Lecture 8 - Open/Closed Loop Systems, PID Controllers

ECOR1044: Control Systems Open/Closed Loop Systems PID Controllers Gain Tuning 1 Control Systems Control systems: useful field with a lot of multi-disciplinary applications. – Electrical Engineering – Communications Engineering – Mechanical Engineering – Ci...

ECOR1044: Control Systems Open/Closed Loop Systems PID Controllers Gain Tuning 1 Control Systems Control systems: useful field with a lot of multi-disciplinary applications. – Electrical Engineering – Communications Engineering – Mechanical Engineering – Civil Engineering – Industrial Engineering – Aerospace Engineering, etc. 2  Control Systems › In the absence of feedback, the control system is highly sensitive to disturbances and to both knowledge of and variations in parameters of 𝐺(𝑠). › If the open-loop system does not provide a satisfactory response, then a suitable cascade controller, 𝐺𝑐 (𝑠), can be inserted preceding the process, 𝐺(𝑠). 3  Control Systems › The advantages of closed-loop feedback control systems: – Decrease sensitivity of the system to variations in the parameters. – Reject the disturbances. – Attenuate measurement noise. – Reduce the steady-state error of the system. – Easy control and adjustment of the transient response of the system. 4  Stability › If the cone is resting on its base and is tipped slightly, it returns to its original equilibrium position. This position and response are said to be stable. › If the cone rests on its side and is displaced slightly, it rolls with no tendency to leave the position on its side. This position is designated as the neutral stability. › if the cone is placed on its tip and released, it falls onto its side. This position is said to be unstable. 5  Linear Systems › The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. › The homogeneity principle states that the output of a linear system is always directly proportional to the input, so if we put twice as much into the system we will, in turn, get out twice as much. 6  Linear Approximation › A linear approximation is as accurate as the assumption of small signals is applicable to the specific problem. › This is useful for a linear approximation to the nonlinear function. 7 Mass on a nonlinear spring  Example: Pendulum › Consider the pendulum oscillator: › The torque is › The equilibrium condition for the mass is. Linearize the model? › The approximation at the equilibrium point, › where, › This approximation is reasonably accurate for 8  Performance Measures › The parameter 𝑝 may minimize the performance measure 𝑀2 if we select 𝑝 as a very small value. However, this results in large measure 𝑀1 , an undesirable situation. › If the performance measures are equally important, the crossover point at 𝑝𝑚𝑖𝑛 provides the best compromise. › The performance measures help to answer the question, How well does the system perform the task for which it was designed? 9  Test Input Signals › The standard test input signals commonly used are the step input, the ramp input, and the parabolic input. › The ramp signal is the integral of the step input, and the parabola is the integral of the ramp input. 10  Performance Indices › A system is considered an optimum control system when the system parameters are adjusted so that the index reaches an extremum, commonly a minimum value. › A performance index must be a number that is always positive or zero. It › A common performance index is the integral of the square of the error, ISE, which is defined as, › It is convenient to choose 𝑇 as the settling time 𝑇𝑠. 11  Performance Indices › This criterion will penalize excessively overdamped and underdamped systems. › The minimum value of the integral occurs for a compromise value of the damping. 12  Performance Indices › Three other performance indices we might consider include › The ITAE is able to reduce the contribution of any large initial errors, as well as to emphasize errors occurring later in the response. › The performance index ITAE provides the best selectivity of the performance indices; that is, the minimum value of the integral is readily discernible as the system parameters are varied. › The general form of the performance integral is › where 𝑓 is a function of the error, input, output, and time. 13 14 Open-Loop vs. Closed-Loop Control Systems The control action is totally independent of the output of the system: System Input Input Plant Output Controller Or Process Disturbances System Input Plant Input + Error Output − Controller Or Process Sensor 15  Control Systems › The alteration or adjustment of a control system to provide a suitable performance is called compensation. Cascade Compensation Feedback Compensation Output Compensation Input Compensation 16 Simple Open-loop Controller ON-OFF controller – Temperature control of a chamber using a timer switch – ON-OFF controller will switch the output on and off according to a set timer System Timer Input Output Input ON/OFF Heater – Pros Switch Simple and economical Easy to maintain Stable – Cons Inaccurate Unreliable Change due to disturbances cannot be corrected 17 Simple Closed-loop Controller ON-OFF controller – Temperature control of a chamber using a temperature sensor and a switch. – ON-OFF controller will switch the output when the output crosses a setpoint. System Input + Error ON/OFF Input Output Switch Heater − Temperature Sensor – A common problem with such type of controllers is oscillations in output. – The drawbacks of an ON-OFF controller can be overcome by using a more sophisticated controller, e.g. a PID controller. 18 Closed-loop System: Terminologies Process Variable (PV) - the system parameter that needs to be controlled, such as temperature (ºC), pressure (psi), flow rate (L/min) Sensor - used to measure the process variable and provide feedback to the control system. The Set Point (SP) - the desired or command value for the process variable, such as 100 ºC in the case of a temperature control system Error 𝑒 - the difference between the PV and the SP. Used by the control system to determine the action to get desired output Plant SP + e = SP-PV System input Or Output Controller − Process PV Sensor 19 Closed-loop System: Example Speed SP + e = SP-PV PWM Signal Output Calculation DC Motor − (Pi) Rotary PV Encoder 20 Closed-loop System: Terminologies (cont.) – Rise Time - time to go from 10% to 90% of the steady-state or final value. – Overshoot - the maximum amount exceeding final value (target). – Settling time - time to settle to within a certain percentage of final value. – Steady-State Error - difference between the process variable and target. 21 Disturbance Rejection › Express the output Y in terms of reference R and disturbances D1 and D2 : 22 Disturbance Rejection › Suppose C is a large positive gain. What happens as C tends to infinity? Too good to be true! 23  PID Controller A PID controller applies a correction based on Proportional, Integral, and Derivative terms (hence the name PID). It is simply the sum of three parallel actions to generate a control output 𝑢(𝑡). Set-point (SP) Process Variable (PV) Sensor 24 PID Controller The output of a PID controller has the following general form: – 𝑢(𝑡): Control signal sent to the system – 𝑒(𝑡): Error Value (SP-PV) – 𝐾𝑝: Proportional constant that accounts for the present error value. – 𝐾𝑖: Integral constant that accounts for historical error values. – 𝐾𝑑: Derivative constant that accounts for future error values. 25 Advantages of PID Controller Accurate Eliminates Steady-State Error Reduces Overshoot and Oscillations Widely Applicable 26 Proportional Controller A Proportional Controller provides a control input that is proportional to e(t). Action: Applies a correction proportional to the error. Effect: As the error increases, the corrective action increases. Strength: Fast initial response. 27 Disturbance Rejection: Proportional Control › The control action: 28 Disturbance Rejection: Proportional Control 29 Advantages of Proportional Controller Advantages: Simplicity: Easy to implement and understand. Fast Response: Quickly responds to large errors. Stabilization: Provides consistent corrective effort based on real-time errors. Limitations: Steady-State Error: In some cases, the system may not reach the setpoint, leaving a residual error. Oscillations: If the proportional gain is too high, it can cause the system to oscillate or become unstable. 30 Proportional Integral (PI) Controller A PI controller is a feedback controller combining proportional and integral actions to correct errors in a system. Features: ▪ Proportional (P): Reacts to the current error. ▪ Integral (I): Reacts to the accumulation of past errors. 31 Disturbance Rejection: Integral Control › The control action: › At steady-state: › The integral action eliminates the effect of disturbance at steady-state. 32 Disturbance Rejection: Integral Control › Any value of K2 will eliminate the steady-state error. 33 Advantages of Proportional Integral Controller Advantages: Simplicity: Easy to implement and understand. Steady-State Error: Eliminate steady-state error. Tracking: Suitable for many real-world applications. Limitations: Slow Response: Slower response due to integral action. Oscillations: If the integral gain is too high, it can cause the system to oscillate or become unstable. 34 Application of Proportional Integral Controller PI controller needs tuning to eliminate the steady-state error Choosing low and high values for proportional and integral gains cause weak performance or instability 35 Proportional Integral Controller Integral control: 36 PID Controller Derivative control: – Derivative control is proportional to the rate of change of the error signal. 𝑑 𝐷𝑜𝑢𝑡 = 𝐾𝑑 𝑒(𝑡) 𝑑𝑡 – It introduces an element of ‘prediction’ into the control action. – It has a damping effect and reduces the oscillations caused by a large gain 𝐾𝑝 and improves settling time. – Introducing a derivative control gain 𝐾𝑑 will have the effect of: Increasing the stability of the system. Reducing the overshoot, and Improving the transient response. 37 PID Controller: Tuning 38 Example: Spring Damper System How change in 𝐾𝑝 affects the response 𝐾𝑝 = 24 𝐾𝑝 = 10 𝐾𝑝 = 5 𝐾𝑝 = 1 39 Example: Spring Damper System How change in 𝐾𝑑 affects the response 𝐾𝑑 = 0 𝐾𝑑 = 1 keeping 𝐾𝑝 = 24 𝐾𝑑 = 2 𝐾𝑑 = 4 40 PID Variants For a given control task, it is obviously not always necessary to adopt all the three actions of PID controller. This gives rise to variations of the PID controller by setting respective gains to be zero: P, PI, PD and PID Controllers. PID Controller Tuning: – The selection of the PID gains (Kp, Ki and Kd ) of the PID controllers is not trivial. – The process is called “Tuning the Loop”. – Simplest way of PID tuning is through trial and error using the following table: 41 Summary of P, PI, and PID Algorithm Controller Advantages Disadvantages Use Cases - Simple and fast - Steady-state error - When response response remains speed is more P (Proportional) - Easy to tune (one - Cannot correct all important than parameter) errors accuracy Eliminates steady- - Slower than P - Systems requiring state error controller steady-state PI (Proportional- - Suitable for slower - More complex to accuracy, e.g., Integral) processes tune (two temperature parameters) control PID (Proportional- - Fast response - Most complex to - High-performance Integral-Derivative) - Eliminates steady- tune (three systems requiring state error parameters) stability, e.g., - Reduces overshoot - Can lead to motion control and oscillations instability if tuned improperly 42 PID Controller 43 Designing a PID Controller Steps: Obtain an open-loop response and determine what needs to be improved. Add a proportional control to improve the rise time. Add an integral control to eliminate the steady-state error. If needed, add a derivative control to improve the overshoot. Adjust each of Kp, Ki, and Kd until you obtain a desired overall response. 44  Ziegler–Nichols Tuning › First, set 𝐾𝐼 = 0 and 𝐾𝐷 = 0. Increase the gain 𝑲𝒑 until the output of the closed-loop system oscillates just on the edge of instability. › Once the value of 𝐾𝑝 (with 𝐾𝐼 = 0 and 𝐾𝐷 = 0) is found that brings the closed-loop system to the edge of stability, you reduce the gain 𝑲𝒑 to achieve what is known as the quarter amplitude decay. That is, the amplitude of the closed-loop response is reduced approximately to one- fourth of the maximum value in one oscillatory period. › A rule-of-thumb is to start by reducing the proportional gain 𝐾𝑝 by half. › The next step of the design process is to increase 𝑲𝑰 and 𝑲𝑫 manually to achieve a desired step response. 45  Example: › Step 1: set 𝐾𝐼 = 0 and 𝐾𝐷 = 0 and increase 𝐾𝑝 until the closed-loop system has sustained oscillations. › when 𝑲𝒑 = 𝟖𝟖𝟓. 𝟓, we have a sustained oscillation of magnitude 𝐴 = 1.9 › and period 𝑃 = 0.83 𝑠. 46  Example: › Reduce 𝑲𝒑 = 𝟖𝟖𝟓. 𝟓 by half as a first step to achieving a step response with approximately a quarter amplitude decay. › To accomplish this reduction, we refined the value of 𝐾𝑝 by slowly reducing the value from 𝑲𝒑 = 𝟒𝟒𝟐. 𝟕𝟓 to 𝑲𝒑 = 𝟑𝟕𝟎. 47 Proportional Integral Controller: Tuning 48 PID: Limitations 49 Questions? 50