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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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**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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**Loop transfer functions of IMC-PID Controllers**

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

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**IMC-PID for FOPDT process**

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**Loop transfer functions of IMC-PID Controllers**

FOPDT processes: SOPDT processes: the resulting loop transfer function becomes:

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**Inverse-based Controller Design**

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

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**Loop transfer functions of Inverse-based Controllers**

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= a= GM = PM = 60 o

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**PI/PID Controllers Based on FOPDT Model**

ko=0.65 , a=0.4 A direct synthesis approach is used PI controller ko=0.5 Controller parameters (actual PID)

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**PID Controller Based on SOPDT Model**

Controller parameters (ideal PID) ko=0.5

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

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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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**A: Ku > 1 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**

ko=0.65 , a=0.4 A direct synthesis approach is used PI controller ko=0.5 Controller parameters (actual PID) In terms of ultimate data (Ku = kp kcu)

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**Defaulted value: ko=0.55 a=0.4**

PI controller Defaulted value: ko= 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 kp and need to be estimated in advance ko=0.5 In terms of ultimate data

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

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**Estimation of process gain kp**

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 kp and kcu 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. 3 Ex. 1 Ex. 2

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**Ideal PID controller with an extra filter**

The value of kp 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 = 55o PID control gain margin = 2.1, phase margin = 60o

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**The optimal IAE value occurs at a phase margin about 30o to 50o **

Optimal System for Disturbance Rejection 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 30o to 50o trade-off between disturbance performance and phase margin is not always needed PI control PID control

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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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