ModelingToolkit.jl state-space representations -- what is the recommendation?

New to Julia
1

What is the best way to represent a state space system in ModelingToolkit? For example

@variables t x1(t) x2(t) u(t) y(t)
@constants a11 = -1 a12 = 0.1a21 = 0 a22 = -1 b1 = 1 b2 = 0 c1 = 1 c2 = 1
D = Differential(t)
eqs = [D(x1) ~ a11*x1 + a12*x2 + b1*u,
		D(x2) ~ a21*x1 + a22*x2 + b2*u]
  1. When I try to add the y = c1x1 + c2x2 equation, I run into an error (The LHS cannot contain nondifferentiated variables. Please run structural_simplify) and following the suggestion to use structural_simplify gives me an ExtraVariablesSystemException. Is this not intended to be supported; therefore, without any output feedback, I would compute y separately?
  2. I’ve seen elsewhere that the equations can be written as a matrix differential equation, but I get an error about unknown axes for x.

Are there any simple examples of doing this?

2

It looks like you’re talking about linear statespace systems? If that’s the case, check out GitHub - JuliaControl/ControlSystemsMTK.jl: Interface between ControlSystems and ModelingToolkit

3

The answer from @baggepinnen above is the correct “I want to solve a problem answer”. Here is the “I want to build this myself” answer.

The reason you get an error is that u(t) is unspecified. When you apply structural_simplify, all the pieces of the system must be in place.

The way to debug this is to run

partial_sys = structural_simplify(sys; check_consistency=false)

You can then compare states(partial_sys) to equations(partial_sys), and immediately notice that there is no expression for u.

Here is one possible approach:

function build_system()
    sts = @variables t x(t)[1:2] u(t) y(t)
    ps = @parameters a[1:2,1:2]=[-1 0.1; 0 -1] b[1:2]=[1 0] c[1:2]=[1 1]
    D = Differential(t)

    eqs = [D.(x) .~ a * x + b .* u
           y ~ c' * x
           u ~ sin(t)
          ]

    sys = ODESystem(eqs, t; name=:test)

    sys = structural_simplify(sys)
end
4

Thank you guys so much! This is exactly what I needed.