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> using Jusdl # hdie

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

julia> histfunc(out, u, t)
ERROR: UndefVarError: histfunc not defined

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

julia> hist = History(histfunc, conslags, ())
ERROR: UndefVarError: History not defined

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

julia> ds = DDESystem(nothing, Bus(), statefunc, outputfunc, state, hist, 0.)
ERROR: UndefVarError: state not defined

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> task = launch(ds)
ERROR: UndefVarError: ds not defined

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> drive(ds, 1.)
ERROR: UndefVarError: ds not defined

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> approve(ds)
ERROR: UndefVarError: ds not defined

We can continue to drive ds.

julia> for t in 2. : 10.
           drive(ds, t)
           approve(ds)
       end
ERROR: UndefVarError: ds not defined

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

julia> task
ERROR: UndefVarError: task not defined

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> terminate(ds)
ERROR: UndefVarError: ds not defined

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> ds.output[1].buffer.data
ERROR: UndefVarError: ds not defined

Full API

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

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

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

julia> const out = zeros(1);

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

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

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

julia> tau = 1;

julia> conslags = [tau];

julia> ds = DDESystem(nothing, Bus(), (statefunc, histfunc), outputfunc, [1.],  0.)
DDESystem(state:[1.0], t:0.0, input:nothing, output:Bus(nlinks:1, eltype:Link{Float64}, isreadable:false, iswritable:false))