python-control · murrayrm · May 28, 2023 · May 23, 2023 · May 23, 2023 · May 23, 2023
diff --git a/doc/examples.rst b/doc/examples.rst
@@ -27,6 +27,8 @@ other sources.
    phaseplots
    robust_siso
    robust_mimo
+   scherer_etal_ex7_H2_h2syn
+   scherer_etal_ex7_Hinf_hinfsyn
    cruise-control
    steering-gainsched
    steering-optimal

diff --git a/doc/scherer_etal_ex7_H2_h2syn.py b/doc/scherer_etal_ex7_H2_h2syn.py
@@ -0,0 +1 @@
+../examples/scherer_etal_ex7_H2_h2syn.py
diff --git a/doc/scherer_etal_ex7_H2_h2syn.rst b/doc/scherer_etal_ex7_H2_h2syn.rst
@@ -0,0 +1,15 @@
+H2 synthesis, based on Scherer et al. 1997 example 7
+----------------------------------------------------
+
+Code
+....
+.. literalinclude:: scherer_etal_ex7_H2_h2syn.py
+   :language: python
+   :linenos:
+
+
+Notes
+.....
+
+1. The environment variable `PYCONTROL_TEST_EXAMPLES` is used for
+testing to turn off plotting of the outputs.
diff --git a/doc/scherer_etal_ex7_Hinf_hinfsyn.py b/doc/scherer_etal_ex7_Hinf_hinfsyn.py
@@ -0,0 +1 @@
+../examples/scherer_etal_ex7_Hinf_hinfsyn.py
diff --git a/doc/scherer_etal_ex7_Hinf_hinfsyn.rst b/doc/scherer_etal_ex7_Hinf_hinfsyn.rst
@@ -0,0 +1,15 @@
+Hinf synthesis, based on Scherer et al. 1997 example 7
+------------------------------------------------------
+
+Code
+....
+.. literalinclude:: scherer_etal_ex7_Hinf_hinfsyn.py
+   :language: python
+   :linenos:
+
+
+Notes
+.....
+
+1. The environment variable `PYCONTROL_TEST_EXAMPLES` is used for
+testing to turn off plotting of the outputs.
diff --git a/examples/scherer_etal_ex7_H2_h2syn.py b/examples/scherer_etal_ex7_H2_h2syn.py
@@ -0,0 +1,84 @@
+"""H2 design using h2syn.
+
+Demonstrate H2 desing for a SISO plant using h2syn. Based on [1], Ex. 7.
+
+[1] Scherer, Gahinet, & Chilali, "Multiobjective Output-Feedback Control via
+LMI Optimization", IEEE Trans. Automatic Control, Vol. 42, No. 7, July 1997.
+
+[2] Zhou & Doyle, "Essentials of Robust Control", Prentice Hall,
+Upper Saddle River, NJ, 1998.
+"""
+# %%
+# Packages
+import numpy as np
+import control
+
+# %%
+# State-space system.
+
+# Process model.
+A = np.array([[0, 10, 2],
+              [-1, 1, 0],
+              [0, 2, -5]])
+B1 = np.array([[1],
+               [0],
+               [1]])
+B2 = np.array([[0],
+               [1],
+               [0]])
+
+# Plant output.
+C2 = np.array([[0, 1, 0]])
+D21 = np.array([[2]])
+D22 = np.array([[0]])
+
+# H2 performance.
+C1 = np.array([[0, 1, 0],
+               [0, 0, 1],
+               [0, 0, 0]])
+D11 = np.array([[0],
+                [0],
+                [0]])
+D12 = np.array([[0],
+                [0],
+                [1]])
+
+# Dimensions.
+n_u, n_y = 1, 1
+
+# %%
+# H2 design using h2syn.
+
+# Create augmented plant.
+Baug = np.block([B1, B2])
+Caug = np.block([[C1], [C2]])
+Daug = np.block([[D11, D12], [D21, D22]])
+Paug = control.ss(A, Baug, Caug, Daug)
+
+# Input to h2syn is Paug, number of inputs to controller,
+# and number of outputs from the controller.
+K = control.h2syn(Paug, n_y, n_u)
+
+# Extarct controller ss realization.
+A_K, B_K, C_K, D_K = K.A, K.B, K.C, K.D
+
+# %%
+# Compute closed-loop H2 norm.
+
+# Compute closed-loop system, Tzw(s). See Eq. 4 in [1].
+Azw = np.block([[A + B2 @ D_K @ C2, B2 @ C_K],
+                [B_K @ C2, A_K]])
+Bzw = np.block([[B1 + B2 @ D_K @ D21],
+                [B_K @ D21]])
+Czw = np.block([C1 + D12 @ D_K @ C2, D12 @ C_K])
+Dzw = D11 + D12 @ D_K @ D21
+Tzw = control.ss(Azw, Bzw, Czw, Dzw)
+
+# Compute closed-loop H2 norm via Lyapunov equation.
+# See [2], Lemma 4.4, pg 53.
+Qzw = control.lyap(Azw.T, Czw.T @ Czw)
+nu = np.sqrt(np.trace(Bzw.T @ Qzw @ Bzw))
+print(f'The closed-loop H_2 norm of Tzw(s) is {nu}.')
+# Value is 7.748350599360575, the same as reported in [1].
+
+# %%
diff --git a/examples/scherer_etal_ex7_Hinf_hinfsyn.py b/examples/scherer_etal_ex7_Hinf_hinfsyn.py
@@ -0,0 +1,57 @@
+"""Hinf design using hinfsyn.
+
+Demonstrate Hinf desing for a SISO plant using h2syn. Based on [1], Ex. 7.
+
+[1] Scherer, Gahinet, & Chilali, "Multiobjective Output-Feedback Control via
+LMI Optimization", IEEE Trans. Automatic Control, Vol. 42, No. 7, July 1997.
+"""
+# %%
+# Packages
+import numpy as np
+import control
+
+# %%
+# State-space system.
+
+# Process model.
+A = np.array([[0, 10, 2],
+              [-1, 1, 0],
+              [0, 2, -5]])
+B1 = np.array([[1],
+               [0],
+               [1]])
+B2 = np.array([[0],
+               [1],
+               [0]])
+
+# Plant output.
+C2 = np.array([[0, 1, 0]])
+D21 = np.array([[2]])
+D22 = np.array([[0]])
+
+# Hinf performance.
+C1 = np.array([[1, 0, 0],
+               [0, 0, 0]])
+D11 = np.array([[0],
+                [0]])
+D12 = np.array([[0],
+                [1]])
+
+# Dimensions.
+n_x, n_u, n_y = 3, 1, 1
+
+# %%
+# Hinf design using hinfsyn.
+
+# Create augmented plant.
+Baug = np.block([B1, B2])
+Caug = np.block([[C1], [C2]])
+Daug = np.block([[D11, D12], [D21, D22]])
+Paug = control.ss(A, Baug, Caug, Daug)
+
+# Input to hinfsyn is Paug, number of inputs to controller,
+# and number of outputs from the controller.
+K, Tzw, gamma, rcond = control.hinfsyn(Paug, n_y, n_u)
+print(f'The closed-loop H_inf norm of Tzw(s) is {gamma}.')
+
+# %%