Linear-Quadratic Optimal Control of Differential-Algebraic Systems: The Infinite Time Horizon Problem with Zero Terminal State

Reis, Timo; Voigt, Matthias

doi:10.1137/18m1189609

Cited by 25 publications

(18 citation statements)

References 39 publications

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“…As for the theory of control of DAEs, an immediate strengthening of the results could be achieved by removing the assumption on impulse controllability. A more general framework will also consider indefinite cost functionals and suitable replacements for the Riccati equations, as they are used recent works on singular feedback control [5] or infinite time horizon problems [29].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems

Heiland¹,

Zuazua²

2020

Preprint

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The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of the turnpike is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike in the nondetectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems, we establish existence and convergence of solutions to a generalized differential Riccati equation.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems

Heiland¹,

Zuazua²

2020

Preprint

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“…One way to work directly with the DAE is to adapt the approach by Reis and Voigt (2019) to model predictive control: their results allow to characterize the optimal value and optimal solution using so-called Lur'e equations for the DAE. In order to use these findings for MPC, it is necessary to characterize the positive definiteness of the optimal value in terms of the DAE-OCP.…”

Section: Conclusion and Open Problemmentioning

confidence: 99%

“…There has been a lot of research in the related field of optimal control for DAEs, both in an analytical context using Riccati (Cobb, 1983;Kunkel & Mehrmann, 2008;Lamour, März, & Tischendorf, 2013;S. L. Campbell, Kunkel, & Mehrmann, 2012) or Lur'e quations (Bankmann, 2016;Reis & Voigt, 2019) as well as in a numerical context (Gerdts, 2011). However, the analytical results do not encompass state or input constraints and to our knowledge, none of these results has been extended yet to the stability analysis of MPC schemes.…”

Section: Introductionmentioning

confidence: 99%

Model predictive control for singular differential-algebraic equations

Ilchmann,

Witschel,

Worthmann

2022

Preprint

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We study model predictive control for singular differential-algebraic equations with higher index. This is a novelty when compared to the literature where only regular differential-algebraic equations with additional assumptions on the index and/or controllability are considered. By regularization techniques, we are able to derive an equivalent optimal control problem for an ordinary differential equation to which well-known model predictive control techniques can be applied. This allows the construction of terminal constraints and costs such that the origin is asymptotically stable w.r.t. the resulting closed-loop system.

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“…Like eigenvalue problems associated with sys-ccLODE, there are existing efforts for developing accurate and efficient algorithms for solving generalized eigenvalue problems which are computationally expensive for large systems [27]. Sys-LODAEs are also studied from a controls perspective for controllability and observability and to design optimal controls [25,28,29].…”

Section: Future Directionsmentioning

confidence: 99%

Analytic Methods for Differential Algebraic Equations

Alkhairy¹

2021

Preprint

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We introduce methods for deriving analytic solutions from differential-algebraic systems of equations (DAEs), as well as methods for deriving governing equations for analytic characterization which is currently limited to very small systems as it is carried out by hand. Analytic solutions to the system and analytic characterization through governing equations provide insights into the behaviors of DAEs as well as the parametric regions of operation for each potential behavior. DAEs are a mixture of algebraic, ordinary differential, and/or partial differential equations which are multivariate constitutive equations that arise from fundamental laws of a discipline which relate multiple dependent variables. For each system (DAEs), and choice of dependent variable, there is a corresponding governing equation which is univariate ODE or PDE that is typically higher order than the constitutive equations of the system. We first introduce a direct formulation for representing systems of linear differential-algebraic scalar constitutive equations that are homogeneous or nonhomogeneous with constant or nonconstant coefficients. Unlike state space formulations, our formulation follows very directly from the system of constitutive equations without the need for introducing state variables or singular matrices. Using this formulation for the system of constitutive equations (DAEs), we develop methods for deriving analytic expressions for the full solution (complementary and particular) for all dependent variables of systems that consist of constant coefficient ordinary-DAEs and special cases of partial-DAEs. We also develop methods for deriving the governing equation for a chosen dependent variable for the constant coefficient ordinary-DAEs and partial-DAEs as well as special cases of variable coefficient DAEs. The methods can be automated with symbolic coding environments thereby allowing for dealing with systems of any size while retaining analytic nature. This is relevant for interpretable modeling, analytic characterization and estimation, and engineering design in which the objective is to tune parameter values to achieve specific behavior. Such insights cannot directly be obtained using numerical simulations.

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Linear-Quadratic Optimal Control of Differential-Algebraic Systems: The Infinite Time Horizon Problem with Zero Terminal State

Cited by 25 publications

References 39 publications

Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems

Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems

Model predictive control for singular differential-algebraic equations

Analytic Methods for Differential Algebraic Equations

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