Efficient Robust Optimization for Robust Control with Constraints.
File(s)
Accepted version
Author(s)
Goulart, PJ
Kerrigan, EC
Ralph, D
Type
Journal Article
Abstract
This paper proposes an efficient computational technique for the
optimal control of linear discrete-time systems subject to bounded disturbances
with mixed linear constraints on the states and inputs. The problem of computing
an optimal state feedback control policy, given the current state, is non-convex.
A recent breakthrough has been the application of robust optimization
techniques to reparameterize this problem as a convex program. While the
reparameterized problem is theoretically tractable, the number of variables is
quadratic in the number of stages or horizon length N and has no apparent
exploitable structure, leading to computational time of O(N6) per iteration of
an interior-point method. We focus on the case when the disturbance set is ∞-
norm bounded or the linear map of a hypercube, and the cost function involves
the minimization of a quadratic cost. Here we make use of state variables to
regain a sparse problem structure that is related to the structure of the original
problem, that is, the policy optimization problem may be decomposed into a
set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primaldual
interior-point methods in which each iteration requires O(N3) time. This
cubic iteration time can be guaranteed using a Riccati-based block factorization
technique, which is standard in discrete-time optimal control. Numerical results
are presented, using a standard sparse primal-dual interior point solver, that
illustrate the efficiency of this approach.
optimal control of linear discrete-time systems subject to bounded disturbances
with mixed linear constraints on the states and inputs. The problem of computing
an optimal state feedback control policy, given the current state, is non-convex.
A recent breakthrough has been the application of robust optimization
techniques to reparameterize this problem as a convex program. While the
reparameterized problem is theoretically tractable, the number of variables is
quadratic in the number of stages or horizon length N and has no apparent
exploitable structure, leading to computational time of O(N6) per iteration of
an interior-point method. We focus on the case when the disturbance set is ∞-
norm bounded or the linear map of a hypercube, and the cost function involves
the minimization of a quadratic cost. Here we make use of state variables to
regain a sparse problem structure that is related to the structure of the original
problem, that is, the policy optimization problem may be decomposed into a
set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primaldual
interior-point methods in which each iteration requires O(N3) time. This
cubic iteration time can be guaranteed using a Riccati-based block factorization
technique, which is standard in discrete-time optimal control. Numerical results
are presented, using a standard sparse primal-dual interior point solver, that
illustrate the efficiency of this approach.
Date Issued
2007-05-25
Date Acceptance
2006-12-04
Citation
Mathematical Programming, 2007, 114 (1), pp.115-147
ISSN
1436-4646
Publisher
Springer Verlag (Germany)
Start Page
115
End Page
147
Journal / Book Title
Mathematical Programming
Volume
114
Issue
1
Copyright Statement
The final publication is available at Springer via https://dx.doi.org/10.1007/s10107-007-0096-6
Subjects
Constrained Control
Robust Control
Robust Optimization
Optimal control
receding horizon control
predictive control
Publication Status
Published