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Linear Analysis

Linear analysis refers to the process of linearizing a nonlinear model and analysing the resulting linear dynamical system. To facilitate linear analysis, ModelingToolkit provides the concept of an AnalysisPoint, which can be inserted in-between two causal blocks (such as those from ModelingToolkitStandardLibrary.Blocks sub module). Once a model containing analysis points is built, several operations are available:

  • get_sensitivity get the sensitivity function (wiki), $S(s)$, as defined in the field of control theory.
  • get_comp_sensitivity get the complementary sensitivity function $T(s) : S(s)+T(s)=1$.
  • get_looptransfer get the (open) loop-transfer function where the loop starts and ends in the analysis point. For a typical simple feedback connection with a plant $P(s)$ and a controller $C(s)$, the loop-transfer function at the plant output is $P(s)C(s)$.
  • linearize can be called with two analysis points denoting the input and output of the linearized system.
  • open_loop return a new (nonlinear) system where the loop has been broken in the analysis point, i.e., the connection the analysis point usually implies has been removed.

An analysis point can be created explicitly using the constructor AnalysisPoint, or automatically when connecting two causal components using connect:

connect(comp1.output, :analysis_point_name, comp2.input)

A single output can also be connected to multiple inputs:

connect(comp1.output, :analysis_point_name, comp2.input, comp3.input, comp4.input)
Causality

Analysis points are causal, i.e., they imply a directionality for the flow of information. The order of the connections in the connect statement is thus important, i.e., connect(out, :name, in) is different from connect(in, :name, out).

The directionality of an analysis point can be thought of as an arrow in a block diagram, where the name of the analysis point applies to the arrow itself.

┌─────┐         ┌─────┐
│     │  name   │     │
│  out├────────►│in   │
│     │         │     │
└─────┘         └─────┘

This is signified by the name being the middle argument to connect.

Of the above mentioned functions, all except for open_loop return the output of ModelingToolkit.linearize, which is

matrices, simplified_sys = linearize(_...)
# matrices = (; A, B, C, D)

i.e., matrices is a named tuple containing the matrices of a linear state-space system on the form

\[\begin{aligned} \dot x &= Ax + Bu\\ y &= Cx + Du \end{aligned}\]

Example

The following example builds a simple closed-loop system with a plant $P$ and a controller $C$. Two analysis points are inserted, one before and one after $P$. We then derive a number of sensitivity functions and show the corresponding code using the package ControlSystemBase.jl

using ModelingToolkitStandardLibrary.Blocks, ModelingToolkit
using ModelingToolkit: t_nounits as t

@named P = FirstOrder(k = 1, T = 1) # A first-order system with pole in -1
@named C = Gain(-1)             # A P controller

eqs = [connect(P.output, :plant_output, C.input)  # Connect with an automatically created analysis point called :plant_output
       connect(C.output, :plant_input, P.input)]
sys = System(eqs, t, systems = [P, C], name = :feedback_system)

matrices_S = get_sensitivity(sys, :plant_input)[1] # Compute the matrices of a state-space representation of the (input)sensitivity function.
matrices_T = get_comp_sensitivity(sys, :plant_input)[1]
(A = [-2.0;;], B = [-1.0;;], C = [-1.0;;], D = [0.0;;])

Continued linear analysis and design can be performed using ControlSystemsBase.jl. We create ControlSystemsBase.StateSpace objects using

using ControlSystemsBase, Plots
S = ss(matrices_S...)
T = ss(matrices_T...)
bodeplot([S, T], lab = ["S" "" "T" ""], plot_title = "Bode plot of sensitivity functions",
    margin = 5Plots.mm)
Example block output

The sensitivity functions obtained this way should be equivalent to the ones obtained with the code below

using ControlSystemsBase
P = tf(1.0, [1, 1]) |> ss
C = 1                      # Negative feedback assumed in ControlSystems
S = sensitivity(P, C)      # or feedback(1, P*C)
T = comp_sensitivity(P, C) # or feedback(P*C)
ControlSystemsBase.StateSpace{ControlSystemsBase.Continuous, Float64}
A = 
 -2.0
B = 
 1.0
C = 
 1.0
D = 
 0.0

Continuous-time state-space model

We may also derive the loop-transfer function $L(s) = P(s)C(s)$ using

matrices_L = get_looptransfer(sys, :plant_output)[1]
L = ss(matrices_L...)
ControlSystemsBase.StateSpace{ControlSystemsBase.Continuous, Float64}
A = 
 -1.0
B = 
 -1.0
C = 
 1.0
D = 
 0.0

Continuous-time state-space model

which is equivalent to the following with ControlSystems

L = P * (-C) # Add the minus sign to build the negative feedback into the controller
ControlSystemsBase.StateSpace{ControlSystemsBase.Continuous, Float64}
A = 
 -1.0
B = 
 -1.0
C = 
 1.0
D = 
 -0.0

Continuous-time state-space model

To obtain the transfer function between two analysis points, we call linearize

matrices_PS = linearize(sys, :plant_input, :plant_output)[1]
(A = [-2.0;;], B = [1.0;;], C = [1.0;;], D = [0.0;;])

this particular transfer function should be equivalent to the linear system P(s)S(s), i.e., equivalent to

feedback(P, C)
ControlSystemsBase.StateSpace{ControlSystemsBase.Continuous, Float64}
A = 
 -2.0
B = 
 1.0
C = 
 1.0
D = 
 0.0

Continuous-time state-space model

Obtaining transfer functions

A statespace system from ControlSystemsBase can be converted to a transfer function using the function tf:

tf(S)
ControlSystemsBase.TransferFunction{ControlSystemsBase.Continuous, ControlSystemsBase.SisoRational{Float64}}
1.0s + 1.0
----------
1.0s + 2.0

Continuous-time transfer function model

Gain and phase margins

Further linear analysis can be performed using the analysis methods from ControlSystemsBase. For example, calculating the gain and phase margins of a system can be done using

margin(P)
(wgm = [NaN;;], gm = [Inf;;], wpm = [NaN;;], pm = [Inf;;])

(they are infinite for this system). A Nyquist plot can be produced using

nyquistplot(P)
Example block output

Index

    ModelingToolkit.get_comp_sensitivity — Function
    get_comp_sensitivity(sys, ap::AnalysisPoint; kwargs)
get_comp_sensitivity(sys, ap_name::Symbol; kwargs)

    Compute the complementary sensitivity function in analysis point ap. The complementary sensitivity function is obtained by introducing an infinitesimal perturbation d at the output of ap, linearizing the system and computing the transfer function between d and the input of ap.

    Arguments:

    • kwargs: Are sent to ModelingToolkit.linearize

    See also get_sensitivity, get_looptransfer.

    source
    ModelingToolkit.get_comp_sensitivity_function — Method
    get_comp_sensitivity_function(
    sys::ModelingToolkitBase.AbstractSystem,
    aps;
    kwargs...
) -> Tuple{ModelingToolkit.LinearizationFunction{DI, AI, _A, P, _B, _C, J1, J2, J3, J4, IA, @NamedTuple{abstol::Float64, reltol::Float64, nlsolve_alg::Nothing}} where {DI<:AbstractVector{Int64}, AI<:AbstractVector{Int64}, _A, P<:ODEProblem, _B, _C, J1<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing}}, ModelingToolkit.var"#uff#12"{var"#64#fun"}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#uff#12"), _A, _B}), var"#64#fun", _A}}, J2<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing}}, _A, _B, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:ForwardDiff.JacobianConfig, _A, _B}}, J3<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing, Nothing}}, ModelingToolkit.var"#pff#13"{var"#65#fun", var"#66#setter"}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#pff#13"), _A, _B}), var"#65#fun", var"#66#setter", _A}}, J4<:(ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing, Nothing}}, ModelingToolkit.var"#hpf#11"{fun, setter}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#hpf#11"), _A, _B}), fun, setter, _A}), IA<:Union{SciMLBase.NoInit, SciMLBase.OverrideInit{Nothing, Nothing, Nothing}}}, Any}

    Return the complementary sensitivity function for the analysis point(s) aps, and the modified system simplified with the appropriate inputs and outputs.

    Keyword Arguments

    • loop_openings: A list of analysis points whose connections should be removed and the outputs set to the input as a part of the linear analysis.

    • system_modifier: A function taking the transformed system and applying any additional transformations, returning the modified system. The modified system is passed to linearization_function.

    All other keyword arguments are forwarded to linearization_function.

    source
    ModelingToolkit.get_looptransfer — Function
    get_looptransfer(sys, ap::AnalysisPoint; kwargs)
get_looptransfer(sys, ap_name::Symbol; kwargs)

    Compute the (linearized) loop-transfer function in analysis point ap, from ap.out to ap.in.

    Negative feedback

    Feedback loops often use negative feedback, and the computed loop-transfer function will in this case have the negative feedback included. Standard analysis tools often assume a loop-transfer function without the negative gain built in, and the result of this function may thus need negation before use.

    Arguments:

    • kwargs: Are sent to ModelingToolkit.linearize

    See also get_sensitivity, get_comp_sensitivity, open_loop.

    source
    ModelingToolkit.get_looptransfer_function — Method
    get_looptransfer_function(
    sys::ModelingToolkitBase.AbstractSystem,
    aps;
    kwargs...
) -> Tuple{ModelingToolkit.LinearizationFunction{DI, AI, _A, P, _B, _C, J1, J2, J3, J4, IA, @NamedTuple{abstol::Float64, reltol::Float64, nlsolve_alg::Nothing}} where {DI<:AbstractVector{Int64}, AI<:AbstractVector{Int64}, _A, P<:ODEProblem, _B, _C, J1<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing}}, ModelingToolkit.var"#uff#12"{var"#64#fun"}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#uff#12"), _A, _B}), var"#64#fun", _A}}, J2<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing}}, _A, _B, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:ForwardDiff.JacobianConfig, _A, _B}}, J3<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing, Nothing}}, ModelingToolkit.var"#pff#13"{var"#65#fun", var"#66#setter"}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#pff#13"), _A, _B}), var"#65#fun", var"#66#setter", _A}}, J4<:(ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing, Nothing}}, ModelingToolkit.var"#hpf#11"{fun, setter}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#hpf#11"), _A, _B}), fun, setter, _A}), IA<:Union{SciMLBase.NoInit, SciMLBase.OverrideInit{Nothing, Nothing, Nothing}}}, Any}

    Return the loop-transfer function for the analysis point(s) aps, and the modified system simplified with the appropriate inputs and outputs.

    Keyword Arguments

    • loop_openings: A list of analysis points whose connections should be removed and the outputs set to the input as a part of the linear analysis.

    • system_modifier: A function taking the transformed system and applying any additional transformations, returning the modified system. The modified system is passed to linearization_function.

    All other keyword arguments are forwarded to linearization_function.

    source
    ModelingToolkit.get_sensitivity — Function
    get_sensitivity(sys, ap::AnalysisPoint; kwargs)
get_sensitivity(sys, ap_name::Symbol; kwargs)

    Compute the sensitivity function in analysis point ap. The sensitivity function is obtained by introducing an infinitesimal perturbation d at the input of ap, linearizing the system and computing the transfer function between d and the output of ap.

    Arguments:

    • kwargs: Are sent to ModelingToolkit.linearize

    See also get_comp_sensitivity, get_looptransfer.

    source
    ModelingToolkit.get_sensitivity_function — Method
    get_sensitivity_function(
    sys::ModelingToolkitBase.AbstractSystem,
    aps;
    kwargs...
) -> Tuple{ModelingToolkit.LinearizationFunction{DI, AI, _A, P, _B, _C, J1, J2, J3, J4, IA, @NamedTuple{abstol::Float64, reltol::Float64, nlsolve_alg::Nothing}} where {DI<:AbstractVector{Int64}, AI<:AbstractVector{Int64}, _A, P<:ODEProblem, _B, _C, J1<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing}}, ModelingToolkit.var"#uff#12"{var"#64#fun"}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#uff#12"), _A, _B}), var"#64#fun", _A}}, J2<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing}}, _A, _B, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:ForwardDiff.JacobianConfig, _A, _B}}, J3<:Union{Nothing, ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing, Nothing}}, ModelingToolkit.var"#pff#13"{var"#65#fun", var"#66#setter"}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#pff#13"), _A, _B}), var"#65#fun", var"#66#setter", _A}}, J4<:(ModelingToolkit.PreparedJacobian{true, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, C, Tuple{Nothing, Nothing, Nothing}}, ModelingToolkit.var"#hpf#11"{fun, setter}, _A, ADTypes.AutoForwardDiff{nothing, Nothing}} where {C<:(ForwardDiff.JacobianConfig{T, _A, _B, <:Tuple{Any, Any}} where {T<:(ForwardDiff.Tag{F} where F<:ModelingToolkit.var"#hpf#11"), _A, _B}), fun, setter, _A}), IA<:Union{SciMLBase.NoInit, SciMLBase.OverrideInit{Nothing, Nothing, Nothing}}}, Any}

    Return the sensitivity function for the analysis point(s) aps, and the modified system simplified with the appropriate inputs and outputs.

    Keyword Arguments

    • loop_openings: A list of analysis points whose connections should be removed and the outputs set to the input as a part of the linear analysis.

    • system_modifier: A function taking the transformed system and applying any additional transformations, returning the modified system. The modified system is passed to linearization_function.

    All other keyword arguments are forwarded to linearization_function.

    source
    ModelingToolkit.linearization_ap_transform — Method
    sys, input_vars, output_vars =
    linearization_ap_transform(
    sys,
    inputs::Union{Vector{Symbol}, Vector{AnalysisPoint}, Symbol, AnalysisPoint},
    outputs,
    loop_openings
) -> Tuple{Any, Vector{SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymReal}}, Vector{SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymReal}}}

    Apply analysis-point transformations to prepare a system for linearization.

    Returns

    • sys: The transformed system.
    • input_vars: A vector of input variables corresponding to the input analysis points.
    • output_vars: A vector of output variables corresponding to the output analysis points.
    source