---
readstatus: false
dateread:
title: "Feedback Control Theory"
year: 2009
authors:

  
  - "Doyle, John"
  
  - "A, Francis"
  
  - "Tannenbaum, Allen"
  

citekey: "doyleFeedbackControlTheory2009"







---
# Indexing Information
## DOI
[](https://doi.org/)
## ISBN
[](https://www.isbnsearch.org/isbn/)
## Tags:


>[!Abstract]
>In any system, if there exists a linear relationship between two variables, then it is said that it is a linear system.

>[!note] Markdown Notes
>None!

>[!seealso] Related Papers
>[Guaranteed margins for LQG regulators](doyleGuaranteedMarginsLQG1978)

# Annotations

>[!attention] Highlight
> *The book is addressed to students in engineering who have had an undergraduate course insignals and systems, including an introduction to frequency-domain methods of analyzing feedbackcontrol systems, namely, Bode plots and the Nyquist criterion.*
> 

>[!attention] Highlight
> *The simplest objective might be to keep y small(or close to some equilibrium point)—a regulator problem—or to keep y − r small for r, a referenceor command signal, in some set—a servomechanism or servo problem.*
> 

>[!attention] Highlight
> *Uncertainty arises from twosources: unknown or unpredictable inputs (disturbance, noise, etc.) and unpredictable dynamics.*
> 

>[!attention] Highlight
> *Ideally, the model should cover the data in the sense that it should be capable of producingevery experimentally observed input-output pair. (Of course, it would be better to cover not just the data observed in a finite number of experiments, but anything that can be produced by the realphysical system. Obviously, this is impossible.)*
> 

>[!attention] Highlight
> *Very rarely is the exogenous input w a fixed, known signal. One of these rare instances is wherea robot manipulator is required to trace out a definite path, as in welding. Usually, w is not fixed but belongs to a set that can be characterized to some degree. Some examples:• In a thermostat-controlled temperature regulator for a house, the reference signal is alwayspiecewise constant: at certain times during the day the thermostat is set to a new value. The temperature of the outside air is not piecewise constant but varies slowly within bounds.• In a vehicle such as an airplane or ship the pilot’s commands on the steering wheel, throttle, pedals, and so on come from a predictable set, and the gusts and wave motions have amplitudesand frequencies that can be bounded with some degree of confidence.  • The load power drawn on an electric power system has predictable characteristics.Sometimes the designer does not attempt to model the exogenous inputs.*
> 

>[!attention] Highlight
> *transfer function fromreference input r to tracking error e is denoted S, the sensitivity function*
> 

>[!attention] Highlight
> *Lemma 1 The 2-norm of Gˆ is finite iff Gˆ is strictly proper and has no poles on the imaginaryaxis; the ∞-norm is finite iff Gˆ is proper and has no poles on the imaginary axis.*
> 

>[!attention] Highlight
> *A stronger notion of well-posedness that makes sense when P, C, and F are proper is thatthe nine transfer functions above are proper. A necessary and sufficient condition for this is that1 + PCF not be strictly proper [i.e., PCF(∞) 6= −1].*
> 

>[!attention] Highlight
> *Nyquist Criterion Construct the Nyquist plot of PCF, indenting to the left around poles on the imaginary axis. Let n denote the total number of poles of P, C, and F in Res ≥ 0. Then the feedbacksystem is internally stable iff the Nyquist plot does not pass through the point -1 and encircles itexactly n times counterclockwise.*
> 

>[!attention] Highlight
> *Define the loop transfer function Lˆ := PˆCˆ. The transfer function from reference input r totracking error e isSˆ :=11 + Lˆ ,called the sensitivity function—*
> 

>[!quote] Other Highlight
> *Here we used Table 2.1: the maximum amplitude of e equals the ∞-norm of the transfer function.  Or if we define the (trivial, in this case) weighting function W1(s) = 1/ǫ, then the performance specification is kW1Sk∞ < 1.The situation becomes mo*
> 
> >[!note] Note
> >Ladies and gentlemen, we got him.

>[!attention] Highlight
> *Various performance specifications could be made using weighted versions of the transfer functions above. Note that a performance spec with weight W on P S is equivalent to the weight W P on S. Similarly, a weight W on CS = T /P is equivalent to the weight W/P on T . Thus performance specs that involve e result in weights on S and performance specs on u result in weights on T . Essentially all problems in this book boil down to weighting S or T or some combination, and the tradeoff between making S small and making T small is the main issue in design.*
> 

>[!attention] Highlight
> *Thus one type of structured set is parametrized by a finite number of scalar parameters (one parameter, a, in this example). Another type of structured uncertainty is a discrete set of plants, not necessarily parametrized explicitly.*
> 

>[!attention] Highlight
> *For us, unstructured sets are more important, for two reasons. First, we believe that all models used in feedback design should include some unstructured uncertainty to cover unmodeled dynamics, particularly at high frequency. Other types of uncertainty, though important, may or may not arise naturally in a given problem.*
> 

>[!attention] Highlight
> *The multiplicative perturbation model is not suitable for every application because the disk covering the uncertainty set is sometimes too coarse an approximation. In this case a controller designed for the multiplicative uncertainty model would probably be too conservative for the original uncertainty model. The discussion above illustrates an important point. In modeling a plant we may arrive at a certain plant set. This set may be too awkward to cope with mathematically, so we may embed it in a larger set that is easier to handle. Conceivably, the achievable performance for the larger set may not be as good as the achievable performance for the smaller; that is, there may exist—even though we cannot find it—a controller that is better for the smaller set than the controller we design for the larger set. In this sense the latter controller is conservative for the smaller set.*
> 

>[!quote] Other Highlight
> *A controller C is robust with respect to this characteristic if this characteristic holds for every plant in P.*
> 

>[!attention] Highlight
> *Better stability margins are obtained by taking explicit frequency-dependent perturbation models: for example, the multiplicative perturbation model, P ̃ = (1 + ∆W2)P . Fix a positive number β and consider the family of plants  {P ̃ : ∆ is stable and ‖∆‖∞ ≤ β}.  Now a controller C that achieves internal stability for the nominal plant P will stabilize this entire family if β is small enough. Denote by βsup the least upper bound on β such that C achieves internal stability for the entire family. Then βsup is a stability margin (with respect to this uncertainty model).*
> 

### Imported: 2024-10-16 9:25 am


