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Merge pull request #895 from jrforbes/main
Add H2 and Hinf synthesis examples
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‎doc/examples.rst‎

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phaseplots
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robust_siso
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robust_mimo
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scherer_etal_ex7_H2_h2syn
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scherer_etal_ex7_Hinf_hinfsyn
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cruise-control
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steering-gainsched
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steering-optimal

‎doc/scherer_etal_ex7_H2_h2syn.py‎

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../examples/scherer_etal_ex7_H2_h2syn.py

‎doc/scherer_etal_ex7_H2_h2syn.rst‎

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H2 synthesis, based on Scherer et al. 1997 example 7
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----------------------------------------------------
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Code
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....
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.. literalinclude:: scherer_etal_ex7_H2_h2syn.py
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:language: python
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:linenos:
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Notes
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.....
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1. The environment variable `PYCONTROL_TEST_EXAMPLES` is used for
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testing to turn off plotting of the outputs.

‎doc/scherer_etal_ex7_Hinf_hinfsyn.py‎

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../examples/scherer_etal_ex7_Hinf_hinfsyn.py

‎doc/scherer_etal_ex7_Hinf_hinfsyn.rst‎

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Hinf synthesis, based on Scherer et al. 1997 example 7
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------------------------------------------------------
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Code
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....
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.. literalinclude:: scherer_etal_ex7_Hinf_hinfsyn.py
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:language: python
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:linenos:
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Notes
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.....
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1. The environment variable `PYCONTROL_TEST_EXAMPLES` is used for
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testing to turn off plotting of the outputs.

‎examples/scherer_etal_ex7_H2_h2syn.py‎

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"""H2 design using h2syn.
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Demonstrate H2 desing for a SISO plant using h2syn. Based on [1], Ex. 7.
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[1] Scherer, Gahinet, & Chilali, "Multiobjective Output-Feedback Control via
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LMI Optimization", IEEE Trans. Automatic Control, Vol. 42, No. 7, July 1997.
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[2] Zhou & Doyle, "Essentials of Robust Control", Prentice Hall,
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Upper Saddle River, NJ, 1998.
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"""
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# %%
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# Packages
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import numpy as np
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import control
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# %%
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# State-space system.
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# Process model.
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A = np.array([[0, 10, 2],
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[-1, 1, 0],
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[0, 2, -5]])
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B1 = np.array([[1],
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[0],
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[1]])
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B2 = np.array([[0],
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[1],
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[0]])
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# Plant output.
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C2 = np.array([[0, 1, 0]])
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D21 = np.array([[2]])
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D22 = np.array([[0]])
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# H2 performance.
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C1 = np.array([[0, 1, 0],
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[0, 0, 1],
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[0, 0, 0]])
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D11 = np.array([[0],
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[0],
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[0]])
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D12 = np.array([[0],
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[0],
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[1]])
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# Dimensions.
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n_u, n_y = 1, 1
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# %%
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# H2 design using h2syn.
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# Create augmented plant.
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Baug = np.block([B1, B2])
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Caug = np.block([[C1], [C2]])
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Daug = np.block([[D11, D12], [D21, D22]])
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Paug = control.ss(A, Baug, Caug, Daug)
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# Input to h2syn is Paug, number of inputs to controller,
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# and number of outputs from the controller.
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K = control.h2syn(Paug, n_y, n_u)
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# Extarct controller ss realization.
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A_K, B_K, C_K, D_K = K.A, K.B, K.C, K.D
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# %%
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# Compute closed-loop H2 norm.
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# Compute closed-loop system, Tzw(s). See Eq. 4 in [1].
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Azw = np.block([[A + B2 @ D_K @ C2, B2 @ C_K],
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[B_K @ C2, A_K]])
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Bzw = np.block([[B1 + B2 @ D_K @ D21],
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[B_K @ D21]])
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Czw = np.block([C1 + D12 @ D_K @ C2, D12 @ C_K])
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Dzw = D11 + D12 @ D_K @ D21
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Tzw = control.ss(Azw, Bzw, Czw, Dzw)
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# Compute closed-loop H2 norm via Lyapunov equation.
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# See [2], Lemma 4.4, pg 53.
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Qzw = control.lyap(Azw.T, Czw.T @ Czw)
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nu = np.sqrt(np.trace(Bzw.T @ Qzw @ Bzw))
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print(f'The closed-loop H_2 norm of Tzw(s) is {nu}.')
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# Value is 7.748350599360575, the same as reported in [1].
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# %%

‎examples/scherer_etal_ex7_Hinf_hinfsyn.py‎

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"""Hinf design using hinfsyn.
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Demonstrate Hinf desing for a SISO plant using h2syn. Based on [1], Ex. 7.
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[1] Scherer, Gahinet, & Chilali, "Multiobjective Output-Feedback Control via
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LMI Optimization", IEEE Trans. Automatic Control, Vol. 42, No. 7, July 1997.
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"""
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# %%
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# Packages
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import numpy as np
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import control
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# %%
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# State-space system.
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# Process model.
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A = np.array([[0, 10, 2],
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[-1, 1, 0],
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[0, 2, -5]])
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B1 = np.array([[1],
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[0],
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[1]])
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B2 = np.array([[0],
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[1],
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[0]])
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# Plant output.
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C2 = np.array([[0, 1, 0]])
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D21 = np.array([[2]])
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D22 = np.array([[0]])
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# Hinf performance.
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C1 = np.array([[1, 0, 0],
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[0, 0, 0]])
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D11 = np.array([[0],
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[0]])
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D12 = np.array([[0],
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[1]])
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# Dimensions.
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n_x, n_u, n_y = 3, 1, 1
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# %%
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# Hinf design using hinfsyn.
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# Create augmented plant.
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Baug = np.block([B1, B2])
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Caug = np.block([[C1], [C2]])
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Daug = np.block([[D11, D12], [D21, D22]])
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Paug = control.ss(A, Baug, Caug, Daug)
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# Input to hinfsyn is Paug, number of inputs to controller,
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# and number of outputs from the controller.
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K, Tzw, gamma, rcond = control.hinfsyn(Paug, n_y, n_u)
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print(f'The closed-loop H_inf norm of Tzw(s) is {gamma}.')
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# %%

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