Controller (Notes)

Chapter 6

Analysis and Design of Control Systems in Time Domain Dr. Franjo Cecelj Cecelja a1 Faculty of Engineering and Physical Sciences University of Surrey

6.1 6. 1

Intr In trodu oduct ctio ion n

The discussi discussion on in thi thiss Lec Lecture ture is lim limite ited d to tim time-d e-doma omain in ana analys lysis is and design based on the transienttransi ent-response response analysis analysis and partiall partially y on the analysis related to the pole positio p osition. n. As we will be focusing on the feedb feedback ack control, control, or close closed d loop lo op control, all the definit definitions ions that we introduced so far fully apply. Still, here are some of them just presented as a reminder.

6.1.1 6.1 .1

Closed Clo sed Loop Loop vers versus us Open Open Loop Loop Syste Systems ms

An advantage of the closed loop system is the fact that the use of feedback makes the system response relatively relatively insensitive insensitive to the exter external nal distur disturbances bances and int internal ernal variation variation in syste system m parameters. parame ters. This makes makes it possible to develop relatively relatively inaccurate inaccurate mathematical mathematical model, mo del, but also to use inaccurate and inexpensive components to still obtain sufficiently accurate control. From the point of view of stability, it is much more difficult to obtain a stable system with a closed loop system as the closed loop system sometimes tend to overcorrect errors and hence to cause the oscillations and even a full instability of the system.

6.1.2 6.1 .2

Genera Gen erall Require Requiremen ments ts of Contro Controll Systems Systems

Any control system must be stable and this is the primary requirement. In addition to absolute stability, the control system must have a relative stability; that is the response must be fast enough to respond to the disturbances, but should show no or very limited oscillations - there will always be a trade-off between these two requirements. In the case of the second-order system, 1

These lecture notes have been compiled from the literature stated in the Bibliography Section

97

98 CHAPTER CHAPTER 6.

ANALYSIS ANAL YSIS AND DESIGN DESIGN OF CONTROL SYSTEMS SYSTEMS IN TIME DOMAIN

in many practical applications the compromise is found by taking the damping ratio ζ = 0.707, ζ = for which the poles are located on the line at an angle of θ 0.65 [rad rad]. ]. Then, the response is fast enough with and acceptable overshot to the step response, but without oscillation, as shown in Figure 6.1. In Figure 6.1, the left side shows shows the response response of a sec second ond-or -order der system system to the unity step input function with the following parameters; ζ = 0.707, ω = 1 [rad rad], ], K K = 1. 1. Th Thee right side of the figure shows the concomitant pole location in the s s-plane. -plane.

≈

Figure 6.1: Response of an ’ideal’ second order system In the case of the first-order system, which is always considered to be an underdamped system as it never has any overshoot in its response to the input step function, the poles are normally expected to be as far from the imaginary jω axis as possible as this guaranties that the response will be as fast as possible. In turn, this means that the time constant τ τ is is expected to be as small as possible. Additionally, placing the poles far from the j ω axis ensures a higher stability stabil ity of the system. Figure 6.2 shows the response response of a first order syste system m for tw twoo different values of the time constant; τ constant; τ = 0.1 [s] and τ and τ = 1 [s] for which the poles are placed at s = 10 and s and s = 1, respectively. Also, along with the stability, the system must be able to reduce the errors, particularly the steady state error as we saw with the first-order system responding to the ramp input function, to a small, tolerable value.

−

6.2 6.2. 6. 2.1 1

Autom Aut omati aticc Co Cont ntrol roller lerss Conc Co ncep eptt

An automatic controller compares the actual value Y Y ((s) of the process or plant output with ¯ the desired values Y Y ((s), determines deviation and produces a control signal U U ((s) to reduce the deviation or error E error E (s) to zero or to a small and tolerable value. The way the controller produces the control signal is called control action, as shown in Figure 6.3 where the transfer function D(s) denotes the controller.

6.2. AUTOMATIC CONTROLLERS

99

Figure 6.2: The response of the first-order system

Figure 6.3: Feedback control action The automatic controllers, in literature sometimes called industrial automatic controllers, can be classified according to their control action as: 1. Proportional controllers; 2. Integral controllers; 3. Proportional-plus-integral controllers; 4. Proportional-plus-derivative controllers; 5. Proportional-plus-integral-plus-derivative controllers. These are also called the control strategies. A full schematic of a control system is shown in Figure 6.4. Here, sensors are used to measure controllable or output variable of the process, and sometimes the actuators are placed between the process and the controller to manipulated controllable variable (sensors and actuators will be explained in more details in the next Chapter of these notes). For the simplicity reasons, we will assume that these variables are directly accessible and comparable. Hence, the sensor that measure the variable using the transducer, controller that provides the control action and the actuators will all be presented in a single block called the controller.

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ANALYSIS AND DESIGN OF CONTROL SYSTEMS IN TIME DOMAIN

Figure 6.4: Block diagram of an automatic control system

6.2.2

Practical Issues

In reality, the controller is of the form given in Figure 6.5. It contains the sensor that ’measures’ the output variable Y (s) with the measurement constant K m (Y  (s) = Y (s)K m ) of the process and converts its values into the signal, often into an electrical, pneumatic or hydraulic signals ¯ which are comparable with the desired output Y (s) signal. It is only then, that the error signal  ¯ E (s) (= Y (s) Y (s)) is created, on which the control strategy D(s) is applied. Also, the output signal U  (s) of the control strategy D(s) is very often incompatible with the input or manipulated variable U (s) of the process. Actuators are used to provide this compatibility. Consequently, the sensors are devices that convert physical quantities of the output variable into a usable signal proportional to it. On the other side, actuators are devices that convert a signal into the physical quantities of the input variable into the process or plant.

−

Figure 6.5: Practical realisation of an automatic controller Let’s take a car cruising controller the task of which is to keep the speed of the car ¯ constant and stable whatever the conditions. The desired output Y (s) is usually an electric signal proportional to the required speed of the car. The actual speed of the car is measured by a sensor called tachometer that converts the angular speed of the wheels, hence the linear speed of the car, into an electric signal Y  (s) that is proportional to it, but also comparable to ¯ desired output Y (s). The difference of these signals, E (s), is then processed by an appropriate controller D(s), which provides the electrical signal U  (s). More precisely, the controller D(s) is either an analogue or a digital computer processing one electrical signal into another. Finally, the actuator uses the electric signal U  (s) and converts it into the valve opening that changes the flow of the air-fuel mixture into the engine. Since the sensors and actuators will both be dealt with in more details in one of the follow-on lectures, here we will assume that the process output Y (s) is directly comparable to ¯ the desired output Y (s), and also to the process input U (s), so the whole scheme simplifies to

101


that given in figure Figure 6.3.

6.2.3

Proportional Controllers

The proportional controller produces the output signal u(t) (pressure in the case of pneumatic controller, current or voltage in the case of electronic controller) that is proportional to the error e(t) (= y¯(t) y(t)), which gives u(t) = K c e(t) + k (6.1)

−

where u(t) is the output of the controller, hence the input to the process; K c is the controller gain or sensitivity; e(t) is the error signal; k is a constant. In practice, the gain K c is adjustable so that the response of the system can be tuned. Also, the constant k is the value of the output of the controller when the error e(t) = 0 and in practical situations it is adjusted so that the output of the process is at required value for e(t) = 0. For instance, this may be required to remove the steady state error from the system. Since, this is better done by introducing integral controller, here we will assume that the constant k = 0 unless specified otherwise. Consequently, applying the Laplace transform on the equation (6.1), the transfer function of the proportional controller is D(s) =

U (s) = K c E (s)

(6.2)

The term proportional is commonly used among the control population instead of gain, although the proportional controller is obviously a simple amplifier with a gain K c . The First-order System with Proportional Controller

Applying the proportional controller to a first-order system (Figure 6.6), the overall transfer function is K D(s)H (s) τ G(s) = = (6.3) 1 + D(s)H (s) s + 1+τ K c

c

It is evident from equation (6.3) that the overall amplification of the system is K c , but also the pole of the system is shifted from original position s 1 = 1/τ to new position s 1 = (1 + K c )/τ . As a consequence, if K c > 0, the pole is placed further from the imaginary jω axis in s-plane, hence reducing the time constant of the system, and improving the overall response (faster system). For K c < 0, the response is slower as the pole is closer to the imaginary jω axis and the system may even become unstable for K c < 1 as the pole would be in the right-hand side of the s-plane. Figure 6.7 shows the time domain impulse response of the first order system H (s) = τ s1+1 without the controller (K c = 0) and the system with proportional controller and gain K c = 4.

−

−

−

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Figure 6.6: Proportional controller with the first-order system

Figure 6.7: Response of the first order system for two different gains Kc of the proportional controller The response of the closed-loop system with proportional controller to the unity step ¯ input function (¯ y (t) = 1 Y (s) = 1/s), can be obtained in the following way. Applying the partial-fraction expansion to the equation (6.3) we get:

⇒

K D(s)H (s) K 1 K 2 τ = = + 1 + D(s)H (s) s s s + K τ +1 s + K τ +1 c

¯ Y (s) = Y (s)





c

c

Using the partial-fraction expansion it is not difficult to obtain the coefficients as K 1 = and K 1 = K K +1 , hence the equation (6.4) becomes:

−

(6.4) K c K c +1

c

c

Y (s) =

K c K c +1

s

K c K c +1

− s+

(6.5)

K c +1 τ

So, the time-domain response is from the Laplace transform table: y(t) =

K c 1 K c + 1

−

e

t(K c +1) τ



(6.6)

103


For t , the system output with proportional controller will never reach the expected output y¯(t) = 1. Instead, it will reach the level that follows from equation (6.6) and which is:

→∞

y( ) =

∞

K c K c + 1

(6.7)

hence the steady state error is e(t) = y¯(t)

− y(t)|

t→∞

=

K c K c + 1

(6.8)

So, the proportional controller introduces a steady state error which is proportional to the gain K c . For K c = 1, the steady-state error is as high as 50%, as shown in Figure 6.8, and it decreases as K c increases: higher the proportional controller gain K c , lover the steady-state error. This is the main disadvantage of introducing a proportional controller.

Figure 6.8: Steady-state error with the proportional controller Similarly, for the proportional controller applied to a second-order system (Figure 6.9), the overall transfer function is G(s) =

D(s)H (s) K c ωn2 = 2 1 + D(s)H (s) s + 2ζωn s + ωn2 (K c + 1)

(6.9)

Figure 6.9: The second-order system with proportional controller Effectively, the proportional controller changes the natural frequency of the second-order system from ω n to ω n = ω n K c + 1, while maintaining the same damping ratio. Consequently,

√

104CHAPTER 6.


the increase in K c , improves the response (faster system) and, similarly to the first-order system, reduces the steady state error as long as K c > 0.

6.2.4

Proportional-Integral Controller

The integral controller (Figure 6.10) produces the output signal u(t) (pressure in the case of pneumatic controller, current or voltage in the case of electronic controller) that is proportional to the integral of the error e(t) (= y¯(t) y(t)) as

−

K c u(t) = T i



e(t)dt

⇒

D(s) =

U (s) K c = E (s) T i s

(6.10)

where u(t) is the output of the controller (input to the process or plant), K c is the gain or sensitivity, e(t) is the error signal, T i is the integral time, and 1/T i is a measure of the speed of response and is referred to as the reset time. The primary virtue of this controller is that it

Figure 6.10: The integral-proportional controller can provide a finite value of control signal u(t) with no error signal e(t). This is because, as an integrator, this controller does not take into account only the current value of the error e(t), but also the past history of the e(t) values. To support this theory, let’s look into the response of ≥0 this controller u(t) to the step function of error signal e(t). For e(t) = 10 tt< 0 , in the s-domain



it becomes E (s) = 1s , then from equation (6.8) the controller output signal is U (s) = E (s)

K c K c = T i s T i s2

(6.11)

and concomitant time response is

K c t (6.12) T i So, even if the error signal e(t) seizes to exist (e(t) = 0), the output of the controller u(t) will not be zero but the value achieved in previous time which depends on the previous history of values. Consequently, this controller tends to remove a steady state error e(t) by forcing the process to rectify it. u(t) =

Example 6.1 Let’s take the first-order process as shown in Figure 6.11. The overall transfer

function is easily found to be K c

G(s) =

D(s)H (s) = 2 1τ T 1 + D(s)H (s) s + τ s + i

K c τ Y i

(6.13)

105


which, according to the equation of general format for a second order transfer function (Chapter 4) gives the damping ratio ζ =  1 and natural frequency ωn = τK T . Also,



T i τK c

2

c

i

overall transfer function (6.13) is the second-order transfer function with the time response to unity step input function given as g(t) = 1

−

1

 − 1

e−ζω t sin tωn n

ζ 2

  − 1

ζ 2

− arctg

  −  ζ 2

1

ζ

(6.14)

If the time approaches infinity (t ), the steady state error in the system response is zero since e( ) = y¯( ) y( ) = 0. Note that the steady state error for the first order system with proportional controller and unity step input function, according to the equation (6.8), was 50%. So, the introduction of the integral controller removes the steady state error and this is the main usage of the proportional-integral controller.

∞

∞− ∞

→∞

Figure 6.11: Proportional-Integral controller in action 1 Figure 6.12 shows the response of the first-order system ( H (s) = s+1 ) with the proportional8 integral controller (D(s) = 2s ) to the unity step input function (blue curve) in comparison with the response of the first-order system without controller (red curve). It is apparent that the introduction of the integral controller changes the response similar to that of the second-order system, but more importantly removes the steady state error (e( ) = 0).

∞

6.2.5

Proportional-Derivative Controller

Derivative controller has the time domain form u(t) = T D

de(t) dt

(6.15)

Therefore, the s-domain transfer function of this controller becomes D(s) =

U (s) = T D s E (s)

(6.16)

where T D is called derivative time. The derivative controller is always used in conjunction with proportional and/or integral controller to increase the damping and generally improves the stability of the system. In practice, pure derivative controller is not practical to implement for the reason that if the error e(t) remains constant, the output of the derivative controller u(t) would be zero (derivative of a constant function is zero), so the control action would not take place. Proportional or integral

106CHAPTER 6.


Figure 6.12: Response of the first-order system with the proportional-integral controller and unity step input function term is therefore needed to provide a control signal at this time. Consequently, the practical proportional-derivative controller has the form D(s) = K c + K c T D s = K c (1 + T D s)

(6.17)

where K c is the proportional gain. With the derivative controller the corrections depend on the rate of change of the error signal e(t). As a result, the derivative controller performs an anticipatory response. Example 6.2 Let’s take the example of an electrical circuitry as given in Figure 6.13 with

the capacitor voltage being the output variable (y(t) = u C (t)) and source voltage being the input variable (u(t)). The transfer function has the form U C (s) = 2 U (s) s +

1

CL R 1 s + CL L

(6.18)

which for C = 100 [µF ], L = 1000 [H ] and R = 6 [kΩ] gives the transfer function of the form U C (s) 10 = 2 U (s) s + 6s + 10

(6.19)

√

with natural frequency ωn = 10 = 3.16 [ rad ], the damping ratio ζ = 2√ 610 = 0.95, and s pole location s1,2 = 3 j, as shown in Figure 6.15. Applying the proportional-derivative controller with this plant, as shown in Figure 6.14, the overall transfer function is obtained as Y (s) D(s)H (s) 10K c (1 + T D s) = = 2 (6.20) G(s) = ¯ 1 + D(s)H (s) s + s (6 + 10K c T D ) + 10(1 + K c ) Y (s)

− ±

which for K c = T D = 1 becomes G(s) =

10 (s + 1) s + 16s + 20 2

(6.21)


107

Figure 6.13: An electronic circuit with two real poles s1,2 = 8 6.6, the damping ratio ζ = 1.79 and natural frequency ωn = 4.47. The increase in the damping ratio from original ζ = 0.95 to ζ = 1.79 indicates that the stability of the system has improved. This is supported by the p ole positions, which for the closed-loop system have moved further from the imaginary jω axis, hence improved the stability of the system, as indicated in Figure 6.15. Also the introduced zero in equation

− ±

Figure 6.14: Proportional-derivative controller in action (6.21) contributes to better system performance, but this will be the topic of further and advanced control in the follow-on module. The system response to the unity step input function is shown in Figure 6.16, where the red curve is response of the system without controller (open-loop system), the blue graph is response of the system with derivative-proportional controller with K c = 1 and T i = 1, and the green graph is the response of the system with derivative-proportional controller with K c = 10 and T i = 1. It is evident that proportional-derivative controller also introduces the steady state error, which decreases as the proportional gain K c increases. The reason for the existence of the steady state error is similar as to that of proportional only controller; it does not have the integral part that would take into account previous values of the error e(t). Also, it is evident from Figure 6.16 that derivative action improves the system response by making it much faster, still maintaining the system stability. This is the main advantage of introducing the derivative action.

6.2.6

Proportional-Integral-Derivative (PID) Controller

For control over steady-state and transient errors we can combine all three control strategies we have discussed in this Chapter to get proportional-integral-derivative (PID) control as a linear combination of proportional, integral and derivative action. Normally, all three gain constants are adjustable. The PID combination is often able to provide an acceptable degree of error (steady state error) together with acceptable stability and damping. More precisely, PID controllers are so effective that PID control is a standard in processing and some other

108CHAPTER 6.


Figure 6.15: Pole location of the second-order system industries. The time response of the PID controller is u(t) = K c



1 e(t) + T i

or in s-domain U (s) = E (s)K c which gives the controller transfer function as

 

de(t) e(t)dt + T D dt

1 1+ + T D s T i s







U (s) 1 D(s) = = K c 1 + + T D s E (s) T i s



(6.22)

(6.23)

(6.24)

The complete control system is shown in Figure 6.17. Example 6.3 Taking the first order system of the form

H (s) =

1 s+1

(6.25)

the response of the system with PID controller is shown in Figure 6.18 where the red curve is response of the system without controller (open-loop system), the blue graph is response of the system with PID controller with K c = 5, T D = 0.1 and T i = 0.5, and the green graph is the response of the system with derivative-proportional controller with K c = 1, T D = 0.1 and T i = 0.5. It is evident that PID controller removes the steady state error and that the system is faster than the open-loop system without controller.


109

Figure 6.16: Response of the system with derivative-proportional controller

Figure 6.17: PID control strategy

6.2.7

Tuning PID Controllers

To design a particular control loop, the engineer has to adjust the constants K c , T i and T D to provide an acceptable performance of the closed loop system. This process is called tuning the controller. The criteria for tuning is based on the ideas presented earlier in this Chapter; increasing K c and T 1 tends to reduce system errors but may affect the system stability, while increasing T D improves stability. To develop a controller that will meet steady-state and transient specifications, together with appropriate stability of the system, is a daunting task as it requires a complete knowledge of the system and its mathematical model. This is mainly based on the system analysis in the frequency ( s) domain, as you will see during the follow-on module. Sophisticated methods are available to develop and in particular to tune the gain constants K c , T i and T D of a PID controller for a particular process. One of the most popular methods today is Ziegler-Nichols method, which recognises that the step response of a process contains sufficient information about the system for most practical cases. The step response of any system has a general form as shown in Figure 6.19, and is called the process reaction curve. This curve can be generated by experimental data from the step input function applied on the i

110CHAPTER 6.


Figure 6.18: Response of a first-order system with PID controller process or plant, but in many cases it can be recorded from a normal operation of the process as the step input is a very often occurred type of inputs. The S-shape of the process reaction

Figure 6.19: Process reaction curve curve is characteristic of many high-order systems, but it may be approximated by a first-order transfer function and associated delay as H (s) =

Y (s) Ke−λs = U (s) τs + 1

(6.26)

where the all constants can be determined from the system response as shown in Figure 6.19. If the tangent is drawn at the inflection point of the reaction curve, then the slope of the line is R =

K τ

(6.27)

111


and the intersection of the tangent line with the time axis defines the time delay L = λ. Ziegler and Nichols approached the tuning problem in such a way that the decay ratio is to be approximately 0.25 which means that the exponential decay to the impulse input achieves a quarter value after one period of oscillation, as shown in Figure 6.20. This roughly corresponds to ζ = 0.21 for a second-order system which was fount to be a good compromise between quick response and appropriate stability margins (how far the poles are from the imaginary jω axis) for a system with delays.

Figure 6.20: Quarter decay ratio Simulation of the above requirements provided the PID parameters as shown in Table 6.1. Type of Controller

Optimum Gain 1

Proportional

K c =

RL

Proportional-integral

K c =

0.9 RL

T i =

0.3

K c =

1.2 RL

Proportional-integral-derivative

L

T i = 2L T D = 0.5L Table 6.1: Ziegler-Nichols tuning for the PID controller and decay ratio of 0.25

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Example 18

Lets take a first-order system with delay λ = 2 [s] of the form e−2s H (s) = 5s + 1

(6.28)

The system time response to the unity step input function is shown in Figure 21. From the

Figure 6.21: Response of the first order system with delay to the unity step input function response we can find that L = 3, τ = 5, K = 1 and consequently R = K = 0.2. The parameters τ 1.2 of the PID controller, from the Table 6.1 are; K c = RL , T i = 2L = 6 and T D = 0.5L = 1.5, so the transfer function of the controller is D(s) =

4.32s + 0.72 6s

(6.29)

The time-domain response to the unity step input function is given in Figure 6.22 where the blue curve is the response of the open-loop system without controller and the green curve is the response of the closed-loop system with the PID controller as given by equation 6.29.

6.3 6.3.1

Skill-Assessment Exercise Review Questions

1. Specify advantages, disadvantages and other specifics related to close-loop control systems; 2. Specify and explain major requirements that a control system has to satisfy and which are important guidelines for control system design;

6.3. SKILL-ASSESSMENT EXERCISE

113

Figure 6.22: Response of the system with PID controller 3. With the help of a diagram or otherwise explain the concept of automatic controllers. What are known control strategies? 4. Explain the roles of sensors and actuators in a control system; 5. With the help of a diagram or otherwise explain the concept of proportional controller. What parameters of the first orders system are affected by introduction of a proportional controller and how? 6. How does the proportional controller affect the second order system? 7. With the help of a diagram or otherwise explain the concept of proportional-integral controller. What parameters of the first orders system are affected by introduction of a proportional-integral controller and how? 8. Explain the terms integral time and reset time. 9. Explain how a proportional-integral controller eliminates the steady state error of a control system. 10. With the help of a diagram or otherwise explain the concept of proportional-derivative controller. What parameters of the first orders system are affected by introduction of a proportional-derivative controller and how? 11. Briefly explain the term derivative time. 12. Explain why derivative controllers are not used on their own.

114CHAPTER 6.


13. With the help of a diagram or otherwise explain the concept of proportional-integralderivative controller.

6.3.2

Solving Problems

Task 5.1 (s) 2 Given the first-order transfer function of a control system as H (s) = Y = s+1 . Calculate U (s) and plot the position of the system pole(s) in the s-plane. Sketch the time-domain response to the unity step input function and calculate the steady state error. Close the loop with the proportional controller D(s) = K c . For two values of the gain K c = 1 and K c = 10, calculate the pole positions of the overall system G(s) and plot them in s-plane. For both values of K c calculate the steady state error of the response to the unity step input function.

Task 5.2 5 For the DC motor described by the transfer function H (s) = U Ω((ss)) = s(s+10) find the value of the proportional controller gain K c that would provide ’the best’ time domain response of the closed-loop system. Sketch the position of the pole(s) is s-domain, as well as the time domain response to the unity step input function y¯(t). a

Task 5.3

Given the response to the unity step input function of the system H (s) = U ((s)) as in the Figure 6.23, derive the transfer function that would provide the ’best’ fit. Determine the parameters of the PID controller that would provide a suitable closed-loop system. Sketch the block diagram of the overall system. Y s

Task 5.4

Given the control system H (s) = to the input function u(t) =



t

0

Y (s) U (s)

t≥0 t<0

=

1

s+1

. Determine the steady state error of its response

6.3. SKILL-ASSESSMENT EXERCISE

Figure 6.23: The system response

115

116CHAPTER 6.


Bibliography [1] Ogata K; Sysatem Dynamics - third edition , Prentice-Hall International (1998) - Chapter 6 [2] Fraklin GF, Powell JD, Emami-Naeini A; Feedback Control of Dynamic Systems - fourth edition, Prentince Hall 2002 [3] Coughanowr DR; Process Systems Analysis and Control - secod edition, McGraw-Hill inc, (1991)

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BIBLIOGRAPHY

Controller (Notes)

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