Linear Time-varying Discrete-time Systems Research Articles

Model predictive control for constrained uncertain, time-varying systems

In this paper, a new state feedback model predictive control for polytopic uncertain and/or time-varying linear discrete-time systems with input and state constraints is proposed. The main idea is to incorporate state decomposition principle and model predictive control in order to obtain a large domain of attraction using a short prediction horizon. At each time instant a quadratic programming problem is solved on-line. Proofs of recursive feasibility and asymptotic stability are given.

Positive Descriptor Time-varying Discrete-time Linear Systems and Their Asymptotic Stability

The positivity and asymptotic stability of the descriptor time-varying discrete-time linear systems are addressed. The Weierstrass-Kronecker theorem on the decomposition of the regular pencil is extended to the time-varying discrete-time descriptor linear systems. Using the extension necessary and sufficient conditions for the positivity of the systems are established. Sufficient conditions for asymptotic stability of the positive systems are presented. The effectiveness of the tests is demonstrated on the example.

구간 시변 지연시간을 갖는 양의 시변 이산 구간 시스템의 새로운 안정 조건

음이 아닌 입력에 대하여 음이 아닌 초기상태에서 출발한 모든 상태변수 값들이 시간에 대하여 항상 음이 아닌 값을 유지하는 시스템은 양의 시스템으로 정의된다. 본 논문에서는 상태변수에 시변 지연시간이 있는 양의 시변 선형 이산 구간 시스템의 안정조건을 새롭게 제안한다. 시변 지연시간은 최소와 최대 지연시간 범위에서 변하는 것으로 고려된다. 제안된 안정조건은 리아프노프 방정식의 상한 해 한계를 이용하여 유도되며, 매우 간단한 부등식의 형태로 표현된다. 수치예제를 통하여 새로운 안정조건들이 안정성 판단에 간단하고 효과적으로 적용될 수 있음을 확인한다. A dynamic system is called positive if any trajectory of the system starting from non-negative initial states remains forever non-negative for non-negative controls. In this paper, new sufficient conditions for asymptotic stability of the interval positive time-varying linear discrete-time systems with time-varying delay in states are considered. The considered time-varying delay time has an interval-like bound which has minimum and maximum delay time. The proposed conditions are established by using a solution bound of the Lyapunov equation and they are expressed by simple inequalities which do not require any complex numerical algorithms. An example is given to illustrate that the new conditions are simple and effective in checking stability for interval positive time-varying discrete systems.

Sensor Fault Estimation Filter Design for Discrete-time Linear Time-varying Systems

This paper proposes a sensor fault diagnosis method for a class of discrete-time linear time-varying (LTV) systems. In this paper, the considered system is firstly formulated as a descriptor system representation by considering the sensor faults as auxiliary state variables. Based on the descriptor system model, a fault estimation filter which can simultaneously estimate the state and the sensor fault magnitudes is designed via a minimum-variance principle. Then, a fault diagnosis scheme is presented by using a bank of the proposed fault estimation filters. The novelty of this paper lies in developing a sensor fault diagnosis method for discrete LTV systems without any assumption on the dynamic of fault. Another advantage of the proposed method is its ability to detect, isolate and estimate sensor faults in the presence of process noise and measurement noise. Simulation results are given to illustrate the effectiveness of the proposed method.

Sufficient conditions for simultaneous stabilization of three linear systems within the framework of nest algebras

In this paper, we study the problem of the simultaneous stabilization of three time-varying discrete-time linear systems within the framework of nest algebras. From the perspective of strong transitivity, we establish sufficient condition for the existence of time-varying simultaneously stabilizing controllers based on the coprime factorization. In particular, we also derive a sufficient condition for simultaneous stabilizability of three systems, where the systems are pairwise simultaneously stabilizable. Additionally, the sufficient conditions given in this paper lead to a constructive controller design to stabilize simultaneously three systems. These results hold as well in the time-invariant case. Two examples are included for illustration.

Commutativity of cascade connected discrete-time linear time-varying systems

After summarizing the commutativity results for analogue linear time-varying systems, the commutativity of cascade connected discrete-time linear time-varying systems is introduced. Some general open questions on the subject are stated and new results are presented, intended to answer some of them. Two examples are given to reinforce the concept and to show up its possible benefits in engineering to reduce noise effects and design robust systems.

Robust stabilisation of discrete-time time-varying linear systems with Markovian switching and nonlinear parametric uncertainties

In this paper, the problem of the robustness of the stability of a discrete-time linear stochastic system is addressed. The nominal plant is described by a discrete-time time-varying linear system subject to random jumping according with a non-homogeneous Markov chain with a finite number of states. The class of admissible uncertainties consists of multiplicative white noise type perturbations with unknown intensity. It is assumed that the intensity of white noise type perturbations is modelled by unknown nonlinear functions subject to linear growth conditions. The class of admissible controls consists of stabilising state feedback control laws. We show that the best robustness performance is achieved by the stability provided by a state feedback design based on the stabilising solution of a suitable discrete-time Riccati-type equation.

Active fault detection and isolation of discrete-time linear time-varying systems: a set-membership approach

Active fault detection and isolation (AFDI) is used for detection and isolation of faults that are hidden in the normal operation because of a low excitation signal or due to the regulatory actions of the controller. In this paper, a new AFDI method based on set-membership approaches is proposed. In set-membership approaches, instead of a point-wise estimation of the states, a set-valued estimation of them is computed. If this set becomes empty the given model of the system is not consistent with the measurements. Therefore, the model is falsified. When more than one model of the system remains un-falsified, the AFDI method is used to generate an auxiliary signal that is injected into the system for detection and isolation of faults that remain otherwise hidden or non-isolated using passive FDI (PFDI) methods. Having the set-valued estimation of the states for each model, the proposed AFDI method finds an optimal input signal that guarantees FDI in a finite time horizon. The input signal is updated at each iteration in a decreasing receding horizon manner based on the set-valued estimation of the current states and un-falsified models at the current sample time. The problem is solved by a number of linear and quadratic programming problems, which result in a computationally efficient algorithm. The method is tested on a numerical example as well as on the pitch actuator of a benchmark wind turbine.

Implicit improved vertex control for uncertain, time-varying linear discrete-time systems with state and control constraints

The problem of regulating an uncertain and/or time-varying linear discrete-time system with state and control constraints to the origin is addressed. It is shown that feasibility and a robustly asymptotically stable closed loop can be achieved using an interpolation technique. The design method can be seen as an alternative to optimization-based control schemes such as Robust Model Predictive Control. Especially for problems requiring complex calculations to find the optimal solution, the present method can provide a straightforward suboptimal solution. A simulation demonstrates the performance of this class of constrained controllers.

Temporal and one‐step stabilisability and detectability of discrete‐time linear systems

In a past study the authors drew attention to the fact that time-varying discrete-time linear systems may be temporarily uncontrollable and unreconstructable and that this is vital knowledge for both control engineers and system scientists. Describing and detecting the temporal loss of controllability and reconstructability requires considering discrete-time systems with variable dimensions and the j-step, k-step Kalman decomposition. In this study for linear discrete-time systems with variable dimensions measures of temporal and one-step stabilisability and detectability are developed. These measures indicate to what extent the temporal loss of controllability and reconstructability may lead to temporal instability of the closed-loop system when designing a static state or dynamic output feedback controller. The measures are calculated by solving specific linear quadratic cheap control problems by means of standard linear quadratic control algorithms. The importance of our developments for control system design is illustrated by means of two numerical examples.

Constrained control for linear time-varying systems in the presence of disturbances

The aim of this paper is twofold. In the first part, we revisit the robust invariance for ellipsoidal sets with respect to discrete-time linear time-varying and uncertain systems with bounded additive disturbances. In this context we provide an extension of the invariance condition of the previously published results in the literature. In the second part a novel robust interpolation based control method involving several local unconstrained robust optimal controls is proposed. Among them, one controller plays the role of a performance candidate, while the remaining controllers are used in the interpolation scheme for enlarging the domain of attraction. By using an appropriate objective function, recursive feasibility and a robustly asymptotically stable closed loop behavior are guaranteed.

Generalised Fractional Indexes Approximation with Application to Discrete-Time Generalised Weyl Symbol Computation

Unique description of discrete-time, linear time-invariant systems on infinite time horizon requires only definition of finite number of coefficients, usually relatively small. In contradistinction the description of discrete-time linear time-varying systems requires in general definition of an infinite number of coefficients. Nevertheless neither analyzing nor processing data with infinite dimensional size is impossible. The main aim of the paper is to develop new generalised fractional indexes – computational method which allows to determine generalised Weyl symbol for arbitrary real a not only for a=±0.5 (integer indexes) and a=0. Parameter a allow to shape the set of parameterised impulse responses. The selection of the parameter a in the generalised Weyl symbol enable selection of the best accuracy region for the time-frequency transform. Numerical examples illustrates how the approximation of the system response with generalised fractional indexes increase accuracy for the computation of the discrete-time, time-frequency transformation calculated on finite time horizon. Ill. 4, bibl. 18 (in English; abstracts in English and Lithuanian).DOI: http://dx.doi.org/10.5755/j01.eee.123.7.2367

Infinite horizon [formula omitted] control for discrete-time time-varying Markov jump systems with multiplicative noise

In this paper we consider the infinite horizon H2/H∞ control problem for discrete-time time-varying linear systems subject to Markov jump parameters and state-multiplicative noise. A stochastic version of a bounded real lemma is firstly developed for a general class of discrete-time time-varying Markov jump systems with state- and disturbance-multiplicative noise. By which we present a necessary and sufficient condition for the solvability of the H2/H∞ control problem in terms of four coupled discrete-time Riccati equations. Moreover, the obtained design is applied to a macroeconomic problem to verify its effectiveness.

Interpolation based control for constrained linear time-varying or uncertain systems in the presence of disturbances

In this paper, the robust control of constrained discrete-time linear time-varying or uncertain systems affected by bounded additive disturbance is considered. The main idea is to use an interpolation technique between several local unconstrained robust optimal controls. From the implementation point of view, the scheme is shown to be computationally attractive, at each time instant a quadratic programming (QP) problem being solved on-line. The proofs of recursive feasibility of the optimization procedure and ISS stability of the closed loop system are given.

Variational Bayesian Two-Stage Kalman Filter for Systems with Unknown Inputs

In this paper, we consider the probabilistic approach in adaptive state estimation for discrete-time linear stochastic time-varying systems with unknown inputs. When lack knowledge of noise, we regard the system state, unknown inputs and time-varying noise parameters as hidden variables with conjugate priors, and use variational Bayes method to side-step the complicated Bayesian inference. The proposed algorithm is based on optimal two-stage Kalman filtering technique that enables us to obtain marginal posterior distributions of hidden variables. A numerical example is provided to demonstrate the state estimation performance of proposed algorithm.

Optimal linear estimator for discrete-time systems with random delays

In this paper, optimal estimation for discrete-time linear time-varying systems with randomly state and measurement delays is considered. By introducing a set of binary random variables, the system is converted into the one with both multiplicative noises and constant delays. Then, an estimator which includes the cases of smoothing and filtering, is derived via the projection formula, and the solution is given in terms of a partial difference Riccati equation with boundary conditions. A predictor for such systems is also presented based on the proposed filter and smoother. The obtained estimators have the same dimension as the original state. Conditions for existence, uniqueness, and stability of the steady-state optimal estimators are studied for time-invariant cases. In this case, the obtained estimators are very easy to implement and all calculations can be performed off line, leading to a linear time-invariant estimator.

Modelling of time-varying discrete-time systems

A theory to analyse linear time-varying (LTV) discrete-time systems from the point of digital signal processing of view is established here. The LTV discrete-time system is changing with time slowly in comparison with the sampling frequency of the input signal is supposed. Three models were set up to simulate and analyse the LTV system. Models 1 and 2 can be used to design and analyse the LTV system, while the filter bank representation relates the LTV system with the time-varying filter bank. Based on the established models, the input–output relationship of a time-varying system in the frequency-domain was analysed in detail.

Infinite Horizon H2/H∞ Control for Discrete-Time Time-Varying Markov Jump Systems with Multiplicative Noise

AbstractIn this paper we consider the infinite horizon H2/H∞ control problem for discrete-time time-varying linear systems subject to Markov jump parameters and state-multiplicative noises. A stochastic bounded real lemma is firstly developed for a class of discrete-time time-varying Markov jump systems with state- and disturbance-multiplicative noises. Based on which, a necessary and sufficient condition is provided for the existence of linear state feedback H2/H∞ controller in terms of four coupled discrete-time Riccati difference equations. Finally, a concluding remark and some open topics are proposed.

SIMPLIFIED DESIGN OF LOW-PASS, LINEAR PARAMETER-VARYING, FINITE IMPULSE RESPONSE FILTERS

The main aim of the paper is to develop simplified tools and methods for design and analysis of linear parameter-varying (LPV) finite impulse response filters (FIR). FIR filters with constant coefficients have comprehensive theoretical foundations and design methods with the main advantages: good linearity of phase diagram, guaranteed stability, simple practical implementation. Although filters with constant coefficients guarantee particular properties in frequency domain, i.e. noise damping, they also increase rise time for rapid signal changes. In order to avoid such blurring effects a simplified design method for low-pass LPV FIR filters is developed. To synthesize the filter, two cut-off frequencies are needed accompanied with given filter order, shape tuning function and threshold detection condition for sequential operation. Quantitatively assess the filter quality and properties of the tuning functions are analyzed using both time and frequency dependent criteria. In the first case, difference Euclidean norm is used, while the frequency approach for filter analysis takes advantage of SVD-DFT transformation of linear time-varying discrete-time system, as previously defined by the author, employing singular value decomposition, discrete Fourier transformation and power spectral density properties.

A structured matrix approach to efficient calculation of LQG repetitive learning controllers in the lifted setting

This article discusses the use of repetitive control for output reference tracking in linear time-varying discrete time systems with both repetitive and non-repetitive noise components. The design of such controllers is formulated as a lifted linear stochastic output feedback problem on which the mature techniques of discrete linear control may be applied. In many modern applications, the large size of the system matrices in such a control problem inhibits the application of standard solvers and optimisation techniques. For linear quadratic Gaussian (LQG) problems, the matrices of the lifted feedback problem can be fitted into the recently developed sequentially semi-separable structure. Innovative numerical solutions are developed that have 𝒪(N) computational complexity (where N is the trial length) in both controller synthesis and implementation, comparable to that of many non-lifted and Fourier transform based learning control methods. Moreover, within this formulation, the system is allowed to vary over the learning cycle, closed-loop stability is guaranteed, and stochastic noise and disturbances are handled in an LQG sense.