where ys(r) is the zero-input response, or that part of the response due to the initial conditions(or states)only, and y()is the zero-state response, or that part of the response due to the input f(o) only Here f(r)=0, and thus(100. 11)becomes P+ ao)y(t)=0 That is D(p)y(r) The roots of D(p)=0 can be categorized as either distinct or multiple. That is, in general, (p)=I(p-x,)II(p-λ where there are r distinct roots and q sets of multiple roots(each set has multiplicity ki). Note that r+o=n, whereo2yi=k, Each distinct root contributes a term to ys(o)of the form c;, where g is a constant, while each set of multiple roots contributes a set of terms to ys(n)of the form 2-o-lci telit, where ci i is some constant. Thus, the zero-input response is given by ys(t) Ci, 100.15 The coefficients cij and o+i are selected to satisfy the initial conditions. Special case If all the roots of D(p)=0 are distinct and the initial conditions for(100.11)are given by y(o, dy(o) then the coefficients of (100.15) are given by the solution of y(0) dy(0) dt dy(o) Zero-State Response: ydt Here the initial conditions are made identically zero. Observing(100.11), let bnp"+…+bp+b p+an-1p+…+a1p+ao e 2000 by CRC Press LLC
© 2000 by CRC Press LLC where yS(t) is the zero-input response, or that part of the response due to the initial conditions (or states) only, and yI (t) is the zero-state response, or that part of the response due to the input f(t) only. Zero-Input Response: yS(t) Here f(t) = 0, and thus (100.11) becomes (pn + an–1pn–1 + … + a1p + a0)y(t) = 0 (100.14) That is, D(p)y(t) = 0 The roots of D(p) = 0 can be categorized as either distinct or multiple. That is, in general, where there are r distinct roots and q sets of multiple roots (each set has multiplicity ki ). Note that r + s = n, where s D = Âq i=1ki . Each distinct root contributes a term to yS(t) of the form cie lit, where ci is a constant, while each set of multiple roots contributes a set of terms to yS(t) of the form Âj=0ki–1ci,jt j elit, where ci,j is some constant. Thus, the zero-input response is given by (100.15) The coefficients ci,j and cs+i are selected to satisfy the initial conditions. Special Case If all the roots of D(p) = 0 are distinct and the initial conditions for (100.11) are given by then the coefficients of (100.15) are given by the solution of (100.16) Zero-State Response: yI(t) Here the initial conditions are made identically zero. Observing (100.11), let D p p p i k i q q i i r i ( ) = - ( ) ( - ) = + = ’ ’ l l 1 1 y t c t e c e S i j j t j k i q i t i r i i i ( ) = + , = - = + = Â Â Â l + s l s 0 1 1 1 y dy dt d y dt n n ( ), ( ) , ( ) 0 0 0 1 1 . . . , - - Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ Ô y dy dt d y dt c c c n n n n n n n n ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 1 1 1 1 1 1 2 1 1 2 1 1 1 2 M L L M MMM L M - - - - - È Î Í Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ ˙ = È Î Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ È Î Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ l l l l l l H p b p b p b p a p a p a m m n n n ( ) = + + + + - + + + - L L 1 0 1 1 1 0
denote a rational function in the p operator. Consider using partial-fraction expansion on H(p)as H(P g p when the first term corresponds to the sets of multiple roots and the second term corresponds to the distinct Note the constant residuals are computed as P (P-A)H(p (k;-)!d Then h(t)= (100.18) (-1)! is the impulse response of the system(100. 11). Then the zero-state response is given by y,(t)= f(t)h(t-t)d (100.19) that is, y(r) is the time convolution between input f(r)and impulse response h(n). In some instances, it may be easier to find y (t)and y (r)using Laplace Transform methods Measures of the Dynamic System Response Several measures may be employed to investigate dynamic response performance. These include 1. Speed of the response--how quickly does the system reach its final value 2. Accuracy--how close is the final response to the desired response 3. Relative stability-how stable is the system or how close is the system to instability 4. Sensitivity--what happens to the system response if the system parameters change Objectives 3 and 4 may be analyzed by frequency domain methods(Section 100.3). Time-domain measures classically analyze the dynamic response by partitioning the total response into its steady-state (objective 2) and transient(objective 1)components. The steady-state response is that part of the response which remains as time approaches infinity, the transient response is that part of the response which vanishes as time approaches infinity. Measures of the Steady-State Response In the steady state, the accuracy of the time response is an indication of how well the dynamic response follows a desired time trajectory. Usually a test signal (reference signal)is selected to measure accuracy. Consider Fig. 100.5. In this configuration, the objective is to force y(t) to track a reference signal r(t) as close as possible e 2000 by CRC Press LLC
© 2000 by CRC Press LLC denote a rational function in the p operator. Consider using partial-fraction expansion on H(p) as (100.17) when the first term corresponds to the sets of multiple roots and the second term corresponds to the distinct roots. Note the constant residuals are computed as gs+i = [(p – lq+i )H(p)]p = lq+i and Then (100.18) is the impulse response of the system (100.11). Then the zero-state response is given by (100.19) that is, yI (t) is the time convolution between input f(t) and impulse response h(t). In some instances, it may be easier to find ys (t) and yI (t) using Laplace Transform methods. Measures of the Dynamic System Response Several measures may be employed to investigate dynamic response performance. These include: 1. Speed of the response—how quickly does the system reach its final value 2. Accuracy—how close is the final response to the desired response 3. Relative stability—how stable is the system or how close is the system to instability 4. Sensitivity—what happens to the system response if the system parameters change Objectives 3 and 4 may be analyzed by frequency domain methods (Section 100.3). Time-domain measures classically analyze the dynamic response by partitioning the total response into its steady-state (objective 2) and transient (objective 1) components. The steady-state response is that part of the response which remains as time approaches infinity; the transient response is that part of the response which vanishes as time approaches infinity. Measures of the Steady-State Response In the steady state, the accuracy of the time response is an indication of how well the dynamic response follows a desired time trajectory. Usually a test signal (reference signal) is selected to measure accuracy. Consider Fig. 100.5. In this configuration, the objective is to force y(t) to track a reference signal r(t) as close as possible. H p g p g p i j i j j k i q i i q i i r ( ) ( ) , = - + = = - + = + Â Â Â l l s 1 1 1 g k j d dp p H p i j i k j k j i k p i i i i , ( ) ( ) ( )! = ( ) ( ) - { } - - - = 1 l l h t g j t e g e i j j t j k i q i i r t i i i ( ) ( )! , = - + - = = + = Â Â Â + 1 1 1 1 1 l s ls y t f h t d I t ( ) = - ( ) ( ) Ú t t t 0
(t) f(t) FIGURE 100.5 A tracking controller configuration. TABLE 100.2 Steady-State Error Constants Error Ru(r): step function K。=limG(s)H(s) Rtu(r ): ramp function k K,= lim sG(s)H(s) Rru(t) parabolic function R K= lim s"GsH(s The steady-state error is a measure of the accuracy of the output y(o) in tracking the reference input r(t). Other configurations with different performance measures would result in other definitions of the steady-state error between two signals. From Fig. 100.5, the error e(t)is e(t)=r(t)-y(t (100.20) and the steady-state error is ess(t)= lim e(t)= lim sE(s) (100.21) assuming the limits exists, where E(s) is the Laplace transform of e(n), and s is the Laplacian operator. with G(s)the transfer function of the system and H(s) the transfer function of the controller, the transfer function between y(r)and r(t) is found to b T(s) G(S)H(s) (100.22) 1+G(s)H(s) E(s) R(s) (100.23) 1+G(s)H(s) Direct application of the steady-state error for various inputs yields Table 100.2. Note u(r) is the function. This table can be extended to an mth-order input in a straightforward manner. Note that for ess(t) to go to zero with a reference signal Cru(n), the term G(s)H(s) must have at least m poles at the origin(a type e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The steady-state erroris a measure of the accuracy of the output y(t) in tracking the reference input r(t). Other configurations with different performance measures would result in other definitions of the steady-state error between two signals. From Fig. 100.5, the error e(t) is e(t) = r(t) – y(t) (100.20) and the steady-state error is (100.21) assuming the limits exists, where E(s) is the Laplace transform of e(t), and s is the Laplacian operator. With G(s) the transfer function of the system and H(s) the transfer function of the controller, the transfer function between y(t) and r(t) is found to be (100.22) with (100.23) Direct application of the steady-state error for various inputs yields Table 100.2. Note u(t) is the unit step function. This table can be extended to an mth-order input in a straightforward manner. Note that for eSS(t) to go to zero with a reference signal Ctmu(t), the term G(s)H(s) must have at least m poles at the origin (a type m system). FIGURE 100.5 A tracking controller configuration. TABLE 100.2 Steady-State Error Constants r(t) Error Test Signal eSS(t) Constant Ru t Rtu t R t u t ( ) ( ): ( ): : step function ramp function parabolic function 2 2 R K R K R K p v a 1 + K G s H s K sG s H s K s G s H s p s v s a s lim ( ) ( ) lim ( ) ( ) lim ( ) ( ) 0 0 0 2 = = = Æ Æ Æ e t e t sE s SS t s ( ) = = ( ) ( ) Æ • Æ • lim lim T s G s H s G s H s ( ) ( ) ( ) ( ) ( ) = 1 + E s R s G s H s ( ) ( ) ( ) ( ) = 1 +
Unit step overshoot 0,50 FIGURE 100.6 Step response. Measures of the Transient Response In general, analysis of the transient response of a dynamic system to a reference input is difficult. Hence rmulating a standard measure of performance becomes complicated. In many cases, the response is dominated by a pair of poles and thus acts like a second-order system. Consider a reference unit step input to a dynamic system( Fig. 100.6). Critical parameters transient response include: 1. M: maximum overshoot 2. overshoot= M/AX 100%, where A is the final value of the time response 3. t: delay time-the time required to reach 50%of A 4. tr: rise time-the time required to go from 10%of A to 90%of A 5. t: settling time-the time required for the response to reach and stay within 5% of A To calculate these measures, consider a second-order system T(s)= (100.24) s2+25os+o2 where E is the damping coefficient and o, is the natural frequency of oscillation. For the range 0<5<1, the system response is underdamped, resulting in a damped oscillatory output. For a unit step input, the response is given by St-tan (0<5<1)(100.25) The eigenvalues(poles)of the system [roots of the denominator of T(s)] provide some measure of the time constants of the system. For the system under study, the eigenvalues are at 0,±j0 where e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Measures of the Transient Response In general, analysis of the transient response of a dynamic system to a reference input is difficult. Hence formulating a standard measure of performance becomes complicated. In many cases, the response is dominated by a pair of poles and thus acts like a second-order system. Consider a reference unit step input to a dynamic system (Fig. 100.6). Critical parameters that measure transient response include: 1. M: maximum overshoot 2. % overshoot = M/A ¥ 100%, where A is the final value of the time response 3. td: delay time—the time required to reach 50% of A 4. tr: rise time—the time required to go from 10% of A to 90% of A 5. ts : settling time—the time required for the response to reach and stay within 5% of A To calculate these measures, consider a second-order system (100.24) where x is the damping coefficient and wn is the natural frequency of oscillation. For the range 0 < x < 1, the system response is underdamped, resulting in a damped oscillatory output. For a unit step input, the response is given by (100.25) The eigenvalues (poles) of the system [roots of the denominator of T(s)] provide some measure of the time constants of the system. For the system under study, the eigenvalues are at FIGURE 100.6 Step response. T s s s n n n ( ) = + + w xw w 2 2 2 2 y t e t nt n ( ) – = + ( ) - - - - Ê Ë Á Á ˆ ¯ ˜ ˜ < < - 1 1 1 1 0 1 2 2 1 xw 2 x w x x x sin tan x – –xw w x n n ± - j 1 1 j = - 2 where D
FIGURE 100.7 Effect of the damping coefficient on the dynamic response. FIGURE 100.8 Effect of the natural frequency of oscillation on the dynamic response. From the expression of y(r), one sees that the term So, affects the rise time and exponential decay time The effects of the damping coefficient on the transient response are seen in Fig. 100.7. The effects of the natural frequency of oscillation o, of the transient response can be seen in Fig. 100.8 O increases,the frequency of oscillation increases For the case when 0<5<1, the underdamped case, one can analyze the critical transient response parameters e 2000 by CRC Press LLC
© 2000 by CRC Press LLC From the expression of y(t), one sees that the term xwn affects the rise time and exponential decay time. The effects of the damping coefficient on the transient response are seen in Fig. 100.7. The effects of the natural frequency of oscillation wn of the transient response can be seen in Fig. 100.8. As wn increases, the frequency of oscillation increases. For the case when 0 < x < 1, the underdamped case, one can analyze the critical transient response parameters. FIGURE 100.7 Effect of the damping coefficient on the dynamic response. FIGURE 100.8 Effect of the natural frequency of oscillation on the dynamic response