Editing Dissipative system (section)

== Dissipative systems in control theory ==
[[Jan Camiel Willems|Willems]] first introduced the concept of dissipativity in systems theory<ref>{{cite journal |last1=Willems |first1=J.C. |title=Dissipative dynamical systems part 1: General theory |journal=Arch. Rational Mech. Anal. |date=1972 |volume=45 |issue=5 |page=321 |doi=10.1007/BF00276493 |bibcode=1972ArRMA..45..321W |hdl=10338.dmlcz/135639 |s2cid=123076101 |url=https://homes.esat.kuleuven.be/~sistawww/smc/jwillems/Articles/JournalArticles/1972.1.pdf }}</ref> to describe dynamical systems by input-output properties. Considering a dynamical system described by its state <math> x(t) </math>, its input <math>u(t)</math> and its output <math>y(t)</math>, the input-output correlation is given a supply rate <math> w(u(t),y(t))</math>. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function <math> V(x(t))</math> such that <math>V(0)=0</math>, <math>V(x(t))\ge 0 </math> and

:<math> \dot{V}(x(t)) \le w(u(t),y(t))</math>.<ref>{{cite book |last1=Arcak |first1=Murat |last2=Meissen |first2=Chris |last3=Packard |first3=Andrew |title=Networks of Dissipative Systems |date=2016 |publisher=Springer International Publishing |isbn=978-3-319-29928-0 }}</ref>

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate <math> w(u(t),y(t)) = u(t)^Ty(t) </math>.

The physical interpretation is that <math>V(x)</math> is the energy stored in the system, whereas <math>w(u(t),y(t))</math> is the energy that is supplied to the system.

This notion has a strong connection with [[Lyapunov stability]], where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by [[Vasile M. Popov|V.M. Popov]], [[Jan Camiel Willems|J.C. Willems]], D.J. Hill, and P. Moylan. In the case of linear invariant systems{{clarify|reason=Is this the same as a "linear time-invariant system" as in the Wikipedia articles "LTI system theory"?|date=April 2015}}, this is known as positive real transfer functions, and a fundamental tool is the so-called [[Kalman–Yakubovich–Popov lemma]] which relates the state space and the frequency domain properties of positive real systems{{clarify|reason=What is a positive real system?|date=April 2015}}.<ref>{{cite book|url=https://www.springer.com/978-1-84628-892-0|title=Process Control - The Passive Systems Approach| last1=Bao| first1=Jie| last2=Lee| first2=Peter L.| author-link2=Peter Lee (engineer)| publisher=[[Springer Science+Business Media|Springer-Verlag London]]|year=2007|doi=10.1007/978-1-84628-893-7|isbn=978-1-84628-892-0}}</ref> Dissipative systems are still an active field of research in systems and control, due to their important applications.