Bilinear Dynamical Systems Research Articles

Accelerated algorithm for calculating Gramians of bilinear models of electric power systems

Bilinear models allow to characterize more precisely the behavior of physical processes in electric power systems and thereby improve the quality of their monitoring and control. However, the use of even medium size bilinear models usually requires a lot of computations. In order to improve the efficiency of bilinear analysis of power systems, this paper proposes an accelerated algorithm for calculating the Gramians of bilinear dynamical systems, which also takes into account the effects of sparse matrices. The conventional algorithms iterates the whole matrix. In contrast, in the proposed algorithm we continue iterate only large enough components of the solution matrix and exclude from iterations those components of the Gramian for which corrections become small. The algorithm was implemented and tested for four bilinear models.

On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that the Gramians of controllability and observability of a bilinear system can be divided into parts (sub-Gramians) that characterize the measure of these generalized modes and their interactions. Furthermore, the properties of sub-Gramians were investigated in relation to modal controllability and observability. We also propose an algorithm for computing the Gramians and sub-Gramians based on the element-wise computation of the solution matrix. Based on the proposed algorithm, a novel criterion for the existence of solutions to the generalized Lyapunov equation is proposed, which allows, in some cases, to expand the domain of guaranteed existence of a solution of bilinear equations. Examples are provided that illustrate the application and practical use of the considered spectral decompositions.

Structure-preserving interpolation for model reduction of parametric bilinear systems

In this paper, we present a projection-based interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis.

Spectral decompositions for the solutions of Lyapunov equations for bilinear dynamical systems

In this paper, novel spectral decompositions are obtained for the solutions of generalized Lyapunov equations, which are observed in the study of controllability and observability of the state vector in deterministic bilinear systems. The same equations are used in the stability analysis and stabilization of stochastic linear control systems. To calculate these spectral decompositions, an iterative algorithm is proposed that uses the residues of the resolvent of the dynamics matrix. This algorithm converges for any initial guess, for a non-singular and stable dynamical system. The practical significance of the obtained results is that they allow one to characterize the contribution of individual eigen-components or their pairwise combinations to the asymptotic dynamics of the perturbation energy in deterministic bilinear and stochastic linear systems. In particular, the norm of the obtained eigen-components increases when frequencies of the corresponding oscillating modes approximate each other. Thus, the proposed decompositions provide a new fundamental approach for quantifying resonant modal interactions in a large and important class of weakly nonlinear systems.

Spectral Decompositions for the Solutions of Lyapunov Equations for Bilinear Dynamical Systems

Spectral Decompositions for the Solutions of Lyapunov Equations for Bilinear Dynamical Systems

A bilinear mathcal {H}_{2} model order reduction approach to linear parameter-varying systems

This paper focuses on the model reduction problem for a special class of linear parameter-varying systems. This kind of systems can be reformulated as bilinear dynamical systems. Based on the bilinear system theory, we give a definition of the mathcal {H}_{2} norm in the generalized frequency domain. Then, a model reduction method is proposed based on the gradient descent on the Grassmann manifold. The merit of the method is that by utilizing the gradient flow analysis, the algorithm is guaranteed to converge, and further speedup of the convergence rate can be achieved as well. Two numerical examples are tested to demonstrate the proposed method.

Inexact Linear Solves in Model Reduction of Bilinear Dynamical Systems

The bilinear iterative rational Krylov algorithm (BIRKA) is a very popular, standard, and mathematically sound algorithm for reducing bilinear dynamical systems that arise commonly in science and engineering. This reduction process is termed as a model order reduction (MOR) and leads to a faster simulation of such systems. An efficient variant of the BIRKA, Truncated BIRKA (TBIRKA) has also been recently proposed. Like for any MOR algorithm, these two algorithms also require solving multiple linear systems as part of the model reduction process. For reducing the MOR time, these linear systems are often solved by an iterative solver, which introduces approximation errors (implying inexact solves). Hence, stability analysis of the MOR algorithms with respect to inexact linear solves is important. In our past work, we have shown that under mild conditions, the BIRKA is stable. Here, we look at the stability of the TBIRKA in the same context. Besides deriving the conditions for a stable TBIRKA, our other novel contribution is the more intuitive methodology for achieving this. The stability analysis techniques that we propose here can be extended to many other methods for doing the MOR of bilinear dynamical systems, e.g., using balanced truncation or the ADI methods.

Interpolatory model reduction of parameterized bilinear dynamical systems

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this has not yet occurred for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function are matched tangentially by those of the corresponding reduced-order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including additional vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian.

$H_{2}$ model order reduction based on gramians of discrete-time bilinear systems

Abstract In this paper, on the basis of gramians, $H_{2}$-optimal model order reduction of discrete-time bilinear dynamic systems is investigated. First, based on the reachability gramian and the observability gramian, the $H_{2}$-optimal necessary conditions of the discrete-time bilinear system are derived and the corresponding algorithm is proposed. When the algorithm is convergent, the resulting reduced system satisfies the necessary conditions. Then, for the single-input and single-output system, the $H_2$-optimal necessary conditions based on the generalized cross gramian are presented and the related algorithm is accordingly generated. Similarly, the convergent algorithm can make the reduced system satisfy the necessary conditions. Finally, two numerical examples are employed to illustrate the effectiveness of the proposed methods.

On the positivity, excitability and transparency properties of a class of time-varying bilinear dynamic systems under multiple point internal and external delays

AbstractThis paper investigates formally the internal and external positivity, together with its excitability and transparency properties, of a class of bilinear time-varying continuous-time dynami...

Stability analysis of Bilinear Iterative Rational Krylov Algorithm

Models coming from different physical applications are very large in size. Simulation with such systems is expensive so one usually obtains a reduced model (by model reduction) that replicates the input-output behavior of the original full model. A recently proposed algorithm for model reduction of bilinear dynamical systems, Bilinear Iterative Rational Krylov Algorithm (BIRKA), does so in a locally optimal way. This algorithm requires solving very large linear systems of equations. Usually these systems are solved by direct methods (e.g., LU), which are very expensive. A better choice is iterative methods (e.g., Krylov). However, iterative methods introduce errors in linear solves because they are not exact. They solve the given linear system up to a certain tolerance. We prove that under some mild assumptions BIRKA is stable with respect to the error introduced by the inexact linear solves. We also analyze the accuracy of the reduced system obtained from using these inexact solves and support all our results by numerical experiments.

Structural control of single-input rank one bilinear systems

A bilinear dynamical system can be used to represent the model of a network in which the state obeys linear dynamics and the input is the edge weight of certain controlled edges in the network. We present algebraic and graph-theoretic conditions for the structural controllability of a class of bilinear systems with a single control where the input matrix is rank one. Subsequently, we use these conditions, given a system state graph, to develop an algorithm to design the location of controlled edges (the input matrix) such that the system is structurally controllable.

Fault accommodation in bilinear systems – a logic-dynamic approach

Purpose – The purpose of this study is to investigate the problem of fault accommodation in bilinear dynamic systems. Design/methodology/approach – Solution to this problem is related to constructing the control law which provides full decoupling with respect to the fault effects. The so-called logic-dynamic approach will be used to solve this problem. The main steps of this approach are: replacing the initial bilinear system by certain linear one, solving the problem under consideration for this linear system by well-known linear methods with some restrictions, taking into account the bilinear term to correct the obtained linear solution. Findings – Existing conditions of the fault accommodation problem in a form of rank equalities and inequalities are formulated. Calculating relations for the control law and the auxiliary systems are given. Practical implications – The suggested method allows determining such a control law that preserves the main performances of the system in the faulty case, while the minor performances may degrade. Originality/value – The main advantage of the logic-dynamic approach is a possibility to solve the problem of fault accommodation for nonlinear systems by linear methods without decreasing the main properties of the obtained solution.

Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints

We consider the optimal control problem of minimizing an objective function that is quadratic in the control over a fixed interval for a multi-input bilinear dynamical system in the presence of control constraints. Such models are motivated by and applied to mathematical models for cancer chemotherapy over an a priori specified fixed therapy horizon. The necessary conditions for optimality of the Pontryagin maximum principle are easily evaluated and give a functional description of optimal controls as continuous functions of states and multipliers. However, there is no a priori guarantee that a numerically computed extremal controlled trajectory is locally optimal. In this paper, we formulate sufficient conditions for strong local optimality that are based on the existence of a bounded solution to a matrix Riccati differential equation. The theory is applied to a $3$-compartment model for multi-drug cancer chemotherapy with cytotoxic and cytostatic agents. The numerical results are compared with those for a corresponding optimal control problem when the objective is taken linear in the controls.

Controllability and observability of Boolean control networks with time-variant delays in states.

This brief investigates the controllability and observability of Boolean control networks with (not necessarily bounded) time-variant delays in states. After a brief introduction to converting a Boolean control network to an equivalent discrete-time bilinear dynamical system via the semi-tensor product of matrices, the system is split into a finite number of subsystems (constructed forest) with no time delays by using the idea of splitting time that is proposed in this brief. Then, the controllability and observability of the system are investigated by verifying any so-called controllability constructed path and any so-called observability constructed paths in the above forest, respectively, which generalize some recent relevant results. Matrix test criteria for the controllability and observability are given. The corresponding control design algorithms based on the controllability theorems are given. We also show that the computing complexity of our algorithm is much less than that of the existing algorithms.

Response of SDOF bilinear elasto-dynamical models of cracked concrete beams for El Centro earthquake

Investigated herein are the dynamic responses of three different single degree of freedom (SDOF) bilinear dynamical models of a cracked concrete beam subjected to the El Centro earthquake. The first model is an equivalent unilinear model while the other two models have different stiffness values but differ from each other in the dependence of their damping coefficients on the sense of amplitude. All three models have the same natural period. The three models have been found to predict different response spectra for the same bilinear dynamical system. The effect of constant loading on the dynamical response of the concrete structures subjected to sinusoidal and random loading has been found to be significant for more flexible structures. Implications of these theoretical predictions for seismic design of concrete structures are discussed.

Identification of discrete-time bilinear systems through equivalent linear models

This paper presents a novel method for identification of discrete-time, time-invariant state-space models of bilinear dynamical systems using the steady-state portion of a single input/multiple output time-history measurements. These measurements are recorded by exciting the system with a linear combination of sine and cosine functions of user-selected frequencies enriched by a subtle amount of random component. The proposed method relies on conversion of the bilinear system into an equivalent linear model (ELM) by an accurate approximation of the state in the bilinear term using a set of sine and cosine basis functions whose frequencies are obtained as combinations of the input frequencies. Observer/Kalman Filter Identification (OKID), a linear time invariant (LTI) system identification algorithm, is used to identify the aforementioned ELM from which the original bilinear model is recovered. A numerical example is also provided.

Bilinear elastodynamical models of cracked concrete beams

Concrete structures are generally cracked in flexural tension at working loads. Concrete beams with asymmetric section details and crack patterns exhibit different flexural rigidity depending upon the sense of the applied flexural moment. In this paper, three different models, having the same natural period, of such SDOF bilinear dynamical systems have been proposed. The Model-I and Model-II have constant damping coefficient, but the latter is characterized by two stiffness coefficients depending upon the sense of vibration amplitude. The Model-III, additionally, has two damping coefficients as well. In this paper, the dynamical response of Model-III to sinusoidal loading has been investigated and compared with that of Model-II studied earlier. It has been found that Model-III exhibits regular and irregular sub-harmonics, jump phenomena and strong sensitivity to initial conditions, forcing frequency, system period as well as the sense of peak sinusoidal force. The constant sustained load has been found to affect the natural period of the dynamical system. The predictions of Model-I have been compared with those of the approximate linear model adopted in present practice. The behaviour exhibited by different models of the SDOF cracked elastic concrete structures under working loads and the theoretical and practical implications of the approach followed have been critically evaluated.

Bayesian deconvolution fMRI data using bilinear dynamical systems

In Penny et al. [Penny, W., Ghahramani, Z., Friston, K.J. 2005. Bilinear dynamical systems. Philos. Trans. R. Soc. Lond. B Biol. Sci. 360(1457) 983–993], a particular case of the Linear Dynamical Systems (LDSs) was used to model the dynamic behavior of the BOLD response in functional MRI. This state-space model, called bilinear dynamical system (BDS), is used to deconvolve the fMRI time series in order to estimate the neuronal response induced by the different stimuli of the experimental paradigm. The BDS model parameters are estimated using an expectation–maximization (EM) algorithm proposed by Ghahramani and Hinton [Ghahramani, Z., Hinton, G.E. 1996. Parameter Estimation for Linear Dynamical Systems. Technical Report, Department of Computer Science, University of Toronto]. In this paper we introduce modifications to the BDS model in order to explicitly model the spatial variations of the haemodynamic response function (HRF) in the brain using a non-parametric approach. While in Penny et al. [Penny, W., Ghahramani, Z., Friston, K.J. 2005. Bilinear dynamical systems. Philos. Trans. R. Soc. Lond. B Biol. Sci. 360(1457) 983–993] the relationship between neuronal activation and fMRI signals is formulated as a first-order convolution with a kernel expansion using basis functions (typically two or three), in this paper, we argue in favor of a spatially adaptive GLM in which a local non-parametric estimation of the HRF is performed. Furthermore, in order to overcome the overfitting problem typically associated with simple EM estimates, we propose a full Variational Bayes (VB) solution to infer the BDS model parameters. We demonstrate the usefulness of our model which is able to estimate both the neuronal activity and the haemodynamic response function in every voxel of the brain. We first examine the behavior of this approach when applied to simulated data with different temporal and noise features. As an example we will show how this method can be used to improve interpretability of estimates from an independent component analysis (ICA) analysis of fMRI data. We finally demonstrate its use on real fMRI data in one slice of the brain.

A model-order reduction method based on Krylov subspaces for mimo bilinear dynamical systems

In this paper, we present a Krylov subspace based projection method for reduced-order modeling of large scale bilinear multi-input multioutput (MIMO) systems. The reduced-order bilinear system is constructed in such a way that it can match a desired number of moments of multivariable transfer functions corresponding to the kernels of Volterra series representation of the original system. Numerical examples report the effectiveness of this method.