Example for Linear Quadratic Gaussion control design

control

#1

Hello,

My colleague and me wrote the following example for LQG design of a speed controller of a DC motor:

using ControlSystems
"""
Example for designing an LQG speed controller for an electrical DC motor.
"""

# Constants
Ke = 0.006156                                # electromotive force constant in V/rpm
Kt                      = 0.0728             # Torque constant (Nm/A)
J                       = 2.8*700e-6;        # Inertia of motor plus load (kgm^2)
Rel                     = 0.11;              # DC motor resistance (Ω)
L                       = 34e-6;             # DC motor inductance (H)

# helper to convert a scalar into an 1x1 Matrix
mat(sc) = reshape([sc],1,1)

# Create an instance of a DC motor model in state space form.
# Ke: electromotive force constant in V/rpm
# Kt: torque constant (Nm/A)
# L: Inductance; R: Resistance, J: Inertia; b: viscous friction coefficient
function motor(Ke, Kt, L, R, J, b=1e-3)
    Ke1 = Ke*60.0/2π   # change unit to rad/s
    A = [-b/J     Kt/J
         -Ke1/L   -R/L];
    B = [0
        1/L];
    C = [1     0];
    D = 0;
    sys = ss(A, B, C, D);
end

p60 = motor(Ke, Kt, L, Rel, J)
stepplot(p60, 0.2, 0.001)
bodeplot(p60)

# LQR control
Q = [1.     0;
     0      1]

R = 20.
K = lqr(p60.A, p60.B, Q, R)
# needs to be modified if Nbar is not a scalar
Nbar = 1. / (p60.D - (p60.C - p60.D*K) * inv(p60.A - p60.B*K) * p60.B)

# Kalman filter based observer
Vd = [10.     0   # covariance of the speed estimation
     0       10]; # covariance of the current estimation
Vn = 0.04;       # covariance for the speed measurement (radians/s)^2
G = LQG(p60, Q, mat(R), Vd, mat(Vn))
Gcl = G[:cl]
T = G[:T]
S = G[:S];
f1 = sigmaplot([S,T],logspace(-3,3,1000))
f2 = stepplot(Gcl, label=["Closed loop system using LQG"])
Plots.plot(f1,f2)

It requires to use the master branch of ControlSystems.jl.

Pkg.add("ControlSystems.jl")
Pkg.checkout("ControlSystems.jl")

Any suggestions how to simplify or improve this code?
For example, is the gain Nbar also available using the LQG type?

Extra question: Is it possible to get more then one plot window?

Uwe