Merge pull request #84 from JuliaControl/minor_correction_doc

Minor correction doc

Author: franckgaga

docs/src/manual/nonlinmpc.md

@@ -338,25 +338,27 @@ the [`LinMPC`](@ref) instance based on repeated online linearization.
 The [`setmodel!`](@ref) method allows online adaptation of a linear plant model. Combined
 with the automatic linearization of [`linearize`](@ref), a successive linearization MPC can
 be designed with minimal efforts. The [`SteadyKalmanFilter`](@ref) does not support
-[`setmodel!`](@ref), so we need to use the time-varying [`KalmanFilter`](@ref) instead:
+[`setmodel!`](@ref) so we need to use the time-varying [`KalmanFilter`](@ref), and we
+initialize it with a linearization at ``θ = ω = τ = 0``:
 
 ```@example 1
-kf   = KalmanFilter(linmodel; σQ, σR, nint_u, σQint_u)
+linmodel = linearize(model, x=[0, 0], u=[0])
+kf = KalmanFilter(linmodel; σQ, σR, nint_u, σQint_u)
 mpc3 = LinMPC(kf; Hp, Hc, Mwt, Nwt, Cwt=Inf, optim=daqp)
 mpc3 = setconstraint!(mpc3; umin, umax)
 ```
 
 We create a function that simulates the plant and the adaptive controller:
 
 ```@example 1
-function test_slmpc(nonlinmodel, mpc, ry, plant; x_0=plant.xop, y_step=0)
-    N = 35
+function sim_adapt!(mpc, nonlinmodel, N, ry, plant, x_0, x̂_0, y_step=[0])
     U_data, Y_data, Ry_data = zeros(plant.nu, N), zeros(plant.ny, N), zeros(plant.ny, N)
     setstate!(plant, x_0)
-    u, y = [0.0], plant()
-    x̂ = initstate!(mpc, u, y)
+    initstate!(mpc, [0], plant())
+    setstate!(mpc, x̂_0)
+    x̂ = x̂_0
     for i = 1:N
-        y = plant() .+ y_step
+        y = plant() + y_step
         u = moveinput!(mpc, ry)
         linmodel = linearize(nonlinmodel; u, x=x̂[1:2])
         setmodel!(mpc, linmodel)
@@ -376,7 +378,8 @@ The [`SimResult`](@ref) object is for plotting purposes only. The adaptive [`Lin
 performances are similar to the nonlinear MPC, both for the 180° setpoint:
 
 ```@example 1
-res_slin = test_slmpc(model, mpc3, [180], plant, x_0=[0, 0]) 
+x_0 = [0, 0]; x̂_0 = [0, 0, 0]; ry = [180]
+res_slin = sim_adapt!(mpc3, model, N, ry, plant, x_0, x̂_0)
 plot(res_slin)
 savefig("plot10_NonLinMPC.svg"); nothing # hide
 ```
@@ -386,7 +389,8 @@ savefig("plot10_NonLinMPC.svg"); nothing # hide
 and the 10° step disturbance:
 
 ```@example 1
-res_slin = test_slmpc(model, mpc3, [180], plant, x_0=[π, 0], y_step=[10]) 
+x_0 = [π, 0]; x̂_0 = [π, 0, 0]; y_step = [10]
+res_slin = sim_adapt!(mpc3, model, N, ry, plant, x_0, x̂_0, y_step)
 plot(res_slin)
 savefig("plot11_NonLinMPC.svg"); nothing # hide
 ```