Smith predictor

This example designs a controller for a plant with a time delay using the internal-model principle, which in this case implies the use of a Smith predictor. The plant is given by $ \dfrac{1}{s + 1}e^{-s\tau} = P_0 e^{-s\tau}$

and the control architecture looks like this

                ┌──────┐              ┌─────────────┐
r               │      │          u   │             │
───+──+────────►│  C0  ├───────────┬─►│ P0*exp(-st) ├─┐y
   ▲  ▲         │      │           │  │             │ │
  -│  │-        └──────┘           │  └─────────────┘ │
   │  │                            │                  │
   │  │ ┌──────────┐    ┌──────┐   │                  │
   │  │ │          │    │      │   │                  │
   │  └─┤1-exp(-st)│◄───┤  P0  │◄──┘                  │
   │    │          │    │      │                      │
   │    └──────────┘    └──────┘                      │
   │                                                  │

The benefit of this approach is that the controller $C_0$ can be designed for the nominal plant $P_0$ without time delay, and still behave well in the presence of the delay. We also see why we refer to such a controller as using an "internal model", due to the presence of a model of $P_0$ in the inner feedback path.

We now set up the nominal system and PI controller

using ControlSystemsBase, Plots
P0 = ss(-1, 1, 1, 0) # Nominal system
StateSpace{Continuous, Int64}
A = 
B = 
C = 
D = 

Continuous-time state-space model

We design a PI controller for nominal system using placePI. To verify the pole placement, use, e.g., dampreport(feedback(P0, C0))

ω0 = 2
ζ  = 0.7
C0, _ = placePI(P0, ω0, ζ)
(TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}
1.7999999999999998s + 4.0

Continuous-time transfer function model, 1.7999999999999998, 0.44999999999999996)

We then setup delayed plant + Smith predictor-based controller

τ = 8
P = delay(τ) * P0
C = feedback(C0, (1.0 - delay(τ))*P0) # form the inner feedback connection in the diagram above
DelayLtiSystem{Float64, Float64}

P: StateSpace{Continuous, Float64}
A = 
 0.0  -2.0
 2.0  -2.8
B = 
 2.0                 -2.0
 1.7999999999999998  -1.7999999999999998
C = 
 2.0  -1.7999999999999998
 0.0  -1.0
D = 
 1.7999999999999998  -1.7999999999999998
 0.0                  0.0

Continuous-time state-space model

Delays: [8.0]

We now plot the closed loop responses. The transfer function from $r$ to $y$ is given by $PC_r/(1+PC_r)$ = feedback(P*C,1), and from a load disturbance entering at $u$ the transfer function is $P/(1+PC_r)$ = feedback(P, C)

using ControlSystems # Load full ControlSystems for delay-system simulation
G = [feedback(P*C, 1) feedback(P, C)] # Reference step at t = 0 and load disturbance step at t = 15
fig_timeresp = plot(lsim(G, (_,t) -> [1; t >= 15], 0:0.1:40),  title="τ = $τ")

Plot the frequency response of the predictor part and compare to a negative delay, which would be an ideal controller that can (typically) not be realized in practice (a negative delay implies foresight).

C_pred = feedback(1, C0*(ss(1.0) - delay(τ))*P0)
fig_bode = bodeplot([C_pred, delay(-τ)], exp10.(-1:0.002:0.4), ls=[:solid :solid :dash :dash], title="", lab=["Smith predictor" "" "Ideal predictor" ""])
plot!(yticks=[0.1, 1, 10], sp=1)
plot!(yticks=0:180:1080, sp=2)

Check the Nyquist plot. Note that the Nyquist curve encircles -1 for τ > 2.99

fig_nyquist = nyquistplot(C * P, exp10.(-1:1e-4:2), title="τ = $τ")

A video tutorial on delay systems is available here:

Additional design methods for delay systems

Many standard control-design methods fail for delay systems, or any system not represented as a rational function. In addition to using the Smith predictor outlined above, there are however several common tricks that can be applied to make use of these methods.

  • Approximate the delay using a pade approximation, this will result in a standard rational model. The drawbacks include zeros in the right half plane and a failure to capture the extreme phase loss of the delay for high frequencies.
  • Discretize the system with a sample time that fits an integer multiple in the delay time. A delay can be represented exactly in discrete time, but if the sample time is chosen small in relation to the delay time, a large number of extra states will be introduced.
  • Neglect the delay and design the controller with large phase and delay margins. This is perhaps not a terribly sophisticated method, but nevertheless useful in practice.
  • Neglect the delay, but model it as uncertainty. See Modeling uncertain time delays in the RobustAndOptimalControl.jl extension package. This can help you get a feeling for the margin with which you must design your controller when you have neglected to model the delay.
  • Frequency-domain methods such as manual loop shaping, and some forms of optimization-based tuning, handle time delays natively.

Whatever method is used to design in the presence of delays, the robustness and performance of the design should preferably be verified using a model of the plant where the delay is included, uncertain or not.