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سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي - دکتر سید مجید اسما عیل زاده

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2 :review

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4 Control Configurations

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6 Controll er Type

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Historical note The first application of PID controller was in 1922 by Minorsky on ship steering. Minorsky (1922) “Directional stability of automatically steered bodies”, J. Am. Soc. Naval Eng., 34, p.284. This was the first mathematical treatment of the type of controller that is now used to control almost all industrial processes.

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The current situation Despite the abundance of sophisticated tools, including advanced controller design techniques, PID controllers are still the most widely used controller structure in modern industry, controlling more that 95% of closed-loop industrial processes. Different PID controllers differ in the way how their parameters be tuned, manually, or automatically. Most of the DCS systems have built-in routines to perform auto- tuning of PID controllers based on the loop characteristics. They are often called: auto-tuners.

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The PID Algorithm The PID algorithm is the most popular feedback controller algorithm used. It is a robust easily understood algorithm that can provide excellent control performance despite the varied dynamic characteristics of processes. As the name suggests, the PID algorithm consists of three basic modes: the Proportional mode, the Integral mode & the Derivative mode.

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PID controllers

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P, PI or PID Controller When utilizing the PID algorithm, it is necessary to decide which modes are to be used (P, I or D) and then specify the parameters (or settings) for each mode used. Generally, three basic algorithms are used: P, PI or PID. Controllers are designed to eliminate the need for continuous operator attention. Cruise control in a car and a house thermostat are common examples of how controllers are used to automatically adjust some variable to hold a measurement (or process variable) to a desired variable (or set-point)

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Controller Output The variable being controlled is the output of the controller (and the input of the plant): The output of the controller will change in response to a change in measurement or set-point (that said a change in the tracking error) provides excitation to the plantsystem to be controlled

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PID Controller In the s-domain, the PID controller may be represented as: In the time domain: proportional gainintegral gainderivative gain

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PID Controller In the time domain: The signal u(t) will be sent to the plant, and a new output y(t) will be obtained. This new output y(t) will be sent back to the sensor again to find the new error signal e(t). The controllers takes this new error signal and computes its derivative and its integral gain. This process goes on and on.

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Definitions In the time domain: proportional gainintegral gain derivative gain derivative time constantintegral time constant

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PID structures Standard PID controllers have the following structures: Proportional only: Proportional plus Integral: Proportional plus derivative: Proportional, integral and derivative:

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What is an Op-Amp? An Operational Amplifier is an electronic device used to perform mathematical operations in a circuit – they are generally abbreviated as “Op-Amps” Op-Amps are high gain devices that amplify a signal using an external power supply They are composed of multiple transistors, resistors, and capacitors Common types of op-amps: Inverting Non-Inverting Integrating Differential Summing

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What is an Op-Amp? All op-amps use a voltage supply (V cc ) to amplify the signal The supply voltages can either have equal value but opposite signs, or the low side is grounded and the high side has a value of twice the voltage input Some common applications of op-amps: Low Pass Filters Strain Gauges PID Controllers V - Inverting Input V + Non-Inverting Input V-V- V+V+ V out V+V+ V-V- +V cc -V cc

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Typical 8 Pin Op-Amp

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Amplifier Gain All op-amps can be represented by the formula: Where K is the gain, and is a property of the individual op-amp This gain should be distinguished from the gain of the op-amp circuit which is generally denoted by A v A potential source of confusion comes from failing to properly distinguish between the op-amp and the op-amp circuit V out = K (V + - V - ) A v = V out / V in V-V- V+V+ V out Op-Amp Op-Amp Circuit

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Comparator A comparator is an example of an open-loop op-amp If V + > V - V out = V sat ≈ V cc If V + < V - V out = -V sat ≈ - V cc i - = 0A i + = 0A +V cc - V cc V ou t + - V-V- V+V+ V in - V in + V out V in+ - V in- + V sat - V sat

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Summing Op-Amp Application of a non-inverting op- amp and Millman‘s theorem V out = V inA + V inB + V inC V in A V out + - RARA R2R2 V-V- V+V+ V in B V in C RBRB RCRC R1R1 (1)Millman’s theorem: if R A = R B = R C = R (1) (2)Non-inverting Op-Amp: (2) V’

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Substraction (Differential) Op-Amp V out = V inA - V inB Applying Kirchhoff’s Rules and Op-Amp Calculation Rules yields: V in B V out + - R4R4 V-V- V+V+ V in A R1R1 R2R2 R3R3 if R 1 = R 2 = R 3 = R 4 Note: if R 1 = R 3 = R and R 2 = R 4 = a R

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Derivative Op-Amp Applying Kirchhoff’s Rules and Op-Amp Calculation Rules yields: V in V out + - R V-V- V+V+ R C V in V out

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Integrating Op-Amp V in V out + - R V-V- V+V+ R C V in V out Applying Kirchhoff’s Rules and Op-Amp Calculation Rules yields:

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Goal is to have V SET = V OUT Remember that V ERROR = V SET – V SENSOR Output Process uses V ERROR from the PID controller to adjust V out such that it is ~V SET P I D Output Process Sensor V ERROR V SET V OUT V SENSOR PID Controller – System Block Diagram

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Applications PID Controller – System Circuit Diagram Calculates V ERROR = -(V SET + V SENSOR ) Signal conditioning allows you to introduce a time delay which could account for things like inertia System to control -V SENSOR

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Applications PID Controller – PID Controller Circuit Diagram V ERROR AdjustChange KpKp RP1, RP2 KiKi RI, CI KdKd RD, CD V ERROR PID

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Controller Effects A proportional controller (P) reduces error responses to disturbances, but still allows a steady-state error. When the controller includes a term proportional to the integral of the error (I), then the steady state error to a constant input is eliminated, although typically at the cost of deterioration in the dynamic response. A derivative control typically makes the system better damped and more stable.

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Closed-loop Response Rise timeMaximum overshoot Settling time Steady- state error PDecreaseIncreaseSmall change Decrease I Increase Eliminate DSmall change Decrease Small change Note that these correlations may not be exactly accurate, because P, I and D gains are dependent of each other.

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Example problem of PID Suppose we have a simple mass, spring, damper problem. The dynamic model is such as: Taking the Laplace Transform, we obtain: The Transfer function is then given by:

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Example problem (cont’d) Let By plugging these values in the transfer function: The goal of this problem is to show you how each of contribute to obtain: fast rise time, minimum overshoot, no steady-state error.

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Ex (cont’d): No controller The (open) loop transfer function is given by: The steady-state value for the output is:

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Ex (cont’d): Open-loop step response 1/20=0.05 is the final value of the output to an unit step input. This corresponds to a steady-state error of 95%, quite large! The settling time is about 1.5 sec.

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Ex (cont’d): Proportional Controller The closed loop transfer function is given by:

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Ex (cont’d): Proportional control Let The above plot shows that the proportional controller reduced both the rise time and the steady-state error, increased the overshoot, and decreased the settling time by small amount.

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Ex (cont’d): PD Controller The closed loop transfer function is given by:

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Ex (cont’d): PD control Let This plot shows that the proportional derivative controller reduced both the overshoot and the settling time, and had small effect on the rise time and the steady-state error.

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Ex (cont’d): PI Controller The closed loop transfer function is given by:

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Ex (cont’d): PI Controller Let We have reduced the proportional gain because the integral controller also reduces the rise time and increases the overshoot as the proportional controller does (double effect). The above response shows that the integral controller eliminated the steady-state error.

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Ex (cont’d): PID Controller The closed loop transfer function is given by:

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Ex (cont’d): PID Controller Let Now, we have obtained the system with no overshoot, fast rise time, and no steady-state error.

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Ex (cont’d): Summary PD P PI PID

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PID Controller Functions Output feedback from Proportional action compare output with set-point Eliminate steady-state offset (=error) from Integral action apply constant control even when error is zero Anticipation From Derivative action react to rapid rate of change before errors grows too big

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Effect of Proportional, Integral & Derivative Gains on the Dynamic Response

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Proportional Controller Pure gain (or attenuation) since: the controller input is error the controller output is a proportional gain

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Change in gain in P controller Increase in gain: Upgrade both steady- state and transient responses Reduce steady-state error Reduce stability!

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high P Controller with high gain

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Integral Controller Integral of error with a constant gain increase the system type by 1 eliminate steady-state error for a unit step input amplify overshoot and oscillations

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Change in gain for PI controller Increase in gain: Do not upgrade steady- state responses Increase slightly settling time Increase oscillations and overshoot!

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Derivative Controller Differentiation of error with a constant gain detect rapid change in output reduce overshoot and oscillation do not affect the steady-state response

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Effect of change for gain PD controller Increase in gain: Upgrade transient response Decrease the peak time and rise time Increase overshoot and settling time!

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Changes in gains for PID Controller

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