Group Codes and Behaviors

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.

Interval observer design for unknown input estimation of linear time-invariant discrete-time 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.

An algebraic theory of linear periodic discrete-time (LPDT) systems is developed in terms of a periodic polynomial matrix description (PMD) and in a dynamics (module) structure defined over a non-commutative, non-integral and non-principal ideal ring /spl Rscr/ (the ring of periodic polynomials). Various properties of matrices and modules over /spl Rscr/ are explored, and links between algebraic properties of LPDT systems such as reachability, controllability are established directly in the periodic PMD and /spl Rscr/-module descriptions; i.e., without using the technique of lifting a LPDT system to its associated linear time-invariant (LTI) ones. The advantage of such characterizations is to show that LPDT systems are not index-invariant only for a class-that we characterize-of non-reversible time systems and that, except for this class of systems, the parametrization of all stabilizing controllers can be constructed similarly to LTI systems. Furthermore, one shows that for the class of non-reversible time systems above-mentioned, it is not possible to extract a canonical part like for LTI systems.

This paper is concerned with the decentralized H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> controller synthesis problem for discrete-time linear time-invariant (LTI) systems. In spite of intensive research efforts over the last several decades, this problem is believed to be non-convex and still outstanding in general. Therefore most existing approaches resort to heuristic optimization algorithms that do not allow us to draw any definite conclusion on the quality of the designed controllers. To get around this difficult situation, in this paper, we propose convex optimization procedures for computing lower bounds of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance that is achievable via decentralized LTI controllers of any order. In particular, we show that sharpened lower bounds can be obtained by making good use of structures of the LTI plant typically observed in the decentralized control setting. We illustrate via numerical examples that these lower bounds are indeed useful to ensure the good quality of decentralized controllers designed by a heuristic optimization.

We present a couple of data-based methods to analyze the state controllability and state observability of discrete-time Linear time-Invariant (LTI) systems with time-delays, which have unknown parameter matrices. They first augment the system into a high dimensional LTI model, then apply the measured state and output data to directly build the controllability and observability matrices respectively of this high dimensional model, whose ranks are used as the criteria of the corresponding properties of the system before augmentation. These data-based methods have low computational load and calculation complexity.

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.

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.

Chapter 4 - Frequency Analysis: The Fourier Series

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

A discrete-time time-invariant linear system (DILS) is considered which has resulted from the discretization of a continuous-time time-invariant linear system (CILS) by the zero order hold. In this system, the system and input matrices are given by exp(AT) and $ where A and B are the system and input matrices of the CILS, and T is a sampling interval. Since they involve the matrix exponential, their computations are not easy. First, on the assumption that the Kalman canonical form of the CILS is sought, a formal Kalman canonical form of the DILS is determined. Second, with the use of this form and the decomposition of A into the projectors onto the generalized eigenspaces, an algorithm is presented which computes the system and input matrices of the DlLS.

This paper studies the problem of stability analysis and controller design for discrete-time linear time invariant (LTI) systems with state saturation. Both full state saturation and partial state saturation are considered. Firstly, a new system model is constructed for solving the key problem. Then, a new method is presented for estimating the domain of attraction of the origin for a system with state saturation. Based on this method, an LMI-based ( linear matrix inequality) iterative algorithms is proposed for determining if a given ellipsoid is contractively invariant. Moreover, an LMI-based algorithm is developed for designing dynamic output-feedback controllers which guarantee that the domain of attraction of the origin for the closed-loop system is as large as possible. An example is given to illustrate the efficiency of the design method.

There is a well-developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller. This theory describes limitations on achievable performance present when the plant has non-minimum phase zeros, unstable poles, and/or time delays. The parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants. This paper develops a theory of design limitations for sampled-data feedback systems wherein the response of the analogue system output is considered. This is done using the fact that the steady-state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows fundamental sensitivity and complementary sensitivity functions that re...

This paper presents two mathematical conceptualizations of driving-point system-function measurement of linear time-invariant (LTI) systems with lumped-parameter description. Whereas the formalization of this dual pair of Source-Measure Units (SMU) has been - for the twin purposes of ease of demonstration and ready cognizance - effected with a specific relevance to electrical quantities, the underlying concept of system-function is sufficiently general to be universally applicable to all classes of LTI systems. The formulation has been done in the time-domain, thus retaining full immediacy with the real system variables. The outcome of the measurement, therefore, upon due rationalization, emerges as a rationic polynomial. The natural frequencies of the system are then extractible from the characteristic polynomial, which latter is contained in the measured rationic. The `naturality' of the system-under-test as observed from a specified port-of-access is thus laid bare. The method is illustrated through examples, which also show the invariance of the system characteristic polynomial with respect to the point of view of the observer, except in cases of pole-zero cancellation. An alternative interpretation of the singularities of the system-function has also been furnished.