# Module 08 Controller Designs: Compensators and PIDs

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1 Module 08 Controller Designs: Compensators and PIDs Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Webpage: taha March 31, 2016 Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 1 / 34

2 Introduction Readings: , Ogata; 7.6,10.3,10.5 Dorf & Bishop In Module 7, we learned to sketch the RL for any TF We saw how poles change as a function of the gain K K was a controller a constant controller Many times, K as a controller is not enough Example: system cannot be stabilized with a choice of K-gain Or, settling time is still high, overshoot still bad Today, we ll learn how to design more complicated controllers Objective: find G c (s) such that CLTF has desired properties such as settling time, maximum overshoot,... Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 2 / 34

3 Typical RL Plots (a) Root-locus plot of a single-pole system; (b) root-locus plot of a two-pole system; (c) root-locus plot of a three-pole system (a) Root-locus plot of a three-pole system; (b), (c), and (d) root-locus plots showing effects of addition of a zero to the three-pole system. Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 3 / 34

4 Effects of Adding Poles and Zeros on RL Adding poles pulls the RL to the right Systems become less stable, settling is slower Adding zeros pulls the RL to the left Systems become more stable (this is tricky), settling is faster Question: Can we conclude that a compensator (controller) G c (s) should always be a combination of zeros? Since, you know, it makes system more stable and settling is faster? Not really. Why?? Because adding a zero amplifies the high frequency noise So, we can t add a zero alone (i.e., G c (s) = s + z), and we can t add a pole alone either (G c (s) = 1 s+p ). Solution? Solution Add a compensator of this form: G c (s) = K s + z Objective: find K, z, p given certain desired properties s + p Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 4 / 34

5 Two Controller Choices: Lead and Lag Compensators G c (s) = K s + z s + p Objective: find K, z, p given certain desired properties For G c (s) above, K, z, p are all real +ve values to be found 2 combinations: (a) lead controller; (b) lag controller: Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 5 / 34

6 Lead and Lag Compensators Lead compensator provides a +ve angle contribution: Gc ld (s) = K s + z ld Gc (s) = (s+z) (s+p) = θ z θ p = θ lead > 0 s + p Speeds up transients by lowering rise time & decreasing overshoots Lag compensator provides a -ve angle contribution: G ld c (s) = K s + z s + p G ld c (s) = (s+z) (s+p) = θ z θ p = θ lag < 0 Improves the steady-state accuracy of the system for tracking inputs What if p = z? That s a constant gain (pole & zero cancel out) Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 6 / 34

7 Lead Compensator Example Initially, the above system has ζ = 0.5 and ω n = 2 Obj: design G ld c (s) = C(s) = K s+z s+p, such that ζ d = 0.5, ω nd = 4 Can we do that via gain K? No, see the RL below for C(s) = K Hence, we can never reach s d via a constant gain, need compensator s d = ζ d ω nd ± 1 ζd 2ω nd = 2 ± j2 3 Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 7 / 34

8 Lead Compensator Example (Cont d) Objective: design G ld c (s) = K s+z s+p, such that ζ d = 0.5, ω nd = 4 To find K, z, p, follow this algorithm: 0. Find s d for s 2 d + 2ζ dω nd s d + ω 2 nd = 0, s d = 2 + j Find angle of deficiency φ, as follows: θ = G(s d ) = G( 2 + j2 3) = 210 deg φ = 180 (θ) = 30 deg 2. Connect s d to the origin OK 3. Draw a horizontal line to the left from s d OK 4. Find the bisector of the above two lines OK 5. Draw 2 lines that make angles φ/2 & φ/2 with the bisector OK 6. Their intersections with the real lines are p and z OK Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 8 / 34

9 Lead Compensator Example Finding K We now know that z = 2.9, p = 5.4 G ld c (s) = C(s) = K s+2.9 s+5.4 We know that all points on the RL satisfy 1 + KG(s)G ld c (s) = 0 1 = KG(s)G ld c (s) We know for sure that s d belongs to the RL, so solve for K: s d ld 4K = 1 K = 4.68 Gc (s) = C(s) = 4.68 s s d (s d + 2)(s d + 5.4) s Compensated RL plot: Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 9 / 34

10 Step Response: Old vs. New Response Compensated system reaches SS faster (shorter rise, settling times), although it has a higher M p. That said, we designed the compensator according to the design specs. Design specs weren t so smart, perhaps. Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 10 / 34

11 Lead Compensator Example 2 10 ld Given G(s) =, find Gc (s) such that the CLTF has s(s + 1) ζ d = 0.5 and ω nd = 3 Figures: (a) uncompensated control system (b) uncompensated root-locus plot Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 11 / 34

12 Lead Compensator Example 2 (Cont d) 10 ld Given G(s) =, find Gc (s) such that the CLTF has s(s + 1) ζ d = 0.5 and ω nd = 3 Figures: (a) compensated system, (b) desired closed-loop pole location Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 12 / 34

13 Lead Compensator Example 2 (Cont d) Objective: design G ld c (s) = K s+z s+p, such that ζ d = 0.5, ω nd = 3 0. Find s d for s 2 d + 2ζ dω nd s d + ω 2 nd = 0, s d = 1.5 ± j Find angle of deficiency φ, as follows: θ = G(s d ) = G( 1.5+j2.58) = 138 deg φ = 180 (138) = deg 2. Connect s d to the origin OK 3. Draw a horizontal line to the left from s d OK 4. Find the bisector of the above two lines OK 5. Draw 2 lines that make angles φ/2 & φ/2 with the bisector OK 6. Their intersections with the real lines are p and z OK Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 13 / 34

14 Compensated System, Example 2 Unit-step response and RL plot for the compensated system: Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 14 / 34

15 Lag, Lead-Lag Compensators Now that we understand lead compensators, we can discuss lag and lead-lag compensators Recall that lead compensators: improve transient response and stability But they do not typically reduce SSE Lag compensators G lg c (s): reduce SSE, so sometimes we want smaller SSE rather than shorter rise and settling time as in a lead compensator Optimal solution: lead-lag (LL) compensator Gc ll (s) = Gc ld (s)gc lg (s) LL compensators provides the benefits of both lead and lag compensators Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 15 / 34

16 Lead-Lag Compensators Unfortunately, we don t have time to cover LL compensator design Design procedure is simple, please read more about it from your textbooks But we ll show a figure that illustrates the difference in performance: Left figure (step response), right figure (ramp response) Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 16 / 34

17 PID Control Definitions and Basics Proportional Integral Derivative controller PID control Without a doubt the most widely used controllers in industry today Bread and butter of control, 90% of control loops use PID control Proportional: G c (s) = K, Integration: G c (s) = 1 Ts, Derivative: G c (s) = Ks Can have combinations of the above controllers: P,I,D,PI,PD,ID,PID Major objectives for designing G c (s): 1 Stability the most important objective: CLTF is stable 2 Steady-state error (SSE) minimize this as much as we could 3 Time-specs M p, t r, t s,... Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 17 / 34

18 PID Controller Device You can either use some tuning rules (which we will learn about during this module), or use an auto-tune function that figures out the parameters to a PID controller. Check for examples. Prices for common PID controllers range from \$20 to \$200, depending on size, quality, and performance. Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 18 / 34

19 Example 1: P controller for FOS Assume G(s) = 1 Ts+1 first order system (FOS) We can design a P controller (i.e., G c (s) = K) Result: Larger K will increase the response speed SSE is present no matter how large K is recall the SSE Table ;) Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 19 / 34

20 Example 2: Integral (I) controller for FOS Assume G(s) = 1 Ts+1 first order system (FOS) We can design an I controller (i.e., G c (s) = K/s) Result: SSE for step input is completely eliminated But transients are bad can cause instability for some K Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 20 / 34

21 PID Controller PID (Proportional-Integral-Derivative) controller takes this form: ( G c (s) = K p ) T i s + T ds Design objective: find parameters K p, T i, T d given required specs This process is called PID tuning process of adjusting K p, T i, T d Many different tuning rules exist Ziegler-Nichols Rule: first PID tuning rules (first and second method) After finding these parameters, input them on the PID device Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 21 / 34

22 More on PID Controllers Proportional term, i.e., G P c (s) = K p : Proportional term responds immediately to the current tracking error. Typically, however, it cannot achieve the desired tracking accuracy without excessively large gain. Integral term, i.e., G I c(s) = K p T i s : Integral term yields a zero steady-state error in tracking a step function (a constant set-point). This term is slow in response to the current tracking error. Derivative term, i.e., G D c (s) = K p T d s: Derivative term is effective for plants with large dead-time Reduces transient overshoots, but amplifies higher frequencies sensor noise Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 22 / 34

23 Ziegler-Nichols Rule: First Method Step 1 Obtain plant s unit step response experimentally 1 Unit step response is S-shaped for many plants Only valid if the step-response is S-shaped Step 2 Obtain delay time L from the experimental plot Step 3 Obtain time constant T from the experimental plot Step 4 Use tuning rule table to determine K p, T i, T d given L, T (next slide) 1 In industrial applications, control engineers usually specify the performance of the controlled system based on the system step response. Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 23 / 34

24 Obtaining L, T from Experimental Plot Of course, this is an approximation, but you have to be accurate with your computation of L and T Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 24 / 34

25 Obtaining K p, T i, T d via Tuning Method 1 ( G c (s) = K p ) T i s + T ds If you want a PID controller, choose the third row and compute the parameters: ( ( G PID (s) = G c (s) = K p ) s + 1 ) 2 T i s + T L ds = 0.6T s Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 25 / 34

26 PID Tuning, Method 1 Example Given a plant G(s) = 10 s 2, find K, L, T first + 6s + 5 Given the procedure, we find that G(s) 2e 0.05s 0.8s + 1 Plug these values in the table to obtain G c (s) = G PID (s) Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 26 / 34

27 Ziegler-Nichols Rule: Second Method Step 1 Set T i =, T d = 0 (above left figure) and increase K p until step response of the closed-loop system has sustained oscillations If no oscillation occurs for all values of K p, this method is not applicable Step 2 Record K cr (critical value of gain K p ) and P cr (period of the oscillation); see above right figure Step 3 Use tuning rule table to determine K p, T i, T d given K cr, P cr (next slide) Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 27 / 34

28 Obtaining K p, T i, T d via Tuning Method 2 ( G c (s) = K p ) T i s + T ds ( ( G PID (s) = G c (s) = K p ) s + 4 T i s + T ds = 0.075K cr P cr s P cr ) 2 Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 28 / 34

29 Method 2 Example 1 Given a plant G(s) = s(s+1)(s+5), find the PID parameters using the second PID design method Solution: Experimentally, we plot the step response till we have sustained oscillations (Step 1) We can record K cr = 30, P cr = Step Response Amplitude Time (seconds) Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 29 / 34

30 Method 2 Example Looking at the table, we can find K p, T i, T d : ( G c (s) = ) 1.4s s Note: we can find K cr by applying the RH table for the CP (s 3 + 6s 2 + 5s + K p ) Then, you can find K p that would make the CP unstable K cr = K max p = 30 Then find the frequency ω cr that solves this equation (jω cr ) 3 +6(jω cr ) 2 +5jω cr +30 = 0 ω cr ω cr = 5 P cr = 2π ω cr = 2.8 Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 30 / 34

31 Step Response after PID Design Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 31 / 34

32 PID Control Summary We only covered one type of PID control, called Type A PID control: ( G PID A (s) = G c (s) = K p ) T i s + T (s + β)2 ds = α s where α and β are the PID constants that depend on the plant s performance So, when do we use P,PI,PD, or PID control? Well, it depends on what you want Parameter SSE Response Speed Stability Oscillations Overshoot K p K i = 1 T i K d = T d Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 32 / 34

33 Course Progress Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 33 / 34

34 Questions And Suggestions? Thank You! Please visit engineering.utsa.edu/ taha IFF you want to know more Ahmad F. Taha Module 08 Controller Designs: Compensators and PIDs 34 / 34

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