Stegink, T.W. (2014) Robust synchronization of multiplicatively perturbed multi-agent systems and an LMI-based approach to the robust stabilization problem. Master's Thesis / Essay, Mathematics.
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Abstract
The main topic of this master thesis is robust synchronization of uncertain multi-agent systems using observer based protocols. For a given network where the dynamics of each agent is the same, we consider multiplicative perturbations on the transfer matrix of the agents. These perturbations are assumed to be stable and bounded in H-infinity norm by some a priori given tolerance. The problem of robust synchronization is to synchronize the network for all perturbations that are bounded by this tolerance. It is shown that a protocol achieves robust synchronization if and only if all controllers in a finite set of observer based controllers robustly stabilize a given, single linear system. A solution to this problem is given for the case of undirected network graphs and heterogeneous perturbations on the agents. Furthermore a similar solution is given for the case of directed graphs and homogeneous perturbations. For both cases, robustly synchronizing protocols are expressed in terms of the solutions of certain algebraic Riccati equations. It will be shown that an upper bound for the guaranteed achievable tolerance of the perturbations is given in terms of the spectral radius of the solutions of these Riccati equations and in terms of the ratio between the second smallest and the largest eigenvalue of the Laplacian matrix. The second part of this thesis consists of an LMI-based approach to the H-infinity control problem and an application of this theory to the robust stabilization problem. Necessary and sufficient conditions for the solvability of the robust stabilization problem are established and are expressed in terms of the solvability of certain linear matrix inequalities (LMI's). An algorithm is provided to compute controllers that solve the H-infinity control problem for any given tolerance. The connection of the H-infinity control problem and the robust stabilization problem is made via the small-gain theorem. In the robust stabilization problem also the special well-known cases of additive, coprime factor and multiplicative perturbations are analyzed. In these cases, necessary and sufficient conditions for the solvability of the robust stabilization problem are given in terms of the solvability of algebraic Riccati equalities and inequalities. Furthermore, a condition for the maximum achievable tolerance can be isolated and is expressed in terms of the spectral radius of the solutions of the Riccati (in)equalities.
| Item Type: | Thesis (Master's Thesis / Essay) |
|---|---|
| Degree programme: | Mathematics |
| Thesis type: | Master's Thesis / Essay |
| Language: | English |
| Date Deposited: | 15 Feb 2018 07:57 |
| Last Modified: | 15 Feb 2018 07:57 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/11942 |
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