Dec 14, 2012

Compressed Sensing: A New Trend in Control


I am now in Maui to attend IEEE CDC 2012. IEEE CDC (Conference on Decision and Control) is the most important conference of control, and presentations in CDC show recent trends in control.

I have found researches in this CDC that pertain to sparsity or compressed sensing (CS). The number of such researches is increasing in control.

Today I introduce researches related to sparsity or CS found in the final program, including one by myself.

Packetized Predictive Control for Rate-Limited Networks via Sparse Representation
Nagahara, Masaaki,  Kyoto Univ.
Quevedo, Daniel E.,  The Univ. of Newcastle
Ostergaard, Jan,  Aalborg Univ.
Abstract: We study a networked control architecture for linear time-invariant plants in which an unreliable data-rate limited network is placed between the controller and the plant input. The distinguishing aspect of the situation at hand is that an unreliable data-rate limited network is placed between controller and the plant input. To achieve robustness with respect to dropouts, the controller transmits data packets containing plant input predictions, which minimize a finite horizon cost function. In our formulation, we design sparse packets for rate-limited networks, by adopting an L0 optimization, which can be effectively solved by an orthogonal matching pursuit method. Our formulation ensures asymptotic stability of the control loop in the presence of bounded packet dropouts. Simulation results indicate that the proposed controller provides sparse control packets, thereby giving bit-rate reductions for the case of memoryless scalar coding schemes when compared to the use of, more common, quadratic cost functions, as in linear quadratic (LQ) control.

Sparse Multiple Kernels for Impulse Response Estimation with Majorization Minimization Algorithms
Chen, Tianshi,  Linköping Univ. Sweden
Ljung, Lennart,  Linkoping Univ.
Andersen, Martin,  Linköping Univ.
Chiuso, Alessandro,  Univ. di Padova
Carli, Francesca, P,  Univ. of Padova
Pillonetto, Gianluigi,  Univ. of Padova
Abstract: This contribution aims to enrich the recently introduced kernel-based Bayesian approach for linear system identification. Instead of a single kernel, we use multiple kernels, which can be instances of any existing kernels for the impulse response estimation of linear systems. We also introduce a new class of kernels constructed based on output error (OE) model estimates. In this way, a more flexible and richer representation of the kernel is obtained. Due to this representation the associated hyper-parameter estimation problem has two good features. First, it is a difference of convex functions programming (DCP) problem. While it is still nonconvex, it can be transformed into a sequence of convex optimization problems with majorization minimization (MM) algorithms and a local minima can thus be found iteratively. Second, it leads to sparse hyper-parameters and thus sparse multiple kernels. This feature shows the kernel-based approach with multiple kernels has the potential to tackle various problems of finding sparse solutions in linear system identification. As an illustration the sparse dynamic network identification problem is studied.

Topology Identification of a Sparse Dynamic Network
Seneviratne, Akila,  The Univ. of New South Wales
Solo, Victor,  Univ. of New South Wales
Abstract: This paper addresses the problem of identifying the topology of a sparsely connected network of dynamic systems. The goal is to identify the links, the direction of information flow and the transfer function of each dynamic system. The output of each system is affected by the incoming data of the directly connected systems and noise. In contrast to the related existing work we use causal Laguerre basis functions to expand the transfer functions. Since the network is sparsely connected we estimate the system topology using an algorithm which optimizes the l0 penalized least squares criterion with grouped variables. This also contrasts with previous work which usually uses and l1 penalty. The l0 penalty has the potential to generate greater sparsity. We present simulation results to demonstrate the effectiveness of this method.

Data-Driven Graph Reconstruction Using Compressive Sensing
Chang, Young Hwan Univ. of California, Berkeley
Tomlin, Claire J. UC Berkeley
Abstract: Modeling of biological signal pathways forms the basis of systems biology. Also, network models have been important representations of biological signal pathways. In many biological signal pathways, the underlying networks over which the propagations spread are unobserved so inferring network structures from observed data is an important procedure to study the biological systems. In this paper, we focus on protein regulatory networks which are sparse and where the time series measurements of protein dynamics are available. We propose a method based on compressive sensing (CS) for reconstructing a sparse network structure based on limited time-series gene expression data without any a priori information. We present a set of numerical examples to demonstrate the method. We discuss issues of coherence in the data set, and we demonstrate that incoherence in the sensing matrix can be used as a performance metric and a guideline for designing effective experiments.

Stability Analysis of Non-Vector Space Control Via Compressive Feedbacks
Zhao, Jianguo,  Michigan State Univ.
Xi, Ning,  Michigan State Univ.
Sun, Liang,  Michigan State Univ.
Song, Bo,  Michigan State Univ.
Abstract: A new non-vector space control method with compressive feedback is presented in this paper. Two problems are discussed. The first is the development of the non-vector space control method. It formulates the control problem in the space of sets instead of the vector space, and it provides a convenient way to model physical phenomenon such as shapes and images. For the non-vector space dynamical systems, the issues related to stability analysis and design of controllers are discussed in the paper. The second problem is to design controllers for non-vector space dynamic systems using compressive feedback, which is a new concept proposed in this paper. Compressive feedback, obtained from the original feedback, reduces the amount of data for the original feedback but retains the essential information to ensure the stability and other key performance measures for a control system. The design methods for controllers based on compressive feedback are developed in this paper. Furthermore, the non-vector space control system with compressive feedback has been successfully applied to visual servoing, and simulations results have clearly demonstrated the superb performance and advantages.

Schuler, Simone,  Univ. of Stuttgart
Zelazo, Daniel,  Univ. Stuttgart
Allgower, Frank,  Univ. of Stuttgart
Abstract: This paper considers the problem of designing sparse relative sensing networks (RSN) subject to a given Hinf-performance constraint. The topology design considers homogeneous and heterogeneous agents over weighted graphs. We develop a computationally efficient formulation of the sparse topology design via a convex l1-relaxation. This makes the proposed algorithm attractive for practical applications. We also demonstrate how this relaxation can be used to embed additional performance criteria, such as maximization of the algebraic connectivity of the RSN.

Sparse Coloured System Identification with Guaranteed Stability
Seneviratne, Akila,  The Univ. of New South Wales
Solo, Victor,  Univ. of New South Wales
Abstract: We develop a method for sparse transfer function estimation in the presence of coloured noise. Unlike most previous sparse transfer function estimation methods our approach via Laguerre polynomials guarantees stability. Also unlike previous methods the new procedure can handle coloured noise. We discuss both l1 and l0 penalized procedures. Simulation results are presented to demonstrate the effectiveness of this approach and to compare the performance of the two penalties that are considered.

Pan, Wei,  Imperial Coll. London
Yuan, Ye,  Univ. of Cambridge
Goncalves, Jorge,  M. Univ. of Cambridge
STAN, Guy-Bart Vincent,  Imperial Coll. London
Abstract: Reconstruction of biochemical reaction networks (BRN) and genetic regulatory networks (GRN) in particular is a central topic in systems biology which raises crucial theoretical challenges in system identification. Nonlinear Ordinary Differential Equations (ODEs) that involve polynomial and rational functions are typically used to model biochemical reaction networks. Such nonlinear models make the problem of determining the connectivity of biochemical networks from time-series experimental data quite difficult. In this paper, we present a network reconstruction algorithm that can deal with ODE model descriptions containing polynomial and rational functions. Rather than identifying the parameters of linear or nonlinear ODEs characterised by predefined equation structures, our methodology allows us to determine the nonlinear ODEs structure together with their associated parameters. To solve the network reconstruction problem, we cast it as a compressive sensing (CS) problem and use sparse Bayesian learning (SBL) algorithms as a computationally efficient and robust way to obtain its solution.

Comparison of the Sparse-Grid Quadrature Rule and the Cubature Rule in the Nonlinear Filtering
Jia, Bin,  Mississippi State Univ.
Xin, Ming,  Mississippi State Univ.
Cheng, Yang,  Mississippi State Univ.
Abstract: In this paper, the recently developed sparse-grid quadrature filter is compared with the cubature Kalman filter. The relation between the sparse-grid quadrature rule and the cubature rule is revealed. It can be shown that arbitrary degree cubature rules can be obtained by the projection of the sparse-grid quadrature rule. Since both rules can achieve an arbitrary high degree of accuracy, they are more accurate than the conventional third-degree cubature rule and the unscented transformation. In addition, they are computationally more efficient than the Gauss-Hermite quadrature rule and the Monte-Carlo method when they are used to calculate the Gaussian type integrals in the nonlinear filtering. The comparison of these rules is demonstrated by a benchmark numerical integration example.

Sanandaji, Borhan M.,  Univ. of California at Berkeley
Vincent, Tyrone L.,  Colorado School of Mines
Poolla, Kameshwar,  Univ. of California at Berkeley
Wakin, Michael,  Colorado School of Mines
Abstract: In this tutorial, we review some of the recent results concerning Compressive System Identification (CSI) (identification from few measurements) of sparse channels (and in general, Finite Impulse Response (FIR) systems) when it is known a priori that the impulse response of the system under study is sparse (high-dimensional but with few non-zero entries) in an appropriate basis. For the systems under study in this tutorial, the system identification problem boils down to an inverse problem of the form Ax = b, where the vector x is high-dimensional but with few non-zero entries and the matrix A is underdetermined. Over the past few years, several algorithms with corresponding recovery conditions have been proposed to perform such a recovery. These conditions provide the number of measurements sufficient for correct recovery. In this note, we review alternate approaches to derive such recovery conditions concerning CSI of FIR systems whose impulse response is known to be sparse.

Shah, Parikshit,  Massachusetts Inst. of Tech.
Bhaskar, Badri,  Narayan Univ. of Wisconsin - Madison
Tang, Gongguo,  Univ. of Wisconsin-Madison
Recht, Benjamin,  Univ. of Wisconsin-Madison
Abstract: This paper proposes a new algorithm for linear system identification from noisy measurements. The proposed algorithm balances a data fidelity term with a norm induced by the set of single pole filters. We pose a convex optimization problem that approximately solves the atomic norm minimization problem and identifies the unknown system from noisy linear measurements. This problem can be solved efficiently with standard, free software. We provide rigorous statistical guarantees that explicitly bound the estimation error (in the H_2-norm) in terms of the stability radius, the Hankel singular values of the true system and the number of measurements. These results in turn yield complexity bounds and asymptotic consistency. We provide numerical experiments demonstrating the efficacy of our method for estimating linear systems from a variety of linear measurements.

Tóth, Roland,  Eindhoven Univ. of Tech.
Hjalmarsson, Håkan,  Royal Inst. of Tech.
Rojas, Cristian R.,  KTH Royal Inst. of Tech.
Abstract: Accurate parametric identification of Linear Parameter-Varying (LPV) systems requires an optimal prior selection of model order and a set of functional dependencies for the parameterization of the model coefficients. In order to address this problem for linear regression models, a regressor shrinkage method, the Non-Negative Garrote (NNG) approach, has been proposed recently. This approach achieves statistically efficient order and structural coefficient dependence selection using only measured data of the system. However, particular drawbacks of the NNG are that it is not applicable for large-scale over-parameterized problems due to computational limitations and that adequate performance of the estimator requires a relatively large data set compared to the size of the parameterization used in the model. To overcome these limitations, a recently introduced L_1 sparse estimator approach, the so-called SPARSEVA method, is extended to the LPV case and its performance is compared to the NNG.

Siraj, Muhammad Mohsin,  Eindhoven Univ. of Tech.
Tóth, Roland,  Eindhoven Univ. of Tech.
Weiland, Siep,  Eindhoven Univ. of Tech.
Abstract: In the reduction of Linear Parameter-Varying (LPV) models, decreasing model complexity is not only limited to model order reduction (state reduction), but also to the simplification of the dependency of the model on the scheduling variable. This is due to the fact that the concept of minimality is strongly connected to both types of complexities. While order reduction of LPV models has been deeply studied in the literature resulting in the extension of various reduction approaches of the LTI system theory, reduction of the scheduling dependency still remains to be a largely open problem. In this paper, a model reduction method for LPV state-space models is proposed which achieves both state-order and scheduling dependency reduction. The introduced approach is based on an LPV Ho-Kalman algorithm via imposing a sparsity expectation on the extended Hankel matrix of the model to be reduced. This sparsity is realized by an L1-norm based minimization of a priori selected set of dependencies associated sub-Markov parameters. The performance of the proposed method is demonstrated via a representative simulation example.

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