The addResponses Method

In this example, the objective is to find a design such that the RMS values of displacement, velocity, and acceleration are minimized. The RMS signals are computed as:(1)

$$\begin{array}{l}R1={\displaystyle \underset{0}{\overset{T}{\int }}{x}^{2}dt}\\ R2={\displaystyle \underset{0}{\overset{T}{\int }}{\dot{x}}^{2}dt}\\ R3={\displaystyle \underset{0}{\overset{T}{\int }}{\ddot{x}}^{2}dt}\end{array}$$

We also create a constraint (based on response type ResponseExpression) to make sure that the gain of PID controller would not increase infinitely. The constraint has the form:(2)

$$\phi \triangleq \text{1}-k_{P:}^{\text{2}}-k_I^{\text{2}}-k_D^{\text{2}}<0$$

  def addResponse(self, time):
    """
    We minimize the influence of disturbance on the block
    R1: Integral(acceleration**2,time)
    R2: Integral(velocity**2,time)
    R3: Integral(Displacement**2,time)
    R4: Constraint (1 - kp**2 - ki**2 - kd**2 < 0)
    """
    # Define the desired profile
    zero_vector = [0.0 for i in time]
    targetValue = zip(time,zero_vector)

    # Measure the RMS acceleration deviation from desired
    expr = 'ACCZ({I},{J})'.format(I=self.block.cm.id, J=self.rm.id)
    self.R1 = RMS2 (
      label         ='acceleration',
      targetValue   = targetValue, 
      measuredValue = expr,
      )

    # Measure the RMS velocity deviation from desired
    expr = 'VZ({I},{J})'.format(I=self.block.cm.id,J=self.rm.id)
    self.R2 = RMS2 (
      label         = 'velocity',
      targetValue   = targetValue,
      measuredValue = expr,
      )

    # Measure the RMS displacement deviation from desired
    expr = 'DZ({I},{J})'.format(I=self.block.cm.id,J=self.rm.id)
    self.R3 = RMS2 (
      label         = 'displacement',
      targetValue   = targetValue, 
      measuredValue = expr,
      )

    # The constraint
    self.R4 = ResponseExpression(
      label='sum of squares',
      function = '1 - kp**2 - kd**2 - ki**2',
      symbols = ["kp", "kd", "ki"],
      variables = [self.dv_kp, self.dv_kd, self.dv_ki]
      )