Optimisation and adaptation of synchronisation controllers for networked second-order infinite-dimensional systems

ABSTRACT

The adaptation and optimisation of synchronisation control for networked second-order distributed parameter systems are considered. The objective is to design output feedback controllers guaranteeing agreement between the states of the N systems that individually track a reference model. The controller structure involves feedback terms consisting of the pairwise difference of the measurements of the second-order systems as well as coupling terms that enforce consensus. Placing the closed-loop networked systems in aggregate form allows for further optimisation of the synchronisation gains. Using the aggregate systems ‘closed-loop energy’ as a suitable optimisation measure, the search for the synchronisation gains reduces to the minimisation of the optimisation index, which eventually is described by the trace of the solution to a parameterised Lyapunov operator equation. Considering the adaptation of the synchronisation gains offers an alternative to optimisation. The adaptation is based on Lyapunov redesign-methods and utilises a parameter-dependent Lyapunov functional to extract them. Due to its structure, all network topology information is handled at the local level, thereby relaxing the graph topology conditions in the adaptive case. Numerical studies are included to provide an insight on the effects of synchronisation control of networked second-order distributed parameter systems.

KEYWORDS:

• Second-order infinite-dimensional systems
• spatially distributed controllers
• adaptive control
• synchronisation

Notes

1. Following Demetriou (2004b, 2009), the parameter γ must satisfy $$\begin{array}{c}\hfill {\displaystyle \gamma >max\left\{1,c_1^2\frac{1+{\lambda }_{\ell }}{{\kappa }_{\ell }},\frac{c_2^2}{{\delta }_{\ell }-c_2^2{\lambda }_{\ell }}\right\}\text{or}\gamma >\frac{c_2^2}{{\delta }_{\ell }-c_2^2{\lambda }_{\ell }},}\end{array}$$to ensure that $$\begin{array}{c}\hfill \begin{array}{c}min\{\gamma {\kappa }_{\ell }-c_1,\gamma -1,1\}\left({\parallel E\parallel }_{{\mathbb{V}}_1}^2+|\dot{E}{|}_{\mathbb{H}}^2+{\left|{\mathcal{L}}_p\right|}_{\Theta }^2\right.\hfill \\ +\left.|{\mathcal{L}}_v{|}_{\Theta }^2\right)\le V\le max\{\gamma +1,\gamma {\kappa }_u+c_1^2+{\delta }_u{c}_{12}^2\}\hfill \\ \times \left({\parallel E\parallel }_{{\mathbb{V}}_1}^2+|\dot{E}{|}_{\mathbb{H}}^2+|{\mathcal{L}}_p{|}_{\Theta }^2+{\left|{\mathcal{L}}_v\right|}_{\Theta }^2\right)\hfill \end{array}\end{array}$$and $$\begin{array}{ccc}\hfill \dot{V}& \le & -2min\left\{\gamma \frac{{\delta }_{\ell }}{c_2^2}-\gamma {\lambda }_{\ell }-1,{\kappa }_{\ell },{\sigma }_p,{\sigma }_v\right\}\hfill \\ & & \times \left({\parallel E\parallel }_{{\mathbb{V}}_1}^2+|\dot{E}{|}_{\mathbb{H}}^2+|{\mathcal{L}}_p{|}_{\Theta }^2+{\left|{\mathcal{L}}_v\right|}_{\Theta }^2\right).\hfill \end{array}$$