Anna University
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
Subject Code/Name: 147405: CONTROL SYSTEMS
SEM / YEAR: IV/ II
2 MARK ANSWERS
UNIT I - CONTROL SYSTEM MODELLING
PART-A
1. Write Mason’s gain formula
The overall gain or transfer function
k – no. of forward paths in the signal flow graph
P_{k} – forward path gain of k^{th} forward path
Δ – 1- (sum of individual loop gains) + (sum of gain products of all possible combinations of two non-touching loops) – (sum of gain products of all possible combinations of three non-touching loops) + etc.,
Δ_{k} – Δ for that part of the graph which is not touching k^{th} forward path.
2. What is mathematical model of a system?
The set of mathematical equations describing the dynamic characteristics of the system is called mathematical model of a system.
3. What do you meant by sensitivity of the control system?
Sensitivity is a measure of the effectiveness of feedback in reducing the influence of variations on the system performance.
4. What is system?
When a number of elements or components are connected in a sequence to perform a specific function is called a system.
5. What is control system?
When the output quantity is controlled by varying the input quantity, then the system is called control system. The output quantity is called controlled variable or response, the input quantity is called command signal or excitation.
6. What are the two major types of control systems?
Open loop control systems, and
Closed loop control systems
7. Define open loop and closed loop systems.
Open loop system
Any physical system which does not automatically correct the variation in its output is called an open loop control system.
Closed loop system
A system in which the controlling action or input is dependent on the output or changes in output is called closed loop control system.
8. What is feedback? What type of feed back is employed in control system?
Feedback is a property of the system by which it permits the output to be compared with input so that appropriate controlling action can be decided.
Negative feedback is employed in control systems.
9. Why negative feedback is preferred in control systems?
Negative feedback results in better stability in steady state and rejects any disturbance signals. It also has low sensitivity to parameter variations. Hence, negative feedback is preferred in control systems.
10. Distinguish between open loop and closed loop systems.
Open loop system | Closed loop system |
Inaccurate and unreliable | Accurate and reliable |
Simple and economical | Complex and costly |
Changes in output due to external disturbances are not corrected automatically | Changes in output due to external disturbances are corrected automatically |
They are generally stable | Great efforts are needed to design a stable system. |
11. State principle of superposition theorem.
Superposition principle implies that if a system model has responses y_{1}(t) and y_{2}(t) to any two inputs x_{1}(t) and x_{2}(t) respectively, then the system response to the linear combination of these inputs
a_{1}x_{1}(t) + a_{2}x_{2}(t) is given by the linear combination of individual outputs, a_{1}y_{1}(t) + a_{2}y_{2}(t) where a_{1}, a_{2 }are constants.
12. What is time variant and Time invariant?
If the parameters of the system are varying with time, then the control system is time-variant. Otherwise, system is time-invariant.
Example of time-variant: Space vehicle whose mass decreases with time as it leaves earth.
Example of time-invariant: Different electrical networks consisting of resistance, inductance and capacitance.
13. Define transfer function.
The transfer function of a linear time invariant system is defined as the ratio of Laplace transform of the output variables to the Laplace transform of the input variables under the assumption that all initial conditions are zero.
Transfer function = with zero initial conditions.
14. Write force balance equation of ideal spring, ideal mass.
Let a force F be applied to an ideal spring with spring constant K. The spring will offer an opposing force S_{K} which is proportional to displacement.
Let a force F be applied to an ideal mass M, the mass will offer an opposing force S_{M} which is proportional to acceleration.
15. Name the two types of electrical analogous for mechanical system.
The two types of analogies for the mechanical system are force-voltage and force-current analogy.
16. What is signal flow graph?
A signal flow graph is a diagram that represents that represents a set of simultaneous algebraic equations. By taking Laplace transform, the time domain differential equations governing a control system can be transferred to a set of algebraic equations in s-domain.
17. Define non-touching loop.
The loops are set to be non-touching if they do not have common nodes.
18. Write down force-voltage analogous equations relating the mechanical and electrical system for a simple mars spring-damper system.
Mechanical system F(s) = M s^{2} X(s) + B s X(s) + K X(s)
V(s) = Ls^{2} Q(s) + R s Q(s) + 1/C Q(s) – force-voltage analogy.
I(s) = C s^{2} Φ(s) + 1/R s Φ(s) + 1/L Φ(s) – force-current analogy.
19. What are the two assumptions to be made while deriving transfer function of electrical systems?
1. The system should be approximated by a linear lumped constant parameters model by making suitable assumptions.
2. It is assumed that there is no loading. i.e. no power is drawn at the output of the system.
UNIT II - TIME RESPONSE ANALYSIS
PART-A
1. What is time response?
The time response is the output of closed loop system of a function of time. It is denoted by c(t). it is given by inverse laplace of the product of input and transfer function of the system.
The close d loop transfer function, C(s)/ R(s) = G(s)/ 1+ G(s) H(s)
Response in s domain, C(s)=R(s)G(s)/1+ G(s) H(s)
Response in timedomain,c(t)=inverae L.T{ R(s)G(s)/1+ G(s) H(s)}
2. What is transient and steady state response?
The transient response is the response of the system when the input changes from one state to another. The response of the system as t tends to infinity is called steady state response.
3. Name the test signals used in time response analysis.
Step signal, impulse signal, Ramp signal, parabolic signal, acceleration and sinusoidal signals.
4. Define step signal.
The step signal is a signal whose value changes from zero to A at t= 0 and remains constant at A for t>0.
r(t) = A, t ≥ 0
0, t < 0
5. Define Ramp signal and parabolic signal.
The ramp signal is a signal whose value increases linearly with time from an initial value of zero at t=0.the ramp signal resembles a constant velocity.
r(t) = At, t ≥ 0
0, t < 0
Parabolic signal is a signal in which the instantaneous value varies as the square of the time from an initial value of zero at t=0. The mathematical representation of parabolic signal is
r(t) = t ≥ 0
0, t < 0
6. What is impulse signal?
A signal which is available for a very short duration is called impulse signal and is denoted by
7. How is system classified depending on the value of damping?
Depending on the value of damping, the system is classified into following four cases.
Case i) Un-damped system, ζ = 0
Case ii) Under-damped system,0 < ζ < 1
Case iii) Critically damped system ζ = 1
Case iv) Over-damped system ζ > 1
8. What is damped frequency of oscillation?
In under-damped system, the response is damped oscillatory. The frequency of damped oscillation is given by ω_{d} = ω_{n} √(1- ζ^{2})
9. List the time domain specifications.
rise time, delay time, peak time, peak over shoot, Settling time.
10. Define rise time, delay time, peak time, peak over shoot.
The time taken for response to reach 50% of final value for the very first time is delay time.
The time taken for response to raise from 0% to 100% for the very first time is rise time.
The time taken for the response to reach the peak value for the first time is peak time.
Peak overshoot is defined as the ratio of maximum peak value measured from the Maxmium value to final value
Settling time is defined as the time taken by the response to reach and stay within specified error
11. What is steady state error?
The steady state error is defined as the value of error as time tends to infinity.
12. What are static error constants?
Positional error constant Velocity error constant Acceleration error constant
13. Define position, velocity and acceleration error constants.
Position error constant K_{p} = . The steady state error in type-0 system when the input is unit step is given by .
The velocity error constant K_{v} = . The steady state error in type-1 system when the input is unit ramp is given by .
The acceleration error constant K_{a} = . The steady state error in type-2 system when the input is unit parabolic is given by .
14. What are generalized error constants?
They are the coefficients of generalized series. The generalized error series is given by
e(t) = C_{0}r(t) + C_{1}dr(t)/dt + ( C_{2 }/ 2! ) dr^{2}(t)/dt^{2} + ………….. + (C_{n} / n!) dr^{n}(t)/dt^{n}…
The coefficients C_{0}, C_{1}, C_{2},…,C_{n} are called generalized error coefficients or dynamic error coefficients.
15. List the advantages of generalized error constants.
i)Steady state is function of time.
ii)Steady state can be determined from any type of input.
16. What are the effects of damping ratio on the time response of a second order system?
The system breaks in to continuous oscillations for ζ = 0,as damping ratio is increased, the response becomes progressively less oscillatory till it becomes critically damped for ζ = 1 and becomes over damped for ζ >1.
UNIT III - FREQUENCY RESPONSE ANALAYSIS
PART-A
1. What is frequency response analysis?
A frequency responses the steady state response of a system when the input to the system is a sinusoidal signal.
2. What is Nichol’s chart?
Nichol’s chart is a frequency response plot of the open loop transfer function of a system. It is a graph between magnitude of G(jω) in dB and the phase of G(jω) in degree, plotted on a ordinary graph sheet.
3. What are the two contours of Nichols chart?
Closed loop contour and phase contour.
4. What are the advantages of Nichol’s chart?
The advantages are:
i) It is used to find closed loop frequency response from open loop frequency response.
ii) The frequency domain specifications can be determined from Nichol’s chart.
iii) The gain of the system can be adjusted to satisfy the given specifications.
5. What is polar plot?
The polar plot of a sinusoidal transfer function G(jω) is a plot of the magnitude of G(jω) versus the phase angle of G(jω) on polar or rectangular co-ordinates as ω is varied from zero to infinity.
6. Define gain cross over frequency.
The gain cross over frequency ω gc is the frequency at which the magnitude of the open loop transfer function is unity..
7. Define Phase cross over frequency.
The frequency at which, the phase of open loop transfer functions is called phase cross over frequency ωpc
8. Define Phase Margin.
The phase margin ,γ is the amount of phase lag at the gain cross over frequency required to bring system to the verge of instability.
9. Define Gain Margin.
The gain margin,kg is defined as the reciprocal of the magnitude of the open loop transfer function at phase cross over frequency. Gain margin kg = 1 / G(jωpc)
10. How do you calculate the gain margin from the polar plot?
The reciprocal of the magnitude of the open loop transfer function at phase cross over frequency. Gain margin kg = 1 / G(jωpc)
11. How do you find the stability of the system by using polar plot?
Stability depends on the comparision of magnitudes of ωgc and ωpc.
ωgc < ωpc, G.M and P.M are positive, system is stable.
ωgc > ωpc, G.M and P.M are negative, system is unstable.
ωgc = ωpc, G.M and P.M are zerto, system is marginally stable.
12. What are the advantages of Bode plot?
The main advantages are:
i) Multiplication of magnitude can be in to addition.
ii) A simple method for sketching an approximate log curve is available.
iii) It is based on asymptotic approximation. Such approximation is sufficient if rough information on the frequency response characteristic is needed.
iv) The phase angle curves can be easily drawn if a template for the phase angle curve of 1+ jω is available.
13. List the Frequency domain specifications?
Frequency domain specification are i)Resonant peak. ii)Resonant frequency
14. What is minimum phase system?
A transfer function which has the least phase angle for a given magnitude curve is called minimum phase system.
15. What is non-minimum transfer function?
A transfer function which has one or more zeros in the right half s-plane is known as non-minimum transfer function.
16. What is cut off frequency?
The frequency at which the magnitude | G(jω)| is 1/√2 times of its maximum value is known as cut off frequency
17. What is Band width?
The bandwidth is the range of frequencies for which the system gain Is more than 3 dbB. The bandwidth is a measure of the ability of a feedback system to reproduce the input signal ,noise rejection characteristics and rise time.
UNIT IV – STABILITY ANALYSIS
PART-A
1. State Nyquist stability Criterion.
If the Nyquist plot of the open loop transfer function G(s) corresponding to the nyquist control in the S-plane encircles the critical point –1+j0 in the counter clockwise direction as many times as the number of right half S-plane poles of G(s),the closed loop system is stable.
2. What is root locus?
The path taken by the roots of the open loop transfer function when the loop gain is varied from 0 to α are called root loci.
3. What is the necessary condition for stability?
The necessary and sufficient condition for stability is that all of the elements in the first column of the routh array should be positive.
4. What is characteristic equation?
The denominator polynomial of C(s)/R(s) is the characteristic equation of the system.
5. How the roots of characteristic are related to stability?
If the roots of the characteristic equation has positive real part, then the impulse response of the system is not bounded, hence the system will be unstable. If the roots has negative real part, then the impulse response is bounded, hence the system is stable.
6. Define stability.
For a bounded input signal, if the output has constant amplitude oscillations may be stable.
7. What do you mean by dominant pole?
The dominant pole is a pair of complex conjugate poles which decides the transient response of the system. In higher order systems, the dominant poles are very close to origin and all other poles of the system are widely separated and so they have less effect of transient response of the system.
8. What are break away points?
At break away point, the root locus breaks from the real axis to enter into the complex plane. To find break away points, from an equation for K from the characteristic equation and differentiate the equation of K with respect to s. Then find the roots of the equation dK/ds = 0.
9. How will you find the root locus on real axis?
To find root locus on real axis, choose a test point on real axis. If the total number of poles and zeros on the real axis to the right of this test point is odd number, then the test point lies on the root locus. If it is even, then the test point does not lie on the root locus.
10. What is centroid? How the centroid is calculated?
The meeting point of asymptotes with real axis is called centroid. The centroid is given by
11. State BIBO stability criterion.
A linear relaxed system is said to have BIBIO stability if every bounded input results in a bounded output.
UNIT-V
STATE VARIABLE ANALYSIS & DIGITAL CONTROL SYSTEMS
PART-A
1. What is sampled data control system?
In a control system,if the signals in any part of the system is discrete then the entire system is said to be sampled data system.
2. Write the advantages and disadvantages of sampled data control system.
Advantages:
High accuracy, minimum cost, The complex control calculations can be performed very easily, Data processing with the help of digital controllers is straight and fast,
Disadvantages:
The sampling process tend to result in more errors, which may affect the system performance. There are limitations on the signal resolution due to the finite wordlength of the digital processors, Designing the digital controllers is very much complex than designin the analog controller for an equivalent level of performance.
3. State sampling theorem.
A continuous time signal can be cinpletely represented in its samples and recovered back if the sampling frequency Fs≥2Fmax where Fs is the sampling frequency and Fmax is the maximum frequency present in the signal.
4. What is periodic sampling?
Sampling of a signal at uniform equal intervals is called periodic sampling.
5. What are hold circuits & explain it.
The function of the hold circuit is to reconstrust the signal which is applied as input to the sampler. The simplest holding device. Holds the signal between two consecutive instants at its preceded value.till next sampling instant is reached.
6. What are the problems encountered in a practical hold circuits?
Hold mode may drop occure, nonlinear variation during sampling aperture,error in the periodicity of sampling.
7. What are the methods available for the stability analysis of sampled data control system?
8. What are the advantages of state space analysis?
It can be applied to non-linear as well as time varying systems.
Any type of input can be considered for designing the system
It can be conveniently applied to multiple input multiple output systems.
The state variables selected need not necessarily be the physical quantities of the system
9. What are state variables?
The state of the system is a minimal set of variables known as state variables.
10. What are phase variables?
The phase variables are defined as the state variables which are obtained from one of the system variables and its derivatives.