Control studies of time-delayed dynamical systems with the method of continuous time approximation

Optimal feedback control strategies for periodic delayed systems

“…is originally controllable as shown in [11]. Practically, this is not a problem since all we require is to control the current state y 1 (t) = x(t) and not y i (t), i = 2, .…”

“…The matrices A 1 , A 2 , and B in Eq. (1) take the form of In this strategy, which extends the delayed feedback control strategies in [2,11,15] to include periodic control gains, the stability of the closed-loop response may be investigated in the parameter space of available control gains using Floquet theory, including the optimization of the control gains by minimizing the spectral radius of the monodromy operator which must be less than unity for asymptotic stability. We note that rather than obtaining the monodromy matrix associated with the closed-loop system of Eqs.…”

“…These include the Chebyshev spectral continuous time approximation (CSCTA) and reduced LyapunovFloquet transformation (RLFT). Other methods that convert a system of DDEs to a non-delayed system of ODEs are finite differences, which was employed for feedback control of delayed systems in [11], and the Galerkin-projection [12] technique. By using the CSCTA technique [13], a periodic DDE is transformed into a system of periodic ODEs using Chebyshev spectral collocation where the dimension of the resulting system depends on the number of collocation points used.…”

“…In the first strategy, which extends the constant-gain delayed feedback control strategy in [2,11,15], delayed state feedback control with discrete delay is implemented and the stability of the closed-loop response is investigated in the parameter space of available control gains using Floquet theory, including the optimization of the control gains for minimum spectral radius of the closed-loop response. In the second strategy, optimal (time-varying LQR) delayedfeedback control with distributed delay is used with the CSCTA-approximated system by directly solving the periodic Riccati equation (PRE) backwards in time or by solving the corresponding algebraic Riccati equation (ARE) from which the periodic control gains are obtained.…”

Delay, state, and parameter estimation in chaotic and hyperchaotic delayed systems with uncertainty and time-varying delay

“…The main idea behind CTA is that the infinite-dimensional vectors Y (t) and G (Y (t) , t, a) and the operator A(t) can be approximated by finite-dimensional ones. Further information on the relation between the domain and the spectrum of the solution operator can be found in [30][31][32][33][34]. As shown and discussed in [35], spectral differentiation has a major advantage over finite difference differentiation [32,34] in its "spectrally accurate" exponential convergence characteristics.…”