u(t) Fig. 3. Weighting factors unit composed of four 8m 3 cells in series. he concentrates of the cells are combined, thus forming the rougher bank concentrate, while the tail of the last cell is the circuit tail. he model is based on the usual assumptions of first order flotation kinetics in perfectly mixed tanks, with additional non selective entrainment of particles in the concentrate. he particle population is divided into four classes having different mineral contents and flotation rates: the mineral - rich particles (fast kinetics) the mineral - poor particles (slow kinetics) the gangue particles, containing a small amount of locked mineral, which are divided into two sub-classes: the slow-flotation particles and the no-flotation particles. All the particles are entrained by the water going to the concentrate. he three rate constants, as well as the proportion of slowly floating gangue and the entrainment coefficients, are related to the two manipulated variables, the collector and air flowrates, by sigmoïd functions coupled to transfer functions simulating the particle conditioning stage. Disturbances are generated by changing the ore and water feedrates and the proportions and mineral contents of the various classes of particles, thus simulating feed grade and liberation degree variations. he simulator has been used to characterize the steady-state and dynamic behaviours of the process. First, the steady-state response of the process was characterized within a given rectangular window ABCD in the u-plane, while maintaining the feed characteristics at their nominal values (see Figure 4). he domain in the u-plane was selected sufficiently small in order to not excite severe nonlinearities such as gain sign inversions. he y-plane is defined by the usual grade-recovery coordinates (Figure 5). he steady-state process nonlinearity is indicated by the distortion of the ABCD parallelogram in the y- plane, and is particularly strong along the lines BC and DA corresponding to the collector flowrate variation. he steady-state process response in the same ABCD u-plane window was also plotted for different feed characteristics. Figure 5 shows the responses for the four feed characteristics described in able, in comparison to the nominal feed characteristics response. It is clear form Figure 5 that a common reachable setpoint does not exist for the five feed characteristics, when the u variations are limited to the specified ABCD window of Figure 4. Areas F and F 2, corresponding to high feedrate and low feed grade, exhibit lower grade-recovery performances than areas F 3 and F 4, corresponding to low feedrate Collector feedrate (%) 50 45 40 35 30 25 20 Fig. 4. u-plane Mineral recovery 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 Fig. 5. y-plane D A u 3 u 2 u C B 20 25 30 35 40 45 50 55 Air flowrate (%) C F F 2 F N F 3 F 4 0.35 0.5 0.2 0.25 0.3 0.35 0.4 Concentrate grade B and high feedgrade. his is quite logical, since, at the same time the ore is poorer and the mean residence time in the rougher bank lower. From areas F to F 2 and F 3 to F 4, the mineral liberation degree increases, thus improving the circuit performance as it should do. D A
Concentrate grade 0.28 0.27 0.26 Nonlinear: Linear model : Linear model 2: 0 20 40 60 80 00 20 ime (min) Fig. 6. Linear and nonlinear GlobPC control - Nominal feed characteristics Concentrate grade 0.36 0.35 0.34 0.33 Nonlinear: Linear model : Linear model 2: 0 20 40 60 80 00 ime (min) 20 Fig. 7. Linear and nonlinear GlobPC control - Feed characteristics F 4 able Rougher feed characteristics Feed F F 2 F 3 F 4 F N (Nominal) Feedrate (kg/min) Feedgrade (%) Liberation degree 3600.8 low 3600 0.5 high 2400 6.3 low 2400 5.7 high 3000 3. medium Finally the transfer matrices were identified at the four corners ABCD of the five areas of Figure 5, by simulating step responses of the process to very small variations of u, and using the transfer function structure of Equation 4. he parameters of the transfer functions do not much vary as a function of the feed characteristics, however they exhibit very significant variations within the u-plane window ABCD. As already observed for the process gains, the nonlinearities are stronger in the direction of the collector flowrate. For instance, the three local models, identified at locations u, u 2 and u 3 of Figure 4 for the nominal feed characteristics, have the following parameter values : P = -0.0008, P 2 = -5.03, P 3 = 0, P 4 = P 5 = 0.94 P 2 = -0.00697, P 22 = -0.9537, P 32 = 0, P 42 = P 52 = 0.837 P 3 = -0.0046, P 23 = 0, P 33 = 0, P 43 = P 53 = 0.8965. he zero of the system presents a very important variations. 5. RESULS Since the air flowrate has only a weak effect on the local models, the illustration of the GlobPC is limited to the mono-input-mono-output process: collector flowrate/concentrate grade. Furthermore the feedforward controller is not illustrated since changes in the feed characteristics require at the same time setpoint variations, a feature which would blur the controller behaviour that is to be shown. he GlobPC controller is tested using a series of setpoint steps, selected such as the collector flowrate variations goes slightly outside the range defined by Figure 4. he air flowrate is kept constant to 38%. he tests were performed for two different feed characteristics: F N (nominal) and F 4. Figures 6 and 7 compare the performances of linear and nonlinear
GlobPC. For linear GlobPC all three local models, 2, 3 were tested. Results obtained with linear control using model 3 are not shown since unstability occurs at about t = 50 for both F N and F 4. Linear control using model is not unstable but its time response is sometimes significantly longer than the one of nonlinear control. When using model 2, linear control becomes unstable before the end of the test. In both linear and nonlinear cases, the controllers settings are: sampling period = 0.5 minute, H = 20, H R =5, λ = λ R = 0, Q (q - ) = Q R (q - ) = - q -, K = K R =, F (q - ) = F R (q - ) =, G (q - ) = 0.394/( - 0.606 q - ), G R (q - ) =, and G P 3 ( q ) = ( - 0.4q - ) 3 /(A(q - ) ( - q - )) where A(q - ) is the denominator of G P ( q ) at the given sampling period. Since nonlinear GlobPC makes use of more information about the process, it exhibits better performances and robustness than linear GlobPC. 6. CONCLUSION he key issue in GlobPC is to use three different controllers for tracking, disturbance rejection and feedforward, thus leading to perfect decoupling of all modes of control. he GlobPC structure, based on the internal model loop, cannot be implemented to control unstable processes. However, this structure exhibits many advantages, such as a clean separation of ) the tracking, regulation and feedforward dynamics 2) the stochastic and deterministic predictions 3) the stochastic and deterministic parts of the controllers, and 4) the reference models for tracking, regulation and feedforward. All the well separated functions of the various parts of GlobPC make easier student training, strategy design, computer object-programming and controller tuning. he results obtained with the flotation simulator have shown that regime-based modelling allows good operation over a wide range of operating conditions. he authors are grateful to the following funding organizations : CRM (Centre de Recherche Minérale), FCAR (Fonds pour la Formation de Chercheurs et l Aide à la Recherche), CANME (Canadian Centre for Mineral and Energy echnology), NSERC (Natural Science and Engineering Research Council of Canada) and 4 Canadian mining companies. 8. REFERENCES Clarke, D.W., C. Mohtadi and P.S. uffs (987). Generalized predictive control - Part I. he basic algorithm, Automatica, 23 (2), 37-48. Desbiens, A., D. Hodouin, K. Najim and F. Flament (994). Long-Range Predictive Control of a Rougher Flotation Unit, Minerals Eng., 7 (), 2-37. Foss, B.A.,.A. Johansen and A.V. Sørensen (995). Nonlinear Predictive Control Using Local Models - Applied to a Batch Fermentation Process, Control Eng. Practice, 3 (3), 389-396. Garcia, C.E. and M. Morari (982). Internal model control.. A unifying review and some new results, Ind. Eng. Chem. Process Des. Dev., 2 (2), 308-323. Hodouin, D., E. Gagnon and A. Pomerleau (995). Autostop : a unified software for simulation of automatic stochastic optimal predictive control loops, Proc. of 3rd Canadian conf. on computer applications in the mineral industry, Montreal, 796-805. Irving, E., C.M. Falinower and C. Fonte (986). Adaptive generalized predictive control with multiple reference model, Proc. IFAC workshop on adapt. systems in control and signal process., Lund, Sweden. Johansen,.A. and B.A. Foss (993). Constructing NARMAX Models using ARMAX Models, Int. J. Control, 58 (5), 25-53. McKee, D.J., 99, Automatic Flotation Control - A Review of 20 Years of Effort, Minerals Eng., 4, 653-666. M Saad, M., M. Duque and I.D. Landau (986). Practical implications of recent results in robustness of adaptive control schemes, Proc. Of the 25th CDC, Athens, Greece. Plamondon, É., A. Desbiens and D. Hodouin (998). Global Predictive Control : a Structure Including GPC, PSRM and Other Predictive Controllers, in preparation. 7. ACKNOWLEDGMENS