Advanced Control Systems ACS Lecture8 Splane Design Dr

Advanced Control Systems (ACS) Lecture-8 S-plane Design Dr. Imtiaz Hussain Assistant Professor email: imtiaz. hussain@faculty. muet. edu. pk URL : http: //imtiazhussainkalwar. weebly. com/ 1

Lecture Outline 2

Lead Compensation • Generally Lead compensators are represented by following transfer function • or 3

Lead Compensation 4

Example-1 • Consider the position control system shown in following figure. • It is desired to design an Electronic lead compensator Gc(s) so that the dominant closed poles have the damping ratio 0. 5 and undamped natural frequency 3 rad/sec. 5

Step-1 (Example-1) • Determine the characteristics of given system using root loci. • The damping ratio of the closed-loop poles is 0. 158. • The undamped natural frequency of the closed-loop poles is 3. 1623 rad/sec. • Because the damping ratio is small, this system will have a large overshoot in the step response and is not desirable. 6

Step-2 (Example-1) • From the performance specifications, determine the desired location for the dominant closed-loop poles. • Desired performance Specifications are: § It is desired to have damping ratio 0. 5 and undamped natural frequency 3 rad/sec. 7

Step-3 (Exampl-1) • To calculate the angle of deficiency apply Angle Condition at desired closed loop pole. Desired Closed Loop Pole -2 -1 100. 8 o -2 120 o -1 8

Step-5 (Exampl-1) Solution-1 • Solution-1 – If we choose the zero of the lead compensator at s = -1 so that it will cancel the plant pole at s =-1, then the compensator pole must be located at s =-3. 9

Step-5 (Example-1) Solution-1 • The pole and zero of compensator are determined as 10

Step-6 (Example-1) Solution-1 • The Value of Kc can be determined using magnitude condition. 11

Final Design Check Solution-1 12

Step-5 (Exampl-1) Solution-2 • Solution-2 -2 -1 90 o 49. 2 o -3 -2 -1 13

Step-5 (Exampl-1) Solution-2 • Solution-2 -2 -1 90 o 49. 2 o -3 -2 -1 14

Exampl-2 • Design a lead compensator following system. • The damping ratio of closed loop poles is 0. 5 and natural undamped frequency 2 rad/sec. It is desired to modify the closed loop poles so that natural undamped frequency becomes 4 rad/sec without changing the damping ratio. 15

Example-3 • Consider the model of space vehicle control system depicted in following figure. • Design lead compensator such that the damping ratio and natural undamped frequency of dominant closed loop poles are 0. 5 and 2 rad/sec. 16

Lag Compensation • 17

Lag Compensation 18

Lag Compensation • 19

Lag Compensation • 20

Lag Compensation • 21

Design Procedure • The procedure for designing lag compensators by the rootlocus method may be stated as follows. • We will assume that the uncompensated system meets the transient-response specifications by simple gain adjustment. • If this is not the case then we need to design a lag-lead compensator which we will discuss in next few classes. 22

Example-4 • 23

Example-4 (Step-1) • The dominant closed-loop poles of given system are s = -0. 3307 ± j 0. 5864 24

Example-4 (Step-3) • 25

Example-4 (Step-3) 26

Example-4 (Step-4) • 27

Example-4 (Step-4) Solution-1 • Place the zero and pole of the lag compensator at s=– 0. 05 and s=– 0. 005, respectively. • The transfer function of the lag compensator becomes • Open loop transfer function is given as 28

Example-4 (Step-5) Solution-1 • Root locus of uncompensated and compensated systems. • New Closed Loop poles are 29

Example-4 (Step-6) Solution-1 • 30

Example-1 (Step-6) Solution-1 • Then the compensator transfer function is given as 31

Example-4 (Final Design Check) Solution-1 • The compensated system has following open loop transfer function. • Static velocity error constant is calculated as 32

Example-4 (Step-4) Solution-2 • Place the zero and pole of the lag compensator at s=– 0. 01 and s=– 0. 001, respectively. • The transfer function of the lag compensator becomes • Open loop transfer function is given as 33

Example-4 (Step-5) Solution-2 • Root locus of uncompensated and compensated systems. • New Closed Loop poles are 34

Example-5 • 35

Lag-Lead Compensation • Lag-Lead compensators are represented by following transfer function • Where Kc belongs to lead portion of the compensator.

Lag-Lead Compensation

Example-6 (Case-1) • Consider the control system shown in following figure • The damping ratio is 0. 125, the undamped natural frequency is 2 rad/sec, and the static velocity error constant is 8 sec– 1. • It is desired to make the damping ratio of the dominant closed-loop poles equal to 0. 5 and to increase the undamped natural frequency to 5 rad/sec and the static velocity error constant to 80 sec– 1. • Design an appropriate compensator to meet all the performance specifications.

Example-6 (Case-1) • From the performance specifications, the dominant closed-loop poles must be at • Since • Therefore the phase-lead portion of the lag–lead compensator must contribute 55° so that the root locus passes through the desired location of the dominant closed-loop poles.

Example-6 (Case-1) •

Example-6 (Case-1) •

Example-6 (Case-1) •

Example-6 (Case-1) • Now the transfer function of the designed lag–lead compensator is given by

PID • PID Stands for – P Proportional – I Integral – D Derivative 44

Four Modes of Controllers • Each mode of control has specific advantages and limitations. • On-Off (Bang) Control • Proportional (P) • Proportional plus Integral (PI) • Proportional plus Derivative (PD) • Proportional plus Integral plus Derivative (PID) 45

On-Off Control • This is the simplest form of control. Set point Error Output 46

Proportional Control (P) • In proportional mode, there is a continuous linear relation between value of the controlled variable and position of the final control element. - • Output of proportional controller is • The transfer function can be written as 47

Proportional Controllers (P) • As the gain is increased the system responds faster to changes in set-point but becomes progressively underdamped and eventually unstable. 48

Proportional Plus Integral Controllers (PI) • Integral control describes a controller in which the output rate of change is dependent on the magnitude of the input. • Specifically, a smaller amplitude input causes a slower rate of change of the output. 49

Proportional Plus Integral Control (PI) + + - 50

Proportional Plus Integral Control (PI) • The transfer function can be written as 51

Proportional Plus derivative Control (PD) + + - 52

Proportional Plus derivative Control (PD) • The transfer function can be written as 53

Proportional Plus derivative Control (PD) • The stability and overshoot problems that arise when a proportional controller is used at high gain can be mitigated by adding a term proportional to the time-derivative of the error signal. The value of the damping can be adjusted to achieve a critically damped response. 54

Proportional Plus derivative Control (PD) • The higher the error signal rate of change, the sooner the final control element is positioned to the desired value. • The added derivative action reduces initial overshoot of the measured variable, and therefore aids in stabilizing the process sooner. • This control mode is called proportional plus derivative (PD) control because the derivative section responds to the rate of change of the error signal 55

Proportional Plus Integral Plus Derivative Control (PID) + - + + 56

Proportional Plus Integral Plus Derivative Control (PID) 57

Proportional Plus Integral Plus Derivative Control (PID) • Although PD control deals neatly with the overshoot and ringing problems associated with proportional control it does not cure the problem with the steady-state error. Fortunately it is possible to eliminate this while using relatively low gain by adding an integral term to the control function which becomes 58

The Characteristics of P, I, and D controllers CL RESPONSE RISE TIME OVERSHOOT SETTLING TIME S-S ERROR Kp Decrease Increase Small Change Decrease Ki Decrease Increase Eliminate Kd Small Change Decrease Small Change 59

Empirical Tuning (Zeigler-Nichol’s First Method) • In the first method, we obtain experimentally the response of the plant to a unit-step input. • If the plant involves neither integrator(s) nor dominant complexconjugate poles, then such a unit-step response curve may look S-shaped 60

Zeigler-Nichol’s First Method • This method applies if the response to a step input exhibits an S-shaped curve. • Such step-response curves may be generated experimentally or from a dynamic simulation of the plant. Table-1 61

Zeigler-Nichol’s Second Method • 62

Zeigler-Nichol’s Second Method • Thus, the critical gain Kcr and the corresponding period Pcr are determined. Table-2 63

Example-7 1 L t 64

Example-7 65

Example-8 • 66

Example-8 • Transfer function of the plant is • Since plant has an integrator therefore Ziegler-Nichol’s first method is not applicable. • According to second method proportional gain is varied till sustained oscillations are produced. • That value of Kc is referred as Kcr. 67

Example-8 • 68

Example-8 • 69

Example-8 • 70

Example-8 • 71

Example-8 • Transfer function of PID controller is thus obtained as 72

Example-8 73

To download this lecture visit http: //imtiazhussainkalwar. weebly. com/ END OF LECTURE-8 74