# pvtol-nested.py - inner/outer design for vectored thrust aircraft

# RMM, 5 Sep 09

#

# This file works through a fairly complicated control design and

# analysis, corresponding to the planar vertical takeoff and landing

# (PVTOL) aircraft in Astrom and Mruray, Chapter 11. It is intended

# to demonstrate the basic functionality of the python-control

# package.

#

from matplotlib.pyplot import * # Grab MATLAB plotting functions

from control.matlab import * # MATLAB-like functions

import numpy as np

# System parameters

m = 4; # mass of aircraft

J = 0.0475; # inertia around pitch axis

r = 0.25; # distance to center of force

g = 9.8; # gravitational constant

c = 0.05; # damping factor (estimated)

# Transfer functions for dynamics

Pi = tf([r], [J, 0, 0]); # inner loop (roll)

Po = tf([1], [m, c, 0]); # outer loop (position)

# Use state space versions

Pi = tf2ss(Pi);

Po = tf2ss(Po);

#

# Inner loop control design

#

# This is the controller for the pitch dynamics. Goal is to have

# fast response for the pitch dynamics so that we can use this as a

# control for the lateral dynamics

#

# Design a simple lead controller for the system

k = 200; a = 2; b = 50;

Ci = k*tf([1, a], [1, b]); # lead compensator

# Convert to statespace

Ci = tf2ss(Ci);

# Compute the loop transfer function for the inner loop

Li = Pi*Ci;

# Bode plot for the open loop process

figure(1);

bode(Pi);

# Bode plot for the loop transfer function, with margins

figure(2);

bode(Li);

# Compute out the gain and phase margins

#! Not implemented

# (gm, pm, wcg, wcp) = margin(Li);

# Compute the sensitivity and complementary sensitivity functions

Si = feedback(1, Li);

Ti = Li * Si;

# Check to make sure that the specification is met

figure(3); gangof4(Pi, Ci);

# Compute out the actual transfer function from u1 to v1 (see L8.2 notes)

# Hi = Ci*(1-m*g*Pi)/(1+Ci*Pi);

Hi = parallel(feedback(Ci, Pi), -m*g*feedback(Ci*Pi, 1));

figure(4); clf; subplot(221);

bode(Hi);

# Now design the lateral control system

a = 0.02; b = 5; K = 2;

Co = -K*tf([1, 0.3], [1, 10]); # another lead compensator

# Convert to statespace

Co = tf2ss(Co);

# Compute the loop transfer function for the outer loop

Lo = -m*g*Po*Co;

figure(5);

bode(Lo); # margin(Lo)

# Finally compute the real outer-loop loop gain + responses

L = Co*Hi*Po;

S = feedback(1, L);

T = feedback(L, 1);

# Compute stability margins

#! Not yet implemented

# (gm, pm, wgc, wpc) = margin(L);

figure(6); clf; subplot(221);

bode(L, logspace(-4, 3));

# Add crossover line

subplot(211);

loglog([1e-4, 1e3], [1, 1], 'k-')

# Replot phase starting at -90 degrees

(mag, phase, w) = freqresp(L, logspace(-4, 3));

phase = phase - 360;

subplot(212);

semilogx([1e-4, 1e3], [-180, -180], 'k-')

semilogx(w, np.squeeze(phase), 'b-')

axis([1e-4, 1e3, -360, 0]);

xlabel('Frequency [deg]'); ylabel('Phase [deg]');

# set(gca, 'YTick', [-360, -270, -180, -90, 0]);

# set(gca, 'XTick', [10^-4, 10^-2, 1, 100]);

#

# Nyquist plot for complete design

#

figure(7); clf;

axis([-700, 5300, -3000, 3000]);

nyquist(L, (0.0001, 1000));

axis([-700, 5300, -3000, 3000]);

# Add a box in the region we are going to expand

plot([-400, -400, 200, 200, -400], [-100, 100, 100, -100, -100], 'r-')

# Expanded region

figure(8); clf; subplot(231);

axis([-10, 5, -20, 20]);

nyquist(L);

axis([-10, 5, -20, 20]);

# set up the color

color = 'b';

# Add arrows to the plot

# H1 = L.evalfr(0.4); H2 = L.evalfr(0.41);

# arrow([real(H1), imag(H1)], [real(H2), imag(H2)], AM_normal_arrowsize, \

# 'EdgeColor', color, 'FaceColor', color);

# H1 = freqresp(L, 0.35); H2 = freqresp(L, 0.36);

# arrow([real(H2), -imag(H2)], [real(H1), -imag(H1)], AM_normal_arrowsize, \

# 'EdgeColor', color, 'FaceColor', color);

figure(9);

(Yvec, Tvec) = step(T, linspace(1, 20));

plot(Tvec.T, Yvec.T);

(Yvec, Tvec) = step(Co*S, linspace(1, 20));

plot(Tvec.T, Yvec.T);

#TODO: PZmap for statespace systems has not yet been implemented.

figure(10); clf();

# (P, Z) = pzmap(T, Plot=True)

# print "Closed loop poles and zeros: ", P, Z

# Gang of Four

figure(11); clf();

gangof4(Hi*Po, Co, linspace(-2, 3));