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From: fisico32 on 30 Jun 2010 15:03
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



From: Tim Wescott on 30 Jun 2010 18:01
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
From: fisico32 on 1 Jul 2010 09:53
>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.

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?

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?

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....

From: Tim Wescott on 1 Jul 2010 10:23
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
From: fisico32 on 1 Jul 2010 10:35
>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

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