a note on parameterization of the class of deadbeat controllers the problem of parameterizing the class of deadbeat controllers for a given discrete time system through the minimum number of parameters was solved by schlegel 1 this note shows how to utilize the above solution to study some problems in designing deadbeat controllers first an algorithm is developed to compute a controller which minimizes in an average sense a given objective function second a necessary and sufficient condition is given for the existence of an output deadbeat controller finally the problem of parameterizing the set of deadbeat controllers for those systems transformable to the phase variable block canonical form is reconsidered

sextic formalism in anisotropic elasticity for almost non semisimple matrix n abstract the sextic formalism of stroh for anisotropic elasticity leads to the eigen relation n p in which n is a 6 6 real matrix the orthogonality and closure relations as well as many other relations involving the eigenvalues p and the eigenvectors x are based on the assumption that n is simple or semisimple so that the six eigenvectors span a six dimensional space problems arise when n is non semisimple in fact there are problems even when n is almost non semisimple we present a modified formalism which is valid regardless of whether n is simple almost non semisimple or non semisimple the modified formalism does not apply when n is semisimple

the design of deadbeat controllers by state transition graph two kinds of deadbeat control problems are considered one is the state deadbeat control problem and the other is the pointwise minimum time deadbeat control problem a simple graph called the state transition graph of a matrix is introduced and simple algorithms based on it giving deadbeat controllers are presented the set of pointwise minimum time deadbeat controllers is characterized the set of output feedback deadbeat controller is also considered

comments on design of optimal dead beat controllers it is shown that the system considered in the numerical example of the above paper belongs to a class of systems with n states and r inputs for which the set of all minimum time dead beat mtdb controllers is parameterized by an arbitrary matrix of order lambda 1 r n x n lambda r where is the greatest integer less than the ratio n r since the example treats the case of n 3 r 2 the mdtb controller is parameterized by a single scalar element

hamiltonian interpretation of the stroh formalism in anisotropic elasticity stroh s sextic formalism for static problems or steady motions in anisotropic elasticity is a formulation in which the equation of equilibrium motion is written as a system of first order differential equations for the displacement and traction in terms of one of the spatial variables the so called fundamental elasticity matrix n appearing in this formulation has the property that when partitioned as a 2 2 block matrix its 12 and 21 blocks are symmetric matrices and its 11 block is the transpose of its 22 block this property gives rise to a large number of orthogonality and closure relations and is fundamental to the success of the stroh formalism in solving a large variety of problems in general anisotropic elasticity first we show that the matrix n is guaranteed to have the above property by the fact that the stroh formulation is in fact a hamiltonian formulation with one of the spatial variables acting as the time like variable this interpretation provides a much desired guide in dealing with other problems for which the governing equations are different such as incompressible elasticity and problems associated with anisotropic elastic plates as described by the kirchhoff plate theory we show that for the last two problems the hamiltonian interpretation simplifies the derivations significantly leading to a stroh formulation in each case which is equivalent to but much simpler than what is available in the existing literature

simultaneous deadbeat tracking controller synthesis abstract a direct and simple design algorithm for the problem of simultaneously deadbeat tracking a prespecified class of inputs by only one controller is presented this novel approach is based on polynomial algebra and the principle of system realizability without solving linear polynomial diophantine equations the proposed procedure consists of solving a set of linear equations depending on system dynamics and yields the deadbeat controller in a compact form well suited for system with inaccessible states and changeable reference signals since all solutions are described in a parametric form some performance criteria can be easily incorporated as well examples are presented to illustrate the usefulness and computational easiness of this simple design algorithm

octet formalism for kirchhoff anisotropic plates an octet formalism is established for stretching and bending deformations of an anisotropic elastic plate based on the kirchhoff theory the plate is inhomogeneous in the thickness direction and thus includes the laminated plate as a special case by defining a 4 eigenvector associated with two in plane displacements and two rotational angles of a normal of the mid plane and a 4 eigenvector associated with four stress functions the three equilibrium equations of the plate reduce to a standard eight dimensional eigenrelation the fundamental elastic plate matrix defined in this work is found to place the two eigenvectors in reverse order to form the left and right 8 eigenvectors of the matrix the same capability has been observed for the fundamental elasticity matrix in stroh s sextic formalism for generalized plane strain elasticity thus our octet formalism symbolically preserves extensive elegant properties and identities pertaining to the capability e g the orthogonality and closure relations which have been established in the stroh sextic formalism having shown that the eigenvalues cannot be real we provide the eigenvalue representations for special cases of material and geometry the non semisimple case including an isotropic material is briefly discussed

on the adomian decomposition method and comparisons with picard s method abstract this paper shows that the adomian decomposition method stochastic systems academic press new york 1983 is not to be identified with any of the earlier methods since it solves a wide class of nonlinear and stochastic equations without the often unphysical assumptions which have become customary similarities between the picard method and the decomposition method are purely superficial

an improved optimal homotopy analysis algorithm for nonlinear differential equations abstract this paper presents an improved optimal homotopy analysis algorithm to deal with nonlinear first order differential equations we discuss the optimal selection of the auxiliary linear operator in the application of the homotopy analysis method that will accelerate the convergence of series solutions then a powerful algorithm that employed the optimal auxiliary linear operator of the homotopy analysis method is presented to facilitate the calculations numerical illustrations are examined to show the efficiency and the pertinent features of the proposed algorithm this algorithm could lead to a promising approach for many nonlinear problems in applied sciences

extended meshfree analysis of transverse and inplane loading of a laminated anisotropic plate of general planform geometry abstract an extended meshfree method is presented for the analysis of a laminated anisotropic plate under elastostatic loading the plate may be of any planform shape with its thickness profile composed of perfectly bonded uniform thickness layers of distinct anisotropic materials both transverse and inplane loads are considered using a first order shear deformation theory for flexural behavior and generalized plane stress for the membrane behavior in this extended meshfree method a rectangular domain is initially considered with the plate of arbitrary geometry inscribed within it a particular solution in the form of an analytic generalized navier solution a compound double fourier series is used to capture the response due to the loading within the rectangular domain then a homogeneous solution by meshfree analysis is added to treat the augmented boundary conditions on the actual contour of the plate these augmented conditions are composed of the prescribed values and that of the particular solution evaluated around the plate s contour concentrated transverse and inplane loads in the form of uniform loads over a very small patch are considered with this generalized navier solution representation when a meshfree portion is added to account for the boundary conditions such solutions constitute the green s functions for the plate the viability of these double fourier series representations is shown by the convergence rates for the kinematic and force moment fields an additional example of a two layer 30 angleply circular plate is given to illustrate the capability of this extended meshfree method

optimal control of drug therapy in cancer treatment abstract we present an optimal control strategy for nonlinear systems with application to the drug therapy of cancer the tumour growth model is represented by a system of equations from population dynamics which is based on the competition between normal cells and tumour cells the effect of the immune system in the presence of cancer is also included in the model a linear time varying approximation technique is proposed to construct an optimal control strategy for the nonlinear system which is valid not only within small perturbations around the equilibrium point but also for global dynamics of the system the optimal control method is applied to eliminate the tumour cells whilst minimizing the amount of drug in the simulated model simulation results show that cancer cells can be eradicated 1 in a very short time with a small amount of drug using an optimal administration of the therapy

new exact solutions to the fitzhugh nagumo equation abstract by the first integral method a series of new exact solutions of the fitzhugh nagumo equation have been obtained it is shown that this method is one of the most effective approaches to obtain the exact solutions of the nonlinear evolution equations especially for nonintegrable models

parameterization of the class of deadbeat controllers this technical note deals with the problem of parameterizing the class of deadbeat controllers for a given system through the minimum of parameters it is shown that if n and m are dimensions of the state vector and control vector respectively and the i s are the controllability indexes then almost the whole class of deadbeat controllers is explicitly parameterizable through mn mu 1 3 mu 2 2m 1 mu m parameters

the explicit series solution of sir and sis epidemic models in this paper the sir and sis epidemic models in biology are solved by means of an analytic technique for nonlinear problems namely the homotopy analysis method ham both of the sir and sis models are described by coupled nonlinear differential equations a one parameter family of explicit series solutions are obtained for both models this parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models our analytic results agree well with the numerical ones this analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology

activation of effector immune cells promotes tumor stochastic extinction in this article we provide homotopy solutions of a cancer nonlinear model describing the dynamics of tumor cells in interaction with healthy and effector immune cells we apply a semi analytic technique for solving strongly nonlinear systems the step homotopy analysis method sham this algorithm based on a modification of the standard homotopy analysis method ham allows to obtain a one parameter family of explicit series solutions by using the homotopy solutions we first investigate the dynamical effect of the activation of the effector immune cells in the deterministic dynamics showing that an increased activation makes the system to enter into chaotic dynamics via a period doubling bifurcation scenario then by adding demographic stochasticity into the homotopy solutions we show as a difference from the deterministic dynamics that an increased activation of the immune cells facilitates cancer clearance involving tumor cells extinction and healthy cells persistence our results highlight the importance of therapies activating the effector immune cells at early stages of cancer progression

parametric time optimal control of linear discrete time systems by state feedback part 2 irregular kronecker invariants in an earlier paper part 1 a parametric state feedback time optimal control law was devised for the regular kronecker invariants on the basis of parametrization of the vector companion form the present paper describes a straight forward technique for obtaining the parametric feedback matrix by exploiting the parametric vector companion forms in the case of irregular kronecker invariants hidden properties of vector companion forms are also given which result in entirely different and interesting techniques for parametrization of the control law an illustrative example is presented

minimal time minimal order deadbeat regulator with internal stability this paper is concerned with the output deadbeat finite settling time regulation with internal stability for linear discrete time multivariable systems the two cases are treated separately the one is the regulation by state feedback and the other is the regulation through function observers for both cases the basic solvability conditions the minimal settling time and the characterizations of the minimal time regulators are derived throughout the paper the well posedness condition plays a fundamental role a deterministic version of the so called separation theorem in lqg problem is derived namely the minimal time state feedback regulator combined with the minimal time minimal order function observer yields the overall minimal time minimal order regulator by output feedback

parameterization of the class of deadbeat controllers via the theory of decoupling a novel approach is presented for parameterizing the class of minimum time deadbeat controllers mtdc through the minimum number of parameters the approach is based on the theory of decoupling and the properties of square decouplable systems the main result is a compact parametric form for the class of mtdc of a discrete time system contrary to many existing techniques no special assumption on the invertibility of the transition matrix is required

on the general solution of the state deadbeat control problem in this note the state deadbeat control problem is considered it is shown that after appropriate change of basis of input and state spaces the general solution of the state deadbeat control problem can be expressed completely by the rows of the powers of system matrix this result yields a very simple procedure for the calculation of a state feedback deadbeat control gain it also provides the number of free parameters which could be used for further design purposes the results are illustrated by an example at the end of the note

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

on deadbeat controllers the possibility of achieving a deadbeat regulation and tracking in linear multivariable systems independently of their initial conditions is investigated necessary and sufficient conditions for the existence of various deadbeat controllers are established in a constructive way a detailed analysis of the control sequence generated by any deadbeat controller is then presented

stroh like formalism for the coupled stretching bending analysis of composite laminates due to the two dimensional nature of thin plates the lamination theory considering the composite laminates with in plane and plate bending problems coupling each other is treated in this paper by using complex variable formulation by following the steps of stroh formalism for two dimensional linear anisotropic elasticity a displacement complex variable formalism developed by the other researchers was introduced and re derived in a different but more stroh like way in addition a brand new mixed formalism mixed use of displacements and stresses as basic functions is established to compensate the displacement formalism in order to transfer all the related formulae and mathematical techniques of the stroh formalism to these two formalisms the general solutions for the basic equations of lamination theory and their associated eigenrelations have been purposely arranged in the form of stroh formalism moreover by using the presently developed mixed formalism the explicit expressions for the fundamental matrix and eigenvectors are obtained first time for the most general composite laminates furthermore letting the coupling stiffness vanish the formalism has been reduced to the case of symmetric laminates and checked by a recently developed stroh like formalism for the plate bending problems the comparison between stroh formalism for two dimensional problem stroh like formalism for plate bending problem displacement formalism and mixed formalism is then made at the end of this paper and through their connection some useful relations are obtained

the discrete linear time invariant time optimal control problem an overview a review of two decades of investigation into one of the fundamental problems of control theory the discrete linear time invariant time optimal control problem is presented two classes of multi input time optimal controller one based on system controllability and the other on eigenvalue assignment are considered in some detail also discussed are the dual state reconstruction problem the deadbeat control of linear systems with inaccessible state and other extensions the exposition is a unifying one and involves only that matrix algebra commonly encountered in control theory

an exact three dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness an exact three dimensional solution is presented for the deformation and stress distribution in an elliptical plate under uniform normal loading of the lateral surfaces and clamped along its edge the plate is assumed to be of constant moderate thickness and composed of anisotropic elastic material which is inhomogeneous in the through thickness direction but symmetric about the mid plane the only material symmetry assumed is that of reflectional symmetry in planes parallel to the mid plane a transfer matrix method is used which without making any further assumptions gives the exact solution at each point in the plate in terms of the stress and displacement at the mid plane the two dimensional differential equations governing these mid plane values are found to be the same as those for an equivalent homogeneous plate whose constant elastic moduli are determined by appropriate through thickness weighted averages of the inhomogeneous moduli the solution of the two dimensional problem is known for such a plate when subject to the specified surface and edge conditions and yields a closed form analytical solution that satisfies all the governing equations and surface conditions of the full three dimensional elasticity problem with edge displacement conditions satisfied on the mid plane the important special case of an anisotropic laminated plate is given by assuming piecewise constant properties through the thickness

some explicit expressions of stroh like formalism for coupled stretching bending analysis abstract since stroh like formalism for coupled stretching bending analysis preserves essential features of stroh formalism for two dimensional anisotropic elasticity it becomes important to get the corresponding explicit expressions of some important matrices frequently appeared in stroh like formalism in this paper explicit expressions are obtained for the fundamental elasticity matrix n the generalized fundamental elasticity matrix n the material eigenvector matrices a and b and barnett lothe tensors l s and h in these expressions n a and b are presented for the general laminates whereas the other matrices are presented only for the symmetric laminates to show the applicability to the degenerate laminates these expressions are further reduced to the isotropic plates finally they are applied to get the explicit real component form solutions of the green s functions for an infinite isotropic plate

boundary integral equations for general laminated plates with coupled stretching bending deformation the boundary integral equations for the stretching bending coupling analysis of general laminated plates are derived in this paper with the aid of the reciprocal theorem of betti and raleigh no restriction is placed on the location of the point load or moment and so it can be an internal or external point or a smooth or non smooth boundary point the fundamental solutions derived from green s function for an infinite laminated plate subjected to concentrated forces moments are presented in the complex matrix form through the use of some identities converting complex form to real form the free term coefficients which are important for the boundary integral equations are obtained explicitly and expressed in the real form alternative formulae for calculating the free term coefficients are also derived using five rigid body movements by using the explicit real expressions obtained recently for the stroh like formalism of coupled stretching bending analysis the free term coefficients are further reduced to the cases of isotropic plates and are then compared with known expressions published in the literature since stretching and bending decouple for isotropic plates comparison is made separately for the in plane problem and plate bending problem and it is shown that the boundary integral equations derived in this paper agree with previous results for these two special cases

pole assignment by state transition graph the pole assignment problem is considered using the graph representation of a matrix the parametrization of controllers which have a specified characteristic polynomial is given a simple algorithm based on graphs is presented and two examples are given