Consequence of Lemmas 17.1 on page 494 and 17.4 on page 502.
The canonical decomposition described in Theorem 17.5 on the preceding page
leads to
Lemma 17.6. Consider the transfer function matrix H(s) given by
Y (s) = H(s)U(s) (17.8.4)
Then
H = C(sI A)1B
(17.3.10)
Using (17.3.4) and (17.3.10) we obtain the similar state space description given
by
486 Linear State Space Models Chapter 17
A=
5 0 0
0 3 0
0 0 2
;B=
0.0
1.414
0.0
; (17.3.11)
C=
_
0.5345 1.4142 0.7071
_
D = 0 (17.3.12)
17.4 Transfer Functions R
x(t) = Aox(t) + Bou(t) + JCo(x(t) x(t) + Jv(t) (18.8.3)
x(t) = (Ao JCo)x(t) Jv(t) (18.8.4)
We then see, on applying the Laplace transform to (18.8.4), that
X
(s) = [sI Ao + JCo]1x(0) [sI Ao + JCo]1JV (s) (18.8.5)
Equation (18.8.5) reveals a trade-off in
0
xno
=
0
xno
(17.9.5)
and
_
Co 0
_
0
xno
= 0 (17.9.6)
(ii) Similar to part (i) using properties of controllability.
We will next use the above result to study the system properties of cascaded
systems.
Consider the cascaded system in Figure 17.3.
We ass
control law of the form:
u(t) = Koxo(t) d(t) + r(t) (18.7.1)
We observe that this law ensures asymptotic cancellation of the input disturbance
from the input signal, provided that the estimate of the disturbance; d(t), is
stable and unbiased. We will show
= Bo(s)
det(sI Ao + BoK)
= Bo(s)
F(s)
where Bo(s) and Ao(s) are the numerator and denominator of the nominal
loop,r espectively. P(s) and L(s) are the polynomials defined in (18.5.12) and
(18.5.13),r espectively. F(s) is the polynomial defined in (18.2.16
H
(
1 + hd(t)
H
K
H
_
1 + hd(t)
2H
_
since
528 Synthesis via State Space Methods Chapter 18
1+&1+&
2
(18.3.24)
then
K
H
_
1 + hd(t)
2H
_
=K
H+K
2
H
hd(t) (18.3.25)
=K
H+K
2
H
(h1(t) h2(t) H)
=K
H+K
2
H
(h1(t) h2(t) K
2
H
H
=K
H
2
+K
2
H
(h1(t) h2(t)
This
556 Introduction to Nonlinear Control Chapter 19
x1Q = yQ
_=
lim
tx1(t) = 1.13 and x2Q
_=
lim
tx2(t) = 0 (19.2.19)
The values for uQ, x1Q and x2Q are then used,in conjunction with the MATLAB
commands linmod and ss2tf,to obtain the linearized model Go(s),w
The interaction is quantified by the third fundamental result presented in the
chapter: the nominal poles of the overall closed loop are the union of the
observer poles and the closed loop poles induced by the feedback gains if all
states could be measur
number of inputs, in their injection points, in the number of measurements
and in the choice of variables to be measured may yield different properties.
A transfer function can always be derived from a state space model.
A state space model can be built
the closure of its range is a strict subset of .
Remark 19.2. To provide motivation for the above concept of a non-minimum
phase nonlinear operator,we specialize to the case of linear time invariant systems.
In this case,if f has a non-minimum phase zero,
function.
Problem 16.6. A disturbance feedforward scheme (see Figure 10.2 on page 272)
is to be used in the two d.o.f. control of a plant with
Go1(s) =
s+ 3
(s + 1)(s + 4)
(16.6.7)
It is also known that the disturbance dg(t) is a signal with energy distri
Nevertheless, nonlinearities are frequently encountered and are a very important
consideration.
Smooth static nonlinearities at input and output
are frequently a consequence of nonlinear actuator and sensor characteristics
are the easiest form of nonl
for the i th controller, Ci(s). Thus, in a sense, Ci(s) is a controller-controller.
This controller-controller is relatively easy to design since, at least, the plant in this
case is well known as it is actually the ith controller. For this controller-con
(18.2.8)
Now,i f one uses state feedback of the form
Section 18.2. Pole Assignment by State Feedback 521
u(t) = Kcxc(t) + r(t) (18.2.9)
where
Kc
_=
_
kcn
1 kcn
2 . . . kc0
_
(18.2.10)
then,t he closed loop state space representation becomes
x c(t) =
an1 .
Lemma 18.4. Consider the state space model (18.3.1)(18.3.2) and an associated
observer of the form (18.3.3).
Then the estimation error x(t),defi ned by (18.3.5), satisfies
x(t) = (Ao JCo)x(t) (18.3.6)
Moreover,pr ovided the model is completely observable,
model takes the form
483
484 Linear State Space Models Chapter 17
x (t) = Ax(t) + Bu(t) x(to) = xo (17.2.1)
y(t) = Cx(t) + Du(t) (17.2.2)
where x Rn is the state vector, u Rm is the control signal, y Rp is the output,
xo Rn is the state vector at time t =
_
K1 K2
_
x
x2
_
More will be said about this (linear) strategy in section 22.4 of Chapter 21.
As in the case of the observer, there are several ways we could design the feedback
gain
_
K1 K2
. Three possibilities are:
1. Base the design on a fixed set of
0
(17.6.31)
495
A=
S1
S2
A[T1 T2] =
S1AT1 S1AT2
0 S2AT2
(17.6.32)
The zero entries in (17.6.31) follow since the values of B belong to the range
space of c and S2T1 = 0. Similarly,t he zero elements in (17.6.32) are a consequence
of the columns of AT1 bel
x1
x2
(17.9.12)
Thus using the PBH test the composite system is not completely observable.
Part (b) is established similarly by choosing x so that
Section 17.10. Summary 511
xT =
_
xT1
xT2
_
(17.9.13)
0 = xT2
(I A2)
xT1
= xT2
B2C1 (I A1)1
Then
_
xT1
xT2
; G2(s) =
2s + 3
(s 1)(s + 5)
(17.12.12)
Prove that,for nonzero initial conditions, the output of System 2 includes an
unbounded mode et which is not affected by the input, u(t),to the cascaded system.
Problem 17.11. Consider a system composed by a contin
n + n1n1 + + 1 + 0 = det(I A) is the characteristic polynomial
of A.
Proof
Using (17.6.26) and the controllability property of the SISO system,we note that
there exists a unique similarity transformation to take the model (17.6.41) into the
model (17.6.38
following example illustrates that in discrete time,th e two concepts are different.
Consider
x[k + 1] = 0 x[0] = xo (17.7.1)
y[k] = 0 (17.7.2)
This system is clearly reconstructible for all T 1,si nce we know for certain
that x[T] = 0 for T 1. However,it
This means that if we define
FQ_
_= [p()]1_ (19.6.14)
then
y(t) = FQ_t
(19.6.15)
Finally, to obtain a perfect inverse at d.c., we set p0 = 1. The control strategy
(19.6.12) is commonly known as input-output feedback linearization, because, as seen
from (1
(17.5.8)
This can be expressed as
Yq(z) =
_n
i=1
bi1Vi(z) where Vi(z) = zi1
Aq(z)Uq(z) (17.5.9)
From the above definitions, we have that
vi[k] = Z1 [V (z)] = qvi1[k] for i = 1, 2, . . . , n (17.5.10)
where q is the forward shift operator.
We can then choo
Remark 17.1. We observe that G(s) can be expressed as
G(s) =
CAdj(sI A)1B
det(sI A)
(17.4.8)
Hence the poles of G(s) are eigenvalues of A,i.e. they are roots of the characteristic
polynomial of matrix A
det(sI A) =
n
i=1
(s i) (17.4.9)
where 1, 2, . . .
Example 18.1 (Tank Level Estimation). As a simple application of a linear
observer to estimate states we consider the problem of two coupled tanks in which
only the height of the liquid in the second tank is actually measured but where we
are also intere
lim
t
d(t) = (18.7.10)
The transient in d(t) is a linear combination of the observer modes.
Also
x(t) = L1
_
(sj1 + j2)
E(s) Y (s) + s
E(s)U(s)
_
(18.7.11)
We see that the transfer function from U(s) to X (s) is 1
s+1 as required. Also,
choosing K to pla