Linear Time-varying Research Articles

A Simple Method for Orthogonal Polynomial Approximation of Linear Time-Varying System Gramians

Abstract The paper considers the problem of reachability and observability finite interval gramians computation and approximation for linear time-varying systems. Both gramians are obtained from system trajectories. The reachability gramian is derived from the adjoint state impulse response and the observability gramian is obtained from the zero-input output response. The application of the adjoint system is discussed and the two basic impulse response characteristics, namely the regular and the adjoint state impulse responses are presented. The relation with the linear time-invariant case is also discussed and the role of the state transition matrix for computing the gramians is shown. An algorithm for derivation of the state transition matrix is proposed, which is based on the integration of the state equation by using the method of Runge-Kutta. The gramians are approximated in terms of Legendre orthogonal series representations of system trajectories.

Automated assessment of knowledge hierarchy evolution: comparing directed acyclic graphs

Automated construction of knowledge hierarchies from huge data corpora is gaining increasing attention in recent years, in order to tackle the infeasibility of manually extracting and semantically linking millions of concepts. As a knowledge hierarchy evolves with these automated techniques, there is a need for measures to assess its temporal evolution, quantifying the similarities between different versions and identifying the relative growth of different subgraphs in the knowledge hierarchy. In this paper, we focus on measures that leverage structural properties of the knowledge hierarchy graph to assess the temporal changes. We propose a principled and scalable similarity measure, based on Katz similarity between concept nodes, for comparing different versions of a knowledge hierarchy, modeled as a generic directed acyclic graph. We present theoretical analysis to depict that the proposed measure accurately captures the salient properties of taxonomic hierarchies, assesses changes in the ordering of nodes, along with the logical subsumption of relationships among concepts. We also present a linear time variant of the measure, and show that our measures, unlike previous approaches, are tunable to cater to diverse application needs. We further show that our measure provides interpretability, thereby identifying the key structural and logical difference in the hierarchies. Experiments on a real DBpedia and biological knowledge hierarchy showcase that our measures accurately capture structural similarity, while providing enhanced scalability and tunability. Also, we demonstrate that the temporal evolution of different subgraphs in this knowledge hierarchy, as captured purely by our structural measure, corresponds well with the known disruptions in the related subject areas.

Blind identification of fully observed linear time-varying systems via sparse recovery

Discrete-time linear time-varying (LTV) systems form a powerful class of models to approximate complex dynamical systems with nonlinear dynamics for the purpose of analysis, design and control. Motivated by inference of spatio-temporal dynamics in breast cancer research, we propose a method to efficiently solve an identification problem for a specific class of discrete-time LTV systems, in which the states are fully observed and there is no access to system inputs. In addition, it is assumed that we do not know on which states the inputs act, which can change between time steps, and that the total number of inputs is sparse over all states and over time. The problem is formulated as a compressive sensing problem, which incorporates the effect of measurement noise and which has a solution with a partially sparse support. We derive sufficient conditions for the unique recovery of the system model and input values, which lead to practical conditions on the number of experiments and rank conditions on system outputs. Synthetic experiments analyze the method’s sensitivity to noise for randomly generated models.

Explicit model predictive control for linear time-variant systems with application to double-lane-change maneuver.

Explicit model predictive control (eMPC) has been proposed to reduce the huge computational complexity of MPC while maintaining the performance of MPC. Therefore, this control method has been more widely employed in the automotive industry than MPC. In this paper, an eMPC is designed to perform a double-lane-change (DLC) maneuver. This task has been employed to demonstrate the efficacy of controllers in an autonomous driving situation. In this sense, the proposed controller shows better performance than a driver model designed in CarSim at a high vehicle longitudinal velocity. The main contribution of this paper is to present an eMPC for discrete-time linear time-variant (LTV) systems so that the proposed controller can be robust against parameter variation. In a state-space representation of the vehicle, the longitudinal velocity of the vehicle is assumed to be a constant so that the whole system is linear time-invariant (LTI). However, it is inevitable that this velocity varies in an actual driving situation. Therefore, an eMPC controller is designed using an add-on unit to consider the varying parameter without modification of the eMPC solution. The CarSim simulation results of eMPC show enhanced performance compared to that of eMPC for the LTI system.

Finite horizon robustness analysis of LTV systems using integral quadratic constraints

The goal of this paper is to assess the robustness of an uncertain linear time-varying (LTV) system on a finite time horizon. The uncertain system is modeled as an interconnection of a known LTV system and a perturbation. The input/output behavior of the perturbation is described by time-domain, integral quadratic constraints (IQCs). Typical notions of robustness, e.g. nominal stability and gain/phase margins, can be insufficient for finite-horizon analysis. Instead, this paper focuses on robust induced gains and bounds on the reachable set of states. Sufficient conditions to compute robust performance bounds are formulated using dissipation inequalities and IQCs. The analysis conditions are provided in two equivalent forms as Riccati differential equations and differential linear matrix inequalities, and an algorithm is developed leveraging both forms.

New backward whirl phenomena in intact and cracked rotor systems

Based on several published studies and relevant Campbell diagrams, it is well known that the passage through the critical backward whirl (BW) rotational speeds in rotor systems should theoretically precede the passage through the critical forward whirl (FW) rotational speeds. Theoretically, this means that the location of the zone/zones of the BW orbits should be captured before acquiring and exceeding the critical FW rotational speed under the effect of an unbalanced force excitation. However, for rotor systems that exhibit recurrent startup and coast down operations, the associated equations of motion take the form of linear time-varying (LTV) equations for which the traditional Campbell diagram is no longer valid for predicting the related critical FW and BW rotational speeds. Accordingly, herein, it is numerically and experimentally verified that new zones of BW rotational speeds appear immediately after the critical FW rotational speed is acquired and exceeded during startup and coast down operations in the cases of intact and cracked rotor systems. These numerical and experimental investigations have been carefully conducted to provide robust evidence about the appearance of these new BW zones. These discovered BW zones are found to be affected by the crack damage in the shaft and were accompanied with a stiffness asymmetry in the bearings. Unlike the case where the critical FW rotational speed is acquired and exceeded, the vibration during the case at which speeds acquire and exceed these new BW zones of rotational speeds was found to be associated with an abrupt reduction in whirl amplitudes to minimum values for both start-up and coast down operations.

Multi-Objective Design of Optimal Sliding Mode Control for Trajectory Tracking of SCARA Robot Based on Genetic Algorithm

This paper focuses on robust optimal sliding mode control (SMC) law for uncertain discrete robotic systems, which are known by their highly nonlinearities, unmodeled dynamics, and uncertainties. The main results of this paper are divided into three phases. In the first phase, in order to design an optimal control law, based on the linear quadratic regulator (LQR), the robotic system is described as a linear time-varying (LTV) model. In the second phase, as the performances of the SMC greatly depend on the choice of the sliding surface, a novel method based on the resolution of a Sylvester equation is proposed. The compensation of both disturbances and uncertainties is ensured by the integral sliding mode control. Finally, to solve the problem accompanying the LQR synthesis, genetic algorithm (GA) is used as an offline tool to search the two weighting matrices. The main contribution of this paper is to consider a multi-objective optimization problem, which aims to minimize not only the chattering phenomenon but also other control performances. A novel dynamically aggregated objective function is proposed in such a way that the designer is provided, once the optimization is achieved, by a set of nondominated solutions and then he selects the most preferable alternative. To show the performance of the new controller, a selective compliance assembly robot arm robot (SCARA) is considered. The results show that the manipulator tracing performance is considerably improved with the proposed control scheme.

Stability analysis of linear time-varying time-delay systems by non-quadratic Lyapunov functions with indefinite derivatives

This paper presents a stability analysis of linear time-varying (LTV) time-delay systems by using non-quadratic Lyapunov functions and functionals. Two types of sufficient conditions are proposed for testing several different kinds of stability, such as asymptotic stability, exponential stability and uniformly exponential stability, for LTV systems without delay. Then, by constructing suitable non-quadratic Lyapunov functions (functionals) that are respectively the time-varying weighted L1 and L∞ norms of the state variables and by using properties of the uniformly asymptotically stable (UAS) function and the recently established improved Razumikhin and Krasovskii stability theorems, both delay-dependent and delay-independent conditions are proposed for testing uniformly exponential stability of a class of LTV time-delay systems. The time derivatives of the non-quadratic Lyapunov functions (functionals) along the solutions are allowed to be indefinite, namely, to take both negative and positive values. Numerical examples show that the non-quadratic Lyapunov functions (functionals) based methods are more efficient than the existing ones that are based on quadratic functions (functionals).

On strong structural controllability and observability of linear time-varying systems: a constructive method

In this paper, we consider the controllability and observability of generalized linear time-varying (LTV) systems whose coefficients are not exactly known. All that is known about these systems is the placement of non-zero entries in their coefficient matrices $(A,B)$. We provide the characterizations in order to judge whether the placements can guarantee the controllability/observability of such LTV systems, regardless of the exact value of each non-zero coefficient. We also present a direct and efficient algorithm with an associated time cost of $O(n+m+\nu)$ to verify the conditions of our characterizations, where $n$ and $m$ denote the number of columns of $A$ and $B$, respectively, and $\nu$ is number of non-zero entries in $(A,B)$.

Electromagnetic Nonreciprocity

This paper aims at providing a global perspective on electromagnetic nonreciprocity and clarifying confusions that arose in the recent developments of the field. It provides a general definition of nonreciprocity and classifies nonreciprocal systems according to their linear time-invariant (LTI), linear time-variant (LTV) or nonlinear nonreciprocal natures. The theory of nonlinear systems is established on the foundation of the concepts of time reversal, time-reversal symmetry, time-reversal symmetry breaking and related Onsager- Casimir relations. Special attention is given to LTI systems, as the most common nonreciprocal systems, for which a generalized form of the Lorentz reciprocity theorem is derived. The delicate issue of loss in nonreciprocal systems is demystified and the so-called thermodynamics paradox is resolved from energy conservation considerations. The fundamental characteristics and applications of LTI, LTV and nonlinear nonreciprocal systems are overviewed with the help of pedagogical examples. Finally, asymmetric structures with fallacious nonreciprocal appearances are debunked.

Recursive approximate solution to time-varying matrix differential Riccati equation: linear and nonlinear systems

ABSTRACTAn approximate solution is proposed for linear-time-varying (LTV) systems based on Taylor series expansion in a recursive manner. The intention is to present a fast numerical solution with reduced sampling time in computation. The proposed procedure is implemented on finite-horizon linear and nonlinear optimal control problem. Backward integration (BI) is a well known method to give a solution to finite-horizon optimal control problem. The BI performs a two-round solution: first one elicits an optimal gain and the second one completes the answer. It is very important to finish the backward solution promptly lest in practical work, system should not wait for any action. The proposed recursive solution was applied for mathematical examples as well as a manipulator as a representative of complex nonlinear systems, since path planning is a critical subject solved by optimal control in robotics.

Improved stability criteria for linear time-varying systems on time scales

ABSTRACT In this paper, asymptotic stability problems of linear time-varying (LTV) systems on time scales are considered based on a less conservative Lyapunov inequality, whose right side is not required to be necessarily negative. It is shown that the Lyapunov inequality covers not only the corresponding trivial (continuous and discrete) ones but also nontrivial ones. Based on this inequality, some necessary and sufficient conditions for asymptotic stability, exponential stability, uniformly exponential stability of LTV systems on time scales are obtained. An example about nontrivial systems is given for illustrating the effectiveness of the proposed results.

Envelope-Domain Analysis and Modeling of Super-Regenerative Oscillators

An envelope-domain methodology for the numerical modeling of super-regenerative oscillators (SROs) is presented. The main advantage is its generality of application to transistor-based oscillators with arbitrary topology. Initially, a stability analysis of the nonoscillatory steady-state solution, forced by the quench signal, is performed. It is based on the calculation of a linear-time-variant (LTV) transfer function, obtained by linearizing the circuit envelope-domain equations about the nonoscillatory regime. Under moderate quench frequencies, it will be possible to estimate the SRO normalized envelope and sensitivity function from the detected dominant pair of complex-conjugate poles. In the general case, the SRO oscillatory response is modeled with a numerical method, valid under linear operation with respect to the input signal. This is based on the calculation of the LTV impulse response from a time–frequency transfer function obtained under a small-signal sinusoidal excitation. The LTV impulse response enables a straightforward determination of the sensitivity time interval and time distance to the envelope maximum. An integral expression, in terms of the LTV transfer function, will provide the SRO response to any small-signal input with any arbitrarily carrier frequency and modulation. The methodology has been successfully validated through its application to an SRO at 2.7 GHz, which has been manufactured and measured.

Tube-based robust output feedback MPC for constrained LTV systems with applications in chemical processes

Robust output feedback model predictive control (MPC) of linear time-varying (LTV) systems subject to bounded state and output disturbances is considered in this paper. The output feedback controller is designed in such a way that the observer is time-varying so the unknown state and the observer state are bounded by the tubes whose center is the state of the nominal LTV system that is steered to the origin. The on-line computational complexity is similar to that required for nominal MPC for LTV systems. Robust stability is ensured by using the notion of Lyapunov functions. The controller design is illustrated via two examples in chemical processes. The results show that the proposed output feedback controller can guarantee robust stability and constraint satisfaction of the LTV systems in the presence of bounded state and output disturbances.

Multi-scale simulation of nonlinear thin-shell sound with wave turbulence

Thin shells --- solids that are thin in one dimension compared to the other two --- often emit rich nonlinear sounds when struck. Strong excitations can even cause chaotic thin-shell vibrations, producing sounds whose energy spectrum diffuses from low to high frequencies over time --- a phenomenon known as wave turbulence. It is all these nonlinearities that grant shells such as cymbals and gongs their characteristic "glinting" sound. Yet, simulation models that efficiently capture these sound effects remain elusive. We propose a physically based, multi-scale reduced simulation method to synthesize nonlinear thin-shell sounds. We first split nonlinear vibrations into two scales, with a small low-frequency part simulated in a fully nonlinear way, and a high-frequency part containing many more modes approximated through time-varying linearization. This allows us to capture interesting nonlinearities in the shells' deformation, tens of times faster than previous approaches. Furthermore, we propose a method that enriches simulated sounds with wave turbulent sound details through a phenomenological diffusion model in the frequency domain, and thereby sidestep the expensive simulation of chaotic high-frequency dynamics. We show several examples of our simulations, illustrating the efficiency and realism of our model.

Determination of Optimum Drying Temperature Profile by Iterative Learning Control (ILC) Method to Obtain a Desired Moisture Content in Tablets

<p>The paper presents an industrial case study example to evaluate the performance of the linear time varying (LTV) perturbation model based iterative learning control (ILC) in a pilot scale batch system. The operating data based strategy applied here is based on utilizing the repetitive nature of batch processes to update the operating trajectories using process knowledge obtained from previous runs and thereby providing a convergent batch-to-batch improvement of the process performance indicator. The method was applied to determine the required drying temperature of Paracetamol granules to obtain desired moisture content at the end of the batch. After granulation operations, Paracetamol granules were dried in a fluid bed dryer in the pilot plant laboratory of GlaxoSmithKline Bangladesh Limited, Chittagong, Bangladesh. These results demonstrate the potential of the ILC approach for controlling batch processes without rigorous process models.</p><p>Chemical Engineering Research Bulletin 20(2018) 1-7</p>

Finite‐time distributed global optimal control for linear time‐varying multi‐agent systems: a dynamic output‐feedback perspective

This study presents a systematic procedure to solve the problem of finite-time distributed global optimal control for the leaderless and leader-follower homogeneous multi-agent systems (MASs). Each agent's dynamics is given in general form of continuous-time linear time-varying (LTV) systems. The communication network among the agents is also assumed to be undirected and directed under fixed topology. First, based on asymptotic stability principle, inverse optimal results of general LTV systems are constructed. Second, the necessary and sufficient conditions are given for the existence of the distributed optimal state feedback and optimal dynamic output feedback control that solves a global optimal control problem for MASs. Then, the sufficient condition for finite-time stability of both the state and the output feedback problems for MASs are stated. Finally, some numerical example results are provided to illustrate the effectiveness and applicability of the proposed scheme.

Adaptive vibration attenuation with globally convergent parameter estimation

Abstract Parameter estimation problems can be nonlinear, even if the dynamics are expressed by a linear model. The extended Kalman filter (EKF), even though it is one of the most popular nonlinear estimation techniques, may not converge without sufficient a priori information. This paper utilizes a globally convergent nonlinear estimation method—the double Kalman filter (DKF)—for a vibrating cantilever beam. A globally valid linear time-varying (LTV) model is required by the first stage of the DKF depending on some conditions on input and output excitation. Without considering noise, this LTV model provides the first stage and is globally equivalent to the nonlinear system. Since the neglected input and output noises can degrade the quality of estimation, the second stage linearizes the nonlinear dynamics, utilizing the nominally globally convergent estimate of the first stage, and improves the quality of estimation. Both estimation methods were applied to a cantilever beam setup in real-time. An adaptive linear quadratic regulator utilizes the estimated parameters to attenuate unknown transient disturbances. Different scenarios have been explored, providing a fair comparison between EKF and DKF. These methods have been implemented on an embedded ARM-based microcontroller unit and illustrates improved convergent properties of the DKF over the EKF. The global stability of the DKF is verified and it has been observed that it needs twice the computational cost of the EKF.

Exponential Stability of Coupled Linear Delay Time-Varying Differential–Difference Equations

Coupled linear delay time-varying differential–difference equations are considered. Explicit criteria for exponential stability of such equations are presented. The obtained results are used to derive sufficient conditions for exponential stability in linear time-varying differential equations of neutral type and in singular linear delay time-varying differential equations. Some illustrative examples are given.

Time-varying linear control for tiltrotor aircraft

Tiltrotor aircraft have three flight modes: helicopter mode, airplane mode, and transition mode. A tiltrotor has characteristics of highly nonlinear, time-varying flight dynamics and inertial/control couplings in its transition mode. It can transit from the helicopter mode to the airplane mode by tilting its nacelles, and an effective controller is crucial to accomplish tilting transition missions. Longitudinal dynamic characteristics of the tiltrotor are described by a nonlinear Lagrange-form model, which takes into account inertial/control couplings and aerodynamic interferences. Reference commands for airspeed velocity and attitude in the transition mode are calculated dynamically by visiting a command library which is founded in advance by analyzing the flight envelope of the tiltrotor. A Time-Varying Linear (TVL) model is obtained using a Taylor-expansion based online linearization technique from the nonlinear model. Subsequently, based on an optimal control concept, an online optimization based control method with input constraints considered is proposed. To validate the proposed control method, three typical tilting transition missions are simulated using the nonlinear model of XV-15 tiltrotor aircraft. Simulation results show that the controller can be used to control the tiltrotor throughout its operating envelop which includes a transition flight, and can also deal with vertical gust disturbances.