Zsolt Péter Biró

All controllers for an LPV robust control problem

Abstract As an extension of the robust H ∞ method, the time domain design based on linear matrix inequalities (LMI) presented in Scherer [2001] is an appealing and conceptually simple framework to obtain robust LPV controllers. This paper provides a numerical reliable method to compute and parametrize all LPV controllers that corresponds to a given multiplier matrix.

Möbius transform and efficient LPV synthesis

As an extension of the robust H∞ method, the time domain design based on linear matrix inequalities (LMI) presented in [10] is an appealing and conceptually simple framework to obtain robust LPV controllers, however, the resulting controller is not always suitable for implementation. This paper shows that in a variant of the S-procedure how the multiplier is transformed if the parameter domains are related through a Möbius transform. Based on this result, if certain conditions are fulfilled, the paper provides a numerical reliable method to compute and parametrize all the scheduling variables of the LPV controllers that correspond to a given multiplier matrix and can be expressed in terms of the scheduling variables of the plant.

Quadratic separators in the input-output setting

The paper provides a formulation of the quadratic separator result in a fairly general input-output operator framework and it focuses on the specific points that are different compared to the usual finite dimensional formulations. The major contribution of the paper is to emphasize the role of the causality in obtaining the stability result having a quadratic separator and the fact that the plant and the boundedness constant in the formulation of the well-posedness condition determine the robust stability domain associated to the given plant, i.e., the set of those systems {K} that stabilizes the given plant and produce the given “disturbance rejection” level γ Concerning the causality issue the necessity of the homotopy argument in the IQC results is also explained and its relation with a general result concerning nest algebras is pointed out.

Elimination lemma and contractive extensions

The Elimination lemma provides analysis conditions that constitute a fundamental tool in obtaining state space solutions of robust control design problems. Using operator theoretical concepts, the paper provides a derivation of the Elimination lemma based on a contractive extension principle and to give a Krein space geometric interpretation of the result. In this way the statement of the lemma is related to general techniques, like interpolation and commutant lifting theory. This result is considered to be an initial step in the control relevant formulation of the lemma in an infinite dimensional setting.

Controller Scheduling Blocks in Non-Conservative LPV Design

As an extension of the robust H∞ theory, the time domain design based on linear matrix inequalities (LMI) presented in Scherer [2001] is a conceptually simple framework to obtain LPV controllers. However the constructed scheduling variables are not suitable for an efficient implementation, in general. This paper investigates the possibility of constructing the scheduling block of a qLPV controller explicitly, i.e., in the form of a linear fractional transformation (LFT). It is shown that if both the primary and dual multiplier LMI equations leads to maximal indefinite subspaces and a coupling condition holds, then the problem is solvable. If it is the case, a constructive algorithm was also provided to build the desired scheduling variables.

Numerical interface curves for some nonlinear diffusion equations

An initial value problem for the generalized Kolmogorov-Petrowsky-Piscunov (nonlinear degenerate reaction-diffusion) equation is studied numerically by the help of a slightly modified finite difference scheme of Douglas-Yanenko-Mimura type. If the initial function has compact support, the solution also will have compact support and an interface appears between the domains where the solution is positive and where it is zero. We present some examples for different parameter values where the numerical solution as well as numerical interfaces behave according to the analytical theory.