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