Identification of Linear, Parameter Varying, and Nonlinear Systems: Theory, Computation, and Applications
Organizers: Wallace E. Larimore (Adaptics, Inc.), Pepijn B. Cox and Roland Tóth (Eindhoven University of Technology).
In this workshop, first powerful subspace identification methods (SIM) are described for the well understood case of data-driven modeling of linear time-invariant (LTI) systems. Recent extensions are then developed to linear parameter-varying (LPV), quasi-LPV, and general nonlinear (NL) systems with polynomial nonlinearities. The presentation, following the extended tutorial paper (Larimore, ACC2013), includes detailed conceptual development of the theory and computational methods with references to the research literature for those interested. Numerous applications are discussed including aircraft wing flutter (LPV), chemical process control (LTI), automotive engine modeling (quasi-LPV, NL), and the Lorenz attractor (NL). An emphasis is placed on conceptual understanding of the subspace identification method to allow effective application to system modeling, control, and fault diagnosis.
Over the past decade, major advances have been made in LTI system identification with data gathered in the open loop (Larimore, ACC1999) and closed-loop settings (Larimore, DYCOPS2004; Chiuso, TAC2010). However, efficiently identifying state-space models of LPV and NL systems remains an open question, as the required computational complexity for subspace methods grows exponentially with the number of system inputs, outputs, and states while prediction error methods correspond to nonlinear parameter optimization problems prone to local minima and also often leading to infeasible computational requirements.
The workshop presents a statistical approach using the fundamental canonical variate analysis (CVA) method for subspace identification of LTI systems, with detailed extensions to LPV and NL systems. The LTI case includes basic concepts of reduced rank modeling of ill-conditioned data to obtain the most appropriate statistical model structure and order using optimal maximum likelihood methods. The fundamental statistical approach gives expressions of the multistep-ahead likelihood function for subspace identification of LTI systems. This leads to direct estimation of parameters using singular value decomposition type methods that avoid iterative nonlinear parameter optimization. This results in statistically optimal maximum likelihood parameter estimates and likelihood ratio tests of hypotheses. The parameter estimates have optimal Cramer-Rao lower bound accuracy, and the likelihood ratio hypothesis tests on model structure, model change, and process faults produce optimal decisions. The LTI CVA method is compared to different system identification methods - including other subspace, prediction error, and maximum likelihood approaches - and show considerably less computational time and higher parameter accuracy.
The LTI subspace methods are extended to LPV systems, which are represented in the state-space by matrices that are no longer constant, but are functions of the system operating point or other external “exogenous” variables called the scheduling signals. Often the underlying dependence of these matrices are captured in terms of linear plus constant functions of the scheduling. For example, this allows the identification of constant underlying structural stiffness parameters while wing flutter dynamics vary with scheduling functions of speed and altitude. This is further extended to quasi-LPV systems where the scheduling functions may be functions of dynamic variables, i.e. of the inputs and/or outputs of the system directly (Larimore, Cox and Tóth, ACC2015). This includes the NL case of bilinear and general polynomial systems that are universal approximators. The developed subspace identification method for parameter-varying systems avoids the exponential growth in computational characteristics exhibited by other SIMs.
The workshop is continued with introducing alternative LPV subspace identification approaches, including a novel basis reduced realization scheme on identified FIR models (Cox, Tóth and Petreczky, LPVS2015), predictor based subspace approaches, and other alternatives. Maximum likelihood refinement of the subspace estimates is also presented in terms of an expectation-maximization method and the gradient-based optimization of the prediction error. Computational efficiency and estimation performance of the presented approaches is analyzed and compared.
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