The aim of this paper is to connect irreversible thermodynamics and the passivity theory of nonlinear control.
A process system with input \(u\), output \(y\) and internal states \(\overline z\), defined on a domain \({\mathcal V}\) with smooth boundary \({\mathcal B}\) is said to be passive if there exist a nonnegative constant \(\beta\) and a functional \(V: Z\to\mathbb{R}_+\) (\(Z\) is a Banach space) so that \(V(0)= 0\) and for all \(t<\infty\) the following holds \[ V(\overline z(t))- V(\overline z(0))\leq \int^t_0 \langle y,u\rangle_\partial ds- \beta \int^t_0\|\overline z\|^2_{L_2({\mathcal V};\mathbb{R}^n)}ds. \] If \(\beta> 0\) then the system is called strictly passive.
It is shown that a classical non-equilibrium system is passive when the local equilibrium hypothesis and Onsager-Casimir-type relations are used for diffusive/conductive-type phenomena.
A strictly passive system is stable and stable invertible in the following sense: if either or both of \(y\) and \(u\) are equal to zero then the internal states \(\overline z\) converge to a passive state, i.e. a state where \(V(\overline z)= 0\). Sufficient conditions for passivity of process systems and convergence to stationary solutions are given.
The available storage is also defined at the state \(z\) relative to a reference state \(z^*\) as \[ a(z, z^*)= \varepsilon(z)- [\varepsilon(z^*)+ w(z^*)^T(z- z^*)]\geq 0, \] where \(\varepsilon(z):Z\to\mathbb{R}\) is a convex extension, i.e. the symmetric \(n\times n\) matrix \(M\) with elements \(M_{ij}= \partial^2\varepsilon(z)/\partial z_i \partial z_j\) is positive definite, and \(w\) is the directional derivative of \(\varepsilon\) so that \(w^T= \partial_z\varepsilon\).
The storage function is derived from the convexity of the entropy and is closely related to the thermodynamic availability. The authors present a new thermodynamic potential that can be related to the thermodynamic availability.
It is shown also that chemical processes described by diffusion and heat conduction are dissipative. Such processes are therefore open loop stable. Any chemical process can be stabilized by distributed control provided that the sensor and actuator locations are suitable.