From differential-algebraic equations to state-space representation

This demonstration is taken from Ghiaus (2013).

Differential-algebraic equations (DAE)

Let’s consider the system of differential-algebraic equations (DAE) obtained from a thermal circuit:

\[C \dot{\theta} = -A^T G A \theta + A^T G b + f\]

We want to transform this system of equations into state-space representation:

\[\begin{split}\begin{cases} \dot{\theta}_C = A_s \theta_C + B_s u\\ y = C_s \theta_C + D_s u \end{cases}\end{split}\]

DAE in block matrices

We write the DAE as:

\[C \dot{\theta} = K \theta + K_b b + f\]

where \(K = -A^T G A\) and \(K_b = A^T G\).

We write this equation in block matrices corresponding to nodes without capacities, \(\theta_0\), and to nodes with capacities, \(\theta_C\):

\(\begin{bmatrix} 0 & 0\\ 0 & C_C \end{bmatrix}\) \(\begin{bmatrix} \dot{\theta}_0\\ \dot{\theta}_C \end{bmatrix}=\) \(\begin{bmatrix} K_{11} & K_{12}\\ K_{21} & K_{22} \end{bmatrix}\) \(\begin{bmatrix} \theta_0\\ \theta_C \end{bmatrix} +\) \(\begin{bmatrix} K_{b1}\\ K_{b2} \end{bmatrix}b +\) \(\begin{bmatrix} I_{11} & 0\\ 0 & I_{22} \end{bmatrix}\) \(\begin{bmatrix} f_0\\ f_C \end{bmatrix}\)

where:

\(C = \begin{bmatrix} 0 & 0\\ 0 & C_C \end{bmatrix}\) - matrix of capacities separated in blocks of capacities equal to zero and capacities different of zero, grouped in the diagonal matrix \(C_C\).

\(K = -A^T G A = \begin{bmatrix} K_{11} & K_{12}\\ K_{21} & K_{22} \end{bmatrix}\)

\(K_b = A^T G = \begin{bmatrix} K_{b1}\\ K_{b2} \end{bmatrix}\)

\(I_{11}\) and \(I_{22}\) are identity matrices of size of \(\theta_0\) and \(\theta_C\),, respectively.

Elimination of nodes without capacities

By multiplying the first row with \(-K_{21}K_{11}^{-1}\), we obtain:

\(\begin{bmatrix} 0 & 0\\ 0 & C_C \end{bmatrix}\) \(\begin{bmatrix} \dot{\theta}_0\\ \dot{\theta}_C \end{bmatrix}=\) \(\begin{bmatrix} -K_{21} & -K_{21} K_{11}^{-1} K_{12}\\ K_{21} & K_{22} \end{bmatrix}\) \(\begin{bmatrix} \theta_0\\ \theta_C \end{bmatrix}+\) \(\begin{bmatrix} -K_{21} K_{11}^{-1} K_{b1}\\ K_{b2} \end{bmatrix}b +\) \(\begin{bmatrix} -K_{21} K_{11}^{-1} & 0\\ 0 & I_{22} \end{bmatrix}\) \(\begin{bmatrix} f_0\\ f_C \end{bmatrix}\)

By adding the two rows, we obtain:

\(\begin{bmatrix} 0 & C_C \end{bmatrix}\) \(\begin{bmatrix} \dot{\theta}_0\\ \dot{\theta}_C \end{bmatrix} =\) \(\begin{bmatrix} 0 & -K_{21} K_{11}^{-1} K_{12} + K_{22} \end{bmatrix}\) \(\begin{bmatrix} \theta_0\\ \theta_C \end{bmatrix}+\) \(\begin{bmatrix} -K_{21} K_{11}^{-1} K_{b1} + K_{b2} \end{bmatrix}b+\) \(\begin{bmatrix} -K_{21} K_{11}^{-1} & I_{22} \end{bmatrix}\) \(\begin{bmatrix} f_0\\ f_C \end{bmatrix}\)

State-space representation

Since the matrix \(C_C\) is invertible, the above equation becomes:

\(\dot{\theta}_C = C_C^{—1}\) \(\begin{bmatrix} -K_{21} K_{11}^{-1} K_{12} + K_{22} \end{bmatrix}\theta_C\) + \(C^{-1}(-K_{21} K_{11}^{-1} K_{b1} + K_{b2})b\) + \(\begin{bmatrix} -K_{21} K_{11}^{-1} & I_{22} \end{bmatrix}\) \(\begin{bmatrix} f_0\\ f_C \end{bmatrix}\)

By grouping all sources into a vector of inputs, \(u = \begin{bmatrix} b \\ f_0 \\ f_C \end{bmatrix}\),

we obtain:

\(\dot{\theta}_C = C_C^{—1}\) \(\begin{bmatrix} -K_{21} K_{11}^{-1} K_{12} + K_{22} \end{bmatrix}\theta_C\)+ \(C_C^{-1}\) \(\begin{bmatrix} -K_{21} K_{11}^{-1} K_{b1} + K_{b2} & -K_{21} K_{11}^{-1} & I_{22} \end{bmatrix}\) \(\begin{bmatrix} b\\ f_0\\ f_C \end{bmatrix}\)

which represents the state equation of the state-space representation:

\[\begin{split}\begin{cases} \dot{\theta}_C = A_s \theta_C + B_s u\\ \theta_0 = C_s \theta_C + D_s u \end{cases}\end{split}\]

where:

\(A_s= C_C^{—1}[-K_{21} K_{11}^{-1} K_{12} + K_{22}]\)

\(B_s = C_C^{-1}\) \(\begin{bmatrix} -K_{21} K_{11}^{-1} K_{b1} + K_{b2} & -K_{21} K_{11}^{-1} & I_{22} \end{bmatrix}\)

\(C_s = -K_{11}^{-1} K_{12}\)

\(D_s = -K_{11}^{-1} \begin{bmatrix} K_{b1} & I_{11} & 0 \end{bmatrix}\)

\(\theta_C\) - vector of temperatures in nodes with capacities.

\(\theta_0\) - vector of temperatures in nodes without capacities.

\(u = \begin{bmatrix} b \\ f_0 \\ f_C \end{bmatrix}\), 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.

After solving for temperatures, the flows can be obtained:

\[q = G(-A \theta + b)\]

where:

• \(\theta = \begin{bmatrix} \theta_0\\ \theta_C \end{bmatrix}\) is the vector of temperatures;

• \(A, G, b\) - incidence and conductance matrices and temperature source vector.

The transformation from the system of differential-algebraic equations (DAE) to state-space representation is implemented in the function tc2ss of module dm4bem.

Time integration of state equation

Numerical integration of the state equation is done by finding the value of temperature at time \(t + \Delta t\) given the value at time \(t\):

\[\frac{\theta_{k + 1} - \theta_k}{\Delta t} = f \cdot (A_s \theta_{k+1} + B_s u_{k+1}) + (1 - f) \cdot (A_s \theta_k + B_s u_k)\]

or

\[(I - f \Delta t A) \theta_{k+1} = (I + (1 - f)\Delta t A_s) \theta_k + \Delta t B (f u_{k+1} + (1 - f) u_k)\]

where \(0 \leq f \leq 1\) is a weighing factor between current and future values.

• If \(f = 0\), the scheme is Euler explicit:

\[ \theta_{k+1} = (I + \Delta t A_s) \theta _k + \Delta t B_s u_k \]

• If \(f = 1\), the scheme is Euler implicit

\[\theta_{k+1} = (I - \Delta t A_s)^{-1} ( \theta _k + \Delta t B_s u_{k+1} )\]

• If \(f = 0.5\), the scheme is Crank-Nicholson’s or trapezoidal rule:

\[\theta_{k+1} = (I - 0.5 \Delta t A_s)^{-1} ((I + 0.5 \Delta t A_s) \theta _k + 0.5 \Delta t B_s (u_{k+1} + u_k))\]

References

Ghiaus, C. (2013). Causality issue in the heat balance method for calculating the design heating and cooling loads, Energy 50: 292-301, HAL 03605823

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 HAL 03578578