## PID Passivity-Based Control: Application to Energy and Mechanical Systems

Motivated by current practice, in this course we explore the possibility of applying theindustry-standard PID controllers to regulate the behavior of nonlinear systems. As is wellknown, PID controllers are highly successful when the main control objective is to drive a given output signal to a constant value. PIDs, however, have two main drawbacks, first, the task of tuning the gains is far from obvious when the systems operating region is large; second, in some practical applications the control objective cannot be captured by the behaviour of output signals. We show that, for a wide class of physical systems, these two difficulties can be overcome exploiting the property of passivity of the system.

Passivity is a fundamental property of dynamical systems, which in the case of physical

systems captures the universal feature of energy conservation. It is well-known that PID

controllers are passive systems—for all positive PID gains—and that the feedback interconnection of two passive systems is stable. Therefore, wrapping the PID around a passive output trivialises the gain tuning task. Clearly, the first step in the design is to identify all passive outputs of the system. It turns out that this task is achievable for a large class of physical systems described by port-Hamiltonian models.

In many applications the desired values for the outputs are different from zero, whence

the PID is wrapped around the error signal. In this case, it is necessary to investigate

whether the system is passive with respect to this error signal—a property called passivity

of the incremental model, which is studied in the course.

If the control objective is to stabilize (in the Lyapunov sense) a constant equilibrium

it is necessary to build a Lyapunov function. In the course we identify—via some easily

verifiable integrability conditions—a class of systems for which this more ambitious objective

is achieved.