A POD projection method for large-scale algebraic Riccati equations

Krämer, Boris; Singler, John R.

doi:10.3934/naco.2016018

Cited by 8 publications

(3 citation statements)

References 64 publications

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“…As in the LQR case, suitable model reduction methods [1], [2], [17] are essential to forming a solution methodology for distributed parameter control problems with quadratic nonlinearities. First of all, the Riccati equation is still needed to compute the linear term [7], [25], [26] and the curse-of-dimensionality still appears with higher-order polynomial approximations of the feedback law.…”

Section: Motivationmentioning

confidence: 99%

The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems

Borggaard¹,

Zietsman²

2019

Preprint

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Feedback control problems involving autonomous quadratic systems are prevalent, yet there are only a limited number of software tools available for approximating their solution due to the complexity of the problem. This paper represents a step forward in the special case where both the state equation and the control costs are quadratic. As it represents the natural extension of the linear-quadratic regulator (LQR) problem, we describe this setting as the quadraticquadratic regulator (QQR) problem. This is significantly more challenging and holds the LQR as special case that must be solved along the way. We describe an algorithm that exploits the structure of the QQR problem that arises when implementing Al'Brekht's method. This approach is amenable to feedback laws with low degree polynomials but have a relatively modest model dimension that could be achieved by modern model reduction methods. This problem has an elegant formulation and a solution that introduces several linear systems where the structure suggests modern tensor-based linear solvers. We demonstrate this algorithm on a suite of random test problems then apply it to a distributed parameter control problem that fits the QQR framework. Comparisons to linear feedback control laws show a modest benefit using the QQR formulation.

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Section: Motivationmentioning

confidence: 99%

The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems

Borggaard¹,

Zietsman²

2019

Preprint

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“…have been successful [42,7,16,6,43]. The rank of a matrix is defined by the maximum number of linearly independent rows or columns.…”

Section: Problem Formulationmentioning

confidence: 99%

Feedback Control for Systems with Uncertain Parameters Using Online-Adaptive Reduced Models

Krämer¹,

Peherstorfer²,

Willcox³

2017

SIAM J. Appl. Dyn. Syst.

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Abstract. We consider control and stabilization for large-scale dynamical systems with uncertain, time-varying parameters. The time-critical task of controlling a dynamical system poses major challenges: using large-scale models is prohibitive, and accurately inferring parameters can be expensive, too. We address both problems by proposing an offline-online strategy for controlling systems with timevarying parameters. During the offline phase, we use a high-fidelity model to compute a library of optimal feedback controller gains over a sampled set of parameter values. Then, during the online phase, in which the uncertain parameter changes over time, we learn a reduced-order model from system data. The learned reduced-order model is employed within an optimization routine to update the feedback control throughout the online phase. Since the system data naturally reflects the uncertain parameter, the data-driven updating of the controller gains is achieved without an explicit parameter estimation step. We consider two numerical test problems in the form of partial differential equations: a convection-diffusion system, and a model for flow through a porous medium. We demonstrate on those models that the proposed method successfully stabilizes the system model in the presence of process noise.

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“…First of all, the Riccati equation must be solved to compute the linear term, e.g. Benner et al (2013); Singler (2008); Singler and Kramer (2016) and the curse-of-dimensionality still appears with higher-order polynomial approximations of the feedback law.…”

mentioning

confidence: 99%

On Approximating Polynomial-Quadratic Regulator Problems

Borggaard¹,

Zietsman²

2020

Preprint

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Feedback control problems involving autonomous polynomial systems are prevalent, yet there are limited algorithms and software for approximating their solution. This paper represents a step forward by considering the special case of the regulator problem where the state equation has polynomial nonlinearity, control costs are quadratic, and the feedback control is approximated by low-degree polynomials. As this represents the natural extension of the linear-quadratic regulator (LQR) and quadratic-quadratic regulator (QQR) problems, we denote this class as polynomial-quadratic regulator (PQR) problems. The present approach is amenable to feedback approximations with low degree polynomials and to problems of modest model dimension. This setting can be achieved in many problems using modern model reduction methods. The Al'Brekht algorithm, when applied to polynomial nonlinearities represented as Kronecker products leads to an elegant formulation. The terms of the feedback control lead to large linear systems that can be effectively solved with an N-way generalization of the Bartels-Stewart algorithm. We demonstrate our algorithm with numerical examples that include the Lorenz equations, a ring of van der Pol oscillators, and a discretized version of the Burgers equation. The software described here is available on Github.

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A POD projection method for large-scale algebraic Riccati equations

Cited by 8 publications

References 64 publications

The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems

The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems

Feedback Control for Systems with Uncertain Parameters Using Online-Adaptive Reduced Models

On Approximating Polynomial-Quadratic Regulator Problems

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