Apps

State Space Model Object

Apps.LinearStateSpace History

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Also see [[Apps/DiscreteStateSpace|Discrete State Space]] and [[https://apmonitor.com/pdc/index.php/Main/StateSpaceModel|State Space Introduction]]
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Also see:

*
[[https://apmonitor.com/pdc/index.php/Main/StateSpaceModel|State Space Introduction]]
* [[Apps/DiscreteStateSpace|Discrete State Space]]
* [[Apps/ARXTimeSeries | ARX Time Series
]]
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(:title State Space Models:)
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(:title State Space Model Object:)
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(:title APMonitor and GEKKO State Space Models:)
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(:title State Space Models:)
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Also see [[Apps/DiscreteStateSpace|Discrete State Space]]
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Also see [[Apps/DiscreteStateSpace|Discrete State Space]] and [[https://apmonitor.com/pdc/index.php/Main/StateSpaceModel|State Space Introduction]]
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!! Linear Model Predictive Control
* %list list-page% [[Attach:fir.apm | Linear State Space]]

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(:title APMonitor and GEKKO State Space Models:)
(:keywords linear, state space, stability, dynamic, multiple input, multiple output, MIMO, model predictive control:)
(:description Linear Time Invariant (LTI) state space models are a linear representation of a dynamic system in either discrete or continuous time. Examples show how to use continuous LTI state space models in APMonitor and GEKKO.:)

%width=50px%Attach:apm.png [[Main/Objects|APMonitor Objects]]

 Type: Object
 Data: A, B, C, and D matrices
 Inputs: Input (u)
 Outputs: States (x), Output (y)
 Description: LTI State Space Model
 APMonitor Usage: sys = lti
 GEKKO Usage: x,y,u = m.state_space(A,B,C,D=None)

Linear Time Invariant (LTI) state space models are a linear representation of a dynamic system in either discrete or continuous time. Putting a model into state space form is the basis for many methods in process dynamics and control analysis. Below is the continuous time form of a model in state space form.

{$\dot x = A x + B u$}

{$y = C x + D u$}

with states {`x in \mathbb{R}^n`} and state derivatives {`\dot x = {dx}/{dt} in \mathbb{R}^n`}. The notation {`in \mathbb{R}^n`} means that {`x`} and {`\dot x`} are in the set of real-numbered vectors of length {`n`}. The other elements are the outputs {`y in \mathbb{R}^p`}, the inputs {`u in \mathbb{R}^m`}, the state transition matrix {`A`}, the input matrix {`B`}, and the output matrix {`C`}. The remaining matrix {`D`} is typically zeros because the inputs do not typically affect the outputs directly. The dimensions of each matrix are shown below with {`m`} inputs, {`n`} states, and {`p`} outputs.

{$A \in \mathbb{R}^{n \, \mathrm{x} \, n}$}
{$B \in \mathbb{R}^{n \, \mathrm{x} \, m}$}
{$C \in \mathbb{R}^{p \, \mathrm{x} \, n}$}
{$D \in \mathbb{R}^{p \, \mathrm{x} \, m}$}

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* %list list-page% [[Attach:fir.apm | Linear State Space]]
November 17, 2018, at 07:29 AM by 174.148.88.122 -
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Also see [[Apps/DiscreteStateSpace|Discrete State Space]]
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!!!! Example Model Predictive Control in GEKKO
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!!!! Example Model Predictive Control in [[https://gekko.readthedocs.io/en/latest/|GEKKO]]
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!! Example Model

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!!!! Example Model in APMonitor
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!!!! 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]])

#%% Build GEKKO State Space model
m = GEKKO()
x,y,u = m.state_space(A,B,C,D=None)

# customize names
# MVs
mv0 = u[0]
mv1 = u[1]
# Feedforward
ff0 = u[2]
# 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

## CV tuning
# tau = first order time constant for trajectories
y[0].tau = 5
y[1].tau = 8
# tr_init = 0 (dead-band)
#        = 1 (first order trajectory)
#        = 2 (first order traj, re-center with each cycle)
y[0].tr_init = 0
y[1].tr_init = 0
# targets (dead-band needs upper and lower values)
# SPHI = upper set point
# 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

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

m.solve() # (GUI=True)

# 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:)
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 File *.mpc.txt
to:
 File mpc.txt
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 File *.mpc.a.txt
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 File mpc.a.txt
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 File *.mpc.b.txt
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 File mpc.b.txt
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 File *.mpc.c.txt
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 File mpc.c.txt
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 File *.mpc.d.txt
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 File mpc.d.txt
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Model control
  Objects
    mpc = lti
  End Objects
End Model
to:
 Model control
   Objects
     mpc = lti
   End Objects
 End Model
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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
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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
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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
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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
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File *.mpc.d.txt
  1  1  0.2
End File
to:
 File *.mpc.d.txt
   1  1  0.2
 End File
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!!! Example Model


! new linear time-invariant object
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!! Example Model


 ! new linear time-invariant object
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! 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]
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! A matrix (row, column, value)
to:
 ! A matrix (row, column, value)
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! B matrix (row, column, value)
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 ! B matrix (row, column, value)
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! C matrix (row, column, value)
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 ! C matrix (row, column, value)
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! D matrix (row, column, value)
to:
 ! D matrix (row, column, value)
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Attach:lti_step_response.png
to:
Attach:lti_step_response.png

!!! 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 -
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These models are typically in the finite impulse response form or linear state space form.  Either model form can be converted to an %blue%A%red%P%black%Monitor 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 %blue%A%red%P%black%Monitor for a linear MPC upgrade.  Once in %blue%A%red%P%black%Monitor 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 %blue%A%red%P%black%Monitor 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 %blue%A%red%P%black%Monitor 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 -
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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 %blue%A%red%P%black%Monitor uses for estimation and control.
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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 %blue%A%red%P%black%Monitor 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 -
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!! Linear State Space Model
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!! Linear Model Predictive Control
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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 %blue%A%red%P%black%Monitor uses for estimation and control.
November 04, 2008, at 04:06 PM by 158.35.225.231 -
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Attach:linear_ss.png
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Attach:lti_step_response.png
November 04, 2008, at 04:06 PM by 158.35.225.231 -
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* %list list-page% [[Attach:fir.apm | Linear State Space]]
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* %list list-page% [[Attach:fir.apm | Linear State Space]]

Attach:linear_ss.png
November 04, 2008, at 03:49 AM by 98.199.241.177 -
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!! Linear State Space Model

* %list list-page% [[Attach:fir.apm | Linear State Space]]