Apps
~~!! Linear Model Predictive Control~~

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

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

Also see [[Apps/DiscreteStateSpace|Discrete State Space]]

!!!! 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:)
~~!~~ 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]
~~Attach:lti_step_response.png~~
~~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~~.

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.

## State Space Model Object

## Apps.LinearStateSpace History

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GEKKO Usage: x,y,u = m.state_space(A,B,C)

~~Optional~~ arguments: ~~D~~=None,~~E=None, ~~discrete=False,dense=False

to:

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

Other optional arguments: E=None,discrete=False,dense=False

Other optional arguments: E=None,discrete=False,dense=False

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GEKKO Usage: x,y,u = m.state_space(A,B,C~~,~~D=None,E=None,~~\~~

discrete=False,dense=~~False)~~

to:

GEKKO Usage: x,y,u = m.state_space(A,B,C)

Optional arguments: D=None,E=None, discrete=False,dense=False

Optional arguments: D=None,E=None, discrete=False,dense=False

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Data: A, B, C, ~~and ~~D matrices

to:

Data: A, B, C, D, and E matrices

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

to:

GEKKO Usage: x,y,u = m.state_space(A,B,C,D=None,E=None,\

discrete=False,dense=False)

discrete=False,dense=False)

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

to:

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

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{$E \in \mathbb{R}^{n \, \mathrm{x} \, n}$}

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

to:

Also see:

* [[https://apmonitor.com/pdc/index.php/Main/StateSpaceModel|State Space Introduction]]

* [[Apps/DiscreteStateSpace|Discrete State Space]]

* [[Apps/ARXTimeSeries | ARX Time Series]]

* [[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~~:)

to:

(:title State Space Model Object:)

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(:title~~ APMonitor and GEKKO~~ State Space Models:)

to:

(:title State Space Models:)

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Also see [[Apps/DiscreteStateSpace|Discrete State Space]]

to:

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

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

to:

(: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.

{$A \in \mathbb{R}^{n \, \mathrm{x} \, n}$}

{$B \in \mathbb{R}^{n \, \mathrm{x} \, m}$}

{$C \in \mathbb{R}^{p \, \mathrm{x} \, n}$}

{$D \in \mathbb{R}^{p \, \mathrm{x} \, m}$}

(: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]]

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Also see [[Apps/DiscreteStateSpace|Discrete State Space]]

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!!!! Example Model Predictive Control in ~~GEKKO~~

to:

!!!! Example Model Predictive Control in [[https://gekko.readthedocs.io/en/latest/|GEKKO]]

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!! Example Model

to:

!!!! 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

to:

File mpc.a.txt

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File ~~*.~~mpc.b.txt

to:

File mpc.b.txt

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File ~~*.~~mpc.c.txt

to:

File mpc.c.txt

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File ~~*.~~mpc.d.txt

to:

File mpc.d.txt

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Model control

Objects

mpc = lti

End Objects

End Model

Objects

mpc = lti

End Objects

End Model

to:

Model control

Objects

mpc = lti

End Objects

End Model

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

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

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

1 1 0.9

2 2 0.1

3 3 0.5

End File

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File *.mpc.a.txt

1 1 0.9

2 2 0.1

3 3 0.5

End File

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

1 1 1.0

2 2 1.0

3 1 0.5

3 2 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

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

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2 2 1.0

3 3 2.0

End File

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File *.mpc.c.txt

1 1 0.5

2 2 1.0

3 3 2.0

End File

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

1 1 0.2

End File

to:

File *.mpc.d.txt

1 1 0.2

End File

1 1 0.2

End File

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!!~~!~~ Example Model

! new linear time-invariant object

! new linear time-invariant object

to:

!! Example Model

! new linear time-invariant object

! new linear time-invariant object

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!

! discrete form

! 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]

! 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)

to:

! B matrix (row, column, value)

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! C matrix (row, column, value)

to:

! C matrix (row, column, value)

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! D matrix (row, column, value)

to:

! D matrix (row, column, value)

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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

!!! 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

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 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.

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.

Changed lines 4-6 from:

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 %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.

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.

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!! Linear ~~State Space Model~~

to:

!! 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.

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Attach:~~linear~~_~~ss~~.png

to:

Attach:lti_step_response.png

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

to:

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

Attach:linear_ss.png

Attach:linear_ss.png