Optimal control paper-medical science

Optimal control paper-medical science

Problem 1: Constrained, Free Final Time Optimal Control. (30 points)
Generate the free final time optimal control for the system ̇
=
with the initial state zero with the final
state constraint ()
= 1. In Simulink use a time step of 0.05 seconds and initial guess at the
control of 0.1 for all time and initial guess of final time of 1 sec.
a) Impose a control (input) constraint of +/-1 and set the lower bound on final time at 0.1 seconds
and the MAXIMUM value of tfinal to be 10 seconds. Find the optimal control when the
performance index is
=
+ ∫ (

− )

. Plot the final state and control time
histories and show the final cost value.
b) Keep the input and final time constraints the same but change the performance to
=
+
∫ (

− 2.0)

. Plot the final state and control time histories and show the final cost
value.
c) Now impose a tfinal maximum constraint of 20 sec and find the optimal control, leaving the
minimum value of tfinal to be 0.1 seconds. Plot the final state and control time histories and
show the final cost value.
Problem 2: Full-State Feedback, Observers, and Kalman Filter State Estimation (50 points)
Modify the Simulink diagram you developed for Homework 6 to implement the following dynamics and
initial conditions: ̈+
0.7̇
+
= ,
̇(0)
= 1, (0)
= 0, and have the output be
= .
Set the random
noise variance to 0.1.
a) Develop the feedback controller using the LQR function with Q identity and R identity. Using
the true full-state feedback, plot the state and control time histories along with the output time
history. Leave the controller gain fixed for the next parts.
b) Develop an observer using Q=10*eye and R identity. Plot the observer state estimates. If they
look close to the true states (and they should), then use those estimated states in the feedback
controller. Plot the true states and the control time history. Leave the observer gain fixed for the
next part.
c) Implement the Kalman filter to estimate the states. First, use Q as identity and R as identity.
Compare the Kalman filter state estimates when using the true states in the feedback control.
Assuming they look reasonable, close the loop using the Kalman filter state estimates in the
feedback and plot the time histories of the states and controls.
d) Finally, after using both the observer-reconstructed states and the Kalman filter-reconstructed
states, which of the two produce the best results when used in the feedback control (i.e. which
estimate the states better and which has a better quality control signal).
Problem 3: Dynamic Programming Minimum Cost Path through a Grid (20 points)
Given the grid shown below, determine the optimal path when starting at the goal (x = p) and working
backward to the beginning point (x = a), taking the lowest cost path at each decision point. What is the
final cost taking that approach? Now look for a lower cost by trying different paths. Is there a lower
cost? Complete the table below which specifies the control action to take at each step, k, when taking the
lowest cost route. Note that u = 1 means going up and u = -1 means going down.
k = 0 x = a u = _
k = 1 x = _ u = _
k = 2 x = _ u = _
k = 3 x = _ u = _
k = 4 x = _ u = _
k = 5 x = _ u = _
k = 6 x = p
Bonus question (5 points)
Describe in words and potentially a MATLAB pseudo-script.
When imposing control constraints we’ve been using the lower bound and upper bound arrays in the
fmincon function. However, we sometimes need to impose state constraints, such as in a situation where
we want to do a min time solution going between stoplights, starting from zero velocity and coming to a
full stop at the next light. But we don’t want to exceed the speed limit. Describe in words or pseudo
scripts how you would implement such a state constraint in a numerical optimization, similar to what
we’ve been doing already.

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