# Power Law re-boot (PCT Side Bar)

[From Rick Marken (2016.09.18.1230)]

···

RM: I see everyone is off talking about Roger’s model (or non-model, as Martin would have it) so I’m probably just talking to myself but, what the heck, this is supposed to be a list about PCT so I’ll sprinkle in a little just for my own satisfaction.

Martin Taylor (2016.09.17.22.56) –

``````MT: True. Apart from misrepresenting what Roger wrote, they referred to
``````

Roger’s words as if he had proposed a model,

RM: Well, it looks like a model to me so I’ll treat it as though it is one. In fact, this is a perfectly fine PCT model of a control system that controls velocity, Vp. A linear version of the system side of Roger’s model looks like this:

Vm = k*(Vr-Vp)

So the output of the control system, Vm, is proportional to the difference between the controlled perception, Vp, and the reference for that perception, Vr. What’s missing is a description of the perceptual function that produces Vp and the environmental feedback connection from system output, Vm, to controlled input. I will assume that Vp is the perception of the tangential velocity of the object that is moving. So I’ll define Vp is a function of the X, Y position of that object. So

Vp = k.i*Sqrt((dX/dt)^2 +(dY/dt)^2)

RM: That is, the perception of the velocity of the moved object is proportional (by the input constant k.i) to V as calculated in the power law studies. Now we have to figure out how Vm relates to Vp. In order to have an effect on Vp, the system must be able to affect dX/dt and dY/dt. That is, the system has to be able to change the position of the object in the X and Y dimensions independently. This means that the output of the control system can’t be a single variable, Vm. And a single control system can’t independently vary two variables. So it looks like we’re going to have to model control of velocity using two control systems, one controlling movement in the X dimension and the other controlling it in the Y dimension. But this would mean each system could control only a component of Vp, one controlling dX/dt and the other dY/dt. But that looks like the way it has to be. So velocity control (in two dimensions) requires two control systems, that could be defined as follows:

Px = dX/dt : Py = dY/dt

Ox = kx* (Rx-Px) : Oy = dY/dt

dX/dt = Ox= dx; dY/dt = Oy + dy

RM: So we have two control systems, one controlling movement in the X direction (Px) and the other controlling movement in the Y dimension (Py). The state of the variables controlled by each system is affected by the output of each system (Ox, Oy) and environmental disturbances (dx,dy).

RM: Neither of these two control systems alone controls tangential velocity, Vp. This has to be done by a higher level system that sends references to these systems controlling dX/dt and dY/dt). But a single higher level control system cannot vary the references for two lower level system independently and we know that people make movements where the movements in the X and Y dimensions are different. So how can we get the model to behave this way.

RM: This is where curvature might come in. Curvature, like tangential velocity,Vp, is a function of dX/dt and dY/dt. And it is a variable aspect of the movement that might also be controlled! So why not have curvature as a second, higher level controlled variable, Cp. So now we have two higher level control systems, one controlling Vp and the other Cp, where both Vp and Cp are functions of the perceptions controlled by the two lower level systems, Px and Py.

Vp =kv* sqrt(Px^2+Py^2)

Cp = kc* [|PxdY2/dt-dX2/dtPy)/sqrt(Px^2+Py^2)]

RM: Note that the perception of both velocity and curvature depend on Px and Py but curvature also depends on dX2/dt and dY2/dt. This suggests that I may need a couple more lower level systems to control these variables. Bad news. Clearly this requies more thought. But I think I will try to build this two level system and see what happens.

RM: But there are some important things that we have learned from this exercise. First, we need at least two control system in order to control the velocity of movement. In order to make the movement move at the desired velocity in the right direction we need another system to control curvature (or, now that I think of it, perhaps just a system that controls instantaneous direction rather than curvature). And, finally, curvature is a consequence of the same outputs (movement in the X and Y dimension) that affect velocity. So curvature is not a disturbance to the movement; it’s a consequence of output, as is velocity.

RM: I’ll try to make a diagram of this model and write a simulation asap. Again, just for my benefit; I now return you to your regularly schedule high integrity PCT program.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

RKM:

``````                      a)
``````

Let Vr be the intended (reference) velocity.

``````                      b)
``````

Let Vp be the perceived velocity.

``````                      c)
``````

Let e=Vr-Vp

``````                      d).
``````

Let Vm be the output velocity, where Vm is
some function of e.

``````                      e)
``````

If Vp=Vr, then e=0 and Vm will remain
unchanged.

``````                      f)
``````

If Vp>Vr, then Vm will be reduced until
e=0.

``````          RM: I guess my comments on Roger's model didn't make
``````

much of an impression in this new, high integrity world of
PCT.

[Martin Taylor 2016.09.18.17.03]

[From Rick Marken (2016.09.18.1230)]

``````It's actually a suggestion that if there is a velocity control loop,
``````

it has a perceptual function that produces Vp as a
function of Vm and curvature rather than as a function of
just Vm.

``````Well, it might be if it had an output function that influenced
``````

something in the environment, and an environmental feedback path
that influenced Vm, and a perceptual function that took Vm
and curvature as input and produced Vp as its perceptual
value. If you add all those it would become a model of a control
system.

``````So you asserting that Roger has a model that doesn't include the
``````

usual integrative output function. Roger didn’t suggest that or any
other output function, but if you are going to produce a model
incorporating his curvature suggestion, why not? Your model can have
your own choice of output function. The necessary integration might
come in when you decide how you want the output to influence Vm.

``````X, Y position AND the curvature, please. Inclusion of the curvature
``````

in the velocity perceptual function was all that Roger suggested,
after all. If you omit it, why link this message to Roger’s? It
would be your invention entirely. What’s missing in Roger’s
“perfectly fine PCT model” of a velocity control loop is everything
except the perceptual function and the comparator. If you change the
key property of his perceptual function, what’s left of his
suggestion but his bog-standard comparator?

``````Remainder omitted.

Martin
``````
···

RM: I see everyone is off talking
about Roger’s model (or non-model, as Martin would have it)
so I’m probably just talking to myself but, what the heck,
this is supposed to be a list about PCT so I’ll sprinkle in
a little just for my own satisfaction.

Martin Taylor (2016.09.17.22.56) –

``````            MT: True. Apart from misrepresenting what Roger
``````

wrote, they referred to Roger’s words as if he had
proposed a model,

``````          RM: Well, it looks like a model to me so I'll treat it
``````

as though it is one.

RKM:

``````                                  a)
``````

Let Vr be the intended (reference)
velocity.

``````                                  b)
``````

Let Vp be the perceived velocity.

``````                                  c)
``````

Let e=Vr-Vp

``````                                  d).
``````

Let Vm be the output velocity,
where Vm is some function of e.

``````                                  e)
``````

If Vp=Vr, then e=0 and Vm will
remain unchanged.

``````                                  f)
``````

If Vp>Vr, then Vm will be
reduced until e=0.

``````                        RM: I guess my comments on Roger's model
``````

didn’t make much of an impression in this
new, high integrity world of PCT.

``````          In fact, this is a perfectly fine PCT model of a
``````

control system that controls velocity, Vp.

``````          A linear version of the system side of Roger's model
``````

looks like this:

Vm = k*(Vr-Vp)

``````          So the output of the control system, Vm, is
``````

proportional to the difference between the controlled
perception, Vp, and the reference for that perception, Vr.

``````          What's missing is a description of the perceptual
``````

function that produces Vp and the environmental feedback
connection from system output, Vm, to controlled input. I
will assume that Vp is the perception of the tangential
velocity of the object that is moving. So I’ll define Vp
is a function of the X, Y position of that object.

[Roger K. Moore 2016.09.19.11.21)]

Sorry, too overwhelmed (with students) to respond fully, but a couple of remarks on my suggested solution …

• control can be in cartesian or vector space - two parameters in both cases - and I suspect the latter is more appropriate here (i.e. velocity and direction)

• I imagined the error term controlling the acceleration (i.e.Vmdot), just as in cruise control on a vehicle - so one control loop for velocity (Vr in, Vm out)

• the direction sequence could be predetermined or use another control loop to steer (i.e. to follow a path)

• the key unknown (in my suggested solution) is indeed the perceptual feedback function

Cheers

Roger

···

On 18 September 2016 at 22:30, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2016.09.18.17.03]

[From Rick Marken (2016.09.18.1230)]

``````It's actually a suggestion that if there is a velocity control loop,
``````

it has a perceptual function that produces Vp as a
function of Vm and curvature rather than as a function of
just Vm.

``````Well, it might be if it had an output function that influenced
``````

something in the environment, and an environmental feedback path
that influenced Vm, and a perceptual function that took Vm
and curvature as input and produced Vp as its perceptual
value. If you add all those it would become a model of a control
system.

``````So you asserting that Roger has a model that doesn't include the
``````

usual integrative output function. Roger didn’t suggest that or any
other output function, but if you are going to produce a model
incorporating his curvature suggestion, why not? Your model can have
your own choice of output function. The necessary integration might
come in when you decide how you want the output to influence Vm.

``````X, Y position AND the curvature, please. Inclusion of the curvature
``````

in the velocity perceptual function was all that Roger suggested,
after all. If you omit it, why link this message to Roger’s? It
would be your invention entirely. What’s missing in Roger’s
“perfectly fine PCT model” of a velocity control loop is everything
except the perceptual function and the comparator. If you change the
key property of his perceptual function, what’s left of his
suggestion but his bog-standard comparator?

``````Remainder omitted.

Martin
``````
``````        RM: I see everyone is off talking
``````

about Roger’s model (or non-model, as Martin would have it)
so I’m probably just talking to myself but, what the heck,
this is supposed to be a list about PCT so I’ll sprinkle in
a little just for my own satisfaction.

Martin Taylor (2016.09.17.22.56) –

``````            MT: True. Apart from misrepresenting what Roger
``````

wrote, they referred to Roger’s words as if he had
proposed a model,

``````          RM: Well, it looks like a model to me so I'll treat it
``````

as though it is one.

RKM:

``````                                  a)
``````

Let Vr be the intended (reference)
velocity.

``````                                  b)
``````

Let Vp be the perceived velocity.

``````                                  c)
``````

Let e=Vr-Vp

``````                                  d).
``````

Let Vm be the output velocity,
where Vm is some function of e.

``````                                  e)
``````

If Vp=Vr, then e=0 and Vm will
remain unchanged.

``````                                  f)
``````

If Vp>Vr, then Vm will be
reduced until e=0.

``````                        RM: I guess my comments on Roger's model
``````

didn’t make much of an impression in this
new, high integrity world of PCT.

``````          In fact, this is a perfectly fine PCT model of a
``````

control system that controls velocity, Vp.

``````          A linear version of the system side of Roger's model
``````

looks like this:

Vm = k*(Vr-Vp)

``````          So the output of the control system, Vm, is
``````

proportional to the difference between the controlled
perception, Vp, and the reference for that perception, Vr.

``````          What's missing is a description of the perceptual
``````

function that produces Vp and the environmental feedback
connection from system output, Vm, to controlled input. I
will assume that Vp is the perception of the tangential
velocity of the object that is moving. So I’ll define Vp
is a function of the X, Y position of that object.

[From Rick Marken (2016.09.19.1200)]

RM: While mulling over this modeling project it struck me that there is a PCT explanation of the power law if the law is taken at face value; as a decrease in velocity in response to an increase in curvature. The power law is:

V =k* R^1/3

RM: The PCT explanation of this law simply assumes that the controlled variable is a perception of the relationship between V and R, specifically:

P = log(V) - 1/3 log®.

RM: The output variable of the system that controls this perception is V, the reference for P is 0 and the disturbance to the controlled variable is R. In order to keep log(V) - 1/3 log® equal to the reference value of 0 the control system has to vary V so that V = R^1/3. I have set up a control system that does this and it works like a charm; when the system controls P as defined above, a regression analysis of log ® on log (V) results in a power coefficient estimate of approximately .33, where the closeness of the approximation to .33 depends on the gain of the control system.

RM: So this could be considered the PCT explanation of the power law: What is being controlled when there is an observed power law relationship between V and R is the perceived difference between log(V) and 1/3 log® (or, equivalently, the perceived ratio V/R^1/3). This model will fit the data perfectly.

RM: What is wrong with this model is invisible as long as we don’t include in the model the mechanism that produces the variation in the variables V and R. The model treats V as an output produced by the control system and R and an environmental disturbance. But in fact, both V and R are consequences of system outputs. The system outputs that produce variations in V also produces variations in R. For example, when you move your finger in an elliptical trajectory the muscle forces – outputs – that affect the speed with which your finger is moving, V, at any instant, are, at the same time, affecting the degree of curvature, R, through which your finger is moving at that instant. So both V and R are variable consequences of your outputs (muscle forces); variations in R are not independent of these outputs so they cannot be considered disturbances to a controlled variable that is also affected by V.

RM: So an important aspect of modeling control is representing the physical situation correctly. In modeling things like finger movements it is important to see how the movement variables are physically related to each other; in finger movement it is important to recognize that both the velocity and curvature of movement are consequences of how the finger is moved by muscle forces – the mechanism that produces variations in both V and R

Best

Rick

···

On Sun, Sep 18, 2016 at 12:32 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.09.18.1230)]

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

RM: I see everyone is off talking about Roger’s model (or non-model, as Martin would have it) so I’m probably just talking to myself but, what the heck, this is supposed to be a list about PCT so I’ll sprinkle in a little just for my own satisfaction.

Martin Taylor (2016.09.17.22.56) –

``````MT: True. Apart from misrepresenting what Roger wrote, they referred to
``````

Roger’s words as if he had proposed a model,

RM: Well, it looks like a model to me so I’ll treat it as though it is one. In fact, this is a perfectly fine PCT model of a control system that controls velocity, Vp. A linear version of the system side of Roger’s model looks like this:

Vm = k*(Vr-Vp)

So the output of the control system, Vm, is proportional to the difference between the controlled perception, Vp, and the reference for that perception, Vr. What’s missing is a description of the perceptual function that produces Vp and the environmental feedback connection from system output, Vm, to controlled input. I will assume that Vp is the perception of the tangential velocity of the object that is moving. So I’ll define Vp is a function of the X, Y position of that object. So

Vp = k.i*Sqrt((dX/dt)^2 +(dY/dt)^2)

RM: That is, the perception of the velocity of the moved object is proportional (by the input constant k.i) to V as calculated in the power law studies. Now we have to figure out how Vm relates to Vp. In order to have an effect on Vp, the system must be able to affect dX/dt and dY/dt. That is, the system has to be able to change the position of the object in the X and Y dimensions independently. This means that the output of the control system can’t be a single variable, Vm. And a single control system can’t independently vary two variables. So it looks like we’re going to have to model control of velocity using two control systems, one controlling movement in the X dimension and the other controlling it in the Y dimension. But this would mean each system could control only a component of Vp, one controlling dX/dt and the other dY/dt. But that looks like the way it has to be. So velocity control (in two dimensions) requires two control systems, that could be defined as follows:

Px = dX/dt : Py = dY/dt

Ox = kx* (Rx-Px) : Oy = dY/dt

dX/dt = Ox= dx; dY/dt = Oy + dy

RM: So we have two control systems, one controlling movement in the X direction (Px) and the other controlling movement in the Y dimension (Py). The state of the variables controlled by each system is affected by the output of each system (Ox, Oy) and environmental disturbances (dx,dy).

RM: Neither of these two control systems alone controls tangential velocity, Vp. This has to be done by a higher level system that sends references to these systems controlling dX/dt and dY/dt). But a single higher level control system cannot vary the references for two lower level system independently and we know that people make movements where the movements in the X and Y dimensions are different. So how can we get the model to behave this way.

RM: This is where curvature might come in. Curvature, like tangential velocity,Vp, is a function of dX/dt and dY/dt. And it is a variable aspect of the movement that might also be controlled! So why not have curvature as a second, higher level controlled variable, Cp. So now we have two higher level control systems, one controlling Vp and the other Cp, where both Vp and Cp are functions of the perceptions controlled by the two lower level systems, Px and Py.

Vp =kv* sqrt(Px^2+Py^2)

Cp = kc* [|PxdY2/dt-dX2/dtPy)/sqrt(Px^2+Py^2)]

RM: Note that the perception of both velocity and curvature depend on Px and Py but curvature also depends on dX2/dt and dY2/dt. This suggests that I may need a couple more lower level systems to control these variables. Bad news. Clearly this requies more thought. But I think I will try to build this two level system and see what happens.

RM: But there are some important things that we have learned from this exercise. First, we need at least two control system in order to control the velocity of movement. In order to make the movement move at the desired velocity in the right direction we need another system to control curvature (or, now that I think of it, perhaps just a system that controls instantaneous direction rather than curvature). And, finally, curvature is a consequence of the same outputs (movement in the X and Y dimension) that affect velocity. So curvature is not a disturbance to the movement; it’s a consequence of output, as is velocity.

RM: I’ll try to make a diagram of this model and write a simulation asap. Again, just for my benefit; I now return you to your regularly schedule high integrity PCT program.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

RKM:

``````                      a)
``````

Let Vr be the intended (reference) velocity.

``````                      b)
``````

Let Vp be the perceived velocity.

``````                      c)
``````

Let e=Vr-Vp

``````                      d).
``````

Let Vm be the output velocity, where Vm is
some function of e.

``````                      e)
``````

If Vp=Vr, then e=0 and Vm will remain
unchanged.

``````                      f)
``````

If Vp>Vr, then Vm will be reduced until
e=0.

``````          RM: I guess my comments on Roger's model didn't make
``````

much of an impression in this new, high integrity world of
PCT.

[From Rick Marken (2016.09.21.2145)]

Roger K. Moore (2016.09.19.11.21)

RKM: Sorry, too overwhelmed (with students) to respond fully, but a couple of remarks on my suggested solution ...

- control can be in cartesian or vector space - two parameters in both cases - and I suspect the latter is more appropriate here (i.e. velocity and direction)

RM: The problem with themodel isn't how the space within which the movement takes place is represented. The problem is that you have not specified what is being moved and how it is being moved. Velocity and curvature are measures of the change in position over time of some object, such as a hand or a cursor, a fly larvae or a planet. The instantaneous position of the moving object can be represented in Cartesian or polar coordinates. The velocity and curvature of the movement can then be derived from changes in these measures of instantaneous position.
RM: Power law researchers base their measures of velocity and curvature on changes in instantaneous position represented in Cartesian (X, Y) coordinates. But these measures could also have been derived from changes in instantaneous position represented in polar (r, Theta) coordinates. And regardless of how the changes in instantaneous position are represented, the model must also explain what causes these changes in position. When the object moved is a finger, the cause of changes in instantaneous position are muscle forces. When the object moved is a planet, the cause of changes in instantaneous position is gravity.
RM: If the model fails to say what is moved and how it is moved then you end up with a pseudo- model (that can look quite convincing) like the one I described in Rick Marken (2016.09.19.1200). That model was described completely in terms of measures of velocity, V, and curvature, C, and it works perfectly until the basis of these measures -- what is being moved and how -- is taken into account. Here's a quick repeat of my pseudo-model explanation of the power law:

RM: While mulling over this modeling project it struck me that there is a PCT explanation of the power law if the law is taken at face value; as a decrease in velocity in response to an increase in curvature. The power law is:

V =k* R^1/3

RM: The PCT explanation of this law simply assumes that the controlled variable is a perception of the relationship between V and R, specifically:
P = log(V) - 1/3 log(R).

RM: The output variable of the system that controls this perception is V, the reference for P is 0 and the disturbance to the controlled variable is R. In order to keep log(V) - 1/3 log(R) equal to the reference value of 0 the control system has to vary V so that V = R^1/3. I have set up a control system that does this and it works like a charm; when the system controls P as defined above, a regression analysis of log (R) on log (V) results in a power coefficient estimate of approximately .33, where the closeness of the approximation to .33 depends on the gain of the control system.

RM: So this could be considered the PCT explanation of the power law: What is being controlled when there is an observed power law relationship between V and R is the perceived difference between log(V) and 1/3 log(R) (or, equivalently, the perceived ratio V/R^1/3). This model will fit the data perfectly.

Best
Rick

···

--
Richard S. Marken
"The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves." -- William T. Powers

[From Bruce Abbott (2016.09.22.0820 EDT)]

Rick Marken (2016.09.21.2145) –

Roger K. Moore (2016.09.19.11.21)

RKM: Sorry, too overwhelmed (with students) to respond fully, but a couple of remarks on my suggested solution …

• control can be in cartesian or vector space - two parameters in both cases - and I suspect the latter is more appropriate here (i.e. velocity and direction)

Hmmm – I didd not receive the cited post from Roger.Â There was some sort of interruption of service from CSGnet that I experienced on Tuesday (a post I sent to CSGnet did not show up in my inbox until over six hours later), so perhaps this is another result of that problem.

Bruce

[From Rick Marken (2016.09.22.1410)]

Bruce Abbott (2016.09.22.0820 EDT)--

Roger K. Moore (2016.09.19.11.21)

Â

RKM: Sorry, too overwhelmed (with students) to respond fully, but a couple of remarks on my suggested solution ...

Â - control can be in cartesian or vector space - two parameters in both cases -Â and I suspect the latter is more appropriate here (i.e. velocity and direction)

Â

BA: Hmmm  I did not receive the cited post from Roger.Â There was somee sort of interruption of service from CSGnet that I experienced on Tuesday (a post I sent to CSGnet did not show up in my inbox until over six hours later), so perhaps this is another result of that problem.

RM: You probably have it by now but in case not here it is:Â

[Roger K. Moore 2016.09.19.11.21)]
Sorry, too overwhelmed (with students) to respond fully, but a couple of remarks on my suggested solution ...
Â - control can be in cartesian or vector space - two parameters in both cases -Â and I suspect the latter is more appropriate here (i.e. velocity and direction)
Â - I imagined the error term controlling the acceleration (i.e.Vmdot), just as in cruise control on a vehicle - so one control loop for velocity (Vr in, Vm out)
Â - the direction sequence could be predetermined or use another control loop to steer (i.e. to follow a path)Â
Â - the key unknown (in my suggested solution) is indeed the perceptual feedback function
Cheers

Best
Rick

···

--
Richard S. MarkenÂ
"The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves." -- William T. Powers

[From Bruce Abbott (2016.09.22.1805 EDT)]

Rick Marken (2016.09.22.1410)–

Bruce Abbott (2016.09.22.0820 EDT)–

Roger K. Moore (2016.09.19.11.21)

RKM: Sorry, too overwhelmed (with students) to respond fully, but a couple of remarks on my suggested solution …

• control can be in cartesian or vector space - two parameters in both cases - and I suspect the latter is more appropriate here (i.e. velocity and direction)

BA: Hmmm – I did not receive the cited post from Roger. ; There was some sort of interruption of service from CSGnet that I experienced on Tuesday (a post I sent to CSGnet did not show up in my inbox until over six hours later), so perhaps this is another result of that problem.

RM: You probably have it by now but in case not here it is:

BA:Â Yes, I do, but thanks anyway!

[Roger K. Moore 2016.09.19.11.21)]
Sorry, too overwhelmed (with students) to respond fully, but a couple of remarks on my suggested solution …

• control can be in cartesian or vector space - two parameters in both cases - and I suspect the latter is more appropriate here (i.e. velocity and direction)
• I imagined the error term controlling the acceleration (i.e.Vmdot), just as in cruise control on a vehicle - so one control loop for velocity (Vr in, Vm out)
• the direction sequence could be predetermined or use another control loop to steer (i.e. to follow a path)
• the key unknown (in my suggested solution) is indeed the perceptual feedback function
Cheers

Bruce