Part III Performance Limitations in Feedback Control with
8.1 Introduction to Fundamental Limitation in Feedback Control
Performance limitation analysis is an important part of any control system design process. This analysis reveals, a priori, whether the control objectives are achievable or not and whether there are fundamental or unavoidable de- sign trades-offs. These limitations and trade-offs, arise due to dynamic and structural characteristics of the feedback system, and are expressed in terms of functions that quantify system performance in various senses. Performance limitation analysis also has implications in determining applicability, i.e. in deciding whether a particular control structure or strategy is worth applying to a particular problem. In cases where the desired performance is unachiev- able, the result of this analysis may suggest how the system can be rearranged (by adding sensors and actuators, or even by re-designing the controlled sys- tem) so as to relax the trade-offs.
Performance limitation analysis generally yields results that can be clas- sified into the following two groups:
• Limits on performance that hold for all possible designs in a given control architecture.
• The best achievable performance for a particular scenario.
The first group of results has evolved from the work of Bode during the 1940s [35]. This has become the foundation of the frequency-domain approach to the analysis of fundamental limitations in feedback control system design.
The frequency domain approach has, since then, been thoroughly studied and extended by the work of various researchers, including Horowitz [108], Freudenberg and Looze [74], and others. The main idea behind this approach is that the closed-loop transfer functions of a linear feedback system (e.g.the sensitivity and complementary sensitivity functions) are analytic in the closed- right hand side of the complex plane if the system is closed-loop stable. This property, combined with interpolation constraints that must be satisfied at certain frequencies due to the structure of the feedback loop, leads to integral
8.1 Introduction to Fundamental Limitation in Feedback Control Systems 147 relations, which can be used to quantify fundamental performance limitations and design trade-offs.
The second group of results is often chosen to be a special case that can be used as a benchmark against which other more realistic designs are compared.
One of these methods—reviewed in a stochastic form in this chapter—relies on cheap limiting optimal control. The key idea is that the optimal controller will attain its best performance if the control effort is not penalized in the cost function to be minimised. Allowing arbitrarily large control signals is ob- viously impractical. However, the fact that, if under these conditions, the cost associated with the optimisation cannot be reduced to zero exposes the pres- ence of fundamental limitations to any type of controller. This approach has been extensively studied for linear systems [128, 129, 71, 194, 184, 48] and has been extended to unconstrained nonlinear systems [201]. Also, a characteri- sation for stable-SISO systems tracking and rejecting step signals with input constraints can be found in [176, 177].
Whichever approach is used to study performance limitations, the knowl- edge of these limitations allows the designer to judge a potential design before going deep into it. This can save incurring expenses in terms of design time or effort.
To introduce the topic, let us consider a simple motivation example. Con- sider the linear feedback system shown in Figure 8.1, in which G(s) is the transfer function of the plant or system to be controlled, and C(s) is the transfer function of the controller.
r C(s) G(s)
di do
u y
n –
Fig. 8.1.Single-loop output feedback control system.
The scheme shown in Figure 8.1 can be taken as a simplified version of the block diagram of a control system of a stabiliser. Indeed, under linearity assumptions, we can consider, for the purpose of analysis, that the wave- induced motion is an output disturbance. Then, for fin stabilisers, we could have that
• ris the desired roll angle,i.e.zero.
• yis the roll angle φand/or its time derivatives.
• do is the wave-induced motion in roll and its time derivatives.
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• di accounts for unmodelled dynamics and other effects diminishing the control action such as local flows due to the wave-particle orbits.
• uis the desired mechanical angle of the fins.
The double objective of rudder stabilisers can be considered as a single-input two-output (SITO) problem as illustrated in Figure 8.2. In this case,
• r= [0, ψd]t is the of vector desired roll and yaw angle .
• y= [φ, ψ]tis the vector of roll and yaw angles; this may also include their time derivatives.
• do = [dφ, dψ]t is the wave-induced motion in roll (and yaw) and its time derivatives.
• di is similar to the case of fins, accounting for unmodelled dynamics and other effects diminishing the control action.
• u=αd is the desired mechanical angle of the rudder.
0
ψd
φ αd α
Gα(s)
ψ dφ
dψ
Gφα(s)
Gψα(s) Cψ(s)
Cφ(s)
Fig. 8.2.Simplified rudder stabiliser control scheme
The transfer functionsCφ(s) and Cψ(s) represent the roll and yaw con- trollers respectively. The transfer function Gα(s) describes the dynamics of the steering machinery, i.e.it maps the desired rudder angleαd into the ac- tual rudder angleα. The transfer functionsGφα(s) andGψα(s) map the actual rudder angle into the roll and yaw angles respectively.
For ease of exposition, let us introduce the topic of performance limitations using SISO systems. The same ideas, however, extend to the multivarible case, and as we will see, conclusions drawn from a SISO analysis are also relevant to rudder stabilisers.
One way of interpreting the effect of control is by analyzing the properties of the feedback structure depicted in Figure 8.1 in the frequency domain.
Indeed, using the Laplace transform, we can establish the effect of the different magnitudes on the output of the system via
8.1 Introduction to Fundamental Limitation in Feedback Control Systems 149 y(s) =T(s)r(s) +S(s)do(s) +Si(s)di(s) +Sn(s)n(s). (8.1) For the feedback system shown in Figure 8.1, it follows that:
S(s) = 1
1 +G(s)C(s) T(s) = G(s)C(s) 1 +G(s)C(s) Si(s) = G(s)
1 +G(s)C(s) Sn(s) = −G(s)C(s) 1 +G(s)C(s).
(8.2)
The transfer functionS(s) is called the sensitivity transfer function, and maps the output disturbance into the output. The functionT(s) is called the complementary sensitivity transfer function, and maps the reference into the output. The functionSi(s) is called theinput sensitivity transfer function and maps the input disturbance into the output. Finally, the functionSn(s) is the transfer function that maps the measurement noise into the output.
Having defined the sensitivity functions, let us qualitatively discuss the most basic performance limitations that are associated with the feedback sys- tem through a simple example.
Example 4 (Perfect control and basic design trade-offs) Perfect con- trol would be achieved if the output y (variable of interest) followed the ref- erence r as closely as possible for all frequencies in a desired range Ω (i.e., ω∈Ω) despite the disturbances present on the system; namely,do,di andn.
This ambitious goal for the design of the controllerC(s) can be quantita- tively stated by the following specifications in the frequency domain:
|S(jω)|≈0, |T(jω)|≈1, |Si(jω)|≈0, |Sn(jω)|≈0 ∀ω∈Ω.
However, from the definitions of the sensitivity functions (cf. (8.2)) it follows that it is impossible, in general, to satisfy such specifications. Indeed, from (8.2), we have that
S(s) +T(s) = 1, Si(s) =G(s)S(s), Sn(s) =−T(s). (8.3) Therefore, for example, it is impossible to reduce the effect of the mea- surement noise on the output, and at the same time obtain a good tracking response (makingy followr) because|Sn(jω)|=|T(jω)|.This is a fundamen- tal limitation associated with the structure of the control system. This could be alleviated by pre-filtering the measurement. This is just one example of how to modify the system to relax the limitation.
On the other hand, the algebraic constraintS(s) +T(s) = 1, indicates that if we reduce|T(jω)|to avoid the influence of the noisenat some frequencies, then |S(jω)| increases at these frequencies (at any frequency, either |S(jω)|
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or|T(jω)|must be greater or equal to 0.5), and both input and output distur- bances may affect the output if they have enough energy at these frequencies.
Furthermore, the ability to reject both disturbances depends on the frequency separation between di(jω) anddo(jω) and the plantG(jω). All these are ex- amples of trade-offsassociated with the design. ◦ ◦ ◦ The above motivation example evidences the delicate interplay between the different parts of the control system, and the basic limitations and trade-offs the designer usually faces. Note that no particular characteristics of the plant (e.g. delays, unstable poles and non-minimum phase zeros) were mentioned.
These characteristics and limited control action (constraints imposed by the actuators) can indeed aggravate the limitations and design trade-offs. For the case of stabiliser design for displacement ships, the non-minimum phase characteristic of the plant is one of the main effects imposing limitations on the performance of the control system.