frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

low order integral action controller synthesis abstract a simple controller synthesis method is developed for certain classes of linear time invariant multi input multi output plants the number of poles in each entry of these controllers depends on the number of right half plane plant zeros and is independent of the number of poles of the plant to be stabilized furthermore these controllers have integral action so that they achieve asymptotic tracking of step input references with zero steady state error the designed controller s poles and zeros are all in the stable region with the exception of one pole at the origin for the integral action design requirement the freedom available in the design parameters may be used for additional performance objectives although the only goal here is stabilization and tracking of constant references

formulas for hankel singular values and vectors for a class of input delay systems this paper gives explicit formulas for hankel singular values and vectors for a class of multiple input delay systems the class consists of systems whose gramian is reducible to an integral operator with semi separable kernel this structure is exploited in deriving a transcendental equation for singular values

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

frequency weighted model reduction via quasi kalman decomposition a frequency weighted model reduction method based on quasi kalman decomposition is proposed for discrete systems the method is essentially an extension of the technique presented by lu and lee 1985 to the frequency weighted case and can handle both single sided as well as double sided weights two illustrative examples show the effectiveness of the method and results are compared with the method developed by enns 1984

balanced approximation can be optimal it is shown by example that there exist systems whose balanced approximations are optimal specifically for various norms including the hankel norm and the 2 norm it is shown that the approximation error can achieve the minimum possible kevwords balanced approximation model reduction hankel norm finite differences hamiltonian systems let g s c sl a tb a b c be an n state stable minimal system with controllability and observability gramians p and q the hankel singular values of g are ai 2 i 2 pq i 1 2 nandareorderedat a2 a 0 as usual we use the following standard norms of g to measure approximation errors

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

theory of regular m band wavelet bases orthonormal m band wavelet bases have been constructed and applied by several authors this paper makes three main contributions first it generalizes the minimal length k regular 2 band wavelets of daubechies 1988 to the m band case by deriving explicit formulas for k regular m band scaling filters several equivalent characterizations of k regularity are given and their significance explained second two approaches to the construction of the m 1 wavelet filters and associated wavelet bases are described one relies on a state space characterization with a novel technique to obtain the unitary wavelet filters the other uses a factorization approach third this paper gives a set of necessary and sufficient condition on the m band scaling filter for it to generate an orthonormal wavelet basis the conditions are very similar to those obtained by cohen 1990 and lawton 1990 for 2 band wavelets

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

short communication on balancing a cart pole system using t s fuzzy model this paper presents two easily implementable control schemes for balancing a cart pole system using intelligent control tools the proposed schemes use the takagi sugeno t s fuzzy model of a nonlinear system the concept of network inversion is used to design the controller for such a system in one of the control schemes the control input necessary to achieve a desired output is computed directly through iterative inversion of the fuzzy model in the other scheme a parallel distributed compensator form is chosen for the controller and the parameters or the feedback gains of the controller are updated through network inversion the updating laws are derived using both continuous and discrete time lyapunov approaches the inversion based parameter update avoids the need of any sufficient condition or prerequisite constraint as in existing t s fuzzy model based control designs like lmi techniques and robust control techniques the proposed controllers have been implemented on the cart pole system both in simulation and real time a comparative performance analysis for different control algorithms is presented the performance of the proposed controller is compared with the well established lqr control in real time the proposed control algorithms work for a wide range of operating regions compared to lqr and are more robust in the sense that they can tolerate an output disturbance of higher magnitude

identification and model reduction of time varying discrete time systems abstract linear discrete time variable systems are considered and an important class of uniform realizations is defined the necessary and sufficient conditions for a pulse response to be uniformly realizable are obtained two model reduction schemes one via balanced realizations and the other via hankel matrix are proposed the latter approach can be considered as an identification algorithm for the two dimensional pulse response sequence h k i it is shown that the two approaches yield an identical reduced model which is always asymptotically stable

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

approximation of fir by iir digital filters an algorithm based on balanced model reduction an algorithm for the approximation of finite impulse response fir filters by infinite impulse response iir filters is presented the algorithm is based on a concept of balanced model reduction the matrix inversions normally associated with this procedure are notoriously error prone due to ill conditioning of the special matrix forms required this difficulty is circumvented here by directly formulating a reduced state space system description which is input output equivalent to the system that would more conventionally be obtained following the explicit step of constructing an interim balanced realization examples of fir by iir filter approximations are included

on hankel singular values and reflected zeros of linear dynamical systems this note discusses a relationship between the hankel singular values and reflected zeros of linear systems our main result proves that the hankel singular values of a linear continuous time system increase decrease pointwise when one or more zeros of the transfer function are reflected with respect to the imaginary axis that is move from the left right half to the right left half of the complex plane we also derive a similar result for linear discrete time systems

on the use of modal derivatives for nonlinear model order reduction summary modal derivative is an approach to compute a reduced basis for model order reduction of large scale nonlinear systems that typically stem from the discretization of partial differential equations in this way a complex nonlinear simulation model can be integrated into an optimization problem or the design of a controller based on the resulting small scale state space model we investigate the approximation properties of modal derivatives analytically and thus lay a theoretical foundation of their use in model order reduction which has been missing so far concentrating on the application field of structural mechanics and structural dynamics we show that the concept of modal derivatives can also be applied as nonlinear extension of the craig bampton family of methods for substructuring we furthermore generalize the approach from a pure projection scheme to a novel reduced order modeling method that replaces all nonlinear terms by quadratic expressions in the reduced state variables this complexity reduction leads to a frequency preserving nonlinear quadratic state space model numerical examples with carefully chosen nonlinear model problems and three dimensional nonlinear elasticity confirm the analytical properties of the modal derivative reduction and show the potential of the proposed novel complexity reduction methods along with the current limitations copyright 2016 john wiley sons ltd

parametrization and optimization of reduced order filters this paper deals with the design of reduced order filters for linear signal models to ensure that the transfer function from the noise to the filtering error is not only stable but also has a minimum h sub 2 norm one of the major results in the paper is the discovery and parametrization of a set of filters of fixed order which lead to stable filtering error transfer functions such a parametrization is given in terms of an arbitrary orthogonal projection matrix and a given full order filter as a consequence the problem of minimizing the h sub 2 norm of the filtering error transfer function over the set of reduced order filters can be formulated as an equivalent unconstrained parametric optimization problem over a compact manifold a gradient flow based algorithm is proposed to find an optimal solution to the problem nice properties of the algorithm including the convergence property are established theoretically as well as demonstrated numerically with an example

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

on the structure of balanced and other principal representations of siso systems a new matrix w co which can be considered as a cross gramian matrix which contains information about both controllability and observability is defined for single input single output linear systems using this matrix the structural properties of linear systems are studied in the context of principal component analysis the matrix w co can be used in obtaining balanced and other principal representations without computation of the controllability and the observability gramians the importance of this matrix in model order reduction is highlighted

ltv model reduction with upper error bounds a new approach for computing upper error bounds for reduced order models of linear time varying systems is presented it is based on a transformation technique of the hankel singular values using positive real odd incremented functions by applying such time varying functions the singular values to be removed can be forced to become equal and constant so that they can be reduced two variations of this method are proposed one for finite time horizons and the other for infinite time problems including periodic systems

balanced realizations of discrete time stable all pass systems and the tangential schur algorithm in this paper the connections are investigated between two different approaches towards the parametrization of multivariable stable all pass systems in discrete time the first approach involves the tangential schur algorithm which employs linear fractional transformations it stems from the theory of reproducing kernel hilbert spaces and enables the direct construction of overlapping local parametrizations using schur parameters and interpolation points the second approach proceeds in terms of state space realizations in the scalar case a balanced canonical form exists that can also be parametrized by schur parameters this canonical form can be constructed recursively using unitary matrix operations here this procedure is generalized to the multivariable case by establishing the connections with the first approach it gives rise to balanced realizations and overlapping canonical forms directly in terms of the parameters used in the tangential schur algorithm

asynchronous optimized schwarz methods with and without overlap an asynchronous version of the optimized schwarz method for the solution of differential equations on a parallel computational environment is studied in a one way subdivision of the computational domain with and without overlap the method is shown to converge when the optimal artificial interface conditions are used convergence is also proved for the laplacian operator under very mild conditions on the size of the subdomains when approximate non optimal interface conditions are utilized numerical results are presented on a large three dimensional problem on modern parallel clusters and supercomputers illustrating the efficiency of the asynchronous approach

improved design method for nearly linear phase iir filters using constrained optimization a new optimization method for the design of nearly linear phase iir digital filters that satisfy prescribed specifications is proposed the group delay deviation is minimized under the constraint that the passband ripple and stopband attenuation are within the prescribed specifications and either a prescribed or an optimized group delay can be achieved by representing the filter in terms of a cascade of second order sections a non restrictive stability constraint characterized by a set of linear inequality constraints can be incorporated in the optimization algorithm an additional feature of the method which is very useful in certain applications is that it provides the capability of constraining the maximum gain in transition bands to be below a prescribed level experimental results show that filters designed using the proposed method have much lower group delay deviation for the same passband ripple and stopband attenuation when compared with corresponding filters designed with several state of the art competing methods

an output error bound for time limited balanced truncation abstract model order reduction mor is often applied to spatially discretized partial differential equations to reduce their order and hence decrease computational complexity a reduced system can be obtained e g by time limited balanced truncation a method that aims to construct an accurate reduced order model on a given finite time interval 0 t this particular balancing related mor technique is studied in this paper an l t 2 error bound based on the truncated time limited singular values is proved and is the main result of this paper the derived error bound converges as t to the well known h error bound of unrestricted balanced truncation a scheme that is used to construct a good reduced system on the entire time line the techniques within the proofs of this paper can also be applied to unrestricted balanced truncation so that a relatively short time domain proof of the h error bound is found here