Discussion on: "A Stacked Model Structure for Off-Line Parameter Variation Estimation in Multi-equilibria Nonlinear Systems"

Preventive diagnostics and maintenance of complex dynamic systems such as chemical plants and aircraft engines benefit from knowledge of the system components. The conditions of the components are often represented by the parameters of the system, in which case the current conditions of the components can be deduced from the variations of the system parameters about their nominal values. This paper considers nonlinear dynamic systems that tend to operate about equilibrium points and focuses on off-line estimation of the variations of important parameters about their nominal values. It uses the common practice of treating the parameter variation estimation vector as a subset of a large state vector, which here is formed by stacking state-space plant descriptions corresponding to selected equilibrium points. A fundamental problem for parameter estimation, no matter which parameter estimation method is used, is that the parameters of interest may not be observable unless data from an appropriate number and combination of equilibrium points is used in the estimation process. Using analysis based on linearized models, this paper establishes criteria for selecting the combinations of equilibrium points that, when stacked, render the parameter vector observable in the sensor data collected about these equilibrium points. A nonlinear filter is then applied to estimate the parameter variation vector as part of the augmented (or stacked) state vector. Numerical simulation results of a Continuously Stirred Tank Reactor verify the theory. In particular, it is seen, as predicted by the theory, that data from one, two or three equilibrium points is not sufficiently rich to yield information about each parameter variation. It is also seen that not all arbitrary combinations of four equilibrium points can provide the information required for estimation of the parameter variation vector; only combinations that satisfy the proposed criteria were able to yield successful estimation of all parameters.