Sparse System Identification Based On Orthonormal Rational Function Bases

Since orthonormal rational function?ORF?basis has many advantages in structure,it has attracted attention from a large number of scholars in the field of system identification.The priori knowledge of poles contained in the ORF basis functions means that the representations of rational transfer functions under the ORF basis have the sparsity.Because the representation coefficient of transfer function under the ORF basis is infinite dimensional,the definition of sparsity in compressed sensing is not applicable for rational transfer functions.Extending the concept of sparsity from the finite dimensional vector space to the infinite dimensional function space is the key problem of sparse representation of transfer function.To solve this problem,this paper proposes the definition of-sparsity for the infinite sequence.On this basis,the sparse representation and identification of transfer functions under a single ORF basis are studied.Because the introduction of redundant basis can improve the sparsity of representation,the identification problem is further extended to the sparse system represented under pairs of ORF bases.To solve the sparse identification problem under pairs of ORF bases,the guarantee for the uniqueness of sparse representation is indispensable.In theory,the uncertainty principle of rational transfer function is established.Then the uniqueness theorem of joint sparse representation under pairs of ORF bases is given.For the reconstruction of the joint sparse representation,a frequency domain identification method for sparse systems represented under pairs of ORF bases is proposed.A sufficient condition for accurate identification of the joint sparse representation with high probability and quantitative analysis of identification performance are given.The algorithms of the sparse identification for both single and paired ORF bases are implemented with convex optimization,and the effectiveness and advantages of the proposed approaches are shown via numerical examples.The main research works of this thesis are as follows: e?1?To solve the problem of sparse representation in infinite-dimensional space,the definition of-sparsity for the infinite sequence is proposed.On this basis,a method of frequency domain identification for sparse system under a single ORF basis is proposed.We proof that optimization can efficiently recover the sparse coefficient of the etransfer function under a single ORF basis with a small number of random observations on the unit circle.Simulation experiments show that l₁ optimization can efficiently reconstruct the sparse coefficients of transfer functions represented by Takenaka-Malmquist?TM?basis.l₁?2?For the joint sparse representation of transfer functions under pairs of ORF bases,the uncertainty principle concerning pairs of compressible representation of rational transfer functions in different ORF bases in the infinite dimensional function space is presented.The uniqueness of sparse representation using such pairs is provided as a direct consequence of uncertainty principle.The innovation is to extend the joint sparse signal representation problem under a pair of standard orthogonal bases in finite dimensional vector space to the joint sparse representation of rational functions in infinite dimensional function space,which lay a theoretical foundation for the reconstruction of joint sparse representation.?3?For the reconstruction problem of the joint sparse representation of transfer functions,an identification method of the joint sparse representation of the rational transfer function is proposed from the frequency domain perspective.The lower bound of the number of measurements required to accurately reconstruct the joint sparse representation with high probability is given.The quantitative analysis of identification performance is given as well.?4?The reconstruction of joint sparse representation for a rational transfer function under FIR and TM pairs is considered.The uniqueness of the sparse representation under such pairs is proved and the analytic formula of the bound for the uniqueness is provided.The sufficient condition for solving the sparse representation by l₁ optimization is obtained,when using a limited number of random frequency domain samples on the upper unit circle.The simulations verify that l₁ optimization can reconstruct the coefficient under such pairs with high probability.