Outline. Robust MPC and multiparametric convex programming. A. Bemporad C. Filippi. Motivation: Robust MPC. Multiparametric convex programming

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1 Robust MPC and multiparametric convex programming A. Bemporad C. Filippi D. Muñoz de la Peña CC Meeting Siena /4 September 003 Outline Motivation: Robust MPC Multiparametric convex programming Kothares 96 Examples

2 Model Predictive Control past future Predicted outputs y(t+k t) t t+ Manipulated Inputs u(t+k) t+n At time t: Get new measurements y(t) Estimate the current state x(t) Solve with respect to the QP prob. (=finite-horizon open-loop optimal control) Apply only and discard the remaining optimal inputs Go to time t+ Robust MPC Robust MPC are based on uncertain models: UNCERTAIN PREDICTIONS past future Predicted outputs y(t+k t) Manipulated Inputs u(t+k) t t+ t+n

3 Robust MPC Robust MPC formulation: Worst case minimization Robust constraint satisfaction Robust MPC Robust MPC computation: Maximum convex function HARD HIGH COMPUTATINAL BURDEN Minimum convex function Tractable 3

4 Robust MPC Robust constraint handling: X K+ X k+ x K X K+3 One constraint for each vertex on the convex hull Exponential number of constraints with the prediction horizon Robust MPC Open loop MPC: Minimize over a single trajectory Number of free variables linear with the prediction horizon Feedback: Minimize a set of trajectories Number of free variables exponential with the prediction horizon Apply dynamic programming 4

5 Conclusion MPC ROBUST (in general) ARE COMPUTATIONALY DEMANDING Robust MPC & mp Convex Robust MPC controllers often are defined as convex optimization problems Tractable but not solvable on line Linear Cost function Explicit solution using Dynamic programming [] A. Bemporad, F. Borreli, M.Morari Min-max control of constrained uncertain discrete time linear systems,ieee Trans. Automatic Control 003. Quadratic cost function Not extended Multiparametric Convex Programming 5

6 Multiparametric Convex Programming Computation offline Suboptimal solution Online efficienty IMPLEMENTATION Multiparametric Convex Programming where: 6

7 Basic Result on Convex MP (Mangasarian, Rosen, 964) V * and Θ f may not be easy to express analytically Bounds on the Value Function Consider the m-dimensional simplex: Optimizers at the vertices: Linear upper-bound: Define. is feasible for all Convex upper-bound: Convex PWA lower-bound: (s i i ) Result: 7

8 Linear feedback solution θ 0 θ θ S θ LINEAR SOLUTION Max Error-Bounds Computation Max absolute error inside simplex S: not very easy to compute in general... Define another error measure: this is a convex program! Result: 8

9 Recursive Approximation θ 0 Recursive Algorithm. Compute ε CP (S) and let (x CP,θ CP ) the corresponding worst point. If ε CP (S)ε then for all i=0,,...,m θ S S S θ CP S 0 θ.. Get a new smaller simplex S i by replacing θ i θ CP.. If S i is full dimensional, call this algorithm on S i Approximate MP-Convex Solver For all simplices S,..., S Nm, apply the recursive splitting algorithm to get an approximation with overall error ε The approximate multi-parametric solution can be conveniently evaluated via the recursion tree simplices recursion tree 9

10 Piecewise Linear Optimal Control laws Multi-parametric programming is a useful tool for getting Piecewise Linear (sub)optimal Control laws Multi-parametric convex nonlinear programming is a promising tool for obtaining robust MPC laws of limited on-line complexity [] A. Bemporad and C. Filippi, ``Approximate Multiparametric Convex Programming, submitted CDC 003. Paper download: Robust MPC & mp Convex Robust MPC controllers often are defined as convex optimization problems Multiparametric Convex PERFORMANCE CONSTRAINT HANDLING STABILITY?? 0

11 A case study: Kothare s 96 [] M.V.Kothare, V. Balakrishna, M.Morari ``Robust Constrained Model Predictive Control using Linear Matrix Inequalities, Automatica 33, 996 LTV Systems: Upper bound maximum infinity cost Robust constraints Linear feedback law A case study: Kothare s 94 - II Kothare s optimization problem = SDP (Conservative solution) MP CONVEX (Unconstrained case)

12 Properties of the controller Stability Stability proof:

13 Examples Examples.5 89 regions 0 tree levels

14 Relative Error bound Aproximation with maximum relative error In this case, is not a SDP APROXIMATION Stability DUAL CONTROL Stability guaranteed for Kothare s properties 4

15 Relative error bound regions 0 tree levels regions 7 tree levels Examples 5

16 6 Error bounds Absolute error Relative error Complexity Reg tree xmax nu nx

17 Conclusions Mp Convex is a tool to take into account for MPC Robust fast implementation BUT Is computing demanding Case dependent So if you have: A good uncertain model Complex control objectives only solvable with a complex robust MPC Small sampling time Take a look on MpConvex THE END 7