Robust Tuning of PI and PID Controllers

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Robust Tuning of PI and PID Controllers

• By
• Birgitta Kristiansson and Bengt Lennartson
• Presented By
• Pratyush Singh
• Naga. C. Pemmaraju
• Sandeep Panjala
• Srinivas Malipeddi
• Prasanth Bandi
• February 13, 2006.

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Presentation Overview

• Introduction
• Evaluation of Controllers
• Control and Plant Models
• Tuning PI and PID Controllers for Stable Plants
• Tuning PID Controllers for Plants with Integral
  Action
• Conclusion

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Introduction

• Proportional Integral (PI) controller and
  Proportional Integral Derivative controller
  (PID)-most commonly used controllers.
• Have been in use in different forms since ancient
  history.
• Numerous design and tuning strategies in
  industrial use but with limited access to the
  user.

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Objective of this Paper

• Poorly tuned controllers- Suffering from
  lack-of-control costs.
• Using a Low pass filter with derivative action.
• Proper tuning provides process disturbance
  rejection without control action or sensitivity
  to sensor noise.
• Investigates the influence of controller
  parameters, especially damping factor and time
  constant of the controller zeros.

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EVALUATION OF CONTROLLERS

• The Evaluation procedure is based on 4 criteria
  representing LF, MF, MHF, HF performance and
  robustness properties.
• The objective is to keep the MF, MHF, HF criteria
  equal, to evaluate the improvement in LF
  performance.
• EVALUTION CRITERIA
• control
  error e(t) r(t) - y(t).
• loop transfer function L(s) G(s)K(s)

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EVALUATION OF CONTROLLERS contd..

• The controller K(s) can be either strictly proper
  or exactly proper. Including integral action in
  K(s) implies the asymptotic properties.
• where ki - integral gain,
• k8 - HF gain,
• m - rolloff rate of the controller.
• These circles are shown for default values
• Ms 1.7, Mt 1.3. These values represent an
• empirical compromise
• Large Value of Ms faster response
• Smaller Value of Ms Slower system
• Small Gain Theorem Plan with significant model
• uncertainty can be rendered closedloop stable by
  
• t(s) small.

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Evaluation Procedure

• Evaluation of different controllers is obtained
  when of the 4 criteria are kept equal, while the
  tunning parameters are modified to minimize the
  fourth criterion
• By this optimization procedure, completely
  different controllers can be compared under equal
  conditions

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CONTROLLER AND PLANT MODELS

• Parameterization of Controllers
• The Goal is to design PID Controllers to reject
  process disturbances
• The traditional PID Controller KPID(s) Kp
  Ki/s Kds, has the draw back of being improper.
  To Limit the HF gain, The PID Controller is often
  augmented by a Lowpass Filter on the derivative
  part.
• Kp Proportional gain
• Ti Integral time
• Td Derivative time
• Tf Filter Time Constant
• Advantage is that the change in the gain module
  requires adjustment of only Kp

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Description of Plant Dynamics

• To Obtain useful timing rules for PI and PID
  controllers, the requirements for plant knowledge
  must be moderate
• The parameter K is used to capture the dynamics
  of a controlled process and is defined as
• W180c Frequency at which the plant has a
  phase margin of 180
• For Higher values of K1, the plant is more
  complex and harder to control.

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Tuning PI and PID Controllers for Stable Plants

• Parameters
• HF Gain k8
• Damping factor ?
• Zero Time Constant t
• Integral gain ki
• Tuning rules for PID Controllers
• PI Controllers
• PI Vs PID

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HF Gain k8

• The figure illustrate the trade off between
• Jv Vs Ju When The MF Robustness
• GMs is fixed
• From the Figure it is clear that even if
• we increase Ju beyond certain value
• say Jue there is no advantage
• For the PID controllers Ju is typically
• equal to k8 which means that the
• required level of Ju determines the HF
• gain k8 .

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Damping Factor ?

• The damping factor ? has no clear dependence
  either on the plant dynamics, as measured by
  ?150, or on GMS or Ju. When Ju is chosen near
  Juec, the optimal value of ? normally ranges from
  0.70 to 0.85.
• Figure shows that ? 0.75 is often a good
  choice.

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Zero Time Constant t

• The optimal zero time constant t depends on the
  plant dynamics.
• t is often proportional to T63, the length of
  time that a step response takes to reach 63 of
  its final value.

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Integral Gain ki

• ki 1/Jv
• The controller parameters ? , t , and ß are given
  by tuning rules, the integral gain ki is also the
  parameter best suited for manual tuning of the
  desired tradeoff between rise time (quickness)
  and damping (robustness) of the step response.
• Traditional Tuning, a 4, b 10 it is desirable
  to view a comparison of the optimal Jv/Ju
  relationship for different values of ?.
• a Ti/Td 4 b Td/Tf 10

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Tuning rules for PID Controllers

• Tuning Rules Based on K150
• This requires knowledge of k150 and w150G which
  is suitable for plants with poles on negative
  real axis.
• Tuning rules for stable non-oscillating plants of
  the second order
• An alternative to the expression for k8 is
• ß 2 14K150.
• When all four control parameters are tuned, GMs
  is in the range 1.65-1.80, generally near 1.7

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Tuning rules for PID Controllers Contd..

• Tuning Rules Based on T63
• Approximately one third of T63 serves as an
  appropriate value for t.
• Tuning rules for PID controller zeros
• The remaining parameters k8 and ki can be used to
  tune the tradeoff between output performance and
  control activity.
• The reference and disturbance step responses are
  optimal.

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Tuning rules for PID Controllers Contd..

• PID Tuning rules for first order plants with time
  delay
• Reference and disturbance step responses for a
  first order plant with moderate time delay.

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Tuning rules for PI Controllers

• ß ? 1.
• The goal of PI tuning rules to see that the
  demand on the MF robustness is fulfilled.
• PI tuning rules for plants of at least the second
  order
• PI tuning rules for first order plants with time
  delay.

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PI Vs PID

• The rules discussed shows that it is not
  difficult to design PID controllers.
• ß 1 and ? 1 gt Optimal control activity Ju is
  determined by the minimum in Jv/Ju .
• By introducing derivative action, performance can
  be achieved without excessive sensitivity to
  sensor noise and HF model uncertainties.

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Tuning of PID Controllers for Plants with
Integral Action

• For plants with integral action, the optimal
  value of ? is often close to one (corresponding
  to a double zero), but ? tends to grow with
  increasing complexity in terms of ?i150.
• Figure 13 illustrates the relationship Jv f
  (Ju) with various values of ? as well as with
  fixed a 4 and b 10. For values of Ju larger
  than 6.5, the required GMS cannot be reached.
• By observing the graph we can know that a value
  of ? slightly above the the optimal point is a
  better choice.

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Contd

• The fallowing graph shows the relationship
  between Jv/Ju for various PID zero time constants
  t for a plant with integral action
• By observing this graph, we can see that if t is
  more than marginally lower than the optimal
  value, Jv increases drastically and the
  requirement on GMS cannot be met, as shown in
  Figure. Except for the lowest values of Ju, the
  optimal value of t is almost independent of Ju.
• Figure (b) also illustrates that the improvement
  in output performance Jv is significant when
  derivative action is included, compared to the
  case in which a PI controller is used. The
  control activity in the latter case is Ju 0.45.

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HF Gain k8, Filter Factor ß, and Integral Gain ki

• Without adverse consequences, ß may for most
  plants (except when ?i150 lt 0.7) be fixed to 20,
  a value corresponding to the well-known value b
  10
• The integral gain must be normalized twice with
  respect to a time or frequency parameter, since
  both ki and the plant integral gain have units of
  inverse time

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Additional Filter Action

• To achieve a strictly proper controller, the PI
  controller can be augmented by a lowpass filter,
  or the first-order filter in the PID controller
  can be replaced by a higher order filter. This
  modification is advantageous in cases of
  significant sensor noise or unmodeled HF
  resonances .
• When a first-order lowpass filter is included,
  the PI controller becomes a PIF controller with
  transfer function
• When the first-order filter in the PID controller
  (4) is replaced by a second-order filter, the
  resulting PIDF controller has the form

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Contd

• Figure shows an example of a plant controlled by
  three exactly proper and four strictly proper
  controllers. All controllers are optimized with
  GMS 1.7.
• An extra lowpass filter in a PI or a PID
  controller can reduce the HF gain, and hence the
  sensitivity to sensor noise, with little
  deterioration of the LF performance. This
  property is illustrated by the controller gain
  K and the disturbance step responses for three
  exactly proper controllers and four strictly
  proper controllers, where k8 is limited to 20 and
  80, respectively

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Tuning rules for PIDF controllers

• ß 10, ? 0.8, t T63/3, ?f 0.4. If more or
  less control activity is preferable, adjust ß.
  Finally, adjust ki to the desired damping of a
  closedloop step response.
• The simple design method for the PIDF controller
  can be compared to the H8 loop-shaping strategy
  .The main idea in H8 loop-shaping is to augment
  the plant with a weight function W and modify W
  until a desired open loop shape is obtained for
  the augmented plant G WG. The resulting
  controller is then combined with the weight
  function to obtain the final controller. When a
  PID filter is used for the weight function, the
  results of the PIDF controller and the H8
  controller are almost equal.

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Conclusion

• The article proposes simple methods for close to
  optimal tuning of PI and PID controllers.
• Shows that tuning PID is as easy as PI with
  higher control activity and better output
  performance. Use of low-pass filter with
  derivative action is recommended for HF roll off
  of the controller.
• The major advantage of these tuning rules,
  compared to the previous rules, is the use of
  derivative action without introducing high
  sensitivity to sensor noise in the control
  signal, due to the inclusion of the low-pass
  filter in the design procedure.