Perturbation bounds for root-clustering of linear continuous-time systems

Sufficient conditions are obtained for the eigenvalues of a continuous system to stay in a specified subregion of the left half complex plane in spite of perturbations in the nominal system. Measures are determined for linear state space models using Lyapunov stability analysis. The bounds are derived for both highly and weakly structured perturbations. Bounds are also found assuming a number of uncertain parameters of the system perturbed independently and thus various elements of the system matrix are perturbed dependently. The results of the structured perturbations arc then extended to interval matrices.     

Perturbation bounds for root-clustering of linear systems in aspecified second order subregion

Sufficient bounds for structured and unstructured uncertainties for root-clustering in a specified second order subregion of the complex plane, for both continuous-time and discrete-time systems, are given using the generalized Lyapunov theory. Furthermore, for unstructured uncertainties, a still less conservative result is obtained by shifting the center or focus of the subregion along the real axis to the origin and by applying root-clustering to the “shifted eigenvalue” system matrix, which is obtained by shifting the eigenvalues of the system matrix correspondingly     

Improved bounds for linear discrete-time systems with structuredperturbations

A time-domain analysis of the stability robustness of linear discrete-time systems subject to time-varying structured perturbations is considered. The Lyapunov stability theory is used to obtain bounds on the perturbation such that the systems remain stable. It is shown that these bounds are less conservative than the existing ones. This is illustrated via two numerical examples     

Covariance bounds for discrete-time linear systems with time-varying parameter uncertainty

This paper deals with the computation of upper bounds for the state covariance matrix of discrete-time linear systems subject to stochastic excitation and additive time-varying uncertainty in the system dynamic matrix. Such upper bounds are obtained as the stabilizing solutions of suitable H ∞ -type Riccati equations. A necessary and sufficient condition for the existence of such solutions is given in terms of the H ∞ -norm of a suitable transfer function. As for the computation of the optimal bound, it is demonstrated that the bounds are a convex function of a scalar parameter, so that efficient numerical schemes can be worked out.     

Comments on “Perturbation bounds for root-clustering oflinear systems in a specified second order subregion”

This paper comments on the results of the above mentioned paper by Bakker-Luo-Johnson (ibid. vol.40 (1995)). We note that Theorems 3.1 and 4.1 are incorrectly stated, i.e., they are not valid for the non-Ω-transformable regions. The results cannot cover the ride quality region listed in Table 1 since it is a non-Ω-transformable region     

Stability robustness bounds for discrete-time linear regulators

Stability robustness analysis and design for linear multivariable discrete-time systems with bounded uncertainties are discussed. Robust stability of the full-state feedback linear quadratic (LQ) regulator in the presence of perturbations (modelling errors) of the system matrices is investigated. These results are based on a recently developed bound on elemental (structured) time-varying perturbations of an asymptotically stable linear time-invariant discrete-time system. Lyapunov theory and singular value decomposition techniques are employed in deriving these bounds. Extensions of these results to linear stochastic systems with the Kalman filter as the stale estimator (LQG regulators) and to reduced-order dynamic compensator feedback are described. A state feedback control design method is presented for LQ regulators, using a quantitative measure called the Stability Robustness Index. Simple examples illustrate these new results.     

Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems

The stability robustness of linear discrete-time systems in the time domain is addressed using the Lyapunov approach. Bounds on linear time-varying perturbations that maintain the stability of an asymptotically stable linear time-invariant discrete-time nominal system are obtained for both structured and unstructured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structured perturbation case is extended to the stability analysis of interval matrices.     

New robust stability bounds of linear discrete-time systems with time-varying uncertainties

This paper presents new uncertain parameter variation bounds for linear discrete-time systems to preserve asymptotic stability. The Lyapunov method is utilized to treat both structured and unstructured uncertainties, and the results are optimized with respect to a parameter in the inequality used. When applied to examples considered by previous authors, our results give less conservative bounds.     

On Lyapunov stability robustness bounds for linear discrete-time systems with structured time-varying uncertainty

In this paper we analyse the stability robustness of linear discrete-time systems which are described by a state-space model but are perturbed with structured time-varying uncertainty. We present new Lyapunov stability robustness bounds in which the freedom of the matrix Q is utilized more effectively than that used by Kolla et al. (1989) to obtain a larger bound of tolerable time-varying uncertainty, and the similarity transformation is employed more directly and usefully than that proposed by Kolla and Farison (1990) to reduce conservation. Further, the relationship between the matrix Q and the similarity transformation matrix M is given. Improvements are illustrated by an application of our proposed method to a macroeconomic system     

Almost sure estimation and synthesis of ultimate bounds for discrete-time Markov jump linear systems

This paper is concerned with the problems of almost sure ultimate bound estimation and controller design for Markov jump linear systems with bounded stochastic disturbances. By utilising a Lyapunov-based scheme proposed in this paper, an almost sure estimation of ellipsoidal ultimate bound (EUB) of the system is obtained through tractable matrix inequalities. On the basis of the estimation results, the problem of designing mode-dependent state feedback controllers that make the closed-loop system admit a prescribed ellipsoid as an EUB is considered. The obtained results on estimation and synthesis are then extended to the case of systems with deficient mode information. Finally, a practical example in DC motor devices is presented to demonstrate the applicability of the obtained results.     

Correlation bounds for discrete-time systems with banded dynamics

We consider the steady-state error covariance for a discrete-time system with banded dynamics. Such systems frequently arise from the spatial and temporal discretization of partial differential equations. In such systems, the magnitudes of the entries of the steady-state covariance matrix typically decrease as the distance from the diagonal increases. We obtain a bound on the entries of the covariance matrix beyond a given distance from the diagonal.     

Receding-horizon estimation for discrete-time linear systems

The problem of estimating the state of a discrete-time linear system can be addressed by minimizing an estimation cost function dependent on a batch of recent measure and input vectors. This problem has been solved by introducing a receding-horizon objective function that includes also a weighted penalty term related to the prediction of the state. For such an estimator, convergence results and unbiasedness properties have been proved. The issues concerning the design of this filter are discussed in terms of the choice of the free parameters in the cost function. The performance of the proposed receding-horizon filter is evaluated and compared with other techniques by means of a numerical example.     

Reduced-order observers for linear discrete-time systems

Luenberger's observer is considered as an alternate to the Kalman filter for obtaining state estimates in linear discrete-time stochastic systems. An interesting new solution to the problem of constructing optimal and suboptimal reduced-order observers is presented. The solution contains as special cases both Kalman's optimal filter and the optimal minimal-order observer of Leondes and Novak. Also, the Tse and Athans observer is obtained as a special case of the reduced-order observer solution.     

Implicit linear discrete-time systems

This paper studies various properties of implicit linear discrete-time systems given by a linear difference equationEx _k+1 =Fx _k +Gu _k . The topics considered include a basic characterization of these subspaces which describe acceptance of all input sequences, the uniqueness property and regularity, and the notion of controllability. This work was performed under the auspices of Fund RP.I.02: Teoria sterowania i optymalizacji ciągłych układów dynamicznych i procesów dyskretnych.     

Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems

This paper considers the robustness of stochastic stability of Markovian jump linear systems in continuous- and discrete-time with respect to their transition rates and probabilities, respectively. The continuous-time (discrete-time) system is described via a continuous-valued state vector and a discrete-valued mode which varies according to a Markov process (chain). By using stochastic Lyapunov function approach and Kronecker product transformation techniques, sufficient conditions are obtained for the robust stochastic stability of the underlying systems, which are in terms of upper bounds on the perturbed transition rates and probabilities. Analytical expressions are derived for scalar systems, which are straightforward to use. Numerical examples are presented to show the potential of the proposed techniques.     

Perturbation bounds for robust stability in linear state space models

This paper addresses the aspect of stability robustness of linear systems in the time domain. Upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability, both for structured as well as unstructured perturbations using the Lyapunov approach. For structured perturbation the resulting bound is such that it garners the structural information about the nominal (as well as the perturbation) matrix into a single unified expression. In the case of unstructured perturbations, special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound (without the need to solve the Lyapunov equation). Improvement of the proposed measures is illustrated with the help of examples.     

Robust linear filtering for discrete-time hybrid Markov linear systems

In this paper we consider the robust linear filtering of hybrid discrete-time Markovian jump linear systems. We assume that only an output of the system is available, and therefore the values of the jump parameter are not known. It is desired to design a dynamic linear filter such that the closed loop system is mean square stable and minimizes the stationary expected value of the square error. We consider uncertainties on the parameters of the possible modes of operation of the system. A linear matrix inequalities (LMI) formulation is proposed to solve the problem. For the case in which there are no uncertainties on the modes of operation of the system, we show that the LMI formulation provides a filter with the same stationary mean square error as the one obtained from the Riccati equation approach.     

Nonpassivity of linear discrete-time systems

A strictly causal linear discrete-time system cannot be passive unless it is the zero system. An intrinsically unstable system (defined in the text) can not be passive. These statements are true whether the system is time-invariant or time-varying.     

All covariance controllers for linear discrete-time systems

The authors characterize the set of covariances that a linear discrete-time plant with a specified-order controller can have. The controllers that assign such covariances to any linear discrete-time system are given explicitly in closed form. The freedom in these covariance controllers is explicit and is parameterized by two orthogonal matrices. By appropriately choosing these free parameters, additional system objectives can be achieved without altering the state covariance, and the stability of the closed-loop system is guaranteed