Solving the infinite-horizon constrained LQR problem using splitting techniques

Stathopoulos, Georgios T.; Korda, Milan; Jones, Colin N.

doi:10.3182/20140824-6-za-1003.00650

Cited by 8 publications

(6 citation statements)

References 13 publications

(13 reference statements)

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“…with the last inequality following directly from point (iii). (v) The proof is already presented in [9], but we repeat it for completeness. First note that for the statement to hold it is sufficient to show that…”

Section: Convergence Resultsmentioning

confidence: 91%

“…The reason for considering the condensed formulation is the need for strong convexity of the primal objective, which implies the Lipschitz continuity of ∇h (·) [7,Corollary 18.16]. By using the condensed form we avoid the restrictive assumption of Q 0 required in [9].…”

Section: Problem Statement and Dualizationmentioning

confidence: 99%

“…This work is based on our recent result in [9], where the same problem is solved by employing a primal-dual scheme, the Alternating Minimization Algorithm (AMA) [10], which is equivalent to FBS. The approach is equivalent to splitting the infinite-horizon constrained LQR problem into an unconstrained LQR problem and a proximal minimization problem.…”

Section: Introductionmentioning

confidence: 99%

“…In the current work, the result is developed in both theoretical and practical directions. From the theoretical perspective, we (i) drop the limiting assumption for positive-definiteness of the state penalty matrix Q that existed in [9], and, more importantly, we (ii) prove convergence of an accelerated counterpart of the original FBS method, a fact that enables a (significantly) faster convergence rate. From the practical perspective, we provide a fully implementable method, competitive for real-time control.…”

Section: Introductionmentioning

confidence: 99%

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Solving the Infinite-Horizon Constrained LQR Problem Using Accelerated Dual Proximal Methods

Stathopoulos

Korda

Jones

2017

IEEE Trans. Automat. Contr.

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This work presents an algorithmic scheme for solving the infinite-time constrained linear quadratic regulation problem. We employ an accelerated version of a popular proximal gradient scheme, commonly known as the Forward-Backward Splitting (FBS), and prove its convergence to the optimal solution in our infinite-dimensional setting. Each iteration of the algorithm requires only finite memory, is computationally cheap, and makes no use of terminal invariant sets; hence, the algorithm can be applied to systems of very large dimensions. The acceleration brings in 'optimal' convergence rates O(1/k 2 ) for function values and O(1/k) for primal iterates and renders the proposed method a practical alternative to model predictive control schemes for setpoint tracking. In addition, for the case when the true system is subject to disturbances or modelling errors, we propose an efficient warm-starting procedure, which significantly reduces the number of iterations when the algorithm is applied in closed-loop. Numerical examples demonstrate the approach. Index TermsConstrained LQR, Alternating minimization, Operator splitting I. INTRODUCTION An important extension of the famous result of [1] on the closed form solution of the infinitehorizon linear quadratic regulation (LQR) problem is the case where input and state variables are constrained. This problem is computationally significantly more difficult and has been by and large addressed only approximately. A prime example of an approximation scheme is model predictive control (MPC), which approximates the infinite-time constrained problem by a finitetime one. Stability of such MPC controllers is then typically enforced by adding a suitable terminal constraint and a terminal penalty. The inclusion of a terminal constraint limits the feasible region of the MPC controller, and, consequently, the region of attraction of the closedloop system. In practical applications, this problem is typically overcome by simply choosing a "sufficiently" long horizon based on process insight (e.g., dominant time constant). Closed-loop behavior is then analyzed a posteriori, for instance by exhaustive simulation.There have been few results addressing directly the infinite-horizon constrained LQR (CLQR) problem. Among the most well-known efforts are the works

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Section: Convergence Resultsmentioning

confidence: 91%

Section: Problem Statement and Dualizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solving the Infinite-Horizon Constrained LQR Problem Using Accelerated Dual Proximal Methods

Stathopoulos

Korda

Jones

2017

IEEE Trans. Automat. Contr.

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“…Infinite-dimensional problems We can roughly state that all the theoretical results stated for the three algorithms presented in this work are originally derived in Hilbert-space settings [7], [128], [5], [85], [21], [30]. The authors of [125] use these converegence guarantees to solve the constrained linear quadratic regulator problem, i.e., an infinite-horizon MPC regulation problem by means of the AMA and FAMA [126].…”

Section: Extensions and Other Directionsmentioning

confidence: 99%

Sensor Fault Diagnosis

Stathopoulos

Shukla

Szűcs

et al. 2016

FnT in Systems and Control

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The significant progress that has been made in recent years both in hardware implementations and in numerical computing has rendered real-time optimization-based control a viable option when it comes to advanced industrial applications. More recently, the need for control of a process in the presence of a limited amout of hardware resources has triggered research in the direction of embedded optimization-based control. At the same time, and standing at the other side of the spectrum, the field of big data has emerged, seeking for solutions to problems that classical optimization algorithms are incapable to provide. This triggered some interest to revisit the family of first order methods commonly known as decomposition schemes or operator splitting methods. Although it is established that splitting methods are quite beneficial when applied to large-scale problems, their potential in solving small to medium scale embedded optimization problems has not been studied so extensively. Our purpose is to study the behavior of such algorithms as solvers of control-related problems of that scale. Our effort focuses on identifying special characteristics of these problems and how they can be exploited by some popular splitting methods.

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Linear quadratic regulation for discrete-time antilinear systems: An anti-Riccati matrix equation approach

Qian

Liu

et al. 2016

Journal of the Franklin Institute

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Solving the infinite-horizon constrained LQR problem using splitting techniques

Cited by 8 publications

References 13 publications

Solving the Infinite-Horizon Constrained LQR Problem Using Accelerated Dual Proximal Methods

Solving the Infinite-Horizon Constrained LQR Problem Using Accelerated Dual Proximal Methods

Sensor Fault Diagnosis

Linear quadratic regulation for discrete-time antilinear systems: An anti-Riccati matrix equation approach

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