This lecture presents an introduction to a very powerful design method, Backstepping.
Emphasis is upon the need for Backstepping Technique, instead of presenting a procedural
form of the technique. A more procedural form will be provided by ECE 874.
1
Remember the state equations of anti-angiogenic tumor growth model:

p
p   p 1  
q 

q  bq  dp
2/3
q  G uq
p  R  : tu m o r vo lu m e [ m m 3]
1
q  R  : carryin g cap acity o f vascu lar n etw o rk [ m m 3]
1
  R  : tu m o r g ro w th p aram eter
b , d  R  : g ro w th co n stan ts
G  R  : an ti  an g io g en ictu m o r k illin g p aram eter
u  R  : m an ip u lated co n tro l in p u t d ep icts th e d ru g d o se
 co n c. 
2

p
p   p 1  
q 

q  bq  dp
2/3
q  G uq
Last week, we designed a controller to drive the carrying capacity of the vascular network, q,
to a desired trajectory, by using the control input, u. Remember that the equilibrium point
analysis shows that tumor volume, p, follows q at equilibria.
What if we want to stabilize directly p, instead of stabilizing q?
Note that the control input signal appears in the second state equation but we want to
stabilize the first state variable.
How can we drive p to a desired trajectory by directly or indirectly using u?
3

p
p   p 1  
q 

q  bq  dp
2/3
q  G uq
Let’s first forget the second state equation, and assume that q is the control input for the
first state equation.
To observe the performance of the controller to be designed, an error signal can be defined
as
e  pd  p
where pd is the desired trajectory for tumor volume. Investigating the error system dynamics
yields
e  pd  p
 pd   p  
p
2
q
4
e  pd  p
p
 pd   p  
2
(*)
q
If q had been the actual control input for this system, then we would design it to drive the
tracking error to zero by using the following Lyapunov-based approach.
V 
1
e
2
2
Does this function satisfy Lyapunov Stability Conditions ?
 V (0)  0 only at e  0
 V ( e )  0  e  D   0
 V (e)  0
5
To make the third one to happen, take time derivative of the Lyapunov function:
d 1 2
V 
 e 
dt  2


1
2 ee
2
 ee
Substituting e equation investigated in (*) yields
2

p 
V  e  pd   p  

q


To make this function negative (semi) definite, design q as


1
q  p 


p


p

K
e
d
e 

2
where Ke is a positive control gain. Then the final expression for the time derivative of
Lyapunov function will be
V   K ee
2
6
Note that


2
  V  V
2
V   K ee 

V 
1
e
2
w here   2 K e and   R 
Solution of this differential equation is
V ( t )  V (0) exp    t 
This means that V goes to zero exponentially fast and so does e.
Then the best stability result we can achieve for this system is exponential stability.
7
In reality, q is not a control input signal. Actually it is a state variable.
But, if we force q to follow a desired trajectory by considering it as a virtual control input,
then we will be able to manipulate the trajectory of p to drive the trajectory a desired
trajectory. This is the main idea of Backstepping Technique.
Let’s demonstrate this idea by using a simple system.
Consider the following system: x1   x1  x 2
2
x2  u
Control objective is to drive x1 to a desired trajectory, x1d , by using the control input signal, u.
The control input signal appears at the second state equation while we are trying to stabilize
the first state variable. But if we drive x2 to a desired profile, say x2d , we may stabilize x1 by
using the actual control input, u. To test this, let’s add and subtract the virtual control input,
x2d , to the error dynamics.
e  x1  x 1 d
e   x1  x 2  x 2 d  x 2 d  x 1 d
2
At this step, define a new error variable, h , can be defined to quantify the measure of ability
to track x2d .
h @ x2  x2 d
The dynamics becomes
e   x1  h  x 2 d  x1 d
2
x2  u
e   x1  h  x 2 d  x1 d
2
x2  u
Now we can design the virtual control input (or, in other words, the desired trajectory for x2) as
x 2 d  x1  x 1 d  K e e
2
The error dynamics becomes e   K e e  h
Now we can backstep on h by investigating its dynamics to reach the actual control input signal:
h  x2  x2 d
h  x2  x2 d
u
d
x

dt
2
1
 x1 d  K e e 
 u   2 x1 x1  x1 d  K e e 
2
 u   2 x1   x1  x 2   x 1 d  K e   K e e  h  


 u  2 x1  2 x1 x 2  x 1 d  K e e  K e h
3
2
h  u  2 x1  2 x1 x 2  x1 d  K e e  K e h
3
2
Note that we have reached the actual control input signal, u. Now we can design u to achieve
the control objective.
So far we selected the Lyapunov function as V 
1
e
2
2
But, for this time, we have some additional variables and we must prove the boundedness of
all signals in the closed loop system. For this reason, we will select the Lyapunov function as
V 
1
2
e 
2
1
2
h
2
V 
1
2
e 
2
1
h 
2
2
V  ee  hh
3
2
 e   K e e  h   h  u  2 x1  2 x1 x 2  x1 d  K e e  K e h 
Design the control input signal as
u   2 x1  2 x1 x 2  x1 d  K e e  K e h  e  K h h
3
2
Feed-forward term
V   K ee  K hh
2
2
Cross Feed-back
term
term
V 
1
e 
2
2
1
h
2
2
V   K ee  K hh
2
2
V  0 exponentially
 e, h  0
e  L   x1  L  since e  x1  x1 d
x1  L   x 2 d  L  since x 2 d  x1  x1 d  K e e
2
x 2 d  L   x 2  L  since h  x 2  x 2 d 3
T hen u  L  since u   2 x1  2 x1 x 2  x1 d  K e e  K e h  e  K h h
3
All signal in the closed-loop system are bounded.
2
Challenge
Remember the tumor growth model:

p
p   p 1  
q 

q  bq  dp
2/3
q  G uq
Note that q appears at denominator.
Can you still apply the backstepping procedure to this system?
If no, why?
If yes, how?