1. Control Systems EEE F242/INSTR F242

2. Resources Text book: “ Control System Design” by Graham C. Goodwin, Stefan F. Graebe and Mario E. Salgado, Prentice Hall, 2001. (download from internet) References: 1. MIT OPEN COURSEWARE, Feedback control systems, fall 2010 (mainly used for control system tutorials on MATLAB discussed in common hour)

3. Resources http://www.prenhall.com/goodwin http://csd.newcastle.edu.au/control/ (for Power Point slides of text book and solved examples of text book) * Others (conveyed during semester if any)

4. Course coverage Introduction:  Motivation of control engineering  types of control system design  the principal goal of control  open and closed loop architectures

5. Course coverage Modeling:  Building models  model structure  state space models  solution of continuous time state space models  modeling errors  linearization  case studies

6. Course coverage Continuous time signals and systems:  Linear continuous time model  Laplace transform  transfer functions  stability of transfer functions  impulse and step response  poles, zeros and time response  frequency response  Fourier transform

7. Course coverage Analysis of SISO control loops:  Feedback structures  nominal sensitivity functions  closed loop stability based on the characteristic polynomial  stability and polynomial analysis  root locus, nominal stability using frequency response  relative stability: stability margins and sensitivity peaks  robustness

8. Course coverage Classical PID control:  PID structure  empirical tuning  Ziegler-Nichols oscillation method  reaction curve based method  lead lag compensators

9. Course coverage SISO control design:  Fundamental limitations in SISO control  frequency domain design limitations  architectural issues in SISO control  dealing with constraint

10. Course coverage Digital computer control:  Models for sampled data systems (sampling, linear discrete time models, Z-transform, discrete transfer functions, discrete system stability, discrete model for sampled continuous system)  Digital control (discrete time sensitivity functions, zeros of sampled data systems)

11. Course coverage Advance SISO control:  SISO controller parameterization (affine parameterization, PID synthesis using the Affine parameterization, undesirable closed loop poles, discrete time systems)  Linear state space model (controllability and stabilizability, observability and delectability, canonical decomposition, pole zero cancellation and system properties)

12. Evaluation schemes Mid-semester test: 25 Marks Comprehensive exam: 50 Marks Assignments: 25 Marks

13. History of automatic control Industrial revolution in late 18 th and early 19 th centuries prompted more complex automatic control systems The most famous of the industrial revolution control devices is Watt Flyball Governer, invented by James Watt to control the speed of steam engines

14. Watt Fly ball Governor

15. Fly ball Governor Flyball governor used on a steam engine in a cotton factory near Manchester in the United Kingdom. Actually, this cotton factory is still running today

16. The theory of automatic control developed rapidly during second world war(1939-45) leading to advanced weapon control systems like anti-aircraft gunsights, ballistic missile control systems etc. Pioneering work of Bode, Nyquist, Nichols, Evans and others resulted in simple graphical means for analyzing SISO feedback control systems (classical control theory)

17. State space approach to control developed in 60’s followed by Wiener, Kalman and others work on optimal estimation and control. This allowed multivariable problem to be treated in unified manner(modern control theory)

18. In 80’s sophisticated control dealing with issues related to the effect of model error on the performance of feedback controllers was developed (Robust control theory) Non-linear control theory was developed in parallel Process plant design also developed during the early part of 20 th century Chemical production largely occurred in batch process

19. Processes become more and more sophisticated and continuous with time and advancements in control systems Process plants automatic control was based around pneumatic systems (the controller was a mechanical device which used air pressure as communication and calculation medium)

20. In late fifties the transistor started to come into industrial use, followed by integrated circuit technology Transition started from pneumatic controllers to electronic systems (except in applications in highly flammable areas) Early 80’s computer based control system became cheap and powerful enough to use for process control system

21. Current:  Adaptive control  Auto tuning  Intelligent control  Almost every equipment has some controller embedded in it

22. System Integration Plant: the process to be controlled Objectives Sensors Actuators Communications Computing Architectures and interfacing Algorithms Accounting for disturbances and uncertainty

23. Flatness control set-up for a rolling mill

24. Introduction to principles of feedback Goal: Design a controller Gc(s) so that the system has some desired characteristics Typical objectives: Stabilize the system (stabilization) Regulate the system about some design point (regulation) Follow a given set of commands signals (tracking) Reduce response to disturbances (disturbance rejection)

25. For automatic control, we need to interface the system with a controller The controller itself has got its own physical realization and behavior Controller can be realized in a chip, analog electronics, a PLC, or a computer

26. Aim of a control engineer To find technically, environmentally, and economically feasible ways to control the desired output of a system This is possible only when we understand a process, analyze mathematically, model it, select sensors and actuators, instruct actuators with correct algorithm and so on..

27. The overall system must be stable Feedback is a key tool which can change the system behaviour Feedback systems with controllers and other components may force the system towards instability

28. Example (process schematic of a bloom caster) Large container acting as a reservoir for molten steel

29. Simplified block diagram

30. Modeling It starts with defining relevant process variables

31. Mathematical model for the process Mould level is proportion to the integral of difference between in-and outflow For, for unit cross section If mould level sensors are corrupted by noise, measured level is

32. With noise If mould level sensors are corrupted by noise, measured level is, Block diagram is given by,

33. Feedback and Feedforward simplest controller with constant gain K This controller features joint feedback and a preemptive action (feedforward).

34. Response with K=1,5

35. Conceptual controller If we know the effect an action at the input of a system produces at the output and If we have a desired behavior for the system output, then we simply need to invert relationship between input and output to determine what input action is necessary to achieve the desired output behavior

36. We have to generate u in such a way that y = r

37. Open loop controller

38. Closed loop controller

39. Closed loop controller with sensors

40. Requirement of a measurement system Reliability: should operate within necessary range Accuracy: for a variable with constant value, the measurement should settle to the correct value Responsiveness: If the variable changes, the measurement should be able to follow the change

41. Noise immunity: The measurement system, including the transmission should not be effected by noise Linearity: Non intrusive: The measuring device should not affect the behavior of the plant

42. Modeling How to select model complexity How to build model for a plant How to describe model errors How to linearize non-linear models

43. Model complexity To build an exact model of a plant is near impossible task Models are kept simple keeping essential features in mind Various models are: 1. Nominal model: Approximate description of plant used for control design

44. 2. Calibration model: this is more comprehensive description of the plant. It includes other features not used for design but have a direct bearing on achieved performance 3. Model error: this is difference between nominal model and calibration model

45. Building models Black box approach: model parameters are varied either by trial-and-error or by algorithms, until the dynamic behavior of model and plant match sufficiently well Using physical laws like conservation of mass, energy and momentum

46. State space model State space description provide the dynamics as a set of coupled first-order differential equations in a set of internal variables known as state variables, together with a set of algebraic equations that combine the state variables into physical output variables

47. State space model

48. Linear state-space model A, B, C, D are matrices of appropriate dimensions

49. Example

50. If the state variables are chosen to be The state vector becomes,

52. Model of a dc motor

54. State space model Introducing

55. Solution of continuous time state space model Solution in terms of exponential matrix which is defined as The solution is given by

56. Linearization Within certain operating range, non linear systems can be described by linear models Consider,

57. if, is a given set of trajectory that satisfies Then,

58. The trajectory may correspond to an equilibrium point of state space model and xQ, uQ,yQ will not depend on time and

59. First order Taylor series approximation gives,

60. In the matrix form,

61. In terms of increment,

62. This linearization procedure produces a model which is linear in the incremental components of inputs and outputs around a chosen operating point.

63. Example Consider a continuous time model given by, If the input fluctuates around u = 2, find an operating point with uQ = 2 and the linear model around it

64. With uQ = 2, and dx(t)/dt = 0 The linearized model is given by,

65. Output with variable u(t)

66. Continuous time signals Linear high order differential equation models Laplace transform, which converts linear differential equations to algebraic equations Methods for assessing the stability of linear dynamic systems Frequency response

67. Linear continuous time models Advantage is, superposition is applied. Output due to different inputs can be added with individual outputs

68. Laplace transform

69. Higher order function For zero initial condition, Y(s) = G(s) U(s) Transfer function is given by, G(s) = B(s)/A(s)

70. Assuming A(s) and B(s) are not zero simultaneously for any value of s The roots of B(s) = 0 are called system zeros The roots of A(s) = 0 are called system poles If A(s) =0 has nk roots at s= λk, the pole λk is said to have multiplicity nk

71. The difference in degree between A(s) and B(s) is called relative degree If m < n, the model is said to be strictly proper(positive relative degree) If m = n, model is said to be biproper (zero relative degree) If m ≤ n, model is said to be proper If m > n, model is said to be improper (negative relative degree)

72. Remarks Real systems are always strictly proper. However some controller design methods lead to biproper or even improper transfer functions. To be implemented, these controllers are usually made proper Often practical systems have a time delay between input and output which is associated with the transport of material from one point to another.

73. Heat transfer system Transfer function from output to input is given by,

74. Transfer function of state space model

75. For zero initial condition, Laplace transform of output Y(t) is related to the transform input U(s) as

76. Stability of transfer function Response of a system having transfer function G(s) is of the form, Where, βki is a function of initial conditions, and every pole at s = λk, has multiplicity nk N1+n2+n3+…………+np = n

77. The system is stable if bounded input produces a bounded output for all bounded initial conditions Partial fraction expansion is used to decompose the total response into responses of each pole taken separately For continuous time systems, stability requires that the poles have negative real parts, i.e., they need to be in the open left half plane (OLHP) of complex plane ‘s’

78. Impulse and step responses of continuous time linear systems >Linear system response to Dirac delta function(Δ 0)is of Special Interest >Laplace transform of Dirac delta is 1 >For zero initial condition, output response Y(s) = G(s)U(s) = G(s)

79.  The transfer function of a continuous time system is the Laplace transform of its response to an impulse(Dirac delta) with zero initial condition

80. Step response It is the response due to step function U(s) = 1/s For which Y(s) = G(s)*1/s Steady state response for a unit step is given by

81. Step response(typical parameters)

82. Steady state value y∞: the final value of the step response Rise time, tr: the time elapsed up to the instant at which the step response reaches, for the first time the value Kr y∞. The constant Kr is 0.9 or 1 Overshoot, Mp: the maximum instantaneous amount by which the step response exceeds its final value. Usually expressed as a percentage of y∞

83. Undershoot, Mu: the maximum instantaneous amount by which the step response falls below zero Settling time, ts: The time elapsed until the step response enters(without leaving it afterwards) a specified deviation band, ±δ, around the final value. It is usually defined as either 2% or 5% of steady state value

84. Poles, zeros and time responses b1, b2,………bm and a1, a2,…..an are the zeros and poles of the transfer function

85. Interest will be on those zeros which lie on or in the neighborhood of the imaginary axis and in those poles which lie in the RHP. Poles and zeros in these locations decide the dynamic behavior of the system When all poles and zeros of a transfer function lie in the left half of complex plane (s), it is called minimum phase transfer function

86. If a transfer function is referred to as a stable transfer function, all its poles are in open LHP If it is said to be unstable it has at least one pole in the closed RHP

87. poles Any transfer function can be expanded into partial fraction expansion Each term contains either a single real pole, a complex conjugate pair or multiple combinations with repeated poles The effect of these poles on transient behavior can be understood by first and second order poles and their interactions

88. First order pole

89. Plot Parameters y∞ and time constant t can be computed graphically

90. Second order poles Where, Ψ(0<Ψ<1) is known as damping factor and wn as the natural or undamped natural frequency Damped natural frequency is given by

91. The system has two complex conjugate poles s1 and s2 given by Laplace transform of its unit step response is given by,

93. response

94. Rise time and overshoot For kr = 1;

95. Maximum overshoot

97. Comments Small damping factor ᵩ will lead to small rise time, at the expense of high overshoot Every pole generates a special component or natural mode in the system response to an impulse input Fast poles are the poles which are much farther away from the stability boundary than the other system poles

98. Transients associated with fast poles extinguish faster than those associated with other poles Dominant poles or slow poles are the OLHP system poles which are closer to the stability boundary than the rest of the system poles For a system with poles (-1, -2±j6, -4, -5±j3), the dominant pole is -1 and fast poles are - 5±j3

99. Zeros Effect of zeros on the response of a transfer function is more difficult to analyze Poles are associated with the states in isolation, zeros arise from additive interaction amongst the states associated with different poles Zeros of a transfer function depend on where the input is applied and how the output is formed as a function of the states

100. Poles and zeros effect

101. The poles of G determines whether G is stable or unstable, as well as the decay rate and oscillation frequencies of the initial condition response The poles of G do not depend on either the input matrix B or the output matrix C The zeros are determined by the dynamics matrix A as well as B and C

102. Hence, the zeros of G depend on the physical placement of the sensors and actuators relative to the underlying dynamics The concept of zero distinguishes control theory from dynamical system theory

103. Example There are two natural modes, e-t and e-2t from the two poles at -1 and -2 respectively

104. Effect of zero A fast zero (|c|>>1) has no significant impact on the transient response

105. When the zero is slow and stable, a significant overshoot is observed When the zero is slow and unstable, a significant undershoot is observed

106. Frequency response This is the response of the system to a sine wave Response to sine wave also contains information about the system response to other signals Any signal defined in the interval [t 0, t f ], can be represented as a linear combination of sine waves of frequencies 0, ω of, 2ω 0f, 3ω 0f, ……., using Fourier analysis, where ω of = 2π/(t f -t 0 ) is known as fundamental frequency

107. Response Let the transfer function be Steady state response to sinωt is given by y(t) = |H(jω)|sin(ωt +  (ω)) Where,

108. A sine wave input forces a sine wave at the output with the same frequency. Moreover, the amplitude of the output sine wave is modified by a factor equal to the magnitude of H(jω) and the phase is shifted by a quantity equal to the phase of H(jω).

109. Bode plot Bode diagrams consist of a pair of plots. 1. One of these plots depicts the magnitude of the frequency response as a function of the angular frequency, and 2. the other depicts the angle of the frequency response, also as a function of the angular frequency.

110. Bode plot axes The abscissa axis is linear in log(ω) where the log is base 10. This allows a compact representation of the frequency response along a wide range of frequencies. The unit on this axis is the decade, where a decade is the distance between ω 1 and 10ω 1 for any value of ω 1.

111. The magnitude of the frequency response is measured in decibels [dB], i.e. in units of 20log|H(jω)|. This has several advantages, including good accuracy for small and large values of |H(jω)|, facility to build simple approximations for 20log|H(jω)|, and the fact that the frequency response of cascade systems can be obtained by adding the individual frequency responses.

112. The angle is measured on a linear scale in radians or degrees

113. Plot details and approximations A simple gain K has constant magnitude and phase Bode diagram. The magnitude diagram is a horizontal line at 20log|K|[dB] and the phase diagram is a horizontal line at 0[rad](when K is real positive) or at π[rad] (when K is real negative) The factor s k has a magnitude diagram which is a straight line with slope equal to 20k[dB/decade] and constant phase, equal to k  /2. This line crosses the horizontal axis (0[dB]) at ω = 1.

114. The factor as + 1 has a magnitude Bode diagram which can be asymptotically approximated as : – for |aω|<<1, 20 log|ajω + 1|  20 log(1) = 0[dB], i.e. for low frequencies, this magnitude is a horizontal line. This is known as the low frequency asymptote. – For |aω|>>1, 20 log|ajω+ 1|  20 log(|aω|) i.e. for high frequencies, this magnitude is a straight line with a slope of 20[dB/decade] which crosses the horizontal axis (0[dB]) at ω = |a| -1. This is known as the high frequency asymptote

115. the phase response is more complex. It roughly changes over two decades. One decade below |a| -1 the phase is approximately zero. One decade above |a| -1 the phase is approximately sign(a)0.5  [rad]. Connecting the points (0.1|a| -1, 0) and (10|a| -1, 0) by a straight line, gives sign(a)0.25  for the phase at w = |a| -1. This is a very rough approximation. For a = a 1 + ja 2, the phase Bode diagram of the factor as + 1 corresponds to the angle of the complex number with real part 1 - ωa 2 and imaginary part a 1 ω.

116. Example In standard form,

117. Magnitude plot

118. Phase plot

119. Filtering In an ideal amplifier, the frequency response would be H(jω) = K, for all ω, that is every frequency component would pass through the system with equal gain and no phase shift Practically H(jω) cannot be constant for all ω System filters inputs of different frequencies to produce the output, that is the system deals with different sine-wave component selectively, according to their individual frequencies

120. Three frequency sets The pass band: in which all the frequency components pass through the system with approximately the same amplification (or attenuation) and a phase shift which is approximately proportional to ω The stop band: in which all the frequency components are stopped. In this band, |H(jω)| is small compared to the value of |H(jω)| in the pass band Transition band: which is intermediate band between pass band and stop band

121. Filter properties Cut-off frequency, ω c this is the value of ω such that where is respectively |H(0)| for low pass filters and band reject filters |H(  )| for high pass filters the maximum value of |H(jω)| in the pass band, for band pass filters

122. Band width: This is a measure of the frequency width of the pass band (or the reject band). It is defined as B ω = ω c2 - ω c1, where ω c2 > ω c1  0. In this definition, ω c1 and ω c2 are cut-off frequencies on either side of the pass band or reject band (for low pass filters, ω c1 = 0)

123. Frequency response of a band pass filter Lower cut-off frequency is ωc1 = 50 [rad/s] and upper cut-off frequency is ωc2 = 200[rad/s] The bandwidth, Bω = 150[rad/s]

124. All pass filter A system with a constant frequency response amplitude is known as an all pass filter A pure time delay is an all pass filter

125. Distortion and fidelity Let this signal be the input to a linear stable system, The system processes this signal with fidelity if the amplitude of all sine-wave components are amplified (or attenuated) by approximately the same factor and The individual responses have the same time delay Through the system

126. Frequently encountered model

129. Analysis of SISO control loops For a given controller and plant connected in feedback, we will try to observe A) is the loop stable? B) what are the sensitivities to various disturbances? C) what is the impact of linear modeling errors? D) how do small nonlinearities impact on the loop

130. Analysis tools Root locus Nyquist stability analysis

131. Feedback structures Capacity to reduce the effect of disturbances Decrease sensitivity to model errors or to stabilize an unstable system Poorly applied feedback can make a previously stable system unstable, add oscillatory behavior into previously smooth response or result in high sensitivity to measurement noise

132. Simple feedback control system

133. C(s) and G 0 (s) denote the transfer functions of the controller and the nominal plant model respectively, Represented by,

134. P(s), L(s), B 0 (s) and A 0 (s) are polynomials in s R(s), U(s), and Y(s) denote the Laplace transforms of setpoint, control signal and plant output respectively D i (s), D 0 (s) and D m (s) denote the Laplace transforms of input disturbance, output disturbance and measurement noise, respectively

135. Laplace transform of system input and output

136. Another structure

137. Feedback control system characteristics H(s) G(s) R(s)E(s) Y(s) + - Output of the open loop system is, Y(s) = G(s) R(s) Output of closed loop system is Y(s) =[ G(s)/{1+GH(s)}] R(s)

138. Actuating error signal is E(s) = [1/GH(s)] R(s) So, to reduce the error, the magnitude of [1+GH(s)] must be greater than 1 over the range of s under consideration

139. Sensitivity of control system to parameter variations For a closed loop system, with G(s) H(s) >> 1, Output Y(s) ≈ [1/H(s)] R(s), that is the output is only affected by H(s) which may be a constant With G(s) H(s) >> 1, the system response may be highly oscillatory and even unstable With increasing the magnitude of loop transfer function G(s) H(s), effect of G(s) on the output is reduced

140. Effect of parameter variations Let the change in process leading to new process G(s) + ΔG(s) For the open loop case, ΔY(s) = ΔG(s) R(s) In the closed loop case, Y(s) +ΔY(s) = [G(s) + ΔG(s)/{1+ G(s) + ΔG(s)}H(s)]R(s)

141. Change in output is, ΔY(s) = [ΔG(s)/{1+GH(s)+ΔGH(s)}] R(s) For, GH(s) >> ΔG H(s) ΔY(s) = [ΔG(s)/{1+GH(s)}] R(s) Change in the output of the closed loop system is reduced by a factor of [1+GH(s)]

142. System sensitivity System sensitivity is defined as the ratio of the percentage change in the system transfer function to the percent change of the process transfer function for system transfer function T(s) = Y(s)/R(s), Sensitivity is defined as, S = [ΔT(s)/T(s)]/[ΔG(s)/G(s)] For small incremental changes, S = [∂T/T]/[∂G/G]

143. Sensitivity of open loop system is 1 For closed loop system with T(s) = [G/{1+GH}] S = [∂T/∂G] G/T = [1/{GH(s)}] So, the sensitivity of the system may be reduced below that of open loop system by increasing GH(s) over the frequency range of interest

144. Sensitivity with changes in feedback element H(s) S(T,H) = (∂T/∂H)H/T = -GH/[1+GH] When GH is large, the sensitivity approaches unity and the changes in H(s) directly affect the output response Important to keep feedback elements which does not change with time

146. Output and input

147. Nominal sensitivity functions

148. Names T 0 (s):Nominal complementary sensitivity S 0 (t):Nominal sensitivity S i0 (s):Nominal input disturbance sensitivity S u0 (s):Nominal control sensitivity Nominal closed loop characteristic polynomial is defined by

149. Input and output in terms of sensitivities Effect of initial conditions on plant output and controller output is given by

150. Output and input in matrix form

151. Closed loop stability The nominal closed loop is internally stable if and only if the roots of the nominal closed loop characteristic equation, All lie in the in the open left half plane

152. Example T 0 (s) is stable but the nominal input disturbance sensitivity is unstable since, And does not satisfy the condition of all the roots of Characteristic polynomial lie in left half plane

153. Stability and polynomial analysis This polynomial contains any root with nonnegative part? Polynomials having all roots in the closed left plane ( with nonpositive real parts) are known as Hurwitz polynomials

154. Some properties of this polynomial The coefficient a n-1 satisfies the condition The coefficient a 0 satisfies the condition

155. If all roots of p(s) have negative real parts, it is necessary that a i > 0 Routh’s algorithm: This algorithm determines whether the polynomial is strictly Hurwitz or not?

156. Routh’s algorithm Let a polynomial p(s) of degree n, defined as The numerical array for Routh’s algorithm is defined as

159. Examples

161. Special cases

168. Steady-state error How to find the steady-state error for a unity feedback system How to specify a system’s steady-state error performance How to find steady-state error for a nonunity feedback system How to find the steady-state error for systems represented in state-space

169. Depends on the test signal Steady-state error is the difference between the input and output for a prescribed test input as t goes to infinity Since we are concerned with the difference between the input and output of a feedback control system after steady- state has been reached, our discussion is limited to stable systems

170. Test inputs

171. Concept of steady-state error Output 1 has zero steady state error and output 2 has a finite steady state error

172. Ramp input Output 1 has zero steady state error, output 2 has finite steady state error and output 3 has infinite steady state error

173. Error E(s) is the difference between output C(s) and the input R(s)

174. Open loop transfer function E(s) = R(s)/[1-T(s)] Using final value theorem

175. example

176. Open loop transfer function

177. Steady state error

178. Error for disturbance

180. Non-unity feedback

182. Root locus Let a system defined with the transfer function Let this system be controlled with a very simple proportional controller with gain K

183. System with controller Closed loop transfer function is given by

184. System behavior with various values of K For K = 1

185. K= 10

186. K = 100

187. Observations with K = 1: response is slow With K = 10: system response has overshoot With K = 100: response is very oscillatory

188. Another approach Finding the poles of closed loop transfer function This will give overdamped response for 9>4K, underdamped response for 9<4K, and critically damped response for 9=4K

189. Underdamped response with ξ = 1/√2 Magnitude of real and imaginary parts of the roots equal to each other

190. Response Response is fast with less overshoot

191. More complicated systems Another approach is needed to analyze the system behavior with varying values of K

192. Plotting the locus at various values of K KRoots 1-2.62, -0.38 2-2, -1 4-1.5 ± j1.32 10-1.5 ± j2.78 20-1.5 ± j4.21 40-1.5 ± j6.14 100-1.5 ± j9.89 For small values of K system is overdamped and becomes underdamped as K increses

193. MATLAB HELP >> G=tf(1,[1 3 0]) Transfer function: 1/(s^2 + 3 s) >> rlocus(G) >> axis([-4 0 -10 10])

194. Root locus

195. Information from the transfer function Closed loop transfer function T(s) Characteristic equation is

196. Characteristic equation

197. Magnitude and angle condition For the characteristic equation, Magnitude condition is expressed as, The phase angle is expressed as,

198. Rules Rule 1: the root locus is symmetric about the real axis Rule 2: The number of branches of the root locus is equal to the order of the characteristic equation Rule 3: the locus starts (when K = 0) at poles of the loop gain and ends (when K →∞ ) at the zeros

199. Rule 4: The locus exists on real axis to the left of an odd number of poles and zeros on the axis Rule 5: asymptotes intersect real axis at, q = n-m Asymptotes radiate out with angles Where, r=1,3,5..

200. Rule 6: there are break-away and break-in points of the locus on the real axis where, Rule 7: angle of departure from complex pole

201. The angle of departure from a complex pole, p j, is 180 degrees + (sum of angles between p j and all zeros) - (sum of angles between p j and all other poles).

202. Rule 8: The angle of arrival at a complex pole, z j, is 180 degrees + (sum of angles between z j and all other zeros) - (sum of angles between z j and all poles).

203. Rule 9: Use Routh-Horwitz to determine where the locus crosses the imaginary axis.

204. Example We have n=3 poles at s = -2, -1 ± 1j. We have m=1 finite zero at s = -1. So there exists q=2 zeros as s goes to infinity (q = n-m = 3-1 = 2). Characteristic Equation is 1+KG(s)H(s)=0, or 1+KN(s)/D(s)=0, or D(s)+KN(s) = s 3 + 4 s 2 + 6 s + 4+ K( s + 1 ) = 0

205. Root locus

206. the locus is symmetric about the real axis The open loop transfer function, G(s)H(s), has 3 poles, therefore the locus has 3 branches. Each branch is displayed in a different color.

207. Start and end points Root locus starts (K=0) at poles of open loop transfer function, G(s)H(s). These are shown by an "x" on the diagram above As K→∞ the location of closed loop poles move to the zeros of the open loop transfer function, G(s)H(s). Finite zeros are shown by a "o" on the diagram above. Don't forget we have we also have q=n-m=2 zeros at infinity. (We have n=3 finite poles, and m=1 finite zero).

208. Locus on real axis