A convex parameterization for solving constrained min-max problems with a quadratic cost

Kerrigan, Eric C.; Álamo, T.

doi:10.23919/acc.2004.1383791

Cited by 11 publications

(11 citation statements)

References 4 publications

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“…Some results along these lines are already available for problems with ∞-norm bounded disturbances [10] and the minimization of the finite-horizon 2 gain of a system [14].…”

Section: Discussionmentioning

confidence: 99%

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Optimization over state feedback policies for robust control with constraints

2006

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This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.

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“…Some results along these lines are already available for problems with ∞-norm bounded disturbances [10] and the minimization of the finite-horizon 2 gain of a system [14].…”

Section: Discussionmentioning

confidence: 99%

“…where the vector z i ∈ R a represents the dual variables associated with the i th row of the maximization in (14). By combining these into a matrix Z := [ z1 ... zN ] ∈ R a×t , one can rewrite Π df N (x) in terms of purely affine constraints:…”

Section: Computation Of Admissible Policiesmentioning

confidence: 99%

Optimization over state feedback policies for robust control with constraints

2006

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“…A set of necessary ingredients ensuring robust exponential stability has been identified. The proposed scheme is computationally simpler than the schemes proposed in (Löfberg, 2003;van Hessem and Bosgra, 2003;Kerrigan and Alamo, 2004;Langson et al, 2004) and it has an advantage over schemes proposed in (Kouvaritakis et al, 2000;Chisci et al, 2001;Mayne and Langson, 2001;Smith, 2004;Mayne et al, 2005) because the feedback component of control policy, which is piecewise affine, results in a smaller tube cross-section.…”

Section: Numerical Examplementioning

confidence: 98%

“…, µ N −1 (·)} of control laws. Determination of a feedback control policy is usually prohibitively difficult and various simplifying approximations have been proposed in the literature (Kouvaritakis et al, 2000;Chisci et al, 2001;Mayne and Langson, 2001;Löfberg, 2003;van Hessem and Bosgra, 2003;Kerrigan and Alamo, 2004;Smith, 2004 loop and feedback model predictive controller generate a tube of trajectories when uncertainty is present. Feedback model predictive control reduces the spread of predicted trajectories resulting from uncertainty.…”

Section: Introductionmentioning

confidence: 99%

A Simple Tube Controller for Efficient Robust Model Predictive Control of Constrained Linear Discrete Time Systems Subject to Bounded Disturbances

Raković

Mayne

2005

IFAC Proceedings Volumes

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We present a simple tube controller for efficient robust model predictive control of constrained linear, discrete-time systems in the presence of bounded disturbances. The proposed control policy ensures that controlled trajectories are confined to a given tube despite uncertainty. The robust optimal control problem that is solved on-line is a standard quadratic programming problem of marginally increased complexity compared with that required for model predictive control in the deterministic case. We show how to optimize the tube cross section, how to construct an adequate tube terminal set and we establish robust exponential stability of a suitable robust control invariant set (the 'origin' for uncertain system) with enlarged domain of attraction. Copyright

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“…157-8]. Alternatively, the cost function for the finite horizon control problem may require the minimization of the finite-horizon 2 gain of a system [28,34]. In all of these cases, there is a strong possibility that the underlying problem structure may be exploited to realize a substantial increase in computational efficiency.…”

Section: Discussionmentioning

confidence: 99%

Efficient robust optimization for robust control with constraints

2007

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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(N 6 ) 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 primal-dual interior-point methods in which each iteration requires O(N 3 ) 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.

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A convex parameterization for solving constrained min-max problems with a quadratic cost

Cited by 11 publications

References 4 publications

Optimization over state feedback policies for robust control with constraints

Optimization over state feedback policies for robust control with constraints

A Simple Tube Controller for Efficient Robust Model Predictive Control of Constrained Linear Discrete Time Systems Subject to Bounded Disturbances

Efficient robust optimization for robust control with constraints

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