# Examples

## LQR design

The infinite-horizon LQR controller is derived as the linear state-feedback $u = -Lx$ that minimizes the following quadratic cost function

$$$L = \text{arg\;min}_L \int_0^\infty x^T Q x + u^T R u \; dt$$$

where $x$ is the state vector and $u$ is the input vector.

The example below performs a simple LQR design for a double integrator in discrete time using the function lqr. In this example, we will use the method of lsim that accepts a function $u(x, t)$ as input. This allows us to easily simulate the system both control input and a disturbance input. For more advanced LQR and LQG design, see the LQGProblem type in RobustAndOptimalControl.

using ControlSystemsBase
using LinearAlgebra # For identity matrix I
using Plots

# Create system
Ts      = 0.1
A       = [1 Ts; 0 1]
B       = [0; 1]
C       = [1 0]
sys     = ss(A,B,C,0,Ts)

# Design controller
Q       = I # Weighting matrix for state
R       = I # Weighting matrix for input
L       = lqr(Discrete,A,B,Q,R) # lqr(sys,Q,R) can also be used

# Simulation
u(x,t)  = -L*x .+ 1.5(t>=2.5) # Form control law (u is a function of t and x), a constant input disturbance is affecting the system from t≧2.5
t       = 0:Ts:5              # Time vector
x0      = [1,0]               # Initial condition
y, t, x, uout = lsim(sys,u,t,x0=x0)
plot(t,x', lab=["Position" "Velocity"], xlabel="Time [s]")

save_docs_plot("lqrplot.svg"); # hide

To design an LQG controller (LQR with a Kalman filter), see the functions

## PID design functions

A basic PID controller can be constructed using the constructor pid. In ControlSystems.jl, we often refer to three different formulations of the PID controller, which are defined as

• Standard form: $K_p(1 + \frac{1}{T_i s} + T_ds)$
• Series form: $K_c(1 + \frac{1}{τ_i s})(τ_d s + 1)$
• Parallel form: $K_p + \frac{K_i}{s} + K_d s$

Most functions that construct PID controllers allow the user to select which form to use.

A tutorial on PID design is available here:

The following examples show basic workflows for designing PI/PID controllers.

### PI loop shaping example

By plotting the gang of four under unit feedback for the process

$$$P(s) = \dfrac{1}{(s + 1)^4}$$$
using ControlSystemsBase, Plots
P = tf(1, [1,1])^4
gangoffourplot(P, tf(1))

we notice that the sensitivity function is a bit too high around frequencies ω = 0.8 rad/s. Since we want to control the process using a simple PI-controller, we utilize the function loopshapingPI and tell it that we want 60 degrees phase margin at this frequency. The resulting gang of four is plotted for both the constructed controller and for unit feedback.

using ControlSystemsBase, Plots
P = tf(1, [1,1])^4
ωp = 0.8
C,kp,ki,fig = loopshapingPI(P,ωp,phasemargin=60,form=:parallel, doplot=true)
fig

We could also consider a situation where we want to create a closed-loop system with the bandwidth ω = 2 rad/s, in which case we would write something like

ωp = 2
C60,kp,ki,fig = loopshapingPI(P,ωp,rl=1,phasemargin=60,form=:standard,doplot=true)
fig

Here we specify that we want the Nyquist curve L(iω) = P(iω)C(iω) to pass the point |L(iω)| = rl = 1, arg(L(iω)) = -180 + phasemargin = -180 + 60 The gang of four tells us that we can indeed get a very robust and fast controller with this design method, but it will cost us significant control action to double the bandwidth of all four poles.

### PID loop shaping

Processes with inertia, like double integrators, require a derivative term in the controller for good results. The function loopshapingPID allows you to specify a point in the Nyquist plane where the loop-transfer function $L(s) = P(s)C(s)$ should be tangent to the circle that denotes $|T| = |\dfrac{PC}{1 + PC}| = M_t$ The tangent point is specified by specifying $M_t$ and the angle $\phi_t$ between the real axis and the tangent point, indicated in the Nyquist plot below.

using ControlSystemsBase, Plots
P  = tf(1, [1,0,0]) # A double integrator
Mt = 1.3            # Maximum magnitude of complementary sensitivity
ϕt = 75             # Angle of tangent point
ω  = 1              # Frequency at which the specification holds
C, kp, ki, kd, fig = loopshapingPID(P, ω; Mt, ϕt, doplot=true)
fig

To get good robustness, we typically aim for a $M_t$ less than 1.5. In general, the smaller $M_t$ we require, the larger the controller gain will be.

Since we are designing a PID controller, we expect a large controller gain for high frequencies. This is generally undesirable for both robustness and noise reasons, and is commonly solved by introducing a lowpass filter in series with the controller. The example below passes the keyword argument Tf=1/20ω to indicate that we want to add a second-order lowpass filter with a cutoff frequency 20 times faster than the design frequency.

Tf = 1/20ω
C, kp, ki, kd, fig, CF = loopshapingPID(P, ω; Mt, ϕt, doplot=true, Tf)
fig

As we can see, the addition of the filter increases the high-frequency roll-off in both $T$ and $CS$, which is typically desireable.

To get better control over the filter, it can be pre-designed and supplied to loopshapingPID with the keyword argument F:

F = tf(1, [Tf^2, 2*Tf/sqrt(2), 1])
C, kp, ki, kd, fig, CF = loopshapingPID(P, ω; Mt, ϕt, doplot=true, F)

A video tutorial on pole placement is available here:

The following example illustrates how we can perform advanced pole-zero placement using the function rstc (rstd in discrete time). The task is to make the process $P$ a bit faster and damp the poorly damped poles.

Define the process

ζ = 0.2
ω = 1

B = [1]
A = [1, 2ζ*ω, ω^2]
P = tf(B,A)

Define the desired closed-loop response, calculate the controller polynomials and simulate the closed-loop system. The design utilizes an observer poles twice as fast as the closed-loop poles. An additional observer pole is added in order to get a casual controller when an integrator is added to the controller.

using ControlSystems
import DSP: conv
# Control design
ζ0 = 0.7
ω0 = 2
Am = [1, 2ζ0*ω0, ω0^2]
Ao = conv(2Am, [1/2, 1]) # Observer polynomial, add extra pole due to the integrator
AR = [1,0] # Force the controller to contain an integrator ( 1/(s+0) )

B⁺  = [1] # The process numerator polynomial can be facored as B = B⁺B⁻ where B⁻ contains the zeros we do not want to cancel (non-minimum phase and poorly damped zeros)
B⁻  = [1]
Bm  = conv(B⁺, B⁻) # In this case, keep the entire numerator polynomial of the process

R,S,T = rstc(B⁺,B⁻,A,Bm,Am,Ao,AR) # Calculate the 2-DOF controller polynomials

Gcl = tf(conv(B,T),zpconv(A,R,B,S)) # Form the closed loop polynomial from reference to output, the closed-loop characteristic polynomial is AR + BS, the function zpconv takes care of the polynomial multiplication and makes sure the coefficient vectores are of equal length

plot(step(P, 20))
plot!(step(Gcl, 20)) # Visualize the open and closed loop responses.
save_docs_plot("ppstepplot.svg") # hide
gangoffourplot(P, tf(-S,R)) # Plot the gang of four to check that all tranfer functions are OK
save_docs_plot("ppgofplot.svg"); # hide

## Stability boundary for PID controllers

The stability boundary, i.e., the surface of PID parameters where the transfer function $P(s)C(s)$ equals -1, can be plotted with the command stabregionPID. The process can be given in function form or as a regular LTIsystem.

P1 = s -> exp(-sqrt(s))
doplot = true
form = :parallel
kp, ki, f1 = stabregionPID(P1,exp10.(range(-5, stop=1, length=1000)); doplot, form); f1
P2 = s -> 100*(s+6).^2. /(s.*(s+1).^2. *(s+50).^2)
kp, ki, f2 = stabregionPID(P2,exp10.(range(-5, stop=2, length=1000)); doplot, form); f2
P3 = tf(1,[1,1])^4
kp, ki, f3 = stabregionPID(P3,exp10.(range(-5, stop=0, length=1000)); doplot, form); f3

save_docs_plot(f1, "stab1.svg") # hide
save_docs_plot(f2, "stab2.svg") # hide
save_docs_plot(f3, "stab3.svg"); # hide

## PID plots

This example utilizes the function pidplots, which accepts vectors of PID-parameters and produces relevant plots. The task is to take a system with bandwidth 1 rad/s and produce a closed-loop system with bandwidth 0.1 rad/s. If one is not careful and proceed with pole placement, one easily get a system with very poor robustness.

using ControlSystemsBase
P = tf([1.], [1., 1])

ζ = 0.5 # Desired damping
ws = exp10.(range(-1, stop=2, length=8)) # A vector of closed-loop bandwidths
kp = 2*ζ*ws .- 1 # Simple pole placement with PI given the closed-loop bandwidth, the poles are placed in a butterworth pattern
ki = ws.^2

ω = exp10.(range(-3, stop = 2, length = 500))
pidplots(
P,
:nyquist;
params_p = kp,
params_i = ki,
ω = ω,
ylims = (-2, 2),
xlims = (-3, 3),
form = :parallel,
)
save_docs_plot("pidplotsnyquist1.svg") # hide
pidplots(P, :gof; params_p = kp, params_i = ki, ω = ω, legend = false, form=:parallel, legendfontsize=6, size=(1000, 1000))
# You can also request both Nyquist and Gang-of-four plots (more plots are available, see ?pidplots ):
# pidplots(P,:nyquist,:gof;kps=kp,kis=ki,ω=ω);
save_docs_plot("pidplotsgof1.svg"); # hide

Now try a different strategy, where we have specified a gain crossover frequency of 0.1 rad/s

kp = range(-1, stop=1, length=8) #
ki = sqrt.(1 .- kp.^2)/10

pidplots(P,:nyquist,;params_p=kp,params_i=ki,ylims=(-1,1),xlims=(-1.5,1.5), form=:parallel)
save_docs_plot("pidplotsnyquist2.svg") # hide
pidplots(P,:gof,;params_p=kp,params_i=ki,legend=false,ylims=(0.08,8),xlims=(0.003,20), form=:parallel, legendfontsize=6, size=(1000, 1000))
save_docs_plot("pidplotsgof2.svg"); # hide