NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 31. Observer Design

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1 Observer Design Dr. Bishakh Bhattacharya h Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD

2 This Lecture Contains Full State Feedback Control for System in Non canonical form Ackermann s algorithm Bass Gura formulation Numerical Example Joint Initiative of IITs and IISc - Funded by MHRD

3 Introduction o to Observer e Design The design of full-state feedback control assumes the accessibility (possibility of sensing) of the complete state vector. However, in reality one may have only a subset of them available for direct sensing while the other states are to be estimated via simulation. Accepting that there will be finite error in this process, the focus is whether the error could be driven to zero at a faster rate than the plant-dynamics. Obviously, such a strategy is feasible only if the states are observable.

4 Observer e in a block-diagram daga reference u Plant Output (y) u B + + A Observer + Controller Estimated States () C - + Y L

5 Design of an Observer The governing equation for a dynamic system (Plant) in state- space representation may be written as: X AX Bu, Y CX The governing equation for the Observer based on the block diagram is shown below. The superscript ^ refers to estimation. Xˆ AXˆ Bu L(Y Yˆ CX ˆ Y) ˆ Define the error in estimation of state vector as e X (X X) ˆ Joint Initiative of IITs and IISc Funded by MHRD 5

6 Observer e Design based on Error Dynamics The error dynamics could be derived now from the observer governing equation and state t space equations for the system as: e Y (A LC) X e X Yˆ Ce X. The corresponding characteristic equation may be written as: si (A LC) You need to design the observer gains such that the desired error dynamics is obtained.

7 Case A: Observer design for canonical system Case Obse e des g o ca o ca syste Suppose, the system [A, C] is available in observer canonical form: a n l ˆ, ˆ, ˆ 2 C L A l l n a a

8 Observer Design for Case A The first thing you need to check is whether the system is fully observable or not. This can be done by checking whether the rank of the observability matrix equals the order of the system as stated earlier. Once the answer is affirmative you may proceed for the observer design. Whenever, the desired eigen-values related to the error-dynamics are specified, one can construct the desired characteristic equation identical to controller design. The observer gain matrix for such cases may be obtained from the simple relationship l d a i i ni ni Here d and a refer to the vector coefficients of the desired and the open-loop characteristic polynomial. n Joint Initiative of IITs and IISc Funded by MHRD 8

9 Observer design for system in non- canonical form If a system is not in observer canonical form, then one needs to transform the system matrices first into the particular canonical form. The transformation matrix required for such cases has been derived as T O O ˆ Here, O and Oˆ are the observability matrices related to the non-canonical and canonical forms respectively. After obtaining the observer gains in observer canonical form, one can transform the gain vector to the original non-canonical form as: L TLˆ O Joint Initiative of IITs and IISc Funded by MHRD 9

10 Assignment: The system matrices for a plant (A, B and C) are as follows: 4 A 2, B and C Design an observer for the plant, where the desired characteristic polynomial is given by: s 3 2 s 2 25s 5 Joint Initiative of IITs and IISc Funded by MHRD

11 Special References for this lecture Control System Design, Bernard Friedland, Dover Control Systems Engineering Norman S Nise, John Wiley & Sons Design of Feedback Control Systems Stefani, Shahian, Savant, Hostetter Oxford Joint Initiative of IITs and IISc Funded by MHRD

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