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((statefunc, histfunc), outputfunc, [1.],  0., nothing, Outport())
DDESystem(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) @0x00007ffa50413340

julia> task2 = @async while true
           all(take!(iport) .=== NaN) && break
           end
Task (runnable) @0x00007ffa54cd5390

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) @0x00007ffa50413340

julia> task2
Task (runnable) @0x00007ffa54cd5390

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

Jusdl.DDESystemType
DDESystem(input, output, statefunc, outputfunc, state, t, modelargs=(), solverargs=(); 
    alg=DDEAlg, modelkwargs=NamedTuple(), solverkwargs=NamedTuple())

Constructs a DDESystem with input and output. statefunc is the state function and outputfunc is the output function of DDESystem. state is the initial state and t is the time. modelargs and modelkwargs are passed into ODEProblem and solverargs and solverkwargs are passed into solve method of DifferentialEquations. alg is the algorithm to solve the differential equation of the system.

The DDESystem is represented by

\[ \begin{array}{l} \dot{x} = f(x, h, u, t) \\ y = g(x, u, t) \end{array}\]

where $t$ is the time t, $x$ is the value of state, $u$ is the value of input, $y$ is the value of output. $f$ is statefunc, $g$ is outputfunc. $h$is the history function of history. solver is used to solve the above differential equation.

The syntax of statefunc must be of the form

function statefunc(dx, x, u, t)
    dx .= ... # Update dx
end

and the syntax of outputfunc must be of the form

function outputfunc(x, u, t)
    y = ... # Compute y 
    return y
end

Example

julia> const out = zeros(1);

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

julia> function sfuncdde(dx, x, h, u, t)
           h(out, u, t - tau) # Update out vector
           dx[1] = out[1] + x[1]
       end;

julia> ofuncdde(x, u, t) = x;

julia> tau = 1;

julia> conslags = [tau];

julia> DDESystem((sfuncdde, histfunc), ofuncdde, [1.],  0., nothing, Outport())
DDESystem(state:[1.0], t:0.0, input:nothing, output:Outport(numpins:1, eltype:Outpin{Float64}))
Info

See DifferentialEquations for more information about modelargs, modelkwargs, solverargs solverkwargs and alg.

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