When does the algebraic Riccati equation have a negative semi-definite solution?

Find a reasonable necessary and sufficient frequency domain condition, i.e, a condition in terms of the rational matrix ∂W, or possibly in terms of the two-variable rational matrix W, for the existence of a real symmetric negative semi-definite solution of the algebraic Riccati equation

Strong Structural Controllability and Zero Forcing

In this chapter, we study controllability and output controllability of systems defined over graphs. Specifically, we consider a family of state-space systems, where the state matrix of each system has a zero/non-zero structure that is determined by a given directed graph. Within this setup, we investigate under which conditions all systems in this family are controllable, a property referred to as strong structural controllability. Moreover, we are interested in conditions for strong structural output controllability. We will show that the graph-theoretic concept of zero forcing is instrumental in these problems. In particular, as our first contribution, we prove necessary and sufficient conditions for strong structural controllability in terms of so-called zero forcing sets. Second, we show that zero forcing sets can also be used to state both a necessary and a sufficient condition for strong structural output controllability. In addition to these main results, we include interesting results on the controllability of subfamilies of systems and on the problem of leader selection.</p

Implementation of behavioral systems

In this chapter, we study control by interconnection of a given linear differential system (the plant behavior) with a suitable controller. The problem formulations and their solutions are completely representation free, and specified only in terms of the system dynamics. A controller is a system that constrains the plant behavior through a certain set of variables. In this context, there are two main situations to be considered: either all the system variables are available for control, i.e., are control variables (full control) or only some of the variables are control variables (partial control). For systems evolving over a time domain (1D) the problems of implementability by partial (regular) interconnection are well understood. In this chapter, we study why similar results are not valid in themultidimensional (nD) case. Finally, we study two important classes of controllers, namely, canonical controllers and regular controllers