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 show for linear parameter-varying (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, Quanser 3 DOF gyroscope identification (Quasi-LPV, NL), 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. In addition, an overview is provided on other LPV subspace identification approaches and iterative nonlinear optimization based identification schemes.

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). Unfortunately, LPV and NL subspace identification approaches suffer heavily from the curse of dimensionality where computational complexity grows exponentially with linear increase in the number of inputs, outputs, or states and can result in ill-conditioned estimation problems with high parameter variances for real-world applications. In contrast, nonlinear optimization approaches based on the prediction error minimization (PEM) framework result in a heavy nonlinear optimization problem, prone to local minima and convergence problems. The workshop presents a first principles statistical approach using the fundamental canonical variate analysis (CVA) method for subspace identification of linear time-invariant systems, with detailed extensions to linear parameter-varying 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 multistepahead 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 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 (Larimore, Cox and Tóth, ACC2015). Quasi-LPV systems include bilinear and general polynomial systems that are universal approximators. The developed CVA subspace identification method for parameter-varying systems avoids the exponential growth in computation characteristic of many other SIM methods.

The workshop is continued with introducing alternative LPV subspace identification approaches, including novel basis reduced realization scheme on the impulse response (Cox, Tóth and Petreczky, LPVS2015), predictor based subspace approaches, and other alternatives. To improve the model estimate of SIMs in terms of the prediction error, it will be shown how this model estimate can be used as an efficient initialization for iterative nonlinear optimization techniques to overcome convergence problems and to decrease the computational load of the nonlinear optimization. The workshop will cover the two most popular nonlinear optimization techniques: the expectation-maximization method and the gradient-based optimization of the prediction error. Comparisons are made between system identification methods including subspace, prediction error, and maximum likelihood, for which CVA achieves considerably less computation time and higher accuracy.