Välkomna till TSRT15 Reglerteknik Föreläsning 4. Summary of lecture 3 Root locus More specifications Zeros (if there is time)

Transcription

1 Välkomna till TSRT15 Reglerteknik Föreläsning 4 Summary of lecture 3 Root locus More specifications Zeros (if there is time)

2 Summary of last lecture 2 We introduced the PID-controller (Proportional Integrating Differentiating) P-part controls the speed I-part reduces/removes stationary control error D-part reduces/removes oscillations We defined three important transfer functions: loop gain, closed-loop system and the sensitivity function

3 Summary of last lecture 3 Error coefficients were defined as the remaining stationary control error when the referens 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 Root-locus 4 In the last lecture, we derived a controller a levitating ball, and obtained the following closed-loop system for a PID-controller The dynamics of the closed-loop system is charcterized 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

5 Root-locus 5 The standard problem: We study polynomials in the following form Example: 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 och K I =0.1 and compute plles for 0 K D <

6 Root-locus 6 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 =10000: {-99.9, ±0.001i}

7 Root-locus 7 Root-locus have common features, and we will now learn how to sketch these without actually computing a lot of roots We assume P(s) and Q(s) are given in the following form Additionally, we assume n m and K 0

8 Root-locus 8 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 Ifm<nthereare asymptotic branches tending towards infinity Since complex poles always appear in complex conjugated pairs, the root-locus 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

9 Root-locus 9 Non-obvious properties: Asymptotes The n-m roots not tending towards the end-points move along asymptotes starting in the point real point and directed along rays with angle

10 Rotort 10 Non-obvious properties: Real axis Those parts of the real axis that have an odd number of real startand end-points to the right are part of the root-locus Let p i be real start-points and q i real end-points. We then have The statement follows from a sign-analysis of the ratio, and the fact that the left-hand side is positive for sufficiently large positive real s (we do not care about the proof though)

11 Root-locus 11 Non-obvious properties: Breakaway from real axis When the root-locus leaves the real axis, it makes an 90º angle with the real axis

12 Root-locus 12 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

13 Rotort 13 Starting-points (n=3): End-points (m=1): Asymptotes: Intersection between asymptotes and real axis

14 Rotort 14 Inclusion of real axis in root-locus Intersection with imaginary axis? Hence, K>0.5 leads to stable poles

15 Root-locus 15 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

16 Specifications 16 So where do we want to have our poles? We have previously seen that we want to have all poles in the left half-plane for stability, complex poles leads to oscillations and the distance to the origin controls the speed of the system We will now make these statements more precise, and relate to stepresponses

17 Specifications 17

18 Specifications 18 Overshoot M: Largest output divided by final value (sometimes in %) Rise-time T r : Time required to go from 10% to 90% of final value Solution-time T s : Time before output signal stays within 5% from final value

19 Specifications 19 First-order system: The specifications can easily be translated to requirements on the pole in 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 final value

20 Specificationer 20 Second-order system The specifications for a 2:nd order system is slightly more involved The details are not important

21 Specifications 21 What to remember: The solution-time is roughly 3/(distance to the origin) for systems with real poles and oscillating systems with a reasonable relative damping ξ (between 0.5 and 1) A relative damping 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

22 Root-locus 22 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

23 Summary 23 Summary of todays lecture A root-locus describes how the poles move in the complex plane when a parameter in the pole polynomial is varying Simple rules helps us to draw the root-locus without actually computing a lot of roots The solution-time for a step is roughly 3/(distance to the origin for the pole closest to the origin) A relative damping of 0.7 gives an overshoot of roughly 5%

24 Summary 24 Important words Root-locus: Position of the poles in the complex plane as a function of a parameter in the pole polynomial Solution time: The time it takes for a step response until it stays within 5% of the final value Rise-time: The time it takes for a step-response to go from 10% to 90% of the final value Overshoot: Largest output value divided by the final value