nth-order activity of continuous systems a,b,c rodrigo castro and c ernesto kofman a eth zürich,...
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nth-order Activity of Continuous Systemsa,b,cRodrigo Castro and cErnesto Kofman
aETH Zürich, SwitzerlandbUniversity of Buenos Aires & cCIFASIS-CONICET, Argentina
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 2
• Introduction– Activity: Original definition– Need for an nth-order extension
• nth-Order Quantization– Zero (static), First and Second Order– nth-Order Quantization– Quantized State Systems (QSS)
• nth-Order Activity– The error perspective– nth-Order error dynamics– Definition of nth-Order Activity
• Examples– Example I: 1st. Order Non-stiff system– Example II: 2nd. order Stiff system
• Conclusions & Future Work
Agenda
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 3
Activity: Original definition Introduction
• The original definition of activity takes into account changes only in the signal values.
x1(t)
x2(t)
x3(t)
x1(t0)=x2(t0)=x3(t0) x1(tf)=x2(tf)=x3(tf)
t0 tf
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 4
Activity: Original definition Introduction
• The original definition of activity takes into account changes only in the signal values.
• As a consequence, for a monotonically increasing or decreasing signal– the activity can be fully determined only
by the distance between the final and the initial value,
x1(t)
x2(t)
x3(t)
x1(t0)=x2(t0)=x3(t0) x1(tf)=x2(tf)=x3(tf)
t0 tf
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 5
Activity: Original definition Introduction
• The original definition of activity takes into account changes only in the signal values.
• As a consequence, for a monotonically increasing or decreasing signal– the activity can be fully determined only
by the distance between the final and the initial value,
– without using at all the information about how it goes from the initial to the final value.
x1(t)
x2(t)
x3(t)
x1(t0)=x2(t0)=x3(t0) x1(tf)=x2(tf)=x3(tf)
t0 tf
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 6
Activity: Original definition Introduction
• The original definition of activity takes into account changes only in the signal values.
• As a consequence, for a monotonically increasing or decreasing signal– the activity can be fully determined only
by the distance between the final and the initial value,
– without using at all the information about how it goes from the initial to the final value.
x1(t)
x2(t)
x3(t)
x1(t0)=x2(t0)=x3(t0) x1(tf)=x2(tf)=x3(tf) A1=A2=A3
t0 tf
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 7
Activity: Original definition Introduction
• The original definition of activity takes into account changes only in the signal values.
• As a consequence, for a monotonically increasing or decreasing signal– the activity can be fully determined only
by the distance between the final and the initial value,
– without using at all the information about how it goes from the initial to the final value.
x1(t)
x2(t)
x3(t)
x1(t0)=x2(t0)=x3(t0) x1(tf)=x2(tf)=x3(tf) A1=A2=A3
t0 tf
A1=A2=A3
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 8
x2(t)
Activity: Original definition Introduction
• When a continuous signal is quantized with a zero-order quantization function– we obtain the well-known
piecewise constant trajectory
tf1tf2tf3t0
m1
m2
m3
∆Qi
xi(t) qi(t)
∆Qi
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 9
x2(t)
Activity: Original definition Introduction
• When a continuous signal is quantized with a zero-order quantization function– we obtain the well-known
piecewise constant trajectory
tf1tf2tf3t0
m1
m2
m3
– For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is:
∆Qi
xi(t) qi(t)
∆Qi
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 10
x2(t)
Activity: Original definition Introduction
• When a continuous signal is quantized with a zero-order quantization function– we obtain the well-known
piecewise constant trajectory
tf1tf2tf3t0
m1
m2
m3
– For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is:
∆Qi
xi(t) qi(t)
∆Qi
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 11
x2(t)
Activity: Original definition Introduction
• When a continuous signal is quantized with a zero-order quantization function– we obtain the well-known
piecewise constant trajectory
tf1tf2tf3t0
x1(t)
m1
m2
m3
– For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is:
∆Qi
xi(t) qi(t)
∆Qi
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 12
x2(t)
Activity: Original definition Introduction
• When a continuous signal is quantized with a zero-order quantization function– we obtain the well-known
piecewise constant trajectory
tf1tf2tf3t0
x1(t)x3(t)
m1
m2
m3
– For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is:
∆Qi
xi(t) qi(t)
∆Qi
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 13
x2(t)
Activity: Original definition Introduction
• When a continuous signal is quantized with a zero-order quantization function– we obtain the well-known
piecewise constant trajectory
x1(t0)=x2(t0)=x3(t0) x1(tf1)=x2(tf2)=x3(tf3) A1=A2=A3
tf1tf2tf3t0
x1(t)x3(t)
m1
m2
m3
– For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is:
∆Qi
xi(t) qi(t)
∆Qi
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 14
• Zero-order quantization functions are those used in first-order accurate QSS numerical integration methods– QSS1, LIQSS1, CQSS, BQSS
Activity: Original definition Introduction
– For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is:
xi(t) qi(t)
∆Qi
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 15
• Zero-order quantization functions are those used in first-order accurate QSS numerical integration methods– QSS1, LIQSS1, CQSS, BQSS– For these methods, the number of
signal quantum crossings can establish a lower bound for the number of integration steps• required to approximate the analytical solution • with an accuracy (maximum error) bounded by the quantum size
Activity: Original definition Introduction
– For each interval of time at which x(t) is monotonic, the number of signal quantum crossings is:
xi(t) qi(t)
∆Qi
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 16
• “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational
effort (# integration steps)
Need for an nth-order extension Introduction
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 17
• “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational
effort (# integration steps)
• Nice features:– Convenient and intuitive visual relation between the solution x(t) and its quantized
version q(t)– Can be easily expressed in terms of maxs and mins:
Need for an nth-order extension Introduction
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 18
• “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational
effort (# integration steps)
• Nice features:– Convenient and intuitive visual relation between the solution x(t) and its quantized
version q(t)– Can be easily expressed in terms of maxs and mins:
• Disadvantages:– It works only for first order accurate methods.– Not valid for higher order accurate methods
• Existing QSS methods: QSS 1 to 4, LIQSS 1 to 4, DQSS 1 to 3
Need for an nth-order extension Introduction
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 19
• “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational
effort (# integration steps)
• Nice features:– Convenient and intuitive visual relation between the solution x(t) and its quantized
version q(t)– Can be easily expressed in terms of maxs and mins:
• Disadvantages:– It works only for first order accurate methods.– Not valid for higher order accurate methods
• Existing QSS methods: QSS 1 to 4, LIQSS 1 to 4, DQSS 1 to 3
– Intuition: QSS1 vs. QSS2• For a given ∆Q,
1st. Order Activity is the same# of Steps is NOT the same
Need for an nth-order extension Introduction
QSS1 QSS2zero order quantizationq(t) piecewise constant
first order quantizationq(t) piecewise linear
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 20
• “Classical”, “First order” Activity – Offers then a link between integration accuracy (Quantum size) and computational
effort (# integration steps)
• Nice features:– Convenient and intuitive visual relation between the solution x(t) and its quantized
version q(t)– Can be easily expressed in terms of maxs and mins:
• Disadvantages:– It works only for first order accurate methods.– Not valid for higher order accurate methods
• Existing QSS methods: QSS 1 to 4, LIQSS 1 to 4, DQSS 1 to 3
– Intuition: QSS1 vs. QSS2• For a given ∆Q,
1st. Order Activity is the same# of Steps is NOT the same
• A formal extension forActivity of nth-order is required.
Need for an nth-order extension Introduction
QSS1 QSS2zero order quantizationq(t) piecewise constant
first order quantizationq(t) piecewise linear
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 21
• Quantization: the key “error-driven” process– Zero-order (static)
Zero (static) and First Order nth-Order Quantization
k1=A1/∆Q
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 22
• Quantization: the key “error-driven” process– Zero-order (static)
Zero (static) and First Order nth-Order Quantization
k1=A1/∆Q
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 23
• Quantization: the key “error-driven” process– Zero-order (static)
Zero (static) and First Order nth-Order Quantization
err(t) k1=A1/∆Q
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 24
• Quantization: the key “error-driven” process– Zero-order (static)
Zero (static) and First Order nth-Order Quantization
err(t)
q(t)piecewise constant
k1=A1/∆Q
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 25
• Quantization: the key “error-driven” process– Zero-order (static)
– First-order
Zero (static) and First Order nth-Order Quantization
err(t)
q(t)piecewise constant
k1=A1/∆Q
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 26
• Quantization: the key “error-driven” process– Zero-order (static)
– First-order
Zero (static) and First Order nth-Order Quantization
err(t)
q(t)piecewise constant
k1=A1/∆Q
k2<k1
1
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 27
• Quantization: the key “error-driven” process– Zero-order (static)
– First-order
Zero (static) and First Order nth-Order Quantization
err(t)
q(t)piecewise constant
k1=A1/∆Q
k2<k1
1
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 28
• Quantization: the key “error-driven” process– Zero-order (static)
– First-order
Zero (static) and First Order nth-Order Quantization
err(t)
err(t)
q(t)piecewise constant
k1=A1/∆Q
k2<k1
1
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 29
• Quantization: the key “error-driven” process– Zero-order (static)
– First-order
Zero (static) and First Order nth-Order Quantization
err(t)
err(t)
q(t)piecewise constant
q(t)piecewise linear
k1=A1/∆Q
k2<k1
1
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 30
• Quantization: the key “error-driven” process– Zero-order (static)
– First-order
Zero (static) and First Order nth-Order Quantization
err(t)
err(t)
q(t)piecewise constant
q(t)piecewise linear
• No visual “quantization grid” available anymore• Now also “how” the signal grows matters (e.g. only one event needed to quantize x(t)=k.t)
k1=A1/∆Q
k2<k1
1
polynomial segments j=0,1,2,…
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 31
• Quantization: the key “error-driven” process– Second-order
Second Order nth-Order Quantization
k3<k2
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 32
• Quantization: the key “error-driven” process– Second-order
Second Order nth-Order Quantization
err(t)
q(t)piecewise parabolic
k3<k2
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 33
• Quantization: the key “error-driven” process– Second-order
– The quantization scheme directly determinesthe number of “polynomial segments” (steps) required
Second Order nth-Order Quantization
err(t)
q(t)piecewise parabolic
k3<k2
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 34
• Quantization: the key “error-driven” process– Second-order
– The quantization scheme directly determinesthe number of “polynomial segments” (steps) required
– We will start with a definition of quantization of order nwhich will lead us to a definition of activity of order n
Second Order nth-Order Quantization
err(t)
q(t)piecewise parabolic
k3<k2
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 35
• Quantization: the key “error-driven” process– nth-order (with n>0)
nth-Order nth-Order Quantization
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 36
• Quantization: the key “error-driven” process– nth-order (with n>0)
nth-Order nth-Order Quantization
err(t)
q(t)piecewise nth-order
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 37
• Quantization: the key “error-driven” process– nth-order (with n>0)
• m=1,2,…,n
nth-Order nth-Order Quantization
err(t)
q(t)piecewise nth-order
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 38
• The quantization process keeps track of the dynamics of the error between an input signal and its quantized version– It is the key mechanism used by QSS integrators for error control– QSS: Already discussed in previous presentations. Quick recap:
Quantized State Systems (QSS) nth-Order Quantization
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 39
• The quantization process keeps track of the dynamics of the error between an input signal and its quantized version– It is the key mechanism used by QSS integrators for error control– QSS: Already discussed in previous presentations. Quick recap:
Quantized State Systems (QSS) nth-Order Quantization
quantized integratorpure
integrator
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 40
• The quantization process keeps track of the dynamics of the error between an input signal and its quantized version– It is the key mechanism used by QSS integrators for error control– QSS: Already discussed in previous presentations. Quick recap:
Quantized State Systems (QSS) nth-Order Quantization
quantized integratorpure
integrator
Event={c0} QSS1
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 41
• The quantization process keeps track of the dynamics of the error between an input signal and its quantized version– It is the key mechanism used by QSS integrators for error control– QSS: Already discussed in previous presentations. Quick recap:
Quantized State Systems (QSS) nth-Order Quantization
quantized integratorpure
integrator
Event={c0} QSS1QSS2Event={c0,c1}
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 42
• The quantization process keeps track of the dynamics of the error between an input signal and its quantized version– It is the key mechanism used by QSS integrators for error control– QSS: Already discussed in previous presentations. Quick recap:
Quantized State Systems (QSS) nth-Order Quantization
quantized integratorpure
integrator
Event={c0} QSS1QSS2Event={c0,c1}
QSS3Event={c0,c1,c2}
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 43
• The original definition of Activity integrates the rate of change of the signal x(t):
The error perspective nth-Order Activity
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 44
• The original definition of Activity integrates the rate of change of the signal x(t):
• When q(t) is the result of a zero-order quantization, the rate of change of the signal x(t) coincides with the rate of growth of the error |q(t)-x(t)|
The error perspective nth-Order Activity
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 45
• The original definition of Activity integrates the rate of change of the signal x(t):
• When q(t) is the result of a zero-order quantization, the rate of change of the signal x(t) coincides with the rate of growth of the error |q(t)-x(t)|
– Consequently this formula works:
The error perspective nth-Order Activity
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 46
• The original definition of Activity integrates the rate of change of the signal x(t):
• When q(t) is the result of a zero-order quantization, the rate of change of the signal x(t) coincides with the rate of growth of the error |q(t)-x(t)|
– Consequently this formula works:
The error perspective nth-Order Activity
• But …
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 47
• When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different:
nth-Order error dynamics nth-Order Activity
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 48
• When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different:
nth-Order error dynamics nth-Order Activity
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 49
• When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different:
nth-Order error dynamics nth-Order Activity
=
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 50
• When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different:
• Let us replace x(t) by its Taylor series expansion:
nth-Order error dynamics nth-Order Activity
=
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 51
• When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different:
• Let us replace x(t) by its Taylor series expansion:
nth-Order error dynamics nth-Order Activity
=
=
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 52
• When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different:
• Let us replace x(t) by its Taylor series expansion:
• Then, the dynamics of the error can be expressed as:
nth-Order error dynamics nth-Order Activity
=
=
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 53
• When q(t) is the result of a high order quantization, the rate of growth of the error |q(t)-x(t)| looks different:
• Let us replace x(t) by its Taylor series expansion:
• Then, the dynamics of the error can be expressed as:
nth-Order error dynamics nth-Order Activity
=
=
=0 in all *QSSn methods
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 54
Definition of nth-Order Activity nth-Order Activity
• So, we have error
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 55
Definition of nth-Order Activity nth-Order Activity
• So, we have error
• The discontinuities in the polynomial segments (i.e., the integration steps) occur when:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 56
Definition of nth-Order Activity nth-Order Activity
err(tj+1)=∆Q
• So, we have error
• The discontinuities in the polynomial segments (i.e., the integration steps) occur when:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 57
Definition of nth-Order Activity nth-Order Activity
err(tj+1)=∆Q
holds, producing “one new step”:
• So, we have error
• The discontinuities in the polynomial segments (i.e., the integration steps) occur when:
• Therefore, at each new discontinuity instant tj+1
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 58
Definition of nth-Order Activity nth-Order Activity
err(tj+1)=∆Q
holds, producing “one new step”:
• So, we have error
• The discontinuities in the polynomial segments (i.e., the integration steps) occur when:
• Therefore, at each new discontinuity instant tj+1
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 59
Definition of nth-Order Activity nth-Order Activity
err(tj+1)=∆Q
holds, producing “one new step”:
• So, we have error
• The discontinuities in the polynomial segments (i.e., the integration steps) occur when:
• Therefore, at each new discontinuity instant tj+1
• Adding up these steps throughout all segments:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 60
Definition of nth-Order Activity nth-Order Activity
err(tj+1)=∆Q
holds, producing “one new step”:
• So, we have error
• The discontinuities in the polynomial segments (i.e., the integration steps) occur when:
• Therefore, at each new discontinuity instant tj+1
• Adding up these steps throughout all segments:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 61
Definition of nth-Order Activity nth-Order Activity
err(tj+1)=∆Q
define
Activity of order n
holds, producing “one new step”:
• So, we have error
• The discontinuities in the polynomial segments (i.e., the integration steps) occur when:
• Therefore, at each new discontinuity instant tj+1
• Adding up these steps throughout all segments:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 62
Definition of nth-Order Activity nth-Order Activity
err(tj+1)=∆Q
define
Activity of order n
holds, producing “one new step”:
• So, we have error
• The discontinuities in the polynomial segments (i.e., the integration steps) occur when:
• Therefore, at each new discontinuity instant tj+1
• Adding up these steps throughout all segments:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 63
• The new definition retains the notions that Activity is – a property related to the inherent dynamics a signal– independent of the accuracy (quantum size) of choice
Definition of nth-Order Activity nth-Order Activity
nth-Order Activity
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 64
• The new definition retains the notions that Activity is – a property related to the inherent dynamics a signal– independent of the accuracy (quantum size) of choice
Definition of nth-Order Activity nth-Order Activity
First Order Activitynth-Order Activity
n=1
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 65
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 66
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 67
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
xa(t) “analytical” or “exact”
x(0)=1
tf=5
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 68
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 69
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 70
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 71
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 72
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
• Practical and theoretical number of steps match reasonably close
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 73
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
• Practical and theoretical number of steps match reasonably close =∆Qb/∆Qa ; =kb/ka = =()1/2 =()1/3
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 74
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
QSS1Zoom in: t=[1,4]
xa(t) “analytical” or “exact”
q(t)
x(t)
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 75
• 1st order system
Example I: Non-stiff system Examples
Solution:System:
nth-order Activity:
xa(t) “analytical” or “exact”
q(t)
x(t)
QSS2Zoom in: t=[1,4]
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 76
• 2nd order system
Example II: Stiff system Examples
System:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 77
• 2nd order system
Example II: Stiff system Examples
System:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 78
• 2nd order system
Example II: Stiff system Examples
Analytical solution:
System:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 79
• 2nd order system
Example II: Stiff system Examples
Analytical solution:
System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 80
• 2nd order system
Example II: Stiff system Examples
Analytical solution:
System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 81
• 2nd order system
Example II: Stiff system Examples
Analytical solution:
System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 82
• 2nd order system
Example II: Stiff system Examples
Analytical solution:
System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 83
• 2nd order system
Example II: Stiff system Examples
Analytical solution:
System:
nth-order Activity:
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 84
• 2nd order system
• We simulate with non-stiff (QSS) and stiff (LIQSS) solvers
Example II: Stiff system Examples
System:
nth-order Activity:
QSS1 LIQSS1
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 85
• 2nd order system
• We simulate with non-stiff (QSS) and stiff (LIQSS) solvers
Example II: Stiff system Examples
System:
nth-order Activity:
High frequency spurious oscillations in q2(t)
QSS1 LIQSS1
spurious oscillations avoided
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 86
• 2nd order system
• First Order Activity
Example II: Stiff system Examples
System:
nth-order Activity:
• Practical and theoretical # of Steps match closely for LIQSS
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 87
• 2nd order system
• First Order Activity
Example II: Stiff system Examples
System:
nth-order Activity:
• Practical and theoretical # of Steps match closely for LIQSS
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 88
• 2nd order system
• First Order Activity
Example II: Stiff system Examples
System:
nth-order Activity:
• Practical and theoretical # of Steps match closely for LIQSS For q2(t) with QSS, the practical # of Steps is obviously unacceptable
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 89
• 2nd order system
• Higher Order Activity (2nd , 3rd): Expected results verified.– Results are shown below only for the “conflictive variable” q2 (t)
Example II: Stiff system Examples
• Practical and theoretical # of Steps match closely for LIQSS
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 90
• 2nd order system
• Higher Order Activity (2nd , 3rd): Expected results verified.– Results are shown below only for the “conflictive variable” q2 (t)
Example II: Stiff system Examples
• Practical and theoretical # of Steps match closely for LIQSS For q2(t) with QSS, the practical # of Steps is obviously unacceptable
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 91
• We have presented a generalization of the concept of activity for continuous time signals.
Conclusions
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 92
• We have presented a generalization of the concept of activity for continuous time signals.
• The classical definition of activity measures the rate of change of the signal
Conclusions
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 93
• We have presented a generalization of the concept of activity for continuous time signals.
• The classical definition of activity measures the rate of change of the signal
• The new definition of activity of nth-order takes into account the rate of change of the derivatives.– Now the “how” [a continuous signal evolves] matters.– Not only the signal’s maxima and minima– Less intuitive, less visualizable
Conclusions
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 94
• We have presented a generalization of the concept of activity for continuous time signals.
• The classical definition of activity measures the rate of change of the signal
• The new definition of activity of nth-order takes into account the rate of change of the derivatives.– Now the “how” [a continuous signal evolves] matters.– Not only the signal’s maxima and minima– Less intuitive, less visualizable
• Activity of order n provides a means by which establishing an ideal lower bound for the number of integration steps– against which comparing the performance of a (suitably selected)
Quantization-based integration method of order n.
Conclusions
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 95
• We have presented a generalization of the concept of activity for continuous time signals.
• The classical definition of activity measures the rate of change of the signal
• The new definition of activity of nth-order takes into account the rate of change of the derivatives.– Now the “how” [a continuous signal evolves] matters.– Not only the signal’s maxima and minima– Less intuitive, less visualizable
• Activity of order n provides a means by which establishing an ideal lower bound for the number of integration steps– against which comparing the performance of a (suitably selected)
Quantization-based integration method of order n.
• Activity of order n is, so far, almost exclusively of theoretical relevance– For calculating A(n) we need to know the analytical solution of a system… – … but that is exactly what we can’t know prior to simulation !!!
(for most cases of practical interest)
Conclusions
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 96
• Explore how the knowledge of activity measures for each variable in a given system can be exploited to:– derive optimal model partitions into multiple parallel processing
nodes (cores, processors, servers) in order to maximize speedups as compared against a serial (single node) simulation.
Ongoing work
Dr. Rodrigo Castro Activity-based Modeling & Simulation (ACTIMS'2014) ETH Zurich , Switzerland, Jan. 2014 97
Q&A
"!Thanks")(int s
rcastro@dc.uba.arrodrigo.castro@usys.ethz.ch
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