Control Systems Lecture 4 Constantinos Siettos Feedback Control
Control Systems Lecture (4) Constantinos Siettos
Feedback Control Systems Control of a System’s Response Transient Steady State
Why Feedback? Steady-State Error Y(s) For R(s)=1/s The difference between input and output for a predetermined input for
Steady-State Error- Proportional Control error, E(s) reference input, R(s) + - error, u(s) Control Gc(s) feedback H(s)=1 system Gp(s) output, Y(s)
Steady-State Error with Proportional Controller - Step Input Α Constant of steady state error
Steady-State Error with Proportional Controller-Ramp Input Α Constant of velocity error
Steady-State Error with Proportional Controller error, E(s) reference input, R(s) + - error, u(s) Control Gc(s)=kc feedback H(s)=1 E. g. for a 1 st order process system Gp(s) output, Y(s)
Steady-State Error, Use of Matlab clear all Kc=1; K=1; a=0. 5; sys 1=tf(1, 1); sys 2=tf(K, [1 a]); sysh=tf(1, 1); sys 3=series(sys 1, sys 2); sys 4=feedback(sys 3, sysh) step(sys 4) T=0: 0. 01: 4 U=ones(length(T), 1); LSIM(sys 4, U, T) U=0. 35*T; LSIM(sys 4, U, T)
Steady-State Error. Simple Design Example Find Κ so that the steady-state error is 10% 1. For R(s) step ?
Steady-State Error. Simple Design Example Find Κ so that the steady-state error is 10% 1. For R(s) ramp ? So Κ=672
Control System of the Direction of Automotive Robot y 1. For R(s) step and Κ 2=0: 2. For R(s) step and Κ 2>0:
Control System of the Direction of Automotive Robot y 3. For R(s) ramp and Κ 2>0:
Control System of the Direction of Automotive Robot y 3. For R(s) ramp and Κ 2>0:
Control System of the Direction of Automotive Robot-Use of Matlab clear all K 1=1; K 2=0. 5; Tau=0. 1 sys 1=tf([Κ 1 Κ 2], [1 0]); sys 2=tf(K, [tau 1]); sysh=tf(1, 1); sys 3=series(sys 1, sys 2); sys 4=feedback(sys 3, sysh) T=0: 0. 01: 10 U=sin(T); LSIM(sys 4, U, T)
Performance of Feedback System in Time Field Percent Overshoot is defined as: P. O. = [(Mpt – fv) / fv] * 100% Mpt = The peak value of the time response fv = Final value of the response
Performance of Feedback System in Time Field Oscillation Period: Time that the output takes the maximum value : The first time that the edge appears for n=1:
Performance of Feedback System in Time Field-MATLAB clear all t=[0: 0. 1: 12]; Num=1; For zeta=[0. 1 0. 2 0. 5 0. 9 1 2]; Den 1=[1 2*zeta 1]; Sys=tf(num, den); [y, tout]=step(sys, t); plot(tout, y); Pause(1) hold on; end ζ
An example: Reading the Hard-Disc - R(s) + Κ 5 - + D(s) Y(s), Rotation Velocity H(s)=1 Design: the maximum overshoot <5%, Τs<250, Overshoot in step disturbance <5 x 10 -3
An example: Reading the Hard-Disc R(s) + - Κ 5 - D(s) + H(s)=1 Y(s), Rotation Velocity Design: the maximum overshoot <5%, Τs<250, Overshoot in step disturbance <5 x 10 -3 maple( 'solve(100*exp(10*pi/(sqrt(5*x))/sqrt(1 -100/(5*x)))<5), x' )
An example: Reading the Hard-Disc R(s) + - Κ 5 H(s)=1 - D(s) + Y(s), Rotation Velocity Design: the maximum overshoot <5%, Τs<250, Overshoot in step disturbance <5 x 10 -3 clear all t=[0: 0. 01: 1]; K=25; Num=[5*K]; Den 1=[1 20 5*K]; Sys=tf(num, den); [y, tout]=step(sys, t); plot(tout, y), grid; Pause(1) hold on; end
Control of the Insouline injection R(s) + E(s) - insouline Pump Κ Y(s) Sugar concentration Body H(s)=1 What is the value of Κ so that the maximum value of response in a unit step disturbance do not exceed 5 x 10 -3, Time period of settling 250 ms
Control of the Insouline injection R(s) + E(s) Pump Κ insouline Body Y(s) Sugar concentration - H(s)=1 What is the value of Κ so that the maximum value of response in unit step disturbance does not exceed 2%, settling time 4 ms clear all t=[0: 0. 1: 12]; Num=[K 2 K]; Den 1=[1 (1+K) 2*K]; Sys=tf(num, den); [y, tout]=step(sys, t); plot(tout, y);
Integral Indices of Performance – Optimal Design • Criteria that optimize the closed loop response of set-point changes or disturbances. • The optimal values of the control parameters minimize the integral criteria • The 3 most used criteria are the following: 1. Integral of the absolute value of the error (IAE) e(t) denotes the difference between set point and output
Integral Indices of Performance – Optimal Design • Criteria that optimize the closed loop response of set-point changes or disturbances. • The optimal values of the control parameters minimize the integral criteria • The 3 most used criteria are the following: 2. Integral of the squared error (ISE) 3. Integral of the time-weighted absolute error (ITAE)
Integral Indices of Performance – Optimal Design-Matlab Gain R(s) + Activation of the Wind Slope (BANK) Plane Dynamics Y(s), bank angle Κ H(s)=1 Gyroscope
Integral Indices of Performance – Optimal Design-Matlab Gain R(s) + - Κ Activation of the Plane Dynamics Wind Slope Y(s), bank angl H(s)=1 Gyroscope Aim: Optimization of the closed loop response in a step change with IAE criterion clear all kiter=0; for K=0. 1: 0. 001: 2; sys 1=tf(K, 1); sys 2=tf(10, [1 10]); sys 3=tf(12, [1 2 0]); sys 12=series(sys 1, sys 2); sysopen=series(sys 12, sys 3); sysclosed=feedback(sysopen, 1); %H=get(sysclosed); y=step(sysclosed); kiter=kiter+1; Ksave(kiter)=K; e(kiter)=sum(abs(y-1)); [K e(kiter)] end plot(Ksave, e)
Integral Indices of Performance – Optimal Design-Matlab Aim: Optimization of the closed loop response in a step change with IAE criterion
Feedback Control loop
Characteristic Equation The transient response of the system is determined from the place of poles which are the roots of the characteristic equation A state of a system is stable if bounded inputs result to bounded outputs The complex s-plane seperates in two regions: one stable whchcorresponds to the left semi-plane and the unstable region which corresponds to the right semi-plane j s-plane x x x x x stable unstable
Poles and Stability
Linear Control System– PID Controllers
Linear Control Systems– PID Controllers e(t)- the error from setpoint [e(t)=Rs-ys] Kc- the controller gain is a tuning parameter t. I- the reset time is a tuning parameter and determines the amount of integral action. t. D- the derivative time is a tuning parameter and determines the amount of derivative action.
Steady-State Error
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