Disturbance modeling and rejection with LQG controllers

This example will demonstrate how you can add disturbance models to ta plant model and achieve effective disturbance rejection using an LQG controller. For simplicity, we will consider a simple first-order system $G$

\[\begin{aligned} \dot{x} &= -ax + b(u + d) \\ y &= cx \end{aligned}\]

where a load disturbance $d$ is acting on the input of the system. This is a simple and very common model for load disturbances. In this example, we will let $d$ be a unit step at time $t=10$, this will effectively create a LQG controller with integral action.

We will begin by setting up the LQG problem and solve it without andy disturbance model. For details regarding the setup of an LQG problem, see, the LQGProblem documentation.

We start by defining the process model and discretize it using zero-order hold.

using RobustAndOptimalControl, ControlSystems, Plots, LinearAlgebra
Ts = 1 # Sample time
G = c2d(ss(tf(1, [10, 1])), Ts) # Process model
ControlSystems.StateSpace{ControlSystems.Discrete{Int64}, Float64}
A = 
B = 
C = 
D = 

Sample Time: 1 (seconds)
Discrete-time state-space model

We then choose the parameters of the LQG controller, i.e., the cost matrices $Q_1, Q_2$ as well as the covariance matrices $R_1, R_2$

nx  = G.nx
nu  = G.nu
ny  = G.ny
x0  = zeros(G.nx) # Initial condition

Q1 = 100diagm(ones(G.nx)) # state cost matrix
Q2 = 0.01diagm(ones(nu))  # control cost matrix

R1 = 0.001I(nx) # State noise covariance
R2 = I(ny)      # measurement noise covariance
prob = LQGProblem(G, Q1, Q2, R1, R2)

disturbance = (x, t) -> t * Ts ≥ 10 # This is our load disturbance, a step at ``t = 10``
Gcl = G_PS(prob)  # This forms the transfer function from load disturbance to output
res = lsim(Gcl, disturbance, 100)

As we can see, the controller appears to do next to nothing to suppress the disturbance. The problem is that the Kalman filter does not have a model for such a disturbance, and its estimate of the state will thus be severely biased.

The next step is to add a disturbance model to the plant model. Since the disturbance if of low-frequency character (indeed, its transfer function is $1/s$), we make use of the function add_low_frequency_disturbance

Gd = add_low_frequency_disturbance(G, ϵ = 1e-6) # The ϵ moves the integrator pole slightly into the stable region
nx = Gd.nx

There is no point trying to penalize the disturbance state in the LQR problem, it's not controllable, we thus penalize the output only, which we can write as

\[y^T Q_1 y = (Cx)^T Q_1 Cx = x^T (C^T Q_1C) x\]

C  = Gd.C
Q1 = 100C'diagm(ones(G.nx)) * C # state cost matrix
x0 = zeros(nx)

We also provide new covariance matrices for the Kalman filter where the entry of the state-covariance matrix that corresponds to the disturbance state (the second and last state) determines how fast the Kalman filter integrates the disturbance. We choose a large value (1), implying fast integration

R1 = diagm([0.001, 1])
R2 = I(ny)
prob = LQGProblem(Gd, Q1, Q2, R1, R2)
Gcl  = [G_PS(prob); -comp_sensitivity(prob)] # -comp_sensitivity(prob) is the same as the transfer function from load disturbance to control signal
res  = lsim(Gcl, disturbance, 100)
plot(res, ylabel=["y" "u"]); ylims!((-0.05, 0.3), sp = 1)

This time, we see that the controller indeed rejects the disturbance and the control signal settles on -1 which is exactly what's required to counteract the load disturbance of +1.

Before we feel confident about deploying the LQG controller, we investigate its closed-loop properties.

w = exp10.(LinRange(-3, log10(pi / Ts), 200))
gangoffourplot(prob, w, lab = "", legend = :bottomright)

We see that our design led to a system with a rather high peak in sensitivity. This is an indication that we perhaps added too much "integral action" by a too fast observer pole related to the disturbance state. Let's see how a slightly more conservative design fares:

R1 = diagm([0.001, 0.2]) # Reduce the noise on the integrator state from 1 to 0.2
R2 = I(ny)
prob = LQGProblem(Gd, Q1, Q2, R1, R2)

Gcl = [G_PS(prob); -comp_sensitivity(prob)]
res = lsim(Gcl, disturbance, 100)
f1 = plot(res, ylabel=["y" "u"]); ylims!((-0.05, 0.3), sp = 1)
f2 = gangoffourplot(prob, w, lab = "", legend = :bottomright)

plot(f1, f2, titlefontsize=10)

We see that we now have a slightly larger disturbance response than before, but in exchange, we lowered the peak sensitivity and complimentary sensitivity from (1.5, 1.31) to (1.25, 1.11), a more robust design. We also reduced the amplification of measurement noise ($CS = C/(1+PC)$). To be really happy with the design, we should probably add high-frequency roll-off as well.