Update documentation

Author: hersle

docs/src/tutorials/SampledData.md

@@ -97,7 +97,8 @@ H(z) = \dfrac{b_2 z^2 + b_1 z + b_0}{a_2 z^2 + a_1 z + a_0}
 may thus be modeled as
 
 ```julia
-@variables t y(t) [description = "Output"] u(t) [description = "Input"]
+t = ModelingToolkit.t_nounits
+@variables y(t) [description = "Output"] u(t) [description = "Input"]
 k = ShiftIndex(Clock(t, dt))
 eqs = [
     a2 * y(k) + a1 * y(k - 1) + a0 * y(k - 2) ~ b2 * u(k) + b1 * u(k - 1) + b0 * u(k - 2)

src/discretedomain.jl

@@ -98,7 +98,7 @@ $(FIELDS)
 ```jldoctest
 julia> using Symbolics
 
-julia> @variables t;
+julia> t = ModelingToolkit.t_nounits
 
 julia> Δ = Sample(t, 0.01)
 (::Sample) (generic function with 2 methods)
@@ -166,7 +166,9 @@ The `ShiftIndex` operator allows you to index a signal and obtain a shifted disc
 # Examples
 
 ```
-julia> @variables t x(t);
+julia> t = ModelingToolkit.t_nounits;
+
+julia> @variables x(t);
 
 julia> k = ShiftIndex(t, 0.1);
 

src/systems/abstractsystem.jl

@@ -2237,11 +2237,10 @@ This example builds the following feedback interconnection and linearizes it fro
 
 ```julia
 using ModelingToolkit
-@variables t
+using ModelingToolkit: t_nounits as t, D_nounits as D
 function plant(; name)
     @variables x(t) = 1
     @variables u(t)=0 y(t)=0
-    D = Differential(t)
     eqs = [D(x) ~ -x + u
            y ~ x]
     ODESystem(eqs, t; name = name)
@@ -2250,7 +2249,6 @@ end
 function ref_filt(; name)
     @variables x(t)=0 y(t)=0
     @variables u(t)=0 [input = true]
-    D = Differential(t)
     eqs = [D(x) ~ -2 * x + u
            y ~ x]
     ODESystem(eqs, t, name = name)

src/systems/alias_elimination.jl

@@ -388,8 +388,10 @@ Use Kahn's algorithm to topologically sort observed equations.
 
 Example:
 ```julia
-julia> @variables t x(t) y(t) z(t) k(t)
-(t, x(t), y(t), z(t), k(t))
+julia> t = ModelingToolkit.t_nounits
+
+julia> @variables x(t) y(t) z(t) k(t)
+(x(t), y(t), z(t), k(t))
 
 julia> eqs = [
            x ~ y + z

src/systems/dependency_graphs.jl

@@ -15,8 +15,9 @@ Example:
 
 ```julia
 using ModelingToolkit
+using ModelingToolkit: t_nounits as t
 @parameters β γ κ η
-@variables t S(t) I(t) R(t)
+@variables S(t) I(t) R(t)
 
 rate₁ = β * S * I
 rate₂ = γ * I + t

src/systems/diffeqs/odesystem.jl

@@ -10,10 +10,10 @@ $(FIELDS)
 
 ```julia
 using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @parameters σ ρ β
-@variables t x(t) y(t) z(t)
-D = Differential(t)
+@variables x(t) y(t) z(t)
 
 eqs = [D(x) ~ σ*(y-x),
        D(y) ~ x*(ρ-z)-y,

src/systems/diffeqs/sdesystem.jl

@@ -10,10 +10,10 @@ $(FIELDS)
 
 ```julia
 using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @parameters σ ρ β
-@variables t x(t) y(t) z(t)
-D = Differential(t)
+@variables x(t) y(t) z(t)
 
 eqs = [D(x) ~ σ*(y-x),
        D(y) ~ x*(ρ-z)-y,
@@ -321,10 +321,10 @@ experiments. Springer Science & Business Media.
 
 ```julia
 using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @parameters α β
-@variables t x(t) y(t) z(t)
-D = Differential(t)
+@variables x(t) y(t) z(t)
 
 eqs = [D(x) ~ α*x]
 noiseeqs = [β*x]

src/systems/jumps/jumpsystem.jl

@@ -34,9 +34,10 @@ $(FIELDS)
 
 ```julia
 using ModelingToolkit, JumpProcesses
+using ModelingToolkit: t_nounits as t
 
 @parameters β γ
-@variables t S(t) I(t) R(t)
+@variables S(t) I(t) R(t)
 rate₁   = β*S*I
 affect₁ = [S ~ S - 1, I ~ I + 1]
 rate₂   = γ*I

src/systems/pde/pdesystem.jl

@@ -11,8 +11,8 @@ $(FIELDS)
 ```julia
 using ModelingToolkit
 
-@parameters x
-@variables t u(..)
+@parameters x t
+@variables u(..)
 Dxx = Differential(x)^2
 Dtt = Differential(t)^2
 Dt = Differential(t)

src/utils.jl

@@ -345,7 +345,8 @@ Return a `Set` containing all variables in `x` that appear in
 Example:
 
 ```
-@variables t u(t) y(t)
+t = ModelingToolkit.t_nounits
+@variables u(t) y(t)
 D  = Differential(t)
 v  = ModelingToolkit.vars(D(y) ~ u)
 v == Set([D(y), u])