Merge pull request #793 from JuliaControl/analysisdocs

add analysis examples to the docs

docs/make.jl

@@ -54,6 +54,7 @@ makedocs(modules=[ControlSystems, ControlSystemsBase],
         ],
         "Examples" => Any[
             "Design" => "examples/example.md",
+            "Analysis" => "examples/analysis.md",
             "Smith predictor" => "examples/smith_predictor.md",
             "Iterative Learning Control (ILC)" => "examples/ilc.md",
             "Properties of delay systems" => "examples/delay_systems.md",

docs/src/examples/analysis.md

@@ -0,0 +1,67 @@
+# Analysis of linear control systems
+From classical control, we get robustness measures such as gain and phase margins. These provide a quick and intuitive way to assess robustness of single-input, single-output systems, but also have a number of downsides, such as optimism in the presence of simultaneous gain and phase variations as well as limited applicability for MIMO systems.
+
+Gain and phase margins can be computed using the functions [`margin`](@ref) and [`marginplot`](@ref)
+
+## Example: Gain and phase margins
+```@example
+using ControlSystemsBase, Plots
+P = tf(1, [1, 0.2, 1])
+C = pid(0.2, 1)
+loopgain = P*C
+marginplot(loopgain)
+```
+
+## Sensitivity analysis
+More generally applicable measures of robustness include analysis of sensitivity functions, notably the peaks of the sensitivity function
+```math
+S(s) = (I + P(s)C(s))^{-1}
+```
+and the complementary sensitivity function
+```math
+T(s) = I - S(s) = (I + P(s)C(s))^{-1}P(s)C(s)
+```
+
+### Examples
+We can plot all four sensitivity functions referred to as the "gang of four" using [`gangoffourplot`](@ref).
+```@example SENS
+using ControlSystemsBase, Plots
+P = tf(1, [1, 0.2, 1])
+C = pid(0.2, 1)
+gangoffourplot(P, C)
+```
+
+The peak value of the sensitivity function, ``M_S``, can be computed using [`hinfnorm`](@ref)
+```@example SENS
+S = sensitivity(P, C)
+Ms, ωMs = hinfnorm(S)
+```
+
+And we can plot a circle in the Nyquist plot corresponding to the inverse distance between the loop-transfer function and the critical point:
+```@example SENS
+w = exp10.(-1:0.001:2)
+nyquistplot(P*C, w, Ms_circles=[Ms], xlims=(-1.2, 0.5), ylims=(-2, 0.3))
+```
+
+``M_S`` is always ``≥ 1``, but we typically want to keep it below 1.3-2 for robustness reasons. For SISO systems, ``M_S`` is linked to the classical gain and phase margins through the following inequalities:
+```math
+\begin{aligned}
+\phi_m &≥ 2 \sin^{-1}\left(\dfrac{1}{2M_S}\right) \text{rad}\\
+g_m &≥ \dfrac{M_S}{M_S-1}
+\end{aligned}
+```
+
+We can also obtain individual sensitivity function using the low-level function [`feedback`](@ref) directly, or using one of the higher-level functions
+- [`sensitivity`](@ref)
+- [`comp_sensitivity`](@ref)
+- [`G_PS`](@ref)
+- [`G_CS`](@ref)
+- [`gangoffour`](@ref)
+- [`extended_gangoffour`](@ref)
+- [`feedback_control`](@ref)
+
+
+## Further reading
+A modern robustness measure is the [`diskmargin`](https://juliacontrol.github.io/RobustAndOptimalControl.jl/dev/#Diskmargin-example), that analyses the robustness of a SISO or MIMO system to simultaneous gain and phase variations.
+
+In the presence of structured uncertainty, such as parameter uncertainty or other explicitly modeled uncertainty, the structured singular value (often referred to as $\mu$), provides a way to analyze robustness with respect to the modeled uncertainty. See the [RobustAndOptimalControl.jl](https://juliacontrol.github.io/RobustAndOptimalControl.jl/dev/) package for more details.