Part III Performance Limitations in Feedback Control with
8.3 Deterministic SISO Performance Limitations of RRS
8.3.1 Sensitivity Integrals–Frequency Domain Approach
Theorem 8.1 (Poisson Integral for Sensitivity Function—SISO Case). Consider the feedback system shown in Figure 8.1, and let S(s) be the sensitivity function defined in (8.2). Assume the open loop system L(s) = G(s)C(s) has a set of poles {pi : i = 1, . . . , np} ∈ C+ (open right hand side of the complex plane). Then L(s)can be factorized as
L(s) = ˜L(s)BS−1(s),
156 8 Linear Performance Limitations
whereL(s)˜ is a proper rational transfer function with no poles inC+, and the all-pass factor or Blaschke product given by
BS(s) = 'np i=1
pi−s pi+s.
Provided the closed loop system is stable, then for each zeroq=σq+jωq ∈C+
(open right hand side of the complex plane) of L(s), the following integral constraint holds:
∞
−∞
ln|S(jω)| σq
σq2+ (ωq−ω)2dω=πln|BS−1(q)|. (8.12)
Proof. See [89, 199]
This result represents a weighted balance of area under the curve of ln|S(jω)|. If, for simplicity, the open-loop system is stable, then the right- hand side of (8.12) is zero. This means that if the output disturbance is to be attenuated, i.e. ln|S(jω)| < 0 (or equivalently |S(jω)| < 1) in a range of frequencies ω ∈ Ω, then there must be amplification of disturbances at frequencies outsideΩ,i.e. forω /∈ Ω, ln|S(jω)| >0 (or |S(jω)|>1) so the area balance is zero. Furthermore, due to the weighting factor in the above integral, this balance of area has to be achieved over a limited band of frequen- cies, which depend on the position of the MNP zeros. For example, Figure 8.7 shows the form of the weighting function for a real NMP zero. In the case of unstable systems, the trade off is worse because the right-hand side is positive.
ω
1 σq
1 2σq
σ2q
Fig. 8.7.Weighting function of the sensitivity integral constraint.
Let us see what implications the above result have in the design of a control system. Consider a stable system G(s) with a real NMP zeroq. Then, for an interval [ω1, ω2] withω2> ω1>0, let us consider the integral of the weighting function, i.e.,
8.3 Deterministic SISO Performance Limitations of RRS 157
Θq(ω1, ω2) = ω2
ω1
q q2+ω2dω
= arctanω2
q −arctanω1
q .
(8.13)
Now, suppose that the feedback loop is to be designed to achieve
|S(jω)| ≤α1<1, ∀ω∈Ω1= [ω1, ω2]. (8.14) Dividing the range of integration in (8.12), and using the inequality (8.14) and also the fact that|S(jω)| ≤ S(jω)∞ for allω, we obtain that
lnα1 Θq(ω1, ω2) + lnS(jω)∞[π−Θq(ω1, ω2)]≥0. (8.15) By exponentiating both sides of (8.15), it then follows that
S(jω)∞≥ 1
α1
πΘq(ω1,ω2)
−Θq(ω1,ω2)
(8.16) Thus, the right-hand side of (8.16) is a lower bound on the sensitivity peak.
It is immediate from (8.16) that the lower bound on the sensitivity peak is strictly greater than one: this follows from the fact that α1 < 1 and Θq(ω1, ω2)< π. Furthermore, the more the sensitivity is pushed down, i.e., the lower is α1, and the bigger is the interval [ω1, ω2], then the biggerS(jω)∞ will be at frequencies outside that interval.
Expression (8.16) can also be used to analyse the worst location of the interval [ω1, ω2] of sensitivity reduction with respect to the NMP zero. This occurs at the logarithmic average of the frequenciesω1 andω2,i.e.
q=√
ω1ω2. (8.17)
Expression (8.16) does not give any indication about the location of the S(jω)∞, i.e. the frequency at which this maximum is located. However, for robustness purposes and high-frequency noise rejection, it follows from (8.2) that it is necessary to constrain the complementary sensitivity |T(jω)| at high frequency. This can be characterised by the following constraint:
|T(jω)| ≤α2<1 ∀ω∈Ω3= [−∞,−ω3]∪[ω3,∞]. (8.18) Note that (8.18) implies |S(jω)| ≤ 1 +α2, ∀ω ∈ Ω3. Therefore, using this additional information, and following a similar procedure as above, we obtain
S(jω)∞≥ 1
α1
Γq(ωΓq(ω3)+Γq(ω2)−Γq(ω1)
1)−Γq(ω2)
× 1
1 +α2
Γq(ω3)+Γq(ωπ−Γq(ω3)
1)−Γq(ω2)
(8.19)
158 8 Linear Performance Limitations where, Γq(ωi) = arctanωiσ−ωq
q , for i = 1,2,3. From this last expression it is clear that the lower the bandwidth ofT(s) and the smallerα2, the bigger the contribution to the lower bound on S(jω)∞, and more importantly, that the maximum sensitivity peak will be in the range of frequencies
ΩM = (0, ω1)∪(ω2, ω3) (8.20) Figure 8.8 shows a schematic of the situation described above. This type of result is typical of classical linear controllers, PID andH∞—see [199].
Fig. 8.8.Sensitivity trade-off.
Note that the results discussed above are very general; the only thing we have asked from the controller is to be stabilising,i.e.be designed such that the closed loop is stable.
The above description of the disturbance attenuation problem has been formulated from a deterministic point of view. The use of frequency response is particularly attractive to consider sinusoidal disturbances.The result depicted in Figure 8.8 can be interpreted as a robustness trade-off with respect to the knowledge of the frequency of the disturbance.Indeed, if the frequency of the disturbance were known exactly, the sensitivity could be reduced to zero (in theory because no input constraints or limits apply in the above analysis).
For example, we could use the Internal Model Principle (see [89]) to design a controller that completely rejects that sinusoidal disturbance. However, if the
8.3 Deterministic SISO Performance Limitations of RRS 159 frequency of the disturbance is not known exactly or the disturbance energy is distributed over a range of frequencies, the reduction of the sensitivity should be considered over that range of frequencies. The price to pay for doing this is an increase of sensitivity outside this range of reduction, and the risk of disturbance amplification if the disturbance is indeed outside the reduction range.
The above discussion evidences limitations of controllers that seek large sensitivity reduction at a range of frequency when there is a NMP zero and uncertainty about the actual frequencies of the disturbance. From the above analysis, it follows that this becomes particularly critical when the reduction is sought close to the frequency of an MNP zero. We will next see how this affects the design of RRS.
8.3.2 Performance Trade-offs of Non-adaptive Feedback