A basic feedback control loop is depicted in Figure 4.1. It consists of a plant and a compensator through which an error signaleis processed to generate a control signalu.
This error signal is the difference between reference inputrand the measured outputy1 which is corrupted by noisen. Further, a disturbancedis present. The control system consists only of a single compensator and is hence referred to as asingle-degree-of-freedom control system. The only “degree of freedom” is the choice of how the error is mapped into a control signal.
Compensator Plant Disturbance
Model
r e
d
u y
n
−
Figure 4.1:Schematic of a single-degree-of-freedom control loop.
1Compatible dimensions are assumed without loss of generality. In case only the firstnrmeasured output signals are to be controlled, i. e.,r(t)∈Rnr is the reference for a subset of the outputsy(t)∈Rny with nr< ny, the signalris elongated asyref:=Inr
0
rand the error is calculated ase=yref−y.
A more general control loop is depicted in Figure 4.2. The control system there consists of two different subsystems. The reference signal is processed by a feedforward compensator and the measured signal is processed by a feedback compensator. While the first loop only operates on the difference of the measurement and the reference signal, the second loop processes both signals independently. The second loop is hence referred to as a two-degrees-of-freedom control loop. It is superior to a single-degree-of-freedom control loop which can be recovered as a special case if the feedforward and feedback compensator are selected to be equal.
Feedforward
Compensator Plant
Disturbance Model
Feedback Compensator r
d
u y
n
−
Figure 4.2: Schematic of a two-degrees-of-freedom control loop.
A yet more complex control loop is shown in Figure 4.3. It consists of five different subsystems. The reference signal is still passed through a feedforward compensator, but the output of this feedforward compensator is compared to the output of the feedback compensator to form a generalized error which is then processed by a cascade compensator to calculate a control signal. Additional contributions to the control signal are further directly calculated from the reference command and the measurement signal, bypassing the cascade compensator. The naming convention for these subsystems is inconsistent and ambiguous in the literature and actually not important. The important point is that the control loop still has only two degrees of freedom as there are only two independent signals: the reference command and the measurement.
Feedforward
Bypass Disturbance
Model Feedforward
Compensator Cascade
Compensator Plant
Feedback Bypass Feedback
Compensator
r y
−
n d
−
Figure 4.3: A more complex two-degrees-of-freedom control loop.
Hence, any five subsystems can be combined into the structure of Figure 4.2 without altering any properties of the control loop [e. g., Horowitz 1963, § 6.1, Lunze 1988, Sec. 2.3].
The following example illustrates this.
Example 4.1 (Two-Degrees-of-Freedom PID Control). To illustrate the use of differ-ent two-degrees-of-freedom control loop architectures, a standard proportional-integral-derivative (PID) controller [e. g., Åström & Hägglund 1995] is considered. In short, the proportional part is used to specify the bandwidth of the control loop, the integral part guarantees steady-state accuracy and the derivative part adds phase lead where required.
The PID controller is described as the series interconnection of a PI stageCPI(s) and a lead compensator stageCD(s) as
CPI(s) = kps+ki critical for robustness and to achieve a sufficient bandwidth for satisfactory disturbance rejection, but it is usually adverse for following step changes in the reference signal:
the well-known derivative kick causes overshoot. Assuming a controller was designed to satisfy requirements on robustness and disturbance rejection (e. g., using classical loopshaping as in Section 4.2.1), there are several ways of mitigating this effect by using a two-degrees-of-freedom control structure. Four completely equivalent possibilities are shown in Figure 4.4.
(a)Feedforward/feedback compensators (r, y).
PI(s)D(s) P
(c)Feedforward/feedback compensators (r, e).
PI(s) P
Figure 4.4: Some equivalent two-degrees-of-freedom PID controllers.
The loop shown in Figure 4.4a uses the structure of Figure 4.2. The controller is imple-mented such that the feedback properties are determined by the full PID controller, while the feedforward compensator lacks the derivative part. The configuration in Figure 4.4b is sometimes referred to as using a prefilter. The lead compensator dynamics are canceled in the forward path by first processing the reference signal through a lag compensator. In Figure 4.4c, the additional feedforward compensator is used to subtract the control signal that corresponds to the derivative kick. In Figure 4.4d, the controller is implemented as a cascade in which only the output measurement is processed through the lead compensator and the resulting control signal bypasses the PI-part. All of these loops represent the exact same two-degrees-of-freedom controller and yield identical input-output maps.
In this example, the only purpose of the two-degrees-of-freedom structure is to avoid the derivative kick. Similar phenomena are also common with more complex high-order controllers, although they are far less obvious in that case. The feedforward path of a properly designed two-degrees-of-freedom controller will therefore typically lack the lead action of the feedback path or adequately compensate for it. Two-degrees-of-freedom controllers can also be used for a variety of different purposes. For example, it is possible to achieve different feedforward and feedback bandwidths or to apply specific signal modifications such as notching out frequencies in the feedforward path. 4 In this thesis, the structure of Figure 4.2 is used for all control systems. The mathematical model to describe this structure is shown in Figure 4.5. The plant model is denotedP (ny
outputs,nu inputs) and a disturbance modelPd (ny outputs,nd inputs) is included. The disturbance model can encompass output disturbances (Pd=Iny) but also provides the possibility that disturbances are filtered through part of the plant dynamics. In this case, a common realization ofP andPdis realistic, with the limiting case of load disturbances that occur directly at the plant input (Pd=P). Feedforward compensatorCFF (nu outputs, nr inputs) and feedback compensatorCFB (nu outputs,ny inputs) are subsumed in the controllerK, i. e.,
u=
CFF CFB
| {z }
K
r
−y
. (4.1)
CFF P
Pd
CFB
r
d
u y
− K n
Figure 4.5:Block diagram of two-degrees-of-freedom control loop.
One particular advantage of this structure is that feedback properties solely depend onCFB and that the single-degree-of-freedom case is trivially recovered by settingCFF=CFB=:C.
This simplifies notation while the discussion of feedback properties remains valid both for two-degrees-of-freedom and single-degree-of-freedom control systems.
The input-output maps for a control loop with such a controller are
y= How controllers are designed to specifically adjust these input-output maps is discussed in the subsequent sections. First, the terminology of sensitivity functions is introduced. The el-ementary sensitivity functions of a feedback loop areoutput sensitivitySo= (I+P CFB)−1 andinput sensitivitySi= (I+CFBP)−1. Further, thecomplementary output sensitivity To= (I+P CFB)−1P CFBandcomplementary input sensitivityTi= (I+CFBP)−1CFBP are defined based on the relationsSo+To=I andSi+Ti=I. If SISO loops are consid-ered,Si =So=:S andTi=To=:T. The concatenationSoPd of the disturbance model with the output sensitivity is referred to asdisturbance sensitivity. IfPd=P, i. e., load disturbances at the plant input are assumed, the additional relationSoP =P Si holds andSiCFBP is the same asTi. All of these sensitivity functions depend solely on the feedback compensator. The concatenationSiC•• of a compensator (either feedforward or feedback) and the input sensitivity function is referred to ascontrol sensitivity. In a two-degrees-of-freedom setting, reference tracking is governed by areference transmission functionR=SoP CFF such that atracking error function Sr=Inr
0
−R can also be defined.2 For a single-degree-of-freedom control loop, Equations (4.2) simplify to
y=
In this case, the relation SiC = C So for the control sensitivity holds and tracking is governed by the complementary sensitivity function.
2Again, it is without loss of generality assumed that the firstnroutputs are to be tracked.