Chapter%206%20Model%20Predictive%20Control

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Chapter 6Model Predictive Control

• Prof. Shi-Shang Jang
• National Tsing-Hua University
• Chemical Engineering Department

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Historical Development

• Questions
• Given a system what are the absolute limitations
  to output control?
• How can one make use of a process model and
  account for model error?
• Smith Predictor(1955)
• Feedforward Control
• Inferential Control (1975, 1979)
• Dynamic Matrix Control (Shell, 1979)
• Model Algorithmic Control (France, 1979)
• Internal Model Control (Garcia and Morari, 1982)

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Criteria for Controller Quality

• Regulatory Behavior Compensation for
  (unmeasured disturbances)
• Servo Behavior- Follow set point changes (fast,
  smooth, no offset)
• Robustness-Controller should be effective when
  there are modeling errors (both structure and
  parameters)

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Criteria for Controller Quality-Continued

• Constraints- Ability to deal with constraints on
  inputs and states (no windup)
• Remark 90 of loops can be handled by PID type
  controllers.

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Internal Model Control (IMC) Structure
eys-yym
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Analysis of Internal Model Structure

• From the block diagram

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Properties of IMC (Principles of Internal Model
Control)

• 1. (Dual Stability) If the model is perfect,
  stability of controller and plant is sufficient
  for overall system stability
• Proof If GpGm, then
• yGpGI(ys-d)d
• u(ys-d) GI
• Use IMC only on stable systems, unstable systems
  can be stabilized by feedback control
• Constraints on inputs has no effect on stability

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Properties of IMC (Principles of Internal Model
Control)- continued

• 2. (Prefect Control) If model is perfect and
  invertible, and GIGp-1, then yys for any d
• Notes (1) This is optimal control.
• (2) Suppose Gp-1 is not
  realizable, then it is recommended to factor this
  transfer function into two terms
  Gp(s)G(s)G-(s) where G(s) is not realizable
  contains all time delays and RHP zeros. In this
  case the best controller possible is
  GI(s)G--1(s), this controller minimizes sum of
  the square of the errors in output.
• (3) This suggests the design of
  F(s) as GIGp-1 F(s) such that GI(s) realizable.

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Example
Which is realizable if we choose
Where ndegree of D(s)-degree of N(s)gt0, ? is
chosen
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Properties of IMC (Principles of Internal Model
Control)- continued

• 3. Zero Offset
• There is no offset if we choose
• GI(0)1/Gm(0)
• Pf

• This means the output will attain the set point
  exactly in presence of persistent disturbances
  and set-point changes - integral feedback
• b. Note that this is true even if the model is
  imperfect.

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Properties of IMC (Principles of Internal Model
Control)- continued

• 4. Comparison with Feedback Controller
• If we choose
• Then we get classical output feedback control.
  Note that GI(s) in the closed loop transfer
  function of the following

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Properties of IMC (Principles of Internal Model
Control)- continued

• Joining this with the IMC block diagram we get
• which reduced to a classical feedback control

d

ys(s)
Gc
Gp
-
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Properties of IMC (Principles of Internal Model
Control)- continued

• Notes
• (a)
• may be looked upon as a poor approximation to
  Gm-1(s). For large Kc, this approximation gets
  better.
• (b) Note the similarity of this structure to
  Smith Predictor also approximates the invertible
  part of Gm-1(s).
• (c) This feedback structure implies use of a
  filter

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On-Line Tuning of IMC

1. Choose a process model through plant tests
2. Choose a filter
   to make GI(s) realizable
3. Decrease ? untill system becomes oscillatory.
   Brosilow recommends the following time constant
   (where ?umin. filter constant Puperiod of
   oscillation)
4. If ?/Dlt1, then IMC will yield better performance
   than a PID controller. If 1lt ?/Dlt2 then IMC and
   PID are competitive.

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Computation of Approximate Inverses

• In practice, it is easier to find an approximate
  inverse of the process transfer function in the
  time domain (using discrete models).
• Time Domain View
• Given the past history of inputs to the process
  and current estimate of the disturbance, compute
  the current and future inputs which will make the
  output follow the desired set point.
• Limitation in Practice
• The future will be limited to a finite time
  horizon (3-4 times time-constant of system.
• Attention must be limited to values of output at
  discrete times.

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Summary of MPC

• MPC consists of three blocks
• Process model
• A controller (approximate inverse)
• A filter
• Advantages
• Quality of response depends on controller design
• Robustness depends on filter
• Stability is not an issue
• Implementation is straight forward
• On-line tuning can be provided by the filter
  time constant

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Computation of Approximate Inverses - Continued

• 3. Values of all future inputs may be limited
  to a few in the immediate future.
• 4. Problem must be solved every so often (at
  discrete sampling times) when new estimates of
  disturbance become available.
• 5. We must limit the size and velocity of control
  input variations.
• 6. On-line computations should be kept to a
  minimum.
• 7. Smooth transfer between auto/manual should be
  possible.
• 8. It should be recognize constraints on inputs.
• 9. There should be operator adjustable
  constant(s) to account for plant/model mismatch.

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Review of least-square problem

• Given a set of equations
• Axbe
• We seed a solution which minimizes
• ?iei2
• The solution is given by
• x(ATA)-1ATb
• We term (ATA)-1AT to be pseudo inverse of matrix A

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A Discrete Input Plant Model
Note that N/D is actually the impulse response of
the system m(z)1 without delay
y(t)
h3
h4
h5
h1
h2
h6
h7
t
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Example G(s)1/(s1)3
global m m1TSPAN0 1 Y00 0
0 Y_realY_sampleT_real T_sample0
Y_model0 for i129 T,Y
ODE45('model_3',TSPAN,Y0) TSPANTSPAN(2),TSPAN(2
)1 Y0(1)Y(end,1) Y0(2)Y(end,2) Y0(3)Y(end,
3) TTT(end) Y_realY_realY(,1)T_realT_re
alT m0 T_sampleT_sample,TT Y_modelY_mod
el,Y0(1) end function dymodel_3(t,y) global
m dy(1)y(2) dy(2)y(3) dy(3)-3y(3)-3y(2)-y(1
)m dydy'
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Example G(s)1/(s1)3 - continued
TSPAN0 1mmzeros(1,29) Y00 0
0 Y_realY_sampleT_real T_sample0
Y_pred0 for i150 mrandn(1,1) for
i128 mm(29-i1)mm(29-i) end
mm(1)m T,Y ODE45('model_3',TSPAN,Y0) TSPAN
TSPAN(2),TSPAN(2)1 Y0(1)Y(end,1) Y0(2)Y(end
,2) Y0(3)Y(end,3) TTT(end) Y_realY_realY(
,1)T_realT_realT ypY_modelmm' Y_predY_
pred,yp T_sampleT_sample,TT end
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A Discrete Input Plant Model
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A Discrete Input Plant Model-Continued
This expresses y in terms of past inputs m i.e.
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Approximate Inversion
Since we cannot make y(t)yd(t) exactly, we pose
the following least square minimization problem
subject to the above process model
No control changes beyond M
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The Solution

• The previous problem can be solved based on a
  Quadratic Programming solver or using previous
  pseudo-inverse of matrix approach.

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MPC-Servo Control (A Feed-forward Approach) Want
yk1yk2yd
P4 M4
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MPC-Servo Control (A Feed-forward Approach)
-Example
Y_real
Y_sample
time
time
P4 M4
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MPC-Servo Horizon Control (A Feed-forward
Approach) Want yk1yk2yd, but mk1mk2mk3
P4 M2
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MPC-Servo Horizon Control (A Feed-forward
Approach) Want yk1yk2yd, but mk1mk2mk3
Response
Time
P4 M2
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MPC-Regulation Control (A Feedback Approach)
P4 M4
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MPC-Regulation Control (A Feedback Approach)
P4 M2
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MPC-Regulation Control (A Feedback Approach)
Response
Time
M2
M4
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Multi-variable Discrete Input Plant Model
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Examples of Multivariable Control Control of a
Mixing Tank
MVs Flow of Hot Stream CVs Level in the
tank Flow of Cold Stream
Temperature in the tank
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Example- Mixing Tank Problem
Height
Time
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Example- Mixing Tank Problem
Temperature
Time
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Dynamic Matrix Control (DMC)
Response
Response
a4,.
a3
h2
h3
h4,.
a2
h1
a1
Time
Time
Step response
Pulse response
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Dynamic Matrix Control (DMC)- Continued
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Dynamic Matrix Control (DMC)- Continued
disturbance
Effect of the past
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Dynamic Matrix Control (DMC)- Continued
P4 M4
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Dynamic Matrix Control (DMC)- Continued
P4 M2
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Tuning Procedures

• Sampling time (T) stability is not affected by
  T. Larger T leads to less variations in m, but
  deteriorates system performance in presence of
  frequent disturbances
• Horizon for m (M) Choosing MP (perfect
  control) leads to severe oscillation in m(t).
  Reducing M, leads to a more desired response

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Tuning Procedures - Continued

• Input penalty parameter ? Increasing ? makes
  system more sluggish and nonzero ? lead to
  offset, but can be compensated by adding integral
  control algorithm itself.
• Optimization horizon, P increasing P gets
  better inverse for system of order n, Pgt2n is
  generally sufficient.

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Summary - Continued

• Model Predictive Control (MPC) is the major
  existed advanced process control in chemical
  engineering industrial
• The modeling in the MPC is crucial
• The tuning of MPC using M (horizon of
  suppression) is the most effective for stability.
  All other parameters may also frequently
  implement to improve the control quality