Optimal feedback control strategies for periodic delayed systems

Abstract

In this study, three strategies based on infinitedimensional Floquet theory, Chebyshev spectral collocation, and the Lyapunov-Floquet transformation (LFT) are proposed for optimal feedback control of linear time periodic delay differential equations using periodic control gains. First, a periodic-gain discrete-delayed feedback control is implemented where optimization of the control gains is included to obtain the minimum spectral radius of the closedloop response. Second, a large set of ODEs is obtained using the Chebyshev spectral continuous time approximation, after which optimal (time-varying LQR) control is used to obtain a periodic-gain distributed-delayed feedback control. The third strategy involves the use of both CSCTA and the reduced LFT, along with either pole-placement or time-invariant LQR used on a linear time invariant auxiliary system, to obtain a periodic-gain non-delayed feedback control that asymptotically stabilizes the original system. The delayed Mathieu equation is used as an illustrative example for all three control strategies.

H∞ control of periodic piecewise vibration systems with actuator saturation

“…To solve this problem, a procedure consisting of the Lyapunov–Floquet transformation, the backstepping technique and Floquet theory was proposed in Deshmuhk and Sinha (2004). The optimal control with solution of a periodic Riccati equation is used in helicopter vibration control (Arcara et al., 2000) and for periodic delayed systems, optimal controls separately based on Floquet theory, Chebyshev spectral collocation and the Lyapunov–Floquet transformation were studied in Nazari et al. (2014).…”

Decentralized Consensus Control of a Rigid-Body Spacecraft Formation with Communication Delay

“…In general, the abstract representation of Eq. (29) is the evolution of the history function φ in a Banach space [39,40], i.e.…”

Continuous Thrust Stationkeeping in Earth-Moon L1 Halo Orbits Based on LQR control and Floquet Theory

“…To solve the periodic ARE in this study, the period T was discretized into 1000 intervals and the ARE was solved for each time interval. That is, the infinite horizon time-periodic LQR approach is considered here in order to make the periodic closed-loop matrix (A(t) − BK(t)) stable according to Floquet theory [21][22][23][24][25]. Alternatively, one can use the method of harmonic balance to solve the periodic ARE via expansion of the elements of P(t) in a Fourier series with an arbitrary number of terms and the Fourier coefficients obtained by solving systems of nonlinear algebraic equations.…”