11/12/2012 Due: 2 weeks Homework 7 1. The linearized suspended ball is described by 01 010 1x xu  = +    . a) Use state feedback to stabilize the system producing closed loop eigenvalues at -1, -1/2. b) The ball position 1x can be measured using a photocell, but the velocity 2x is more difficult to obtain. Suppose, therefore, that 1yx=. 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 26 112 32 112 21 10x xu−−    = −− + −    −−  . a) What eigenvalues can be obtained using state feedback? Now suppose 1 121 1 11yx−−=. 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 Ewy Cx=++= Suppose an observer is constructed in the following manner: ( )ˆˆˆˆˆx Ax Bu K y yy Cx=++ −= a) is the observer estimating the state, i.e. does ( ) ( )ˆxt xt→? b) Suppose w is an unknown constant. Suggest how an observer can be used to estimate w as well as the state. (Hint: 0w=. Append w to the state vector, forming a new state xzw=, and estimate z.) 4. Consider again the magnetically suspended ball. Suppose m = 0.02 kg, 58 10c−= × (mks units), g = 10 (for numerical convenience). a) It is desired that the nonlinear ball system have an equilibrium position at 10.02X =. Find the corresponding equilibrium current. b) Obtain a linearized state model based upon a). c) Suppose 1yxδ= 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, ˆexx= − ii) settling time for the closed loop system is less than 0.25 seconds, and overshoot to an initial offset in position deviation 1xδ 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 α. Use the following dynamics model as the nominal plant model: 10eeeZZVVqqMMδααδααδ = +   and use data for 16α= (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