Instructions
Tutorial problems
1∗. Find the transfer function H(s) = Y(s) for the feedback system shown in Figure 1. R(s)
ELEN90055 Control Systems Worksheet 3
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Semester 2
This worksheet covers materials about block diagram algebra, stability criterion and steady state error. Solutions to starred problems in part 1 (tutorial problems) will be provided to students after the tutorial. Additional solved problems are provided in part 2. Students are encouraged to work on the solutions to the problems in parts 3 and ?? by themselves.
Figure 1: Problem 1.
2∗. Consider the system shown in Figure 2. This system contains a delay, indicated by the non-rational transfer function e−sT .
Figure 2: Feedback system with a delay.
(a) Determine the closed loop characteristic equations. Note: this will include the delay term e−sT .
v a r i a t i o n o f M p v s ζ à e ≈ 1 + T2 s 2à For the closed-loop system in Figure , compute
August 22, 2à
(b) Use the following approximation to the delay function and determine values for T à For a second-order system, Þnd and sketch the allowable region in the s-plane for the
and A such that the system is stable.
poles of a transfer function with system response requirements: tr ≤ àà9sec, Mp ≤
5à, and t ≤ 4secà Hint: refer to Fig 3à23 in FTranklin, Powell, Emami-Naeni for s −sT 1−2s
a The DC gain, initial value and Þnal valueà
3 . Consider the closed-loop system in Figure 3 representing the block diagram of a DC motor where k, K, and J are positive, and let r(t) = L −1{R(s)} be a unit step function. Compute
b K, k so that the maximum overshoot for a unit-step response is 5à and tp = the initial value and the final value of the output.
3secà Let J = kg-m2à
5à DeÞne the type for a feedback systemà For a general feedback system in Figà 5, characterise the system type if,
Figure : a
G s = s à H s =
Figure 3: Problem 3
3à Determine whether the following characteristic polynomials represent stable or un-
stable systemsà
Gs =ss+3àHs =s+2 For the unity feedback system in Figure 4 with
b s4+s3−s− d
c s4 +2s3 +3s2 +6s+ 4 find the steady-state error when theGinspu=t is à H s =
a 2s4 +8s3 +às2 +às+2à
4 ( s +à H 1 s ) = s + 5 G(s)= s2(s+2)
4à For what values of K do the following polynomials have negative real parts
(a) r(6tà)F=or1t.he unity feedback system in Figà 6 with (b) r(3t)=t. 2 4 s+
a s + 4 + K s + 6s + 2 G s = 2
2 ss+2 (c) r(4t)=t3. 2
b s +6s +s +6s+K Þnd
a The position, velocity, and acceleration error constantsà
Figure 4: Problem 3
5∗. ConsiderthesystemshowninFigure5whereD(s)=s+k,R(S)isthereferenceinputand W (s) is a disturbance.
(a) Find the transfer functions from R to Y and from W to Y .
(b) What conditions must k satisfy such that the system is stable.
(c) Let R(s) = W (s) = 1/s. Is it possible to choose the parameter k such that the steady state error of the system from r(t) = 1 and also the steady state output from the disturbance w(t) = L −1{W(s)} are zero?
Figure 5: Problem 5
6. Supposethatunityfeedbackistobeappliedaroundtheopen-loopsystemslistedbelow(so the closed-loop system is similar to Figure 4. Use Routh’s stability criterion to determine whether the resulting closed-loop systems will be stable.
G(s) = 4(s + 2)
s(s3 +2s2 +3s+4)
G(s)= 2(s+4) s2(s+1)
G(s) = 4(s3 +2s2 +s+1) s2(s3 +2s2 −s−1)
Additional solved problems
1. Solved Problems 4.3, 4.9, 5.5 from the solved problems of the web site for Goodwin, Graebe, Salgado ’Control System Design’ – see Example problems with worked solutions at http://www.csd.elo.utfsm.cl/.
3 Previous exam questions
1. Q2(ii) part (a); Control system final exam 2013.
2. Q3(d) the fist part and Q3(e); Control system final exam 2015.
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