Radar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D.

Transcription

1 Radar Dish ME 304 CONTROL SYSTEMS Mechanical Engineering Deartment, Middle East Technical University Armature controlled dc motor Outside θ D outut Inside θ r inut r θ m Gearbox Control Transmitter θ D dc amlifier Control Transformer Prof. Dr. Y. Samim Ünlüsoy Prof. Dr. Y. Samim Ünlüsoy 1

2 COURSE OUTLINE I. INTRODUCTION & BASIC CONCEPTS II. MODELING DYNAMIC SYSTEMS III. CONTROL SYSTEM COMPONENTS IV. STABILITY V. TRANSIENT RESPONSE VI. STEADY STATE RESPONSE VII. DISTURBANCE REJECTION CH VIII VIII. CONTROL ACTIONS & CONTROLLERS IX. FREQUENCY RESPONSE ANALYSIS X. SENSITIVITY ANALYSIS xi. ROOT LOCUS ANALYSIS Prof. Dr. Y. Samim Ünlüsoy 2

3 CONTROL ACTIONS & CONTROLLERS OBJECTIVES In this chater : Basic control actions and controllers, and their characteristics ti will be examined. Their alications will be introduced. Prof. Dr. Y. Samim Ünlüsoy 3

4 CONTROL ACTIONS & CONTROLLERS A controller is a device which roduces a control signal that results in the reduction of the deviation of controlled outut from the desired value. Prof. Dr. Y. Samim Ünlüsoy 4

5 CONTROL ACTIONS & CONTROLLERS The inut to the controller is the actuating error E a (s), which is the difference between the system resonse B(s), as measured by a sensor, and the reference signal R(s), which reresents the desired d system resonse. R(s) + E a (s) M(s) C(s) Controller G(s) - B(s) H(s) Prof. Dr. Y. Samim Ünlüsoy 5

6 CONTROL ACTIONS & CONTROLLERS Controller roduces the maniulated inut at its outut to be alied to the actuator. t R(s) + - E a (s) Controller M(s) G(s) C(s) B(s) H(s) Prof. Dr. Y. Samim Ünlüsoy 6

7 CONTROL ACTIONS & CONTROLLERS A controller is required to shae the error signal such that certain control criteria or secifications, are satisfied. These criteria may involve : - Transient resonse characteristics, -Steady-state error, - Disturbance rejection, - Sensitivity to arameter changes. Prof. Dr. Y. Samim Ünlüsoy 7

8 CONTROL ACTIONS & CONTROLLERS Basic Control Actions : A controller roduces a control signal (maniulated inut) according to its inut which is the actuating error. The algorithm that relates the error and the control signal is called the control action (law)(strategy). Control Action Actuating error Controller Control signal Prof. Dr. Y. Samim Ünlüsoy 8

9 CONTROL ACTIONS & CONTROLLERS The most commonly used Control Actions are : 1. Two osition (on-off, off, bang-bang), bang), 2. Proortional (P-control), 3. Derivative (D-control), 4. Integral (I-control). Controllers to rovide these control actions (with the excetion of D-control) and the combinations of the last three are commercially available. Prof. Dr. Y. Samim Ünlüsoy 9

10 TWO POSITION CONTROL (Ogata ) 64) Two Position Control For this tye of control, the outut of the controller can have only one of the two ossible values, M or M 1 2. e(t) M 1 m(t ) M 2 e(t) controller m(t) m(t)= M1 for e(t)>0 M2 for e(t)<0 Prof. Dr. Y. Samim Ünlüsoy 10

11 TWO POSITION CONTROL The value of M 2 either is usually i) zero, in which case the e(t) M 1 m(t) controller is called the 0 on-off off controller, or On-off controller ii) equal to -M 1, in which M case, the controller is e(t) m(t) called the bang-bang -M controller. Bang-bang controller Prof. Dr. Y. Samim Ünlüsoy 11

12 TWO POSITION CONTROL In ractice, the two osition controllers exhibit somewhat significant hysteresis (usually due to friction), sometimes introduced intentionally to revent remature wear-out of the switching hardware. In this case the outut e(t) 1 m(t) switches to M only 1 M after the actuating error 2 becomes ositive by an amount d. Similarly it Differential ga switches back to M only 2 after the actuating error becomes equal to -d. M 1 d Prof. Dr. Y. Samim Ünlüsoy 12

13 TWO POSITION CONTROL The existence of a differential ga reduces the accuracy of the control system, but it also reduces the frequency of switching which results in longer oerational life. e(t) M 1 m(t) M 2 Differential ga Prof. Dr. Y. Samim Ünlüsoy 13

14 TWO POSITION CONTROL EXAMPLE a Consider the water level control system illustrated in the figure. Solenoid valve Inut Switch + _ v q i h Float R C q o Prof. Dr. Y. Samim Ünlüsoy 14

15 TWO POSITION CONTROL EXAMPLE b Assume at first that the tank is emty. In this case, the solenoid will be energized oening the valve fully. h R Hence rovided that q >q i o, the water level will rise. q i C + _ v q o dh q-q i o =C h=rqo dt RC dh dt +h=rq i Prof. Dr. Y. Samim Ünlüsoy 15

16 TWO POSITION CONTROL EXAMPLE c The system is of first order and the variation of water level versus time is shown by the filling curve. h(t) Filling curve (q i >q o ) Emtying curve (q i =0) RC dh dt +h=rq i t If, at some time t o, the solenoid is de-energized energized closing the valve comletely, q i =0, then the water in the tank will drain off. The variation of the water level in the tank is now shown by the emtying curve. dh RC +h= 0 dt Prof. Dr. Y. Samim Ünlüsoy 16

17 TWO POSITION CONTROL EXAMPLE d If the switch is adjusted for a desired water level, the inut q i will be on or off (either a ositive constant or zero) deending on the difference between the desired and the actual water levels. If the clearances and friction in the ivot oints are considered, the switching characteristics are likely to have a certain amount of differential ga. q i 2d e(t)=h d -h Prof. Dr. Y. Samim Ünlüsoy 17

18 TWO POSITION CONTROL EXAMPLE e Therefore during the actual oeration, inut will be on until the water level exceeds the desired level by half the differential ga. h d q i 0 h(t) 2d Filling curve Emtying (q i >q o ) curve (q i =0) Then the solenoid valve will be shut off until the water level dros below the desired level by half the differential ga. The water level will continuously oscillate about the desired level. t Prof. Dr. Y. Samim Ünlüsoy 18

19 TWO POSITION CONTROL EXAMPLE f It should be noted that, the smaller the differential ga is, the smaller is the deviation from the desired level. But on the other hand, the number of switch on and off s increases. Differential ga small large Δt Prof. Dr. Y. Samim Ünlüsoy 19

20 PROPORTIONAL CONTROL Ogata.65 Proortional Control (P-control) With this tye of control action, a control signal roortional to error signal is roduced. m(t)=k e (t) a M(s) =K E (s) a R(s) + E a (s) M(s) C(s) K G(s) - B(s) H(s) K : roortional gain. Prof. Dr. Y. Samim Ünlüsoy 20

21 PROPORTIONAL CONTROL A roortional controller is essentially an amlifier with an adjustable gain. The value of K should be selected to satisfy the requirements of stability, accuracy, and satisfactory transient resonse, as well as satisfactory disturbance rejection characteristics. Prof. Dr. Y. Samim Ünlüsoy 21

22 PROPORTIONAL CONTROL In general, For small values of K, the corrective action is slow articularly for small errors. For large values of K, the erformance of the control system stem is imroved. But this may lead to instability. Usually, a comromise is necessary in selecting a roer gain. If this is not ossible, then roortional control action is used with some other control action(s). Prof. Dr. Y. Samim Ünlüsoy 22

23 DERIVATIVE CONTROL Derivative Control (D-control) In this case the control signal of the controller is roortional to the derivative (sloe) of the error signal. m(t)=kk d de (t) () a dt M(s) =Kd s E (s) a Derivative control action is never used alone, since it does not resond to a constant error, however large it may be. K d : derivative gain (adjustable). Prof. Dr. Y. Samim Ünlüsoy 23

24 DERIVATIVE CONTROL Derivative control action resonds to the rate of change of error signal and can roduce a control signal before the error becomes too large. As such, derivative control action anticiates the error, takes early corrective action, and tends to increase the stability of the system. Prof. Dr. Y. Samim Ünlüsoy 24

25 DERIVATIVE CONTROL Derivative control action has no direct effect on steady state t error. But it increases the daming in the system and allows a higher value for the oen loo gain K which reduces the steady state error. Prof. Dr. Y. Samim Ünlüsoy 25

26 DERIVATIVE CONTROL Derivative control, however, has disadvantages d as well. It amlifies noise signals coming in with the error signal and may saturate the actuator. It cannot be used if the error signal is not otd differentiable. e tabe Thus derivative control is used only together with some other control action! Prof. Dr. Y. Samim Ünlüsoy 26

27 INTEGRAL CONTROL Ogata.65 Integral Control (I-control) With this tye of control action, control signal is roortional to the integral of the error signal. M(s) K = m(t)= K i e (t)dt a E (s) a s i R(s) + E a (s) K M(s) C(s) i G(s) - s B(s) H(s) K i : integral gain (adjustable). Prof. Dr. Y. Samim Ünlüsoy 27

28 INTEGRAL CONTROL It is obvious that even a small error can be detected, t d since integral control roduces a control signal roortional to the area under the error signal. Hence, integral control increases the accuracy of the system. Integral control is said to look at the ast of the error signal. e(t) Large error Small error small area large area t Prof. Dr. Y. Samim Ünlüsoy 28

29 INTEGRAL CONTROL Remember that each s term in the denominator of the oen loo transfer function increases the tye of the system by one, and thus reduces the steady state error. The use of integral controller will increase the tye of the oen loo transfer function by one. Note, however, that for zero error signal, the integral control may still roduce a constant control signal which may in turn lead to instability. Prof. Dr. Y. Samim Ünlüsoy 29

30 PROPORTIONAL + DERIVATIVE (PD) CONTROL Control action with roortional + derivative (PD) control is given by m(t)=k e(t)+k T d de(t) M(s) =K E(s) ( ) dt 1+T d s R(s) + - E(s) K ( 1+T s) d M(s) G(s) C(s) B(s) H(s) T d : derivative time. Prof. Dr. Y. Samim Ünlüsoy 30

31 PROPORTIONAL + INTEGRAL (PI) CONTROL Control action with roortional + integral (PI) control is given by m(t)=k e(t)+k 1 T i e(t)dt M(s) 1 =K 1+ E(s) Ti s R(s) + - E(s) K 1 1+ Ts i M(s) G(s) C(s) T i B(s) H(s) : integral time Prof. Dr. Y. Samim Ünlüsoy 31

32 PROPORTIONAL + INTEGRAL + DERIVATIVE (PID) CONTROL Control action with roortional + integral + derivative (PID) control is given by K de(t) m(t)=ke(t)+kt d + e(t)dt dt T i R(s) + E(s) 1 M(s) K 1+T d s+ - Ts i M(s) 1 =K 1+Tds+ E(s) T i s B(s) H(s) Prof. Dr. Y. Samim Ünlüsoy 32

33 REALIZATION OF CONTROLLERS Comonents in most control systems can be mechanical, electrical, hydraulic or neumatic. In some cases, sensors, controllers, and actuators may be obtained from combinations such as electro- mechanical or electro-hydraulic comonents. See Ogata for various realizations of electrical, hydraulic, and neumatic controllers. Prof. Dr. Y. Samim Ünlüsoy 33

34 CONTROL ACTIONS - Proortional Control Position Control of an Inertial Load θ r (s) + - Disturbance Controller + + K + - Dc motor+load C 1 Js+b s θ(s) K b K : motor torque constant, t R a : armature resistance, C=K/ R a K b : motor back emf constant, J : motor + load inertia, b : viscous daming coefficient. Prof. Dr. Y. Samim Ünlüsoy 34

35 CONTROL ACTIONS - Proortional Control Initially neglect disturbance, i.e. D(s)=0. θ r (s) + - Controller K + - Dc motor+load C 1 Js+b s θ(s) K b KC θ(s) = J θ r(s) 2 b+k K C bc s + s+ J J 2 K C b+k bc ω n = 2ξω n= J J b+kbc 1 ξ = 2 JC K Prof. Dr. Y. Samim Ünlüsoy 35

36 2 KC n ω = J b+kbc 1 Proortional Control CONTROL ACTIONS ξ = 2 JC K β ω t It is noted that increasing r the roortional gain : i) Undamed natural t frequency of the system is increased, thus the seed of resonse is increased. π β 1 d = β= tan ω ξω d π = ω 2 ω =ω ξ d d n 1 n ξ -π ξ 1-ξ 2 M =e ii) The daming ratio is decreased, thus the relative stability of the system is ts ( 5% ) reduced. Note : ξω n 3 t 5% = ξωn 4 t 2% = ξωn is not affected by K! s ( ) Prof. Dr. Y. Samim Ünlüsoy 36

37 CONTROL ACTIONS - Proortional Control Now, let us examine the steady state resonse. θ r (s) + - Controller K + - Dc motor+load C Js+b 1 s θ(s) K b θ r (s) + - Js 2 K C ( ) + b+k C s b θ(s) Oen loo transfer function : K C G(s)= s Js+ b+k b C ( ) Prof. Dr. Y. Samim Ünlüsoy 37

38 CONTROL ACTIONS - Proortional Control Classify the oen loo transfer function. First convert to standard form. KC G(s)= s Js+ b+kb C Thus, ( ) System tye is 1. The oen loo gain is KC b+kbc G(s)= J s s+1 b+kb C KC K= b+k C b Prof. Dr. Y. Samim Ünlüsoy 38

39 CONTROL ACTIONS - Proortional Control System tye is 1. The oen loo gain is K C K= b+k C b Thus the steady state t error due to unit ste inut will be zero. The steady state error due to unit ram inut will be finite. 1 b+kbc 1 e ss = = K C K The steady state error due to unit ram inut can be minimized by using a larger value of the roortional gain. Prof. Dr. Y. Samim Ünlüsoy 39

40 CONTROL ACTIONS - Proortional Control Now let us consider disturbance inut only. Disturbance Controller K Dc motor+load C Js+b K b 1 s θ(s) D(s) + - Js 2 C ( ) + b+k C s b θ(s) In this case, the outut itself K will be the error. Prof. Dr. Y. Samim Ünlüsoy 40

41 CONTROL ACTIONS - Proortional Control In this case, the outut itself will be the error. D(s) C θ(s) ) 2 Js + + ( b+kbc) s - K C 1 1 e ss=lims 0s = 2 ( ) b s K Js + b+k C s+kc It is observed that for minimum steady state error due to a ste disturbance, and a ram inut, K must be as large as stability requirements allow. Prof. Dr. Y. Samim Ünlüsoy 41

42 CONTROL ACTIONS Proortional + Derivative Control If a roortional + derivative controller is used : θ r (s) + - Controller ( ) K T s+1 d Disturbance + + Js Dc motor+load 2 C ( ) + b+k C s b θ(s) Neglecting disturbance ( ) θ(s) K C T s+1 = θ (s) Js + b+k C+K T C s+k C d θ (s) 2 disturbance r ( ) b d Prof. Dr. Y. Samim Ünlüsoy 42

43 CONTROL ACTIONS Proortional + Derivative Control KC 1+T d s ( ) θ(s) = J θ r (s) 2 b+kb C+K Td C K C s + s+ J J Note that : 2 KC n ω = J ( ) 2ξω = n b+ K b+ktd C b+k CK bc 1 1 ξ = + T 2 JC K 2 J Derivative control does not affect the undamed natural frequency. (Meaning?) Derivative time T d can be selected to imrove daming ratio. J d Prof. Dr. Y. Samim Ünlüsoy 43

44 CONTROL ACTIONS Proortional + Derivative Control ( ) ( ) K C 1+T s K C 1+T ds b+kbc G(s)= = s Js+ ( b+kbc) J s s+1 b+kb C System tye is still 1, thus the steady state error for unit ste inut is zero. The steady state error for unit ram inut is given as : d 1 b+kbc 1 e ss= = K C K Therefore, the derivative control has no effect on the steady state error. Prof. Dr. Y. Samim Ünlüsoy 44

45 CONTROL ACTIONS Proortional + Derivative Control If the disturbance is taken to be the only inut, then the outut θ(s) will be the error. D(s) + - Js Dc motor+load 2 C ( ) + b+k C s b ( ) K T s+1 d Controller θ(s) C 1 1 e ss =lims 0s = 2 Js + ( b+k ) s K bc+ktdc s+kc Prof. Dr. Y. Samim Ünlüsoy 45

46 CONTROL ACTIONS Proortional + Derivative Control It is observed that, if the disturbance is taken to be the only inut, the steady state error (equal to the steady state value of the outut) for a unit ste inut also is unaffected by the inclusion of the derivative control. e ss unit ste 1 = K Prof. Dr. Y. Samim Ünlüsoy 46

47 CONTROL ACTIONS Proortional + Derivative Control In conclusion, the addition of derivative control action to roortional control action - is useful to rovide additional control on daming and hence imroves relative stability, - has no direct effect on steady state error. It may, however, be useful in reducing steady state error indirectly by allowing the use of larger values of K without causing stability roblems. Prof. Dr. Y. Samim Ünlüsoy 47

48 CONTROL ACTIONS Velocity (Tachometer ) Feedback Velocity or tachometer feedback can be emloyed together with roortional control to rovide characteristics similar to that of PD-control. θ r (s) + _ Controller Disturbance + K + _ Dc motor+load C Js+ b+k C ( ) b 1 s θ(s) K t Tachometer Prof. Dr. Y. Samim Ünlüsoy 48

49 CONTROL ACTIONS Velocity (Tachometer ) Feedback Consider the case without disturbance. Controller Dc motor+load θ r (s) θ(s) + C 1 ( ) K Js+ ( b+k ) s - - bc Kt Tachometer θ(s) K C = θ (s) 2 Js + b+k C+K K C s+k C ( ) r b t Prof. Dr. Y. Samim Ünlüsoy 49

50 CONTROL ACTIONS Velocity (Tachometer ) Feedback Clearly, if you relace T by K d t, the characteristic equation is of the same form as that of the PD-control case. Thus the undamed natural frequency and daming ratio exressions are again the same, if one relaces T by d K t. KC θ(s) = 2 ( ) θ (s) Js + b+k C+K K C s+k C r b t 2 KC b+kbc 1 1 CK ω n = ξ = + K J 2 JC K 2 J t Prof. Dr. Y. Samim Ünlüsoy 50

51 CONTROL ACTIONS Velocity (Tachometer ) Feedback KC KC b+kbc+kktc G(s)= ( ) b t = s Js+ b+k C+K K C J s s+1 b+kbc+kktc System tye is 1, thus the steady state error for unit ste inut is zero. The steady state error for unit ram inut is given as : 1 b+k bc+k K tc 1 b+k b C 1 e ss= = = +K K C K C K t Prof. Dr. Y. Samim Ünlüsoy 51

52 CONTROL ACTIONS Velocity (Tachometer ) Feedback Now let us consider disturbance inut only. - Disturbance Controller + K - + Dc motor+load C Js+ ( b+kbc) 1 s θ(s) Kt Tachometer D(s) + - Js 2 K C ( ) + b+k C s b ( 1+K s) t θ(s) In this case, the outut itself will be the error. Prof. Dr. Y. Samim Ünlüsoy 52

53 CONTROL ACTIONS Velocity (Tachometer ) Feedback In this case, D(s) C θ(s) the outut 2 Js + itself will be + ( b+kbc) s - the error. K ( 1+K s) C 1 1 e ss=lims 0s = 2 Js + b+c( K ) b +K Kt s+k C s K It is observed that disturbance resonse of roortional control + velocity feedback is the same as that of roortional + derivative control. t Prof. Dr. Y. Samim Ünlüsoy 53

54 CONTROL ACTIONS Proortional + Integral Control If a roortional + integral controller is used : θ r (s) + - Disturbance Controller Dc motor+load + 1 C θ(s) K 1+ Ts 2 i + Js + ( b+kbc) s θ(s) KC( Ts+1 i ) = 3 2 disturbance ( ) Neglecting disturbance θ (s) TJs +b+k Cs +K TC Cs+K C r i b i Prof. Dr. Y. Samim Ünlüsoy 54

55 CONTROL ACTIONS Proortional + Integral Control K C( Ts i +1 ) ( ) θ(s) = θ (s) T Js +T b+k C s +K T Cs+K C 3 2 r i i b i Note that : Addition of integral control action to roortional control action increases the order of the system. Thus, transient and steady resonse as well as the stability and disturbance rejection characteristics of the system will be changed. Prof. Dr. Y. Samim Ünlüsoy 55

56 CONTROL ACTIONS Proortional + Integral Control ( ) ( ) K C Ts+1 G(s)= = 2 s Ti Js+ b+kbc ( i ) ( ) K C Ts+1 T b+k C i i b 2 J s s+1 b+kb C System tye is now 2, thus the steady state error for unit ste and ram inuts are zero. The steady state error for unit acceleration inut is given as : ( ) 1 Ti b+kbc 1 e ss= = K C K Therefore, the integral control can be used to reduce steady state error. Prof. Dr. Y. Samim Ünlüsoy 56

57 CONTROL ACTIONS Proortional + Integral Control If the disturbance is taken to be the only inut, then the outut θ(s) will be the error. Dc motor+load D(s) C 2 Js ( ) + b - K + b+k C s 1 1+ Ts i Controller θ(s) TCs i 1 e ss =lims 0s = ( ) TJs i +Ti b+kbc s +K s TCs+K i C TCs i 1 e ss =lims 0s = TJs i +Ti( b+kbc) s +KTCs+K i C s T K i Prof. Dr. Y. Samim Ünlüsoy 57

58 CONTROL ACTIONS Proortional + Integral Control If the disturbance is taken to be the only inut, the steady state error (equal to the steady state value of the outut) for a unit ste disturbance inut is zeroed and the steady state error for a unit ram disturbance inut becomes finite by the inclusion of the integral control. e =0 ss ste e ss unit ram T = K i Prof. Dr. Y. Samim Ünlüsoy 58

59 CONTROL ACTIONS Proortional + Integral Control In conclusion, it is observed that the addition of integral control action to roortional control action - is useful in reducing the steady state error by increasing the tye of the oen loo transfer function by one, -changes the transient resonse characteristics of the system by increasing the order of the system, making the system more suscetible to instability. Prof. Dr. Y. Samim Ünlüsoy 59