Pseudo-linear identification: optimal joint parameter and state estimation of linear stochastic MIMO systems
Hopkins, Mark A.
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This dissertation presents a new method of simultaneous parameter and state estimation for linear, stochastic, discrete—time, multiple-input, multiple-output (MIMO) (B systems. This new method is called pseudo·Iinear identification (PLID), and extends an earlier method to the more general case where system input and output measurements are corrupted by noise. PLID can be applied to completely observable, completely controllable systems with known structure (i.e., known observability indexes) and unknown parameters. No assumptions on pole and zero locations are required; and no assumptions on relative degree are required, except that the system transfer functions must be strictly proper. Under standard gaussian assumptions on the various noises, for time-invariant systems in the class described above, it is proved that PLID is the optimal estimator (in the mean-square·error sense) of the states and the parameters, conditioned on the output measurements. It is also proved, under a reasonable assumption of persistent excitation, that the PLID parameter estimates converge a.e. to the true parameter values of the unknown system. For deterministic systems, it is proved that PLID exactly identifies the states and parameters in the minimum possible time, so—called deadbeat identification. The proof brings out an interesting relation between the estimate error propagation and the observability matrix of the time-varying extended system (the extended system incorporates the unknown parameters into the state vector). This relation gives rise to an intuitively appealing notion of persistent excitation. Some results of system identification simulations are presented. Several different cases are simulated, including a two-input, two-output system with non-minimum-phase zeros, and an unstable system. A comparison of PLID with the widely used extended Kalman filter is presented for a single-input, single·output system with near cancellation of a pole-zero pair. Results are also presented from simulations of the adaptive control of an unstable. two-input, two-output system In these simulations, PLID is used in a se1f—tuning regulator to identify the parameters needed to compute the feedback gain matrix, and (simultaneously) to estimate the system states, for the state feedback
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