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Unformatted text preview: Controllability
and Observability Controllability and observability are important structural properties of dy
namic system. First identiﬁed and studied by Kalman (1960) and later by
Kalman et al. (1962), these properties have continued to be examined dur
ing the last four decades. We will discuss only a few of the known results for
linear constants systems with one input and one output. In the text we dis
cuss these concepts in connection with control law and estimator designs. For
example, in Section 7.2 we suggest that if the square matrix given by e = [G FG F2 G FHG] (D1)
is nonsingular, then by transformation of the state we can convert the given description into control canonical form. We can then construct a control law
that will give the closedloop system an arbitrary characteristic equation. 849 850 Appendix D Controllability and Observability Definition  Definition ll D.1 Controllability We begin our formal discussion of controllability with the ﬁrst of four deﬁni
tions: The system (F, G) is controllable if for any given nthorder polynomial 016(5)
there exists a (unique) control law 14 = —Kx such that the characteristic poly
nomial of F — GK is ac(s). From the results of Ackermann’s formula (see Appendix E), we have the fol
lowing mathematical test for controllability: (F, G) is a controllable pair if
and only if the rank of G is n. Deﬁnition I based on pole placement is a
frequency—domain concept. Controllability can be equivalently deﬁned in the
time domain. The system (F, G) is controllable if there exists a (piecewise continuous) control
signal u(t) that will take the state of the system from any initial state x0 to any
desired ﬁnal state x,» in a ﬁnite time interval. We will now show that the system is controllable by this deﬁnition if and only
if ('3 is fullrank. We ﬁrst assume that the system is controllable but that rank[G FG FZG F"‘1G] < 11. (D2)
We can then ﬁnd a vector v such that
v[G FG FZG ... F”‘1G] = 0, (D3) or
VG = vFG = vFZG = ... : vF"_1G = 0. (D4) The Cayley—Hamilton theorem states that F satisﬁes its own characteristic equa
tion, namely,
—F" = alF"_1 + azFH + ~  ~ + anI. (D5) Therefore,
—vF"G = alvF”_1G + ava"_2G + .  . + anvG = 0. (D6) Byinduction, vF”+kG = 0fork = 0, 1, 2, . . .,oer’” G = 0form = 0, 1, 2, . . .,
and thus 1
veFIG = v (I + Ft + 5W? + . . ) G = 0 (D7) for all t. However, the zero initialcondition response (x0 = 0) is t
x(t) =/ veF("’)Gu(r)dr
0 t
:5?th e'F’Gu(t)dt. (D.8)
0 Definition lil Section D.1 Controllability 851 Using Eq. (D.7), Eq. (D.8) becomes
t
vx(t) = / veF(’_’)Gu(t)dr = 0 (D9)
0 for all u(t) and t > 0. This implies that all points reachable from the origin are
orthogonal to v. This restricts the reachable space and therefore contradicts
the second deﬁnition of controllability. Thus if 6’ is singular, (F, G) is not
controllable by Deﬁnition 11. Next we assume that 6’ is fullrank but (F, G) is uncontrollable by Deﬁnition
11. This means that there exists a nonzero vector v such that tf
V/ eF(’f’T)Gu(1)dr = 0, (D10)
0 because the whole state space is not reachable. But Eq. (D.10) implies that
veF(’f_T)G = 0, 0 5 r 5 if. (D.l 1) If we set I = If, we see that VG = 0. Also, differentiating Eq. (DH) and
letting r = If gives vFG : 0. Continuing this process, we ﬁnd that VG = vFG = VFZG = ... = vFHG = 0, (D.12) which contradicts the assumption that G is fullrank. We have now shown that the system is controllable by Deﬁnition II if
and only if the rank of C” is n, exactly the same condition we found for pole
assignment. Our third deﬁnition comes closest to the structural character of
controllability: The system (F, G) is controllable if every mode of F is connected to the control
input. Because of the generality of the modal structure of systems, we will only treat
the case of systems for which F can be transformed to diagonal form. (The
doubleintegration plant does not qualify.) Suppose we have a diagonal matrix
Fd and its corresponding input matrix Gd with elements g). The structure of
such a system is shown in Fig. D.l. By deﬁnition, for a controllable system the
input must be connected to each mode so that the g, are all nonzero. However,
this is not enough if the poles (ii) are not distinct. Suppose, for instance, that
M = A2. The ﬁrst two state equations are then 561d = )‘le1d + gilt, X201 = szd + gzu. (D.13) 852 Appendix D Controllability and Observability Figure D.1 Block diagram of a system with a diagonal matrix H l D
g1 s4—A1
'— __
l‘ 8 2 _’ 1 g7] _.IY"A If we deﬁne a new state, S 2 gym — guy, the equation for S is 5: £25610]  5'15ch = gzkmd + 8281”  8111M: — gigzu = MS, (D14) which does not include the control n; hence g is not controllable. The point is
that if any two poles are equal in a diagonal Fd system with only one input, we
effectively have a hidden mode that is not connected to the control, and the
system is not controllable [Fig D.2(a)]. This is because the two state variables
move together exactly, so we cannot independently control x 1,1 and xzd. There
fore, even in such a simple case, we have two conditions for controllability: 1. All eigenvalues of F d are distinct. 2. No element of Gd is zero. Now let us consider the controllability matrix of this diagonal system. By direct
computation, 81 81M 81M 6’: £2 £212 g,1 gnkn gnkn nl
2 ~l
gl 0 1 A1 A1 A;
82 2 5
= . 1 Ag A2 . . (D15)
0 g" 1 A" xi A24
1* l l
M Y +1 s —l . " (b) (c) Figure D.2 Examples of uncontrollable systems Section D.1 Controllability 853 Note that the controllability matrix C’ is the product of two matrices and is
nonsingular if and only if both of these matrices are invertible. The ﬁrst matrix
has a determinant that is the product of the gi, and the second matrix (called
a Vandermonde matrix) is nonsingular if and only if the A, are distinct. Thus
Deﬁnition III is equivalent to having a nonsingular ('3 also. Important to the subject of controllability is the Popov—Hautus—Rosenbrock
(PHR) test (see Rosenbrock, 1970, and Kailath, 1980), which is an alternate
way to test the rank (or determinant) of G. The system (F, G) is controllable if
the system of equations vT[s1 — F G] = 0T (D16) has only the trivial solution VT = 0T, that is, if the following matrix pencil is
fullrank for all s rank[sI — F G]: n, (D.17) or if there is no nonzero vT such that1 VTF = svT, (D18)
vTG = o. (D.19) This test is equivalent to the rankofG test. It is easy to show that if such a
vector v exists, then ('3 is singular. For, if a nonzero v exists such that vTG = 0, then by Eqs. (D18) and (D.19) we have
VTFG = svTG = 0. (D20)
Then, multiplying by F G, we ﬁnd that
VTFZG = svTFG = 0, (13.21) and so on. Thus we determine that VTG 2 0T has a nontrivial solution, that 6’
is singular, and that the system is not controllable. To show that a nontrivial VT
exists if C is singular requires more development, which we will not give here
(see Kailath, 1980). We have given two pictures of uncontrollability. Either a mode is phys
ically disconnected from the input [Fig. D.2(b)], or else two parallel subsys
tems have identical characteristic roots [Fig D.2(a)]. The control engineer
should be aware of the existence of a third simple situation, illustrated in
Fig. D.2(c) namely, a polezero cancellation. Here the problem is that the
mode at s = 1 appears to be connected to the input but is masked by the zero
at s = 1 in the preceding subsystem; the result is an uncontrollable system. 1 VT is a left eigenvector of F. 854 Appendix D Controllability and Observability Definition IV This can be conﬁrmed in several ways. First let us look at the controllability
matrix. The system matrices are —1 0 —2
Fl1 ll Gl1 i
so the controllability matrix is e=[G FG]=[_12 3]] (D22) which is clearly singular. The controllability matrix may be computed using the
ctrb command in MATLAB [cc] = ctrb(F,G). If we compute the transfer function
from u to xz, we ﬁnd s—l I I
H(S):s+1<s—1)=s+1' (D23) Because the natural mode at s = l disappears from the input—output descrip
tion, it is not connected to the input. Finally, if we consider the PHR test, (D24) [SI—F G]=[SJr1 0 4]. —1 S—1 1 and let s = 1, then we must test the rank of 20—2
—101’ which is clearly less than 2. This result means, again, that the system is
uncontrollable. The asymptotically stable system (F, G) is controllable if the controllability
Gramian, the square symmetric matrix 01.” given by the solution to the follow
ing Lyapunov equation Fe, + egFT + GGT = 0, (D25) is nonsingular. The controllability Gramian is also the solution to the following
integral equation 00
e, = / eTFGGTe’FTdr. (D26)
0 One physical interpretation of the controllability Gramian is that if the input
to the system is white gaussian noise, then 0,. is the covariance of the state. The
controllability Gramian (for an asymptotically stable system) can be computed
with the following command in MATLAB, [cg] = gram(F,G). Section D.2 Observability 855 In conclusion, the four deﬁnitions for controllability—pole assignment (Deﬁni
tion I), state reachability (Deﬁnition II), mode coupling to the input (Deﬁnition
III), and controllability Gramian (Deﬁnition IV)—are equivalent. The tests for
any of these four properties are found in terms of the rank of the controllability
or controllability Gramian matrices or the rank of the matrix pencil [sI  F G].
If G is nonsingular, then we can assign the closedloop poles arbitrarily by state
feedback, we can move the state to any point in the state space in a ﬁnite time,
and every mode is connected to the control input.2 We have shown the latter
for diagonal F only, but the result is true in general. D.2 Observability So far we have discussed only controllability. The concept of Observability is
parallel to that of controllability, and all of the results we have discussed thus far
may be transformed to statements about Observability by invoking the property
of duality, as discussed in Section 7.5.1. The Observability deﬁnitions analogous
to those for controllability are as follows: 1. Deﬁnition I . The system (F, H) is observable if, for any n th—order polynomial
OMS), there exists an estimator gain L such that the characteristic equation
of the state estimator error is 059(5). 2. Deﬁnition [1. The system (F, H) is observable if, for any x(0), there is a ﬁnite
time I such that x(0) can be determined (uniquely) from u(t) and y(t) for
0 5 t 5 r. 3. Deﬁnition [11. The system (F, H) is observable if every dynamic mode in F
is connected to the output through H. 4. Deﬁnition IV. The asymptotically stable system (F, H) is observable if the
Observability Gramian is nonsingular. As we saw in the discussion for controllability, mathematical tests can be de
veloped for Observability. The system is observable if the Observability matrix H HF
o = _ (D27) HFn~l is nonsingular. If we take the transpose of (9 and let HT 2 G and FT = F, then
we ﬁnd the controllability matrix of (F, G), another manifestation of duality.
The Observability matrix, (9, may be computed using the obsv command 1n 2 WC have shown the latter for diagonal F only, but the result is true in general. 6 Appendix D Controllability and Observability Ackermann’s Formula
for Pole Placement MATLAB [oo] = obsv(F, H). The system (F, H) is observable if the following matrix
pencil is fullrank for all s: rank [511; F] = n. (D28) The observability Gramian, (9g, which is a symmetric matrix, and the solution
to the integral equation 00 T
(9g = / eTF HTHeTFdr, (13.29)
0 as well as the Lyapunov equation
FTog + ogF + HTH = 0, (D30) can also be computed (for an asymptotically stable system) using the gram
command in MATLAB [og] = gram(F’,H’). The observability Gramian has an in terpretatlon as the mformation matrix 1n the context of estimatlon. Given the plant and statevariable equation X=Fx+Gu, (El)
our objective is to ﬁnd a state feedback control law
it = —KX (E2)
such that the closed—loop characteristic polynomial is
(16(5) = det(sI — F + GK). (E3) First we have to select ac (s), which determines where the poles are to be shifted;
then we have to ﬁnd K such that Eq. (E3) will be satisﬁed. Our technique is
i based on transforming the plant equation into control canonical form.
We begin by considering the effect of an arbitrary nonsingular transforma
tion of the state, ‘ X = Ti, (E4)  where i is the new transformed state. The equations of motion in the new
coordinates, from Eq. (E4), are X=T§=FX+GM =FTi+Gu, (E.5)
i = T’IFTi—kT’lGu = Fi—l—Cu. (E6)
857 ...
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