JUNE 2014 Solved Question Paper 1 a: Explain with examples open loop and closed loop control systems. List merits and demerits of both. Jun. 2014, 10 Marks Open & Closed Loop System - Advantages & Disadvantages [Control System]. In control systems engineering, a system is actually a group of objects or elements capable of performing individual tasks. They are connected in a specific sequence to perform a specific function. A system is of 2 types: 1. Open loop system which is also called as Manual control system. 2. Closed loop system which is also named as automatic control system. In this post, we will be discussing various advantages and disadvantages of the 2 types of control systems. Open Loop System: Advantages: 1. Simplicity and stability: they are simpler in their layout and hence areeconomical and stable too due to their simplicity. 2. Construction: Since these are having a simple layout so are easier to construct. Disadvantages: 1. Accuracy and Reliability: since these systems do not have a feedback mechanism, so they are very inaccurate in terms of result output and hence they are unreliable too. 2. Due to the absence of a feedback mechanism, they are unable to remove the disturbances occurring from external sources. Closed Loop System: Advantages: 1. Accuracy: They are more accurate than open loop system due to their complex construction. They are equally accurate and are not disturbed in the presence of nonlinearities. 2. Noise reduction ability: Since they are composed of a feedback mechanism, so they clear out the errors between input and output signals, and hence remain unaffected to the external noise sources. Disadvantages: 1. Construction: They are relatively more complex in construction and hence it adds up to the cost making it costlier than open loop system.
2. Since it consists of feedback loop, it may create oscillatory response of the system and it also reduces the overall gain of the system. 3. Stability: It is less stable than open loop system but this disadvantage can be striked off since we can make the sensitivity of the system very small so as to make the system as stable as possible. Open Loop Control System A control system in which the control action is totally independent of output of the system then it is called open loop control system. Manual control system is also an open loop control system. Fig - 1 shows the block diagram of open loop control system in which process output is totally independent of controller action. Open-loop Motor Control So for example, assume the DC motor controller as shown. The speed of rotation of the motor will depend upon the voltage supplied to the amplifier (the controller) by the potentiometer. The value of the input voltage could be proportional to the position of the potentiometer. If the potentiometer is moved to the top of the resistance the maximum positive voltage will be supplied to the amplifier representing full speed. Likewise, if the potentiometer wiper is moved to the bottom of the resistance, zero voltage will be supplied representing a very slow speed or stop. Then the position of the potentiometers slider represents the input, θi which is amplified by the amplifier (controller) to drive the DC motor (process) at a set speed N representing the output,θo of the system. The motor will continue to rotate at a fixed speed determined by the position of the potentiometer. As the signal path from the input to the output is a direct path not forming part of any loop, the overall gain of the system will the cascaded values of the individual gains from the potentiometer, amplifier, motor and load. It is clearly desirable that the output speed of the motor
should be identical to the position of the potentiometer, giving the overall gain of the system as unity. However, the individual gains of the potentiometer, amplifier and motor may vary over time with changes in supply voltage or temperature, or the motors load may increase representing external disturbances to the open-loop motor control system. But the user will eventually become aware of the change in the systems performance (change in motor speed) and may correct it by increasing or decreasing the potentiometers input signal accordingly to maintain the original or desired speed. The advantages of this type of open-loop motor control is that it is potentially cheap and simple to implement making it ideal for use in well-defined systems were the relationship between input and output is direct and not influenced by any outside disturbances. Unfortunately this type of open-loop system is inadequate as variations or disturbances in the system affect the speed of the motor. Then another form of control is required. Practical Examples of Open Loop Control System 1. Electric Hand Drier Hot air (output) comes out as long as you keep your hand under the machine, irrespective of how much your hand is dried. 2. Automatic Washing Machine This machine runs according to the pre-set time irrespective of washing is completed or not. 3. Bread Toaster - This machine runs as per adjusted time irrespective of toasting is completed or not. 4. Automatic Tea/Coffee Maker These machines also function for pre adjusted time only. 5. Timer Based Clothes Drier This machine dries wet clothes for pre adjusted time, it does not matter how much the clothes are dried. 6. Light Switch lamps glow whenever light switch is on irrespective of light is required or not. 7. Volume on Stereo System Volume is adjusted manually irrespective of output volume level. Closed Loop Control System Control system in which the output has an effect on the input quantity in such a manner that the input quantity will adjust itself based on the output generated is called closed loop control system. Open loop control system can be converted in to closed loop control system by providing a feedback. This feedback automatically makes the suitable changes in the output due to external disturbance. In this way closed loop control system is called automatic control system. Figure below shows the block diagram of closed loop control system in which feedback
is taken from output and fed in to input. Closed-loop Motor Control Any external disturbances to the closed-loop motor control system such as the motors load increasing would create a difference in the actual motor speed and the potentiometer input set point. This difference would produce an error signal which the controller would automatically respond too adjusting the motors speed. Then the controller works to minimize the error signal, with zero error indicating actual speed which equals set point. Electronically, we could implement such a simple closed-loop tachometer-feedback motor control circuit using an operational amplifier (op-amp) for the controller as shown. Closed-loop Motor Controller Circuit
Practical Examples of Closed Loop Control System 1. Automatic Electric Iron Heating elements are controlled by output temperature of the iron. 2. Servo Voltage Stabilizer Voltage controller operates depending upon output voltage of the system. 3. Water Level Controller Input water is controlled by water level of the reservoir. 4. Missile Launched & Auto Tracked by Radar The direction of missile is controlled by comparing the target and position of the missile. 5. An Air Conditioner An air conditioner functions depending upon the temperature of the room. 6. Cooling System in Car It operates depending upon the temperature which it controls. Comparison of Closed Loop And Open Loop Control System Sr. No. Open loop control system Closed loop control system 1 The feedback element is absent. The feedback element is always present. 2 An error detector is not present. An error detector is always present. 3 It is stable one. It may become unstable. 4 Easy to construct. Complicated construction. 5 It is an economical. It is costly. 6 Having small bandwidth. Having large bandwidth. 7 It is inaccurate. It is accurate.
8 Less maintenance. More maintenance. 9 It is unreliable. It is reliable. 10 Examples: Hand drier, tea maker Examples: Servo voltage stabilizer, perspiration 1 b: Draw the electrical network based on torque-current analogy give all the perfor-mance equations for Figure. 1 Jul. 2014, 10 Marks T θ 1 k 1 θ 2 k 2 J1 J2 f 1 f 2 Figure 1:
3 a: Draw the transient response characteristics of a control system to a unit step input and define the following: i) Delay time; ii) Rise time; iii) Peak time; iv)maximum overshoot; v) Settling time 5 Marks
3 b: Derive the expressions for peak time t p for a second order system for step input. 5 Marks
3 c: The response of a servo mechanism is c(t) = 1 + 0.2e 60t +1.2e 10t to a unit step input. Obtain an expression for closed loop transfer function. Determine the undamped natural frequency and damping ratio. 6Marks
3 d: The open loop transfer function of a unity feedback system is given by G(s) =K/S(ST+1) where K and T are positive constant. By what factor should the amplifier, gain K be reduced so that the peak overshoot of unit step response of the system is reduced from 75% to 25%. 4 Marks
4 a: Explain Routh-Hurwitz criterion in stability of a control system. What are the disadvantages of RH criterion on stability of control system? Jun. 2014, 4+4 Marks
4 b: The characteristics equation for certain feedback control system is given below. Deter-mine the system is stable or not and find the value of K for a stable system s 3 + 3Ks 2 + (K + 2)s + 4 = 0. Jun. 2014, 6 Marks
4 c: The open-loop TF of a unity negative feedback system is given by K(s + 3) G(s) = s(s2 + 2s + 3)(s + 5)(s + 6) Find the value of K of which the closed loop system is stable. Jun. 2014, 6 Marks
5 a: For a unity feedback system, the open-loop transfer function is given by K G(s) = s(s + 2)(s 2 + 6s + 25). Jun. 2014, 15 Marks (i) Sketch the root locus for K. (ii) At what value of K the system becomes unstable. (iii) At this point of instability, determine the frequency of oscillation of the system.
K 5 b: Consider the system with G(s)H(s) = s(s + 2)(s + 4) whether s = 0.75 and s = 1 + j4 are on the root locus or not. using angle condition find Jun. 2014, 5 Marks
(or)
s = 0.75 and s = 1 + j4 are not on the root locus. 6 a: Explain the procedure for investigating the stability using Nyquist criterion. Jun. 2014, 8 Marks, 6 Marks Nyquist Stability Criterion The Nyquist plot allows us also to predict the stability and performance of a closed-loop system by observing its open-loop behavior. The Nyquist criterion can be used for design purposes regardless of open-loop stability (remember that the Bode design methods assume that the
system is stable in open loop). Therefore, we use this criterion to determine closed-loop stability when the Bode plots display confusing information. The Nyquist diagram is basically a plot of G(j* w) where G(s) is the open-loop transfer function and w is a vector of frequencies which encloses the entire right-half plane. In drawing the Nyquist diagram, both positive and negative frequencies (from zero to infinity) are taken into account. We will represent positive frequencies in red and negative frequencies in green. The frequency vector used in plotting the Nyquist diagram usually looks like this (if you can imagine the plot stretching out to infinity): follows: However, if we have open-loop poles or zeros on the jw axis, G(s) will not be defined at those points, and we must loop around them when we are plotting the contour. Such a contour would look as Please note that the contour loops around the pole on the jw axis. As we mentioned before, the Matlab nyquist command does not take poles or zeros on the jw axis into account and therefore produces an incorrect plot. The Cauchy criterion The Cauchy criterion (from complex analysis) states that when taking a closed contour in the complex plane, and mapping it through a complex function G(s), the number of times that the plot of G(s) encircles the origin is equal to the number of zeros of G(s) enclosed by the frequency
contour minus the number of poles of G(s) enclosed by the frequency contour. Encirclements of the origin are counted as positive if they are in the same direction as the original closed contour or negative if they are in the opposite direction. When studying feedback controls, we are not as interested in G(s) as in the closed-loop transfer function: G(s) --------- 1 + G(s) If 1+ G(s) encircles the origin, then G(s) will enclose the point -1. Since we are interested in the closed-loop stability, we want to know if there are any closed-loop poles (zeros of 1 + G(s)) in the right-half plane. Therefore, the behavior of the Nyquist diagram around the -1 point in the real axis is very important; however, the axis on the standard nyquist diagram might make it hard to see what's happening around this point To view a simple Nyquist plot using Matlab, we will define the following transfer function and view the Nyquist plot: 0.5 ------- s - 0.5 Closed Loop Stability Consider the negative feedback system
Remember from the Cauchy criterion that the number N of times that the plot of G(s)H(s) encircles -1 is equal to the number Z of zeros of 1 + G(s)H(s) enclosed by the frequency contour minus the number P of poles of 1 + G(s)H(s) enclosed by the frequency contour (N = Z - P). Keeping careful track of open- and closed-loop transfer functions, as well as numerators and denominators, you should convince yourself that: the zeros of 1 + G(s)H(s) are the poles of the closed-loop transfer function the poles of 1 + G(s)H(s) are the poles of the open-loop transfer function. The Nyquist criterion then states that: P = the number of open-loop (unstable) poles of G(s)H(s) N = the number of times the Nyquist diagram encircles -1 clockwise encirclements of -1 count as positive encirclements counter-clockwise (or anti-clockwise) encirclements of -1 count as negative encirclements Z = the number of right half-plane (positive, real) poles of the closed-loop system The important equation which relates these three quantities is: Z = P + N Note: This is only one convention for the Nyquist criterion. Another convention states that a positive N counts the counter-clockwise or anti-clockwise encirclements of -1. The P and Z variables remain the same. In this case the equation becomes Z = P - N. Throughout these tutorials, we will use a positive sign for clockwise encirclements. Another way of looking at it is to imagine you are standing on top of the -1 point and are following the diagram from beginning to end. Now ask yourself: How many times did I turn my head a full 360 degrees? Again, if the motion was clockwise, N is positive, and if the motion is anti-clockwise, N is negative. Knowing the number of right-half plane (unstable) poles in open loop (P), and the number of encirclements of -1 made by the Nyquist diagram (N), we can determine the closed-loop stability of the system. If Z = P + N is a positive, nonzero number, the closed-loop system is unstable. We can also use the Nyquist diagram to find the range of gains for a closed-loop unity feedback system to be stable. The system we will test looks like this:
where G(s) is : s^2 + 10 s + 24 --------------- s^2-8 s + 15 This system has a gain K which can be varied in order to modify the response of the closed-loop system. However, we will see that we can only vary this gain within certain limits, since we have to make sure that our closed-loop system will be stable. This is what we will be looking for: the range of gains that will make this system stable in the closed loop. Gain Margin Gain Margin is defined as the change in open-loop gain expressed in decibels (db), required at 180 degrees of phase shift to make the system unstable. Now we are going to find out where this comes from. First of all, let's say that we have a system that is stable if there are no Nyquist encirclements of -1, such as : 50 ----------------------- s^3 + 9 s^2 + 30 s + 40 Looking at the roots, we find that we have no open loop poles in the right half plane and therefore no closed-loop poles in the right half plane if there are no Nyquist encirclements of -1. Now, how much can we vary the gain before this system becomes unstable in closed loop? Let's look at the following figure: The open-loop system represented by this plot will become unstable in
closed loop if the gain is increased past a certain boundary. The negative real axis area between - 1/a (defined as the point where the 180 degree phase shift occurs...that is, where the diagram crosses the real axis) and -1 represents the amount of increase in gain that can be tolerated before closed-loop instability. Phase Margin We have defined the phase margin as the change in open-loop phase shift required at unity gain to make a closed-loop system unstable. Let's look at the following graphical definition of this concept to get a better idea of what we are talking about. Let's analyze the previous plot and think about what is happening. From our previous example we know that this particular system will be unstable in closed loop if the Nyquist diagram encircles the -1 point. However, we must also realize that if the diagram is shifted by theta degrees, it will then touch the -1 point at the negative real axis, making the system marginally stable in closed loop. Therefore, the angle required to make this system marginally stable in closed loop is called the phase margin (measured in degrees). In order to find the point we measure this angle from, we draw a circle with radius of 1, find the point in the Nyquist diagram with a magnitude of 1 (gain of zero db), and measure the phase shift needed for this point to be at an angle of 180 deg. 6 b: Using Nyquist stability criterion, investigate the closed loop stability of a negative feedback control system whose open loop transfer function is given by K(sT a + 1) G(s)H(s) = ; K, T a > 0. Jun. 2014, 12 Marks s 3
7 b: List the limitations of lead and lag compensations. Jun. 2014, 5 Marks Disadvantages of Phase Lead Compensation Some of the disadvantages of the phase lead compensation - 1. Steady state error is not improved. Effect of Phase Lead Compensation 1. The velocity constant K v increases. 2. The slope of the magnitude plot reduces at the gain crossover frequency so that relative stability improves & error decrease due to error is directly proportional to the slope. 3. Phase margin increases. 4. Response becomes faster.
Disadvantages of Phase Lag Compensation Some of the disadvantages of the phase lag compensation - 1. Due to the presence of phase lag compensation the speed of the system decreases. Effect of Phase Lag Compensation 1. Gain crossover frequency increases. 2. Bandwidth decreases. 3. Phase margin will be increase. 4. Response will be slower before due to decreasing bandwidth, the rise time and the settling time become larger. 7 c: State the properties of state transition matrix and derive them. Jun. 2014, 5 Marks