I What is root locus. I System analysis via root locus. I How to plot root locus. Root locus (RL) I Uses the poles and zeros of the OL TF


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1 EE C28 / ME C34 Feedback Control Systems Lecture Chapter 8 Root Locus Techniques Lecture abstract Alexandre Bayen Department of Electrical Engineering & Computer Science University of California Berkeley Topics covered in this presentation I What is root locus I System analysis via root locus I How to plot root locus September 0, 203 Bayen (EECS, UCB) Feedback Control Systems September 0, 203 / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Lecture outline Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 History interlude Definitions, [, p. 388] Walter Richard Evans I I American control theorist I 948 Inventor of the root locus method I 988 Richard E. Bellman Control Heritage Award Root locus (RL) I Uses the poles and zeros of the OL TF (product of the forward path TF and FB path TF) to analyze and design the poles of a CL TF as a system (plant or controller) parameter, K, thatshowsupas a gain in the OL TF is varied I Graphical representation of I Stability (CL poles) I Range of stability, instability, & marginal stability I Transient response I T r, T s,&%os I Solutions for systems of order > 2 Figure: ±FB system Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39
2 The control system problem, [, p. 388] The control system problem, [, p. 388] I Forward TF I OL TF KG(s)H(s) I OL TF poles una ected by the one system gain, K I Feedback TF I FB CL TF G(s) = N G(s) D G (s) H(s) = N H(s) D H (s) Figure: a. FB CL system; b. equivalent function T (s) = KN G (s)d H (s) D G (s)d H (s)+kn G (s)n H (s) Figure: a. FB CL system, b. equivalent function Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Vector representation of complex numbers, [, p. 388] Vector representation of complex numbers, [, p. 390] I Cartesian, + j! I Polar, M\ I Magnitude, M I Angle, I Function, F (s) I Example, (s + a) I Vector from the zero, a, of the function to the point s Figure: Vector representation of complex numbers: a. s = + j!, b. (s + a); c. alternate representation of (s + a), d.(s + 7) s!5+j2 I Function, F (s) I Complicated my (s + z i ) Y i= numerator s complex factors F (s) = = Y ny denominator s complex factors (s + p j ) I I Magnitude Angle j= M = = X zero angles Y zero lengths Y pole lengths = my (s + z i) i= ny (s + p j) j= X X m pole angles = \(s + z i) i= nx \(s + p j) i=j Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / Defining the RL 8.3 Properties of the RL Bayen (EECS, UCB) Feedback Control Systems September 0, 203 / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39
3 8.3 Properties of the RL FB CL poles, [, p. 394] T (s) = KG(s) +KG(s)H(s) The angle of the complex number is an odd multiple of 80 KG(s)H(s) = =\(2k + )80 k =0, ±, ±2, ±3,... Figure: FB system The system gain, K, satisfies magnitude criterion angle criterion KG(s)H(s) = \KG(s)H(s) =(2k + )80 and thus G(s) H(s) Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Basic rules for sketching FB RL, [, p. 397] Basic rules for sketching FB RL, [, p. 397] I Number of branches: Equals the number of CL poles I Symmetry: About the real axis I Realaxis segments: On the real axis, for K>0, therlexiststothe left of an odd number of realaxis, finite OL poles and/or finite OL zeros I Starting and ending points: The RL begins at the finite & infinite poles of G(s)H(s) and ends at the finite & infinite zeros of G(s)H(s) I Behavior at : The RL approaches straight lines as asymptotes as the RL approaches. Further, the equation of the asymptotes is given by the realaxis intercept, a, and angle, a, as follows P P finite poles finite zeros a = a = #finite poles #finite zeros (2k + ) #finite poles #finite zeros where k =0, ±, ±2, ±3,... and the angle is given in radians with respect to the positive extension of the realaxis Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Example, [, p. 400] Example (FB RL with asymptotes) I Problem: Sketch the RL Figure: RL & asymptotes for system Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39
4 Additional rules for refining a RL sketch, [, p. 402] Di erential calculus procedure, [, p. 402] I Realaxis breakaway & breakin points: At the breakaway or breakin point, the branches of the RL form an angle of 80 /n with the real axis, where n is the number of CL poles arriving at or departing from the single breakaway or breakin point on the realaxis. I The j!axis crossings: The j!crossing is a point on the RL that separates the stable operation of the system from the unstable operation. I Angles of departure & arrival: The value of! at the axis crossing yields the frequency of oscillation, while the gain, K, at the j!axis crossing yields the maximum or minimum positive gain for system stability. I Plotting & calibrating the RL: All points on the RL satisfy the angle criterion, which can be used to solve for the gain, K, at any point on the RL. Procedure I Maximize & minimize the gain, K, using di erential calculus: The RL breaks away from the realaxis at a point where the gain is maximum and breaks into the realaxis at a point where the gain is minimum. For all points on the RL G(s)H(s) For points along the realaxis segment of the RL where breakaway and breakin points could exist, s =. Di erentiating with respect to and setting the derivative equal to zero, results in points of maximum and minimum gain and hence the breakaway and breakin points. Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Transition procedure, [, p. 402] The j!crossings, [, p. 405] Procedure I Eliminates the need to di erentiate. Breakaway and breakin points satisfy the relationship mx i= + z i = nx j= + p j where z i and p i are the negative of the zero and pole values, respectively, of G(s)H(s). Procedures for finding j!crossings I Using the RouthHurwitz criterion, forcing a row of zeros in the Routh table will yield the gain; going back one row to the even polynomial equation and solving for the roots yields the frequency at the imaginaryaxis crossing. I At the j!crossing, the sum of angles from the finite OL poles & zeros must add to (2k + )80.Searchthej!axis for a point that meets this angle condition. Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Angles of departure & arrival, [, p. 407] Plotting & calibrating the RL, [, p. 40] The RL departs from complex, OL poles and arrives at complex, OL zeros I Assume a point close to the complex pole or zero. Add all angles drawn from all OL poles and zeros to this point. The sum equals (2k + )80. The only unknown angle is that drawn from the close pole or zero, since the vectors drawn from all other poles and zeros can be considered drawn to the complex pole or zero that is close to the point. Solving for the unknown angle yields the angle of departure or arrival. Search a given line for a point yielding X zero angles X pole angles =(2k + )80 or \G(s)H(s) =(2k + )80 The gain at that point on the RL satisfies Q finite pole lengths G(s)H(s) = Q finite zero lengths Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39
5 Example, [, p. 42] Example (FB RL & critical points) I Problem: Sketch RL & find I =0.45 line crossing I j!axis crossing I The breakaway point I The range of stable K Figure: RL Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Conditions justifying a 2 nd order approximation, [, p. 45] I Higherorder poles are much farther (rule of thumb: > 5 ) into the LHP than the dominant 2 nd order pair of poles. I CL zeros near the CL 2 nd order pole pair are nearly canceled by the close proximity of higherorder CL poles. I CL zeros not canceled by the close proximity of higherorder CL poles are far removed from the CL 2 nd order pole pair. Figure: Making 2 nd order approximation Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Higherorder system design, [, p. 46] 8.8 Generalized RL Procedure. Sketch RL 2. Assume the system is a 2 nd order system without any zeros and then find the gain to meet the transient response specification 3. Justify your 2 nd order assumptions 4. If the assumptions cannot be justified, your solution will have to be simulated in order to be sure it meets the transient response specification. It is a good idea to simulate all solutions, anyway Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39
6 8.8 Generalized RL Example, [, p. 49] Example (FB RL with a parameter pole) I Problem: Create an equivalent system whose denominator is +p G(s)H(s) and sketch the RL Figure: RL Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 +FB CL poles, [, p. 394] Basic rules for sketching +FB RL, [, p. 42] T (s) = KG(s) KG(s)H(s) The angle of the complex number is an even multiple of 80 KG(s)H(s) ==\k360 k =0, ±, ±2, ±3,... Figure: +FB system The system gain, K, satisfies magnitude criterion angle criterion and thus KG(s)H(s) = \KG(s)H(s) =k360 G(s) H(s) I Number of branches: Equals the number of CL poles (same as FB) I Symmetry: About the real axis (same as FB) I Realaxis segments: On the real axis, for K>0, therlexiststothe left of an even number of realaxis, finite OL poles and/or finite OL zeros I Starting and ending points: The RL begins at the finite & infinite poles of G(s)H(s) and ends at the finite & infinite zeros of G(s)H(s) (same as FB) Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Basic rules for sketching +FB RL, [, p. 422] I Behavior at : The RL approaches straight lines as asymptotes as the RL approaches. Further, the equation of the asymptotes is given by the realaxis intercept, a, and angle, a, as follows P P finite poles finite zeros a = a = #finite poles #finite zeros k2 #finite poles #finite zeros where k =0, ±, ±2, ±3,... and the angle is given in radians with respect to the positive extension of the realaxis. Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39
7 Definitions, [, p. 424] Example, [, p. 425] Root sensitivity I The ratio of the fractional change in a CL pole to the fractional change in a system parameter, such as a gain. Sensitivity of a CL pole, s, to gain, K Approximated as where S s: K s s K s s = s(s s:k ) K s K is found by di erentiating the CE with respect to K Example (root sensitivity of a CL system to gain variations) I Problem: Find the root sensitivity of the system at s = 5+j5 (for which 50) andcalculatethe change in the pole location for a 0% change in K Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bayen (EECS, UCB) Feedback Control Systems September 0, / 39 Bibliography Norman S. Nise. Control Systems Engineering, 20. Bayen (EECS, UCB) Feedback Control Systems September 0, / 39
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