Parameter-dependent Constraints Research Articles

Stable model reduction for linear variational inequalities with parameter-dependent constraints

We consider model reduction for linear variational inequalities with parameter-dependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameter-dependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-disks with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry.

Gain‐scheduled constrained state feedback control of DC–DC buck power converter

A gain-scheduled constrained state feedback controller (SFC) for a SiC MOSFET direct current (DC)-DC buck power converter with a non-synchronous control strategy is presented. First, the synthesis process of a gain-scheduled SFC for a control system with a state-dependent parameter is proposed. Next, constraints of important states and control variables are a posteriori introduced into control algorithm to provide satisfactory dynamic behaviour and safe operation of the plant. The boundary values of control signals which provide permissible values of the future state-space variables are derived from the models of power converters that take into account two modes of operation (i.e. charging and discharging mode). The novelty of the proposed approach lies in efficient control algorithm that utilises the gain-scheduled SFC with the parameter-dependent constraint handling method. The designed control algorithm is extensively tested through simulation and experiments at a switching frequency of 36 kHz. The proposed solution is further compared with a conventional cascade control structure based on two proportional integral controllers for voltage and current loops as well as with a non-constrained state feedback control structure to illustrate its performance and low computational complexity.

Preconditioners for Saddle Point Systems with Trace Constraints Coupling 2D and 1D Domains

We study preconditioners for a model problem describing the coupling of two elliptic subproblems posed over domains with different topological dimension by a parameter dependent constraint. A pair of parameter robust and efficient preconditioners is proposed and analyzed. Robustness and efficiency of the preconditioners is demonstrated by numerical experiments.

An Asymptotically Exact Approach to Robust Semidefinite Programming Problems with Function Variables

This technical note provides an approximate approach to a semidefinite programming problem with a parameter-dependent constraint and a function variable. This problem covers a variety of control problems including a robust stability/performance analysis with a parameter-dependent Lyapunov function. In the proposed approach, the original problem is approximated by a standard semidefinite programming problem through two steps: first, the function variable is approximated by a finite-dimensional variable; second, the parameter-dependent constraint is approximated by a finite number of parameter-independent constraints. Both steps produce approximation error. On the sum of these approximation errors, this technical note provides an upper bound. This bound enables quantitative analysis of the approach and gives an efficient way to reduction of the approximation error. Moreover, this technical note discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem.

Reorganization of the configuration manifold and critical systems

When the dynamics of holonomic mechanical systems are described in redundant coordinates with parameter-dependent constraints, some special features arise. At certain (critical) values of the parameters, the configuration manifold of the system may have, for example, self-intersections; in the neighbourhood of such a singular submanifold one cannot introduce Lagrangian coordinates of the system and it becomes difficult (even impossible) to define the orbits of the “critical” system correctly by taking limits with respect to the parameter. In such cases the classical Poincaré theory of bifurcation of equilibria in conservative systems also needs some adjustment. As an example, a crank-gear mechanism is considered.

Hopf bifurcation with constraints

IN THIS study we give conditions for the existence of periodic orbits of a strongly continuous semiflow which branch off from a steady state for a certain parameter value. The semiflow is defined in a closed subset of a Banach space which contains the steady state. The steady state is kept fixed whereas both the semiflow and its domain may depend strongly continuously on the parameter. It will be shown that an asymptotic @-contraction hypothesis together with a transversality and no-resonance hypothesis for the eigenvalues of the infinitesimal generator of the linearized semiflow imply Hopf bifurcation. The technique consists in a generalized Lyapunov-Schmidt decomposition method using spectral projections which depend strongly continuously on the bifurcation parameter. The existence of a continuum of bifurcating periodic orbits follows from a degree argument. The theory can be applied to partial functional differential equations when the bifurcation parameter changes transformed arguments or the initial or boundary values underly parameter-dependent constraints. In general such semiflows do not exhibit the smoothness hypotheses which are necessary for the application of available Hopf bifurcation theorems. There are. a lot of such theorems either for special types of functional differential equations, partial differential equations, Volterra-integral equations, or for a general semiflow in Banach space but under considerable stronger hypotheses than in the present paper. The reader is referred to [l], [2], [4], [5], [7]-[12], [15] and to the references therein. Of course the stronger hypotheses in these papers allow to get more detailed information on the structure of the branch of the bifurcating periodic orbits such as smootheness with respect to the parameter, local stability, and the direction of bifurcation. In contrast to them the hypotheses in our paper guarantee only local existence and they are too weak to ensure smoothness of the branch nor the existence of a center manifold. But even the local existence problem was still unsolved if the semiflow and its domain depend merely strongly continuously on the parameter. The difficulty is due to the fact that the usual argument basing on the implicit function theorem will not work. Some examples which demonstrate the application of our theory will be given in a subsequent paper. Before we will explain the main hypotheses we shall introduce some notations. B denotes a real Banach space and C C Iw’ a compact neighborhood of 0 E R*, the domain of the bifurcation-parameter. Norms and the modulus of a complex number are always indicated by the symbol 1.1. To every c E C we join a closed subset X’ C B, the domain of the semiflow.