1. Mathematical modeling of many physical systems calls for high-order dynamic equations. The presence of certain parameters such as spring constant, mass, moments of inertia and Reynold’s number to name a few, are the cause of stiffness and increased order of these equations. It is highly difficult to arrive at exact analytical solutions of these nonlinear governing equations with known and sometimes unknown variable coefficients and therefore an approximate solution is computed. Singular perturbation theory is an scheme used to simplify systems which inherently possess both fast and slow dynamics. Such systems are characterised by a small parameter multiplying the highest derivative. Suppression of this small parameter reduces the order of the system, and thus the label ‘singularly perturbed’. The commencement of singular perturbation theory dates back to the 1904 work of Prandtl1on fluid boundary layers. But it was not until the 19th century2-4that applications of perturbation methods were explored for control design.
2. Tracking properties of general singularly perturbed systems were first studied by Grujic5in 1982. This work laid down the foundations of tracking theory in a Lyapunov sense. Later in 1988 this work was extended for nonlinear time-varying singularly perturbed systems.6However, it is assumed that separate controls are available for both; the slow and the fast subsystems and the algebraic set of equations have a trivial solution. Jayasuriya7designed a controller structure for a linear singularly perturbed model consisting of a precompensatortoachievethetrackingobjectiveandastabilizingcompensatorinconjunctionwithaLuenberger observer to estimate the slow subsystem dynamics. Asymptotic tracking was guaranteed as long as the closed loop system possessed two clusters of eigenvalues with their ratio less than a predetermined upper bound. Christofides et.al8developed robust controller design for systems with stabilizable fast subsystem, and; input/output linearizable slow subsystem with input-to-state stable inverse dynamics. This work considered a general class of nonlinear time-varying singularly perturbed systems that have dynamics linear in the fast states.
3. Another approach to tracking was presented in an article by Heck9in 1991. This paper addressed the design of sliding-mode controllers for a class of linear time-invariant systems where tracking of slow variables is desired. Forboththeslowandthe fastsubsystems, a sliding mode controller is designed and a composite of these controls is then implemented on the full-order system. Concept of composite control, that is designing separate controller for each of the subsystems and then implementing their cumulative to the full higher order system was initiated by Suzuki and Miura10in 1976 and since then this concept has been extensively used by researchers for robust stabilization of systems with time-scale properties.11-13
4. In the aerospace community, tracking of slow variables is achieved by assuming the fast variables to be control variables to the slow subsystem, which avoids the problem of finding a root of the nonlinear algebraic equations. Pioneering work in this area was published by Menon et.al16in 1987. Ref.16 designed flight test trajectory control systems using dynamic inversion. The output variables to be tracked were total velocity, angle-of-attack, sideslip and altitude. Once the desired angular rates were calculated, dynamic inversion is applied to the fast subsystem to compute the aerodynamic control surface deflections. This work was extended for application to over actuated systems by Snell et.al.17More recently the same concept has been employed to design longitudinal wind shear flight control laws18and for control of generic reentry vehicles.19
5. Noticeable from the above study of literature are certain facts. Although all the systems studied fall under the category of Eqs.1, different design methodologies have been developed for varied physical systems, for which, further, several control techniques have been employed. Moreover, the control laws developed for general form of physical systems, assume existence of unique solution of the transcendental equation. For general dynamical system models, the existence of isolated roots for the fast states is not guaranteed. Study of singularly perturbed systems with such behaviour has been the focus of what is called the ‘Geometric Singular Perturbation Theory’, foundation of which, was laid down by Fenichel in 1979.20Since then, this theory has been employed for transforming multiple-time scale systems into standard form (Eq.1).21,22and to develop reduced-order models.23Geometric methods study the reduced-order models (Eqs.2,3) through perspective of integral manifolds. Work by Sharkey and O’Reilly24designed stabilizing control laws for a special of class of singularly perturbed systems wherein the control appears only in fast dynamics. The global nature of the above stabilization results was proved by Chen25later on n 1998.