Apps

Linear State Space

Apps.LinearStateSpace History

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November 17, 2018, at 07:29 AM by 174.148.88.122 -
Added lines 152-153:
Changed line 65 from:

Example Model Predictive Control in GEKKO

to:

Example Model Predictive Control in GEKKO

Changed lines 10-12 from:

Example Model

to:

Example Model in APMonitor

Added lines 64-151:

Example Model Predictive Control in GEKKO

(:source lang=python:) import numpy as np from gekko import GEKKO

A = np.array([[-.003, 0.039, 0, -0.322],

              [-0.065, -0.319, 7.74, 0],
              [0.020, -0.101, -0.429, 0],
              [0, 0, 1, 0]])

B = np.array([[0.01, 1, 2],

              [-0.18, -0.04, 2],
              [-1.16, 0.598, 2],
              [0, 0, 2]]
            )

C = np.array([[1, 0, 0, 0],

              [0, -1, 0, 7.74]])
  1. Build GEKKO State Space model

m = GEKKO() x,y,u = m.state_space(A,B,C,D=None)

  1. customize names
  2. MVs

mv0 = u[0] mv1 = u[1]

  1. Feedforward

ff0 = u[2]

  1. CVs

cv0 = y[0] cv1 = y[1]

m.time = [0, 0.1, 0.2, 0.4, 1, 1.5, 2, 3, 4] m.options.imode = 6 m.options.nodes = 3

u[1].lower = -5 u[1].upper = 5 u[1].dcost = 1 u[1].status = 1

u[1].lower = -5 u[1].upper = 5 u[1].dcost = 1 u[1].status = 1

  1. CV tuning
  1. tau = first order time constant for trajectories

y[0].tau = 5 y[1].tau = 8

  1. tr_init = 0 (dead-band)
  2. = 1 (first order trajectory)
  3. = 2 (first order traj, re-center with each cycle)

y[0].tr_init = 0 y[1].tr_init = 0

  1. targets (dead-band needs upper and lower values)
  2. SPHI = upper set point
  3. SPLO = lower set point

y[0].sphi= -8.5 y[0].splo= -9.5 y[1].sphi= 5.4 y[1].splo= 4.6

y[0].status = 1 y[1].status = 1

  1. feedforward

u[2].status = 0 u[2].value = np.zeros(np.size(m.time)) u[2].value[3:] = 2.5

m.solve() # (GUI=True)

  1. also create a Python plot

import matplotlib.pyplot as plt

plt.subplot(2,1,1) plt.plot(m.time,mv0.value,'r-',label=r'$u_0$ as MV') plt.plot(m.time,mv1.value,'b--',label=r'$u_1$ as MV') plt.plot(m.time,ff0.value,'g:',label=r'$u_2$ as feedforward') plt.subplot(2,1,2) plt.plot(m.time,cv0.value,'r-',label=r'$y_0$') plt.plot(m.time,cv1.value,'b--',label=r'$y_1$') plt.show() (:sourceend:)

Changed line 32 from:
 File *.mpc.txt
to:
 File mpc.txt
Changed line 40 from:
 File *.mpc.a.txt
to:
 File mpc.a.txt
Changed line 47 from:
 File *.mpc.b.txt
to:
 File mpc.b.txt
Changed line 55 from:
 File *.mpc.c.txt
to:
 File mpc.c.txt
Changed line 62 from:
 File *.mpc.d.txt
to:
 File mpc.d.txt
Changed lines 14-19 from:

Model control

  Objects
    mpc = lti
  End Objects

End Model

to:
 Model control
   Objects
     mpc = lti
   End Objects
 End Model
Changed lines 32-38 from:

File *.mpc.txt

  sparse, continuous  ! dense/sparse, continuous/discrete
  2      ! m=number of inputs
  3      ! n=number of states
  3      ! p=number of outputs

End File

to:
 File *.mpc.txt
   sparse, continuous  ! dense/sparse, continuous/discrete
   2      ! m=number of inputs
   3      ! n=number of states
   3      ! p=number of outputs
 End File
Changed lines 40-45 from:

File *.mpc.a.txt

  1  1  0.9
  2  2  0.1
  3  3  0.5

End File

to:
 File *.mpc.a.txt
   1  1  0.9
   2  2  0.1
   3  3  0.5
 End File
Changed lines 47-53 from:

File *.mpc.b.txt

  1  1  1.0
  2  2  1.0
  3  1  0.5
  3  2  0.5

End File

to:
 File *.mpc.b.txt
   1  1  1.0
   2  2  1.0
   3  1  0.5
   3  2  0.5
 End File
Changed lines 55-60 from:

File *.mpc.c.txt

  1  1  0.5
  2  2  1.0
  3  3  2.0

End File

to:
 File *.mpc.c.txt
   1  1  0.5
   2  2  1.0
   3  3  2.0
 End File
Changed lines 62-64 from:

File *.mpc.d.txt

  1  1  0.2

End File

to:
 File *.mpc.d.txt
   1  1  0.2
 End File
Changed lines 10-13 from:

Example Model

new linear time-invariant object

to:

Example Model

 ! new linear time-invariant object
Changed lines 20-31 from:

Model information

continuous form

dx/dt = A * x + B * u

y = C * x + D * u

dimensions

(nx1) = (nxn)*(nx1) + (nxm)*(mx1)

(px1) = (pxn)*(nx1) + (pxm)*(mx1)

discrete form

x[k+1] = A * x[k] + B * u[k]

y[k] = C * x[k] + D * u[k]

to:
 ! Model information
 ! continuous form
 ! dx/dt = A * x + B * u
 !     y = C * x + D * u
 !
 ! dimensions
 ! (nx1) = (nxn)*(nx1) + (nxm)*(mx1)
 ! (px1) = (pxn)*(nx1) + (pxm)*(mx1)
 !
 ! discrete form
 ! x[k+1] = A * x[k] + B * u[k]
 !   y[k] = C * x[k] + D * u[k]
Changed line 39 from:

A matrix (row, column, value)

to:
 ! A matrix (row, column, value)
Changed line 46 from:

B matrix (row, column, value)

to:
 ! B matrix (row, column, value)
Changed line 54 from:

C matrix (row, column, value)

to:
 ! C matrix (row, column, value)
Changed line 61 from:

D matrix (row, column, value)

to:
 ! D matrix (row, column, value)
Changed lines 8-64 from:
to:

Example Model

new linear time-invariant object

Model control

  Objects
    mpc = lti
  End Objects

End Model

Model information

continuous form

dx/dt = A * x + B * u

y = C * x + D * u

dimensions

(nx1) = (nxn)*(nx1) + (nxm)*(mx1)

(px1) = (pxn)*(nx1) + (pxm)*(mx1)

discrete form

x[k+1] = A * x[k] + B * u[k]

y[k] = C * x[k] + D * u[k]

File *.mpc.txt

  sparse, continuous  ! dense/sparse, continuous/discrete
  2      ! m=number of inputs
  3      ! n=number of states
  3      ! p=number of outputs

End File

A matrix (row, column, value)

File *.mpc.a.txt

  1  1  0.9
  2  2  0.1
  3  3  0.5

End File

B matrix (row, column, value)

File *.mpc.b.txt

  1  1  1.0
  2  2  1.0
  3  1  0.5
  3  2  0.5

End File

C matrix (row, column, value)

File *.mpc.c.txt

  1  1  0.5
  2  2  1.0
  3  3  2.0

End File

D matrix (row, column, value)

File *.mpc.d.txt

  1  1  0.2

End File

November 04, 2008, at 07:54 PM by 158.35.225.231 -
Changed line 6 from:

These models are typically in the finite impulse response form or linear state space form. Either model form can be converted to an APMonitor for a linear MPC upgrade. Once the linear MPC model is converted, nonlinear elements can be added to avoid multiple model switching, gain scheduling, or other ad hoc measures commonly employed because of linear MPC restrictions.

to:

These models are typically in the finite impulse response form or linear state space form. Either model form can be converted to an APMonitor for a linear MPC upgrade. Once in APMonitor form, nonlinear elements can be added to avoid multiple model switching, gain scheduling, or other ad hoc measures commonly employed because of linear MPC restrictions.

November 04, 2008, at 07:54 PM by 158.35.225.231 -
Changed line 6 from:

These models are typically in the finite impulse response form or linear state space form. Either model form can be converted to an APMonitor for a linear MPC upgrade. Once the linear MPC model is converted, nonlinear elements can be added to avoid multiple model switching, gain scheduling, or other ad hoc measures commonly employed because of linear MPC shortcomings.

to:

These models are typically in the finite impulse response form or linear state space form. Either model form can be converted to an APMonitor for a linear MPC upgrade. Once the linear MPC model is converted, nonlinear elements can be added to avoid multiple model switching, gain scheduling, or other ad hoc measures commonly employed because of linear MPC restrictions.

November 04, 2008, at 07:54 PM by 158.35.225.231 -
Changed lines 4-6 from:

Linear model predictive controllers are based on models in the finite impulse response form or linear state space form. Either model form can be converted to a form that APMonitor uses for estimation and control.

to:

Model Predictive Control, or MPC, is an advanced method of process control that has been in use in the process industries such as chemical plants and oil refineries since the 1980s. Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification.

These models are typically in the finite impulse response form or linear state space form. Either model form can be converted to an APMonitor for a linear MPC upgrade. Once the linear MPC model is converted, nonlinear elements can be added to avoid multiple model switching, gain scheduling, or other ad hoc measures commonly employed because of linear MPC shortcomings.

November 04, 2008, at 07:38 PM by 158.35.225.231 -
Changed lines 1-2 from:

Linear State Space Model

to:

Linear Model Predictive Control

Added lines 3-4:

Linear model predictive controllers are based on models in the finite impulse response form or linear state space form. Either model form can be converted to a form that APMonitor uses for estimation and control.

November 04, 2008, at 04:06 PM by 158.35.225.231 -
Changed line 5 from:
to:
November 04, 2008, at 03:49 AM by 98.199.241.177 -
Added lines 1-3:

Linear State Space Model