Exercise 
 Suppose that the nominal model of a plant is Gs 

s 

and the uncertainty in the model is parameterized by multiplicative uncertainty with the weight wI s 

 s  

 s 

Call the resulting set 
Now nd the extreme parameter values in each of the plants ag below so that each plant belongs to the set 
All parameters are assumed to be positive
One approach is to plot lI   jG
G  j in 

 for each G
Ga
Gb etc
 and adjust the parameter in question until lI just touches jwI jj

a Neglected delay Find the largest  for Ga  Ge
s Answer  

b Neglected lag Find the largest  for Gb  G 

s  Answer  

c Uncertain pole Find the range of a for Gc  
s a Answer  to  

d Uncertain pole time constant form Find the range of T for Gd  
Ts 
Answer  to 

e Neglected resonance Find the range of for Ge  G 
s


s

 
Answer  to 

f  Neglected dynamics Find the largest integer m for Gf  G 


s  m Answer 

g Neglected RHPzero Find the largest z for Gg  G

zs 

zs  Answer 

These results imply that a control system which meets given stability and performance requirements for all plants in  is also guaranteed to satisfy the same requirements for the above plants Ga Gb
  
Gg h Repeat the above with a new nominal plant G  s  and with everything else the same except Gd  Ts  
Answer Same as above

i Repeat the above with a new weight wI s 
s  
 s 
We end this section with a couple of remarks on uncertainty modelling
 We can usually get away with just one source of complex uncertainty for SISO systems
 With an H
uncertainty description it is possible to represent time delays corresponding to an innitedimensional plant and unmodelled dynamics of innite order  even though the nominal model and associated weights have nite order