Hossain, Sumon; Trenn, Stephan Model reduction for switched differential-algebraic equations with known switching signal Unpublished 2024, (submitted). @unpublished{HossTren24pp,Building on our recently proposed model reduction methods for switched ordinary linear systems we propose a comprehensive model reduction method for linear switched differential-algebraic equations (DAEs). In contrast to most other available model reduction methods for switched systems we consider the switching signal as a given time-variance of the system. This allows us to exploit certain linear subspaces in the reduction process and also provide in general significantly smaller reduced models compared to methods which consider arbitrary switching signals. Model reduction for switched DAEs has some unique features which makes a generalization of the available methods nontrivial; in particular, the presence of jumps and Dirac impulses in response to switches have to be carefully treated. Furthermore, due the algebraic constraints, the reachability subspaces cannot be the full space, hence a straightforward application of balanced truncation is not possible (because the corresponding reachability Gramians will be structurally non-invertible). We resolve this problem by first apply an exact model reduction which reduces the switched DAE to a switched ordinary systems with jumps and carefully keep track of the impulsive effects. As a second step we then apply a midpoint balanced truncation approach to further reduce the switched system. In addition to the challenge to appropriately take into account the Dirac impulses, another novel challenge was the occurrence of input-dependent state-jumps. We propose to deal with input-dependent jumps by combining certain discrete-time reachability Gramians with continuous time reachability Gramians. We provide corresponding Matlab implementations of the proposed algorithms and illustrate their effectiveness with some academic examples. |
Wijnbergen, Paul; Trenn, Stephan Impulse-free linear quadratic optimal control of switched differential algebraic equations Unpublished 2024, (provisionally accepted in "Communications in Optimization Theory"). @unpublished{WijnTren24pp,In this paper the finite horizon linear quadratic regulator (LQR) problem for switched linear differential algebraic equations is studied. It is shown that for switched DAEs with a switching signal that induces locally finitely many switches, the problem can be solved by recursively solving several LQR problems for non-switched DAE. First, it is shown how to solve the non-switched problems for index-1 DAEs followed by an extension of the results to higher index DAEs. The resulting optimal control can be computed based on the solution of a Riccati differential equation expressed in terms of the differential system matrices. The paper concludes with the extension of the results to the LQR problem for general switched DAEs. |
Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan Output contraction analysis of nonlinear systems Unpublished 2023, (submitted). @unpublished{YinJaya23ppb,This paper introduce the notion of output contraction that expands the contraction notion to the time-varying nonlinear systems with output. It pertains to the systems’ property that any pair of outputs from the system converge to each other exponentially. This concept exhibits a more expansive nature when contrasted with another generalized contraction framework known as partial contraction. The first result establishes a connection between the output contraction of a time-varying system and the output exponential stability of its variational system. Subsequently, we derive a sufficient condition for achieving output contraction in time-varying systems by applying the output contraction Lyapunov criterion. Finally, we apply the results to analyze the output exponential stability of nonlinear time-invariant systems. |
Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan Contraction analysis of time-varying DAE systems via auxiliary ODE systems Unpublished 2023, (submitted). @unpublished{YinJaya23ppa, |
Submitted
Model reduction for switched differential-algebraic equations with known switching signal Unpublished 2024, (submitted). |
Impulse-free linear quadratic optimal control of switched differential algebraic equations Unpublished 2024, (provisionally accepted in "Communications in Optimization Theory"). |
Output contraction analysis of nonlinear systems Unpublished 2023, (submitted). |
Contraction analysis of time-varying DAE systems via auxiliary ODE systems Unpublished 2023, (submitted). |