Discrete-time Lyapunov Equations Research Articles

On the stability of delayed linear discrete-time systems with periodic coefficients

Stability estimates are obtained for delayed linear periodic discrete-time systems. Bounds on the decay of the solution are derived via a suitable Lyapunov–Krasovskii-type functional and the solvability of some periodic discrete-time Lyapunov equations.

Periodically refreshed quantum thermal machines

We introduce unique class of cyclic quantum thermal machines (QTMs) which can maximize their performance at the finite value of cycle duration τ where they are most irreversible. These QTMs are based on single-stroke thermodynamic cycles realized by the non-equilibrium steady state (NESS) of the so-called Periodically Refreshed Baths (PReB) process. We find that such QTMs can interpolate between standard collisional QTMs, which consider repeated interactions with single-site environments, and autonomous QTMs operated by simultaneous coupling to multiple macroscopic baths. We discuss the physical realization of such processes and show that their implementation requires a finite number of copies of the baths. Interestingly, maximizing performance by operating in the most irreversible point as a function of τ comes at the cost of increasing the complexity of realizing such a regime, the latter quantified by the increase in the number of copies of baths required. We demonstrate this physics considering a simple example. We also introduce an elegant description of the PReB process for Gaussian systems in terms of a discrete-time Lyapunov equation. Further, our analysis also reveals interesting connections with Zeno and anti-Zeno effects.

Consensus-based distributed filtering with fusion step analysis

For consensus on measurement-based distributed filtering (CMDF), through infinite consensus fusion operations during each sampling interval, each node in the sensor network can achieve optimal filtering performance with centralized filtering. However, due to the limited communication resources in physical systems, the number of fusion steps cannot be infinite. To deal with this issue, the present paper analyzes the performance of CMDF with finite consensus fusion operations. First, by introducing a modified discrete-time algebraic Riccati equation and several novel techniques, the convergence of the estimation error covariance matrix of each sensor is guaranteed under a collective observability condition. In particular, the steady-state covariance matrix can be simplified as the solution to a discrete-time Lyapunov equation. Moreover, the performance degradation induced by reduced fusion frequency is obtained in closed form, which establishes an analytical relation between the performance of the CMDF with finite fusion steps and that of centralized filtering. Meanwhile, it provides a trade-off between the filtering performance and the communication cost. Furthermore, it is shown that the steady-state estimation error covariance matrix exponentially converges to the centralized optimal steady-state covariance matrix with fusion operations tending to infinity during each sampling interval. Finally, the theoretical results are verified with illustrative numerical experiments.

Stability Analysis for Input Saturated Discrete-Time Switched Systems With Average Dwell-Time

This paper studies the stability analysis of the input saturated discrete-time switched systems with average dwell-time based on the parametric discrete-time Riccati equation. The state feedback controller and the observer-based output feedback controller are designed to guarantee the exponential stability of the closed-loop system. The proposed method is simple and easy to operate in practice. The designed controllers can be computed easily by solving the parametric discrete-time Lyapunov equation. The main advantages of this paper are that the stability analysis is based on the properties of the saturation function and limited structure information of the actuator saturation and that the designed controllers have good robustness to the structure uncertainty of the input saturation. The simulation results illustrate the effectiveness of the proposed methods.

Distributed Solver for Discrete-Time Lyapunov Equations Over Dynamic Networks With Linear Convergence Rate

The problem of solving discrete-time Lyapunov equations (DTLEs) is investigated over multiagent network systems, where each agent has access to its local information and communicates with its neighbors. To obtain a solution to DTLE, a distributed algorithm with uncoordinated constant step sizes is proposed over time-varying topologies. The convergence properties and the range of constant step sizes of the proposed algorithm are analyzed. Moreover, a linear convergence rate is proved and the convergence performances over dynamic networks are verified by numerical simulations.

A Theoretical Framework for Stability Regions for Standing Balance of Humanoids Based on Their LIPM Treatment

The aim of this paper is to construct a theoretical framework for stability analysis relevant to standing balance of humanoids on top of the linear inverted pendulum model, in which their dynamics between the center of mass (CoM) and the zero moment point (ZMP) is dealt with. Based on the well-known sufficient condition that the contact between the ground and the support leg is stable if the corresponding ZMP is always inside the supporting region, this paper aims at characterizing three types of the associated stability regions. More precisely, assuming no external force disturbances affecting the motion of the humanoids, the stability region of the initial CoM position and velocity values can be explicitly computed by solving a finite number of linear inequalities. The stability regions of time-invariant force disturbances such as impulsive force and constant force disturbances are also dealt with in this paper, where the former is exactly obtained through a finite number of linear inequalities while the latter is approximately derived by using an idea of truncation. Furthermore, time-varying force disturbances of finite energy and finite amplitude are concerned with, and their maximum admissible ${l} _{2}$ and ${l} _{\infty }$ norms are computed in this paper, where the former can be exactly obtained by solving the discrete-time Lyapunov equation while the latter is approximately derived through an idea of truncation. It is further shown for both the truncation ideas that the approximately obtained stability regions converge to the exact stability regions with an exponential order of ${N}$ , where ${N}$ is the truncation parameter. Finally, the effectiveness of the computation methods proposed in this paper is demonstrated through some simulation results.

Comparative Performance Evaluation of an Accuracy-Enhancing Lyapunov Solver

Lyapunov equations are key mathematical objects in systems theory, analysis and design of control systems, and in many applications, including balanced realization algorithms, procedures for reduced order models, Newton methods for algebraic Riccati equations, or stabilization algorithms. A new iterative accuracy-enhancing solver for both standard and generalized continuous- and discrete-time Lyapunov equations is proposed and investigated in this paper. The underlying algorithm and some technical details are summarized. At each iteration, the computed solution of a reduced Lyapunov equation serves as a correction term to refine the current solution of the initial equation. The best available algorithms for solving Lyapunov equations with dense matrices, employing the real Schur(-triangular) form of the coefficient matrices, are used. The reduction to Schur(-triangular) form has to be done only once, before starting the iterative process. The algorithm converges in very few iterations. The results obtained by solving series of numerically difficult examples derived from the SLICOT benchmark collections for Lyapunov equations are compared to the solutions returned by the MATLAB and SLICOT solvers. The new solver can be more accurate than these state-of-the-art solvers and requires little additional computational effort.

Efficient Techniques for Solving the Periodic Projected Lyapunov Equations and Model Reduction of Periodic Systems

We have presented the efficient techniques for the solutions of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form. These types of problems arise in model reduction and state feedback problems of periodic descriptor systems. Two most popular techniques to solve such Lyapunov equations iteratively are the low-rank alternating direction implicit (LR-ADI) method and the low-rank Smith method. The main contribution of this paper is to update the LR-ADI method by exploiting the ideas of the adaptive shift parameters computation and the efficient handling of complex shift parameters. These approaches efficiently reduce the computational cost with respect to time and memory. We also apply these iterative Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. We illustrate numerical results to show the performance and accuracy of the proposed methods.

On minimal rank solutions to symmetric Lyapunov equations in Euclidean Jordan algebra

The minimal rank solutions of continuous-time and discrete-time symmetric Lyapunov equations, which have important applications in dynamical systems, are generally difficult to achieve due to the involved rank minimization. By employing the decomposition techniques of Euclidean Jordan algebra and the symmetric Lyapunov operators, we show that in the setting of Euclidean Jordan algebra, these minimal rank solutions of both symmetric continuous-time and discrete-time Lyapunov equations are unique and can be exactly solved by the corresponding Schatten -norm () relaxation problems under some easy-to-check conditions. Moreover, both the upper and lower bounds for the minimal ranks are proposed.

Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

We discuss the numerical solution of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

A low-rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations

The squared Smith method is adapted to solve large-scale discrete-time Lyapunov matrix equations. The adaptation uses a Krylov subspace to generate the squared Smith iteration in a low-rank form. A restarting mechanism is employed to cope with the increase of memory storage of the Krylov basis. Theoretical aspects of the algorithm are presented. Several numerical illustrations are reported.

Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

This paper studies the iterative solutions of Lyapunov matrix equations associated with Itô stochastic systems having Markovian jump parameters. For the discrete-time case, when the associated stochastic system is mean square stable, two iterative algorithms with one in direct form and the other one in implicit form are established. The convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system. For the continuous-time case, a transformation is first performed so that it is transformed into an equivalent discrete-time Lyapunov equation. Then the iterative solution can be obtained by applying the iterative algorithm developed for discrete-time Lyapunov equation. Similar to the discrete-time case, an implicit iteration is also proposed for the continuous case. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms are analyzed and compared. Numerical examples are worked out to validate the effectiveness of the proposed algorithms.

Synthesis and operation of an FFT-decoupled fixed-order reversed-field pinch plasma control system based on identification data

Recent developments and applications of system identification methods for the reversed-field pinch (RFP) machine EXTRAP T2R have yielded plasma response parameters for decoupled dynamics. These data sets are fundamental for a real-time implementable fast Fourier transform (FFT) decoupled discrete-time fixed-order strongly stabilizing synthesis as described in this work. Robustness is assessed over the data set by bootstrap calculation of the sensitivity transfer function worst-case -gain distribution. Output tracking and magnetohydrodynamic mode m = 1 tracking are considered in the same framework simply as two distinct weighted traces of a performance channel output-covariance matrix as derived from the closed-loop discrete-time Lyapunov equation. The behaviour of the resulting multivariable controller is investigated with dedicated T2R experiments.

Iterative solutions of coupled discrete Markovian jump Lyapunov equations

In this paper, two iterative methods are given to solve coupled discrete Markovian jump Lyapunov equations arising from the stability analysis of discrete-time Markovian jump systems. When such equations have unique positive definite solutions, a sufficient condition for the convergence of iterates for one iterative method is derived. When such equations only have unique solutions, but not necessarily positive definite, the above iterative method may fail and an alterative iterative method is provided by using the solutions of discrete-time Lyapunov equations as its intermediate steps. A sufficient condition is also presented for the convergence of the iterates.

Norm estimates of the Fourier series coefficients of the matrix resolvent

The resolvent over the unit circle of a matrix A can be decomposed in a general form of a Fourier series. This note shows that the Fourier coefficients decrease to zero with a rate depending essentially on the norm of the solution of the discrete-time Lyapunov equation applied to A .

A note on the Lyapunov stability of periodic discrete-time systems

We characterize the stability of discrete-time Lyapunov equations with periodic coefficients. The characterization can be seen as the analog of the classical stability theorem of Lyapunov equations with constant coefficients. It involves quantities readily computable with good accuracy.

UDU Factored Discrete-Time Lyapunov Recursions Solve Optimal Reduced-Order LQG Problems

A new algorithm is presented to solve both the finitehorizon time-varying and infinite-horizon timeinvariant discrete-time optimal reduced-order linear quadratic Gaussian (LQG) problem. In both cases the first order necessary optimality conditions can be represented by two non-linearly coupled discrete-time Lyapunov equations, which run forward and backward in discrete time. The algorithm iterates these two equations forward and backward in discrete time, respectively, until they converge. In the finite-horizon time varying case the iterations start from boundary conditions and the forward and backward in time recursions are repeated until they converge. The discrete-time recursions are suitable for UDU factorisation. It is illustrated how UDU factorisation may increase both the numerical efficiency and accuracy of the recursions. By means of several numerical examples and the benchmark problem proposed by the European Journal of Control, the results obtained with the new algorithm are compared to results obtained with algorithms that iterate the strengthened discrete-time optimal projection equations forward and backward in time. The convergence properties are illustrated to be comparable. Especially when the reduced compensator dimensions are significantly smaller than those of the controlled system, the algorithm presented in this paper is more efficient.

Estimates from the discrete-time Lyapunov equation

The solutions of the discrete-time Lyapunov equation applied to a matrix A with no eigenvalues on a circle are characterized. The equation is used to derive bounds for the powers of A.

Numerically robust delta-domain solutions to discrete-time Lyapunov equations

A problem of numerical conditioning of a special kind of discrete-time Lyapunov equations is considered. It is assumed that a discretisation procedure equipped with the zero-order holder mechanism is utilised that leads to the data matrices that are affinely related to the sampling period and matrices that are independent or linearly related to the squared sampling period. It is shown that common forward shift operator techniques for solving these equations become ill-conditioned for a sufficiently small sampling period and that numerical robustness and reliability of computations can be significantly improved via utilising the so-called delta operator form of the origin equations.

On solution bounds for three types of Lyapunov matrix equations: continuous, discrete and unified equations

An approach is proposed to obtain a series of solution bounds for the unified algebraic Lyapunov matrix equation by making use of its feature, i.e., a mediator between the continuous-time and discrete-time Lyapunov equations. Due to the approach, every existing solution bound for both the continuous and discrete equations becomes intact in the sense that it yields a corresponding bound for the unified equation. The approach is endorsed by this simple fact: the three types of Lyapunov equations share the solution, provided certain relationships are maintained among corresponding coefficient matrices of the equations. Some examples are presented to show that the proposed method actually produces new bounds and builds bridges over existing bounds.