Controlled and conditioned invariant varieties for polynomial control systems
- Bedingt regelungsinvariante Varietäten für polynomielle Kontrollsysteme
Schilli, Christian; Zerz, Eva Barbara (Thesis advisor); Walcher, Sebastian (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2016
The main goal of this thesis is the generalisation of the notion of “controlled and conditioned invariant subspaces for linear control systems”, introduced by G. Basile and G. Marro in the late sixties. In view of this, we mostly treat input-afﬁne control systems with output, which are deﬁned over a commutative, multivariate, polynomial ring with real or complex ground ﬁeld. A given variety is called “controlled invariant” for such a system if we can ﬁnd a feedback law that causes the closed loop system to have this variety as an invariant set, i.e. all trajectories that start on the variety remain there for all time. Several approaches for the feedback law are made, namely polynomial and rational state feedback as well as polynomial and rational output feedback. If it is indeed possible to ﬁnd an output feedback which makes the variety invariant for the closed loop system, then we call the variety “controlled and conditioned invariant”.The present work begins by giving some mathematical foundation, introducing basic deﬁnitions and results of ordinary differential equations, algebraic geometry and the theory of Gröbner bases. We develop computer algebraic methods, for instance for the determination of the intersection of an afﬁne module over a polynomial ring with a free module over a subalgebra of this ring or of a fractional module with a vector space, which help us to decide the properties described above for a given control system and a variety.One crucial object in the context of this thesis is the set of polynomial vector ﬁelds which leaves a variety invariant. The elements of this set are called polynomial vector ﬁelds on the variety. In fact, this set has a module structure over the considered polynomial ring as a characterisation of the invariance of a variety for a polynomial vector shows, which also gives rise to an algorithmic approach for ﬁnding a ﬁnite generating system of this module. Furthermore, we investigate the structure of this module: Some submodules will be derived, relations between these submodules as well as conditions on which they already coincide with the whole set of polynomial vector ﬁelds on the variety. Moreover, the module of polynomial vector ﬁelds on a variety helps us to compare our notion with one made by A. Isidori in the nineties, called distributional invariance, and to characterise the invariance of a variety even for rational vector ﬁelds.From this point on, it is easy to ﬁnd an equivalent condition for a variety being controlled invariant for a polynomial control system with polynomial state feedback, in terms of the given control matrix and the module of polynomial vector ﬁelds. We may use techniques from the theory of Gröbner bases to check this condition and, in the afﬁrmative case, to derive the set of all polynomial state feedback making the variety invariant. It turns out that this set is an afﬁne module over the polynomial ring. In view of controlled and conditioned invariance, one has to decide if the intersection of this set with a free module over the subalgebra generated by the individual components of the output of the system is non-empty. The above mentioned algorithms will do this task. Similar considerations can be done to ﬁnd methods to decide the controlled (and conditioned) invariance of a variety even for rational systems with rational state/output feedback.