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- 1. Chapter 1 Control Problem and Control Actions 1.1 Control problem In any control system, where the dynamic variable has to be maintained at the desired set point value, it is the controller which enables the requirement of the control objective to be met. The control design problem is the problem of determining the characteristics of the controller so that the controlled output can be: 1. Set to prescribed values called reference 2. Maintained at the reference values despite the unknown disturbances 3. Conditions (1) and (2) are met despite the inherent uncertainties and changes in the plant dynamic characteristics. 4. Maintained within some constrains. The first requirement above is called Tracking or stabilization depending on whether the set- point continuously changes or not, The second condition is called disturbance rejection. The third condition is called Robust tracking/stabilization and disturbance rejection. The fourth condition is called optimal tracking/stabilization and disturbance rejection. 1 1 Liquid level control system in a buffer tank
- 2. 1.1.1 Control assessment framework Control systems may be assessed in terms of 1. Stability, 2. Reference tracking and 3. Disturbance rejection. Consider the system in the figure below; We want to discuss and assess control system performance in terms of stability, reference tracking and disturbance rejection System response as a function of inputs: θo = θd + G1U but U = KE and E = θi − θo Therefore θo = θd + G1K(θi − θo) So θo(s) = G1(s)K(s) 1 + G1(s)K(s) θi(s) + 1 1 + G1(s)K(s) θd(s) Three properties of interest are: • Closed-loop system stability:- Depends upon making sure that the poles of 1 + G1(s)K(s) are in LHP of s-plane • Reference tracking performance:- Looks at the shape of time response (Speed, overshoot and steady state errors) • Disturbance rejection performance:- Looks at speed of response , size of peak disturbance and make sure that no steady state error due to disturbance. Consider the following transfer function which represents a heating system θo(s) = G1(s)U(s) Where G1(s) = 0.3 2s + 1 2
- 3. The input signal U(s) is the power in KW from the heater and the output signal θo(s) is the resulting temperature. The time constant of the system is 2 hours We wish to control the system behaviour. So we use proportional controller and unity feedback. Assume that the reference signal is a unit step input and the disturbance is given by θd(t) = 0.5 Control assessment of the system • Closed-loop system stability and response:- Open-loop transfer function Go(s) = G1(s)Kp = 0.3Kp 2s + 1 There is single open-loop pole at s = −0.5 Closed-loop transfer function GCL(s) = G1(s)K(s) 1 + G1(s)K(s) = 0.3Kp 2s + 1 + 0.3Kp So there is single closed-loop pole at s = −0.5 − 0.15Kp The System is stable for positive values of Kp • Reference tracking performance:- Output response to reference input G1(s)K(s) 1 + G1(s)K(s) θi(s) θo(s) = GCL(s)θi(s) = Final value theorem gives θo ss = lim s→0 sθo(s) = lim s→0 s G1(s)K(s) 1 + G1(s)K(s) θi(s) For this system , the steady state response θo ss = lim s→0 s 0.3Kp 1 2s + 1 + 0.3Kp s = 0.3Kp 1 + 0.3Kp 3
- 4. This implies that the steady state error reduces as Kp increases. In reality , there will be physical limitations on gain. ie actuator may not deliver (Actuator saturation.) Pole plot and response is as below The response speeds up with increasing Kp • Disturbance rejection performance:- Output response to disturbance input θo(s) = 1 1 + G1(s)K(s) θd(s) For this system: θo(s) = 2s + 1 2s + 1 + Kp 0.5 s Taking inverse transforms gives the disturbance time response θo(s) = 0.25Kp 0.5 + 0.15Kp 1 + 0.3Kpe−(0.5+0.15Kp)t As K becomes larger, the steady state value of the disturbance response gets smaller and the speed of response increases 4
- 5. 1.2 Control actions Given a general plant as shown Fig. 1.1 The manner in which the automatic controller produces the Figure 1.1: A general plant. control signal is called the control action. The control signal is produced by the controller, thus a controller has to be connected to the plant. The configuration may take either Close loop or Open loop as shown in Fig.1.2 and 1.3. Figure 1.2: Close loop Controlled system. Figure 1.3: Open loop Controlled system. 5
- 6. 1.2.1 Basic control actions Control actions may be further classified into control modes namely: • Discontinuous control mode: In discontinuous controllers, the manipulated variable y (Con- trol signal) changes between discrete values. Depending on how many different states the manipulated variable can assume, a distinction is made between two-position, three-position , multi-position controllers and floating controller. Compared to continuous controllers,discontinuous controllers operate on very simple switching of final controlling elements. If the system contains energy storing components, the controlled variable responds continuously, despite the step changes in the manipulated variable. If the corresponding time constants are large enough, good control results at small errors can even be reached with discontinuous controllers and simple control elements. • Continuous control mode:In continuous controllers, the manipulated variable can assume any value within the controller output range. The characteristic of continuous controllers usually exhibits proportional (P), integral (I) or differential (D) action • Composite mode: The characteristic of Composite controllers usually exhibits combinations of proportional (P), integral (I) or differential (D) action. Figure 1.4: Control actions. 6
- 7. € ‚ ƒ ƒ „ † ‡€ ˆ ‡ ‚‰ Š ‹ † † Controller O/P P ( ) ON 100% OFF 0% Neutral Zone error e ( ) 1 Two point control mode (Bang-Bang controller) 1.3 DISCONTINUOUS CONTROLLER MODES 7 1 100% 50% 0 – e1 + e1 0 error (e) Controller 0/P P e2
- 8. €‚ ƒ „ † ‡ˆ‰ ˆ‰ cc p KF – KF error (e) dp dt 0 0 € ‚ ƒ ‚„ ‚ 1 2 3 4 Floating Action Floating Action –KF Floating Action Neutral Zone Error (e) Controller O/P P(0) +KF 8
- 9. Application Controller O/p + K2 + K1 0 – K1 – K2 – e2 – e1 0 e1 e2 error 9 1.4 CONTINUOUS CONTROLLER MODES 1.4.1 Proportional Controller Mode €‚ ƒ€„ ‚ † ‡ ˆ‚‰ † † Š Kp P S ( ) E S ( )
- 10. Controller o/p P 100% 0% 100 % Saturation 0% Saturation error (e) P.B 7 t t t = t1 error (e) Controller o/p P = t t1 €‚ ƒ 10 Controller Process R s ( ) E s ( ) P s ( ) kp C s ( ) 1 1 + TS Fig. 2.3
- 11. m dd p dd pp l ll d€ l l lm pp l e d pb bpp lpble d ld ‚ ƒ ƒ „ ƒ † € ‚ 1 ƒ „ Controller o/p kF – kF 0 o error e ( ) dp dt e P ƒ † ‡ †ˆ ‡ 11
- 12. € ‚ ‚ƒ„ ‚ † ‚ ‚ ‚ † 1 T s i E s ( ) P s ( ) ‡ ˆ ‰ ‚Š‚ ‹ ‚ Œ 12 A Ti t error e A Controller o/p P = t t1 t
- 13. R s ( ) P s ( ) 1 1 + Ts C s ( ) 1 Tis (Integral Mode) (Process) 1 Tis Kp 1 + P S ( ) E S ( ) error (e) P-only I-only + P I only P + I Action { t = t1 t t t t t = t1 t = t1 t = t1 A t P K A = + p AKp Ti € ‚ 13
- 14. R( ) 1 1 + T C( ) + P I controller (Process) { { 1 T Kp 1 + € 14 13 Derivative Unlike P-only and I-only controls, D-control is a form of feed forward control. D-control anticipates the process conditions by analyzing the change in error. It functions to minimize the change of error. The primary benefit of D controllers is to resist change in the system, the most important of these being oscillations. The control output is calculated based on the rate of change of the error with time. The larger the rate of the change in error, the more pronounced the controller response will be. D-control correlates the controller output to the derivative of the error. This D-control behavior is mathematically illustrated in Equation p(t) = Td de dt Where • p(t) = controller output • Td = derivative time constant • de = change in error • dt = change in time
- 15. Mathematically, derivative control is the opposite of integral control. Although I-only controls exist, D-only controls do not exist. D- controls measure only the rate of change in error. D-controls do not know where the setpoint is, so it is usually used in conjunction with another method of control, such as P-only or a PI combination control. D-control is usually used for processes with rapidly changing process outputs. However, like the I-control, the D control is mathematically more complex than the P-control. Since it will take a computer algorithm longer to calculate a derivative or an integral than to simply linearly relate the input and output variables, adding a D- control slows down the controller’s response time. Unlike proportional and integral controllers, derivative controllers do not guide the system to a steady state because they do not act when error does not change even in presence of the error. Because of this property, D controllers must be coupled with P, I or PI controllers to properly control the system. € ‚ƒ „ „ † Kp (1 + d T S) P S ( ) E S ( )
- 16. R( ) (1 + K T p ) 1 1 + T Controller Process € ‚ ƒ„ € † ‡ ƒ ˆ error (e) P-only D-only + P D Action P + D Action { t t t t At t = t1 t = t1 t = t1 t = t1 15
- 17. Characteristics of + + controller € € ‚ƒ ‚ „ †‡ ˆ ‰ † † Š‹ Œ 1.5.3 Proportional + Integral + Derivative Controller Mode [+ + Mode] 1 T S i Kp 1 + P S ( ) E S ( ) + d T S 16
- 18. €‚ƒ„ €ƒ† ‡ ˆ € ‰ Š 17