`ControlSystemsBase.append`

`ControlSystemsBase.c2d`

`ControlSystemsBase.delay`

`ControlSystemsBase.feedback`

`ControlSystemsBase.feedback2dof`

`ControlSystemsBase.minreal`

`ControlSystemsBase.pade`

`ControlSystemsBase.parallel`

`ControlSystemsBase.series`

`ControlSystemsBase.seriesform`

`ControlSystemsBase.sminreal`

`ControlSystemsBase.ss`

`ControlSystemsBase.ssdata`

`ControlSystemsBase.tf`

`ControlSystemsBase.thiran`

`ControlSystemsBase.zpk`

See also Connecting named systems together.

# Constructing systems

`ControlSystemsBase.append`

— Function`append(systems::StateSpace...), append(systems::TransferFunction...)`

Append systems in block diagonal form

`ControlSystemsBase.c2d`

— Function```
sysd = c2d(sys::AbstractStateSpace{<:Continuous}, Ts, method=:zoh; w_prewarp=0)
Gd = c2d(G::TransferFunction{<:Continuous}, Ts, method=:zoh)
```

Convert the continuous-time system `sys`

into a discrete-time system with sample time `Ts`

, using the specified `method`

(:`zoh`

, `:foh`

, `:fwdeuler`

or `:tustin`

).

`method = :tustin`

performs a bilinear transform with prewarp frequency `w_prewarp`

.

`w_prewarp`

: Frequency (rad/s) for pre-warping when using the Tustin method, has no effect for other methods.

See also `c2d_x0map`

**Extended help**

ZoH sampling is exact for linear systems with piece-wise constant inputs (step invariant), i.e., the solution obtained using `lsim`

is not approximative (modulu machine precision). ZoH sampling is commonly used to discretize continuous-time plant models that are to be controlled using a discrete-time controller.

FoH sampling is exact for linear systems with piece-wise linear inputs (ramp invariant), this is a good choice for simulation of systems with smooth continuous inputs.

To approximate the behavior of a continuous-time system well in the frequency domain, the `:tustin`

(trapezoidal / bilinear) method may be most appropriate. In this case, the pre-warping argument can be used to ensure that the frequency response of the discrete-time system matches the continuous-time system at a given frequency. The tustin transformation alters the meaning of the state components, while ZoH and FoH preserve the meaning of the state components. The Tustin method is commonly used to discretize a continuous-time controller.

The forward-Euler method generally requires the sample time to be very small relative to the time constants of the system, and its use is generally discouraged.

Classical rules-of-thumb for selecting the sample time for control design dictate that `Ts`

should be chosen as $0.2 ≤ ωgc⋅Ts ≤ 0.6$ where $ωgc$ is the gain-crossover frequency (rad/s).

```
Qd = c2d(sys::StateSpace{Continuous}, Qc::Matrix, Ts; opt=:o)
Qd, Rd = c2d(sys::StateSpace{Continuous}, Qc::Matrix, Rc::Matrix, Ts; opt=:o)
Qd = c2d(sys::StateSpace{Discrete}, Qc::Matrix; opt=:o)
Qd, Rd = c2d(sys::StateSpace{Discrete}, Qc::Matrix, Rc::Matrix; opt=:o)
```

Sample a continuous-time covariance or LQR cost matrix to fit the provided discrete-time system.

If `opt = :o`

(default), the matrix is assumed to be a covariance matrix. The measurement covariance `R`

may also be provided. If `opt = :c`

, the matrix is instead assumed to be a cost matrix for an LQR problem.

Measurement covariance (here called `Rc`

) is usually estimated in discrete time, and is in this case not dependent on the sample rate. Discretization of the measurement covariance only makes sense when a continuous-time controller has been designed and the closest corresponding discrete-time controller is desired.

The method used comes from theorem 5 in the reference below.

Ref: "Discrete-time Solutions to the Continuous-time Differential Lyapunov Equation With Applications to Kalman Filtering", Patrik Axelsson and Fredrik Gustafsson

On singular covariance matrices: The traditional double integrator with covariance matrix `Q = diagm([0,σ²])`

can not be sampled with this method. Instead, the input matrix ("Cholesky factor") of `Q`

must be manually kept track of, e.g., the noise of variance `σ²`

enters like `N = [0, 1]`

which is sampled using ZoH and becomes `Nd = [1/2 Ts^2; Ts]`

which results in the covariance matrix `σ² * Nd * Nd'`

.

**Example:**

The following example designs a continuous-time LQR controller for a resonant system. This is simulated with OrdinaryDiffEq to allow the ODE integrator to also integrate the continuous-time LQR cost (the cost is added as an additional state variable). We then discretize both the system and the cost matrices and simulate the same thing. The discretization of an LQR contorller in this way is sometimes refered to as `lqrd`

.

```
using ControlSystemsBase, LinearAlgebra, OrdinaryDiffEq, Test
sysc = DemoSystems.resonant()
x0 = ones(sysc.nx)
Qc = [1 0.01; 0.01 2] # Continuous-time cost matrix for the state
Rc = I(1) # Continuous-time cost matrix for the input
L = lqr(sysc, Qc, Rc)
dynamics = function (xc, p, t)
x = xc[1:sysc.nx]
u = -L*x
dx = sysc.A*x + sysc.B*u
dc = dot(x, Qc, x) + dot(u, Rc, u)
return [dx; dc]
end
prob = ODEProblem(dynamics, [x0; 0], (0.0, 10.0))
sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)
cc = sol.u[end][end] # Continuous-time cost
# Discrete-time version
Ts = 0.01
sysd = c2d(sysc, Ts)
Qd, Rd = c2d(sysd, Qc, Rc, opt=:c)
Ld = lqr(sysd, Qd, Rd)
sold = lsim(sysd, (x, t) -> -Ld*x, 0:Ts:10, x0 = x0)
function cost(x, u, Q, R)
dot(x, Q, x) + dot(u, R, u)
end
cd = cost(sold.x, sold.u, Qd, Rd) # Discrete-time cost
@test cc ≈ cd rtol=0.01 # These should be similar
```

`c2d(G::DelayLtiSystem, Ts, method=:zoh)`

`ControlSystemsBase.feedback`

— Function```
feedback(sys)
feedback(sys1, sys2)
```

For a general LTI-system, `feedback`

forms the negative feedback interconnection

```
>-+ sys1 +-->
| |
(-)sys2 +
```

If no second system is given, negative identity feedback is assumed

```
feedback(sys1::AbstractStateSpace, sys2::AbstractStateSpace;
U1=:, Y1=:, U2=:, Y2=:, W1=:, Z1=:, W2=Int[], Z2=Int[],
Wperm=:, Zperm=:, pos_feedback::Bool=false)
```

*Basic use* `feedback(sys1, sys2)`

forms the (negative) feedback interconnection

```
┌──────────────┐
◄──────────┤ sys1 │◄──── Σ ◄──────
│ │ │ │
│ └──────────────┘ -1
│ |
│ ┌──────────────┐ │
└─────►│ sys2 ├──────┘
│ │
└──────────────┘
```

If no second system `sys2`

is given, negative identity feedback (`sys2 = 1`

) is assumed. The returned closed-loop system will have a state vector comprised of the state of `sys1`

followed by the state of `sys2`

.

*Advanced use* `feedback`

also supports more flexible use according to the figure below

```
┌──────────────┐
z1◄─────┤ sys1 │◄──────w1
┌─── y1◄─────┤ │◄──────u1 ◄─┐
│ └──────────────┘ │
│ α
│ ┌──────────────┐ │
└──► u2─────►│ sys2 ├───────►y2──┘
w2─────►│ ├───────►z2
└──────────────┘
```

`U1`

, `W1`

specifies the indices of the input signals of `sys1`

corresponding to `u1`

and `w1`

`Y1`

, `Z1`

specifies the indices of the output signals of `sys1`

corresponding to `y1`

and `z1`

`U2`

, `W2`

, `Y2`

, `Z2`

specifies the corresponding signals of `sys2`

Specify `Wperm`

and `Zperm`

to reorder the inputs (corresponding to [w1; w2]) and outputs (corresponding to [z1; z2]) in the resulting statespace model.

Negative feedback (α = -1) is the default. Specify `pos_feedback=true`

for positive feedback (α = 1).

See also `lft`

, `starprod`

, `sensitivity`

, `input_sensitivity`

, `output_sensitivity`

, `comp_sensitivity`

, `input_comp_sensitivity`

, `output_comp_sensitivity`

, `G_PS`

, `G_CS`

.

The manual section From block diagrams to code contains higher-level instructions on how to use this function.

See Zhou, Doyle, Glover (1996) for similar (somewhat less symmetric) formulas.

`ControlSystemsBase.feedback2dof`

— Function```
feedback2dof(P,R,S,T)
feedback2dof(B,A,R,S,T)
```

- Return
`BT/(AR+ST)`

where B and A are the numerator and denominator polynomials of`P`

respectively - Return
`BT/(AR+ST)`

`feedback2dof(P::TransferFunction, C::TransferFunction, F::TransferFunction)`

Return the transfer function `P(F+C)/(1+PC)`

which is the closed-loop system with process `P`

, controller `C`

and feedforward filter `F`

from reference to control signal (by-passing `C`

).

```
+-------+
| |
+-----> F +----+
| | | |
| +-------+ |
| +-------+ | +-------+
r | - | | | | | y
+--+-----> C +----+----> P +---+-->
| | | | | |
| +-------+ +-------+ |
| |
+--------------------------------+
```

`ControlSystemsBase.minreal`

— Function`minreal(tf::TransferFunction, eps=sqrt(eps()))`

Create a minimal representation of each transfer function in `tf`

by cancelling poles and zeros will promote system to an appropriate numeric type

`minreal(sys::T; fast=false, kwargs...)`

Minimal realisation algorithm from P. Van Dooreen, The generalized eigenstructure problem in linear system theory, IEEE Transactions on Automatic Control

For information about the options, see `?ControlSystemsBase.MatrixPencils.lsminreal`

See also `sminreal`

, which is both numerically exact and substantially faster than `minreal`

, but with a much more limited potential in removing non-minimal dynamics.

`ControlSystemsBase.parallel`

— Function`parallel(sys1::LTISystem, sys2::LTISystem)`

Connect systems in parallel, equivalent to `sys2+sys1`

`ControlSystemsBase.series`

— Function`series(sys1::LTISystem, sys2::LTISystem)`

Connect systems in series, equivalent to `sys2*sys1`

`ControlSystemsBase.sminreal`

— Function`sminreal(sys)`

Compute the structurally minimal realization of the state-space system `sys`

. A structurally minimal realization is one where only states that can be determined to be uncontrollable and unobservable based on the location of 0s in `sys`

are removed.

Systems with numerical noise in the coefficients, e.g., noise on the order of `eps`

require truncation to zero to be affected by structural simplification, e.g.,

```
trunc_zero!(A) = A[abs.(A) .< 10eps(maximum(abs, A))] .= 0
trunc_zero!(sys.A); trunc_zero!(sys.B); trunc_zero!(sys.C)
sminreal(sys)
```

In contrast to `minreal`

, which performs pole-zero cancellation using linear-algebra operations, has an 𝑂(nₓ^3) complexity and is subject to numerical tolerances, `sminreal`

is computationally very cheap and numerically exact (operates on integers). However, the ability of `sminreal`

to reduce the order of the model is much less powerful.

See also `minreal`

.

`ControlSystemsBase.ss`

— Function```
sys = ss(A, B, C, D) # Continuous
sys = ss(A, B, C, D, Ts) # Discrete
```

Create a state-space model `sys::StateSpace{TE, T}`

with matrix element type `T`

and TE is `Continuous`

or `<:Discrete`

.

This is a continuous-time model if `Ts`

is omitted. Otherwise, this is a discrete-time model with sampling period `Ts`

.

`D`

may be specified as `0`

in which case a zero matrix of appropriate size is constructed automatically. `sys = ss(D [, Ts])`

specifies a static gain matrix `D`

.

To associate names with states, inputs and outputs, see `named_ss`

.

`ControlSystemsBase.tf`

— Function```
sys = tf(num, den[, Ts])
sys = tf(gain[, Ts])
```

Create as a fraction of polynomials:

`sys::TransferFunction{SisoRational{T,TR}} = numerator/denominator`

where T is the type of the coefficients in the polynomial.

`num`

: the coefficients of the numerator polynomial. Either scalar or vector to create SISO systems

or an array of vectors to create MIMO system.

`den`

: the coefficients of the denominator polynomial. Either vector to create SISO systems

or an array of vectors to create MIMO system.

`Ts`

: Sample time if discrete time system.

The polynomial coefficients are ordered starting from the highest order term.

Other uses:

`tf(sys)`

: Convert`sys`

to`tf`

form.`tf("s")`

,`tf("z")`

: Create the continuous-time transfer function`s`

, or the discrete-time transfer function`z`

.`numpoly(sys)`

,`denpoly(sys)`

: Get the numerator and denominator polynomials of`sys`

as a matrix of vectors, where the outer matrix is of size`n_output × n_inputs`

.

`ControlSystemsBase.zpk`

— Function```
zpk(gain[, Ts])
zpk(num, den, k[, Ts])
zpk(sys)
```

Create transfer function on zero pole gain form. The numerator and denominator are represented by their poles and zeros.

`sys::TransferFunction{SisoZpk{T,TR}} = k*numerator/denominator`

where `T`

is the type of `k`

and `TR`

the type of the zeros/poles, usually Float64 and Complex{Float64}.

`num`

: the roots of the numerator polynomial. Either scalar or vector to create SISO systems

or an array of vectors to create MIMO system.

`den`

: the roots of the denominator polynomial. Either vector to create SISO systems

or an array of vectors to create MIMO system.

`k`

: The gain of the system. Obs, this is not the same as`dcgain`

.`Ts`

: Sample time if discrete time system.

Other uses:

`zpk(sys)`

: Convert`sys`

to`zpk`

form.`zpk("s")`

: Create the transferfunction`s`

.

`ControlSystemsBase.delay`

— Function```
delay(tau)
delay(tau, Ts)
delay(T::Type{<:Number}, tau)
delay(T::Type{<:Number}, tau, Ts)
```

Create a pure time delay of length `τ`

of type `T`

.

The type `T`

defaults to `promote_type(Float64, typeof(tau))`

.

If `Ts`

is given, the delay is discretized with sampling time `Ts`

and a discrete-time StateSpace object is returned.

**Example:**

Create a LTI system with an input delay of `L`

```
L = 1
tf(1, [1, 1])*delay(L)
s = tf("s")
tf(1, [1, 1])*exp(-s*L) # Equivalent to the version above
```

`ControlSystemsBase.pade`

— Function`pade(τ::Real, N::Int)`

Compute the `N`

th order Padé approximation of a time-delay of length `τ`

.

See also `thiran`

for discretization of delays.

`pade(G::DelayLtiSystem, N)`

Approximate all time-delays in `G`

by Padé approximations of degree `N`

.

`ControlSystemsBase.thiran`

— Function`thiran(τ::Real, Ts)`

Discretize a potentially fractional delay $τ$ as a Thiran all-pass filter with sample time `Ts`

.

The Thiran all-pass filter gives an a maximally flat group delay.

If $τ$ is an integer multiple of $Ts$, the Thiran all-pass filter reduces to $z^{-τ/Ts}$.

Ref: T. I. Laakso, V. Valimaki, M. Karjalainen and U. K. Laine, "Splitting the unit delay [FIR/all pass filters design]," in IEEE Signal Processing Magazine, vol. 13, no. 1, 1996.

`ControlSystemsBase.ssdata`

— Function`A, B, C, D = ssdata(sys)`

A destructor that outputs the statespace matrices.

`ControlSystemsBase.seriesform`

— Function`Gs, k = seriesform(G::TransferFunction{Discrete})`

Convert a transfer function `G`

to a vector of second-order transfer functions and a scalar gain `k`

, the product of which equals `G`

.