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 nonzero 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 nonzero 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 pseudolinear 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 nonzero 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 pseudolinear 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 pseudolinear, 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 timevarying 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 timevarying 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 nonzero 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 pseudolinear 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 pseudolinear, 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 timevarying 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 >timevarying 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 smallsignal behavior of a nonlinear autonomous (timeinvariant) system. In other words, a nonlinear (timeinvariant) 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 (zeroinput) response. LTI system is the firstdegree (linear) homogenous term, LTV is the seconddegree (linear) homogenous term, and so on. In that sense, the delta function is a homogeneous linear term of zerodegree. The constant term is also called the zerodegree impulse response, the corresponding LTV term is called a firstdegree impulse response, and the LTV term is called a seconddegree impulse response and so on. Using this terminology, the unitimpulse ffunction is called a minusone homogeneous linear term. It is said that the LTV system is the least order (of seconddegree) system that can satisfy the requirements of a nonlinear autonomous (timeinvariant) system. Note that an autonomous LTV system is clasified as timeinvariant 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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