add input-affine equation decomposition

src/ModelingToolkit.jl

@@ -224,6 +224,7 @@ include("structural_transformation/StructuralTransformations.jl")
 
 @reexport using .StructuralTransformations
 include("inputoutput.jl")
+include("input_affine_form.jl")
 
 include("adjoints.jl")
 include("deprecations.jl")
@@ -259,6 +260,7 @@ export JumpProblem
 export alias_elimination, flatten
 export connect, domain_connect, @connector, Connection, AnalysisPoint, Flow, Stream,
        instream
+export input_affine_form
 export initial_state, transition, activeState, entry, ticksInState, timeInState
 export @component, @mtkmodel, @mtkcompile, @mtkbuild
 export isinput, isoutput, getbounds, hasbounds, getguess, hasguess, isdisturbance,

src/input_affine_form.jl

@@ -0,0 +1,55 @@
+"""
+    input_affine_form(eqs, inputs)
+
+Extract control-affine (input-affine) form from symbolic equations.
+
+Given a system of equations of the form `ẋ = F(x, u)` where `x` is the state
+and `u` are the inputs, this function extracts the control-affine representation:
+`ẋ = f(x) + g(x)*u`
+
+where:
+- `f(x)` is the drift term (dynamics without inputs)
+- `g(x)` is the input matrix
+
+# Arguments
+- `eqs`: Vector of symbolic equations representing the system dynamics
+- `inputs`: Vector of input/control variables
+
+# Returns
+- `f`: The drift vector f(x)
+- `g`: The input matrix g(x)
+
+# Example
+```julia
+using ModelingToolkit, Symbolics
+
+@variables x1 x2 u1 u2
+state  = [x1, x2]
+inputs = [u1, u2]
+
+# Define system equations: ẋ = F(x, u)
+eqs = [
+    -x1 + 2*x2 + u1,
+    x1*x2 - x2 + u1 + 2*u2
+]
+
+# Extract control-affine form
+f, g = input_affine_form(eqs, inputs)
+```
+
+# Notes
+The function assumes that the equations are affine in the inputs. If the equations
+are nonlinear in the inputs, the result may not be meaningful.
+"""
+function input_affine_form(eqs, inputs)
+    # Extract the input matrix g(x) by taking coefficients of each input
+    g = [Symbolics.coeff(Symbolics.simplify(eq, expand = true), u)
+         for eq in eqs, u in inputs]
+    g = Symbolics.simplify.(g, expand = true)
+
+    # Extract the drift term f(x) by substituting inputs = 0
+    f = Symbolics.substitute.(eqs, Ref(Dict(inputs .=> 0)))
+    f = Symbolics.simplify.(f, expand = true)
+
+    return f, g
+end

test/input_affine_form.jl

@@ -0,0 +1,135 @@
+using ModelingToolkit, Test
+using Symbolics
+using StaticArrays
+
+@testset "input_affine_form" begin
+    # Test with simple linear system
+    @testset "Simple linear system" begin
+        @variables x1 x2 u1 u2
+        state = [x1, x2]
+        inputs = [u1, u2]
+
+        eqs = [
+            -x1 + 2*x2 + u1,
+            x1*x2 - x2 + u1 + 2*u2
+        ]
+
+        f, g = input_affine_form(eqs, inputs)
+
+        # Verify reconstruction
+        eqs_reconstructed = f + g * inputs
+        @test isequal(Symbolics.simplify.(eqs_reconstructed), Symbolics.simplify.(eqs))
+
+        # Check dimensions
+        @test length(f) == length(eqs)
+        @test size(g) == (length(eqs), length(inputs))
+    end
+
+    # Test with Segway dynamics example
+    @testset "Segway dynamics" begin
+        # Segway parameters
+        grav = 9.81
+        R = 0.195
+        M = 2 * 2.485
+        Jc = 2 * 0.0559
+        L = 0.169
+        m = 44.798
+        Jg = 3.836
+        m0 = 52.710
+        J0 = 5.108
+        Km = 2 * 1.262
+        bt = 2 * 1.225
+
+        # Dynamics of Segway in Euler-Lagrange form
+        D(q) = [m0 m*L*cos(q[2]); m*L*cos(q[2]) J0]
+        function H(q, q̇)
+            return SA[
+                -m * L * sin(q[2]) * q̇[2] + bt * (q̇[1] - R * q̇[2]) / R,
+                -m * grav * L * sin(q[2]) - bt * (q̇[1] - R * q̇[2])
+            ]
+        end
+        B(q) = SA[Km / R, -Km]
+
+        # Convert to control affine form
+        function f_seg(x)
+            q, q̇ = x[SA[1, 2]], x[SA[3, 4]]
+            return [q̇; -D(q) \ H(q, q̇)]
+        end
+        function g_seg(x)
+            q, q̇ = x[SA[1, 2]], x[SA[3, 4]]
+            return [SA[0, 0]; D(q) \ B(q)]
+        end
+
+        # Trace dynamics symbolically
+        @variables q1 q2 qd1 qd2 u
+        x = [q1; q2; qd1; qd2]
+        inputs = [u]
+        eqs = f_seg(x) + g_seg(x) * u
+
+        # Extract control-affine form
+        fe, ge = input_affine_form(eqs, inputs)
+
+        # Test reconstruction
+        eqs2 = fe + ge * inputs
+        diff = Symbolics.simplify.(eqs2 - eqs, expand = true)
+
+        # The difference should be zero or very close to zero symbolically
+        # We test numerically since symbolic simplification might not be perfect
+        f2, _ = build_function(fe, x, expression = false)
+        g2, _ = build_function(ge, x, expression = false)
+
+        for i in 1:10
+            x_val = rand(length(x))
+            @test f2(x_val) ≈ f_seg(x_val) rtol=1e-10
+            @test g2(x_val) ≈ g_seg(x_val) rtol=1e-10
+        end
+    end
+
+    # Test with multiple inputs
+    @testset "Multiple inputs" begin
+        @variables x1 x2 x3 u1 u2
+        state = [x1, x2, x3]
+        inputs = [u1, u2]
+
+        eqs = [
+            x2,
+            x3,
+            -x1 - 2*x2 - x3 + u1 + 3*u2
+        ]
+
+        f, g = input_affine_form(eqs, inputs)
+
+        # Expected results
+        f_expected = [x2, x3, -x1 - 2*x2 - x3]
+        g_expected = [0 0; 0 0; 1 3]
+
+        @test isequal(Symbolics.simplify.(f), Symbolics.simplify.(f_expected))
+
+        # Test g matrix elements
+        for i in 1:size(g, 1), j in 1:size(g, 2)
+
+            @test isequal(Symbolics.simplify(g[i, j]), g_expected[i, j])
+        end
+    end
+
+    # Test with nonlinear state dynamics
+    @testset "Nonlinear state dynamics" begin
+        @variables x1 x2 u
+        state = [x1, x2]
+        inputs = [u]
+
+        eqs = [
+            x2,
+            -sin(x1) - x2 + u
+        ]
+
+        f, g = input_affine_form(eqs, inputs)
+
+        # Expected results
+        f_expected = [x2, -sin(x1) - x2]
+        g_expected = reshape([0, 1], 2, 1)
+
+        @test isequal(Symbolics.simplify.(f), Symbolics.simplify.(f_expected))
+        @test isequal(g, g_expected)
+    end
+end

test/runtests.jl

@@ -34,6 +34,7 @@ end
             @safetestset "Direct Usage Test" include("direct.jl")
             @safetestset "System Linearity Test" include("linearity.jl")
             @safetestset "Input Output Test" include("input_output_handling.jl")
+            @safetestset "Input Affine Form Test" include("input_affine_form.jl")
             @safetestset "Clock Test" include("clock.jl")
             @safetestset "ODESystem Test" include("odesystem.jl")
             @safetestset "Dynamic Quantities Test" include("dq_units.jl")