Leonard System Research Articles

Control Design for the Ward–Leonard System in Wind Turbines

A robust optimal controller for the Ward-Leonard system in a wind turbine is used to meet the performance and stability requirements when the system parameters change. However, designing according to the robust optimal method often leads to a high-order controller. This study investigated the application of order reduction algorithms to simplify the controller and help it better meet the real control problem. Comparative evaluation of the order reduction controller methods showed that the second-order reduction controller according to Moore's balanced truncation algorithm was the most suitable to replace the higher-order controller. The step response quality of the system was better when using a second-order reduction controller than a higher-order controller.

Notes on the Leonard System Classification

Around 2001 we classified the Leonard systems up to isomorphism. The proof was lengthy and involved considerable computation. In this paper we give a proof that is shorter and involves minimal computation. We also give a comprehensive description of the intersection numbers of a Leonard system.

Idempotent systems

In this paper we introduce the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. We focus on a family of idempotent systems, said to be symmetric. A symmetric idempotent system is an abstraction of the primary module for the subconstituent algebra of a symmetric association scheme. We describe the symmetric idempotent systems in detail. We also consider a class of symmetric idempotent systems, said to be P-polynomial and Q-polynomial. In the topic of orthogonal polynomials there is an object called a Leonard system. We show that a Leonard system is essentially the same thing as a symmetric idempotent system that is P-polynomial and Q-polynomial.

Abundance of Strange Attractors Near an Attracting Periodically Perturbed Network

We study the dynamics of the periodically forced May--Leonard system. We extend previous results on the field, and we identify different dynamical regimes depending on the strength of attraction $\delta$ of the network and the frequency $\omega$ of the periodic forcing. We focus our attention in the case $\delta\gg1$ and $\omega \approx 0$, where we show that, for a positive Lebesgue measure set of parameters (amplitude of the periodic forcing), the dynamics are dominated by strange attractors with fully stochastic properties, supporting Sinai--Ruelle--Bowen (SRB) measures. The proof is performed by using the Wang and Young's theory of rank-one strange attractors. This work ends the discussion about the existence of observable and sustainable chaos in this scenario. We also identify some bifurcations occurring in the transition from an attracting two-torus to rank-one strange attractors, whose existence has been suggested by numerical simulations.

The Terwilliger algebra of the Grassmann scheme Jq(N,D) revisited from the viewpoint of the quantum affine algebra [formula omitted

Let Δ be the set of all quadruples (ν,μ,d,e) of integers that satisfy0≤D−d2≤ν≤μ≤D−d≤D,e+d+Dis even,|e|≤2ν−D+d,d∈{e+D−2ν,min⁡{D−μ,e+D−2ν+2(N−2D)}}. In [6], it is shown for the Terwilliger algebra T of the Grassmann scheme Jq(N,D), N≥2D, that the isomorphism classes of irreducible T-modules W are determined by their endpoint ν, dual endpoint μ, diameter d, and auxiliary parameter e, that come from the Leonard system attached to W, and it is claimed without proof that the quadruples (ν,μ,d,e) belong to Δ, if d≥1.Let Λ be the set of triples (α,β,ρ) of non-negative integers that satisfy0≤α≤D−ρ2,0≤β≤N−D−ρ2,0≤α+β≤D−ρ. We construct a mapping from Λ to Δ which is bijective if N>2D and 2:1 if N=2D.We show that the set Λ naturally parameterizes the isomorphism classes of irreducible T-modules, by embedding the standard module of Jq(N,D) in a bigger space that allows a Uq(slˆ2)-module structure [9].As a byproduct we have the following: for a fixed ρ, 0≤ρ≤D, set N′=N−2ρ, D′=D−ρ, and Λρ={(α,β)|(α,β,ρ)∈Λ}. Then Λρ is precisely the set that parameterizes the isomorphism classes of irreducible T-modules for the Johnson scheme J(N′,D′)[3].

Quasi-Steady-State Models of Three Timescale Systems: A Bond Graph Approach

Modelling in bond graph to obtain reduced models of systems with singular perturbations is applied. This singularly perturbed system is characterized by having three timescales, i.e., slow, medium, and fast dynamics. From a bond graph whose storage elements have an integral causality assignment (BGI), the mathematical model of the complete system can be determined. By assigning a derivative causality assignment to the storage elements for the fast dynamics and maintaining an integral causality assignment for the slow and medium dynamics on the bond graph, reduced models for the slow and medium dynamics are obtained. When a derivative causality to the storage elements for the fast and medium dynamics is assigned and an integral causality assignment to the slow dynamics is applied, the most reduced model is determined. Finally, the proposed methodology to the Ward Leonard system is applied.

Leonard pairs having specified end-entries

Fix an algebraically closed field F and an integer d≥3. Let V be a vector space over F with dimension d+1. A Leonard pair on V is an ordered pair of diagonalizable linear transformations A:V→V and A⁎:V→V, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let {vi}i=0d (resp. {vi⁎}i=0d) be such an eigenbasis for A (resp. A⁎). For 0≤i≤d define a linear transformation Ei:V→V such that Eivi=vi and Eivj=0 if j≠i(0≤j≤d). Define Ei⁎:V→V in a similar way. The sequence Φ=(A,{Ei}i=0d,A⁎,{Ei⁎}i=0d) is called a Leonard system on V with diameter d. With respect to the basis {vi}i=0d, let {θi}i=0d (resp. {ai⁎}i=0d) be the diagonal entries of the matrix representing A (resp. A⁎). With respect to the basis {vi⁎}i=0d, let {θi⁎}i=0d (resp. {ai}i=0d) be the diagonal entries of the matrix representing A⁎ (resp. A). It is known that {θi}i=0d (resp. {θi⁎}i=0d) are mutually distinct, and the expressions (θi−1−θi+2)/(θi−θi+1), (θi−1⁎−θi+2⁎)/(θi⁎−θi+1⁎) are equal and independent of i for 1≤i≤d−2. Write this common value as β+1. In the present paper we consider the “end-entries” θ0, θd, θ0⁎, θd⁎, a0, ad, a0⁎, ad⁎. We prove that a Leonard system with diameter d is determined up to isomorphism by its end-entries and β if and only if either (i) β≠±2 and qd−1≠−1, where β=q+q−1, or (ii) β=±2 and Char(F)≠2.

The end-parameters of a Leonard pair

Fix an algebraically closed field F and an integer d≥3. Let V be a vector space over F with dimension d+1. A Leonard pair on V is a pair of diagonalizable linear transformations A:V→V and A⁎:V→V, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. There is an object related to a Leonard pair called a Leonard system. It is known that a Leonard system is determined up to isomorphism by a sequence of scalars ({θi}i=0d,{θi⁎}i=0d,{φi}i=1d,{ϕi}i=1d), called its parameter array. The scalars {θi}i=0d (resp. {θi⁎}i=0d) are mutually distinct, and the expressions (θi−2−θi+1)/(θi−1−θi), (θi−2⁎−θi+1⁎)/(θi−1⁎−θi⁎) are equal and independent of i for 2≤i≤d−1. Write this common value as β+1. In the present paper, we consider the “end-parameters” θ0,θd,θ0⁎,θd⁎,φ1,φd,ϕ1,ϕd of the parameter array. We show that a Leonard system is determined up to isomorphism by the end-parameters and β. We display a relation between the end-parameters and β. Using this relation, we show that there are up to isomorphism at most ⌊(d−1)/2⌋ Leonard systems that have specified end-parameters. The upper bound ⌊(d−1)/2⌋ is best possible.

Perturbations of May–Leonard system

In this paper we study the quadratic homogeneous perturbations of the 3-dimensional May–Leonard system with α+β=2. It is shown that there are perturbed systems having exactly one or two limit cycles bifurcated from the periodic orbits of May–Leonard system. This is proved by estimating the number of zeros of the first and the second order Melnikov functions.

Dual polar graphs, the quantum algebra [formula omitted], and Leonard systems of dual q-Krawtchouk type

In this paper we consider how the following three objects are related: (i) the dual polar graphs; (ii) the quantum algebra Uq(sl2); (iii) the Leonard systems of dual q-Krawtchouk type. For convenience we first describe how (ii) and (iii) are related. For a given Leonard system of dual q-Krawtchouk type, we obtain two Uq(sl2)-module structures on its underlying vector space. We now describe how (i) and (iii) are related. Let Γ denote a dual polar graph. Fix a vertex x of Γ and let T=T(x) denote the corresponding subconstituent algebra. By definition T is generated by the adjacency matrix A of Γ and a certain diagonal matrix A*=A*(x) called the dual adjacency matrix that corresponds to x. By construction the algebra T is semisimple. We show that for each irreducible T-module W the restrictions of A and A* to W induce a Leonard system of dual q-Krawtchouk type. We now describe how (i) and (ii) are related. We obtain two Uq(sl2)-module structures on the standard module of Γ. We describe how these two Uq(sl2)-module structures are related. Each of these Uq(sl2)-module structures induces a C-algebra homomorphism Uq(sl2)→T. We show that in each case T is generated by the image together with the center of T. Using the combinatorics of Γ we obtain a generating set L,F,R,K of T along with some attractive relations satisfied by these generators.

Coexistence in the two-dimensional May-Leonard model with random rates

We employ Monte Carlo simulations to numerically study the temporal evolution and transient oscillations of the population densities, the associated frequency power spectra, and the spatial correlation functions in the (quasi-)steady state in two-dimensional stochastic May--Leonard models of mobile individuals, allowing for particle exchanges with nearest-neighbors and hopping onto empty sites. We therefore consider a class of four-state three-species cyclic predator-prey models whose total particle number is not conserved. We demonstrate that quenched disorder in either the reaction or in the mobility rates hardly impacts the dynamical evolution, the emergence and structure of spiral patterns, or the mean extinction time in this system. We also show that direct particle pair exchange processes promote the formation of regular spiral structures. Moreover, upon increasing the rates of mobility, we observe a remarkable change in the extinction properties in the May--Leonard system (for small system sizes): (1) As the mobility rate exceeds a threshold that separates a species coexistence (quasi-)steady state from an absorbing state, the mean extinction time as function of system size N crosses over from a functional form ~ e^{cN} / N (where c is a constant) to a linear dependence; (2) the measured histogram of extinction times displays a corresponding crossover from an (approximately) exponential to a Gaussian distribution. The latter results are found to hold true also when the mobility rates are randomly distributed.

Nonlinear Stability for Diffusion Models in Biology

The reduction method for studying optimal nonlinear stability of constant solutions to some ecological models with diffusion, which include the Cantrell–Cosner and the May–Leonard systems, is given. A new canonical energy (Lyapunov function) is introduced, and it is proved that the regions of linear and nonlinear stability coincide with a known radius of attraction for the initial data. Attention is focused on a May–Leonard system with circular symmetry, an asymmetric May–Leonard system with diffusion, and a system for aggregation of glia in the brain.

Boundedness and partial survival of species in nonautonomous Lotka–Volterra systems

Consider the partial survival and extinction of species in models governed by the following nonautonomous Lotka–Volterra system: d x i ( t ) d t = x i ( t ) c i ( t ) - ∑ j = 1 n ∑ l = 0 m a ij l ( t ) x j ( t - τ l ) , t ⩾ t 0 , 1 ⩽ i ⩽ n , x i ( t ) = φ i ( t ) , t ⩽ t 0 and φ i ( t 0 ) > 0 , 1 ⩽ i ⩽ n , where each φ i ( t ) is a continuous for t ⩽ t 0 , each c i ( t ) and a ij l ( t ) are continuous functions on [ t 0 , + ∞ ) , c i ( t ) , a ii 0 ( t ) , a ij l ( t ) , 1 ⩽ i , j ⩽ n , 1 ⩽ l ⩽ m are bounded on [ t 0 , + ∞ ) and inf t ⩾ t 0 a ii 0 ( t ) > 0 , a ij l ( t ) ⩾ 0 , 1 ⩽ i , j ⩽ n , 1 ⩽ l ⩽ m , τ 0 = 0 and τ l ⩾ 0 , 0 ⩽ l ⩽ m . This paper generalizes results of the well-known May–Leonard system and the recent work of Redheffer (Nonlinear Anal.: Real World Appl. 4 (2003) 301) to the above nonautonomous Lotka–Volterra systems with delays. We establish conditions of the boundedness and partial survival for the solution x ( t ) = [ x i ( t ) ] of system in the sence that max 1 ⩽ i ⩽ n x i ( t ) is bounded away from 0 even though inf t ⩾ t 0 x i ( t ) = 0 for some i.

Mean values and the nonautonomous May–Leonard equations

This paper is concerned with a nonautonomous version of the well-known May–Leonard system, that is, a system like theirs but with time-varying rather than constant coefficients. One consequence of their work is that the solution x exhibits partial survival, in the sense that max i x i(t) is bounded away from 0 even though inf x i(t) =0 for each i. This aspect of the subject is developed here under much weaker assumptions than those in the pioneering paper of May and Leonard. The analysis leads to a surprising resurrection of a hypothesis that was introduced by Vance and Coddington in 1989.

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A:V→V and A *:V→V satisfying both conditions below: 1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A * is irreducible tridiagonal. 2. [(ii)] There exists a basis for V with respect to which the matrix representing A * is diagonal and the matrix representing A is irreducible tridiagonal. We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A * on V, there exists a sequence of scalars β,γ,γ *,ϱ,ϱ * taken from K such that both 0= [A,A 2A *−βAA *A+A *A 2−γ(AA *+A *A)−ϱA *], 0= [A *,A *2A−βA *AA *+AA *2−γ *(A *A+AA *)−ϱ *A], where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.

Singularity Structure of Third‐Order Dynamical Systems. I

A general third‐order dynamical system with polynomial right‐hand sides of finite degrees in the dependent variables is analyzed to unravel the singularity structure of its solutions about a movable singular point. To that end, the system is first transformed to a second‐order Briot–Bouquet system and a third auxiliary equation via a transformation, similar to one used earlier by R. A. Smith in 1973–1974 for a general second‐order dynamical system. This transformation imposes some constraints on the coefficients appearing in the general third‐order system. The known results for the second‐order Briot–Bouquet system are used to explicitly write out Laurent or psi‐series solutions of the general third‐order system about a movable singularity. The convergence of the relevant series solutions in a deleted neighborhood of the singularity is ensured. The theory developed here is illustrated with the help of the May–Leonard system.

Singularity Structure of Third‐Order Dynamical Systems. II

The singularity structure of the solutions of a general third‐order system, with polynomial right‐hand sides of degree less than or equal to two, is studied about a movable singular point. An algorithm for transforming the given third‐order system to a third‐order Briot–Bouquet system is presented. The dominant behavior of a solution of the given system near a movable singularity is used to construct a transformation that changes the given system directly to a third‐order Briot–Bouquet system. The results of Horn for the third‐order Briot–Bouquet system are exploited to give the complete form of the series solutions of the given third‐order system; convergence of these series in a deleted neighborhood of the singularity is ensured. This algorithm is used to study the singularity structure of the solutions of the Lorenz system, the Rikitake system, the three‐wave interaction problem, the Rabinovich system, the Lotka–Volterra system, and the May–Leonard system for different sets of parameter values. The proposed approach goes far beyond the ARS algorithm.

Development of a high-speed elevator controlled by current source inverter system with sinusoidal input and output

An inverter elevator (2.5-4 m/s rated speed) which is controlled by a current source inverter system with sinusoidal input and output is developed. In comparison with a conventional DC elevator controlled by the thyristor Leonard system, it has the following features: (1) a reduction of higher harmonic currents to less than 5%; (2) more than a 30% reduction of power source capacity; (3) more than a 10% reduction of energy consumption, and (4) a 40% reduction of traction machine weight. The sinusoidal pulsewidth modulation control of the converter and inverter, performed by one-chip microprocessors, is described along with a detection and protection system for abnormal overvoltage. A general description of the elevator's performance is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Full armature voltage drop compensated and automatic voltage regulated static leonard control system

Full armature voltage drop compensated and automatic voltage regulated static leonard control system

PARAMETER PLANE SYNTHESIS OF A DUAL CONVERTER FED VARIABLE SPEED D.C. DRIVE SYSTEM

ABSTRACT This paper describes the design, analysis and experimental studies of a variable speed d.c. drive fed from a single phase dual converter, the d.c. motor field is separately excited. The dual converter is operating in circulating current mode. It incorporates cosine firing technique using integrated circuitry. A speed loop with PI controller maintains the desired speed irrespective of the load variation on the motor. An inner current control loop again incorporating a PI controller, protects the thyristors from overcurrents and also provides fast response overcoming the effect: of disturbances. The method effectively replaces the Ward Leonard system. The parameters of the PI controllers are designed on the basis of system relative stability and response of the drive system D-decomposition technique is used for finding the region of parameters for stable drive system and this is checked using frequency scanning technique. Experimental results are also given.