EE16B M. Lustig, EECS UC Berkeley
EE16BDesigning Information
Devices and Systems IILecture 7B
Outputs and Observability
EE16B M. Lustig, EECS UC Berkeley
Intro
• Last time– State feedback control– Eigen value assignment
• Today:– Example of cooperative, adaptive cruise control– Output and observability– (MRI as a dynamical system – maybe)
EE16B M. Lustig, EECS UC Berkeley
GOOGLE PROJECT LOON BALLOONS
https://x.company/loon/
EE16B M. Lustig, EECS UC Berkeley
EE16B M. Lustig, EECS UC Berkeley
EE16B M. Lustig, EECS UC Berkeley
Cooperative Adaptive Cruise control
Example:
EE16B M. Lustig, EECS UC Berkeley
Cooperative Adaptive Cruise control
Q) What eigen-values will you want here?
EE16B M. Lustig, EECS UC Berkeley
Let’s look at input more closely…
But leader chooses his own acceleration ul(t)
Q) What does the follower need to know to implement?
A) Cooperative (vehicle2vehicle comm.) range sensor (for distance and velocity)
EE16B M. Lustig, EECS UC Berkeley
Outputs
Can’t always measure state directly or all states…
Define output:p x n matrix for p outputs
EE16B M. Lustig, EECS UC Berkeley
Outputs
Can’t always measure state directly or all states…
Define output:p x n matrix for p outputs
EE16B M. Lustig, EECS UC Berkeley
Observability
A system is “observable” if, by watching y(0),y(1),y(2),… we can determine the full state
Two stage approach:1) Determine initial state x(0) from y(0),y(1),….
2) Ignore input:
EE16B M. Lustig, EECS UC Berkeley
Observability
Q: What conditions on Ot, to determine x(0) uniquely?
A: Ot must have n independent rows strictly On-1 has full rank null-space is {0}
Observability has rank = n
EE16B M. Lustig, EECS UC Berkeley
Observability
With input:
EE16B M. Lustig, EECS UC Berkeley
Example
rotation matrix
EE16B M. Lustig, EECS UC Berkeley
Example
rotation matrix
EE16B M. Lustig, EECS UC Berkeley
Example
rotation matrix
EE16B M. Lustig, EECS UC Berkeley
Example
rotation matrix
Q: What is ϴ = 179o?
EE16B M. Lustig, EECS UC Berkeley
State Feedback Control
w/o observer
EE16B M. Lustig, EECS UC Berkeley
A Common Observer Algorithm
Start with initial guess Update estimate each time using:
Copy of system model correction
observer
EE16B M. Lustig, EECS UC Berkeley
Kalman Filter• Accounts for noise and errors in our system model and inputs
Copy of system model correction
A more elaborate form of the observer where the matrix L is also updated at each time, is known as the Kalman Filter and is the industry standard in navigation. The Kalman Filter takes into account the statistical properties of the noise that corrupts measurements and minimizes the mean square error between x(t) and xˆ(t)
EE16B M. Lustig, EECS UC Berkeley
Control Recap
• Controllability:
If Rn is full rank then we can move to any target valueSame rank test for continuous time
• Open loop control:Can use the above equation to design an input sequence – and apply it blindly. Accuracy of result will depend on accuracy of model.
EE16B M. Lustig, EECS UC Berkeley
Control Recap – State Feedback
Closed-loop system:
If controllable, can assign eigenvalues for A+BK arbitrarily
If not, some eigenvalues of A can not be changed!(could be OK, if stable, bad news if not)
Must choose K s.t. A+BK has eigenvalues inside the unit circle (or left half-plane for continuous time)
EE16B M. Lustig, EECS UC Berkeley
Control Recap - Observers
Not all state variable are measured, but we get “outputs”
On-1 must have n independent rows (full rank) to determine x(0) uniquely from output
EE16B M. Lustig, EECS UC Berkeley
How Does MRI Work? (some today – more later!)
•Magnetic Polarization -- Very strong uniform magnet• Excitation
-- Very powerful RF transmitter• Acquisition
-- Location is encoded by gradient magnetic fields-- Very powerful audio amps
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