# Figure 1.18: Air–fuel controller based on selectors

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Chapter | Introduction |
---|---|

Figure number | 1.18 |

Figure title | Air–fuel controller based on selectors |

GitHub URL | https://github.com/murrayrm/fbs2e-python/blob/main/figure-1.18-airfuel selectors.py |

Requires | python-control |

**Figure 1.18:** Air–fuel controller based on selectors. The left figure shows the system architecture. The letters R and Y in the PI controller denote the input ports for reference and measured signal respectively. The right figure shows a simulation where the power reference r is changed stepwise at t = 1 and t = 15. Notice that the normalized air flow is larger than the normalized fuel flow both for increasing and decreasing reference steps.

# figure-1.18-airfuel_selectors.py - Air-fuel control example # RMM, 20 Jun 2021 # # Air–fuel controller based on selectors. The right figure shows a # simulation where the power reference r is changed stepwise at t = 1 and t # = 15. Notice that the normalized air flow is larger than the normalized # fuel flow both for increasing and decreasing reference steps. # Package import import numpy as np import matplotlib.pyplot as plt import control as ct import cruise # # Air and fuel (oil) dynamics and controllers # # These dynamics come from Karl Astrom and are embedded in a SIMULINK # diagram used for the initial part of the book. The basic structure for # both the air and fuel controllers is a PI controller with output feedback # for the proportional term and integral feedback on the error. This cuts # the feedthrough term for the proportional feedback and gives a smoother # step response (see Figure 11.1b for the basic structure). # # Min selector for oil PI controller input min_block = ct.NonlinearIOSystem( updfcn=None, outfcn=lambda t, x, u, params: min(u), name='min', inputs=['u1', 'u2'], outputs='y') # Max selector for air PI controller input max_block = ct.NonlinearIOSystem( updfcn=None, outfcn=lambda t, x, u, params: max(u), name='max', inputs=['u1', 'u2'], outputs='y') # Oil and air flow dynamics (from KJA SIMULINK diagram) Po = ct.tf([1], [1, 1]) Pa = ct.tf([4], [1, 4]) # PI controller for oil flow kpo = 2; kio = 4 Cio = ct.tf([kio], [1, 0]) Cpo = kpo oil_block = ct.LinearIOSystem( ct.tf2ss(Po * Cio / (1 + Po * (Cio + Cpo))), name="oil", inputs='r', outputs='y') # PI controller for air flow kpa = 1; kia = 1 Cia = ct.tf([kia], [1, 0]) Cpa = kpa air_block = ct.LinearIOSystem( ct.tf2ss(Pa * Cia / (1 + Pa * (Cia + Cpa))), name="air", inputs='r', outputs='y') # # Air-fuel selector dynamics # # The selector dynamics are based on the diagram Figure 1.18a, where we have # already pre-computing the transfer function around the process/controller # pairs (so the air and oil blocks have input 'R' and output 'Y' from the # diagram). We use the interconnect function along with named signals to # set everything up. # airfuel = ct.interconnect( [min_block, max_block, oil_block, air_block], connections = ( ['oil.r', 'min.y'], ['air.r', 'max.y'], ['min.u2', 'air.y'], ['max.u1', 'oil.y']), inplist = [['min.u1', 'max.u2']], inputs='ref', outlist = ['air.y', 'oil.y'], outputs=['air', 'oil']) # # Input/output response # # Finally, we simulate the dynamics with an input singla as showin in Figure # 1.18b, consisting of a step increase from 0 to 1 at time t = 1 sec and # then a decrease from 1 to 0.5 at time t = 15 sec. # T = np.linspace(0, 30, 101) ref = np.array([ 0 if t <= 1 else 1 if t <= 15 else 0.5 for t in T]) t, y = ct.input_output_response(airfuel, T, ref) # Plot the results plt.subplot(2, 2, 1) plt.plot(t, ref, t, y[0], t, y[1]) plt.legend(['ref', 'air', 'fuel']) plt.xlabel('Time $t$ [sec]') plt.ylabel('Normalized signals')