This subsection starts this book’s main journey through results concerning complementary mappings, by which we mean mappings relating signals of the loop and such that their sum is a constant mapping. We consider here the sensitivity function,S, and the complementary sensitivity function, T, of the classical unity feedback configuration of linear feedback control theory, represented in Figure 3.2.
b - i- -b
6
FIGURE 3.2. Feedback control system.
These two mappings satisfy the complementarity constraint
S(s) +T(s) =1 . (3.7)
This represents an algebraic trade-off (Freudenberg and Looze, 1985), since it constrains the properties of the closed-loop system at each frequency.
It implies, for example, that the magnitudes ofS and T cannot both be smaller than1/2at the same frequency (Doyle et al., 1992).
Assuming minimality,Shas zeros at the unstable open-loop plant poles andT has zeros at the nonminimum phase open-loop plant zeros. These are generally known asinterpolation constraints. More precisely, let the open-
52 3. SISO Control loop system beL and assume that it is free of unstable hidden modes2. Then, the sensitivity and complementary sensitivity functions have the forms
S(s) = 1
1+L(s) , and T(s) = L(s)
1+L(s) . (3.8) The following lemma formalizes the interpolation constraints thatSand T must satisfy at the CRHP poles and zeros ofL.
Lemma 3.1.3 (Interpolation Constraints).Assume that the open-loop sys- temL is free of unstable hidden modes. Then S andT must satisfy the following conditions.
(i) Ifp∈ +is a pole ofL, then
S(p) =0, and T(p) =1. (3.9) (ii) Ifq∈ +is a zero ofL, then
S(q) =1, and T(q) =0. (3.10)
Proof. Follows immediately from (3.8).
This result states that the CRHP zeros ofS and T are determined by those ofL−1 andL. We introduce for convenience the following notation for the ORHP zeros ofSandT.3
ZS,{s∈ +:S(s) =0},
ZT ,{s∈ +:T(s) =0}. (3.11)
Lemma 3.1.3 then establishes the fact thatZSis the set of ORHP poles ofL andZT is the set of ORHP zeros ofL. In other words, we have translated the open-loop characteristics of instability and “nonminimum phaseness”
into properties that the functionsSandTmust satisfy in the CRHP. In par- ticular, note that if the open-loop system is formed by the cascade of the plant,G, and the controller,K, i.e.,L= GK, then the constraints imposed by CRHP poles and zeros ofGholdirrespectiveof the choice ofK, provided thatLhas no unstable zero-pole cancelation.
As a first extension of Bode’s results to integrals on sensitivity functions, we will derive Horowitz’s formula (3.1) and a complementary result forT using Theorem 3.1.1 and Corollary 3.1.2.
2Recall from Chapter 2 that is free of unstable hidden modes if there are no cancela- tions of CRHP zeros and poles between the plant and controller whose cascade connection forms .
3In the sequel, when defining a set of zeros of a transfer function, the zeros are repeated according to their multiplicities.
3.1 Bode Integral Formulae 53 Example 3.1.1 (Horowitz’s formula forS). Suppose thatLis aproper ratio- nal function without poles in the ORHP. Assume that the closed-loop system is stable (i.e., the numerator of1+Lis Hurwitz andL(∞)6= −1) and con- sider the corresponding sensitivity functionS. It follows thatZSin (3.11) is empty and that the functionH=logSsatisfies assumptions (i) and (ii) of §3.1.1. Thus, we can apply Theorem 3.1.1 to the real part, log|S(jω)|, to obtain
Z
0
[log|S(jω)|−log|S(j∞)|]dω= −π 2 Res
s= logS(s).
If we further assume that the open-loop plant hasrelative degree two or more, then4S(j∞) =1and Ress= logS(s) =0. This recovers Horowitz’s
area formula given in (3.1). ◦
Example 3.1.2 (Horowitz’s formula forT). Assume thatL(s)is aminimum phase rational function such thatL(0)6=0. Then, if we consider the comple- mentary sensitivity functionT, we have thatZT in (3.11) is empty. Next, notice that the functionT(1/ξ)can be written as
T(1/ξ) = 1
1+1/L(1/ξ) . (3.12) Then, under the assumption of closed-loop stability (which impliesL(0)6=
−1), the function H(1/ξ) = logT(1/ξ)satisfies conditions (i) and (ii) of
§3.1.1. We thus obtain, from Corollary 3.1.2, Z
0
[log|T(jω)|−log|T(0)|]dω ω2 = π
2 1 T(0)lim
s 0
dT(s) ds .
If we further assume thatLhas at least two integratorsthenT(0) = 1and lims 0dT(s)/ds=0, which yields
Z
0
log|T(jω)|dω ω2 =0 .
◦ It is instructive at this point to reflect on the complementarity of these formulae forSandT, as well as the hypotheses required on the open-loop systemLto derive them. Under appropriate conditions, SandT exhibit symmetry with respect to frequency inversion (Kwakernaak, 1995). For example, the condition thatLbe proper andL(j∞)6= −1(or alternatively, thatLbe strictly proper) imply the analyticity ofSand logSat infinity. On the other hand, the requirement thatL(0)6= {0,−1}(or thatLhas poles at zero) gives the analyticity ofT and logT at zero.
4See Example A.9.2 in Appendix A.
54 3. SISO Control The extensions to unstable open-loop plants and to plants with time de- lay were derived by Freudenberg and Looze 1985; 1987. The complemen- tary result forT was obtained by Middleton and Goodwin (1990). These two area formulae will be given next under the name of theBode Integrals forSandT. We choose these names since the results are natural extensions of Bode’s integral (3.2) to the sensitivity and complementary sensitivity functions.
Theorem 3.1.4 (Bode Integral forS). LetSbe the sensitivity function de- fined by (3.8). Let{pi:i=1, . . . , np}be the set of poles in the ORHP of the open-loop systemL. Then, assuming closed-loop stability,
(i) ifLis a proper rational function, Z
0
log
S(jω) S(j∞)
dω= π 2 lim
s
s[S(s) −S(∞)]
S(∞) +π
np
X
i=1
pi; (3.13) (ii) ifL(s) =L0(s)e−sτ, whereL0(s)is a strictly proper rational function
andτ > 0,
Z
0
log|S(jω)|dω=π
np
X
i=1
pi. (3.14)
Proof. Consider the contour of Figure 3.3, where the indentationsC1, C2, . . . , Cnp into the right half plane avoid the branch cuts of logS corre- sponding to the zeros ofS(see Figure A.19 in Appendix A). LetC0consist of the remaining portions of the imaginary axis and letCRbe the semicir- cle of radiusR.
FIGURE 3.3. Contour for Bode Sensitivity Integral.
Since log[S(s)/S(∞)]is analytic on and inside the total contour, denoted byC, then, by Cauchy’s integral theorem,
lim 0 R
Z
C
log S(s)
S(∞)ds=0 .
3.1 Bode Integral Formulae 55 Note that the computation of the portions of the integral on the imaginary axis and on the indentations is the same for both cases (i) and (ii) (the only distinction being that in (ii) we have thatS(∞) =1). However, the integral on the large semicircle requires a different analysis in each case.
The portion of the integral onC0gives5
lim 0 R
Z
C0
log S(s)
S(∞)ds=2j Z
0
log
S(jω) S(j∞)
dω .
For the integral on the branch cut indentations we have, using (A.85) of Appendix A,
np
X
i=1
Z
Ci
log S(s)
S(∞)ds= −j2π
np
X
i=1
Repi= −j2π
np
X
i=1
pi,
where the last equality follows since complex zeros must appear in conju- gate pairs.
It remains to compute the integral onCRfor each of the cases (i) and (ii).
(i) IfLis a proper rational function, the assumption of closed-loop sta- bility guarantees that the function logS(s)is analytic at infinity. We can then use Example A.9.2 of Appendix A to compute the integral onCRin the limit whenR→ ∞as
Rlim
Z
CR
log S(s)
S(∞)ds=jπRes
s= log S(s) S(∞)
=jπ 1 S(∞) lim
s s[S(∞) −S(s)],
(3.15)
where we have used (A.84) from Appendix A.
(ii) IfL =L0e−sτ, withτ > 0andL0strictly proper, we have that there is anr > 0such that, for allswith Res≥0and|s|> r, the modulus of L(s) satisfies|L(s)| < 1. Then, a similar use of the expansion of log(1+s)as in Example A.9.2 of Appendix A yields
logS(s) = −L0(s)e−sτ+ L20(s)e−2sτ 2 +ã ã ã ,
in{s:|s|> rand Res≥0}. SinceL0 has relative degree at least one, then the above expansion has the form
logS(s) = c−1
s e−sτ+ã ã ã , in {s:|s|> rand Res≥0}.
5Under the convention that the phase of a negative real number is equal to for ≥
and for .
56 3. SISO Control Then, using the result in Example A.4.3 of Appendix A, we finally have that
Rlim
Z
CR
logS(s)ds=0 .
The result then follows by combining the integrals over all portions of the total contour for each of the cases (i) and (ii).
Theorem 3.1.4 shows that the presence of open-loop unstable poles fur- ther restricts the compromise between areas of sensitivity attenuation and amplification imposed by the integral. Specifically, since the termPnp
i=1pi
is nonnegative, the contribution to the integral of those frequencies where
|S(jω)|< 1is clearly reduced ifLhas unstable poles. Moreover, the farther from thejω-axis the poles are, the worse will be their effect.
The first term on the RHS of (3.13) appears only if the plant has relative degree0or1and no time delay, and admits an interpretation in terms of the time response ofL. Indeed, ifl(t)denotes the response of the plant to a unitary step input, we can alternatively write
slim
s[S(s) −S(∞)]
S(∞) = lim
s s
L(∞) −L(s) 1+L(s)
= − l(0˙ +) 1+L(∞) ,
where the last step follows from the Initial Value Theorem of the Laplace transform (e.g., Franklin et al., 1994). Thus, ˙l(0+)is the initial slope ofl(t), as illustrated in Figure 3.4.
arctan ˙
FIGURE 3.4. Step response of a plant of relative degree .
If the plant has relative degree0or1, a large positive value for ˙l(0+)will ameliorate the integral constraint forS. This, as we will see in more detail in §3.1.3, is justified by the fact that a larger value for ˙l(0+)is associated with larger bandwidth of the system.
3.1 Bode Integral Formulae 57 If the plant has relative degree greater than1, then ˙l(0+)vanishes, and (3.13) reduces to (3.14). We will see in §3.1.3, however, that larger band- width still alleviates the trade-offs induced by this integral constraint.
Naturally, an equivalent relation holds for the complementary sensitiv- ity function, as stated in the following theorem.
Theorem 3.1.5 (Bode Integral forT). LetT be the complementary sensi- tivity function defined by (3.8). Let{qi :i =1, . . . , nq}be the set of zeros in the ORHP of the open-loop systemL, and suppose thatL(0)6=0. Then, assuming closed-loop stability,
(i) ifLis a proper rational function Z
0
log
T(jω) T(0)
dω ω2 = π
2 1 T(0)lim
s 0
dT(s) ds +π
nq
X
i=1
1 qi
; (3.16) (ii) ifL(s) =L0(s)e−sτ, whereL0(s)is a strictly proper rational function
andτ > 0, Z
0
log
T(jω) T(0)
dω ω2 = π
2 1 T(0) lim
s 0
dT(s) ds +π
nq
X
i=1
1 qi
+π
2τ . (3.17) Proof. The proof follows along the same lines as that of Theorem 3.1.4 with inverted symmetry in the roles played by the pointss=0ands=∞. The interested reader may find the details in Middleton (1991).
It is interesting to note that the integral forT is in fact the same as that forSunder frequency inversion. Indeed, by lettingν=1/ω, we can alter- natively express
Z
0
log
T(jω) T(0)
dω ω2 =
Z
0
log
T(1/jν) T(0)
dν .
Accordingly, as seen in (3.16) and (3.17), nonminimum phase zeros ofL play forT an entirely equivalent role to that of unstable poles forS; i.e., they worsen the integral constraint. In this case, zeros in the ORHP that arecloserto thejω-axis will pose a greater difficulty in shapingT.
The first term on the RHSs of (3.16) and (3.17) has an interpretation in terms of steady-state properties of the plant. Consider the feedback loop shown in Figure 3.5. The Laplace transform of the error signale=r−yis given by
E(s) =
1−T(s) T(0)
R(s).
Hence, ifr is a unitary ramp, i.e.,R(s) = 1/s2, then the corresponding steady-state value ofe(t)(see Figure 3.6) can be computed using the Final
58 3. SISO Control
i b
b
i b
- -
6 - -
- 6
-
FIGURE 3.5. Type-1 feedback system.
Value Theorem (e.g., Franklin et al., 1994) and L’Hospital’s rule (p. 260 Widder, 1961, e.g., ) as
ess= lim
t e(t)
= lim
s 0sE(s)
= lim
s 0
1− T(s) T(0) s
= − 1 T(0)lim
s 0
dT(s) ds .
Thus the constant1/T(0)dT/ds|s=0 plays a role similar to that played by the reciprocal of the velocity constant in a Type-1 feedback system (e.g., Truxal, 1955, p. 286). Consequently, the corresponding term on the RHSs of (3.16) and (3.17) can ameliorate the severity of the design trade-offonly if the steady-state error to a ramp input is large and positive, so that the output lags the reference input significantly.
FIGURE 3.6. Steady-state error to ramp.
Finally, the last term on the RHS of (3.17) shows that the trade-off also worsens if the plant has a time delay. In a sense this is not surprising, since
3.1 Bode Integral Formulae 59 a time delay may be approximated by nonminimum phase zeros6, which already appear in connection with the Bode integral forT.
In the following section we will discuss more detailed design implica- tions of the integral constraints given by Theorem 3.1.4 and Theorem 3.1.5.