Passivity is a particular instance of dissipativity, a system-theoretic property that relates different notions of stability (see the diagram).
Within a passivity framework, the control problem is approached by
using the notion of energy-shaping as the main design principle and
exploiting the physical properties of the plant.
The basic steps are:
Derive an energy-based model (e.g., Euler-Lagrange, port-Hamiltonian, Brayton-Moser, etc). Energy-based models are simpler and intuitive; they also provide a dissipation inequality that can be used as a starting point.
Using the energy conservation property, design a controller that “shapes” the system's energy as required. (The requirement of having a physical interpretation of the controller usually simplifies this task.)
Add damping to improve the transient response.
It is well known that many mechanical and electro-mechanical systems can be described using port-Hamiltonian models. Besides, a large class of RLC nonlinear circuits can also be put in port-Hamiltonian form, provided their graphs satisfy some conditions [CJOGC09]. This class of systems can be asymptotically stabilized with a nonlinear proportional-integral controller [JOGCC07]. Implicit port-Hamiltonian models are developed in [CGHM13], and the interconnection and damping assignment (IDA) control technique is developed for implicit port-Hamiltonian models in [CG16]. A linear matrix inequality framework for under-actuated mechanical systems is presented in [CCR19].
Energy shaping can be accomplished via standard passivity-based techniques like energy balance [CO09] or interconnection and damping assignment [OvdSCA08, CG15]. Alternatively, one may reformulate the control problem in a control-by-interconnection setting [COvdSA09], where the original system and the controller are viewed as energy ports that are interconnected to produce the desired behavior.
Jan Willems' behavioral approach to systems and control is particularly well suited for developing passivity-based control theory. The results presented in the thesis are stated within the behavioral framework. Romeo Ortega supervised the thesis.
Passivity-based control schemes usually take advantage of the following property: the interconnection of two passive systems is again passive. This property ceases to hold when delays are part of the interconnection structure (e.g., as in the case of two systems that are interconnected employing a communication channel).
A linear transformation can be applied to the Telegrapher's equations (which define a passive system) in order to obtain a pair of uncoupled transport equations (which define a pair of delays). In the context of robot telemanipulation, Anderson and Spong proposed to modify a communication channel, modeled as a pair of delays, by applying the inverse transformation, so that the transformed channel is passive and the overall interconnected system is again passive.
In the context of set-point regulation, Anderson's transformation performance is evaluated in terms of achievable σ-stability. Using a frequency domain approach, analytic formulae for optimal control parameters are derived [CEMR18].
The renowned robustness of sliding-mode control can be understood in terms of the signum function's multivalued nature. According to Filippov's theory, robustness originates from the fact that a signum function appearing on the right-hand side of a differential equation takes values on the power set of the real line (as opposed to the real line itself). It is shown in [MC17, MC15, MC14] that it is possible to achieve similar (or sometimes better) robust controllers with the use of other multivalued functions. In the references mentioned above, the robust output regulation of passive linear systems is achieved using set-valued, maximally monotone passive controllers (see [MCB25a] as well). The results are extended to nonlinear Lagrangian systems in [MBC17a] and formulated in the port-Hamiltonian framework in [Cas22].
There are systems that possess a local but non-global port-Hamiltonian representation, non-globality being a consequence of the phase-space topology. Locally port-Hamiltonian systems may transgress the dissipation inequality satisfied by their global counterparts, but far from being a defect, non-globality extends the modeling power of port-Hamiltonian systems by providing a framework to model cyclic physical processes for energy conversion and extraction [CG21]. An example of such a physical process is the Stirling engine described in [GC20].
