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### 30.7 System Analysis-Time Domain

— Function File: c2d (sys, opt, t)
— Function File: c2d (sys, t)

> Converts the system data structure describing:

>               .
>               x = Ac x + Bc u

> into a discrete time equivalent model:

>               x[n+1] = Ad x[n] + Bd u[n]

> via the matrix exponential or bilinear transform.

> Inputs

> sys
>   system data structure (may have both continuous time and discrete time subsystems)
> opt
>   string argument; conversion option (optional argument; may be omitted as shown above)
>   `"ex"`
>     use the matrix exponential (default)
>   `"bi"`
>     use the bilinear transformation

>                                 2(z-1)
>                             s = -----
>                                 T(z+1)

>     FIXME: This option exits with an error if sys is not purely continuous. (The `ex` option can handle mixed systems.)

>   `"matched"`
>     Use the matched pole/zero equivalent transformation (currently only works for purely continuous SISO systems).
> t
>   sampling time; required if sys is purely continuous.

>   Note that if the second argument is not a string, `c2d()` assumes that the second argument is t and performs appropriate argument checks.

> Output

> dsys
>   Discrete time equivalent via zero-order hold, sample each t sec.

> This function adds the suffix `_d` to the names of the new discrete states.

— Function File: d2c (sys, tol)
— Function File: d2c (sys, opt)

> Convert a discrete (sub)system into a purely continuous one. The sampling time used is `sysgettsam(`sys`)`.

> Inputs

> sys
>   system data structure with discrete components
> tol
>   Scalar value. Tolerance for convergence of default `"log"` option (see below)
> opt
>   conversion option. Choose from:
>   `"log"`
>     (default) Conversion is performed via a matrix logarithm. Due to some problems with this computation, it is followed by a steepest descent algorithm to identify continuous time a, b, to get a better fit to the original data.

>     If called as `d2c (`sys`,` tol`)`, with tol positive scalar, the `"log"` option is used. The default value for tol is `1e-8`.

>   `"bi"`
>     Conversion is performed via bilinear transform z = (1 + s T / 2)/(1 - s T / 2) where T is the system sampling time (see `sysgettsam`).

>     FIXME: bilinear option exits with an error if sys is not purely discrete

> Output
> csys
>   continuous time system (same dimensions and signal names as in sys).

— Function File: [dsys, fidx] = dmr2d (sys, idx, sprefix, ts2, cuflg)

> convert a multirate digital system to a single rate digital system states specified by idx, sprefix are sampled at ts2, all others are assumed sampled at ts1 = `sysgettsam (`sys`)`.

> Inputs

> sys
>   discrete time system; `dmr2d` exits with an error if sys is not discrete
> idx
>   indices or names of states with sampling time `sysgettsam(`sys`)` (may be empty); see `cellidx`
> sprefix
>   list of string prefixes of states with sampling time `sysgettsam(`sys`)` (may be empty)
> ts2
>   sampling time of states not specified by idx, sprefix must be an integer multiple of `sysgettsam(`sys`)`
> cuflg
>   "constant u flag" if cuflg is nonzero then the system inputs are assumed to be constant over the revised sampling interval ts2. Otherwise, since the inputs can change during the interval t in [k ts2, (k+1) ts2], an additional set of inputs is included in the revised B matrix so that these intersample inputs may be included in the single-rate system. default cuflg = 1.

> Outputs

> dsys
>   equivalent discrete time system with sampling time ts2.

>   The sampling time of sys is updated to ts2.

>   if cuflg=0 then a set of additional inputs is added to the system with suffixes _d1, ..., _dn to indicate their delay from the starting time k ts2, i.e. u = [u_1; u_1_d1; ..., u_1_dn] where u_1_dk is the input k*ts1 units of time after u_1 is sampled. (ts1 is the original sampling time of the discrete time system and ts2 = (n+1)*ts1)

> fidx
>   indices of "formerly fast" states specified by idx and sprefix; these states are updated to the new (slower) sampling interval ts2.

> WARNING Not thoroughly tested yet; especially when cuflg == 0.

— Function File: damp (p, tsam)

> Displays eigenvalues, natural frequencies and damping ratios of the eigenvalues of a matrix p or the A matrix of a system p, respectively. If p is a system, tsam must not be specified. If p is a matrix and tsam is specified, eigenvalues of p are assumed to be in z-domain.

> See also: eig.

— Function File: dcgain (sys, tol)

> Returns dc-gain matrix. If dc-gain is infinite an empty matrix is returned. The argument tol is an optional tolerance for the condition number of the A Matrix in sys (default tol = 1.0e-10)

— Function File: [y, t] = impulse (sys, inp, tstop, n)

> Impulse response for a linear system. The system can be discrete or multivariable (or both). If no output arguments are specified, `impulse` produces a plot or the impulse response data for system sys.

> Inputs

> sys
>   System data structure.
> inp
>   Index of input being excited
> tstop
>   The argument tstop (scalar value) denotes the time when the simulation should end.
> n
>   the number of data values.

>   Both parameters tstop and n can be omitted and will be computed from the eigenvalues of the A Matrix.

> Outputs
> y
>   Values of the impulse response.
> t
>   Times of the impulse response.

> See also: step.

— Function File: [y, t] = step (sys, inp, tstop, n)

> Step response for a linear system. The system can be discrete or multivariable (or both). If no output arguments are specified, `step` produces a plot or the step response data for system sys.

> Inputs

> sys
>   System data structure.
> inp
>   Index of input being excited
> tstop
>   The argument tstop (scalar value) denotes the time when the simulation should end.
> n
>   the number of data values.

>   Both parameters tstop and n can be omitted and will be computed from the eigenvalues of the A Matrix.

> Outputs
> y
>   Values of the step response.
> t
>   Times of the step response.

> When invoked with the output parameter y the plot is not displayed.

> See also: impulse.