Singular perturbations and order reduction in control theory — An overview

Preservation of controllability in linear time-invariant perturbed systems†

“…Furthermore, the controllability of the slow subsystem of a singularly perturbed system is shown to be invariant to a class of fast feedback controls, and hence we can neglect the fast subsystem if it is stable. These results clarify those obtained by Kokotovic and Haddad (1975), Kokotovic and Yackel (1972) and Chow and Kokotovic (1976 b). The presentation in this paper is aimed at giving a structural interpretation of the controllability (Lin 1974) of perturbed systems.…”

“…We first define the slow and fast subsystems of the singularly perturbed system (6). It is known (Kokotovic and Haddad 1975) that system (6) possesses slow modes with n l small eigenvalues of magnitude 0(1) and fast modes with n. large eigenvalues of magnitude O(l/JL)' Assuming that the transient of the fast modes is instantaneous, that is, letting JL = 0, we obtain from (6) the reduced order system 'Ii. = AllY.…”

Linear Quadratic Regulation and Tracking using Output Feedback with Direct Feedthrough

“…Apart from strictly proper practical systems, semi-proper systems or systems with direct feedthrough have been studied for various reasons and situations in control system synthesis and analysis, e.g., inclusion of the measured acceleration of an aircraft in the system model, the proportional-integral-derivative controller, and the phase advance filter in various control systems [2]. In the process of model order reduction to eliminate highfrequency effects, direct feedthrough terms can be generated in the measurement equation [3,4] by using the concept of singular perturbations in control [5].…”

PI/PID Control for Nonlinear Systems via Singular Perturbation Technique

“…model in the form of singularly perturbed system (Kokotović et al, 1976;Naidu & Calise, 2001;Naidu, 2002;Saksena et al, 1984). The second one is that the singularly perturbed dynamical systems can also appear as the result of a high gain in feedback (Meerov, 1965;Young et al, 1977).…”

“…The main notions of singularly perturbed systems can be considered based on the following continuous-time system:Ẋ = f (X, Z),( 1 ) µŻ = g(X, Z), ( 2 ) where µ is a small positive parameter, X ∈ R n , Z ∈ R m ,a n d f and g are continuously differentiable functions of X and Z. The system (1)- (2) is called the standard singularly perturbed system (Khalil , 2002;Kokotović et al, 1976;Kokotović & Khalil, 1986).…”