Disturbance decoupling for singulars systems by proportional and derivate feedback and proportional and derivate outputm injection

We study the disturbance decoupling problem for linear time invariant singular systems. We give necessary and su±cient conditions for the existence of a solution to the disturbance decoupling problem with or without stability via a proportional and derivative feedback and proportional and derivative output injection that also makes the resulting closed-loop system regular and/or of index at most one. All results are based on canonical reduced forms that can be computed using a complete system of invariants that can be implemented in a numerically stable way.Postprint (published version

Asymptotic control theory for a system of linear oscillators

We present an asymptotic control theory for a system of an arbitrary number of linear oscillators under a common bounded control. We suggest a design method of a feedback control for this system. By using the DiPerna-Lions theory of singular ODEs, we prove that the suggested control law correctly defines the motion of the system. The obtained control is asymptotically optimal: the ratio of the motion time to zero under this control to the minimum one is close to 1 if the initial energy of the system is large. The results are partially based on a new perturbation theory of observable linear systems.Comment: 34 pages; published versio

Rigid systems of second-order linear differential equations

We say that a system of differential equations d^2x(t)/dt^2=Adx(t)/dt+Bx(t)+Cu(t), in which A and B are m-by-m complex matrices and C is an m-by-n complex matrix, is rigid if it can be reduced by substitutions x(t)=Sy(t), u(t)=Udy(t)/dt+Vy(t)+Pv(t) with nonsingular S and P to each system obtained from it by a small enough perturbation of its matrices A,B,C. We prove that there exists a rigid system if and only if m<n(1+square_root{5})/2, and describe all rigid systems.Comment: 22 page

Geometric reduction in optimal control theory with symmetries

A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of system. Then, Noether's theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in the regular and singular cases, which are previously described.Comment: 24 pages. LaTeX file. The paper has been reorganized. Additional comments have been included in Section 3. The example in Section 5.2 has been revisited. Some references have been adde

Contact systems and corank one involutive subdistributions

We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on Jn(R,Rm) in a way analogous to that how Kumpera-Ruiz normal form generalizes the canonical contact system on Jn(R,R), which is also called Goursat normal form.Comment: LaTeX2e, 29 pages, submitted to Acta applicandae mathematica