Nonlinear Quadratic Dynamic Matrix Control with State Estimation

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Nonlinear Quadratic Dynamic Matrix Control with
State Estimation

• Hao-Yeh Lee
• Process System Engineering Laboratory
• Department of Chemical Engineering National
  Taiwan University

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Reference

• Gattu, G., and E. Zafiriou, Nonlinear Quadratic
  Dynamic Matrix Control with State Estimation,
  Ind. Chem. Eng. Res., 31, 1096-1104 (1992).
• Ali, Emad, and E. Zafiriou, On The Tuning of
  Nonlinear Model Predictive Control Algorithms,
  American Control Conference, 786-790 (1993)
• Henson, M. A., and D. E. Seborg, Nonlinear
  Process Control, Prentice-Hall PTR (1997).

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Outline

• Introduction
• Linear and Nonlinear QDMC
• Algorithm Formulation with State Estimation
• Example
• Tuning parameters
• Conclusions

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Introduction

• Model predictive control (MPC)
• Dynamic matrix control (DMC Cutler and baker ,
  1979)
• An extension of DMC to handle constraints
  explicitly as linear inequalities was introduced
  by Garcia and Morshedi (1986) and denoted as
  quadratic dynamic matrix control (QDMC).
• Garcia (1984) proposed an extension of QDMC to
  nonlinear processes.

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Linear and nonlinear QDMC

• Linear QDMC utilizes a step or impulse response
  model of the process, and NLQDMC utilizes the
  model of the process represented by nonlinear
  ordinary differential equations.
• These approximations are necessary in order for
  the on-line optimization to be a single QP at
  each sampling point.

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Algorithm formulation with state estimation

• For the general case of MIMO systems, consider
  process and measurement models of the form
• where x is the state vector, y is the output
  vector, u is the vector of manipulated variables,
  and w (0, Q) and u (0, R) are white noise. Q
  and R are covariance matrices associated with
  process and measurement noise

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Algorithm formulation with state
estimation(contd)

• Know at Sampling Instant k y(k) the plant
  mea-surement, the estimate of
  state vector at k based on information at k-1,
  and u(k-1) the manipulated variable.

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Effect of future manipulated variables

• Step 1 Linearize the at
  and u(k-1) to obtain
• where

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Effect of future manipulated variables(contd)

• Step 2 Discretize above equations to obtain
• where Fk and Gk are discrete state space matrices
  (e.g., Åström and Wittenmark, 1984), obtained
  from Ak, Bk, and the sampling time.

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Effect of future manipulated variables(contd)

• Step 3 Compute the step response coefficients
  Si,k (i 1, 2, ..., P) where P is the prediction
  horizon. Si,k can be obtained from
• Step response coefficients can also be obtained
  by numerical integration of the linearized model
  over P sampling intervals with u 1.0 and x (tk)
  0.0 where tk is the time at any sampling point
  k.

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Computation of filter gain

• Step 4 Compute the steady-state Kalman gain
  using the recursive relation (Åström and
  Wittenmark, 1984)
• where Pjk is the state covariance at iteration j
  for the model obtained by linearization at
  sampling point k. P8k is the steady-state value
  of state covariance for that model.

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Effect of past manipulated variables

• Step 5 The effect of past inputs on future
  output predictions, y(k1), y(k2), ...,
  y(kP) is computed follows. Here the superscript
  indicates that input values in the future are
  kept constant and equal to u(k-1).
• Set
• Define
• Assume
• For i 1, 2, ..., P, successively integrate
  over one sampling time from
  , with
  and then add to obtain
  Addition of Kkd provides
  correction to the state. We can then write

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Output prediction

• Step 6 The predicted output is computed as the
  sum of the effect of past and future manipulated
  variables and the future predicted disturbances.

Future disturbances
Past effect
Future effect
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Optimization

• Optimization.
• where P is the prediction horizon
• M is the number of future moves
• It is assumed that u(kM-1) u(kM) ...
  u(kP-1).
• G and L are diagonal weight matrices.

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Optimization(contd)

• The above optimization problem with constraints
  can be written as a standard quadratic
  programming problem

Subject to
where
and D and b depend on the constraints on
manipulated variables, change in manipulated
variables, and outputs.
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Estimation of state

• Step 7 The M future manipulated variables are
  computed, but only the first move is implemented
  (Garcia and Morshedi, 1986).
• Estimation of State.
• Step 8 Integrate from
  and u(k) over one sampling time and add
  Kkd to obtain

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Example

• For the reaction A B ? P the rate of
  decomposition of B is
• The system is described by a dynamic model of the
  form

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Example(contd)

• Isothermal CSTR

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Example(contd)

• u1 is the inlet flow rate with condensed B,
• u2 is the inlet flow rate with dilute B,
• x1 is the liquid height in the tank,
• x2 is the concentration of B in the reactor.
• The control problem is simulated with the values
• k1 1.0, k2 1.0,
• CB1 24.9, and CB2 0.1.

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Multi-equilibrium points at steady state

• Multi-equilibrium points of CB

At u11.0, u21.0
Lower steady state a 100, 0.6327 Middle
steady state ß 100, 2.7870 Upper steady state
? 100, 7.0747
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Simulation results

• A setpoint change from an initial condition of
  xl0 40.00 and x20 0.1 to the unstable
  steady-state point with values at x1 100.00 and
  x2 2.787.
• The lower bounds on u1 and u2 are kept at zero
• The upper bounds varied from 5, 10 to 8
• A sampling time Ts 1.0 min
• Tuning parameter values P 5 and M 5
• For the tuning parameter L diag0.0,0.0, G
  diag1.0,1.0

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Simulation results(contd)

• The plant is running at the unstable steady
  state. Consider a step disturbance of 0.5 unit in
  u1.
• A sampling time Ts 1.0 min
• the tuning parameter values P 5.0, M 5.0, L
  0.0, u10 1.0 and u20 1.0 are used in the
  simulations.
• The lower bounds on u1 and u2 are kept at zero,
  and there are no upper bounds.

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Tuning parameters

• System parameter
• Sampling time
• Tuning parameters
• Prediction horizon
• Longer horizons tend to produce more aggressive
  control action and more sensitive to
  disturbances.
• Control horizon
• Shortening the control horizon relative to the
  prediction horizon tend to produce less
  aggressive controllers, slower system response
  and less sensitive to disturbances
• Penalty weights

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Some problems of NLQDMC

• Truncation error in NLQDMC
• Different sampling times
• If system has large different responses in each
  loop
• Tuning problem in NLQDMC

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Optimization based of tuning method
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Optimization based of tuning method(contd)
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Conclusion

• The proposed algorithm stabilizes open-loop
  unstable plants and The incorporation of a Kalman
  filter also results in better disturbance
  rejection when compared to Garcia's algorithm.
• The major advantage of the proposed algorithm
  compared to the nonlinear programming approaches
  is that only a single quadratic program is solved
  on-line at each sampling time.
• The use of the software package CONSOLE can
  obtain solution to an off-line optimization to
  tune the NLQDMC parameters.