CDELSSM – Continuous state space model with time delay

Block SymbolLicensing group: ADVANCED

Function Description
The CDELSSM block (Continuous State Space Model with time DELay) simulates behavior of a linear system with time delay $d e l$ :

\begin{array}{rcl} \frac{d x (t)}{d t} & = & A_{c} x (t) + B_{c} u (t - d e l), x (0) = x 0 \\ y (t) & = & C_{c} x (t) + D_{c} u (t), \end{array}

where $x (t) \in ℝ^{n}$ is the state vector, $x 0 \in ℝ^{n}$ is the initial value of the state vector, $u (t) \in ℝ^{m}$ is the input vector, $y (t) \in ℝ^{p}$ is the output vector. The matrix $A_{c} \in ℝ^{n \times n}$ is the system dynamics matrix, $B_{c} \in ℝ^{n \times m}$ is the input matrix, $C_{c} \in ℝ^{p \times n}$ is the output matrix and $D_{c} \in ℝ^{p \times m}$ is the direct transmission (feedthrough) matrix. If UD=off, the matrix $D_{c}$ is not used during simulation (it behaves as if it were zero).

All matrices are specified in the same format as in Matlab, i.e. the whole matrix is placed in brackets, elements are entered by rows, elements of a row are separated by spaces (blanks), rows are separated by semicolons. The $x 0$ vector is a column, therefore the elements are separated by semicolons (each element is in a separate row).

The simulated system is first converted to the discrete (discretized) state space model:

\begin{array}{rcl} x ((k + 1) T) & = & A_{d} x (k T) + B_{d 1} u ((k - d) T) + B_{d 2} u ((k - d + 1) T), x (0) = x 0 \\ y (k T) & = & C_{c} x (k T) + D_{c} u (k T), \end{array}

where $k \in {1, 2, \dots}$ is the simulation step, $T$ is the execution period of the block in seconds and $d$ is a delay in simulation step such that $(d - 1) T < d e l \leq d . T$ . The period $T$ is not entered in the block, it is determined automatically as a period of the task (TASK, QTASK nebo IOTASK) containing the block.

Inputs of the simulated system u1..u16 represent the input vector u(t). For a given simulation, the first $m$ inputs are used, where $m$ is the number of columns of the matrix Bc. If the input $u (t)$ is changed only in the moments of sampling and between two consecutive sampling instants is constant, i.e. $u (t) = u (k T)$ for $t \in [k T, (k + 1) T)$ , then the matrices $A_{d}$ , $B_{d 1}$ and $B_{d 2}$ are determined by:

\begin{array}{rcl} A_{d} & = & e^{A_{c} T} \\ B_{d 1} & = & e^{A_{c} (T - Δ)} \int_{0}^{Δ} e^{A_{c} τ} B_{c} d τ \\ B_{d 2} & = & \int_{0}^{T - Δ} e^{A_{c} τ} B_{c} d τ, \end{array}

where $Δ = d e l - (d - 1) T$ .

Computation of discrete matrices $A_{d}$ , $B_{d 1}$ and $B_{d 2}$ is based on a method described in [7], which uses Padé approximations of matrix exponential and its integral and scaling technique.

During the real-time simulation, single simulation step of the above discrete state space model is computed in each execution time instant. Outputs of the simulated system y1..y16 represent the state of the system x(t) and for a given simulation, the first $p$ outputs are used, where $p$ is the number of rows of the matrix Cc.

The output iE is an integer and contains information about the simulation progress:

0: everything is OK, the block simulates correctly
-213: incompatibility of the dimensions of the state space model matrices
-510: the task is ill-conditioned (one of the working matrices is singular or close to a singular matrix)
xxx: error code xxx of the REXYGENsystem, see more in Appendix C

This block propagates the signal quality. More information can be found in the 1.4 section.

Input

R1	Block reset	Bool
HLD	Hold current model state	Bool
u1..u16	Analog input of the block	Double (F64)

Parameter

UD	Matrix Dc usage	Bool
del	Model delay [s] $↓$ 0.0	Double (F64)
is	Pade approximation order $↓$ 0 $↑$ 4 $⊙$ 2	Long (I32)
eps	Approximation accuracy $↓$ 0.0 $↑$ 1.0 $⊙$ 1e-15	Double (F64)
Ac	Matrix A of the continuous model $⊙$ [-0.36 -1.24 -0.18; 1 0 0; 0 1 0]	Double (F64)
Bc	Matrix B of the continuous model $⊙$ [0.5; 0; 0]	Double (F64)
Cc	Matrix C of the continuous model $⊙$ [0.12 0.48 0.36]	Double (F64)
Dc	Matrix D of the continuous model $⊙$ [0]	Double (F64)
x0	Initial value of the state x $⊙$ [0; 0; 0]	Double (F64)

Output

iE	Error code	Error
y1..y16	Analog output of the block	Double (F64)

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