In this workshop, the powerful subspace identification method (SIM) is described for the well understood case of linear time-invariant (LTI) systems. Recent extensions are then developed to linear parametervarying (LPV), Quasi-LPV, and general nonlinear (NL) systems such as polynomial systems. 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 including aircraft wing flutter (LPV), chemical process control (LTI), automotive engine (Quasi-LPV, NL) modeling, and the Lorenz attractor (NL) are discussed. 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 system identification for the LTI cases of no feedback (Larimore, ACC1999) and unknown feedback (Larimore, 2004; Chiuso, TAC2010). However, for LPV and NL systems limitations remain including, for subspace methods the required computation grows exponentially with the number of system inputs, outputs, and states, and for maximum likelihood methods iterative nonlinear parameter optimization may not convergence, leading often to infeasible computation.

The workshop presents a first principles statistical approach using the fundamental canonical variate analysis (CVA) method for subspace identification of linear time-invariant (LTI) systems, with detailed extensions to linear parameter-varying (LPV) and nonlinear 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. The result is 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. Comparisons made between system identification methods including subspace, prediction error, and maximum likelihood, and show considerably less computation and higher accuracy.

The LTI subspace methods are extended to LPV systems that are expressible in the LTI form where the constant LTI parameters are multiplied by parameter-varying scheduling functions depending on the system operating point. For example, this allows for the identification of constant underlying structural stiffness parameters while wing flutter dynamics vary with scheduling functions of speed and altitude operating point variables. This is further extended to Quasi-LPV systems where the scheduling functions may be functions of the inputs and/or outputs of the system. Quasi-LPV systems include bilinear and general polynomial systems that are universal approximators. The developed subspace identification method for parametervarying systems avoids the exponential growth in computations characteristic of previous SIM methods. Applications are discussed to monitoring and fault detection in closed-loop chemical processes, identification of vibrating structures under feedback, adaptive control of aircraft wing flutter, identification of the chaotic Lorenz attractor, and identification and monitoring of Quasi-LPV automotive engines.