norm based robust control of state constrained discrete time linear systems the author presents a theoretical framework for analyzing the stability properties of constrained discrete time systems in the presence of uncertainty and it is shown that this formalism provides a unifying approach including as a particular case the well known technique of estimating robustness bounds from the solution of a lyapunov equation these results are applied to the problem of designing feedback controllers capable of stabilizing a family of systems while at the same time satisfying state space constraints

minimality and local state decompositions of a nonlinear state space realization using energy functions in this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state space realization of a formal power series is minimal specifically it is shown that positivity of these functions in addition to a stability requirement and a few technical conditions implies minimality using the nonlinear analogue of the kalman decomposition connections are then established between minimality singular value functions balanced realizations and various notions of reachability and observability for nonlinear systems

an algorithm for robust performance tests based on frequency dependent lmis in this paper an algorithm is introduced for feasibility problems associated with frequency dependent linear matrix inequalities these problems correspond to h 2 or h robust performance tests for a given controller in the face of feedback perturbations to this effect an auxiliary h 2 cost functional is introduced and a sequence of h 2 problems with a single linear constraint is considered in which the h 2 cost functional is kept unchanged while the linear constraints are iteratively modified it is shown that in each step the auxiliary optimal solution moves closer in the sense of a weighted quadratic norm to the target feasible set conditions are also established under which the sequence of solutions to the auxiliary problems yields a solution to the original feasibility problem the method is illustrated by a numerical example corresponding to a h robust performance test for a multivariable system

numerical algorithms for the nevanlinna pick problem based on the formulation in 14 efficient algorithms are devised for two variants of the nevanlinna pick problem their implementation is validated by several examples the method depends on establishing the equivalence of two different operator theoretic approaches to the solution of classical interpolation problems 1 14

single input single output h control with time domain constraints standard h optimization cannot handle specifications or constraints on the time response of a closed loop system exactly in this paper the problem of h optimization subject to time domain constraints over a finite horizon is considered following an idea from helton and sideris 1989 ieee trans aut control ac 34 427 434 the problem is transformed into a finite dimensional optimization program that is shown to be convex although generically nondifferentiable the objective function is constructed via state space methods and several properties of the optimal constrained controller are discussed together with a robust algorithm for computation it is shown how satisfying the constraints over a finite horizon can imply overall good behavior

balanced realization and model order reduction for nonlinear systems based on singular value analysis this paper discusses balanced realization and model order reduction for both continuous time and discrete time general nonlinear systems based on singular value analysis of the corresponding hankel operators singular value analysis clarifies the gain structure of a given nonlinear operator here it is proved that singular value analysis of smooth hankel operators defined on hilbert spaces can be characterized by simple equations in terms of their states a balanced realization and model order reduction procedure is derived based on it and several important properties such as stability balanced form hankel norm controllability and observability of the original system are preserved the work improves the earlier results of k fujimoto and j m a scherpen ieee trans automat control 50 2005 pp 2 18 and then continues with new balancing and model reduction results

synthesis of a robust broadband duct anc system using convex programming approach a robust active controller using spatially feedforward structure is proposed for broadband attenuation of noise in ducts to meet the requirements of performance and robust stability in the presence of plant uncertainties an h2 cost function and an h constrain are employed in the synthesis of the controller the design is then converted into a convex programming problem using q parametrization and frequency discretization an optimal controller that satisfies the quadratic cost functions and linear inequality constraints can be found by sequential quadratic programming the optimal controller was implemented via a digital signal processor dsp and verified by experiments experiment results showed that the system attained 16 5 db maximal attenuation and 5 9 db total attenuation in the frequency band 200 600 hz

duals and propagators a canonical formalism for nonlinear equations a new formalism for solving general nonlinear equations is proposed given a nonlinear operator equation a dual operator is constructed canonically similar to the adjoint operator in linear theory by using this dual operator advanced and retarded propagators analogs of the green s functions in linear theory are generated these propagators satisfy the customary reciprocity and semigroup properties and yield the formal solution of the original nonlinear problem it is further shown that these propagators can be obtained either by solving a linear equation that still contains an implicit dependence on the problem s solution or by solving a closed form nonlinear integral equation from which the problem s solution itself is completely eliminated the formalism is canonical in the sense that its applicability is not affected by the particularities of the nonlinear operator boundary conditions and underlying phase space all aspects of applying this formalism to nonlinear problems are illustrated analy

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

effective elastic properties of suspensions of radially symmetric particles abstract in this work we study the exact solution to the single inclusion problem in elastostatics for inclusions which have a spherically symmetric elastic profile we present a systematic scheme by which the so called intrinsic moduli which determine the response of a particle to an external field can be calculated for that purpose a transfer matrix method is developed which enables us to express the intrinsic moduli of the particle in terms of the moduli of its constituent layers this method works in both two and three dimensions in the case of two layers the results of the generalised self consistent scheme of christensen and lo could be easily rederived the calculated intrinsic moduli can be substituted in previously derived mean field expressions thus providing us with expressions for the effective elastic moduli of suspensions of layered spheres or circles

convergence of optimal control problems with an h norm constraint this paper considers the problem of computing a stabilizing controller for a multivariable plant so that the performance objective defined on a closed loop transfer matrix is minimized while the h norm of possibly another transfer matrix is constrained in other words the problem is one of keeping the induced l2 to l2 norm from an input vector to an output vector below some pre specified level e g for guaranteeing robust stability against unstructured perturbations while achieving the best possible performance as measured by a function defined over a closed loop transfer matrix such a problem is referred to as a mixed optimization problem and includes h2 h l1 h and time domain constrained h control as special cases the approach in this paper is to approximate the mixed problem which is infinite dimensional by one involving a finite number of variables the performance achieved by a solution to the finite problem can be arbitrarily close to the optimal performance of the mixed problem moreover given a level of suboptimality the number of variables involved in the optimization can be bounded a priori in terms of the data the theory is illustrated by briefly discussing two examples

nonlinear and semilinear spectrum for asymptotically linear or positively homogeneous operators we prove some properties for the nonlinear spectrum f and semilinear spectrum l n when f and n are asymptotically linear or positively homogeneous thus close to a linear operator the results generalize a previous result which required n to be a linear operator and l to be the identity map applying the theorems we prove a result on the existence of a positive eigenvalue and eigenvector for a compact positive operator examples of applications to the study of a three point boundary value problem are also given

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

cluster expansion for the dielectric constant of a polarizable suspension we derive a cluster expansion for the electric susceptibility kernel of a dielectric suspension of spherically symmetric inclusions in a uniform background this also leads to a cluster expansion for the effective dielectric constant it is shown that the cluster integrals of any order are absolutely convergent so that the dielectric constant is well defined and independent of the shape of the sample in the limit of a large system we compare with virial expansions derived earlier in statistical mechanics for the dielectric constant of a nonpolar gas in these expansions the virial coefficients are given by integrals which are only conditionally convergent

nonlinear input normal realizations based on the differential eigenstructure of hankel operators this paper investigates the differential eigenstructure of hankel operators for nonlinear systems first it is proven that the variational system and the hamiltonian extension with extended input and output spaces can be interpreted as the ga spl circ teaux differential and its adjoint of a dynamical input output system respectively second the ga spl circ teaux differential is utilized to clarify the main result the differential eigenstructure of the nonlinear hankel operator which is closely related to the hankel norm of the original system third a new characterization of the nonlinear extension of hankel singular values are given based on the differential eigenstructure finally a balancing procedure to obtain a new input normal output diagonal realization is derived the results in this paper thus provide new insights to the realization and balancing theory for nonlinear systems

envelope constrained h sub spl infin filter design an lmi optimization approach in this correspondence we solve the envelope constrained h sub spl infin filter design problem by minimizing the h sub spl infin norm of the filtering error transfer function subject to the constraint that the filter output is contained in a prescribed envelope the filter design is transformed into a standard linear matrix inequality lmi optimization problem

non causal fliess operators and their shuffle algebra fliess operators as a class of non linear operators have been well studied in several respects they have a well developed realization theory and convenient representations in terms of directed infinite products of exponential lie series their interconnection as subsystems has been studied as has their relationship to rational systems they find applications in such diverse areas as discretization methods for controls systems optimal control neural network analysis and the numerical solution of stochastic differential equations one issue concerning fliess operators however that has received little attention is their possible generalization to the non causal case examples of such operators appear implicitly in the literature addressing hilbert adjoints of causal non linear operators and system inversion for the purpose of output tracking but a general systematic treatment of the subject has not appeared in this paper a non causal extension of a fliess operator is developed with the primary focus being on local convergence continuity the associated shuffle algebra and computing adjoint operators

balanced canonical forms for minimal systems a normalized coprime factor approach abstract canonical forms are derived for the set of minimal systems of given order from a canonical form for a class of coinner transfer functions one of these canonical forms is in terms of so called riccati balanced coordinates the application of this work to model reduction is discussed

hamiltonian realizations of nonlinear adjoint operators this paper addresses the issue of state space realizations for nonlinear adjoint operators in particular the relationships between nonlinear hilbert adjoint operators hamiltonian extensions and port controlled hamiltonian systems are established then characterizations of the adjoints of controllability observability and hankel operators are derived from this analysis the state space realizations of such adjoint operators provide new insights on singular value analysis and duality issues in nonlinear control systems theory finally a duality between the controllability and observability energy functions is proved

on non linear systems additive feedback and rationality abstract the problem of stabilizing a nonlinear system by the application of additive output feedback is considered it is shown that if an infective recursive system can be so stabilized then it must be rational namely it must be possible to express as a ratio y pq l of two stable and recursive systems p and q

on the nonuniqueness of singular value functions and balanced nonlinear realizations the notion of balanced realizations for nonlinear state space model reduction was first introduced in 1993 analogous to the linear case the so called singular value functions of a system describe the relative importance of each state component from an input output point of view in this paper it is shown that the usual procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case specifically it appears that the singular value functions as currently defined are dependent on a particular factorization of the observability function it is shown by example that in a fixed coordinate frame this factorization is not unique and thus other distinct definitions for the singular value functions and balanced realizations are possible one method relating singular value functions from different factorizations is presented

effective elasticity of a solid suspension of spheres ii effective moduli in lowest multipole approximation abstract we evaluate explicitly through terms of second order in volume fraction the effective elastic moduli derived formally in a preceeding article for a random suspension of spherically symmetric inclusions in a background matrix by interpreting the earlier formulae in terms of induced stresses on an inclusion we reduce the calculation to the treatment of a one inclusion and a two inclusion problem only to treat the one inclusion problem we derive a faxen type theorem for the total stress induced on the inclusion by an incident displacement field for the two inclusion problem we use a method of reflections whereby we obtain the contribution of lowest order in the separation of inclusion centres by introducing a point stress model of the inclusions we sum a subset of reflections to all orders in inclusion separation we find that in the point stress model the effective moduli are given in terms of only two of the scattering coefficients introduced elsewhere for the one inclusion problem for the special case of uniform inclusions we give numerical results for the effective moduli which are compared with exact upper and lower bounds

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

h2 and h approximations for eigenvalues vector functions of transfer matrices h2 and h weighting sequence approximations to rational and irrational functions trefethen 1981 helton et al 1989 are pertinent to characteristic locus design the most effective technique combines inverse discrete fourier transforms with what amount to h2 approximations in this paper we discuss the theoretical background and develop some new h2 and h techniques the resulting algorithms can be applied to general rational irrational function and vector function approximation problems they are shown to be particularly effective in the approximation of eigenvalue vector functions of transfer function matrices the derived techniques provide some very useful tools for the characteristic locus design method

h sub spl infin optimization with time domain constraints standard h sub spl infin optimization cannot handle specifications or constraints on the time response of a closed loop system exactly in this paper the problem of h sub spl infin optimization subject to time domain constraints over a finite horizon is considered more specifically given a set of fixed inputs w sup i it is required to find a controller such that a closed loop transfer matrix has an h sub spl infin norm less than one and the time responses y sup i to the signals w sup i belong to some prespecified sets spl omega sup i first the one block constrained h sub spl infin optimal control problem is reduced to a finite dimensional convex minimization problem and a standard h sub spl infin optimization problem then the general four block h sub spl infin optimal control problem is solved by reduction to the one block case the objective function is constructed via state space methods and some properties of h sub spl infin optimal constrained controllers are given it is shown how satisfaction of the constraints over a finite horizon can imply good behavior overall an efficient computational procedure based on the ellipsoid algorithm is also discussed

hankel singular value functions from schmidt pairs for nonlinear input output systems this paper presents three results in singular value analysis of hankel operators for nonlinear input output systems first the notion of a schmidt pair is defined for a nonlinear hankel operator this makes it possible to define a hankel singular value function from a purely input output point of view and without introducing a state space setting however if a state space realization is known to exist then a set of sufficient conditions is given for the existence of a schmidt pair and the state space provides a convenient representation of the corresponding singular value function finally it is shown that in a specific coordinate frame it is possible to relate this new singular value function definition to the original state space notion used to describe nonlinear balanced realizations

approximation of delay systems a case study abstract the approximation of delay systems is studied a numerically well behaved method for computing hankel optima rational approximations for delay systems is presented based on certain properties of hankel matrices and the theory of laguerre polynomials the cf method of trefethen and certain pade approximations of delay systems are also considered the importance of a certain wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the l norm and the hankel norm is shown completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions a case study and numerical comparison is presented for the approximation of the parametric family of first order stable delay systems g s exp s ts 1 numerical experience indicates that l optimized cf approximations based on short truncated maclaurin series give in general somewhat smaller l approximation errors than hankel optim

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

a polynomial approach to minimax frequency domain optimization of multivariable feedback systems the properties of a feedback system where the plant has rational transfer matrix h and the compensator has transfer matrix g can be characterized through the system functions s i hg and t g i hg 1 good disturbance attenuation robustness limited bandwidth and compensator roll off may be obtained by minimizing a criterion of the form z where z v s t w1 wls t w2 w2t v with respect to the compensator transfer matrix g here v w1 and w2 are suitable rational weighting matrices the solution of the problem can be reduced to a pair of matrix polynomial equations since the optimal solution is highly non unique special solutions with additional optimality properties are considered as well the paper includes a discussion of the numerical solution of the polynomial equations and of the question how to choose the weighting matrices to ensure required feedback system properties an example illustrates the results

h control with time domain constraints the infinite horizon case abstract it has been shown recently that time domain constraints can be incorporated into h optimal control problems when the constraints are enforced over a finite horizon i e over a finite number of sample instants this is done in sideris and rotstein 1993 rotstein and sideris to appear where the finite horizon problem is reduced into a finite dimensional convex but generically nondifferentiable optimization program in this paper the infinite horizon case is addressed it is shown that when the horizon length goes to infinity the solutions to the finite horizon problems converge to a solution of the infinite horizon problem moreover a simple modification of the finite horizon problems guarantees convergence to a solution that is easily approximated by a finite dimensional transfer function

normalized coprime factorizations and balancing for unstable nonlinear systems in this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state space point of view in order to do so we introduce the notion of inner and co inner nonlinear systems secondly we deal with balancing for unstable nonlinear systems for linear systems there are several ways to approach this problem and we generalize the ideas of balancing the normalized coprime factorization and of lqg balancing to the nonlinear case lqg balancing can be seen as measuring the influence of a state component on a certain future and past energy function we extend this interpretation to the nonlinear case furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system we give the relation between these two methods of balancing which is similar to the relation in the linear case

on a class of approximate solutions to multi objective h 2 h problems a class of matrix function optimization problems is considered in which the cost functional is the h 2 norm of an affine function of the decision variables and there are several constraints involving h 2 and h x norms of other affine functions such problems arise naturally in feedback control theory as closed loop transfer functions are affine functions of the socalled youla parameter and control objectives can often be expressed as prescribed upper bounds on the h 2 and h x norms of these functions a procedure for obtaining approximate solutions to such problems is described which involves solving a sequence of h 2 problems with a single weighted h 2 constraint this procedure hinges upon a weighting updating scheme in which roughly speaking a multiplication operator is applied to an additively corrected version of the weighting function at each step with both the operator and the additive correction being defined on the basis of the solution of the current auxiliary quadratic problem it is shown

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

frequency response algorithms for h sub infinity optimization with time domain constraints a very broad framework for control system design is considered that encompasses frequency response methodologies for h sub infinity optimization that solve various aspects of the control design problem and that are less well known that state space methods the focus is on linear programming lawson s algorithm and trefethen s algorithm a modified lawson s algorithm is proposed and related to trefethan s method the modified algorithm is shown to be significantly faster than linear programming and lawson s algorithm it is also shown how to extend the modified lawson s algorithm so as to handle time domain constraints in addition to frequency domain specifications which distinguishes it from other h sub infinity optimization methods some steps are taken toward dealing with time domain constraints within an h sub infinity optimization framework