Uncertain Linear Systems Research Articles

Random and interval uncertain symplectic methods for linear Birkhoff equations and their comparison study

With the increasing complexity and design requirement of engineering structures, the numerical computation plays an increasingly important role in the structural analysis. The symplectic algorithm for Birkhoffian systems has advantages over the traditional algorithm in accuracy and stability, and general mechanical systems can be transformed into Birkhoffian systems in theory. Additionally, the influence of uncertainties in practical engineering cannot be ignored. However, the research foundation of symplectic algorithms for Birkhoffian systems considering uncertainties is still weak. In this paper, the random and interval uncertain symplectic algorithms for linear Birkhoffian systems are proposed, respectively, which can be applied to the numerical calculation of the structural dynamic response with uncertainties. Furthermore, the compatibility between the two methods is studied. Based on the small parameter perturbation theory, the uncertain perturbation equation of the uncertain linear Birkhoffian system is established. According to different types of uncertainties, based on the probability theory and interval mathematical method, expressions of the mean and variance of random Birkhoffian systems, as well as the median value and radius of interval Birkhoffian systems are derived. By combining aforementioned methods with Birkhoffian symplectic algorithms, the random and interval perturbation methods for calculating the uncertain response of linear Birkhoffian systems are proposed, respectively. Since the uncertainty exists objectively, the uncertain features of the response should not be incompatible when characterization and analysis methods for uncertainties are different. Based on the Chebyshev inequality, this paper deduces the probabilistic and interval boundaries from the result calculated by proposed methods and the comparison between the two kinds of boundaries. It is mathematically proved that the probabilistic upper bound is less than the interval upper bound and the probabilistic lower bound is larger than the interval lower bound. Finally, examples are given to demonstrate the effectiveness of the proposed method in calculating the response of uncertain dynamics and its superiority over the traditional numerical methods.

Robust Controller Design for Uncertain Linear Systems with Finite-frequency Specifications: A Polynomially Parameter-dependent Approach

This paper investigates the robust controller design for polytopic-type uncertain linear systems disturbed by external disturbances in restricted frequency ranges. The primary goal is to design a state-feedback controller, ensuring that the closed-loop system is robustly stable and keeps a prescribed finite-frequency (FF) disturbanceattenuation performance. For this purpose, a robust generalized Kalman-Yakubovich-Popov (KYP) lemma is applied to describe the finite-frequency specification, aiming at improving the disturbance-attenuation performance over the given frequency range. In this setting, more relaxed analysis conditions for robust stability and finitefrequency specifications are derived by introducing additional slack variables, which contain some existing conditions as special cases. Based on the homogeneous polynomially parameter-dependent (HPPD) technique, new controller design conditions in terms of linear matrix inequalities (LMIs) are developed. It is shown that the proposed finite-frequency design scheme can achieve a better disturbance- suppression performance in restricted frequency ranges than the existing ones, which is illustrated by an example about the satellite system.

Distributed state estimation for uncertain linear systems: A regularized least-squares approach

This paper addresses the state estimation problem for a discrete-time uncertain system with a network of sensors, where the system is not necessarily observable by each sensor and deterministic uncertainties exist in the system matrices. A new robust estimator is designed for each sensor, using only its own and neighbor’s information, which is fully distributed. Moreover, a novel information fusion strategy is developed to guarantee the estimation performance, based on the collective observability of the sensor network, which greatly relaxes the technical assumption of the proposed estimator. Theoretically, it can be ensured that if the observed system is time-varying, the gains of the estimator will be bounded. Furthermore, if the system is time-invariant, these gains will be convergent. Subsequently, the estimation error covariance will be ultimately bounded if the observed system is quadratically bounded. In the end, the superiority of the proposed robust distributed state estimation algorithm is illustrated by several numerical simulation examples.

Design and Analysis of Event-Triggered Neuro-Adaptive Controller (ETNAC) for Uncertain Systems

In this paper, six new event-triggered neuro-adaptive control (ETNAC) schemes are presented for uncertain linear systems. Novelty of this paper lies in (i) the construction of the proposed ETNAC schemes, (ii) the design of event-triggering conditions, and (iii) the design of an observer called the modified state observer (MSO). In the proposed schemes, the MSO, the controller, and the event-triggering mechanisms are constructed and organized in a way such that they provide the control system designer with flexibility to choose between the one-way or two-way data exchange and also between the dynamic or static triggering conditions. The event-triggering conditions are designed on the basis of real performance parameters, such as the estimation/tracking errors that render control updates more on actual system events instead of the often-used extended time sampling. Another unique feature of ETNAC is its online uncertainty approximation capability even during inter-event times, which makes the controller robust and efficient. This part is developed with the help of an artificial neural network (ANN) and a polynomial regression-based MSO. The MSO formulations have two tunable gains, which allow fast uncertainty estimation without inducing high frequency oscillations, even while the system is in a transient state. Lyapunov analysis is used to show the stability of the system as well as to develop the event-triggering conditions. Effectiveness of the proposed controllers is demonstrated using benchmark numerical examples.

Robust Finite-Time Control of Linear System with Non-Differentiable Time-Varying Delay

Practical systems such as hybrid power systems are currently implemented around the world. In order to get the system to work properly, the systems usually require their behavior to be maintained or state values to stay within a certain threshold. However, it is difficult to form a perfect mathematical model for describing behavior of the practical systems since there may be some information (uncertainties) that is not observed. Thus, in this article, we studied the stability of an uncertain linear system with a non-differentiable time-varying delay. We constructed Lyapunov-Krasovskii functionals (LKFs) containing several symmetric positive definite matrices to obtain robust finite-time stability (RFTS) and stabilization (RFTU) of the uncertain linear system. With the controller and uncertainties in the considered system, there exist nonlinear terms occurring in the formulation process. Past research handled these nonlinear terms as new variables but this led to some difficulty from a computation point of view. Instead, we applied a novel approach via Cauchy-like matrix inequalities to handle these difficulties. In the end, we present three numerical simulations to show the effectiveness of our proposed theory.

Interpolation based predictive control by ellipsoidal invariant sets

A new interpolating control scheme for uncertain linear systems is developed. The interpolation is done between two saturated control laws, each given with its associated invariant set. A linear search is performed to compute a sub-optimal interpolation coefficient. Two algorithms are suggested — each incorporating a different optimization objective. The method guarantees robust stability and recursive feasibility, also in presence of persistent disturbances.

Tracking with prescribed performance for linear non-minimum phase systems

We consider tracking control for uncertain linear systems with known relative degree which are possibly non-minimum phase, i.e., their zero dynamics may have an unstable part. For a given sufficiently smooth reference signal we design a low-complexity controller which achieves that the tracking error evolves within a prescribed performance funnel. We present a novel approach where a new output is constructed, with respect to which the system has a higher relative degree, but the unstable part of the zero dynamics is eliminated. Using recent results in funnel control, we then design a controller with respect to this new output, which also incorporates a new reference signal. We prove that the original output stays within a prescribed performance funnel around the original reference trajectory and all signals in the closed-loop system are bounded. The results are illustrated by some simulations.

Characteristic Model-Based Adaptive Control for a Class of MIMO Uncertain Nonaffine Nonlinear Systems Governed by Differential Equations

This paper addresses the difficulty of designing a controller for a class of multi-input multi-output uncertain nonaffine nonlinear systems governed by differential equations. We first derive the first-order characteristic model composed of a linear time-varying uncertain system for such nonaffine systems and then design an adaptive controller based on this first-order characteristic model for position tracking control. The designed controller exhibits a simple structure that can effectively avoid the controller singularity problem. The stability of the closed-loop system is analyzed using the Lyapunov method. The effectiveness of our proposed method is validated with a numerical example.

Sliding Mode Control for Robust Consensus of General Linear Uncertain Multi-agent Systems

This paper investigates the robust consensus problem for general high-order linear multi-agent systems with external disturbances and internal uncertainties in coefficient matrices. Distributed protocols based on sliding mode control are proposed to realize the robust consensus under matched uncertainties.

Impulsive finite-time observer design for uncertain positive linear systems with L2-gain analysis

This paper is concerned with the design of a finite-time positive observer (FTPO) for continuous-time positive linear systems, which is robust regarding the L2-gain performance. In positive observers, the estimation of the state variables is always nonnegative. In contrast to previous positive observers with asymptotic convergence, an FTPO estimates positive state variables in a finite time. The proposed FTPO observer, using two Identity Luenberger observers and based on the impulsive framework, estimates exactly the state variables of positive systems in a predetermined time interval. Furthermore, sufficient conditions are given in terms of linear matrix inequalities (LMIs) to guarantee the L2-gain performance of the estimation error. Finally, the performance and robustness of the proposed FTPO are validated using numerical simulations.

Predictor Feedback for Uncertain Linear Systems With Distributed Input Delays

While the existing literature on delay-adaptive control concentrates on uncertain plants with discrete input delays, this article proposes a predictor feedback for stabilization of uncertain linear systems with distributed input delays. A finite-dimensional linear system with distributed actuator delay may come with five types of uncertainties: unknown delay, unknown delay kernel, unknown plant parameters, unmeasurable finite-dimensional plant state, and unmeasurable infinite-dimensional actuator state. For different combinations of uncertainties, distinct control designs are developed in the article, from which the users can make selections to address a vast range of problems they face.

Event‐triggered sliding mode control for time‐delay uncertain systems

AbstractThis paper addresses the event‐triggered sliding mode control to solve the stabilization problem of linear uncertain systems with time delays. First, the sufficient conditions for the existence of linear sliding surface for uncertain systems are derived in terms of linear matrix inequalities. Then, the sufficient conditions for the existence of linear sliding surface for time‐delay uncertain systems are given. The robust controller is designed based on the event‐triggering mechanism to stabilize the time‐delay uncertain systems. Moreover, it is proved that under the proposed event‐triggered sliding mode control, the corresponding triggering instants does not exhibit accumulation of triggering instants. Finally, a numerical example is given to illustrate the efficiency and feasibility of the proposed results.

Robust Finite-Time Tracking for Uncertain Linear Systems with Actuator Faults

In this paper, a robust finite-time fault-tolerant control (FTC) scheme is developed for uncertain linear systems in the presence of actuator faults. Since the system uncertainties and actuator faults are unknown, the controller parameters are updated online by the adaptive laws without the need for fault detection and isolation. It is proved that the proposed state-feedback model reference adaptive finite-time FTC scheme can guarantee that the tracking error converges to a small neighborhood of the origin in finite time. An application example for an aircraft lateral-directional dynamic system is presented to show the effectiveness of the proposed control scheme.

ANN Based Solution of Uncertain Linear Systems of Equations

Linear systems of equations have many applications in the area of engineering sciences, mathematics, operations research and statistics. It is worth mentioning that the coefficient matrix of the linear systems of equations may not be always crisp due to various uncertainties. These uncertainties may be in the form of interval. Likewise, the solution set and right hand side vector may also be in interval. In this respect, a fully interval linear system of equations \( (\tilde{P}\tilde{z} = \tilde{q}) \) is one where the coefficient matrix, the unknown vector and the right hand side vector all are in the form of interval. Although various authors proposed different methods to handle the fully linear systems of equations but those are sometimes problem specific etc. As such, in this paper \( n \times n \) fully interval linear systems of equations has been solved based on artificial neural network (ANN) model. In this regard, step by step algorithm has been included. Further, a convergence theorem has also been discussed for choosing suitable learning parameter. Few numerical examples and an application problem related to electrical circuit have been solved using the proposed method. Detail procedure has been discussed with numerical results to show the efficacy and powerfulness of the method.

Output feedback robust MPC of uncertain norm-bounded linear systems with disturbance

The output feedback robust model predictive control (MPC) for the linear systems with norm-bounded uncertainty and disturbance is studied. A new state estimator is formulated where both estimator dynamic matrix and estimator gain are decision variables being optimised on-line. The infinite horizon control moves are parameterised as a feedback control law, and the resultant non-convex optimisation problem is solved through a linearisation method. The recursive feasibility and closed-loop stability are guaranteed. The proposed approach can achieve better performance and include previous approaches as special cases. Two numerical examples are given to illustrate the effectiveness of the proposed approach.

Global Quadratic Stabilization in Probability for Switched Linear Stochastic Systems

Global quadratic stabilization in probability is considered for both switched linear certain stochastic systems and switched linear uncertain stochastic systems where there are norm bounded uncertainties. Under the assumption that every single subsystem is NOT globally quadratically stable in probability (GQS-P), we propose both static and dynamic output based switching laws such that the switched system on hand is GQS-P. In the case of static output based switching, the condition is expressed by a set of matrix inequalities, while the design of dynamic output based switching is proposed with a convex combination of subsystems and a robust Luenberger observer for each subsystem. Numerical examples are presented to show validity of the design conditions and switching algorithms.

An LMI-based algorithm for static output-feedback stabilization of continuous-time positive polytopic linear systems

This paper presents a new technique to design static output-feedback controllers for continuous-time positive uncertain linear systems. The design is performed through an iterative algorithm based on parameter-dependent linear matrix inequality conditions, solved by means of relaxations, with local convergence guaranteed. A qualified feasible solution provides a stabilizing output-feedback controller that also assures the positivity of the closed-loop system. The main advantage of the proposed methodology is that the control gain is handled directly as an optimization variable, that is, no change of variables is needed to recover the gain and no particular structure (e.g., diagonal) is imposed on the Lyapunov or slack variable matrix to guarantee closed-loop positivity. This particular feature also facilitates the design of decentralized or element-wise bounded gains, as illustrated by numerical experiments.

A New Data-Driven Approach of Reference Shaping

Abstract In this paper, we propose a data-driven method of reference shaping to improve the tracking performance of uncertain linear systems. The proposed method can be implemented on a system based on its input-output data; the finite-time L2-norm of the tracking error, estimated using the data, can be minimized. Moreover, the proposed algorithm can be extended for cases where the control inputs are constrained owing to actuation limits. The effectiveness of the proposed method is demonstrated through a numerical simulation and an experiment conducted using the cart system.

A Robust State Estimator With Adaptive Factor

This article proposes a robust state estimator with adaptively adjusted observation noise covariance for uncertain linear systems. Since the residuals between observation data and estimation data satisfying the normal distribution, the Mahalanobis distance are known to be Chi-square distributed. We use Chi-square tests to timely distinguish outliers that beyond the confidence interval and adjust the estimated observation noise covariance to a more likely value. Combined with the robust estimation method, an improved algorithm is derived to deal with uncertainties in both system parameters and observation noise covariance. Based on the proposed prediction form, we test the obtained robust state estimator with different deterioration conditions of observation noise covariance and compare it with the estimators without adaptive factor. The simulation results show that the derived state estimator may reduce the accumulation of estimation errors, smooth the estimated state curve, and the performance of the proposed estimator can be significantly improved as the deterioration enlarging.

Balanced Stochastic Optimal Control of Uncertain Linear Systems with Constraints

This paper addresses finite-time horizon optimal control for discrete-time dynamics with additional stochastic disturbances. In contrast to most existing approaches to this problem, we also minimize the uncertainty of future states arising from stochastic disturbances and from an uncertain initial state. Thus, the optimal control strategy balances the minimization of the expected distances to a reference signal, and the minimization of the uncertainty respectively. As opposed to prior work, the optimization is formulated subject to possible disturbance feedback policies. This enables to solve one semi-definite program over H steps, instead of solving H problems over one step, and the resulting reduced complexity allows one to use the scheme in online and predictive control. The proposed method is applicable to time-varying state constraints (in the sense of chance constraints) as well as time-invariant input constraints.