A Sequential Convex Programming Approach to Solving Quadratic Programs and Optimal Control Problems With Linear Complementarity Constraints

Hall, Jonas; Nurkanović, Armin; Messerer, Florian; Diehl, Moritz

doi:10.1109/lcsys.2021.3083467

Cited by 11 publications

(7 citation statements)

References 26 publications

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“…Some of the appealing properties of the constraints (27) are given by the next lemma. In our notation θ n,m,i is the i-th component of the vector θ n,m .…”

Section: Continuity Of λ(•) and µ(•)mentioning

confidence: 99%

“…. , n s }, k = j, then θ n,m,i λ n,k,i > 0 which violates (27), thus all λ n,m ′ ,i = 0, m ′ ∈ {0, . .…”

Section: Continuity Of λ(•) and µ(•)mentioning

confidence: 99%

“…The conditions (27) are given in their sparsest form. Due to the nonnegativity of Λ n and Θ n there are many equivalent formulations of this condition, e.g., all conditions above can be summed up for a single finite element or even for all finite elements on the regarded control interval.…”

Section: Continuity Of λ(•) and µ(•)mentioning

confidence: 99%

“…Using FESD to discretize OCP with DCS results in mathematical programs with complementarity constraints which can be solved efficiently with smooth optimization techniques by solving few several Nonlinear Programming (NLP) problems. [8,27,32,33,41]. Thus, we avoid the nonsmooth difficulty encountered within direct multiple shooting with switch-detecting integrators.…”

Section: Introductionmentioning

confidence: 99%

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Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems

Nurkanović¹,

Mario²,

Albrecht³

et al. 2022

Preprint

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This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We regard the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart [44]. FESD is based on solving of nonlinear complementarity problems and able to automatically detect the nonsmooth events in time. If standard Runge-Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step-size to be a degree of freedom. Additional complementarity conditions, that we call cross complementarities, enable exact switch detection, hence FESD is able to recover the high order accuracy which the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step-size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD in demonstrated on several simulation and optimal control examples.

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“…Some of the appealing properties of the constraints (27) are given by the next lemma. In our notation θ n,m,i is the i-th component of the vector θ n,m .…”

Section: Continuity Of λ(•) and µ(•)mentioning

confidence: 99%

“…. , n s }, k = j, then θ n,m,i λ n,k,i > 0 which violates (27), thus all λ n,m ′ ,i = 0, m ′ ∈ {0, . .…”

Section: Continuity Of λ(•) and µ(•)mentioning

confidence: 99%

Section: Continuity Of λ(•) and µ(•)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems

Nurkanović¹,

Mario²,

Albrecht³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The resulting discretized OCP are Mathematical Programs with Complementarity Constraints (MPCC). With appropriate reformulations the MPCC can often be solved by only a few NLP solves [15]- [17], i.e., the highly nonsmooth and nonlinear OCP are solved by purely derivative based algorithms. A time optimal control problem of a hybrid system with hysteresis and a nonsmooth objective illustrates theoretical and algorithmic developments.…”

Section: Introductionmentioning

confidence: 99%

Continuous Optimization for Control of Hybrid Systems with Hysteresis via Time-Freezing

Nurkanović,

Diehl

2022

Preprint

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This article regards numerical optimal control of a class of hybrid systems with hysteresis using solely techniques from nonlinear optimization, without any integer variables. Hysteresis is a rate independent memory effect which often results in severe nonsmoothness in the dynamics. These systems are not simply Piecewise Smooth Systems (PSS); they are a more complicated form of hybrid systems. We introduce a time-freezing reformulation which transforms these systems into a PSS. From the theoretical side, this reformulation opens the door to study systems with hysteresis via the rich tools developed for Filippov systems. From the practical side, it enables the use of the recently developed Finite Elements with Switch Detection [1], which makes high accuracy numerical optimal control of hybrid systems with hysteresis possible.

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Finite Elements with Switch Detection for direct optimal control of nonsmooth systems

Nurkanović,

Sperl,

Albrecht

et al. 2024

Numer. Math.

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This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We consider the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart (Numer Math 58(1):299–328, 1990). FESD is based on solving nonlinear complementarity problems and can automatically detect nonsmooth events in time. If standard time-stepping Runge–Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step size be a degree of freedom. Additional complementarity conditions, which we call cross complementarities, enable exact switch detection, hence FESD can recover the high order accuracy that the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD is demonstrated in several simulation and optimal control examples. In an optimal control problem benchmark with FESD, we achieve up to five orders of magnitude more accurate solutions than a standard time-stepping approach for the same computational time.

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A Sequential Convex Programming Approach to Solving Quadratic Programs and Optimal Control Problems With Linear Complementarity Constraints

Cited by 11 publications

References 26 publications

Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems

Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems

Continuous Optimization for Control of Hybrid Systems with Hysteresis via Time-Freezing

Finite Elements with Switch Detection for direct optimal control of nonsmooth systems

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