Lyapunov Differential Equation Research Articles

Matrix Pontryagin principle approach to controllability metrics maximization under sparsity constraints

This study considers optimization problems for time-varying network topologies that maximize the controllability of linear network systems under sparsity constraints. The major theoretical differences from the existing findings on control node scheduling problems is that our method is based on the matrix-valued Pontryagin maximum principle applied to the controllability Lyapunov differential equation. This allows us to handle various controllability metrics that can only be approximated by existing methods. Finally, we perform a numerical evaluation of our proposed method and compare results with those of our previous approximation method.

Optimal sequential fusion estimation for sampled‐data systems with random delays and multiplicative noise

AbstractThis article is concerned with the optimal sequential fusion filter design issue for a class of sampled‐data systems subject to random transmission delays and multiplicative noise. The considered delay is modeled as a multistate Markov chain, while the multiplicative noise corrupting the observation measurements is assumed be a white noise sequence with known statistical properties. First, by reorganizing the original measurements, the solvability of the addressed problem is cast into the feasibility of an optimal estimation issue for the hybrid Markovian jump system without delays. Then, an optimal sequential fusion filter is constructed by utilizing a dynamic model with finite jumps on basis of the arrival order of measurements from different sensors, guaranteeing estimation performances both at sampling instants and within the sampling intervals. Moreover, the desired estimator gains can be computed via solving a series of coupled differential Riccati equations with jumps and coupled differential Lyapunov equations. Further, a stationary sequential fusion filter is achieved with the guarantee of the exponentially mean‐square stability. Finally, two simulation examples are provided to verify the validity of the proposed algorithms.

Multi-Performance Economical Optimization of Base Isolation System with Tuned Inerter Negative Stiffness Damper Subjected to Non-Stationary Seismic Excitation

Base isolation proves to be an effective vibration control strategy for buildings. However, the base-isolation floor (BIF) may undergo substantial displacements. To improve the seismic performance of base-isolation system, this paper proposes a multi-performance economical optimization procedure (MPEOP) for exploring the optimal parameters of base isolation system with tuned inerter negative stiffness damper (BIS-TINSD) subjected to non-stationary seismic excitation. The simplified analysis model for BIS-TINSD is developed, followed by the formulation of the state-space representation. The non-stationary seismic excitation is represented as a stationary Gaussian process with a time-modulating function based on the Clough–Penzien spectrum, and combining the equations of motion with seismic excitation results in the augmented state-space representation of structure-damper-excitation, followed by the formulation of differential Lyapunov equation. The multi-objective performance economical index is proposed, in which the performance optimization objective is defined as a weighted combination of base isolation floor displacement and superstructure acceleration, and the economic optimization objective is set as the support stiffness. The Pareto optimal fronts (POF) are adopted to deal with optimal objectives. The examination of optimal design parameters is extended under real earthquake records. The results demonstrate the effectiveness of the MPEOP, and indicate the advantages of the BIS-TINSD compared with the inerter amplify damper (IAD) and negative stiffness inerter damper (NSID).

Hierarchy of Quadratic Lyapunov Functions for Linear Time-Varying and Related Systems

A method for constructing homogeneous polynomial Lyapunov functions is presented for linear time-varying or switched-linear systems and the class of nonlinear systems that can be represented as such. The method uses a simple recursion based on the Kronecker product to generate a hierarchy of related dynamical systems, whose first element is the system under study and the second element is the well-known Lyapunov differential equation. It is then proven that a quadratic Lyapunov function for the system at one level in the hierarchy, which can be found via semidefinite programming, is a homogeneous polynomial Lyapunov function for the system at the base level in the hierarchy. Searching for Lyapunov functions of the foregoing kind is equivalent to searching for homogeneous polynomial Lyapunov functions via the formulation of sum-of-squares programs. The quadratic perspective presented in this paper enables the easy development of procedures to compute bounds on pointwise-in-time system metrics, such as peak norms, system stability margins, and many other performance measures. The applications of the theory to analyzing an aircraft model, on the one hand, and an experimental aerospace vehicle, on the other hand, are presented. The theory can be comprehended with a first course on state-space control systems and an elementary knowledge of convex programming.

Continuous-time balanced truncation for time-periodic fluid flows using frequential Gramians

Reduced-order models for flows that exhibit time-periodic behavior (e.g., flows in turbomachinery and wake flows) are critical for several tasks, including active control and optimization. One well-known procedure to obtain the desired reduced-order model in the proximity of a periodic solution of the governing equations is continuous-time balanced truncation. Within this framework, the periodic reachability and observability Gramians are usually estimated numerically via quadrature using the forward and adjoint post-transient response to impulses. However, this procedure can be computationally expensive, especially in the presence of slowly-decaying transients. Moreover, it can only be performed if the periodic orbit is stable in the sense of Floquet. In order to address these issues, we use the frequency-domain representation of the Gramians, which we henceforth refer to as frequential Gramians. First, these frequential Gramians are well-defined for both stable and unstable dynamics. In particular, we show that when the underlying system is unstable, these Gramians satisfy a pair of allied differential Lyapunov equations. Second, they can be estimated numerically by solving algebraic systems of equations that lend themselves to heavy computational parallelism and that deliver the desired post-transient response without having to follow physical transients. The computational gains that we can achieve by using the frequency domain are demonstrated on a simple three-dimensional toy model that exhibits time-periodic dynamics. We then demonstrate this method on a periodically-forced axisymmetric jet at Reynolds numbers Re=1250 and Re=1500. At the lower Reynolds number, the flow strongly amplifies subharmonic perturbations and exhibits vortex pairing about a Floquet-stable T-periodic solution. At the higher Reynolds number, the underlying T-periodic orbit is unstable and the flow naturally settles onto a 2T-periodic limit cycle characterized by pairing vortices. At both Reynolds numbers, we compute a reduced-order model and we use it to design a feedback controller and a state estimator capable of suppressing vortex pairing.

On Quaternion Modeling and Kalman Filtering from Vector Observations

This paper presents novel results on quaternion state-space modeling and estimation from rate gyro measurements and vector observations. The spectral properties of the Lyapunov differential equation for the quaternion second-order moment are investigated and expressions for its solution are developed. The properties of a four-dimensional rank-degenerate measurement model of the quaternion are analyzed. Four different approaches are suggested in order to reduce the dimension of the measurement equation, hinging on information optimization or on the spectral properties of the measurement matrix. While the measurement dimension’s reduction saves computations, it does not impair the exact statistical treatment of state-dependent process and measurement noises. As a result, the proposed filters are optimal and statistically consistent without tuning. The performance of the maximum information rate quaternion filter is verified via extensive Monte Carlo simulations. The test cases of spinning and rapidly tumbling rotations of a small satellite on a low Earth orbit are simulated. The filters process measurements of the sun vector, of the Earth magnetic field, and of rate gyros. The novel filters converge where a standard extended Kalman filter fails to do so under similar conditions.

A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations

In this paper, we apply the Bernstein collocation method to construct the solution set of the Sylvester matrix differential equation (Sy-MDE) which involves the Lyapunov matrix differential equation. The method depends on the collocation method and Bernstein polynomials. The main advantage of the proposed method is that by using this method Sy-MDE reduces to a linear system of algebraic equations which can be solved by using an appropriate iterative method. We analyze the error and give a theorem that bounds the error. We also give the residual correction procedure to estimate the error. By using the procedure, we obtain a new approximate solution, namely a corrected Bernstein collocation solution. To illustrate how the proposed method is applied, several examples are given. Numerical experiments show the effectiveness and accuracy of the method for solving such types of Sy-MDE.

Stochastic Optimization of Dissipation Structures Based on Lyapunov Differential Equations and the Full Stress Design Method

This article presents a Lyapunov precise integral-based analysis method for seismic structures with added viscous fluid dampers. This study uses the full stress algorithm as the optimization method, considering the mean square of interstory drifts as the optimization objective, the position of the damper as the optimization object, and the random vibration analysis method as the calculation method to optimize seismic frame structures with viscous dampers. A precise integral solution is derived for the Lyapunov equation based on the general expression of the Lyapunov differential equation for the damping system under the excitation of a nonstationary stochastic process using two types of modulation functions: g(t)=1 and g(t)=t. Finally, the optimal damping arrangement is achieved using this method with a six-layer non-eccentric planar frame. In addition, the optimization results of this study are verified with those in the literature using time-history analysis, which verifies the feasibility and effectiveness of the proposed method. This study provides a method for the optimal configuration of dampers for seismic response of structures, which is beneficial for engineering applications and the protection of seismic structures.

Stochastic Optimization and Sensitivity Analysis of the Combined Negative Stiffness Damped Outrigger and Conventional Damped Outrigger Systems Subjected to Nonstationary Seismic Excitation

In recent years, various kinds of damped outrigger systems, particularly with the negative stiffness devices, have been shown to effectively suppress excessive vibration induced by earthquake and wind. To gain insight into such systems, this paper proposes a stochastic optimization method to investigate optimal configurations of combined negative stiffness damped outriggers (NSDOs) and conventional damped outriggers (CDOs) subjected to nonstationary stochastic seismic excitation. The simplified analysis model of various combinations of damped outriggers is developed, and the state-space representation of outrigger systems is then formulated. The nonstationary seismic excitation is modeled as a uniformly modulated stationary Gaussian process with a time-modulating function following Clough–Penzien spectrum, and combining equations of motions of outrigger systems and seismic excitation gives rise to the augmented state-space representation of structure-damper-excitation and subsequently the differential Lyapunov equation. The optimal designs of damped outriggers are defined via the solution of the differential Lyapunov equation. The multiobjective optimization with Pareto optimal fronts is adopted to deal with conflicting objectives between harmful interstory drift and floor absolute acceleration. The sensitivity of negative stiffness devices on optimal objectives is also investigated. The optimal designs are further examined under real typical and near-fault earthquake records. The results demonstrate the efficacy of the proposed method and provide insight into the systems subjected to nonstationary seismic excitation. While the purely NSDO systems could have better performance with adequate negative stiffness devices, the proposed combined NSDO and CDO systems prove to be more effective with limited negative stiffness devices and thus provide more design flexibilities tailored to different application situations.

Controllability and observability for fractal linear dynamical systems

In the present paper, we investigate the controllability, and observability of a non-homogeneous continuous-time fractal dynamical system (NC-TFDS). We show that the controllability is equivalent to a controllability matrix that has a full rank. We also show a relationship between controllability and fractal differential Lyapunov equation (F-DLE). We give some theorems for the observability of a NC-TFDS. Finally, we offer two examples are provided to illustrate the effectiveness of our results and compare it with the two papers Al-Zhour (2022) and (Sadek et al., 2022).

Computing the Lyapunov operator φ-functions, with an application to matrix-valued exponential integrators

In this paper, we develop efficient and accurate evaluation for the Lyapunov operator function φl(LA)[Q], where φl(⋅) is the function related to the exponential, LA is a Lyapunov operator and Q is a symmetric and full-rank matrix. An important application of the algorithm is to the matrix-valued exponential integrators for matrix differential equations such as differential Lyapunov equations and differential Riccati equations. The method is exploited by using the modified scaling and squaring procedure combined with the truncated Taylor series. A quasi-backward error analysis is presented to determine the value of the scaling parameter and the degree of the Taylor approximation. Numerical experiments show that the algorithm performs well in both accuracy and efficiency.

Lyapunov equation in open quantum systems and non-Hermitian physics

The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic Hamiltonian, linear equations of motion) open quantum systems, typically from quantum master equations. Despite this, the Lyapunov equation is seldom considered a fundamental formalism for linear open quantum systems. In this work we aim to change that. We establish the Lyapunov equation as a fundamental and efficient formalism for linear open quantum systems that can go beyond the limitations of various standard quantum master equation descriptions, while remaining of much less complexity than general exact formalisms. This also provides valuable insights for non-Hermitian quantum physics. In particular, we derive the Lyapunov equation for the most general number conserving linear system in a lattice of arbitrary dimension and geometry, connected to an arbitrary number of baths at different temperatures and chemical potentials. Three slightly different forms of the Lyapunov equation are derived via an equation of motion approach, by making increasing levels of controlled approximations, without reference to any quantum master equation. Then we discuss their relation with quantum master equations, positivity, accuracy and additivity issues, the possibility of describing dark states, general perturbative solutions in terms of single-particle eigenvectors and eigenvalues of the system, and quantum regression formulas. Our derivation gives a clear understanding of the origin of the non-Hermitian Hamiltonian describing the dynamics and separates it from the effects of quantum and thermal fluctuations. Many of these results would have been hard to obtain via standard quantum master equation approaches.

Finite-time Annular Domain Stability and Stabilization of Itô-type Stochastic Time-Varying Systems with Wiener and Poisson noises

This paper investigates finite time annular domain (FTAD) stability and stabilization for Itô-type stochastic time-varying systems with continuous Wiener and discontinuous Poisson noises (STVSWPNs). First, using Itô-Levy formula and time-varying multiple quadratic Lyapunov functions, two less conservative FTAD-stability conditions based generalised differential Lyapunov equations (GDLEs) and differential linear matrix inequalities (DLMIs) are obtained. Second, the FTAD-stabilization is studied and some new sufficient conditions for the existence of state feedback and static output feedback controllers are presented by tractable differential linear matrix inequalities. Moreover, a new numerical algorithm is given. Finally, a numerical example and a real-world example are utilized to show the effectiveness of the proposed methods.

Non-stationary approximate response of non-linear multi-degree-of-freedom systems subjected to combined periodic and stochastic excitation

An approximate method is presented for determining the non-stationary response of multi-degree-of-freedom non-linear systems subjected to combined periodic and stochastic excitation. Specifically, first, decomposing the system response as a combination of a periodic and of a zero-mean stochastic component, transfers the equation of motion into two sets of equivalent coupled differential sub-equations, governing the deterministic and the stochastic component, respectively. Next, the derived stochastic sub-equations under non-stationary stochastic excitation are cast into equivalent linear equations by resorting to the non-stationary statistical linearization method. Further, the related Lyapunov differential equations governing the second moment of the linear stochastic response, and the deterministic sub-equations governing the periodic response, are solved simultaneously using standard numerical algorithms. Pertinent Monte Carlo simulation demonstrates the applicability and accuracy of the proposed semi-analytical method.

Lyapunov differential equations and inequalities for stability and stabilization of linear time-varying systems

This paper studies exponential stability and stabilization of linear time-varying systems without assuming that the coefficient matrix is bounded, which is generally necessary in the existing literature. First, by introducing the notions of uniformly exponentially stable and uniformly exponentially expanding functions, some Lyapunov differential inequalities based characterizations for exponential stability of linear time-varying systems are established. Second, it is shown that the linear time-varying system is both uniformly exponentially stable and uniformly exponentially expanding if and only if the corresponding Lyapunov differential equation has a unique positive definite solution. Third, some degenerated Lyapunov differential equations, where the Q matrix is only positive semi-definite, are also re-examined. Finally, a weighted controllability Gramian, which contains a weighting time-varying function that can be properly designed to reveal a trade-off between the convergence rate/regulation time of the closed-loop system and the control effort, based stabilization approach is established to ensure that the closed-loop system is both uniformly exponentially stable and uniformly exponentially expanding. The effectiveness of the proposed approach is illustrated by the design of the spacecraft attitude control system with magnetic actuation.

Fast calibration of two‐factor models for energy option pricing

AbstractEnergy companies need efficient procedures to perform market calibration of stochastic models for commodities. If the Black framework is chosen for option pricing, the bottleneck of the market calibration is the computation of the variance of the asset. Energy commodities are commonly represented by multifactor linear models, whose variance obeys a matrix Lyapunov differential equation. In this article, analytical and methods to derive the variance are discussed: the Lyapunov approach is shown to be more straightforward than ad hoc derivations found in the literature and can be readily extended to higher dimensional models. A case study is presented, where the variance of a two‐factor mean‐reverting model is embedded into the Black formulae and the model parameters are calibrated against listed options. The analytical and numerical methods are compared, showing that the former makes the calibration 14 times faster. A Python implementation of the proposed methods is available as open‐source software on GitHub.

The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations

In this paper, we present a new approach for solving large-scale differential Lyapunov equations. The proposed approach is based on projection of the initial problem onto an extended block Krylov subspace by using extended nonsymmetric block Lanczos algorithm then, we get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix equation is solved by the Backward Differentiation Formula method (BDF) or Rosenbrock method (ROS), the obtained solution allows to build a low-rank approximate solution of the original problem. Moreover, we also give some theoretical results. The numerical results demonstrate the performance of our approach.

Modified Douglas splitting method for differential matrix equations

In this paper, we consider a modified Douglas splitting method for a class of differential matrix equations, including differential Lyapunov and differential Riccati equations. The method we consider is based on a natural three-term splitting of the equations. The implementation of the algorithm requires only the solution of a linear algebraic system with multiple right-hand sides in each time step. It is proved that the method is convergent of order two and it preserves the symmetry and positive semidefiniteness of solutions of differential Lyapunov equations. Moreover, we show how the method can be handled in a low-rank setting for large-scale computations. We also provide a theoretical a priori error analysis for the low-rank algorithms. Numerical results are presented to verify the theoretical analysis.

Lyapunov-Equation-Based Stability Analysis for Switched Linear Systems and Its Application to Switched Adaptive Control

This article investigates the stability of continuous-time switched linear systems with dwell-time constraints. A fresh insight into this established problem is provided via novel stability conditions that require the solution to a family of differential Lyapunov equations and algebraic Lyapunov equations. The proposed analysis, which leads to a peculiar Lyapunov function that is decreasing in between and at switching instants, enjoys the following properties: it achieves the same dwell time as the well-known result in the research “stability and stabilization of continuous time switched linear systems” by Geromel and Colaneri; it removes the increasing computational complexity of the linear interpolation method; it leads to a straightforward counterpart for discrete-time switched linear systems.We show the application of this methodology to the problem of adaptive control of switched linear systems with parametric uncertainties.

Necessary/sufficient conditions for Pareto optimality in finite horizon mean-field type stochastic differential game

This paper is concerned with necessary and sufficient conditions for the existence of Pareto solutions in finite horizon mean-field type stochastic cooperative differential game. Based on the equivalent characterization of Pareto optimality, the problem is transformed into a set of constrained mean-field type stochastic optimal control problems with a special structure. Utilizing the mean-field type stochastic minimum principle, the necessary conditions are put forward. Under certain convex assumptions, it is shown that the necessary conditions are also sufficient ones. Next, the indefinite linear quadratic (LQ) case is studied. It is pointed out that the solvability of two related generalized differential Riccati equations (GDREs) provides a sufficient condition under which Pareto efficient strategies are equivalent to weighted sum optimal controls. In addition, all Pareto solutions are obtained based on the solutions of two generalized differential Lyapunov equations (GDLEs). At last, an example sheds light on the effectiveness of the theoretical results.