on the structure of h sup infinity control systems and related extensions an existence condition of a h sup infinity controller which achieves a prescribed norm bound of the closed loop transfer function is derived in the frequency domain based on a generalization of the notion of j lossless systems this condition is regarded as a frequency domain representation of the well known existence condition in the state space represented in terms of the two algebraic riccati equations a notion of j orthogonal complement which is introduced as a generalization of the usual orthogonal complement play an important role in clarifying the fundamental frequency domain structure of the model matching problem and simplifying the computation of controllers the results are extended to the nonstandard case where the direct feedthrough from the input to the error or from the exogenous signal to the output is no longer of full rank it is shown that the proper controller achieving the prescribed norm bound exists even in this case in nonstandard cases the controller order can be smaller than the plant order

poncelet s theorem paraorthogonal polynomials and the numerical range of compressed multiplication operators abstract there has been considerable recent literature connecting poncelet s theorem to ellipses blaschke products and numerical ranges summarized for example in the recent book 16 we show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle opuc and in turn can provide new insights to the theory of opuc

adaptive observers design for a class of linear descriptor systems in this paper we propose a new approach to an adaptive observer design for a class of multi input multi output linear descriptor systems global exponential convergence for joint state parameter estimation is established for noise free systems for noise corrupted descriptor systems it is proved that the state and parameter estimation errors are bounded and converge in the mean to zero when the noises are bounded and have zero means potential applications of the presented adaptive observer are online system identification fault detection and adaptive control of descriptor systems numerical example is presented to illustrate the performance of the proposed adaptive observer

adaptive observer for multiple input multiple output mimo linear time varying systems for joint state parameter estimation in linear time varying ltv multiple input multiple output mimo systems an approach to the design of adaptive observers is proposed it is conceptually simple and computationally efficient its global exponential convergence is established for noise free systems in the presence of noises it is proved that the estimation errors are bounded and converge in the mean to zero if the noises are bounded and have zero means potential applications are fault detection and isolation and adaptive control

identification with nonparametric uncertainty an identification technique that is robust to nonparametric uncertainty i e model mismatch is presented the identifier produces either a parameter set estimate or a frequency response set estimate the estimates result from the inclusion of a model of the nonparametric uncertainty in the plant model the frequency response set estimate is shown to always contain the frequency response of the plant as long as certain modeling conditions are met this type of identifier would be useful in applications such as control where a property such as stability or performance level must be achieved in the face of low order modeling and its associated nonparametric uncertainty

adaptive observers with arbitrary exponential rate of convergence for nonlinear systems concerns the same class of linearly parameterized single output nonlinear systems that the authors previously identified in 1992 in terms of differential geometric conditions when persistency of excitation conditions are satisfied the adaptive observers presented in this note guarantee arbitrarily fast exponential convergence both of parameter and state estimates to actual parameters and states while previous adaptive observers guarantee only exponential not arbitrarily fast convergence this extends earlier results for linear systems

simultaneous identification of nominal model parametric uncertainty and unstructured uncertainty for robust control abstract the problem attacked in this paper is to obtain the smallest model set which is described by nominal model parametric uncertainty bound unstructured uncertainty bound and consistent with a number of noise free input output data under the condition that the denominator of the nominal model is prescribed for the compatibility with the available robust control theory the unstructured uncertainty is measured by h norm while the parametric uncertainty is by parameter variation interval it is shown that the unstructured uncertainty bound will generally increase if the nominal model error uncertainty is completely regarded as unstructured moreover the identification problem is reduced to a convex optimization problem which is computationally tractable

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

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

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

a class of algorithms for identification in h abs traet in this paper a class of algorithms for the problem of system identification in are investigated these algorithms are characterized by a two stage structure and involve a class of window functions some conditions in terms of properties of the window functions are derived which guarantee robust convergence of the algorithms identification errors are analyzed for several common window functions this leads to some insights into the trade off between the error induced by approximation and that due to noise

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

stable adaptive observers for nonlinear time varying systems an adaptive observer identifier for single input single output observable nonlinear systems that can be transformed to a certain observable canonical form is described sufficient conditions for stability of this observer are provided these conditions are in terms of the structure of the system and canonical form the boundedness of the parameter variations and the sufficient richness of some signals the scope of the canonical form and the use of the observer identifier is motivated by the presentation of applications to time invariant bilinear systems nonlinear systems in phase variable form a biotechnological process and a robot manipulator in each case the specific stability conditions are presented

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

adaptive quantification of model uncertainties by rational approximation an adaptive rational approximation approach for quantifying the effect of uncertainty is presented it is shown that a tight frequency upper bound on the uncertainty is obtainable by adaptive rational approximation the identifier which consists of the plant identifier and the uncertainty identifier is discussed the plant identifier gives a nominal model which has lower complexity than that of the true plant errors between the estimated nominal model and the true plant are characterized by a sequence of rational functions which converges to the accurate upper bound of the uncertainty in the frequency domain since approximation and identification are grouped together the whole procedure is completely automatic this allows robust control and adaptive control to be combined

time domain identification for robust control abstract in this paper we are concerned with a problem of robust control oriented system identification in the time domain based on the well known schur takagi aak theorem we propose a linear algorithm to obtain the nominal model of the plant to be identified and the minimal bound of the uncertainty of the nominal model error which is measured by h norm it is also shown that in the model set defined by the nominal model and the uncertainty bound there exists at least one model which matches the prescribed input output data given in the time domain

design of observers for descriptor systems in this note simple and straightforward methods to design full and reduced order observers for linear time invariant descriptor systems are presented the approach for the reduced order observer design is based on the generalized sylvester equation sufficient conditions for the existence of the observers are given an illustrative example is included

adaptive observer for continuous descriptor systems we design an adaptive observer for a stable single input and single output continuous descriptor generalized state space or singular system under the assumption that the descriptor system is observable it is transformed into what we call descriptor observer canonical form this form also allows the descriptor variable to be represented by a linear combination of the 2n unknown parameters and the filtered input and output measurement we also show that the estimates converge to the true values when the bounded input satisfies the richness condition under the assumption that the system is controllable

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

unitary interpolants factorization indices and infinite hankel block matrices abstract the existence uniqueness and construction of unitary n n matrix valued functions j j j in wiener like algebras on the circle with prescribed matrix fourier coefficients j j for j 0 are studied in particular if j then such an exists with j if and only if 0 1 where v denotes the infinite block hankel matrix j k v j k 0 1 acting in the sequence space l n 2 one of the main results is that the nonnegative factorization indices of every such are uniquely determined by the given data in terms of the dimensions of the kernels of i v v whereas the negative factorization indices are arbitrary it is also shown that there is a unique such if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for in terms of certain schmidt pairs of 0 are given the results depend upon a fine analysis of the structure of the kernels of i v v and of the one step extension problem of adamjan arov and krein funct anal appl 2 1968 1 18 isometric interpolants for the nonsquare case are also considered

a sliding mode observer for infinitely unobservable descriptor systems in existing work of sliding mode observers smos for descriptor systems a necessary condition is that the system must be infinitely observable this paper presents a scheme that circumvents that condition by reformulating the system as a reduced order system where certain structures in the system matrix are manipulated and certain states are treated as unknown inputs following that an smo is implemented on the reduced order system where state and fault estimation can be achieved necessary and sufficient conditions of this scheme are also presented finally a simulation example shows the effectiveness of the proposed scheme

structure of model uncertainty for a weakly corrupted plant in this paper we investigate the structure of the transfer function set which includes all the transfer functions deduced from the plant available information it is shown that when an upper bound of the plant transfer function s h sup spl infin norm has been supplied and the noise contaminating the time domain identification experiment data is not too significant such a transfer function set can be parameterized by a linear fractional transformation of two transfer function matrices one of them is a fixed transfer function matrix which is completely determined by the plant available information and the noise magnitude the other is a norm bounded structure fixed free transfer function matrix moreover it is shown that the problem of analytically obtaining the fixed complexity nominal model that best approximates this transfer function set is as difficult as the spl mu synthesis problem

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

on adaptive observers for state affine systems in this paper the problem of adaptive observer design for the class of state affine systems is discussed the discussion is based on recent results on adaptive observer with exponential rate of convergence obtained for multi input multi output linear time varying systems on the one hand and the well known kalman like design for state affine systems on the other in particular the relationship between both designs is emphasized showing how they can even be equivalent the interest of such an adaptive observer for state affine systems is illustrated by the example of state and parameter estimation for the lorenz chaotic system the observer performances are illustrated via simulation

remarks on nonlinear adaptive observer design abstract a solution to the problem of state estimation for systems with unknown parameters is to use some on line adaptation of the observer parameters on the basis of various existing results for such adaptive observer designs a unifying adaptive observer form is proposed in this paper this form indeed emphasizes properties allowing some asymptotic state estimation in spite of unknown parameters as well as additional properties which further allow parameter estimation as an example it is shown how an adaptive observer can be designed for a class of state affine systems