Bilinear Systems Research Articles

Loewner functions for bilinear systems

This work brings together the moment matching approach based on Loewner functions and the classical Loewner framework based on the Loewner pencil in the case of bilinear systems. New Loewner functions are defined based on the bilinear Loewner framework, and a Loewner equivalent model is produced using these functions. This model is composed of infinite series that needs to be truncated in order to be implemented in practice. In this context, a new notion of approximate Loewner equivalence is introduced. In the end, it is shown that the moment matching procedure based on the proposed Loewner functions and the classical interpolatory bilinear Loewner framework both result in κ-Loewner equivalent models, the main difference being that the latter preserves bilinearity at the expense of a higher order.

Finite Spectrum Assignment Problem and Stabilization of Bilinear Systems With Both Lumped and Distributed Delay

We consider a bilinear control system defined by a linear time-invariant system of differential equations with both lumped and distributed delay in the state variable. We study the arbitrary finite spectrum assignment problem. One needs to construct a control vector such that the characteristic function of the closed-loop system becomes a polynomial with arbitrary preassigned coefficients. We obtain conditions on coefficients of the system under which the criterion was found for solvability of this finite spectrum assignment problem. This criterion is expressed in terms of rank conditions for matrices of the special form. Corollaries on stabilization of bilinear systems with delay are obtained.

Edge Selections in Bilinear Dynamic Networks

We develop some basic principles for the design and robustness analysis of a continuous-time bilinear dynamical network, where an attacker can manipulate the strength of the interconnections/edges between some of the agents/nodes. We formulate the edge protection optimization problem of picking a limited number of attack-free edges and minimizing the impact of the attack over the bilinear dynamical network. In particular, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> -norm of bilinear systems is known to capture robustness and performance properties analogous to its linear counterpart and provides valuable insights for identifying which edges are most sensitive to attacks. The exact optimization problem is combinatorial in the number of edges, and brute-force approaches show poor scalability. However, we show that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> -norm as a cost function is supermodular and, therefore, allows for efficient greedy approximations of the optimal solution. We illustrate and compare the effectiveness of our theoretical findings via numerical simulations.

Near-Controllability of Bilinear Systems Through Near-Transitivity: A New Lie-Algebraic Notion

Near-Controllability of Bilinear Systems Through Near-Transitivity: A New Lie-Algebraic Notion

Output Feedback Tracking Control for a Class of Bilinear Systems: Application to Power Electronics Devices

Output Feedback Tracking Control for a Class of Bilinear Systems: Application to Power Electronics Devices

Optimal control problems governed by a class of nonlinear systems

&lt;abstract&gt;&lt;p&gt;This article suggested a solution to a flow control problem governed by a class of nonlinear systems called bilinear systems. The problem was initially well-posed, and after it was established that an optimal control solution existed, its characteristics were stated. After that, we demonstrated how to use various bounded feedback controls to make a plate equation's flow close to the required profile. As an application, we resolved the plate equation-governed partial flow control issue. The findings bring up a variety of system applications, which can be employed in engineering advancement.&lt;/p&gt;&lt;/abstract&gt;

A brief review on stability investigations of numerical methods for systems of stochastic differential equations

&lt;abstract&gt;&lt;p&gt;A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by $ m $-dimensional Wiener processes $ W = (W^1, W^2, ..., W^m) $ is presented in $ \mathbb{R}^d $. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of $ \mathbb{R}^{d \times d} $, instead of classic stability functions with values in $ \mathbb{C}^1 $, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size $ h $ is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions $ d \ge 1 $.&lt;/p&gt;&lt;/abstract&gt;

Painlevé integrable property, Bäcklund transformations, Lax pair, and soliton solutions of a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid

In this paper, we focus our attention on a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid with symbolic computation. The Painlevé integrable property is derived. Via the Ablowitz–Kaup–Newell–Segur procedure, we obtain a Lax pair under the coefficient constraints. Based on the Hirota method, we obtain a bilinear form and a bilinear Bäcklund transformation under the coefficient constraints. We derive the auto-Bäcklund transformations based on the truncated Painlevé expansions. According to the bilinear form, we give the two-soliton solutions under the coefficient constraints. We also discuss the relation between the variable coefficients and soliton solutions, i.e., how the two solitons present different types with the different forms of the variable coefficients.

A reduced-order adaptive state observer for DC–DC converters with unknown constant power load

In this paper, main objective is to develop a reduced-order adaptive state observer (RASO) for a large class of DC–DC converters with constant power load (CPL) to estimate their unavailable states and unknown parameter, which facilitates achieving an output feedback control (OFC) scheme of the systems. Its construction contains two main procedures. First, by using a generalized parameter estimation based observer (GPEBO) technique, state reconstruction problem is reformulated as the estimate of parameter after the system reparameterization. A linear regressor equation is then obtained. Second, we apply GPEBO technique for it again and then dynamic regressor extension and mixing (DREM) method are adopted such that a RASO for the converters described by a class of bilinear system is designed, which is called as 2G+D observer. The comparison study among the proposed observer and other existing observers is conducted. An obvious merit of the former is that the observability of the systems is only verified for ensuring exponential convergence without needing any extra excitation condition. Subsequently, the presented observer is applied to boost converter feeding unknown CPL for achieving an OFC. Finally, the simulation and experimental results are provided.

Fractional output stabilization of distributed bilinear systems with a time delay

Fractional output stabilization of distributed bilinear systems with a time delay

Exponential stabilization of Riesz-spectral bilinear boundary control systems

Exponential stabilization of Riesz-spectral bilinear boundary control systems

A unifying framework for tangential interpolation of structured bilinear control systems

A unifying framework for tangential interpolation of structured bilinear control systems

Nonlinear instantaneous characteristics and asymmetry of the bilinear system with application to the structural damage estimation

Common structural damage usually exhibits bilinear behavior in the vibration. To identify such damage, the nonlinearity and asymmetry shown in the free vibration of the bilinear system are studied. The instantaneous characteristics of the free vibration are analyzed by employing the well-known HT method. The modulated instantaneous parameters and softening stiffness are used as damage indicators. Three new indices are proposed for ‘bilinear-type’ damage detection. For further damage severity estimation, two bilinear-stiffness estimation methods are developed based on the asymmetric restoring force and the asymmetric phase trace, respectively. Numerical study shows that the methods can accurately estimate the bilinear stiffness of the cracked beam with strong noise robustness and low sensitivity to initial weak nonlinearity. Experimental study was carried out on a steel cantilever beam with loose-connection. The damage can be effectively detected by the proposed indices and the stiffness coefficient ratios can be estimated by the two proposed methods.

Guided Deep Learning Manifold Linearization of Porous Media Flow Equations

Summary Integrated reservoir studies for performance prediction and decision-making processes are computationally expensive. In this paper, we develop a novel linearization approach to reduce the computational burden of intensive reservoir simulation execution. We achieve this by introducing two novel components: (1) augmention of the state-space to yield a bilinear system and (2) an autoencoder based on a deep neural network to linearize physics reservoir equations in a reduced manifold using a Koopman operator. Recognizing that reservoir simulators execute expensive Newton-Raphson iterations after each timestep to solve the nonlinearities of the physical model, we propose “lifting” the physics to a more amenable manifold where the model behaves close to a linear system, similar to the Koopman theory, thus avoiding the iteration step. We use autoencoder deep neural networks with specific loss functions and structure to transform the nonlinear equation and frame it as a bilinear system with constant matrices over time. In such a way, it forces the states (pressures and saturations) to evolve in time by simple matrix multiplications in the lifted manifold. We also adopt a “guided” training approach, which is performed in three steps: (1) We initially train the autoencoder, (2) then we use a “conventional” model order reduction (MOR) as an initializer for the final (3) full training, when we use reservoir knowledge to improve and to lead the results to physically meaningful output. Many simulation studies exhibit extremely nonlinear and multiscale behavior, which can be difficult to model and control. Koopman operators can be shown to represent any dynamical system through linear dynamics. We applied this new framework to a 2D two-phase (oil and water) reservoir subject to a waterflooding plan with three wells (one injector and two producers) with speedups around 100 times faster and accuracy in the order of 1% to 3% on the pressure and saturation predictions. It is worthwhile noting that this method is a nonintrusive data-driven method because it does not need access to the reservoir simulation internal structure; thus, it is easily applied to commercial reservoir simulators and is also extendable to other studies. In addition, an extra benefit of this framework is to enable the plethora of well-developed tools for MOR of linear systems. To the authors’ knowledge, this is the first work that uses the Koopman operator for linearizing the system with controls. As with any MOR method, this can be directly applied to a well-control optimization problem and well-placement studies with low computational cost in the prediction step and good accuracy.

Feedback stabilization of non-homogeneous distributed bilinear systems with varying time delay

This paper deals with the feedback stabilization for a class of non-homogeneous distributed bilinear systems, with periodically varying time delay, evolving in Hilbert state space. Sufficient conditions for polynomial and exponential stabilization are given. Explicit energy decay rates are provided in both cases. Illustrative examples are displayed.

Effectiveness of a TRM solution for rammed earth under in-plane cyclic loads

To evaluate the effectiveness of a TRM-strengthening solution for rammed earth walls subjected to in-plane cyclic loads, an experimental program was conducted on a strengthened mock-up previously damaged. The experimental results are discussed in comparison with the previous unstrengthened model in terms of cracking pattern, damage identification, displacements, base shear coefficient, stiffness degradation, and energy dissipation; in addition, simplified equivalent linear and bi-linear systems are inferred to assess the performance. The outcomes highlighted the effectiveness of the TRM solution in improving the in-plane shear capacity, the ductility and the dissipated energy of the mock-up.

An observer design for Takagi-Sugeno fuzzy bilinear control systems

An observer design for Takagi-Sugeno fuzzy bilinear control systems

Ideal observability for bilinear discrete-time systems with and without delays in observation

Ideal observability for bilinear discrete-time systems with and without delays in observation

Optimal control problem for unbounded bilinear systems and applications

In this paper, we will investigate the optimal control problem for unbounded bilinear systems, with -admissible control operators. We will first study the case of finite time-horizon with unconstrained or constrained endpoint. This result is further applied to build the optimal control for infinite time horizon. Finally, we solve the bilinear optimal control for the transport equation. Then we consider the fractional diffusion equation, for which we prove the strong stabilisation by an optimal time-varying feedback control.

Geometric criteria for controllability and near‐controllability of driftless discrete‐time bilinear systems

AbstractGeometric approaches have been well developed for both linear and nonlinear systems, which are useful in the analysis and design of control systems. However, geometric approaches have been less considered for discrete‐time bilinear systems. In this paper, controllability and near‐controllability of driftless discrete‐time bilinear systems are dealt with from a geometric point of view. More specifically, geometric characterizations are presented for controllability and near‐controllability as well as for controllable subspaces and nearly controllable subspaces of the systems. Differently from the classical geometric approach for linear systems, it is shown that, for the bilinear systems, a controllable subspace has to be determined by two linear subspaces, while a nearly controllable subspace is determined by one linear subspace. As a result, geometric criteria for controllability, near‐controllability, controllable subspaces, and nearly controllable subspaces of the systems are respectively derived, which are also applied to multi‐input discrete‐time bilinear systems and to linear time‐invariant systems with a multiplicative perturbation. Examples are given to illustrate the derived geometric criteria.