Nonlinear Control Systems: Analysis and Design

Properties of double (or two-dimensional) Laplace transform (2DLT) for analysis of linear time-varying (LTV) systems are explored. The one-dimensional to two-dimensional expansion formulas, as well as the two-dimensional to one-dimensional reduction formulas, are outlined. The merit of the proposed technique for frequency analysis of LTV systems is demonstrated.

This paper introduces a linear time invariant system analysis tool, the singular value sensitivity function, which can be used in conjunction with singular value analyses to provide more complete estimates of system feedback properties. An important feature of this tool is the ability to analyze the effect of simultaneous variations in several system parameters.

This technical note is concerned with exponential stability analysis of linear time-varying systems represented by the second-order vector differential equations with the singular leading coefficient matrix. Using bounding techniques on the trajectories of the linear time-varying systems, the stability problem of the time-varying systems is transformed to that of the time-invariant systems and a new sufficient condition for exponential stability is derived. The obtained condition can be applied to the nonsingular case and it is proven that the condition for the nonsingular case is superior to a test presented in the recent literature. Analogously, the proposed method is also extended to the stability analysis for a class of linear time-varying systems with delay. By illustrative examples, better results are obtained by the proposed criteria as compared with some results in the literature.

Error-bounds are developed for balanced truncation of linear time-varying systems, leading to an extension of the twice the sum of the tail formulas, well known in the time-invariant case. The approach relies on an operator-theoretic framework for analysis of linear time-varying systems. This provides a multivariable notion of frequency for such systems, which are thus characterized by rational functions of many complex variables, allowing the problem to be formulated in the linear-fractional framework. Using a time-varying version of standard necessary conditions for reduced-order modeling, based on convex operator inequalities, we show that these error-bounds for balanced truncation are related to the closest possible reduced-order modeling error in a sense which parallels the time-invariant case.

The stability of a particular class of shaping filters, namely, the periodic shaping filters, is considered in this paper. The main objective of the present paper is to show that by considering the output covariance function of this periodic shaping filter it is possible to determine the stability of this class of shaping filters in the bounded input-bounded output sense. Furthermore, it is demonstrated that if γ(t, τ) is separable in t and τ, and the components of the separating vectors that result in the scalar covariant function are jointly linearly independent over one period of the time interval, then the resulting shaping filter is uniformly asymptotically stable. The results of this paper are presented in a series of theorems that, in effect, enable the determination of stability properties of such systems by using their structural properties only. The approach taken in this paper is thus parallel to that of Lyapunov's second method in stability analysis of linear time-varying systems.

Solution of multipoint boundary value problems via block-pulse functions

A new approach has been developed in this paper to solve time-independent Schrödinger wave equation for any arbitrary potential and space varying mass as well. The method is based on the state transition matrix used in the analysis of linear time-varying systems, and can determine both bound states and reflection and transmission coefficients associated with scattering problems. Numerical examples for the computation of eigenvalues and eigenmodes associated with bound states are presented for quadratic potential, quartic potential, constant potential well and arbitrary potential well with both constant and space-varying or position-dependent masses. Similarly, transmission coefficients for scattering problems without any infinite potential, and time delays for scattering problems with an infinite potential are computed for arbitrary potential wells.

The direct method presented in this paper for the calculation of the mean square ensemble average of nonstationary functions can be of great use in the analysis of linear time-varying systems with periodical random phase inputs. The results can be obtained either by a closed-form solution or by a single computer run instead of numerous Monte Carlo simulations. The method can be equally applied for adjoint analysis.

The published literature for stability analysis of linear time-delay systems has attracted much interest among the researchers over the last five decades. In general, the stability for linear time-delay systems can be checked exactly only by eigenvalue considerations. Two types of stability conditions are delay-independent type, mainly used for arbitrarily large delays, and delay-dependent type, which is mostly useful in applications that consider maximum delay. In this paper, we consider a linear time-invariant system with delay differential equation form for quiet-standing case study as a derivative feedback control. When it is reasonably chosen with intentional delays, the effects of time delay for ankle torque can be used to improve and stabilize the response of the close-loop systems.

The response at time t (R(t)) of a (causal linear time invariant) system to an input A(t) is represented by: [equation: see text] where K(t) is called the unit impulse response function of the system, and the integration on the right side of the equation (above) is called the convolution (from the latin cum volvere: to interwine) of A(t) and K(t). The system described by this equation is at zero (initial conditions) when t = 0. Although it does not even begin to describe the incredible variety of possible responses of biological systems to inputs, this representation has large applicability in biology. One of the most frequently used applications is known as deconvolution: to deinterwine R(t) given a known K(t) (or A(t)) and observations of R(t), to obtain A(t) (or K(t)). In this paper attention is focused on a greater variety of aspects associated with the use of linear systems to describe biological systems. In particular I define causal linear time-invariant systems and their properties and review the most important classes of methods to solve the deconvolution problem, address. The problem of model selection, the problem of obtaining statistics and in particular confidence bands for the estimated A(t) (and K(t)), and the problem of deconvolution in a population context is also addressed, and so is the application of linear system analysis to determine fraction of input absorbed (bioavailability). A general model to do so in a multiinput-site linear system is presented. Finally the application of linear system analysis to control a biological system, and in particular to target a desired response level, is described, and a general method to do so is presented. Applications to simulated, endocrinology, and pharmacokinetics data are reported.

related to the use of Fractional-order (FO) differential equations in modeling and control. FO differential equations are found to provide a more realistic, faithful, and compact representations of many real world, natural and manmade systems. FO controllers, on the other hand, have been able to achieve a better closed-loop performance and robustness, than their integer-order counterparts. In this paper, we provide a systematic and rigorous time and frequency domain analysis of linear FO systems. Various concepts like stability, step response, frequency response are discussed in detail for a variety of linear FO systems. We also give the state space representations for these systems and comment on the controllability and observability. The exercise presented here conveys the fact that the time and frequency domain analysis of FO linear systems are very similar to that of the integer-order linear systems.

The usual invariance results for asymptotic stability for continuous autonomous finite-dimensional dynamical systems involve a positive definite Lyapunov function whose time derivative along the system motions is negative semidefinite (along with certain invariance conditions). This is equivalent to requiring that along the system motions, the Lyapunov function is nonincreasing at all times with increasing time. In this paper we establish an invariance result for asymptotic stability for continuous and discontinuous nonautonomous finite-dimensional dynamical systems, which requires a positive definite Lyapunov function which when evaluated along the system motions is nonincreasing only on certain unbounded discrete time sets E with increasing time. This allows that between the time instants determined by E, the Lyapunov function may increase (i.e., over some finite time intervals, the system may exhibit unstable behavior). We also show that the usual invariance theorem for asymptotic stability reduces to the invariance theorem for continuous dynamical systems presented herein. In addition, we establish a variant to the above result involving estimates of the asymptotic behavior of the system's motions. We apply our results to four examples. One of these involves the stabilization of conservative mechanical systems using energy dissipation intermittently.

Analysis of Linear Time Invariant Systems with Time Delay

A matrix-operator representation of parameter sensitivities is used to provide an algebraic "structural" analysis of local parameter identifiability in linear time-invariant ordinary differential equation systems. Necessary conditions for identifiability depend only upon the system matrices, no integrals must be computed, and arbitrary parametrization may be used. Relations to insensitivity are discussed, and design techniques are suggested which use the nonidentifiable (insensitive) subspace to systematically reduce the number of exciting parameters in individually designed parameter identification experiments. Finally, sufficiency conditions for zero-state identifiability are examined in terms of control inputs which continually excite the natural modes of the parameter sensitivities.

The paper deals with the identification of linear time invariant (LTI) systems by a special observer. An observer emitting an frequency modulated continuous wave (FMCW) signal and having a stretch processor as receiver will be considered for system identification. A thorough derivation of the gathered baseband signal for arbitrary LTI systems will be given. It is shown, that the received signal is approximately given by the transfer function of the LTI system over the frequency sweep of the FMCW signal. The proof relies on an infinite large time-bandwidth product of the transmit signal, such that errors remain in practical applications with a finite time-bandwidth product. Monte–Carlo simulations are conducted to verify the approximation and to quantify its accuracy and remaining errors. The findings are important for e.g. calibration or derivation of a device model in FMCW radar applications.