Automatic Control (TSRT15): Lecture 4


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1 Automatic Control (TSRT15): Lecture 4 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Phone: Office: Bhouse extrance 2527
2 Review of the last lecture 2 We introduced the PIDcontroller (Proportional Integral Derivative) Ppart controls the speed Ipart reduces/removes steady state error Dpart reduces/removes oscillations We defined three important transfer functions: loop gain, closedloop system and the sensitivity function where G(s) and F(s) are transfer functions of the system and the controller.
3 Review of the last lecture 3 Error coefficients were defined as the remaining steady state error when the reference signal is a step, ramp, etc The number of integrators in the loopgain G O (s) = F(s)G(s) decides the number of error coefficients that are zero
4 Outline 4 Design Specifications Root locus
5 Design Specifications 5 R(s) Σ E(s) F(s) U(s) G(s) Y(s) The specifications, which are stated in terms of the measures of performance, indicate the quality of the system to the designer. 1 The specifications include several time response indices for a specified input test signal and the steady state error. We will only consider the step signal. The specifications usually have connections with the positions of the poles on the complex plane and can thus be achieved by placing the poles in suitable region on the complex plane.
6 Design Specifications 6
7 Design Specifications 7 Let s first identify y f is the steady state. r is the magnitude of the step input signal. e 0 is the steady state error for the step input signal. Overshoot: Rise time T r : Time required to go from 10% to 90% of y f. Settling time T s : Time required for y(t) to settle within a certain percentage δ of y f. δ is typically set to be 5%.
8 Design Specifications 8 Firstorder system The specifications can easily be translated to requirements on the pole for a firstorder system (or a system dominated by one pole) Also remember the timeconstant (1/a) which defines the time it takes to reach 63% of the steady state y f.
9 Design Specifications 9 Secondorder system The specifications for a second order system is slightly more involved The details are not important
10 Design Specifications 10 What to remember: The settlingtime is roughly 3/ Re(p) for first and second order systems where p is the pole of the system. A damping ratio ξ of 0.7 gives an overshoot around 5%, which typically is what we want. Im In the complex plane, it means that we want the poles to be in the shadowed area (in a cone with an angle 45º corresponding to ξ=0.7) 45º Re
11 Effects of a third pole 11 One real pole: A pair of complex conjugate poles: Im Im Re Re
12 Effects of a third pole 12 a=10, ω 0 =1, ξ=0.7 a=1, ω 0 =10, ξ=0.7 The rightmost pole(s) (real or complex conjugate) on the strictly left half plane are said to be dominant pole(s) if the absolute value of its real part is less than one tenth of the absolute value of the real part of all the other poles. The response of a high order system can be approximated by a low order system with the dominant pole(s) as pole(s).
13 Rootlocus 13 In the last lecture, we derived a controller for a levitating ball, and obtained the following closedloop system for a PIDcontroller The dynamics of the closedloop system is characterized by the poles of the transfer function,.i.e., the roots of the pole polynomial Question today how do poles depend on parameters in the polynomial. Here, three parameters and three roots, but we will only study the case when only one parameter is allowed to vary.
14 Rootlocus 14 K P and K I fixed, K D varying Since roots are simple to compute in MATLAB, we can compute them for varying K D, and plot them in the complex plane We try with K P =1 and K I =0.1 and compute poles for 0` K D < `
15 Rootlocus 15 Poles for K D =0: {0.09, 0.049` i} Poles for K D =2: {1.28, 0.59,0.13} Poles for K D =100: {99.9, ` 0.031i}
16 Rootlocus 16 Rootlocus is the locus or path of the roots traced out on the splane as a parameter changes. We study the roots of the following equation as K changes We assume P(s) and Q(s) are given in the following form where n ` m and K ` 0 Rootlocus have common features, and we will now learn how to sketch these without actually computing a lot of roots.
17 Rootlocus 17 Simple properties For any K, there are n roots. The rootlocus is said to have n branches. The roots for K=0 are the roots of P(s)=0. These roots are called startingpoints. The roots for K=` are the roots of Q(s)=0. These roots are called the endpoints. Since complex poles always appear in complex conjugate pairs, the rootlocus is symmetric w.r.t the real axis. A stability border can be found by solving an equation for crossing the imaginary axis P(iω) + KQ(iω)=0.
18 Rootlocus 18 Nonobvious properties: Asymptotes If m<n, nm loci will end at zeros at infinity. The nm loci proceed to the zeros at infinity along asymptotes as K approaches infinity. These linear asymptotes are centered at a point on the real axis The angle of the asymptotes with respect to the real axis is Nonobvious properties: Real axis The root locus on the real axis always lies in a section of the real axis to the left of an odd number of poles and zeros.
19 Rootlocus 19 Example: Levitating ball We sketch the rootlocus for the levitating ball controlled using a PIDcontroller where the Ipart has been fixed to K I =2 and Dpart fixed to K D =4. The poles are thus given by We identify our start and endpolynomials
20 Rootlocus 20 Startingpoints (n=3): Endpoints (m=1): Asymptotes: Intersection between asymptotes and real axis
21 Rootlocus 21 Inclusion of real axis in rootlocus Intersection with imaginary axis? Hence, K>0.5 leads to stable poles
22 Rootlocus 22 True rootlocus Note that the fact that the two complex roots become real for an interval not can be seen using our methodology. The two complex poles could just as well have gone directly towards the asymptotic directions, according to our rules.
23 Use rootlocus to achieve the specifications 23 It seems possible to pick K P to place the poles in the desired area, when K I and K D are fixed at 2 and 4
24 Summary of this lecture 24 A rootlocus is the path of the poles traced out in the splane as a system parameter varies from 0 to `. Simple rules helps us to sketch the rootlocus without actually computing a lot of roots. The settling time for a step is roughly 3/ Re(p) where p is the pole closest to the origin. A damping ratio of 0.7 gives an overshoot of roughly 5%. The response of a high order system can be approximated by a low order system with the dominant pole(s) as pole(s).
25 Summary of this lecture 25 Important concepts Rootlocus: Position of the poles in the complex plane as a function of a parameter in the pole polynomial. Settling time: The time it takes for a step response until it stays within 5% of the steady state. Rise time: The time it takes for a stepresponse to go from 10% to 90% of the steady state. Overshoot: Largest output subtracts the steady state and then is divided by the steady state. Dominant pole(s) : The rightmost pole(s) (real or complex conjugate) on the strictly left half plane are said to be dominant pole(s) if the absolute value of its real part is less than one tenth of the absolute value of the real part of all the other poles.
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