DDESystem

Construction of DDESystem

A DDESystem is represented by the following state equation

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

and by the output equation

where $y$ is the value of the output.

As an example, consider a system with the state equation

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) @0x00007fe0d82cd390

julia> task2 = @async while true
           all(take!(iport) .=== NaN) && break
           end
Task (runnable) @0x00007fe0e277c280

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) @0x00007fe0d82cd390

julia> task2
Task (runnable) @0x00007fe0e277c280

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}:
 436466.8181121704
 121540.54530401707
  33844.744019167716
   9424.577256277034
   2624.411352804969
    730.8063984951228
    203.50408965546976
     56.667448342646395
     15.778109354779708
      4.436563289075523
      ⋮
      0.0
      0.0
      0.0
      0.0
      0.0
      0.0
      0.0
      0.0
      0.0

Full API

@def_dde_system ex

@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

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

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

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();

DDESystem(; constantlags, depslags, righthandside, history, readout, state, input, output)

DDESystem(; constantlags, depslags, righthandside, history, readout, state, input, output)