Control Systems Lect. 4 PID Basil Hamed

Design Specifications often use design specifications to describe what the system should do and how it is done. These specifications are unique to each individual application and often include specifications about relative stability, steady-state accuracy (error), transient-response characteristics In some applications there may be additional specifications on sensitivity to parameter variations, that is, robustness, or disturbance rejection. Basil Hamed

Design Specifications The design of linear control systems can be carried out in either the time domain or the frequency domain. For instance, steady-state accuracy is often specified with respect to a step input, a ramp input, or a parabolic input, and the design to meet a certain requirement is more conveniently carried out in the time domain. Other specifications such as maximum overshoot, rise time, and settling time are all defined for a unit-step input and, therefore, are used specifically for time-domain design Basil Hamed

Design Specifications Thus, the choice of whether the design should be conducted in the time domain or the frequency domain depends often on the preference of the designer. We should be quick to point out, however, that in most cases, time-domain specifications such as maximum overshoot, rise time, and settling time are usually used as the final measure of system performance. Basil Hamed

Various Controller Configurations Various controller configurations in control-system compensation, Series or cascade compensation, Feedback compensation, (c) State-feedback control, (d) Series-feedback compensation, (e) Forward compensation with series compensation, (f) Feedforward compensation. Basil Hamed

Fundamental Principles of Design After a controller configuration is chosen, the designer must choose a controller type that, with proper selection of its element values, will satisfy all the design specifications. Engineering practice usually dictates that one choose the simplest controller that meets all the design specifications. In most cases, the more complex a controller is, the more it costs, the less reliable it is, and the more difficult it is to design. Choosing a specific controller for a specific application is often based on the designer's past experience and sometimes intuition, and it entails as much art as it does science. Basil Hamed

PID This chapter examines a particular control structure that has become almost universally used in industrial control. It is based on a particular fixed structure controller family, the so-called PID controller family. These controllers have proven to be robust and extremely beneficial in the control of many important applications. PID stands for: P (Proportional) I (Integral) D (Derivative) Basil Hamed

PID One of the commonly used controllers in the compensation schemes is a PID controller PID applies a signal to the process that is proportional to the actuating signal in addition to adding integral and derivative of the actuating signal. These signal components are easily realized and visualized in the time domain, PID controllers are commonly designed using time-domain methods. Basil Hamed

Historical Note PID control is one of the earlier control strategies Early feedback control devices used the ideas of proportional, integral and derivative action in their structures. Its early implementation was in pneumatic devices, followed by vacuum and solid state analog electronics, before arriving at today’s digital implementation of microprocessors.. Basil Hamed

Historical Note It has a simple control structure which was understood by plant operators and which they found relatively easy to tune. According to a survey for process control systems conducted in 1989, more than 95 of the control loops were of the PID type Basil Hamed

Why PID Control Simple, easy to use Wide Application: Petrochemical, Pharmaceuticals, Food, Chemical, Aerospace and Semiconductor, etc. Robust: Insensitive to changes to plant parameter and disturbance. Over 95% control loops are PID with two exceptions: On/off control for those with low control requirement loops Advanced control for those difficult systems and with high control quality.

General Control System Basil Hamed

PID Block Diagram Basil Hamed

PID Mathematically Consider the input error variable, e(t): Let p(t) = Kp*e(t) {p proportional to e } Let i(t) = Ki*∫e(t)dt {i integral of e } Let d(t) = Kd* de(t)/dt {d derivative of e} Basil Hamed

Four Modes of Controllers Four modes of control commonly used for most applications are: Proportional (P) Proportional plus Reset (PI) Proportional plus Rate (PD) Proportional plus Reset plus Rate (PID) Basil Hamed

Four Modes of Controllers Basil Hamed

Tips for Designing a PID Controller When you are designing a PID controller for a given system, follow the steps shown below to obtain a desired response. 1. Obtain an open-loop response and determine what needs to be improved 2. Add a proportional control to improve the rise time 3. Add a derivative control to improve the overshoot 4. Add an integral control to eliminate the steady-state error Adjust each of Kp, Ki, and Kd until you obtain a desired overall response. Lastly, please keep in mind that you do not need to implement all three controllers (proportional, derivative, and integral) into a single system, if not necessary. For example, if a PI controller gives a good enough response, then you don't need to implement derivative controller to the system. Keep the controller as simple as possible.

The Characteristics of P, I, and D controllers A proportional controller (Kp) will have the effect of reducing the rise time and will reduce, but never eliminate, the steady-state error. An integral control (Ki) will have the effect of eliminating the steady-state error, but it may make the transient response worse. A derivative control (Kd) will have the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response.

Proportional Control By only employing proportional control, a steady state error occurs. Proportional and Integral Control The response becomes more oscillatory and needs longer to settle, the error disappears. Proportional, Integral and Derivative Control All design specifications can be reached.

The Characteristics of P, I, and D controllers Basil Hamed

PI Equations 11/13/2018 Basil Hamed

PI Characteristics Basil Hamed

PI Control Basil Hamed

PD Controller 11/13/2018 Basil Hamed

PD Controller Basil Hamed

PD Controller Basil Hamed

PID Controller Basil Hamed

PID Controller Characteristics Basil Hamed

Design of PID Controllers Based on the knowledge of P, I and D – trial and error – manual tuning – simulation 11/13/2018 Basil Hamed

Example Problem Suppose we have a simple mass, spring, and damper problem. Basil Hamed

Example Problem The modeling equation of this system is Taking the Laplace transform of the modeling equation, we get The transfer function between the displacement and the input then becomes Let M = 1 kg b = 10 N s/m k = 20 N/m F = 1 N Plug these values into the above transfer function Basil Hamed

Example Problem The goal of this problem is to show you how each of kp, ki , and kd contributes to obtain Fast rise time Minimum overshoot No steady-state error Basil Hamed

Open-Loop Control - Example num=1; den=[1 10 20]; step(num,den) The DC gain of the plant transfer function is 1/20, so 0.05 is the final value of the output to an unit step input. This corresponds to the steady-state error of 0.95, quite large indeed. Furthermore, the rise time is about one second, and the settling time is about 1.5 seconds. Let's design a controller that will reduce the rise time, reduce the settling time, and eliminate the steady-state error

Proportional Control - Example The proportional controller (Kp) reduces the rise time, increases the overshoot, and reduces the steady-state error. MATLAB Example Kp=300; num=[Kp]; den=[1 10 20+Kp]; t=0:0.01:2; step(num,den,t)   K=300 K=100

Proportional - Derivative - Example Now, let's take a look at a PD control. From the table shown above, we see that the derivative controller (Kd) reduces both the overshoot and the settling time. The closed-loop transfer function of the given system with a PD controller is: Kp=300; Kd=10; num=[Kd Kp]; den=[1 10+Kd 20+Kp]; t=0:0.01:2; step(num,den,t) Kd=10 Kd=20 This plot shows that the derivative controller reduced both the overshoot and the settling time, and had a small effect on the rise time and the steady-state error

Proportional - Integral - Example The integral controller (Ki) decreases the rise time, increases both the overshoot and the settling time, and eliminates the steady-state error MATLAB Example Kp=30; Ki=70; num=[Kp Ki]; den=[1 10 20+Kp Ki]; t=0:0.01:2; step(num,den,t) Ki=70 Ki=100 We have reduced the proportional gain (Kp) 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

Proportional-Integral-Derivative Example Now, let's take a look at a PID controller. The closed-loop transfer function of the given system with a PID controller is: After several trial and error runs, the gains Kp = 350, ki = 300, and kd = 50 provided the desired response Now, we have obtained a closed-loop system with no overshoot, fast rise time, and no steady-state error Basil Hamed

Proportional-Integral-Derivative(PID) Control For control over steady-state and transient errors we can combine all three control strategies we have discussed is PID control. PID combination is sometimes able to provide an acceptable degree of error reduction simultaneously with stability and damping. Basil Hamed

PID by MATLAB Implement Example Consider a system with transfer function T=10K/[(1*2)s^2+ (1+2)s+1+AK] Change it manually Basil Hamed

PID by MATLAB Implement Example 2 Consider a system with transfer function T=(K*S+Ki)/[S^3+ 3*S^2+(2+K)*S+Ki] Basil Hamed

Basil Hamed

Tuning a PID Automatic PID Tuning Ziegler and Nichols Tuning There is a fairly standard procedure for tuning PID controllers: Trial and Error Tuning Automatic PID Tuning Ziegler and Nichols Tuning Cohen-Coon Tuning Basil Hamed

Trial and Error Tuning The trial and error tuning method is based on guess-and-check. In this method, the proportional action is the main control, while the integral and derivative actions refine it. The controller gain, Kp, is adjusted with the integral and derivative actions held at a minimum, until a desired output is obtained. Basil Hamed

Automatic PID Tuning MATLAB provides tools for automatically choosing optimal PID gains which makes the trial and error process described above unnecessary. You can access the tuning algorithm directly using pidtune or through a nice graphical user interface (GUI) using pidtool. Basil Hamed

Ziegler-Nichols Tuning of PID Regulators J. G. Ziegler and N. B. Nichols recognized that the step responses of a large number of processes control systems exhibits a process reaction curve Ziegler & Nichols gave two methods for tuning the controller Open-Loop Tuning Method or Process Reaction Method Based on a stability boundary Basil Hamed

Ziegler-Nichols Open-Loop Tuning Method or Process Reaction Method This method remains a popular technique for tuning controllers that use proportional, integral, and derivative actions. The Ziegler-Nichols open-loop method is also referred to as a process reaction method, because it tests the open-loop reaction of the process to a change in the control variable output. This basic test requires that the response of the system be recorded, preferably by a plotter or computer. Once certain process response values are found, they can be plugged into the Ziegler-Nichols equation with specific multiplier constants for the gains of a controller with either P, PI, or PID actions. Basil Hamed

Ziegler-Nichols Open-Loop Tuning Method or Process Reaction Method To use the Ziegler-Nichols open-loop tuning method, you must perform the following steps: 1. Make an open loop step test 2. From the process reaction curve determine the transportation lag =L or dead time= td, the time constant or time for the response to change = Basil Hamed

Ziegler-Nichols Open-Loop Tuning Method or Process Reaction Method Basil Hamed

Ziegler-Nichols Open-Loop Tuning Method or Process Reaction Method Basil Hamed

Sample of MATLAB Implement Consider a system with transfer function T=2/[(S+2)*(0.18*S^2+0.6*S+1)] With L=0.38;R=1; 1 Basil Hamed

4 2 3 5 Basil Hamed

Basil Hamed

Ziegler-Nichols closed-loop tuning method The Ziegler-Nichols closed-loop tuning method allows you to use the ultimate gain value, Ku, and the ultimate period of oscillation, Pu, to calculate K . It is a simple method of tuning PID controllers and can be refined to give better approximations of the controller. You can obtain the controller constants K , Ti , and Td in a system with feedback. The Ziegler-Nichols closed-loop tuning method is limited to tuning processes that cannot run in an open-loop environment Determining the ultimate gain value, Ku, is accomplished by finding the value of the proportional-only gain that causes the control loop to oscillate indefinitely at steady state Basil Hamed

Ziegler-Nichols closed-loop tuning method Basil Hamed

Z.-N. Tuning of PID Regulators Method Two Based on a stability boundary Basil Hamed

Worked out Example Problem You're a controls engineer working for Flawless Design company when your optimal controller breaks down. As a backup, you figure that by using coarse knowledge of a classical method, you may be able to sustain development of the product. After adjusting the gain to one set of data taken from a controller, you find that your ultimate gain is 4.3289. From the adjusted plot below, determine the type of loop this graph represents; then, please calculate K, Ti, and Td for all three types of controllers. Basil Hamed

Worked out Example Solution From the fact that this graph oscillates and is not a step function, we see that this is a closed loop. Thus, the values will be calculated accordingly. We're given the Ultimate gain, Ku = 4.3289. From the graph below, we see that the ultimate period at this gain is Pu = 6.28 Basil Hamed

Worked out Example From this, we can calculate the Kc, Ti, and Td for all three types of controllers. The results are tabulated below. (Results were calculated from the Ziegler-Nichols closed-loop equations.) Basil Hamed

Numerical Example Consider a plant with a model given by Find the parameters of a PID controller using the Z-N oscillation method. Obtain a graph of the response to a unit step input reference and to a unit step.

Example Basil Hamed

Example Hence, from Table , we have K= 0.6 ku=4.8, Ti=1/2 Pu=1.81, Td=1/8 Pu=0.45 Ku = 8 and Pu =3.62 Basil Hamed

Example Kp = 4.8, Ki = 2.64, Kd = 2.16 Basil Hamed

General comments about controller tuning The different methodologies of controller tuning, known as Ziegler-Nichols method have been illustrated. It is to be remembered that the recommended settings are obtained from extensive experimentation with number of different processes; there is no theoretical basis behind these selections. As a result, a better combination of the P, I, D values may always be found, that will give less oscillation and better settling time. But with no a-priori knowledge of the system, it is always advisable to perform the experimentation and select the controller settings, obtained from Ziegler-Nichols method. But there is always scope for improving the performance of the controller by fine-tuning. So, Ziegler-Nichols method provides initial settings that will give satisfactory, result, but it is always advisable to fine-tune the controller further for the particular process and better performance is expected to be achieved Basil Hamed

General Comments about Controller Tuning Nowadays digital computers are replacing the conventional analog controllers. P-I-D control actions are generated through digital computations. Digital outputs of the controllers are converted to analog signals before they are fed to the actuators. In many cases, commercial software are available for Auto tuning the process. Basil Hamed

General Comments about Controller Tuning Why is the proportional gain Kp for PI control is less than the value for P-only control? Why Kp for PID control is more than that PI? Solution Addition of integral control action to P-only control tends to make the closed loop system more oscillatory; in order to overcome this problem, the suggested value of Kp with ZN tuning is reduced. Addition of derivative action again damps down the oscillation; as a result larger value of Kp in a PID controller is permissible. Basil Hamed