Research On Theory And Method Of System Identification Based On Low-dimensional Structural Constraints

The dynamical system in the real world is difficult to model because of its growing scale and complexity.Especially in modern science and engineering,dynamical system often has a large number of unknown parameters,whose number is far greater than the number of input and output samples,leading to dimensionality disaster.In this context,classical identification theories and methods are no longer applicable,and the potential sparse and low-rank low-dimensional structural characteristics of the system should be utilized.In this thesis,the system identification theory based on low-dimensional structural constraints is established,and the system identification method with low complexity and high robustness is developed.The main research contents of this thesis are as follows:The sparse identification of block-oriented nonlinear systems is studied.Aiming at the problem of block-oriented nonlinear system identification,an implicit sparse model driven by time series data is established,and a system identification algorithm based on alternating direction method is proposed and its convergence is proved.Simulation experiments are carried out on two kinds of block-oriented nonlinear systems.The results show that,compared with the classical methods,the proposed algorithm can accurately select the dominant items and estimate the system parameters using a small amount of data.Sparse identification of spatio-temporal data-driven partial differential equations is studied.Aiming at the identification problem of partial differential equations,a sparse model of nonlinear system driven by spatio-temporal data is established,a sparse Bayesian identification algorithm is proposed,and the convergence analysis and statistical analysis of the algorithm are given.Simulation experiments are carried out on four classical physics systems.The results show that,compared with the traditional algorithms,the proposed algorithm can reconstruct the spatio-temporal dynamics model with a small amount of data.An online sparse identification algorithm with low complexity is proposed.To solve the problem of high computational complexity of batch identification algorithm,an online sparse identification algorithm based on alternating minimization was developed by applying sparse constraints to recursive least squares.The convergence rate of the algorithm was given.Simulation experiments are carried out on three datasets.The results show that,compared with the traditional online identification algorithms,the proposed algorithm reduces the computational complexity and improves the computational accuracy.A robust sparse identification algorithm based on reweighted alternating minimization is proposed.A class of non-convex and non-smooth programming is designed for system identification under impulse noise,and a robust system identification algorithm is developed by using linearization and acceleration strategies,and its convergence is proved.Simulation experiments are carried out under two kinds of impulse noise.The results show that,compared with the classical system identification algorithms,the proposed algorithm improves the computational accuracy and has stronger robustness.The problem of non-asymptotic identification for linear dynamic systems is studied.Aiming at the failure of classical identification method in high dimensional setting,a multivariate linear regression model with low-rank constraint was constructed by collecting multiple sample trajectories and sub-Gaussian input excitation,and a nuclear norm heuristic method was proposed and its statistical analysis was given.In the noiseless case,it is proved that the proposed method can recover the dynamic system with high probability from a finite number of sample trajectories by introducing the weak restricted isometric property.In the noisy case,it is proved that the proposed method can obtain parameter estimation with high probability and small estimation error from a finite number of sample trajectories by introducing the restricted curvature condition.