# Document 11696275

```ROSE&shy;HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 506
Final Project Due 19 November 2015
1. Control of Ball Plate balancing system using LQ and classical LQG methods This project will take you through the full process of control design including linearization, controllability and observability analysis, feedback control and Kalman filter design.
Consider the problem of balancing a ball on a plate where the plate is rotated along two perpendicular axes by a set of nested gimbals. We will use the coordinate definitions such that the outer gimbal rotates the platform an angle of  about the y axis, causing a motion in the positive x direction. The inner gimbal rotates the platform subsequently through an angle of  about the x axis, resulting in a motion in the negative y direction. The differential equations of motion are as follows:
x  t  
5
g sin    d1
7
(1)
5
y (t )   g sin   cos   d 2
7
Where d1 and d2 are disturbances on the ball. The disturbances are zero&shy;mean, mutually independent, white gaussian random processes with intensities defined by cov d1  t  d1  k    0.16  t  k 
cov d 2  t  d 2  k    0.25  t  k 
(2)
cov d1  t  d 2  k    0
1. Show that Taylor series linearization gives the same result as a small angle 
approximation when linearizing about x , x  0 , y , y  0 ,  ,  0
2. Find the state space description of the linear model with states x, x , y , y , and controls


 , , and outputs x, y. 3. Determine whether the system is observable and controllable. Also, determine whether the system would be observable if only x were measured.
4. LQR Design with Perfect Sensors. As a prelude to stochastic LQG design let us carry out
a classical LQR design assuming for the time being that we can measure, and feedback, all four state variables exactly, i.e. without any sensor noise. Design an LQ regulator using the following quadratic cost functional:



J   x 2  t   y 2  t   101 / 2  2  t   101 / 2 2  t  dt
0
(3)
Note that the use of this cost functional implies that we are concerned much more with the amount of control effort as compared with fast convergence of the position states.
Homework 9
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ROSE&shy;HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 506
Design the LQ regulator and compute its closed&shy;loop poles. Carry out a deterministic simulation of the closed&shy;loop LQ regulator for the initial plant state. Be certain to use the non&shy;linear plant dynamics shown above for all your simulations.
x(0) = 10 cm ( = 0.1 m); all others zero
and plot the states and the two controls. Repeat with the initial condition
y(0) = 10 cm ( = 0.1 m)
x(0) = 10 cm and y(0) = 10 cm, all others zero
(3)
5. LQG Design with Two Sensors. Assume that you can measure both position states, but not the velocity states. Solve the LQG problem. Calculate the closed&shy;loop poles of the LQG design and relate to those of the Kalman filter and of the LQ regulator of part 4. Assume that both sensors had additive white zero&shy;mean Gaussian sensor noise, and the noise intensity is 1x10&shy;6 m2 for each sensor.
Set the disturbance torques and sensor noises to zero and carry out a deterministic simulation of
the closed&shy;loop LQG regulator for the initial plant state (3) and plot the states and the two controls. Be sure to use the non&shy;linear plant equations for the plant (linear internal model for the Kalman filter). You may set the initial estimator states to zero, or small random values (do not match the plant initial conditions). Compare with those found in part 4.
Your report should include at least the following plots: Time response of all four states for the each of the LQR simulations (3 cases).
Actual and estimated states for each of the LQG simulations (3 cases).
Homework 9
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