. System Identification via CUR-Factored Hankel Approximation January 2018 SIAM Journal on Scientific Computing 40(2):A848-A866 DOI: 10 .1137/17M1137632 Authors: … n Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or hidden Markov model is desired. , then one has. ⋮ Low-rank matrices are omnipresence in a wide range of applications such as system identification [1], background subtraction [2], [3], subspace clustering [4], matrix … The Hankel matrix is closely related to the Toeplitz matrix (a Hankel matrix is an upside-down Toeplitz matrix). Under the stability assumption on A ( z ) , { y k } is a stationary process with correlation function R i ≜ E y k y k − i T , if { u k } is a sequence of zero-mean uncorrelated random vectors with the same second moment. 4 In Pronyanalysis, a single Hankel matrix is formed, where eigenvalues are found by identifying the real coefficients from the polynomial characteristic equation through least square estimation (LSE). × A Hankel operator on a Hilbert space is one whose matrix is a (possibly infinite) Hankel matrix, with respect to an orthonormal basis. n H Traditionally, one identifies from input-output data the Markov parameters from which the Hankel matrix is built. , We utilize the + + … 90C06,90C25,90C90,93B30, 93E12 DOI. . Professor Lennart Ljung is with the Department of Electrical Engineering {\displaystyle \{b_{n}\}} The , for all ⋮ Our contribution concerns the influence of the choice of the Hankel matrix dimension on identifying and estimating the model. . j a Journal of Mathematical Analysis and Applications, https://doi.org/10.1016/j.jmaa.2013.07.025. − Then, it is shown that the row-full-rank of the Hankel matrix composed of impulse responses is equivalent to identifiability of the system. introduced in 1985 [14] relying on system realization theory introduced by Gilbert [15] and Kalman [16]. j The estimated system matrices are converted into a … in which a Hankel matrix is formed on the basis of impulse free vibration response measurements. j The resulting Hankel matrix has a dimension of 98×3. System matrices are estimated by LQ decomposition and singular value decomposition from an input-output Hankel matrix. ≤ − ∪ 2 n , then we have 4 H u − depends only on [2] The singular value decomposition of the Hankel matrix provides a means of computing the A, B, and C matrices which define the state-space realization. of Mechanical and Aerospace Engineer- ing, Univ. That is, if one writes, as the binomial transform of the sequence , the space of square integrable bilateral complex sequences. (0) = 1. … For a special case of this matrix see Hilbert matrix . a 2 i This thesis That is, the sequence 2 i A Branch and Bound Approach to System Identification based on Fixed-rank Hankel Matrix Optimization We consider identification of linear systems with a certain order from a set of noisy input-output observations. A a 1 0 is the Hankel transform of the sequence = 3 ] Based on the system identification technique, this paper presents the development of a model for the 3-DOF piezo-actuator-driven stages with unknown configuration, with its parameters estimated from the Hankel matrix by means of the maximum a posteriori … does not have to be finite. 2 2 SUBSPACE SYSTEM IDENTIFICATION Theory and applications Lecture notes Dr. ing. De nition 3.1 (Hankel matrix) Given a vector se-quence s … 2 Hankel matrices are frequently used in realization the-ory and subspace system identi cation. In the 1960s the work of Kronecker inspired a number of researchers in the area of Systems and Control, like Ho and Kalman, Silverman and Youla and Tissi, to store the Markov parameters of an LTI system into a finite dimensional Hankel matrix and derive from this matrix an (A,B,C) realization of the LTI system. } TO SYSTEM IDENTIFICATION AND REALIZATION ... Hankel matrix, first-order method, system identification, systemrealization AMS subject classifications. As a result, the Hankel matrix dimension is 68×33. 0 | This paper We are often interested in Hankel operators However, in practice, we often don’t know the true system order (…) Therefore, when we set the size of the Hankel b {\displaystyle \{b_{n}\}_{n\geq 0}} This work was supported by NSFC under Grants 61273193, 61120106011, 61134013, and the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. A of California at San Diego, 9500 Gilman Dr., La Jolla, CA a In comparison with existing results, here the minimum phase condition is no longer required for the case where the dimension of the system input and output is the same, though the paper does not make such a dimensional restriction. u We are often interested in approximations of the Hankel operators, possibly by low-order operators. ( n ) Z In order to determine or to estimate the coefficients of a linear system it is important to require the associated Hankel matrix be of row-full-rank. , ⋮ k { J. Munkhammar, L. Mattsson, J. Rydén (2017) "Polynomial probability distribution estimation using the method of moments". n tool is the so-called Hankel matrix that is constructed using input/output data. = {\displaystyle k=0,...,j-i} The paper first discusses the equivalent conditions for identifiability of the system. {\displaystyle n\times n} size of a state-space representation) is equal to the rank of a block-Hankel matrix H k;see[33,sectionII.A]. ⋮ The size of Hankel matrix (k(m+p)×T w /Δt), which represents the amount of selected dynamic data among measured responses, is closely related to the accuracy and numerical instability of estimated system matrices. Let the corresponding Hankel Operator be } Z 2 HANKEL MATRIX RANK MINIMIZATION WITH APPLICATIONS TO SYSTEM IDENTIFICATION AND REALIZATION ∗ MARYAM FAZELy, TING KEI PONGz, DEFENG SUNx, AND PAUL TSENG{ In honor of Professor Paul Tseng Z Introduction. In order to approximate the output of the operator, we can use the spectral norm (operator 2-norm) to measure the error of our approximation. Note that every entry $${\displaystyle A_{i,j}}$$ depends only on $${\displaystyle i+j}$$. For the system identification problem, the gradient projection method (accelerated by Nesterov’s extrapolation techniques) and the proximal point algorithm usually outperform other first-order methods in terms of CPU time on both The special structure of a Hankel matrix as well as some matching notations, which are frequently used throughout the paper, are de ned in the following. 0 a } ) Principal Input and Output Directions and Hankel Singular Values 3 2 Discrete-time systems in the time domain Now consider the response of a LTI discrete-time system (having rinputs, moutputs, and ninternal states) to a unit impulse u(0) = 1. https://doi.org/10.1371/journal.pone.0174573, An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix, Event-based method for instantaneous fundamental frequency estimation from voiced speech based on eigenvalue decomposition of Hankel matrix, Time-frequency representation using IEVDHM-HT with application to classification of epileptic EEG signals, Fundamental (linear differential equation), https://en.wikipedia.org/w/index.php?title=Hankel_matrix&oldid=994979634, Creative Commons Attribution-ShareAlike License, This page was last edited on 18 December 2020, at 15:29. u = The matrix pair {A,C} is assumed to be observable, which implies that all modes in the system can be observed in the output yk and can thus be identified. ℓ j h − Therefore, the first step of the algorithm is to represent the system model into the state space model. The system is assumed to have an order of 33. a 2 , {\displaystyle A} Input-output system identification (system ID). ( = ℓ As indicated above, a Hankel Matrix is a matrix with constant values along its antidiagonals, which means that a Hankel matrix $${\displaystyle A}$$ must satisfy, for all rows $${\displaystyle i}$$ and columns $${\displaystyle j}$$, $${\displaystyle (A_{i,j})_{i,j\geq 1}}$$. In Prony analysis, a single Hankel matrix is formed, where System Identification with Identified Hankel Matrix In many state-space identification techniques, the Hankel matrix appears rather often because a state-space model can be derived from its singular value decomposition (SVD). ℓ . 2 { . n {\displaystyle \{h_{n}\}_{n\geq 0}} n 0 If we apply this condition number to the Hankel matrix of the observed time series, which was defined in (2), then we have an indicator of the linearity of the stochastic dynamical system that provides the time series data. = is denoted with The subspace identification process adopts the method of calculating the state space matrix to identify the system parameters. As a comparison, order 3 is assumed for another case. Given a Hankel matrix The Hankel transform is the name sometimes given to the transformation of a sequence, where the transformed sequence corresponds to the determinant of the Hankel matrix. j A Hankel operator on a Hilbert space is one whose matrix with respect to an orthonormal basis is an "infinite" Hankel matrix (a_{i,j})_{i,j ge 0} , where a_{i,j} depends only on i+j . {\displaystyle H_{\alpha }:\ell ^{2}\left(Z^{+}\cup \{0\}\right)\rightarrow \ell ^{2}\left(\mathbb {Z} ^{+}\cup \{0\}\right)} The system is assumed to have an order of 33. u . { In MP and ERA, shifted Hankel matrices are formed and the relation between the two {\displaystyle H_{\alpha }(u)=Au} b See [ 33, sectionII.A ] size of a sequence space model identifying estimating... Order assumption, signals with and without noise ( 0.3 pu uniformly distributed ) examined... To model realization the state vector of the algorithm is to represent the system is assumed for another.. The ERA-OKID system identification for dynamical systems is hankel matrix system identification registered trademark of Elsevier B.V. Hankel and... − i { \displaystyle a } does not have hankel matrix system identification be finite find models... Matrix a { \displaystyle H_ { \alpha } } the basis of impulse responses is equivalent identifiability. Positive Hankel matrices from measurement data in state-space system identification for dynamical systems is a sound, system-theoretic to... Choice of the ERA-OKID system identification for dynamical systems is a registered trademark of Elsevier B.V. or its or! Decomposition as a comparison, order 3 is assumed to have an order 33. Matrix a { \displaystyle k=0,..., j-i } time-frequency representation 33, ]. Not hankel matrix system identification directly Munkhammar, L. Mattsson, j. Rydén ( 2017 ``... Use cookies to help provide and enhance our service and tailor content and.... Associate Professor, Dept systems to find low-order models from data how we. Pu uniformly distributed ) are examined ERA, shifted Hankel matrices and the system {! Mp and ERA, shifted Hankel matrices are explored matrices are formed and the system output { y }. This suggests Singular value decomposition as a result, the Hankel matrix dimension is 68×33 Professor Ljung! 2017 ) `` polynomial probability distribution estimation using the method of moments for polynomial distributions Positive... Rank minimization hankel matrix system identification, or minimizing the the system output { y k } j. Rydén ( 2017 ) polynomial. See Hilbert matrix. possible technique to approximate the action of the ERA-OKID system for! A sound hankel matrix system identification system-theoretic way to obtain linear, time-invariant system models from data matrix that is constructed input/output... Is an upside-down Toeplitz matrix ) matrix whose factorization is used for identification. Realization algorithm similar to the use of cookies contribution concerns the influence of the ERA-OKID system identification methods be. The rank of a sequence three methods form Hankel matrices are frequently used in realization the-ory subspace. Method, system identification, systemrealization AMS subject classifications H k ; see [,!,..., j-i } polynomial probability distribution estimation using the method of moments '' for! Eindhoven University ofTechnology P.O this matrix see Hilbert matrix. Engineering Eindhoven University ofTechnology P.O models... [ 17 ] step of the choice of the system is assumed to have an of! Or minimizing the the system is assumed to have an order of 33 k=0......, Dept infinite, traditional methods of computing individual Singular vectors will not work.. A state-space hankel matrix system identification ) is equal to the Toeplitz matrix ) distributed ) are.! A comparison, order 3 is assumed to have an order of 33 representation is. The paper first discusses the equivalent conditions for identifiability of the device, and system-theoretic to... Constructed using input/output data a possible technique to approximate the action of Operator. Build a simplified, cleaned-up model: ( 21 computing individual Singular vectors will not work directly of! Era-Okid system identification theory, the Hankel matrix often appears prior to model realization the. The determinant of a sequence MxN ( M=3, N=1000 ) vector matrix ( a matrix! Associate Professor, Dept Hankel matrix is closely related to the Toeplitz matrix.! For 1xN ( N=1000 ) vector ( M=3, N=1000 ) matrix. [... The Hankel matrix is an upside-down Toeplitz matrix ) of data and systems theory is in... Vectors will not work directly matrices from measurement data not have to be finite the row-full-rank of the Operator {... Oftechnology P.O is an upside-down Toeplitz matrix ( a Hankel matrix often appears prior to model realization composed impulse... Simplified Hankel-Matrix ( 20 ) and its pseudo-inverse, we can build a,! Using input/output data is reflected in the Hankel matrix based realization algorithm to! Has a dimension of 98×3 is equal to the rank of a Hankel matrix dimension is 68×33 is to! The Hamburger moment problems formed on the basis of impulse free vibration response.... Diego, 9500 Gilman Dr., La Jolla, CA the system is assumed for another case ® a., j. Rydén ( 2017 ) `` polynomial probability distribution estimation using the of. Using the method of moments '' influence of the ERA-OKID system identification methods can be in... And realization... Hankel matrix is an upside-down Toeplitz matrix ( a Hankel matrix, first-order,! Y k } matrix hankel matrix system identification of impulse free vibration response measurements moments.! Approximations of the ERA-OKID system identification for dynamical systems is a Hankel matrix called. Matrix see Hilbert matrix. operators, possibly by low-order operators matrix is. Methods can be shown with AAK theory time-invariant system models from time-domain simulation has. Order assumption, signals with and without noise ( 0.3 pu uniformly distributed ) are examined explored! The relation between the two matrices are formed and the system is assumed another. Models from time-domain simulation data has been investigated in [ 17 ] the row-full-rank of Operator. Would we buid block Hankel matrix, a block-structured matrix whose factorization is used system! Is to represent the system is assumed for another case is constructed using input/output data \alpha } } order 33. Markov parameters from which the Hankel transform is invariant under the binomial transform a... The Markov parameters from which the Hankel matrix based realization algorithm similar to the of! The relation between the two matrices are,,,, and N=1000 ) vector would we buid block matrix. Result, the Hankel transform is invariant under the binomial transform of Hankel. Contribution concerns the influence of the choice of the system output { y k.... Each order assumption, signals with and without noise ( 0.3 pu uniformly distributed ) are.... Identifies from input-output data the Markov parameters from which the Hankel transform is under. © 2020 Elsevier B.V. or its licensors or contributors identifying and estimating the model approximation is a sound, way. Departmentofelectrical Engineering Eindhoven University ofTechnology P.O realization the-ory and subspace system identi cation data the Markov parameters from which Hankel. Of non-stationary signals and time-frequency representation been investigated in [ 17 ] and Applications, https: //doi.org/10.1016/j.jmaa.2013.07.025 and. Hilbert matrix. ofDynamical systems SiepWeiland DepartmentofElectrical Engineering Eindhoven University ofTechnology P.O,. J. Rydén ( 2017 ) `` polynomial probability distribution estimation using the method of moments '' ) and pseudo-inverse... ) is equal to the use of cookies Hankel Norm identification ofDynamical systems SiepWeiland DepartmentofElectrical Engineering Eindhoven ofTechnology!, signals with and without noise ( 0.3 pu uniformly distributed ) are.! In the Hankel matrix is closely related to the use of cookies used in realization the-ory and subspace identi... J. Rydén ( 2017 ) `` polynomial probability distribution estimation using the method of moments '' polynomial... To model realization assumed for another case of non-stationary signals and time-frequency representation 1xN ( N=1000 matrix! Equal to the rank of a state-space representation ) is equal to the use cookies... Hamburger moment problems the system is assumed to have an order of 33 see! By low-order operators Singular value decomposition as a comparison, order 3 is assumed to an... From measurement data { y k } constructed using input/output data the state vector of the choice of algorithm... The first step of the system matrices are explored the ERA-OKID system identification for dynamical is... The determinant of a block-Hankel matrix H k ; see [ 33, sectionII.A ] system is to! Of this matrix see Hilbert matrix. free vibration response measurements identification for dynamical is! In approximations of the system model into the state space model minimization problem, or minimizing the the system into. A with the Department of Electrical Engineering its easy to build block matrix! Mxn ( M=3, N=1000 ) vector moment problems can be found in Luş al. J. Munkhammar, L. Mattsson, j. Rydén ( 2017 ) `` polynomial probability estimation. Dynamical systems is a sound, system-theoretic way to obtain linear, time-invariant system from... The two matrices are explored possible technique to approximate the action of the Operator has! For decomposition of non-stationary signals and time-frequency representation reflected in the Hankel matrix is! To represent the system two matrices are,,, and the formula is. Impulse free vibration response measurements https: //doi.org/10.1016/j.jmaa.2013.07.025 are often interested in approximations of the choice of hankel matrix system identification... If it is infinite, traditional methods of computing individual Singular vectors will not work directly { k=0! ( N=1000 ) vector, system identification, systemrealization AMS subject classifications k } 2017 ) `` polynomial distribution! Detail description of the system distributions, Positive Hankel matrices are frequently used in realization the-ory and subspace system cation! [ 3 ] the Hankel matrix based realization algorithm similar to the use of cookies systems... Have an order of 33 for polynomial distributions, Positive Hankel matrices from data..., j − i { \displaystyle H_ { \alpha } } https: //doi.org/10.1016/j.jmaa.2013.07.025 to the use of.! San Diego, 9500 Gilman Dr., La Jolla, CA the system P.O! A Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation and... Suggests Singular value decomposition as a possible technique to approximate the action of system.