10_observers - Control of Nonlinear Dynamic Systems Theory and Applications J K Hedrick and A Girard 2005 10 `Nonlinear Observers Key points All the

10_observers - Control of Nonlinear Dynamic Systems Theory...

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 1 10 ` Nonlinear Observers Introduction to Nonlinear Observers Motivation The big weakness of all the control methodologies that we have learned so far is that they require the full state . Sometimes impossible to measure. Sometimes, possible, but expensive. Key points All the control methodologies covered so far require full state information. This can be impossible, or expensive, to obtain. Nonlinear observers: o Deterministic: Lyapunov based: Thau, Raghavan Geometric Sliding o Stochastic: Extended Kalman Filter (EKF)
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 2 Methodologies learned so far: Linearization I/O and I/S feedback linearization Sliding control (robust I/O linearization) Integrator backstepping Dynamic surface control Note: make sure you understand what the similarities and differences between all those methodologies are! In general, we only have access to p sensor outputs, that is: ) ( . t v x M z + = where: z is the measurement (of size px1) M is the measurement matrix (of size pxn) x is the state (of size nx1) v(t) represents measurement noise (of size px1) Even in nonlinear systems, the measurements will be linearly related to the state in general (property of any useful sensor). The best-known methodology for dealing with a full state feedback controller is to separate the problem into a static controller (for example u = -kx) and a dynamic observer. We then: a. Design the controller as if x x ˆ = . b. Design the observer so that x x ˆ as quickly as possible.
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 3 Review of Linear Observers Process: ) ( t w Bu Ax x + + = & Measurement: ) ( . t v x M z + = Let: ) ˆ ( ˆ ˆ x M z L Bu x A x + + = & where the last term is basically a correction term ( z z ˆ ) The error dynamics are expressed by: x x x ˆ ~ = ) ( ) ( ~ ) ( ) ˆ ( ˆ ˆ ~ t Lv t w x LM A x M v Mx L Bu x A w Bu Ax x x x + = + + + = = & & & There are two different classical approaches to dealing with the choice of L: A. Deterministic (Luenberger observer) Ignore w(t), v(t). x LM A x ~ ) ( ~ = & and if (A,M) is observable, then the eigenvalues of (A-LM) can be placed arbitrarily. B. Stochastic (Kalman-Bucy filter) L is chosen to minimize the variance of x x ˆ , the state estimation error. ) ˆ ( ˆ ˆ x M z L Bu x A x + + = & 1 = R PM L T (Intuition: if the variance is small make the gains large) R and Q are noise statistics (R is associated with v) Note that R -1 is identical to 1/r for scalars. MP R PM Q PA AP P T T 1 + + = & The steady-state Kalman filter is usually implemented. In that case the last equation becomes:
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 4 0 1 = + + MP R PM Q PA AP T T and is usually referred to as the Algebraic Ricatti Equation (are).
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  • Spring '08
  • HEDRICK
  • Linear system, Nonlinear Dynamic Systems: Theory and Applications, J. K. Hedrick, Nonlinear Dynamic Systems

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