# Lorenz System VSSM controller

Hello guys I am working on a small project “VSSM controller” with Julia programming I only have two questions

I have exact model parameters

the function of time so it can be written in x(t), y(t), z(t) form, and u is a control signal.

The Nominal Trajectory is: Asin(ωt), where t is time in seconds, A, ω are control parameters, which is in the three direction:A1 = 2, ω1 = 0.5, A2 =3.8, ω2 = 0.7, A3 = 1.5, ω3 = 0.9.

I want to make the figures for the :

• nominal and realized trajectories
• tracking errors
• phase spaces

that is already done in my code, I just don’t know how to make the PyPlot for:

• control signals
• velocities

for this model.

Thank you in advance

you can check my Julia code below, and thank you in advance.

``````############################
# Lorenz System  (MIMO)    #
# by VSSM (all directions) #
############################

###########################################
# Controll parameters for VSSM controller #
###########################################

K=500
w=1

#############################################
# Parameters to design a Nominal Trajectory #
#############################################

A₁=2
ω₁=0.5
A₂=3.8
ω₂=0.7
A₃=1.5
ω₃=0.9

#################
# Time variable #
#################

# cycle time in seconds
δt=1e-3
# The lenght of the simulation in seconds
LONG=Int(2e4)

Λ=12

##########################
# Exact model parameters #
##########################
exact_a=1
exact_b=100
exact_c=0.4

############################
# Approx. model parameters #
############################
approx_a=0.8
approx_b=90
approx_c=0.3

#Arrays

xN=zeros(LONG)
xN_p=zeros(LONG)
yN=zeros(LONG)
yN_p=zeros(LONG)
zN=zeros(LONG)
zN_p=zeros(LONG)

x_p_des=zeros(LONG)
y_p_des=zeros(LONG)
z_p_des=zeros(LONG)

u_x=zeros(LONG)
u_y=zeros(LONG)
u_z=zeros(LONG)

t=zeros(LONG)

S_x=zeros(LONG)
S_p_x=zeros(LONG)
S_y=zeros(LONG)
S_p_y=zeros(LONG)
S_z=zeros(LONG)
S_p_z=zeros(LONG)

x=zeros(LONG)
y=zeros(LONG)
z=zeros(LONG)
x_p=zeros(LONG)
y_p=zeros(LONG)
z_p=zeros(LONG)

#initial conditions
hint_x=0
hint_y=0
hint_z=0

x[1]=A₁*sin(ω₁*δt)
y[1]=A₂*sin(ω₂*δt)
z[1]=A₃*sin(ω₃*δt)

##############
# SIMULATION #
##############

for i=1:LONG-1 #i counts the cycles of the simulation
global hint_x,hint_y,hint_z
t[i]=δt*i
#create Nominal Trajectory
xN[i]=A₁*sin(ω₁*t[i])
xN_p[i]=A₁*ω₁*cos(ω₁*t[i])
#xN_p[i]=-A₁*ω₁^2*sin(ω₁*t[i])

yN[i]=A₂*sin(ω₂*t[i])
yN_p[i]=A₂*ω₂*cos(ω₂*t[i])
#yN_p[i]=-A₂*ω₂^2*sin(ω₂*t[i])

zN[i]=A₃*sin(ω₃*t[i])
zN_p[i]=A₃*ω₃*cos(ω₃*t[i])
#zN_p[i]=-A₃*ω₃^2*sin(ω₃*t[i])
# Compute  error (h) and error integral (hint) and the first derivative of the error (h_p)
h_x=xN[i]-x[i]
h_y=yN[i]-y[i]
h_z=zN[i]-z[i]

hint_x=hint_x+δt*h_x
hint_y=hint_y+δt*h_y
hint_z=hint_z+δt*h_z

h_p_x=xN_p[i]-x_p[i]
h_p_y=yN_p[i]-y_p[i]
h_p_z=zN_p[i]-z_p[i]

#Error metric (S), and the first derivative of the Error metric (S_p)
S_x[i]=Λ*hint_x+h_x
S_p_x[i]=Λ*h_x+h_p_x
S_y[i]=Λ*hint_y+h_y
S_p_y[i]=Λ*h_y+h_p_y
S_z[i]=Λ*hint_z+h_z
S_p_z[i]=Λ*h_z+h_p_z
#Kinematic Block (PID type feedback)
x_p_des[i]=xN_p[i]+Λ*h_x+K*tanh(S_x[i]/w)
y_p_des[i]=yN_p[i]+Λ*h_y+K*tanh(S_y[i]/w)
z_p_des[i]=zN_p[i]+Λ*h_z+K*tanh(S_z[i]/w)

# Create the control signal
u_x[i]=x_p_des[i]-approx_a*(0.4*z[i]+z[i])
u_y[i]=y_p_des[i]-approx_b*x[i]*z[i]-y[i]
u_z[i]=z_p_des[i]-approx_c*x[i]+y[i]

# Get the system "reaction" (the exact model of the system)
x_p[i]=exact_a*(0.4*z[i]+z[i])+u_x[i]
y_p[i]=exact_c*x[i]*z[i]-y[i]+u_y[i]
z_p[i]=-exact_b*x[i]+y[i]+u_z[i]
# Integrate back to x, y, z
x[i+1]=x[i]+δt*x_p[i]
y[i+1]=y[i]+δt*y_p[i]
z[i+1]=z[i]+δt*z_p[i]
end #for

using PyPlot

figure(1)
grid("on")
title("Nominal and Realized trajectorys")
xlabel(L"Time \$[s]\$")
ylabel(L"q \$[m]\$")
plot(t[3:LONG-1],xN[3:LONG-1],color="#7684FF")
plot(t[3:LONG-1],yN[3:LONG-1],color="#2A40FF")
plot(t[3:LONG-1],zN[3:LONG-1],color="#3B427F")

plot(t[3:LONG-1],x[3:LONG-1],color="#FFAA41","r--")
plot(t[3:LONG-1],y[3:LONG-1],color="#FF9F28","r--")
plot(t[3:LONG-1],z[3:LONG-1],color="#B2690E","r--")

figure(2)
grid("on")
title("Tracking Error")
xlabel(L"Time \$[s]\$")
ylabel(L"error \$[m]\$")
plot(t[3:LONG-1],xN[3:LONG-1]-x[3:LONG-1],color="red")
plot(t[3:LONG-1],yN[3:LONG-1]-y[3:LONG-1],color="green")
plot(t[3:LONG-1],zN[3:LONG-1]-z[3:LONG-1],color="blue")

figure(3)
grid("on")
title("Phase Space of Error Metric (S_p vs S)")
plot(S_x[3:LONG-1],S_p_x[3:LONG-1],color="red")
plot(S_y[3:LONG-1],S_p_y[3:LONG-1],color="green","r--")
plot(S_z[3:LONG-1],S_p_z[3:LONG-1],color="blue","r-.")

figure(4)
grid("on")
title("Phase Space of motion (x_p vs x)")
plot(xN[1:LONG-1],xN_p[1:LONG-1],color="red")
plot(x[1:LONG-1],x_p[1:LONG-1],color="blue","r--")
plot(yN[1:LONG-1],yN_p[1:LONG-1],color="#7684FF")
plot(y[1:LONG-1],y_p[1:LONG-1],color="#FFAA41","r--")
plot(zN[1:LONG-1],zN_p[1:LONG-1],color="#2A40FF")
plot(z[1:LONG-1],z_p[1:LONG-1],color="#FF9F28","r-.")

PyPlot.display_figs()

``````