Computation of transfer function matrices of periodic systems

We present a numerical approach to evaluate the transfer function matrices of a periodic system corresponding to lifted state-space representations as constant systems. The proposed pole-zero method determines each entry of the transfer function matrix in a minimal zeros-poles-gain representation. A basic computation is the minimal realization of special single-input single-output periodic systems, for which both balancing-related as well as orthogonal periodic Kalman forms based algorithms can be employed. The main computational ingredient to compute poles is the extended periodic real Schur form of a periodic matrix. This form also underlies the solution of periodic Lyapunov equations when computing minimal realizations via balancing-related techniques. To compute zeros and gains, numerically stable fast algorithms are proposed, which are specially tailored to particular single-input single-output periodic systems. The new method relies exclusively on reliable numerical computations and is well suited for robust software implementations. Numerical examples computed with MATLAB-based implementations show the applicability of the proposed method to high-order periodic systems.     

Computation of minimal periodic realizations of transfer-function matrices

We present a numerical approach to compute a minimal periodic state-space realization of a transfer-function matrix corresponding to a lifted state-space representation. The proposed method determines a realization with time-varying state dimensions by using exclusively orthogonal transformations. The new method is numerically reliable, computationally efficient and thus well suited for robust software implementations.     

Balanced truncation reduced models of quadratic-bilinear systems in time interval

Accumulative error along with time impedes the application of many computational methods on high-precision simulation; therefore, time-interval methods emerge. In this paper, a new model reduction method of quadratic-bilinear (QB) systems based on time-interval Gramians (TIGs) is presented. The conditions for generalized Lyapunov equations whose solutions are exactly TIGs to be solvable are derived. Lyapunov stability and error bound are discussed to demonstrate the succession and advance of time-interval balanced truncation (TIBT). The accuracy and robustness are improved, as is illustrated in numerical results.     

Bifurcation and chaos of a non-autonomous rotational machine systems

In this paper, complex dynamic behaviors of the centrifugal flywheel governor systems are studied. We go deeper investigating the stability of the equilibrium points in the centrifugal flywheel governor system. These systems have a rich variety of non-linear behaviors, which are investigated here by numerically integrating the Lagrangian equations of motion. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of non-linear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems. In addition, the methods and conclusions would be useful for rotational machines designers.     

Model matching for discrete-time periodic systems with time-varying relative degree and order

This paper extends the well-known solution for the linear time invariant model matching problem to discrete-time periodic systems with time-varying relative degree and order. It is shown that a key step to the design of a periodic output feedback controller is to compute the stable inverse of the periodic system. Using input–output equations, this problem is solved and model matching is achieved with system internal stability.     

An Improved Eulerian Approach for the Finite Time Lyapunov Exponent

We propose a new Eulerian numerical approach to compute the Jacobian of flow maps in continuous dynamical systems and subsequently the so-called finite time Lyapunov exponent (FTLE) for Lagrangian coherent structure extraction. The original approach computes the flow map and then numerically determines the Jacobian of the map using finite differences. The new algorithm improves the original Eulerian formulation so that we first obtain partial differential equations for each component of the Jacobian and then solve these equations to obtain the required Jacobian. For periodic dynamical systems, based on the time doubling technique developed for computing the longtime flow map, we also propose a new efficient iterative method to compute the Jacobian of the longtime flow map. Numerical examples will demonstrate that our new proposed approach is more accurate than the original one in computing the Jacobian and thus the FTLE field, especially near the FTLE ridges.     

A novel extended precise integration method based on Fourier series expansion for the H2-norm of linear time-varying periodic systems

A new reliable algorithm for computing the H₂-norm of linear time-varying periodic (LTP) systems via the periodic Lyapunov differential equation (PLDE) is proposed. By taking full advantage of the periodicity, the transition matrix of the underlying LTP system associated with the PLDE is effectively computed by developing a novel extended precise integration method based on Fourier series expansion, where the time-consuming work for the computation of the matrix exponential and its related integrals in every sub-interval is avoided. Then, a highly accurate and efficient algorithm for the PLDE is derived using the block form of the transition matrix. Thus, the H₂-norm is evaluated by solving a simple first-order ordinary differential equation. Finally, two numerical examples are presented and compared with other algorithms to verify the numerical accuracy and efficiency of the proposed algorithm.     

Reconstruction of the Fourier expansion of inputs of linear time-varying systems

In this paper we propose a general method to estimate periodic unknown input signals of finite-dimensional linear time-varying systems. We present an infinite-dimensional observer that reconstructs the coefficients of the Fourier decomposition of such systems. Although the overall system is infinite dimensional, convergence of the observer can be proven using a standard Lyapunov approach along with classic mathematical tools such as Cauchy series, Parseval equality, and compact embeddings of Hilbert spaces. Besides its low computational complexity and global convergence, this observer has the advantage of providing a simple asymptotic formula that is useful for tuning finite-dimensional filters. Two illustrative examples are presented.     

Balanced truncation of linear time-varying systems

In this paper, balanced truncation of linear time-varying systems is studied in discrete and continuous time. Based on relatively basic calculations with time-varying Lyapunov equations/inequalities we are able to derive both upper and lower error bounds for the truncated models. These results generalize well-known time-invariant formulas. The case of time-varying state dimension is considered. Input-output stability of all truncated balanced realizations is also proven. The method is finally successfully applied to a high-order model.     

New results on robust H ∞ filtering design for discrete-time piecewise linear delay systems

This paper revisits the problem of robust H _∞ filtering design for a class of discrete-time piecewise linear state-delayed systems. The state delay is assumed to be time-varying and of an interval-like type, which means that both the lower and upper bounds of the time-varying delay are available. The parameter uncertainties are assumed to have a structured linear fractional form. Based on a novel delay-dependent piecewise Lyapunov–Krasovskii functional combined with Finsler's Lemma, a new delay-dependent sufficient condition for robust H _∞ performance analysis is first derived and then the filter synthesis is developed. It is shown that by using a new linearisation technique, a unified framework can be developed so that both the full-order and reduced-order filters can be obtained by solving a set of linear matrix inequalities (LMIs), which are numerically efficient with commercially available software. Finally, a numerical example is provided to illustrate the effectiveness and less conservatism of the proposed approach.     

Fuzzy robust H ∞ filter design for nonlinear discrete-time systems with interval time delays

This article deals with the problem of H _∞ filter design for nonlinear discrete-time systems with norm-bounded parameter uncertainties and time-varying delays. A new Lyapunov function and free-weighting matrix method are used for filtering design, consequently, a delay-dependent design method is first proposed in terms of linear matrix inequalities, which produces a less conservative result. Finally, numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method.     

Exponential stability and robust H∞ control of a class of discrete-time switched non-linear systems with time-varying delays via T-S fuzzy model

This paper deals with stability and robust H_∞ control of discrete-time switched non-linear systems with time-varying delays. The T-S fuzzy models are utilised to represent each sub-non-linear system. Thus, with two level functions, namely, crisp switching functions and local fuzzy weighting functions, we introduce a discrete-time switched fuzzy systems, which inherently contain the features of the switched hybrid systems and T-S fuzzy systems. Piecewise fuzzy weighting-dependent Lyapunov–Krasovskii functionals (PFLKFs) and average dwell-time approach are utilised in this paper for the exponentially stability analysis and controller design, and with free fuzzy weighting matrix scheme, switching control laws are obtained such that H_∞ performance is satisfied. The conditions of stability and the control laws are given in the form of linear matrix inequalities (LMIs) that are numerically feasible. The state decay estimate is explicitly given. A numerical example and the control of delayed single link robot arm with uncertain part are given to demonstrate the efficiency of the proposed method.     

New delay-interval stability condition

The delay-dependent stability problem for systems with time-delay varying in an interval is addressed in this article. The new idea in this article is to connect two very efficient approaches: the discretised Lyapunov functional for systems with pointwise delay and the convex analysis for systems with time-varying delay. The proposed method is able to check the stability interval when the time-varying delay d(t) belongs to an interval [r,?τ]. The case of unstable delayed systems for r?=?0 is also treatable. The resulting criterion, expressed in terms of a convex optimisation problem, outperforms the existing ones in the literature, as illustrated by the numerical examples.     

How non-zero initial conditions affect the minimality of linear discrete-time systems

From the state-space approach to linear systems, promoted by Kalman, we learned that minimality is equivalent with reachability together with observability. Our past research on optimal reduced-order LQG controller synthesis revealed that if the initial conditions are non-zero, minimality is no longer equivalent with reachability together with observability. In the behavioural approach to linear systems promoted by Willems, that consider systems as exclusion laws, minimality is equivalent with observability. This article describes and explains in detail these apparently fundamental differences. Out of the discussion, the system properties weak reachability or excitability, and the dual property weak observability emerge. Weak reachability is weaker than reachability and becomes identical only if the initial conditions are empty or zero. Weak reachability together with observability is equivalent with minimality. Taking the behavioural systems point of view, minimality becomes equivalent with observability when the linear system is time invariant. This article also reveals the precise influence of a possibly stochastic initial state on the dimension of a minimal realisation. The issues raised in this article become especially apparent if linear time-varying systems (controllers) with time-varying dimensions are considered. Systems with time-varying dimensions play a major role in the realisation theory of computer algorithms. Moreover, they provide minimal realisations with smaller dimensions. Therefore, the results of this article are of practical importance for the minimal realisation of discrete-time (digital) controllers and computer algorithms with non-zero initial conditions. Theoretically, the results of this article generalise the minimality property to linear systems with time-varying dimensions and non-zero initial conditions.     

H∞ control problem of linear periodic piecewise time-delay systems

This paper investigates the H_∞ control problem based on exponential stability and weighted L₂-gain analyses for a class of continuous-time linear periodic piecewise systems with time delay. A periodic piecewise Lyapunov–Krasovskii functional is developed by integrating a discontinuous time-varying matrix function with two global terms. By applying the improved constraints to the stability and L₂-gain analyses, sufficient delay-dependent exponential stability and weighted L₂-gain criteria are proposed for the periodic piecewise time-delay system. Based on these analyses, an H_∞ control scheme is designed under the considerations of periodic state feedback control input and iterative optimisation. Finally, numerical examples are presented to illustrate the effectiveness of our proposed conditions.     

Strict Lyapunov functions for time-varying systems

Uniformly asymptotically stable periodic time-varying systems for which is known a Lyapunov function with a derivative along the trajectories non-positive and negative definite in the state variable on non-empty open intervals of the time are considered. For these systems, strict Lyapunov functions are constructed.     

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

On the stability of continuous-time adaptive controllers for pure delay systems

We analyse the stability properties of the linear retarded differential-difference equation (RDDE) that arises in the study of adaptive control of pure delay systems. Three different results are given. Firstly, using Floquet theory necessary and sufficient conditions for stability are established for the case of periodic signals of specified frequencies. Secondly, the theory of averaging is used to derive stability-instability conditions under slow adaptation. Finally, a globally stable modified adaptive regulator for systems with known, possibly time-varying, time delay is presented. Two alternative proofs of the latter results are given. One based on the method of Lyapunov functionals and the second one using a Razumikhin-type theorem.     

Almost sure stability of stochastic neutral Cohen–Grossberg neural networks with Lévy noise and time-varying delays

The almost sure stability for the stochastic neutral Cohen–Grossberg neural networks (SNCGNNs) with Lévy noise, time-varying delays, and Markovian switching would be deliberated in this article. By means of the nonnegative semimartingale convergence theorem (NSCT), the neutral Itô formula, M-matrix method, and selecting appropriate Lyapunov function, several almost sure stability criterions for the SNCGNNs could be derived. Moreover, according to the M-matrix theory, the upper bounds of the coefficients at any mode are given. Finally, two examples and numerical simulations verify the correctness of theoretical analysis for the stability criterions proposed in the article.