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Optimization-based PI/PID control for SOPDT process

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Summary on optimization-based PI/PID control

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Best achievable IAE performance by PI/PID control of FOPDT process

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Optimal rise-time vs, IAE in PI/PID control of SOPDT process

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FOPDT

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SOPDT

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According to the IMC theory, nominal loop transfer function of a control system that has an inverse-based controller will be of the following: Loop transfer functions of IMC-PID Controllers

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IMC-PID for FOPDT process

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Loop transfer functions of IMC-PID Controllers the resulting loop transfer function becomes: FOPDT processes: SOPDT processes:

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We should learn what happens to the Z-N tuned controllers? How inverse-based controllers are synthesized? Inverse-based Controller Design

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Inverse-based synthesis approach is used –Target loop transfer function (LTF) –This LTF has satisfactory control performance as well as reasonable stability robustness ko and a are selected to meet desired control specification Defaulted value: ko=0.65 a=0.4 GM = 2.7 PM = 60 o Loop transfer functions of Inverse-based Controllers

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PI/PID Controllers Based on FOPDT Model A direct synthesis approach is used –PI controller ko=0.5 Controller parameters (actual PID) –PID controller ko=0.65, a=0.4

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PID Controller Based on SOPDT Model Controller parameters (ideal PID) k o =0.5

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Gain margin vs. phase margin at a=0.4 Phase margin Gain margin

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Auto-tune Autotuning via relay feedback: Astrom and Hagglund (1984) Referred as autotune variation (ATV): Luyben (1987) Main advantage: under closed-loop

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Apply Z-N or T-L tuning

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MODEL-BASED CONTROLLERS DESIGN Reduced order models –FOPDT Monotonic step response For zero offset, PI or PID controller is considered Usage of PI or PID controller depend on: –The application occasions –The dynamic characteristics of given process Processes are classified into two groups for controller tuning - Underdamped SOPDT Oscillatory step response

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Criterion for Classifying model order In general, processes with overdamped or slightly underdamped SOPDT dynamics can be identified with FOPDT models for controller tuning Q: When an SOPDT process could be reduced to an FOPDT parameterization? A: Ku > 1

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PI/PID Controllers Based on FOPDT Model A direct synthesis approach is used –PI controller ko=0.5 Controller parameters (actual PID) –PID controller ko=0.65, a=0.4 In terms of ultimate data ( K u = k p k c u )

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PI controller Defaulted value: k o =0.55 a=0.4 In terms of ultimate data

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PID Controller Based on SOPDT Model Only PID controller is used for significant underdamped SOPDT dynamics, i.e. Controller parameters (ideal PID) The values of k p and need to be estimated in advance k o =0.5 In terms of ultimate data

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Dynamic Process FOPDT Model SOPDT Model PID Controller Group I Group II PI Controller PID Controller

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Estimation of process gain k p –Start the ATV test with a temporal disturbance to setpoint or process input –Define – and have cycling responses

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Estimation from is subject to error, sometimes as high as 20% From Fourier series expansion Ultimate gain is computed exactly as:

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Estimation of Apparent Deadtime In an ATV test, two measured quantities are used to characterize the effect of the apparent deadtime – – For SOPDT process, this two quantities are functions of and

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Underdamped SOPDT processes

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Algorithm for estimation of apparent deadtime –Starting from a guessed value of –Calculate and, and feed them into networks to compute and –Check if the eq. holds –If not, increase the value of until the above eq. holds. At that time, is the estimated apparent deadtime

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In ATV test, it provides and which are functions of and Locate on this figure Zone I: FOPDT parameterization Zone II: SOPDT parameterization

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Initiate ATV test by a short period of manual disturbance and record y(t) and u(t) until constant cycling is attained –Compute k p and k cu –Estimate the apparent deadtime –Classify the process by the location of –If the process belongs to Group I, tune PI or PID controller based on FOPDT model parameterization –If the process belongs to Group II, tune PID controller based on SOPDT model parameterization

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Examples

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Ex. 1 Ex. 2 Ex. 3

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Ideal PID controller with an extra filter The value of k p and need to be known in advance

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Optimal IAE Value for Set-point Tracking –PI control –PID control These optimal systems have reasonable stability robustness –PI control gain margin = 2.6 For unit step set-point change (Huang and Jeng, 2002 ) –phase margin = 55 o –PID control gain margin = 2.1, phase margin = 60 o

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Control systems designed for optimal input disturbance response will give smaller gain margin and phase margin than those designed for optimal set-point response. The optimal IAE value occurs at a phase margin about 30 o to 50 o –trade-off between disturbance performance and phase margin is not always needed PI controlPID control Optimal System for Disturbance Rejection

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Optimal IAE Value for Disturbance Rejection The smaller the gain margin is (i.e. less robust), the lower the optimal IAE value can achieve. – trade-off between disturbance performance and gain margin is needed PI control PID control PI control PID control Gain margin

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