Robust stabilization of singularly perturbed systems

We discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models. In [1], it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the ‘true’ system containing both fast and slow modes. This phenomenon was then expanded to include the case where the simplified system is strictly proper in [2]. The objectives of this note are threefold: (i) to show that, given any proper system and any stabilizing controller for it that is proper but not strictly proper, there exists a singular perturbation of the system that is destabilized by that controller, (ii) to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter, and (iii) to provide a characterization, in the same spirit as [3,4], of the set of all strictly proper controllers that stabilize a given proper plant. By combining these results, it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system.     

H∞ control of two-time-scale systems

H_∞ control of linear time-invariant singularly perturbed systems is considered. A sequential procedure is described to decompose the problem into slow and fast subproblems. The fast problem is solved first. Then the slow problem is solved under a constraint on the value of the compensator at infinity. A composite compensator is formed as the parallel connection of the fast compensator with the strictly proper part of the slow compensator. The asymptotic validity of the composite compensator is established.     

Stabilization of singularly perturbed strictly bilinear systems

The stabilization problem via state feedback, for the class of strictly bilinear singularly perturbed systems, is considered. It is shown that the stabilizing controller of the overall system can be determined by simply. using the quadratic stabilizing controllers of the fast and slow subsystems. The procedure for computing the two matrix-gains of the controller is given and some illustrative examples are worked out.     

Two-time-scale distributions and singular perturbations

Two-time-scale (TTS) distributions are introduced. For a class of stable systems, it is shown that every TTS distribution has a two-frequency-scale (TFS) Laplace transform. Conversely, it is shown that the impulse response of any stable TFS transfer function, and hence any stable (standard) singularly perturbed system, can be characterized in terms of a stable TTS distribution. A time domain decomposition for TTS distributions is obtained which parallels the slow and fast decomposition of singularly perturbed systems and also the frequency domain decomposition of TFS transfer functions. It is shown that every stable TTS distribution can be decomposed in terms of two simpler distributions represented in two different time scales. A composite distribution is constructed from these two which approximates the TTS distribution arbitrarily closely in the L ₁ norm.     

Composite Fuzzy Control of Nonlinear Singularly Perturbed Systems

This paper presents the composite fuzzy control to stabilize the nonlinear singularly perturbed (NSP) systems with guaranteed H^infin control performance. We use the Takagi-Sugeno (T-S) fuzzy model to construct the singularly perturbed fuzzy (SPF) systems. The corresponding fuzzy slow and fast subsystems of the original SPF system are also obtained. At first, a set of common positive-define matrices and the controller gains are determined by the Lyapunov stability theorem and linear matrix inequality (LMI) approach. Then, a sufficient condition is derived for the robust stabilization of NSP systems. The composite fuzzy control will stabilize the original NSP systems for all epsivisin(0,epsiv^*) and the allowable perturbation bound epsiv^* can be determined via some algebra inequalities. A practice example is adopted to demonstrate the feasibility and effectiveness of the proposed control scheme     

H∞-norms and disturbance attenuation for systemswith fast transients

It is shown that for a singularly perturbed system, the H_∞ norm of the transfer matrix function tends to the largest of the H_∞ norms for the boundary layer system and for the reduced slow model. As an application complementing the results of Pan and Basar (1993, 1994), it is proven that fast transients cannot always be neglected when designing a stabilizing controller with disturbance attenuation     

Balanced realizations of singularly perturbed systems

Balanced realizations of linear time-invariant singularly perturbed systems are studied. The reduced model of a singularly perturbed system that is obtained, based on the balanced realization, is compared to that obtained by time-scale considerations. An approximate balancing transformation for singularly perturbed systems is constructed based on balancing transformations of the approximating slow and fast subsystems.     

Robust stability of non-standard nonlinear singularly perturbed discrete systems with uncertainties

In the past several decades, the singularly perturbed discrete systems have received much attention for the stability analysis and controller design. Recently, there are some results about the nonlinear singularly perturbed discrete systems. Compared with the existing result, we consider the robust stability of the uncertain nonlinear singularly perturbed discrete systems with the less conservative assumption via the Lyapunov function method. Moreover, the previous results of the singularly perturbed discrete system are only applied to the system, which is composed of the slow part and the fast part, separately. However, we consider the non-standard nonlinear singularly perturbed discrete system in which the slow part and the fast part coexist, that is, a general case of the nonlinear singularly perturbed discrete systems. Then, by using the lower-order subsystems from two standard systems, we present the robust stability of the non-standard nonlinear singularly perturbed discrete system with uncertainties.     

Multirate and composite control of two-time-scale discrete-time systems

The design of stabilizing feedback control of singularly perturbed diserete-time systems is decomposed into the design of slow and fast controllers which are combined to form the composite control. Composite control strategies are developed for the case of single rate measurements (all variables are measured at the same rate) as well as for the case of multirate measurements (slow variables are measured at a rate slower than that of fast variables).     

On the reachable set of inflated singularly perturbed differential equations

Singularly perturbed differential equations with slow and fast subsystems are under consideration. It is well-known that the invariant probability measures of the unperturbed fast subsystems produce a limiting system for the slow subsystem. Unfortunately, the limiting system is non-smooth in general and might be too big since it also generates trajectories that are not related to the singularly perturbed system. However, if the flows produced by the unperturbed fast subsystems are chain transitive an inflation of the singularly perturbed system can be used in order to approximate all trajectories of the limiting system. Moreover, it turns out that the reachable sets of the limiting system are contained in the reachable sets of the slightly inflated singularly perturbed system.     

The observer-based controller design of discrete-time singularly perturbed systems

A class of linear shift-invariant discrete-time singularly perturbed systems with inaccessible states is considered. A design technique is formulated by which the stabilizing controller can be formed through the controllers of the slow and fast subsystems. Sufficient conditions for stability of the closed-loop system under this composite controller are given.     

D-stability problem of discrete singularly perturbed systems

The D-stability problem of discrete time-delay singularly perturbed systems is examined. A two-stage method is first developed to analyse the stability relationship between a discrete time-delay singularly perturbed system and its corresponding slow and fast subsystems. Finally, the upper bound of a singular perturbation parameter is derived such that D-stability of the slow and fast subsystems can imply that of the original system, provided that the singular perturbation parameter is within this bound. This fact enables us to investigate the D-stability of the original system by establishing that of its corresponding slow and fast subsystems.     

Robust state estimation for singularly perturbed systems

This paper deals with the design of interval observers for singularly perturbed linear systems. The full-order system is first decoupled into slow and fast subsystems. Then, using the cooperativity theory, an interval observer is designed for the slow and fast subsystems assuming that the measurement noise and the disturbances are bounded and the singular perturbed parameter is uncertain. This decoupling leads to two observers that estimate the lower and upper bounds for the feasible state domain. A numerical example shows the efficiency of the proposed technique.     

Global stability of systems with amplitude and rate saturation compensation

Rate and amplitude saturation of the actuator is common in practical control systems. When the actuator is rate and amplitude saturated, the control performance can deteriorate rapidly, and in the worse case, the closed‐loop system can become unstable. It is therefore important that both types of saturations are properly compensated. Following the approach for compensating amplitude constraints, a scheme for compensating systems with both rate and amplitude saturation is proposed in this paper. The conditions for the compensated system to be globally stable are derived, and from this result, a procedure for designing the rate and amplitude saturation compensators is devised. As it is difficult to design both the rate and the amplitude saturation compensators simultaneously, a two‐step approach is adopted. In the proposed compensator design procedure, the amplitude saturation compensator is designed first, followed by the rate saturation compensator. As the compensators designed using the proposed procedure satisfy the conditions for global stability, the compensated system is therefore globally stable. It is also shown that these compensators can be designed using the LMI technique. The implementation of the design procedure is demonstrated by a simulation example. Copyright © 2004 John Wiley & Sons, Ltd.     

Stabilization of non-linear systems†

The problem of stabilizing a discrete-time non-linear system is considered. For a rather large class of common stabilizable non-linear systems, a procedure leading to the stabilization of a given non-linear system Σ belonging to that class is derived. In this procedure, a pair of compensators is constructed, consisting of a precompensator and an output feedback compensator, which, when connected in closed loop around the system Σ, yield a closed-loop system that is internally stable for bounded input sequences. The procedure allows the construction of infinitely many different pairs of such compensators, thus facilitating the choice of a convenient one.     

Composite control of non-linear singularly perturbed systems: a geometric approach

Using a geometric approach, a composite control—the sum of a slow control and a fast control—is derived for a general class of non-linear singularly perturbed systems. A new and simpler method of composite control design is proposed whereby the fast control is completely designed at the outset. The slow control is then free to be chosen such that the slow integral manifold of the original system approximates a desired design manifold to within any specified order of ε accuracy.     

Preservation of controllability in linear time-invariant perturbed systems†

The controllability of systems with weak connections is studied. A necessary and sufficient condition for a singularly perturbed system to be strongly controllable is obtained. The controllability invariance of the slow subsystem of a singularly perturbed system due to a class of fast feedback controls is shown.     

Proportional-plus-multiintegral stabilizing compensators for aclass of MIMO feedback systems with infinite-dimensional plants

A method is presented for designing proportional-plus-multiintegral stabilizing compensators for a class of feedback systems with exponentially stable infinite-dimensional plants. These compensators enable the feedback system to asymptotically track polynomial inputs and suppress polynomial disturbances of corresponding order. When used in design, the effectiveness of the construction procedure presented can be increased considerably by combining it with the computational stability criterion and semiinfinite optimization. In this manner, the coefficient matrices of a compensator designed by the method can be further modified to ensure better feedback-system performance with respect to various performance requirements, without destroying stability, tracking, and disturbance-rejection properties     

H2 guaranteed cost control for singularly perturbeduncertain systems

The H₂ guaranteed cost control problem for a singularly perturbed norm-bounded uncertain system is addressed using the quadratic stabilizability framework. After defining the corresponding slow and fast uncertain subsystems, the set of quadratic stabilizing composite controls is characterized. Two Riccati equations have to be solved, one for the slow subsystem and the other for the fast subsystem. Choosing appropriately the weighting matrices, it is shown how to pick up in the set of quadratic stabilizing composite controls, a control minimizing an upper bound on the H₂ norm of a certain transfer matrix. The case of the guaranteed cost control problem for the reduced system is also investigated     

Robust disturbance attenuation with stability for a class of uncertain singularly perturbed systems

In this paper, the problem of robust stability and robust disturbance attenuation is investigated for a class of singularly perturbed linear systems with norm-bounded parameter uncertainties in both state and output equations. Based on the slow and fast subsystems, a composite linear controller is designed such that both robust stability and a prescribed H infinity performance for the full-order system are achieved, irrespective of the uncertainties. Our results show that the above problem can be converted to an H infinity control problem for a related singularly perturbed linear system without parameter uncertainty. Thus, the existing results on H infinity control of singularly perturbed systems can be applied to obtain solutions to the problem of robust H infinity control for the uncertain systems, which is independent of the singular perturbation epsilon when epsilon is sufficiently small. An example is given to show the potential of the proposed technique.