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Problem 7.63PP
Suppose we wish the closed-loop system to behave like a desired model, called the implicit 
model
Z = A/77Z.
We may minimize a modified LQR performance index
j f “ [ (J - A ^ / q , » - A .y ) + a .
Show that this performance Index Is > 
weighting term between the control s 
Show that this performance Index Is equivalent to the standard one with the addition of a cross­
weighting term between the control and the state of the form
where 
" J 1
0 = (CA - A«C)’'Q i (CA - A .C), 
S = (CA - A«C),
f i = R + B’'C’’Q,CB.
Step-by-step solution
step 1 of 1
Consider the general state variable form of vector equations.
x = A i+ B i / .......(1)
y = C r ...... (2)
> = C i ...... (3)
Consider the general form of LQR design.
J = J (* ’'Qx + u’'R « ) * (4)
0
Consider the modified form of LQR performance index.
J = J ( ( y - A . y f Q , ( y - A . y ) + (5)
Substitute equations (2), (3) In equation (5).
J = J^ (C x - A ^ C x ) Q j (C x - A ^C x) +
Substitute equation (1) in equation (6).
Rearrange the equation.
[ x ^ ( C A -A .C ) ’ -Q , ( C A - A .C ) x ] + 
y = J [x ^ (C A -A .c y Q, CBH+i(’ (C B y Q, C B(CA -A .C )x]+ dt 
[ ii’' (C B y -Q , -CB b + ii’'R ii]
Here, in term 2, both the equations are in scalar quantities and are equal. Rewrite the equation.
[ x ^ ( C A -A .C ) ’ -Q , - (C A - A .C ) x ] +
(6)
y = [ (7)[2 « 4 (C B y Q , C B ( C A - A .C ) x ] +
[ii’' (C B ^ -Q , •CBu + ii’'Rb ]
Now substitute the following vectors in equation (7).
Q -[(C A -A .C y Q, (CA-A.C)]
S - Q , C B (C A - A ,C ) 
R - R + B ^ C ’ Q ,C B 
Rewrite equation (7).
y = J(x ’^Qx+2a^Sx+i/^Ru)