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From: Fred Marshall on 1 Jul 2010 11:17
fisico32 wrote:
>> On 07/01/2010 06:53 AM, fisico32 wrote:
>>>> On 06/30/2010 12:03 PM, fisico32 wrote:
>>>>> Hello Forum
>>>>>
>>>>> given a LTI system, I want to make sure I understand the meaning of
>>>>> "zero state response" and "zero input response".
>>>>>
>>>>> The ZSR is the impulse response only existing if a system is acted
> upon
>>> by
>>>>> an external force. That assume that if there is not external force
> the
>>>>> system does nothing.
>>>> Correct.
>>>>
>>>> As an aside, you don't have to make your assumption: if the system is
>>>> LTI and the states are all zero then by definition the system does
>>> nothing.
>>>>> The ZIR is the natural response due to some initial conditions that
>>> then
>>>>> instantaneously disappear. The system may decay or not depending on
>>>>> absorption.....
>>>>>
>>>>> If this difference is correct, is an initial condition could be seen
> an
>>>>> external forcing function that exist only for an instant of time, so
> it
>>> can
>>>>> be considered as a special case of ZSR.... correct?
>>>> Almost. It is possible for a system to have states, or linear
>>>> combinations of states (called modes), that cannot be affected by the
>>>> input. These uncontrollable modes can still affect the output, and if
>>>> they have non-zero initial values then you'll see that in the output.
>>>>
>>>> You'll never see these modes in a transfer function -- transfer
>>>> functions more or less by definition only show controllable and
>>>> observable modes. But they can be there if you describe the system in
>>>> state space.
>>>>
>>>> --
>>>>
>>>> Tim Wescott
>>>> Wescott Design Services
>>>> http://www.wescottdesign.com
>>>>
>>>> Do you need to implement control loops in software?
>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>>>
>>>
>>> Thanks Tim.
>>>
>>> So, for example, a system like y(t)=x(t)+5
>>> It is pseudo-linear and not a zero response state system. If x(t)=0,
> the
>>> output is not zero but y(t)=0.
>> The term you're looking for isn't pseudo-linear, it is "affine". With a
>> shift of origin the system is linear. Alternately, you can model this
>> as a linear system with an extra input that you happen to set to 5, or
>> as a linear system with an extra integrator whose initial condition you
>> set to 5.
>>
>> Check your math. If x(t) = 0 then y(t) must be 5 -- and normally you
>> would take y(t) as the output.
>>> How about the system y(t)=x(t) +3t^2? Even if the input does not exist,
> the
>>> output seems to have its own existance and increase as t^2...What type
> of
>>> system is that? It is not nonlinea, but not even linear. It is not zero
>>> state..is it a zero input response system?
>> Strictly speaking it is a nonlinear system, because it doesn't pass the
>> superposition test. But it's also an affine system like the one above,
>> just with a time-varying offset (or input, or with a chain of three
>> integrators with appropriate initial values).
>>
>>> I would say that a if a system output is nonzero even if x(t)=0, then
> it
>>> must have some memory. Would it correspond to a LTI system of IIR type?
>> Does the system y(t) = 0 * x(t) + 1 have memory? Does it exhibit
>> superposition?
>>
>>> It is said that the "total" response is the sum of ZSR+ZIR. This is
> because
>>> the could have both an initial condition and a persisiting external
> force
>>> applied to the system....
>> For a linear system, yes (note that this is true even for a linear
>> time-varying system). If you want to pay attention to observability and
>> controllability, then say "visible response" instead of "response" and
>> you've covered your bases.
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Services
>> http://www.wescottdesign.com
>>
>> Do you need to implement control loops in software?
>> "Applied Control Theory for Embedded Systems" was written for you.
>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>
>
>
> Tim,
> by the way, since you mention LTV systems, a professor of mine said that
>
> ".....AN LTV system is a small-signal behavior of a nonlinear autonomous
> (time-invariant) system. In other words, a nonlinear (time-invariant)
> system can be written as a composit function y(t)=h o x(t) and if the
> system is analytical, can be expanded as an infinte sum of regular
> homogeneous linear terms. The constant term in this series is interpreted
> as the initial state (zero-input) response. LTI system is the first-degree
> (linear) homogenous term, LTV is the second-degree (linear) homogenous
> term, and so on. In that sense, the delta function is a homogeneous linear
> term of zero-degree. The constant term is also called the zero-degree
> impulse response, the corresponding LTV term is called a first-degree
> impulse response, and the LTV term is called a second-degree impulse
> response and so on. Using this terminology, the unit-impulse ffunction is
> called a minus-one homogeneous linear term. It is said that the LTV
> system is the least order (of second-degree) system that can satisfy the
> requirements of a nonlinear autonomous (time-invariant) system.
> Note that an autonomous LTV system is clasified as time-invariant because
> the system behaviour depends on |t1- t2|, i.e., the distance between the
> system time (t1) and the signal time (t2) (this is called a norm in
> mathematics) not the variations of individual time arguments!...."
>
>
> I am still trying to get my head around it. Do you agree and understand
> what he is stating?
> thanks
>

I don't think that I agree if I interpret the first sentence literally.
We had a very long thread here years ago debating whether time varying
systems can be linear systems. The answer is yes because the tests for
linearity can be met.
An example is a 4-quadrant multiplier with one input a fixed sinusoid
making it a time-varying system relative to the other input.
The linearity test is done with the other input vs. the output.
That "new frequencies" are introduced doesn't cause it to fail the test.

What would "small signal behavior" have to do with any of this? So,
perhaps that's part of the confusion / disagreement. Sure, one can
often observe small signal behavior and see that the system behaves much
like a linear system (time invariant or time varying as in the case of
the multiplier/modulator above). But that seems a departure from the
discussion here. I don't see what it adds.

I once had a friend (who was very smart and well educated) who suggested
that one cannot build a modulator without there being a "nonlinear
element or device" .... That may have been true with respect to the
components that were available but I never "got it" in a theoretical
sense. What is a "nonlinear element or device" anyway? How do you know
without performing the test for linearity? Can you perform the test for
linearity while considering multiple inputs? Maybe that has something
to do with it but it's not something that I know. I'll be that Tim does
though....

Fred
From: Tim Wescott on 1 Jul 2010 12:15
On 07/01/2010 07:35 AM, fisico32 wrote:
>> On 07/01/2010 06:53 AM, fisico32 wrote:
>>>> On 06/30/2010 12:03 PM, fisico32 wrote:
>>>>> Hello Forum
>>>>>
>>>>> given a LTI system, I want to make sure I understand the meaning of
>>>>> "zero state response" and "zero input response".
>>>>>
>>>>> The ZSR is the impulse response only existing if a system is acted
> upon
>>> by
>>>>> an external force. That assume that if there is not external force
> the
>>>>> system does nothing.
>>>>
>>>> Correct.
>>>>
>>>> As an aside, you don't have to make your assumption: if the system is
>>>> LTI and the states are all zero then by definition the system does
>>> nothing.
>>>>
>>>>> The ZIR is the natural response due to some initial conditions that
>>> then
>>>>> instantaneously disappear. The system may decay or not depending on
>>>>> absorption.....
>>>>>
>>>>> If this difference is correct, is an initial condition could be seen
> an
>>>>> external forcing function that exist only for an instant of time, so
> it
>>> can
>>>>> be considered as a special case of ZSR.... correct?
>>>>
>>>> Almost. It is possible for a system to have states, or linear
>>>> combinations of states (called modes), that cannot be affected by the
>>>> input. These uncontrollable modes can still affect the output, and if
>>>> they have non-zero initial values then you'll see that in the output.
>>>>
>>>> You'll never see these modes in a transfer function -- transfer
>>>> functions more or less by definition only show controllable and
>>>> observable modes. But they can be there if you describe the system in
>>>> state space.
>>>>
>>>> --
>>>>
>>>> Tim Wescott
>>>> Wescott Design Services
>>>> http://www.wescottdesign.com
>>>>
>>>> Do you need to implement control loops in software?
>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>>>
>>>
>>>
>>> Thanks Tim.
>>>
>>> So, for example, a system like y(t)=x(t)+5
>>> It is pseudo-linear and not a zero response state system. If x(t)=0,
> the
>>> output is not zero but y(t)=0.
>>
>> The term you're looking for isn't pseudo-linear, it is "affine". With a
>> shift of origin the system is linear. Alternately, you can model this
>> as a linear system with an extra input that you happen to set to 5, or
>> as a linear system with an extra integrator whose initial condition you
>> set to 5.
>>
>> Check your math. If x(t) = 0 then y(t) must be 5 -- and normally you
>> would take y(t) as the output.
>>>
>>> How about the system y(t)=x(t) +3t^2? Even if the input does not exist,
> the
>>> output seems to have its own existance and increase as t^2...What type
> of
>>> system is that? It is not nonlinea, but not even linear. It is not zero
>>> state..is it a zero input response system?
>>
>> Strictly speaking it is a nonlinear system, because it doesn't pass the
>> superposition test. But it's also an affine system like the one above,
>> just with a time-varying offset (or input, or with a chain of three
>> integrators with appropriate initial values).
>>
>>> I would say that a if a system output is nonzero even if x(t)=0, then
> it
>>> must have some memory. Would it correspond to a LTI system of IIR type?
>>
>> Does the system y(t) = 0 * x(t) + 1 have memory? Does it exhibit
>> superposition?
>>
>>> It is said that the "total" response is the sum of ZSR+ZIR. This is
> because
>>> the could have both an initial condition and a persisiting external
> force
>>> applied to the system....
>>
>> For a linear system, yes (note that this is true even for a linear
>> time-varying system). If you want to pay attention to observability and
>> controllability, then say "visible response" instead of "response" and
>> you've covered your bases.
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Services
>> http://www.wescottdesign.com
>>
>> Do you need to implement control loops in software?
>> "Applied Control Theory for Embedded Systems" was written for you.
>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>
>
>
> Tim,
> by the way, since you mention LTV systems, a professor of mine said that
>
> ".....AN LTV system is a small-signal behavior of a nonlinear autonomous
> (time-invariant) system. In other words, a nonlinear (time-invariant)
> system can be written as a composit function y(t)=h o x(t) and if the
> system is analytical, can be expanded as an infinte sum of regular
> homogeneous linear terms. The constant term in this series is interpreted
> as the initial state (zero-input) response. LTI system is the first-degree
> (linear) homogenous term, LTV is the second-degree (linear) homogenous
> term, and so on. In that sense, the delta function is a homogeneous linear
> term of zero-degree. The constant term is also called the zero-degree
> impulse response, the corresponding LTV term is called a first-degree
> impulse response, and the LTV term is called a second-degree impulse
> response and so on. Using this terminology, the unit-impulse ffunction is
> called a minus-one homogeneous linear term. It is said that the LTV
> system is the least order (of second-degree) system that can satisfy the
> requirements of a nonlinear autonomous (time-invariant) system.

The key term that's buried in there is "small-signal behavior".
"Small-signal behavior" means "in the limit as our input signal goes to
zero" (with emphasis on the zero), or perhaps "There's a bear! I bet we
can poke it if we don't poke it hard!"

So just as you can model a transistor amplifier as a linear system by
looking at it's "small signal AC behavior" (i.e. by noting that for
small inputs it looks like an affine system, and with blocking caps on
both input and output it looks pretty linear), if you have a nonlinear
system that's doing something (like oscillating) you can -- sometimes --
treat that as an affine time-varying system.

An example of this is a regenerative radio receiver. These clever
little gadgets were invented by Major Edwin Armstrong
(http://en.wikipedia.org/wiki/Regenerative_receiver). They work by
feeding an oscillator circuit with a weak signal from an antenna. When
you're receiving Morse code the signal from the antenna gets multiplied
to the oscillator's signal in the oscillator's amplifying element (tube
or transistor), and rectified to low frequencies, where it can be
amplified and applied to a speaker. In this case the whole system is
quite nonlinear, yet for a small enough input signal (so you can ignore
blocking, Jerry) its behavior is that of a linear time-varying system.

OTOH, when you waltz up to your new car and push the little button on
the key fob to make the doors unlock, chances are high that the receiver
in the car that senses the key fob is a superregenerative receiver.
There isn't a useful description of a superregen that doesn't take the
nonlinearities into account -- the way that a superregen achieves it's
astounding sensitivity is by turning an oscillator on and off (with a
nonlinear element), and rectifying its output (which is a nonlinear
behavior). The oscillator starts faster in the presence of a signal on
the antenna, which makes the output bigger right before quench; because
the signal is effectively amplified a bazzilion times by the oscillator
before quench vast sensitivity is achieved with just a few really cheap
components.

> Note that an autonomous LTV system is clasified as time-invariant because
> the system behaviour depends on |t1- t2|, i.e., the distance between the
> system time (t1) and the signal time (t2) (this is called a norm in
> mathematics) not the variations of individual time arguments!...."
>
>
> I am still trying to get my head around it. Do you agree and understand
> what he is stating?
> thanks

I don't quite get that last part -- but the first part does make some
sense, and in fact there's a lot of useful work that you can do with it.
Where I've used that approximation most is in controlling a nonlinear
system -- you close one eye and claim that the system's apparent
time-variance is slow enough that it won't affect the moment-to-moment
behavior of the system (yea, right), then you make a family of linear
system approximations over the state space of the nonlinear system, then
you design a controller that will achieve your desired performance over
the whole state space (or you find out that you can't, and go back to
the drawing board).

When I was in the process of getting out of school and for the first few
years after I was really pissed off about the emphasis on small-signal
behavior, because it is emphasized so much yet no one really goes into
depth into just how often the assumption is violated. Sometimes "small
signal" is uselessly teeny. Sometimes (e.g. with a push-pull amp) a
device shows its nonlinearities _more_ with small signals, and less with
large ones, at least in proportion to the input signal.

So I probably haven't paid as much attention to anything that invokes
"small signal" behavior as I should. Now I do accept that it is useful
-- but you have to be exquisitely aware of the limitations of the model
at all times, so that you know when a small-signal assumption is
appropriate, and when it is just paving on the garden path that you're
leading yourself down.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
From: HardySpicer on 1 Jul 2010 16:23
On Jul 1, 7:03 am, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com>
wrote:
> Hello Forum
>
> given a LTI system, I want to make sure I understand the meaning of
> "zero state response" and "zero input response".
>
> The ZSR is the impulse response only existing if a system is acted upon by
> an external force. That assume that if there is not external force the
> system does nothing.
>
> The ZIR is the natural response due to some initial conditions that then
> instantaneously disappear. The system may decay or not depending on
> absorption.....
>
> If this difference is correct, is an initial condition could be seen an
> external forcing function that exist only for an instant of time, so it can
> be considered as a special case of ZSR.... correct?
>
> thanks
> fisico32

Zero Input response is the response from initial conditions on the
state only with no input.

Hardy
From: fisico32 on 2 Jul 2010 10:01
>On 07/01/2010 07:35 AM, fisico32 wrote:
>>> On 07/01/2010 06:53 AM, fisico32 wrote:
>>>>> On 06/30/2010 12:03 PM, fisico32 wrote:
>>>>>> Hello Forum
>>>>>>
>>>>>> given a LTI system, I want to make sure I understand the meaning of
>>>>>> "zero state response" and "zero input response".
>>>>>>
>>>>>> The ZSR is the impulse response only existing if a system is acted
>> upon
>>>> by
>>>>>> an external force. That assume that if there is not external force
>> the
>>>>>> system does nothing.
>>>>>
>>>>> Correct.
>>>>>
>>>>> As an aside, you don't have to make your assumption: if the system
is
>>>>> LTI and the states are all zero then by definition the system does
>>>> nothing.
>>>>>
>>>>>> The ZIR is the natural response due to some initial conditions that
>>>> then
>>>>>> instantaneously disappear. The system may decay or not depending on
>>>>>> absorption.....
>>>>>>
>>>>>> If this difference is correct, is an initial condition could be
seen
>> an
>>>>>> external forcing function that exist only for an instant of time,
so
>> it
>>>> can
>>>>>> be considered as a special case of ZSR.... correct?
>>>>>
>>>>> Almost. It is possible for a system to have states, or linear
>>>>> combinations of states (called modes), that cannot be affected by
the
>>>>> input. These uncontrollable modes can still affect the output, and
if
>>>>> they have non-zero initial values then you'll see that in the
output.
>>>>>
>>>>> You'll never see these modes in a transfer function -- transfer
>>>>> functions more or less by definition only show controllable and
>>>>> observable modes. But they can be there if you describe the system
in
>>>>> state space.
>>>>>
>>>>> --
>>>>>
>>>>> Tim Wescott
>>>>> Wescott Design Services
>>>>> http://www.wescottdesign.com
>>>>>
>>>>> Do you need to implement control loops in software?
>>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>>>>
>>>>
>>>>
>>>> Thanks Tim.
>>>>
>>>> So, for example, a system like y(t)=x(t)+5
>>>> It is pseudo-linear and not a zero response state system. If x(t)=0,
>> the
>>>> output is not zero but y(t)=0.
>>>
>>> The term you're looking for isn't pseudo-linear, it is "affine". With
a
>>> shift of origin the system is linear. Alternately, you can model this
>>> as a linear system with an extra input that you happen to set to 5, or
>>> as a linear system with an extra integrator whose initial condition
you
>>> set to 5.
>>>
>>> Check your math. If x(t) = 0 then y(t) must be 5 -- and normally you
>>> would take y(t) as the output.
>>>>
>>>> How about the system y(t)=x(t) +3t^2? Even if the input does not
exist,
>> the
>>>> output seems to have its own existance and increase as t^2...What
type
>> of
>>>> system is that? It is not nonlinea, but not even linear. It is not
zero
>>>> state..is it a zero input response system?
>>>
>>> Strictly speaking it is a nonlinear system, because it doesn't pass
the
>>> superposition test. But it's also an affine system like the one
above,
>>> just with a time-varying offset (or input, or with a chain of three
>>> integrators with appropriate initial values).
>>>
>>>> I would say that a if a system output is nonzero even if x(t)=0, then
>> it
>>>> must have some memory. Would it correspond to a LTI system of IIR
type?
>>>
>>> Does the system y(t) = 0 * x(t) + 1 have memory? Does it exhibit
>>> superposition?
>>>
>>>> It is said that the "total" response is the sum of ZSR+ZIR. This is
>> because
>>>> the could have both an initial condition and a persisiting external
>> force
>>>> applied to the system....
>>>
>>> For a linear system, yes (note that this is true even for a linear
>>> time-varying system). If you want to pay attention to observability
and
>>> controllability, then say "visible response" instead of "response" and
>>> you've covered your bases.
>>>
>>> --
>>>
>>> Tim Wescott
>>> Wescott Design Services
>>> http://www.wescottdesign.com
>>>
>>> Do you need to implement control loops in software?
>>> "Applied Control Theory for Embedded Systems" was written for you.
>>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>>
>>
>>
>> Tim,
>> by the way, since you mention LTV systems, a professor of mine said
that
>>
>> ".....AN LTV system is a small-signal behavior of a nonlinear
autonomous
>> (time-invariant) system. In other words, a nonlinear (time-invariant)
>> system can be written as a composit function y(t)=h o x(t) and if the
>> system is analytical, can be expanded as an infinte sum of regular
>> homogeneous linear terms. The constant term in this series is
interpreted
>> as the initial state (zero-input) response. LTI system is the
first-degree
>> (linear) homogenous term, LTV is the second-degree (linear) homogenous
>> term, and so on. In that sense, the delta function is a homogeneous
linear
>> term of zero-degree. The constant term is also called the zero-degree
>> impulse response, the corresponding LTV term is called a first-degree
>> impulse response, and the LTV term is called a second-degree impulse
>> response and so on. Using this terminology, the unit-impulse ffunction
is
>> called a minus-one homogeneous linear term. It is said that the LTV
>> system is the least order (of second-degree) system that can satisfy
the
>> requirements of a nonlinear autonomous (time-invariant) system.
>
>The key term that's buried in there is "small-signal behavior".
>"Small-signal behavior" means "in the limit as our input signal goes to
>zero" (with emphasis on the zero), or perhaps "There's a bear! I bet we
>can poke it if we don't poke it hard!"
>
>So just as you can model a transistor amplifier as a linear system by
>looking at it's "small signal AC behavior" (i.e. by noting that for
>small inputs it looks like an affine system, and with blocking caps on
>both input and output it looks pretty linear), if you have a nonlinear
>system that's doing something (like oscillating) you can -- sometimes --
>treat that as an affine time-varying system.
>
>An example of this is a regenerative radio receiver. These clever
>little gadgets were invented by Major Edwin Armstrong
>(http://en.wikipedia.org/wiki/Regenerative_receiver). They work by
>feeding an oscillator circuit with a weak signal from an antenna. When
>you're receiving Morse code the signal from the antenna gets multiplied
>to the oscillator's signal in the oscillator's amplifying element (tube
>or transistor), and rectified to low frequencies, where it can be
>amplified and applied to a speaker. In this case the whole system is
>quite nonlinear, yet for a small enough input signal (so you can ignore
>blocking, Jerry) its behavior is that of a linear time-varying system.
>
>OTOH, when you waltz up to your new car and push the little button on
>the key fob to make the doors unlock, chances are high that the receiver
>in the car that senses the key fob is a superregenerative receiver.
>There isn't a useful description of a superregen that doesn't take the
>nonlinearities into account -- the way that a superregen achieves it's
>astounding sensitivity is by turning an oscillator on and off (with a
>nonlinear element), and rectifying its output (which is a nonlinear
>behavior). The oscillator starts faster in the presence of a signal on
>the antenna, which makes the output bigger right before quench; because
>the signal is effectively amplified a bazzilion times by the oscillator
>before quench vast sensitivity is achieved with just a few really cheap
>components.
>
>> Note that an autonomous LTV system is clasified as time-invariant
because
>> the system behaviour depends on |t1- t2|, i.e., the distance between
the
>> system time (t1) and the signal time (t2) (this is called a norm in
>> mathematics) not the variations of individual time arguments!...."
>>
>>
>> I am still trying to get my head around it. Do you agree and understand
>> what he is stating?
>> thanks
>
>I don't quite get that last part -- but the first part does make some
>sense, and in fact there's a lot of useful work that you can do with it.
> Where I've used that approximation most is in controlling a nonlinear
>system -- you close one eye and claim that the system's apparent
>time-variance is slow enough that it won't affect the moment-to-moment
>behavior of the system (yea, right), then you make a family of linear
>system approximations over the state space of the nonlinear system, then
>you design a controller that will achieve your desired performance over
>the whole state space (or you find out that you can't, and go back to
>the drawing board).
>
>When I was in the process of getting out of school and for the first few
>years after I was really pissed off about the emphasis on small-signal
>behavior, because it is emphasized so much yet no one really goes into
>depth into just how often the assumption is violated. Sometimes "small
>signal" is uselessly teeny. Sometimes (e.g. with a push-pull amp) a
>device shows its nonlinearities _more_ with small signals, and less with
>large ones, at least in proportion to the input signal.
>
>So I probably haven't paid as much attention to anything that invokes
>"small signal" behavior as I should. Now I do accept that it is useful
>-- but you have to be exquisitely aware of the limitations of the model
>at all times, so that you know when a small-signal assumption is
>appropriate, and when it is just paving on the garden path that you're
>leading yourself down.
>
>--
>
>Tim Wescott
>Wescott Design Services
>http://www.wescottdesign.com
>
>Do you need to implement control loops in software?
>"Applied Control Theory for Embedded Systems" was written for you.
>See details at http://www.wescottdesign.com/actfes/actfes.html
>

Ok, Tim, forgive my low level of understanding: I understand what small
signal behavior is: approximating a nonlinear system as a linear one by
linearizing it;
I understand what a composite function is and how it could be expanded into
a series. In this case the functions in the composite function are
the input x(t) and h.
But I don't get why the 2nd term of this expansion is the LTV system....
What are the basis functions used in this series expansion?

How is it possible that the nonlinear system is time-invariant while the
second term in this expansion is the linear "time variant"? That seems a
contradiction...what am i missing?
From: Tim Wescott on 2 Jul 2010 12:05
On 07/02/2010 07:01 AM, fisico32 wrote:
>> On 07/01/2010 07:35 AM, fisico32 wrote:
>>>> On 07/01/2010 06:53 AM, fisico32 wrote:
>>>>>> On 06/30/2010 12:03 PM, fisico32 wrote:
>>>>>>> Hello Forum
>>>>>>>
>>>>>>> given a LTI system, I want to make sure I understand the meaning of
>>>>>>> "zero state response" and "zero input response".
>>>>>>>
>>>>>>> The ZSR is the impulse response only existing if a system is acted
>>> upon
>>>>> by
>>>>>>> an external force. That assume that if there is not external force
>>> the
>>>>>>> system does nothing.
>>>>>>
>>>>>> Correct.
>>>>>>
>>>>>> As an aside, you don't have to make your assumption: if the system
> is
>>>>>> LTI and the states are all zero then by definition the system does
>>>>> nothing.
>>>>>>
>>>>>>> The ZIR is the natural response due to some initial conditions that
>>>>> then
>>>>>>> instantaneously disappear. The system may decay or not depending on
>>>>>>> absorption.....
>>>>>>>
>>>>>>> If this difference is correct, is an initial condition could be
> seen
>>> an
>>>>>>> external forcing function that exist only for an instant of time,
> so
>>> it
>>>>> can
>>>>>>> be considered as a special case of ZSR.... correct?
>>>>>>
>>>>>> Almost. It is possible for a system to have states, or linear
>>>>>> combinations of states (called modes), that cannot be affected by
> the
>>>>>> input. These uncontrollable modes can still affect the output, and
> if
>>>>>> they have non-zero initial values then you'll see that in the
> output.
>>>>>>
>>>>>> You'll never see these modes in a transfer function -- transfer
>>>>>> functions more or less by definition only show controllable and
>>>>>> observable modes. But they can be there if you describe the system
> in
>>>>>> state space.
>>>>>>
>>>>>> --
>>>>>>
>>>>>> Tim Wescott
>>>>>> Wescott Design Services
>>>>>> http://www.wescottdesign.com
>>>>>>
>>>>>> Do you need to implement control loops in software?
>>>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>>>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>>>>>
>>>>>
>>>>>
>>>>> Thanks Tim.
>>>>>
>>>>> So, for example, a system like y(t)=x(t)+5
>>>>> It is pseudo-linear and not a zero response state system. If x(t)=0,
>>> the
>>>>> output is not zero but y(t)=0.
>>>>
>>>> The term you're looking for isn't pseudo-linear, it is "affine". With
> a
>>>> shift of origin the system is linear. Alternately, you can model this
>>>> as a linear system with an extra input that you happen to set to 5, or
>>>> as a linear system with an extra integrator whose initial condition
> you
>>>> set to 5.
>>>>
>>>> Check your math. If x(t) = 0 then y(t) must be 5 -- and normally you
>>>> would take y(t) as the output.
>>>>>
>>>>> How about the system y(t)=x(t) +3t^2? Even if the input does not
> exist,
>>> the
>>>>> output seems to have its own existance and increase as t^2...What
> type
>>> of
>>>>> system is that? It is not nonlinea, but not even linear. It is not
> zero
>>>>> state..is it a zero input response system?
>>>>
>>>> Strictly speaking it is a nonlinear system, because it doesn't pass
> the
>>>> superposition test. But it's also an affine system like the one
> above,
>>>> just with a time-varying offset (or input, or with a chain of three
>>>> integrators with appropriate initial values).
>>>>
>>>>> I would say that a if a system output is nonzero even if x(t)=0, then
>>> it
>>>>> must have some memory. Would it correspond to a LTI system of IIR
> type?
>>>>
>>>> Does the system y(t) = 0 * x(t) + 1 have memory? Does it exhibit
>>>> superposition?
>>>>
>>>>> It is said that the "total" response is the sum of ZSR+ZIR. This is
>>> because
>>>>> the could have both an initial condition and a persisiting external
>>> force
>>>>> applied to the system....
>>>>
>>>> For a linear system, yes (note that this is true even for a linear
>>>> time-varying system). If you want to pay attention to observability
> and
>>>> controllability, then say "visible response" instead of "response" and
>>>> you've covered your bases.
>>>>
>>>> --
>>>>
>>>> Tim Wescott
>>>> Wescott Design Services
>>>> http://www.wescottdesign.com
>>>>
>>>> Do you need to implement control loops in software?
>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>>>
>>>
>>>
>>> Tim,
>>> by the way, since you mention LTV systems, a professor of mine said
> that
>>>
>>> ".....AN LTV system is a small-signal behavior of a nonlinear
> autonomous
>>> (time-invariant) system. In other words, a nonlinear (time-invariant)
>>> system can be written as a composit function y(t)=h o x(t) and if the
>>> system is analytical, can be expanded as an infinte sum of regular
>>> homogeneous linear terms. The constant term in this series is
> interpreted
>>> as the initial state (zero-input) response. LTI system is the
> first-degree
>>> (linear) homogenous term, LTV is the second-degree (linear) homogenous
>>> term, and so on. In that sense, the delta function is a homogeneous
> linear
>>> term of zero-degree. The constant term is also called the zero-degree
>>> impulse response, the corresponding LTV term is called a first-degree
>>> impulse response, and the LTV term is called a second-degree impulse
>>> response and so on. Using this terminology, the unit-impulse ffunction
> is
>>> called a minus-one homogeneous linear term. It is said that the LTV
>>> system is the least order (of second-degree) system that can satisfy
> the
>>> requirements of a nonlinear autonomous (time-invariant) system.
>>
>> The key term that's buried in there is "small-signal behavior".
>> "Small-signal behavior" means "in the limit as our input signal goes to
>> zero" (with emphasis on the zero), or perhaps "There's a bear! I bet we
>> can poke it if we don't poke it hard!"
>>
>> So just as you can model a transistor amplifier as a linear system by
>> looking at it's "small signal AC behavior" (i.e. by noting that for
>> small inputs it looks like an affine system, and with blocking caps on
>> both input and output it looks pretty linear), if you have a nonlinear
>> system that's doing something (like oscillating) you can -- sometimes --
>> treat that as an affine time-varying system.
>>
>> An example of this is a regenerative radio receiver. These clever
>> little gadgets were invented by Major Edwin Armstrong
>> (http://en.wikipedia.org/wiki/Regenerative_receiver). They work by
>> feeding an oscillator circuit with a weak signal from an antenna. When
>> you're receiving Morse code the signal from the antenna gets multiplied
>> to the oscillator's signal in the oscillator's amplifying element (tube
>> or transistor), and rectified to low frequencies, where it can be
>> amplified and applied to a speaker. In this case the whole system is
>> quite nonlinear, yet for a small enough input signal (so you can ignore
>> blocking, Jerry) its behavior is that of a linear time-varying system.
>>
>> OTOH, when you waltz up to your new car and push the little button on
>> the key fob to make the doors unlock, chances are high that the receiver
>> in the car that senses the key fob is a superregenerative receiver.
>> There isn't a useful description of a superregen that doesn't take the
>> nonlinearities into account -- the way that a superregen achieves it's
>> astounding sensitivity is by turning an oscillator on and off (with a
>> nonlinear element), and rectifying its output (which is a nonlinear
>> behavior). The oscillator starts faster in the presence of a signal on
>> the antenna, which makes the output bigger right before quench; because
>> the signal is effectively amplified a bazzilion times by the oscillator
>> before quench vast sensitivity is achieved with just a few really cheap
>> components.
>>
>>> Note that an autonomous LTV system is clasified as time-invariant
> because
>>> the system behaviour depends on |t1- t2|, i.e., the distance between
> the
>>> system time (t1) and the signal time (t2) (this is called a norm in
>>> mathematics) not the variations of individual time arguments!...."
>>>
>>>
>>> I am still trying to get my head around it. Do you agree and understand
>>> what he is stating?
>>> thanks
>>
>> I don't quite get that last part -- but the first part does make some
>> sense, and in fact there's a lot of useful work that you can do with it.
>> Where I've used that approximation most is in controlling a nonlinear
>> system -- you close one eye and claim that the system's apparent
>> time-variance is slow enough that it won't affect the moment-to-moment
>> behavior of the system (yea, right), then you make a family of linear
>> system approximations over the state space of the nonlinear system, then
>> you design a controller that will achieve your desired performance over
>> the whole state space (or you find out that you can't, and go back to
>> the drawing board).
>>
>> When I was in the process of getting out of school and for the first few
>> years after I was really pissed off about the emphasis on small-signal
>> behavior, because it is emphasized so much yet no one really goes into
>> depth into just how often the assumption is violated. Sometimes "small
>> signal" is uselessly teeny. Sometimes (e.g. with a push-pull amp) a
>> device shows its nonlinearities _more_ with small signals, and less with
>> large ones, at least in proportion to the input signal.
>>
>> So I probably haven't paid as much attention to anything that invokes
>> "small signal" behavior as I should. Now I do accept that it is useful
>> -- but you have to be exquisitely aware of the limitations of the model
>> at all times, so that you know when a small-signal assumption is
>> appropriate, and when it is just paving on the garden path that you're
>> leading yourself down.
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Services
>> http://www.wescottdesign.com
>>
>> Do you need to implement control loops in software?
>> "Applied Control Theory for Embedded Systems" was written for you.
>> See details at http://www.wescottdesign.com/actfes/actfes.html
>>
>
> Ok, Tim, forgive my low level of understanding: I understand what small
> signal behavior is: approximating a nonlinear system as a linear one by
> linearizing it;
> I understand what a composite function is and how it could be expanded into
> a series. In this case the functions in the composite function are
> the input x(t) and h.
> But I don't get why the 2nd term of this expansion is the LTV system....
> What are the basis functions used in this series expansion?

It sounds like the guy is using something very like a plain ol' Taylor's
series expansion of the state evolution (or state transition) function.
So if the system is defined by

dx/dt = f(x)

then you can do a Taylor's series expansion around x0:

dx/dt = f(x0) + (x * d/dx f(x0))/2 + ...
+ (x^n * d^n/dx^n f(x0))/n!

The first term is constant, the second term appears to be a simple gain,
and we close our eyes and ignore the rest of the terms. So, one way to
linearize a system would be to choose some x0 and take the linearization
as gospel.

> How is it possible that the nonlinear system is time-invariant while the
> second term in this expansion is the linear "time variant"? That seems a
> contradiction...what am i missing?

But assume that you've got a nonlinear system that you're going to
perturb ever so slightly, and that you want to know it's behavior over a
time period that's long enough so that your two-term Taylor's series
expansion isn't going to be good enough. One thing you could do is just
accept the rest of the terms in the Taylor's series, but they you've
forked yourself back into full-on nonlinear analysis.

Another thing you could do -- if your perturbations are small enough --
is to find the system's "ideal" response. Then, instead of taking x0 as
a constant, take x0 as a function over time. This makes those first two
terms of your Taylor's expansion into functions of time -- but you still
have a description of a system that's affine (which makes it trivial to
turn into a linear system by subtracting f(x0(t))) and time varying
(because x0(t) is now a function of time). So you can use linear
time-varying analysis, which, while harder than linear time-invariant
analysis, can still be loads better than nonlinear analysis.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
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