Design of controllers chapter 7, Exams of Aerospace Engineering

Download Design of controllers chapter 7 and more Exams Aerospace Engineering in PDF only on Docsity! Chapter 7 Design of Control System What we have learned… •How to build the mathematical model of a control system •Write differential equations based on physical laws •Transform the differential equations to the s- domain by Laplace transform •Obtain the transfer function transient response of the closed-loop system What we have learned… • Stability condition of a linear system • The roots of its characteristic equation (CE) must all be located in the left-half s- plane (LHP). • Methods of determining stability • Routh-Hurwitz criterion • Test whether any of the roots of CE lie in RHP • Indicate the number of roots that lie on the jw- axis and in RHP What we have learned… •Methods of determining stability •Nyquist criterion •Semi-graphical method -- Nyquist plot •Analyze closed-loop stability based on the loop transfer function G(s)H(s) •Closed-loop stability criterion: N=-P •For minimum-phase loop function: N=0 •Bode diagram •Plot the magnitue of the loop transfer function G(jw)H(jw) in dB and the phase in degrees versus frequency w •Closed-loop stability can be determined by observing the behavior mathematical model What we have learned… To change the performance of a control system Vary the gain K Add poles Add zeros What we have learnt • Introduction • Controllable form • Diagonal/canonical variable/normal forms • Controllability • Observability automatic controller actuating ! Controller ot error signal e(t) output u(t) detector Output c(t) reference input r(t) Automatic Controllers • The controller detects the actuating error signal, which is usually at a very low power level, and amplifies it to a sufficiently high level. • The output of an automatic controller is fed to an actuator, such as an electric motor, a hydraulic motor, or a pneumatic motor or valve. ✓The actuator is a power device that produces the input to the plant according to the control signal so that the output signal will approach the reference input signal. Classifications of Industrial Controllers Most industrial controllers may be classified according to their control actions as: a. Two-position or on–off controllers b. Proportional controllers c. Integral controllers d. Proportional-plus-integral controllers e. Proportional-plus-derivative controllers f. Proportional-plus-integral-plus-derivative controllers Two-Position or On–Off Control Action •The actuating element has only two fixed positions, which are, in many cases, simply on and off Two-position or on– off control •This form of control is relatively simple and inexpensive and, for this reason, is very widely used in both industrial and domestic control systems. •Let the output signal from the controller be u(t) and the actuating error signal be e(t). • In two-position control, the signal u(t) remains at either a maximum or minimum value, depending on whether the actuating error signal is positive or negative, • This is the simplest form of control, used by almost all domestic thermostats. Two-Position or On–Off Control Action • for for • where U1 and U2 are constants. The minimum value U2 is usually either zero or –U1 . • Two-position controllers are generally electrical devices, and an electric solenoid- operated valve is widely used in such controllers. • Pneumatic proportional controllers with very high gains act as two-position controllers and are sometimes called pneumatic two position controllers. Proportional Control Action •• For a controller with proportional control action, the relationship between the output of the controller u(t) and the actuating error signal e(t) is or, in Laplace-transformed quantities, = where Kp is termed the proportional gain. • The proportional controller is essentially an amplifier with an adjustable gain. • Depends on current/present errror actuating Controller error signal e(t) output u(t) reference input r(t) feed back signal b(t) Figure 3 Block diagram of a proportional controller Integral Control Action •• In a controller with integral control action, the value of the controller output u(t) is changed at a rate proportional to the actuating error signal e(t).That is, Or where is an adjustable constant. The transfer function of the integral controller is Proportional-plus-Integral Control Action ▪ The control action of a proportional-plus-integral controller is defined by • Where, Kp is the proportional gain, Ti is the integral time which are adjustable. actuating Controller error signal e(t) output u(t) reference input r(t) feed back signal b(t) Figure 5 Block diagram of a proportional-plus-integral control system  P-I control action u(t) K, (Koocecbe-------- proportional only  actuating Controller cimor error signal e(t) output u(t) detector reference input r(t) feed back signal b(t)  u(t) P-D control action . proportional only 0 t Figure 8 Response of PD controller to unit actuating error signal CONTROL ACTIONS ➢ The proportional term looks at where my value is currently. ➢ Integral looks at where I’ve been over time. ➢ Derivative tries to predict the future value. ➢ Derivative tries to work opposite of where proportional and integral are trying to drive the process. P and I are trying to drive one way, and D is trying to counteract that. ➢ Derivative has its largest effect when the process is changing rapidly in one direction KKP + I s Proportional-Derivative (PD) control: KP + KDs or the transfer function is ay" Ke tete]  PID control action PD control action ud) Pe proportional only 0 t Figure 10 Response of PID controller to unit actuating error signal D  Ziegler-Nichols Rules for Tuning PID Controllers G (s) = K + KI + K s = K  1 + 1 + T s  c P s D P   TI s  ▪ The process of selecting the controller parameters to meet given performance specifications is known as controller tuning. • Ziegler and Nichols proposed rules for determining values of the Th p e ro re po a r r t e io t n w a o lg m a e in th K o d , s in c t a e ll g r d al Z t i i e m g e le T r- i N a ic n h d ol d s e t r u iv n a in ti g ve ru t l i e m s e : t T h d e b fi a rst ed m o e n th t o h d e hibit s be ap First method --Obtain the response of the plant to a unit- step input --If the plant involves neither integrator nor dominant complex-conjugate poles, then it steunpitr- estseppoinnpsu et ex Plan t an SS-- sshhaappeerecsuprovnese, the first method can y(t) plied . The S-shape curve may be characterized by two constants: Delay First method. Table I Type of Control le r KP TI TD P T L  0 PI 0.9 T L L 0.3 0 PI D 1.2 T L 2L 0.5L ue pply . or whatev thod P Second method. -- SetK I = KD = 0 and use the proportKional control only Kcr --Increase from 0 to ∞ to a critical value at which the output first exhibits KP sustained oscillations --If the output does not exhibit sustained oRs(csi)llatioE(nss) f erUvc (sa)l mY (as)y take, then this me KP not a Plant Second method. y(t) Pcr 0 t R(s) E(s) U (s)c Y (s) PlantPID controller 1 s(s + 1)(s + 5) Gc (s) Exampl e Consider the following control system Design a PID controller to make the maximum overshoot of the system to Sboel utaiop n p W. reosxtaimrt adets eiglyn t2h5e %PIDocronletrsolsle.r by applying Ziegler-Nichols rules. Here the transfer function of the plant is known, we can use analytical method instead of experimental method. D  R(s) E(s) U (s)c Y (s) PlantPID controller 1 s(s + 1)(s + 5) Gc (s) The PID controller has the transfer function G (s) = K + KI + K s =  + 1 + T s  c P s D KP 1  TI s  Since the plant has a integrator, we use the second method. By s etting The Routh’s array s3 1 5 s2 6 K s1 30  KP 6 s0 K P p R(s) E(s) U (s)c Y (s) PlantPID controller 1 s(s + 1)(s + 5) Gc (s) 5 With Kp set to Kcr(=30), the CE becomes s3 + 6s2 + 5s + 30 = 0 To find the frequency of the sustained oscillation, we substitute s=jw into the CE as follows: ( j)3 + 6( j)2 + 5( j) + 30 = 0 or 6(5  2 ) + j(5  2 ) = 0  = = 5 Hence the period of the sustained oscillation is 2  Pcr  = 2  = 2.8099 vershoo t. >>num = [6.3223 18 12.811]; >>den = [1 6 11.3223 18 12.811]; >>step(num,den) R(s) E(s) U (s)c Y (s) PlantPID controller 1 s(s + 1)(s + 5) Gc (s) The closed-loop transfer function with the PID controller is Y (s) = 6.3223s2 + 18s + 12.811 R(s) s4 + 6s3 + 11.3223s2 + 18s + 12.811 Now let us examine the unit-step response of the closed-loop system to see if it exhibits approximately 25% maximum o 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 UnitStep Response 0.2 0 0 5 10 15 Time (sec) The maximum overshoot is about 62% and is excessive with respect to the requirement of 25%. A m p lit u d e   The amount of maximum overshoot can be reduced by fine tuning the parameters of the PID controller. Such fine tuning can be made on the computer.  1  6.3223 ( s + 1.4235) 2 Gc (s) = 18   1 + 1.405s + 0.35124s   = s Move the pole locations Gc (s) = 18  1 +  1 0.3077s + 0.7692s   = 13.836 ( s + 0.65) 2 s The maximum overshoot is reduced to 18%. Increase the gain 2 Note that the Ziegler-Nichols’ tuning rule has provided a starting point for fine tuning. The Characteristics of P, I, and D Controllers ✓• A proportional controller ( ) will have the effect of reducing the rise time and will reduce but never eliminate the steady-state error. ✓An integral control () will have the effect of eliminating the steady-state error for a constant or step input, but it may make the transient response slower. ✓ A derivative control () will have the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response. Used to obtain a controller with high sensitivity. -this form of control anticipates the increasing error, initiates an early correction action and tends to increase the stability of a system The Characteristics of P, I, and D Controllers CLOSED LOOP RESPONSE Rise Time Overshoot Settling time Steady-state errors Kp Decrease Increase Small change Decrease Ki Decrease Increase Increase Eliminate Kd Small change Decrease Decrease No change •Plug these values into the above transfer function The goal of this problem is to show you how each of , and contributes to obtain • Fast rise time • Minimum overshoot • No steady-state error •Open-Loop Step Response •Let's first view the open-loop step response. Create a new m-file and run the following code: s = tf('s'); P = 1/(s^2 + 10*s + 20); step(P) ▪ The DC gain of the plant transfer function is 1/20, so 0.05 is the final value of the output to an unit step input. ▪ This corresponds to the steady-state error of 0.95, quite large indeed. Furthermore, the rise time is about one second, and the settling time is about 1.5 seconds. ▪ Let's design a controller that will reduce the rise time, reduce the settling time, and eliminate the steady-state error. Proportional Control • proportional controller (Kp) reduces the rise time, increases the overshoot, and reduces the steady-state error. • The closed-loop transfer function of the above system with a proportional controller is: Let the proportional gain (Kp) equal 300 and change the m-file to the following: •The above plot shows that the proportional controller reduced both the rise time and the steady-state error, increased the overshoot, and decreased the settling time by small amount. Proportional-Derivative Control •From the table shown , we see that the derivative controller (Kd) reduces both the overshoot and the settling time. The closed- loop transfer function of the given system with a PD controller is: Let Kp equal 300 as before and let Kd equal 10. Enter the following commands into an m-file and run it in the MATLAB command window. Kp = 300; Kd = 10; C = pid(Kp,0,Kd) T = feedback(C*P,1) t = 0:0.01:2; step(T,t) C = Kp + Kd * s with Kp = 300, Kd = 10 Continuous-time PD controller in parallel form. . T = 10 s + 300 s^2 + 20 s + 320 •Continuous-time transfer function •This plot shows that the derivative controller reduced both the overshoot and the settling time, and had a small effect on the rise time and the steady-state error. Automatic PID Tuning •MATLAB provides tools for automatically choosing optimal PID gains which makes the trial and error process described above unnecessary. • You can access the tuning algorithm directly using pidtune or through a nice graphical user interface (GUI) using pidtool. •The MATLAB automated tuning algorithm chooses PID gains to balance performance (response time, bandwidth) and robustness (stability margins). •By default the algorthm designs for a 60 degree phase margin. •The pidtool GUI window, like that shown below, should appear. END OF THE COURSE