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## Chapter 5 Design using Transformation Technique – Classical Method

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**Chapter 5**Design using Transformation Technique – Classical Method**time**Transient response**D(s)**G(s) + - unity feedback Steady-state response**Rule of Thumb**ex)**Design by Emulation**• Design specifications: • Overshoot to a step input less than 16%. • Settling time to 1% to be less than 10sec. • Tracking error to a ramp input of slope 0.01 rad/sec to be • less than 0.01 rad. • 4. Sampling time to give at least 10 samples in a rise time.**0**-0.46 forbidden region**R**E C c(t) u(k) r(t) e(k) e(t) T=0.2**Bode plot of the continuous design for the antenna control**1 10 0 10 Magnitude -1 10 -2 10 -1 0 1 10 10 10 -80 -100 -120 Phase, degrees -140 -160 -180 -1 0 1 10 10 10**T=0.2**T=1**Direct Design by Root Locus**in the z-plane • The method for continuous-time systems can be extended w/o modification • The effects of the system gain and/or sampling period can be investigated Performance 0 -1 1**Overshoot**0 -1 1 Settling time**-1**1**Velocity error constant**Acceleration error constant**pole**-1 0 1 zero Large overshoot Poor dynamic response Errors are decreased Small steady-state error against Good transient response**Discrete root locus with and without compensation**1 0.5 p /T 0.6 p /T 0.4 p /T 0.8 0.1 0.7 p /T 0.3 p /T 0.2 0.3 0.6 0.8 p /T 0.4 0.2 p /T 0.5 0.4 0.6 0.7 0.9 p /T 0.1 p /T 0.8 0.2 0.9 p /T Imaginary Axis 0 p /T -0.2 0.9 p /T 0.1 p /T -0.4 0.8 p /T 0.2 p /T -0.6 0.7 p /T 0.3 p /T -0.8 0.6 p /T 0.4 p /T 0.5 p /T -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real Axis**-1**1 0 0.61**Root locus for antenna design**1 0.5 p /T 0.6 p /T 0.4 p /T 0.8 0.1 0.7 p /T 0.3 p /T 0.2 0.3 0.6 0.4 0.8 p /T 0.2 p /T 0.5 0.4 0.6 0.7 0.9 p /T 0.8 0.1 p /T 0.2 0.9 p /T Imaginary Axis 0 p /T -0.2 0.9 p /T 0.1 p /T -0.4 0.8 p /T 0.2 p /T -0.6 0.7 p /T 0.3 p /T -0.8 0.6 p /T 0.4 p /T 0.5 p /T -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real Axis**Root locus for compensated Antenna Design**1 0.5 p /T 0.6 p /T 0.4 p /T 0.8 0.1 0.7 p /T 0.3 p /T 0.2 0.3 0.6 0.4 0.8 p /T 0.2 p /T 0.5 0.4 0.6 0.7 0.9 p /T 0.8 0.1 p /T 0.2 0.9 p /T Imaginary Axis 0 p /T -0.2 0.9 p /T 0.1 p /T -0.4 0.8 p /T 0.2 p /T -0.6 0.7 p /T 0.3 p /T -0.8 0.6 p /T 0.4 p /T 0.5 p /T -1 -1 -0.5 0 0.5 1 Real Axis**Step Response of Compensated Antenna**1.5 1 0.5 0 OUTPUT, Y and CONTROL, U/10 -0.5 -1 -1.5 0 2 4 6 8 10 12 14 16 18 20 TIME (SEC)**Root locus for Compensated Antenna Design**0.76 0.64 0.5 0.34 0.16 1 0.86 0.8 0.6 0.94 0.4 0.985 0.2 1.4 1.2 1 0.8 0.6 0.4 0.2 Imaginary Axis 0 -0.2 0.985 -0.4 0.94 -0.6 -0.8 0.86 -1 0.64 0.5 0.34 0.16 0.76 -1 -0.5 0 0.5 1 Real Axis**Step response of compensated Antenna Design**1.5 1 0.5 OUTPUT, Y and CONTROL, U/10 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 TIME (SEC)**Root locus for compensated Antenna Design**1.2 0.64 0.5 0.34 0.16 1 1 0.76 0.8 0.8 0.86 0.6 0.6 0.4 0.94 0.4 0.2 0.985 0.2 Imaginary Axis 0 -0.2 0.985 0.2 -0.4 0.94 0.4 -0.6 0.6 0.86 -0.8 0.8 0.76 -1 1 0.64 0.5 0.34 0.16 1.2 -1 -0.5 0 0.5 1 Real Axis**Step response for compensated antenna Design**2 1.5 1 0.5 OUTPUT, Y and CONTROL, U/10 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 TIME (SEC)**Frequency Response Methods**1. The gain/ phase curve can be easily plotted by hand. 2. The frequency response can be measured experimentally. 3. The dynamic response specification can be easily interpreted in terms of gain/ phase margin. • The system error constants and can be read directly from the low frequency asymptote of the gain plot. 5. The correction to the gain/phase curves can be quickly computed. 6. The effect of pole/ zero gain changes of a compensator can be easily determined. Note : 1, 5, 6 above are less true for discrete frequency response design using z - transform.**Nyquist Stability Criterion**Continuous case zeros of the closed-loop characteristic equation, n(s) + d(s) =poles of the closed-loop system, n(s)+d(s) open-loop system known closed-loop system characteristic equation**Z (unknown)= # of unstable zeros (same direction)**of 1 + K D(s) G(s) ( or # of unstable poles of H(s) ) P(known) = # of unstable poles (opposite direction) of 1 + K D(s) G(s) ( or # of unstable poles of KD(s)G(s)) N(known after mapping) = # of encirclement (same direction) of the origin of 1+KD(s)G(s) ( or -1 of KD(s)G(s) ) Z must be zero for stability**-1**-1/K S-plane 1+KD(s)G(s)-plane KD(s)G(s)-plane unstable poles 0 D(s)G(s)-plane Z – P =N orZ = P + N**Discrete case ( The ideas are identical )**• Unstable region of the z-plane is the outside of the unit circle • Consider the encirclement of the stable region. • N = { # of stable zeros } - { # of stable poles} • = { n– Z } – { n– P } • = P – Z Z= P– N • In summary, • Determine the number, P, of unstable poles of KDG. • 2. Plot KD(z)G(z) for the unit circle, and . • 3. Set N equal to the net number of CCW encirclements of the point • -1 on the plot • 4. Compute Z = P – N. This system is stable iffZ =0.**ex) p. 241 (Franklin’s)**The unit feedback discrete system with the plant transfer function with sampling rate ½Hz and zero-order hold**Nyquist plot from Example 1 using contour**1 0.8 0.6 0.4 0.2 0 Imaginary Axis -0.2 -0.4 -0.6 -0.8 -1 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 Real Axis**Design Spec. in the Frequency Domain**Gain Margin (GM) : The factor by which the gain can be increased before the system to go unstable Phase Margin (PM): A measure of how much additional phase lag or time delay can be tolerated in the loop before instability results. ex) p. 243**GM=1.8, PM=18**Bode plot Nyquist plot