1Steady-State Error For Non-Unity FeedbackSystems (H(s) 6= 1)A. OverviewFor unity feedback systems with H(s) = 1, the output of the summing junction in the normal single-loopfeedback configuration is the difference between the reference input R(s) and the output C(s). Assumingthat the reference input is the desired value of the output, then the output of the summing junction is thesystem error E(s). The expression for the steady-state error, ess= limt→∞[e(t)] is easily derived for the unityfeedback case; this development is shown on the web page http://ece.gmu.edu/˜gbeale/ece 421/ess 01.html.When the system does not have unity feedback, that is, H(s) 6= 1, then the output of the summing junctionis not the difference between the reference input and the actual output. However, we will still define the erroras E(s) = R(s) − C(s). Unfortunately, that signal does not appear in the non-unity feedback blo ck diagram,shown in the top diagram of Fig. 1. The signal E(s) is shown in the second diagram in the figure.Two methods will be used to derive the expression for steady-state error for the non-unity feedback case.The first method will proceed directly from the second block diagram in the figure. The second method will useblock diagram manipulation from the top diagram in the figure to convert the system into an equivalent unityfeedback system. We will make the same assumptions on the types of input signals that will be consideredthat were made for unity feedback systems, namely that R(s) = A/sq.B. Method 1The system is described by the top diagram in Fig. 1. The forward transfer function G(s) is given byG(s) =KNg(s)Dg(s)=KxNg1(s)sNDg1(s),Ng1(0) = 1Dg1(0) = 1(1)where N is the number of open-loop poles in G(s) that occur at s = 0. The feedback transfer function H(s)is given byH(s) =KF BNh(s)Dh(s)=KhNh1(s)Dh1(s),Nh1(0) = 1Dh1(0) = 1(2)The feedback block H(s) is assumed to have no poles or zeros at the origin. The final forms shown for bothG(s) and H(s) are the time constant form, so the only terms that will impact the steady-state error will bethe gains Kxand Khand the N poles of G(s) at the origin.As shown in the second diagram in Fig. 1, the error signal is given by E(s) = R(s) − C(s), which becomesE(s) = R(s) − TCL(s)R(s) =·1 −G(s)1 + G(s)H(s)¸R(s) =·1 + G(s)H(s) − G(s)1 + G(s)H(s)¸R(s) (3)Assuming that the closed-loop system is stable, the steady-state error can be obtained by applying the FinalValue Theorem to E(s). This isess= limt→∞[e(t)] = lims→0[sE(s)] = lims→0½s·1 + G(s)H(s) − G(s)1 + G(s)H(s)¸R(s)¾(4)With the reference input restricted to the class of signals represented by R(s) = A/sq, the expression for esscan be written asess= A · lims→0½s(1−q)·1 + G(s)H(s) − G(s)1 + G(s)H(s)¸¾(5)2If we substitute the time constant forms for G(s) and H(s) from (1) and (2) into (5), we get the followingresults.ess= A · lims→0s(1−q)1 +KxNg1(s)sNDg1(s)·KhNh1(s)Dh1(s)−KxNg1(s)sNDg1(s)1 +KxNg1(s)sNDg1(s)KhNh1(s)Dh1(s)= A · lims→0(s(1−q)"1 +KxKhsN−KxsN1 +KxKhsN#)(6)where the last expression makes use of the fact that Ng1(0) = Dg1(0) = Nh1(0) = Dh1(0) = 1. Multiplyingnumerator and denominator of (6) by sN, we getess= A · lims→0½s(1−q)·sN+ KxKh− KxsN+ KxKh¸¾= A · lims→0"s(N+1−q)sN+ KxKh+Kx(Kh− 1) s(1−q)sN+ KxKh#(7)Equation (7) is the general expression for steady-state error with non-unity feedback.Clearly, if Kh= lims→0[H(s)] = 1, the expression for essin (7) is identical to the expression for the unityfeedback case. Thus, the non-unity feedback configuration is equivalent—from the standpoint of steady-stateerror—to the unity feedback configuration if lims→0[H(s)] = 1.If the reference input is a step input, q = 1 . If N ≥ 1, the steady-state error becomesess= A · lims→0·sN+ KxKh− KxsN+ KxKh¸= AµKh− 1Kh¶(8)so ess= 0 when Kh= 1, as expected. If Kh< 1, the steady-state error is negative, which means that theoutput is larger than the input. If Kh= 0, the steady-state error becomes infinity; the feedback loop is brokenin steady-state, G(s) has N integrators, so the output is unbounded for the step input.If q > 1, the steady-state error isess= A · lims→0"s(N+1−q)sN+ KxKh+1s(q−1)·Kx(Kh− 1)sN+ KxKh#(9)so ess= ∞ for q > 1 unless Kh= 1, regardless of the value of N. Thus, non-unity feedback systems willalways have infinite steady-state error if the reference input is any signal of higher order than a step functionif Kh6= 1.C. Method 2A simple identity can be used to convert a non-unity feedback system into an equivalent unity feedbacksystem. This identity is H(s) = H(s) − 1 + 1, so the first diagram in Fig. 1 can be converted into the thirddiagram. The inner loop has a feedback block of H(s) − 1, and the outer loop has unity feedback. The errorsignal is the output of the left-most summing junction in the third diagram. The inner loop can be reducedto a single block with transfer functionG1(s) =G(s)1 + G(s) [H(s) − 1](10)The equivalent system, shown in the bottom diagram of Fig. 1 has the standard unity feedback configuration,with G1(s) being the forward transfer function. Once this conversion has been done, the normal unity feedbackequations for steady-state error can be applied directly to G1(s), that is,ess= A · lims→0"s(N1+1−q)sN1+ Kx1#(11)where N1is the number of poles of G1(s) at the origin andKx1= lims→0[G1(s)] (12)3G sH sG1sG sG sH sH sR sR sR sC sC sC sE sE sE sR s C sFig. 1. Block diagrams for non-unity feedback steady-state