Spl Lscr Research Articles

Fuzzy H/sub /spl infin// output feedback control design for singularly perturbed systems with pole placement constraints: an LMI approach

This paper examines the problem of designing an H/sub /spl infin// output feedback controller with pole placement constraints for singular perturbed Takagi-Sugeno (TS) fuzzy models. We propose a fuzzy H/sub /spl infin// output feedback controller that not only guarantees the /spl Lscr//sub 2/-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, but also ensures closed-loop poles of each subsystem are in a prespecified linear matrix inequality (LMI) region. In order to alleviate the numerical stiffness caused by the singular perturbation /spl epsiv/, the design technique is formulated in terms of a family of /spl epsiv/-independent linear matrix inequalities. The proposed approach can be applied both standard and nonstandard singularly perturbed nonlinear systems. A numerical example is provided to illustrate the design developed in this paper.

Design of /spl lscr//sub 1/-optimal controllers with flexible disturbance rejection level

This note presents a new design methodology that allows for flexible management of the tradeoff between the ability of a system to attenuate disturbance signals versus its expected worst peak-to-peak amplification. The proposed strategy applies to linear time-invariant systems which are subject to persistent disturbance signals and combines a recently developed quasi-robust linear programming concept with a well-known /spl lscr//sub 1/-optimal controller synthesis approach. The benefit of the resulting technique is demonstrated using an example.

How to fool an unbounded adversary with a short key

The symmetric encryption problem which manifests itself when two parties must securely transmit a message m with a short shared secret key is considered in conjunction with a computationally unbounded adversary. As the adversary is unbounded, any encryption scheme must leak information about m; in particular, the mutual information between m and its ciphertext cannot be zero. Despite this, a family of encryption schemes is presented that guarantee that for any message space in {0,1}/sup n/ with minimum entropy n-/spl lscr/ and for any Boolean function h:{0,1}/sup n/ /spl rarr/ {0,1}, no adversary can predict h(m) from the ciphertext of m with more than 1/n/sup /spl omega/(1)/ advantage; this is achieved with keys of length /spl lscr/+/spl omega/(logn). In general, keys of length /spl lscr/+s yield a bound of 2/sup -/spl Theta/(s)/ on the advantage. These encryption schemes rely on no unproven assumptions and can be implemented efficiently. Applications of this to cryptosystems based on complexity-theoretic assumptions are discussed and, in addition, a simplified proof of a fundamental "elision lemma" of Goldwasser and Micali is provided.

Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.

Constrained quadratic programming techniques for control allocation

The paper considers the objective of optimally specifying redundant control effectors under constraints, a problem commonly referred to as control allocation. The problem is posed as a mixed /spl lscr//sub 2/-norm optimization objective and converted to a quadratic programming formulation. The implementation of an interior-point algorithm is presented. Alternative methods including fixed-point and active set methods are used to evaluate the reliability, accuracy and efficiency of a primal-dual interior-point method. While the computational load of the interior-point method is found to be greater for problems of small size, convergence to the optimal solution is also more uniform and predictable. In addition, the properties of the algorithm scale favorably with problem size.

On minimizing distortion and relative entropy

A common approach for estimating a probability mass function w when given a prior q and moment constraints given by Aw/spl les/b is to minimize the relative entropy between w and q subject to the set of linear constraints. In such cases, the solution w is known to have exponential form. We consider the case in which the linear constraints are noisy, uncertain, infeasible, or otherwise soft. A solution can then be obtained by minimizing both the relative entropy and violation of the constraints Aw/spl les/b. A penalty parameter /spl sigma/ weights the relative importance of these two objectives. We show that this penalty formulation also yields a solution w with exponential form. If the distortion is based on an /spl lscr//sub p/ norm, then the exponential form of w is shown to have exponential decay parameters that are bounded as a function of /spl sigma/. We also state conditions under which the solution w to the penalty formulation will result in zero distortion, so that the moment constraints hold exactly. These properties are useful in choosing penalty parameters, evaluating the impact of chosen penalty parameters, and proving properties about methods that use such penalty formulations.

Decoding by Linear Programming

This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f/spl isin/R/sup n/ from corrupted measurements y=Af+e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the /spl lscr//sub 1/-minimization problem (/spl par/x/spl par//sub /spl lscr/1/:=/spl Sigma//sub i/|x/sub i/|) min(g/spl isin/R/sup n/) /spl par/y - Ag/spl par//sub /spl lscr/1/ provided that the support of the vector of errors is not too large, /spl par/e/spl par//sub /spl lscr/0/:=|{i:e/sub i/ /spl ne/ 0}|/spl les//spl rho//spl middot/m for some /spl rho/>0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work. Finally, underlying the success of /spl lscr//sub 1/ is a crucial property we call the uniform uncertainty principle that we shall describe in detail.

Control-oriented model validation and errors quantification in the /spl lscr//sub 1/ setup

A priori information required for robust synthesis includes a nominal model and a model of uncertainty. The latter is typically in the form of additive exogenous disturbance and plant perturbations with assumed bounds. If these bounds are unknown or too conservative, they have to be estimated from measurement data. In this paper, the problem of errors quantification is considered in the framework of the /spl lscr//sub 1/ optimal robust control theory associated with the /spl lscr//sub /spl infin// signal space. The optimal errors quantification is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion. For model with unstructured uncertainty entering the system in a linear fractional manner, the optimal errors quantification is reduced to quadratic fractional programming. For system under coprime factor perturbations, the optimal errors quantification is reduced to linear fractional programming.

Application of structured total least squares for system identification and model reduction

The following identification problem is considered: Minimize the /spl lscr//sub 2/ norm of the difference between a given time series and an approximating one under the constraint that the approximating time series is a trajectory of a linear time invariant system of a fixed complexity. The complexity is measured by the input dimension and the maximum lag. The question leads to a problem that is known as the global total least squares problem and alternatively can be viewed as maximum likelihood identification in the errors-in-variables setup. Multiple time series and latent variables can be considered in the same setting. Special cases of the problem are autonomous system identification, approximate realization, and finite time optimal /spl lscr//sub 2/ model reduction. The identification problem is related to the structured total least squares problem. This paper presents an efficient software package that implements the theory. The proposed method and software are tested on data sets from the database for the identification of systems DAISY.

A sparse signal reconstruction perspective for source localization with sensor arrays

We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the /spl lscr//sub 1/-norm. A number of recent theoretical results on sparsifying properties of /spl lscr//sub 1/ penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum that exhibits super-resolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a second-order cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations and introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Crame/spl acute/r-Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques, including increased resolution, improved robustness to noise, limitations in data quantity, and correlation of the sources, as well as not requiring an accurate initialization.

VLSI implementation of MIMO detection using the sphere decoding algorithm

Multiple-input multiple-output (MIMO) techniques are a key enabling technology for high-rate wireless communications. This paper discusses two ASIC implementations of MIMO sphere decoders. The first ASIC attains maximum-likelihood performance with an average throughput of 73 Mb/s at a signal-to-noise ratio (SNR) of 20 dB; the second ASIC shows only a negligible bit-error-rate degradation and achieves a throughput of 170 Mb/s at the same SNR. The three key contributing factors to high throughput and low complexity are: depth-first tree traversal with radius reduction, implemented in a one-node-per-cycle architecture, the use of the /spl lscr//sup /spl infin//-instead of /spl lscr//sup 2/-norm, and, finally, the efficient implementation of the enumeration approach recently proposed in . The resulting ASICs currently rank among the fastest reported MIMO detector implementations.

Image reconstruction by linear programming

One way of image denoising is to project a noisy image to the subspace of admissible images derived, for instance, by PCA. However, a major drawback of this method is that all pixels are updated by the projection, even when only a few pixels are corrupted by noise or occlusion. We propose a new method to identify the noisy pixels by l1-norm penalization and to update the identified pixels only. The identification and updating of noisy pixels are formulated as one linear program which can be efficiently solved. In particular, one can apply the upsilon trick to directly specify the fraction of pixels to be reconstructed. Moreover, we extend the linear program to be able to exploit prior knowledge that occlusions often appear in contiguous blocks (e.g., sunglasses on faces). The basic idea is to penalize boundary points and interior points of the occluded area differently. We are also able to show the upsilon property for this extended LP leading to a method which is easy to use. Experimental results demonstrate the power of our approach.

On the Computation of the Linear Complexity and the<tex>$k$</tex>-Error Linear Complexity of Binary Sequences With Period a Power of Two

The linear Games-Chan algorithm for computing the linear complexity c(s) of a binary sequence s of period /spl lscr/=2/sup n/ requires the knowledge of the full sequence, while the quadratic Berlekamp-Massey algorithm requires knowledge of only 2c(s) terms. We show that we can modify the Games-Chan algorithm so that it computes the complexity in linear time knowing only 2c(s) terms. The algorithms of Stamp-Martin and Lauder-Paterson can also be modified, without loss of efficiency, to compute analogs of the k-error linear complexity for finite binary sequences viewed as initial segments of infinite sequences with period a power of two. We also develop an algorithm which, given a constant c and an infinite binary sequence s with period /spl lscr/=2/sup n/, computes the minimum number k of errors (and an associated error sequence) needed over a period of s for bringing the linear complexity of s below c. The algorithm has a time and space bit complexity of O(/spl lscr/). We apply our algorithm to decoding and encoding binary repeated-root cyclic codes of length /spl lscr/ in linear, O(/spl lscr/), time and space. A previous decoding algorithm proposed by Lauder and Paterson has O(/spl lscr/(log/spl lscr/)/sup 2/) complexity.

Unsupervised classification of polarimetric synthetic aperture Radar images using fuzzy clustering and EM clustering

Five clustering techniques are compared by classifying a polarimetric synthetic aperture radar image. The pixels are complex covariance matrices, which are known to have the complex Wishart distribution. Two techniques are fuzzy clustering algorithms based on the standard /spl lscr//sub 1/ and /spl lscr//sub 2/ metrics. Two others are new, combining a robust fuzzy C-means clustering technique with a distance measure based on the Wishart distribution. The fifth clustering technique is an application of the expectation-maximization algorithm assuming the data are Wishart. The clustering algorithms that are based on the Wishart are demonstrably more effective than the clustering algorithms that appeal only to the /spl lscr//sub p/ norms. The results support the conclusion that the pixel model is more important than the clustering mechanism.

Distributed Control of Heterogeneous Systems

This work considers control design for distributed systems, where the controller is to adopt and preserve the distributed spatial structure of the nominal system. The specific feature of this work is that it does not require the underlying system dynamics to be homogeneous (shift invariant) with respect to spatial or temporal variables. Operator theoretic tools for working with these systems are developed, and lead to sufficient convex conditions for analysis and synthesis with respect to the /spl lscr//sub 2/-induced norm.

Almost Security of Cryptographic Boolean Functions

The propagation criterion, PC(/spl lscr/) of order k, is one of the most general cryptographic criteria of secure Boolean functions f. In this paper, we formalize its /spl epsiv/-almost version. The new definition requires that f(X)+f(X+/spl Delta/) is almost uniformly distributed while in the original definition, it must be strictly uniformly distributed. Better parameters are then obtained than the strict PC(/spl lscr/) of order k functions. To construct /spl epsiv/-almost PC(/spl lscr/) of order k functions, we introduce a notion of domain distance.

Sparse Bayesian Learning for Basis Selection

Sparse Bayesian learning (SBL) and specifically relevance vector machines have received much attention in the machine learning literature as a means of achieving parsimonious representations in the context of regression and classification. The methodology relies on a parameterized prior that encourages models with few nonzero weights. In this paper, we adapt SBL to the signal processing problem of basis selection from overcomplete dictionaries, proving several results about the SBL cost function that elucidate its general behavior and provide solid theoretical justification for this application. Specifically, we have shown that SBL retains a desirable property of the /spl lscr//sub 0/-norm diversity measure (i.e., the global minimum is achieved at the maximally sparse solution) while often possessing a more limited constellation of local minima. We have also demonstrated that the local minima that do exist are achieved at sparse solutions. Later, we provide a novel interpretation of SBL that gives us valuable insight into why it is successful in producing sparse representations. Finally, we include simulation studies comparing sparse Bayesian learning with basis pursuit and the more recent FOCal Underdetermined System Solver (FOCUSS) class of basis selection algorithms. These results indicate that our theoretical insights translate directly into improved performance.

Model Invalidation in<tex>$ell_1$</tex>Using Frequency-Domain Data

In this note, we study the problem of invalidating uncertain models with an additive uncertainty. The problem is to check the existence of an uncertainty and a measurement noise which fit to the given model structure and the uncertainty/noise description, as well as the experimental data used for invalidation. We consider a mixed setting in which the uncertainty is characterized in time domain by the /spl lscr//sub 1/ induced system norm, while the available data are frequency response samples of the system. We show that this problem, which by formulation poses an infinite-dimensional primal optimization problem, can be solved in a dual, finite-dimensional space with finitely many constraints.

A Network Information Theory for Wireless Communication: Scaling Laws and Optimal Operation

How much information can be carried over a wireless network with a multiplicity of nodes, and how should the nodes cooperate to transfer information? To study these questions, we formulate a model of wireless networks that particularly takes into account the distances between nodes, and the resulting attenuation of radio signals, and study a performance measure that weights information by the distance over which it is transported. Consider a network with the following features. I) n nodes located on a plane, with minimum separation distance /spl rho//sub min/>0. II) A simplistic model of signal attenuation e/sup -/spl gamma//spl rho////spl rho//sup /spl delta// over a distance /spl rho/, where /spl gamma//spl ges/0 is the absorption constant (usually positive, unless over a vacuum), and /spl delta/>0 is the path loss exponent. III) All receptions subject to additive Gaussian noise of variance /spl sigma//sup 2/. The performance measure we mainly, but not exclusively, study is the transport capacity C/sub T/:=sup/spl Sigma/on/sub /spl lscr/=1//sup m/R/sub /spl lscr///spl middot//spl rho//sub /spl lscr//, where the supremum is taken over m, and vectors (R/sub 1/,R/sub 2/,...,R/sub m/) of feasible rates for m source-destination pairs, and /spl rho//sub /spl lscr// is the distance between the /spl lscr/th source and its destination. It is the supremum distance-weighted sum of rates that the wireless network can deliver. We show that there is a dichotomy between the cases of relatively high and relatively low attenuation. When /spl gamma/>0 or /spl delta/>3, the relatively high attenuation case, the transport capacity is bounded by a constant multiple of the sum of the transmit powers of the nodes in the network. However, when /spl gamma/=0 and /spl delta/<3/2, the low-attenuation case, we show that there exist networks that can provide unbounded transport capacity for fixed total power, yielding zero energy priced communication. Examples show that nodes can profitably cooperate over large distances using coherence and multiuser estimation when the attenuation is low. These results are established by developing a coding scheme and an achievable rate for Gaussian multiple-relay channels, a result that may be of interest in its own right.

Performance Analysis of Minimum&amp;lt;tex&amp;gt;$ell_1$&amp;lt;/tex&amp;gt;-Norm Solutions for Underdetermined Source Separation

Results of the analysis of the performance of minimum /spl lscr//sub 1/-norm solutions in underdetermined blind source separation, that is, separation of n sources from m(<n) linearly mixed observations, are presented in this paper. The minimum /spl lscr//sub 1/-norm solutions are known to be justified as maximum a posteriori probability (MAP) solutions under a Laplacian prior. Previous works have not given much attention to the performance of minimum /spl lscr//sub 1/-norm solutions, despite the need to know about its properties in order to investigate its practical effectiveness. We first derive a probability density of minimum /spl lscr//sub 1/-norm solutions and some properties. We then show that the minimum /spl lscr//sub 1/-norm solutions work best in a case in which the number of simultaneous nonzero source time samples is less than the number of sensors at each time point or in a case in which the source signals have a highly peaked distribution. We also show that when neither of these conditions is satisfied, the performance of minimum /spl lscr//sub 1/-norm solutions is almost the same as that of linear solutions obtained by the Moore-Penrose inverse. Our results show when the minimum /spl lscr//sub 1/-norm solutions are reliable.