The filtering sensitivitiesPandMare not the direct counterpart of the con- trol sensitivitiesSandT. This is because the estimation problem depicted in Figure 7.1, when set in a control framework, is more general than the one-degree of freedom control loop whereSandTare defined. Indeed, the
“filtering” versions ofS andT would correspond to the output-filtering
170 7. General Concepts sensitivities, i.e., the sensitivities arising in the problem of estimating the system noise-free output (z=y−w), as discussed in Example 7.2.1.
We will next derive a complementarity constraint for linear feedback control that parallels the filtering constraint given in (7.6). Consider the control loop shown in Figure 7.6, whereGis the plant transfer function given by
G,
Gzv Gzu
Gyv Gyu
, (7.12)
andKis a controller that stabilizes the feedback loop.
j j
b b
b -
- ? ?
-
-
FIGURE 7.6. Control Loop.
This configuration is standard, e.g., in theH control literature.2 The plantGhas two sets of inputs: the external inputs,v, and the control in- puts,u. Also, it has a set outputs, y0, that are available to the controller Kafter being corrupted with output disturbances,d, and sensor noise,n, and a set of signals of interest,z, which are generally the signals to be controlled or regulated.
The “classical” sensitivity and complementary sensitivity,SandT, are defined for a simpler unity feedback control loop (see Figure 3.2 in Chap- ter 3), where they are directly connected with performance, disturbance rejection and robustness properties (cf. §2.2.2 in Chapter 2). However, as we see next, they can also be defined for the feedback loop in Figure 7.6.
Introduce first the notationHba to indicate the total mapping – possi- bly nonphysical – from signalato signalb, i.e., after combining (adding, composing, etc.) all the ways in whichais a function ofband solving any feedback loops that may exist.3Then, we have that
S=Hyd = (I+GyuK)−1,
T = −Hyn=GyuK(I+GyuK)−1. (7.13) As seen in Chapter 3,SandTsatisfy the complementarity constraint
S+T =I.
2In the standard configuration, in fact, and are absorbed into . Here we choose to draw them explicitly to emphasize the way in which they affect the output .
3Cf. the notation introduced at the end of §7.1.
7.2 Sensitivity Functions 171 This relation involves mappings from external inputs that are directly in- jected into the loop (i.e., without intermediate dynamics), to system sig- nals that are fed-back in the same loop (see Figure 7.6). A more general result, comparable to the filtering relation given in (7.7), can be devel- oped for the configuration of Figure 7.6. This structural constraint involves mappings from external inputs that are injected dynamically in the loop, to some internal variables, e.g., the system (combination of) states,z, which are not directly fed-back. This result is stated below.
Theorem 7.2.2. Consider the general control loop of Figure 7.6, where the plant Gis given by (7.12). The total mappings, Hzv and Hzd, from the external inputsv andd to the internal variablesz, satisfy the following structural relation:
Hzv+ (−Hzd)Gyv=Gzv. (7.14) Proof. From Figure 7.6, and the definitions in (7.12) and (7.13), we have, in the Laplace transform domain
Y0=SGyvV+TD, and
Z=GzvV+GzuK(D−Y0)
= (Gzv−GzuKSGyv)V+GzuKSD. (7.15) Then
Hzv=Gzv−GzuKSGyv, Hzd=GzuKS,
from which (7.14) follows.
Notice that, when considering mappings toz, the sensor noise, n, in Figure 7.6 can be set to zero since its influence onzis the same as that ofd.
A complementarity constraint that parallels (7.6) can be readily obtained by multiplying through both sides of (7.14) byG−1zv and making similar definitions to those in (7.4).
The relation given in (7.14) is clearly analogous to that of (7.7). This in- dicates that the filtering problem considered in §7.2 fits into the structure of the control configuration of Figure 7.6. In the search for generality, how- ever, we have gone too far: we now need to select the particular choices of GandKthat will give the exact control counterpart of (7.7).
Making the choices of inputs and outputs as shown in Figure 7.7, we conclude that the controllerKcorresponds to the filterF, whilst the plant Gcorresponds to the matrix
Gf,
Gzv −I Gyv 0
.
172 7. General Concepts
j
b - b
- - ?
-
˜
FIGURE 7.7. Equivalent Filtering Loop.
Observe that setting the entry (2,2) ofGfequal to zero amounts to open- ing the loop (S=I, T =0in (7.13)), but this is the way in which we have set up the filtering problem in §7.2. In the context of bounded error fil- tering, however, we will soon see thatF satisfies conditions that ensure boundedness of ˜z. There is then, ideally, no need for feedback if these con- ditions hold.
Although a feedback loop can be made explicit by considering the filter driven by the innovations, as we have seen in §7.2.1, note that the mere presence of feedback does not make the observer a closed loop in general (i.e., when the signal to be estimated is not just the disturbance-free out- put, i.e.,z6=y−w). Indeed, it was shown by Bhattacharyya (1976) that an observer is a closed-loop system if and only if it can be expressed as a sys- temdriven by the estimation errorz˜=z− ^z. Moreover, Bhattacharyya (1976) proved that an observer must be a closed-loop system if it is to provide observer action despite arbitrarily small perturbations of the system state space matrices.