Review & today's topics State feedback - UBC Mechanical ...

MECH468
Modern Control Engineering
MECH550P
Foundations in Control Engineering
Lecture 21
State feedback design
Lyapunov equation method
Dr. Ryozo Nagamune
Department of Mechanical Engineering
University of British Columbia
Review & today’s s topics
• In the last two lectures
• State feedback
• Pole placement theorem: : Arbitrary pole placement is
possible by a state feedback u=-Kx
if and only if (A,B)
is controllable
• Direct method & canonical form method to compute
the feedback gain K (“place.m(
place.m”)
• Today’s s topics
• Lyapunov equation method
• Where to place closed-loop loop poles
• Stabilizability
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State feedback (review)
• Block diagram
Plant
Pole placement theorem (review)
• If (A,B) is controllable, the eigenvalues of (A-BK)
can be placed arbitrarily (provided that they are
symmetric w.r.t. . the real axis).
Im
• Open-loop and closed-loop loop systems
Re
K
: Closed-loop loop poles (design parameters)
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Review & today’s s topics
• In the last two lectures
• State feedback
• Pole placement theorem: Arbitrary pole placement is
possible by a state feedback u=-Kx
if and only if (A,B)
is controllable
• Direct method & canonical form method to compute
the feedback gain K (“place.m(
place.m”)
• Today’s s topics
• Lyapunov equation method
• Where to place closed-loop loop poles
• Stabilizability
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Lyapunov equation method
Step 0: Check if (A,B) is controllable. If it is, go to
Step 1.
Step 1: Select a matrix F with desired eigenvalues
s.t.
Step 2: Select a matrix
Step 3: Solve the Lyapunov equation w.r.t. . T
Step 4: If T is singular, redo from Step 3 with
different
Step 5: If T is nonsingular,
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Remarks on Lyapunov eq. . method
• How to construct F
• Suppose that the desired closed-loop loop poles are
Idea of Lyapunov eq. . method
• From Steps 3 & 5,
Then, F can be chosen as
• If T is nonsingular
• How to construct : Select randomly!
• Condition for the unique solution of the
Lyapunov equation:
• Conditions
are necessary for the Lyapunov equation to
have a nonsingular solution. (Proof omitted.)
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• SS model
An example
DC motor position control
http://www.engin.umich.edu/group/ctm/examples
www.engin.umich.edu/group/ctm/examples/
• CL poles
• Step 1:
• Step 2:
• Step 3:
• Step 5:
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DC motor position control (cont’d)
• State-space model
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DC motor position control (cont’d)
• Specifications: For
• r(t)=0
• Settling time < 40 ms
• Overshoot < 16%
• Zero steady state error
• Open-loop system
• Poles = 0, -59.226
-1.4545E+6
• Not asymptotically stable!
1
0 0.02 0.04 0.06 0.08 0.1
Time (sec)
Feedback control for stability & performance!
Position (rad)
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1.02
1.015
1.01
1.005
y(t) ) does not
go to zero!
3

Position (rad)
1
0.5
0
State feedback position control
-0.5
0 0.02 0.04 0.06 0.08 0.1
Time (sec)
Control input (volt)
10
0
-10
0 0.02 0.04 0.06 0.08 0.1
Time (sec)
Position (rad)
1
0.5
0
-0.5
0 0.02 0.04 0.06 0.08 0.1
Time (sec)
Control input (volt)
As the pole is moved away from the real axis, the overshoot
becomes larger and the response becomes faster.
10
0
-10
0 0.02 0.04 0.06 0.08 0.1
Time (sec)
Exercise
• Try the simulation by yourselves! (Matlab(
code is
on the course homepage.) Change the pole
location, and get a feeling how responses are
affected by the pole location.
• For the system below, design a state feedback
u=-Kx
so that the closed-loop loop system has -11 and
-22 as its eigenvalues, , by Lyapunov eq. . method.
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Review & today’s s topics
• In the last two lectures
• State feedback
• Pole placement theorem: Arbitrary pole placement is
possible by a state feedback u=-Kx
if and only if (A,B)
is controllable
• Direct method & canonical form method to compute
the feedback gain K (“place.m(
place.m”)
• Today’s s topics
• Lyapunov equation method
• Where to place closed-loop loop poles
• Stabilizability
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Rules of thumbs
• Move the poles for improvement of stability and
performance.
• Do not try to move poles farther than necessary.
• Control input should not be too large.
(desired CL) – (OL)
• Place the poles in similar distances from the
origin, , to make the control effort efficient.
• Control effort depends on the farthest poles, while
speed depends on the closest poles to the origin.
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Butterworth configuration
• 1930 Stephen Butterworth
Review & today’s s topics
• In the last two lectures
• State feedback
• Pole placement theorem: Arbitrary pole placement is
possible by a state feedback u=-Kx
if and only if (A,B)
is controllable
• Direct method & canonical form method to compute
the feedback gain K (“place.m(
place.m”)
• Today’s s topics
• Lyapunov equation method
• Where to place closed-loop loop poles
• Stabilizability
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Stabilizability
• Suppose that (A,B) is NOT controllable.
• If the “uncontrollable part” of A-matrix A
is stable,
then the system is called stabilizable.
Controllable pair
Eigenvalues of this cannot be changed by state feedback.
Summary
• State feedback
• Lyapunov eq. . method to design state feedback gain
• Where to place closed-loop loop poles (one always needs
trial-and
and-error)
• Rules of thumbs
• Butterworth configuration
• Optimal method (We will learn more later in LQR.)
• Stabilizability
• Next,
• Tracking (servo) control
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Homework
• 4-1: For the inverted pendulum model in
Homework 2-32
3 (see slides for Lecture 10),
design a state feedback controller.
• Initial condition
• The cart stops at y=0 with the upright pendulum.
• The settling time should be at most 5 sec.
• The input should be less than 15N all the time.
Plot the trajectories of states and input by, for
example, Simulink or lsim.m.
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