Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard

Transcription

1 Control Systems II ETH, MAVT, IDSC, Lecture 4 17/03/2017

2 Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design Lecture 1 Recalls, Introductory case study Lecture 2 Cascaded Control Lecture 3 Case study: Ball on Wheel (BoW) (Mathworks Guest) Controller Implementation Lecture 4 Control design for time delayed systems (Smith predictor), Robustness criteria Lecture 5 Real PID, Gain scheduling, Emulation, Aliasing, Anti-reset windup MIMO System Analysis Lecture 6 MIMO Introduction Lecture 7 Singular Values Lecture 8 Frequency domain MIMO Controller Synthesis Lecture 9 State feedback (LQR), BoW Lecture 10 Finite horizon LQR, Model Predictive Control (MPC) Lecture 11 State observer Lecture 12 Output feedback (LQG, LTR) Lecture 13 Case studies, Exam preparation 2

3 Lecture 4 Class Content 1. Control design for systems with time delays: the Smith Predictor 2. Robustness and performance criteria 3

4 Lecture 4 Class Content 1. Control design for systems with time delays: the Smith Predictor 2. Robustness criteria 4

5 System with time delay: example 1 Control inputs System output Delay computed as : Delay due to the distance between the location of the control action ( ) and the location where the material height ( ) is taken. 5

6 System with time delay: example 2 Control inputs: To cold-water mass flow Uc Hot-water mass flow Uh System output Output water temperature To Th U h U c Delay due to the distance between the control action (Uc, Uh) and the location where the water temperature is taken. Tc 6

7 Definition of a time delay Laplace transform of a delayed function f t s ft () Example with a step function 0 t 7

8 Frequency response of e s Σ s = e s Delay u t = cos 2π f t Excitation frequency 0.1 Hz 1 Hz 10 Hz Response amplitude Response phase 8

9 Transfer functions with time delay R s Y s System output delay Ys R s Y s System input delay Ys 9

10 Time constant vs. time delay Pure time constant Combined time constant and time delay Pure time delay 10

11 Many plants encountered in practice are well approximated by the combination of: First order system with time constant τ And a pure delay element with delay T Plant with combined time constant and time delay P( s) K e 1 s T s In particular thermal systems, such as Heat exchangers Boilers Radiators can often be modeled with this approach. K=1, T=1s, τ=1s 11

12 Challenges of controlling time-delayed systems Plants with substantial delays are hard to control. T T + τ > 0.3 T: time delay (Totzeit), τ time constant (Zeitkonstante) 1. The controller is forced to use of old information (information about the output at some time in the past, rather than in the present), to compute the control signals. 2. Usually, if you are able design a controller for a delayed system it will likely result in a disappointing slow response. 3. Regular PID controllers are not well suited for that class of problems (in particular the D term is not very useful is such a case. The derivative signal does not bring any information of the future trajectory of a system, if that system contains pure delays. ). Question : What can be done to cope more efficiently with delays in the loop? Need for a predictive element in the controller 12

13 Predictive controllers: a recipe 1. Indication: when to use a predictive controller? T T + τ > 0.3 T: time delay (Totzeit) τ time constant (Zeitkonstante) 2. Limits: what can be achieved by the use of a predictive controller? The prediction allows to design the controller for the plant without delay, i.e., with much less phase drop. The delay cannot be removed by a predictor! Every transfer function (T, S) is affected by the same delay as the plant. 3. Prerequisites: when can a predictive controller be used? The plant must be asymptotically stable. A good model of the plant must be available. 13

14 Predictive controllers: one method One type of predictive controller for systems with pure time delay is the Smith predictor, invented by Otto J. M. Smith in 1957 Otto J. M. Smith : Born in 1917, in Urbana, Illinois, USA B.S. in Electrical Engineering, University of Oklahoma, Norman, 1938 Ph.D. in Power and High Voltage, Stanford University, 1941 Most of his career: Professor at University of California Berkeley Passed away in 2009 (91 yrs old) The Smith Predictor: A Process Engineer's Crystal Ball,' Control Engineering, May

15 Smith predictor Ps () ˆy The main idea of a predictive controller is: to use a linear model of the plant to compute an estimate of the non-delayed plant output y r (t). The feeback signal is the sum of: 1. the non-delayed plant output (estimate or prediction) y r t 2. and the correction signal ε, which represents the difference between y(t) and y(t). 15

16 Smith predictor Ps () In the ideal case of - no plant/model mismatch: - no external input disturbance: The output writes: ˆy Ys () In the non ideal case of - plant/model mismatch: - external input disturbance: Remark: in this case the result is nice because the time delay does appear in the feedback loop. The system behavior is rather complicated, see following examples. 16

17 Example: second order system + 1 s delay Goal: Design a controller for the plant described by P s = Static gain = Poles : Delay : Damping: 0.5 s 2 + 2s + 1 e s 17

18 Example: second order system + 1 s delay Controller design without prediction Control design constraints PM (phase margin) = Steady state error: Overshoot: Cross over frequency: Using Matlab sisotool, a loop-shaping approach will result in the following PI controller: Cs () s s Conclusion: Step response is slow. The fast drop of the phase due to the delay prevents a high bandwidth (see next slide when there is no delay). 18

19 Controller with no delay Hint: Compare the cross-over frequencies for the open-loop gain with/without delay, and the time response. Conclusion: without the delay in the system, the controller can be designed much faster. 19

20 Controller comparison Advantages of prediction Disadvantages of prediction 20

21 Smith predictor

22 Model accuracy: effects of delay uncertainties Conclusion If the time delay in the internal model is not accurate, the Smith predictor may not perform well, and thus the control performance deteriorates. 22

23 Lecture 4 Class Content 1. Control design for systems with time delays: the Smith Predictor 2. Robustness and performance criteria a) Robust Nyquist theorem b) Nominal performance c) Robust performance 23

24 Lecture 4 2. Robustness and performance Recall: uncertainty and the Nyquist stability theorem 1 + L jω > L jω W 2 jω ω 0, 24

25 Lecture 4 2. Robustness and performance A. Robust Nyquist theorem The robust Nyquist theorem represents can be interpreted as upper bound for the complementary sensitivity: or 25

26 Lecture 4 2. Robustness and performance Recap: sensitivity and performance The sensitivity S s is the transfer function from r e and from d y. Small sensitivity = good performance 26

27 Lecture 4 2. Robustness and performance B. Nominal performance A good performance condition would be an upper bound for the sensitivity: or 27

28 Lecture 4 2. Robustness and performance B. Nominal performance The nominal performance condition can be Investigated in the Nyquist diagram: 28

29 Lecture 4 2. Robustness and performance C. Robust performance Robust performance represents a combination of the : robust Nyquist theorem, and the nominal performance condition. 29