Stabilization of distributed systems using irreversible thermodynamics (Q5950248)

The aim of this paper is to connect irreversible thermodynamics and the passivity theory of nonlinear control.NEWLINENEWLINENEWLINEA 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 NEWLINE\[NEWLINEV(\overline z(t))- V(\overline z(0))\leq \int^t_0 \langle y,u\rangle_\partial ds- \beta \int^t_0\|\overline z\|^2_{L_2({\mathcal V};\mathbb{R}^n)}ds.NEWLINE\]NEWLINE If \(\beta> 0\) then the system is called strictly passive.NEWLINENEWLINENEWLINEIt 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.NEWLINENEWLINENEWLINEA 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.NEWLINENEWLINENEWLINEThe available storage is also defined at the state \(z\) relative to a reference state \(z^*\) as NEWLINE\[NEWLINEa(z, z^*)= \varepsilon(z)- [\varepsilon(z^*)+ w(z^*)^T(z- z^*)]\geq 0,NEWLINE\]NEWLINE 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 \(w^T= \partial_z\varepsilon\).NEWLINENEWLINENEWLINEThe 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.NEWLINENEWLINENEWLINEIt 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.