Sparse Optimization for System Identification

Varanasi, Santhosh Kumar and Jampana, Phanindra Varma (2019) Sparse Optimization for System Identification. PhD thesis, Indian Institute of Technology Hyderabad.

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Abstract

Identification of accurate dynamic models is essential in analysis and design of control systems. A first-principles model provides accurate physical description of the system. However, the accuracy is achieved at the cost of detailed analysis involving thorough physical understanding which makes the method impractical for most industrial applications. In contrast, black-box models are built from measured input-output data. The main advantage is that these approaches typically yields relatively simpler models that can describe the system's behaviour well within a defined operational regime. Such models can be identified in discrete-time (DT) (deference equation models) or in continuous time (CT) (ordinary differential equation models). Identification of DT models is a well studied area where as identification of continuous linear time-invariant (CLTI) models from discrete data has gained importance in recent years as there are several advantages such as (i) CLTI models can be identified using non-uniformly sampled data (ii) CLTI models provide physical insight into the system as most processes encountered in chemical engineering are continuous in nature and (iii) CLTI models can effectively handle stiff systems i.e., system with time constants of different orders of magnitude so that the system contains both fast and slow dynamics. The main area of research delineated in this thesis deals with various aspects of identification of CLTI models namely: (i) parameter estimation in CLTI output-error models (ii) topology identification of sparse networks of CLTI systems and (iii) subspace identification of CLTI state-space models with missing output-data. The enabling framework in all these studies is sparsity constrained optimization. For parameter estimation, the thesis proposes a sparsity constraint on the solution vector (i.e., most of the elements of the solution vector are equal to zero) to identify both the model order and the coefficients simultaneously. Topology identification is achieved by assuming the network is densely connected and again using a sparsity constraint to obtain the true connects and the dynamic models of all the nodes in the network. The aforementioned studies are for the case of SISO systems. For continuous MIMO systems, the fundamental problem is the determination of the dimension of the state space from input-output data. In this thesis, this is realized using a nuclear norm constraint, which is a convex relaxation of the minimum rank constraint. Another important problem in identification that effects the accuracy of parameter estimation is input design. It is well-known that the covariance of the parameter estimates obtained by using a particular input is lower bounded by the inverse of Fisher information matrix - this bound is also known as the Cramer-Rao lower bound. Since the Fisher information matrix depends on the true parameters also, existing methodologies employ an iterative procedure for estimating the inputs as well as the parameters. However, in practice this is very time consuming. This thesis proposes a method which can obtain the optimal input by applying a step signal in two directions. The method is based on expanding the transfer function in a known orthonormal basis for the Hardy space of stable transfer functions. Models in the discrete form are also considered when the underlying dynamics is not linear. In this case, the thesis proposes NARX (nonlinear auto-regressive with exogenous inputs) models as good candidates for model and topology identification. The thesis includes case studies for identification of an industrial three phase separator unit and simultaneous topology and model estimation in rat neuronal networks. Finally, dynamic models in the form of partial differential equations (PDEs) are also considered. In the discrete and ODE setting, the parameter vector is definite dimensional. However, in the PDEs observed in EIT (Electrical Impedance Tomography) the parameter vector (here the conductivity) is infinite dimensional. The main problem in EIT is to reconstruct the conductivity map from boundary measurements of voltages and currents. Using the observation that in most chemical process systems, the difference in conductivity is sparse, this thesis proposes a novel image reconstruction algorithm based on sparse optimization.

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