1.7. Model predictive control
The development of MHE, although proposed much earlier in many different forms throughout the literature, was pushed in the nineties motivated by the success of Model Predictive Control (MPC), its counterpart for control. Model Predictive Control (MPC) has gained widespread interest in both academia (see textbooks [31, 121, 128, 156]) and industry (see [145] for a survey) over the past decades. In a wide range of indus- tries it has become the method of choice for advanced process control.
The ultimate goal in optimal control is to find a feedback law that minimizes a certain control objective over an infinite horizon, starting from the current state x0and subject to a process model (as described in Section 1.2) and constraints. Typically, but not necessarily, the objective is quadratic
J(x, u) = ∞
∑
k=0 kxkk2Qk+ kukk 2 Rk.where the weightings Qk and Rk are tuning parameters. The optimal solution can
be obtained from the solution of an infinite dimensional partial differential equation, called Hamilton-Jacobi-Bellman (HJB) equation. In general, a closed-form expres- sion for the solution of the HJB equation does not exist. One exception is linear unconstrained systems with quadratic objectives. In this case, the solution follows from a matrix equation, i.e. a Riccati equation, and the resulting feedback controller is called LQR.
Another class of solution methods is based on Pontryagin’s Maximum Principle [143] and proceed by maximizing the Hamiltonian matrix. Pontryagin’s maximum principle is closely related to the HJB equation and provides conditions that an optimal trajec- tory must satisfy. However, while the HJB equation provides sufficient conditions for optimality, the minimum principle provides only necessary conditions. The maximum principle typically leads to an intricate multi-point boundary value problem.
Alternatively, and similarly to the MHE case, the infinite-horizon control problem can be replaced by an equivalent finite-horizon problem, due to the Markov property of the state-space model.
minx,u ∑Nk=0−1kxkk2Qk+ kukk 2 Rk+
V
(xN) s.t. x0 = ¯x0, xk+1 = fk(xk, uk, pk), k = 0, . . . , N − 1, 0 ≥ gk(xk, uk) k= 0, . . . , N − 1, 0 ≥ gN(xN), (1.24)where
V
(xN) is the terminal cost or end cost and ¯x0is the fixed initial state. The mini- mization is with regards to the state and control sequences{x0, . . . , xN} and {u0, . . . , uN−1}Since a closed-form expression of the terminal cost rarely exists, it should be approxi- mated F(xN) = ˆ
V
(xN). Mayne et al [131] derived conditions for stability of the MPCapproximation. One popular strategy for approximation of the terminal cost is to as- sume that after the horizon the system can be controlled using LQR. In this case the (approximate) terminal cost is
ˆ
V
(xN) = kxNk2PN.where PNis the solution to the corresponding LQR discrete-algebraic Riccati equation.
This approach, sometimes called dual-mode MPC, guarantees asymptotic stability for linear systems in the absence of disturbances [131].
The strategy of MPC is to solve the open-loop fixed-size optimization problem (i.e. (1.24) with approximated terminal cost), apply only the first element of the optimal input sequence to the process, obtain a new state estimate and repeat the procedure.
The unique combination of several important features distinguishes MPC from other control methods. First, analogous to MHE, it is possible to incorporate constraints and impose multivariate nonlinear models in a natural way. Constraints are even more relevant for the control problem than for the estimation problem, because safety limita- tions, environmental regulations and economic objectives force companies to operate their processes at the constraints. Second, the extensive research on MPC has led to formulations with guaranteed stability [131]. Finally, the ability to control processes proactively is a key feature of MPC. When disturbances are known in advance (e.g. grade changes in chemical processes), significant performance gains can be obtained in comparison with pure feedback control by incorporating these future disturbances into the control problem. A common motivation for the importance of this feature is by the example of driving a car; in the event of an upcoming turn one already takes this information into account by slowing down and changing to the outer lane in order to follow an efficient path.
In order to fully exploit the potential of MPC it is required that the underlying model and its parameters are constantly updated to take disturbances and plant-model mis- match into account. The performance of the closed loop system is directly influenced by the quality of the estimates. The combination of MHE and MPC yields a pow- erful and versatile strategy for advanced process control. States and parameters are adapted based on incoming measurements leading to improved prediction accuracies in turn leading to improved control performance. In addition, empirical studies [117] show that the MPC problem becomes easier to solve when estimates are more accurate because the predicted behavior resembles the true plant behavior more closely.