transfer functions of regular linear systems part ii the system operator and the lax phillips semigroup this paper is a sequel to a paper by the second author on regular linear systems 1994 referred to here as part i we introduce the system operator of a well posed linear system which for a finite dimensional system described by x ax bu y cx du would be the s dependent matrix s s a si c b d in the general case s s is an unbounded operator and we show that it can be split into four blocks as in the finite dimensional case but the splitting is not unique the upper row consists of the uniquely determined blocks a si and b as in the finite dimensional case but the lower row is more problematic for weakly regular systems which are introduced and studied here there exists a special splitting of s s where the right lower block is the feedthrough operator of the system using s 0 we give representation theorems which generalize those from part i to well posed linear systems and also to the situation when the initial time is we also introduce the lax phillips semigroup t induced by a well posed linear system which is in fact an alternative representation of a system used in scattering theory our concept of a lax phillips semigroup differs in several respects from the classical one for example by allowing an index r which determines an exponential weight in the input and output spaces this index allows us to characterize the spectrum of a and also the points where s s is not invertible in terms of the spectrum of the generator of for various values of the system is dissipative if and only if with index zero is a contraction semigroup

the ito clifford integral abstract we present a theory of non commutative stochastic integration analogous to the ito theory it is shown that wick products of fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted operatorvalued square integrable integrands for square integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined and a doob meyer decomposition for supermartingales obtained

optimum design of multiple constraint layered systems for vibration control an analytical formulation is presented for predicting the stiffness and damping of constrained layered beams that have multiple viscoelastic damping layers a model is derived for symmetrical setups using variational methods the equations to evaluate the stiffness and damping are derived in closed form and can be evaluated for different boundary conditions the complex modulus approach is used to model the elastic and shear moduli of the viscoelastic material a parametric analysis has been conducted to study the effects of different parameters on the damping and stiffness of the system under simply supported boundary conditions the optimization process to obtain structural and material parameters maximizing system damping for a given temperature and frequency range is also presented nomenclature e y oung s modulus of the constraining layer ec complex modulus of the viscoelastic material g shear modulus of the constraining material gc complex shear modulus of the viscoelastic material h layer thickness l length of the beam m system mass q ij element of the transformed reduced stiffness matrix u longitudinal displacement of a point in the neutral plane of a layer w transverse displacement of a point in the neutral plane of a layer wi imaginary part of the natural frequency wr real part of the natural frequency zk distance from the point in the kth layer to the neutral axis of the layer e normal strain loss factor of the mulitlayered system c loss factor of the viscoelastic material material density slope of transverse displacement shear strain in the viscoelastic material frequency of vibration rad s

h_ infty fuzzy control for a class of nonlinear coupled ode pde systems with input constraint this paper deals with the problem of h infty fuzzy control design with an input constraint for a class of coupled systems which consist of an n dimensional nonlinear subsystem of ordinary differential equations odes and a scalar linear parabolic subsystem of partial differential equation pde connected in feedback initially the nonlinear coupled system is represented by a takagi sugeno t s fuzzy coupled ode pde model then based on the fuzzy model and parallel distributed compensation scheme a fuzzy state feedback control design is developed via lyapunov s direct method such that the resulting closed loop fuzzy coupled system is exponentially stable and a prescribed h infty performance of disturbance attenuation is satisfied the existing condition of the proposed h infty fuzzy controllers is given in terms of linear matrix inequalities lmis moreover in order to make the attenuation level as small as possible while the input constraint is respected to avoid the high magnitude a suboptimal h infty constrained fuzzy control problem is also addressed which is formulated as an lmi optimization problem finally the proposed method is applied to the control of a hypersonic rocket car to illustrate its effectiveness

boundary stabilization of a coupled wave ode system with internal anti damping in this article the backstepping method is employed to stabilize a coupled wave ode system with internal anti damping by means of decoupling them into a stable cascaded wave ode system at the same time the existence of the kernels in backstepping transformation and inverse transformation is proved by iterative method

vibration analysis of a multiple layered viscoelastic structure using the biot damping model the viscoelastic damping treatment is a very popular method for attenuating vibration and noise problems it provides the designer much flexibility in adjusting the system damping to eliminate resonances in the structures a survey of the previous research on material damping models found inadequacies in developing the numerical models of multiple layer constrained layers and the introduction of the combination of different viscoelastic damping materials through advanced frequency domain damping models in this paper a detailed numerical analysis and design of a multiple layer constrained damping structure using the biot damping model are presented to investigate the vibration performance the nonlinear curve fitting of the complex shear modulus and the transient response solution of the numerical viscoelastic damping structure are also discussed by the proposed approach the simple beam problem that is presented can be easily extended to the industrial problem with arbitrary geometries and boundary conditions the validation of this numerical approach is proven by comparison with the previous closed form results

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

exact boundary controllability of a hybrid system of elasticity a hybrid control system is presented an elastic beam governed by a partial differential equation linked to a rigid body which is governed by an ordinary differential equation and to which control forces and torques are applied the entire system elastic beam plus rigid body is proved to be exactly controllable by smooth open loop controllers applied to the rigid body only and in arbitrarily short durations this system is modeled as a two dimensional space structure

time orthogonal unitary dilations and noncommutative feynman kac formulae an analysis of feynman kac formulae reveals that typically the unperturbed semigroup is expressed as the expectation of a random unitary evolution and the perturbed semigroup is the expectation of a perturbation of this evolution in which the latter perturbation is effected by a cocycle with certain covariance properties with respect to the group of translations and reflections of the line we consider generalisations of the classical commutative formalism in which the probabilistic properties are described in terms of non commutative probability theory based on von neumann algebras examples of this type are generated by means of second quantisation from a unitary dilation of a given self adjoint contraction semigroup called the time orthogonal unitary dilation whose key feature is that the dilation operators corresponding to disjoint time intervals act nontrivially only in mutually orthogonal supplementary hilbert spaces

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

controllability and observability of a well posed system coupled with a finite dimensional system we consider coupled systems consisting of a well posed and strictly proper hence regular subsystem and a finite dimensional subsystem connected in feedback the external world interacts with the coupled system via the finite dimensional part which receives the external input and sends out the output under several assumptions we derive well posedness regularity exact or approximate controllability and exact or approximate observability results for such coupled systems

classical and quantum probability we follow the development of probability theory from the beginning of the last century emphasizing that quantum theory is really a generalization of this theory the great achievements of probability theory such as the theory of processes generalized random fields estimation theory and information geometry are reviewed their quantum versions are then described

quantum stochastic calculus operation valued stochastic processes and continual measurements in quantum mechanics the physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process ovsp in this article it is shown how the formalism of quantum stochastic calculus of hudson and parthasarathy allows in a simple way for constructing a large class of ovsp s that in particular contains the quantum counting processes of davies and srinivas and continual gaussian measurements this result is obtained by means of a stochastic dilation of the ovsp s at the level of the enlarged system probabilities turn out to be expressed in terms of projection valued measures associated with certain time dependent commuting self adjoint operators

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

feynman kac representation of some noncommutative elliptic operators abstract gaussian averages of automorphisms of a von neumannn algebra yield markov semigroups by the well known procedure of subordination we construct operator valued martingales to realise perturbations of such semigroups through feynman kac formulae the perturbations are noncommutative vector fields and the martingales are operator families which are determined by an ito equation on each vector and satisfy cocycle relations with respect to a randomised flow on the algebra in particular this gives a probabilistic representation of some symmetric markov semigroups considered by davies and lindsay

algebraic theory of product integrals in quantum stochastic calculus motivated by the search for solutions of the quantum yang baxter equation an algebraic theory of quantum stochastic product integrals is developed the product integrators are formal power series in an indeterminate h whose coefficients are elements of the lie algebra l labelling the usual integrators of a many dimensional quantum stochastic calculus the product integrals are also formal power series in h whose coefficients are finite iterated additive stochastic integrals which act on the exponential domain in the fock space of the calculus and which represent elements of the universal enveloping algebra u of l they obey a multiplication rule suggested by the quantum ito product formula and are characterized among all such formal power series by a grouplike property

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

well posedness and controllability of a class of coupled linear systems we consider coupled systems consisting of an infinite dimensional part and a finite dimensional part connected in feedback such as for example the well known scole system a beam with a rigid body attached at one end the external world interacts with the coupled system via the finite dimensional part which receives the external input and sends out the output the infinite dimensional part is assumed to be such that it becomes well posed and strictly proper when connected in cascade with an integrator under several assumptions we derive well posedness and exact controllability results for such coupled systems the first main result concerns the case when the input signal of the finite dimensional part is the difference between the external input and the feedback signal the second main result allows a more general structure for the finite dimensional part we also prove a result for the approximate controllability of coupled systems

nonlinear control of a coupled pde ode system modeling a switched power converter with a transmission line we consider an infinite dimensional system modelling a boost converter connected to a load via a transmission line the governing equations form a system coupling the telegraph partial differential equation with the ordinary differential equations model ing the converter the coupling is given by the boundary conditions and the nonlinear controller we introduce we design a nonlinear saturating control law using a lya punov function for the averaged model of the system the main results give the well posedness and stability properties of the obtained closed loop system

exponentially stabilizing fuzzy controller design for a nonlinear ode beam cascaded system and its application to flexible air breathing hypersonic vehicle abstract this paper addresses fuzzy control design for a class of nonlinear systems which are described by nonlinear ordinary differential equations odes cascaded with an euler bernoulli beam ebb equation two design difficulties are involved in the control design addressed in this paper the first one is caused by the ebb equation whose spatiotemporal dynamics is affected by the output of the nonlinear ode subsystem through its differential equation rather than boundary conditions a state differential transformation is introduced for the ebb equation to bring the output of the ode subsystem in the differential equation to its boundary conditions the second one comes from the nonlinear ode subsystem to overcome this difficulty it is assumed that an exact takagi sugeno t s fuzzy ode model is utilized by the sector nonlinearity approach to describe the dynamics of nonlinear ode subsystem based on the t s fuzzy model a composite lyapunov function is constructed to develop a fuzzy controller via the ode state feedback and boundary feedback of transformed ebb equation to exponentially stabilize the nonlinear ode ebb cascaded system the design procedure is presented in terms of bilinear matrix inequalities bmis moreover a simple design method is also provided in terms of linear matrix inequalities lmis from the obtained design procedure finally numerical simulations on flight control with vibration suppression of a flexible air breathing hypersonic vehicle are given to illustrate the effectiveness of the proposed design method