Estimation and control in time-delayed dynamical systems using the chebyshev spectral continuous time approximation and reduced liapunov-floquet transformation

In this chapter, a detailed description is provided of the authors’ recent work on estimation and control of retarded time-delayed systems with (possibly) timeperiodic coefficients using the Chebyshev spectral continuous time approximation (CSCTA) and reduced Liapunov-Floquet transformation (RLFT). First, these specialized computational tools are outlined and illustrated with several examples including various forms of the delayed Mathieu equation with single and multiple discrete delays as well as discontinuous distributed delay. Specifically, it is demonstrated how a time-periodic linear delayed system may be transformed into a set of constant-coefficient ODEs of low dimension in which the eigenstructure (including the dominant characteristic exponents of the delayed system) is exactly preserved in the transformed domain. Next, the control problem of (possibly time-periodic) time-delayed systems is explored, in which three strategies are suggested for designing closed-loop linear feedback controllers. Specifically, it is shown that using CSCTA and RLFT enables the control design to take place in the transformed coordinate system in which both the delay and periodic coefficients are effectively removed, thereby enabling the use of traditional control design tools for linear time-invariant systems. Next, the problem of estimation of states and parameters (including the delay) for nonlinear delayed systems using optimal stochastic filtering with noise-corrupted, possibly incomplete measurements is explored using CSCTA. After discretizing the delayed system with a set of ordinary differential equations, the stochastic estimation problem is implemented via the use of state augmentation and an extended Kalman-Bucy filter. This estimator can also be employed in an observer-based feedback controller, and for an illustrative example in a practical engineering problem, the problem of spacecraft attitude estimation and multi-actuator regulation control in the presence of time delay in one actuator is discussed and illustrated.