DDESystem

Construction of DDESystem

A DDESystem is represented by the following state equation

\[ \dot{x} = f(x, h, u, t) \quad t \geq t_0\]

where $t$ is the time, $x$ is the value of the state, $u$ is the value of the input. $h$ is the history function for which

\[ x(t) = h(t) \quad t \leq t_0\]

and by the output equation

\[ y = g(x, u, t) \]

where $y$ is the value of the output.

As an example, consider a system with the state equation

\[ \begin{array}{l} \dot{x} = -x(t - \tau) \quad t \geq 0 \\ x(t) = 1. -\tau \leq t \leq 0 \\ \end{array}\]

First, we define the history function histfunc,


julia> const out = zeros(1)
1-element Array{Float64,1}:
 0.0

julia> histfunc(out, u, t) = (out .= 1.);

Note that histfunc mutates a vector out. This mutation is for performance reasons. Next the state function can be defined

julia> function statefunc(dx, x, h, u, t)
           h(out, u, t - tau) # Update out vector
           dx[1] = out[1] + x[1]
       end
statefunc (generic function with 1 method)

and let us take all the state variables as outputs. Thus, the output function is

julia> outputfunc(x, u, t) = x
outputfunc (generic function with 1 method)

Next, we need to define the history for the system. History is defined by specifying a history function, and the type of the lags. There may be two different lag: constant lags which are independent of the state variable $x$ and the dependent lags which are mainly the functions of the state variable $x$. Note that for this example, the have constant lags. Thus,

julia> tau = 1
1

julia> conslags = [tau]
1-element Array{Int64,1}:
 1

At this point, we are ready to construct the system.

julia> ds = DDESystem(righthandside=statefunc, history=histfunc, readout=outputfunc, state=[1.],  input=nothing, output=Outport(), constlags=conslags, depslags=nothing)
DDESystem(righthandside:statefunc, readout:outputfunc, state:[1.0], t:0.0, input:nothing, output:Outport(numpins:1, eltype:Outpin{Float64}))

Basic Operation of DDESystem

The basis operaiton of DDESystem is the same as those of other dynamical systems. When triggered from its trigger link, the DDESystem reads its time from its trigger link, reads input, solves its differential equation, computes its output and writes the computed output to its output bus. To drive DDESystem, we must first launch it,

julia> iport, trg, hnd = Inport(), Outpin(), Inpin{Bool}()
(Inport(numpins:1, eltype:Inpin{Float64}), Outpin(eltype:Float64, isbound:false), Inpin(eltype:Bool, isbound:false))

julia> connect!(ds.output, iport)
1-element Array{Link{Float64},1}:
 Link(state:open, eltype:Float64, isreadable:false, iswritable:false)

julia> connect!(trg, ds.trigger)
Link(state:open, eltype:Float64, isreadable:false, iswritable:false)

julia> connect!(ds.handshake, hnd)
Link(state:open, eltype:Bool, isreadable:false, iswritable:false)

julia> task = launch(ds)
Task (runnable) @0x00007f1fca575d50

julia> task2 = @async while true
           all(take!(iport) .=== NaN) && break
           end
Task (runnable) @0x00007f1fcf7f3340

When launched, ds is drivable. To drive ds, we can use the syntax drive(ds, t) or put!(ds.trigger, t) where t is the time until which ds is to be driven.

julia> put!(trg, 1.)

When driven, ds reads the time t from its trigger link, (since its input is nothing, ds does nothing during its input reading stage), solves its differential equation, computes output and writes the value of its output to its output bus. To signify, the step was taken with success, ds writes true to its handshake which must be read to further drive ds. For this, we can use the syntax approve!(ds) or take!(ds.handshake).

julia> take!(hnd)
true

We can continue to drive ds.

julia> for t in 2. : 10.
           put!(trg, t)
           take!(hnd)
       end

When launched, we constructed a task whose state is running which implies that ds can be driven.

julia> task
Task (runnable) @0x00007f1fca575d50

julia> task2
Task (runnable) @0x00007f1fcf7f3340

As long as the state of the task is running, ds can be driven. To terminate task safely, we need to terminate the ds.

julia> put!(trg, NaN)

Note that the state of task is done which implies that ds is not drivable any more.

Note that the output values of ds is written to its output bus.

julia> iport[1].link.buffer
64-element Buffer{Cyclic,Float64,1}:
 436466.81811256823
 121540.54530408025
  33844.744019186095
   9424.577256280585
   2624.41135280592
    730.8063984953665
    203.5040896555394
     56.66744834264986
     15.77810935478092
      4.436563289075519
      ⋮
      0.0
      0.0
      0.0
      0.0
      0.0
      0.0
      0.0
      0.0
      0.0

Full API

Causal.@def_dde_system — Macro

@def_dde_system ex

where ex is the expression to define to define a new AbstractDDESystem component type. The usage is as follows:

@def_dde_system mutable struct MyDDESystem{T1,T2,T3,...,TN,OP,RH,RO,ST,IP,OP} <: AbstractDDESystem
    param1::T1 = param1_default                 # optional field 
    param2::T2 = param2_default                 # optional field 
    param3::T3 = param3_default                 # optional field
        ⋮
    paramN::TN = paramN_default                 # optional field 
    constlags::CL = constlags_default           # mandatory field
    depslags::DL = depslags_default             # mandatory field
    righthandside::RH = righthandside_function  # mandatory field
    history::HST = history_function             # mandatory field
    readout::RO = readout_function              # mandatory field
    state::ST = state_default                   # mandatory field
    input::IP = input_defauult                  # mandatory field
    output::OP = output_default                 # mandatory field
end

Here, MyDDESystem has N parameters. MyDDESystem is represented by the righthandside and readout function. state, input and output is the state, input port and output port of MyDDESystem.

Warning

righthandside must have the signature

function righthandside(dx, x, u, t, args...; kwargs...)
    dx .= .... # update dx 
end

and readout must have the signature

function readout(x, u, t)
    y = ...
    return y
end

Warning

New DDE system must be a subtype of AbstractDDESystem to function properly.

Example

julia> _delay_feedback_system_cache = zeros(1)
1-element Array{Float64,1}:
 0.0

julia> _delay_feedback_system_tau = 1.
1.0

julia> _delay_feedback_system_constlags = [1.]
1-element Array{Float64,1}:
 1.0

julia> _delay_feedback_system_history(cache, u, t) = (cache .= 1.)
_delay_feedback_system_history (generic function with 1 method)

julia> function _delay_feedback_system_rhs(dx, x, h, u, t, 
           cache=_delay_feedback_system_cache, τ=_delay_feedback_system_tau)
           h(cache, u, t - τ)  # Update cache 
           dx[1] = cache[1] + x[1]
       end
_delay_feedback_system_rhs (generic function with 3 methods)

julia> @def_dde_system mutable struct MyDDESystem{RH, HST, RO, IP, OP} <: AbstractDDESystem
           constlags::Vector{Float64} = _delay_feedback_system_constlags
           depslags::Nothing = nothing
           righthandside::RH = _delay_feedback_system_rhs
           history::HST = _delay_feedback_system_history
           readout::RO = (x, u, t) -> x 
           state::Vector{Float64} = rand(1)
           input::IP = nothing 
           output::OP = Outport(1)
       end

julia> ds = MyDDESystem();

source

Causal.DDESystem — Type

mutable struct DDESystem{CL, DL, RH, HST, RO, ST<:(AbstractArray{var"#s301",1} where var"#s301"<:Real), IP<:(Union{var"#s300", var"#s299"} where var"#s299"<:Nothing where var"#s300"<:Inport), OP<:(Union{var"#s298", var"#s297"} where var"#s297"<:Nothing where var"#s298"<:Outport), var"253", var"254", var"255", Symbol, var"256", Float64, var"257", var"258", var"259", var"260", var"261", var"262"} <: AbstractDDESystem

Construct a generic DDE system

Fields

constlags::Any
Constant lags
depslags::Any
Dependent lags
righthandside::Any
Right-hand-side function
history::Any
History function
readout::Any
Readout function
state::AbstractArray{var"#s301",1} where var"#s301"<:Real
State
input::Union{var"#s300", var"#s299"} where var"#s299"<:Nothing where var"#s300"<:Inport
Input. Expected to be an Inport or Nothing
output::Union{var"#s298", var"#s297"} where var"#s297"<:Nothing where var"#s298"<:Outport
Output port
trigger::Any
handshake::Any
callbacks::Any
name::Any
id::Any
t::Any
modelargs::Any
modelkwargs::Any
solverargs::Any
solverkwargs::Any
alg::Any
integrator::Any

source

Causal.DelayFeedbackSystem — Type

mutable struct DelayFeedbackSystem{CL<:(AbstractArray{var"#s301",1} where var"#s301"<:Real), RH, HST, RO, ST<:(AbstractArray{var"#s300",1} where var"#s300"<:Real), IP<:(Union{var"#s299", var"#s298"} where var"#s298"<:Nothing where var"#s299"<:Inport), OP<:(Union{var"#s297", var"#s296"} where var"#s296"<:Nothing where var"#s297"<:Outport), var"253", var"254", var"255", Symbol, var"256", Float64, var"257", var"258", var"259", var"260", var"261", var"262"} <: AbstractDDESystem

Constructs DelayFeedbackSystem

Fields

constlags::AbstractArray{var"#s301",1} where var"#s301"<:Real
Constant lags
depslags::Nothing
Dependent lags
righthandside::Any
Right-hand-side function
history::Any
History function
readout::Any
Readout function
state::AbstractArray{var"#s300",1} where var"#s300"<:Real
State
input::Union{var"#s299", var"#s298"} where var"#s298"<:Nothing where var"#s299"<:Inport
Input, Expected to be an Inport of Nothing
output::Union{var"#s297", var"#s296"} where var"#s296"<:Nothing where var"#s297"<:Outport
Output port
trigger::Any
handshake::Any
callbacks::Any
name::Any
id::Any
t::Any
modelargs::Any
modelkwargs::Any
solverargs::Any
solverkwargs::Any
alg::Any
integrator::Any

source