Time-invariant Case Research Articles

Approximate adjoint-based iterative learning control

This paper characterises stochastic convergence properties of adjoint-based (gradient-based) iterative learning control (ILC) applied to systems with load disturbances, when provided only with approximate gradient information and noisy measurements. Specifically, conditions are discussed under which the approximations will result in a scheme which converges to an optimal control input. Both the cases of time-invariant step sizes and cases of decreasing step sizes (as in stochastic approximation) are discussed. These theoretical results are supplemented with an application on a sequencing batch reactor for wastewater treatment plants, where approximate gradient information is available. It is found that for such case adjoint-based ILC outperforms inverse-based ILC and model-free P-type ILC, both in terms of convergence rate and measurement noise tolerance.

Time-varying controller based on flatness for nonlinear anti-lock brake system

It is shown that by the use of flatness the problem of pole placement, which consists in imposing closed-loop system dynamics, can be related to track desired trajectories in the finite-dimensional linear time-invariant case. Polynomial two-degree-of-freedom controller can then be designed with the use of an exact observer and without resolving the Bézout's equation. In this paper, an extension of these developments is proposed in the linear time-varying (LTV) framework. The proposed approach is illustrated with the control of nonlinear model of an anti-lock brake system. The time-varying controller obtained from the LTV model ensures the trajectory tracking of the nonlinear model.

Model reduction of linear time-varying systems over finite horizons

We consider the problem of approximating a linear time-varying p×m discrete-time state space model S of high dimension by another linear time-varying p×m discrete-time state space model Sˆ of much smaller dimension, using an error criterion defined over a finite time interval. We derive the gradients of the norm of the approximation error and show how this can be solved via a fixed point iteration. We compare this to the classical H2 norm approximation problem for the infinite horizon time-invariant case and show that our solution extends this to the time-varying and finite horizon case.

Permanental Partition Models and Markovian Gibbs Structures

We study both time-invariant and time-varying Gibbs distributions for configurations of particles into disjoint clusters. Specifically, we introduce and give some fundamental properties for a class of partition models, called permanental partition models, whose distributions are expressed in terms of the α-permanent of a similarity matrix parameter. We show that, in the time-invariant case, the permanental partition model is a refinement of the celebrated Pitman–Ewens distribution; whereas, in the time-varying case, the permanental model refines the Ewens cut-and-paste Markov chains (J. Appl. Probab. 43(3):778–791, 2011). By a special property of the α-permanent, the partition function can be computed exactly, allowing us to make several precise statements about this general model, including a characterization of exchangeable and consistent permanental models.

VIBRATION ANALYSIS OF PLANETARY GEAR SYSTEM

In this work a dynamic model of a planetary gear transmission is developed to study the sensitivity of the natural frequencies and vibration modes to system parameters in perturbed situation. Parameters under consideration include component masses ,moment of inertia , mesh and support stiff nesses .The model admits three planar degree of freedom for planets ,sun, ring, and carrier. Vibration modes are classified into translational, rotational and planet modes .Well-defined modal properties of tuned (cyclically symmetric) planetary gears for linear ,time-invariant case are used to calculate eigensensitivities and express them in simple formulae .These formulae provide efficient mean to determine the sensitivity to stiffness ,mass and inertia parameters in perturbed situation.

Characterization of [formula omitted] index for linear time-varying systems

The increasing interest in ℋ− index in fault detection has led to some recent development in time domain investigation. This paper at first presents a necessary and sufficient condition of the ℋ− index in finite time horizon for linear time-varying systems. It is shown that the feasibility of the ℋ− index, which is given when the ℋ− index is greater than a predefined value, is equivalent to the existence of a certain backward differential Riccati equation under a constraint. The square system as a special case is discussed. The corresponding results are extended to systems with unknown initial condition with a modified definition of the ℋ− index. The condition for weighted index to emphasize the signal effect at different time instant is also derived by transformation. The result is also compared with the famous bounded real lemma. The connection with the linear time-invariant (LTI) case is also discussed. Finally, examples are given to illustrate the main results.

Performance Evaluation of a Multi-User MIMO System With Prediction of Time-Varying Indoor Channels

In this paper, the performance of a multi-user multiple-input multiple-output (MIMO) system in time-varying channels is evaluated using measurement data. We consider the multi-user MIMO system using a block diagonalization (BD) scheme and an eigenbeam-space division multiplexing (E-SDM) technique. In an ideal case, the BD scheme eliminates inter-user interference, and the E-SDM technique suppresses inter-stream interference. In actual radio environments, however, channels change over time. This causes interference in the multi-user MIMO system even though the BD scheme and the E-SDM technique are used. To overcome this problem, the authors have developed a simple channel prediction scheme on the basis of a linear extrapolation and have demonstrated its effectiveness by computer simulations assuming the Jakes' model. To verify the performance of the channel prediction scheme in actual environments, we conducted a measurement campaign in indoor environments and measured a large amount of channel data. Using these data, we examined the channel transition and channel tracking with the prediction method. Then we obtained the bit-error rate (BER) performance. The prediction technique was shown to track the channel and improve the BER performance almost to that in the ideal time invariant case.

CONTRAPOSITION 2-REGULARIZED NYQUIST STABILITY CRITERIA FOR LINEAR CONTINUOUS-TIME PERIODIC SYSTEMS

By employing the 2-regularized determinant technique about Hilbert-Schmidt operators and creating what we call the contraposition return difference relationship for finite-dimensional linear continuous-time periodic (FDLCP) feedback systems, a contraposition 2-regularized Nyquist stability criterion is developed for asymptotic stability analysis in this study. The stability conditions of the contraposition 2-regularized Nyquist stability criterion are necessary and sufficient, which can be implemented graphically in a way similar to what we do in linear time-invariant cases but without involving open-loop poles distribution and encircling orientation counting of the Nyquist loci. A numeric implementation algorithm for the contraposition 2-regularized Nyquist criterion is also proposed via truncating the harmonic transfer operator of the FDLCP systems.

Reconstructibility of time-invariant and periodic behavioural systems

In this article, the properties of behavioural reconstructibility and forward-observability for systems over the whole time axis ℤ are introduced. These properties are characterised in terms of appropriate rank conditions, for the time-invariant case. A comparison is made with the existing results in the behavioural setting as well as in the classical state space framework. In the particular case of a periodic system, it is shown that there exists an equivalence between the reconstructibility of the periodic system and its associated lifted system, which is time-invariant. Furthermore, we prove that, for a classical state space system, state reconstructibility is equivalent to behavioural reconstructibility, regardless of the time varying or time-invariant nature of the system. This allows deriving rank tests for the cases of time-invariant and of periodic systems, rediscovering the already known results for state reconstructibility from an alternative perspective. The obtained results contribute to establishing links between two different settings, thus providing a better insight into the considered systems properties.

Uncertainties Quantification for Subspace Identification of Rotating Systems

The dynamics of rotating systems such as helicopters and wind turbines show periodically time-varying behavior. Differently from the linear time-invariant case, such systems cannot be characterized by the classical modal parametrization. An alternative description is made possible with the Floquet theory which extends the modal analysis and the notions of frequencies and damping ratios to the periodic case. Based on this description, the present paper suggests a new output-only subspace identification algorithm able to extract the modal parameters. Besides, in operational modal analysis, the identified modes may be subjected to statistical uncertainty from ambient vibration measurements. Hence, the suggested identification algorithm will allow as well an efficient estimation of the uncertainty bounds on the identified modal parameters.

A Design Method for Pole Placement and Observer of Linear Time-Varying Discrete MIMO Systems

It is well known that, the pole placement controller can be designing for linear time-varying systems using the Frobenius canonical form, as for the time invariant case. This paper presents the new approach to the design of the pole placement controller for linear time-varying discrete multivariable systems. The concept of the relative degrees of multivariable system plays an important role, and the time-varying feedback gain can be simply calculated without transforming the system into any canonical form, which is regarded as a discrete Ackerman's method. This method is applied in order to calculate the observer gain for linear time-varying systems.

Optimal linear estimator for discrete-time systems with random delays

In this paper, optimal estimation for discrete-time linear time-varying systems with randomly state and measurement delays is considered. By introducing a set of binary random variables, the system is converted into the one with both multiplicative noises and constant delays. Then, an estimator which includes the cases of smoothing and filtering, is derived via the projection formula, and the solution is given in terms of a partial difference Riccati equation with boundary conditions. A predictor for such systems is also presented based on the proposed filter and smoother. The obtained estimators have the same dimension as the original state. Conditions for existence, uniqueness, and stability of the steady-state optimal estimators are studied for time-invariant cases. In this case, the obtained estimators are very easy to implement and all calculations can be performed off line, leading to a linear time-invariant estimator.

Optimal experimental design for sampling voltage on dendritic trees in the low-SNR regime

Due to the limitations of current voltage sensing techniques, optimal filtering of noisy, undersampled voltage signals on dendritic trees is a key problem in computational cellular neuroscience. These limitations lead to voltage data that is incomplete (in the sense of only capturing a small portion of the full spatiotemporal signal) and often highly noisy. In this paper we use a Kalman filtering framework to develop optimal experimental design methods for voltage sampling. Our approach is to use a simple greedy algorithm with lazy evaluation to minimize the expected square error of the estimated spatiotemporal voltage signal. We take advantage of some particular features of the dendritic filtering problem to efficiently calculate the Kalman estimator's covariance. We test our framework with simulations of real dendritic branching structures and compare the quality of both time-invariant and time-varying sampling schemes. While the benefit of using the experimental design methods was modest in the time-invariant case, improvements of 25-100% over more naïve methods were found when the observation locations were allowed to change with time. We also present a heuristic approximation to the greedy algorithm that is an order of magnitude faster while still providing comparable results.

Darboux Polynomials for Lie Algebras

AbstractDarboux polynomials and relative characteristic polynomials extend to time-invariant nonlinear systems the concept of eigenvector-eigenvalue pair for linear systems, and are, therefore, very useful to depict the behavior of the system. In this article, using tools such as Lie algebras, it is shown how Darboux polynomials can be characterized and computed in closed-form for time-varying nonlinear systems, thus extending similar results valid in the time-invariant case.

Fractional‐order iterative learning control for fractional‐order linear systems

AbstractIn this paper, we discuss in time domain the convergence of the iterative process for fractional‐order systems. Fractional order iterative learning updating schemes are considered. For the linear time invariant (LTI) system case, the convergence conditions of the fractional‐order and integer‐order iterative learning schemes are proved to be equivalent for D=0. It has been proved by theory and verified by MATLAB/SIMULINK that the tracking speed is the fastest when the system and iterative learning scheme have the same fractional order.Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Correlation-Based and Model-Based Blind Single-Channel Late-Reverberation Suppression in Noisy Time-Varying Acoustical Environments

This paper considers suppression of late reverberation and additive noise in single-channel speech recordings. The reverberation introduces long-term correlation in the observed signal. In the first part of this work, we show how this correlation can be used to estimate the late reverberant spectral variance (LRSV) without having to assume a specific model for the room impulse responses (RIRs) while no explicit estimates of RIR model parameters are needed. That makes this <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">correlation-based</i> approach more robust against RIR modeling errors. However, the correlation-based method can follow only slow time variations in the RIRs. Existing <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">model-based</i> methods use statistical models for the RIRs, that depend on one or more parameters that have to be estimated blindly. The common statistical models lead to simple expressions for the LRSV that depend on past values of the spectral variance of the reverberant, noise-free, signal. All existing model-based LRSV estimators in the literature are derived assuming the RIRs to be <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">time-invariant realizations</i> of a stochastic process. In the second part of this paper, we go one step further and analyze <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">time-varying</i> RIRs. We show that in this case the reverberance tends to become decorrelated. We discuss the relations between different RIR models and their corresponding LRSV estimators. We show theoretically that similar simple estimators exist as in the time-invariant case, provided that the reverberation time <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">60</sub> and direct-to-reverberation ratio (DRR) of the RIRs remain nearly constant during an interval of the order of a few frames. We show that the reverberation time can be taken frequency-bin independent in DFT-based enhancement algorithms. Experiments with time-varying RIRs validate the analysis. Experiments with additive nonstationary noise and time-invariant RIRs show the influence of blind estimation of the reverberation time and the DRR.

Stabilization of Time-varying Stochastic Port-Hamiltonian Systems Based on Stochastic Passivity

The authors have introduced stochastic port-Hamiltonian systems and have clarified some of their properties. Stochastic port-Hamiltonian systems are extension of deterministic port-Hamiltonian systems, which are used to express various deterministic passive systems. However, since only time-invariant case has been considered in our previous results, the aim of this paper is to extend them to time-varying case. Finally, we propose a stabilization method based on passivity and the stochastic generalized canonical transformation, which is a pair of coordinate and feedback transformations preserving the stochastic Hamiltonian structure.

Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems

The output feedback pole placement problem is solved in an input–output algebraic formalism for linear time-varying (LTV) systems. The recent extensions of the notions of transfer matrices and poles of the system to the case of LTV systems are exploited here to provide constructive solutions based, as in the linear time-invariant (LTI) case, on the solutions of diophantine equations. Also, differences with the results known in the LTI case are pointed out, especially concerning the possibilities to assign specific dynamics to the closed-loop system and the conditions for tracking and disturbance rejection. This approach is applied to the control of nonlinear systems by linearization around a given trajectory. Several examples are treated in detail to show the computation and implementation issues.

Online Adaptive Estimation of Sparse Signals: Where RLS Meets the $\ell_1$-Norm

Using the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm to regularize the least-squares criterion, the batch least-absolute shrinkage and selection operator (Lasso) has well-documented merits for estimating sparse signals of interest emerging in various applications where observations adhere to parsimonious linear regression models. To cope with high complexity, increasing memory requirements, and lack of tracking capability that batch Lasso estimators face when processing observations sequentially, the present paper develops a novel time-weighted Lasso (TWL) approach. Performance analysis reveals that TWL cannot estimate consistently the desired signal support without compromising rate of convergence. This motivates the development of a time- and norm-weighted Lasso (TNWL) scheme with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm weights obtained from the recursive least-squares (RLS) algorithm. The resultant algorithm consistently estimates the support of sparse signals without reducing the convergence rate. To cope with sparsity-aware recursive real-time processing, novel adaptive algorithms are also developed to enable online coordinate descent solvers of TWL and TNWL that provably converge to the true sparse signal in the time-invariant case. Simulated tests compare competing alternatives and corroborate the performance of the novel algorithms in estimating time-invariant signals, and tracking time-varying signals under sparsity constraints.

On output feedback control of singularly perturbed systems

We treat the problem of robustness of output feedback controllers with respect to singular perturbations. Given a singularly perturbed control system whose boundary layer system is exponentially stable and whose reduced order system is exponentially stabilizable via a (possibly dynamical) output feedback controller, we present a sufficient condition which ensures that the system obtained by applying the same controller to the original full order singularly perturbed control system is exponentially stable for sufficiently small values of the perturbation parameter. This condition, which is less restrictive than those previously given in the literature, is shown to be always satisfied when the singular perturbation is due to the presence of fast actuators and/or sensors. Furthermore, we show explicitly that, in the linear time-invariant case, if this condition is not satisfied then there exists an output feedback controller which stabilizes the reduced order system but destabilizes the full order system.