Application of general discrete orthogonal polynomials to optimal-control systems

General orthogonal polynomials are introduced to analyze and approximate the solution of a class of scaled systems. Using the operational matrix of integration, together with the operational matrix of linear transformation, the dynamical equation of a scaled system is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the general orthogonal polynomials can be determined recursively by the derived algorithm. An illustrative example is given to demonstrate the validity and applicability of the orthogonal polynomial approximations.

Difference equations for discrete classical multiple orthogonal polynomials

Based on a hybrid orthogonal function, a new linear, continuous and bounded operator, piecewise general orthogonal polynomials operator (PGOPO), is proposed. Its main properties and operational rules have been strictly constructed based on the theory of convergence in the mean square, and are useful to discuss the orthogonal polynomial approach in the proper mathematics frame. Then applying the PGOPO method to solve the identification problem of nonlinear systems, the convergence analysis of PGOPO approximation method is given. Finally, using the PGOPO method to solve the parameter identification of two kinds of time-varying systems, the simulation studies show that the algorithm is simple and more effective than that of general orthogonal polynomials.

We derive and factorize the fourth-order q-difference equations satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coefficients of q-classical orthogonal polynomials. These perturbations include the rth associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified q-classical orthogonal polynomials. Moreover we find a basis of four linearly independent solutions of these fourth-order q-difference equations and express the modified families in terms of the starting ones.

The operational matrix consisting of the product of two time functions, and the operational matrices for forward or backward integration consisting of general orthogonal polynomials are derived, respectively, for the analysis and optimal control of linear time-varying systems with a quadratic performance measure. The present results include results obtained via Chebyshev, Legendre, Laguerre, Jacobi, Hermite and ultraspherical polynomials as special cases.

A very effective method of using the generalized orthogonal polynomials (GOP) for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-varying coefficients in the form of finite-order polynomials is presented. It is based on the differentiation operational matrix of the GOP, which can represent all kinds of individual orthogonal polynomials. The main advantage of using the differentiation operational matrix is that parameter estimation can be made starting at any time of interest, i.e. without the restriction of starting at zero time. In addition, the present computation algorithm is simpler than that of the integration operational matrix. Using the concept of GOP expansion for state and control functions, the differential input-output equation is converted into a set of linear algebraic equations. The unknown parameters are evaluated by a weighted least-squares estimation method. Very satisfactory results for illustrative example are...

Definition 10.1 (State variables). Consider a system with a known dynamical behaviour, and suppose that its inputs are known from an arbitrary time instant t on. The state variables of the system are those in a set of variables, with as few elements as possible, such that, knowing them at instant t, it is possible to calculate the system's future behaviour. Remark 10.1. As is well known, a system's state variables are not unique. If x(t) is an n x 1 vector with the system's state variables, and P is an n x n invertible matrix, then the variables in vector w(t) = Px(t) also are state variables. Fractional order systems have no state variables, as we shall see below; but it is possible to obtain for them representations similar to those that use the state variables of integer systems. This chapter addresses first the general case of multipleinput, multiple-output (MIMO) systems, and then the particular case of single-input, single-output (SISO) systems. The discretisation of these representations is also addressed.

Arnoldi-based model order reduction for linear systems with inhomogeneous initial conditions

If the nineteenth century was a period in which orthogonal polynomials were treated as special functions, for which all manner of identities had to be established, then the twentieth century has been a period in which they have been viewed as objects worthy of analysis, especially their asymptotic behaviour as the degree tends to infinity. The research monograph under review presents a polished part of the asymptotic theory.

Pade-Type Approximation and General Orthogonal Polynomials.

exist for n=0,1,…. We normalize μ by m0=1. The distribution functionFμ is right continuous and defined by

We introduce the combinatorial notion of heaps of pieces, which gives a geometric interpretation of the Cartier-Foata's commutation monoid. This theory unifies and simplifies many other works in Combinatorics : bijective proofs in matrix algebra (MacMahon Master theorem, inversion matrix formula, Jacobi identity, Cayley-Hamilton theorem), combinatorial theory for general (formal) orthogonal polynomials, reciprocal of Rogers-Ramanujan identities, graph theory (matching and chromatic polynomials). Heaps may bring new light on classical subjects as poset theory. They are related to other fields as Theoretical Computer Science (parallelism) and Statistical Physics (directed animals problem, lattice gas model with hard-core interactions). Complete proofs and definitions are given in sections 2, 3,4,5. Other sections give a summary of possible applications of heaps.

In 1999, Grunbaum, Haine and Horozov defined a large family of commutative algebras of ordinary differential operators which have orthogonal polynomials as eigenfunctions. These polynomials are mutually orthogonal with respect to a Laguerre-type weight distribution, thus providing solutions to Krall's problem. In the present paper we give a new proof of their result which establishes a conjecture, concerning the explicit characterization of the dual commutative algebra of eigenvalues. In particular, for the Koornwinder's generalization of Laguerre polynomials, our approach yields an explicit set of generators for the whole algebra of differential operators. We also illustrate how more general Sobolev-type orthogonal polynomials fit within this theory.

The paper proposes derivative-free nonlinear Kalman Filtering for MIMO nonlinear dynamical systems. The considered nonlinear filtering scheme which is based on differential flatness theory extends the class of systems to which Kalman Filtering can be applied without the need for calculation of Jacobian matrices. To deduce if a dynamical system is differentially flat, the following should be examined: (i) the existence of the flat output, which is a variable that can be written as a function of the system's state variables (ii) the system's state variables and the input can be written as functions of the flat output and its derivatives. Nonlinear systems satisfying the differential flatness property can be written in the Brunovsky form via a transformation of their state variables and control inputs. After transforming the nonlinear system to the canonical form it is straightforward to apply the standard Kalman Filter recursion. The performance of the proposed derivative-free nonlinear filtering scheme is tested through simulation experiments on benchmark nonlinear multi-input multi-output dynamical systems, such as robotic manipulators.

Multiple orthogonal polynomials