EE611
Deterministic Systems
State Feedback
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
State Feedback
Block diagram of a system with state feedback:
ẋ
x
ut 
1
rt 
b
c
+
+
s
-
A
Reference input
Given original system:
ẋ=A xb ut 
y t =c x
Show that state feedback
y t 
k
State feedback
ẋ= A−bk xbr t 
system becomes: yt =c x
Controllability and Feedback
Can the application of state feedback to a system result
in a loss of controllability?
Theorem 8.1
Given pair (A-bk, b) and 1xn feedback vector k, then
(A-bk, b) is controllable iff (A, b) is controllable.
Controllable Canonical Form
Given controllable system ẋ=A xb ut 
y t =c x
with characteristic polynomial
 s=∣s I−A∣=sn1 s n−1 2 sn−2... n−1 s− n
system can be transformed into controllable canonical
form by x =P x
[ ]
1
0
Q=P−1=[ b A b A 2 b A 3 b ... A n−1 b ] 0
⋮
0
1
1
0
⋮
0
2
1
1
⋮
0
...  n−1
...  n−2
...  n−3
⋮ ⋮
... 1
Controllable Canonical Form
[ ][
̇x 1
−1 − 2
̇x 2
1
0
⋮ = ⋮
⋮
0
0
̇x n−1
0
0
̇x n
][ ] [ ]
... − n−1 − n x 1
1
0
x 2
...
0
0
⋱
⋮
⋮
⋮  ⋮ ut 
0
...
0
0 ̇x n−1
0
...
1
0
x n
y t =[ 1 2 ...  n−1 n ]
n−1
[]
n−2
x 1
x 2
⋮
x n−1
x n
1 s 2 s ⋯n−1 sn
y  s
g  s=
=
u
 s sn  1 sn−1 2 sn−2 ⋯ n−1 s n
Eigenvalue Placement
System behavior is modified by changing its
eigenvalues through feedback.
Theorem 8.3
Given n-dimensional controllable state equation:
ẋ=A xb ut 
yt =c xd ut
then state feedback u = r – kx with 1xn real-valued
feedback vector k, the eigenvalues of A-bk can be
arbitrarily assigned provided that complex conjugate
eigenvalues are assigned in pairs.
Example
For unstable system below, find a feedback vector k to
place eigenvalues at -4 and -2:
[ ] []
ẋ=
2 24
1
x ut 
1 0
0
y t =[ 1 2 ] x
Show k = [ 8 32] and that observability is lost as a
results of the feedback.
Example
Use controllable canonical form to find feedback vector
k to place eigenvalues at -1±j1 and -(2)0.5
[
] []
2 1 0
0
ẋ= 0 2 0 x 1 ut 
0 0 −1
1
y t =[ 2 −1 −1 ] x
Show k = [ 6.41, 4.83, -1.77]P = [11.38, 6.37, .05]
And for controllable canonical form transformation:
[
0
1
1
Q= 1 −1 −2
−1 −4 4
]
Regulation and Tracking
Regulator Problem: Design feedback system to bring
response to zero in response to a disturbance/initial
condition.
Tracking Problem: Given a step change in reference r
to value a for t > 0, design a feedback system so that
response y(t) approaches a as t approaches infinity
(asymptotic tracking).
Regulation
As long as real parts of poles/eigenvalues are negative
regulation is achieved. In cases where eigenvalues are
too close to the jω axis, response can be improved by
moving eigenvalues into desired range.
y t=c exp   A−bk  t  x 0
Tracking
Poles/eigenvalues must be negative as in the case of
regulation. In addition a gain (scaling) on reference
signal must be applied to eliminate steady-state error in
the tracking problem.
ut = p rt −kx
Gain p needs to be the reciprocal of the transfer function
under the following limit:
n
1
p=lim
=
n
s  0 g f s
Example
Design a tracking system with state feedback such that
system will settle in less than 2.5 seconds (within 2% of
steady-state error) after a step change at time t=0, with
0 steady state error.
[
][]
−0.75
1.25
−1.25
2
ẋ= −0.075 −1.375 0.5375 x 0.5 u t 
−0.25 −1.25 0.125
1
y t =[ 1 0 1 ] x
Show eigenvalues −3.2  2 , −3.2± j 3.2 result in a
k = [122.7, 273.2, -373.0] and p= 141.2
Stabilization
If an unstable system is not controllable, it may still be
possible to stabilize it with feedback if the unstable state
variables are associated with the controllable part of the
system
[ ][
][ ] [ ]
c A
 12 x

x
A
bc
̇ c
c
=

ut 
 c x
x c
0 A
0
̇
 c
A partitioned state feedback vector can be introduced
such that:
x

u=r−k x=r−k x=r−[ k c k c ] c
xc
[]
and
[ ][
 c− 
 12 − 
A
bc k c A
bc k c
̇x c
=
 c
x
0
A
̇ c
][ ] [ ]

x
bc
c

ut 
x
0
 c
Example
Determine if the unstable system can be stabilized. If so, find the
feedback vector to stabilize the system.
[
0
0
0
ẋ= 0
0
0
0
0 0
0
0
0
1 0
0
0
0
0 −2 0
0
0
0 0 −1 1
0
0 0 −1 −1 0
0 0
0
0
1
0 0
0
0 −2
] []
0
1
0
1
0
0
0 x 0 ut 
0
0
2
1
1
1
y t =[ 1 −1 2 0 1 0 1 ] x
Show system is not controllable, but can be stabilized. If
unstable modes have their real parts shifted left by 2, the
feedback vector is k = [-2, -12, 0, 0, 0, 4, -6], this leaves the
originally stables modes unaffected.
Lecture Note Homework U11.1
Implement the system below in Simulink before and after
feedback compensation (problem 8.7 in text) for a unit step
response. Find feedback vector k such that resulting eigenvalues
are at -2, -1±j and will track asymptotically any step reference
input.
1 1 −2
1
y t =[ 2 0 0 ] x
ẋ= 0 1 1 x 0 u t
0 0 1
1
[
] []
Use the state space model for the original system, however add a
state feedback loop using summers, gains, products, and
constants. Hand in a copy of the simulation schematic with the
corresponding plot of their outputs. Assume the system is
relaxed at t = 0.