Stability Of Linear Dynamical Systems Research Articles

Quantitative Aeroelastic Stability Prediction of Wings Exhibiting Nonlinear Restoring Forces

In engineering practice, eigen-solution is used to assess the stability of linear dynamical systems. However, the linearity assumption in dynamical systems sometimes implies simplifications, particularly when strong nonlinearities exist. In this case, eigen-analysis requires linerisation of the problem and hence fails to provide a direct stability estimation. For this reason, a more reliable tool should be implemented to predict nonlinear phenomena such as chaos or limit cycle oscillations. One method to overcome this difficulty is the Lyapunov Characteristic Exponents (LCEs), which provide quantitative indications of the stability characteristics of dynamical systems governed by nonlinear time-dependent differential equations. Stability prediction using Lyapunov Characteristic Exponents is compatible with the eigen-solution when the problem is linear. Moreover, LCE estimations do not need a steady or equilibrium solution and they can be calculated as the system response evolves in time. Hence, they provide a generalization of traditional stability analysis using eigenvalues. These properties of Lyapunov Exponents are very useful in aeroelastic problems possessing nonlinear characteristics, which may significantly alter the aeroelastic characteristics, and result in chaotic and limit cycle behaviour. A very common nonlinearity in flexible systems is the nonlinear restoring force such as cubic stiffness, which would substantially benefit from using LCEs in stability assessment. This work presents the quantitative evaluation of aeroelastic stability indicators in the presence of nonlinear restoring force. The method is demonstrated on a two-dimensional aeroelastic problem by comparing the system behaviour and estimated Lyapunov Exponents.

Estimation and stochastic control of nonlinear dynamic systems over the AWGN channel: Application in tele‐presence and tele‐operation of autonomous vehicles

AbstractThis paper is concerned with tele‐presence and tele‐operation of autonomous vehicles over the Additive White Gaussian Noise (AWGN) channel. This wireless communication channel is subject to transmission noise and transmission power constraint. An encoder, decoder and controller are proposed by implementing a novel linearization method for linearizing the nonlinear dynamic systems at operating points. For the linearized systems, the encoder, decoder and controller are implemented that were previously proposed for output tracking as well as stability of linear dynamic systems. Two novel linearization methods are proposed and their satisfactory performances are proved for output tracking as well as stability of nonlinear dynamic system. The first method is based on the fixed linearization rate and the second one is based on the variable (optimal) linearization rate. The decoder that is based on the second method is in fact the extended Kalman filter with the optimal linearization rate. The performances of these two methods are compared with each other and the other proposed methods by implementing them to the problems of the tele‐presence and tele‐operation of autonomous vehicles over the AWGN channel. Their satisfactory performances are illustrated. It is illustrated that the second method has higher computational complexity with slightly better performance and is applicable to a larger class of reference trajectories.

Efficient Learning of a Linear Dynamical System With Stability Guarantees

We propose a principled method for projecting an arbitrary square matrix to the non-convex set of asymptotically stable matrices. Leveraging ideas from large deviations theory, we show that this projection is optimal in an information-theoretic sense and that it simply amounts to shifting the initial matrix by an optimal linear quadratic feedback gain, which can be computed exactly and highly efficiently by solving a standard linear quadratic regulator problem. The proposed approach allows us to learn the system matrix of a stable linear dynamical system from a single trajectory of correlated state observations. The resulting estimator is guaranteed to be stable and offers statistical bounds on the estimation error.

Implications of Regret on Stability of Linear Dynamical Systems

The setting of an agent making decisions under uncertainty and under dynamic constraints is common for the fields of optimal control, reinforcement learning, and recently also for online learning. In the online learning setting, the quality of an agent's decision is often quantified by the concept of regret, comparing the performance of the chosen decisions to the best possible ones in hindsight. While regret is a useful performance measure, when dynamical systems are concerned, it is important to also assess the stability of the closed-loop system for a chosen policy. In this work, we show that for linear state feedback policies and linear systems subject to adversarial disturbances, linear regret implies asymptotic stability in both time-varying and time-invariant settings. Conversely, we also show that bounded input bounded state stability and summability of the state transition matrices imply linear regret.

Clustering Techniques for Stable Linear Dynamical Systems with applications to Hard Disk Drives

In Robust Control and Data Driven Robust Control design methodologies, multiple plant transfer functions or a family of transfer functions are considered and a common controller is designed such that all the plants that fall into this family are stabilized. Though the plants are stabilized, the controller might be sub-optimal for each of the plants when the variations in the plants are large. This paper presents a way of clustering stable linear dynamical systems for the design of robust controllers within each of the clusters such that the controllers are optimal for each of the clusters. First a k-medoids algorithm for hard clustering will be presented for stable Linear Time Invariant (LTI) systems and then a Gaussian Mixture Models (GMM) clustering for a special class of LTI systems, common for Hard Disk Drive plants, will be presented.

把持タスクにおける仮想力と力学系を用いた軌道修正による遠隔操作支援

This paper presents a dynamical system based approach to assistive teleoperation for object grasping. According to the target position and the current user input, a virtual desired goal is generated for a stable second order linear dynamical system via linear interpolation. In order to achieve collision avoidance between the target object and the robot finger, a virtual force is generated as an input to the dynamical system for online trajectory modification. Experimental results of an object grasping task are presented to demonstrate the effectiveness of the proposed assistive approach in a simulated robot teleoperation system.

Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations.

We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.

A Convex Parameterization of Robust Recurrent Neural Networks

Recurrent neural networks (RNNs) are a class of nonlinear dynamical systems often used to model sequence-to-sequence maps. RNNs have excellent expressive power but lack the stability or robustness guarantees that are necessary for many applications. In this letter, we formulate convex sets of RNNs with stability and robustness guarantees. The guarantees are derived using incremental quadratic constraints and can ensure global exponential stability of all solutions, and bounds on incremental ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain (the Lipschitz constant of the learned sequence-to-sequence mapping). Using an implicit model structure, we construct a parametrization of RNNs that is jointly convex in the model parameters and stability certificate. We prove that this model structure includes all previously-proposed convex sets of stable RNNs as special cases, and also includes all stable linear dynamical systems. Numerical experiments illustrate the utility of the proposed model class in the context of non-linear system identification.

Kernel-based identification of asymptotically stable continuous-time linear dynamical systems

ABSTRACT In many engineering applications, continuous-time models are preferred to discrete-time ones, in that they provide good physical insight and can be derived also from non-uniformly sampled data. However, for such models, model selection is a hard task if no prior physical knowledge is given. In this paper, we propose a non-parametric approach to infer a continuous-time linear model from data, by automatically selecting a proper structure of the transfer function and guaranteeing to preserve the system stability properties. By means of benchmark simulation examples, the proposed approach is shown to outperform state-of-the-art continuous-time methods, also in the critical case when short sequences of canonical input signals, like impulses or steps, are used for model learning.

On stability of linear dynamic systems with hysteresis feedback

The stability of linear dynamic systems with hysteresis in feedback is considered. While the absolute stability for memoryless nonlinearities (known as Lure’s problem) can be proved by the well-known circle criterion, the multivalued rate-independent hysteresis poses significant challenges for feedback systems, especially for proof of convergence to an equilibrium state correspondingly set. The dissipative behavior of clockwise input-output hysteresis is considered with two boundary cases of energy losses at reversal cycles. For upper boundary cases of maximal (parallelogram shape) hysteresis loop, an equivalent transformation of the closed-loop system is provided. This allows for the application of the circle criterion of absolute stability. Invariant sets as a consequence of hysteresis are discussed. Several numerical examples are demonstrated, including a feedback-controlled double-mass harmonic oscillator with hysteresis and one stable and one unstable poles configuration.

Distributed Actuator Selection: Achieving Optimality via a Primal-Dual Algorithm

This letter addresses the actuator selection problem, i.e., given an interconnection of asymptotically stable linear dynamical systems on a network and m possible actuators choose ν among them to achieve a certain objective. In general, this is a combinatorial optimization problem which is hard to solve; convex relaxations do not usually yield an optimal solution for the original problem. In this letter we focus on a particular instance of the actuator selection problem, namely the formulation with the trace of the controllability Gramian matrix as the optimization metric, and show that such a choice gives rise to an integer linear program (LP). Using properties of integral polyhedra, we show through a sequence of reformulations that the optimal solution of this problem can be determined by means of an LP without introducing any relaxation gap. This allows us to obtain the optimal solution using a primal-dual distributed algorithm, thus providing a scalable approach to the problem of actuator placement which has been up to now performed in a centralized manner enumerating all possible placement alternatives. We illustrate the main features of our approach by means of a case study involving a simplified model of the European power grid.

Discrete dynamical stabilization of a naturally diverging mass in a harmonically time-varying potential

We numerically investigate the stability and linear oscillatory behavior of a naturally diverging mass whose potential energy is harmonically modulated. It is known that in the Kapitza limit, i.e. when the period of modulation is much smaller than the diverging time, the collapsing mass can be dynamically stabilized and behave like an effective classic harmonic oscillator. We find that in the regime where the period of modulation is larger than the collapsing time of the mass, dynamical stabilization is still possible but in a discrete fashion. Only almost-periodic vibrational modes, or Floquet forms (FFs), are allowed that are located in independent stability stripes in the modulation parameter space. Reducing the FFs to their periodic eigenfunctions, one can transform the original equation of motion to a dimensionless Schrödinger stationary wave equation with a harmonic potential. This transformation allows for an analytical prediction of the stability stripes and the modal shapes of the vibrating mass. These results shed new light on the stability of linear dynamical systems, analytical solutions of Mathieu equations and on the relations between Initial and Boundary Value Problems.

Finite time identification in unstable linear systems

Identification of the parameters of stable linear dynamical systems is a well-studied problem in the literature, both in the low and high-dimensionalsettings. However, there are hardly any results for the unstable case, especially regarding finite time bounds. For this setting, classical results on least-squares estimation of the dynamics parameters are not applicable and therefore new concepts and technical approaches need to be developed to address the issue. Unstable linear systems arise in key real applications in control theory, econometrics, and finance. This study establishes finite time bounds for the identification error of the least-squares estimates for a fairly large class of heavy-tailed noise distributions, and transition matrices of such systems. The results relate the time length (samples) required for estimation to a function of the problem dimension and key characteristics of the true underlying transition matrix and the noise distribution. To establish them, appropriate concentration inequalities for random matrices and for sequences of martingale differences are leveraged.

Maximum likelihood identification of stable linear dynamical systems

This paper concerns maximum likelihood identification of linear time invariant state space models, subject to model stability constraints. We combine Expectation Maximization (EM) and Lagrangian re ...

Чисельне моделювання d-області стійкості дробових лінійних динамічних систем

Чисельне моделювання d-області стійкості дробових лінійних динамічних систем

Isolation of domains of stability of linear dynamic systems with fractional regulators

Isolation of domains of stability of linear dynamic systems with fractional regulators

Tensor logarithmic norm and its applications

SummaryMatrix logarithmic norm is an important quantity, which characterize the stability of linear dynamical systems. We propose the logarithmic norms for tensors and tensor pairs, and extend some classical results from the matrix case. Moreover, the explicit forms of several tensor logarithmic norms and semi‐norms are also derived. Employing the tensor logarithmic norms, we bound the real parts of all the eigenvalues of a complex tensor and study the stability of a class of nonlinear dynamical systems. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Sufficient conditions for the stabilization of linear dynamical systems

We consider the output stabilization problem for a linear time-independent system. We show that if the system output is zero, y(t) ≡ 0, then, for the stabilization of the entire state vector, the sufficient condition of stability of the zero dynamics can be replaced by the boundedness of the system input (provided that the characteristic polynomial of the zero dynamics has no eigenvalues on the imaginary axis).

Stability of linear dynamic systems over the packet erasure channel: a co-design approach

This paper is concerned with the stability of linear time-invariant dynamic systems over the packet erasure channel subject to minimum bit rate constraint when an encoder and a decoder are unaware of the control signal. This assumption results in co-designing the encoder, decoder and controller. The encoder, decoder, controller and conditions relating transmission rate to packet erasure probability and eigenvalues of the system matrix A are presented for almost sure asymptotic stability of linear time-invariant dynamic systems over the packet erasure channel with feedback acknowledgment. When the eigenvalues of the system matrix A are real valued, it is shown that the obtained condition for stability is tight. Simulation result illustrates the satisfactory performance of the proposed encoder, decoder and controller for almost sure asymptotic stability.

A note on “A class of linear differential dynamical systems with fuzzy initial condition”

In this note, we show that the proposed approach in J. Xu et al. (2007) [1] fails to obtain the stable solutions to the stable linear dynamical systems with fuzzy initial conditions. As it will be explained, this shortcoming is caused by neglecting to move the stability property of the fuzzy system to the quasi-level-wise system. Moreover, to handle this, another approach is proposed similar to the previous one for the stable and unstable systems.