A03_DAE2SS.ipynb

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    "# From differential-algebraic equations to state-space representation"
   ]
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    "This demonstration is taken from [Ghiaus (2013)](https://doi.org/10.1016/j.energy.2012.10.024).\n",
    "\n",
    "Let's consider the system of diferential-algebraic equations (DAE) obtained from a thermal circuit:\n",
    "\n",
    "$$C \\dot{\\theta} = -A^T G A \\theta + A^T G b + f$$\n",
    "\n",
    "We want to transform these equation into [state-space representation](https://en.m.wikipedia.org/wiki/State-space_representation):\n",
    "\n",
    "$$\\begin{cases}\n",
    "\\dot{\\theta}_C = A_s \\theta_C + B_s u\\\\\n",
    "y = C_s \\theta_C + D_s u\n",
    "\\end{cases}$$\n",
    "\n",
    "We write the DAE as:\n",
    "\n",
    "$$C \\dot{\\theta} = K \\theta + K_b b + f$$\n",
    "\n",
    "where $K = -A^T G A$ and $K_b = A^T G$.\n",
    "\n",
    "We write this equation in block matrices corresponding to nodes without capacities, $\\theta_0$, and to nodes with capacities, $\\theta_C$:\n",
    "\n",
    "$\\begin{bmatrix}\n",
    "0 & 0\\\\ \n",
    "0 & C_C\n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "\\dot{\\theta}_0\\\\ \n",
    "\\dot{\\theta}_C\n",
    "\\end{bmatrix}$\n",
    "=\n",
    "$\\begin{bmatrix}\n",
    "K_{11} & K_{12}\\\\ \n",
    "K_{21} & K_{22} \n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "\\theta_0\\\\ \n",
    "\\theta_C\n",
    "\\end{bmatrix} +$\n",
    "$\\begin{bmatrix}\n",
    "K_{b1}\\\\ \n",
    "K_{b2} \n",
    "\\end{bmatrix}b +$\n",
    "$\\begin{bmatrix}\n",
    "I_{11} & 0\\\\ \n",
    "0 & I_{22} \n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "f_0\\\\ \n",
    "f_C \n",
    "\\end{bmatrix}$\n",
    "\n",
    "where:\n",
    "\n",
    "$C = \\begin{bmatrix}\n",
    "0 & 0\\\\ \n",
    "0 & C_C\n",
    "\\end{bmatrix}$ - matrix of capacities separated in blocks of capacities equal to zero and capacities different of zero, grouped in the diagonal matrix $C_C$.\n",
    "\n",
    "$K = -A^T G A = \\begin{bmatrix}\n",
    "K_{11} & K_{12}\\\\ \n",
    "K_{21} & K_{22} \n",
    "\\end{bmatrix}$\n",
    "\n",
    "$K_b = A^T G = \\begin{bmatrix}\n",
    "K_{b1}\\\\ \n",
    "K_{b2} \n",
    "\\end{bmatrix}$\n",
    "\n",
    "$I_{11}$ and $I_{22}$ are [identity matrices](https://en.m.wikipedia.org/wiki/Identity_matrix) of size of $\\theta_0$ and $\\theta_C$,, respectively.\n",
    "\n",
    "By multiplying the first row with $-K_{21}K_{11}^{-1}$, we obtain:\n",
    "\n",
    "$\\begin{bmatrix}\n",
    "0 & 0\\\\ \n",
    "0 & C_C\n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "\\dot{\\theta}_0\\\\ \n",
    "\\dot{\\theta}_C\n",
    "\\end{bmatrix}=$\n",
    "$\\begin{bmatrix}\n",
    "-K_{21} & -K_{21} K_{11}^{-1} K_{12}\\\\ \n",
    "K_{21} & K_{22} \n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "\\theta_0\\\\ \n",
    "\\theta_C\n",
    "\\end{bmatrix}+$\n",
    "$\\begin{bmatrix}\n",
    "-K_{21} K_{11}^{-1} K_{b1}\\\\ \n",
    "K_{b2} \n",
    "\\end{bmatrix}b +$\n",
    "$\\begin{bmatrix}\n",
    "-K_{21} K_{11}^{-1} & 0\\\\ \n",
    "0 & I_{22} \n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "f_0\\\\ \n",
    "f_C \n",
    "\\end{bmatrix}$\n",
    "\n",
    "By adding the two rows, we obtain:\n",
    "\n",
    "$\\begin{bmatrix} \n",
    "0 & C_C\n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "\\dot{\\theta}_0\\\\ \n",
    "\\dot{\\theta}_C\n",
    "\\end{bmatrix} =$\n",
    "$\\begin{bmatrix}\n",
    "0 & -K_{21} K_{11}^{-1} K_{12} + K_{22} \n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "\\theta_0\\\\ \n",
    "\\theta_C\n",
    "\\end{bmatrix}+$\n",
    "$\\begin{bmatrix} \n",
    "-K_{21} K_{11}^{-1} K_{b1} + K_{b2} \n",
    "\\end{bmatrix}b+$\n",
    "$\\begin{bmatrix} \n",
    "-K_{21} K_{11}^{-1} & I_{22} \n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "f_0\\\\ \n",
    "f_C \n",
    "\\end{bmatrix}$\n",
    "\n",
    "Since the matrix $C_C$ is [invertible](https://en.m.wikipedia.org/wiki/Invertible_matrix),\n",
    "the above equation becomes:\n",
    "\n",
    "$\\dot{\\theta}_C = C_C^{—1}$\n",
    "$\\begin{bmatrix}\n",
    "-K_{21} K_{11}^{-1} K_{12} + K_{22}\n",
    "\\end{bmatrix}\\theta_C$ + \n",
    "$C^{-1}(-K_{21} K_{11}^{-1} K_{b1} + K_{b2})b$ +\n",
    "$\\begin{bmatrix} \n",
    "-K_{21} K_{11}^{-1} & I_{22} \n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "f_0\\\\ \n",
    "f_C \n",
    "\\end{bmatrix}$\n",
    "\n",
    "By grouping all sources into a vector of inputs,\n",
    "$u = \\begin{bmatrix}\n",
    "b \\\\ f_0 \\\\ f_C\n",
    "\\end{bmatrix}$,\n",
    "\n",
    "we obtain:\n",
    "\n",
    "$\\dot{\\theta}_C = C_C^{—1}$\n",
    "$\\begin{bmatrix}\n",
    "-K_{21} K_{11}^{-1} K_{12} + K_{22}\n",
    "\\end{bmatrix}\\theta_C$+ \n",
    "$C_C^{-1}$\n",
    "$\\begin{bmatrix} \n",
    "-K_{21} K_{11}^{-1} K_{b1} + K_{b2} &\n",
    "-K_{21} K_{11}^{-1} & I_{22}\n",
    "\\end{bmatrix}$\n",
    "$\\begin{bmatrix}\n",
    "b\\\\\n",
    "f_0\\\\ \n",
    "f_C \n",
    "\\end{bmatrix}$\n",
    "\n",
    "which represents the state equation of the [state-space representation](https://en.m.wikipedia.org/wiki/State-space_representation):\n",
    "\n",
    "$$\\begin{cases}\n",
    "\\dot{\\theta}_C = A_s \\theta_C + B_s u\\\\\n",
    "\\theta_0 = C_s \\theta_C + D_s u\n",
    "\\end{cases}$$\n",
    "\n",
    "where:\n",
    "\n",
    "$A_s= C_C^{—1}[-K_{21} K_{11}^{-1} K_{12} + K_{22}]$\n",
    "\n",
    "$B_s = C_C^{-1}$\n",
    "$\\begin{bmatrix} \n",
    "-K_{21} K_{11}^{-1} K_{b1} + K_{b2} &\n",
    "-K_{21} K_{11}^{-1} & I_{22}\n",
    "\\end{bmatrix}$\n",
    "\n",
    "$C_s = -K_{11}^{-1} K_{12}$\n",
    "\n",
    "$D_s = -K_{11}^{-1} \\begin{bmatrix} K_{b1} & I_{11} & 0 \\end{bmatrix}$\n",
    "\n",
    "$\\theta_C$ - vector of temperatures in nodes with capacities.\n",
    "\n",
    "$\\theta_0$ - vector of temperatures in nodes without capacities.\n",
    "\n",
    "$u = \\begin{bmatrix}\n",
    "b \\\\ f_0 \\\\ f_C\n",
    "\\end{bmatrix}$, vector of inputs, where $b$ is the vector of temperature sources, $f_0$ and $f_C$ are the vector of flow-rate sources in the nodes without capacities and the nodes with capacities, respectively.\n",
    "\n",
    "After solving for temperatures, the flows can be obtained:\n",
    "\n",
    "$$q = G(-A \\theta + b)$$\n",
    "\n",
    "The transformation from the system of differential-algebraic equations (DAE) to state-space representation is implemented in the function `tc2ss` of module `dm4bem`."
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   "source": [
    "## References\n",
    "\n",
    "Ghiaus, C. (2013) Causality issue in the heat balance method for calculating the design heating and cooling loads, *Energy* 50: 292-301, https://doi.org/10.1016/j.energy.2012.10.024, [hal-03605823](https://hal.archives-ouvertes.fr/hal-03605823/document)\n",
    "\n",
    "C. Ghiaus (2021). Dynamic Models for Energy Control of Smart Homes, in *S. Ploix M. Amayri, N. Bouguila (eds.) Towards Energy Smart Homes*, Online ISBN: 978-3-030-76477-7, Print ISBN: 978-3-030-76476-0, Springer, pp. 163-198 (ref.)\n",
    "[DOI 10.1007/978-3-030-76477-7_5](https://doi.org/10.1007/978-3-030-76477-7_5), [HAL 03578578](https://hal.archives-ouvertes.fr/hal-03578578/document)"
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