(s2 + 5s + 1)/(s + 2)(s + 3)
+ 5s + 1)/s(s + 3) (? + 5s + 1)/s(s + 2)
(s 2
'
j
r
f
l'' (
1188
APPENDIXC
Thus:
I
are the requ ired coefficients. This is the parti al fraction expa nsio n resul t used in Example 3.3 of Chap ter 3.
CASE 2: Rep eate d roots. In this case, we cons ider that Q(s) has q- k disti nct root s and one root , rw is repe ated k time s, thus: R(s) =
ew_ = (-AI - + ... + Q(s) s-r 1
Aq-k
)
s-rq -k
+
P*(s ) k (s-r m)
(C.47a)
or
= RO(s)
R(s)
+
R*(s )
(C.47b)
whe re R0(s) em resp onds to the sum of unre peat ed factors in wing ed brac kets. Obs erve that the coefficients Ai, i = 1, 2, ... , q- k can be foun d as in Case 1. Now expa nd the rem aini ng term as follows:
Mul tiply ing by (s- r,i in Eq. (C.47a),
Bk
usin g Eq. (C.48), and setti ngs = r give s: 711
= }~n;. 0
[
(
s-
rm
)k ..fl& Q(s) .J
(C.49)
To simp lify the nota tion , if we defin e:
= (s- r m)" P(s) = Q(s)
(s- rm)k R(s)
(C.50)
i.e., wha t is left of R(s} after (s- r )k has been rem oved , th~n we have: 711 (C.5 l)
The othe r Bi coef ficie nts are obta ined by diffe rent iatio n. Eqs. (C.47) and (C.48) that: (s-rm )kR( s) = (s-rm )kR 0(s) +Bk
+
Obs erve from
(s-r m)B k-!+ (s-r m) 2Bk_ 2
(s- r711 )k- 2B2 + (s-r m)k -lB 1
+ ... + (C.52)
I
LAPLACE AND z-TRANSFORMS
1189
And now, by differentiating with respect to s and settin gs =rm' we obtain: (C.53)
or (C.54)
The others are determ ined similarly by repea ted differentiation in Eq. (C.52). The result is:
__1_[
4i*(s)J
tfk-j)
Bj- (k-j) !
J.s(k-J)
s=rm
(C.55)
It is straig htforw ard to generalize these result s to the case with several sets of repea ted
roots.
Examp le C.3
PART IAL FRAC TION EXPANSION: REPEATED ROOT S.
Obtain a partial fraction expans ion for the function: Rs
-
( ) -
s(s 2
I
+ 6s +
9)
Soluti on: Factoring the denom inator quadra tic gives: I R(s) = ~(s + 3)2
which we wish to expan d as:
s(s + 3) 2
In this case, the repeat ed root is at s = 3, and: P(s) = Q(s)
= s(s + 3)2
Also: I
Thus, we obtain, in the usual manne r (multiplying by
sand setting s= 0):
1190
APPENDIX C
Next, we obtain:
"l
I
and finally, since:
l
then we have:
l
9
The complete partial fraction expansion is therefore given by: s(s
1
1/9
+ 3) 2
-.;-
-113
II
-119
+ (s + 3 ) 2 + (s + 3)
CASE 3: Complex conjugate roots. In this case: R(s)
M
cl
Cz
= Q()s = -(s-r -) +s-r -( *) + R+(s)
(C.56)
where r
= a±bj
(C.57)
and R+(s) is the remaining portion of R(s). H we let
=
(C.58a) (C.58b)
then C1 and C2 are obtained in the usual manner, by multiplying in tum first by (s- r), and settings =r, and then by (s- ,.) and settings =,.; the result is:
cI -[~] (r-1'*)
(C.59)
-[~] (I'*- r)
(C.60)
Cz -
It is easy to show that C2 will always be the complex conjugate of C1 , so that in actual fact, only one of them need be evaluated. Example C.4
PARTIAL FRACI"ION EXPANSION: COMPLEX ROOTS.
Obtain a partial fraction expansion for the function: R (s) -- s(s 2 + 42s + 2)
1191
LAPLACE AND z-TRANSFORMS
Solution: Factoring the denominator quadratic gives: 4
R( s ) --
s(s - r)(s - r*)
with r
= - (1 + ;);
r*
= - (1 -f)
and we wish to expand this as: C2 Cl AI -; + (s + I + f) + (s + i - j)
4 s(s - r)(s - r*)
In this case: 4
s
and since r- r" =-2j, we obtain from Eq. (C.59) that:
and from here that
c2
= -1
+j
We obtain A1 in the usual manner (multiply by s, sets= 0) to be 2. The complete partial fraction expansion is therefore given by: 2
4 s(s - r)(s - r*)
= -; +
(-1-i)
(-l+j)
(s + I + j) + (s + I - j)
Note from Eqs. (C.59) and (C.60), and also from the example, that C1 and C2 will naturally occur as complex numbers. This is awkward if the end result is not merely partial fraction expansion, but the Laplace inversion to obtain R(t). For this reason, Eq. (C.56) is often consolidated to: (C.61)
where we have made use of the fact that C2 = C1*, the complex conjugate of C1• It is now easy to show that Eq. (C.61) simplifies to: R( )
s
(s - a)
= 2 a (s- a)2 + b2
-
2{3
b
(s- a)z + bz +
R+( )
s
(C.62)
where
a
[~]
Re(C 1) -- Re (r- r*)
(C.63a)
1192
APPENDIX C
f3 -
Im(C 1)
--
[~]
Im (r _ r*)
(C.63b)
The Laplace inversion of Eq. (C.62) will therefore be:
f3
R(t) = 1e'l1 (a cos bt -
sin bt) + R+(t)
Thus, for the function in Example C.4, a= -1, b =-1, a= -1, inverse transform is:
(C.64)
f3 = -1,
and the
2 + 2 e-t [sin (-t)- cos (-t)]
R(t)
or R(t)
=2-
2 e-t (sin t + cos t)
CASE 4: R(s) not strictly proper.
When p = q (i.e., the numerator and denominator polynomials of R(s) are of equal order), R(s) is now merely proper. The procedure for handling this case calls for expressing R(s) as: R(s) = A 0
+ R 1(s)
(C.64)
where R1(s) is now strictly proper and can therefore be expanded as before. It is easy to show that A 0 is given by: A0
=
lim R(s)
(C.65)
s->~
Example C.5
Obtain a
pa~tial
PARTIAL FRACTION EXPANSION: R(s) NOT STRICTLY PROPER. fraction expansion for the function: (s 2 +4s+3)
R (s)
= (s 2 + 9s +
20)
Solution: Factoring both the numerator and the denominator quadratics we obtain: R(s)
=
(s (s
+ 3)(s +I) + 4)(s + 5)
which is not strictly proper. First we obtain A 0 by dividing numerator and denominator by s2, before taking the limit as s ~ oo; giving the resultA0 = 1. Thus, we may expand R(s) as follows: (s
(s
+ 3)(s + I) + 4)(s + 5)
!+ _AI _ _ + _A2 __ (s + 4) (s + 5)
obtaining in the usual manner that A 1 =3, and A2 =-8. The complete partial fraction expansion is now given by:
LAPLACE AND z-TRANSFORMS
1193 R() s
=
1 +_3_ _ _8_ (s + 4) (s + 5)
The transfer function of the lead/lag system is an example of a rational function which is not strictly proper. Reference [1] should be consulted for more information about the Laplace . transform and its application in engineering, A table of Laplace transforms of some common functions is provided below in Table C.1. References [2,3] contain more extensive tables of Laplace transforms.
Table C. I. Table of Laplace Transforms of Some Common Functions
J(s)
J(t)
1
1.
-
2.
-
3.
-
1
5
1
t
52
1 s"
t" -1 (n- 1)!
2{f
4.
5 -3/2
5.
r~) (k~ O)
f-1
6.
~(k~O)
tk -1e-at
7a.
-s+a
e....,,
--
1 +1 1 s(ts + 1) 1 (s + a) 2 1 (ts + 1)2 1 s(-rs + 1?
-1 e -1/r
'tS
'r
s
(s + a)k 1
7b. 8.
9a. 9b.
10. lla.
llb. 12.
1 - 11
(s +a)
(n = 1,2, ... )
1 )" (n = 1, 2, ... ) (-rs + 1 1 (s + a)(s + b)
1-e-ttr te-• 1 t -e-tlr
r-
1 - (1 + t/-r)e -tlr
1 - t" -1 e-at -(n- 1)! 1
tn-1
e -1/r
-r"(n- 1)!
_l_(e-at- e-bt) (b- a)
1194
APPENDIXC
Table<:L Continued
13.
14. 15.
J(s)
f(t)
s (s + a)(s +b)
ae-•') ~be_,,) (
1 s(s + a)(s + b)
1 - {be_., - ae-b') _!_ [ 1 + -(a-b) ab
b b s2-b2
17.
52+ b2
18.
s2-b2 b2 s (5 2 + b2)
19.
sin bt
s2+~
16.
J
sinh bt
s
cosbt
s
cosh bt (1-cos bt)
20.
b (s + a) 2 + b2
e-~~ 1
21.
s +a (s + a) 2 + b 2
e"""' cos bt
22.
J(s +a)
e_,.'J(t)
23.
e-li•J(s)
f(t- b); withf(t) =0 fort< 0
C.4
sinbt
THE z-TRANSFORM
Just as the Laplace transform arises from specializing the Fourier transform for a one-sided (semi-infinite) independent variable, we will now show that the z-transform derives from specializing the Laplace transform for sampled functions that take nonzero values only at discrete points in time. Such a preamble is necessary if the apparent nonsymmetry between the z-transform and its inversion formula is to be understood properly.
C.4.1 Definition and Inverse Formula Consider the situation in which the continuous function f{t) in the Laplace transform expression Eq. (C.12) is sampled so that it takes on finite values f(tk} at discrete points, t()l t 1, •••, f;, ••• in time, at regular intervals of size !J.t. Let this sampled version of f(t) be .f(t). Then .f(t) =f(tk) only at the sample point tk = kilt, for k =0, 1, 2, ... ; it is zero elsewhere. It therefore appears as a "train"
l
1195
LAPLACE AND z-TRANSF ORMS
of spikes at the discrete sample points t0, t1, represen ted mathema tically as:
t
••• , 1, ••• ,
and may therefore be
(C.66)
where 0( t - tk) is the delta function. We now wish to find the Laplace transform of this function by introducing Eq. (C.66) into Eq. (C.l2): L{f'(t)) =
f~ e_., [k~~(tk) o(t- tk) Jdt
(C.67)
0
Now, by the linearity property of Laplace transforms, we know that the transform of a sum is the sum of the individu al transforms; thus Eq. (C.67) becomes: (C.68)
The integral is easily evaluate d by invoking the property of the delta function, and Eq. (C.68) becomes:
or, by replacing tk with k fl.t, we obtain finally: (C.69)
us This should now be compare d with the Laplace transform of the continuo j(t): function, (C.12)
where we observe that Eq. (C.69) is a discretized version of Eq. (C.l2). we If now for convenience (to simplify the expressi on in Eq. (C.69)), introduce the notation: (C.70)
and refer to the resulting L[f"(t)] as the "z-transform" of f(t), then Eq. (C.69) becomes: (C.71)
1196
APPENDIXc
which is the "formal" definition of the z-transform of the sampled signal f'(t}. Because Eq. (C.69} is a valid Laplace transform, its inverse transform is given by Eq. (C.13); and upon introducing the transformation in Eq. (C.70), we obtain:
J(t1,)
= f*(t)
.
where
f"J
1 = -2 . 19 ,,
''2I = =
1
!
l
· dz (C.72)
j
e-{a-jm)/>1
l
e-{a+jt»)/>1
Thus unlike the other integral transforms for which the transform and its inverse appear symmetric, the z-transform and its inverse appear to be nonsymmetri c. However, this is merely an artifact of the notational convenience employed: in its "fundamental " (if not entirely practical} form, the transform is actually as shown in Eq. (C.67), and it is as symmetrical with the inverse in Eq. (C.72), as Eqs. (C.12) and (C.13) are. As with Laplace transforms, it should be noted that the inverse transform formula is hardly ever used for inverting a z-transform. It is customary to use Tables of z-transforms, an example of which is presented at the end of this section.
C.4.2 Properties and Useful Theorems of z -Transforms
r
I f
The following are some properties and theorems of the z-transform that are particularly useful in engineering applications. Some of them have precise parallels with the Laplace transform, and some do not Because of the peculiar nature of discrete signals, some of these properties have no equivalent for the Laplace transform; by the same token some of the properties of the Laplace transform also have no precise z-transform equivalent. 1. Linearity
Extension to linear combination of n functions is straightforward. 2. Time Shift to the Left H Z{f(k)J = J(z}, then: Z{J(k+ 1)} = if
and if f{O)
(C.74)
= 0, then: Z{j(k + 1)}
= if
(C.75)
l
1
LAPLACE AND z-TRANSFORMS
1197
Also: Z(J(k + 2)} = i-J(z)- z2fl0) - z.f\1)
(C.76)
and in general: Z(J(k+m)) = znfJ(z)-zm.f\0)-zm-l.f\1)- ... -l/(m-1)
(C.77)
or
(C.78)
3. Time Shift to the Right (C.79)
Z(j(k-m)) = z-mJ(z)
4. Transform of (Forward and Backward) Differences
Define the forward difference operator, h., as: IJ.fl.k) = .1\k + 1)- .1\k)
(C.80)
then we may make use of Eq. (C.74) to obtain: Z{IJ.fl.k)) = (z- l'ff(z)- 1/{0)
(C.81)
Similarly, since from Eq. (C.80): 11~k)
= 11(11j{k)) = j{k + 2)- 2.1\k + 1) + j{k)
(C.82)
then
(C.83) In general:
m-1
Z{t1"'.1\k)} = (z-1)mJ(z)-z L,(z-1)m-i-It1:t\:O)
(C.84)
i= 0
Define the backward difference operator, V, as: V.f{k) =
.1\k)-.f{k- 1)
(C.85)
Then we may make use of Eq. (C.79) to obtain: (C.86) Similarly, since from Eq. (C.85): "'12.1\k)
= V(V.f{k))
= .1\k)- 2.1\k- 1) + .1\k- 2)
(C.86)
APPENDIX C
1198 then
(C.87)
and in general: (C.88)
5. Compl ex Shift
j(k), then the If the functio n j(t) is sample d with sample time ~t to obtain where: ri'j(k) is e-<~l_f(t) n sample d version of the functio
We now have the result: (C.89)
and the converse:
z-1{J(rz)} Typical application: since
= r~k)
(C.90)
z-1{1/(1- z-1)} = 1, then by Eq. (C.90):
z-1 {1-1pz
} = (p-Jrk = k P
1
(C.91)
(d. Examp le 24.3)
6.
Transform of Sums (C.92)
7. Compl ex Differe ntiatio n
Z{ k ftk)} = -z
£J\.z)
(C.93)
and in general: (C.94)
n f(k) = 1{k) is Typical application: since the z-transform of the unit step functio unit ramp the of form z-trans the 1 (C.93), Eq. by known to be 1/(1 - z- ), then as: d obtaine is k = function j(k)
1199
LAPLACE AND z-TRANSFORMS
Z{k} = Z{k l(k)} =
-z£(t ~ z-I)
z-1
=
-~....,--..,.
(1-z;~l)-2
8. Complex Integration (C.95)
9. Fina l-Va lue Theo rem lim lim j*(t) = lim f(k) = z-+1
t-+•
((1 - z- 1:J(z)]
(C.96)
k-+..
ide of the unit disk. (The theo rem is prov ided (1- z-1)}(z) has no poles outs the h the limit as k ~ co exists.) As with applicable only to functions for whic the g inin obta for ul usef very resu lt is Lapl ace tran sfor m equi vale nt, this func tion sfer tran e puls its from ess proc stea dy-s tate gain of a (stable) (incorporating the ZOH element). 10. Initi al-V alue Theo rem
lim j*(t)
t-+0
=
rf(z;)] lim f(k) == lim .....
k-+0
(C.97)
,
the limit as z ~ oo exists. applicable only to functions for which 11. The Discrete Convolution Theorem
: If Z{t( k)} = }(z), and Z{g(k)} = g(z), then
~if
=
~if
(C.98)
of the conv oluti on theo rem of Laplace This is a precise discrete equi vale nt le previous section. The result is applicab transforms give n as Property 12 in the from pulse transfer functions. in obtaining impulse-response models
ations C.4.3 Sol utio n of Linear Difference Equ z-tra nsfo rm find s appl icati on in the As with the Laplace tran sfor m, the Consider the linear difference equation: solu tion of linear difference equations. b u(k- 1) + ... + bmu (k- m) y(k) + a 1y(k- 1) + ... + a,Y( k- n) = 1
(C.99)
drop ping the" for convenience), we By takin g z-transforms of both side s {and obta in:
1200
APPENDIX c
which is solved for y(z) to yield:
y(z) (C.IOO)
Thus the solution to Eq. (C.99) for any given (z-transformable) sequence u(k) is given by that function y(k) whose z-transform is given in Eq. (C.lOO). Thus, as with the Laplace transform, the main challenge in finding such solutions lies in inverti ng the indicated z-transform. The technique of z-trans form inversion by long divisio n (or, equivalently expansion as an infinite series) was discussed in Chapte r 24. It is not as popular as the technique of partial fraction expansion.
C.4.4 Partia l Fracti on Expan sion The proced ure for obtaining partial fraction expansions is valid for all rational polyno mials R( •) regardless of the argume nt. Thus, what was presen ted in Section C.3.4 applies directl y once we replace s by z (or by z-1 ). We will therefore merely illustrate with some specific examples. Exampl e C.6
SOLUT ION OF A FIRST-ORDER DIFFERENCE EQUATION BY z-TRANSFORMS, AND PARTIAL FRACT ION EXPANSION.
Use z-transfoxms to solve the first-order difference equation: y(k)
= 0.7y(k- 1)
+ 1.5u(k- 1)
(C.101)
given that u(k) = 1 for all k. (That is, find the unit step respons e of the first-order system whose model is as in Eq. (C.lOl).) Solutio n: By taking z-transforms of both sides and rearranging, we obtain: y(z)
I.sz- 1 = 1 - 0.7z-1 u(z)
or, introducing the z-transform of the unit step function for u(z): 1.5z- 1
(C.102)
We now wish to expand the rational function of z-1 as:
(I
1.5z-1 z- 1)(1- 0.7z- 1)
AI
A2
= (1- z- 1) +(I- 0.7z- 1)
(C.103)
Since the roots of the denomi nator polynomial are real and distinct (they are located at z = 1 and z = 0.7), we proceed as in Section C.3.4: multiplying in Eq. (C.l03) by (1-z-1)
LAPLACE AND z-TRANSFORMS
1201
and setting z = 1 we obtain immediately A 1 = 5. Similarly, multiplying by (1- 0.7z-l) and settings = 0.7 gives A2 = -5. Thus, by partial fraction expansion Eq. (C.102) becomes: (C.l04)
from where the required solution is obtained from the inverse z-transform of Eq. (C.104) as: y(k)
=
5[1 - (0. 7)k]
(C.l05)
The next example illustrates how to handle repeated roots. Example C.7
PARTIAL FRACTION EXPANSION: REPEATED ROOTS IN z-TRANSFORM S.
Obtain a partial fraction expansion of the function: 0.8- 0.3z- 1
(C.l06)
Solution: Because of the repeated pole at z = 1, we wish to expand the RHS of Eq. (C.106) as: 0.8- 0.3z-l (I - z-IJ2(1 - o.sz-1)
AI
= 0- o.sz-I)
81 B2 +(I - z-1)2 + (1 - z-1)
We obtain A 1 = 0.2 in the usual manner: multiply by (1 - O.Sz-1) and set z = 0.5. And now: *() _ 0.8- 0.3z- 1 z - (I- o.sz-I)
and from it we obtain B2 immediately as 1.0 by setting z =1. Next, by differentiation, we obtain: d
dz
*
(z)
=
-0.1
(z - 0.5 )2
from where we now obtain B1 as..:. 0.4 by setting z =1. The required complete partial fraction expansion is therefore: 0.8- 0.3z- 1
0.2
1
-0.4
(1- z-1)2(1- o.sz-1) = (1- o.sz-1) + (1- z-1)2 +(I- z-1)
(C.l07)
which the reader may verify by longhand calculation.
The references given below should be consulted for more information about the theory of the z-transform, and its various applications. A table of z-transforms of some common functions is provided below in Table C.2; Table C.3 contains some common s transfer functions and the correspondin g pulse transfer functions incorporating the zero-order hold element.
1202
APPENDIX c Table c.2 Table of z -Transforms of Some Common Functions (Sample Time = M)
J(s)
f(t)
f(k)
1.
-1 s
1(t)
1(k)
1 1- z-1
2.
;2
t
kM
.M.z--1 (I- z-1)2
(n -1)!
sn
t"- 1
(kl:J.t)" -1
1 s+a
e-at
e-akt>t
1 1- e-a~>tz-1
-
pk
1 1- pz-1
3.
4a.
4b.
1
-
J
lim
1
a-+0 ( - )
~~ z)
n-1
5.
1 (s + a) 2
te-at
kMe-akAt
6.
s (s + a) 2
(1- at)e-at
(1 - akllt)e-ak4t
1 - (1 + aM)e-a~>tz- 1 {1 _ e-at>tz-1)2
7a.
a s(s +a)
1-e-<''
1-e-akl>t
{1 - e-a~>~~z- 1 (1- z-1)(1- e-aMz- 1)
-
1-pc
(1 -E)z-1 (1- z- 1)(1-pz-1)
e-<1'- e-bt
,..w_ e--bkl>t
-
(pl)k - (p2)k
(Pt- P2)z-l (1- Ptz-1)(1- p2z-l)
7b.
Sa.
Sb.
-
b-a (s + a)(s +b) -
Llte-•~>~z- 1
(1 _ e-al>tz-1 )2
(e-al>t_ e-bl>t~z-1 (1 - e-al>tz-1)(1- e-bl>tz-l)
9.
b (s2 + b2)
sin bt
sin bkM
z-1 sinbM (1 - 2r1cosMt + z-2)
10.
s (s2 + b2)
cos bt
cos bkllt
1- z-1cosbM (1 - 2z-1cosMt + z-2)
l
~
Table C.3. Some Common Continuous Transfer Functions and Equivalent Pulse Transfer Functions with Zero-Order Hold Sampling. (Sample Time = M)
g(z) =
g(s)
Q
~
~
b z-1 + b z-2
2 1 1 + a1 z-1 + a2 z-2
~
g(k)
'T1
1.
1 s
2.
1 s2
b1 a1
flt 2
a1 1 s"
3.
4.
1 'tS
+1
flt2
=
<1 - z-1)
bl
= (1
a1 =
kflt2
(k -1) flt 2
2
2
-+
·b b1---2 -2 , 2-2; a2
~
fltlk-1
= flt; b2 =0 = -1; az = 0
=1
~
I
(-1)" CJ"
--;:;! Cla"
- e-Af/'9; b2
-e-Ath; a2 = 0
=0
(
z z -
e-
(kflt)" --n!
[(k - 1) flt]" n!
bl pk-1 bl = (1 - p); p
= e-Af/t
or: (pk- 1 - pk)
....
N 0
w
-.. . . . . .
r~..........-~·--
.a
~Q:.,~~-I\IIi-14
$
($W.
I
..
I
•
0
u
?:::
Table C.3. Continued
@
~ ()
gz
g(s)
5.
1 s ('tS + 1)
b1 = 't
f'
=
":t -
b2
= 't
a1
= -(1
~
= e-Atl-c
b1 z-1 + b2 z-2 1 + a1 z-1 + a2 z-2
~
1 + e-At/-r ;
1 - e-At/-r -
~t e-At/~
+ e-At/-c)
g(k)
b1 b2 - - (1 - pk) + - - (1 -.pk-1) 1-p 1-p
-:c e· ~ b2='t(1-p-~t bl = 't
p
1 + p
or: lit - 't {pk- 1 -
6.
ab (s + a)(s + b)
b1 =
(a *-b)
bz al
a2
""
0
[b {1 -
e-<~At)
p
= e-Atl-c pk)
- a(1 - e-bAt)]
b1 ( b2 - - P1k- Pl) + (b- a) P1 - Pz P1 - Pz 1 [tze"dl' (1 - e-b<\1) - IJe-bAt (1 - e-<141)] b _ [b (1 - p1) - a (1 - p2)) = 1 (b -a) (b- a) b _ [apt (1 - P2) - bpz {1 - Pt~ = -(e--
--(_p k-1- p2k-?
N ,....,
R
o
~
0
........
-·-·
R
•
-
-
-
-
-
•
rr••
U444W*.....-:T
A
~~-c-.~
~
Table C.3. Continu ed
g(s)
g(z) =
~
7.
bt
(s + a'f
1 -r;2s2
+
2~u
(1 + a.dt) e-
+1
b2
at a2 ----
-
pi] I
'C
b1
b2 = p (p + at:..t - 1) p = e-<~At
or: [(1 - aM) pk- 1
~ ;~<1
I
btlql'-t + b2(k- 1)pk- 2 b1 = 1 - (1 + at:..t) p
= (e-aM)2
Ql=
~
g(k)
1 + at z-t + a2 z-2
b2 = e-<~"" (e-a"" + a.dt - 1) a1 = -'2.e-«41 a2
8.
=1 -
~
bt z-1 + b2 z-2
= 1 - e-f.IJI~ [cos (mM)
pk-1 [cos m (k - 1) t:..t + ysin m (k - 1) t:..t]i
t
+ CO't sin (coot)]
= (e-f.IJI~'f + ~~~ [~ sin (coot) = -2e-/:4tl~ cos (coot) = (~1~2
- pk [cos cokM + ysin cokM]
cos (coot~
0):::
~ ;~<1 'C
~ 'Y = -
p =
CO't e-{41/'t
---~~
-----
----
-
--------
.... "'
Q
tro
-
----
----- --
----
--~
-
,......
@$.
•W,.,
w;:;R+M !'.4l·_4W.. Pl.~~~-:.~.--.
.,....,... .• ,~-~~..,...
.........Jii+<
u
Table C.3. Continued
>< iS
i1i
§: ~
g(s)
bt z-t + bz z-2
g(z) =
s
bl = e--<'41 (s + a)Z b2 = -e-ttAI at = -2e-ttAt az = (e"'""At)2 [ab (e"'""At - e-!>At) + be (1 - e-<~At) - ac (1 - e-bAt)] (s + c) 10. (s + a) (s + b) b1 = (b -a) c) e-bAt] [c (b - a) e-aAt e-bAt + a (c - b) e""AI + b (a (a* b) bz = ab (b -a)
9.
g(k)
1 + at z-1 + az z-2
al = -(e_,At _ e-I>AI)
kpk - (k - 1) pk- 1 p =
e-aAt
l_pk -2) _b_ t_ (pk- pk) + _b_z _ (pk2 1 2 1 Pt - Pz Pt - Pz [ ab (p1 - pz) + be (1 - p1) - ac (1 - p2) ] bl = (b - a) [ c (b - a) p1 p2 + a (c - b) p1 + b (a - c) p2 J bz = ab (b - a) Pt =
a2 = (e"'""AI) (e-I'AI)
e-
Pz = e-bAt -
I.C
<::>
,.... "'
-
--
·-
-
--
·-
--
LAPLACE AND z-TRANSFORMS
1207
REFERENCES AND SUGGEST ED FURTHER READING A general reference that covers several aspects of Laplace transforms theory and applications is: 1.
Churchill, R. V., Modern Operational Mathematics in Engineering, McGraw-Hill, New York (1958)
More extensive tables of Laplace transforms may be found, for example, in: 2. 3.
CRC Standard Mathematical Tables, Chemical Rubber Co., Cleveland (1978) Abramowi tz, M. and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, D. C. (1964)
A detailed treatment of the theory and application of z-transforms may be found in Ref. [4); a more abbreviated but useful discussion may be found in Ref. [5]. Each of these references contains more extensive tables of z-transforms Ref. [6] contains many worked examples of partial fraction expansions for rational polynomials in the z variable. 4. 5. 6.
Jury, E. I., Theory and Application of the z-transform Method, J. Wiley, New York (1964) Kuo, B. C., Analysis and Synthesis of Sampled-Data Control Systems, Prentice-H all, Englewood Cliffs, NJ (1963) Ogata, K., Discrete-Time Control Systems, Prentice-Hall, Englewood Cliffs, NJ (1987)
i ~
j
APJPJENDJIX
D REV IEW OF MAT RIX ALGEBRA The convenience of representing systems of several equations in compac t, vectormatrix form- and how such represe ntations facilitate further analys iscombin e to make a workin g knowle dge of matrix method s virtuall y indispensable to a proper study of multivariable control systems. Althoug h most readers will have encoun tered matrix method s before, it is perhaps useful to briefly review the essential concepts here. Those readers who feel confide nt of their skills in this area may prefer to skip over this materia l and move on to another topic.
D.l
DEFINITION, ADDI TION AND SUBTRACTION
A matrix is an array of number s arranged in rows and columns, written with the following notation:
(D.l)
Here because A contains 2 rows and 3 columns, it is called a 2 x 3 matrix. The elements lltt' a12, •••, etc., have the notation that a is the elemen t of the ith row 11 and jth column of A. In general, a matrix with m rows and n columns is said to have dimens ion m x n. In the special case when n = m (i.e., the number of rows and column s are identica l) the matrix, for obvious reasons, is said to be square; otherwi se the matrix is said to be rectangular or nonsquare. As in the case of scalar quantities, matrices can also be added, subtrac ted, multiplied, etc., but special rules of algebra must be followed. Only matrices having the same dimensions (i.e., same number of rows and columns) can be added or subtracted. For example, given the matrices A, B, C:
1209
APPENDIXD
1210
B only A and C can be added or subtracted; because its dimensions are different, can D = c + A of swn The C. or A either from, d cannot be added to, or subtracte be written:
(D.3)
(or Thus addition (or subtracti on) of two matrices is accomplished by adding s operation These matrices. subtracti ng) the correspo nding elements of the two In order. same the of matrices of number any to may, of course, be extended particula r observe that adding k identica l matrices A (equival ent to scalar the multiplic ation of A by k) results in a matrix in which each element is merely to is matrix a when Thus, k. scalar the by d multiplie A of correspo nding element each be multipli ed by a scalar, the required operatio n is done by multiply ing element of the matrix by the scalar; for example, for A given in Eq. (0.2): (D.4)
Example D.l
MATRIX ADDmO N, SUBTRACTION, AND SCALAR MULTIPLICATION.
n, The following are some simple numerical examples of matrix addition, subtractio and scalar multiplication:
[ :
: ] + [ :
: ] = [
;5 :4 ]
4[ 6
: ] - 3[ :
: ] = [
~: :I ]
1211
REVIEW OF MATRIX ALGEBRA
MULTIPLICATION
D.2
Matrix multiplication can be performed on two matrices A and B to form the product AB only if the number of columns of A is equal to the number of rows in B. H this condition is satisfied, the matrices A and B are said to be conformable in the order AB. Let us consider that A is an m x n matrix (m rows and n columns); then to form the product P= AB, B must bean nxr matrix (n rows and rcolumns). The resulting product Pis then an m x r matrix (m rows and r columns) whose elements Pij are determine d from the elements of A and B according to the following nile: n
Py··
L a 1bk.~ i =1, 2, ... , m; j =1, 2, ... , r = k=l
(D.5)
I
For example, if A and B are defined by Eq. (D.2}, then A is a 2 x 3 matrix, B a 3 x 3 matrix, and the product P = AB a 2 x 3 matrix is obtained as:
= [ ( auhu + al2b2l + al3b3l) (aubl2 + al~22 + al3b32) (aubl3 + al~23 + al3b33) ]
( ~. bll + anb2l + a23b3l) ( ~lb12 + ~22 + auh32) ( ~lbl3 + Onbn + ~3b33) (D.6)
ExampleD.2
MATRIX MULTIPU CATION.
The following are some simple examples to illustrate how matrix multiplication is carried out. 1
(3Xl+4X2 )
(3X2+4X 5}] [ 11
[ =
(-2xl-lx 2) (-2x2-lx 5)
=
-4
26
-9
J
APPEN DIXD
1212 2
,
3.
that BA exists. Unlike in scalar algebra, the fact that AB exists does not imply m x r, but order of is and exists AB then r, For example, if A is m x n and B is n x n. x m with able conform not is r x n ion dimens the since BA does not exist Thus, exist. BA and AB But if instead B had been an n x m matrix, then both the equals case) this in (n A of s column of only in cases where the numbe r the equals case) this (min A of rows of r numbe the and 8, of numbe r of rows Note, BA. and AB ts produc numbe r of column s of B, can we form both the BA will be of however, that in this case, the order of AB will be m x m while : general in that, so n, order n x (D.7)
cation In matrix algebra, therefore, unlike in scalar algebra, the order of multipli nt. importa is very or to be permutable. Those special matrices for which AB = BA are said to commute Exampl eD.3
NON COMM UTATIV ITY OF MATRI X MULTI PLICAT ION.
Let us illustrat e Eq. (D.7) with the following two matrices:
s one may comput e both Since A and 8 have the same number of rows and column AB:
=[ and BA:
19 43
22 50
J
II
1213
REVIEW OF MATRIX ALGEBRA
BA
= [ s7
2 ] = [ (5 + 18)
6 ] [ 1 8
4
3
(10 + 24}
J
(7 + 24) (14 + 32)
=[23 34] 31
from where we see that, indeed, AB
D.3
46
* BA.
TRANSPO SE OF A MATRIX
The transpose of the matrix A, denoted Ar, is the matrix formed by interchanging the rows with the columns. For example, if:
: :l then a 11 a21 AT = [
all
an
al3
a23
]
(D.8)
Thus if A is an m x n matrix, then Jll is an n x m matrix. Observe therefore that both products ATA and AA T can be formed, and that each will be a square matrix. However, we note again, that in general: (D.9)
The transpose of a matrix made up of the product of other matrices may be expressed by the relation: (D.lO)
which is valid for any number of matrices A, B, C, D, ...
ExampleD.4
TRANSPOSE OF A MATRIX PRODUCT.
To illustrate relation Eq. (0.10), let us define:
and from Example 0.3 we recall that:
APPENDIX.D
1214 [ 19 43
AD
22] 50
so that (AB)T
=[
19 43 22
J
50
Now:
so that
and Eq. (0.10) may be seen to hold.
D.4
SOME SCALAR PROPERTIES OF MATRICES
Associate d with each matrix are several sCillar quantities that are quite useful for characteri zing matrices in general; the following are some of the most importan t
0.4.1 Trace of a Matrix If A is a square, n x n matrix, the trace of A, sometime s denoted Tr(A), is defined as the sum of the elements on the main diagonal, i.e.: n
Tr(A)
= .'Ea6 •••
For example, given:
we have that: Tr(A) = (1 +4) =5
and
Tr(C) = (0+2+1) =3
(D.ll)
REVIEW OF MATRIX ALGEBRA
1215
0.4.2 Determinants and Cofactors Of all the scalar properties of a matrix, none is perhaps more important than the determinant. By way of definition, the "abs"olute value" of a square matrix A. denoted I A I , is called its determinant. For a simple 2 x 2 matrix:
(0.12)
the determinant is defined by: (0.13)
For higher order matrices, the determinant is more conveniently defined in terms of cofactors.
Cofactors The cofactor of the element aij is defined as the determinant of the matrix left when the ith row and the jfh column are removed, multiplied by the factor (-l)i+i. Forexample,letusconsiderthe3 x3matrix:
(0.14)
The cofactor, c12' of element a12 is:
(0.15)
Similarly, the cofactor C23 is:
(0.16)
The General n x n Detenninant For the general n x n matrix A whose elements a;i have corresponding cofactors cii' the determinant I A I is defined as: n
L a;Fij
IAI = i
~t
(for any column j)
(0.17)
1216
APPENDIXD or n
IAI =
ft,a;F;i
(for any row i)
(D.l8)
Eqs. (D.l7) and (D. IS) are known as the Laplace expansion formulas; they imply that to obtain I A I :
·
1. 2
Choose any row or column of A Evaluate the n cofactors corresponding to the elements in the chosen row or column. 3. The sum of the n produ cts of eleme nts and cofact ors will give the requir ed determ inant. (It is impor tant to note that the same answer is obtained regardless of the row or column chosen.) To illustr ate, let us evalua te the determ inant of the 3 x 3 matrix given by Eq. (D.14) by expan ding in the first row. In this case, we have that:
That is: a,,
al2
al3
~I
0 22
~3
a31
a32
0 33
~
£ln
~I
a23
-a,z
all 0 32
a33
~I
~2
a3t
a32
+ al3 ~I
a33
(D.I9)
It is easily shown that the expansion by any other row or column yields the same result. Examp leD.S
DETER MINAN T OF A 3 x 3 MATRIX.
The determ inant of the 3 x 3 matrix:
may be evaluat ed as indicat ed above: by expand ing in
IAI =
1
I : : I -21 : : I
+
row 1, for exampl e, we have:
31 : : I
1 (45 -48)- 2 (36 -42) + (32- 35) -3+12 -9 = 0
Thus for this matrix,
IA I
= 0. We
will have cause to refer to this matrix later on.
REVIEW OF MATRIX ALGEBRA
1217
Properties of Determinants The following are some important properties of determinants frequently used to simplify determinant evaluation.
1.
Interchanging the rows with the columns leaves the value of a determinant unchanged. The main imrlications of this property are twofold: (a) I A I = AT (b) Since rows and columns of a determinant are interchangeab le, any
I;
statement tluzt holds true for rows is also true for columns, and vice versa.
2.
If any two rows (columns) are interchanged, the sign of the determinant is reversed.
3. If any two rows (columns) are identical, the determinant is zero. 4.
Multiplying the elements of any one row (column) by the same constant results in the value of the determinant being multiplied by this constant factor; ie., if, for example:
Dl
au
all
al3
~I
~
~
~I
~2
0 33
=
and
Dl
=
kau
all
al3
~I
~
~
~I
all
"l3
thenD2 =kD1 5.
If, for any i and j ( i "# j) the elements of row (column) i are related to the
elements of row (column) j by a constant multiplicativ e factor, the determinant is zero; i.e., if, for example:
where we observe that column 1 is k times column 2, then D 1 = 0. 6.
Adding a constant multiple of the elements of one row (column) to correspondin g elements of another row (column) leaves the value of the determinant unchanged; ie., if, for example:
Dl
=
thenD2 =Dl'
au
a12
al3
~I
a22
~
a31
~2
a33
and
Dl
=
(au+ kal~
0 1l
all
(~~+~
a22
a23
("JI +~)
a32
a33
..
1218
APPENDIXD
7.
The value of a determinant that contains a whole row (column) of zeros is zero.
8.
The determinant of a diagonal matrix (a matrix with nonzero elements only on the main diagonal; see Section D.5 below) is the product of the elements on the main diagonal. The determinant of a triangular matrix (a matrix for which all elements below the main diagonal, or above the main diagonal are zero; see Section D.S below) is also equal to the product of the elements on the main diagonal.
Alien Cofactors As indicated in Eqs. (D.17) and (D.18), any expansion in which the elements of any row (colurrm) in a matrix are combined with their corresponding cofactors (i.e., a Laplace expansion) gives rise to the determinant of the matrix in question. Any other expansion in which the elements of one row (column) are combined with co factors of elements of another row (colurrm) is called an expansion by alien cofactors, and its value is always zero. Thus, for example, while the Laplace expansion using row 1 of Eq. (D.14) gives:
the expansion combining the elements of row 1 with the cofactors of row 2 gives:
and similarly:
In general, therefore, for any column j: j=k (D.20)
i*k and for any row i: n
I,a..c1.
J= 1 ' 1
~
__
{IAI;
i=k
0;
i*k
(D.21)
Thus Laplace expansion by corresponding cofactors gives I A I ; expansion by alien cofactors always gives zero. A very important implication of these facts is that if we form a matrix C from the cofactors of A, then:
0 0 IAI 0
(D.22)
l
I
REVIEW OF MATRIX ALGEBRA
1219
a diagonal matrix with the determinant of A on the main diagonal. We shall refer to this result later.
D.4.3 Minors and Rank of a Matrix LetA be a general m x n matrix with m < n. Many m xm determinants may be formed from the m rows and any m of the n columns of A. These m x m determinants are called minors (of order m) of the matrix A. We may similarly obtain several (m- 1) x (m - 1) determinants from the various combinations of (m -1) rows and (m -1) columns c-hosen from them rows and n columns of A. These determinants are also minors, but of order ( m- 1). Minors may also be categorized as 1st minor, 2nd minor, etc. The minor of the highest order is referred to the 1st minor, the next highest minor is the 2nd rninor, etc., until we get to the lowest order minor, the one containing a single element. In the special case of a square n x n matrix, there is, of course, one and only one minor of order n, and there is no minor of higher order than this. Observe, therefore, that the 1st minor of a square matrix is what we have referred to as the determinant of the matrix. The second minor will be of order (n - 1). Note, therefore, that the cofactors of the elements of a square matrix are intimately related to the 2nd minors of the matrix.
Rank of a Matrix The rank of a matrix is defined a5 the order of the highest nonvanishing minor of that matrix (or, equivalently, the order of the highest square submatrix having a nonzero determinant). In Example D5, for instance, the 1st minor (of order 3) vanished, but none of the 2nd minors (of order 2) vanished; the single elements of the matrix- nine in all- are themselves minors (of order 1), and all are nonzero. Of the non vanishing minors, therefore, the highest order is 2. Thus, A in Example D.5 is of rank 2. A square n x n matrix is said to be of full rank if its rank= n; the matrix is said to be rank deficient if its rank is less than n. In several practical applications of matrix methods, the rank of the matrix involved provides valuable information about the nature of the problem at hand. For example, in the solution of a system of linear algebraic equations by matrix methods (see Section D.7 below), the number of independent solutions that can be found is directly related to the rank of the matrix involved.
D.S
SOME SPECIAL MATRICES
1. The Diagonal Matrix H all the elements of a square matrix are zero except those on the main diagonal from the top left-hand comer to the bottom right-hand comer, the matrix is said to be diagonal. Some examples are:
1220
APPENDIXD
2.
The Tria ngul ar Matrix A squa re matrix in which all the elem ents below the main diagonal are zero is called an upper triangular matrix. By the sam e token, a lower triangular matrix is one for which all the elements above the main diagonal are zero. Thus:
and
are, respectively, examples of uppe r and
lower triangular matrices.
3. The Iden tity Matrix A diagonal matrix in whic h the nonzero elements on the main diagonal are all unity is called the identity matrix; it is frequ ently denoted by I, i.e.: 0 1 0
0 0
0
0
The significance of the identity matrix will be obvious later, but for now , we note that it has the unique property that IQ= Q= QI
(D.23)
for any general matrix Q of conforma ble orde r with the mult iplyi ng iden tity matrix. 4.
Orth ogon al Matrices
HATA = I, the identity matrix, then A is an orthogonal (or orthonormal, or unitary) matrix. For example, as the reader may easily verify:
is an orthogonal matrix. 5. Sym metr ic and Skew -Sym metr ic Matr ices Whe n every pair of (real) elements that are symmetrically placed in a matrix with respect to the main diagonal are equal, the matrix is said to be a (real) symmetric matrix. That is~ for a (real) symmetric matr ix: (D.24a)
Some examples of symmetric matrices are:
I
REVIEW OF MATRIX ALGEBRA
1221
B=r ~ ~ ~2 J ... 5
Thu s a sym met ric mat rix and its
-2
4
transpose are identical, i.e.: (D.24b)
A skew-symmetric mat rix is one for
which: (D.25a)
The following are som e examples
of such matrices:
For skew -sym met ric matrices: (D.25b)
By noti ng that aij can be exp ress
ed as:
(D.26a)
it is easily esta blis hed that every squ are mat rix A can be reso lved into the sum of a symmetric mat rix A 0 and a skew -symmetric mat rix A', i.e.: (D.26b)
6. Her mit ian Mat rice s Wh en sym met rica lly situ ated elem ents in a mat rix of com plex num bers are complex conjugates, i.e.: (D.27)
the mat rix is said to be Hermitian.
For example: [
3
(2-3 j)
J
(2+ 3j) 5 is a Her miti an matrix. Not e that the (real) symmetric mat rix is a spec ial form of a Her miti an mat rix in whi ch the coef ficients of j are absent.
7. Sin gula r Mat rice s A mat rix who se dete rmi nan t is zero is said to be singular. Thu s, for exa mpl e, the mat rix enc oun tere d in Exa mpl e D.S is a sing ular mat rix. A mat rix who se dete rmi nan t is not zero is said to be nonsingular.
1222
APPENDIXD
8. Idempotent Matrices A matrix A is said to be idempotent if: AA =A
(D.28)
It is easily verified, for example, that the matrix:
is idempotent. Note that the identity matrix is a special type of an idempotent matrix. It is easy to establish that the only nonsingular idempotent matrix is the identity matrix; all other idempotent matrices are singular.
0.6
INVERSE OF A MATRIX
The matrix counterpart of division is the inverse operation. Thus, corresponding to the reciprocal of a number in scalar algebra is the inverse of the matrix A, denoted by K 1, (read as" A inverse"): a matrix for which both AK1 and A -IA give I, the identity matrix, i.e.: (D.29)
The inverse matrix A -I does not exist for all matrices; it exists only if: 1. 2.
A is square, t and Its determinant is not zero (i.e., the matrix is non-singular).
The inverse of A is defined by: A-t =
C _ adj (A) IAI-
IAI
(D.30)
where C is the matrix of cofactors, i.e.:
C=
c" c,z [ Cz, . c22
and, as defined previously,
c, c,.] 3
•••
Cn
...
... en, cn2 cn3
C;j
Cz,.
... c,..
is the cofactor of the element aij of the matrix A.
cT, the transpose of this matrix of cofactors, is termed the adjomt of A and often abbreviated adj (A). This expression for the matrix inverse may be seen to follow directly from Eq. (0.22) and the definition of the inverse in Eq. (0.29). +Generalizing the concept of an inverse to include nonsquare matrices will be discussed shortly.
ij •
REVIEW OF MATRIX ALGEBRA
1223
Clearly because I A I = 0 for singular matrices, the matrix inverse as expressed in Eq. (D.30) does not exist for such matrices. It should be noted that 1.
The inverse of a matrix composed of a product of other matrices may be written in the form:
2
For the simple 2 x 2 matrix, the general definition in Eq. (0.30) simplifies to a form easy to remember. Thus given:
we have that IAI = ai 1a22 A-I =
1 [
IAI
ExampleD .6
a 1 ~21 ,
and it is easily shown that:
lin
(0.31)
~I
OBTAININ G MATRIX INVERSES.
Let us illustrate the foregoing discussion by obtaining the inverses of the following three matrices:
The first is a 2 x 2 matrix whose determinan t is easily obtained as according to Eq. (0.31) therefore, we easily have that
I A I =3- 2 =1; 1
from where the reader may want to verify by direct multiplica tion that AI (Airl = (AirlAI = l Next we observe that determinan t of the 3 x 3 matrix A was obtained in Example 0.5 as zero; thus it is a singular matrix, and therefore (~)3t does not exist. The determinan t of~ may be obtained by expanding in the cofactors of the first row, i.e.:
~~~
=
2
1: : 1-2 1: : I + 3 1: : I
2 (45-48)-2 (36-42) + 3 (32-35)= -3
Thus the matrix is nonsingula r, and we may proceed to compute the inverse. We recall from the definition of cofactors that the nine cofactors of A are obtained as:
1224 Cn=
APPENDIX D
I: : I I: : I
I: : I I: : I
=-3; c12=(-l)
~I =(-1)
=6; C22=
=6; C13=
I: : I I: : I
=-3; C23=(-l)
=-3
=-2
Thus the matrix of cofactors is:
so that the adjoint of ~ is:
~ (~) ~. [ "
The inverse ( ~r1 is then:
::]
To check, we see by matrix multiplication that
r
and the reader may in similar fashion verify that (A 3 1As is also equal to I.
D.7
LINEAR ALGEBRAIC EQUATIONS
Much of the motivation behind developing matrix operations is due to their particular usefulness in handling sets of linear algebraic equations. For example, the three linear equations: auxl + aiM + aiM
= bl
"21x1
+ "22l2 + "23X3
= b2
D:liXI
+ "n1i + a33'%3 = b3
(D.32)
I
REVIEW OF MATRIX ALGEBRA
1225
containing the unknowns .,, '2, .13 can be represented by the single matrix equation: Ax= b
(D.33)
where
The solution to these equations can be obtained very simply by premultiplyin g
Eq. (0.33) by A-1 to yield:
which becomes: (D.35)
upon recalling the property of a matrix inverse, that K 1A =I, and recalling
Eq. (0.23) for the property of the identity matrix.
There are several standard, computer-ori ented numerical procedures for performing the operation A - 1 to produce the solution given in Eq. (0.35). Indeed, the availability of these computer subroutines makes the use of matrix techniques very attractive from a practical viewpoint It should be noted that the solution Eq. (0.35) exists only if A is nonsingular, as only then will A- 1 exist When A is singular, K 1 ceases to exist and Eq. (0.35) will be meaningless. The singularity of A of course implies that I A I =0, and from property 5 given for determinants in Section 0.4, this indicates that some of the rows and columns of A are linearly dependent. Thus the singularity of A means that only a subset of the equations we are trying to solve are truly linearly independent; as such, linearly independent solutions can be found only for this subset It can be shown that the actual number of such linearly independent solutions is equal to the rank of A. Example0.7
SOLUTION OF UNEARALGE BRAIC EQUATIONS.
To illustrate the use of matrices in the solution of linear algebraic equations, let us consider the set of equations:
2xt
+ 2x2 + :h3
=
4x1 + 5x2 + 6x3
3
7x,_ + &:2 + 9x 3
5
which in matrix notation is given by: Ax
=
b
APPENDIX D
1226 where A, b, x are:
Notice that A is simply the third matrix whose inverse was computed in Example D.6, i.e.:
Thus, from Eq. (0.35), the solution to this set of algebraic equations is obtained as:
Ii
I
Thusx 1 =0, x 2 =1, x 3 =-1/3 is the required solution. Let us now alter the coefficient of xt in the first equation from 2 to 1 so that we now have the following set of equations to solve for the three unknowns: x 1 + 2x 2 + 3x 3 4x1 + 5x 2 + 6x 3 7x1 + Sx2 + 9x 3
1 3 5
Observe in this case that
a matrix whose determinant had previously been obtained as 0 and whose rank was later identified as 2. The immediate implication is that since A is singular, we cannot solve this new set of equations for all the three unknowns. Observe now that by adding the first and third equations we obtain an equation that is exactly twice the second equation; thus, the three equations are in fact not completely independent . It is easily shown that we can solve these equations independent ly only for two of the unknowns; the third unknown can only be obtained as a linear combination of these two. Thus there are only two linearly independent solutions to this new set of equations, exactly the same as the rank of A.
l
1227
REVIEW OF MATRIX ALGEBRA
D.S
GENERALIZED INVERSES
Consider the general problem of solving m linear algebraic equations:
(D.36)
containing the n unknowns J1, x 2 , x 3 , ••• , x,.. Observe that this system of equations, as in Eq. (D.32) can also be represented by the single matrix equation: Ax= b
(D.33)
except that in this case, A is an m x n matrix, xis ann-dimensional vector, and b is an m-dimensional vector: a,2
a,.
0 21
On
~
a3l
a32
~
am!
am2
amn
au
A=
..
.
·-[i} ·=[i]
(D.37)
1hree distinct cases of this general problem can be identified: CASEl: m=n • Equal number of equations and unknowns (well-defined system). • if A is nonsingular then n independent solutions exist. 1his is the case discussed in Section D.7.
CASE2: m >n • More equations than unknowns (overdefined system). • No "regular" solutions exist (in the sense that no set of values for the unknowns will exactly satisfy all the equations simultaneously). CASE3: m < n • Fewer equations than unknowns ( underdefined system). • An infinite set of solutions exist (in the sense that the excess n - m unknowns can be assigned any arbitrary set of values and the remaining m unknowns solved for in terms of these arbitrary values).
1228
APPENDIXD
For the Case 1 prob lem, as disc usse d in Section D.7, whe neve r I A I"# O, the requ ired solu tion is obtained as given in Eq. (D.35): X
= A- 1b
(D.35)
(It is easy to establish that whe neve r the n x n matrix A is singular, the prob lem redu ces to a Case 3 prob lem with m -no w less than n- being equal to the rank of the rank-deficient A.) In each of the othe r two cases, solutions of the form given in Eq. (D.35) can be obta ined but only after impo sing addi tiona l conditions as we now show.
Overdefined Systems The prob lem here is that there is no vector x for whic h Ax exactly equa ls b. How ever , it is possible to obta in a solu tion in whic h Ax is "closest" tob in the "leas t squa re" sense by finding a vector Q that minimizes: ¢J
= (Ax -bl( Ax -b)
(D.38)
By appl ying the usua l tech niqu es of calculus, it is easy to show that provi ded (ATA) is nonsingular, this "least squa res solution" vector is given by: (D.39a)
or (D.39b)
whic h is of the form in Eq. (D.35) with AL
AL
defined by:
= (ATAriAT
(0.40 )
now play ing the role of A-1. . Note the following abou t AL defined by Eq. (D.40): 1.
ALi sann xmm atrix whil e Ais mxn
2.
ALA = In, the n x n identity matrix, but AA L "#I
3.
AA L = ~ an m x m idempotent matr ix for which I,_ I,_ = I,_
4.
Premultiplying in (2) above by A gives : AALA =A
(D.41)
while post mult iplyi ng by A L gives: (0.42 )
Because only whe n A is multiplied on the ieft by If do we get an identity matr ix, A L is called a left inverse.
REVIEW OF MATRIX ALGEBRA
1229
Underdefined Syst ems The prob lem here is that there is an infinite set of vectors x for whic h Ax exactly equa ls b. How ever, it is possible to obtain a solut ion vector x of minl lnum norm by posin g the following constrained optimizatio n problem:
Min
cp =
t
(XI"x)
subject to: Ax = b
By the usua l meth ods of solvi ng such probl ems (for exam ple, using Lagrange multi plier s) we find that provided (AAT) is nonsingular the requi red "min imum norm solut ion" is: (D.43a) or X
= Ai1>
whic h, once again , is of the form in E"q. (0.35
(D.43b)
) with A R defin ed by:
AR = AT(A A!tl
(D.44)
now playi ng the role of A-1. Again, we note the following abou t AR defin ed by Eq. (0.44): 1.
A R is ann x m matri x while A is m x n
2
AA R =In, them x m identity matrix, but A RA ;i;l
3.
AR A = L:z anoth er n x n idempotent matr ix for whic h, as usual ,
4.
Post multi plyin g the expression in item 2 abov e
Lz Lz =Lz
by A gives: (D.45)
while pre multi plyin g by AR gives: (D.46)
In this case, we obtai n an ident ity matri x only when A is mult iplie d on the right by K; therefore AR is called a right inverse. Noti ng the simil aritie s betw een Eqs. (0.41 ) and (D.45), and also the simil aritie s betw een Eqs. (D.42) and (D.46 ), we can now state the following results: The gene ral syste m of linear algebraic equat ions in Eq. (0.33): Ax= b
(0.33 )
with A as a gene ral m x n matri x (for whic h m is not necessarily equa l to n) has the solution:
1230
APPENDIXD (0.47)
Here A+ is referred to as a generalized (or pseudo) inverse of A, and it satisfies the following conditions:
3.
J
(for the left inverse) (for the right inverse)
Either: A+A= I AA+=I or:
Observe that A-t, the regular inverse of a square, nonsingular matrix satisfies all these conditions. Thus A-t is actually a special case of A+, and t."l)e above is therefore the most general description of the inverse of any matrix. Note, finally, that only A"" 1 is both a left and a right inverse; if Eq. (D.33) is a..1. overdefined system of equations, then a least squares solution can be found only via a left inverse; for an underdefined system of equations, a minimum norm solution can be found only via a right inverse.
ExampleD.S
SOLVING AN OVERDEFINED SYSTEM OF EQUATIONS BY GENERALIZED INVERSES.
To illustrate the use of generalized inverses in the solution of a system of linear algebraic equations, let us consider the set of equations: x 1 + 2x 2
24 - x 2 -2x 1 - x 2
25 -25 -25
a set of three equations in two unknowns. Observe right away that the system is overdefined by one equation, and there is no single set of values for xt and x 2 that will simultaneously satisfy all three equations. These equations may be written in the form Eq. (0.33): Ax = b
(D.33)
In this case:
(D.48)
and from Eq. (0.47), the "least squares" solution is obtained from: (D.47)
where, recalling Eq. (0.40):
*t t '
1231
REVIEW OF MATRIX ALGEBRA
Now, ATA is obtained by direct multiplication as:
a nonsingular matrix whose determinant is 50; thus:
I [ 6 50 -2
=
(ATArl
~2 ]
so that A+ is given by: 2 A+
14
;0 [ I6 -I3
-10
-5
]
(0.49)
and
14
2 I [ + x=Ab= 50
16 -13
-10 -5
J[:J
which, upon evaluation, gives:
=
X
[:J = [~!]
(050)
We now note that the solution in Eq. (0.50) "satisfies" the system of equations only in the least squares sense; i.e., of all possible x choices, this particular x given in Eq. (0.50) provides an Ax that comes closest to b in the least squares sense. We may also observe that, as asserted earlier:
+
1 [
A A = 50
2
14
16 -13
a 2 x 2 identity matrix; also:
-10 ]
-5
1232
[
rI
APPENDIXD
0~ ~.N ~4]
--0.24
0.82
--0.3
--0.4
--0.3
0.5
(D.5I)
It will be a particularly worthwhile exercise for the reader to carefully multiply this matrix given in Eq. (0.51) by itself and therefore confirm that it is indeed an idempotent matrix.
Example D.9
SOLVIN G AN UNDERDEFINED SYSTEM OF EQUAT IONS BY GENERALIZED INVERSES.
Let us now consider the problem of solving the following set of equation
s:
2
a set of two equations in four unknowns, clearly underdefined. Observe right away that we can arbitrarily assign any values to, say, x , and ll• and solve 1 the resulting equations for Xg and x4 • There is thus an infinite number of solutions. H these equations are written in the form Eq. (033):
Ax
=
b
¥
ti
(D.33)
we would have: 2 A= [
II
3
;J
·=[]
b= [ 1: J
(D.52)
i'
and from Eq. (0.47), the "minimu m norm" solution is obtained from: (D.47)
By direct multiplication, we obtain AAr as:
and hence:
1 [ 4 20 -10
I f
where we now recall from Eq. (0.44) that:
(M1)-1
'[i
-10 ] 30
REVIEW OF MATRIX ALGEBRA
1233
Thus, we have that A+ is given by:
-10 ] 30
or -{),3
A+ =
[
-{).1
0.1
(D.53)
0.3
and therefore:
X= A+b = [
~:~ 0.1
0.3
00.5 ]
[ 120
J
-{).5
which, upon evaluation, gives:
(0.54)
The reader may now wish to verify that Eq. (0.54) does in fact satisfy the system of equations exactly, and that for the A given in Eq. (0.52) and A+ given in Eq. (0.53), AA + results in the 2 x 2 identity matrix, while A+ A results in the 4 X 4 idemp otent matrix:
[~:: ~:~ ~:~ :: ] 0.1
-{),2
D.9
0.2 0.3 0.1 0.4
0.4 0.7
EIGENVALUES, EIGENVECTORS, AND MATRIX DIAGONALIZATION
The eigen value / eigenvector problem arises in the determ inatio n of the values of a const ant A. for which the following set of n linear algebraic equat ions has nontri vial solutions:
APPENDI XD
1234
1 '
(0.55)
'This can be reexpressed as:
or
=0
(A-AI)x
(0.56)
Being a system of linear homogene ous equations, the solution to Eq. (0.55) or equivalently Eq. (0.56) will be nontrivial (i.e., x 0) if and only if:
"*
=o
IA-A.II
(0.57)
Expandin g this determina nt results in a polynomial of order n in A.:
Do"'111 +a 1....111-1
111-2
+~"'
1 ... +a11 _ 1 .... +a, =
0
(0.58)
This equation (or its determina nt form in Eq. (0.57)) is called the characteristic equation of then x n matrix A; its n roots, A. 11 A. 2, A. 3, •••,A.,.- which maybe real or imaginary, and may or may not be distinct- are the eigenvalues of the matrix. EIGENVALUES OF A 2 x 2 MATRIX.
Example 0.10
Let us illustrate the computatio n of eigenvalues by considering the matrix:
In this case, the matrix (A- AI) is obtained as: (A-A.I)
=[
1 2 3 -4
J_[ A
]. 2 0 ] = [ (1-l) (-4- A) 3 0 A
from which the characteristic equation is obtained as:
lA-AII= -(l-A)(4+ A)-6 = A2 + 3A -10=0 a second-ord er polynomia l (quadratic) with the solution: At.~
= -5,2
1235
REVIEW OF MATRIX ALGEBRA
Thus in this case the eigenvalues are both real. Note that A- 1 + A. 2 = -3, and that ~~ = -10; also note that Tr(A) = 1 - 4 = -3 and I A I = -10. Thus, we see that ~ + ~ = Tr(A) and A- 1 ~ =I AI. If we change A slightly to:
then the characteristic equation becomes:
lA-AII =
I
I- It
2
-3
4 -.?..
=A? - 5A. + 10
0
whose roots are:
which are complex conjugates. Note again that Tr(A) = 5 = A,. + lt 2 and I A I = 10
=
~~·
Finally if we make another small change in A, i.e.:
then the characteristic equation is: .?..2 -
I +6
=0
which has solutions:
so that the eigenvalues are purely imaginary in this case. Once again Tr(A) ~and IAI =5= ~~·
=0 =.:1. 1 +
Returning now to the original problem in Eqs. (D.55) and (D.56), we note that every distinct eigenvalue It when substituted into Eq. (D.56) gives rise to a corresponding nontrivial solution x; for which: (D.59)
These nontrivial solution vectors are called the eigenvectors of A. Thus every distinct eigenvalue ~._of A has associated with it a corresponding eigenvector x;, j = 1, 2, ... , n. When some eigenvalues are repeated, some specialized considerations - which we will not discuss here - are required. (See the bibliography at the end of the appendix for more details.) We shall restrict ourselves here to the case of distinct eigenvalues. The following are some useful properties of eigenvectors:
1236 1.
2
APPENDIXD Eigenvectors associated with separate, distinct eigenvalues are linearly independent.
The eigenvectors xi are solutio ns of a set of homog eneous linear equatio ns Eq. (0.59) and are thus determ ined only up to a scalar multiplier; i.e., xi and kx.. will both be a solution to Eq. (0.59), if k is a scalar constant. We obtain a normalized eigenv ector X; when the eigenvector xi is scaled by the vector norm xi i.e.:
II II,
_2._
xi= llx.;ll 3.
The normal ized eigenvector
~
is a valid eigenvector since it is a scalar
m~ltiple of ~i; it ~jo~s the sometimes desirab le proper ty that it has
II II- 1.
uruty norm, 1.e., xi
4..
The eigenvectors xi may be calculated by resorti ng to method s of solution of homogeneous linear equations.
It can be shown that the eigenvector xi associated with the eigenvalue A.j is given by any nonzero column of adj(A- A.jl); it can also be shown that there will always
be one and only one independent, nonzero column of adj (AExample D.ll
..:til).
EIGENVECTORS OF A 2 x 2 MATRIX.
Let us illustrate the computation of eigenvectOrs by considering
the 2 x 2 matrix:
for which eigenvalues, A.1 =-5, A.2 =2 were comput ed in Exampl
e 0.10.
For~:
and the adjoint is: adj(A-A - 11)=[ I
-3
-
2
6
]
We may now choose either column of the adjoint as the eigenve ctor 'f, since one is a scalar multipl e of the other; let us choose:
as the eigenvector for ~ =-5. Since the norm of this vector is --J 12 + (-3) 2 = normalized eigenvector corresponding to ~ =-5 will be:
,fTif, the
l I
REVIEW OF MATRIX ALGEBRA
1237
XI
[~] {10
Similarly for A. 2
= 2:
(A-A. 21)
[~I
2
]
--{j
and
adj (A -..:1.21) =
[~
-2 -1
]
Thus: x2
=[; J
can be chosen as the eigenvect or for eigenvalue this case is:
~
= 2. The normalized eigenvector in
Let us return again to the original eigenval ue/ eigenvector problem , and choose the calculated n linearly indepen dent eigenvectors xi, j = 1, 2, ... , n to be the columns of a matrix M, ;.e.: (0.60)
M is called a modal matrix. Obviously, if the eigenvalues lti ~re distinct, then the eigenvectors xi are all independ ent, and the modal matrix M will be nonsingu lar, and M""1 will a.tways exist. This modal matrix is very useful in converting a matrix A to diagonal form. Recall the eigenval ue/ eigenvector problem: to determin e the values of It i for which nontrivial solutions vectors xi can be found for: j = 1, 2, ... , n
(D.61)
For each j, the complete set of equations may be written more compactly as: (D.62)
If we define the diagonal matrix A as:
APPENDIXD
1238 0
0 0
i]
(D.63)
a matrix whose diagonal elements are the n eigenvalu es of A, and observe that the square bracket on the left-hand side in Eq. (D.62) is just the modal matrix M, then we see immediate ly that the square bracket on the right-han d side is just MA. Thus Eq. (D.62) may be written as: AM= MA
(D.64)
and premultip lying both sides by M-1 thus yields: M-1AM =A
(D.65)
Then x n matrix A is thus reduced to what is known as the diagonal canonical fonn by using the transform ation Eq. (D.65). By pre- and postrnulti plying both sides of Eq. (D.65) by M and M-1 respectively, one obtains the companion relation: (D.66)
These two relations are very useful in the solution of linear algebraic and differenti al equations as shall be seen later. Note that the latter relation in Eq. (D.66) indicates that a square matrix A can be decompo sed into three matrices involving only its eigenvalu es and eigenvecto rs. Example 0.12
MATRIX DIAGONALIZATION.
To illustrate matrix diagonaliz ation,let us consider the 2 x 2 matrix:
discussed in Examples D.lO and D.ll. Recall that the eigenvalue s and eigenvecto rs were determined to be:
Thus the modal matrix M is:
1239
REVIEW OF MATRIX ALGEBRA
while
[ ]
M""l
2
1 7
--::;
3 7
1 7
and the diagonal eigenvalue matrix is:
Evaluation of the LHS of Eq. (0.65) yields:
M"IAM
[; ;][: [ t -~] [ ~
1
7
7
2 -4 ] [
~3
2
] 0
4 -5 15
2
] =[ :
2
]
which verifies Eq. (0.65). Similarly, evaluation of the RHS of Eq. (0.66) gives:
verifying the equation.
The following are some general properties of eigenvalues worth noting: 1.
The sum of the eigenvalues of a matrix is equal to the trace of that matrix, i.e.: II
Tr(A)
= •L )...J J=l
2.
The product of the eigenvalues of a matrix is equal to the determinant of that matrix, i.e.: n
IAI =
I1 A..
j=l J
1240
APPENDIX D
3.
A singular matrix has at least one zero eigenvalue.
4.
If the eigenvalues of A are~, .:lz, \, ... , A.n, then the eigenv alues of A -1 are 1/ Av 1/ A.2 , 1/ A.3 , •••, 1/ An.
5.
The eigenvalues of a diagonal or a triangular matrix are identical to the elements on the main diagonal.
6.
Given any nonsi ngula r matrix T, the matrices A, and A =T AT-1 have identical eigenvalues. In this case, such matrices A and A are called similar matrices.
D.lO SINGULAR VALUE DECOMPOSITION The concept of matrix diagonalization indicated in Eq. (0.65) and the comp anion relation in Eq. (0.66) can be exten ded to all matrices, squar e and nonsq uare, by singul ar value decomposition (SVD). The main SVD result may be summ arized as follows: For any real m x n matrix A, it is always possible to find orthog onal (i.e., unitary) matrices Wand V such that: (D.67)
Here,
r is the m X n matrix: r
= [ : :J:wi iliS
·[:
0
0
a2
0
0
0
_:
]
(D.68)
o;.
where, for p = min(m,n), the diagonal elements of S,
whose m columns, W;, i = 1, 2, ..., m, are called the left singular vectors of A; these are the norma lized (orthonormal) eigenvectors of AA '· Vis the n x n matrix: (0.70)
whose n columns vi, i = 1, 2, ... ,n, are called the right singular vectors of A; these are the normalized (orthonormal) eigenvectors of AT A. By virtue of their being comp osed of orthon ormal vectors, the matrices W and V are orthogonal (or unitary) matrices, i.e.:
REVIEW OF MATRIX ALGEBRA
1241
so that
w-1
=
wr
Also:
yry = 1 = yyr
so that: y-I = yr
By applying these properties of unitary matrices, we easily obtain from Eq. (D.67) the companion relation: A= wr,y r
(0.71)
It can be show n that analogously to the eigen value / eigenvector expression for squar e matrices in Eq. (D.61), we have the more general pair of expressions: Av; o;w; Arw; = o;v;
(D.72a) (D.72b)
The ratio of the large st to the smallest singu lar value is desig nated the
condition number of A, i.e.:
(0.73)
and it provides the most reliable indication of how close A is to being singular. Note that for a singular matrix, JC(A) =co; thus nearn ess to singularity is indicated by excessively larg e- but finit e- values for IC'(A~ There are several impo rtant applications of SVD, particularly iri numerical linear algebra. Its applications in process contro l (discussed in Chapters 20 and 22) typically involves square, invertible matrices. It shoul d be noted however, that the full benefit of SVD is realized in dealin g with problems requiring the analysis of nonsquare, possibly rank-deficient matri ces. The following example illustrates SVD appli ed to a nonsq uare matrix; keep in mind , however, that efficient comp uter programs are available for SVD calculations. Example 0.13
SINGULAR VALUE DECO MPOS ITION OF A
3 x 2 MATRIX
To illustrate SVD, let us consider the 3 x 2 matrix :
(D.74)
a slightly modified version of which was studie
d in Example 0.8. For this matrix:
APPENDIX D
1242
whose eigenvalues are obtained as 10, and 5; thus the singular values of A are:
ordered such that q > "i as required by SVD analysis. Thus, the 3 x 2 matrix :E in this case is given by:
(0.75)
Right Singular Vectors The first eigenvector of ATA corresponding to .t1 is obtained as usual from adj (ArA- ~1), and since: adj (ATA-l 11)
=[
2 ] 2 -1
-4
a possible choice for the eigenvector is the second column. Normalizing this with
.V 22 + 12 =--15, the norm of the vector, we obtain the first right singular vector v 1 corresponding to a 1 =i l las:
In similar fashion, the second normalized eigenvector corresponding to .t 2 = 5 is obtained as:
., -
[;,]
Thus:
V=[
2
1
-{5
{5
-\
2
{5
{5
] *-
REVIEW OF MATRIX ALGEBRA
1243
and therefore:
[ ] 2
vr
=
15 _J_
15
.::L
15
.1... {5
(D.76)
The reader may want to verify that this Vis truly a unitary matrix by evaluatin g vTv as well as vvT and confirming that each matrix product gives the 2 x 2 identity matrix.
Left Singular Vectors For the given matrix A;
The eigenvalues of this 3 x 3 matrix are obtained from the characte ristic equation which in this case is:
i.e.: ~.Az.A.J
= 10,5,0
Note that the nonzero eigenvalues of AATare identical to the eigenvalues of ATA. For A, =10:
To find the adjoint of this matrix, we first find the cofactors and take the transpose of the matrix of cofactors. In this case, this is easily obtained as:
and by normalizing any of the nonzero columns, we obtain the first left singular vector of A as:
APPENDIXD
1244
By a similar proeedunt the seconcl and third left singular vectors ft!llpectively corresponding 1o the etsenvalues A, • 5, ond .t3 ~ 0 are oblainecl ao:
with the II!SIIlt that: 0I
W• [
...fi
I O
0I
]
...fi
(0.77)
.=!.o....L
...fi
...fi
Once again. the ...ter may wish 10 verify that this Is aloo a unitary matrix. The matrices given in Ecp. (0.75), (0.76), and (0.77) constltuie the singular !lfllue ~liml of the matrix AinEq.(D.71). Wenowoboervelhat
as m Eq. (0.74). As an exercise, the reader may also wish lo verify Eq. (0.67) for this
example.
D.ll SOLUTION OF UNEAR DIFFERENTIAL EQUATIONS AND 1HE MATRIX EXPONENTIAL It is possible to use the results of matrix algebra to provide general solutions to sets of linear onlinary diffen!ntial equations. Let us recall a dynamic system with control and disturbances neglected:
~
• Ax ;
X(O)•"'
(D.78)
Here A is a constant n x n matrix with distinct eigenvalues, and x(t) is a limevarying solution vector. Let us defme a new set of n variables. z(t), as: ll(t) ~ l.lr1ll(t)
so that
(D.79)
REVIEW OF MATRIX ALGEBRA
1245 1(1) • M&(f)
(0.80)
Here M is the modal matrix described above. Substituting Eq. (D.80) into
Eq. (0.78) yields: M ~ = AMll(t) ;
z(O) = M" 1liQ
(0.81)
Premultiplying both sides by M"1 gives:
M"1M
~
= M"1AM.r(t)
or
4!!!l dt =
z(O)- .... -v -
1\z(f).•
M"'""' "V
(0.82)
Because A is diagonal, these equalioas may now be written as:
dt • .t,z1(1)
z1(0) •
;
: 10
Thus the transformation Bq. (0.80) has converted the original Eqs. (0.78) intn n completely decoupled equations, each of which has a solution of the form:
zj<.t> • e.IJ ~ ; j
= I, 2, ... , n
This may be expressed in matrix notation as: l(t)
where
/" = [
=
e"'ro
(0.83)
~· ~ ~ .~. 0
0
0
]
(0.84)
""
is the ttllltrix trponmtilll for the diagonal matrix, A. The solution Eq. (D.83) may be put back in terms of the original variables, 1(1), by using Eq. (0.79):
M" 1x(l)
•
e"'M"'JO
1246
APPENDIXD
or l(tl
= W'M"'"o
ll(t)
=
(D.SS)
This is often written:
l'"o
(D.86)
where the matrix e"', defined as: (D.87)
is called the mtltrix exponential of AI. It is very important to note that Eqs. (0.86) and (0.87) provide the general solution to any set of homogeneous equations Eq. (0.78) provided A has distinct eigenvalues. Example 0.14
SOLUTION OF UNEAR MATRIX DIFFERENTIAL EQUATIONS.
To illustrate the solution of linear, constant coeffiaent, homogeneous differential equations, let us consider the system: (0.88)
where
Observe that A is the same matrix we have analyzed in Examples 0.10, 0.11, and D.12sowealready know llse~genvalues, At•.!:!• and modal matrix, M. Thus, we have
that
Using Eq. (D.87) we have that
-'1#-$t/2e2'] 6<_, .;• 7
Thus the solution given by Eq. (D.86) is:
REVIEW OF MATRIX ALGEBRA
1247
The extensiml of these ideas to provide solutions for the case of nonlumrogtneous 1/nesr dijfomttiAL tqUIIIUms:
!!!jf •
All(t) + BaCt>+ rcl(t):
ll(to>
="o
(0.89)
is straightforward. Here u (t), d(t) represent control variables and distutbances respectively. When the mab'kes A, B, rare constlmt mab'ices, a derivation rompletely similar to the one just given leads to the solution: ll(t)
s
.,.W-•ol:ro+
J,ACI-~[Ba(1)+rd(1)]dr
(0.90)
'o
when! the matrix exponentials, eA(t-~l, ~·~,may be evaluated using the relation
Eq. (0.87).
l!umple 0.15
SOLtmON OF LINEAR NONHOMOGENEOUS MATRIX OIFFEREN11AL EQUATIONS.
To illustrate the solution of nonhomogeneous equations using Eq. (0.90), let us
suppose we have a syolem Eq. (0.89) wilh A lite aame as lite last """"'f'le:
and
lnaddltlonaaaume "t • t1z • O,d 1 •l (aaJNtantdlstwbanee)an d lhat t0 = 0. In this case Eq. (0.90) talcea lite lonn: x(l)
= ,.V"o +
J. .,AU-~rc!(l)dr 0
Substituting for ,AI, and ,.V"o calculated in lite Example 0.14, we obtain the solution:
1248
APPENDIXD
x(t)
=
or
For the situation where the matrices A, B, r may themselves be functions of time, the solution requires a bit more computation. In this case: x(t)
= •ll(:r,to) "o +
r
~(t, to) ~('l;toT1 [Bu("' + rd("'1d-r
(D.91)
'o
~(t, ~) is an n x n time-varying matrix known as the fundamental matrix solution which may be found from:
Here
dCP(t,t0 )
---;It =
A(t)~(t, t 0 ); ~(t0 ,fo) =I
(D.92)
Because A(t), B(t), r(t) may have arbitrary time dependence, the fundamental matrix solution is usually obtained by numerical methods. Nevertheless, once ~(t,fo) is determined, the solution x(t), may be readily calculated for a variety of control and disturbance inputs, u(t ), d(t), using Eq. (D.91).
D.12
SUMMARY
· In this appendix, the techniques of matrix algebra have been reviewed. These methods find particularly useful application in providing very general solutions to sets of linear algebraic and linear ordinary differential equations. The importance of matrices as a tool for analyzing the dynamics of physical processes should not be undervalued by the reader. As an aid to further study, a bibliography is provided below. REFERENCES AND SUGGESTED FURTHER READINGt
J. Wiley, New York (1967),
1.
Kreysig, E., Advanced Engineering Mathematics (2nd ed.), Chap.7
2
Amundson, N. R., Mathematical Methods in Chemical Engineering, Prentice-Hall,
3. 4. 5. 6.
Englewood Oiffs, NJ (1966) Wylie, C. R., Advanced Engineering Mathematics, J. Wiley, New York (1966) Duncan, W. J., A. R. Collar, and R.Q. Frazer, Elementary Matrices, Cambridge University Press (1963) Bellman, R., Introduction to Matrix Analysis, McGraw-Hill, New York (1.960) Gantmacher, F. R., 1'he Theory of Matrices, (Vols. 1 and 2), Chelsea Publishing Company, New York (1960)
+Listed in increasing order of depth.
;i
~
AJPJPENDKX
E COM PUT ER-A IDED CON TRO L SYSTEM. DES IGN The calculations required for the analysis of process dynamic s, the fitting of models to experime ntal data, and the creation of control system designs can be quite tedious and challeng ing. Fortunat ely, there are a host of interacti ve Comput er-Aided Design (CAD) package s which will carry out these calculations with no program ming effort required by the user. These packages are available for reasonab le fee, and thus should be used by students and others studying the material in this book. Some of the more widely used packages are listed in Table E.l below. Reviews of the field of compute r-aided design and a discussio n of these and other CAD program s can be found in Refs. [1-3].
a
1249
APPENDIX E
1250
Table E.L Some Computer-Aided Analysis and Control System Design Packages
ACSL
Scope Primarily Simulation
cc
Comprehensive
CONSYD
Comprehensive
Ctrl-C Model-C
Comprehensive
EASY5
Comprehensive
Program
. MATLAB with Control System, Robust Control, and System Identification Toolboxes MATRIX-X
Comprehensive
Comprehensive
TUTSIM
Primarily Simulation
UC-SIGNAL UC-ONLINE
Primarily Simulation
Source Mitchell and Gauthier Assoc. 200 Baker Avenue Concord MA 01742 Systems Teclmology, Inc. 13766 S. Hawthorne Blvd. Hawthorne CA 90250 Prof. W. Harmon Ray Dept. of Chemical Eng. University of Wisconsin 1415 Johnson Drive Madison. WI 53706 Systems Control Teclmology 1801 Page Mill Road P. 0. Box 10180 Palo Alto, CA 94303 Boeing Computer Services P.O. Box24346 Seattle WA 98124 Mathworks, Inc• 24 Prime Park Way Natick, MA 01760
Integrated Systems, Inc. 3260 Jay Street Santa Clara CA 95054 Tutsim Products 200 Calif. Ave., #212 Palo Alto CA 94306 University of California Office of Tech. Licensing Berkelev, CA
REFERENCES . 1.
2. 3.
Chen, Z.-Y. (Ed.)., Computer-Aided Design in Control Sytems, Proceedings 4th IPAC Symposium, Pergamon Press, New York (1988) Holt, B. R., et al., "CONSYD- Integrated Software for Computer-Aided Control System Design and Analysis," Comput. Chern. Eng., 11, 187 (1987) Linkens, D. A (Ed.), CAD fur Control Systems, Marcel Dekker, New York (1993)
author in de x A Abramowit z, 1207 Aitolahti, 1029 Albert, 62 Altpeter, 620 Alvarez, 807 Amundson, 1248 Ardell, 1146 Aris, 401 Arkun,328 , 1029,1030 Arnold, K. ].,401 Arnold, V.I., 328 Arulalan, 1028 Ash, 980 AstrOm, 558, 586, 639, 877, 943, 980 Ayral, 1029
B Baker, 1094 Bard, 401 Beck, 401 Bellman, 1248 Beudat, 450 Berry, 717 Bhat, 1095 Biegler, 1028, 1094 Simonet, 1030 Bird, 135, 212 Bischoff, 401 Bode, 167, 620 Bodson, 639 Bongiorno, 1031 Box, 843, 1028, 1059 Boyce, 1174 Bracewell, 135, 450 Bristol, 764, 807 Brogan, 10 Brosilow, 1028
Brown, D. W., 1060 Brown,]. W., 1174 Bruns, 808 Bryson, 1094 Buckley, 31, 1146
c Cadzow, 943 Caldwell, 1028, 1029 Camp, 1031 Caracotsios, 401 Carbonell, 1095 Carnahan, 1175 Ceaglske, 31 Champetier, 639 Chang,102 8 Chen, 1250 Chien, 1146 Churchill, 1174, 1207 Cinar, 1147 Clarke, 1028 Coddington, 1174 Coggan,62 Cohen,557 Collar, 1248 Congalidis, 1146 Conle, 1178 Considine, 557 Coon,557 Copson, 1174 Corripio, 240, 557 Coughanow r, 31, 268, 355 Cozewith, 1146 Crosier, 1059 Cutler, 808, 1028
D de Boor, 1175 DelToro, 31 1251
Deshpand~980, 1028,1146 DiPrima, 1174 Doss, 1059 Downs, 807, 1059 Doyle, F.J., 1147 Doyle,]. c., 8C8, 1094 Driver, 1174 Duncan, 1248
E Eckman,31 Economou, 1028 Edgar, 980 . Engell, 1146 English, 1059 Erickson, 1028 Eykhoff, 450
F Flannery, 1175 Francis, 808 Frank, 676, 1028 Franks, 401 Franklin, 877, 943, 980 Frazer, 1248 Friedly, 327, 401
G Gagnepain, 764 Gantmacher, 1248 Garcia, 676, 807, 1028, 1030, 1094 Gay, 1094 Gear, 1175 Georgiou, 1029 Georgakis, 1029 Gerstle, 1029 Gillette, 1030
1252 Goldberg, 1174 Golubitzky, 327 Gould, 1030 Greenberg, 1174 Grosdidier, 764, 1029 Grossmann, 1094 Guckenheimer, 327, 355 Gumowski, 1146
H Hagglund, 558 Halm,355 Hanus,586 Harris, 1059, 1060 Harrison, 62 Hashinloto, 1030, 1031 Hawkins, 1028 Henrici, 1175 Henrotte, 586 Hernandez, 1029, 1030 Hillier, 1094 Hirnrnelblau, 401 Ho, 1094 Hofhnan, 1060 Hokanson, 1029 Holmes, 327, 355 Holt, 764, 1250 Hoo, 639 Hopf,315 Hopfield, 1095 Hornbeck, 1175 Horowitz, 1029 Hubele, 1059 Hulburt, 807 Hummel, 1030 Hunt, 328, 639 Hunter, 1059, 1060 I Iinoya, 620 Ince, 1174 Iooss, 327, 355 Isermann, 980 Isidori, 328
J Jabr, 1031 Jazwinski, 1094 Jenkins, 843, 1028, 1059 Jensen, 807 Jeron1e, 808, 1146 Joseph, 327, 355
INDEX K Kaiser, 1030 Kantor, 639 Kaspar, 1146 Keats, 1059 Kinnaert, 586 Klatt, 1146 Koppel, 31, 268, 355, 764 Kramer, 1094 Kravaris, 328 Kresta, 1059 Kreyszig, 1174, 1248 Krishnan1urthi, 1059 Kuo, 980, 1175, 1207
L Landau, 639 Lapidus, 401, 450 Lasdon, 1094 Lau, 807 Lecamus, 1030 Lectique, 1029 Lee, J. H., 1029, 1030 Lee, P. L., 676, 1029 Lee, w.,31 Lemaire, 764, 808, 1147 LeRoux, 1030 Levinson, 1174 Lieberman, 1094 Lightfoot, 135, 212 Linkens, 1250 Littnaun, 1030 Longwell, 327 Lopez, 557 Lucas, 1059 Luther, 1175 Luyben,31,240 ,327,401, 717, 1029, 1146 Lyung, 450 M Maarleveld, 1147 Maciejowski, 808 Marlin, 1059 Martens, 943 Martin, 1028, 1029 MacGregor, 1029, 1059, 1060, 1146 Mahrnood,808,1029 Marchetti, 1029 Marquardt, 1094 Mason, 1029
McAuley, 1146 McAvoy, 764, 807, 1029, 1095, 1146 Matsko, 1029 Maurath, 1029 Meadows, 1029 Mehra, 808, 1029, 1030 MellichaD1p,62,980, 1029 Meyer, 328, 639 Michalski, 1095 Mitchell, 1095 Milne-Thon1pson, 1174 Mohtadi, 1028 Monopoli, 639 Montague, 1095 Moore, 807, 1094 Morari, 557, 620, 676, 717, 764, 808, 980, 1028, 1029, 1030, 1094, 1147 Moreau, 1030 Morris, 1095 Morshadi, 1029 Murrill, 557 Muske, 1029, 1030
N Narenda, 639 Newell, 676, 1029 Newton, 1030 Nichols, 557 Niederlinski, 764 Niida, 1094 Nisenfeld, 1146 Nygardas, 620
0 Ogata, 877, 943, 980, 1207 Ogunnaike, 450, 639, 764, 808, 1030, 1146, 1147 Olmo, 1030 Ohshlrna, 1030 Otto, 1028, 1030 Owen, 1095 p Page, 1060 Palsson, 1028 Papon,1030 Parher, 31 Pearson, 1147 Penlidis, 1146 Piovoso, 1095
I
INDEX 1253 Pouliq uen, 1030 Powell , 877, 943 Prett, 807, 1029, 1030, 1094 Propoi , 1030
Q Qin, 1095
R Radem acher, 1147 Ragaz zi, 980 Ramake~808,1028,
1030 '
Ranz, 401 Rao, 1028 Rault, 1029, 1030 Rawlin gs, 1028, 1029, 1030 Ray, 31, 62, 401, 557, 676, 717, 764, 808, 1028, 1094, 1095, 1146, 1147 Rebou let, 639 Redhe ffer, 1174 Richalet, 1030 Richar ds, 1146 Ricker , 1030 Rijnsd orp, 1147 Rivera , 557, 1030 Robert s, 1060 Roffe!, 1060 Rosenbrock, 808 Ross, 1059 Rouha ni, 1030 Routh, 504 Rovira , 557
s Sastri, 1059 Sastry , 639 Schaeffer, 327 Schork , 1030 Schule r, 1147 Scott, 1095 Schwe dock, 1147 Seborg , 764, 980, 1028, 1029, 1031 Seema nn,114 6 Seinfeld, 401, 450 Semin o,558 Shewa rt, 1060 Shinskey,240,586,8~/,
1147 Shunta , 1146 Sim, 1030
Simminger, 1030 Skoges tad, 557, 1030, 1147 Smith, C. A., 240, 557 Smith, C. L., 557 Smith, c. R., 808 Smith, 0. J. M., 620, 1030, 1031 Smith, R. E., 557, 1095 Sokolnikoff, 1174 Stein, 1094 Stepha nopou los, 212, 1095 Stephe nson, 1174 Stewar t, 135, 212, 401 Stone, 1095 Su, 328,63 9 Subrarnarrian, 1030 Sulliva n, 676, 1029
T Takam atsu, 1030 Tanne nbaum , 808 Taylor , 1094 Tessie r, 1029 Testud , 1029, 1030, 1031 Teukb lsky, 1175 Tham, 1095 Truxal , 31 Tsing, 1031 Tu, 1031 Tuffs, 1028 Tyreus , 764
u Umeda ,1094
v Vanhamliki, 1029 Vellering, 1175 Venka tsubra manian , 1095 Verhey , 1174 Vinant e, 717 Vogel, 980
w Waller , 620 Wassick, 1031 Weber ,31 Weekman,31 Wethe rill, 1060 Whale n, 1031 Wilkes, 1175
Williams, 31, 1031 Winde s, 1147 Witten mark,5 86,639 ,877, 943, 980 Wood, 717 Woodb urn, 807 Wylie, 1174, 1248
y Yamamoto, 1031 Yeo, 1031
Yocum, 1028 Youla, 1031 Yuan, 1031
z Zadeh , 1031 Zafirio u, 558, 620, 676, 717, 808, 980, 1029, 1031, 1094 Zames , 1031 Zhang , 1147 Zhao, 1028 Ziegler, 557
subject index A abnormality detection, 1088 ACSL, 1250 actuators, 49 adaptive control, 628 AID converter, 43 alarm interpretatio n, 1088 aliasing, 826 amplitude ratio, see also magnitude ratio, 152, 281 analog ruters, 51 analog input/ output, 44 analog-to-digital converter, 43 antiresetlYin dup,585 ARIMA model, 1051 ARMAX model, 999 artificial intelligence, 1091 auctioneerin g control, 582 autoregressive moving average (ARMA) model, 999 autotuning, 555
B biological systems: anaerobic digester, 310, 405 anaesthesia control, 814 bioreactor, 357 cardiovascul ar regulator, 563, 770 hemodynamics model, 222 human pupil servomechanism, 509 immunology, 243 respiratory process, 220
"black box" models, 410 block diagrams, 463, 824, 927 Bode diagram, 153, 281 asymptotic consideration s, 306 classical controllers, 305 for feedback controller design,549 first-order system, 285 inverse-respo nse system,302 lead/lag system, 294, 300 pure capacity system, 291 second-order system,298 time-delay system, 303 Bode plot, see Bode diagram Bode stability criterion, 549 boiler, 229, 562, 811 bounded-input, boundedoutput stability, 337 Bristol array, see relative gain array (RGA),
c capillary, 154 cascade control, 12, 567 catalytic reactors, 231, 1105 cc, 1250 chemical process, defini lion, 5 objectives of operation, 6 variables of, 13 chemical reactors, 101, 138, 141, 171, 217, 252, 271, 313, 315, 331, 358, 365, 381, 395, 407, 456, 583, 591, 641, 681, 816, 984, 1067, 1138
1255
closed loop, block diagram, 463, 824, 926 characteristic equation, 486 performance criteria, 520 stability, 486 transfer functions, 468 multiloop systems, 724 Ctrl-C, 1250 Cohen and Coon tuning rules, 537 comparator, 465, 927 complex variables, 1155 complex s- plane, Nyquist map of, 286 complex z- plane, 902 computer control system, 36-61 computer-aid ed control system design, 1249 computer control, see digital computer control, condition number, 803, 1241 CONSYD, 1250 control limits (in SPC), 1037 control system synthesis, case studies of, 1097 controllabilit y, 686 controller, 16, 465, 518, 927 controller design, 21 commercial, 518 choosing controller parameters, 524 controller type, 523 conventional feedback, 461 model-based, 645 performance criteria, 520 principles, 519 via optimization , 522
1256 con troller tuning, auto tuning, 555 Cohen and Coon rules, 536 direct synthesis, 526, 538 internal model control, 529, 539 pole placement, 529 stability margins, 530, 552 with approximate models,532 with frequency response models,541 without a model, 555 time-integral rules, 525, Ziegler-Nichols approximate model rules, 536 Ziegler-Nichols stability margin rules, 553 control valve, 514 control variables, 13 comer frequency, 289 crisis management, 1088 critical gain, see also ultimate gain, 531 critically damped response, 191 crossover frequency, 550 crude oil, boiling point curve,8 CUSUM charts, 1040 cyclopentanol, 1138
D D I A converter, see digitalto-analog converter Dahlin controller, 973 damping coefficient, 184 data acquisition, 43, 47 deadbeat control, 971 dead time, see time delay, decay ratio, 520 decoupling, 773 design,776 dynamic, 777 feasibility of, 792 generalized, 781 limitations of, 785 partial, 788 simplified, 778 steady-sta te (static), 790 time scale, 755
INDEX delta function, Dirac, 82 Kroneker, 887 derivativ eaction,4 68 derivative time constant, 468 desired reference trajectory, see reference trajectory, deviation variable(s), 23 difference equations, 1166 differential equations, 1162, 1169 difficult dynamics, controller designs fur, 599 digital compulel', architecture, 38 capabilities, 38 peripherals, 38 digital computer control, 37, 821, 951 digital controller, 951 advanced, 966 poles and zeros of, 954 PID,960 digital filters, 829 digital input/output, 44 Dirac delta function, 82 direct digital control (DOC), 52 Direct Synthesis Control, 526, 538, 970, 978 discrete-time models, 107, 832, 873 backward shift form, 893 forward shift form, 894 poles and zeros, 898, 906 discrete-time systems analysis, 835, 847 block diagram. 918 discretization, 835 dynamics, 883 identification, 838 relation to continuous systems, 901, 918 stability, 933, 937 distillation columns, 7, 32, 138, 172, 230, 369, 420, 442,507,5 62,565,59 0, 711, 719, 733, 749, 760, 766, 783, 791, 798, 810, 845, 982, 1097 distributed control system {DCS),54
distributed paramete r controllers, 1083 distributed paramete r systems, 106, 376, 1083 disturban ce variables, 13 DMC, see Dynamic Matrix Control, drug ingestion, 137, 215 Dynamic Matrix Control, 1000, 1012 ·Dynamic Matrix Identification, 1013 dynamic flowsheet simulation, 1089 E EASY 5,1250 economic forecasting, 220 electronic controllers, 37 empirical process models, see also process identification, 409-450 equal-percentage valve, 514 E~,seeexponentiaily
weighted moving average, expert systems, 1091 exponentially weighted moving average, (EWMA), 1043
F fault detection, see also abnormality detection, feedback control, 12, 17 adaptive, 628 basic loop elements, 463, 514, 927 PID,467 tuning rules, 525 feedforward control, 12, 18, 571, 968 filter(s), analog, 51 digital, 829 final control element, 16, 465, 927 first order hold, 831 first order system, 139-154, 884, 912
~;
INDEX first order plus time delay model,253 approximation of higher order dynamics, 256 characte rization of process reaction curve, 533 use in controller design, 536 flow controller, 12 Fourier transform, 1178 frequency: comer,2 89 crossover, 550 Nyquist , 828 resonan t, 298 sampling, 827 frequency response analysis , 275-307 applicat ion in controller design,541 Bode/N yquist plots, 286305 identifica lion, 436 models, 110 pulse testing, 437 furnace, 8, 33, 269, 505, 581
G gain margin, 552 gasoline blending, 219,743 gain scheduling, 628 gas storage tank, 56 general ized inverses, 1227 Generic Model Control, (GMC), 671 grade transitions, 1089 grindin g circuit, 789, 810
H heatexc hanger, l07,257 , 273, 310, 377, 506, 560, 588, 596, 681, 986 heating tank, 14, 85, 95, 462 Heaviside function, 80 higher order systems, 205211 holds,8 30 Honeyw ell Multiva riable Predictive Control, (HMPC ),l016 Horizon Predictive Control, (HPC), 1016
1257
I IDCOM, 1008, 1011 ideal forcing functions, impulse,82 ramp,83 realization of, 85 rectangular pulse, 81 sinusoidal, 85 step, 80 ill-conditioning, condition number, 803 degeneracy,798 SVD, tool for detecting, 806 impulse,B2 impulse response function, in model predictive control, 1008 moments from data, 427 impulse response identification, 422 impulse response model, 111, 833 ingot, 388, 1084 input variable(s), 13 input/ output interfaces, 42 input/o utput pairing, 735 instrumentation, 47 interaction, 735 integral action, 467 interaction compensators, see decouplers, Internal Model Control, (IMC), 529, 539, 665 inverse response cornpensation,613 inverse response systems, 225-240, 608 controller designs for, 608 main characteristics, 226 physical examples, 229231
J Jury array, 938 Jury stability test, 938
K Kalman filter, 1064
L
Laplace transfon n, application to the solution of differential equations, 77, 1185 defmiti on,71,1 180 inversion, 73, 1180 tables, 1193 lead/lag systems, 161-166 lime kiln, 818 lineariz ation, approximate, 626 exact, by variable transformation, 632 liquid/l iquid extraction, 405 loop pairing, 683, 727, 735-758 lumped parameter systems, 368
M magnih lderatio ,288 manipu lated variables, 13 manometer, 185 manual control, 10 MATLAB, 1250 matrices, 1209-1248 MATRIX-X, 1250 measuring device, 465, 927 methanol oxidation, 1107 MIMO systems, see multiinput-multi-output systems, minimum variance control, 1052 Model Algorithmic Control (MAC), see also IDCOM, 1008, 1011 model-based control, 645 MODEL-C, 1250 model identification, (see process identification), model predictive control, (MPC), 991 applications of, 995 commercial software packages for, 1011 model forms for, 997 nonline ar, 1020 model reference adaptive control, 629
1258
MPC, see model predictive control, multi-input, multi-output (MIMO) systems, 683-807 multidelay compensator, 1103 multiloop control systems, controller designs for, 723 ·controller tuning, 759 multiple input, single-output systems, 582 multivariable block diagrams, '775 multivariable control, block diagram, 712, 775 controller design, 759,
773 decoupling, 716 input/output pairing, 683, 727, 735 interactions, 686, 724 process models, 688 multivariable dynamics, 694, 890 multivariable identification, 447 multivariable system poles, 703 multivariable system zeros, 704 N NARMA(X), models, see nonlinear ARMA
models, neural networks, 1091 Niederlinski index, 738 nonlinear ARMA (NARMA) models, 1026 nonlinear control, 625 nonlinear MPC, 106, 1020, 1026 nonlinear systems, 311-327 controller design philosophies, 625 dynamic response, 326 methods of dynamic analysis, 314 stability, 343 nonminirnum phase systems, 601
INDEX
Nyquist diagram. 286, 543 Nyquist frequency, 828 Nyquist plot, see Nyquist diagram, Nyquist stability criterion, 543
0 observability, 686 observers, 1064 offset, 25, 485 oil refinery, 1086 on-line optimization, 1086 open loop, control strategy, 18 stability, 333 open loop unstable systems, 617 optimal control, 674 optimization, 1086 Optimum Predictive Control, (OPC), 1014 output variable(s), 13 overdamped response, 191 overdefined systems, 1227 override control, 581 overshoot, 520
p P controller, see proportional controller, P &: I diagram, 59, 1149 packed bed reactors, 406, 1105 Pade approximations, for time delays, 261 paper machine, 982 parallel computing, 1093 parameter estimation, basic principles of, 381 differential equation models,384 partial decoupling, 788 PO controller, see proportional-plusderivative controller, pendulum, 335 performance criteria, 520 phase angle, 'Zl7 phase lag, 277 phase lead, 292 phase margin, 552
PI controller, see proportional-plusintegral controller, PID controller, see proportional-plusintegral-plusderivative controller, pipe, 246, 561 plant/model mismatch, 1070 pneumatic transmission, 37 pole placement, for continuous controller design,529 for discrete controller design,978 poles, closed loop, 496, 935 transfer function, 132, 933 pollution control, 216, 217, 331 polymerization, p~,58,270,332,
357, 456, 589, 640, 766, 844, 949, 987, 1039, 1113, 1122 predictive control, see model predictive control, Predictive Control, (PC), (commercial software package), 1015 pressing iron, 172 process and instrumentation diagrams, seeP&: I diagrams, process characterization cube, 1133 process dynamics, main objectives, 65 definition, 67 process identification, 409-450 frequency response, 4..~ impulse response, 422 multivariable, 447 pulse testing, 437 step response, 417 process model, formulation of, 92, · 363-401 interrelationships between various forms, 115, 128 various forms of: state-space, 104
INDEX
frequency response, 99 impulse response, 100 transfer function, 128, 131 transform domain, 98, 108 process reaction curve, 533 process variability, 1070 product quality, specifications, 6 product ion rate, 6 proport ional band, 467, 518 proport ional controller, 467, 474 proport ional-pl usderivative controller, 468 proportional-plus-integral controller, 467, 478 proportional-plus-integralplus-derivative controller, 468, 518 pulse testing, 437 comparison with direct sine wave testing, 436 input characteristics, 441 multivariable, 447 pulse transfer function model,867 pure capacity system, 157-161 pure gain system, 153-157 pure time delay process, limit of infinite number of first order systems in series, 262 physical example of, 245, 252, 257 transfer function of, 249 pyrolysis fractionator, 169, 589
Q quality control, see statistical process control, quality control charting, 1037 quarter- decay ratio, 531 quick-opening valve, 514
R ramp function, 83
1259 ratio control, 579 realization, 117 rectangular pulse, 81 reference trajectory, in controller design by direct synthesis, 648 for model predictive control, 996 regulatory control, 19 relative gain array (RGA), 728 relay controller, 555 reset rate, 468 reset windup , see also antireset windup , 585 resonance, 298 reverse action, in controller hardwa re, 517 ringing, 972 rise time, 520 robust controller design, 1069 robust performance, 1070 robust stability, 1070 root locus, 496 Routh stability test, 489
s safety, 6 sample-and-hold, see also zero-order hold, 824 effective time delay contributed by, 832 sampled data system, 821 block diagram analysis for, 917 closed loop block diagrams, 927 stability of, 933, 937 sampler, 823 scrubber, 454, 559 second order system, 183205, 913 selective control schemes, 581 self-tuning regulator, 630 sensors, 16 serial correlation, 1046 serial transmission, 43 servo control, 19 set-point, 19 set-point tracking, see servo problem,
set-poin t trajectory, see reference trajectory, settling time, 520 Shewar t (quality control) chart, 1037 signals, analog voltage, 51 analog current, 51 digital, 52 signal conditioning, 49, 829 amplification, 50 multiplexing, 50 filtering, 51 single-i nput multiple-output systems, 581 singula r value decomposition (SVD), 806 singula r values, 803 sinusoid al inputs, 84 SISO systems, see singleinput-single-output systems Smith predictor,605 SPC, see statistical process control, split-range control, 583 spring-s hock absorber, 185, 193 stability , 333-355 definiti ons,335 discrete time, 933 linear, 337 multivariable syst!!rns, 710 nonline ar, 343 open loop, 349 closedl oop,353 ,486 stability criteria, bilinear, 937 Bode,54 9 direct substitu tion, 494 Jury,93 7 Nyquist, 543 root locus, 496 Routh,4 89 stagewise processes, 369 state estimation, 1063 state-space model, 104 state variables, 13 statistical process control, 1033 steady state decoupler designs, 790
1260
step response , idealize d input, 80 identific ation, 433, 533, 1013 first order system, 143 higher order systems , 204 inverse respons e system, 226 lead/lag system, 164 nonline ar systems , 312 pure capacity system, 159 pure gain system. 155 second order system. 188 system with multiple right half plane zeros, 237 stochast ic process control, 1050 storage tank, 136, 171 structur al instabil ity (of multiloo p systems ), 738 supervi sory control, 52 SVD, see singula r value decomposition,
T
INDEX time series, 997, 1051 transfer function , in generali zed view of modelin g, 128 properti es, 132, 893 traditio nal, 132 transform domain model form, 108, 893 poles of, 132, 898 zeros of, 132, 898 transmit ters, 16 TUTSIM, 1250
u UC-ONUNE, 1250 UC-SIGNAL, 1250 ultimate gain. 531 ultimate period, 531 underda mped response, 191 underde fined systems , 1227 unstable system, 617
v
valve, see control valve, Tank, 21, 33, 140, 146, 158, Vogel-E dgar controll er, 975 163, 176, 313, 318, 323, Volterra series models, 330, 348, 353, 358, 404, 1026 452, 521, 571, 579, 584, 588, 592, 627, 636, 642, w 684,746 ,800,84 0,880 tempera ture controll er, 12 water heater, 136, 168, 403 tempera ture measurement, Western Electric rules, 1039 16 theoreti cal process models, 363-401 thermom eter, 403 z-transf orms, 848 time delay compen sation, applicat ion in discrete 604 time control, 859 time delay systems , 245-268, applicat ion to the 603 solution of difference approxi mations for, 261 equatio ns,873, 1199 controll er designs for, defirriti on,849, 1194 603 effect of samplin g on, 857 dynami c response s, 250, inversio n, 854, 860, 1194 254 relation to Laplace models for, 246, 265 transfor m, 859 physica l example , 246, tables, 861, 1202 252 zeros, time domain models, 832 effect of samplin g on, 906, time-int egral criteria (IAE, 909 ISE, ITAE, ITSE), 525, effect on step response , 537 206-211, 231-239 time scale decoupl ing, 755 right half plane, 231
z
zero-ord er hold (ZOH), 830, 869 Ziegler- Nichols tuning rules, 536, 553