16.06 Principles of Automatic Control Lecture 9 Unity Feedback Control With Noise Consider a typical unity feedback control system d, disturbance r + e’ controller plant K(s) G(s) y + - + + e v sensor noise e1 is the error perceived by the control system; e is the actual error. The important transfer functions are Y 1 psq “ ” Spsq R 1 ` KpsqGpsq E ´1 psq “ ” ´Spsq D 1 ` KpsqGpsq E ´KpsqGpsq psq “ ” ´T psq V 1 ` KpsqGpsq Spsq “Sensitivity transfer function T psq “Complementary Sensitivity transfer function For low sensitivity to disturbances, want: |Spsq| ! 1 1 For good tracking of the reference input, want: |Spsq| ! 1 For low sensitivity to sensor noise or errors, want: |T psq| ! 1 But these goals are mutually exclusive, since Spsq ` T psq “ 1 So there is a fundamental trade-oﬀ between good tracking performance and low sensitivity to sensor noise. How is this trade-oﬀ addressed? In most (not all) systems, want good tracking performance at low frequency, low sensitivity to sensor noise at high frequency: - Reference inputs are low frequency - Sensor noise is usually high frequency So let’s look at the lowest frequency, ω “ 0 ps “ 0q... Steady-State Errors Consider a unity feedback system without sensor noise or disturbance: r + e K(s) G(s) - For stability, deﬁne Lpsq “ KpsqGpsq “ "Loop Gain" What is the steady-state error to a unit step input? 2 y 1.4 1.2 1 y(t) e(t) 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 16 18 20 t Use LTs: Epsq “ SpsqRpsq “ 1 1 1 Rpsq “ 1 ` Lpsq 1 ` Lpsq s To ﬁnd the steady error, use ﬁnal value theorem: lim eptq “ lim sEpsq “ sÑ0 sÑ0 1 1 ` Lp0q If L(0) is ﬁnite, we deﬁne Kp ” Lp0q “ "positive error constant" Furthermore, if L(0) is ﬁnite, we say that a system is a “type 0 system”. So a type 0 system will always have a ﬁnite error in response to a steady input r, but the error can be made small by making the position error constant large. To make the steady error zero, we must have that L(0) is inﬁnite. Suppose we can express L(s) as Lpsq “ L0 p0q s where L0 p0q ‰ 0, L0 p0q is ﬁnite. Then L is a “type 1 system” (one pole at s “ 0). We have that 3 lim eptq “ lim s sÑ0 sÑ0 1 1` L0 psq s 1 s “ lim “0 sÑ0 s s ` L0 psq since L0 p0q ‰ 0. What if we want to track a unit ramp instead? rptq “trptq 1 ñ Rpsq “ 2 s The steady-state error for a type 0 system will be ess “ lim sSpsqRpsq sÑ0 1 1 sÑ0 1 ` Lpsq s2 1 1 “ lim “8 sÑ0 1 ` Lp0q s “ lim s The steady-state error for a type 1 system will be ess “ lim sSpsqRpsq sÑ0 1 “ lim s L0 psq s sÑ0 1` 1 “ lim sÑ0 s ` L0 psq 1 “ L0 psq 1 s2 which is ﬁnite. We deﬁne Kv “ L0 psq “ "velocity error constant" More generally, suppose that L(s) has the form Lpsq “ L0 psq sn L is said to be a type n system, and the error constant is Kp ,or Kv or Ka ... “ L0 p0q. 4 Kp “K0 “ lim Lpsq, n “ 0 sÑ0 Kv “K1 “ lim sLpsq, n “ 1 sÑ0 Ka “K2 “ lim s2 Lpsq, n “ 2 sÑ0 .. . Input Type σptq 1 Type 0 ess “ 1`K p Type 1 0 Type 2 0 2 t tσptq σptq 2 ess “ 8 ess “ 8 1 8 Kv 1 0 Ka Obviously, this generalizes, but we usually care most about Kp and Kv - higher order inputs are rare. 5

16.06 Principles of Automatic Control, Lecture 9

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