Delay differential equation: Difference between revisions
Bluelink 2 books for verifiability.) #IABot (v2.0) (GreenC bot |
Mersenne56 (talk | contribs) →External links: Numerous changes, including the addition of a "Further Reading" section and reorganization of "Notes" as "References" |
||
Line 2: | Line 2: | ||
In [[mathematics]], '''delay differential equations''' ('''DDEs''') are a type of [[differential equation]] in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. |
In [[mathematics]], '''delay differential equations''' ('''DDEs''') are a type of [[differential equation]] in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. |
||
DDEs are also called '''time-delay systems''', systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the [[Functional differential equation|functional state]], i.e. [[partial differential equations]] (PDEs) which are infinite dimensional, as opposed to [[ordinary differential equations]] (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs |
DDEs are also called '''time-delay systems''', systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the [[Functional differential equation|functional state]], i.e. [[partial differential equations]] (PDEs) which are infinite dimensional, as opposed to [[ordinary differential equations]] (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:<ref>{{cite journal |last=Richard |first=Jean-Pierre | title=Time Delay Systems: An overview of some recent advances and open problems| year=2003 |journal=Automatica |volume=39 |issue=10 |pages=1667–1694 | doi=10.1016/S0005-1098(03)00167-5 | ref=refRichard2003}}</ref> |
||
# Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, [[Actuator|actuators]], [[Sensor|sensors]], and [[Telecommunications network|communication networks]] that are now involved in feedback control loops introduce such delays. Finally, besides actual delays, time lags are frequently used to simplify very high order models. Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control engineering. |
|||
# Delay systems are still resistant to many ''classical'' controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring effects which are adequately represented by DDEs is not a general alternative: in the best situation (constant and known delays), it leads to the same degree of complexity in the control design. In worst cases (time-varying delays, for instance), it is potentially disastrous in terms of stability and oscillations. |
|||
# Voluntary introduction of delays can benefit the [[control system]].<ref>{{Cite journal|last=Lavaei|first=Javad|last2=Sojoudi|first2=Somayeh|last3=Murray|first3=Richard M.|date=2010|title=Simple delay-based implementation of continuous-time controllers|url=https://ieeexplore.ieee.org/document/5530439|journal=Proceedings of the 2010 American Control Conference|volume=|pages=5781–5788|doi=10.1109/ACC.2010.5530439|via=}}</ref> |
|||
# In spite of their complexity, DDEs often appear as simple infinite-dimensional models in the very complex area of [[partial differential equations]] (PDEs). |
|||
A general form of the time-delay differential equation for <math>x(t)\in \mathbb{R}^n</math> is |
A general form of the time-delay differential equation for <math>x(t)\in \mathbb{R}^n</math> is |
||
:<math>\frac{\rm d}{{\rm d}t}x(t)=f(t,x(t),x_t),</math> |
:<math>\frac{\rm d}{{\rm d}t}x(t)=f(t,x(t),x_t),</math> |
||
where <math>x_t=\{x(\tau):\tau\leq t\}</math> represents the trajectory of the solution in the past. In this equation, <math>f</math> is a functional operator from |
where <math>x_t=\{x(\tau):\tau\leq t\}</math> represents the trajectory of the solution in the past. In this equation, <math>f</math> is a functional operator from |
||
<math>\mathbb{R}\times \mathbb{R}^n\times C^1(\mathbb{R}, \mathbb{R}^n)</math> to <math>\mathbb{R}^n.\,</math> |
<math>\mathbb{R}\times \mathbb{R}^n\times C^1(\mathbb{R}, \mathbb{R}^n)</math> to <math>\mathbb{R}^n.\,</math> |
||
Line 78: | Line 85: | ||
==The characteristic equation== |
==The characteristic equation== |
||
Similar to [[ordinary differential equation|ODE]]s, many properties of linear DDEs can be characterized and analyzed using the [[characteristic equation (calculus)|characteristic equation]].<ref> |
Similar to [[ordinary differential equation|ODE]]s, many properties of linear DDEs can be characterized and analyzed using the [[characteristic equation (calculus)|characteristic equation]].<ref>{{Cite book|last=Michiels|first=Wim|url=https://epubs.siam.org/doi/book/10.1137/1.9780898718645|title=Stability and Stabilization of Time-Delay Systems|last2=Niculescu|first2=Silviu-Iulian|date=|publisher=Society for Industrial and Applied Mathematics|year=2007|isbn=978-0-89871-632-0|series=Advances in Design and Control|location=|pages=3-32|doi=10.1137/1.9780898718645}}</ref> |
||
The characteristic equation associated with the linear DDE with discrete delays |
The characteristic equation associated with the linear DDE with discrete delays |
||
::<math>\frac{\rm d}{{\rm d}t}x(t)=A_0x(t)+A_1x(t-\tau_1)+\dotsb+A_mx(t-\tau_m)</math> |
::<math>\frac{\rm d}{{\rm d}t}x(t)=A_0x(t)+A_1x(t-\tau_1)+\dotsb+A_mx(t-\tau_m)</math> |
||
Line 86: | Line 93: | ||
The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the [[Spectrum of a matrix|spectrum]]. Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a [[spectral theory|spectral analysis]] more involved. The spectrum does however have some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues to the right of any vertical line in the complex plane.{{citation needed|date=July 2013}} |
The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the [[Spectrum of a matrix|spectrum]]. Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a [[spectral theory|spectral analysis]] more involved. The spectrum does however have some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues to the right of any vertical line in the complex plane.{{citation needed|date=July 2013}} |
||
This characteristic equation is a [[nonlinear eigenproblem]] and there are many methods to compute the spectrum numerically.<ref> |
This characteristic equation is a [[nonlinear eigenproblem]] and there are many methods to compute the spectrum numerically.<ref>{{Cite book|last=Michiels|first=Wim|url=https://epubs.siam.org/doi/book/10.1137/1.9780898718645|title=Stability and Stabilization of Time-Delay Systems|last2=Niculescu|first2=Silviu-Iulian|date=|publisher=Society for Industrial and Applied Mathematics|year=2007|isbn=978-0-89871-632-0|series=Advances in Design and Control|location=|pages=33-56|doi=10.1137/1.9780898718645}}</ref> In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE: |
||
:<math>\frac{\rm d}{{\rm d}t}x(t)=-x(t-1).</math> |
:<math>\frac{\rm d}{{\rm d}t}x(t)=-x(t-1).</math> |
||
The characteristic equation is |
The characteristic equation is |
||
Line 97: | Line 104: | ||
* [[Functional differential equation]] |
* [[Functional differential equation]] |
||
== |
==References== |
||
{{reflist}} |
{{reflist}} |
||
== |
==Further reading== |
||
{{Refbegin}} |
|||
* {{Cite book|last=Bellen|given=Alfredo|url=https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198506546.001.0001/acprof-9780198506546|title=Numerical Methods for Delay Differential Equations|last2=Zennaro|first2=Marino|publisher=Oxford University Press|year=2003|isbn=978-0198506546|series=Numerical Mathematics and Scientific Computation|place=Oxford, UK|pages=|authorlink=}} |
|||
*{{cite book |last=Bellman |first=Richard |last2=Cooke |first2=Kenneth L. |title=Differential-difference equations |url=https://archive.org/details/differentialdiff0000bell |url-access=registration |publisher=[[Academic Press]] | location=New York-London | year=1963 | isbn=978-0-12-084850-8}} |
|||
* {{Cite book|last=Bellman|given=Richard|url=https://www.rand.org/content/dam/rand/pubs/reports/2006/R374.pdf|title=Differential-Difference Equations|last2=Cooke|first2=Kenneth L.|publisher=Academic Press|year=1963|isbn=978-0120848508|series=Mathematics in Science and Engineering|place=New York, NY|pages=|authorlink=}} |
|||
*{{cite book |authorlink=Rod Driver |last=Driver |first=Rodney D. |year=1977 |title=Ordinary and Delay Differential Equations |url=https://archive.org/details/ordinarydelaydif0000driv |url-access=registration |publisher=Springer Verlag |location=New York |isbn=0-387-90231-7 }} |
|||
⚫ | * {{Cite book|last=Briat|given=Corentin|url=https://www.springer.com/gp/book/9783662440490|title=Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control|publisher=Springer-Verlag|year=2015|isbn=978-3662440490|series=Advances in Delays and Dynamics|place=Heidelberg, DE|pages=|authorlink=}} |
||
*{{cite book |author1=Michiels, Wim |author2=Niculescu, Silviu-Iulian | title=Stability and stabilization of time-delay systems. An eigenvalue based approach | year=2007 | isbn=978-0-89871-632-0 | doi=10.1137/1.9780898718645 | ref=refMichiels2007}} |
|||
* {{Cite book|last=Driver|given=Rodney D.|url=https://link.springer.com/book/10.1007/978-1-4684-9467-9|title=Ordinary and Delay Differential Equations|publisher=Springer-Verlag|year=1977|isbn=978-0387902319|series=Applied Mathematical Sciences|place=New York, NY|pages=|authorlink=}} |
|||
⚫ | |||
* Bellen, A., & Zennaro, M. (2013). Numerical methods for delay differential equations. [[Oxford university press]]. |
|||
{{Refend}} |
|||
==External links== |
==External links== |
Revision as of 00:52, 22 March 2020
Differential equations |
---|
Scope |
Classification |
Solution |
People |
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:[1]
- Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, actuators, sensors, and communication networks that are now involved in feedback control loops introduce such delays. Finally, besides actual delays, time lags are frequently used to simplify very high order models. Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control engineering.
- Delay systems are still resistant to many classical controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring effects which are adequately represented by DDEs is not a general alternative: in the best situation (constant and known delays), it leads to the same degree of complexity in the control design. In worst cases (time-varying delays, for instance), it is potentially disastrous in terms of stability and oscillations.
- Voluntary introduction of delays can benefit the control system.[2]
- In spite of their complexity, DDEs often appear as simple infinite-dimensional models in the very complex area of partial differential equations (PDEs).
A general form of the time-delay differential equation for is
where represents the trajectory of the solution in the past. In this equation, is a functional operator from to
Examples
- Continuous delay
- Discrete delay
- for .
- Linear with discrete delays
- where .
- Pantograph equation
- where a, b and λ are constants and 0 < λ < 1. This equation and some more general forms are named after the pantographs on trains.[3][4]
Solving DDEs
DDEs are mostly solved in a stepwise fashion with a principle called the method of steps. For instance, consider the DDE with a single delay
with given initial condition . Then the solution on the interval is given by which is the solution to the inhomogeneous initial value problem
- ,
with . This can be continued for the successive intervals by using the solution to the previous interval as inhomogeneous term. In practice, the initial value problem is often solved numerically.
Example
Suppose and . Then the initial value problem can be solved with integration,
i.e., , where the initial condition is given by . Similarly, for the interval we integrate and fit the initial condition,
i.e.,
Reduction to ODE
In some cases, differential equations can be represented in a format that looks like delay differential equations.
- Example 1 Consider an equation
- Introduce to get a system of ODEs
- Example 2 An equation
- is equivalent to
- where
The characteristic equation
Similar to ODEs, many properties of linear DDEs can be characterized and analyzed using the characteristic equation.[5] The characteristic equation associated with the linear DDE with discrete delays
is
- .
The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the spectrum. Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a spectral analysis more involved. The spectrum does however have some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues to the right of any vertical line in the complex plane.[citation needed]
This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically.[6] In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE:
The characteristic equation is
There are an infinite number of solutions to this equation for complex λ. They are given by
- ,
where Wk is the kth branch of the Lambert W function.
See also
References
- ^ Richard, Jean-Pierre (2003). "Time Delay Systems: An overview of some recent advances and open problems". Automatica. 39 (10): 1667–1694. doi:10.1016/S0005-1098(03)00167-5.
- ^ Lavaei, Javad; Sojoudi, Somayeh; Murray, Richard M. (2010). "Simple delay-based implementation of continuous-time controllers". Proceedings of the 2010 American Control Conference: 5781–5788. doi:10.1109/ACC.2010.5530439.
- ^ Griebel, Thomas (2017-01-01). "The pantograph equation in quantum calculus". Masters Theses.
- ^ "The dynamics of a current collection system for an electric locomotive". royalsocietypublishing.org. 1971. doi:10.1098/rspa.1971.0078. Retrieved 2019-01-26.
- ^ Michiels, Wim; Niculescu, Silviu-Iulian (2007). Stability and Stabilization of Time-Delay Systems. Advances in Design and Control. Society for Industrial and Applied Mathematics. pp. 3–32. doi:10.1137/1.9780898718645. ISBN 978-0-89871-632-0.
- ^ Michiels, Wim; Niculescu, Silviu-Iulian (2007). Stability and Stabilization of Time-Delay Systems. Advances in Design and Control. Society for Industrial and Applied Mathematics. pp. 33–56. doi:10.1137/1.9780898718645. ISBN 978-0-89871-632-0.
Further reading
- Bellen, Alfredo; Zennaro, Marino (2003). Numerical Methods for Delay Differential Equations. Numerical Mathematics and Scientific Computation. Oxford, UK: Oxford University Press. ISBN 978-0198506546.
- Bellman, Richard; Cooke, Kenneth L. (1963). Differential-Difference Equations (PDF). Mathematics in Science and Engineering. New York, NY: Academic Press. ISBN 978-0120848508.
- Briat, Corentin (2015). Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control. Advances in Delays and Dynamics. Heidelberg, DE: Springer-Verlag. ISBN 978-3662440490.
- Driver, Rodney D. (1977). Ordinary and Delay Differential Equations. Applied Mathematical Sciences. New York, NY: Springer-Verlag. ISBN 978-0387902319.
External links
- Skip Thompson (ed.). "Delay-Differential Equations". Scholarpedia.