Parameter Estimation of Sigmoid Superpositions: Dynamical System Approach

Superposition of sigmoid function over a finite time interval is shown to be equivalent to the linear combination of the solutions of a linearly parameterized system of logistic differential equations. Due to the linearity with respect to the parameters of the system, it is possible to design an effective procedure for parameter adjustment. Stability properties of this procedure are analyzed. Strategies shown in earlier studies to facilitate learning such as randomization of a learning sequence and adding specially designed disturbances during the learning phase are requirements for guaranteeing convergence in the learning scheme proposed.Comment: 30 pages, 7 figure

Leaders do not look back, or do they?

We study the effect of adding to a directed chain of interconnected systems a directed feedback from the last element in the chain to the first. The problem is closely related to the fundamental question of how a change in network topology may influence the behavior of coupled systems. We begin the analysis by investigating a simple linear system. The matrix that specifies the system dynamics is the transpose of the network Laplacian matrix, which codes the connectivity of the network. Our analysis shows that for any nonzero complex eigenvalue $\lambda$ of this matrix, the following inequality holds: $\frac{|\Im \lambda |}{|\Re \lambda |} \leq \cot\frac{\pi}{n}$. This bound is sharp, as it becomes an equality for an eigenvalue of a simple directed cycle with uniform interaction weights. The latter has the slowest decay of oscillations among all other network configurations with the same number of states. The result is generalized to directed rings and chains of identical nonlinear oscillators. For directed rings, a lower bound $\sigma_c$ for the connection strengths that guarantees asymptotic synchronization is found to follow a similar pattern: $\sigma_c=\frac{1}{1-\cos\left( 2\pi /n\right)}$. Numerical analysis revealed that, depending on the network size $n$, multiple dynamic regimes co-exist in the state space of the system. In addition to the fully synchronous state a rotating wave solution occurs. The effect is observed in networks exceeding a certain critical size. The emergence of a rotating wave highlights the importance of long chains and loops in networks of oscillators: the larger the size of chains and loops, the more sensitive the network dynamics becomes to removal or addition of a single connection

Adaptive observers for nonlinearly parameterized systems subjected to parametric constraints

We consider the problem of adaptive observer design in the settings when the system is allowed to be nonlinear in the parameters, and furthermore they are to satisfy additional feasibility constraints. A solution to the problem is proposed that is based on the idea of universal observers and non-uniform small-gain theorem. The procedure is illustrated with an example.Comment: 19th IFAC World Congress on Automatic Control, 10869-10874, South Africa, Cape Town, 24th-29th August, 201