ECSE 4440 Control System Engineering Fall 2001 Project 3 Controller Design in Frequency Domain TA 1. Abstract 2. Introduction 3. Controller design in Frequency domain 4. Experiment 5. Colclusion
1. Abstract In this project, controller design in frequency domain will be studied. Not only the performance of a controller but the stability and robustness will be considered in frequency domain. There are many useful graphical tools available in frequency domain such that bode plot and Nyquist plot. Those tools make it possible to design a lead-lag controller in frequency domain. 2. Introduction So far, the performance of a control system has been measured and analyzed by its timedomain response characteristics. In practice, however, many signals to be processed are sinusoidal or can be represented by sinusoidal components. And the difficulties of the time domain design lie in the fact that there are no unified methods of arriving at a designed system given the time-domain specifications, such as peak overshoot, rise time, delay time, settling time, and so on. And for system with poorly known or changing high-frequency resonance, time domain design is not efficient. Therefore the design of feedback control systems is probably carried out using frequencyresponse methods more often than any other. The frequency domain design method gives designer various advantages over the time domain method. The first advantage is the stability margins and robustness of a controller for noise or measurement error. And there are several handy graphical methods available in the frequency domain, suitable for the analysis and design of control systems. And it is important to realize that once the analysis and design are carried out in the frequency domain, the time-domain properties of the system can be interpreted based on the relationships that exist between the timedomain and the frequency-domain characteristics. Therefore, we may consider that the primary motivation of conducting control systems analysis and design in the frequency domain is because of convenience and the availability of the existing analytical tools. 2.1 Stability Margin In the previous project, a system becomes unstable if the gain or time delay in input and feedback loop increases past a certain critical point. In other words, there are some margins in gain and phase from stable state to unstable state.
The stability margins can be described by phase margin and gain margin. The gain margin(gm) is the factor by which the gain is less than the neutral stability value and the phase margin(pm) is the amount by which the phase of j ) KG ( jω) = 1 (figure1). G ( ω exceeds 180 when Figure 1 And the PM determines the time specifications of the controlled system such as damping coefficient ς and the overshoot M p. So, PM is sometimes used directly to specify control system performance. Based on the standard second order systems (with no zero), there are rules of thumb as follows; PM 100 ς and n ω cg where ω (1) ω cg is gain cross over freq and ω n is natural freq.. If we recall the rules of the relationship between specifications can be achieved by using (1). ω n, ς and the specifications, the time
2.2 Sensitivity Frequency domain methods enable a designer to make a control system as insensitive in operating frequency range. One of the main objectives of the control system is to keep the tracking error(e=r-y) small for a reference input excitation and to keep the output y small for a disturbance input d in figure 2. d r - e K( u G( y n Figure 2 Special functions have been defined to show the sensitivity of a closed-loop control system. The sensitivity function is defined as 1 S ( = (1 + G( K( ) (2) The sensitivity function is the transfer function of n to y and d to y (in lecture note). And the complementary sensitivity function is defined as T ( G( K( 1+ G( K( = (3) The complementary sensitivity function is the transfer function of r to e and d to u. Therefore the control objective is to keep S ( small over bandwidth of r and d and to keep T ( small over bandwidth of n while maintaining closed loop stability. The loop gain( G ( K( ) shaping can achieve the control objective if we note that GK GK >> 1 << 1 T S 1, S 1, T 0 1 (4) 2.3 Steady State Error 1 1+ G(0) K(0) If input is a step function, the steady state error will be d(0) from figure 2. Therefore, large DC gain leads the small steady state error. Similarly, if the input is a sinusoidal function having a certain frequency, the steady state error will be ω o
s( jw o 1 ) = 1+ G( jw ) K( jw ). Again large ( jw ) o o o K leads the steady state error at ω o small. The large K jw ), however, make the system less robust. So, proper gain ( o should be chosen for the given specifications. 2.4 Time Delay The pure time delay adds a phase shift of ωt because the frequency response of the delay is given by the magnitude and phase of e st s= jω. Therefore for a Given PM, maximum time delay can be expected by T PM = ω max. cg 3. Controller Design in Frequency Domain 3.1 Control Objective In the previous projects, there are useful rules between ς follows in addition to (1). n 1.8 t ω (5) s 2 b ς = where = ln M ( π ) 2 p / 1+ b 4.6 = ςω n t s b (6) σ (7) ω n and the time specification as To satisfy the specification with the above rules, the below conditions should be satisfied. PM > 70 (8) 5 < ω cg < 20 (9) Once the condition (9) becomes satisfied, the spec for S ( and ( T will be achieved. So, target ω is chosen in the range of (9). = 7rad cg / sec For the given spec of the steady state error, the open loop gain should satisfy
G(0) d <.01 1+ G(0) K(0) F d = I < F c c sgnθ (10) c = 1.57 G(0) <.0064 1+ G(0) K(0) Therefore the objective open loop gain G ( K( should be shaped like I c K(0)G(0)>44dB K(jw)G(jw) 0dB 5<w cg <20 W cg Arg(K(jw)G(jw)) PM>70 o -180 o Figure 3 3.2 Lead Controller A lead filter can be used to add phase lead and increase bandwidth with the form of Ts + 1 K lead = K, α < 1. There are 3 design parameters K, T αts + 1 parameters, a systematic method has been studied in class.,α. To choose the
For the given G( 1 F s( s + I =, the target gain crossover frequency ω cg has chosen to v c ) be 7 rad/sec. From bode plot of G (, K = 14.791(23.4dB) is chosen to make cg KG ( 3.5 rad/sec (7/2) in figure 4. ω of Figure 4 And the PM of KG ( is 34.87. So, 70 + 5 34.87 43 ( 5 is extra pad.) phase lead 1 sin 43 is needed to achieve PM > 70. α = =. 189 (-7.235dB). the 1+ sin 43 ω = 5.57rad / sec should be chosen at the the α (db) in figure 5. Finally, max T = 1 /(5.51*.189) =.413
Figure 5 With the decided parameters, the lead filter is.413s + 1 = 14.791.08177s + 1 ( K ( s K lead (11) The PM and BW of G lead ) should be checked. In figure 6, the PM and BW don t meet the spec.
Figure 6 The procedure should be iterated from adding extra margin. After several tries, the extra pad= 30 is found to be enough. + 30 34.87 65 1 sin 65 70 = =. 0486 1+ sin 65 α (- 13.13dB), ω = 7.99, T = 1/(7.99*.0486). 5677 So, the overall filter is max =.57s + 1 = 14.791.0277s + 1 K lead. The lead filter achieves the spec in figure 7.
Figure 7 3.3 Lag Controller A lag filter can be used to boost DC gain to reduce steady state error but can reduce phase Ts + 1 α Ts + 1 margin. Its transfer function is ( = α, α > 1. There are two parameters for the filter α, T. From (10), K lag G( d 1+ K( G( = 1+ K lead α lag 1 s( s + β ) d T s T s lead + 1 lag + 1 1 α T S + 1α T s + 1 s( s + β ) lead lead lag lag where β F F v c =, d = sgnθ I c Ic reduced to. Set s = 0 for the steady state error, the above equation is K lead d α lag Fc. 01, d < = 1.57 K leadα > 157 I < lag c (12) If the chosen K in 3.2 is used, > 10. 56 lead α will satisfy the steady state error spec. lag
The lag compensation can ruin the system performance satisfying the time response spec. So, the zero of the lag filter ( 1 ) T lag should be chosen one decade below than zero of lead 1 filter ( ). T lead So, the lag filter is 67.44s + 11.83 ( = 67.44s + 1 K (13) 3.4 Lead-Lag Controller The lead controller and lag controller defined in previous section control the system in the range of the given specification. The step function of the lead-lag controlled system is in figure 8. The time response specs have been satisfied. Figure 8 The bode plot of the system and Nyquist plot are in figure 9. The phase margin, gain
margin and the BW spec have been satisfied again. Figure 9
3.5 Sensitivity For the sensitivity spec, the sensitivity function and the complementary sensitivity function is given in figure 10. As mentioned in previous section, S(, T ( functions falls down around the gain crossover frequency ω cg. Therefore the properly chosen ω cg should achieve the sensitivity spec and bandwidth spec. Figure 10 3.6 Time Delay The time delay in feedback loop is allowed up to PM since the time delay adds a phase shift of ω cg T. For the PM= 1.409(rad) of the compensated system, T PM 1.409 8.0645 max = = = ω cg.1747 sec In the linear simulation, it can be verified. % Time delay pm_rd=pm*pi/180;.
Tmax=pm_rd/wcp [n d]=pade(tmax,3);gd=tf(n,d); max(real(pole(feedback(gd*k*g,1)))). real(max_pole) = 2.3280e-006 for Tmax = 0.1747 real(max_pole) =.0017 for Tmax=0.1748 3.6 Nonlinear Simulation The nonlinear simulation controlled by same lead-lag compensator has the step response in figure 11. Figure 11 Most of the spec has been achieved. But the settling time is almost twice of the given spec. It might be caused the disturbance term (Coulomb Friction). Simulation diagram is in figure 12.
Figure 12 For the sinusoidal inputs with freq=6 rad/sec and 10 rad/sec, the responses are in figure 13 and figure 14. Figure 13
Figure 14 As we can see in figure 13 and figure 14, the magnitude of output is decreased as input frequency is increased. In figure 14, the magnitude is reduced substantially. So, the overall system is a lowpass filter with the cutoff frequency= ω. 3.7 Discrete Simulation Using Euler s method(forward difference), descretization of the continuous transfer function can be done with the relationship 15. s z 1 T s cg =. The discretized system is in figure
Figure 15 The quantizer in feedback loop should be pointed out here. Usually, a encoder is used for the measurement of the output for feedback. A encoder transforms the continuous output to discrete value. Better encoder gives better measurement values. measurement error, however, can not be avoidable in figure 16. Figure 16 4. Experiment
5. Conclusion In this project, the frequency domain design method has been studied. It is more general and comprehensive than the time domain design. There are graphical tools for frequency domain such as bode plot and nyquist plot. For the 2 nd order system, lead-lag compensator has been designed with the help of the graphical tools. The performance of the controlled system has met with the given specification in time domain and frequency domain.