MPC Fundamentals Figure 1-2(b)shows MPCs previous"moves, uk-41, .,lk-1, as filled circles. As is usually the case. a"zero-order hold receives each move from mPc and sends that value to the plant continuously until the next sampling instant- note the resulting step-wise variations MPC must now calculate the current move, uk. It does so in two phases 1 Estimation. In order to make an intelligent move, MPC needs to know the current state of the system. This includes the true value of the controlled variable, Jk, and any this MPC uses all past and current measurements and the models uy, dompli internal plant variables that influence the future trend, yk+1,.,jk+P. To accomplish →j,and=→y. For details, see Chapter4,“ Disturbance detection and 2 Optimization. Values of setpoints, measured disturbances, and constraints are specified over a finite "horizon"of future sampling instants, k+1,k+2.,k+P, where P(a finite integer 2 1)is the"prediction horizon"-see Figure 1-2 (a). MPC then computes the M moves ulk,a4+1,…a+Ml, where a(≥1,≤P) is the“ control horizon”- see Figure 1-2 b). In our hypothetical example, P=9 and M=4 Suppose that the optimal sequence of moves is the series of four open circles in Figure 1-2 predicts that this will result in the open circles in Figure 1-2(a) the setpoint clo optimal in the sense that no constraints are violated and the outputs track the setpoint"closely. See section Optimization for a detailed definition of optimality MPC sends only the move uk to the plant. The plant operates with this input until MPCs next sampling instant, Af time units later. MPC then obtains a new set of measurements and revises completely the plan it had formulated at the previous sampling instant, thus compensating for model error and unknown disturbances Prediction and control horizons One might wonder why MPC bothers to optimize over P sampling periods into the future and calculate M moves when it discards all but the first move. Indeed under certain conditions MPC gives identical results for P=M= l as it does for P=M=oO. More often however, the horizons have an important impact. Some examples are:MPC Fundamentals 1-9 Figure 1-2 (b) shows MPC’s previous “moves,” uk-41, ..., uk-1, as filled circles. As is usually the case, a “zero-order hold” receives each move from MPC and sends that value to the plant continuously until the next sampling instant – note the resulting step-wise variations in Figure 1-2 (b). MPC must now calculate the current move, uk . It does so in two phases: 1 Estimation. In order to make an intelligent move, MPC needs to know the current state of the system. This includes the true value of the controlled variable, , and any internal plant variables that influence the future trend, , ..., . To accomplish this MPC uses all past and current measurements and the models , , , and . For details, see Chapter 4, “Disturbance Detection and Estimation”. 2 Optimization. Values of setpoints, measured disturbances, and constraints are specified over a finite “horizon” of future sampling instants, k+1, k+2, ..., k+P, where P (a finite integer ³ 1) is the “prediction horizon” – see Figure 1-2 (a). MPC then computes the M moves uk , uk+1, ... uk+M-1, where M ( ³ 1, £ P) is the “control horizon” – see Figure 1-2 (b). In our hypothetical example, P = 9 and M = 4. Suppose that the optimal sequence of moves is the series of four open circles in Figure 1-2 (b). MPC’s model predicts that this will result in the output sequence shown as the nine open circles in Figure 1-2 (a). These moves are optimal in the sense that no constraints are violated and the outputs track the setpoint “closely.” See section Optimization for a detailed definition of optimality. MPC sends only the move uk to the plant. The plant operates with this input until MPC’s next sampling instant, Dt time units later. MPC then obtains a new set of measurements and revises completely the plan it had formulated at the previous sampling instant, thus compensating for model error and unknown disturbances. Prediction and control horizons One might wonder why MPC bothers to optimize over P sampling periods into the future and calculate M moves when it discards all but the first move. Indeed, under certain conditions MPC gives identical results for P = M = 1 as it does for P = M = ¥. More often, however, the horizons have an important impact. Some examples are: y k y k + 1 y k P + u y ® d y ® w y ® z y ®