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A methodology is proposed for the design of sampled-data fixed-order decentralised controllers for Multiple Input Multiple Output (MIMO) Linear Time-Invariant (LTI) time-delay systems. Imperfections in the communication links between continuous-time plants and controllers arising due to transmission time-delays, aperiodic sampling, and asynchronous sensors and actuators are considered. We model the errors induced due to the control imperfections using an operator approach leading to a simple L 2 stability criterion. A frequency domain-based direct optimisation approach towards controller design is proposed in this paper. This approach relies on the minimisation of cost functions, for stability and robustness against control imperfections, as a function of the controller or design parameters. First, the proposed method towards controller design is applied to generic MIMO LTI systems with time-delays. Second, when the delay system to be controlled has the structure of a network of coupled quasi-identical subsystems, we use a scalable algorithm to design identical decentralised controllers through network structure exploitation. Quasi-identical subsystems are identical subsystems that have non-identical uncertainties or control imperfections. By exploiting the structure, we improve the computational efficiency and scalability with the number of subsystems. The methodology has been implemented in a publicly available software, which supports system models in terms of delay differential algebraic equations. Finally, the effectiveness of the methodology is illustrated using a numerical example. [ABSTRACT FROM AUTHOR]

A novel approach for the design of delayed output-feedback controllers that optimize a quadratic cost function is presented. The proposal is based on the formulation of the optimization of the quadratic performance criterion as a minimization problem whose objective function is given in terms of the so-called delay Lyapunov matrix. The computation of the delay Lyapunov matrix sensitivity with respect to the controller gains allows obtaining the gradient of the objective function and solving the minimization problem within a gradient-based optimization framework. The potential of the approach is shown by some examples of practical relevance. [ABSTRACT FROM AUTHOR]

This paper is concerned with an eigenvalue based stabilization method for linear autonomous delay systems with uncertain parameters, modeled by a random vector. The dependence of the characteristic matrix on the uncertain parameters can be nonlinear; also delay values can be subject to uncertainty. Unlike the stability optimization methods for deterministic problems, which minimize the spectral abscissa, our approach shows better robust properties based on a more realistic model, where the uncertainty is taken into account by minimizing an objective function, consisting of the mean of the spectral abscissa with a variance penalty. The properties of the spectrum of delay differential algebraic equations of retarded type are analyzed and an integration method to compute the mean and variance of the spectral abscissa, which describes the stability property of the system, is presented. Subsequently the analysis of the behavior of the gradient of the characteristic roots is addressed, and a suitable explicit formula of the gradient of the objective function is presented. Finally, for designing the controller a suitable optimization routine, requiring the objective function and its gradient, is presented. The efficacy of the method is illustrated with numerical examples, including a model of an experimental heat-exchanger. The algorithms developed are publicly available. [ABSTRACT FROM AUTHOR]

In this paper, a structure-preserving model reduction approach for a class of delay differential equations is proposed. Benefits of this approach are, firstly, the fact that the delay nature of the system is preserved after reduction, secondly, that input–output stability properties are preserved and, thirdly, that a computable error bound reflecting the accuracy of the reduction is provided. These results are applicable to large-scale linear delay differential equations with constant delays, but also extensions to a class of nonlinear delay differential equations with time-varying delays are presented. The effectiveness of the results is evidenced by means of an illustrative example. [ABSTRACT FROM AUTHOR]

This paper presents a novel projection lemma based linear matrix inequality (LMI) framework to design reduced-order multi-objective H 2 / H ∞ controllers for linear time-invariant systems. This framework relies on a set of full-order H 2 / H ∞ controllers, which are used as parameters in sufficient LMIs for the reduced-order controller design. Continuous-time and discrete-time controller designs are treated in a unified fashion. It is theoretically and numerically demonstrated that the approach allows the computation of reduced-order controllers that are potentially less conservative than full-order designs resulting from well-known LMI approaches. Various comparisons with existing reduced-order controller design approaches illustrate the potential of the proposed framework of sufficient LMIs. [ABSTRACT FROM AUTHOR]

This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: We present a numerical method to plot the root locus of Single-Input–Single-Output (SISO) dead-time systems with respect to the controller gain or the system delay. We compute the trajectories of characteristic roots of the closed-loop system on a prescribed complex right half-plane. We calculate the starting, branch and boundary crossing roots of root-locus branches inside the region. We compute the root locus of each characteristic root based on a predictor–corrector type continuation method. To avoid the high sensitivity of roots with respect to the locus parameter in the neighborhood of branch points, the continuation method relies on a natural parameterization of the root-locus trajectory in terms of a distance in the (characteristic root, locus parameter)-space. The method is numerically stable for high order SISO dead-time systems. [Copyright &y& Elsevier]

Abstract: If is an eigenvalue of a time-delay system for the delay then is also an eigenvalue for the delays , for any . We investigate the sensitivity, periodicity and invariance properties of the root for the case that is a double eigenvalue for some . It turns out that under natural conditions (the condition that the root exhibits the completely regular splitting property if the delay is perturbed), the presence of a double imaginary root for some delay implies that is a simple root for the other delays , . Moreover, we show how to characterize the root locus around . The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue. [Copyright &y& Elsevier]

Abstract: The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important from a robust stability point of view. In this paper we present an accurate method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectrum and the level sets of an appropriately defined complex function. The computation is done in two steps. In the prediction step, an approximation of the pseudospectral is obtained based on a rational approximation of the characteristic matrix and the application of a bisection algorithm. Each step in this bisection algorithm relies on checking the presence of the imaginary axis eigenvalues of a complex matrix, similar to the delay free case. In the corrector step, the approximate pseudospectral abscissa is corrected to any given accuracy, by solving a set of nonlinear equations that characterizes the extreme points in the pseudospectrum contours. [Copyright &y& Elsevier]

Abstract: A novel method for the determination of controller parameters in a broad class of linear control systems affected by time-delays is presented. This method is based on an appropriate shaping of the spectrum of the closed-loop system. Its application follows two steps. First, a number of rightmost poles, smaller than the number of controller parameters, are directly assigned. This leads to constraints on the controller parameters. By using algebraic techniques a complete parametrization of all controllers satisfying these constraints is obtained. In the second step, the remaining degrees of freedom in the parameter space are used to shift the remaining part of the system spectrum as far to the left as possible. This corresponds to an optimization problem involving a nonsmooth, non-convex objective function. An extensive real-plant example is presented to demonstrate the effectiveness and applicability of the procedure. [Copyright &y& Elsevier]

Abstract: This paper points out specific stability problems of the state derivative and proportional–derivative state feedback which may result from the presence of otherwise negligible delays in the control loops. It is shown that the stability problems arise from the neutral character of the closed-loop system with the state derivative action under small delay perturbations. As the main result, a stability criterion is derived for a safe implementation of the state derivative feedback. Furthermore, a scheme including filtered state derivative feedback is proposed to overcome the destabilizing effects of small feedback delays. Two application examples are presented. From a theoretical point of view, the main contribution of the paper lies in the stability theory of linear neutral equations with dependent delays, subjected to delay perturbations. [Copyright &y& Elsevier]

Abstract: Definitions for pseudospectra and stability radii of an analytic matrix function are given, where the structure of the function is exploited. Various perturbation measures are considered and computationally tractable formulae are derived. The results are applied to a class of retarded delay differential equations. Special properties of the pseudospectra of such equations are determined and illustrated. [Copyright &y& Elsevier]

Abstract: An eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented, which extends the recently developed continuous pole placement method for delay equations of retarded type. The approach consists of two steps. First the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots, which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation. No restrictions are made on the dimension of the system and the number of delays. Also the effect of small delay perturbations is explicitly taken into account. As a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots. As a second step, stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi-continuous way, by applying small changes to the controller parameters, and meanwhile monitoring other characteristic roots with a large real part. A numerical example is presented. [Copyright &y& Elsevier]

In this paper the prescribed-time output feedback stabilization of linear systems is addressed using both a functional observer, which includes the reduced-order observer as a special case, and a dual observer. The approach is built upon smooth periodic delayed feedback which is a special case of linear periodic time-varying feedback. In contrast to existing periodic control approaches, the proposed feedback achieves prescribed-time convergence of the state to zero rather than asymptotic convergence. At the same time its synthesis requires solving algebraic equations rather than periodic differential equations. Compared with full-order observer based output feedback, the dimensions of the dynamic equations in the reduced-order observer and dual observer based output feedback are respectively n − p and n − m , where n , m and p are respectively the number of state variables, inputs and outputs, which indicates that the established controllers are less expensive than the full-order observer based controllers in implementations. A numerical example is given to verify the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

We consider a recursion formula for multi-dimensional powers of a finite set of matrices, which can be interpreted as a natural generalization of the celebrated Cayley–Hamilton theorem, and we show how it allows to solve an algebraic decision problem on a semigroup of matrices, which bears similarities to the observability problem of a switched linear system. This problem appears in the computation of the H 2 norm of a stable system described by a class of linear time-invariant delay differential equations (DDAEs) with multiple delays. The H 2 norm of a DDAE may not be finite even if there are seemingly no direct feedthrough terms. We show that necessary and sufficient conditions for a finite H 2 norm consist of an infinite number of linear equations to be satisfied, inducing the algebraic decision problem, and that using the generalized Cayley–Hamilton theorem checking these conditions can be turned into a check of a finite number of equations. We conclude with some comments on the computation of the H 2 norm whenever it is finite and by stating an open problem. [ABSTRACT FROM AUTHOR]

The prediction of chatter-free machining parameters is not reliable for industrial applications without guaranteed robustness against modeling uncertainties. Measurement inaccuracies, fitting uncertainties, simplifications and modeling errors typically lead to a mismatch between the mathematical model and the real physical system. This paper presents the robust stability analysis of milling operations based on a pseudospectral approach to take into account the effect of bounded parametric uncertainties both in the cutting coefficients and in the modal parameters. In order to make the predictions more accurate, the operational modal analysis of the spindle during rotation was conducted. The natural frequencies and damping ratios of the dominant vibration modes were identified from the impact tests of the rotating tool at different spindle speeds. The uncertainties of the fitted parameters were included in the computation of the pseudospectral radius of the monodromy operator of the time-periodic system. The solver is tested in a case study and experimental chatter tests are also included for demonstration purposes. • A robust pseudospectral method is presented for milling operations considering uncertainties in both modal parameters and cutting coefficients. • The stability analysis is based on a collocation method using Chebyshev polynomials. • Modal parameters of the rotating spindle are determined by operational modal analysis. • A case study with experimental chatter tests is included. [ABSTRACT FROM AUTHOR]

In this paper, the stabilization problem of neutral-type linear time-delay systems with both state and input delays is studied. A pseudo-predictor feedback (PPF) approach is established to predict the future states such that the input delay is compensated properly. It is shown that the considered system can be stabilized by the PPF controller if and only if an associated integral delay system is asymptotically stable. Moreover, it is proved that the PPF controller can be safely implemented without involving any input filters. The computation of the feedback gains problem is successfully converted to a simultaneous stabilization problem, which is solved by adopting a direct optimization approach. The proposed PPF approach is also extended to neutral-type time-varying time-delay systems. A numerical example is carried out to illustrate the effectiveness of the proposed controllers. [ABSTRACT FROM AUTHOR]

We study the strong ℋ 2 norm of systems modeled by semi-explicit Delay Differential Algebraic Equations (DDAEs). We recall that the finiteness of the strong ℋ 2 norm is linked to an algebraic decision problem that can be solved by checking a finite numbers of equalities. We first improve the verification of the finiteness condition. In particular, the complexity of our new condition removes a dependency on the number of delays. We also show that, without imposing further conditions on the system, the number of checks cannot be further reduced. The methodology relies on interpreting the verification of the finiteness conditions in terms of a Polynomial Identity Testing problem. Second we show, in a constructive way, that if the strong ℋ 2 norm is finite, the system can always be transformed into a regular neutral-type system with the same ℋ 2 norm, without derivatives in the input or in the output equations. This result closes a gap in the literature as such a transformation was known to exist only under additional assumptions on the system. The transformation enables the computation of the strong ℋ 2 norm using delay Lyapunov matrices. Illustrative examples are provided throughout the paper. [ABSTRACT FROM AUTHOR]