دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده

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

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Control Configurations

Control Configurations

Controll er Type

Controll er Type

Controll er Type

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.

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.

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.

PID controllers

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)

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 plant system to be controlled

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

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.

Definitions In the time domain: integral time constant
derivative time constant derivative gain proportional gain integral gain

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

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

What is an Op-Amp? +Vcc V- V- Vout V- V+ Vout V+ V+ -Vcc
All op-amps use a voltage supply (Vcc) 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 +Vcc V- Inverting Input V- V+ Vout V- Vout Vout V+ Non-Inverting Input V+ -Vcc

Typical 8 Pin Op-Amp

Amplifier Gain V- Vout = K (V+ - V-) Vout V+ Op-Amp Av = Vout / Vin
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 Av A potential source of confusion comes from failing to properly distinguish between the op-amp and the op-amp circuit V- V+ Vout Vout = K (V+ - V-) Op-Amp Av = Vout / Vin Op-Amp Circuit

Comparator A comparator is an example of an open-loop op-amp i- = 0A
+Vcc -Vcc Vout + - V- V+ Vin- Vin+ Vout Vin+ - Vin- + Vsat - Vsat If V+ > V- Vout = Vsat ≈ Vcc If V+ < V- Vout = -Vsat ≈ - Vcc

Summing Op-Amp Application of a non-inverting op-amp and Millman‘s theorem (1) Millman’s theorem: if RA = RB = RC = R (1) R2 (2) Non-inverting Op-Amp: (2) R1 V- + - Vout VinA RA V+ VinB RB To control the output voltage V’ VinC RC Vout = VinA + VinB + VinC 25

Substraction (Differential) Op-Amp
Applying Kirchhoff’s Rules and Op-Amp Calculation Rules yields: Vout = VinA - VinB VinB Vout + - R4 V- V+ VinA R1 R2 R3 if R1 = R2 = R3 = R4 To control the output voltage Note: if R1 = R3 = R and R2 = R4 = a R 26

Derivative Op-Amp R C Vin V- - Vout + V+
To control the output voltage Applying Kirchhoff’s Rules and Op-Amp Calculation Rules yields: 27

Integrating Op-Amp C V- Vin R - Vout + V+
To control the output voltage Applying Kirchhoff’s Rules and Op-Amp Calculation Rules yields: 28

PID Controller – System Block Diagram
VOUT VSET VERROR Output Process D VSENSOR Sensor Goal is to have VSET = VOUT Remember that VERROR = VSET – VSENSOR Output Process uses VERROR from the PID controller to adjust Vout such that it is ~VSET

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

Applications PID Controller – PID Controller Circuit Diagram
Adjust Change Kp RP1, RP2 Ki RI, CI Kd RD, CD VERROR VERROR PID

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.

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

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:

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.

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

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.

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

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.

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

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.

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

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.

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

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

Ex (cont’d): Summary P PD PI PID

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

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

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

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

P Controller with high gain

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

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

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

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!

Changes in gains for PID Controller