Lyapunov Transformations Research Articles

Position and Attitude Tracking Controllers Using Lyapunov Transformations for Quadrotors

In this paper, a novel feedback control strategy for quadrotor trajectory tracking is designed and experimentally tested with proof of exponential stability, using the Lyapunov transformations theory. The controller is derived from an inner-outer loop control structure, namely by considering the position system coupled through an interconnection term with the attitude system. For the design of the position controller, the considered dynamics are worked on the body frame, which is uncommon in the literature, and its synthesis derives from theories such as Pontryagin’s maximum principle, Lyapunov theory, and Linear Quadratic Regulator (LQR), which ensure Input-to-state stability, steady-state optimality, and global exponential stability. The attitude system is based on an error quaternion parameterization via a nonlinear coordinate transformation matrix followed by a state input feedback, rendering the system linear and time-invariant. Under a correct transformation, LQR theory ensures almost exponential stability and steady-state optimality for the overall interconnected closed-loop systems. Experimental and simulation results illustrate the performance of the tracking system onboard a quadrotor.

Some majorization inequalities induced by Schur products in Euclidean Jordan algebras

In a Euclidean Jordan algebra V of rank n, an element x is said to be majorized by an element y, symbolically x≺y, if the corresponding eigenvalue vector λ(x) is majorized by λ(y) in Rn. In this article, we describe pointwise majorization inequalities of the form T(x)≺S(x), where T and S are linear transformations induced by Schur products. Specializing, we recover analogs of majorization inequalities of Schur, Hadamard, and Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as majorization inequalities connecting quadratic and Lyapunov transformations on V. We also show how Schur products induced by certain scalar means (such as arithmetic, geometric, harmonic, and logarithmic means) naturally lead to majorization inequalities.

A Riesz-Thorin type interpolation theorem in Euclidean Jordan algebras

In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element a we associate the eigenvalue vector λ(a) in Rn whose components are the eigenvalues of a written in the decreasing order. For any p∈[1,∞], we define the spectral p-norm of a to be the p-norm of λ(a) in Rn. In a recent paper, based on the K-method of real interpolation theory and a majorization technique, we described an interpolation theorem for a linear transformation on V relative to the same spectral p-norm. In this paper, using complex function theory methods, we describe a Riesz-Thorin type interpolation theorem relative to two spectral p-norms. We illustrate the result by estimating the norms of certain special linear transformations such as Lyapunov transformations, quadratic representations, and positive transformations.

A Hölder type inequality and an interpolation theorem in Euclidean Jordan algebras

In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element x we associate the eigenvalue vector λ ( x ) whose components are the eigenvalues of x written in the decreasing order. For any p ∈ [ 1 , ∞ ] , we define the spectral p -norm of x to be the p -norm of λ ( x ) in R n . In this paper, we show that ‖ x ∘ y ‖ 1 ≤ ‖ x ‖ p ‖ y ‖ q , where x ∘ y denotes the Jordan product of two elements x and y in V and q is the conjugate of p . For a linear transformation on V , we state and prove an interpolation theorem relative to these spectral norms. In addition, we compute/estimate the norms of Lyapunov transformations, quadratic representations, and positive transformations on V .

SOLVING STRONGLY MONOTONE LINEAR COMPLEMENTARITY PROBLEMS

Given a linear transformation L on a finite dimensional real inner product space V to itself and an element q ∈ V we consider the general linear complementarity problem LCP (L, K, q) on a proper cone K ⊆ V. We observe that the iterates generated by any closed algorithmic map will converge to a solution for LCP (L, K, q), whenever L is strongly monotone. Lipschitz constants of L is vital in establishing the above said convergence. Hence we compute the Lipschitz constants for certain classes of Lyapunov, Stein and double-sided multiplicative transformations in the setting of semidefinite linear complementarity problems. We give a numerical illustration of a closed algorithmic map in the setting of a standard linear complementarity problem. On account of the difficulties in numerically implementing such algorithms for general linear complementarity problems, we give an alternative algorithm for computing the solution for a special class of strongly monotone semidefinite linear complementarity problems along with a numerical example.

LINEAR TRANSFORMATIONS AND FLOQUET THEOREM FOR LINEAR IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES

This paper is concerned with Cauchy problem, Lyapunov transformations, Floquet and Lyapunov theorems for linear implicit dynamic equation AtxΔ= Btx with index-1 on time scales. The stability of this equation under the act of these Lyapunov transformations is also considered. The results are a unification and generalization of previous results for differential-algebraic equations and implicit difference systems.

Complementarity properties of Peirce-diagonalizable linear transformations on Euclidean Jordan algebras†

Peirce-diagonalizable linear transformations on a Euclidean Jordan algebra are of the form L(x)=A·x:=∑ a ij x ij , where A=[a ij ] is a real symmetric matrix and ∑ x ij is the Peirce decomposition of an element x in the algebra with respect to a Jordan frame. Examples of such transformations include Lyapunov transformations and quadratic representations on Euclidean Jordan algebras. Schur (or Hadamard) product of symmetric matrices provides another example. Motivated by a recent generalization of the Schur product theorem, we study general and complementarity properties of such transformations.

A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) systemʼs poles in the complex plane. We include several examples to show the utility of this theory.

The stability of a specific class of mechanical systems

A special class of mechanical systems is considered, the linearized equations of which either belong to the class of time-varying systems, reducible to stationary systems using constructive Lyapunov transformations or to systems close to these. A method of decomposing of the matrices of a system, which differs from the traditional method, is proposed for investigating of the stability of motion. It is shown that the conclusions concerning the stability are more complete in the case of this decomposition of the system matrix. A number of problems on the stability of motion of various mechanical systems is considered as examples.

On the Lipschitz Continuity of the Solution Map in Semidefinite Linear Complementarity Problems

In this paper, we investigate the Lipschitz continuity of the solution map in semidefinite linear complementarity problems. For a monotone linear transformation defined on the space of real symmetric n × n matrices, we show that the Lipschitz continuity of the solution map implies the globally uniquely solvable (GUS)-property. For Lyapunov transformations with the Q-property, we prove that the Lipschitz continuity of the solution map is equivalent to the strong monotonicity property. For the double-sided multiplicative transformations, we show that the Lipschitz continuity of the solution map implies the GUS-property.

Stability assessment for linear time-varying discrete-time systems using orthogonal Lyapunov transformations

A method is given for determining point-wise orthogonal Lyapunov transformations for use in stability studies for linear time-varying discrete-time systems. The advantage of such transformations for the non-periodic case over standard Lyapunov transformations is explained. Simple useful sufficient conditions for asymptotic stability and divergence of the original system are given in terms of the transformed system. The application of the results to online stability assessment is demonstrated.

Simultaneous Local Controllability of the Spectrum and the Lyapunov Irregularity Coefficient of Regular Systems

System (3) is homogeneous and has bounded piecewise continuous coefficients, and so the characteristic exponents and other invariants of Lyapunov transformations are defined for this system [1, p. 42]. Our main problem is to find conditions under which the entire spectrum of Lyapunov exponents [1, p. 42] and the Lyapunov irregularity coefficient [1, p. 51] of system (3) can be changed as desired (i.e., can be controlled) in a neighborhood of the entire spectrum and the irregularity coefficient of the nonperturbed system (1) by an appropriate choice of the feedback coefficient U(·) (treated as a control). We introduce some notation. Let R be the n-dimensional Euclidean space with canonical orthonormal basis e1, . . . , en and with norm ‖ · ‖ given by the relation ‖x‖ = √ x∗x (the asterisk stands for transposition), and let Mmn be the space of real m × n matrices with spectral norm, i.e., with the operator norm induced in Mmn by the Euclidean norms in R and R. The space of bounded piecewise continuous mappings U : I → Mmn defined on an arbitrary interval I ⊂ R with the uniform norm ‖U‖C(I) := sup{‖U(t)‖ : t ∈ I} is denoted by KCmn(I). We denote the principal solution matrix of the homogeneous system (1) by X(t, s) and its entire spectrum of Lyapunov exponents by λ1(A) ≤ · · · ≤ λn(A). For an arbitrary piecewise continuous bounded function p : R→ R, we set

Global Reducibility of Linear Control Systems to Systems of Scalar Type

for which characteristic exponents and other invariants of Lyapunov transformations of homogeneous systems [1, p. 42; 2, p. 245] are defined. We encounter the problem as to whether it is possible to construct a control U(·) such that some Lyapunov invariant of system (2) with this U(·) coincides with an arbitrary prescribed value of that invariant. This property of system (2) is referred to as the global controllability of the Lyapunov invariant. In the present paper, we prove the global scalarizability of a uniformly completely controllable [3] system (1), which means that, for an arbitrary given piecewise continuous bounded scalar function p(·) on R, there exists a control U(·) such that system (2) with this control is asymptotically equivalent [4] to the system z = p(t)z, z ∈ R; (3)

Lyapunov Reducibility of a Linear System with Aftereffect

with aftereffect, the problem of the existence of Lyapunov transformations has not been considered yet. In the present paper, we try to construct a Lyapunov transformation reducing the restriction of system (1) to a finite-dimensional subspace of initial conditions to some system of ordinary differential equations. The theorem proved below supplements the paper [1] and provides additional possibilities in the analysis of the asymptotics of solutions of system (1).

Relationship Between Strong Monotonicity Property, P2-Property, and the Gus-Property in Semidefinite Linear Complementarity Problems

In a recent paper on semidefinite linear complementarity problems, Gowda and Song (2000) introduced and studied the P-property, P2-property, GUS-property, and strong monotonicity property for linear transformation L: Sn → Sn, where Sn is the space of all symmetric and real n × n matrices. In an attempt to characterize the P2-property, they raised the following two questions: (i) Does the strong monotonicity imply the P2-property? (ii) Does the GUS-property imply the P2-property? In this paper, we show that the strong monotonicity property implies the P2-property for any linear transformation and describe an equivalence between these two properties for Lyapunov and other transformations. We show by means of an example that the GUS-property need not imply the P2-property, even for Lyapunov transformations.

Nonlinear delay systems, Lie algebras and Lyapunov transformations

Nonlinear delay systems, Lie algebras and Lyapunov transformations

Orthogonal Lyapunov transformations with some applications to stability studies

For continuous-time linear time-varying systems, this paper shows how to determine pointwise orthogonal Lyapunov transformations, reveals their advantage over standard Lyapunov transformations for the nonperiodic case, and outlines some potential applications to stability assessment for both linear and nonlinear systems.

Integral Separation, Lyapunov Transformations, and the Small Change in the Direction of the Solutions of Periodic Linear Systems of Ordinary Differential Equations

First, we discuss the relation between the integral separation of solutions of a linear periodic systemx′=A(t)x(1)and the structure of its characteristic multipliers. Second, we investigate the reducibility of (1) by a complex or a real Lyapunov transformation. Third, we prove the property of the small change in the direction of solutions of ω-periodic system (1) as the result of a small perturbation of (1) under the following assumptions:(a)When the characteristic multipliers of (1) are distinct and the perturbation of (1) is admissible. The perturbationC(t)=B(t)−A(t) is calledadmissibleto the system (1), if the characteristic multipliers ρkand ρkof the systems (1) andy′=B(t)y,(2)respectively, have the same argument, i.e., argρk=argρk,k=1,…,n. In this case, we also say that the perturbed ω-periodic system (2) is admissible to the system (1).(b)When the characteristic multipliers of (1) are real and distinct and the perturbation is arbitrary.

Approximation of bounded solutions and exponential dichotomy on an axis

Lyapunov transformations possessing certain properties are used to construct regular two-point boundary-value problems as approximations to the problem of determining a bounded solution in the general case. The concept of “limiting solutions as t→∞” is defined and the behaviour of solutions of linear ordinary differential equations as t→∞ is investigated. The necessary and sufficient conditions are derived under which a singular boundary-value problem with conditions assigned at infinity is uniquely solvable, and an appropriate approximating problem is constructed.

Criterion for the reducibility in the mean of linear differential systems

A criterion (in terms of the singular numbers of Cauchy operators) is obtained for the reducibility in the mean, i.e., for the approximability (in the mean, on a half-line), of a given system to any degree of accuracy by means of Lyapunov transformations of another given system. It is proved that any two triangular systems with identical diagonals are reducible in the mean to each other.