《自动控制原理》课程教学资源（PPT课件讲稿）Module 7 Higher- Order Systems

Module 7 Higher-Order Systems (I hour) TF in evans or bode form Approximate a higher order system with a first or second order one Effect of zeros and poles on the transient response Origins of zeros

Module 7 Higher – Order Systems (1 hour) • TF in Evans or Bode form • Approximate a higher order system with a first or second order one • Effect of zeros and poles on the transient response • Origins of zeros

7.1 Evans and Bode form Higher order systems: Consider the following transfer function B(s) F(S)A(s)(+P)s+p,).(s+p (Evans form) We can pull out the product of the poles and pre-multiply the conjugate pairs F(s)= B(S A(s)(s+PiXs+p2)s+p2) Pippa s+1 s+1 S+1 p2 Bo (Bode form Pippa s+1 +s+1 P F 0 2

7.1 Evans and Bode form

In general any transfer function can be re-written in bode form F(s)=Fo (Bode form) (s+1)z2s+1) 2S2+2 s+1 2 S-+ s+1 Q: What is the DC gain of F(s)? Example F(s)= S+15 s +35+4/(Evans form) (Bode form 154(0051025205+1 Q:What are the values of @ and 5 in the above example?

7.2 Approximate a higher order system with a first or second order one To approximate the system with a lower order transfer function, we keep only the dc gain and the dominant poles in its Bode form: F(s) 60067+1)0.2552+0.75+1 ( Bode form) (0.067+1)=0→s=-15 0.252+0.75s+1)=0→s=-1.5±1.32j 15 X 1.5 F(s 600.25s2+075s+1

7.2 Approximate a higher order system with a first or second order one

Step res pons e x10 16 pproximate F A 14 10 8 6 Original F 0 0.5 1.5 2 3.5 Time(s ec)

Why do the dominant poles"dominate"? Consider the following example 20 20 +1ls+1010 (s+1)S+1) s+1 Let's look at the step response of f and F"in the time domain 20 220121 F(S)=- e ' t-e (S+lls +10)s 9s+19s+10 3c()=220 222 (t)=2-2e (S+1)sss+1

Effect of zeros on the transient response.3 Consider the two systems below G1(S)=2 s<+2LO,s+a +250nS+ The response of the second system to an r(s ) input is C(S=G2(SR(S)=G(SR(S)+ ISSr(s) In particular when R(S)=1/s C(S)=G(S)/S+iG,(s) eso,t Sin(t+)+了 e Sa,t sin(at 2 Response of the original G, Response due to the zero

7.3

Step Response 1.4 Zeros make 2 the response 1.2 more oscillatory 0.8 三三月 0.6 G1 0.4 LSG 0.2 -0.2 16 Time(sec.)

Step re 1.0 1.2 060 0.0 R 三 Mi Inimum 0.6 phase system 0.4 T=0:0.5;1;1.5 0.2 0.4 16 Time(sec.)

Step res ponse 1.4 d66 0.80.0 R 0.6 -0.5 0.4 Non-minimum 1.0 phase system 0.2 了=0:-0.5:-1:-1 -0.2 -0.4 0.4 -0.6 0 10 16 Time(sec.)