Have fun with Automatic Control

I recently kicked off a new project on the design of a Virtual Lab on Automatic Control. Some first experiments are already running online at vlab.mooo.info. Visitors of the web site will be able to play and experiment with closed and open loop systems involving a PID controller and a dynamical system with a time-delay. I have endeavoured to provide full configurability of the framework and allow the user to explore in depth the dynamics of a PID controller in a closed loop. The Bode and Nyquist Diagrams for the open loop system can be generated as well. The project is still in alpha phase (testing) but soon it will be released with more running virtual experiments (with animation and surprises). So... stay tuned (follow @isToxic on twitter for more news).

Notes on Set Convergence

Sequences of sets over a topological space define analogous convergence and boundness notions as sequences of vectors or numbers. Set convergence in the Painleve-Kuratowski sense provides a rigorous framework for the study of sequences of sets and the basis for the study of convergence properties of sequences of extended-real-valued functions and multi-valued mappings. In these notes we define the notions of inner and outer limits and we characterize them in the light of the underlying topology.

When Lyapunov's Stability Theorem is not enough...

There are cases where Lyapunov's stability theory is not enough in the sense that it can prove merely stability for equilibrium points that are actually asymptotically stable (after days and nights struggling to find a Lyapunov function for our system). The pendulum with linear friction term is the prime example of such a case: The total energy function of the system is used as a Lyapunov function. However, this absolutely natural choice fails to provide the expected asymptotic stability result. At that point La Salle's invariance principle chimes in to accommodate the aforementioned theoretical shortage.

Lyapunov on a stamp

Note on La Salle's invariance principle: 

Download the PDF file from http://users.ntua.gr/chvng/ac2/LaSalle_Invariance_Principle.pdf 
Read also: The Realm of State Space Systems

PBPK modelling and Predictive Control

Traditionally, drug administration scheduling is – at best – designed using average population data such as PK/PD profiles. This common practice yields suboptimal therapies and does not consider the distinct attributes of the patients treating them as a bulk. Outliers of the general distribution are likely to exhibit adverse effects due to violation of toxicity constraints or fail to retain the therapeutic levels. The lack of any feedback contributes even more to the probability of something going wrong. Nowadays, the grounds have shifted and the need for accuracy and efficiency calls for closed-loop practices introducing thus automatic control into the field. Computer-aided drug administration will provide an optimal solution to the problem, with the guarantee that all safety requirements are fulfilled.

The Frobenius Theorem

The Frobenius theorem is of exceptional importance both from the point of view of differential geometry and differential topology but also from the Automatic Control and Nonlinear Systems analysis and design point of view. This text aims to be a self-contained report on the Frobenius theorem. First, some necessary definitions are given and basic facts about distributions are stated. The dual objects of distributions - the codistributions are introduced. We try to describe these notions and engage them to involutiveness and complete integrability, to state and prove the Frobenius Theorem which is of great importance in nonlinear control theory.

Ferdinand Georg Frobenius

Inequalities on Probability Spaces

This essay is concerned with the study of inequalities on probability spaces. Mathematical structures such as equalities incorporate an expliteness that is not consistent with the vagueness that characterizes probabilistic and real phenomena. Such inequalities are a powerful tool for a mathematician, an engineer and everyone dealing with probability theory, statistics, measure spaces, information theory and many other scientific fields.

State observer design for nonlinear systems

The Frobenius theorem of differential geometry provides necessary and sufficient conditions for a distribution to be completely integrable. This has important implications on the design of controllers and observers for nonlinear systems. In this presentation we go through all necessary definitions in order to state the Frobenius theorem. These are the notions of a Lie bracket, distribution, involutive and completely integrable distribution, codistribution and annihilator of a distribution. We then state the Frobenius theorem and explain its logical depth.

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