ModelingToolkitStandardLibrary: Blocks

RealInput(;name, guess)

Connector with one input signal of type Real.

Parameters:

• guess=0: Guess value for u.

States:

• u: Value of the connector which is a scalar.

RealOutput(;name, guess)

Connector with one output signal of type Real.

Parameters:

• guess=0: Guess value for u.

States:

• u: Value of the connector which is a scalar.

RealInputArray(;name, nin, guess)

Connector with an array of input signals of type Real.

Parameters:

• nin: Number of inputs.
• guess=zeros(nin): Guess value for u.

States:

• u: Value of the connector which is an array.

RealOutputArray(;name, nout, guess)

Connector with an array of output signals of type Real.

Parameters:

• nout: Number of outputs.
• guess=zeros(nout): Guess value for u.

States:

• u: Value of the connector which is an array.

SISO(;name, u_start = 0.0, y_start = 0.0)

Single input single output (SISO) continuous system block.

Parameters:

• u_start: Initial value for the input
• y_start: Initial value for the output

MIMO(; name, nin = 1, nout = 1, u_start = zeros(nin), y_start = zeros(nout))

Base class for a multiple input multiple output (MIMO) continuous system block.

Parameters:

• nin: Input dimension
• nout: Output dimension
• u_start: Initial value for the input
• y_start: Initial value for the output

Gain(; name, k)

Output the product of a gain value with the input signal.

Parameters:

• k: Scalar gain

Connectors:

• input
• output

MatrixGain(; K::AbstractArray, name)

Output the product of a gain matrix with the input signal vector.

Structural parameters:

• K: Matrix gain

Connectors:

• input
• output

Sum(; input__nin::Int, name)

Output the sum of the elements of the input port vector. Input port dimension can be set with input__nin

Connectors:

• input
• output

Feedback(; name)

Output difference between reference input (input1) and feedback input (input2).

Connectors:

• input1
• input2
• output

Add(; name, k1 = 1.0, k2 = 1.0)

Output the sum of the two scalar inputs.

Parameters:

• k1: Gain for first input
• k2: Gain for second input

Connectors:

• input1
• input2
• output

Add(; name, k1 = 1.0, k2 = 1.0, k3 = 1.0)

Output the sum of the three scalar inputs.

Parameters:

• k1: Gain for first input
• k2: Gain for second input
• k3: Gain for third input

Connectors:

• input1
• input2
• input3
• output

Product(; name)

Output product of the two inputs.

Connectors:

• input1
• input2
• output

Division(; name)

Output first input divided by second input.

Connectors:

• input1
• input2
• output

UnaryMinus(; name)

Output the product of -1 and the input.

Connectors:

• input
• output

Power(; name)

Output the exponential with base as the first input and exponent as second input i.e u1^u2

Connectors:

• base
• exponent
• output

Modulo(; name)

Output the remainder when the first input is divided by second input.

Connectors:

• dividend
• divisor
• remainder

Floor(; name)

Output the floor rounding of the input.

Connectors:

• input
• output

Ceil(; name)

Output the ceiling rounding of the input.

Connectors:

• input
• output

StaticNonLinearity(func; name)

Applies the given function to the input.

If the given function is not composed of simple core methods (e.g. sin, abs, ...), it has to be registered via @register_symbolic func(u)

Connectors:

• input
• output

Abs(; name)

Output the absolute value of the input.

Connectors:

See StaticNonLinearity

Sign(; name)

Output the sign of the input

Connectors:

See StaticNonLinearity

Sqrt(; name)

Output the square root of the input (input >= 0 required).

Connectors:

See StaticNonLinearity

Sin(; name)

Output the sine of the input.

Connectors:

See StaticNonLinearity

Cos(; name)

Output the cosine of the input.

Connectors:

See StaticNonLinearity

Tan(; name)

Output the tangent of the input.

Connectors:

See StaticNonLinearity

Asin(; name)

Output the arc sine of the input.

Connectors:

See StaticNonLinearity

Acos(; name)

Output the arc cosine of the input.

Connectors:

See StaticNonLinearity

Atan(; name)

Output the arc tangent of the input.

Connectors:

See StaticNonLinearity

Atan2(; name)

Output the arc tangent of the input.

Connectors:

• input1
• input2
• output

Sinh(; name)

Output the hyperbolic sine of the input.

Connectors:

See StaticNonLinearity

Cosh(; name)

Output the hyperbolic cosine of the input.

Connectors:

See StaticNonLinearity

Tanh(; name)

Output the hyperbolic tangent of the input.

Connectors:

See StaticNonLinearity

Exp(; name)

Output the exponential (base e) of the input.

Connectors:

See StaticNonLinearity

Log(; name)

Output the natural (base e) logarithm of the input.

Connectors:

See StaticNonLinearity

Log10(; name)

Output the base 10 logarithm of the input.

Connectors:

See StaticNonLinearity

Constant(; name, k = 0.0)

Generate constant signal.

Parameters:

• k: Constant output value

Connectors:

• output

Sine(; name, frequency, amplitude = 1, phase = 0, offset = 0, start_time = 0,
smooth = false)

Generate sine signal.

Parameters:

• frequency: [Hz] Frequency of sine wave
• amplitude: Amplitude of sine wave
• phase: [rad] Phase of sine wave
• offset: Offset of output signal
• start_time: [s] Output y = offset for t < start_time
• smooth: If true, returns a smooth wave. Defaults to false It uses a default smoothing factor of δ=1e-5, but this can be changed by supplying smooth=δ.

Connectors:

• output

Cosine(; name, frequency, amplitude = 1, phase = 0, offset = 0, start_time = 0,
smooth = false)

Generate cosine signal.

Parameters:

• frequency: [Hz] Frequency of cosine wave
• amplitude: Amplitude of cosine wave
• phase: [rad] Phase of cosine wave
• offset: Offset of output signal
• start_time: [s] Output y = offset for t < start_time
• smooth: If true, returns a smooth wave. Defaults to false It uses a default smoothing factor of δ=1e-5, but this can be changed by supplying smooth=δ.

Connectors:

• output

ContinuousClock(; name, offset = 0, start_time = 0)

Generate current time signal.

Parameters:

• offset: Offset of output signal
• start_time: [s] Output y = offset for t < start_time

Connectors:

• output

Ramp(; name, height = 1, duration = 1, offset = 0, start_time = 0, smooth = false)

Generate ramp signal.

Parameters:

• height: Height of ramp
• duration: [s] Duration of ramp (= 0.0 gives a Step)
• offset: Offset of output signal
• start_time: [s] Output y = offset for t < start_time
• smooth: If true, returns a smooth wave. Defaults to false It uses a default smoothing factor of δ=1e-5, but this can be changed by supplying smooth=δ.

Connectors:

• output

Step(;name, height=1, offset=0, start_time=0, duration=Inf, smooth=true)

Generate step signal.

Parameters:

• height: Height of step
• offset: Offset of output signal
• start_time: [s] Output y = offset for t < start_time and thereafter offset+height.
• duration: [s] If duration < Inf is supplied, the output will revert to offset after duration seconds.
• smooth: If true, returns a smooth wave. Defaults to true It uses a default smoothing factor of δ=1e-5, but this can be changed by supplying smooth=δ.

Connectors:

• output

ExpSine(; name, frequency, amplitude = 1, damping = 0.1, phase = 0, offset = 0, start_time = 0, smooth = false)

Exponentially damped sine signal.

Parameters:

• frequency: [Hz] Frequency of sine wave
• amplitude: Amplitude of sine wave
• damping: [1/s] Damping coefficient of sine wave
• phase: [rad] Phase of sine wave
• offset: Offset of output signal
• start_time: [s] Output y = offset for t < start_time
• smooth: If true, returns a smooth wave. Defaults to false It uses a default smoothing factor of δ=1e-5, but this can be changed by supplying smooth=δ.

Connectors:

• output

Limiter(;name, y_max, y_min = y_max > 0 ? -y_max : -Inf)

Limit the range of a signal.

Parameters:

• y_max: Maximum of output signal
• y_min: Minimum of output signal

Connectors:

• input
• output

DeadZone(; name, u_max, u_min = -u_max)

The DeadZone block defines a region of zero output. If the input is within u_min ... u_max, the output is zero. Outside of this zone, the output is a linear function of the input with a slope of 1.

       y▲
        │     /
        │    /
  u_min │   /
─────|──┼──|───────► u
    /   │   u_max
   /    │
  /     │

Parameters:

• u_max: Upper limit of dead zone
• u_min: Lower limit of dead zone

Connectors:

• input
• output

SlewRateLimiter(; name, y_start, rising = 1.0, falling = -rising, Td = 0.001)

Limits the slew rate of a signal. Initial value of state Y can be set with int.y

Parameters:

• rising: Maximum rising slew rate
• falling: Maximum falling slew rate
• Td: [s] Derivative time constant
• y_start: Initial value of y state of SISO

Connectors:

• input
• output

Integrator(;name, k = 1, x = 0.0)

Outputs y = ∫k*u dt, corresponding to the transfer function $1/s$. Initial value of integrator state $x$ can be set with x

Connectors:

• input
• output

Parameters:

• k: Gain of integrator

Unknowns:

• x: State of Integrator. Defaults to 0.0.

Derivative(; name, k = 1, T, x = 0.0)

Outputs an approximate derivative of the input. The transfer function of this block is

k      k        ks  
─ - ─────── = ────── 
T   sT² + T   sT + 1

and a state-space realization is given by ss(-1/T, 1/T, -k/T, k/T) where T is the time constant of the filter. A smaller T leads to a more ideal approximation of the derivative.

Initial value of the state $x$ can be set with x.

Parameters:

• k: Gain
• T: [s] Time constant (T>0 required; T=0 is ideal derivative block)

Unknowns:

• x: Unknown of Derivative. Defaults to 0.0.

Connectors:

• input
• output

FirstOrder(; name, k = 1.0, T, x = 0.0, lowpass = true)

A first-order filter with a single real pole at s = -1/T and gain k. If lowpass=true (default), the transfer function is given by $Y(s)/U(s) =$

   k
───────
sT + 1

and if lowpass=false, by

sT + 1 - k
──────────
  sT + 1

Initial value of the state x can be set with x

Parameters:

• k: Gain
• T: [s] Time constant (T>0 required)

Connectors:

• input
• output

See also SecondOrder

SecondOrder(; name, k = 1.0, w = 1.0, d = 1.0, x = 0.0, xd = 0.0)

A second-order filter with gain k, a bandwidth of w rad/s and relative damping d. The transfer function is given by Y(s)/U(s) =

      k*w^2
─────────────────
s² + 2d*w*s + w^2

Critical damping corresponds to d=1, which yields the fastest step response without overshoot, d < 1 results in an underdamped filter while d > 1 results in an overdamped filter. d = 1/√2 corresponds to a Butterworth filter of order 2 (maximally flat frequency response). Initial value of the state x can be set with x, and of derivative state xd with xd.

Parameters:

• k: Gain
• w: [rad/s] Angular frequency
• d: Damping

Connectors:

• input
• output

StateSpace(A, B, C, D = 0; x = zeros(size(A,1)), u0 = zeros(size(B,2)), y0 = zeros(size(C,1)), name)

A linear, time-invariant state-space system on the form.

\[\begin{aligned} ẋ &= Ax + Bu \\ y &= Cx + Du \end{aligned}\]

Transfer functions can also be simulated by converting them to a StateSpace form.

y0 and u0 can be used to set an operating point, providing them changes the dynamics from an LTI system to the affine system

\[\begin{aligned} ẋ &= Ax + B(u - u0) \\ y &= Cx + D(u - u0) + y0 \end{aligned}\]

For a nonlinear system

\[\begin{aligned} ẋ &= f(x, u) \\ y &= h(x, u) \end{aligned}\]

linearized around the operating point x₀, u₀, we have y0, u0 = h(x₀, u₀), u₀.

TransferFunction(; b, a, name)

A single input, single output, linear time-invariant system provided as a transfer-function.

Y(s) = b(s) / a(s)  U(s)

where b and a are vectors of coefficients of the numerator and denominator polynomials, respectively, ordered such that the coefficient of the highest power of s is first.

The internal state realization is on controller canonical form, with state variable x, output variable y and input variable u. For numerical robustness, the realization used by the integrator is scaled by the last entry of the a parameter. The internally scaled state variable is available as x_scaled.

To set the initial state, it's recommended to set the initial condition for x, and let that of x_scaled be computed automatically.

Parameters:

• b: Numerator polynomial coefficients, e.g., 2s + 3 is specified as [2, 3]
• a: Denominator polynomial coefficients, e.g., s² + 2ωs + ω^2 is specified as [1, 2ω, ω^2]

Connectors:

• input
• output

See also StateSpace which handles MIMO systems, as well as ControlSystemsMTK.jl for an interface between ControlSystems.jl and ModelingToolkit.jl for advanced manipulation of transfer functions and linear statespace systems. For linearization, see linearize and Linear Analysis.

PI(;name, k = 1.0, T = 1.0, int.x = 0.0)

Textbook version of a PI-controller without actuator saturation and anti-windup measure. The proportional gain can be set with k Initial value of integrator state x can be set with int.x

The PI controller is implemented on standard form:

\[U(s) = k (1 + \dfrac{1}{sT}) E(S)\]

Parameters:

• k: Proportional gain
• T: [s] Integrator time constant (T>0 required)

Connectors:

• err_input
• ctr_output

See also LimPI

LimPI(; name, k = 1.0, T, Ta, int__x = 0.0, u_max = 1.0, u_min = -u_max)

Text-book version of a PI-controller with actuator saturation and anti-windup measure.

The PI controller is implemented on standard form

\[u(t) = sat(k (e(t) + ∫\dfrac{1}{T}e(t) dt) )\]

The simplified expression above is given without the anti-windup protection.

Parameters:

• k: Proportional gain
• T: [s] Integrator time constant (T>0 required)
• Ta: [s] Tracking time constant (Ta>0 required)

Connectors:

• err_input
• ctr_output

PID(;name, k=1, Ti=false, Td=false, Nd=10, int__x=0, der__x=0)

Text-book version of a PID-controller without actuator saturation and anti-windup measure.

Parameters:

• k: Gain
• Ti: [s] Integrator time constant (Ti>0 required). If set to false, no integral action is used.
• Td: [s] Derivative time constant (Td>0 required). If set to false, no derivative action is used.
• Nd: [s] Time constant for the derivative approximation (Nd>0 required; Nd=0 is ideal derivative).
• int__x: Initial value for the integrator.
• der__x: Initial value for the derivative state.

Connectors:

• err_input
• ctr_output

See also LimPID

LimPID(; k, Ti=false, Td=false, wp=1, wd=1, Ni, Nd=12, u_max=Inf, u_min=-u_max, gains = false, name)

Proportional-Integral-Derivative (PID) controller with output saturation, set-point weighting and integrator anti-windup.

The equation for the control signal is roughly

k(ep + 1/Ti * ∫e + Td * d/dt(ed))
e = u_r - u_y
ep = wp*u_r - u_y
ed = wd*u_r - u_y

where the transfer function for the derivative includes additional filtering, see ? Derivative for more details.

Parameters:

• k: Proportional gain
• Ti: [s] Integrator time constant. Set to false to turn off integral action.
• Td: [s] Derivative time constant. Set to false to turn off derivative action.
• wp: [0,1] Set-point weighting in the proportional part.
• wd: [0,1] Set-point weighting in the derivative part.
• Nd: [1/s] Derivative limit, limits the derivative gain to Nd/Td. Reasonable values are ∈ [8, 20]. A higher value gives a better approximation of an ideal derivative at the expense of higher noise amplification.
• Ni: Ni*Ti controls the time constant Ta of anti-windup tracking. A common (default) choice is Ta = √(Ti*Td) which is realized by Ni = √(Td / Ti). Anti-windup can be effectively turned off by setting Ni = Inf.
• gains: If gains = true, Ti and Td will be interpreted as gains with a fundamental PID transfer function on parallel form ki=Ti, kd=Td, k + ki/s + kd*s.

Connectors:

• reference
• measurement
• ctr_output