The double description method for the approximation of explicit MPC control laws

A standard model predictive controller (MPC) can be written as a parametric optimization problem whose solution is a piecewise affine (PWA) map from the measured state to the optimal control input. The primary limitation of this optimal `explicit solution¿ is that the complexity can grow quickly with problem size, and so in this paper we seek to compute approximate explicit control laws that can trade-off complexity for approximation error. This computation is accomplished in a two-phase process: First, inner and outer polyhedral approximations of the the convex cost function of the parametric problem are computed with an algorithm based on an extension to the classic double-description method; a convex hull approach. The proposed method has two main advantages from a control point of view: it is an incremental approach, meaning that an approximation of any specified complexity can be produced and it operates on implicitly-defined convex sets, meaning that the optimal solution of the parametric problem is not required. In the second phase of the algorithm, a feasible approximate control law is computed that has the cost function derived in the first phase. For this purpose, a new interpolation method is introduced based on recent work on barycentric interpolation. The resulting control law is continuous, although non-linear and defined over a non-simplical polytopic partition of the state space. The non-simplical nature of the partition generates significantly simpler approximate control laws than current competing methods, as demonstrated on computational examples

Polytopic Approximation of Explicit Model Predictive Controllers

A model predictive control law (MPC) is given by the solution to a parametric optimization problem that can be pre-computed offline, which provides an explicit map from state to input that can be rapidly evaluated online. However, the primary limitations of these optimal explicit solutions are that they are applicable to only a restricted set of systems and that the complexity can grow quickly with problem size. In this paper we compute approximate explicit control laws that trade-off complexity against approximation error for MPC controllers that give rise to convex parametric optimization problems. The algorithm is based on the classic double- description method and returns a polyhedral approx- imation to the optimal cost function. The proposed method has three main advantages from a control point of view: it is an incremental approach, meaning that an approximation of any specified complexity can be produced, it operates on implicitly-defined convex sets, meaning that the prohibitively complex optimal explicit solution is not required and finally it can be applied to any convex parametric optimization problem. A sub-optimal controller based on barycentric in- terpolation is then generated from this approximate polyhedral cost function that is feasible and stabiliz- ing. The resulting control law is continuous, although non-linear and defined over a non-simplical polytopic partition of the state space. The non-simplical nature of the partition generates significantly simpler approx- imate control laws, which is demonstrated on several examples

Explicit predictive control laws. On the geometry of feasible domains and the presence of nonlinearities.

International audienceThis paper is dealing with the receding horizon optimal control techniques having as main goal the reduction of the computational effort inherent to the use of on-line optimization routines. The off-line construction of the explicit solution for the associated multiparametric optimization problems is advocated with a special interest in the presence of nonlinearities in the constraints description. The proposed approach is a geometrical one, based on the topology of the feasible domain. The resulting piecewise linear state feedback control law has to accept a certain degree of suboptimality, as it is the case for local linearizations or decompositions over families of parametric functions. In the presented techniques, this is directly related to the distribution of the extreme points on the frontier of the feasible domain

Scaling Relations for Collision-less Dark Matter Turbulence

Many scaling relations are observed for self-gravitating systems in the universe. We explore the consistent understanding of them from a simple principle based on the proposal that the collision-less dark matter fluid terns into a turbulent state, i.e. dark turbulence, after crossing the caustic surface in the non-linear stage. The dark turbulence will not eddy dominant reflecting the collision-less property. After deriving Kolmogorov scaling laws from Navier-Stokes equation by the method similar to the one for Smoluchowski coagulation equation, we apply this to several observations such as the scale-dependent velocity dispersion, mass-luminosity ratio, magnetic fields, and mass-angular momentum relation, power spectrum of density fluctuations. They all point the concordant value for the constant energy flow per mass: $0.3 cm^2/sec^3$, which may be understood as the speed of the hierarchical coalescence process in the cosmic structure formation.Comment: 26 pages, 6 figure

Digital VLSI Implementation of Piecewise-Affine Controllers Based on Lattice Approach

This paper presents a small, fast, low-power consumption solution for piecewise-affine (PWA) controllers. To achieve this goal, a digital architecture for very-large-scale integration (VLSI) circuits is proposed. The implementation is based on the simplest lattice form, which eliminates the point location problem of other PWA representations and is able to provide continuous PWA controllers defined over generic partitions of the input domain. The architecture is parameterized in terms of number of inputs, outputs, signal resolution, and features of the controller to be generated. The design flows for field-programmable gate arrays and application-specific integrated circuits are detailed. Several application examples of explicit model predictive controllers (such as an adaptive cruise control and the control of a buck-boost dc-dc converter) are included to illustrate the performance of the VLSI solution obtained with the proposed lattice-based architecture

Explicit predictive control laws with a nonlinear constraints handling mechanism

International audienceThis paper is dealing with the receding horizon optimal control techniques having as main goal the reduction of the computational effort inherent to the use of on-line optimization routines. The off-line construction of the explicit solution for the associated multiparametric optimization problems is advocated with a special interest in the presence of nonlinearities in the constraints description. The proposed approach is a geometrical one, based on the topology of the feasible domain. The resulting piecewise linear state feedback control law has to accept a certain degree of suboptimality, as it is the case for local linearizations or decompositions over families of parametric functions. In the presented techniques, this is directly related to the distribution of the extreme points on the frontier of the feasible domain