Linear Time-varying Systems Research Articles

Dynamic inversion and optimal tracking control on the ball-plate system based on a linearized nonholonomic multibody model

This paper addresses the optimal control of the ball-plate system, a well-known nonholonomic system in the context of nonprehensile manipulation, using a multibody dynamics approach. The trajectory tracking control of a steady-state circular motion of the ball on the plate, for any radius and potentially off-centric with respect to the plate’s pivoting point, is achieved by designing a Linear-Quadratic Regulator. A spatial multibody model of the ball-plate system is considered. A key contribution is the analytical computation of the circular steady motion of the ball by dynamic inversion, including the control actions to achieve this reference solution. This enables the analytical computation of the linearized equations along this reference motion, resulting in a periodic linear time-varying (LTV) system, and the application of linear controllability criteria for LTV systems. A controllable linear system, involving the Cartesian coordinates of the contact point and the yaw angle of the sphere, is obtained using a convenient coordinate partition in the linearization. Compared to existing results on the same problem, closed-loop stability about the desired trajectory is achieved for any radius of the circular trajectory.

Tobit Kalman fusion filtering under dynamic event-triggering protocol with token bucket specification

This paper addresses the multi-sensor fusion filtering problem for a class of linear discrete time-varying systems with censored measurement, described by the Tobit model, and scheduled by dynamic event-triggering protocols with token bucket specification. A dynamic event-triggering mechanism is first used to determine whether to transmit measurements under the token bucket specification, allowing transmission only if there are sufficient tokens and if the event-triggering condition is satisfied. Next, two indicator variables are denoted to represent the combined impact of the dynamic event-triggering protocol and the token bucket specification. A local Tobit Kalman filtering algorithm is then designed for each node by minimizing the trace of the filtering error covariance matrix under censoring and information transmission protocols' influence. Subsequently, all local estimates from each node are transmitted to the fusion center, where global estimates are generated using a federal fusion rule. The global estimates with suitable weights are sent back to every node for predictions at subsequent time instants. Finally, an illustrative simulation example is used to evaluate performance of this fused filtering scheme proposed in this paper.

Reinforcement Learning for Finite-Horizon H∞ Tracking Control of Unknown Discrete Linear Time-Varying System

Reinforcement Learning for Finite-Horizon H∞ Tracking Control of Unknown Discrete Linear Time-Varying System

Minimum Variance Prediction of MIMO Linear Time-varying Systems

Minimum Variance Prediction of MIMO Linear Time-varying Systems

Analytical solution for the skip trajectory of hypersonic vehicles with time-varying aerodynamic coefficients

Abstract An analytical method is proposed for calculating the altitude and flight path angle of hypersonic vehicles’ skip trajectories under the influence of arbitrary time-varying aerodynamic coefficients. First, a longitudinal plane dynamics equation is established based on the inertial Earth model and the specific kinetic energy is introduced as independent variable to simplify the derivation. Subsequently, the solution is obtained through the perturbation method, where the zero-order analytical solution is based on existing literature, and the first-order linear time-varying system is decoupled through matrix linear transformation. Finally, by comparing numerical integration and analytical solutions with constant aerodynamic coefficients, the precision of this method in dealing with time-varying aerodynamic coefficients is verified, demonstrating its advantages over analytical solutions with constant aerodynamic coefficients.

Commutativity of Cascaded Connected Fractional Order Linear Time-Varying Systems

Commutativity of Cascaded Connected Fractional Order Linear Time-Varying Systems

Data-Driven Distributed Predictive Control for Voltage Regulation and Current Sharing in DC Microgrids With Communication Constraints.

The use of nonideal communication networks makes communication constraints become a topical issue in the research of dc microgrids. How to design a distributed secondary control scheme for voltage recovery and accurate current sharing in islanded dc microgrids subject to communication constraints is of interest in this article. In order to restore the bus voltage to the rated value, a nonlinear element is first introduced into the primary control layer. Then, the closed-loop system of primary control is modeled as a data-driven time-varying linear system. Based on the established model, considering communication constraints, a distributed secondary predictive control strategy is developed to achieve accurate current sharing. While actively compensating for network delays and packet losses, the proposed method renders mathematical physical models unnecessary for the traditional predictive control, and simultaneously completes the multitask in dc microgrids. Finally, several case studies are conducted on a hardware microgrid experimental platform, which not only verifies the effectiveness of the designed data-driven predictive control strategy but also tests microgrid properties such as the plug-and-play ability.

A robust design of time-varying internal model principle-based control for ultra-precision tracking in a direct-drive servo stage

This paper proposes a robust design of the time-varying internal model principle-based control (TV-IMPC) for tracking sophisticated references generated by linear time-varying (LTV) autonomous systems. The existing TV-IMPC design usually requires a complete knowledge of the plant I/O (input/output) model, leading to the lack of structural robustness. To tackle this issue, we, in this paper, design a gray-box extended state observer (ESO) to estimate and compensate unknown model uncertainties and external disturbances. By means of the ESO feedback, the plant model is kept as nominal, and hence the structural robustness is achieved for the time-varying internal model. It is shown that the proposed design has bounded ESO estimation errors, which can be further adjusted by modifying the corresponding control gains. To stabilize the ESO-based TV-IMPC, a time-varying stabilizer is developed by employing Linear Matrix Inequalities (LMIs). Extensive simulation and experimental studies are conducted on a direct-drive servo stage to validate the proposed robust TV-IMPC with ultra-precision tracking performance (∼60nm RMSE out of ±80mm stroke).

Unbiased minimum-variance disturbance and state estimation for linear systems with both state and output disturbances

This paper addresses the problem of simultaneously estimating the states, the state disturbances and the output disturbances of a linear time variant system. The objective is to develop a simple but optimal filter in the unbiased minimum-variance sense without depending extra complex system decomposition transformation. The derived filter bases on the assumption that no prior knowledge about the dynamical evolution of these disturbances is available.

Enhancing Stability Criteria for Linear Systems with Interval Time-Varying Delays via an Augmented Lyapunov–Krasovskii Functional

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems.

Event-triggered controllers using contraction analysis

In this article, we introduce an event-triggered control method that relies on contraction analysis for linear time-varying systems (LTV) and then extend it to a certain class of nonlinear systems. Contraction analysis considers stability with respect to a nominal trajectory rather than an equilibrium point. If two neighbouring trajectories of a system are located in a contraction region, then they will tend to each other and to a nominal trajectory. In the event-triggered control algorithm that we introduce, we suggest to update the control law whenever the system trajectory is about to leave the contraction region. We show that such a scheduling of the control law guarantees system stability, and we show that a minimum inter-event time exists between consecutive updates of the control law. We also show how to place the system trajectory in a contraction region and demonstrate that the classical controllability assumption on LTV systems is enough to ensure the existence of the required transformation to perform that.

L1 Adaptive Fault-Tolerant Control for Nonlinear Systems Subject to Input Constraint and Multiple Faults

This paper investigates an L1 adaptive fault-tolerant control scheme for nonlinear systems with input constraint, external disturbances, and multiple faults, which include actuator faults and sensor faults. Faults and input constraint are important factors that affect the stability and performance of a control system. Actuators and sensors are the most vulnerable components, with the former receiving more attention in comparison. In this paper, sensor faults are first transformed into pseudo-actuator faults through the augmented matrix approach, which facilitates their handling together with actuator faults. Saturation constraints on the control signal are not conducive to the design of the controller. The conversion of an input-saturated function to a time-varying linear system is completed based on function approximation and Lagrange’s mean value theorem. Moreover, a nonlinear system with unknown input gain and uncertainties is constructed using these methods. Next, an L1 adaptive fault-tolerant controller is designed to cope with uncertainties, including system uncertainties, external disturbances, faults, and approximation errors. In the L1 adaptive controller, the online estimation of the time-varying parameters allows for updating of the system state, while the combination of the two is transmitted to the control law such that it can compensate for the effects of the uncertainties. The stability and performance boundaries are further derived using the Lyapunov theory and the L1 reference system. Finally, simulations are carried out to demonstrate the effectiveness of the proposed controller.

Stability analysis of LTV systems via gain and phase

Abstract This paper considers the feedback stability analysis of infinite-dimensional discrete-time linear time-varying systems with combined gain and phase information. The framework considers a system to be a causal linear operator described by a lower triangular infinite-dimensional complex matrix. For $r$-sectorial operators, we first derive a sufficient condition for the invertibility of $I+AB$ using the gain and phase information. Finally, we use this invertibility criterion to give a sufficient condition for feedback stability of $r$-sectorial systems.

Mitigating Ill-posedness in Parameter Estimation under Sparse measurement for Linear time-varying systems employing Virtual Sensor Responses

Monitoring linear time-varying (LTV) systems using model-based approaches typically requires dense instrumentation and high-fidelity support models, resulting in significant computational and financial costs. Bayesian filtering-based methods adopt a joint state-parameter estimation approach for LTV system monitoring wherein states/parameters are observed as well as inferred from the measurements. Eventually, sparsity in measurement while dealing with high-fidelity models can aggravate the ill-posedness in the estimation diverging the estimation to impractical or no solutions. To enhance estimation resolution and precision, an alternative approach can be supporting estimation with virtual sensor measurements sampled from future time steps. Virtual measurements are in fact measurements sampled from a future time step relative to the time at which estimation is sought. To leverage virtual measurements, the estimation needs to be time-delayed, enabling the observation of states/parameters through measurements taken at both current and subsequent time steps, thereby alleviating the ill-posedness. This method has been explored for an LTV spring-mass-dashpot system, where the numerical investigation utilizes an interacting filtering environment to estimate states and parameters in the presence of sparse measurements. The study has shown how incorporating additional information can stabilize the estimation process, leading to improved estimates for both states and parameters.

Regret optimal control for uncertain stochastic systems

We consider control of uncertain linear time-varying stochastic systems from the perspective of regret minimization. Specifically, we focus on the problem of designing a feedback controller that minimizes the loss relative to a clairvoyant optimal policy that has foreknowledge of both the system dynamics and the exogenous disturbances. In this competitive framework, establishing robustness guarantees proves challenging as, differently from the case where the model is known, the clairvoyant optimal policy is not only inapplicable, but also impossible to compute without knowledge of the system parameters. To address this challenge, we embrace a scenario optimization approach, and we propose minimizing regret robustly over a finite set of randomly sampled system parameters. We prove that this policy optimization problem can be solved through semidefinite programming, and that the corresponding solution retains strong probabilistic out-of-sample regret guarantees in face of the uncertain dynamics. Our method naturally extends to include satisfaction of safety constraints with high probability. We validate our theoretical results and showcase the potential of our approach by means of numerical simulations.

Invariant properties of linear-iterative distributed averaging algorithms and application to error detection

We consider the problem of average consensus in a distributed system comprising a set of nodes that can exchange information among themselves. We focus on a class of algorithms for solving such a problem whereby each node maintains a state and updates it iteratively as a linear combination of the states maintained by its in-neighbors, i.e., nodes from which it receives information directly. Averaging algorithms within this class can be thought of as discrete-time linear time-varying systems without external driving inputs and whose state matrix is column stochastic. As a result, the algorithms exhibit a global invariance property in that the sum of the state variables remains constant at all times. In this paper, we report on another invariance property for the aforementioned class of averaging algorithms. This property is local to each node and reflects the conservation of certain quantities capturing an aggregate of all the values received by a node from its in-neighbors and all the values sent by said node to its out-neighbors (i.e., nodes to which it sends information directly) throughout the execution of the averaging algorithm. We show how this newly-discovered invariant can be leveraged for detecting errors while executing the averaging algorithm.

How to Improve Robust Control of a Linear Time-Varying System by Using Experimental Data

How to Improve Robust Control of a Linear Time-Varying System by Using Experimental Data

Stability Analysis of Linear Time-Varying Delay Systems via a Novel Augmented Variable Approach

This paper investigates the stability issues of time-varying delay systems. Firstly, a novel augmented Lyapunov functional is constructed for a class of bounded time-varying delays by introducing new double integral terms. Subsequently, a time-varying matrix-dependent zero equation is introduced to relax the constraints of traditional constant matrix-dependent zero equations. Secondly, for a class of periodic time-varying delays, considering the monotonicity of the delay and combining it with an augmented variable approach, Lyapunov functionals are constructed for monotonically increasing and monotonically decreasing delay intervals, respectively. Based on the constructed augmented Lyapunov functionals and the employed time-varying zero equation, less conservative stability criteria are obtained separately for bounded and periodic time-varying delays. Lastly, three examples are used to verify the superiority of the stability conditions obtained in this paper.

Fixed-time solution of inequality constrained time-varying linear systems via zeroing neural networks

Zeroing neural network (ZNN) has been explored and applied in a variety of time-varying problems solving. However, ZNN models for inequality constrained time-varying linear equation (ICTVLE) solving are rarely reported. Generally, numerous practical problems can be mathematically modeled as ICTVLE problems. Motivated by the above mentioned issue, a nonlinear tunable activation function activated ZNN (NTAF-ZNN) model is constructed to effectively solve ICTVLE problems. To further improve the performances of NTAF-ZNN model, a fuzzy nonlinear tunable activation function (FNTAF) is also designed. Based on the FNTAF, a fuzzy nonlinear tunable activation function-based ZNN (FNTAF-ZNN) model is realized. Firstly, the fixed-time convergence property and disturbance rejection ability of the NTAF-ZNN and FNTAF-ZNN models are proved by theoretical analysis. Then, comparative simulation results of the two models with other existing ZNN models for solving the ICTVLE problem are provided to further verify their robustness and effectiveness. Additionally, the two models and other existing ZNN models are also employed to realize inequality constrained manipulator trajectory tracking under ideal and noisy environments for further comparisons. Finally, the superior performances of the FNTAF-ZNN model for manipulator trajectory tracking are further verified by physical experiment results.

A Resetting Observer for Linear Time-Varying Systems With Application to Dynamic Positioning of Marine Surface Vessels

A Resetting Observer for Linear Time-Varying Systems With Application to Dynamic Positioning of Marine Surface Vessels