`ControlSystemsBase.G_CS`

`ControlSystemsBase.G_PS`

`ControlSystemsBase.add_input`

`ControlSystemsBase.add_output`

`ControlSystemsBase.append`

`ControlSystemsBase.array2mimo`

`ControlSystemsBase.bodev`

`ControlSystemsBase.bodev`

`ControlSystemsBase.c2d`

`ControlSystemsBase.c2d_poly2poly`

`ControlSystemsBase.c2d_roots2poly`

`ControlSystemsBase.c2d_x0map`

`ControlSystemsBase.comp_sensitivity`

`ControlSystemsBase.d2c`

`ControlSystemsBase.dab`

`ControlSystemsBase.extended_gangoffour`

`ControlSystemsBase.feedback`

`ControlSystemsBase.feedback`

`ControlSystemsBase.feedback2dof`

`ControlSystemsBase.feedback2dof`

`ControlSystemsBase.freqrespv`

`ControlSystemsBase.freqrespv`

`ControlSystemsBase.freqrespv`

`ControlSystemsBase.input_comp_sensitivity`

`ControlSystemsBase.input_sensitivity`

`ControlSystemsBase.kalman`

`ControlSystemsBase.laglink`

`ControlSystemsBase.leadlink`

`ControlSystemsBase.leadlinkat`

`ControlSystemsBase.leadlinkcurve`

`ControlSystemsBase.lft`

`ControlSystemsBase.loopshapingPI`

`ControlSystemsBase.loopshapingPID`

`ControlSystemsBase.lqr`

`ControlSystemsBase.nyquistv`

`ControlSystemsBase.nyquistv`

`ControlSystemsBase.output_comp_sensitivity`

`ControlSystemsBase.output_sensitivity`

`ControlSystemsBase.parallel`

`ControlSystemsBase.pid`

`ControlSystemsBase.pidplots`

`ControlSystemsBase.place`

`ControlSystemsBase.placePI`

`ControlSystemsBase.place_knvd`

`ControlSystemsBase.rstc`

`ControlSystemsBase.rstd`

`ControlSystemsBase.sensitivity`

`ControlSystemsBase.series`

`ControlSystemsBase.sigmav`

`ControlSystemsBase.sigmav`

`ControlSystemsBase.sminreal`

`ControlSystemsBase.stabregionPID`

`ControlSystemsBase.starprod`

`ControlSystemsBase.zpconv`

# Synthesis

For $H_\infty$ and $H_2$ synthesis as well as more advanced LQG design, see RobustAndOptimalControl.

`ControlSystemsBase.kalman`

— Method```
kalman(Continuous, A, C, R1, R2)
kalman(Discrete, A, C, R1, R2)
kalman(sys, R1, R2)
```

Calculate the optimal Kalman gain

The `args...; kwargs...`

are sent to the Riccati solver, allowing specification of cross-covariance etc. See `?MatrixEquations.arec/ared`

for more help.

`ControlSystemsBase.lqr`

— Method```
lqr(sys, Q, R)
lqr(Continuous, A, B, Q, R, args...; kwargs...)
lqr(Discrete, A, B, Q, R, args...; kwargs...)
```

Calculate the optimal gain matrix `K`

for the state-feedback law `u = -K*x`

that minimizes the cost function:

J = integral(x'Qx + u'Ru, 0, inf) for the continuous-time model `dx = Ax + Bu`

. J = sum(x'Qx + u'Ru, 0, inf) for the discrete-time model `x[k+1] = Ax[k] + Bu[k]`

.

Solve the LQR problem for state-space system `sys`

. Works for both discrete and continuous time systems.

The `args...; kwargs...`

are sent to the Riccati solver, allowing specification of cross-covariance etc. See `?MatrixEquations.arec / ared`

for more help.

To obtain also the solution to the Riccati equation and the eigenvalues of the closed-loop system as well, call `ControlSystemsBase.MatrixEquations.arec / ared`

instead (note the different order of the arguments to these functions).

**Examples**

Continuous time

```
using LinearAlgebra # For identity matrix I
using Plots
A = [0 1; 0 0]
B = [0; 1]
C = [1 0]
sys = ss(A,B,C,0)
Q = I
R = I
L = lqr(sys,Q,R) # lqr(Continuous,A,B,Q,R) can also be used
u(x,t) = -L*x # Form control law,
t=0:0.1:5
x0 = [1,0]
y, t, x, uout = lsim(sys,u,t,x0=x0)
plot(t,x', lab=["Position" "Velocity"], xlabel="Time [s]")
```

Discrete time

```
using LinearAlgebra # For identity matrix I
using Plots
Ts = 0.1
A = [1 Ts; 0 1]
B = [0;1]
C = [1 0]
sys = ss(A, B, C, 0, Ts)
Q = I
R = I
L = lqr(Discrete, A,B,Q,R) # lqr(sys,Q,R) can also be used
u(x,t) = -L*x # Form control law,
t=0:Ts:5
x0 = [1,0]
y, t, x, uout = lsim(sys,u,t,x0=x0)
plot(t,x', lab=["Position" "Velocity"], xlabel="Time [s]")
```

`ControlSystemsBase.place`

— Function```
place(A, B, p, opt=:c)
place(sys::StateSpace, p, opt=:c)
```

Calculate the gain matrix `K`

such that `A - BK`

has eigenvalues `p`

.

```
place(A, C, p, opt=:o)
place(sys::StateSpace, p, opt=:o)
```

Calculate the observer gain matrix `L`

such that `A - LC`

has eigenvalues `p`

.

Uses Ackermann's formula for SISO systems and `place_knvd`

for MIMO systems.

Please note that this function can be numerically sensitive, solving the placement problem in extended precision might be beneficial.

`ControlSystemsBase.place_knvd`

— Method`place_knvd(A::AbstractMatrix, B, λ; verbose = false, init = :s)`

Robust pole placement using the algorithm from

"Robust Pole Assignment in Linear State Feedback", Kautsky, Nichols, Van Dooren

This implementation uses "method 0" for the X-step and the QR factorization for all factorizations.

This function will be called automatically when `place`

is called with a MIMO system.

**Arguments:**

`init`

: Determines the initialization strategy for the iterations for find the`X`

matrix. Possible choices are`:id`

(default),`:rand`

,`:s`

.

`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`

). Note that the forward-Euler method generally requires the sample time to be very small relative to the time constants of the system.

`method = :tustin`

performs a bilinear transform with prewarp frequency `w_prewarp`

.

`w_prewarp`

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

See also `c2d_x0map`

`ControlSystemsBase.c2d_poly2poly`

— Method`c2d_poly2poly(ro, Ts)`

returns the polynomial coefficients in discrete time given polynomial coefficients in continuous time

`ControlSystemsBase.c2d_roots2poly`

— Method`c2d_roots2poly(ro, Ts)`

returns the polynomial coefficients in discrete time given a vector of roots in continuous time

`ControlSystemsBase.c2d_x0map`

— Function`sysd, x0map = c2d_x0map(sys::AbstractStateSpace{<:Continuous}, Ts, method=:zoh; w_prewarp=0)`

Returns the discretization `sysd`

of the system `sys`

and a matrix `x0map`

that transforms the initial conditions to the discrete domain by `x0_discrete = x0map*[x0; u0]`

See `c2d`

for further details.

`ControlSystemsBase.d2c`

— Function`d2c(sys::AbstractStateSpace{<:Discrete}, method::Symbol = :zoh; w_prewarp=0)`

Convert discrete-time system to a continuous time system, assuming that the discrete-time system was discretized using `method`

. Available methods are `:zoh, :fwdeuler´.

`w_prewarp`

: Frequency for pre-warping when usingthe Tustin method, has no effect for other methods.

`ControlSystemsBase.dab`

— Method`X,Y = dab(A,B,C)`

Solves the Diophantine-Aryabhatta-Bezout identity

$AX + BY = C$, where $A, B, C, X$ and $Y$ are polynomials and $deg Y = deg A - 1$.

See Computer-Controlled Systems: Theory and Design, Third Edition Karl Johan Åström, Björn Wittenmark

`ControlSystemsBase.rstc`

— MethodSee `?rstd`

for the discrete case

`ControlSystemsBase.rstd`

— Method```
R,S,T = rstd(BPLUS,BMINUS,A,BM1,AM,AO,AR,AS)
R,S,T = rstd(BPLUS,BMINUS,A,BM1,AM,AO,AR)
R,S,T = rstd(BPLUS,BMINUS,A,BM1,AM,AO)
```

Polynomial synthesis in discrete time.

Polynomial synthesis according to CCS ch 10 to design a controller $R(q) u(k) = T(q) r(k) - S(q) y(k)$

Inputs:

`BPLUS`

: Part of open loop numerator`BMINUS`

: Part of open loop numerator`A`

: Open loop denominator`BM1`

: Additional zeros`AM`

: Closed loop denominator`AO`

: Observer polynomial`AR`

: Pre-specified factor of R,

e.g integral part [1, -1]^k

`AS`

: Pre-specified factor of S,

e.g notch filter [1, 0, w^2]

Outputs: `R,S,T`

: Polynomials in controller

See function `dab`

how the solution to the Diophantine- Aryabhatta-Bezout identity is chosen.

See Computer-Controlled Systems: Theory and Design, Third Edition Karl Johan Åström, Björn Wittenmark

`ControlSystemsBase.zpconv`

— Method`zpc(a,r,b,s)`

form `conv(a,r) + conv(b,s)`

where the lengths of the polynomials are equalized by zero-padding such that the addition can be carried out

`ControlSystemsBase.laglink`

— Method`laglink(a, M; [Ts])`

Returns a phase retarding link, the rule of thumb `a = 0.1ωc`

guarantees less than 6 degrees phase margin loss. The bode curve will go from `M`

, bend down at `a/M`

and level out at 1 for frequencies > `a`

\[\dfrac{s + a}{s + a/M} = M \dfrac{1 + s/a}{1 + sM/a}\]

`ControlSystemsBase.leadlink`

— Function`leadlink(b, N, K=1; [Ts])`

Returns a phase advancing link, the top of the phase curve is located at `ω = b√(N)`

where the link amplification is `K√(N)`

The bode curve will go from `K`

, bend up at `b`

and level out at `KN`

for frequencies > `bN`

The phase advance at `ω = b√(N)`

can be plotted as a function of `N`

with `leadlinkcurve()`

Values of `N < 1`

will give a phase retarding link.

\[KN \dfrac{s + b}{s + bN} = K \dfrac{1 + s/b}{1 + s/(bN)}\]

See also `leadlinkat`

`laglink`

`ControlSystemsBase.leadlinkat`

— Function`leadlinkat(ω, N, K=1; [Ts])`

Returns a phase advancing link, the top of the phase curve is located at `ω`

where the link amplification is `K√(N)`

The bode curve will go from `K`

, bend up at `ω/√(N)`

and level out at `KN`

for frequencies > `ω√(N)`

The phase advance at `ω`

can be plotted as a function of `N`

with `leadlinkcurve()`

Values of `N < 1`

will give a phase retarding link.

See also `leadlink`

`laglink`

`ControlSystemsBase.leadlinkcurve`

— Function`leadlinkcurve(start=1)`

Plot the phase advance as a function of `N`

for a lead link (phase advance link) If an input argument `start`

is given, the curve is plotted from `start`

to 10, else from 1 to 10.

See also `leadlink, leadlinkat`

`ControlSystemsBase.loopshapingPI`

— Method`C, kp, ki, fig, CF = loopshapingPI(P, ωp; ϕl, rl, phasemargin, form=:standard, doplot=false, Tf, F)`

Selects the parameters of a PI-controller (on parallel form) such that the Nyquist curve of `P`

at the frequency `ωp`

is moved to `rl exp(i ϕl)`

The parameters can be returned as one of several common representations chosen by `form`

, the options are

`:standard`

- $K_p(1 + 1/(T_i s) + T_ds)$`:series`

- $K_c(1 + 1/(τ_i s))(τ_d s + 1)$`:parallel`

- $K_p + K_i/s + K_d s$

If `phasemargin`

is supplied (in degrees), `ϕl`

is selected such that the curve is moved to an angle of `phasemargin - 180`

degrees

If no `rl`

is given, the magnitude of the curve at `ωp`

is kept the same and only the phase is affected, the same goes for `ϕl`

if no phasemargin is given.

`Tf`

: An optional time constant for second-order measurement noise filter on the form`tf(1, [Tf^2, 2*Tf/sqrt(2), 1])`

to make the controller strictly proper.`F`

: A pre-designed filter to use instead of the default second-order filter that is used if`Tf`

is given.`doplot`

plot the`gangoffourplot`

and`nyquistplot`

of the system.

See also `loopshapingPID`

, `pidplots`

, `stabregionPID`

and `placePI`

.

`ControlSystemsBase.loopshapingPID`

— Method`C, kp, ki, kd, fig, CF = loopshapingPID(P, ω; Mt = 1.3, ϕt=75, form=:standard, doplot=false, lb=-10, ub=10, Tf = 1/1000ω, F = nothing)`

Selects the parameters of a PID-controller such that the Nyquist curve of the loop-transfer function $L = PC$ at the frequency `ω`

is tangent to the circle where the magnitude of $T = PC / (1+PC)$ equals `Mt`

. `ϕt`

denotes the positive angle in degrees between the real axis and the tangent point.

The default values for `Mt`

and `ϕt`

are chosen to give a good design for processes with inertia, and may need tuning for simpler processes.

The gain of the resulting controller is generally increasing with increasing `ω`

and `Mt`

.

**Arguments:**

`P`

: A SISO plant.`ω`

: The specification frequency.`Mt`

: The magnitude of the complementary sensitivity function at the specification frequency, $|T(iω)|$.`ϕt`

: The positive angle in degrees between the real axis and the tangent point.`doplot`

: If true, gang of four and Nyquist plots will be returned in`fig`

.`lb`

: log10 of lower bound for`kd`

.`ub`

: log10 of upper bound for`kd`

.`Tf`

: Time constant for second-order measurement noise filter on the form`tf(1, [Tf^2, 2*Tf/sqrt(2), 1])`

to make the controller strictly proper. A practical controller typically sets this time constant slower than the default, e.g.,`Tf = 1/100ω`

or`Tf = 1/10ω`

`F`

: A pre-designed filter to use instead of the default second-order filter.

The parameters can be returned as one of several common representations chosen by `form`

, the options are

`:standard`

- $K_p(1 + 1/(T_i s) + T_ds)$`:series`

- $K_c(1 + 1/(τ_i s))(τ_d s + 1)$`:parallel`

- $K_p + K_i/s + K_d s$

See also `loopshapingPI`

, `pidplots`

, `stabregionPID`

and `placePI`

.

**Example:**

```
P = tf(1, [1,0,0]) # A double integrator
Mt = 1.3 # Maximum magnitude of complementary sensitivity
ω = 1 # Frequency at which the specification holds
C, kp, ki, kd, fig, CF = loopshapingPID(P, ω; Mt, ϕt = 75, doplot=true)
```

`ControlSystemsBase.pid`

— Function`C = pid(param_p, param_i, [param_d]; form=:standard, state_space=false, [Tf], [Ts])`

Calculates and returns a PID controller.

The `form`

can be chosen as one of the following

`:standard`

-`Kp*(1 + 1/(Ti*s) + Td*s)`

`:series`

-`Kc*(1 + 1/(τi*s))*(τd*s + 1)`

`:parallel`

-`Kp + Ki/s + Kd*s`

If `state_space`

is set to `true`

, either `kd`

has to be zero or a positive `Tf`

has to be provided for creating a filter on the input to allow for a state space realization. The filter used is `1 / (1 + s*Tf + (s*Tf)^2/2)`

, where `Tf`

can typically be chosen as `Ti/N`

for a PI controller and `Td/N`

for a PID controller, and `N`

is commonly in the range 2 to 20. The state space will be returned on controllable canonical form.

For a discrete controller a positive `Ts`

can be supplied. In this case, the continuous-time controller is discretized using the Tustin method.

**Examples**

```
C1 = pid(3.3, 1, 2) # Kd≠0 works without filter in tf form
C2 = pid(3.3, 1, 2; Tf=0.3, state_space=true) # In statespace a filter is needed
C3 = pid(2., 3, 0; Ts=0.4, state_space=true) # Discrete
```

The functions `pid_tf`

and `pid_ss`

are also exported. They take the same parameters and is what is actually called in `pid`

based on the `state_space`

parameter.

`ControlSystemsBase.pidplots`

— Method`pidplots(P, args...; params_p, params_i, params_d=0, form=:standard, ω=0, grid=false, kwargs...)`

Plots interesting figures related to closing the loop around process `P`

with a PID controller supplied in `params`

on one of the following forms:

`:standard`

-`Kp*(1 + 1/(Ti*s) + Td*s)`

`:series`

-`Kc*(1 + 1/(τi*s))*(τd*s + 1)`

`:parallel`

-`Kp + Ki/s + Kd*s`

The sent in values can be arrays to evaluate multiple different controllers, and if `grid=true`

it will be a grid search over all possible combinations of the values.

Available plots are `:gof`

for Gang of four, `:nyquist`

, `:controller`

for a bode plot of the controller TF and `:pz`

for pole-zero maps and should be supplied as additional arguments to the function.

One can also supply a frequency vector `ω`

to be used in Bode and Nyquist plots.

See also `loopshapingPI`

, `stabregionPID`

`ControlSystemsBase.placePI`

— Method`C, kp, ki = placePI(P, ω₀, ζ; form=:standard)`

Selects the parameters of a PI-controller such that the poles of closed loop between `P`

and `C`

are placed to match the poles of `s^2 + 2ζω₀s + ω₀^2`

.

The parameters can be returned as one of several common representations chose by `form`

, the options are

`:standard`

- $K_p(1 + 1/(T_i s))$`:series`

- $K_c(1 + 1/(τ_i s))$ (equivalent to above for PI controllers)`:parallel`

- $K_p + K_i/s$

`C`

is the returned transfer function of the controller and `params`

is a named tuple containing the parameters. The parameters can be accessed as `params.Kp`

or `params["Kp"]`

from the named tuple, or they can be unpacked using `Kp, Ti, Td = values(params)`

.

See also `loopshapingPI`

`ControlSystemsBase.stabregionPID`

— Function`kp, ki, fig = stabregionPID(P, [ω]; kd=0, doplot=false, form=:standard)`

Segments of the curve generated by this program is the boundary of the stability region for a process with transfer function P(s) The provided derivative gain is expected on parallel form, i.e., the form kp + ki/s + kd s, but the result can be transformed to any form given by the `form`

keyword. The curve is found by analyzing

\[P(s)C(s) = -1 ⟹ \\ |PC| = |P| |C| = 1 \\ arg(P) + arg(C) = -π\]

If `P`

is a function (e.g. s -> exp(-sqrt(s)) ), the stability of feedback loops using PI-controllers can be analyzed for processes with models with arbitrary analytic functions See also `loopshapingPI`

, `loopshapingPID`

, `pidplots`

`ControlSystemsBase.sminreal`

— Method`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.add_input`

— Function`add_input(sys::AbstractStateSpace, B2::AbstractArray, D2 = 0)`

Add inputs to `sys`

by forming

\[x' = Ax + [B \; B_2]u y = Cx + [D \; D_2]u\]

If `B2`

is an integer it will be interpreted as an index and an input matrix containing a single 1 at the specified index will be used.

Example: The following example forms an innovation model that takes innovations as inputs

```
G = ssrand(2,2,3, Ts=1)
K = kalman(G, I(G.nx), I(G.ny))
sys = add_input(G, K)
```

`ControlSystemsBase.add_output`

— Function`add_output(sys::AbstractStateSpace, C2::AbstractArray, D2 = 0)`

Add outputs to `sys`

by forming

\[x' = Ax + Bu y = [C; C_2]x + [D; D_2]u\]

If `C2`

is an integer it will be interpreted as an index and an output matrix containing a single 1 at the specified index will be used.

`ControlSystemsBase.append`

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

Append systems in block diagonal form

`ControlSystemsBase.array2mimo`

— Method`array2mimo(M::AbstractArray{<:LTISystem})`

Take an array of `LTISystem`

s and create a single MIMO system.

`ControlSystemsBase.feedback`

— Method```
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.

*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.feedback`

— Method```
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

`ControlSystemsBase.feedback2dof`

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

- Return
`BT/(AR+ST)`

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

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

`ControlSystemsBase.feedback2dof`

— Method`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.lft`

— Function`lft(G, Δ, type=:l)`

Lower and upper linear fractional transformation between systems `G`

and `Δ`

.

Specify `:l`

lor lower LFT, and `:u`

for upper LFT.

`G`

must have more inputs and outputs than `Δ`

has outputs and inputs.

For details, see Chapter 9.1 in **Zhou, K. and JC Doyle**. Essentials of robust control, Prentice hall (NJ), 1998

`ControlSystemsBase.parallel`

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

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

`ControlSystemsBase.series`

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

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

`ControlSystemsBase.starprod`

— Method`starprod(sys1, sys2, dimu, dimy)`

Compute the Redheffer star product.

`length(U1) = length(Y2) = dimu`

and `length(Y1) = length(U2) = dimy`

For details, see Chapter 9.3 in **Zhou, K. and JC Doyle**. Essentials of robust control, Prentice hall (NJ), 1998

`ControlSystemsBase.G_CS`

— Method`G_CS(P, C)`

The closed-loop transfer function from (-) measurement noise or (+) reference to control signal. Technically, the transfer function is given by `(1 + CP)⁻¹C`

so `SC`

would be a better, but nonstandard name.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.G_PS`

— Method`G_PS(P, C)`

The closed-loop transfer function from load disturbance to plant output. Technically, the transfer function is given by `(1 + PC)⁻¹P`

so `SP`

would be a better, but nonstandard name.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.comp_sensitivity`

— Method```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.extended_gangoffour`

— Function`extended_gangoffour(P, C, pos=true)`

Returns a single statespace system that maps

`w1`

reference or measurement noise`w2`

load disturbance

to

`z1`

control error`z2`

control input

```
z1 z2
▲ ┌─────┐ ▲ ┌─────┐
│ │ │ │ │ │
w1──+─┴─►│ C ├──┴───+─►│ P ├─┐
│ │ │ │ │ │ │
│ └─────┘ │ └─────┘ │
│ w2 │
└────────────────────────────┘
```

The returned system has the transfer-function matrix

\[\begin{bmatrix} I \\ C \end{bmatrix} (I + PC)^{-1} \begin{bmatrix} I & P \end{bmatrix}\]

or in code

```
# For SISO P
S = G[1, 1]
PS = G[1, 2]
CS = G[2, 1]
T = G[2, 2]
# For MIMO P
S = G[1:P.ny, 1:P.nu]
PS = G[1:P.ny, P.ny+1:end]
CS = G[P.ny+1:end, 1:P.ny]
T = G[P.ny+1:end, P.ny+1:end] # Input complimentary sensitivity function
```

The gang of four can be plotted like so

```
Gcl = extended_gangoffour(G, C) # Form closed-loop system
bodeplot(Gcl, lab=["S" "CS" "PS" "T"], plotphase=false) |> display # Plot gang of four
```

Note, the last input of Gcl is the negative of the `PS`

and `T`

transfer functions from `gangoffour2`

. To get a transfer matrix with the same sign as `G_PS`

and `input_comp_sensitivity`

, call `extended_gangoffour(P, C, pos=false)`

. See `glover_mcfarlane`

from RobustAndOptimalControl.jl for an extended example. See also `ncfmargin`

and `feedback_control`

from RobustAndOptimalControl.jl.

`ControlSystemsBase.input_comp_sensitivity`

— Method`input_comp_sensitivity(P,C)`

Transfer function from load disturbance to control signal.

- "Input" signifies that the transfer function is from the input of the plant.
- "Complimentary" signifies that the transfer function is to an output (in this case controller output)

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.input_sensitivity`

— Method`input_sensitivity(P, C)`

Transfer function from load disturbance to total plant input.

- "Input" signifies that the transfer function is from the input of the plant.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.output_comp_sensitivity`

— Method`output_comp_sensitivity(P,C)`

Transfer function from measurement noise / reference to plant output.

- "output" signifies that the transfer function is from the output of the plant.
- "Complimentary" signifies that the transfer function is to an output (in this case plant output)

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.output_sensitivity`

— Method`output_sensitivity(P, C)`

Transfer function from measurement noise / reference to control error.

- "output" signifies that the transfer function is from the output of the plant.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.sensitivity`

— MethodThe output sensitivity function $S_o = (I + PC)^{-1}$ is the transfer function from a reference input to control error, while the input sensitivity function $S_i = (I + CP)^{-1}$ is the transfer function from a disturbance at the plant input to the total plant input. For SISO systems, input and output sensitivity functions are equal. In general, we want to minimize the sensitivity function to improve robustness and performance, but pracitcal constraints always cause the sensitivity function to tend to 1 for high frequencies. A robust design minimizes the peak of the sensitivity function, $M_S$. The peak magnitude of $S$ is the inverse of the distance between the open-loop Nyquist curve and the critical point -1. Upper bounding the sensitivity peak $M_S$ gives lower-bounds on phase and gain margins according to

\[ϕ_m ≥ 2\text{sin}^{-1}(\frac{1}{2M_S}), g_m ≥ \frac{M_S}{M_S-1}\]

Generally, bounding $M_S$ is a better objective than looking at gain and phase margins due to the possibility of combined gain and pahse variations, which may lead to poor robustness despite large gain and pahse margins.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.bodev`

— Method`bodev(sys::LTISystem, w::AbstractVector; $(Expr(:kw, :unwrap, true)))`

For use with SISO systems where it acts the same as `bode`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.bodev`

— Method`bodev(sys::LTISystem; )`

For use with SISO systems where it acts the same as `bode`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.freqrespv`

— Method`freqrespv(G::AbstractMatrix, w_vec::AbstractVector{<:Real}; )`

For use with SISO systems where it acts the same as `freqresp`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.freqrespv`

— Method`freqrespv(G::Number, w_vec::AbstractVector{<:Real}; )`

For use with SISO systems where it acts the same as `freqresp`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.freqrespv`

— Method`freqrespv(sys::LTISystem, w_vec::AbstractVector{W}; )`

For use with SISO systems where it acts the same as `freqresp`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.nyquistv`

— Method`nyquistv(sys::LTISystem, w::AbstractVector; )`

For use with SISO systems where it acts the same as `nyquist`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.nyquistv`

— Method`nyquistv(sys::LTISystem; )`

For use with SISO systems where it acts the same as `nyquist`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.sigmav`

— Method`sigmav(sys::LTISystem, w::AbstractVector; )`

For use with SISO systems where it acts the same as `sigma`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.sigmav`

— Method`sigmav(sys::LTISystem; )`

For use with SISO systems where it acts the same as `sigma`

but with the extra dimensions removed in the returned values.