RobustAndOptimalControl.jl

This package is an extension to ControlSystems.jl that provides methods for robust and optimal analysis and synthesis of linear control systems. Some highlights:

  • Named statespace systems (named_ss) where states, inputs and outputs are accessible by names rather than indices. This also facilitates creating complicated feedback interconnections using connect.
  • An interface to DescriptorSystems.jl. Call dss on a statespace system to get a descriptor system. We also forward some methods to implementations in DescriptorSystems.
  • Robust/optimal design methods such as $H_{\infty}$, $H_{2}$, LQG and Glover-McFarlane.
  • Robustness-related metrics such as nugap ($\nu$-gap), ncfmargin, diskmargin etc.
  • Uncertainty modeling with the $M\Delta$ framework (and more). Analsysis methods for this framework are still limited.
  • Model augmentation.
  • An ExtendedStateSpace type that represents a partitioned statespace system $w,u \rightarrow z,y$.

Installation

pkg> add RobustAndOptimalControl

Named systems

See complicated-feedback example

Named systems can be indexed with their names, e.g.,

G[:y2, :u4]

but also using incomplete names, e.g., if G contains outputs :y1, :y2, :y3, :z1, :z2, the following retrieves the three outputs that has the prefix :y

G[:y, :] # Prefix matching is used if no exact match is found.

Connecting systems together

See complicated-feedback example

Example

The following complicated feedback interconnection

                 yF
              ┌────────────────────────────────┐
              │                                │
    ┌───────┐ │  ┌───────┐ yR   ┌─────────┐    │    ┌───────┐
uF  │       │ │  │       ├──────►         │ yC │  uP│       │    yP
────►   F   ├─┴──►   R   │      │    C    ├────+────►   P   ├────┬────►
    │       │    │       │   ┌──►         │         │       │    │
    └───────┘    └───────┘   │  └─────────┘         └───────┘    │
                             │                                   │
                             └───────────────────────────────────┘

can be created by

F = named_ss(ssrand(1, 1, 2, proper=true), x=:xF, u=:uF, y=:yF)
R = named_ss(ssrand(1, 1, 2, proper=true), x=:xR, u=:uR, y=:yR)
C = named_ss(ssrand(1, 1, 2, proper=true), x=:xC, u=:uC, y=:yC)
P = named_ss(ssrand(1, 1, 3, proper=true), x=:xP, u=:uP, y=:yP)

addP = sumblock("uP = yF + yC") # Sum node before P
addC = sumblock("uC = yR - yP") # Sum node before C (drawn as two arrows into C in the diagram)

connections = [
    :yP => :yP # Output to input
    :uP => :uP # addP's output is called the same as P's input
    :yC => :yC
    :yF => :yF
    :yF => :uR
    :uC => :uC
    :yR => :yR
]
w1 = [:uF] # External inputs

G = connect([F, R, C, P, addP, addC], connections; w1)

If an external input is to be connected to multiple points, use a splitter to split up the signal into a set of unique names which are then used in the connections.

Model augmentation

Add disturbance and performance models to your system model.

$H_\infty$ and $H_2$ design

Examples are available in the example folder.

Example: Glover McFarlane design

This example will design a robust controller using the Glover-McFarlane method. This method requires the user to perform an initial loop-shaping design, i.e., by tuning a standard PI controller etc. The glover_mcfarlane method then takes the loop-shaping controller and the plant model and returns a robustified controller. This is example 9.3 from Skogestad, "Multivariable Feedback Control: Analysis and Design".

using RobustAndOptimalControl, ControlSystems, Plots, Test
G = tf(200, [10, 1])*tf(1, [0.05, 1])^2     |> ss # Plant model
Gd = tf(100, [10, 1])                       |> ss # Disturbance model
W1 = tf([1, 2], [1, 1e-6])                  |> ss # Loop-shaping controller
K, γ, info = glover_mcfarlane(G, 1.1; W1)         # K is robustified controller
@test info.γmin ≈ 2.34 atol=0.005
Gcl = extended_gangoffour(G, K) # Form closed-loop system

fig1 = bodeplot([G, info.Gs, G*K], lab=["G" "" "Initial GK" "" "Robustified GK"])
fig2 = bodeplot(Gcl, lab=["S" "KS" "PS" "T"], plotphase=false) # Plot gang of four

# Simulate the response to a disturbance (Gd*feedback(1, G*K) = Gd*S is the closed-loop transfer function from an additive output disturbance)
fig3 = plot(step(Gd*feedback(1, info.Gs), 3), lab="Initial controller")
plot!(step(Gd*feedback(1, G*K), 3), lab="Robustified controller")
fig4 = nyquistplot([info.Gs, G*K], ylims=(-2,1), xlims=(-2, 1),
    Ms_circles = 1.5,
    lab = ["Initial controller" "Robustified controller"],
    title = "Loop transfers with and without robustified controller"
)
plot(fig1, fig2, fig3, fig4, titlefontsize=9, labelfontsize=9, size=(800, 640))

Example of controller reduction:

The order of the controller designed above can be reduced maintaining at least 2/3 of the robustness margin like this

e,_ = ncfmargin(info.Gs, info.Ks)
Kr, hs, infor = baltrunc_coprime(info.Ks, n=info.Ks.nx)
n = findlast(RobustAndOptimalControl.error_bound(hs) .> 2e/3) # 2/3 e sets the robustness margin
Ksr, hs, infor = baltrunc_coprime(info.Ks; n)
@test ncfmargin(info.Gs, Ksr)[1] >= 2/3 * e
Kr = W1*Ksr
bodeplot([G*K, G*Kr], lab=["L original" "" "L Reduced" ""])

This gives a final controller Kr of order 2 instead of order 5, but a very similar robustness margin. You may also call

controller_reduction_plot(info.Gs, info.Ks)

to help you select the controller order.

Example: Glover McFarlane 2-dof design

In this example, we design a 2 degree-of-freedom controller using the Glover McFarlane method. This design method requires you to specify both a loop-shaping controller as well as a reference model. It's usually a good idea to let the reference model have the same number of poles as the system that is being controlled in order not not differentiate the references and introduce non-robustness.

using RobustAndOptimalControl, ControlSystems, Plots
P = tf([1, 5], [1, 2, 10]) # Plant
W1 = tf(1,[1, 0]) |> ss    # Loop shaping controller

Tref = tf(1, [1, 1])^2 |> ss # Reference model of same order as P

K1dof, γ1, info1 = glover_mcfarlane(ss(P), 1.1; W1)
K2dof, γ2, info2 = glover_mcfarlane_2dof(ss(P), Tref, 1.1, 1.1; W1)

G1 = feedback(P*K1dof)
G2 = info2.Gcl

w = exp10.(LinRange(-2, 2, 200))
fig1 = bodeplot(info2.K1, w, lab="Feedforward filter")
fig2 = plot([step(G1, 15), step(G2, 15), step(Tref, 15)], lab=["1-DOF" "2-DOF" "Tref"])
plot(fig1, fig2)

LQG design

The main functionality for LQG design is exposed through LQGProblem. See the docstring for an example.

System analysis

See also Structured singular value and diskmargin below

Structured singular value and diskmargin

Diskmargin example

The diskmargin can be visualized in several ways, as a region of allowed simultaneous gain and pahse variations:

using RobustAndOptimalControl, ControlSystems, Plots
L = tf(25, [1,10,10,10])
dm = diskmargin(L, 0)
plot(dm) # Plot the disk margin to illustrate maximum allowed simultaneous gain and phase variations.

As a Nyquist exclusion disk:

nyquistplot(L)
plot!(dm, nyquist=true) # plot a nyquist exclusion disk. The Nyquist curve will be tangent to this disk at `dm.ω0`
nyquistplot!(dm.f0*L, lab="perturbed") # If we perturb the system with the worst-case perturbation `f0`, the curve will pass through the critical point -1.

And as a frequency-dependent margin

w = exp10.(LinRange(-2, 2, 500))
dms = diskmargin(L, 0, w)
plot(dms)

Closed-loop analysis

Model reduction

Model transformations