OPTEC - K.U.Leuven Center of Excellence: Optimization in Engineering

"Fast implementation of MPC using Set Membership approximation methodologies"
Lorenzo Fagiano (Politecnico di Torino)

Model Predictive Control (MPC) (see e.g. the survey [1]) is a model based control technique where the control action is computed by solving at each sampling time an optimization problem which uses the current system state as the initial condition. In general the control move ut at time t, for time invariant systems, is a nonlinear static function of the system state x_t, i.e. u_t = f(x_t). One of the main limitations in using MPC techniques is the presence of fast plant dynamics which require small sampling periods that do not allow to perform the optimization problem online: this motivates the research efforts devoted to develop computationally tractable MPC solutions, or suitable approximations of MPC control laws, by exploiting the properties of f(x). These approaches include explicit piece-wise linear solutions ([2]), neural approximations ([3]) and piece-wise linear approximations ([4]) of f(x). An alternative approach is presented here using Set Membership (SM) methodologies for nonlinear function estimation as firstly introduced in [5]. Under this context, no assumption on the functional form of f(x) is made and only assumptions on its regularity properties, given by bounds on its value set and on its gradient, are considered. The use of such methodology enables to compute an approximation of a given predictive control law which guarantees to fulfill input constraints. Moreover, it is possible to compute an upper bound of the approximation error which holds for every state value in the considered set. In particular, a key feature of this approach is the possibility of tuning the approximation error in order to guarantee satisfaction of state constraints. This way, as the control computation is simply reduced to the evaluation of a static nonlinear function, the computational time is significantly reduced leading to a “Fast” Model Predictive Control implementation (FMPC). Two different SM methodologies are presented for the approximation of the nominal MPC controller. Both methodologies are based on the off-line computation of a certain number N of exact MPC control moves. The first method derives an “optimal” approximating function which minimizes, for a given N, the guaranteed accuracy level. However, its computational time grows with N. The second one gives lower guaranteed accuracy for a given N, but its computational time is lower and it is approximately constant with N. Thus, at the cost of increasing the number of off-line computations, any desired level of guaranteed accuracy can be obtained without increasing the on-line computation effort. Two numerical examples are given to show the effectiveness of the presented results.

REFERENCES
[1] D. Q. Mayne, J. B. Rawlings, C. V. Rao and P.O.M. Scokaert,
“Constrained model predictive control: Stability and optimality”,
Automatica, 36, 789–814, 2000.
[2] A. Bemporad, M. Morari, V. Dua and E.N. Pistikopoulos “The explicit
linear quadratic regulator for constrained systems”, Automatica
38, 3–20, 2002.
[3] T. Parisini and R. Zoppoli, “A receding-horizon regulator for
nonlinear systems and a neural approximation”. Automatica 31(10), 1443–
1451, 1995.
[4] T. A. Johansen, “Approximate explicit receding horizon control of
constrained nonlinear systems”. Automatica 40, 293–300, 2004.
[5] M. Canale and M. Milanese, “FMPC: a fast implementation of model
predictive control”, in 16th IFAC World Congress, Prague, Czech
Republic, July 2005.

Slides