Farid Golnaraghi Simon Fraser University Vancouver, Canada ISBN-13: 978-1259643835 ISBN-10: 1259643832 1
2 INTRODUCTION In order to find the time response of a control system, we first need to model the overall system dynamics and find its equation of motion. The system could be composed of mechanical, electrical, or other subsystems. Each sub-system may have sensors and actuators to sense the environment and to interact with it. Using Laplace transforms, we can find the transfer function of all the subcomponents and use the block diagram approach or signal flow diagrams to find the interactions among the system components. Depending on our objectives, we can manipulate the system final response by adding feedback or poles and zeros to the system block diagram. Finally, we can find the overall transfer function of the system and, using inverse Laplace transforms or MATLAB (numerical Simulation), obtain the time response of the system to a test input normally a step input.
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The time response of a control system is usually divided into two parts: the transient response and the steady-state response. (7-1) where y t (t) denotes the transient response, and y ss (t) denotes the steady-state response. In control systems, transient response is defined as the part of the time response that goes to zero as time becomes very large. Thus, y t (t) has the property: (7-2) The steady-state response is simply the part of the total response that remains after the transient has died out. 4
시간응답 (time response): 상태나출력의변화를독립변수인시간의경과에나타내는것 일반적으로시스템의출력을구하는과정에서는라플라스변환이사용되나, 최종판단은결국시간영역에서의응답특성에근거를두게된다. 시간응답은항상과도응답 (transient response) 과정상상태응답 (steady-state response) 부분으로되어있다. 5 y(t) = y t (t) + y ss (t) 여기서 y t (t) 는과도응답부분으로시간이지남에따라소멸하는항으로써, 주로초기치또는입력함수의급격한변동에기인하며, 시스템의응답속도등에주로관계된다. y ss (t) 는정상상태응답부분으로과도응답이소멸된후에남는부분, 즉시스템의고유특성이나입력함수에의해나타나는부분이다. 시스템의정밀성등은주로이특성에의해결정. 입력함수로는보통단위계단함수, 램프함수, 포물선함수가사용되는데, 차수가높은함수일수록 t=0 에서의변화는적은반면, 시간이진행될때의변화속도가빨라지기때문에정상상태오차평가에많이쓰이며, 단위계단함수는 t=0 에서극단적으로급격히변화하지만, 그뒤에는변화가전혀없는특성때문에과도특성평가에서매우중요하다. Step-Function Input r(t) = R u s (t) Ramp-Function Input r(t) = Rt u s (t) Parabolic-Function Input r(t) = Rt 2 /2 u s (t)
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7-3 THE UNIT-STEP RESPONSE AND TIME-DOMAIN SPECIFICATIONS For linear control systems, the characterization of the transient response is often done by use of the unit-step function u s (t) as the input. 7
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과도응답은시간이지남에따라 0 으로소멸되는부분이지만, 그동안나타나는진폭의크기와지속시간등은시스템의허용범위이내로유지시켜야하므로정상상태특성과는또다른측면에서매우중요하다. 과도응답특성해석을위해서주로사용되는입력함수는순간적인변화가가장큰단위계단함수이다. 1. Time response of prototype 2nd order system Prototype 2nd order system 시간응답해석에서가장많이사용되는전형적인 2 차시스템은위의그림과같으며, 시간응답은시스템특성근의위치에따라지수또는진동, 감쇄진동등의형태로나타난다. 특성근의위치표시로는직교좌표계 (s=σ+jω) 형태보다는응답특성과직접관계되는 ζ(damping ratio) 와 ω n (natural frequency) 를사용하는것이일반적이다. 13
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TRANSIENT RESPONSE OF A PROTOTYPE SECOND-ORDER SYSTEM Prototype second-order control system. 15
Closed-Loop system Prototype 시스템의 closed loop transfer function M(s) 과특성근 특성근를로나타내면, : damping factor, : damped frequency Unit step response Prototype 2nd order system 의단위계단함수에대한응답은다음과같다. 이므로, 특성근의형태에따라 : overdamped. 는두실근. : critical damped. 는중근. : oscillation. 는두허근. : underdamped. 는공액복소근. 16
에따른응답특성비교 위의예제에서가작을수록진동이커지며, 가 1 에가까울수록진동은감소되거나늦어지는특성을확인할수있다. 17
18 Figure: Locus of roots of the characteristic equation of the prototype second-order system.
19 Fig. 7-10 Step-response comparison for various characteristic-equation-root locations in the s-plane.
Figure: (a) Constant-natural-undamped-frequency loci. (b) Constant-damping-ratio loci. (c) Constant-damping-factor loci. (d) Constant-conditional-frequency loci. 20
Assuming the final value of y(t) is 1. Maximum Overshoot Maximum overshoot occurs at: 21
T p ( Peak time ) y(t)/dt =0 을만족하는 t 중 0 이아닌첫번째시간. %OS ( Percent Overshoot ) %OS = ( Maximum y max - y ss ) / y ss * 100% 따라서, y max (overshoot) 또는 y min (undershoot) 은 에서 y ss = 1 발생하며, 최대오버슈트는 n=1 인때가된다. 22
7-5-3 지연시간 (Delay Time t d ) 최종치의 50% 에이르기까지걸리는시간. y(t)=0.5 의해로부터구할수있다. 그러나, 실제계산은매우복잡하므로, 일반적으로아래그림과같이 ζ-ωntd 곡선을 0 < ζ < 1 의범위에서 ζ 에대해 ωntd 를근사화한 1 차식또는 2 차식의공식으로부터구한다. Fig. 7-15 ζ-ω n t d curve for prototype second-order system. 23
Rise Time Approximation formulas 24
Settling Time (5%) Approximation formulas 25
7-5-5 Transient Response Performance Criteria 26
Ex. 7-5-1 Position Control of a DC Motor: Using Amplifier gain K CONTROLER 27
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7-6 STEADY-STATE ERROR: Definition Figure: Nonunity feedback control system. 29
Nonunity feedback 인경우 Steady State Error nonunity feedback 시스템에서우리가감소시키고자하는오차는 이다. 이식을적당히변형하면 그러므로, 이런경우는개루프전달함수가 인 unity feedback 시스템으로간주할수있다. 이때의정상상태오차는 step input 인경우 ramp input 인경우 parabolic input 인경우 30
Error series 지금까지설명한 error-constant 방법으로는 e ss 또는 t 에서오차가어떻게되는지는알수있으나그과정에대한정보는알수없다. 이와같이정상상태오차 (steady state error) 의시간특성에대한것은 error series 를이용하면파악할수있다. Ex. 5-4-4 Nonunity feedback 시스템의전달함수가아래와같을때정상상태오차를구하라. 단위피드백시스템으로생각할때, open-loop 전달함수는 이때정상상태오차는 step 입력일때, ramp 입력일때, parabolic 입력일때, 31
Nonunity-Feedback Systems 32
Example 7-6-4 CONTROLER Speed Control of DC Motors 33
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7-6-2 Steady-State Error in Systems with a Disturbance 35
7-6-3 Type of Control Systems: Unity Feedback Systems Consider that a control system with unity feedback can be represented by or simplified to the block diagram with H(s)=1. The steady-state error of the system is written e ss depends on the number of poles G(s) has at s=0. This number is known as the type of the control system or, simply, system type. 36
Steady-State Error 정상상태오차는과도응답이해소된후, 응답의정밀성에관련되는특성이다. 예 : 위치나속도가설정값에얼마나가까운가하는문제일반적으로입력신호의변화가빠를수록출력신호가따라가기어려워지기때문에발생되며, 보통, 단위계단함수, 램프함수, 포물선함수가대표적인성능평가함수로사용된다. Unity-feedback system Steady-State Error, e ss 출력과기준입력의정상상태에서의오차. 즉위와같은단위궤환시스템에서 e(t) = r(t) - y(t), rss 는 r(t) 의정상상태부분 (final value theorem 으로부터 ) 이므로 37 따라서, e ss 는 R(s) 와 G(s) 에포함되어있는 s -1 갯수에따라특성이달라지게된다.
Type of Systems: Loop transfer function G(s) 가다음과같은형태일때, N 을시스템의 Type 이라고하며 (Type N 시스템 ), G(s) 에포함되어있는적분기의수 ( 원점에있는극점의수 ) 에해당한다. 일반적으로, Type 이높을수록정상상태오차는감소하나적분의특성에의해불안정할우려가많아진다. * NOTE * 편의상 G(s) 에서적분기를제거한식에서 s 를영으로한값을 G dc (dc-gain) 으로나타내기로한다. 즉 G dc =s N G(s) s=0, 위의경우에는 G dc =K. 38
39 Figure: Typical steady-state error due to a step input.
System Type 에따른 Steady-State Error 1. Steady-state error for step-function input 여기서, = K p 는 step-error constant 이다. System type 에따른 e ss Type 0 system : Type 1 or higher system : 40
2. Steady-state error for ramp-function input 여기서, 분모항은 ramp-error constant 이며 K v 로나타낸다. 즉, System type 에따른 e ss Type 0 system : Type 1 system : 41 Type 2 or higher system :
3. Steady-state error for parabolic function input 여기서, 분모항은 parabolic-error constant 이며 K a 로나타낸다. 즉, System type에따른 e ss Type 0 system : Type 1 system : Type 2 system : 42 Type 3 or higher system :
Steady-State Error due to Step, Ramp, and Parabolic Inputs TYP E Step input Ramp input Parabolic input 0 R/(1+K p ) 1 0 R/ K p 2 0 0 R/ K p Summary 1. The steady-state error properties are for systems with unity feedback only. 2. The steady-state error with linear combinations of the 3 basic types of inputs can be determined by superposing the errors due to each input components. 3. Unity feedback 이아닌 system 은 unity feedback 인시스템으로단순화하거나, error signal 을구하여최종치정리를적용한다. 4. Error-constant method 는 error 가시간에따라어떻게변화하는지는보여주지않는다. 5. Error-constant method 는 sinusoid input 를가진시스템에게는적용할수없다. ( 최종치정리를적용할수없기때문임.) 43
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EX. 7-6-7 45
7-7 BASIC CONTROL SYSTEMS AND EFFECTS OF ADDING POLES AND ZEROS TO TRANSFER FUNCTIONS In all previous examples of control systems we have discussed thus far, the controller has been typically a simple amplifier with a constant gain K. This type of control action is formally known as proportional control, because the control signal at the output of the controller is simply related to the input of the controller by a proportional constant. Intuitively, one should also be able to use the derivative or integral of the input signal, in addition to the proportional operation. Therefore, we can consider a more general continuous-data controller to be one that contains such components as adders or summers (addition or subtraction), amplifiers, attenuators, differentiators, and integrators. For example, one of the best-known controllers used in practice is the PID controller, which stands for proportional, integral, and derivative. The integral and derivative components of the PID controller have individual performance implications, and their applications require an understanding of the basics of these elements. All in all, what these controllers do is add additional poles and zeros to the openor closed-loop transfer function of the overall system. 46
7-7-1 Addition of a Pole to the Forward-Path Transfer Function: Unity-Feedback Systems Increases the maximum overshoot of the closed-loop system. ymax & tr 47
Addition of a Pole to the Forward-Path Transfer Function: Unity-Feedback Systems Also can make the system unstable 48
7-7-2 Addition of a Pole to the Closed-Loop Transfer Function As the pole at s=1/t p is moved toward the origin in the s-plane, the rise time increases and the maximum overshoot decreases. Thus, as far as the overshoot is concerned, adding a pole to the closed-loop transfer function has just the opposite effect to that of adding a pole to the forward-path transfer function. ymax & tr 49
7-7-3 Addition of a Zero to the Closed-Loop Transfer Function ymax & tr Fig. 5-38 shows why the addition of the zero at s =1/T z reduces the rise time and increases the maximum overshoot, according to Eq. (5-164). In fact, as T z approaches infinity, the maximum overshoot also approaches infinity, and yet the system is still stable as long as the overshoot is finite and z is positive. 50
7-7-4 Addition of a Zero to the Forward-Path Transfer Function: Unity-Feedback Systems The term (1+T z s)in the numerator of M(s) increases the maximum overshoot, but T z appears in the coefficient of the s term in the denominator, which has the effect of improving damping, or reducing the maximum overshoot. 분자 (1+Ts) : ymax 분모 Ts : ymax An important finding from these discussions is that, although the characteristic equation roots are generally used to study the relative damping and relative stability of linear control systems, the zeros of the transfer function should not be overlooked in their effects on the transient performance of the system. 51
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56 Overshoot due to effect of zero added to the closedloop TF
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The poles that are close to the imaginary axis in the left-half s-plane give rise to transient responses that will decay relatively slowly, whereas the poles that are far away from the axis (relative to the dominant poles) correspond to fast-decaying time responses. It has been recognized in practice and in the literature that if the magnitude of the real part of a pole (The distance D between the dominant region and the least significant region ) is at least 5 to 10 times that of a dominant pole or a pair of complex 5-10 Dominant Poles and Zeros of Transfer Functions dominant poles, then the pole may be regarded as insignificant insofar as the transient response is concerned. The zeros that are close to the imaginary axis in the left-half s-plane affect the transient responses more significantly, whereas the zeros that are far away from the axis (relative to the dominant poles) have a smaller effect on the time response. The dominant poles and the insignificant poles should most likely be located in the tinted regions. The desired region of the dominant poles is centered around the line that corresponds to z = 0.707. 60
61 7-8-1 Summary of Effects of Poles and Zeros
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7-8-3 The Proper Way of Neglecting the Insignificant Poles with Consideration of the Steady-State Response 63
7-9 CASE STUDY: TIME-DOMAIN ANALYSIS OF A POSITION-CONTROL SYSTEM Because L a in the armature circuit is very small, t a is neglected. Later on we show that this may not necessarily be a good assumption. 64
TIME-DOMAIN ANALYSIS OF A POSITION-CONTROL SYSTEM Cont. 65 r
Unit-Step Transient Response r 66 The system is Type 1, i.e. e ss to a step input is zero.
Unit-Step Transient Response r 67
Time Response of a Third-Order System Dominant Roots are closer to the imaginary axis 68
69 Time Response of Third-Order System vs. Second Order Approximation using L a = 0
70 Time Response of Third-Order System vs. Second Order Approximation using L a = 0
7-9-3 Time Response of a Third-Order System Dominant Roots are closer to the imaginary axis 71
7-10 THE CONTROL LAB 72