Hermite polynomial analysis of linear optimal control systems

The analysis of optimal control systems incorporating observers has been approached using Hermite polynomials. The method simplifies the system of equations into the successive solution of a set of linear algebraic equations. An illustrative example is given and accurate results obtained.     

NUMERICAL GRID GENERATION OF AN IRREGULAR REGION

A procedure of numerical conformal mapping is established to generate grids of a two-dimensional irregular region for further computations. The approach employs a sequence of Zⁿ transformations to map an irregular region into a quadrilateral region with right angles at each of the four corners. This quadrilateral region is then transformed into a rectangular domain by using the boundary integral element method to solve the Laplace mapping equations. Grids are generated in the transformed rectangular domain and are mapped inversely into the original irregular region. The present technique of grid generation is demonstrated and verified in two benchmark problems. For physical applications, steady-state temperature distributions in a circle and an area bounded by two triangles are calculated and verified, using the present techniques. The present technique of numerical conformal mapping can generate grids for the finite element method or boundary-fitted co-ordinates for the finite difference method. © 1997 by John Wiley & Sons, Ltd.     

A general frequency stability criterion for multi-input-output,lumped and distributed-parameter feedback systems

The original Nyquist criterion is based on the comparison of the encirclement of the frequency plot of the return ratio function with the number of poles and the number of zeros of the same function to determine the closed-loop stability of a feedback system. The extensions of the return ratio idea to the stability study of multi-variable feedback systems have used the same terminology and followed a similar course. For the multi-input—output case, the use of the Nyquist criterion or its extension is by no means a simple matter. This paper establishes a new frequency stability criterion which converts the Nyquist criterion from a return ratio oriented approach to a return difference oriented one. Instead of examining the encirclement of the return ratio function to a critical point, we examine the phase change of the positive frequency of the return difference function, and the number of zeros of the positive frequency of the return difference function. This result simplifies the stability study of multi-input—output lumped systems tremendously, and covers multi-input-output distributed-parameters systems naturally. For illustration, several typical examples—single-input-output feedback systems with minimum phase or non-minimum phase open-loop transfer functions, multi-input-output feedback systems with stable or unstable open-loop transfer matrices, multi-input-output feedback systems with irrational or transcendental type distributed-parameter open-loop transfer matrices—are included.     

Transformations of solvents and spectral factors of matrix polynomials,and their applications†

The relationships between solvents and spectral factors of a high-degree matrix polynomial are explored. Various new transformations are developed to convert right (left) solvents into spectral factors and vice versa. The transformation of a right (left) solvent to a left (right) solvent is also established. The newly established algorithms are then applied to determine the spectral factorization of a matrix polynomial for optimal control problems. The developed algebraic theory enhances the capability of the analysis and synthesis of a system described by a high-degree matrix differential equation.     

A squared magnitude continued fraction expansion for stable reduced models

The direct continued fraction method of model reduction for single-input-output and multi-input—output cases sometimes produces unstable reduced models. A generalization is made by using the squared magnitude continued fraction and factorization technique to obtain reduced models which are always stable.     

Robust observer design for nonlinear uncertain systems with LQ regulators

For a class of uncertain systems with linear nominal dynamics and nonlinear uncertainties, it has been shown (Katayama and Sasaki 1987) that linear quadratic (LQ) state feedback regulators can be used to provide robust asymptotic stability. In this paper, we study the combined observer-controller design problem, based on the linear state feedback regulator proposed by Katayama and Sasaki (1987), so that only output feedback is needed. Both full-order and reduced-order observers are considered. For the full-order observer, we propose an algorithm to synthesize the robust observer gain matrix. It is shown that with the observer it is still possible to achieve robust asymptotic stability. For the reduced-order observer, some conditions are derived to guarantee the robust asymptotic stabilizability of the uncertain systems. The trade-off between the magnitudes of controller and observer gains is clear in our approach. An example is used to illustrate the design process of the robust controller with full-order as well as reduced-order observers.     

Hydrogen-enhanced cracking of 2205 duplex stainless steel

Tensile and fatigue crack growth tests of 2205 duplex stainless steel (DSS) were performed in laboratory air, gaseous hydrogen at 0.2 MPa and saturated H₂S solution. The longitudinal specimen showed a lesser degradation of tensile properties than the transverse ones in saturated H₂S solution. The orientation of specimens with respect to rolling direction had little influence on the fatigue crack growth rate (FCGR) of the alloy in air. Furthermore, 2205 duplex stainless steel was susceptible to hydrogen‐enhanced fatigue crack growth. Transmission electron micrographs, in addition to X‐ray diffraction, revealed that the strain‐induced austenite to martensite transformation occurred near the crack surface within a rather narrow depth. Fatigue fractography of the specimens tested in air showed mainly transgranular fatigue fracture with a small amount of flat facet fracture. Furthermore, extensive quasi‐cleavage fracture of 2205 duplex stainless steel was associated with the hydrogen‐enhanced crack growth.     

Analysis of linear time-varying systems and bilinear systems via Taylor series

Linear time-varying systems and bilinear systems are each analysed via Taylor series. Using the operational matrix for integration and the product operational matrix, the dynamical equation of a linear time-varying system (or a bilinear system) is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the Taylor series expansions can be determined recursively by the algorithm derived. The algorithm proposed here is similar to those already developed for orthogonal functions; however, owing to the simplicity of the operational matrix of integration and the product operational matrix, Taylor series present considerable computational advantages compared with the other polynomial series, provided that both the input and the output signals are analytic functions of t.     

Solution of linear dynamic systems with initial or boundary value conditions by shifted Chebyshev approximations

The shifted Chebyshev polynomial approximation is employed to solve the linear, constant parameter, ordinary differential equations of initial or two-point boundary value problems. An effective recursive algorithm is developed to calculate the expansion coefficients of the shifted Chebyshev series. An effective transformation is proposed to transform the two-point boundary value problem into an initial value problem. An illustrative example is included to show that the computational results are accurate.