Multiparameter Singular Perturbation Research Articles

Enhanced reduced-order extended state observers-based position tracking error constraint control of a pump-controlled hydraulic system via multiparameter singular perturbation theory

In this paper, the position tracking error constraint control is investigated for a pump-controlled hydraulic system (PHS) subject to an unmeasurable velocity signal and disturbances. A barrier Lyapunov function (BLF)-based backstepping controller is designed, and a set of enhanced reduced-order extended state observers (RESOs) is proposed to estimate the velocity and disturbances. Dynamic surface control (DSC) is used to avoid a repeated calculation of the derivatives of virtual control inputs. Multiparameter singular perturbation theory is utilized to clarify the working mechanism of the enhanced RESOs and the closed-loop system stability properties. Theoretical analysis results reveal that the closed-loop system is uniformly ultimately bounded and the position tracking error is always constrained within a prescribed bound by a proper selection of the perturbation parameters. Moreover, the final position tracking error can be arbitrarily small by further tuning these parameters. In addition, the enhanced RESOs are proven capable of improving the convergence performance of the closed-loop system. Experiments are conducted on an actual PHS to validate the superiority of the proposed control strategy.

An Asymptotic Framework for Finite Hydraulic Fractures Including Leak‐Off

The dynamics of hydraulic fracture, described by a system of nonlinear integro‐differential equations, is studied through the development and application of a multiparameter singular perturbation analysis. We present a new single expansion framework which describes the interaction between several physical processes, namely viscosity, toughness, and leak‐off. The problem has nonlocal and nonlinear effects which give a complex solution structure involving transitions on small scales near the tip of the fracture. Detailed solutions obtained in the crack tip region vary with the dominant physical processes. The parameters quantifying these processes can be identified from critical scaling relationships, which are then used to construct a smooth solution for the fracture depending on all three processes. Our work focuses on plane strain hydraulic fractures on long time scales, and this methodology shows promise for related models with additional time scales, fluid lag, or different geometries, such as radial (penny‐shaped) fractures and the classical Perkins–Kern–Nordgren (PKN) model.

Time-scale decomposition of the reachable set of constrained linear systems

We consider a linear control system with a multiparameter singular perturbation representing multiple time scales and with constraints for the control and the slow state. The Hausdorff limit of the reachable set when the small parameters tend to zero is found. The result provides a basis for a time-scale approximation of the reachable set.

Stability of systems with multiple very small and very large parasitics

Stability of linear systems with several very small as well as several very large parasitic parameters is analyzed. Time-scale separation is used in conjunction with the recently introduced notion of strong <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> -stability to prove general asymptotic stability results for this class of systems. Moreover, it is observed that, contrary to the case studied by Desoer and Shensa (1970) which involved a single small parasitic parameter and a single large parasitic parameter, the system cannot be transformed into a standard multiparameter singular perturbation problem. Hence, systems containing both stray and sluggish elements do indeed represent a separate class of problems in asymptotic analysis.

Time Scale Based Modular Synthesis of Controllers to Solve the Robust Decentralized Servomechanism Problem*

In [6], the Robust Decentralized Servomechanism Problem (RDSP) is formulated and solved. The controller existence conditions and construction techniques of [6] are based on a global knowledge of the plant dynamics. In this paper, we cast the RDSP in a multimodel situation. This is done by using the multiparameter singular perturbation characterization of multimodelling introduced in [9]. We consider a more restricted information flow between control agents than previous work in multimodelling [9], [10], [16]. The information flows considered are of two types: canonical and non-canonical. The existence conditions and construction techniques for controllers that solve the RDSP for both information flows are considered.

Simplified Design of Controllers to Solve the Robust Decentralized Servomechanism Problem for Large Systems

In [9], the Robust Decentralized Servomechanisiu Problem (RDSP) was formulated and solved. In [12], [13], and [15] parameter optimization techniques wore developed to design controllers with desirable properties: closed loop asymptotic regulation, fast response, low channel interaction, integrity and tolerance to plant variations. The methods of [12] and [13] produce excellent designs - however they are computationally demanding: this makes them impractical to apply directly to large scale system problems. In this paper, the multiparameter singular perturbation characterization of multimodelling introduced in [17] is used to decompose the complex RDSP into smaller manageable problems. The resulting simplified design yields a controller which behaves similarly to a controller that the complex design would produce.

Decomposition and stability for multiparameter singular perturbation problems

Time-scale separation and stability of linear time-varying and time-invariant multiparameter singular perturbation problems are analyzed. The first problem considered in the paper is that of deriving upper bounds on the small parasitic parameters ensuring the existence of an invertible, bounded transformation exactly separating fast and slow dynamics. This problem is most interesting for the time-varying case. The analysis of this problem in the time-varying case requires the two-time-scale setting introduced by H. K. Khalil and P. V. Kokotovic [16]. This entails that the mutual ratios of the small parameters are bounded by known positive constants. The second problem considered is to derive parameter bounds ensuring that the system in question is uniformly asymptotically stable. The results on decomposition are used to facilitate the derivation of these latter bounds. Fortunately, the analysis of decomposition and stability questions for time-invariant multiparameter singular perturbation problems requires no restriction on the relative magnitudes of the small parameters. A concept of strong D -stability is introduced and shown to greatly simplify the stability analysis of time-invariant multiparameter problems.

On the stability of multiple time-scale systems

The stability of time-invariant multiparameter singular perturbation problems is considered and the implications of two time-scale stability results for multiple time-scale systems are clarified. An example shows that the asymptotic stability of a multiparameter singular perturbation problem under the ‘bounded mutual ratios’ assumption for arbitrary bounds on the ratios of the small parameters does not imply asymptotic stability under the multiple time scales assumption for any ordering of the smallness of the parameters. However, this conclusion does apply when only two small parameters are present and the fast variables are scalar-valued. A multiparameter singularly perturbed system may be asymptotically stable for all sufficiently small (and positive) values of the perturbation parameters, even though the boundary layer system does not satisfy the D-stability criterion. These examples are discussed in the light of the ‘strong D-stability’ condition which must be imposed to obtain results that are robus...

Strong D-stability

A square matrix F is said to be D-stable if the eigenvalues of DF have negative real parts for any diagonal matrix D with positive diagonal elements. The perturbational properties of D-stability and block D-stability, a generalization due to khalil and Kokotovic, are examined. Notions of ‘strong D-stability’ and ‘strong block D-stability’ are introduced, and a general class of strongly block D-stable matrices is identified. Briefly, a matrix is strongly D-stable if is is D-stable and if every sufficiently small perturbation of the matrix is also D-stable. The utility of this new concept is illustrated by proving a stability theorem for time-invariant multiparameter singular perturbation problems. The novelty of this theorem is that is applies to two time scale as well as multiple time scale systems, indeed regardless of the relative magnitudes of the singular perturbation parameters. The crucial assumption of the theorem is the strong block D-stability of an associated boundary layer system.

Singularly perturbed Hopf bifurcation

Hopf bifurcation is the simplest way in which periodic solutions can emerge from an equilibrium point of an autonomous one-parameter family of ordinary differential equations. The phenomenon occurs if the linearized system has a pair of complex-conjugate simple eigenvalues which cross the imaginary axis transversely for some value of the parameter. In this paper, conditions are derived which ensure persistence of the Hopf bifurcation under singular perturbations of the vector field. The results justify use of reduced-order models in the study of nonlinear oscillations via Hopf bifurcation. Both single parameter and multiparameter singular perturbation problems are considered. In the single parameter case, we show how Fenichel's center manifold theorem for singularly perturbed systems can be used to prove regular degeneration of the bifurcated periodic solutions and to study their stability. In the multiparameter case, we obtain a novel asymptotic formula for the eigenvalues of the perturbed system. This formula is valid regardless of the relative magnitudes of the small parameters, and the results on multiparameter singularly perturbed Hopf bifurcation apply to two time scales as well as multiple time-scale systems.

Multiparameter singular perturbation problems: Iterative expansions and asymptotic stability

Recursive formulae are derived which yield asymptotic expansions for the eigenvalues of multiparameter singular perturbation problems. These formulae follow readily from an exact expression for the eigenvalues which involves an implicit matrix function. The implicit function satisfies an algebraic matrix Riccati equation reminiscent of a similar equation of the single parameter theory. The results also explicate the ‘block D-stability’ criterion for asymptotic stability previously introduced by Khalil and Kokotovic.

Asymptotic theory of nonlinear surface waves on a viscous fluid in an inclined channel of arbitrary cross section

An asymptotic theory is developed for the study of three-dimensional nonlinear surface waves on an incompressible, viscous fluid with surface tension in an inclined channel of arbitrary cross section. The method used here is based upon a multiparameter singular perturbation scheme within the framework of long-wave approximation. The nonlinear problem is reduced to a sequence of linear elliptic mixed boundary-value problems, which may be solved by means of known methods. Their solutions are then used to determine the wave speed and evolution equations governing the nonlinear wave motion. The results obtained give a quantitative description of a three-dimensinnal bore structure in an inclined channel of arbitrary cross section, and a critical Reynolds number is also defined as a criterion for the instability of the wave motion.

On two parameter singular perturbation of linear boundary value problems

A two parameter perturbation estimate for solutions of a functional equation in Hilbert space is derived. The estimate is applied to two parameter singular perturbation of elliptic boundary value problems with homogeneous Difichlet boundary data. 1. Consider the boundary value problem eAu + ACu + Bu = ff,,f for 0 2m over a bounded domain D; and C is a linear differential operator of order fo as e J 0 and IA J 0. In particular, bounds of the form |u-uol mDD=O(ET)+o(,) in the Bessel potential space Pm(D) will be derived, assuming a like bound for the P-m(D) norm of f-fo. For the one parameter problem obtained by setting C = 0 above, corresponding bounds have been obtained by Friedman [4], Greenlee [5] and Huet [8], [9]. Extensive studies in multiparameter singular perturbation theory have been carried out by O'Malley, cf. [12]. In Greenlee [6] a two parameter perturbation problem analogous to the above, but with C a quasilinear differential operator of order ?<2m, was considered. In this paper a perturbation theorem for a functional equation in abstract Hilbert space is proven. The theorem is then applied to differential problems of the type described above. The notation and methods of this paper are similar to those used in [5] and [6]. 2. Let V and VO be complex Hilbert spaces with VCeVo, and V dense in VO. Denote by I vI v, (v, W)V, I v O the norms and inner products in V and VO respectively. Let a(v, w), c(v, w) be continuous Hermitian bilinear (sesquilinear) forms on V and let b(v, w) be a continuous Hermitian bilinear form on VO. Further assume that: Received by the editors March 6, 1970. AMS 1969 subject classifications. Primary 3514, 4748; Secondary 3545.

A Two Parameter Perturbation Estimate

with C(0) = 0. The solution u of the above equation is to be compared with the solution u0 of Bu0 =/o where/—»/o as e J, 0 and At J, 0. In particular, bounds of the form | u — w0| m.D = o(eT)-\-0(n) for the norm of u—Uo in the Bessel potential space Pm(D) will be derived, assuming a like bound for the L2(D) norm of /— f0. For the linear problem obtained by setting C=0 above, corresponding bounds have been obtained by Friedman [7], Greenlee [9], and Huet [13], [14]. These results supplement earlier work of Visik and Lyusternik [23], Huet [ll], [12], and Ton [21 ]. Singular perturbation problems for nonlinear elliptic and parabolic equations have been considered by Ton [22]. In [17], [18], [19], O'Malley has studied multiparameter singular perturbation problems. An extensive bibliography of the literature on singular perturbation is contained in O'Malley [20]. In this paper a perturbation theorem for a functional equation in