legend (( 'x', 'y' ), loc = 'lower right' ) # Look at different input weightings Qu1a = np. title ( "Identity weights" ) # plt.plot(T, Y, '-', T, Y, '-') plt. linspace ( 0, 10, 100 )) # Step response for the second input H1ay = ss ( Ay - By * K1a, By * K1a * yd, Cy, Dy ) Yy, Ty = step ( H1ay, T = np.
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matrix ( K ) # Close the loop: xdot = Ax - B K (x-xd) # Note: python-control requires we do this 1 input at a time # H1a = ss(A-B*K1a, B*K1a*concatenate((xd, yd), axis=1), C, D) # (T, Y) = step(H1a, T=np.linspace(0,10,100)) # TODO: The following equations will need modifying when converting from np.matrix to np.array # because the results and even intermediate calculations will be different with numpy arrays # For example: # Bx = B # Will need to be changed to: # Bx = B.reshape(-1, 1) # (if we want it to have the same shape as before) # For reference, here is a list of the correct shapes of these objects: # A: (6, 6) # B: (6, 2) # C: (2, 6) # D: (2, 2) # xd: (6, 1) # yd: (6, 1) # Ax: (4, 4) # Bx: (4, 1) # Cx: (1, 4) # Dx: () # Ay: (2, 2) # By: (2, 1) # Cy: (1, 2) # Step response for the first input H1ax = ss ( Ax - Bx * K1a, Bx * K1a * xd, Cx, Dx ) Yx, Tx = step ( H1ax, T = np. diag () K, X, E = lqr ( A, B, Qx1, Qu1a ) K1a = np. suptitle ( "LQR controllers for vectored thrust aircraft (pvtol-lqr)" ) # LQR design # Start with a diagonal weighting Qx1 = np. # Indices for the parts of the state that we want lat = ( 0, 2, 3, 5 ) alt = ( 1, 4 ) # Decoupled dynamics Ax = ( A ) # ! not sure why I have to do it this way Bx = B Cx = C Dx = D Ay = ( A ) # ! not sure why I have to do it this way By = B Cy = C Dy = D # Label the plot plt. To do this, we define the 'lat' and # 'alt' index vectors to consist of the states that are are relevant # to the lateral (x) and vertical (y) dynamics.
SPREEDER CX EPUB PROBLEM CODE
matrix (,, ,, , ]) # Extract the relevant dynamics for use with SISO library # The current python-control library only supports SISO transfer # functions, so we have to modify some parts of the original MATLAB # code to extract out SISO systems. # The way these vectors are used is to compute the closed loop system # dynamics as # xdot = Ax + B u => xdot = (A-BK)x + K xd # u = -K(x - xd) y = Cx # The closed loop dynamics can be simulated using the "step" command, # with K*xd as the input vector (assumes that the "input" is unit size, # so that xd corresponds to the desired steady state. The matrices Cx and Cy are the # corresponding outputs. matrix (, ]) # Construct inputs and outputs corresponding to steps in xy position # The vectors xd and yd correspond to the states that are the desired # equilibrium states for the system. cos ( xe ) / m ], ] ) # Output matrix C = np. sin ( xe )) / m, 0, - c / m, 0 ], ] ) # Input matrix B = np.
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# Dynamics matrix (use matrix type so that * works for multiplication) A = 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) # State space dynamics xe = # equilibrium point of interest ue = # (note these are lists, not matrices) # TODO: The following objects need converting from np.matrix to np.array # This will involve re-working the subsequent equations as the shapes # See below. # import os import numpy as np import matplotlib.pyplot as plt # MATLAB plotting functions from control.matlab import * # MATLAB-like functions # System dynamics # These are the dynamics for the PVTOL system, written in state space # form. It is intended to demonstrate the # basic functionality of the python-control package.
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# pvtol_lqr.m - LQR design for vectored thrust aircraft # RMM, 14 Jan 03 # This file works through an LQR based design problem, using the # planar vertical takeoff and landing (PVTOL) aircraft example from # Astrom and Murray, Chapter 5.