Module 8 System Type Steady-State Error and Muriple control (3 hours) Definition of the system type The kind of input of system How to identify steady-state error quickly How to eliminate steady-state error
Module 8 System Type: Steady-State Error and Muriple Control (3 hours) • Definition of the system type • The kind of input of system • How to identify steady-state error quickly • How to eliminate steady-state error
Definition of error. 8.1 Consider the following block diagram R(s)→)G(s) H(s) The true error is defined as the difference between the input and the output, while the actuating error is the input to the system g E(S=R(S-C(S)(ture error) E,(S=R(S-H(s)C(S(actuating error) For unit feedback systems(H=1), the two errors coincide
8.1
8.1.1 Why we should consider the actuating error Ea(P125) The true error e=r-c is not AE C G observable. since the output has to be measured by H(s) HC=CLH before its value is known EG I+Gh p/ observable, since C=HC car The actuating error ea=R-C e measure d. For the study convenience, we R now make a important 1+G assumption: H=l, that E= ea
8.1.1 Why we should consider the actuating error Ea (P125) • The true error E=R-C is not observable, since the output has to be measured by H(s) before its value is known. • The actuating error Ea =R-C’ is observable, since C’=HC can be measured. • For the study convenience, we now make a important assumption : H=1, that E= Ea R Ea C HC =C' G H R G GH C E + = = 1 1 R + G = 1 1
8.1.2 Errors of non-unity and unity feedback 只 E G Non-unity feedback sys HECH E=R-HC =H(,R-C) H HE R=RIGE E E |1 C GH Unity feedback system
8.1.2 Errors of non-unity and unity feedback R Ea C HC =C' G H R C H 1 GH ' 1 R R H = E' Non-unity feedback sys. Unity feedback system Ea = R − HC ) 1 ( R C H = H − = HE' Ea H E 1 ' =
8.2 The steady-state error ess (e(oo)) E(S=R(s)C(S) R(s) 1+G(S) e(oo)=lim SE(s)=lm SR(S) 1+G(s) e(oo)is determined by two factors r(t and G(s)
8.2 The steady-state error ess ( ) e() ( ) 1 ( ) 1 ( ) ( ) ( ) R s G s E s R s C s + = − = 1 ( ) 1 ( ) lim ( ) lim ( ) 0 0 G s e sE s sR s s s + = = → → e() is determined by two factors: r(t) and G(s)
8.3 The sys. type n-the integral elements number in the forward path of sys In general any transfer function can be written as: G(S) K(s+z1)(s+z2)…(S+2p )K∏(s+z2) where(n+k>p) S"(S+pi)(s+p2).S+Pk) s"II(s+P,) For unit-feedback systems, the/system types defined as the value of n b ove Example: G(S) (S+1) S(5+3s+4)is of type 2 The system type indicates what order of input signals can the given system track "with zero steady state error. The"order"here refers to the power of s in the laplace transform of the input signal. To see this, we will investigate the steady state error of various system types due to impulse R(S=l, Step r(s=1/s, ramp r(s)=1/s 2 and acceleration R(S)=1/s 3 inputs
8.3 The sys. type n – the integral elements number in the forward path of sys
The system type indicates what order of input signals can the given system track" with zero steady state error. The order here refers to the power of s in the laplace transform of the input signal To see this, we will investigate the steady state error of various system types due to impulse r(s=l, step r(s=1/s, ramp r(s)=1/52 and acceleration R(s)=1/s3 inputs 糸统类型表明什么阶次的输入信号能为糸统无稳态误 差的跟踪。这里的阶次引用的是 Laplace变换中输入信 号的幂次。 为了看清这一点,我们来观察由输入信号为脉冲R(s)=1, 阶跃R()=1/s,斜坡R(s)=1/32和加速度R(s)=1/3时,不 同糸统的稳态误差
The system type indicates what order of input signals can the given system “track” with zero steady state error. The “order” here refers to the power of s in the Laplace transform of the input signal. To see this, we will investigate the steady state error of various system types due to impulse R(s)=1, step R(s)=1/s, ramp R(s)=1/s 2 and acceleration R(s)=1/s 3 inputs. 系统类型表明什么阶次的输入信号能为系统无稳态误 差的跟踪。这里的阶次引用的是Laplace变换中输入信 号的幂次。 为了看清这一点，我们来观察由输入信号为脉冲R(s)=1, 阶跃R(s)=1/s, 斜坡R(s)=1/s 2 和加速度R(s)=1/s 3 时，不 同系统的稳态误差
1. Unit feedback systems subjected to IMPULSE input 在.条件下 G E(S)=R-C=R R=1 R R 1+G 1+G1+G ype 0 system e =lime(s)=lims R(s)= →01+G(s) m s 1=0 =0 1+ K∏(s+2) KITz 1+ s'Il(s+p) SI
在…条件下
Type” system e =lime(S)=lims 1=0 S-)0 K∏(s+z K∏z 1+ and higher systems s∏(s+p) e =lime(S)=lims l=0 =0 KII(S+z) KIlz 1+ 1 III(S+ Pi) 7)01 2An impulse can be tracked by systems of all types!
2. Unit feedback systems subjected to a STEP input Type0” system: eee= lim sE(s)=lims ≠0 s-0 →0 1+ K∏I(s+2)S1+ K∏z1+K STI(S+p) Ks RIIz Type“” and higher systems: e= lime(s)=lims =0 s-0 s01K∏(s+z) =lim Klz 1+oo S 5-0 sTI(s+p,) IStep input can be tracked by systems of types I and higher!