Lyapunov Matrix Inequality Research Articles

An explicit solution to the generalized Lyapunov matrix inequality

We review the generalized Lyapunov matrix inequality for continuous-time systems and we provide a solution based on a modification of the Jordan normal form. The proposed construction allows to obtain explicit bounds to the decay rate and the inertia of the solution to the generalized Lyapunov matrix inequality.

On the hyper-Lyapunov matrix inclusions

By adding a scalar parameter to the classical Lyapunov matrix inequality, the underlying structure turns to be richer: Partial order of stability sets is introduced. This is then used to improve estimates of trajectories associated with differential inclusions.Technically, the spectrum of all matrices, satisfying a given Lyapunov inequality, lies within a special disk in the right-half plane: Under inversion such a disk is mapped onto itself.As a by-product, it is shown that these disks are a natural tool to understanding the Matrix-Sign-Function iteration scheme, used in matrix computations.Hyper-Lyapunov inclusions are formulated through Matrix-Quadratic-Form Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within sub-unit disks.

A dual-gain scaling approach to event-triggered adaptive tracking for large-scale nonlinear systems via output feedback

A novel non-backstepping output feedback control approach, called as the dual-gain scaling approach, is proposed by combining the extended Lyapunov matrix inequalities with dual-domination idea. This approach provides partially decentralised adaptive output feedback controllers with three event-triggered mechanisms to achieve the global practical tracking of a family of uncertain large-scale nonlinear systems subject to unknown control coefficients and strongly interconnected nonlinearities with output-polynomial growth rate. The dual gains are embedded in the observers and controllers, which can effectively compensate for time-varying control coefficients and unknown constant growth rate multiplied by output-polynomial function. Three different threshold schemes including fixed threshold, relative threshold and time-varying threshold are introduced to save communication resources. The stability analysis shows that the solutions of the closed-loop system are globally uniformly bounded and the tracking error converges to a preset compact set after a finite time; also, there will be no zeno behaviour in the closed-loop response.

Gain margin and Lyapunov analysis of discrete‐time network synchronization via Riccati design

SummaryThis paper deals with solution analysis and gain margin analysis of a modified algebraic Riccati matrix equation, and the Lyapunov analysis for discrete‐time network synchronization with directed graph topologies. First, the structure of the solution to the Riccati equation associated with a single‐input controllable system is analyzed. The solution matrix entries are represented using unknown closed‐loop pole variables that are solved via a system of scalar quadratic equations. Then, the gain margin is studied for the modified Riccati equation for both multi‐input and single‐input systems. A disc gain margin in the complex plane is obtained using the solution matrix. Finally, the feasibility of the Riccati design for the discrete‐time network synchronization with general directed graphs is solved via the Lyapunov analysis approach and the gain margin approach, respectively. In the design, a network Lyapunov function is constructed using the Kronecker product of two positive definite matrices: one is the graph positive definite matrix solved from a graph Lyapunov matrix inequality involving the graph Laplacian matrix; the other is the dynamical positive definite matrix solved from the modified Riccati equation. The synchronizing conditions are obtained for the two Riccati design approaches, respectively.

Nonlinear stability analysis of transitional flows using quadratic constraints

The dynamics of transitional flows are governed by an interplay between the non-normal linear dynamics and quadratic nonlinearity in the incompressible Navier-Stokes equations. In this work, we propose a framework for nonlinear stability analysis that exploits the fact that nonlinear flow interactions are constrained by the physics encoded in the nonlinearity. In particular, we show that nonlinear stability analysis problems can be posed as convex feasibility and optimization problems based on Lyapunov matrix inequalities, and a set of quadratic constraints that represent the nonlinear flow physics. The proposed framework can be used to conduct global stability, local stability, and transient energy growth analysis. The approach is demonstrated on the low-dimensional Waleffe-Kim-Hamilton model of transition and sustained turbulence. Our analysis correctly determines the critical Reynolds number for global instability. For local stability analysis, we show that the framework can estimate the size of the region of attraction as well as the amplitude of the largest permissible perturbation such that all trajectories converge back to the equilibrium point. Additionally, we show that the framework can predict bounds on the maximum transient energy growth. Finally, we show that careful analysis of the multipliers used to enforce the quadratic constraints can be used to extract dominant nonlinear flow interactions that drive the dynamics and associated instabilities.

An LMI based approach to stabilize a type of nonlinear uncertain neutral-type delay systems

This paper proposes a linear matrix inequality (LMI) based approach for stability analysis of a type of nonlinear uncertain neutral-type delay systems. The delays in states, state derivatives and inputs are known and constant. By constructing a Lyapunov–Krasovskii functional (LKF), a stability criterion in the form of LMI is derived. In this paper, we treat these nonlinear terms to be norm-bounded and construct a Lyapunov matrix inequality considering system states, delayed states, delayed state derivatives, and delayed inputs. The stability criterion of this nonlinear neutral-type delay system is then obtained. To this end, a numerical example is given to demonstrate the feasibility of the proposed approach. The developed method for stability analysis can potentially be applied to critical real-world applications such as the down-hole drilling system.

Vibration Suppression of a Cantilevered Piezoelectric Laminated Composite Plate Subjected to Hygrothermal Loads

In this paper, the vibration suppression strategy for a cantilevered rectangular piezoelectric laminated composite plate is proposed by using the piezoelectric patches, which are attached to the upper and lower surfaces of the plate as the actuators and sensors, respectively. The main contributions of this study is as follows: Firstly, Based on classical laminated plate theory and considering the action of piezoelectric loadings, the dynamics equation of a piezoelectric laminated composite plate subjected to hygrothermal loads is derived by using Hamilton’s principle. The partial differential equations of the piezoelectric laminated composite plate are discretized to a two-degree-of-freedom control equation by using Galerkin method. Secondly, According to the discrete nominal model, a robust controller for the uncertain systems is proposed. Based on the state space equation, the output feedback controller for the system is designed. In the constructed full-dimensional state observer, the estimated state feedback is introduced to construct a closed-loop feedback system. Using the Lyapunov Matrix Inequality (LMI) method and Lyapunov Stability Theory, the feedback gain matrix and observation gain matrix for the system are designed to make the closed-loop system asymptotically stable. Finally, the accuracy and effectiveness of the controller are verified by numerical simulation.

Data-Driven Wide-Area Control Design of Power System Using the Passivity Shortage Framework

A novel wide-area control design is presented to mitigate inter-area power frequency oscillations. A large-scale power system is decomposed into a network of passivity-short subsystems whose nonlinear interconnections have a state-dependent affine form, and by utilizing the passivity shortage framework a two-level design procedure is developed. At the lower level, any generator control can be viewed as one that makes the generator passivity-short and $L_2$ stable, and the stability impact of the lower-level control on the overall system can be characterized in terms of two parameters. While the system is nonlinear, the impact parameters can be optimized by solving a data-driven matrix inequality (DMI), and the high-level wide-area control is then designed by solving another Lyapunov matrix inequality in terms of the design parameters. The proposed methodology makes the design modular, and the resulting control is adaptive with respect to operating conditions of the power system. A test system is used to illustrate the proposed design, including DMI and the wide-area control, and simulation results demonstrate effectiveness in damping out inter-area oscillations.

On common α-scalar Lyapunov solutions

We present two new characterizations regarding common α-scalar solutions for systems of Lyapunov matrix inequalities so as to bridge and broaden relevant existing results. As applications of such results, we further explore the cases involving a set of conformally partitioned block triangular matrices and a set of Z-matrices, respectively, to obtain new results concerning these cases, which are useful in the areas of control and systems theory and engineering.

Application of Lyapunov matrix inequality based unsymmetrical saturated control to a multi-vectored propeller airship

The problem of the design of a controller for a multi-vectored propeller airship is addressed. The controller includes anti-windup that takes into account unsymmetrical actuator constraints. First, a linear transformation is applied to transform the unsymmetrical constraints into symmetric constraints with an amplitude-bounded exogenous disturbance. Then, a stability condition based on a quadratic Lyapunov function for the saturated closed-loop system is proposed. The condition considers both amplitude-bounded and energy-bounded exogenous disturbances. Thus, the controller design problem is transformed into a convex optimization problem expressed in a bilinear matrix inequality form. Two controller design methods were applied: one-step controller and traditional anti-windup controller. The one-step method obtains the controller and the anti-windup compensator in one step while the anti-windup controller method separates this process into the linear controller design and the compensator design. Simulation results showed that both controllers enlarge the stability zone of the saturation system and have good tracking performance. It is shown that the anti-windup controller design method not only has a larger region of stability, but the demanded actuator output exceeds the constraints less and has a smaller anti-windup coefficient matrix compared to the one-step method.

Common Lyapunov Functions for Some Special Classes of Stable Systems

In this work we consider the problem of existence and determination of a common positive definite solution (common quadratic Lyapunov function) to a set of Lyapunov matrix inequalities. We investigate a parameterized matrix family and a compact (in particular finite) set of 2 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\times$</tex> </formula> 2 matrices.

Stability and assignment of spectrum in systems with discrete time lags

The asymptotic stability with a prescribed degree of time delayed systems subject to multiple bounded discrete delays has received important attention in the last years. It is basically proved that theα-stability locally in the delays (i.e., all the eigenvalues have prefixed strictly negative real parts located inRe⁡s≤−α&lt;0) may be tested for a set of admissible delays including possible zero delays either through a set of Lyapunov's matrix inequalities or, equivalently, by checking that an identical number of matrices related to the delayed dynamics are all stability matrices. The result may be easily extended to check theε-asymptotic stability independent of the delays, that is, for all the delays having any values, the eigenvalues are stable and located inRe⁡s≤ε→0−. The above referred number of stable matrices to be tested is2rfor a set of distinctrpoint delays and includes all possible cases of alternate signs for summations for all the matrices of delayed dynamics. The manuscript is completed with a study for prescribed closed-loop spectrum assignment (or “pole placement”) under output feedback.

An improved mixed H/sub 2//H/sub /spl infin// control design for hard disk drives

In this brief, we propose a new mixed H/sub 2//H/sub /spl infin// control design via a linear matrix inequality (LMI) based approach. By an equivalent transformation of Lyapunov matrix inequality, additional slack variables are introduced to characterize both the H/sub 2/ and H/sub /spl infin// performances. Based on the characterization, a new mixed H/sub 2//H/sub /spl infin// control design is proposed where the introduced slack variables add in additional degree of freedom in optimizing the mixed H/sub 2//H/sub /spl infin// performance, resulting in equal or better performance than existing LMI based methods. The proposed control algorithm is applied in the control design for hard disk drives (HDDs) where the problem of minimizing track misregistration (TMR) in the presence of unmodeled dynamics of voice-coil motor (VCM) is formulated as a mixed H/sub 2//H/sub /spl infin// control problem. The simulation and implementation results of the new control design for the VCM actuator demonstrate the significant performance improvement of TMR over the existing method of while ensuring robust stability under unmodeled dynamics of VCM.

Sufficiency-Type Stability and Stabilization Criteria for Linear Time-Invariant Systems with Constant Point Delays

A set of criteria of asymptotic stability for linear and time-invariant systems with constant point delays are derived. The criteria are concerned with α-stability local in the delays and e-stability independent of the delays, namely, stability with all the characteristic roots in Re s≤−α<0 for all delays in some defined real intervals including zero and stability with characteristic roots in Re s<−e<0 as e→0+ for all possible values of the delays, respectively. The results are classified in several groups according to the technique dealt with. The used techniques include both Lyapunov's matrix inequalities and equalities and Gerschgorin's circle theorem. The Lyapunov's inequalities are guaranteed if a set of matrices, built from the matrices of undelayed and delayed dynamics, are stability matrices. Some extensions to robust stability of the above results are also discussed.

Sufficiency-Type Stability and Stabilization Criteria for Linear Time-Invariant Systems with Constant Point Delays

We derive a set of criteria of asymptotic stability for linear and time-invariant systems with multirate point delays. The criteria are concerned with α-stability local in the delays and e-stability independent of the delays and are classified in several groups according to the technique dealt with. The techniques used include both Lyapunov's matrix inequalities and equalities and Gerschgorin's circle theorem. Lyapunov's inequalities are guaranteed if a set of matrices built from matrices of undelayed and delayed dynamics are stability matrices. Some extensions to robust stability of the above results are also discussed.

Unified initial condition response analysis of Lur'e systems and linear time-invariant systems

A unified framework to the initial condition response (ICR) analysis of non-linear time-varying systems of the Lur'e type and linear time-invariant (LTI) systems is considered. To quantify the transient behaviour resulting from initial conditions, an ICR measure is defined. An appropriate upper bound for the ICR measure can be calculated based upon the condition number of a positive definite matrix, associated with a quadratic Lyapunov function. Owing to the particular structure of the Lur'e systems, bounding the ICR measure is transformed into a minimization problem, constrained by either two simultaneous Lyapunov matrix inequalities or a single algebraic Riccati inequality. In the LTI case, the ICR measure is merely the supremum of the induced norm of the state transition matrix and its calculation involves solving a Lyapunov matrix equation. The ICR measure bound is significant from the engineering standpoint since it enables the calculation of the non-saturating domain of the state variables. In the LTI case, this bound also provides a quantification of impulse and unit step responses. The results are illustrated by several examples.

SUFFICIENT CONDITIONS FOR α-STABILITY OF LINEAR TIME-DELAY SYSTEMS

The asymptotic stability of time-delayed systems subject to multiple bounded point delays has received important attention in past years (see, for instance, Bourles, H. 1994. International Journal of Control , 59(2): 529-541; De la Sen, M. 2000. Electronics Letters , 36(4): 373-374; Xu, B. 2000. Journal of Mathematical Analysis and Applications , 282: 484-494). It is basically proved that the local f -stability in the delays (i.e., all the eigenvalues have prefixed strictly negative real parts located in Re s h - f < 0) may be tested for a set of admissible delays including possible zero delays either through a set of Lyapunov matrix inequalities or, equivalently, by checking that an identical number of matrices related to the delayed dynamics are all stability matrices. The result may be easily extended to check the k -asymptotic stability, independent of the delays; that is, for all the delays having any values, the eigenvalues are stable and located in Re s h k M 0 - (De la Sen, 2000; Xu, 2000). This number is 2 r for a set of r distinct point delays and includes all possible cases of alternate signs for summations for all the matrices of delayed dynamics (Xu, 2000).

On Stability Robustness of 2-D Systems Described by theFornasini–Marchesini Model

This paper studies the stability robustness estimates of two-dimensional (2-D) dynamics in the Fornasini–Marchesini model. It is stressed that the use of 2-D Lyapunov matrix inequality is limited in application to the stability robustness estimate problem. Two alternative ways to estimate the stability robustness of a 2-D dynamics are proposed. These are applicable in any stable 2-D dynamics. The comparison between the stability robustness estimate by the 2-D Lyapunov matrix inequality and those presented in this paper is illustrated by a numerical example.

On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities

A sufficient condition for the stability of two-dimensional (2-D) systems described in the Fornasin-Marchesini model is presented in a succinct form. The condition is expressed as a Lyapunov-like matrix inequality involved with the parallel addition of two positive definite matrices.

A neural network for linear matrix inequality problems

Gradient-type Hopfield networks have been widely used in optimization problems solving. This paper presents a novel application by developing a matrix oriented gradient approach to solve a class of linear matrix inequalities (LMIs), which are commonly encountered in the robust control system analysis and design. The solution process is parallel and distributed in neural computation. The proposed networks are proven to be stable in the large. Representative LMIs such as generalized Lyapunov matrix inequalities, simultaneous Lyapunov matrix inequalities, and algebraic Riccati matrix inequalities are considered. Several examples are provided to demonstrate the proposed results. To verify the proposed control scheme in real-time applications, a high-speed digital signal processor is used to emulate the neural-net-based control scheme.