Cyber Physical Systems Example
Authors: Georgios Bakirtzis https://bakirtzis.net/ and Raul Gonzalez Garcia (raulg@iastate.edu)
The following example is a mechanization from
- Compositional Cyber-Physical Systems Modeling - http://dx.doi.org/10.4204/EPTCS.333.9
- Categorical Semantics of Cyber-Physical Systems Theory - https://doi.org/10.1145/3461669
using AlgebraicDynamics
using Catlab
using LabelledArrays
using DifferentialEquations
using Plots
We use functorial semantics to model a cyper-physical system, namely an unmanned aerial vehicle (UAV). We define a diagram of systems (the composition syntax) that is the architecture of the composition. Then, we apply behaviors of the individual parts of the system to the architecture. This composition produces a complete UAV model.
We first have to define our boxes and specify what the inports and outports are. For example, the sensor box has two inports :e
and :s
and one outport s_prime
.
s = Box(:sensor, [:s, :e], [:s′])
c = Box(:controller, [:d, :s′], [:c])
d = Box(:dynamics, [:c], [:s]);
A wiring diagram has outer inports and outports which define the interface of target system. Then we add the boxes and wires to the diagram and visualize the result.
UAV = WiringDiagram([:e,:d], [:s])
sensor = add_box!(UAV, s)
controller = add_box!(UAV, c)
dynamics = add_box!(UAV, d)
add_wires!(UAV, [
# net inputs
(input_id(UAV), 1) => (sensor, 2),
(input_id(UAV), 2) => (controller, 2),
# connections
(sensor, 1) => (controller, 1),
(controller, 1) => (dynamics, 1),
(dynamics, 1) => (sensor, 1),
# net output
(dynamics, 1) => (output_id(UAV), 1)
]);
to_graphviz(UAV)
Then we assign behaviors to inhabit the boxes.
function 𝗟(𝐖)
𝐿(u, x, p, t) = [ -p.𝓐l * (u[1] - x[1] - x[2]) ] # sc
𝐶(u, x, p, t) = [ -p.𝓐c * (u[1] + p.𝓑c*x[1] - x[2]) ] # sl
𝐷(u, x, p, t) = LVector(α = -0.313*u[1] + 56.7*u[2] + 0.232*x[1],
q = -0.013*u[1] - 0.426*u[2] + 0.0203*x[1],
θ = 56.7*u[2] )
u_𝐿(u,p,t) = [ u[1] ] # outputs sl
u_𝐶(u,p,t) = [ u[1] ] # outputs sc
u_𝐷(u,p,t) = [ u[3] ] # outputs θ
return oapply(𝐖,
Dict(:sensor => ContinuousMachine{Float64}(2, 1, 1, 𝐿, u_𝐿),
:controller => ContinuousMachine{Float64}(2, 1, 1, 𝐶, u_𝐶),
:dynamics => ContinuousMachine{Float64}(1, 3, 1, 𝐷, u_𝐷)))
end
𝑢ᵤₐᵥ = 𝗟(UAV)
Lastly, we compute and plot the solution.
# initial values
xₒ = LVector( e = 0.01, # [e, d] -> [θ offset, 𝛿 control input]
d = 0.05);
uₒ = [0.0, 0, 0, 0, 0]
tspan = (0, 20.0)
params = (𝓐l = 100, # decay constant of sensor
𝓐c = 100, # decay constant of controller
𝓑c = 0) # ratio of velocity to reference velocity
prob = ODEProblem(𝑢ᵤₐᵥ, uₒ, xₒ, tspan, params)
sol = solve(prob, alg_hints=[:stiff]);
plot(sol, vars = [1,2, ((t,y) -> (t, y*1e2), 0, 4), 3, 5],
label = ["sl" "sc" "α" "q" "θ"],
lw = 2, title = "Aircraft pitch behaviour",
xlabel = "time", ylabel = "response"
)