Algorithm of robust stability region for interval plant with time delay using fractional order PID controller

By using PI^λD^μ controller, we investigate the problem of computing the robust stability region for interval plant with time delay. The fractional order interval quasi-polynomial is decomposed into several vertex characteristic quasi-polynomials by the lower and upper bounds, in which the value set of the characteristic quasi-polynomial for vertex quasi-polynomials in the complex plane is a polygon. The D-decomposition technique is used to characterize the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters. We investigate how the fractional integrator order λ and the derivative order μ in the range (0, 2) affect the stabilizability of each vertex characteristic quasi-polynomial. The stability region of interval characteristic quasi-polynomial is determined by intersecting the stability region of each quasi-polynomial. The parameters of PI^λD^μ controller are obtained by selecting the control parameters from the stability region. Using the value set together with zero exclusion principle, the robust stability is tested and the algorithm of robust stability region is also proposed. The algorithm proposed here is useful in analyzing and designing the robust PI^λD^μ controller for interval plant. An example is given to show how the presented algorithm can be used to compute all the parameters of a PI^λD^μ controller which stabilize a interval plant family.     

PIλDμ controller design for integer and fractional plants using piecewise orthogonal functions

This paper deals with the design of fractional PID controller for integer and fractional plants. A new analytic method is proposed, the developments are based on the expansion of the control loop signals as well as a chosen reference model input and output over a piecewise orthogonal functions, namely, Block pulse, Walsh and Haar wavelets. The generalized operational matrices of differentiation related to these bases which are fitting the Riemann–Liouville definition accurately are used to replace the fractional differential calculus by an algebraic one easier to solve. Thereafter, the controller tuning is elaborated simply with a matrix manipulation manner. At first, a least square is drawn to find only the controller gains, then a nonlinear function defined as a matrix norm is minimized to optimize the whole parameters. A variety of examples covering both integer and fractional systems and reference models are presented to show the validity of the technique.     

Hardware-in-the-loop experimental study on a fractional order networked control system testbed

Networked Control Systems (NCS) are of great interest in many industries because of their convenience in data sharing and manipulation remotely. However, there are several problems along with NCS itself due to the uncertainties in network communication. One issue inherent to NCS is the network-induced delays which may deteriorate the performance and may even cause instability of the system. Therefore a controller which can make the plant stable at large values of delay is always desirable in NCS systems. Our past work on Optimal Fractional Order Proportional Integral (OFOPI) controller showed that fractional order PI controllers have larger jitter margin (maximum value of delay for which system is stable) for lag-dominated systems when compared to traditional Proportional Integral Derivative (PID) controllers, whereas integer order PID controllers have larger jitter margin for delay-dominated systems. This paper aims at the design process of a tele-presence controller based on OFOPI tuning rules. To illustrate this, an extensive experimental study on the real-time Smart Wheel networked speed control system is performed using hardware-in-the-loop control. The real-time random delay in the world wide network is collected by pinging different locations, and is considered as the delay in our simulation and experimental systems. Comparisons are made with existing integer order PID controller. It is found that the proposed OFOPI controller is a promising controller and has faster response time than the traditional integer order PID controllers. Since the plant into consideration viz. the Smart Wheel is a delay-dominated system, it is verified that PID achieves larger jitter margin as compared to OFOPI tuning rules. Simulation results and real-time experiments showing comparisons between OFOPI and OPID tuning rules prove the significance of this method in NCS.     

LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index

The continuous and discrete time Linear Quadratic Regulator (LQR) theory has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are formulated as the optimal state-feedback gains, corresponding to the standard quadratic cost function involving the state variables and the controller effort. A real coded Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator design while minimizing another time domain integral performance index, comprising of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. The proposed methodology is extended for a new kind of fractional order (FO) integral performance indices. The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control. Guidelines for the choice of integral order of the performance index are given depending on the characteristics of the process, to be controlled.     

Zeros of Sobolev orthogonal polynomials following from coherent pairs

Let {S_n^λ} denote the monic orthogonal polynomial sequence with respect to the Sobolev inner product$\text{〈f,g〉}{}_{\mathrm{S}}\text{=}\text{∫}\text{−∞}\text{∞}\text{fg}\text{d}\text{ψ}{}_0\text{+λ}\text{∫}\text{−∞}\text{∞}\text{f′g′}\text{d}\text{ψ}{}_1\text{,}$where {dψ₀,dψ₁} is a so-called coherent pair and λ>0. Then S_n^λ has n different, real zeros. The position of these zeros with respect to the zeros of other orthogonal polynomials (in particular Laguerre and Jacobi polynomials) is investigated. Coherent pairs are found where the zeros of S_n−1^λ separate the zeros of S_n^λ.     

Applications of branched values to p-adic functional equations on analytic functions

Let \(\mathbb{K}\) be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. Results on branched values obtained in a previous paper are used to prove that algebraic functional equations of the form g ^q = hf ^q + w have no solution among transcendental entire functions f, g or among unbounded analytic functions inside an open disk, when w is a polynomial or a bounded analytic function and h is a polynomial or an analytic function whose zeros are of order multiple of q. We also show that an analytic function whose zeros are multiple of an integer q inside a disk is the q-th power of another analytic function, provided q is a prime to the residue characteristic.     

Factorization of matrix polynomials with multiple roots

The concept of a multiple root of matrix polynomial L(λ) is introduced, and associated spectral properties of L(λ) are investigated. A statement concerning factorization of L(λ) is presented. Applications are made to factorizations of the matrix polynomial L^α(λ), for any positive integer α.     

Spectral and oscillatory properties of a linear pencil of fourth-order differential operators

The paper deals with the spectral and oscillatory properties of a linear operator pencilA ? λB, where the coefficient A corresponds to the differential expression (py″)″ and the coefficient B corresponds to the differential expression ?y″ + cry. In particular, it is shown that all negative eigenvalues of the pencil are simple and, under some additional conditions, the number of zeros of the corresponding eigenfunctions is related to the serial number of the corresponding eigenvalue.     

Fractional order adaptive high-gain controllers for a class of linear systems

In this paper we show that a fractional adaptive controller based on high gain output feedback can always be found to stabilize any given linear, time-invariant, minimum phase, siso systems of relative degree one. We generalize the stability theorem of integer order controllers to the fractional order case, and we introduce a new tuning parameter for the performance behaviour of the controlled plant. A simulation example is given to illustrate the effectiveness of the proposed algorithm.     

On solutions to coupled multiparameter nonlinear Sturm- Liouville boundary value problems whose state components have a specified nodal structure

In preceding articles ([3] and [5]), we began an examination of the structure of the solution set to the two-parameter system \(\matrix{\qquad\qquad-(p_{1}(x)u^{\prime})^{\prime}+q_1(x)u= \lambda u+f(u,v)u\cr \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad in (a,b)\cr \qquad(*)\ \ -(p_{2}(x)v^{\prime})^\prime+ q_{2}(x)v=\mu v+g(u,v)v\cr \qquad\qquad u(a)=u(b)=0=v(a)=v(b).}\) In this article, we treat the case left uncovered in our previous analysis; namely, we assume f (s,0) = 0 and g(0,t) = 0 for all s, t ∈ ?. In this situation, solutions to (*) of the form (λ, μ, u, 0) or (λ, μ,0,v lie in linear subspaces of ?.² dx (C01 [a, b] 2. As such, they are neither locally expressable as functions of (λ, μ) nor are à priori bounded in terms of (λ, μ), as was crucial to the analysis in [3] and [5]. Nevertheless, we demonstrate that solutions to (*) of the form (λ, μ, u, v) with u having n ? 1 simple zeros in (a, b) and v having m ? 1 simple zeros in (a, b), where n and m are positive integers, arise as global secondary bifurcations from solutions of the form (λ,μ,u,0) with u having n ? 1 simple zeros in (a, b) and from solutions of the form (λ, μ, o, v) with v having m ? 1 simple zeros in (a, b). Moreover, we establish that solutions to (*) of the form (λ,μ,u,v) with u having n ? 1 simple zeros in (a, b) and v having m ? 1 simple zeros in (a, b) serve as a link between solutions of the form (λ, μ, u, 0) with u having n ? 1 simple zeros in (a,b) and solutions of the form (λ, μ, 0, v) with v having m ? 1 simple zeros in (a, b). The analysis in this article when combined with that in [3] and [5] provides a fairly comprehensive examination of the structure of the solution set to (*).     

A direct and inverse problem for a Hill's equation with double eigenvalues

Let Δ(λ) be the discriminant of a Hill's equation with a π-periodic potential q(x). Necessary and sufficient conditions on q(x) are established in order that 2 ? Δ(λ) has double zeros at all eigenvalues except for the lowest one.     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

A difference-analytical method for the computation of eigenvalues of the fourth-order equations with separated boundary conditions

We propose a method for the computation of eigenvalues with odd multiplicities for spectral boundary-value problems of the 4th order with separated boundary conditions. In this method approximate eigenvalues appear as zeros of a certain function f(λ) which admits an explicit representation.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

Irreducible Specht modules for Hecke algebras of type A

Let F be a field, n a non-negative integer, λ a partition of n and S^λ the corresponding Specht module for the Iwahori-Hecke algebra H_F,q(S_n). James and Mathas conjecture a necessary and sufficient condition on λ for S^λ to be irreducible. We prove the sufficiency of this condition in the case where F has infinite characteristic and also in the case where q=1.     

Nonexistence of solutions of the p-adic strings

We discuss mathematical aspects of the nonexistence of continuous (nontrivial) solutions of boundary value problems for equations of p-adic closed and open strings in the one-dimensional case. We find that the number of sign changes of the solution ψ(t) is not equal to the order of zeros of the function ψⁿ(t) and that nonnegative (nonpositive) solutions do not exist. In the case of even n, if a solution ψ exists, then the orders of zeros of the function ψn and the order of its tangency to positive maximums (minimums) are not divisible by four and therefore have the form 2(2 r + 1), r ≥ 0.     

On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields

We prove that the number of limit cycles which bifurcate from a two-saddle loop of an analytic planar vector field X ₀ under an arbitrary finite-parameter analytic deformation X _λ , λ ∈ (? ^N , 0), is uniformly bounded with respect to λ.     

The Falkner-Skan Equation II: Dynamics and the Bifurcations of P- and Q-Orbits

The Falkner-Skan equation is a reversible three-dimensional system of ordinary differential equations with two distinguished straight-line trajectories which form a heteroclinic loop between fixed points at infinity. We showed in the previous paper (1995, J. Differential Equations119, 336-394) that at positive integer values of the parameter λ there are bifurcations creating large sets of periodic and other interesting trajectories. Here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as λ→∞, and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur.     

Kinklike solutions of fourth-order differential equations with a cubic bistable nonlinearity

For the equation y ⁽⁴⁾+2y(y ²?1) = 0, we suggest an analytic construction of kinklike solutions (solutions bounded on the entire line and having finitely many zeros) in the form of rapidly convergent series in products of exponential and trigonometric functions. We show that, to within sign and shift, kinklike solutions are uniquely characterized by the tuple of integers n ₁, …, n _k (the integer parts of distances, divided by π, between the successive zeros of these solutions). The positivity of the spatial entropy indicates the existence of chaotic solutions of this equation.     

Coloring,sparseness and girth

Using the classical analysis resolution of singularities algorithm of [G4], we generalize the theorems of [G3] on Rⁿ sublevel set volumes and oscillatory integrals with real phase function to functions over an arbitrary local field of characteristic zero. The p-adic cases of our results provide new estimates for exponential sums as well as new bounds on how often a function f(x), such as a polynomial with integer coefficients, is divisible by various powers of a prime p when x is an integer. Unlike many papers on such exponential sums and p-adic oscillatory integrals, we do not require the Newton polyhedron of the phase to be nondegenerate, but rather as in [G3] we have conditions on the maximum order of the zeroes of certain polynomials corresponding to the compact faces of this Newton polyhedron.