Linear Discrete Time-invariant Systems Research Articles

Stability Analysis of Multi-Dimensional Linear Discrete System and Root Distribution Using Sign Criterion with Real Coefficients

A new idea was proposed to find out the stability and root location of multi-dimensional linear time invariant discrete system (LTIDS) for real coefficient polynomials. For determining stability the sign criterion is synthesized from the Jury’s method for stability which is derived from the characteristic polynomial coefficients of the discrete system. The number of roots lying inside or outside the unit circle and hence on the unit circle is directly determined from the proposed single modified Jury tabulation and the sign criterion. The proposed scheme is simple and the examples are given to bring out the merits of the proposed scheme which is also applicable for the singular and non-singular cases.

Robust weighted fusion steady-state white noise deconvolution smoothers for multisensor systems with uncertain noise variances

For the linear discrete time-invariant multisensor systems with uncertain noise variances, according to the minimax robust estimation principle, based on the worst-case systems with the conservative upper bounds of noise variances, using Kalman filtering approach, the six robust weighted fusion steady-state white noise deconvolution smoothers (WNDS) are presented, include three robust weighted state fuser weighted by matrices, diagonal matrices and scalars, a modified robust covariance intersection (CI) fuser, and two robust weighted measurement fusers. They have the robustness in the sense that their actual smoothing error variances are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. We prove their robustness based on the Lyapunov equation approach. The concept of the robust accuracy is presented, and the robust accuracy relations among the robust local smoothers and six weighted fusion smoothers are rigorously proved. The equivalence among two robust weighted measurement fusers and robust centralized fuser is proved based on the information filter. Three simulation examples are given to verify the robustness and show the effectiveness of the proposed results.

Stochastic iterative learning control for discrete linear time-invariant system with batch-varying reference trajectories

In this paper, we present adaptive iterative learning control (ILC) schemes for discrete linear time-invariant (LTI) stochastic system with batch-varying reference trajectories (BVRT). If reference trajectories change every batch, ILC shows a different convergence property from that of the identical reference trajectory. First, we derive the convergence property and propose deterministic adaptive ILC combined with iterative learning identification for LTI system with BVRT. If the state noise and measurement noise exist, convergence rate and tracking performance are degraded because the controller considers the difference arising from the noise as tracking error. To deal with such a problem, we propose two approaches. The first is based on a batch-domain Kalman filter, which uses the difference between the current output trajectory and the next reference trajectory as a state vector, while the second is based on a time-domain Kalman filter. In the second approach, the system is identified at the end of each batch in an iterative fashion using the observer/Kalman filter identification (OKID). Then, the stochastic problem is handled using Kalman filter with a steady-state Kalman gain obtained from the identification. Therefore, the second approach can track the reference trajectories of discrete LTI stochastic system using only the input–output information. Simulation examples are provided to show the effectiveness of the proposed schemes.

Quasi-Newton-type optimized iterative learning control for discrete linear time invariant systems

In this paper, a quasi-Newton-type optimized iterative learning control (ILC) algorithm is investigated for a class of discrete linear time-invariant systems. The proposed learning algorithm is to update the learning gain matrix by a quasi-Newton-type matrix instead of the inversion of the plant. By means of the mathematical inductive method, the monotone convergence of the proposed algorithm is analyzed, which shows that the tracking error monotonously converges to zero after a finite number of iterations. Compared with the existing optimized ILC algorithms, due to the superlinear convergence of quasi-Newton method, the proposed learning law operates with a faster convergent rate and is robust to the ill-condition of the system model, and thus owns a wide range of applications. Numerical simulations demonstrate the validity and effectiveness.

FIR-type robust H 2 and H ∞ control of discrete linear time-invariant polytopic systems via memory state-feedback control laws

In this paper, robust H 2 and H ∞ control problems for discrete linear time-invariant (LTI) systems with polytopic uncertainties are addressed. The so-called finite impulse response (FIR) controller incorporating the states over several samples from the past to the present is adopted to design robust control laws with improved performances. For the closed-loop stability, parameter-dependent quadratic Lyapunov functions (PD-QLFs) are employed. Sufficient controller synthesis conditions are derived in the form of linear matrix inequalities (LMIs). Finally, examples are given to demonstrate the usefulness of the proposed methods.

Sparse representation in Szegő kernels through reproducing kernel Hilbert space theory with applications

This paper discusses generalization bounds for complex data learning which serve as a theoretical foundation for complex support vector machine (SVM). Drawn on the generalization bounds, a complex SVM approach based on the Szegő kernel of the Hardy space H2(𝔻) is formulated. It is applied to the frequency-domain identification problem of discrete linear time-invariant system (LTIS). Experiments show that the proposed algorithm is effective in applications.

Auxiliary Signal Design for Active Fault Detection Based on Set-membership

In this paper, a new active fault detection method based on set-membership is proposed. Similar to the existing methods, the normal and faulty modes of the system are assumed to be two linear discrete time-invariant systems with bounded energy uncertainty, which is represented by a bounded ellipsoid under the framework of set-membership. Then the minimal energy auxiliary signal is derived in the case with worst cases of uncertainties by adopting a geometric method, such that desired fault detection performance can be guaranteed. Note that with comparison to the existing methods, the proposed method has a clear physical meaning and the set-membership technique makes it more feasible to calculate the fault detection rates relative to different choices of auxiliary signals with smaller energy. Hence the energy of auxiliary signal can be determined by a trade off between the fault detection rate and the impact on normal operation of the system. Simulation results are provided to show the effectiveness of the proposed method.

Modifications of Finitely-Converging Algorithms of Adaptive Control

Finitely-converging algorithms were originally proposed by V.A.Yakubovich to solve problems of pattern recognition. Then, in his seminal paper “To The Theory Of Adaptive Systems,” published in 1968 in Russian Mathematics Doklady, V.A.Yakubovich applied these algorithms for robot adaptive control. To our knowledge it was the first usage of the word “robot” in scientific literature. The proposed approach consists in reformulation of an adaptive control problem as the solution problem for some countable system of inequalities. A procedure of their solution plays a role of adaptation algorithm. It should converge in a finite time in a closed loop. Initially V.A.Yakubovich developed algorithms to solve a problem of adaptive control of discrete linear time-invariant systems. With the expansion of a set of the systems under consideration those basic algorithms were modified, but theirs main idea remained the same. Namely, each correction of the vector of estimated parameters ensures decreasing of the distance between current vector and unknown vector of true parameters by the positive value separated from zero. Thus there may be only finite number of these corrections. This article describes Modifications of basic V.A.Yakubovich's algorithms in relation to adaptive control of several classes of the plants. Among them are hybrid continuous/discrete systems, nonlinear systems, infinite-dimensional systems, and so on.

A new algorithm for dual-rate systems frequency response computation in discrete control systems

This paper addresses an easy computation of the multiple components of the response to a sinusoidal input of a dual-rate linear time-invariant discrete system from the Bode diagram of LTI systems arising from a lifted representation. Based on those results, a generalized Bode diagram is suggested. Some new conclusions derived from this conceptual interpretation are introduced. This diagram provides a better insight in the frequency-response issues in multivariable control than the standard singular value decomposition of the lifted model. As an application, the output ripple suppression in a multirate control scheme is presented.

Defining a Pseudo-Metric Topology on Linear Dynamic Systems

This paper defines a (pseudo) metric topology on the space of stable discrete linear time-invariant (LTI) dynamic systems. The article seeks to present a solution to the problem of comparing two LTI systems or, more formally, to find a metric for the space of stable discrete LTI dynamic systems. To this effect, by comparing the performance of two Kalman filters designed for two dynamic systems, a distance-like pseudo-norm between two systems is developed. The defined metric topology can be exploited to select the closest model, among several possible models Ps, all of which are known, to an observed data sequence modeled as P*. Numerical simulations are provided illustrating the efficacy of the metric derived.

On the Existence of a Mean-Square Stabilizing Solution to a Modified Algebraic Riccati Equation

In this paper, we investigate a mean-square stabilizing solution to a modified algebraic Riccati equation (MARE), which arises in our previous work on the linear quadratic optimal control for linear time-invariant discrete systems with random input gains. An explicit necessary and sufficient condition ensuring the existence of a mean-square stabilizing solution is given directly in terms of the system parameters with the help of theory of positive operators and the assumption of observability or detectability of certain stochastic systems is no longer needed. Such a necessary and sufficient condition is compatible with that for the existence of a stabilizing solution to the standard definite algebraic Riccati equation (ARE).

Covariance Intersection Fusion Kalman Estimator for Multi-Sensor System with Measurements Delays

To handle the state estimation fusion problem between local estimation errors for the system with unknown cross-covariances and to avoid a large computation complexity of cross-covariances, for a multi-sensor linear discrete time-invariant stochastic system with time-delayed measurements, by the measurement transformation method, an equivalent system without measurement delays is obtained, and then using the covariance intersection (CI) fusion method, the covariance intersection fusion steady-state Kalman estimator is presented. It is proved that its accuracy is higher than that of each local estimator, and is lower than that of optimal Kalman fuser weighted by matrices with known cross-covariances. A Monte-Carlo simulation example shows the above accuracy relations, hence it has good performances.

On the Lyapunov exponents of a class of second-order discrete time linear systems with bounded perturbations

In this note we describe the maximal and the minimal values of Lyapunov exponents for second-order discrete time-invariant linear system perturbed by time-varying bounded perturbations. An interpretation of the results in terms of generalized spectral radius is given. An application of obtained formulas to the robust stability problem is demonstrated on a numerical example.

Anisotropy-based suboptimal filtering for the linear discrete time-invariant systems

Consideration was given to the problem of robust filtering for the finite-dimensional linear discrete time-invariant system with measured and estimated outputs. The system is exposed to a random disturbance with the imprecisely known probability distribution. In the information-theoretical terms, the stochastic uncertainty of the input disturbance is defined by the functional of mean anisotropy. The error of estimation was quantified by the anisotropy norm. A sufficient condition for an estimator to exist and ensure that the error is less than the given threshold value was derived in the form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities.

Covariance Intersection Kalman Fuser for Two-Sensor System with Time-Delayed Measurements

To handle the estimation fusion problem between local estimation errors for the system with unknown cross-covariances and to avoid a large computed burden and computational complexity of cross-covariances, for a two-sensor linear discrete time-invariant stochastic system with time-delayed measurements, by the measurement transformation method, an equivalent system without measurement delays is obtained and then using the Covariance Intersection (CI) fusion method, the covariance intersection steady-state Kalman fuser is presented. It is proved that its accuracy is higher than that of each local estimator and is lower than that of optimal Kalman fuser weighted by matrices with known cross-covariances. A Monte-Carlo simulation example shows the above accuracy relation and indicates that its actual accuracy is close to that of the Kalman fuser weighted by matrices, hence it has good performances.

Stabilization and Optimal Quadratic Regulation of Linear Time-Invariant Discrete Systems with Data Dropout Compensation

This paper discusses the global asymptotic stabilization and least-squares optimal control for a linear system with potential transmission errors subject to Bernouilli statistics from a remote controller to the actuator. The controlled system is able to compensate certain missed data through the use of finite size fixed memory storage arrays which store the last valid data which has been transmitted so as to replace the current missed one in the control structures.

Covariance Intersection Kalman Fuser for Two-sensor System with Time-delayed Measurements

For a two-sensor linear discrete time-invariant stochastic system with time-delayed measurements, by the measurement transformation method, an equivalent system without measurement delays is obtained and then using the Covariance Intersection (CI) fusion method, the covariance intersection steady-state Kalman fuser is presented. It can handle the estimation fusion problem between local estimation errors for the system with unknown cross-covariances and avoid a large computed burden and computational complexity of cross-covariances. It is proved that its accuracy is higher than that of each local estimator and is lower than that of optimal Kalman fuser weighted by matrices with known cross-covariances. A Monte-Carlo simulation example shows the above accuracy relation and indicates that its actual accuracy is close to that of the Kalman fuser weighted by matrices, hence it has good performances.

Model falsification using set-valued observers for a class of discrete-time dynamic systems: a coprime factorization approach

This article introduces a new method for model falsification using set-valued observers, which can be applied to a class of discrete linear time-invariant dynamic systems with time-varying model uncertainties. In comparison with previous results, the main advantages of this approach are as follows: The computation of the convex hull of the set-valued estimates of the state can be avoided under certain circumstances; to guarantee convergence of the set-valued estimates of the state, the required number of previous steps is at most as large as the number of states of the nominal plant; and it provides a straightforward nonconservative method to falsify uncertain models of dynamic systems, including open-loop unstable plants. The results obtained are illustrated in simulation, emphasizing the advantages and shortcomings of the suggested method. Copyright © 2013 John Wiley & Sons, Ltd.

Coding scheme based on spherical polar coordinate for control over packet erasure channel

SUMMARY Like feedback strategy, coding scheme is an important part of networked control system design as well. On the basis of spherical polar coordinate, a novel coding scheme is proposed for stabilization problem of discrete linear time invariant system subject to packet erasure channel with feedback. The coding scheme uses encoder without access to control inputs. In the case that a decoder does not use control inputs, a definite relation between the quantized data and the corresponding quantization error is established, which helps to analyze the stability of system, and a selective quantization method is adopted, by which finite data rate is obtained. In the case that a decoder uses control inputs, instead of quantizing the system state at each time step as usual, the encoder quantizes the initial state all the time by updating quantizer. Sufficient conditions guaranteeing the system stable are presented for two cases, respectively, and the corresponding design methods for coding schemes are given. Copyright © 2012 John Wiley & Sons, Ltd.

Self-tuning weighted measurement fusion white noise deconvolution estimator and its convergence analysis

White noise deconvolution or input white noise estimation has a wide range of applications including oil seismic exploration, communication, signal processing, and state estimation. For the multisensor linear discrete time-invariant stochastic systems with correlated measurement noises, and with unknown ARMA model parameters and noise statistics, the on-line AR model parameter estimator based on the Recursive Instrumental Variable (RIV) algorithm, the on-line MA model parameter estimator based on Gevers-Wouters algorithm and the on-line noise statistic estimator by using the correlation method are presented. Using the Kalman filtering method, a self-tuning weighted measurement fusion white noise deconvolution estimator is presented based on the self-tuning Riccati equation. It is proved that the self-tuning fusion white noise deconvolution estimator converges to the optimal fusion steady-state white noise deconvolution estimator in a realization by using the dynamic error system analysis (DESA) method, so that it has the asymptotic global optimality. The simulation example for a 3-sensor system with the Bernoulli-Gaussian input white noise shows its effectiveness.