Signal and system spaces with non-convergent sampling representation

This paper deals with the question of the existence of weak and strong common
\nquadratic Lyapunov functions (CQLFs) for stable discrete-time linear timeinvariant
\n(LTI) systems. The main result of the paper provides a simple characterisation
\nof pairs of such systems for which a weak CQLF of a given form exists
\nbut for which no strong CQLF exists. An application of this result to second
\norder discrete-time LTI systems is presented.

This study investigates characteristic root equivalency relations between commensurate order and integer order Linear Time Invariant (LTI) systems. Author introduces some useful properties of a special class of commensurate order systems, which is called characteristic root equivalency class of LTI systems. These properties present potential to facilitate design and analysis efforts of this class of commensurate order systems. In this sense, straightforward stability checking procedures and design approaches for commensurate order root equivalent systems of the first and second order LTI systems are demonstrated. Findings of the study are validated by illustrative examples.

Model Reference Adaptive Control (MRAC) of a wide repertoire of stable Linear Time Invariant (LTI) systems is addressed here. Even an upper bound on the order of the finite-dimensional system is unavailable. Further, the unknown plant is permitted to have both minimum phase and nonminimum phase zeros. Model following with reference to a completely specified reference model excited by a class of piecewise continuous bounded signals is the goal. The problem is approached by taking recourse to the time moments representation of an LTI system. The treatment here is confined to Single-Input Single-Output (SISO) systems. The adaptive controller is built upon an on-line scheme for time moment estimation of a system given no more than its input and output. As a first step, a cascade compensator is devised. The primary contribution lies in developing a unified framework to eventually address with more finesse the problem of adaptive control of a large family of plants allowed to be minimum or nonminimum phase. Thus, the scheme presented in this paper is confined to lay the basis for more refined compensators--cascade, feedback and both--initially for SISO systems and progressively for Multi-Input Multi-Output (MIMO) systems. Simulations are presented.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

In this paper, the cooperative stabilization problem of linear time-invariant (LTI) systems is studied based on a class of distributed Luenberger observers. A given LTI system is estimated by a network of observers. In case of observability configuration or external attack, each observer has to possess different measurement output of the LTI system. Classical distributed control strategies cannot be directly used because they are assumed to obtain identical information. In view of this problem, a class of distributed observers are designed under an undirected graph, which can recover the state of the LTI system by limited local information. By using any single channel of the estimated state, a control input is proposed to stabilize the LTI system. By analysing the stability of an augmented system which involves the LTI system and the estimation error system, some necessary and sufficient conditions are derived for the existence of control gains that guarantee the stabilization of the LTI system. In addition, some algorithms are proposed to specify the control gains. Lastly, some simulation examples are given to verify the effectiveness of the theoretical result.

Chapter 4 - Frequency Analysis: The Fourier Series

Over the preceding nearly 20 years, I have taught a "standard" first course in control theory many times. During that time, a way of describing and working with one property of bounded-input, bounded-output (BIBO) stable linear time-invariant (LTI) systems, that such systems forget the distant past, has been developed. Students and instructors may find it helpful to refer to this property as a stable system's "forgetery."

In this paper, we investigate two classes of consensus protocols for networked multi-agent systems: linear time-invariant (LTI) systems and linear time-delay systems. Based on the topology of multi-agent systems, the first-order integrator model is developed. The digraph (directed graph) is employed to show the topology of networked systems, and then a consensus convergence criterion is established. For LTI systems, we prove that their consensus value will converge globally asymptotically to the convex hull of initial states. By solving a set of linear equations, we get the convex combinations of equilibria, and we obtain the convex value of continuous system. If the topology is fixed and time-invariant, the consensus value of the linear time-delay system is also the convex hull of the initial states and is identical to the LTI system. Finally, a network of six agents is presented to show the effectiveness of the results of this paper.

In this paper we study the system approximation process, which naturally emerges when a stable linear time-invariant (LTI) system is applied on the Shannon sampling series, for the case that the samples of the signal are disturbed by the non-linear threshold operator, which sets all samples that are below some threshold to zero. We analyze its behavior for signals in the Paley-Wiener space PW1π of bandlimited signals with absolutely integrable Fourier transform as the threshold tends to zero. We treat the pointwise as well as the global behavior and characterize the systems for which there exist signals such that the approximation error diverges as the threshold tends to zero. We further show that for those systems in a certain topological sense almost all signals lead to divergence and that the divergence can be arbitrarily fast.

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.

Nowaday the linear quadratic optimization subject to positive linear time invariant (LTI) system constitute an interesting study considering it can become a mathematical model of variety of real problem whose variables have to nonnegative and trajectories generated by these variables must be nonnegative. In this paper we propose a method to generate an optimal control of linear quadratic optimization subject to positive linear time invariant (LTI) system. A sufficient condition that guarantee the existence of such optimal control is discussed.

In this paper we analyze the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley–Wiener space \(\mathcal{P}\mathcal{W}_{\pi }^{1}\). We study approximation processes, where the input signal is not directly used to generate the system output but instead a sequence of numbers is used that is generated from the input signal by measurement functionals. We consider classical sampling which corresponds to a pointwise evaluation of the signal, as well as several more general measurement functionals. We show that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling. However, if more general measurement functionals are considered, a stable approximation is possible if oversampling is used. Further, we show that without oversampling we have divergence for a large class of practically relevant measurement procedures.

Chapter 13 - LTI Systems: Convolution, Correlation, Coherence, and the Hilbert Transform

A new implementation of frequency domain online test approach is proposed for general digital linear time-invariant(LTI) systems. An analytical signal processing approach is used to extract the online test relation of LTI systems in frequency domain. An optimal implementation of the developed technique is proposed using sliding Goertzel DFT to reduce area and computational overheads with minimum error detection latency. Finite-precision analysis is done for fixed-point implementation of the proposed technique. The overhead is fixed and does not increase as complexity of LTI system increases. Simulation of fault detection capability is done for different filters using developed FLI fault injector platform.

Stepped chirp signal is widely employed in high resolution radar systems. Conventional spectrum reconstruction methods are sensitive to system errors. In this paper, a novel range profile synthesis algorithm is proposed based on multichannel blind deconvolution (MBD) theory. The received signals are modelled as the outputs of a series of linear time-invariant (LTI) systems with a common unknown input. In the presence of system errors, the impulse responses of the LTI systems are also unknown. Thus, range profile synthesis is formulated as a MBD problem. Regularizations are introduced to solve the ill-posedness. Alternating optimization method is used to solve the problem. Simulation results show that the proposed method is able to provide accurate range profile estimations. (4 pages)