Deploying to gh-pages from @ sympy/sympy@e228082 🚀

dev/_sources/guides/physics/index.rst.txt

@@ -9,7 +9,8 @@ The Python Package SymPy can be used to symbolically solve physics problems
 on Classical Mechanics, Continuum Mechanics, Control Systems, High-Energy
 Physics, Optics, Quantum Mechanics, Units and Vector algebra.
 
-.. toctree::
-   :maxdepth: 1
+For physics tutorials, please check out:
 
-   control_problems.rst
+* `Biomechanics Tutorials <../../tutorials/physics/biomechanics/index.html>`_
+* `Mechanics Tutorials <../../tutorials/physics/mechanics/index.html>`_
+* `Control Tutorials <../../tutorials/physics/control/index.html>`_

dev/_sources/guides/physics/control_problems.rst.txt → dev/_sources/tutorials/physics/control/control_problems.rst.txt

@@ -176,7 +176,7 @@ Solution
     [[[-1, 0], []], [[-3/4], [4, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]]]
     >>> pole_zero_plot(G[0, 0])   # doctest: +SKIP
 
-    .. plot:: guides/physics/generate_plots.py q3_3
+    .. plot:: tutorials/physics/control/generate_plots.py q3_3
 
     Subpart 4
 
@@ -189,18 +189,18 @@ Solution
          (s + 1)
     >>> step_response_plot(tf1)  # doctest: +SKIP
 
-    .. plot:: guides/physics/generate_plots.py q3_4
+    .. plot:: tutorials/physics/control/generate_plots.py q3_4
 
     Subpart 5
 
     >>> tf2 = G[0, 1]
     >>> bode_magnitude_plot(tf2)  # doctest: +SKIP
 
-    .. plot:: guides/physics/generate_plots.py q3_5_1
+    .. plot:: tutorials/physics/control/generate_plots.py q3_5_1
 
     >>> bode_phase_plot(tf2)  # doctest: +SKIP
 
-    .. plot:: guides/physics/generate_plots.py q3_5_2
+    .. plot:: tutorials/physics/control/generate_plots.py q3_5_2
 
 
 
@@ -344,11 +344,11 @@ Solution
     TransferFunction(s**4 + 3*s**3 + 3*s**2 + 3*s + 2, 12*s**5 + 193*s**4 + 873*s**3 + 1644*s**2 + 1484*s + 712, s)
     >>> pole_zero_plot(sys6)  # doctest: +SKIP
 
-    .. plot:: guides/physics/generate_plots.py q5
+    .. plot:: tutorials/physics/control/generate_plots.py q5
 
 
 
 References
-----------
+^^^^^^^^^^
 1. `testbook.com <https://testbook.com/objective-questions/mcq-on-transfer-function--5eea6a1039140f30f369e952>`_
 2. `www.vssut.ac.in <https://www.vssut.ac.in/lecture_notes/lecture1423904331.pdf>`_

dev/_sources/tutorials/physics/control/electrical_problems.rst.txt

@@ -0,0 +1,300 @@
+.. _electrical_problems-physics:
+
+=============================================
+Electrical Problems using StateSpace
+=============================================
+
+The state-space approach is a powerful method used to model and analyze systems in control
+theory. Instead of focusing solely on the input-output relationships like the transfer function
+approach, the state-space approach represents systems as a set of first-order differential
+equations.
+
+The state-space representation of a system can be written as:
+
+        .. math::
+
+            \dot{x}(t) = A x(t) + B u(t) \\
+            y(t) = C x(t) + D u(t)
+
+
+Where :math:`x(t)` is the state vector, :math:`u(t)` is the input vector, :math:`y(t)` is the output vector,
+:math:`A`, :math:`B`, :math:`C`, and :math:`D` are matrices that define the system dynamics.
+
+Below are some examples to demonstrate the use of StateSpace to solve Electrical problems.
+
+Example 1
+---------
+
+        .. image:: Electrical_Problems_Q1.svg
+           :align: center
+
+In a series RLC circuit, we have a resistor :math:`R`, an inductor :math:`L`, and a capacitor :math:`C`
+connected in series with an input voltage :math:`v_{in}(t)`. The state variables are the current through
+the inductor :math:`i(t)` and the voltage across the capacitor :math:`v_C(t)`.
+
+Applying **Kirchhoff's Voltage Law** (KVL) around the loop in the above diagram gives:
+
+        .. math::
+
+            v_{in}(t) = R \cdot i(t) + L \frac{di(t)}{dt} +  V_C(t)
+
+Where: :math:`v_{in}(t)` is the input voltage, :math:`i(t)` is the current through the inductor and
+:math:`v_C(t)` is the voltage across the capacitor.
+
+This equation relates the input voltage to the elements of the RLC circuit.
+
+**Capacitor Voltage Equation**
+
+The voltage across the capacitor can be related to the current by:
+
+        .. math::
+
+            V_C(t) = \frac{1}{C} \int i(t) \, dt
+
+Taking the time derivative of both sides, we obtain the rate of change of the capacitor voltage:
+
+        .. math::
+
+            \dot{v}_C(t) = \frac{d v_C(t)}{dt} = \frac{i(t)}{C}
+
+This equation shows that the rate of change of the capacitor voltage is proportional to the current through the circuit.
+
+From the KVL equation, solving for the derivative of the current gives:
+
+        .. math::
+
+            \frac{di(t)}{dt} = -\frac{R}{L} i(t) - \frac{1}{L} v_C(t) + \frac{1}{L} v_{in}(t)
+
+This is the first-order differential equation that describes the rate of change of the current in terms of the circuit's components and input voltage.
+
+The state-space representation expresses the system in terms of state variables, which are typically the variables that describe the energy stored in the circuit elements (such as current and voltage).
+
+We define the state vector `X(t)` as:
+
+        .. math::
+
+            X(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = \begin{bmatrix} i(t) \\ v_C(t) \end{bmatrix}
+
+Here `x_1(t) = i(t)` is the current through the inductor and `x_2(t) = v_C(t)` is the voltage across the capacitor.
+
+The input vector `U(t)` is the input voltage:
+
+        .. math::
+
+            U(t) = v_{in}(t)
+
+The system of differential equations in terms of the state variables becomes:
+
+1. The derivative of the current:
+
+        .. math::
+
+            \dot{x}_1(t) = -\frac{R}{L} x_1(t) - \frac{1}{L} x_2(t) + \frac{1}{L} v_{in}(t)
+
+2. The derivative of the capacitor voltage:
+
+        .. math::
+
+            \dot{x}_2(t) = \frac{x_1(t)}{C}
+
+
+The matrices for the series RLC circuit are:
+
+        .. math::
+
+            A = \begin{bmatrix}
+            -\frac{R}{L} & -\frac{1}{L} \\
+            \frac{1}{C} & 0
+            \end{bmatrix},
+            B = \begin{bmatrix}
+            \frac{1}{L} \\
+            0
+            \end{bmatrix},
+            C = \begin{bmatrix} 0 & 1 \end{bmatrix},
+            D = \begin{bmatrix} 0 \end{bmatrix}
+
+
+Thus, the state-space representation of the series RLC circuit is:
+
+        .. math::
+
+            \dot{X}(t) = \begin{bmatrix}
+            -\frac{R}{L} & -\frac{1}{L} \\
+            \frac{1}{C} & 0
+            \end{bmatrix}
+            \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}
+            + \begin{bmatrix}
+            \frac{1}{L} \\
+            0
+            \end{bmatrix} V_{in}(t)
+
+            Y(t) = \begin{bmatrix} 0 & 1 \end{bmatrix}
+            \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}
+            + \begin{bmatrix} 0 \end{bmatrix} V_{in}(t)
+
+
+The state-space representation provides a compact way of modeling
+the series RLC circuit by using matrices to describe the system's
+dynamics. The matrices :math:`A`, :math:`B`, :math:`C`, and :math:`D`
+capture the relationships between the circuit's state variables,
+input, and output. This representation is particularly useful for
+analyzing the system's behavior in the time domain and for designing
+control systems.
+
+Solution
+
+    >>> from sympy import Matrix, symbols, pprint
+    >>> from sympy.physics.control import *
+    >>> R, L, C = symbols('R L C')
+    >>> A = Matrix([[-R/L, -1/L], [1/C, 0]])
+    >>> B = Matrix([[1/L], [0]])
+    >>> C = Matrix([[0, 1]])
+    >>> D = Matrix([[0]])
+    >>> ss = StateSpace(A, B, C, D)
+    >>> ss
+    StateSpace(Matrix([
+    [-R/L, -1/L],
+    [ 1/C,    0]]), Matrix([
+    [1/L],
+    [  0]]), Matrix([[0, 1]]), Matrix([[0]]))
+
+    We can convert the StateSpace to TransferFunction by rewrite method.
+
+    >>> tf = ss.rewrite(TransferFunction)[0][0]
+    >>> tf
+    TransferFunction(1, C*L*s**2 + C*R*s + 1, s)
+
+
+Example 2
+---------
+
+        .. image:: Electrical_Problems_Q2.svg
+           :align: center
+
+Obtain the state model for a system represented by an electrical
+system as shown in figure
+
+The system is modeled with two state variables,
+`x_1(t)` and `x_2(t)`, which are related to the physical voltages at the nodes
+`v_1(t)` and `v_2(t)` respectively.
+
+Let the two state variables be defined as:
+
+        .. math::
+
+           v_1(t) = x_1(t)
+
+           v_2(t) = x_2(t)
+
+The governing equations are derived by applying Kirchhoff's Current Law (KCL) at the nodes `v_1(t)` and `v_2(t)`.
+
+Applying KCL at node `v_1(t)`:
+
+        .. math::
+
+           \frac{v_1(t) - u(t)}{R} + C \frac{d v_1(t)}{dt} + \frac{v_1(t) - v_2(t)}{R} = 0
+
+Substituting the state variables:
+
+        .. math::
+
+           \frac{x_1(t) - u(t)}{R} + C \frac{dx_1(t)}{dt} + \frac{x_1(t) - x_2(t)}{R} = 0
+
+Simplifying:
+
+        .. math::
+
+           C \dot{x_1}(t) = -\frac{2x_1(t)}{R} + \frac{x_2(t)}{R} + \frac{u(t)}{R}
+
+Thus, the state equation for `x_1(t)` becomes:
+
+        .. math::
+
+           \dot{x_1}(t) = -\frac{2x_1(t)}{RC} + \frac{x_2(t)}{RC} + \frac{u(t)}{RC}
+
+
+Applying KCL at node `v_2(t)`:
+
+        .. math::
+
+            C \frac{d v_2(t)}{dt} + \frac{v_2(t) - v_1(t)}{R} = 0
+
+Substituting the state variables:
+
+        .. math::
+
+           C \frac{d x_2(t)}{dt} + \frac{x_2(t) - x_1(t)}{R} = 0
+
+Simplifying:
+
+        .. math::
+
+           C \dot{x_2}(t) = \frac{x_1(t)}{R} - \frac{x_2(t)}{R}
+
+Thus, the state equation for `x_2(t)` becomes:
+
+        .. math::
+
+           \dot{x_2}(t) = \frac{x_1(t)}{RC} - \frac{x_2(t)}{RC}
+
+The state-space representation is given by the following matrix equation:
+
+        .. math::
+
+           \begin{bmatrix}
+           \dot{x_1}(t) \\
+           \dot{x_2}(t)
+           \end{bmatrix}
+           =
+           \begin{bmatrix}
+           -\frac{2}{RC} & \frac{1}{RC} \\
+           \frac{1}{RC} & -\frac{1}{RC}
+           \end{bmatrix}
+           \begin{bmatrix}
+           x_1(t) \\
+           x_2(t)
+           \end{bmatrix}
+           +
+           \begin{bmatrix}
+           \frac{1}{RC} \\
+           0
+           \end{bmatrix}
+           u(t)
+
+The output of the circuit is defined as:
+
+        .. math::
+
+           y(t) = v_2(t) = x_2(t)
+
+Thus, the output equation can be written as:
+
+        .. math::
+
+           y(t) = \begin{bmatrix} 0 & 1 \end{bmatrix}
+           \begin{bmatrix}
+           x_1(t) \\
+           x_2(t)
+           \end{bmatrix}
+
+
+Solution
+
+    >>> from sympy import symbols, Matrix
+    >>> from sympy.physics.control import *
+    >>> R, C = symbols('R C')
+    >>> A = Matrix([[-2/(R*C), 1/(R*C)], [1/(R*C), -1/(R*C)]])
+    >>> B = Matrix([[1/(R*C)], [0]])
+    >>> C = Matrix([[0, 1]])
+    >>> ss = StateSpace(A, B, C)
+    >>> ss
+    StateSpace(Matrix([
+    [-2/(C*R),  1/(C*R)],
+    [ 1/(C*R), -1/(C*R)]]), Matrix([
+    [1/(C*R)],
+    [      0]]), Matrix([[0, 1]]), Matrix([[0]]))
+
+
+References
+----------
+1. `bmsce.ac.in <https://bmsce.ac.in/Content/TE/STATE_SPACE_ANALYSIS.pdf>`_

dev/_sources/tutorials/physics/control/index.rst.txt

@@ -0,0 +1,27 @@
+.. _control_tutorial:
+
+=============================
+Control Tutorials
+=============================
+
+The control module in SymPy provides essential tools for symbolic control system
+analysis. The TransferFunction class allows for creating transfer functions and
+analyzing their properties, such as stability is_stable(), poles, and zeros.
+Series, Parallel, and Feedback classes enable constructing and simplifying
+system interconnections. The TransferFunctionMatrix handles multi-input, multi-output
+(MIMO) systems, while MIMOSeries, MIMOParallel, and MIMOFeedback extend these
+functionalities for complex systems.
+
+Additionally, the module includes the StateSpace class, which allows for modeling
+control systems using state variables, inputs, and outputs in matrix form. This
+representation is particularly useful for time-domain analysis and handling complex
+MIMO systems.
+
+This tutorial contains a breif guide on how to solve Control Problems using
+`TransferFunction` and `StateSpace`.
+
+.. toctree::
+   :maxdepth: 1
+
+   control_problems.rst
+   electrical_problems.rst

dev/_sources/tutorials/physics/index.rst.txt

@@ -23,3 +23,4 @@ write to our `mailing list
 
    biomechanics/index.rst
    mechanics/index.rst
+   control/index.rst