Outline
Outline Outline 1 Introduction 2
Prerequisites Block diagram for system modeling Modeling Mechanical Electrical
Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction Background Basic Systems Models/Transfers functions 2
Goal of Control Engineering Background Basic Systems Models/Transfers functions Graham C. Goodwin: The fundamental goal of CE is to find technically, environmentally, and economically feasible ways of acting on systems to control their outputs to a desired level of performance in the face of uncertainty of the process and in the presence of uncontrollable external disturbances acting on the process.
The Fly-Ball Governor Background Basic Systems Models/Transfers functions Historical Periods The Industrial Revolution (186s) World War II (194-1945) The Space Race (196s,197s) Economic Globalization (198s)
Evolution of Control Introduction Background Basic Systems Models/Transfers functions 194 Frequency Domain Mainly useful for SISO systems A fundamental tool for many practicing engineers 196 Modern Control State-space approach to linear control theory Suitable for both SISO and MIMO systems No explicit definition of performance and robustness 197 Optimal Control Minimize a given objective function (fuel, time) Suitable for both open-loop and closed-loop control 198 Robust Control Generalization of classical control ideas into MIMO context Based on operator theory which can be interpreted into frequency domain Nonlinear control, Adaptive Control, Hybrid Control
Basic Continuous Control System Background Basic Systems Models/Transfers functions Disturbances Ref. + Feedforward Actuator Plant Output Feedback + + Sensor Measurement noise
Basic Digital Control System Background Basic Systems Models/Transfers functions Ref. + Disturbances Feedforward D/A Act. Plant Clock Output Feedback A/D Sensor
Continuous System Models Background Basic Systems Models/Transfers functions Numerator and denominator General formula G(s) = Y (s) U(s) = b s m + b 1 s m 1 + + b m a s n + a 1 s n 1 + + a n Second order system (special case) G(s) = K s 2 + 2ζω n s + ω 2 n Matlab sys=tf(num,den)
Continuous System Models Background Basic Systems Models/Transfers functions Zeros and poles General formula G(s) = Y (s) U(s) = K Second order system (special case) m k=1 (s z k) n i=1 (s p i) G(s) = K (s p 1 )(s p 2 ) p 1,2 = ζω n ± jω n 1 ζ 2 Matlab sys=zpk(z,p,k)
Outline Introduction 1 Introduction Background Basic Systems Models/Transfers functions 2
Bounded Input Bounded Output Impulse response Roots of characteristic equation Gain and phase margins Nyquist stability criterion
Stability - Impulse Response Continuous systems The system is BIBO stable if and only if the impulse response h(t) is absolutely integrable Discrete systems The system is BIBO stable if and only if the impulse response h[n] is absolutely summable
Stability - Characteristic Roots Asymptotic internal stability Continuous systems All poles of the system are strictly in the LHP of the s-plane Discrete systems All poles of the system are strictly inside the unit circle of the z-plane
Analyzing System types Continuous system Discrete system Analysis domain Time-domain specifications Frequency-domain specifications Different periods Dynamic (transient) responses Steady-state responses
Continuous Time - Dynamic Response Impulse Response Step Response 1.2 1.9 1.8.8.7.6 Amplitude.6 Amplitude.5.4.4.2.3.2.1.2 2 4 6 8 1 12 Time (sec) 1 2 3 4 5 6 7 8 9 1 Time (sec) num1=[1]; den1=[1 2 1]; num2=[1 2]; den2=[1 2 3]; impulse(tf(num1,den1), b,tf(num2,den2), r ) step(tf(num1,den1), b,tf(num2,den2), r )
Continuous Time - Dynamic Response Step Response 1.4 t s Amplitude 1.2 M p 1% 1.9.6 t p.4.1 t r 2 4 Time (sec) 6 8 1 12 Overshoot (M p ) Rise time (t r ) Settling time (t s ) Peak time (t p )
Natural Frequency and Damping
Poles in the Continuous s-plane 2 1.5.5.34.16 1.6 1.4.64 1.2 1.86.76 1.8.5.5.94.985.985.94.6.4.2.2.4.6 1.86.76.8 1.64 1.2 1.5.5.34.16 1.4 1.6 2 1.6 1.4 1.2 1.8.6.4.2
Poles in the Continuous s-plane
Systems with Different Poles Pole Zero Map 6.984 syslhp1.965 syslhp2 syslhp3 sys.925.87.76.6.35 4 2.996 Imaginary Axis 5 2 15 1 5 2.996 4.984.965.925 6 25 2 15 1 5 Real Axis.87.76.6.35
Step Response of Different Systems Step Response 1.2 1.8 Amplitude.6.4.2 orig syslhp1 syslhp2 syslhp3.5 1 1.5 2 2.5 3 3.5 4 Time (sec)
Bode Plot of Different Systems Bode Diagram 5 Magnitude (db) 5 1 15 Phase (deg) 9 18 orig syslhp1 syslhp2 syslhp3 27 1 2 1 1 1 1 1 1 2 1 3 Frequency (rad/sec)
Systems with Different Zeros Pole Zero Map 6 4.9 syslhz1.8 syslhz2 syslhz3 sys.66.52.4.26.12 2.97 Imaginary Axis 9 8 7 6 5 4 3 2 1 2.97 4.9.8.66 6 9 8 7 6 5 4 3 2 1 Real Axis.52.4.26.12
Step Response of Different Systems Step Response 8 7 orig syslhz1 syslhz2 syslhz3 6 5 Amplitude 4 3 2 1.5 1 1.5 2 2.5 Time (sec)
Bode Plot of Different Systems Bode Diagram 4 3 2 1 Magnitude (db) 1 2 3 4 5 6 Phase (deg) 9 9 orig syslhp1 syslhp2 syslhz3 18 1 2 1 1 1 1 1 1 2 Frequency (rad/sec)
System with Zero in RHP Pole Zero Map 6.36.27.19.12.6 sysrhz.5 5 sys 4 4.66 3 2 2.88 Imaginary Axis 1 1.88 2 2.66 3 4 4.5 5.36.27.19 6 3 2.5 2 1.5 1.5.5 1 Real Axis.12.6
Step Response Introduction Step Response 2 1.5 1.5 Amplitude.5 1 1.5 2 2.5 orig rhz 3.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Time (sec)
Bode Plot of System with RHP Bode Diagram 2 1 1 Magnitude (db) 2 3 4 5 6 7 8 36 orig rhz Phase (deg) 18 18 1 2 1 1 1 1 1 1 2 1 3 Frequency (rad/sec)