Singular Perturbation Model Research Articles

Model-driven system identification of transcritical vapor compression systems

This brief uses an air conditioning system to illustrate the benefits of iteratively combining first principles and system identification techniques to develop control-oriented models of complex systems. A transcritical vapor compression system is initially modeled with first principles and then verified with experimental data. Both single-input-single-output (SISO) and multi-input-multi-output (MIMO) system identification techniques are then used to construct locally linear models. Motivated by the ability to capture the salient dynamic characteristics with low-order identified models, the physical model is evaluated for essentially nonminimal dynamics. A singular perturbation model reduction approach is then applied to obtain a minimal representation of the dynamics more suitable for control design, and yielding insight to the underlying system dynamics previously unavailable in the literature. The results demonstrate that iteratively modeling a complex system with first principles and system identification techniques gives greater confidence in the first principles model, and better understanding of the underlying physical dynamics. Although this iterative process requires more time and effort, significant insight and model improvements can be realized.

A Singular Perturbation Model Order Reduction Method for a Specially Structured Model

A Singular Perturbation Model Order Reduction Method for a Specially Structured Model

Spatial-Domain Design of 2-D Quadrantal Symmetric Digital Filters Using Singular-value Decomposition and Singular Perturbation Model Reduction

In this paper a new technique is presented to design low order 2-D quadrantally symmetric HR digital filters. This technique is based on two steps: First, the 2-D impulse response specification (Hankel matrix) is decomposed into k parallel sections, each consists of two cascaded SISO 1-D filters, using the singular value decomposition. Second, a singular perturbational model reduction algorithm is applied to the 1- D filter to approximate the N-dimensional FIR into n-dimensional IIR filters, where n < N. The approximation step is based on computing the eigenspaces associated with the large eigenvalues of the cross- Gramian matrix Wco. Examples are given to illustrate the proposed technique.

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984–2000. The work done in this area during the period 1905–1984 has already been surveyed by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223] for details. Due to the space constraints we have covered only singularly perturbed one-dimensional problems.

Singularly perturbed problems in partial differential equations: a survey

This survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980–2000. In this review we have covered singularly perturbed partial differential equations. About ODEs the survey has already been done by us [see M.K. Kadalbajoo, K.C. Patidar, Appl. Math. Comput. 130 (2002) 457–510].

On the singular perturbation approach to trajectory control of a multilink manipulator with flexible links and joints

The singular perturbation approach has been shown to be successful in controlling both flexible link manipulators and flexible joint manipulators. In this paper, the technique is extended to control both link and joint flexibility of a multilink manipulator simultaneously while a desired trajectory is being tracked. A comprehensive, closed-form, Euler-Lagrange, assumed-mode model of a multilink manipulator has been formulated in which the effects of the link and joint flexibility are incorporated. This is then converted into a two-time-scale, state-space, singular perturbation model where the slow subsystem involves the link angles and the fast subsystem consists of the joint deflection and the link deflection variables. A composite controller is designed based on the two-time-scale model of the flexible link and joint manipulator. Numerical simulation results confirm that the proposed controller satisfactorily suppresses the tip deflection and joint flexibility of the manipulator while very good trajectory tracking is achieved.

The singular perturbation limit of an elastic structure in a rapidly flowing nearly inviscid fluid

The effective forces that govern the vertical oscillations of an elastic structure in a horizontally flowing fluid are displayed for the limit case where fluid velocity and viscosity tend to singular limits. The derived singular perturbation model is based on available representations that model the coupled dynamics as coupled oscillators. The fast dynamics of the system does not, in general, converge on the fast time scale to a stationary point; thus, the classical singular perturbation methods are not applicable. Rather, a method based on Young measures representation of the fast oscillations, and on averaging, is employed.

Control of state-constrained linear dynamical systems: Anti-reset windup approach

An alternative controller design method for state-constrained control systems is presented, where the concept of the anti-reset windup (ARW) strategies is used. The proposed controller design procedure is composed of two steps. First, a linear controller is designed to show a desirable nominal performance by ignoring state constraints. Then, an additional compensator is designed to mitigate the adverse effects due to state constraints. The paper focuses on the effective design method of the additional compensator. By minimising a reasonable performance index, the proposed compensator is expressed explicitly in plant and controller parameters. The resulting dynamics of the compensated controller exhibits the dominant part of linear closed-loop systems, which can be seen from the singular perturbation model reduction theory. The proposed method guarantees the total stability of overall resulting systems if linear controllers were constructed to meet certain conditions.

Second Order Singular Perturbation Models for Phase Transitions

Singular perturbation models involving a penalization of the first order derivatives have provided a new insight into the role played by surface energies in the study of phase transitions problems. It is known that if $W:{\mathbb R}^d \to [0,+\infty)$ grows at least linearly at infinity and it has exactly two potential wells of level zero at $a, b \in {\mathbb R}^d$, then the $\Gamma(L^1)$-limit of the family of functionals $$ {\mathcal F}_\varepsilon(u):= \begin{cases} \int_{\Omega} \left(\frac{W(u)}{\varepsilon}+\varepsilon \vert \nabla u\vert^2\right) \, dx & \hbox{ if $u\in W^{1,2}(\Omega;{\mathbb R}^d)$, }\\ \\ +\infty & \hbox{ if $u\in L^1(\Omega;{\mathbb R}^d)\setminus W^{1,2}(\Omega;{\mathbb R}^d)$, } \end{cases} $$ where $\Omega$ is a bounded, open set in ${\mathbb R}^N$, is given by $$ {\mathcal F}(u):= \begin{cases} {\rm {\bf m }}\; {\rm Per}_{\Omega} (\{u = a\}) & \hbox{ if $u\in BV(\Omega;\{a,b\})$, }\\ +\infty & \hbox{ otherwise, } $$ for a suitable constant {\bf m} depending on the energy density W. In this paper, and motivated by the study of phase transitions for nonlinear elastic materials, the $\Gamma(L^1)$-limit is obtained in the case where in ${\mathcal F}_\varepsilon(u)$ the penalization term $\varepsilon \vert \nabla u\vert^2$ is replaced by $\varepsilon^3 \vert \nabla^2 u\vert^2$, for $u \in W^{2,2}(\Omega;{\mathbb R}^d)$. The resulting functional is of the same form as $ {\mathcal F}(u)$ above.

Design of linear-phase IIR digital filters using singular perturbational model reduction

A new technique for the design of a passband linear-phase infinite impulse response (IIR) digital filter is presented. This technique is based on computing orthonormal bases, which span the left and right invariant subspaces associated with the large and small eigenvalues of the cross-Gramian matrix W/sub c0/, which tends to be symmetric and easily constructed. The orthonormal bases are computed by using the ordered real Schur form decomposition, which has been proved to be robust and numerically stable. The proposed technique is illustrated by several examples.

Singular perturbation modeling of nonlinear processes with nonexplicit time-scale multiplicity

In this article, a modeling framework is proposed for two-time-scale chemical processes modeled by nonlinear ordinary differential equations (ODEs) with large parameters of the form 1 ε , to obtain a standard singularly perturbed representation where the slow and fast variables are explicitly separated. Initially, a result is derived that provides necessary and sufficient conditions for the existence and the explicit form of an ε-independent coordinate change that transforms the two-time-scale process into a standard singularly perturbed form. Whenever these conditions are not satisfied, it is established that an ε-dependent coordinate change, singular at ε=0, has to be employed to obtain a standard singularly perturbed representation of the original two-time-scale process, and the construction of such a transformation is addressed. The application of the proposed framework in deriving standard singularly perturbed representations and its significance in the synthesis of well-conditioned controllers is demonstrated through chemical reactor applications.

DAEs and PDEs in elastic multibody systems

Elastic multibody systems arise in the simulation of vehicles, robots, air- and spacecrafts. They feature a mixed structure with differential-algebraic equations (DAEs) governing the gross motion and partial differential equations (PDEs) describing the elastic deformation of particular bodies. We introduce a general modelling framework for this new application field and discuss numerical simulation techniques from several points of view. Due to different time scales, singular perturbation theory and model reduction play an important role. A slider crank mechanism with a 2D FE grid for the elastic connecting rod illustrates the techniques.

Singular Perturbation Model of Robots with Elastic Joints and Elastic Links Constrained by Rigid Environment

A robot with distributed flexibility in the links and lumped flexibility in the joints is considered in this paper. First the model of the system in free motion is formulated as a set of ordinary differential equations, adopting a finite number of modes of the link deformation. Then algebraic constraint equations on the generalized coordinates of the system are added, to account for the loss of degrees of freedom due to the contact with rigid environment. A reduced order model, expressed in the residual degrees of freedom is then derived, based on a coordinate partitioning procedure. The singularly perturbed model of the system is finally computed, and the expression of the fast subsystem is given. The special cases of a robot with rigid joints or rigid links are also addressed.

Linear-quadratic tracking of coupled slow and fast targets

The paper examines a singular perturbation model of tracking two targets, one slow and a fast one which is assumed periodic. The plant is linear and the cost is quadratic. A limit problem is displayed. It corresponds to the limit behavior of the perturbed system when the small parameter tends to zero. This limit is not of reduced order, as the fast target is periodic. Rather, an infinite-horizon optimal averaging is performed on the fast scale, and serves as an input to slow-scale optimization. This limit design is used to approximate the optimal solution of the singularly perturbed system. Bounds for the rates of the approximation are given. The optimal limit design is computed in a feedback form in the general case, and for illustrative concrete examples.

Robot link and contact surface dynamic interaction: An experimental study of contact task instability

In this article we explore, both theoretically and experimentally, the effect of multiple structural compliance sources on robot stability during contact task execution. Here we examine the effect of link compliance and compliance of the contact surface on overall system stability. Using the theory associated with singularly perturbed dynamic systems, the stability of the robotic system is examined for three distinct cases: (1) link stiffness much greater than contact surface stiffness; (2) contact surface stiffness much greater than link stiffness; and (3) link stiffness comparable to surface stiffness. Theoretical results lead to a conclusion of stability only for case (1). Experiments indicate stable behavior for all these cases. Through the use of singular perturbation modeling and analysis, we are able to write dynamic models of the robot during contact with a complaint surface such that the complaint behavior of the robot links and contact surface are parameterized with just two variables. In doing so, theoretical examination and experimental study of stability, or the lack thereof, is greatly simplified. Instead of exhaustive tests in which experiments or stability analyses are conducted over a wide range of values of link and contact surface compliance, a select number of cases are examined. Under the assumptions that validate the use of singular perturbation modeling techniques, the resultant stability analysis and experimental results are hence conducted in an efficient manner. © 1997 John Wiley & Sons, Inc.

Reduced order modelling of two-time-scale discrete non-linear systems

Application of two-time scale discrete singular perturbation methods have been limited to the linear systems. The area of discrete-time non-linear systems has received little attention. This paper deals with the singular perturbation model of non-linear discrete-time dynamic systems. We present a specific modelling and a mode decoupling approach well adapted for such systems. A comparison principle is used for grouping slow and fast states of a class of non-linear discrete systems. Finally, an example is given to show the feasibility of the theoretical results.

Singular Perturbation Modeling of Nonlinear Processes with Non Explicit Time-Scale Separation

This work addresses the problem of singular perturbation modeling of two-time-scale chemical processes described by nonlinear ODEs that involve large parameters of the form 1/ε. Initially, a result derived in (Kumar et al., 1995) is reviewed that provides necessary and sufficient conditions for the existence and the explicit form of a coordinate change, which is independent of ε and transforms the two-time-scale process to a standard singularly perturbed form where the separation of fast and slow modes is explicit. Whenever these conditions are not satisfied, it is established that the state-space region in which the system exhibits a two-time-scale property depends on the value of ε and an ε-dependent coordinate change, singular at ε = 0, has to be employed to obtain a standard singularly perturbed form of the system. The significance of the proposed singular perturbation modeling framework in the design of well-conditioned controllers that account for the two-time-scale behavior, is demonstrated on a chemical reactor which exhibits a slightly non-minimum phase behavior.

Bilinear transformation and generalized singular perturbation model reduction

A new general bilinear relationship is found between continuous and discrete generalized singular perturbation reduced-order models. This result is applied to the problem of deriving discrete analogs of continuous singular perturbation and direct truncation model reduction and leads to a new definition of discrete "Nyquist" model reduction. Also, "unit circle" bilinear transformations are used to relate several known facts about continuous and discrete balanced model reduction and incorporate them into a symmetrical, unified framework.

A singular perturbation model of fluid dynamics in the vestibular semicircular canal and ampulla

A matched asymptotic analysis is presented that describes the mechanical response of the vestibular semicircular canals to rotation of the head and includes the fluid–structure interaction which takes place within the enlarged ampullary region of the duct. New theoretical results detail the velocity field in a fluid boundary layer surrounding the cupula. The governing equations were linearized for small perturbations in fluid displacement from the prescribed motion of the head and reduced asymptotically by exploiting the slender geometry of the duct. The results include the pressure drop around the three-dimensional endolymphatic duct and through the transitional boundary layers within the ampulla. Results implicitly include the deflected shape of the cupular partition and provide an expression for the dynamic boundary condition acting on the two surfaces of the cupula. In this sense, the analysis reduces the three-dimensional fluid dynamics of the endolymph to a relatively simple boundary condition acting on the surfaces of the cupula. For illustrative purposes we present specific results modelling the cupula as a simple viscoelastic membrane. New results show that the multi-dimensional fluid dynamics within the enlarged ampulla has a significant influence on the pointwise deflection of the cupula near the crista. The spatially averaged displacement of the cupula is shown to agree with previous macromechanical descriptions of endolymph flow and pressure that ignore the fluid–structure interaction at the cupula. As an example, the model is applied to the geometry of the horizontal semicircular canal of the toadfish, Opsanus tau, and results for the deflection of the cupula are compared to individual semicircular canal afferent responses previously reported by Boyle &amp; Highstein (1990). The cupular-shear-angle gain, defined by the angular slope of the cupula at the crista divided by the angular velocity of the head, is relatively constant at frequencies from 0.01 Hz up to 1 Hz. Over this same range, the phase of the cupular shear angle aligns with the angular velocity of the head. Near 10 Hz, the shear-angle gain increases slightly and the phase shows a lead of as much a 30°. Results are sensitive to the cupular stiffness and viscosity. Comparing results to the afferent responses represented within the VIIIth nerve provides additional theoretical evidence that the macromechanical displacement of the cupula accounts for the behaviour of only a subset of afferent fibres.

Singular-perturbation-model-based output controller for a class of systems via Schur decomposition

An output feedback controller based on the singular perturbation model via the Schur approach is presented. Equivalent singular perturbation models for global and each decoupled subsystem are obtained via the Schur decomposition technique in order to generate global and local output feedback control signals. The effect of subsystem state interactions is minimized by employing a global signal while the transient response of the system is regulated using a local control signal. Stability analysis of the proposed control scheme is discussed. The effectiveness of the proposed control scheme based on the singular perturbation model is studied by considering an interconnected power system comprising two generating units. Simulation results are presented.