optimality approximation and complexity in set membership h sub spl infin identification investigates the set membership identification of time invariant discrete time exponentially stable possibly infinite dimensional linear systems from time or frequency domain data corrupted by deterministic noise the aim is to deliver not a single model but a set of models whose size in h sub spl infin norm measures the uncertainty in the identification the main focus of the note is on the optimality properties for finite data and on the tradeoff between optimality and complexity of approximated low order model sets a method is given for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency such approximations allow one to compute within any desired accuracy the identification error of any identified model and to evaluate an optimal model at any given number of frequencies by suitably approximating these values model sets with nominal models in rh sub spl infin are then derived whose order is selected by trading off between model set complexity and identification accuracy degradation this degradation is evaluated by computing the optimality level defined as the ratio between the reduced model identification error and the optimal one a numerical example demonstrates the effectiveness of the presented results

generalized l sub 2 synthesis introduces a general framework for control design which can be used to solve several open problems in robust and optimal control these include systems subject to independently norm bounded disturbances robust stability for systems subject to element by element bounded structured uncertainty certain classes of robust performance problems and systems subject to deterministic white noise disturbances this is achieved by combining concepts such as h infinity optimization linear matrix inequalities and integral quadratic constraints necessary and sufficient conditions for the general problem to have a solution are in terms of a computationally attractive finite dimensional linear matrix inequality

the robust h sub 2 control problem a worst case design the worst case effect of a disturbance system on the h sub 2 norm of the system is analyzed an explicit expression is given for the worst case h sub 2 norm when the disturbance system is allowed to vary over all nonlinear time varying and possibly noncausal systems with bounded l sub 2 induced operator norm an upper bound for this measure which is equal to the worst case h sub 2 norm if the exogeneous input is scalar is defined some further analysis of this upper bound is done and a method to design controllers which minimize this upper bound over all robustly stabilizing controllers is given the latter is done by relating this upper bound to a parameterized version of the auxiliary cost function studied in the literature

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

theory of one tape linear time turing machines a theory of one tape two way one head off line linear time turing machines is essentially different from its polynomial time counterpart since these machines are closely related to finite state automata this paper discusses structural complexity issues of one tape turing machines of various types deterministic nondeterministic reversible alternating probabilistic counting and quantum turing machines that halt in linear time where the running time of a machine is defined as the length of any longest computation path we explore structural properties of one tape linear time turing machines and clarify how the machines resources affect their computational patterns and power

on the time complexity of worst case system identification in this paper we treat a general worst case system identification problem this problem is worst case with respect to both noise and system modeling uncertainty we consider this problem under various a priori information structures we determine bounds on the minimum duration identification experiment that must be run to identify the plant to within a specified guaranteed worst case error bound our results are algorithm independent we show that this minimum duration is prohibitively long based on our results we suggest that worst case with respect to noise system identification requires unrealistic amounts of experimental data

a set based approach for white noise modeling this paper provides a new framework for analyzing white noise disturbances in linear systems rather than the usual stochastic approach noise signals are described as elements in sets and their effect is analyzed from a worst case perspective the paper studies how these sets must be chosen to have adequate properties for system response in the worst case statistics consistent with the stochastic point of view and simple descriptions that allow for tractable worst case analysis the method is demonstrated by considering its implications in two problems rejection of white noise signals in the presence of system uncertainty and worst case system identification

on the control of a rigid body motion affected by stochastic white gaussian noises the non linear equations of motion of a rigid body affected by white gaussian noises are used to study the stabilization of the rotational motion with the help of internal moving masses the control laws ensuring the tendency of the stochastic motion to the unperturbed motion and the programmed motion are determined as non linear functions of the variables describe the stochastic and programmed motions the programmed control law ensures the asymptotic stability with probability is obtained an estimation of the angular velocity of the body tending to the programmed angular velocity is found

a network that learns strassen multiplication we study neural networks whose only non linear components are multipliers to test a new training rule in a context where the precise representation of data is paramount these networks are challenged to discover the rules of matrix multiplication given many examples by limiting the number of multipliers the network is forced to discover the strassen multiplication rules this is the mathematical equivalent of finding low rank decompositions of the n n matrix multiplication tensor mn we train these networks with the conservative learning rule which makes minimal changes to the weights so as to give the correct output for each input at the time the input output pair is received conservative learning needs a few thousand examples to find the rank 7 decomposition of m2 and 105 for the rank 23 decomposition of m3 the lowest known high precision is critical especially for m3 to discriminate between true decompositions and border approximations

a nonlinear functional approach to lft model validation model validation provides a useful means of assessing the ability of a model to account for a specific experimental observation and has application to modeling identification and fault detection in this paper we consider a new approach to the model validation problem by deploying quadratic functionals and more generally nonlinear functionals to specify noise and dynamical perturbation sets specifically we consider a general linear fractional transformation framework for the model structure and use constraints involving nonlinear functional inequalities to specify model non linearities and unknown perturbations and characteristics of noise and disturbance signals sufficient conditions for invalidation of such models are provided in terms of semidefinite programming problems

optimal asymptotic identification under bounded disturbances the intrinsic limitation of worst case identification of linear time invariant systems using data corrupted by bounded disturbances when the unknown plant is known to belong to a given model set is studied this is done by analyzing the optimal worst case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm it is shown that under some topological conditions on the model set there is an identification algorithm which is asymptotically optimal for any input and the optimal asymptotic error is characterized as a function of the inputs these results which hold for any error metric and disturbance norm are applied to three specific identification problems identification of stable systems in the l sub 1 norm identification of stable rational systems in the h sub infinity norm and identification of unstable rational systems in the gap metric for each of these problems the general characterization of optimal asymptotic error is used to find near optimal inputs to minimize the error

on the structure of logspace probabilistic complexity classes we investigate hierarchical properties and logspace reductions of languages recognized by logspace probabilistic turing machines arthur merlin games and games against nature with logspace probabilistic verifiers each logspace complexity class is decomposed into a hierarchy based on corresponding two way multihead finite state automata and we eventually prove the separation of the hierarchy levels even for languages over a single letter alphabet furthermore we show efficient reductions of each logspace complexity class to or between low levels of its corresponding hierarchy we find probabilistic and probabilistic plus nondeterministic variants of savitch s maze threading problem which are logspace complete for pl the class of languages recognized by logspace probabilistic turing machines and respectively ppp the class of languages recognized by polynomial time deterministic turing machines and which can be recognized by one way non sensing two head or one way one head one counter finite state automata with probabilistic and both probabilistic and nondeterministic states respectively

brief is set modeling of white noise a good tool for robust h2 analysis recently a new deterministic characterization of the h 2 norm has been proposed using a new norm w h based on approximate set membership modeling of white noise the main result shows that under mild conditions for a fixed system the gap between the h 2 and w h norms can be made arbitrarily small motivated by these results it has been argued that the w h norm provides a useful tool for analyzing robust h 2 controllers specially since in this context lmi based necessary and sufficient conditions for robust performance are available unfortunately as we show here with an example involving a very simple plant the worst case w h m norm can be conservative by at least a factor of m where m denotes the dimension of the exogenous signal for the original robust h 2 problem moreover the same example shows that competing state space based bounds also exhibit a similar degree of conservatism thus at this point the problem of finding non conservative bounds on the worst h 2 norm under lti or slowly varying ltv perturbations still remains open

a counterexample in power signals space the present author points out that in the paper by vidyasagar 1993 on nonlinear system analysis some of the results obtained are false the purpose of this paper is to give a counterexample it is shown that the set of power signals is not a vector space

the border rank of the multiplication of 2 2 matrices is seven one of the leading problems of algebraic complexity theory is matrix multiplication the naive multiplication of two n n matrices uses n multiplications in 1969 strassen 20 presented an explicit algorithm for multiplying 2 2 matrices using seven multiplications in the opposite direction hopcroft and kerr 12 and winograd 22 proved independently that there is no algorithm for multiplying 2 2 matrices using only six multiplications the precise number of multiplications needed to execute matrix multiplication or any given bilinear map is called the rank of the bilinear map a related problem is to determine the border rank of matrix multiplication or any given bilinear map first introduced in 6 5 roughly speaking some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small good perturbations of the map these perturbed maps can give rise to fast exact algorithms for matrix multiplication see 7 the border rank made appearances in the literature in the 1980s and early 1990s see e g 6 5 19 8 15 2 10 11 4 3 17 16 18 1 9 21 but to our knowledge there has not been much progress on the question since then more precisely for any complex projective variety x cp pv and point p pv define the x rank of p to be the smallest number r such that p is in the linear span of r points of x define r x the r th secant variety of x to be the zariski closure of the set of points of x rank r and define the x border rank of p to be the smallest r such that p r x the terminology is motivated by the case x seg pa 1 pb 1 p c c the segre variety of rank one matrices then the x rank of a matrix is just its usual rank let a b c be vector spaces and let f a b c be a bilinear map i e an element of a b c let x seg pa pb pc p a b c denote the segre variety of decomposable tensors in a b c the border rank of a bilinear map is its x border rank while for the segre product of two projective spaces border rank coincides with rank here they can be quite different in this paper we prove the theorem stated in the title let mmult c c c denote the matrix multiplication operator for 2 2 matrices strassen 18 showed that mmult 5 p p p our method of proof is to decompose

robust stability with time varying structured uncertainty considers the problem of assessing robust stability in the presence of block diagonally structured time varying dynamic uncertainty it is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition the notion of stability here is finite gain stability over finite energy signals in sharp contrast to the case of time invariant dynamic uncertainty this result is not limited by the number uncertainty blocks these results parallel previous results regarding finite gain stability over persistent bounded signals

convex methods for robust h sub 2 analysis of continuous time systems treats the robust h sub 2 performance question of evaluating the rejection of continuous time white noise in the worst case over structured uncertainty in the system a frequency domain convex condition for robust h sub 2 analysis is presented with analogous properties as in the discrete time case in particular necessary and sufficient results are obtained introducing a continuous time version of the methodology of set descriptions of white noise in addition a state space test in terms of linear matrix inequalities is developed for the robust h sub 2 problem in the case of constant uncertainty scalings which apply to nonlinear or time varying uncertainty thus the problem is rendered finite dimensional in the same situation in which robust h sub spl infin analysis is finite dimensional

partial and total matrix multiplication in 1979 considerable progress was made in estimating the complexity of matrix multiplication here the new techniques and recent results are presented based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication some of the entries of the matrices may be zero can efficiently be utilized to perform multiplication of large total matrices by combining pan s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of n times n matrices over arbitrary fields is possible in time o n beta where beta is a bit smaller than 3 ln 52 ln 110 approx 2 522

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

space efficient deterministic simulation of probabilistic automata given a description of a probabilistic automaton one head probabilistic finite state automaton or probabilistic turing machine and an input string x of length n we ask how much space does a deterministic turing machine need in order to decide the acceptance of the input string by that automaton the question is interesting even in the case of one head one way probabilistic finite state automata 1pfa s we call rational stochastic languages ss rat the class of languages recognized by 1pfa s whose transition probabilities and cutpoints i e recognition thresholds are rational numbers the class ss rat contains context sensitive languages that are not context free but on the other hand there are context free languages not included in ss rat our main results are as follows begin itemize item the proper inclusion of ss rat in zlo which is optimal i e ss rat not subset dspace o log n the previous upper bounds were dspace n obtained by p d dieu in 1972 and linebreak 4 dspace log n log log n obtained by h jung in 1984 item probabilistic turing machines with space bound f n in o log n can be deterministically simulated in space o min c f n log n log n f n log log n where c is a constant depending on the simulated probabilistic turing machine the best previously known simulation uses o log n f n log log n space obtained by h jung in 1984 end itemize to obtain these results we develop and use two space efficient but also parallel time efficient techniques which are of independent interest begin itemize item a technique to compare deterministically using only o log n space large numbers given in terms of their values modulo a sequence of primes p1 p2 cdots pn in o na where a is some arbitrary constant item a technique to compare deterministically a threshold with entries of the inverse of a given banded matrix end itemize

the sample complexity of worst case identification of fir linear systems we consider the problem of identification of linear systems in the presence of measurement noise which is unknown but bounded in magnitude by some 0 we focus on the case of linear systems with a finite impulse response it is known that the optimal identification error is related within a factor of 2 to the diameter of a so called uncertainty set and that the latter diameter is upper bounded by 2 if a sufficiently long identification experiment is performed we establish that for any k 1 the minimal length of an identification experiment that is guaranteed to lead to a diameter bounded by 2k behaves like 2nf 1k when n is large where n is the length of the impulse response and is a positive function known in closed form while the framework is entirely deterministic our results are proved using probabilistic tools

convex and finite dimensional conditions for controller synthesis with dynamic integral constraints in this paper the problem of synthesizing controllers when the allowable disturbances and the cost criterion are defined via a finite number of matrix valued dynamic integral constraints is solved this allows one to design optimal control systems for plants subject to fixed input disturbances such as steps sinusoids and periodic disturbances disturbances of fixed and known spectrum disturbances defined as the set of signals in the kernel of a given operator and combinations thereof to penalize particular frequency components of the error variables in the control design process to penalize the maximum amplitude of an error variable and to consider other general cost criteria in the optimization and to solve a large class of robust control synthesis problems necessary and sufficient conditions for the general problem to have a solution are in terms of a computationally attractive finite dimensional linear matrix inequality

mixed h2 h control in a stochastic framework abstract this paper deals with a mixture of h 2 and h in the following way we have two inputs and one output one input signal is a white noise stochastic process and represents errors e g resulting from measurement noise the other input has a more deterministic character if one has a reference signal e g a step as input one cannot model this as white noise but it fits nicely into this new class of inputs the objective is to minimize the effect of these signals on the output of the system we define a cost function which enables us to combine these two exogenous inputs even though they are structurally different the analysis of this function leads to a standard h riccati equation we motivate this cost function by looking at two theoretical applications the derivation of robust performance bounds and a tracking problem