The realm of State Space Systems

These notes, made available at http://users.ntua.gr/chvng/en, cover a wide range of topics on automatic control of linear time invariant systems. We start with an introduction to state space models and offer a rudimentary classification of control systems (Linear Time Invariant, Linear Time Variant, Input Affine and more) and we juxtapose the Laplace transform approach. 

Notes on Automatic Control

Canonical Forms  
The adventure continues with the study of coordinates transformation and in particular linear changes of coordinates. Thus, we introduce the reader to the notion of equivalent systems and we pave the way towards presenting some of the most important realizations of systems such as:

1. Diagonal realization
2. Jordan Canonical Form
3. Controllable Canonical Form
4. Canonical Observable Form

while in the future I might consider offering a few pages to other realizations such as the Hessenberg form. In particular, I have endeavoured to expound in depth the details of the Jordan decomposition that are necessary for the reader to be able to make the most out of this powerful tool.

   Next, we proceed with the study of solutions of LTI control systems. We state without a proof the Caratheodory theorem on existence and uniqueness of solutions of general nonlinear systems and in the sequel we state and prove the lemma of Gronwall and Bellman which we use to derive important results on the continuity of solutions. This way the reader becomes familiar with essential properties of solutions of LTI systems.

Controllability

   The abilities that state space offers us will not be clear until the sixth chapter where we study the controllability of LTI systems. We first exhaust our understanding on the implications of controllability of LTI systems and then we state and prove the well known rank criterion. We learn that controllability is preserved under feedback and changes of coordinates and we introduce the reader to the controllability space of a system. 

In the second part of this chapter we study controllability from a clearly algebraic point of view and we finally arrive at a very important result: the Kalman input-state decomposition of a system. The last part of this chapter is devoted to cyclicity - a purely algebraic notion. Amongst the various results in this chapter we single out one of particular importance: the Wonham theorem which weaves controllability together with cyclicity. according to which we decompose a system into its controllable and uncontrollable parts. We also spare a few pages on the Controllability Gramian.


Stability

   Next, we introduce the reader to the study of stability of equilibrium points. Stability, Asymptotic Stability and Exponential Stability are approached using moth the classical δ-ε definitions and by means of comparison functions (K-class, KL-class, etc). We prove that an LTI system is asymptotically and exponentially stable if its closed loop matrix is Hurwitz and we also explain the impact of zero eigenvalues on the stability of the system. We employ the canonical controllable form in order to design linear feedback laws which place the closed loop poles of the system at desired locations on the complex plane. Our results boil down to Ackerman's formula which offers a concrete (and easily memorizable) solution to the pole placement problem for systems with  a single input. Afterwards we extend our results to multi-input systems. The next section is about stabilizability, a notion weaker than controllability but equivalent to our ability to determine a linear feedback that stabilizes asymptotically the closed-loop system. The next section of this chapter is devoted to the Lyapunov theory of stability which applies to general nonlinear systems. We then examine the implications of Lyapunov's theory on LTI systems.

Throughout the whole text we provide examples where necessary for the reader to familiarize with new notions and savvy certain details of the theory. Certain proofs are left as exercises to the reader (usually come with a hint). All propositions, lemmas and theorems are accompanied by their proofs except if it is not relevant to the scope of our study (e.g. Caratheodory's theorem). Note that this is still a draft version and might contain typos, mistakes and omissions. Feel free to communicate your remarks to the author either by e-mail or commenting here.

Download the text from http://users.ntua.gr/chvng/en/.