# WUSTL ESE 543 - ESE 543 Homework 8 PP RS and Obs (2 pages)

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## ESE 543 Homework 8 PP RS and Obs

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- Pages:
- 2
- School:
- Washington University in St. Louis
- Course:
- Ese 543 - Control Systems Design by State Space Methods

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11 12 2012 Due 2 weeks Homework 7 1 The linearized suspended ball is described by 0 1 0 x x u 1 0 1 a Use state feedback to stabilize the system producing closed loop eigenvalues at 1 1 2 b The ball position x1 can be measured using a photocell but the velocity x2 is more difficult to obtain Suppose therefore that y x1 Design a full order observer having poles at 4 and 5 and use the observer feedback to produce closed loop eigenvalues at 1 2 1 4 5 c repeat b using a first order observer with pole at 6 Give a block diagram showing the controller as a single transfer function 2 Consider 7 2 6 1 1 x 2 3 2 x 1 1 u 2 2 1 1 0 a What eigenvalues can be obtained using state feedback Now suppose 1 1 2 y x 1 1 11 b Describe the pole placement possibilities using an observer and observer feedback 3 Suppose the LTI system is subjected to an unmeasurable disturbance w x Ax Bu Ew y Cx Suppose an observer is constructed in the following manner x Ax Bu K y y y Cx a is the observer estimating the state i e does x t x t b Suppose w is an unknown constant Suggest how an observer can be used to estimate w as well as the state x Hint w 0 Append w to the state vector forming a new state z and estimate z w 4 Consider again the magnetically suspended ball Suppose m 0 02 kg c 8 10 5 mks units g 10 for numerical convenience a It is desired that the nonlinear ball system have an equilibrium position at X 1 0 02 Find the corresponding equilibrium current b Obtain a linearized state model based upon a c Suppose y x1 for the linearized model Use a reduced order observer and pole placement to meet the following specifications 11 12 2012 Due 2 weeks i Error settling time for the observer is less than 0 08 seconds e x x ii settling time for the closed loop system is less than 0 25 seconds and overshoot to an initial offset in position deviation x1 is less than 20 d Draw a block diagram of your design as implemented on the actual system Show the observer feedback as a SISO controller and indicate the set point as a constant reference input e Suppose the parameter c is actually 20 larger than the value given What set point is obtained for the real nonlinear system If you change the parameter what steady state value will actually be obtained f Repeat this same design problem using the robust servo approach obtaining integral control 5 Consider the design of a missile autopilot Using the robust servo formulation presented in class design a pitch autopilot commanding angle of attack Use the following dynamics model as the nominal plant model Z q V M Z 1 e V e q 0 M e and use data for 16 page 32 of lecture notes a Design the autopilot to track a constant command Place the poles of the closed loop system at 5 6 7 b Design an autopilot to track a sinusoidal command Choose the closed loop poles yourself as long as they are in the left half plane

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