CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
OPTIMAL COMPENSATION
OF
LINEAR TIME-INVARIANT SYSTEMS
A thesis submitted in partial satisfaction of the
requirements for the degree of Haster-of Science in
Electrical Engineering
by
Charles Allen Lee
January 1981
The thesis of Charles Allen Lee is approved:
California State University, Northridge
Decembe-r 1980
ii
TABLE OF CONTENTS
Page
Chapter
1
Abstract
1
Derivation of Compensating Control Required
to Minimize Step Input Response Error .
2
3
4
Solution to the Nonlinear Matrix Equation:
-1 T
KBR B K - KA
15
Applications
28
Physical Interpretation
Scope of Application
Examples
Observations
4
Determination of Q Matrix Elements to Achieve
68
Specified System Characteristics
5
Examples of Pole Placement by Iterative Applications of the Optimization Process .
83
110
6
Summary .
7
Extension of Technique and Areas for Future
. 117
Work
iii
LIST OF TABLES
1.
Third Order System Test Parameters
47
2.
Third Order System Response
51
3.
Fourth Order System Test Parameters
55
4.
Fourth Order System Response
57
5.
Fifth Order System Test Parameters
61
6.
Fifth Order System Test Response
62
7.
Q Matrix Elements for Poles of Predetermined
84
Values; System 1
8.
Q Hatrix Elements for Poles of Predetermined.
88
Values; System 2
Optimization for W : Results
95
10.
Constant Mp' Third Order System; q 22 =0
99
11.
Constant Hp' Fourth Order System; q 22 =0
100
12.
Constant Mp' Fifth Order System; q 22 =0
101
13.
Constant Mp' Third Order System; q 22 =.01q 11
102
14.
Constant Mp' Fourth Order System; q 22 =.01q 11
103
15.
Constant Hp' Fifth Order System; q 22 =.01q 11
104
9.
n
iv
LIST OF FIGURES
1.
Uncompensated System
29
2.
Compensated System
30
3.
Pole Migration of Third Order Plant
48
4.
System Response
50
5.
Response vs. Gain, Third Order
53
6.
Pole Migration of Fourth Order Plant
56
7.
Response vs. Gain, Fourth Order
59
8.
Pole Migration of Fifth Order Plant
63
9.
Response vs. Gain, Fifth Order
64
10.
Third Order System Pole Migration to Target
86
11.
Fourth Order System Pole Migration to Target
89
12.
Asymptotic Approach of Wi
96
13.
Constant M , Third Order System
105
14.
Constant Mp, Fourth Order System
106
15.
Constant M , Fifth Order System
107
16.
Settling Time vs. 1/s
n
toW
n
p
p
v
wn
108
ABSTRACT
OPTIMAL COMPENSATION
OF
LINEAR TIME-INVARIANT SYSTEMS
by
Charles Allen Lee
Master of Science in Electrical Engineering
January, 1981
Optimal control theory can be applied to linear, timeinvariant control systems to determine the compensating
feedback gains necessary to:
1.
Cause zero system state error from a specified
steady state value in response to a step input,
2.
Minimize the transient deviation of the system
state from the desired steady state value.
The state equation of the system to be compensated is:
X
=
Ax + Bu
The final state value,
~(oo)
is expressed as linear in r,
the magnitude of the step input:
1
2
z
-
= -x(oo) =
Cr
where C is an nxl matrix of constants.
The analytical measure of performance chosen to minimize
system response is:
J (u)
= ~
/fJ((x- z)T
0
-
Q(~-~) +
R(u- d})dt
where d is the steady state value of u required to force
x to
~,
and Q and Rare weighting matrices.
An nth order, linear, time-invariant system described by:
x
=
Ax + Bu
is forced to the desired steady state value resp:mse to a
step input while minimizing the measure of performance
if it is compensa·ted to achieve:
x =
(A - BR-l BT K)x + (BR-l BT K - A)Cr
where K satisfies the matrix equation:
3
Transient response of a system compensated in the above
manner is controlled by the Q and R matrix elements.
Us-
ing the rule of thumb:
= 4/sWn
Settling time
and the constant
s
nature of optimally compensated sys-
terns, Q matrix elements necessary to meet conventional
figures of merit, such as settling time, are computed by
iterative applications of the previously described optimization process.
q
i+l
=
q
i
+
4k~
The relationship
.2
ln
l
wi
n
is used to update the Q matrix elements between applications of the optimization process.
The result is a sys-
tern which responds with zero steady state error in response to a step input, minimizes the performance measure
and meets the conventional settling time figure of merit.
CHAPTER 1
DERIVATION OF COMPENSATING CONTROL REQUIRED TO
MINIMIZE STEP INPUT RESPONSE ERROR
Given a linear, time invariant, single input system of the
form:
X
=
Ax + Bu
where:
x(t) is the system state vector of order n.
Note that
the t of time dependent variables is usually
dropped. An underlined variable is a vector.
A
is the n x n system plant matrix.
u(-t)
is the control input to the uncompensated system
and the compensating input to the compensated
system.
B
is the n x 1 control matrix.
It is desired to minimize the general functional
J (u)
00
= f 0 g(x,x,p,u,t)dt
- - =-
where:
g(~,x,E,u,t)
is a scalar function in elements of
~'
and p and u and t.
£(t)
is a vector of Lagrange variables required to
satisfy the system state equation constraints.
and g(!,! 1 £,U 1 t) and x(t} are bounded such that:
x(O)
=
0
4
x
5
The term under the integral is further separated into the
measure of system performance, L(x,u,t), and the Lagrange
multiplier, p, times the system state equation such that:
T
.
( 1-1)
g(x,~,£ 1 U,t) = L(x,u,t) + p (A~ + Bu - x).
In the manner of Kirk (1970, p. 114-122, 143-146) the increment of the performance measure is
J(u + ou) - J(u)
~J(u)
=
~J
= J0
00
(u)
(g(x
+ ox,
x
-
oE, u + ou, t) -
+ ox, p +
where the o term denotes a variation in the appropriate
variable.
The expression
is the measure of the
~J(u)
change in J(u) for a change in function of
A Taylor series expansion about
x,~,E,u
g(x,~,E,u,t).
of the integrand
term containing variations results in:
g(x + ox, ~ + o~, p + op, u + ou, t)
.
.
T
g(x,x,p,u,t) + ag(x,x,p,u,t) . ox
-- -.
q:~---
.
=
T
OD
+ ~(~ 1 ~ 1 J2. 1 U,t)
=ax
.
T
ap
+ ag(x,x,E,u,t) ou + (higher order terms in
au.
+
~(~,~'E,u,t) ·o~
ox,o~,oE,ou)
Note here that, because of the time limits on
J(~),
t is
not a varied variable.
When the Taylor series representation of the varied term
is summed in the integrand, the increment of J(u) becomes
6
oo
•
•
ox + ~(x,x,p,u,t)
. a~ t
.
ax t
+ ag(~,x,E,u,t)
op + ag(~,x,E,u,t)
ou
ap
·
au
+ (higher order terms)
)dt.
t:.J(u)
= f ~ag(~,~,p,u,t) T
T
ox
The increment of a functional is defined (Kirk, 1970, p.
117) as the sum of the variation of the functional which
is linear in the varied term with a second term which is a
function of but not necessarily linear in the varied term.
Choosing y as a dummy variable,
t:.J(y)
=
oJ(y,oy,) + f(y,oy)
·
ll oy[ I
such that
If lim f(y,oy) !loy! I = 0,
!loy! i-TO
J(y) is differentiable on y and oJ(y,oy) is the variation
oJ(y,oy) is linear in oy.
of J(y) evaluated for the function y.
It can be readily seen that if f(y,oy)
·
I /oyl I
is defined
as Taylor series terms of second order and·higher, the
limit lim f~y,oy)
II oy!J +o
.1
loy!
I =
0 is satisfied.
OX,OD, and ou remaining after
=-expanding J(u) in a Taylor series are the variation of
The first order terms in
J(u) on
x,~,~'
o~,
and u.
The variation oJ(u) becomes:
oo
•
T
·
T
oJ(u) = f 0 (ag{x,x,p,u,t)
OX+ ag(X,X 1 £ 1 U,t)
OX
.
ax ~
.
a~ T
+ ag(x,x,p,u,t)
op + og(x,~,p,u,t) . ou ) dt
ap - - au - - -
This variation is defined as a linear approximation to the
7
change in the measure of performance J(u) caused by the
changes in the varied functions.
The fundamental theorem
of the calculus of variations is, of course, that oJ(u)
0 at extrema of J(u)
(Wylie, 1975).
=
This requirement can
be assured by forcing the integrand to be zero for all t.
Applying integration by parts to the second term in the
integrand gives:
oJ(u)
=
.
o~(x,x,p,u,t)
T
joo
oo
.
0~ o+fo((og(~,~,:e_,u,t)
T
ox
T
3x
T
d(og(x,x,p,u,t)
))ox+ og(x,x,p,u,t)
op
dtax_--- T
a:e____
ou ) dt.
+ 3g(x,x,p,u,t)
au - - -
(1-2)
The relationship of equation (1-2) will be used in part to
fix the steady state error of the response of a system to
a step input.
The system initial state is also fixed.
Consequently, the variation in the system state vector at
time zero and infinity is zero, i.e.,
•
T
OX leo = 0
3g(x,x,p,u,t)
0
ax - - Substituting the relationship of (1-1) into (1-2) results
ox(O) = ox(oo) = 0,
so
in
oJ(u)
=
f
co
o((3L(~,u,t)
T
· T
+A E + :e_)
ox
ox T
T
T
+ (Ax + Bu - X)
o:e_ + (3L(x,u,t) + B :e_)
au -
ou )
dt.
To guarantee that oJ(u) = 0, it is sufficient to guarantee
the coefficients of
8~,
o:e_, and
ou
are zero:
8
X
=
Ax + Bu
p
=
- ATE
0
=
(1-3)
-
aL(x,u,t)
dXaL(x,u,t) + BTp
dU-
(1-4)
(1-5)
A measure of response of a linear, time invariant control
system to a step input is extremized by a control input
derived from the requirements of equations (1-3) through
(1-5).
=f
The measure of performance J(u)
00
0
L(x,u,t)dt
-
is chosen to measure the error between actual and desired
steady-state values of the state vector, and to accumulate
the transient state vector error.
A measure of perfor-
mance of the following quadratic form is chosen
J(u)
= ~ J~((~-~)T
Q(x-z) + R(u-d)
2
) dt.
(1-6)
where:
z
is the steady state value of the state vector
,x(t)
d
is the steady state value of u(t)
force x(t)
Q
=
required to
z
is an n x n weighting matrix; assumed for this
work to be diagonal, positive-semidefinite
R
is a positive, non-zero weighting term
This measure quantifies the error between the system state,
~'
and the desired system state,
~'
and the error between
the compensation control, u, and the steady state value of
the system control, d.
9
Note here that:
:.3L(x,u,t)
ax -
~
=
I
aL(x,u,t)
axl-
\\
aL(x,u,t)
ax2-
aL(x,u,t)
axnNote also that if two variables are defined such that:
= w1
w
\
and v
=
w2
w
v
n
n
+
and if h
~
.. wn v n'
then,
ah
aw
=
=
v, and dh
av
=
=
'v·
1
n
i t follows that
3L(x,u,t)
() u -
=
aL(~,u,t)
ax
R(u-d)
=
Q(~-~),
and
(1-6)
(1-7)
Substituting (1-6) into (1-4) and (1-7) into (1-5) results
in:
x=
Ax + Bu
p=
-A E - Qx + Qz
T
( 1-3)
( 1-8)
10
=
u
(1-9)
Substituting (1-9) into (1-3) gives
-1 T
x=
Ax - BR
B
E
E• =
-Qx -AT E +
Q
+ Bd
z
System parameters are being measured in response to a step
input, and the steady-state value of the state vector will
be linearly related to the input.
Thus,
x
(oo)
=
0, and
since
X (oo)
=
then Bd
x =
p
=
=
Ax (00 ) + Bu (00 ) and x (oo)
-Az.
Ax - BR
=
~'
=
and u (00 )
d,
Therefore,
-1 .
BT E. - Az
(1-10)
-Qx -ATE + Q~
(1-11)
In the manner of Kirk (1970, p.221) E is assumed to have a
solution of the form:
E
=
Kx + s where K and
~
are, for the moment, assumed
to be time dependent variables.
E. =
K~ + K~ +
Substituting for
·
K~
x
s
=
-Qx -ATp + Qz
and E.
+ K (A~- BR -1 BT (Kx + s) -Az) +
s =
-Q~
-AT (Kx + s) +
Q~
Collecting terms on the right side of the equal sign
0
=
· + KBR -1 BT Kx -KAx -A.LKx
rn
-Kx
-Qx + KBR -1 BT s -AT s -
KAz + Qz
Collecting terms in
~,~,s
and z
s
+
11
(-K
+ KBR-lBTK -KA -ATK -Q)x + (KBR-lBT -AT)s -
(KA +
=
Q)~
s
+
( 1-12)
0
The coefficient of the x variable may be made equal to
zero by satisfying
K=
KBR-lBTK -KA -ATK -Q.
Kalman
(1960, p.ll7) points out that, for a time invariant systern, K can be computed as a matrix of constants thus for-
.
cing K
=
0.
Accordingly, the first set of conditions to
be satisfied for optimality is:
-1 T
T
KBR B .K -KA -A K -Q = 0
(1-13)
Chapter 2 of this work deals with the solution of (1-13).
Note, however, that K is a symmetric matrix.
Equation (1-12) can be satisfied if:
s =
(KBR
-1 T
T
B -A )s + (KA +
Q)~.
(1-14)
Remembering that system response to a step input is being
measured,
X (oo)
=
0
X
( 00
)
=
Z
U
(oo)
=
d
From equation (1-10) it is observed that E(oo) = 0, and
from equation (1-11) p(oo)
lows that s(oo)
x=
=
0.
=
0.
Since
£ =
K~ + ~'
it fol-
Substituting forE in (1-10) gives:
(A -BR-lBTK)x -·BR-lBTs -Az
(1-15)
12
The coefficient of s in (1-14) can be re-written as:
(KBR -lBT-AT) ==(A -BR-lBTK)T
Equation (1-15) dictates that the differential equation in
x is stable iff the eigenvalues of (A -BR
-1 T
B K) are nega-
tive and the differential equation in s is stable iff the
'
1
e~genva
ues o f
that
s
(oo)
=
that
s
(oo)
=
0 for all t.
( A - BR-l BT K) are
' '
pos~t~ve.
It
'
~s
necessary
0, but stability of equation (1-15) forces
The dichotomy is resolved by setting
00
~
=
The remainder of equation (1-12) is satiss = (AT -KBR-lBT)-l (KA + Q)z
fied by:
From ( 1-13) :
=
ATK -KBR-lBTK
(AT -KBR-lBT)-l
Substituting for
x =
=
~
-(KA + Q) from which it can be deduced:
-K(KA + Q)-l
Therefore, s
=
-Kz.
in equation (1-15) :
(A -BR-lBTK)x + (BR-lBTK -A)z.
If z is a linear function of r, the step input, then z =
Cr
~C
an nxl matrix), and the measure of performance J(u)
is extremized (but not necessarily minimized) when:
x=
(A -BR
-1 T
1 T
B K)x + (BR- B K -A)Cr
where K is
determined from the system state equations and R and Q are
defined in equation {1-6),
The increment of a functional !::.J{u)
=
J(u + ou) - J(u)
defines the maximum or minimum nature of the extremum of
13
J(u)
(Kirk, 1970, p.l20).
Let u* be the value of u for
which an extremum of J(u) exists.
J(u* + ou) - J(u*) >0
If:
then
J(u*) is a relative minimum.
If:
J(u* + ou) - J(u*) <0
then J(u*) is a relative maximum.
As previously derived, the increment of J(u*), \vhen expanded in a Taylor series,
results in
00
! 0 (higher order terms) dt.
chosen to force oJ(u*)
=
0.
~J(u*)
=
oJ(u*) +
Values for u* and E have been
What remains is
~J(u*)
=
00
! 0 (higher order terms) dt.
For sufficiently small values of the varied variables, the
second order terms of the Taylor expansion, sometimes referred to (Sage and White, 1977, p.61) as the second variation,
o 2 J(u), will dominate the expression for ~J(u*).
In the case where:
g(~,x,p,u,t) = ~((x-~)T Q(~-~) + R(u-d) 2 + pT(A~ + Bu-~)),
the second variation becomes:
2
o J(u*)
=!
oo
0
T
(8~
Qo~
T
+ ou Rou +
T
o~
AoE
-8xT0E + O£TAox + opT.Bou- O£TO~ + ouTBTop) dt.
The Lagrange variable, £ 1 is not allowed to vary from the
value E* which forces the extremum of J(u*) to be satis-
14
fied under the constraint
x=
Ax + Bu.
The second varia-
tion becomes:
2
Thus o J(u*) is assured positive for all ou and ox as long
as the matrix
(g
~) is positive-semidefinite, and ~J(u*)
is positive as long as o 2 J(u*) dominates the expression
2
Note that, in this instance, o J(u*) is inde-
for ~J(u*).
pendent of the variables ~,~,p, and u causing onJ{u*)
for all n>3.
paper,
=
0
For the measure of performance used in this
~J(u*)
=
2
8 J(u*).
This term is greater than zero
for all ox and ou as long as the matrix
(g
0
R
) is at least
positive-semidefinite, and the extremum of J(u) at u* is a
minimum for all ox and au.
Summarizing the results of this chapter for the system described by the state equation;
~
=
A~+
Bu, excited by a
step input r, the measure of system performance:
J(u)
=
~
T
2
Joo0 ((xz)
Q(x- z) + R(u- d) ))dt where
-
Q is a diagonal, positive-semidefinite matrix, and R is
positive, is minimized when the system is compensated to
achieve the following:
x =
(A -BR-lBTK)x + (BR-lBTK -A)Cr such that the matrix K
satisfies the non linear matrix equation,
KBR-lBTK -KA -ATK -Q = 0.
Chapter 2 will develop the solution to this matrix equation.
CHAPTER 2
SOLUTION TO THE
NONLINEAR MATRIX EQUATION:
KBR-lBTK -KA -ATK -Q
=
0
The solution matrix, K, to the nonlinear matrix equation:
(2-1)
turns out (Sage and White, 1977, p.78) to be symmetric.
th
n(n+l)
.
Thus, for an n-- order system, there are
nonl1near
2
equations to be solved to determine the elements of the K
matrix:
fl(kl, ... k1)
=
0
f. (kl, ... k,)
=
0
1
1
The Newton-Raphson method of iterative solution (Saaty and
Bram, 1964, p.56) is applied to the set of assumed soluO·
tions (k o1 , ko2 , ••• ki).
A Taylor series expansion is per-
formed at the assumed solution point and the terms containing second and higher order derivatives are dropped.
This expansion results in the following set of equations:
+ i
.l.: 1
J=
+
j
.L: 1
J=
0
0
+ i
f.(kl,
... k.)
1
1
.L: 1
J=
a f (k 01 , ... k<?). (k.
1
J
(lk.
- -1
k<?) = 0
J
J
a f 2 (k 01 , ... k9)
=
o
. (k0 , ... k<?) . (k . - k~) =
1
1
J
J
o
d'k
k.
1
. (k . - k~)
J
J
J
af
dk~
1
15
16
The term: i
0
. I: 1
J=
0
-
a£. (k , ... k. > • (k.
1
J
ak~
~
k'?)
J
J
can be expressed as:
T
0
a£.(k
)
(k "":' ko)
-l3k
where:
k =
k1
k2
a£. (k 0
and lak
'
)
=
a£.
j
(ko)
akl
1
()f.
k.
l
ak~
(ko)
2
£
The terms
0
0
0
1 (~), £ 2 (k) , ... fi(k) can be expressed as the
0
vector f(k ) , and the vector equation of the Newton-Raphson iterative solution to equation (2-1) becomes:
f(k
0
)
+ J(f (k
0
0
)).
{~-~ ) = 0 where J(f{k
Jacobian matrix defined by:
J(!_(~o))
(
=
a£ (ko)T
ak
-1-
\
\
\
-------I
l
\
\
\
3£ (ko)T
-2-
ak
\
I
II
--------
a£. (ko)T
-~-
ak
I
0
)
is the
17
Finally,
k
=
k0
-
J-l(f (k 0
))
.f(kO).
(2-2)
Equation (2-2) describes an iterative process to determine
a solution to the set of
n (n+l)
nonlinear equations resul2
ting from (2-1).
Convergence of the Newton-Raphson iterative solution is
dependent on several factors
(Saaty and Bram, 1964, p.64).
In general terms, some (but certainly not all) of these
requirements are:
1.
The inverse of the Jacobian matrix must be bounded.
2.
The norm of the solution error,
I I~-)£' II ,
must be
bounded.
3.
The second order terms in the Taylor series expansion of f
4.
(k) must be bounded.
Satisfaction of convergence criteria for the
first iteration of equation (2-2) is sufficient
to prove the iterative solution process will converge to a solution.
A rigorous proof of convergence of iterative application
of equation (2-2) to the n(~+l) equations resulting from
(2-1) will not be attempted in this paper.
However, com-
rnents will be made regarding the characters of the Jaco-
18
bian matrix, the solution error and the second order Taylor series terms resulting from application of equation
(2-2).
The inverse of the Jacobian matrix must be bounded.
For
the solution derived for this paper, the initial solution
to equation (2-2) is always assumed to be k 0
=
0.
It can
be seen that the remaining sources for terms in the Jacobian matrix are, from equation (2-1), -KA -ATK-Q
=
0.
The Q matrix is independent of K, so the Jacobian matrix
elements are either zero or a combination of the A matrix
elements.
With the A matrix describing a linear, time
invariant system, the A matrix elements are constant and
the cofactors of the Jacobian matrix must be bounded.
-The existence of the inverse of the Jacobian matrix is,
therefore, dependent on the determinant of the Jacobian
matrix being non-zero.
The order of the Jacobian matrix in equation (2-2) grows
rapidly as the order of the A matrix is increased.
A
third order A matrix results in a sixth order Jacobian
matrix in equation (2-2).
Although the cofactors of the
Jacobian matrix can be shown to be bounded, the size of
the Jacobian matrix for even the simplest of systems
19
makes derivation of general requirements for the existence
of the inverse of the Jacobian matrix impractical.
Examination of the Jacobian matrix resulting from the
application of equation (2-2) to a second order system
provides a possibly useful insight.
the form:
A
Given an A matrix of
I.
=
the Jacobian matrix resulting from equation (2-2)
I
I
J
(f (ko) )
=
-4
I
0
all
a21
al2
all+a22
a21
al2
a22
0
is:
The determinant of the Jacobian is:
=
Thus, in this case, the Jacobian matrix is singular if the
A matrix is singular or if the sum of the A matrix diagonal elements is zero.
This last condition arises when
the system described by the A matrix is marginally stable.
Over sixty solutions using the Newton-Raphson equation
were performed in the preparation of this paper.
No cases
of Jacobian matrix singularity occurred during these solutions.
That the Jacobian matrix remains bounded for sta-
20
ble systems seems a valid assumption.
The norm of the solution error,
I l~-k
0
1
I,
must be bounded.
Since, for the first iteration of equation (2-2), k 0 is
assumed to be zero, the expression for the solution error
norm becomes:
11~11
=
The inverse of the Jacobian matrix has been discussed and
is assumed bounded.
Inspection of equation (2-1) shows
f(O) is made up entirely by Q matrix elements which are
constants.
Thus, as long as the inverse of the Jacobian
exists, the norm of the solution error is bounded.
The second order terms in the Taylor series expansion must
be bounded.
Equation (2-2) is second order in K, so the
Taylor series second order terms are elements of the matrix product BR
-1 T
B .
The R matrix has been previously con-
strained to be a positive definite, diagonal matrix, so
1
the BR- BT terms must exist.
Thus, the second order Tay-
lor series terms are constants and satisfy the criteria
that they be bounded.
Rigorous proof of convergence requires calculation of actual values for the inverse of the Jacobian matrix, the
solution error, and the second order Taylor series terms.
21
The size of the Jacobian matrix makes this impractical for
th
the case of a general n-- order system.
It appears to be
more efficient use of computer assets to assume the requirements for convergence are satisfied, perform the
iterative solution
process and test for convergence within
a reasonable number of iterations.
The iterative process described by equation (2-2) is dependent on finding a general expression for the Jacobian
matrix used in equation (2-2).
The following is a des-
cription of the technique used in the work to evaluate
the Jacobian matrix.
Equation (2-1) , restated here,
KBR-lBTK -KA -ATK -Q
=
0
can be rewritten as:
KPK- KA -(KA)T- Q
=
0
where:
P
=
BR-lBT
and:
(KA)T
K.
=
ATKT
=
ATK
due to the symmetrical nature of
Let the K matrix be described by:
K
=
kll
kl2
. . .
kln
kl2
k22
. . .
k2n
kln
k2n
. . .
k
nn
\
I
I
22
Describe each of the columns of the K matrix as a vector:
k T
-1
I
I
J
K
=
(kl
. . . .
k2
I
k T
-2 .
=
k )
-n
. .. .
\
\ kn
where:
k = ' kll \, k2 =
-1
k~2
\
kl2 \, ~3 =i kl3
I
k22 I
k23
I
I
\
kl nlI
'
\k;n
I
\
k
-n
k33
i
k3n
Let the A matrix be described by:
A
=
(a
1
a
:
- 2
.•
.
.
• ••. a )
.-n
where:
al
= all\
\
a21 \
a nl
~2
~ ( a12
\
a22 \I
I
I
\
\
I :
I
I
\ an2
1
I
I
,
.,
...
a
-n =
a
nn
T
23
KPK -(KA + (KA) T ) -Q
Then,
... ,
=
:~
• n
1
/.-:~,
"(a1: a2.:
! ·_:~·
\
)-.(
.
•.• 1'
k
-n
k T \
Ik
.L
\
•
. . . .-n
:a )
;
\, k-n T j'
:a
) )T
.-n
0
0
~22. • • ~
.
0
0
T
k1 .p.~2
k1 .p.k
T
k2T·P·~:
k2'1'.p.k2
'\
\
I
1
T
\ ~n . p.~
k-n T.p.k
-n /
k T
k T -~2 + -2
-1
a1
T
~1 .a2
2~2
T
.a2
k T .a2+ k-2 T ·~n
-n
0
0
0
q22 .••
0
0
=
0
T
\
T
~1 ·~n + k-n. . -a 1 \
T
I
T
k2 .an + ~n ·~2
2k T .a
-n
-n
(2- 4)
24
Remembering that the vector k is comprised of all of the
n (n+l)
unknown elements of the K matrix, and that k is
2
-n
th
comprised of the elements of the n-- column of the K matrix, the Jacobian matrix to apply in the (j+l)th iteration
of equation (2-2) where:
=
kj
J(f(kj))
=
k (j+l)
ak·T
<ak 1
(ak 1
ak
.P.~l
T
.P ·~2
j
J
-1 (f (kj )).f(kj)
is:
+ ~1
jT . p.
8kl
8k
-
2
j +
jT
kl .P. 8k 1 ak
ak 1 . a ) T
1
ak
~1
ak
T
·~2
-
.·
(8kl
ak
(ak 2
ak
T
T
.P.~n·
.P.k 2
j
j
+ kl
+ k2
jT
jT
.P. 8k
akn
.P. 8k 2 - 2
ak
T
• -n
a )
T
This expression for the Jacobian matrix appears formidable.
Recognize, however, that the elements of the derivative
terms
8k.
8kl
are either one or zero depending on which de-
rivative is being taken.
Also, the elements of each row
of the ,Jacobian matrix are derived from the same two col-
25
urnns of the K matrix.
Kj = (k j
-1
: k j
: -2
If two matrices are defined as:
:
: k j)
:
. -n
the K matrix with elements set to the values derived from
th e J.th so 1 u t'10n, an d :
• • • • : ak
· akn }
-st
the K matrix with all elements zero except those correspending to elements kst' 1
~
s
~
t
,
1 ~ t
2 n ,
which
are set to one, the elements of the Jacobian may be calculated by means of the matrix equation:
()Kst PKj + Kj P ()Kst- oKstA- (()Kst A)T.
(2-5)
The Jacobian matrix is determined for each iteration of
equation (2-2) by first evaluating aKst for s=t=l, and
Kj from the previous iteration of (2-2)
in equation (2-5).
for application
The upper diagonal of the resulting
matrix contains the terms of the first column of the
Jacobian matrix.
The aKst matrix is advanced to s=l, t=2
and applied in equation (2-5) to obtain the second column
of the Jacobian matrix.
The process is repeated until
the solution for s=t=n is obtained at which point the
Jacobian matrix is completely evaluated.
The computer program used in this work to solve equation
(2-1) uses a variation of the process described by equation (2-5).
The process used for solutions presented in
26
this paper computes Jacobian matrix terms one element at
a time.
This variance is due to program mechanization and
the resulting Jacobian matrix is the same for both techniques.
The complete computer program to apply equation (2-2) to
solve equation (2-1) can be described as follows:
First the mechanization equations are summarized:
Fj
=
Kj BR-lBTKj - KjA - ATKj - Q
ClK
BR-lBTKj + KjBR-lBT ClK
- ClK A - AT ClK
st
st
st
st
kj+l = kj - J-l (f(kj)) -!<kj)
(2-6)
(2-7)
(2-8)
where:
kj is a vector comprised of the upper diagonal elements of Kj
f(kj)
is a vector comprised of the upper diagonal
elements of the Fj matrix described by equation (2-6)
The computer program is mechanized to compute kj+l as
follows:
1.
Decompose the elements of the kj vector derived
from the previous iteration (or the zero vector
if j+l=l) into the upper diagonal elements of
the Kj matrix
2.
Compute Fj
27
3.
Derive the f(kj) vector from the upper diagonal
elements of Fj
4.
Derive the Jacobian matrix, J(f(kj)) using the
process described by equation (2-7)
'+1
5.
Use equation (2-8) to compute kJ
6.
Repeat the process until
I jJ-l{f(kj))
.
f(~j)j I
is within the desired radius of convergence
It should be noted that, when applying the above process
to digital computer solution, data resolution can affect
l jJ-l (f(~j))
. f(kj)
I I.
If the value of the least sig-
nificant bit of any term in the K matrix exceeds the desired radius of convergence, the process will appear to
fail to converge.
This chapter has presented a solution process for the
matrix equation:
KBR-lBTK - KA - ATK - Q
=
0
which must be satisfied to minimize the measure of performance:
which describes the state error of the system:
x = Ax
+ Bu
when excited by a step input, r(t).
Chapter 3 will apply
this solution process to several examples and examine the
results.
CHAPTER 3
APPLICATIONS
The results of Chapter 1 show that the response of the systern:
x =
Ax + Bu
( 3-1)
to a step input, r, minimizes the performance measure:
J(u) =
~
when the system is compensated to achieve:
x =
(A - BR-l BTK) x + (BR-l BTK - A) Cr
(3-2)
where K satisfies the matrix equation:
KBR-l BTK - KA - AT K - Q
=
0
(3-3)
These results are physically interpreted in the following
manner.
PHYSICAL INTERPRETATION
Restating some of the relationships of Chapter 1 for convenience:
-1 BT
u
= -
B
=
-Az
E
=
Kx + s
-
s
=
-Kz
R
( 1-9)
E
z = Cr
Combining these relationships in equation (1-9) gives:
u
=
- (BT B) -l BT Az - R-lB T (K (x-z))
u
=
-R-lBT Kx + (R-lBT K -
-
--
28
(BTB) -l BT A) Cr.
(3-4)
29
U'(t)
X (t)
B
Figure 1.
Uncompensated System
30
r(t)
u (t) ......---.
B
(R-lBTK -
x(t)
~___..,
f
(BTB)-l BTA)C
A
Figure 2.
Compensated System
1-.---~(t)
31
Using equation (3-4), equation (3-2) may be expressed in
the following form to show the physical relationship between the uncompensated system detailed in Figure 1, and
the compensated system detailed in Figure 2.
x=
Ax + Bu
x =
Ax+ B(-R
-1 T
B Kx + (R
-1 T
B K -
T
(B B)
-1
B
T
A)Cr
Scope of Application
The optimization process described by equation (3-2)
is
applicable to any linear, time-invariant, controllable
system.
For this work, study of this optimization pro-
cess is limited to the types described below.
Observability
Systems to which the optimization process is applied must
be completely observable for a solution to the nonlinear
matrix equation (3-2) to exist.
Investigating a third
order, all pole plant:
x=
Ax + Bu
where
A
=
0
1
0
0
0
1
B =
0
0
1
32
R
=
(1)
0
Q --
0
0
0
Assume state x
K B
2
is not observable.
To satisfy
R- 1 BTK - KA- ATK - Q = 0,
0
0
0
2 a31k13
k12+a33k13+a31k33
k11+a32k13
k22+k13+a32k33
0
0
!
=
0
The equations to be satisfied are:
2
k13 - 2a~1
k 13 - ql1
j
- = 0
-k11 -a32 k13 = 0
k13 k33 - k12 -a33 k13 -a31 k33 = 0
0
~
'
33
- 2 kl2 -q22 = 0
- k22 - kl3 - a32 k33
2
k33 - 2 a33k33 -q33
Note that k
terms.
12
=
=
0
0
is dependent on two independent sets of
Four of the above constraints cannot be satisfied
simultaneously while allowing the q
dependent.
22
term to remain in-
Thus a solution to
KBR -l B T K - KA - A TK -
Q
=
0
may not exist if any of the system states are not observable.
Stability
The optimization processes derived in this work are limited to stable systems out of concern for the behavior of
the Jacobian matrix solution derived in Chapter 2.
As
was noted for a second order system, the Jacobian becomes
singular at the boundary between stable and unstable operation.
The application of the optimization process to
unstable systems is an area for further study.
Configuration
Examples investigated in this work will be single input,
all pole phase variable plants.
34
Examples
The remainder of this chapter is devoted to application
of the previously der:ived optimization technique to three
all pole plants.
It is helpful to relate terms in the compensated state
equation:
x=
(A - BR-lBTK)x + (BR-l BTK - A)Cr
to values of K elements obtained from:
The matrix product BR-l BT results in a symmetric matrix.
Designate the BR-l BT matrix product a syrr@etric matrix
P where:
p
=
35
BR-l BTK = PK =
n
n
L:
p 1. k2i
I:
p 1' kli
. 1
l
. 1
l
l=
l=
n
n
L:
P2.
. 1
l
l=
kli
n
I:
i=l
kli
P2.k2.
l
l
k3i
. ..
I:
P2.
. 1
l
l=
n
L:
p 3.
. 1
l
l=
k2i
L:
i=l
p 3. k3i
l
. k2l'
n
L:
p
n
2.:
k3i
...
n
n
I:
P3.
. 1
l
l=
n
L.
p 1.
. 1
l
l=
p
. 1 Til
l=
. ..
n
L:
p 1' k ni
. 1
l
l=
n
L:
i=l
P2. k ni
l
n
L:
P3.
. 1
l
l=
lJ
=
. 1 Til
l=
k ..
Jl
The nonlinear matrix equation
KBR-lBTK - KA - ATK - Q
becomes:
ni
n
. k3l'
. . . L:
p . k .
. 1 Til Til
l=
Remember that the matrix K is symmetric so that
k ..
k
(3-5)
36
n
n
L:
L:
J=l i=l
'n
n
'I:
L:
k l.P .. kl.
J Jl
l
.P .. k .
2 J Jl 1 l
j=l i=l
n
k
n
L:
L:
j=l i=l
n
n
L:
L:
j=l i=l
k l .P .. k2.
J Jl
l
k2. p .. k2.
l
J Jl
n
n
k 3.P .. kl.
J Jl
l
j=l i=l
L:
L:
n
n
n
L:
L:
k
p .. k2 .
j=l i=l llj Jl
l
L:
n
n
L:
n
.P .. k .
j=l i=l ll] Jl 1 l
L:
L:
k
j=l i=l
n
k 3 .P .. k2.
J Jl
l
n
n
n
.P .. k .
1 J Jl 3 l
j=l i=l
L:
L:
n
n
n
n
z:
L:
L:
L:
L:
L:
k
.P .. k .
2 J Jl 3 l
j=l i=l
n
k
n
n
n
L:
L:
n
n
n
L:
L:
L:
k
k
.P .. k .
j=l i=l ll] Jl. 3 l
.P .. k .
1 J Jl n1
k
.P .. k ..
2 J Jl n1
k
.P .. k .
3 J Jl nl
j=l i=l
.P .. k .
3 J Jl 3 l
j=l i=l
L:
k
j=l i=l
j=l i=l
L:
n
Z:
k .P .. k .
j=l i=l nJ Jl n1
37
n
22:
k .a.
. 1 1l l 1
l=
n
n
1
~= (k1iai2 + k2iai1)
22:
k .a.
i=1 2 l l 2
n
n
2:
(k .a.
+ k .a. )
i=1 1 l lll
lll l 1
2:
(k .a.
+ k .a. )
2 l lll
i= 1
nl l 2
n
2:
(k1.a'3 + k3iai1)
i=1
l l
n
2:
n
2:
(k2.a.3 + k3.a.2)
l ].
. 1
l l
l=
n
n
22:
k .a.
. 1 3l l 3
l=
n
n
l:
n
2L:
. 1
l=
(k
. a.
+ k . a. )
3 l lll
lll l 3
~ -1
_._-
L:
. 1
l=
l:
. 1
l=
(k
.a.
+ k lll. a l. 1 )
1 l lll
k . a. )
(k .a.
2 l lll + nl l 2 ·
(k .a.
+ k lll.a.3)
l
3 l lll
k .a.
i=1 lll lll
38
qll
0
0
. . .
0
0
q22
0
. . .
0
0
0
q33
0
0
0
•
=
0
•
0
(3-6)
•
For the single input, phase variable plant, the matrix
product BR-lBT where R
=
p, becomes:
0
0
0
• 1/p •
(0 0 0 •
.
.
0 b)
0
b
The expansion of this vector product is:
39
0
0
0 .
0
0
0
0
0
0
0
0
0 .
0
0
0
0 .
0
•
•
0
0
. 0
0
•
0
0
All of the P matrix elements are zero except P
2
b /p.
BR
-1 T
B
nn
Equation (3-5) reduces to:
K
=
PK
=
. 0
0
0
0
0
0
0 •
•
• 0
0
0
0
•
•
•
0
k
nn
The state equation for the compensated system is:
x=
(A -BR-l BT K)x + (BR-l BT K -A)Cr
with plant matrix elements:
(A -BR-l B'l' K) =
which is
40
0
1
0
0
0
0
1
0
0
0
0
0
a
p
p
nn
-k
nn
p
b
p
and input matrix elements
(BR- 1 B.T K -A) ==
0
-1
0
0
0
0
-1
0
0
0
0
0
k
p
2
2n
p
b -a
k
n2
p
2
nn
b -a
nn
p
where the C matrix is selected so that the steady state
value of the state vector is z where:
z
=
Cr .
2
41
The examples in this chapter will:
1.
Have B matrix elements set to 1
2.
Have R set to 1
3.
Compensate to achieve
z =
r0
0
0
0
0
with a step input
r(t)
r
0
at t
=
o.
With the above simplifications, the C matrix becomes:
c =
1
0
0
0
and the compensated input matrix becomes:
42
0
0
For the single input, phase variable system with
P
nn
A
= 1 , all other P.lJ. = 0,
=
0
1
0
0
0
0
1
0
a
a
nl
nn
Equation (3-6) reduces to:
2
k,~n k3 n
. . .
k
k2nk3n
. . .
k
.
k
k
3n nn
k
nn
kln
klnk2n
klnk2n
k2n
klnk3n
k2nk3n
k3n
kl n k nn
k
k
2
2
k
2n nn
k
3n nn
k
ln nn
k
2n nn
2
43
... k
a +k a
1 n nn nn n 1
... k
+k a +k a
2 ,n- 2 2 n nn nn n 2
k
l,n-1
+k
a +k a
ln nn nn n1
k
1n
+k
1 ,n- 1
a +k a
... 2k 1 +2k a
nn n 2 2 n nn
n- ,n
nn nn
0
0
0
0
q33
0
0
+k
. . .
. . .
. . .
0
0
0
=
0
0
(3-7)
Looking at the nth row of the compensated plant matrix
(A -BR- 1 BT K)
, and the nth term of the input matrix
(BR- 1 BT K - A)C , it is evident that the compensation
gains of interest are:
k.
ln
1 < i
< n.
44
From the diagonal elements of equation (3-7) it can be
determined that:
k.
lll
2
2a . k.
ill
lll
-
. - q..
2k.
l- 1 ,l
ll
=
0 ,
2 < i
< n
from which is obtained:
k.
lll
= a nl. +
I
a
.
nl
2
+ 2k.l~
. + q ..
ll
1 ,l
2 < i
< n
With these values of compensating gains, the plant and
input matrices respectively become:
45
(A
i
l
-BR-lBTK)
=
0
1
0
0
0
0
I
!
2
- 1 a nn +2k n-l,n +q nn
0
0
0
I
2
anl +qll
Note that the positive root of k.
must be chosen to
ln
assure stability.
With the above compensated plant and input matrices, the
system transfer function is:
x
1
( s)
R ( s)
=
s
n
+ 1a
nn
2 +2k
n-l,n
+q
nn
sn
from which the equivalent, unity feedback system forward
transfer function is obtained :
46
G(s)
=
s(sn-l+ Ia
nn
+2k
n-l,n
+q
nn
(3-8)
Equation (3-8), the forward transfer function of a Type l
system, demonstrates that systems compensated using the
optimization technique derived herein do indeed achieve
zero steady state error for step inputs.
Looking now to the examples.
Example l
The first
exan~le
is a third order, phase variable system
with state equation:
X
0
l
0
0
0
l
-1.375
-2.75
', -.625
0
X
+
0
u
l
The optimization process is applied using the measure of
performance weighting values listed in Table l.
47
Table 1
Third Order System Test Parameters
Test
1
r
qll
q22
q33
1
0
0
1
0
2
67.667
0
0
1
3
134.333
0
0
1
4
201.0
0
0
1
5
267.667
0
0
1
6
334.333
0
0
1
7
401.0
0
0
1
8
200.0
2.0
0
1
9
200.0
20.0
0
1
10
200.0
200.0
0
1
11
200.0
2000.0
0
1
12
200.0
20000.0
0
1
Figure 3 details the closed loop pole migration resulting
from the above twelve test conditions.
Pole plots from tests 1-7 (circles) clearly show closed
loop pole migration towards Butterworth form asymptotes
as q
11
, and therefore, gain, is increased.
The require-
ment that optimal closed loop poles asymptotically
approach a Butterworth form is a Kalman result quoted by
48
.
'
,.n._ ~E..
..
.
-'
.
.
lm
.
' ' ,.• ''' .... ; ......• ''. J .. ' ,; .... :' ...
E.ti-_L ,_c-'==t~~J I
'
t- --~---
'
I
'
j2
~
0
' :~- ~ ~~----~~
.
1
--------.--.----~-~-~-+-------------
---- _[
~0
\'
~-:- ·----··"-
------
\
I
0
i
,,
2
i
I
-
-
-~-~-~~------~:---~- ----
.\... ..
'
i
\
i
i
jl
i
i
i
r--
\
---··t ----· ------------ -· · ·-· -- -----
\1
-----~-
~--·--
0\
\
--------,~--
\
-----------------
\
__ _!_ __
\
~~~-+-t~~~~~~+-~~~~~-+-+~~~~~L~O~+-~~~~~-+~~o
I'
u
-l
--------------------------------- - ---------------~----- ---------- t
I
i
i
-----
i
-~--
f
I
----
I
---:--Fiqur.e3,.. Po 1~
Migration. __ _
.Qf
I
·--------------
1-
·····;-----~-
I
Third. O.radr.Ftat!L ...'-- _ __________
1!1 Uncompense~ted .
&
eCbm:p.ensate.d.t Jes.t no_..
~---------------.
________ ...
+---~-----~------'-
7
--- - ·---- l- ---
-rI -· - - - -a:,
1
~
I'
I
t
~~- ... ;lli'!h-§. /' ..
:
10
./.
__
0.J
.
'
,,l /
~----
,
~
0::!.
I
0r·:--r-----~----
i ei
I
---------r- -- ,--;-~--:--i
-------r---------L
_________ _
I
I
-------· --------~~----
-
-j2
i.
49
D'Azzo and Houpis (1975, p.SlO).
Pole plots from tests 8-12 show the damping effect of increasing the ratio of q
holding q
11
constant.
would be expected.
to q
22
11
from .01 to 100 while
This result is consistent with what
The q
11
term weights the contribution
to the measure of performance resulting from the error in
state variable x , and q
weights the contribution. from
1
22
state variable x .
2
is r
0
The desired steady state value of x
and the desired steady state value of x
2
1
is 0.
Relatively heavier weighting to the error in x 2 results
in decreased response of x
2
at the expense of x .
1
Figure
4 shows the decrease in activity of state variable x 2 and
corresponding more sluggish response of state variable x
as the ratio of weighting elements q
22
to q
11
is increased
from .01 to 1.0.
Third order system response parameters are listed in
Table 2.
1
50
I
!
- -l-----
l
-
- ,
-
: • -L: : • l
tl·---------:·-~---1 --·-:~~~
-
-
I
- ·-- ·-
I
,
,8
---'---------+
~ *z.rq22~11 01
------L---
d)---~-4- Xz;~l-q;itt.&·
,4
-------------- ----- r--
51
Table 2
Third Order System Response
- -
--'"-
.....
- --
·-
------ ---- -· ----- ---- ----
.
Dominant Polei
I
Time@:
Pair
--Nat.
Damp.
I
'Peak :Forward Rate
Error Freq. ing
q 22 \R~sel Peak :over-jGain
Tlme!Over- shoot:
Coeff (red/ Factor
q
sec. 1 shoot. sec. 'k -a . Kv
sec)
(c; )
11 ,I
,
· 1n
n1
I
l
Test
No.
~
----------
. -·-
--
~
1
'
I
I
1
:
i
i
0
12.70: 1.10 3. 5. 85
1.179
.491
.72
.575
67.667 0
,1.29 1.066 2.73
8.250
.944
1.79
.600
3 134.333 0
!1.13 1.067 2.40
11.607
1.068
2.06
.587
14.191
1.149
2.23
.580
'16.372
1.209
2.36
.574
l
2
l
I
!
4 201.0
0
5 267.667 0
i
;
)1.05' 1.067 2.23
I
i0.99 1.067 2.08
!
i
6 334.333 0
:o.95,
1.068 1.96
.
'
18.295
1.258
2.47
.570
7 401.0
'0.92 1.068 1.90 ;20.035
1.300
2.55
.567
:14.156
1.138
2.24
.587
;1.15' 1.033' 2.38
14.156
1.063
2.35
.647
:1.75,0
14.156
.720
8 200.0
0
.01 :1.06:1.063 2.25
I
I
9 200.0
.1
10 200.0
1
11 200.0
10
,14.156
.296
12 200.0
100
14.156
.099
-
52
Figure 5 plots rise time (t )
r
peak overshoot (M )
1
P
1
and
rate error coefficient (Kv) as a function of forward gain.
Rise time (t ) is the time for system output to transition
r
from 10% to 90% of final steady state value.
Forward gain
is the gain of equivalent unity feedback system open loop
transfer function defined in equation (3-8) .
Note from
equation (3-7) that forward gain
G
=
Figure 5 shows the decreases in t
nificantly increasing M .
p
r
achievable without sig-
The dashed lines in Figure 5
show the decrease in Mp at the expense in increasing tr
achieved by increasing the ratio q
22
/q 11 .
The rate error coefficient curve in Figure 5 shows an increase in K with increasing forward gain.
v
This increase
indicates improved system response to a rate input with
increasing gain.
Example 2
The second example is a fourth order, phase variable systern with state equation:
X
=
0
1
0
0
0
0
1
0
0
0
0
l
\ -2
-7
-8
-4.5
0
X
+
0
0
l
u
53
__
--r
~ :-~::~~__:_----d--.L_---+--+--~--+-_:.._-----_ -+----_:_J_~~.:
!-:-:--:-:--
l-··
__£_~
I ~-
II
I
----t----·-----:_·_-_'
I
at
. -!- --- -~----1
-·--t--··--.
I
I
lo
~
54
The optimization process is applied using the measure of
performance weighting values listed in Table 3.
Figure 6 details the closed loop pole migration resulting
from the twelve test conditions of Table 3.
The migra-
tion of the poles to the Butterworth form can be seen
more graphically with this fourth order system.
Also,
optimizing to conditions of tests 20-24 results in a progressively more sluggish system as the single pole migrates towards the originr
55
Table 3
Fourth Order System Test Parameters
Test
13
r
1
0
0
14
67.667
0
0
15
134.333
0
0
16
201.0
0
0
17
267.667
0
0
18
334.333
0
0
19
401.0
0
0
20
200.0
.01
0
21
200.0
.1
0
22
200.0
1
0
23
200.0
10.0
0
24
200.0
100.0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
Response of the compensated fourth order system to a step
input is listed in Table 4.
56
lm
j2
-.....;
__
:_'-.__
.....
Re
-}
....
-------~------~-
-----!---
/
-·
.
- ;-.- - _;.7"'
....-
/r-.. ~
;,.·
----
./
----.,..---~-
__.;
-·
..,.
····-·-
£~~ore Miq.r:.dtton
.
f
9!
.
.
Emu.th . Or_~ PJ.an:t ____ ________ _
!
------ ---- -
.
.
e-IJfte~mpel.lSdteS
i
-- T
d~
\
i
I
--~1§
q;J '
G ::::::::17'
. -.,_~~,--.
1~. i
. __ _ __ _ . · ~- f- _t
-~---- - - - ~---
~
.0 $-(-omfpfl"rSiOfMc l'e-st*- Mtr-- ---r----- ----
I
- . . . . ! . . -L-~I
.
),
.
.
!
/!
'
--- -------
---~
/-
I
--
i,--
··-·• /- . -t . - -.
---'---------------'----'--~--1----------'-----
57
Table 4
Fourth Order System Response
Test
No. qll
Dominant Po lei
Time@
Pair
__j
Peak Forward Rate Nat. ' Damp- ;
Error Freq.
ing
q22 Rise Peak Over- Gain
. Time Over- shoot
Coeff (red/ Factor
(L; )
sec)
qll sec. shoot sec. k 1 n -an1. Kv
4.15
2.236
.305
67.667 0
1. 79 l. 042 3.90
8.466
.567
1. 59.
.548
15 134.333 0
l. 57 1. 060 3.44
11. 762
.637
l. 72
.514
16 201
0
l. 46 1. 068 3.22
14.318
.679
l. 81
.497
17 267.667 0
l. 39 1. 071 3.08
16.482
.711
l. 88.
.487
18 334.333 0
l. 35 1. 075 2.94
18.394
.735
1. 93
.480
19 401
0
1. 31 1. 077 2.90
20. 125
.756
1. 98
.474
20 200
.01 l. 46 l. 063 3.22
14.283
.677
1. 81
.500
21 200
.1
1. 52 1. 046 3.31
14.283
.657
1. 87
.520
22 200
1
2.26
14.283
.530
23 200
10
6.89
14.283
.269
24 200
lOO
14.283
.097
13
14
1
0
58
Figure 7 plots t
r
, M and K of this fourth order system
p
v
as a function of forward gain.
creasing t
The solid lines show de-
and K with increasing gain.
v
The dashed lines show the attenuation of M at the ex-
r
and increasing M
p
p
pense of increased t r and decreased Kv .
Example 3
The third example is a fifth order phase variable system
with state equation:
X
=
0
l
0
0
0
0
0
l
0
0
0
0
0
l
0
0
0
0
0
l
-5
-19
-25.5
-18
0
I
\
0
X
+
-6.5
0
u
0
l
The optimization process is applied using the measure of
performance weighting values listed in Table 5.
Table 6
lists the results.
Figure 8 details the closed loop pole migration resulting
from the test conditions of Table 5.
Note the rapid mi-
gration of the s=-0.5 pole on the real axis in the negative direction as q 11 (and therefore gain) is increased.
Note also the correspondingly lesser movement of the s=
-0.5 + jl poles as q
11
is increased.
Movement of the s=
59
--T-
!
'
- ,.
i
f---~----~----·
-
--·--·-·----!...--
- l'
-
I
~ rn---t------~--+-------------l---
-
.
.
I - -
--------:------
····~--~
-+i
I
I,.;
ar:----.
"U
'
lo.o
---:-~
-f - .c:---:
...,
lo.o
:>.
0
-- t.i.-------
··--,..----~---
-
I
'
. -t-·--------------'
i
I-
I
--·
(OJ
-t······
.,
-
I
-
60
-2
~
jl poles is still less with increasing q
11
.
Review
of Figures 3 and 6 suggest this phenomenon is fundamental
to this optimization process.
This would be expected
since the Butterworth form of a system with n poles results in n poles equidistant from the origin separated
by an angle n/n.
Since the closer poles have further to
travel to reach the Butterworth circle, poles closer to
the origin should travel further towards the circle with
increasing gain.
61
Table 5
Fifth Order System Test Parameters
Test
25
r
1
0
0
0
1
26
67.667
0
0
1
27
134.333
0
0
1
28
201.0
0
0
1
29
267.667
0
0
1
30
334.333
0
0
1
31
401.0
0
0
1
32
200.0
.01
0
1
33
200.0
.1
0
1
34
200.0
1
0
1
35
200.0
10
0
1
36
200.0
100
0
1
Figure 9 plots tr, Mp and Kv as a function of forward
gain.
The effect of increasing the q
22
;q
11
ratio is not
plotted in Figure 9: its effect can be readily determined
from Table 6.
62
Table 6
Fifth Order System Response
Dominant Pole:
Pair
··-·-••• ---- - -------·••:
!Peak Forward Rate Nat.
Damp- I
Rise Peak ·over Gain
Error Freq.
lng
.
q22 Time Over-. shoot
Coeff (red/ Factori
( l; )
q 11 sec. shoot sec. k -a . Kv
sec)
1 n nl
i Time@
-·--n-·--· -·-
·-~·-
!
·Test
No.
25
26
•
0
4.60
5.099
.266
67.667 0
2.70
9.626
.363
1
I
27 134.333 0
2.81 1.015 5.20
12.623
.406
1.44
.607
28 201.0
0
2.13 1. 027 4.83
15.033
.433
1. 48
.579
29 267.667 0
2.00 1. 035 4.55
17.107
.453
1. 51
.559
30 334.333 0
1. 92 1. 042 4.32
18.956
.469
1. 54
.544
31 401.0
0
1. 86 1. 048 4.25
20.640
.482
1.57
.533
32 200.0
.01 2.13 1. 026 4.79
15.000
.432
1. 48
.580
33 200.0
.1
2.17 1. 020 4. 91
15.000
.427
1. 51
.585
1
2.64
15.000
.386
1. 76
.587
35 200.0
10
6.78
15.000
.242
36 200.0
100
15.000
.098
34 200.0
~--··-
63
lm
j2
jl
--~~-
-+,;
~
-------:------~---~~----------~-,
------
---~----------
.
------
'
···- __.....,_ ------- .
!
-·------r-----__:_-
\-
\
!
'<
--------'-----,-------'---:------------------""'--------------
' -,
Re
--;--·-·
Ftq;uce: 8~ Pole Mlo/dtton .
-----~-------------~----------,~:
f
__;__ _
----"--------------
---
\
------~---------
I
/
I
:
:
/
____Q!_+-----.-------:-------
Fit:!i\. Omer Pl~Qt :
/
\
I
,r
I
__L__ --- ·---------. -------
/
f
/
r
r
/
.I
------------- ----- .r
/!I
-jl
1
I
-r·
r
/
L----------:/·
i
---------- . -----j----i
-----.L~-
1
-
I
..(
------,.- -----. f,L---,------,---- --------------- ----r----· ---·------·-------L
...
/
---~-_,L-
/
//
./l
!
~-----------i-
I
:
___
·-----·
---·-------·------~-----~
I
l
r
t'
.:6~
I :
r
L _____ _
-j2
64
!-- --;--:-·+-- ----- ~.-- - --- -..
---f-
i
·---~---T
~-£-~-
I
r
I..
J --~- .
I
--t-I
i
I~
!
1
_1__________
~
. I
N---.-~-----
1•
!-
I
i
I
!
--t--~
~. -0--~-
---------------.---- --------·-
I
l
r-
r· - -
~---!
l
-t'
-+-----·---·---~--~-----+'
i
-- --
:
. -- ----- --rl ---- -----. ,--
i
-- --------·----- ----- ·-f
--- _L ______
.
.-
I
--------- -r1
--- rI
I
I
\
!
-- ----+-------~-r---------i
'
....
c.
'1J_
i
r-------
t-
!
_ _..._...,f---- _ _ ....(!L_ _
----r-\ ----
. t--·--
,
.
I
.
-I!
-- L ____ -----~--• --
[
r1
!
I
----r----------~---
1
!I -
I
_J __ _
I
I
'
!
'
i
I
----t------ ----~---+------~-----~--------------.,
-f------.
-
~
1
.
l
I
r-----1
117.
I..
::).
C"
-~-------~------------7-------~----~~~----~CL~-----r------,_------+-~QU-~-----
~--
-------------------------
l-
'
------~-+--
;
~--
_ _ ---------------+----------------!
I
·I
!
!
~~----~~--~~----~+-----~----~+------+-----=+---~~----~+-----L ____
'<t
rr>
"t
•
00
N
N
N
N
~
65
Observations
Observations based on the results of examples 1, 2 and 3
are summarized below.
Pole Migration
Poles of systems compensated using this optimization process migrate with increasing gain towards the Butterworth
form: n poles positioned equidistant from the origin
spaced ~ radians apart.
Poles closer to the origin ex-
n
hibit greater movement with gain change than poles farther from the origin.
~'leighting
The q
11
elements
element of the Q matrix determines the gain of
the equivalent unity feedback forward transfer function
according to the relationship
=
G
The q
22
I
a~l
+ qll
element attenuates system response by damping the
activity of the x
2
state.
Peak Overshoot
As q
11
, and therefore gain, is increased, the dominant
complex pole pair asymptotically approaches a line of
constant damping factor
(~).
Since peak overshoot of a
66
second order system is dependent only on
~,
sufficiently
high gain would be expected to result in a system where
a further gain increase would result in improvement in tr
and K without an increase in M .
v
p
This condition is, in fact, achieved in Example 1 for the
third order system (see Figure 5), but not in Examples
2 and 3.
Results plotted in Figure 5 suggest a possible
design technique.
With sufficient gain to place the dom-
inant poles on a nearly constant
q
22
/q
11
~
migration path, the
ratio is adjusted to achieve the desired Mp while
q 11 is increased to retain the desired tr or Kv.
This
technique is explored extensively by D'Azzo and Houpis
(1975).
Rate Error Coefficient
Review of Figures 5, 7 and 9 suggests the ratio of
change of K
v
with change in gain remains fairly constant
over large regions of the gain vs K
v
values of gain.
curve for larger
This fact could prove to be another help-
ful design tool.
Summary
This chapter applies the optimization process derived in
chapters 1 and 2 to three, single input, phase variable,
67
all pole systems.
Results of examples in this chapter
are consistent with those of another source.
Results of
this chapter suggest a possible methodology for applying
this optimization process to the design of linear, time
invariant systems.
CHAPTER 4
DETER~INATION
OF Q MATRIX ELEMENTS TO
ACHIEVE SPECIFIED SYSTEM CHARACTERISTICS
The effect of the size of the
matrix elements qll and
Q
q22 on system closed loop poles was observed in Chapter
3.
This chapter develops an approximate relationship
between the changes in the natural frequency
damping factor
and q
22
(Wn) and
(s) which are caused by changes in the q 11
elements of the measure of performance weighting
matrix, Q.
This approximate relationship may be useful,
through iterative applications of the optimization process described in Chapters 1-3, in designing compensation
to achieve desired W and
n
s
characteristics of system dam-
inant poles.
Re-expressing the closed loop system transfer function
for the all pole, phase variable plant derived in Chapter
3'
/
xl ( s)
R
a
2
nl
+
qll
=
(s)
s
n
+
...
/
2
+. a n2 + 2kl2 + q22
2
s + /a
nl + qll
The coefficients of the denominator polynomial are derived from the optimizing feedback gain matrix, K.
68
The
69
closed loop system characteristic equation, expressed in
terms of the K matrix elements, is:
s
n
+ (k
Let
k~
ln
- a
nn
=
nn
)
s
a
k.
ln
n-1
.
nl
+ (k
n-l,n
- a
n,n-1
)
s
n-2
+ ...
In Chapter 3 it was determined
that
kln
k.
ln
:::::
a
:::::
a
n1
+
ni
+
I
I
a
2
n1 + qll
a
q ..
ni 2k.l- 1 ,l. + ll
2
2 < l
< n.
Thus,
y
k"' =
1n
k~
lll
I
a
2
nl + qll
Ia 2
+ qii
ni + 2k.l- 1 ' l.
=
2 < i
<
n
f
and the system characteristic equation becomes:
S
( 4-1)
n-i-l
n-1 + . . . + k"' .
+ ... + k2n s + kln
nn s
n-l,n s
n + k"'
70
An n
th
order polynomial expressed as the product of a
second order and an (n-2)
th
order polynomial is:
2
n-2
n-3
n-4
+ ...
s + wn ) (s
+ cn-3 s
+ cn-4 s
Let this be an expression for the characteristic equation
of the closed loop transfer function of the optimally compensated system.
The second order polynomial represents
the dominant pole pair.
The (n-2)th order polynomial
represents the characteristic equation of the remaining
system poles.
If the indicated multiplication of the
polynomials is carried out, the resulting system characteristic equation is:
( 4-2)
Equating the coefficients of equations (4-1) and (4-2)
results in:
71
k ..
ln
=
w2 c
n 0
k-"
2n
=
w2
2!:; w c
n 0 + n cl
k-"
3n
=
c
0
+ 2
w cl + w2 c2
n
n
(4-3)
Determination of a direct relationship between the q
and q
22
11
elements of the measure of performance weighting
matrix and the
s
and W - of the dominant pole pair which
n
results from the application of the optimization process
described herein would be difficult.
However, an approx-
imate relationship between the change in the q
elements required to achieve a change in
l;
11
and q
22
and W can be
n
used iteratively to achieve a desired configuration of
the dominant pole pair.
The kin and k2n terms are the only terms directly affected
by the q
dk...
ln
11
and q
= 2~"1 n c o
dY-1
22
n
elements.
+ w2 dC
n
o
Thus,
72
These nonlinear first order differential equations can be
linearized about a point
k~
k~
0
ln
'
w0
0
2n
'
n
such that:
0
0
dk~ = 2W° C dW + W
ln
n o
n
n
2
dC
( 4- 4a)
o
(4-4b)
Equations (4-4) are approximated as follows:
dk~
ln
=
2W° C0 dW
n o
n
(4-Sa)
(4-Sb)
Justification for the approximation of equations (4-4a)
by (4-Sa) and (4-4b) by (4-Sb)
is as follows:
First consider equations (4-4a) and (4-Sa).
The examples
of Chapter 3 demonstrate that, for constant g 22 all poles
migrate away from the origin as q
11
kln term is dependent only on q 11 :
I
2
anl + qll
is increased.
The
73
As the kln term is increased, all of the system closed
loop poles will also increase.
Assuming the compensated
system is stable, W and C can be assumed to be positive.
n
o
With the terms W and C
n
dW
n
and de
o
and dk
o
1n
positive and the derivations
(and therefore
dk ~
1 n ) of like sign,
the following relationships must hold:
2W° C0 dW < dk~
n o
n
ln
If equation (4-Sa) is used to compute the dkln required
to achieve dW
the result will fall short of the actual
n,
If equation (4-Sa) is used itvalue of dkin required.
eratively to achieve a desired W , an asymptotic approach
n
of the value of the kln term to the required value would
be expected.
Note again that all of the above assump-
tions appear valid for the tests of Chapter 3 in which
q 22 was held constant.
Consider now the approximation of equation (4-4b) by
(4-Sb).
Exantination of the results of the examples of
Chapter 3 shows that the
s
and W characteristics of the
n
dominant pole pair are increased with increasing q 22 .
74
It is also observed that the residual poles migrate toward the origin with increasing q
22
thereby causing the
residual poles to become the dominant poles.
The added
assumption underlying the approximation of equation
(4-4b) by (4-5b) is that the contribution from the dC
and dC
0
terms is small. Results of an example where
1
these assumptions hold as well as one where they do not
will be explored in Chapter 5.
0
0
Equations (4-3) can be used to solve for C
terms of k~
ln
0
and k~
2n
k~
2n
0
-
and C0 in
1
0
k~
2£;;0
ln
0
wo
n
Substituting these values of C~ and C~ into equations
(4-5) gives:
dk~
ln
=
2k~ 0
ln
wo
n
dW
n
75
dk:: = 2
.Gn
wo
n
k-' 0
2n
r;ok
ln
wo
n
o
dl'l
II
2k-' 0
ln
n +
wo
n
dr;
In vector form:
dk-'
ln
=
2
wo
n
dk ..
2n
k-' 0
ln
0
\
I
dW
I
n
\
II
k-' 0 -L;ok .. 0
2n
ln
wo
n
k-' olI
ln
(4-6)
)
I
di;;
I
!
Equation (4-6) describes the changes in the gain terms
k -- and k -{and consequently k
and k
) required for a
2n
2n
1n
1n
given change in W and r;.
This differential relationship
n
holds as long as the assumptions underlying equations
(4-5) are valid.
The next step is to determine the
different~al
relation-
ship between dkln and dq 11 and between dk 2 n and dq 22 .
k-'
ln
=
k~
=
ln
I
I
a
2
ni + qll
a
2
q ..
ni + 2k.l- 1 'l. + ]_l
2 < i < n
76
1
=
dk"
1n
2
a
2
. + q
=
2
1
dk~
=
lTI
1
. dq11
2k"
1n
Til
1
=
dk~
lTI
I
. dq11
. ( dq . . + 2 dk . 1 . )
ll
a Til~ + 2k.l -
. + q ..
ll
1 ,l
1
( dq . . + 2dk . 1 . )
ll
l-
'l
l-
'l
2 < i
< n
From Chapter 3 equation (3-7) it can be determined that:
= k2 n k4 n -.k2 n a n 4 - k4 n a n 2
k.1- 1 ,1. = k.1- 1 ,n k.1+ 1 ,n
2 < i
<
n-l
By definition:
k.1- l ,n = a n,i-1 + k"i-l,n
k
+ k"
a
i+l,n = n,i+l
i+1,n
k.1 - l ,n a n,i+l -
k.1+ l ,n a n,1. l
77
Combining these last three equations:
k.l - l
=
.
, l
k"'
k"'
i-l,n
i+ln, - an 1 i-l an,i+l
The differential of the k.
l- 1
dk.l - l
1
l
+ k"'
dk~+l
l
i+l 1 n
1 n
. = k"'
i-l,n
l
. term is:
1
dk~l - l ,n
Using this relationship, the differential of the K matrix
terms becomes:
dk"'
ln
=
l
2k"'
ln
dk~
ln
=
l
2 < i
dk"' =
nn
l
(dq l..l + 2k~+l
l
,n dk:l - l ,n + 2k~l - l ,n dk~+l
l
,n
< n
dq
nn
+ 2dk ...
n-l,n
2k-"
nn
The above equations describe a set of nonlinear first
order matrix differential equations:
78
... 0
0
dk-' :
ln
dqll
k ... -k ln ... 0
2n
-k-'
4n k]n ... 0
0
dk;:.:::n;
dq22
dk) n ::= ~2
dq33
k-'
ln
-k'"
3n
0
0
0
0
0
0
-1
0
'
.
( 4-7)
;
dk
k"
nn
Equation (4-7) can be linearized about ki~, k;~, ... ,
0
k"
nn
Also, the scope of this investigation is limited
.
11 and q 22 terms, so qii = 0 for
, dk "
, ... dk"
all 3 .: :._ i .: :._ n, which makes the dk "
4n
nn
3n
terms dependent on dk"
. Using a linearized expression
2n
in terms
of equation (4-7) to solve for dk)n through dk"
nn
to adjustrrEnts on the q
of dk2n gives:
k" -k"o 0 ... 0
3n
2n
~k . . o k ... o
4n -k3~· .. 0
. 5n
-k ... o k5~ ... 0
0
6n
'
0
dk"
3n
dk"
2n
0
dk"
4n
0
0
dk" i=k"o 0
5n :i 4n
'
l
i
0
0
0
dk"
nn
-1
0
0
0
Define the above matrix of k: terms as the M
ln
matrix.
79
The dk)n term is equal to:
-1
0
dk....
3n = Mll
k ..
0
k ..
0
dk . .
2n
4n
3n
0
-1
where M
11
denotes the first row, first column element of
0
the inverse of the M
matrix.
The substitution of this
expression for dkJn into the linearized portion of equation (4-7) which relates the independent terms dkln and
dk2n to dq 11 , dq 22 and dq
results in :
33
k .. o
ln
dk . .
ln
=
2
-k . . o
3n
k .. o
2n
k .. o
ln
k .. o
4n
k 0
3n
0
-1
Mll
dk . .
2n
( 4-8)
Equation (4-6) and (4-8) may be cowbined to form an iterative equation.
Let
~
and W
n
be desired characteris-
tics of the plant dominant pole pair.
Let terms with
superscript "i" be solution values obtained from the
present iteration.
Let terms with superscript "i + 1"
be values to be applied to the next iteration.
ative equation is:
The iter-
80
qll
i+l
k-'" i
0
ln
4
=
i+l
q22
wi
n
-k-'" i
k ... - k-'" i
2n
ln
3n
k
k ... i
0
ln
w - w
n
n
k ... i
4n
·
3n
. Mi
i
i
(4-9)
;+
k ... i
k ... i
2n
ln
wn i
The process is initialized to
l
=
0
Equation (4-9) can be used to determine the q
11
and q
22
values necessary to cause the optimization process described in Chapter l through 3 to compensate an all pole
plant in such a manner as to result in a dominant pole
pair with desired values of
s
and W
n
.
Some considera-
tions regarding the application of this process are dis-
81
cussed below.
The process is dependent on the assumption that the contributions of dC 0 and dc 1 to dk2n in equation (4-4) is
small.
As can be seen from Chapter 3 results, large
values of q
22
;q
11
causes migration of residual poles to-
ward the origin.
The assumptions underlying the approximation of equation
(4-4a) by (4-Sa)
are less limiting in that direction of
residual pole migration with respect to the origin is
affected only by kin .
Chapter 5 will explore two methods
of applying equation (4-9) and compare results.
The process described by equation (4-9) may result in
negative values of q 11 and/or q
.
22
terms are related to q
k
11
and q
22
The kln and k n gain
2
by the equations:
ln
Sufficiently small q
11
or q
22
terms will result in com-
plex solutions for kln or k 2 n respectively.
It is, there-
82
fore necessary to assure that:
when employing equation (4-9).
Negative values of q
11
and q
22
also violate the require-
ment for a positive semidefinite Q matrix in Chapter 1.
Negative values of q
s
11
and q
22
result in desired Wn and
values, but the optimality of systems compensated in
this manner is unproven.
This, along with the other
considerations just discussed is certainly an area for
further philosophical as well as analytical investigation.
Chapter 5 uses the test examples of Chapter 3 to demonstrate the process described by equation (4-9).
tion of equation (4-9) is used on cases where
s
A variais un-
important and demonstrations of the full equation are
presented.
CHAPTER 5
EXAMPLES OF POLE PLACEMENT BY
ITERATIVE APPLICATIONS OF THE
OPTIMIZATION PROCESS
The iterative optimization process developed in Chapter
4 is applied to the examples of Chapter 3 to demonstrate
two possible applications.
The first set of examples
apply equation (4-9) to determine the q
and q
11
neces-
22
sary to position the dominant poles to achieve a desired
s
and W .
The second set of examples use a modified form
n
of equation (4-9) to achieve a desired value of W
n
of the
closed loop dominant poles.
Example l
This
exarr~le
uses System l to demonstrate the application
of equation (4-9)
in determining the q
11
and q
terms
22
necessary to cause an optimally compensated system to
have a dominant pole pair of predetermined
s
and W .
n
The system:
x
=
0
1
0
0
0
l
,-.625
-1.375
-2.75
0
X
+
0
u
l
is used in the iterative applications of equation (4-9)
83
84
TABLE 7
Q Matrix Elements for Poles of Predetermined
Values; System 1
wn
Goal =
i
Iteration
~
12
Goal
=
I~
%
%
wn
Goal
Goal
0
0
0
.525
37.1
.462
65.3
1
1
0
.720
50.9
.575
81.3
2
6.36
-.97
l. 053
74.4
.588
83.2
3
15.62
l. 65
l . 315
93.0
.650
91.9
4
20.43
5.31
l. 435
101.5
.. 699
98.8
5
19.21
5.65
l . 421
100.5
.708
100.1
6
7
8
85
with predetermined values of ~ and W
n
1--2-- respectively.
of I
0.5
and
The poles resulting from each iterNote
ation of equation (4-9) are plotted in Figure 10.
the relatively small movements of the single pole in the
vicinity of s = -2.3 while the dominant pole pair migrates to s = -1 + jl, marked by the square in Figure 10.
Equation (4-9) should be applied to achieve predetermined
values of
s
and Wn when, as in this example, one pole
pair clearly dominates the remaining system poles.
In
cases such as this, system response is closely approximated by the single pole pair, and the desired
~
and W
n
can be selected to achieve desired system response.
Example 2
This example demonstrates application of equation (4-9)
\
to achieve predetermined values of
~
and W
n
for a system
which, in the uncompensated state, does not have one
clearly dominant pole pair.
In fact, the system chosen
for this example, System 2, has a relatively dominant
single pole at s
=
-0.5.
It is desired to position a
dominant pole pair for System 2 at s
=
-1.5
~
jl.S.
The state equation of the uncompensated system is:
86
··-r
.
-- ·---1
i
!
s
~-- L~
1
-
i
t---~---
-
--~--+!
~
·
+-.
I: . ,__
t---- ---
. - -
;¥-
----r----------
~T=-~~:--~--
--r-----1
---~
------+-----
-- ---~--
-,-~---
1·- _______ ; _____
--t---
'
I
Q----J
--r· -
- i
I
iI
I-
I
--t-i
_,_i
! ..
-
I
+-
'
I
l·
i
-r-
'
-- r-
J
- -___
-
-- -
---:-------
,·
:
;·
1
I
+-- ---
~
I
--'-
I
r--
_l
I
87
x=
0
1
0
0
0
0
1
0
0
0
0
1
0
-2
-7
-8
-4.5
1
0
X
+
0
u
The results of the application of equation (4-9) to
System 2 to achieve a dominant pole pair at s = -1.5 +
jl.5 are tabulated in Table 8 and plotted in Figure 11.
Before considering the results of example 2 refer to
Chapter 3 Figure 6, the pole migration of System 2 in
response to various values of q 11 and q
.
22
Note the
migration of the single pole towards zero as the ratio
q 22 ;q
11
is increased in tests 20 through 24.
Refer now
to Figure 11 and Table 8 to analyze the application of
equation (4-9} to System 2.
Note that, after the first iteration, the dominant pole
is the single pole at s = -0.5.
Equation (4-9) is mech-
anized for these examples to bypass adjustments due to
errors if the dominant pole is single.
s
88
TABLE 8
Q matrix elements for poles
of predetermined values; System 2
·-------------------------------~----·-------·---·-····-----··--··-----------~-~----·---~
---~------··
.
Dominant
Pole Pair
-------------·-··-
---- ---· ------·-
--~----·
Iter-!
ation
i
'
i
q22
1
s
~.
iResidual Pole
~~------~~---,
i
Goal=2.121
Goal= .707 /
actual %goal actual %goal!
;
qll
I
wn
1. 414
66.7
.707 100
.5
0
1. 411
66.5
.702
99.3
.562
0
1. 558
73.4
.558
78. 9 . 1.790
0
1
..
_,
~'Vn
s
.987
2
56.44
3
143.85
134.71 2.190 103.3
.575
81.3
.993
4
815.69
134.71 2.353 111.0
.505
71.5
2.274
.984
5
595.47 .-451. 56 *
.499
70.6
2.375
.982
.638
90.2
2.792
.914
.510
72.1
3.417
.665
6
7
8
9
10
1224.4
189.53 2.493 117.5
699.03 2292.1 *
7838.3
-665.48 *
550.18 -296.81 2.257 106.4
1038.7
210.13 2.486 117.2
* denotes unstable solutions
. _....
lm
·----:--T·----~-
I
I.
.
~ -1I
. - . . - . ---r-·----
-··- I
I
. -·-·--.
I
I
' ....... ] tI
I
I
1
I
10&$
1
:
•
I
g
I
l'
rqet: s= - ).5±j 1.$
0~
1
I
I
,
I. . .: --'I J2
f-- :
-----·-r- . ;. ·u-:
: -+
,--+,
!
o
·i ........ :.....
.
l
-r--+ r
-~~-
f~'· · _j ]
·!
I
. :•
- ;
3
I
I
f
I
I
:
:
i
!
.J, . •
t
,
I
. .!
I
:
'
}
_ _It__l-r---------1'
I
I:
I
I
.I
!
t:.:
I oY
I---
!
Q~
02
Olq
!11Z :
4
0~
.
.
e
I
I
l
i
'.
.
I
I
,
--~--
.·~------ . -.. +-- .... ------·--
I
1
-[-~--
I·
M1grgt1ofl t9 T~rg~t
.
............
l
s~'stem
Orqler
,
.... 1
-- -- + - ---r Pol~
.
I
1
11 •.. 'Fo~r1 h
f;giJre!
1
!
1
I
I
I
f- 1
I
1
I
-3
1
I
1
l
el
I
1
1
1
+
0 1
f 1
-2
1
1
I
1
1
1
I
1 '
-1
3
1
1
1
+I
lo
1 0
, I
1
I0
0
co
1..0
...
90
Table 8 data resulting from the second iteration shows
the dominant pole pair lies at s
=
-.87
~
jl.29.
This
pole pair is adjusted towards the target by the third iteration.
Note, however, after the third iteration the domi-
nant pole pair is further from the origin than the single
residual pole at s
=
-.99.
on the single pole at s
=
The fourth iteration operates
-.99 and results again in the
dominant pole pair further from the origin than the residual poles.
However, since the residual poles resulting
from the fourth iteration are complex, equation (4-9) is
applied to this pair and unstable solutions result.
The application of equation (4-9) could probably be modified to correct for the above behavior.
The applicability
of the process of examples 1 and 2 is somewhat questionable, so effort to correct this anomalous behavior was
not expended.
Example 3
Example 3 applies a modified form of equation (4-9) to
determine the q
11
necessary to achieve a desired Wn of
the closed loop dominant poles.
to the following systems:
This method is applied
91
System 1:
x=
0
1
0
0
0
1
-.625
-1.375
0
X
-
+
0
-2.75
u
1
System 2:
x =
0
1
0
0
0
0
0
1
0
0
0
0
1
0
'-2.0
-7.0
-8.0
-4.5
1
X
-
+
0
u
System 3:
*=
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
1
1
1
-5
-19
-25.5
-18
-6.5
1
X +
-
0
u
92
Equation (4-9) is modified to adjust q
11
to achieve a
predetermined value of Wn while holding q 22 through qnn
at zero:
+ 4klni2
w
(W
n
-
W i)
n
( 5-l)
i
n
The qll term is initialized to one (i.e., qll 0 = 1. 0) .
The qll term is adjusted according to equation ( 5-l)
after each optimization process.
The new value is used
to perform another optimization process after which q
is again adjusted.
The process is performed until:
(W
n
-
W
n
i
)
2
<
(0.01
wn ) 2
or, in the interest of economy, until i > 8.
11
93
The final value of W chosen for each system is the unn
damped natural frequency of the highest frequency pole
of the uncompensated system.
The literature focuses
some degree of attention on the fact that, for sufficiently high compensating gains, M
p
(overshoot of system
response to a step input) remains constant with increasing gain.
In other words, the system dominant pole pair
exhibits relatively constant
s
characteristics and has
a dominant effect on system response.
Graphical tech-
niques exist for determining the minimum gain necessary
to achieve relatively constant M
p
1975, p.528).
(D'Azzo and Houpis,
It was observed in the preparation of
this paper that Mp becomes relatively constant as q
11
becomes large enough to cause the natural frequency of
the dominant pole pair to exceed the natural frequency
of the highest frequency pole of the uncompensated systern.
This example, then, is used to demonstrate the
determination of q
11
necessary to achieve a desired Wn
of the dominant pole pair and the relatively constant
nature of Mp for larger values of q
11
.
Table 9 lists the results of successive applications of
equation (5-l) to systems 1, 2 and 3.
each system is:
The W picked for
n
94
System 1:
wn =
2.265
System 2:
wn =
2.0
System 3:
wn
2.236
=
95
TABLE 9
Optimization for W
Results
n
--T
--------~----------sy~t~-1-~-
_ _ _ ------
1 teration
i
!
q 11
•
:W1
· n
~---
--~-sy-~t~2-
--- -----~
----!--~--~-~~--;------------
% of i
Wi
n
q 11
!
. % of
-------8-y~t~
-·--·---------·-
q 11
3 --
----------;----~----------
Wi
· n
i
% of
1
2.2651
0
.525
23.2
2.0
1.536:76.8
2.236
348.211.547 : 69.2
1
6.17
1.053
46.5
2
36.37
1.569
69.3 ill2. 97 . 1.682 84.1
3
101.58
1.949
86.0
201.38
l. 807' 90.4
2205.4 'l. 907 ; 85.3
4
167.72
2.156
95.2
'289.
00
!
l. 894!94. 7
3743.3 :2.034 : 91.0
5
201.73
2. 236
98.7
:354.64
l. 946! 97.3
5240.6 -2.119.94.8
6
212.25
2.258
99.7
394.62
1.974:98.7
'
l. 987.99.4
6406.8 ,2.171 ' 97.1
49.0
!
7
,415. 98
1013.5
l. 738 : 77.7
7180.2
2.201 • 98.4
7641.9
2.217
I
8
The asymptotic approach of Wi to vJ
n
12 for each of the above systems.
n
99.2 '
is detailed 1n Figure
96
r~!'·
. . I . .
-~-+---:--
-~-
------!
______ ____
........_..
•
.
~
1
______,_ _____ _
'
Sgs$em
Sg s:te11'
1,5
r-··
·- c:
3:
__t_
1.0
-~~---__:____------:------
I/----~---------:-·
I
.
I
I
I
-----·---:-----------•--·-·----
i
.
'----~_z:_ 12
I
Ac:umptot~Ajppraof:b of'
...
l
;
•
w},
I
fo---_Wil-'1'1~--~~
i
.
i
r
!.
l-
i
6
7
8
·----~---! ___;_ --:. ---~-- r-I
0,5
2
3
4
5
(itcrdtionl
3------
97
The response characteristics of Systems 1, 2 and 3 to
a unit step input are tabulated in Tables 10, 11, 12, 13,
14 and 15 for gains at and above that necessary to cause
W of the dominant pole pair to equal the W
n
n
of the high-
est frequency pole of the uncompensated system.
The
response data of Tables 10, 11 and 12 are produced with
q 22 = -, while that of tables 13, 14 and 15 are produced
with q 22 = .01 q 11 .
Figures 13, 14 and 15 plot Mp' Kv
and Tr versus the gain of the equivalent, unity feedback
forward transfer function (see equation 3-8) for Systems
1, 2 and 3 respectively.
Comparing the M
p
curves of Figures 13, 14 and 15 with
those of Figures 5, 6 and 7, note the relative uniformity of M
p
15.
with increasing gain in Figures 13, 14 and
Note also in Figures 13, 14 and 15 that the degree
of uniformity of M
remains relatively unchanged for
p
q
22
= .01 q 11 , although the value of MP is reduced.
Settling time, T , for tests 37 through 72 is plotted as
s
a function of 1/s W
n
ted for each test.
in Figure 16.
Two curves are plot-
One curve is plotted using the ac-
tual s of the dominant pole pair in the 1/s W expresn
sion.
The second curve is plotted using the
s
of the
dominant pole pair asymptote which is computed by:
98
~
=
cos(90(1- 1/n))
where, in this expression, n is the order of the system
being compensated.
The rule of thumb for time to settle
within 2% of final value:
=
wn
4/~
is plotted in Figure 16 for comparison.
The data of Figures 13 through 16 suggest a design technique.
Use system settling time as the primary design
criteria.
Then, using either the actual or computed
~
(note from Figure 16 that computer
~
gives a more con-
servative design) , determine the W necessary to achieve
n
the T
s
criteria.
mine the q
11
Finally, use equation (5-l) to deter-
necessary to meet the T
8
criteria.
resulting system overshoot is not acceptable, q
If the
22
can
be increased until system overshoot is within acceptable
limits.
99
TABLE 10
Constant M , Third Order System;
p
!Time !ForPeak jpeak !ward
Rise; over-jover-!gain
timei shootishoot:k 1
a
;
sec.
'0
:sec.
a n
,Test, q 11
. n1
212.25 1.04:1.068;
2.20 14.58
37
;
!
.
Dominant
Pole Pair
Rate - -------'Damp-=---:
error Freq 'ing
coeff rad/ factor
Kv
.sec
~
Settling
time
to 2%
sec.
···--·
1.16 2.26! .579 ,2.97
.
:38
297.31.98:1.068: 2.06'17.25
1.23 2.41
.572
2.79
'39
416.39 . 912 l. 068
l. 92! 20.42
1.31 2.57
.566
2.62
'40
583.10 .858 1.069: l. 80 24.16
1.39 2.74
.561
2.46
'41
816.49 .807 l. 070
l. 69 28.58
1.47 2.91
.555
2.30
.759 1.071' l. 61 33.82
1.56 3.10
.550
2.17
'
·42
1143.2
100
TABLE 11
Constant M , Fourth Order System;
p
~-- ----·--~noffiin:a:n:t----~~r
Test q 11
---···--
-
------
Peak
ward
Rise over- over- gain
time shoot shoot k n1
- -Pole
- - - - -Pair
- -Rate
Damperror Freq ing
coeff rad/ factor
-
~--­
Settling
time
to 2%
- ~ ~ ~ .- -.~- - --- =~=-=~~~--~---~!11__ ~--_!y_____~_~:____ -~----~ --- ~--~~ . - -
··--··----·-·-·-------------···-
····-·
------·- ---------------------
43
415.98 1.31 1.078 2.84
20.49
.76
1.99
.473
3.71
.44
583.97 1.24 1.081 2.73
24.25
.80
2.08
.464
3.55
819.16 1.17 1.085.2.57 :28.69
.84
2.18
.456
3.39
45
:46
1148.4
1.12 1.088 2.47
33.95
.88
2.28
.449
3.24
47
1609.4
1.06 1.090 2.34
40.17'
.92
2.39
.443
3.08
48
2254.8
1.01 1.092 2.29
47.53·
.967 2.50
.437
2.94
101
TABLE 12
Constant M , Fifth Order System;
p
Time
Peak peak
Rise over- overtime shoot shoot
Test ~~l}_ --~ -~s-=-~-'--~ -~!49
7641.9
Forward
gain
k ln
Dominant
Pole Pair
Rate
- Damperror Freq ing
coeff rad/ factor
Settling
Time
to 2%
______:;:c_~~~~ ~nJ,.______ ~_y_ _ _s~~-1;;---~~=-:___~
1.20 1.101 2.80
87.56
.71
2.22
.407
4.63
50 10709.
1.14 1.103 2.72 103.60
.74
2.31
.399
4.49
51 15002.
1.08 1.105 2.61 122.59
.77
2.40
.391
4.34
52 21013.
1.03 1.107 2.52 145.04
.80
2.50
.384
4.20
53 29428.
.99 1.108 2.38 171.62
.83
2.60
.377
4.05
54 41209.
.96 1.110 2.28 203.06
.86
2.70
.372
3.90
102
TABLE 13
Constant Mv Third Order System;
p
Test q 11
-
Dominant
Time ForPole Pair
Peak peak ward Rate
'Damp· Rise over- over- gain error Freq ing
time shoot shoot k, coeff rad/'factor
sec..
%
sec: __ a~~- __ Kv ___ :_~c j s
___
-------------·----·-·-----~--------------------------------------·-----..-------------
i
Settling
time
to 2%
:_~':_:___
---------.
55
212.25 1.04 1.063 2.18
14.58 1.15
2.27'
.586
2.96
56
297.31
.98 1.063 2.06
17.75 1.22
2.42
i
.581
2.78
57
416.39
.92 1.063 2.45
20.A2 1.29
2.59i
.576
2.61
58
583.10
.87 1.063 2.34
24.16 1.37
2.76
.571
2.44
59
816.49
.82 1.063 2.23
28.58 1.45
2.94;
.567
2.29
.77 1.063 2.11
33.82 1.54
3.13
.563
2.15
l
•
60
1143.2
1
103
TABLE 14
Constant M
p
I
q22
---··-- --·· --------- ---
.
---------·----~--
Test ql1
Fourth Order System;
=
.Olqll
----------~----------.
..
--------·--------
Time ForPeak peak . ~r.rard
Rise over- over- gain
time shoot shoot. k
, ln
%
sec.
sec.
a
nl
··--~
61
415.98 1. 30 1.073 2.90
62
63
...
--
-------------
.
····--------------
Dominant
Pole
Pair
- .
Rate
DampError Freq ing
coeff rad/ factor
K
sec
s
v
-·····--···-·--------~---·
Settling
Time
to 2%
sec.
20.49
.76
2.00
.477
3.71
583.97 1. 25 1.078 2.75
24.25
.79
2.09
.468
3.54
819.16 1.19 1. 080 2.60
28.69
.83
2.19
.460
3.38
'
64
1148.4
1.12 1.083 2.47
33.95
.87
2.29
.454
3.23
65
1609.4
1. 07 1. 085 2.35
40.17
.92
2.40
.448
3.08
66
2254.8
1.01 1. 087 2.24
47.53
.96
2.51
.443
2.93
104
TABLE 15
Constant Mp' Fifth Order System;
q22
=
.Olqll
··-----··-- ·------·-····-·-···-· .. ·---.
Test qll
Time
Peak peak
Rise over- overtime shoot shoot
sec.
%
sec.
67
1. 20 1.096 2.80
Forward
gain
k ln
a
nl
Dominant
Pole Pair
Rate
Damperror Freq ing
coeff red/ factor
K
sec
s
v
·-•··-
Settling
time
to 2%
sec.
- ---
·-------····
87.56
.71
2.23
.410
4.58
68 10709.
1.15 1.101 2.69 103.60
.74
:2.32
.402
4.44
69 15002.
1.10 1.102 2.62 122.59
.76
~2.42
.394
4.30
:70 21013.
1. 05 1.102 2.52 145.04
.79
2.52
.388
4.16
'71 29428.
1. 00 1.10 3 2.41 171.62
.82
2.62
.382
4.02
72 41209.
. 96 1.105 2.33 203.06
.85
2.72
7641.9
....
-
-----------------------·-- -------·---··
-
.
------------------
----~--
3.87
.376
------------------
-·-·
--
-.-
105
'!
---~--~---
-----~----··
.
>'
--- _::.::;_~;_ __
-
_,_ . I -:
.
I
a,;
- -
.
: - --
I
q---,-----I
.
i:E:
1.0
-·.!
,_.
I...·
0
0
IT--+,--+--____._-
- - -- - - -
~i_~---- -J~~-- --I
~
I
r-
~---
I
·,
I
1
i
LI
.....---------.;.
'E.:
~:
--
I
"0:
a." '
--a--
,
I'
'
,--.----;
'
,:-
-
-
.
, _____ ·-·-
I
~---r--
i
!
N-----
-~---­
'-
':t-
t:T'
"s.L
~:.
.
o ..
106
I
-I
r··
•
¢···
~
e.:
I...
:)
r:J'
,: __H~- l,L__
107
. .
- 'j
-
.
.
.
.
.
.
------~--~~~--·
I
-------!
I
i
------ ---i
-I--.-----
-1
E
·----~
+.. ($1
:;n: ____ ,i
(fk
I
<»
;
"0
i
t,--------1
.
0
l
··-·-:----~
_c
~!t;
0
I
__ _;
IL
I
-- -------~--"
~
-- ~
I
0
i
-t--1
::l---0'.
-:!
I
------~-----1
i
!
i
108
4
3
2
I
I
,1---------------
---~
/-E=-- T5 = 4fjW0
Jl
-
u
------------
~
---~---~---
- - - - - --------
c
0
v
- - ' - - - - - - - - - - - ·- - - -_ _ ;_ _ _ c _______ _
"'
U1
1-j------
-----------------
/~~u,. -:,. _-~u-~ct_-1 uniu.ql-'•-Tu;umu"~-"'vo:~$"-·~lw =- .),_ ·
W""AFr-.
·-- --· ---------------·----- _
_c__~~-'----,---;----
0.5
1.5
1/JWn !seconds!
109
In summary, this chapter has applied the derivations of
Chapter 4 to three examples and results and observations
have been discussed.
In addition, a method of using the
optimization process to achieve conventional figures of
merit has been suggested.
CHAPTER 6
SUMMARY
The linear, time invariant, single input system:
x=
Ax + Bu
( 6-1)
can be compensated to cause the state vector,
the value z in the steady state, where the
~
~'
to take
elements are
linear functions of a step input, r, such that:
z
= Cr
For this work, the forward and feedback gains necessary
to achieve the desired system state were determined by
applying optimal control techniques to minimize the
func·tional:
(6-2)
J (u)
where:
d
is the value of u required to force x to z
in the steady state
R
is a positive, nonzero scalar
Q
is a positive-semidefinite diagonal matrix
110
111
The desired steady state performance is achieved when the
system described by equation (6-1) is compensated in the
following manner:
The K matrix in this compensated system is an n x n symmetric matrix which satisfies the following nonlinear
matrix equation:
The ratio of the Q matrix elements to one another as
well as to the R scalar affects the transient response
of the compensated system.
For this study, the R element
was set equal to one and the effect of the Q matrix elements q
served.
1.
11
and q
22
on system transient response was ob-
Results are listed below:
As the q 11 element is made larger (q
= 0), compen22
sated system poles migrate away from the origin increasing
system bandwidth and transient overshoot (M ) , and dep
creasing system rise time and settling time of response
to a step input.
112
2.
For the examples presented in this work it was ob-
served that, as q 11 is made larger (q
22
=
0), compensated
system poles migrate towards the Butterworth form: n
poles positioned equidistant from the origin space~ rr/n
radians apart.
This observation is confirmation of a
result attributed to Kalman (D'Azzo and Houpis, 1975).
3.
It was observed for the examples presented in this
work that, as the q
11
element is made larger (q
= 0),
22
compensated system poles approach lines of constant
~
(sometimes referred to as the Butterworth asymptotes) , a
condition arises where increasing q 11 (q
22
=
0) decreases
system rise and settling times in response to a step input without significantly increasing M .
p
This observa-
tion confirms results reported in the literature (D'Azzo
and Houpis, 1975).
This
to as the constant M
p
4.
condition is sometiines referred
condition.
It was noted that, for the examples presented in this
work, a sufficient condition to assure constant M
p
set the q
11
is to
element large enough to cause the natural
frequency of the compensated system dominant poles to
equal or exceed the natural frequency of the fastest pole
of the uncompensated system.
113
5.
It was observed that as the ratio q
22
;q
11
is in-
creased, compensated system performance becomes more
sluggish.
This observation is in agreement with the re-
sults of D'Azzo and Houpis (1975).
It was further ob-
served in the examples presented in this work that increasing the q 22 ;q 11 ratio causes one compensated system
pole to migrate towards the origin.
Part of this paper is devoted to applying optimal control
techniques to the compensation of linear, time invariant
systems to meet conventional figures of merit.
Two suc-
cessive compensation processes suitable for digital computer solution are presented.
A discussion of the pro-
cesses and results of each follows.
The Q matrix elements determine the transient response
of the compensated system.
Q matrix elements, and con-
sequently, the feedback gains, K, necessary to achieve
conventional transient response figures of merit may be
determined by successively analyzing system pole configuration, determining the change in Q elements necessary to move compensated system dominant poles to the
desired configuration, and then compensating the system
using the new Q matrix values.
114
One successive compensation process derived for this work
relates the change in the q 11 and q
elements of the
22
Q matrix to changes in the
system dominant pole pair.
s
and W of the compensated
n
The algorithm for applying
this method is:
wn - wn i
( 6-3)
s - si
where:
q
i+l
is the value of the Q matrix element to be used for the
(i + l)th
iteration
q
i
is the value of the Q matrix element used for the ith iteration
are characteristics of the compensated system dominant pole pair
f rom t h e l. th lteratlon
.
.
.
resu 1 tlng
wn ' s
are the desired characteristics of
the compensated system dominant
pole pair
is a matrix made up of elements of
the K matrix and
s
and W of the
n
dominant pole pair resulting from
the ith iteration.
115
The results of the application of equation (6-3) are as
follows:
1.
For an example system which, in the compensated and
uncompensated states, was closely approximated by a second order system, equation (6-3) satisfactorily positioned the dominant poles to achieve the desired transient response.
2.
For an example system which had dominant poles hav-
ing a natural frequency close to that of one of the remaining system poles, compensation could not be determined using equation (6-3).
This result was due primar-
ily to the migration of one pole towards the origin as
the q
3.
22
;q
11
ratio was increased.
As a result of the examples presented in this work,
equation (6-3)
is not considered generally applicable
to compensation of linear, time invariant systems, although it may be applicable in some specific cases.
A second successive compensation process uses a modified
form of equation (6-3):
+ H (Ki I
w
n
i) . (W
n
( 6-4)
116
Note that q 22 is not modified by this equation and there
is no attempt to control the
s
of the dominant poles.
The results of the application of equation (6-4) are as
follows:
1.
Equation (6-4) successfully compensated the examples
chosen for this work by determining the value of q
11
necessary to achieve a predetermined value of natural
frequency of the dominant pole pair.
2.
Using the rule of thumb:
Ts
=
4/sWn
a technique for applying equation (6-4) to determine compensation required to meet a T
ible.-
s
design criteria is poss-
The q 22 element can be used to decrease system
step input response overshoot to within acceptable levels.
CHAPTER 7
EXTENSION OF TECHNIQUE AND
AREAS FOR FUTURE
~vORK
The optimization process derivation in Chapters 1 and 2
is limited to single input, single output, observable
systems.
The methods of Chapters 3 through 5 were applied
to all pole, phase variable plants.
Although this paper
will not present results from more generally configured
systems, this chapter will surmnarize steps which can be
taken to convert the general n
th
order pole-zero system
into forms from which system pole and zero information
may be obtained.
The solution equations to the nonlinear matrix equation:
are derived for a plant of unrestricted form.
The first
steps in applying the work of Chapters 4 and 5 to a plant
of unrestricted form is the conversion of the unrestricted
plant matrix to a form from which pole and zero information can easily be obtained.
The derivation ln the fol-
lowing paragraphs is similar to one by Ogata (1967, p.201)
but is modified extensively to provide a state equation
117
118
in pure phase variable form:
~'
-/~
/
1
b
---/
1
n-1
R(s) __-=~s_-___~____
:_-_~--
s
-1
~---·-·------- .-
-a
y ( s)
R(s)
1
n-2
+ ... + b2 s + bl
+ b n-1 s
sn + a s n-1 + ... + a2 s + al
n
bn s
=
n-1
where, if
X.
= Ax + Br
y =
then
ex
A =
0
1
0
0
0
0
1
0
0
0
0
1
-a
1
-a
2
-a
3
-a
n
y ( s)
ll9
B
='
0
0
0
1
c
=
b
n
)
As can be seen from the above, the n
th
row of the A mat-
rix contains the coefficients of the polynomial defining
the characteristic equation of the system transfer function, and the C matrix contains the coefficients of the
polynomial defining the zeros of the system transfer
function.
Given a general n
v
=
th
order, pole-zero system:
Fv + Mr
Kv
y
find the transformation matrix, P, such that:
v
=
Px
120
so that:
*
=
y
=
KPx
where:
p -1 FP
= Phase Variable (Companion matrix) form, and
p -l M
=
0
0
0
l
so that:
KP
=
b
n
)
The elements of the KP matrix product are the coefficients of the numerator polynomial of the transfer function.
If the requirements for P
-1
FP and P
-1
M are
met, then the desired property of the KP product will
follow.
121
The first task is to find the companion matrix, A, to the
general matrix F.
The n
th
row of A contains the poly-
nomial coefficients of the characteristic equation of F.
By the Cayley - Hamilton theorem:
F n + a n Fn-1 + a n- lFn-2 + ... + a2F + a l I
The F matrix is known.
tions in n unknowns.
through a
=
0
Equation (7-1) provides n
(7-1)
2
equa-
The companion matrix elements a
1
can be determined directly.
n
It should be noted at this point that application of the
optimization process described herein results in an nth
order system with n distinct roots.
It is comforting to
note that equation (7-1) must be the minimum polynomial
of F ( Wy 1 i e , 19 7 5 , p . 5 7 9 ·) .
The next task is to determine the elements of the P matrix such that:
p -1 FP
=
A
and:
p -1 M
'
=
0
0
0
1
122
Define the P matrix to be made up of column vectors such
that:
p
n
)
so:
p
-1
FP = A
FP
;
PA = (P 1
=
PA
. . .
p2
p
n
)
Looking at the FP product:
. FP )
n
Equating FP and PA results in:
FP
n
+ a
n
P
n
- P
n-l
)
=
0
0
1
0
0
0
0
1
0
0
0
0
1
123
Each of the P matrix vectors can be computed:
-1 p
pl
= ' -a 1
p2
=
F
p3
=
F
P
=
F-l (P
n
-1
-1
F
n
(P 1
- a2 p )
n
(P 2
- a3 p )
n
n-1
a
n
P )
n
All of the P matrix vectors are dependent on P .
n
vector is computed from the relationship
0
0
1
The P
n
124
Iv1
=
p
0
p
n
)
0
0
0
0
0
1
1
=
p
n
Thus,
p
n
= M,
and:
pl = -a 1 F
p2 = F
p3 = F
p
n
-1
-1
-1
(Pl
M
-
a2 M)
(P2 - a3 M)
= F-l (P
n-1
-a
n
M) = M
(7-2)
With the F and M matrices reduced to the phase variable
form, the product:
b
n
)
(7-3)
125
follows as a natural result.
equations
(7-3)
(7-1)
and (7-2)
An example of the use of
in the computation of equation
follows.
Example
Given the unity feedback system with forward transfer
function containing the elements
G
(s)
=
s
s
1
s
1
+ 3
+ 2
s
)
+ 1
the resulting system transfer function is
y (S)
R(S)
=
s +
3
The state equation describing the above system is:
v =
y = (1
-1
1
1
0
-2
1
-1
0
0
0
Equation (7-1)
0)
is:
0
v
+
0
1
v
r
126
F2 +
F3 +
a3
a2 F + al I
0
-1
1
1
0
-2
1
-1
-1
0
0
0
0
0
6
-3
0
1
0
3
-9
3
0
0
1
0
3
0
0
-3
0
a3 !i-l
4
-2
1
-1
1
i
=
+
a2
\
\
)
I
I
\
al
=
Extracting the diagonal elements
1
-1
0
al
1
-2
4
a2
1
0
-1
·. a3
al
a2
a~
J
= - 1/3
0
9
=
0
2
-1
-4
0
5
-1
-4
9
2
-1
-1
0
3
=
The companion matrix elements are determined:
3
3
\
1.
+
127
A
0
1
0
0
0
1
-3
-3
-3
=
!
I
I
which agrees with the characteristic equation of the
original calculation of the transfer function.
Equation (7-2) is now applied to compute the P matrix
vectors:
F
-1
pl
0
0
-3
1
-1
-1
2
1
-2
1
3
=
=
(!)
-3
3
0
0
-3
0
1
-1
-1
0
2
1
-2
1
3
pl
=
1
2
p2
=
1
3
1
p2
=
1
3
0
0
-3
;3
1
-1
-1
:1
2
1
-2
. '2
0
-3
0
1
128
0
p3
=
=
M
0
1
p
=
3
1
0
1
1
0
2
3
1
1
-1
0
-1
3
0
1
-7
2
As a check:
and
p ~1 FP
~
=
0
1
0
0
0
1
-3
-3
-3
=
as expected.
KP
=
(1
1
-1
0
-1
1
1
3
1
0
-1
3
0
0
-2
1
1
1
0
1
-7
2
-1
0
0
2
3
1
Finally,
0
0)
3
1
0
1
1
0
2
3
1
=
(3
1
0)
129
From the application of equations (7-1) and (7-2), it is
found that:
and
Y(8)
R(8)
s
3
8 + 3
2
+ 38 + 38 3
which is the correct result.
The above derivation is by no means exhaustive.
It is
provided as one possible method to reduce the optimally
compensated general system to a form from which pole
and zero information can be extracted.
The resulting
pole and zero information can then be used in applying
equation (5-l)
(accounting for migration of pole towards
zeros, of course) to achieve desired system response.
This process would be an interesting area for future investigation.
130
LIST OF REFERENCES
D'Azzo, J.J. and Houpis, C.H. 1975.
Linear Control System Analysis and Design:
Conventional and Modern.
New York: McGraw-Hill.
Kalman, R.E.
1960. Contributions to the Theory of
Optimal Control. Boletin de la Sociedad Matematica
Mexicana.
102 - 119.
Kirk, D.
duction.
1970. Optimal Control Theory: An IntraEnglewood Cliffs: Prentice-Hall.
Ogata, K.
1967. State Space Analysis of Control
Systems. Englewood Cliffs: Prentice-Hall.
Saatz, T.L. and Bram, J.
1964. Nonlinear Mathematics. New York: McGraw-Hill.
Sage, A.P. and White, C.C.
1977. Optimum Systems
-Control. Englewood Cliffs: Pren tice-Ha-11.
Wylie, C.R.
1975.
matics. New York:
Advanced Engineering MatheMcGraw-Hill.