# # Linear Approximation of a Multivariable Model
# ## Multivariable Systems
# Most process models consist of more than a single state and a single input. Techniques for linearization extend naturally to multivariable systems. The most convenient mathematical tools involve some linear algebra.
# ### Example: Gravity Drained Tanks
#
# Here we develop a linear approximation to a process model for a system consisting of coupled gravity-drained tanks.
#
# 
# #### Model
#
# \begin{eqnarray*}
# A_{1}\frac{dh_{1}}{dt} & = & q_{in}-C_{1}\sqrt{h_{1}}\\
# A_{2}\frac{dh_{2}}{dt} & = & q_{d}+C_{1}\sqrt{h_{1}}-C_{2}\sqrt{h_{2}}
# \end{eqnarray*}
# #### Nominal Inputs
#
# \begin{eqnarray*}
# \bar{q}_{in} & = & \bar{q}_{in}\\
# \bar{q}_{d} & = & 0
# \end{eqnarray*}
# #### Steady State for nominal input
#
# \begin{eqnarray*}
# \bar{h}_{1} & = & \frac{\bar{q}_{in}^{2}}{C_{1}^{2}}\\
# \bar{h}_{2} & = & \frac{\bar{q}_{in}^{2}}{C_{2}^{2}}
# \end{eqnarray*}
# #### Deviation Variables
#
# States
#
# \begin{eqnarray*}
# x_{1} & = & h_{1}-\bar{h}_{1}\\
# x_{2} & = & h_{2}-\bar{h}_{2}
# \end{eqnarray*}
#
# Control input
# $$u=q_{in}-\bar{q}_{in}$$
#
# Disturbance input
# $$d=q_{d}-\bar{q}_{d}$$
#
# Measured output
# $$y=h_{2}-\bar{h}_{2}$$
# #### Linearization
#
# \begin{eqnarray*}
# \frac{dh_{1}}{dt} & = & \frac{1}{A_{1}}\left(q_{in}-C_{1}\sqrt{h_{1}}\right)=f_{1}(h_{1},h_{2},q_{in},q_{d})\\
# \frac{dh_{2}}{dt} & = & \frac{1}{A_{2}}\left(q_{d}+C_{1}\sqrt{h_{1}}-C_{2}\sqrt{h_{2}}\right)=f_{2}(h_{1},h_{2},q_{in},q_{d})
# \end{eqnarray*}
#
# Taylor series expansion for the first tank
#
# \begin{eqnarray*}
# \frac{dx_{1}}{dt} & = & f_{1}(\bar{h}_{1}+x_{1},\bar{h}_{2}+x_{2},\bar{q}_{in}+u,\bar{q}_{d}+d)\\
# & \approx & \underbrace{f_{1}(\bar{h}_{1},\bar{h}_{2},\bar{q}_{in})}_{0}+\left.\frac{\partial f_{1}}{\partial h_{1}}\right|_{SS}x_{1}+\left.\frac{\partial f_{1}}{\partial h_{2}}\right|_{SS}x_{2}+\left.\frac{\partial f_{1}}{\partial q_{in}}\right|_{SS}u+\left.\frac{\partial f_{1}}{\partial q_{d}}\right|_{SS}d\\
# & \approx & \left(-\frac{C_{1}}{2A_{1}\sqrt{\bar{h}_{1}}}\right)x_{1}+\left(0\right)x_{2}+\left(\frac{1}{A_{1}}\right)u+\left(0\right)d\\
# & \approx & \left(-\frac{C_{1}^{2}}{2A_{1}\bar{q}_{in}}\right)x_{1}+\left(\frac{1}{A_{1}}\right)u
# \end{eqnarray*}
#
# Taylor series expansion for the second tank
#
# \begin{eqnarray*}
# \frac{dx_{2}}{dt} & = & f_{2}(\bar{h}_{1}+x_{1},\bar{h}_{2}+x_{2},\bar{q}_{in}+u,\bar{q}_{d}+d)\\
# & \approx & \underbrace{f_{2}(\bar{h}_{1},\bar{h}_{2},\bar{q}_{in})}_{0}+\left.\frac{\partial f_{2}}{\partial h_{1}}\right|_{SS}x_{1}+\left.\frac{\partial f_{2}}{\partial h_{2}}\right|_{SS}x_{2}+\left.\frac{\partial f_{2}}{\partial q_{in}}\right|_{SS}u+\left.\frac{\partial f_{2}}{\partial q_{d}}\right|_{SS}d\\
# & \approx & \left(\frac{C_{1}}{2A_{2}\sqrt{\bar{h}_{1}}}\right)x_{1}+\left(-\frac{C_{2}}{2A_{2}\sqrt{\bar{h}_{2}}}\right)x_{2}+\left(0\right)u+\left(\frac{1}{A_{2}}\right)d\\
# & \approx & \left(\frac{C_{1}^{2}}{2A_{2}\bar{q}_{in}}\right)x_{1}+\left(-\frac{C_{2}^{2}}{2A_{2}\bar{q}_{in}}\right)x_{2}+\left(\frac{1}{A_{2}}\right)d
# \end{eqnarray*}
#
# Measured outputs
#
# \begin{eqnarray*}
# y & = & h_{2}-\bar{h}_{2}\\
# & = & x_{2}
# \end{eqnarray*}
# #### Summary of the linear model using matrix/vector notation
#
# \begin{eqnarray*}
# \frac{d}{dt}\left[\begin{array}{c}
# x_{1}\\
# x_{2}
# \end{array}\right] & = & \left[\begin{array}{cc}
# -\frac{C_{1}^{2}}{2A_{1}\bar{q}_{in}} & 0\\
# \frac{C_{1}^{2}}{2A_{2}\bar{q}_{in}} & -\frac{C_{2}^{2}}{2A_{2}\bar{q}_{in}}
# \end{array}\right]\left[\begin{array}{c}
# x_{1}\\
# x_{2}
# \end{array}\right]+\left[\begin{array}{c}
# \frac{1}{A_{1}}\\
# 0
# \end{array}\right]u+\left[\begin{array}{c}
# 0\\
# \frac{1}{A_{2}}
# \end{array}\right]d\\
# y & = & \left[\begin{array}{cc}
# 0 & 1\end{array}\right]\left[\begin{array}{c}
# x_{1}\\
# x_{2}
# \end{array}\right]+\left[\begin{array}{c}
# 0\end{array}\right]u+\left[\begin{array}{c}
# 0\end{array}\right]d
# \end{eqnarray*}
# ## Exercises
# ### First Order Reaction in a CSTR
# A first-order reaction
#
# $$A\longrightarrow\mbox{Products}$$
# takes place in an isothermal CSTR with constant volume $V$, volumetric
# flowrate in and out the same value $q$, reaction rate constant $k$,
# and feed concentration $c_{AF}$.
#
# * Write the material balance equation for this system as a first-order ordinary differential equation.
# * Algebraically determine the steady-state solution to this equation.
# * For what values of $q$, $k$, $V$, and $c_{AF}$ is this stable?
# * For a particular reaction $k$ = 2 1/min. We run this reaction in a vessel of volume $V$ =10 liters, volumetric flowrate $q$ = 50 liters/min. The desired exit concentrationof $A$ is $0.1$ gmol/liter. Assume that we can manipulate $c_{AF}$, the concentration of $A$ entering the CSTR, with proportional control, in order to control the concentration of $A$ exiting the reactor. Substitute the appropriate control law into the mathematical model and rearrange it to get a model of the same form as part (a).
# * Determine the steady-state solution to this equation. If we set $K_{c}=100$, what is the absolute value of the offset between the exit concentration of $A$ and the desired exit concentration of $A$?
# ### Two Gravity Drained Tanks in Series
# Consider the two-tank model. Assuming the tanks are identical, and
# using the same parameter values as used for the single gravity-drained
# tank, compute values for all coefficients appearing in the multivariable
# linear model. Is the system stable? What makes you think so?
# ### Isothermal CSTR with second-order reaction
# Consider an isothermal CSTR with a single reaction
# $$A\longrightarrow\mbox{Products}$$
# whose reaction rate is second-order. Assume constant volume, $V$,
# and density, $\rho$, and a time-dependent volumetric flow rate $q(t)$.
# The input to the reactor has concentration $c_{A,in}$.
#
# * Write the modeling equation for this system.
# * Assume that the state variable is the concentration of $A$ in the
# output, $c_{A}$, and the flow rate $q$ is an input variable. There
# is a steady-state operating point corresponding to $q=1$ L/min. Other
# parameters are $k=1$ L/mol/min, $V=2$ L, $c_{A,in}=2$ mol/L. Find
# the linearized model for $\frac{dc'_{A}}{dt}$ valid in the neighborhood
# of this steady state.
#
# In[ ]:
#
# < [Linear Approximation of a Process Model](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/03.02-Linear-Approximation-of-a-Process-Model.ipynb) | [Contents](toc.ipynb) | [Fitting First Order plus Time Delay to Step Response](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/03.04-Fitting-First-Order-plus-Time-Delay-to-Step-Response.ipynb) >