d2c

Method of mss.

Convert a discrete-time MTI system to a continuous-time MTI system.

Source: src/model/@mss/d2c.m

Syntax

sysc = d2c(sysd, Ts)
sysc = d2c(sysd, Ts, method)

Description

d2c is the inverse of c2d: it reconstructs a continuous-time model from the discrete-time explicit model

$x_{k+1} = f(x_k,u_k), \qquad y_k = g(x_k,u_k).$

The model must be discrete (timeStepSize ~= 0), otherwise the call errors with “The mss model is already continuous.”

`method`	Rule	Result
`"forward"` (default)	Forward Euler inverted, $\dot x = (f(x,u) - x)/T_s$	`mss` — stays explicit
`"backward"`	Backward Euler	`mdss` — implicit, warns
`"tustin"`	Trapezoidal	`mdss` — implicit, warns

As in c2d, only forward Euler preserves explicitness; "backward" and "tustin" convert with mss2mdss and delegate to d2c of the mdss class after warning about the changed return type. In this case, if the mtiBase of sysc was the literal base, the model base is not preserved because the mdss class does not yet support literal base models.

How forward Euler is inverted

Undoing $x_{k+1} = x_k + T_s f(x_k,u_k)$ means subtracting the identity term and dividing by the sample time. As in c2d this is done on the CPN matrices: one monomial column per state is appended and the state parameter block becomes $[\,\Phi_F\ \ -I\,]/T_s$ , the output block staying $[\,\Phi_G\ \ 0\,]$ :

$S_c = \big[\,S\ \ \big[\begin{smallmatrix}I\\0\end{smallmatrix}\big]\,\big], \qquad \Phi_c = \begin{bmatrix} \Phi_F/T_s & -I/T_s \\ \Phi_G & 0 \end{bmatrix},$

i.e. $\dot x = \big(f(x,u) - x\big)/T_s$ and $y = g(x,u)$ . Signal names are carried over and the result is cleaned up with trivialReduction.

There is no TTTensor path — a tensor-train model raises “No TT implemented Yet”.

Input arguments

Argument	Description
`sysd`	Discrete-time `mss` model (`timeStepSize ~= 0`).
`Ts`	Sample time that was used for the discretization, `Ts > 0`.
`method`	(optional) `"forward"` (default), `"backward"` or `"tustin"`.

Output arguments

Output	Description
`msys`	Continuous-time model: an `mss` for `"forward"`, an `mdss` for `"backward"` / `"tustin"`.

Example

sysc = rmss(3,1,1,4);            % continuous-time model
sysd = c2d(sysc, 0.01);          % discretize

sysr = d2c(sysd, 0.01);          % reconstruct
sysr.timeStepSize = 0;           % mark it continuous again (see note above)