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In [[electronic  engineering]] and [[control theory]], step response is the time behavior of the outputs of a general [[system]] when its inputs change from zero to a finite value in a very short time. In a more idealized mathematical formulation, the '''step response''' of a system in a given initial state consists of the time evolution of its [[output]]s when its [[input|control inputs]] are proportional to [[Heaviside step function]]s.  
In [[philosophy]] the term '''free will''' refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so.  It is an age-old concern to separate what we can do something about, choose to do, from what we cannot.  The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of ''determinism'', sometimes called ''physical reductionism'',<ref name=Ney/>  and the view that determinism precludes free will is called ''incompatibilism''.<ref name=Vihvelin/> 


From a practical standpoint, knowing how the system responds to a sudden input is important because large and fast input signals may have extreme effects on the component itself and on other portions of the overall system dependent on this component. In some cases, the overall system cannot act until the component's output settles down to some vicinity of its final state, delaying the overall system response. Formally, the step response of a dynamical system gives information about the [[stability theory|stability]] of the system, and about its ability to reach one stationary state when starting from another.  
There are several ways to avoid the incompatibilst position, resulting in various ''compatibilist'' positions.<ref name=Timpe/>  One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.  


== Description of time-domain behavior ==
There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?
{{Image|Aspects of step response.PNG|right|200px|Some terms used to describe step response in time domain.}}
The step response of a system can be described by the following aspects of its time behavior:


*Overshoot: The output swings that extend above the final value
==Science does not apply==
*Undershoot: The output swings that extend below the final value
*Ringing: The alternation of overshoot and undershoot
*Settling time: The time to settle down to remain below a specified amplitude of ringing about the final value
*Rise time: The time to first cross the final value; not necessarily the only time it will cross this value


In the case of [[linear]] dynamic systems, much can be inferred about the system from these characteristics. Below, the step response of a simple two-pole amplifier is presented, and some of these terms are illustrated.
One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect.  For example, [[David Hume]] suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion.  [[Immanuel Kant]] suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains.  Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.


==Step response of feedback amplifiers==
A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise. It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience.  For example, we choose to explain phenomena like the [[Higgs boson]] found by elaborate means like a [[hadron collider]], but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention.
{{Image|Negative feedback amplifier.PNG|right|200px| Ideal negative feedback model; open loop gain is ''A''<sub>OL</sub> and feedback factor is β.}}
As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.  
This section describes the step response of a simple [[negative feedback amplifier]] shown in the figure. The amplifier when the feedback is disconnected is called the '''open-loop amplifier'''. When the amplifier including feedback is meant, it is called the '''closed-loop amplifier'''.  


The feedback amplifier consists of a main open-loop amplifier of gain ''A''<sub>OL</sub> and a feedback loop governed by a '''feedback factor''' β.   This feedback amplifier is analyzed to determine how its step response depends upon  the time constants governing the response of the main amplifier, and upon the amount of feedback used.  
Although not explicitly addressing the issue of free will, it may be noted that [[Ludwig Wittgenstein]] argued that the specialized theories of science, as discussed by [[Rudolf Carnap]] for example, inevitably cover only a limited range of experience.  [[Stephen Hawking|Hawking/Mlodinow]] also noted this fact in in their [[model-dependent realism]],<ref name=Hawking/> the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.
{{quote|“Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”<ref name=Davies/>}}
::: —— E.B. Davies <span style="font-size:88%">''Epistemological pluralism'', p. 4</span>


===Analysis===
Still another approach to this matter is analysis of the mind-brain connection (more generally, the [[mind-body problem]]). As suggested by Northoff,<ref name=Northoff/>  there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks.  But observing our own mental activity is not possible in this way – it is a matter of subjective experiences.  The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:
A negative feedback amplifier has gain given by (see [[negative feedback amplifier]]):
{{quote|“...for each individual there is ''one'' 'mental life' but ''two'' ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the ''observational arrangements'' (the means by which a subject and an external observer access the subject's mental processes).”<ref name=Velmans/> |Max Velmans: |How could conscious experiences affect brains?,  p. 11''
}}


:<math>A_{FB} = \frac {A_{OL}} {1+ \beta A_{OL}} \ , </math>
A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by [[Niels Bohr]] and by [[Erwin Schrödinger]] in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)
{{quote|“...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”<ref name=Pockett/> |Susan Pockett |The neuroscience of movement}}


where ''A''<sub>OL</sub> = '''open-loop''' gain, ''A''<sub>FB</sub> = '''closed-loop''' gain (the gain with negative feedback present) and β = '''feedback factor'''. The step response of such an amplifier is easily handled in the case that the open-loop gain has two poles (two time constants, τ<sub>1</sub>, τ<sub>2</sub>), that is, the open-loop gain is given by:


:<math>A_{OL} = \frac {A_0} {(1+j \omega \tau_1) (1 + j \omega \tau_2)} \ , </math>
In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.


with zero-frequency gain ''A''<sub>0</sub> and angular frequency ω = 2π''f'', which leads to the closed-loop gain:
==Science can be accommodated==
A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do.  Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.


:<math>A_{FB} = \frac {A_0} {1+ \beta A_0}</math> &ensp; • &ensp; <math> \  \frac {1} {1+j \omega \frac { \tau_1 + \tau_2 } {1 + \beta A_0} + (j \omega )^2 \frac { \tau_1 \tau_2} {1 + \beta A_0} } \ . </math>
==’Will’ ''versus'' ‘action’==
{{Image|Conjugate poles.PNG|right|200px| Conjugate pole locations for step response of two-pole feedback amplifier; ''Re''(s) <nowiki>=</nowiki> real axis and ''Im''(s) <nowiki>=</nowiki> imaginary axis.}}
There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,<ref name=Norretranders/> which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, [[addiction|drug addiction]] has been related to alteration of the mechanisms in the brain for [[dopamine]] production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.
{{quote|“Philosophers who distinguish ''freedom of action'' and ''freedom of will'' do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.]<ref name=OConnor/>| Timothy O'Connor |Free Will}}
In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?


The time dependence of the amplifier is easy to discover by switching variables to ''s'' = ''j''ω, whereupon the gain becomes:
==Theology==
The ancient Greeks held the view that the gods ''could'' intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:
{{quote| “The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.”|spoken by Cassius| Julius Caesar (I, ii, 140-141)}}
The [[Stoicism|Stoics]] wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a ''responsibility''), a sequence demanded by our natures as rational beings.<ref name=Bobzien/>


:<math> A_{FB} = \frac {A_0} { \tau_1 \tau_2 }</math> &ensp; • &ensp; <math> \frac {1} {s^2 +s \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) + \frac {1+ \beta A_0} {\tau_1 \tau_2}} </math>
In [http://www.iep.utm.edu/chrysipp/ Chrysippus of Soli's] view (an apologist for Stoicism), ''fate'' precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.<ref name=Bobzien2/> The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.<ref name=Bobzien3/>


The poles of this expression (that is, the zeros of the denominator) occur at:
==References==
{{reflist|refs=
<ref name=Bobzien>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See §6.3.3 ''The cylinder and cone analogy'', pp. 258 ''ff''.
</ref>


:<math>2s = - \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) </math>  
<ref name=Bobzien2>
::::<math>\pm \sqrt { \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right) ^2 -\frac {4 \beta A_0 } {\tau_1 \tau_2 } } \ ,</math>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular pp. 386 ''ff''.
</ref>


<ref name=Bobzien3>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular p. 255.
</ref>


which shows for large enough values of β''A''<sub>0</sub> the square root becomes the square root of a negative number, that is the square root becomes imaginary, and the pole positions are complex conjugate numbers, either ''s''<sub>+</sub> or ''s''<sub>−</sub>; see the figure:


:<math> s_{\pm} = -\rho \pm j \mu \ ,  </math>
<ref name=Davies>
 
{{cite web |title=Epistemological pluralism |author=E Brian Davies |url=http://philsci-archive.pitt.edu/3083/1/EP3single.doc |work=PhilSci Archive |year=2006 }}
with
</ref>
 
::<math> \rho = \frac {1}{2} \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right ) \ , </math>
 
and
::<math> \mu = \frac {1} {2} \sqrt { \frac {4 \beta A_0} { \tau_1 \tau_2} - \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2 } \ . </math>
Using polar coordinates with the magnitude of the radius to the roots given by |''s''| (See figure):
 
:<math> | s | = |s_{ \pm } | =  \sqrt{ \rho^2 +\mu^2} \ , </math>
 
and the angular coordinate φ is given by:
 
: <math> \mathrm {cos}\  \phi = \frac { \rho} { | s | } \ </math> &emsp; <math> \mathrm {sin}\  \phi = \frac { \mu} { | s | } \ .</math>
Tables of [[Laplace transform]]s show that the time response of such a system is composed of combinations of the two functions:
 
::<math> e^{- \rho t} \mathrm {sin} ( \mu t) \ </math> <math> \quad </math> and <math> \quad </math> <math> e^{- \rho t} \mathrm {cos} ( \mu t) \ , </math>
 
which is to say, the solutions are damped oscillations in time. In particular, the unit step response of the system is:<ref name=Kuo/>
:<math>S(t) = 1 -  e^{- \rho t} \  \frac {  \mathrm {sin} \left( \mu t + \phi \right)}{ \mathrm {sin}( \phi )} \ . </math>
 
Notice that the damping of the response is set by ρ, that is, by the time constants of the open-loop amplifier. In contrast, the frequency of oscillation is set by μ, that is, by the feedback parameter through β''A''<sub>0</sub>. Because ρ is a sum of reciprocals of time constants, it is interesting to notice that ρ is dominated by the ''shorter'' of the two.
 
===Results===
{{Image|Step response of negative feedback amplifier.PNG|right|300px| Step-response of a linear two-pole feedback amplifier; time is in units of 1/ρ, that is, in terms of the time constants of ''A''<sub>OL</sub>; curves are plotted for three values of ''mu'' <nowiki>=</nowiki> μ, which is controlled by β.}}
The figure to the right shows the time response to a unit step input for three values of the parameter μ. It can be seen that the frequency of oscillation increases with μ, but the oscillations are contained between the two asymptotes set by the exponentials [ 1 - exp (−ρt) ] and [ 1 + exp (−ρt) ]. These asymptotes are determined by ρ and therefore by the time constants of the open-loop amplifier, independent of feedback.
 
The phenomena of oscillation about final value is called '''[[ringing]]'''. The '''[[overshoot]]''' is the maximum swing above final value, and clearly increases with μ. Likewise, the '''undershoot''' is the minimum swing below final value, again increasing with μ. The '''[[settling time]]''' is the time for departures from final value to sink below some specified level, say 10% of final value.
 
The dependence of settling time upon μ is not obvious, and the approximation of a two-pole system probably is not accurate enough to make any real-world conclusions about feedback dependence of settling time. However, the asymptotes [ 1 - exp (−ρt) ] and [ 1 + exp (−ρt) ] clearly impact settling time, and they are controlled by the time constants of the open-loop amplifier, particularly the shorter of the two time constants. That suggests that a specification on settling time must be met by appropriate design of the open-loop amplifier.
 
The two major conclusions from this analysis are:
#Feedback controls the amplitude of oscillation about final value for a given open-loop amplifier and given values of open-loop time constants,  τ<sub>1</sub> and τ<sub>2</sub>.
#The open-loop amplifier decides settling time. It sets the time scale of Figure 3, and the faster the open-loop amplifier, the faster this time scale.
 
As an aside, it may be noted that real-world departures from this linear two-pole model occur due to two major complications: first, real amplifiers have more than two poles, as well as zeros; and second, real amplifiers are nonlinear, so their step response changes with signal amplitude.
 
===Control of overshoot===
{{Image|Overshoot control.PNG|right|350px|Step response for three values of α. Top: α <nowiki>=</nowiki> 4; Center: α <nowiki>=</nowiki> 2; Bottom: α <nowiki>=</nowiki> 0.5. As α is reduced the pole separation reduces, and the overshoot increases.}}
How overshoot may be controlled by appropriate parameter choices is discussed next.
 
Using the equations above, the amount of overshoot can be found by differentiating the step response and finding its maximum value. The result for maximum step response ''S''<sub>max</sub> is:<ref name=Kuo2/>
 
:<math>S_{max}= 1 \  </math>&ensp;<math>\ +\  \mathrm {exp} \left( - \pi \frac { \rho }{ \mu } \right) \ . </math>
 
The final value of the step response is 1, so the exponential is the actual overshoot itself. It is clear the overshoot is zero if μ = 0, which is the condition:
 
:<math> \frac {4 \beta A_0} { \tau_1 \tau_2} = \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2  \ . </math>
 
This quadratic is solved for the ratio of time constants by setting ''x'' = ( τ<sub>1</sub> / τ<sub>2</sub> )<sup>1 / 2 </sup> with the result
 
:<math>x = \sqrt{ \beta A_0 } + \sqrt { \beta A_0 +1 } \ . </math>
 
Because β ''A''<sub>0</sub> >> 1, the 1 in the square root can be dropped, and the result is
 
:<math> \frac { \tau_1} { \tau_2} = 4 \beta A_0 \ . </math>
 
In words, the first time constant must be much larger than the second. To be more adventurous than a design allowing for no overshoot we can introduce a factor α in the above relation:
 
:<math> \frac { \tau_1} { \tau_2} = \alpha \beta A_0 \ , </math>
 
and let α be set by the amount of overshoot that is acceptable.
 
The figure at right illustrates the procedure. Comparing the top panel (α = 4) with the lower panel (α = 0.5) shows lower values for α  increase the rate of response, but increase overshoot. The case α = 2 (center panel) is the [[Butterworth_filter#Maximal_flatness|''maximally flat'']] design that shows no peaking in the [[Bode plot|Bode gain vs. frequency plot]]. That design has the [[rule of thumb]] built-in safety margin to deal with non-ideal realities like multiple poles (or zeros), nonlinearity (signal amplitude dependence) and manufacturing variations, any of which can lead to too much overshoot. The adjustment of the pole separation (that is, setting α ) is the subject of [[frequency compensation]], and one such method is [[pole splitting]].
 
===Control of settling time ===
The amplitude of ringing in the step response is governed by the damping factor exp ( −ρ t ). That is, if we specify some acceptable step response deviation from final value, say Δ, that is:
 
:<math> S(t)  \le 1 + \Delta  \ ,</math>
 
this condition is satisfied  regardless of the value of β ''A''<sub>OL</sub> provided the time is longer than the settling time, say ''t''<sub>S</sub>, given by:<ref name=note1/>
 
:<math> \Delta = e^{- \rho t_S }</math> &emsp; or &emsp; <math>t_S = \frac { \mathrm{ln} \left( \frac{1} { \Delta} \right) } { \rho } = \tau_2 \  \frac {2 \  \mathrm{ln} \left( \frac{1} { \Delta} \right) } { 1 + \frac { \tau_2 } { \tau_1} } \approx 2 \ \tau_2 \  \mathrm{ln} \left( \frac{1} { \Delta} \right)\ , </math>
 
where the approximation τ<sub>1</sub> >> τ<sub>2</sub> is applicable because of  the overshoot control condition, which makes τ<sub>1</sub> = α β''A''<sub>OL</sub> τ<sub>2</sub>. Often the settling time condition is referred to by saying the settling period is inversely proportional to the unity gain bandwidth,  because 1/( 2π τ<sub>2</sub> ) is close to this bandwidth for an amplifier with typical [[Frequency_compensation#Dominant-pole_compensation|dominant pole compensation]]. However, this result is more precise than this [[rule of thumb]]. As an example of this formula, if Δ = 1/e<sup>4</sup> = 1.8 %, the settling time condition is  ''t''<sub>S</sub> = 8 τ<sub>2</sub>.
 
In general, control of overshoot sets the time constant ratio, and settling time ''t''<sub>S</sub> sets τ<sub>2</sub>.<ref name=Johns/><ref name=Sansen/><ref name=note2/>
 
===Phase margin===
{{Image|Gain Bode plot for two-pole amplifier.PNG|right|200px|Bode gain plot to find phase margin; scales are logarithmic, so labeled separations are multiplicative factors. For example, ''f''<sub>0dB</sub>  <nowiki>=</nowiki> β ''A''<sub>0</sub> × ''f''<sub>1</sub>.}}
Next, the choice of pole ratio τ<sub>1</sub> / τ<sub>2</sub> is related to the phase margin of the feedback amplifier.<ref name=note3/>  The procedure outlined in the [[Bode_plot#Examples_using_Bode_plots|Bode plot]] article is followed. To the right is the Bode gain plot for the two-pole amplifier in the range of frequencies up to the second pole position. The assumption behind the plot is that the frequency ''f''<sub>0dB</sub> lies between the lowest pole at ''f''<sub>1</sub> = 1 / ( 2π τ<sub>1</sub> ) and the second pole at ''f''<sub>2</sub> = 1 / ( 2π τ<sub>2</sub> ). As indicated in the figure, this condition is satisfied for values of α ≥ 1.
 
Using the figure the frequency (denoted by ''f''<sub>0dB</sub> ) is found where the loop gain β''A''<sub>0</sub> satisfies the unity gain or 0 dB condition , as defined by:
 
:<math> | \beta A_{OL} ( f_{0db} ) | = 1 \ . </math>
 
The slope of the downward leg of the gain plot is (20 dB/decade); for every factor of ten increase in frequency, the gain drops by the same factor:<ref name=note4/><ref name=note5/>
 
:<math> f_{0dB} = \beta A_0 f_1 \ . </math>
 
The phase margin is the departure of the phase at ''f''<sub>0dB</sub> from −180°. Thus, the margin is:
 
:<math> \phi_m = 180 ^\circ - \mathrm {atan} (f_{0dB} /f_1) - \mathrm {atan} ( f_{0dB} /f_2)  \ . </math>
 
Because ''f''<sub>0dB</sub> / ''f''<sub>1</sub> = β''A''<sub>0</sub> >> 1, the term in ''f''<sub>1</sub> is 90°. That makes the phase margin:
 
:<math> \phi_m = 90 ^\circ -  \mathrm {atan} ( f_{0dB} /f_2)  </math>
::<math> = 90 ^\circ -  \mathrm {atan} \left( \frac {\beta A_0 f_1} {\alpha \beta A_0 f_1 } \right) </math>
::<math> = 90 ^\circ -  \mathrm {atan} \left( \frac {1} {\alpha } \right) </math> <math> = \mathrm {atan} \left(  \alpha  \right) \ . </math>
 
In particular, for case α = 1,  φ<sub>m</sub> = 45°, and for α = 2, φ<sub>m</sub> = 63.4°. Sansen<ref name=Sansen3/> recommends α = 3, φ<sub>m</sub> = 71.6° as a "good safety position to start with".
 
If α is increased by shortening τ<sub>2</sub>, the settling time ''t''<sub>S</sub> also is shortened. If α is increased by lengthening τ<sub>1</sub>, the settling time ''t''<sub>S</sub> is little altered. More commonly, both τ<sub>1</sub> ''and'' τ<sub>2</sub> change, for example if the technique of [[pole splitting]] is used.


As an aside, for an amplifier with more than two poles, the diagram in the figure still may be made to fit the Bode plots by making ''f''<sub>2</sub> a fitting parameter, referred to as an "equivalent second pole" position.<ref name=Palumbo/>


==References and notes==
<ref name=Hawking>
{{reflist|refs=
{{cite book |author=Hawking SW, Mlodinow L. |title=The Grand Design |isbn=0553805371 |url= http://www.amazon.com/Grand-Design-Stephen-Hawking/dp/0553805371#reader_0553805371 |pages=pp. 42-43 |chapter=Chapter 3: What is reality?|year=2010|publisher=Bantam Books}}
<ref name=Kuo>
{{cite book  
|author=Benjamin C Kuo & Golnaraghi F
|title=Automatic control systems
|year= 2003
|pages=p. 253
|publisher=Wiley
|edition=Eighth Edition
|location=New York
|isbn=0-471-13476-7
|url=http://worldcat.org/isbn/0-471-13476-7}}
</ref>
</ref>


<ref name=Kuo2>
<ref name=Norretranders>
{{cite book  
{{cite book |url=http://www.google.com/search?tbo=p&tbm=bks&q=consciousness%2Bplays%2Ba%2Bsmaller%2Brole%2Bin%2Bhuman%2Blife+intitle:User+intitle:illusion&num=10 |title=The user illusion: Cutting consciousness down to size |quote=Consciousness plays a far smaller role in human life than Western culture has tended to believe |author=Tor Nørretranders |isbn=0140230122 |chapter=Preface |pages=p. ''ix'' |publisher=Penguin Books |year=1998 |edition=Jonathan Sydenham translation of ''Maerk verden'' 1991 ed }}
|author=Benjamin C Kuo & Golnaraghi F
|title=p. 259
|isbn=0-471-13476-7
|url=http://worldcat.org/isbn/0-471-13476-7}}
</ref>
</ref>


<ref name=note1>
<ref name=Ney>
This estimate is a bit conservative (long) because the factor 1 /sin(φ) in the overshoot contribution  to ''S'' ( ''t'' ) has been replaced by  1 /sin(φ)  &asymp; 1.
{{cite web |author=Alyssa Ney |title=Reductionism |work=Internet Encyclopedia of Philosophy |date= November 10, 2008 |url=http://www.iep.utm.edu/red-ism/}}
</ref>
</ref>


<ref name=Johns>
<ref name=Northoff>
{{cite book  
A rather extended discussion is provided in {{cite book |title=Philosophy of the Brain: The Brain Problem |author=Georg Northoff |url=http://books.google.com/books?id=r0Bf3lLys6AC&printsec=frontcover |publisher=John Benjamins Publishing |isbn=1588114171 |year=2004 |edition=Volume 52 of Advances in Consciousness Research}}
|author=David A. Johns & Martin K W
|title=Analog integrated circuit design
|year= 1997
|pages=pp. 234-235
|publisher=Wiley
|location=New York
|isbn=0-471-14448-7
|url=http://worldcat.org/isbn/0-471-14448-7}}
</ref>
</ref>


<ref name=Sansen>
<ref name=OConnor>
{{cite book
{{cite web |title=&thinsp;Free Will |date=Oct 29, 2010 |author=O'Connor, Timothy |url=http://plato.stanford.edu/archives/sum2011/entries/freewill |work=The Stanford Encyclopedia of Philosophy (Summer 2011 Edition) |editor=Edward N. Zalta, ed.}}
|author=Willy M C Sansen
|title=Analog design essentials
|page=§0528 p. 163
|year= 2006
|publisher=Springer
|location=Dordrecht, The Netherlands
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>
</ref>


<ref name=note2>
<ref name=Pockett>
According to Johns and Martin, ''op. cit.'', settling time is significant in [[switched capacitor|switched-capacitor circuits]], for example, where an op amp settling time must be less than half a clock period for sufficiently rapid charge transfer.
{{cite book|title=&thinsp;Does Consciousness Cause Behavior? |chapter=The neuroscience of movement |author=Susan Pockett |url=http://books.google.com/books?id=G5CaTnNksgkC&pg=PA19&lpg=PA19 |pages= p. 19 |editor=Susan Pockett, WP Banks, Shaun Gallagher, eds. |publisher=MIT Press |date =2009 |isbn=0262512572}}
</ref>
</ref>


<ref name=note3>
<ref name=Timpe>
The gain margin of the amplifier cannot be found using a two-pole model, because gain margin requires determination of the frequency ''f''<sub>180</sub> where the gain flips sign, and this never happens in a two-pole system. If we know ''f''<sub>180</sub> for the amplifier at hand, the gain margin can be found approximately, but ''f''<sub>180</sub> then depends on the third and higher pole positions, as does the gain margin, unlike the estimate of phase margin, which is a two-pole estimate.
{{cite web |author=Kevin Timpe |title=Free will |work=Internet Encyclopedia of Philosophy |date= March 31, 2006 |url=http://www.iep.utm.edu/freewill/#H5}}
</ref>
</ref>


<ref name=note4>
For more detail on the use of logarithmic scales, see [[Logarithmic_scale#Slope_of_a_log-log_plot|log scale]].
</ref>


<ref name=note5>  
<ref name=Velmans>
See also [[Pole_splitting#Selection_of_CC|Figure 3 in pole splitting]].
{{cite journal |journal=Journal of Consciousness Studies |volume=9 |issue=11 |year=2002 |pages=pp. 2-29 |author=Max Velmans  |title=How Could Conscious Experiences Affect Brains? |url=http://cogprints.org/2750/ |year=2002}}
</ref>
</ref>


<ref name=Sansen3>
<ref name=Vihvelin>
{{cite book
{{cite web |author=Kadri Vihvelin |title=&thinsp;Arguments for Incompatibilism |work=The Stanford Encyclopedia of Philosophy (Spring 2011 Edition) |editor=Edward N. Zalta, ed. |url= http://plato.stanford.edu/archives/spr2011/entries/incompatibilism-arguments/ |date=Mar 1, 2011}}
|author=Willy M C Sansen
|title=§0526 p. 162
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>
 
<ref name=Palumbo> 
{{cite book
|author=Gaetano Palumbo & Pennisi S
|title=Feedback amplifiers: theory and design
|year= 2002
|pages=§ 4.4 pp. 97-98
|publisher=Kluwer Academic Press
|location=Boston/Dordrecht/London
|isbn=0-7923-7643-9
|url=http://books.google.com/books?id=Xb0W1VsQFe0C&pg=PA98&dq=%22equivalent+two-pole+amplifier%22&lr=&as_brr=0&sig=LVB6t0wlg7WL7U_PB4eavGTP7Aw }}
</ref>
</ref>


}}
}}
== Further reading ==
*Robert I. Demrow ''Settling time of operational amplifiers'' [http://www.analog.com/UploadedFiles/Application_Notes/466359863287538299597392756AN359.pdf]
*Cezmi Kayabasi ''Settling time measurement techniques achieving high precision at high speeds'' [http://www.wpi.edu/Pubs/ETD/Available/etd-050505-140358/unrestricted/ckayabasi.pdf]
*Vladimir Igorevic Arnol'd "Ordinary differential equations", various editions from MIT Press and from Springer Verlag, chapter 1 "Fundamental concepts"

Latest revision as of 03:07, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


In philosophy the term free will refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so. It is an age-old concern to separate what we can do something about, choose to do, from what we cannot. The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of determinism, sometimes called physical reductionism,[1] and the view that determinism precludes free will is called incompatibilism.[2]

There are several ways to avoid the incompatibilst position, resulting in various compatibilist positions.[3] One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.

There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?

Science does not apply

One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect. For example, David Hume suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion. Immanuel Kant suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains. Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.

A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise. It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience. For example, we choose to explain phenomena like the Higgs boson found by elaborate means like a hadron collider, but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention. As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.

Although not explicitly addressing the issue of free will, it may be noted that Ludwig Wittgenstein argued that the specialized theories of science, as discussed by Rudolf Carnap for example, inevitably cover only a limited range of experience. Hawking/Mlodinow also noted this fact in in their model-dependent realism,[4] the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.

‘ “Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”[5] ’
—— E.B. Davies Epistemological pluralism, p. 4

Still another approach to this matter is analysis of the mind-brain connection (more generally, the mind-body problem). As suggested by Northoff,[6] there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks. But observing our own mental activity is not possible in this way – it is a matter of subjective experiences. The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:

‘ “...for each individual there is one 'mental life' but two ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the observational arrangements (the means by which a subject and an external observer access the subject's mental processes).”[7] ’

—Max Velmans: , How could conscious experiences affect brains?, p. 11

A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by Niels Bohr and by Erwin Schrödinger in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)

‘ “...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”[8] ’

—Susan Pockett , The neuroscience of movement


In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.

Science can be accommodated

A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do. Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.

’Will’ versus ‘action’

There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,[9] which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, drug addiction has been related to alteration of the mechanisms in the brain for dopamine production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.

‘ “Philosophers who distinguish freedom of action and freedom of will do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.][10] ’

— Timothy O'Connor , Free Will

In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?

Theology

The ancient Greeks held the view that the gods could intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:

‘ “The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.” ’

—spoken by Cassius, Julius Caesar (I, ii, 140-141)

The Stoics wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a responsibility), a sequence demanded by our natures as rational beings.[11]

In Chrysippus of Soli's view (an apologist for Stoicism), fate precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.[12] The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.[13]

References

  1. ↑ Alyssa Ney (November 10, 2008). Reductionism. Internet Encyclopedia of Philosophy.
  2. ↑ Kadri Vihvelin (Mar 1, 2011). Edward N. Zalta, ed.: Arguments for Incompatibilism. The Stanford Encyclopedia of Philosophy (Spring 2011 Edition).
  3. ↑ Kevin Timpe (March 31, 2006). Free will. Internet Encyclopedia of Philosophy.
  4. ↑ Hawking SW, Mlodinow L. (2010). “Chapter 3: What is reality?”, The Grand Design. Bantam Books, pp. 42-43. ISBN 0553805371. 
  5. ↑ E Brian Davies (2006). Epistemological pluralism. PhilSci Archive.
  6. ↑ A rather extended discussion is provided in Georg Northoff (2004). Philosophy of the Brain: The Brain Problem, Volume 52 of Advances in Consciousness Research. John Benjamins Publishing. ISBN 1588114171. 
  7. ↑ Max Velmans (2002). "How Could Conscious Experiences Affect Brains?". Journal of Consciousness Studies 9 (11): pp. 2-29.
  8. ↑ Susan Pockett (2009). “The neuroscience of movement”, Susan Pockett, WP Banks, Shaun Gallagher, eds.:  Does Consciousness Cause Behavior?. MIT Press, p. 19. ISBN 0262512572. 
  9. ↑ Tor Nørretranders (1998). “Preface”, The user illusion: Cutting consciousness down to size, Jonathan Sydenham translation of Maerk verden 1991 ed. Penguin Books, p. ix. ISBN 0140230122. “Consciousness plays a far smaller role in human life than Western culture has tended to believe” 
  10. ↑ O'Connor, Timothy (Oct 29, 2010). Edward N. Zalta, ed.: Free Will. The Stanford Encyclopedia of Philosophy (Summer 2011 Edition).
  11. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See §6.3.3 The cylinder and cone analogy, pp. 258 ff.
  12. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular pp. 386 ff.
  13. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular p. 255.
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