Stable Linear Time-invariant Systems Research Articles

A new method for calculation of closed‐form response of linear time‐invariant systems to periodic input signals

AbstractA new method for finding closed‐form time‐domain solutions of linear time‐invariant (LTI) systems with arbitrary periodic input signals is presented. These solutions, unlike those obtained based on the conventional Fourier‐phasor method, have a finite number of terms in one period. To implement the proposed method, the following steps are carried out: (1) For a given system, represented by a transfer function, an impulse response, a block diagram etc., the governing differential equation relating the output of the system, , to its input, , is obtained. (2) An auxiliary differential equation is formed by simply replacing with and equating the input side to alone. The auxiliary differential equation is solved for each time segment of the input signal in one period, leaving the constant coefficients associated with the homogeneous solutions as unknowns. For an nth‐order system with an input signal consisting of q segments in one period, there are such unknown coefficients. (3) Continuity of and its derivatives, at the connection points of successive segments and the periodicity conditions for the beginning and end points of the period are implemented. (4) The outcome of step 3 is a system of equations in terms of unknown coefficients described in step 2. Solving this system of equations, the solution for in one period is obtained. (5) Finally, using the linearity and differentiation properties of the system and the coefficients of the input side of the differential equation of the system, the total response, , in one period is constructed in terms of and its derivatives. For stable LTI systems, the proposed method can be used without any limitations.

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.

Robust Admittance Control With Complementary Passivity

This paper studies a robust admittance control problem with a passivity requirement for stable and unstable linear time-invariant systems, motivated by control issues originated from physical human-robot interaction. A complementary admittance control structure is proposed and analyzed, revealing that the nominal performance (admittance tracking and passivity) is decoupled from robustness. Simulations on the admittance control for human arm strength augmentation with a passivity requirement validate the proposed controller design.

Real-time [formula omitted] norm optimization of the linear time-invariant system

H∞ norm is widely adopted in control system design and its model-based computation is well-established. In these applications, the optimization of the H∞ norm follows an offline manner. However, the real-time optimization of the H∞ norm of a (stable) linear time-invariant system remains open, which is of considerable interest in the domains of performance optimization, fault-tolerant control, etc. Driven by this desire, as an alternative to the model-based offline solution of the H∞ control/optimization problem, this paper proposes a real-time H∞ norm optimization approach. It would be beneficial to the real-time reconfiguration of the controller/observer under the H∞ optimization framework.

Metric entropy limits on recurrent neural network learning of linear dynamical systems

One of the most influential results in neural network theory is the universal approximation theorem [1], [2], [3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural networks. The purpose of this paper is to establish a result in this spirit for the approximation of general discrete-time linear dynamical systems—including time-varying systems—by recurrent neural networks (RNNs). For the subclass of linear time-invariant (LTI) systems, we devise a quantitative version of this statement. Specifically, measuring the complexity of the considered class of LTI systems through metric entropy according to [4], we show that RNNs can optimally learn—or identify in system-theory parlance—stable LTI systems. For LTI systems whose input-output relation is characterized through a difference equation, this means that RNNs can learn the difference equation from input-output traces in a metric-entropy optimal manner.

Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization

In this paper, we compute a (local) optimal reduced order model that matches a prescribed set of moments of a stable linear time-invariant system of high dimension. We fix the interpolation points and parametrize the models achieving moment-matching in a set of free parameters. Based on the parametrization and using the H2-norm of the approximation error as the objective function, we derive a nonconvex optimization problem, i.e., we search for the optimal free parameters to determine the model yielding the minimal H2-norm of the approximation error. Furthermore, we provide the necessary first-order optimality conditions in terms of the controllability and the observability Gramians of a minimal realization of the error system. We then propose two gradient-type algorithms to compute the (local) optimal models, with mathematical guarantees on the convergence. We also derive convex semidefinite programming relaxations for the nonconvex Problem, under the assumption that the error system admits block-diagonal Gramians, and derive sufficient conditions to guarantee the block diagonalization. The solutions resulting at each step of the proposed algorithms guarantee the achievement of the imposed moment matching conditions. The second gradient-based algorithm exhibits the additional property that, when stopped, yields a stable approximation with a reduced H2-error norm. We illustrate the theory on a CD-player and on a discretized heat equation.

On the necessity and sufficiency of the Zames–Falb multipliers for bounded operators

This paper analyzes the robust feedback stability of a single-input-single-output stable linear time-invariant (LTI) system against three different classes of nonlinear systems using the Zames–Falb multipliers. The contribution is threefold. Firstly, we identify a class of uncertain systems over which the robust feedback stability is equivalent to the existence of an appropriate Zames–Falb multiplier. Secondly, when restricted to be static (a.k.a. memoryless), such a class of systems coincides with the class of sloped-restricted monotone nonlinearities, and the classical result of using the Zames–Falb multipliers to ensure feedback stability is recovered. Thirdly, when restricted to be LTI, the first class is demonstrated to be a subset of the second, and the existence of a Zames–Falb multiplier is shown to be sufficient but not necessary for the robust feedback stability.

A Robust Control Approach to Event-Triggered Networked Control Systems With Time-Varying Delays

A robust control approach is proposed to solve the asymptotic stabilization problem of networked control systems, where the data transmission is mediated by a triggering function, and is subject to time-varying transmission delays. We show that the triggering mechanism and the delay effects could be regarded as uncertain components interconnected with a stable linear time-invariant system. We then use two operators to describe the uncertainties, and find the corresponding integral quadratic constraints of the operators. The stability of the networked control system is analyzed from a robust control point of view. Beyond the stability analysis, a co-design algorithm is devised to design the control gain and detection parameters jointly to maximize the inter-event interval as well as the maximum allowable time-delay. Theoretically, the proposed scheme guarantees that out-of-order packet delivery does not happen. The robust control framework allows the algorithm developed for time-varying delays to be applied to the constant delay and the delay free cases. Finally, the theoretical results are demonstrated and verified by numerical simulations.

Time-Domain Adaptive Higher Harmonic Control for Rejection of Sinusoidal Disturbances

Abstract This paper presents two new time-domain feedback controllers that reject sinusoidal disturbances with known frequencies acting on an asymptotically stable linear time-invariant (LTI) system. The first controller is time-domain higher harmonic control (TD-HHC), which is effective for uncertain LTI systems. The second controller is time-domain adaptive higher harmonic control (TD-AHHC), which is effective for completely unknown LTI systems. TD-HHC requires an estimate of the control-to-performance transfer function evaluated at the disturbance frequencies. In contrast, TD-AHHC does not require any information regarding the LTI system. We analyze the stability and closed-loop performance of TD-HHC and TD-AHHC. For both TD-HHC and TD-AHHC, we show that the controller asymptotically rejects the disturbance. We present numerical simulations comparing TD-HHC and TD-AHHC with frequency-domain higher harmonic control (FD-HHC), which is an existing frequency-domain controller for rejection of sinusoidal disturbances. We also present results from acoustic disturbance rejection experiments, which demonstrate the practical effectiveness of both TD-HHC and TD-AHHC.

Disturbance Observer-Based Offset-Free Global Tracking Control for Input-Constrained LTI Systems with DC/DC Buck Converter Applications

This paper presents an offset-free global tracking control algorithm for the input-constrained plants modeled as controllable and open-loop strictly stable linear time invariant (LTI) systems. The contribution of this study is two-fold: First, a global tracking control law is devised in such a way that it not only leads to offset-free reference tracking but also handles the input constraints using the invariance property of a projection operator embedded in the proposed disturbance observer (DOB). Second, the offset-free tracking property is guaranteed against uncertainties caused by plant-model mismatch using the DOB’s integral action for the state estimation error. Simulation results are given in order to demonstrate the effectiveness of the proposed method by applying it to a DC/DC buck converter.

Stochastic predictive control under intermittent observations and unreliable actions

We propose a provably stabilizing and tractable approach for control of constrained linear systems under intermittent observations and unreliable transmissions of control commands. A smart sensor equipped with a Kalman filter is employed for the estimation of the states from incomplete and corrupt measurements, and an estimator at the controller side optimally feeds the intermittently received sensor data to the controller. The remote controller iteratively solves constrained stochastic optimal control problems and transmits the control commands according to a carefully designed transmission protocol through an unreliable channel. We present a (globally) recursively feasible quadratic program, which is solved online to yield a stabilizing controller for Lyapunov stable linear time invariant systems under any positive bound on control values and any non-zero transmission probabilities of Bernoulli channels.

Adaptive Harmonic Control for Rejection of Sinusoidal Disturbances Acting on an Unknown System

We present a sampled-data adaptive harmonic control algorithm that rejects sinusoids with known frequencies that act on a completely unknown asymptotically stable linear time-invariant system. We analyze the stability and closed-loop performance for systems with at least as many controls as performance measurements. We show that the closed-loop system is uniformly Lyapunov stable and the adaptive controller asymptotically rejects the disturbances. We present numerical simulations comparing the new sampled-data adaptive controller with an existing sinusoidal disturbance rejection approach (i.e., higher harmonic control). We also present results from an active disturbance rejection experiment in an acoustic environment. These experimental results demonstrate the practical effectiveness of the adaptive harmonic controller.

Approximability models and optimal system identification

This article considers the problem of optimally recovering stable linear time-invariant systems observed via linear measurements made on their transfer functions. A common modeling assumption is replaced here by the related assumption that the transfer functions belong to a model set described by approximation capabilities. Capitalizing on recent optimal recovery results relative to such approximability models, we construct some optimal algorithms and characterize the optimal performance for the identification and evaluation of transfer functions in the framework of the Hardy Hilbert space and of the disk algebra. In particular, we determine explicitly the optimal recovery performance for frequency measurements taken at equispaced points on an inner circle or on the torus.

Multistable Energy Shaping of Passive Linear Systems with Hybrid Mode Selector

Abstract This paper presents a novel control strategy for stable linear time–invariant systems operating with a finite number of set points. Inspired by the theory of passivity-based control, the proposed method aims at simultaneously and asymptotically stabilize all the desired working modes by means of a static nonlinear state feedback law. An asynchronous external signal is then employed to trigger a hybrid controller in order to switch between the different working modes. The proposed approach is validated by means of simulations performed on the ubiquitous mass-spring-damper system.

Inverse Lyapunov Theorem for Linear Time Invariant Fractional Order Systems

This paper investigates the inverse Lyapunov theorem for linear time invariant fractional order systems. It is proved that given any stable linear time invariant fractional order system, there exists a positive definite functional with respect to the system state, and the first order time derivative of that functional is negative definite. A systematic procedure to construct such Lyapunov candidates is provided in terms of some Lyapunov functional equations.

Stochastic boundedness of state trajectories of stable LTI systems in the presence of non-vanishing stochastic perturbation

Abstract This paper studies stochastic boundedness of trajectories of a non-vanishing stochastically perturbed stable linear time-invariant system. First, two definitions on stochastic boundedness are presented, then, the boundedness is analyzed via Lyapunov theory. A theorem is proposed, which shows that under a condition on the Lipchitz constant of the perturbation kernel, the trajectories remain stochastically bounded, and the bounds are calculated. Also, the limiting behaviour of the trajectories is studied. At the end, an illustrative example is presented, which shows the effectiveness of the proposed theory.

A Robust Adaptive Cancellation of Unknown Sinusoidal Disturbance

A robust adaptive controller that consists of a well-known PI controller and a robust frequency estimation in parallel was presented to cancel unknown sinusoidal disturbance in a stable linear time invariant system. The convergence of frequency estimation, the accurate cancellation of disturbance and the asymptotic stability of the overall feedback control system are analyzed by slow integral manifold and averaging method. The key advantages include asymptotically convergent cancellation for disturbance with much low frequency, the asymptotic stability of the whole error feedback system independent to the parameters of the plant and robust applicability in engineering. Simulation results validate the transient convergence performance and the steady-state anti-noise performance as well as the action of parameters of the proposed algorithm.

A novel model reduction approach for linear time-invariant systems via enhanced PSO-DV algorithm and improved MPPA method

In this article, the combination of stochastic search and conventional approaches are used to develop an optimal frequency-domain model order reduction method for determining the stable and accurate reduced-order model for the stable large-scale linear time-invariant systems. The method uses the enhanced particle swarm optimization with differentially perturbed velocity algorithm to determine the denominator polynomial coefficients of the reduced-order model, whereas the numerator polynomial coefficients of the reduced-order model are determined by using an improved multi-point Padé approximation method. The method generates an optimum reduced-order model by minimizing an objective function [Formula: see text], which is formulated using two functions. The first function, [Formula: see text], evaluates the measure of integral squared error between the step responses of the original system and the reduced-order model. And the second function evaluates the measure of retention of full impulse response energy of the original system in the reduced-order model. Therefore, by minimizing the objective function ‘ E’, the proposed method is guaranteed for preserving passivity, stability and the accuracy of the original higher order system in the reduced-order model. The proposed method is extended to the linear time-invariant multi-input multi-output system. In this case, an optimal reduced-order model is determined by minimizing a single objective function [Formula: see text], which is formulated by linear scalarizing of all the objective function [Formula: see text] components. The method is popular for preserving stability, passivity and accuracy of the original system in the reduced-order model. The validation of the method is shown by applying to a sixth-order single-input single-output hydropower system model as well as to the seventh-order two-area multi-input multi-output power system model. The comparison of the simulation results of integral squared error and impulse response energy values of the reduced-order models demonstrates the dominance of the proposed method than the existing reduction methods available in the literature.

Non-Existence of Convolution Sum System Representations

Convolution sum system representations are commonly used in signal processing. It is known that the convolution sum, treated as the limit of its partial sums, can be divergent for certain continuous signals and stable linear time-invariant (LTI) systems, even when the convergence of the partial sums is treated in a distributional setting. In this paper, we ask a far more general question: is it at all possible to define a generalized convolution sum with natural properties that works for all absolutely integrable continuous signals that vanish at infinity and all stable LTI systems? We prove that the answer is “no.” Further, for certain subspaces, we give a sufficient and necessary condition for uniform convergence. Finally, we discuss the implications of our results on the effectiveness of window functions in the convolution sum.

Assessment of severity of nonlinearity for distributed parameter systems via nonlinearity measures

The performance of controlled distributed parameter systems (DPSs) not only depends on the controller, but also on the dynamic nature of the process itself. One of the primary factors affecting DPS control is process nonlinearity. In many situations, the extent and severity of nonlinearity is the crucial characteristic in deciding whether linear system analysis and controller synthesis methods are adequate. However, due to their spatio-temporal coupling, traditional nonlinearity measures cannot be directly applied to nonlinear DPSs. In this study, a nonlinearity measure method to quantify the severity of nonlinearity for a class of DPSs is proposed. First, time/space separation and model reduction are carried out using proper orthogonal decomposition (POD). Thus, an optimal linear time-invariant model with a low-order is obtained through the optimization of a spatio-temporal error while full state feedback is incorporated in order to stabilize the linear model. Finally, nonlinearity quantification for DPSs is calculated using the obtained stable linear time-invariant system. The complexity of the calculations for nonlinearity measures is greatly reduced after the model reduction using POD. This method easily estimates the extent to which the process behavior deviates from linearity, which aids in determining whether a linear system analysis and controller synthesis methods are adequate. The nonlinearity quantification indicates that DPSs with smaller values are better approximated by a linear model than DPSs with larger values in the target time/space domain. The effectiveness of the proposed method is illustrated using two numerical examples.