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: B-house extrance 25-27

2 Review of the last lecture 2 We introduced the PID-controller (Proportional Integral Derivative) P-part controls the speed I-part reduces/removes steady state error D-part reduces/removes oscillations We defined three important transfer functions: loop gain, closed-loop 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 loop-gain 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 First-order system The specifications can easily be translated to requirements on the pole for a first-order system (or a system dominated by one pole) Also remember the time-constant (1/a) which defines the time it takes to reach 63% of the steady state y f.

9 Design Specifications 9 Second-order 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 settling-time 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 Root-locus 13 In the last lecture, we derived a controller for a levitating ball, and obtained the following closed-loop system for a PID-controller The dynamics of the closed-loop 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 Root-locus 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 Root-locus 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 Root-locus 16 Root-locus is the locus or path of the roots traced out on the s-plane 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 Root-locus have common features, and we will now learn how to sketch these without actually computing a lot of roots.

17 Root-locus 17 Simple properties For any K, there are n roots. The root-locus 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 end-points. 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 Root-locus 18 Non-obvious properties: Asymptotes If m<n, n-m loci will end at zeros at infinity. The n-m 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 Non-obvious 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 Root-locus 19 Example: Levitating ball We sketch the root-locus for the levitating ball controlled using a PIDcontroller where the I-part has been fixed to K I =2 and D-part fixed to K D =4. The poles are thus given by We identify our start- and end-polynomials

20 Root-locus 20 Starting-points (n=3): End-points (m=1): Asymptotes: Intersection between asymptotes and real axis

21 Root-locus 21 Inclusion of real axis in root-locus Intersection with imaginary axis? Hence, K>0.5 leads to stable poles

22 Root-locus 22 True root-locus 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 root-locus 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 root-locus is the path of the poles traced out in the s-plane as a system parameter varies from 0 to `. Simple rules helps us to sketch the root-locus 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 Root-locus: 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 step-response 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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