On Overshoot and Nonminimum Phase Zeros
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
On Overshoot and Nonminimum Phase Zeros James Stewart and Daniel E. Davison
Abstract—It is widely known that real nonminimum phase zeros lead to step response undershoot, and that the size of the undershoot necessarily tends to infinity as the settling time tends to zero. In this note, we show that the presence of two or more real nonminimum phase zeros can lead to step response overshoot in addition to undershoot. A lower bound on the overshoot is derived, and it is shown that the overshoot, like the undershoot, necessarily tends to infinity as the settling time tends to zero. The results are derived for single-input–single-output linear time-invariant continuous-time systems, and apply to both open-loop control and general two degree-of-freedom closed-loop control. Index Terms—Nonminimum phase zeros, overshoot, transient response, undershoot.
I. INTRODUCTION This note focuses on the control of continuous-time plants that have real nonminimum phase (NMP) zeros, i.e., zeros on the positive real axis in the complex plane. It is well established that NMP zeros limit the achievable closed-loop performance; representative works that discuss such performance limitations include [1]–[6], and good overviews of these and related results include [7] and [8]. Perhaps the most familiar characteristic of real NMP zeros is that they lead to undershoot in the step response. The purpose of this note is to show that NMP zeros can also lead to overshoot in the step response. The main result is that if the plant has two or more real NMP zeros (and any number of additional zeros located anywhere in the complex plane), then there is overshoot if the speed of response is fast enough. The conclusions of this note complement, and in no way contradict, work on the synthesis of compensators that result in a nonovershooting closed-loop step response. For example, in [9], a control strategy is designed to guarantee a nonovershooting step response, but the results assume there is at most one NMP zero (and also use state feedback, whereas the results in the present note apply to output feedback); in [10], a nonovershooting control scheme is devised that applies even if the plant has multiple NMP zeros, but the scheme relies critically on making the closed-loop response sufficiently slow; and in [11] and [12], nonovershooting controllers are considered where there are no constraints on the speed of response, but the results apply only if the plant is minimum phase. The conclusions of this note also complement other results related to step response overshoot and pole-zero locations. For example, it is well known that a single minimum-phase zero can lead to step response overshoot (e.g., see [7]); of course, a key difference between minimum-phase zeros and NMP zeros is that it is possible to cancel out the former, but not the latter, and still maintain closed-loop stability. Similarly, a real unstable plant pole leads to closed-loop step response overshoot, assuming that a one degree-of-freedom control configuration is used (e.g., see [7]). Throughout this note, let P (s) denote the known plant which has at least one NMP zero, let r denote the reference signal, and let y denote the plant output. The controller structure under investigation is Manuscript received September 21, 2005; revised February 3, 2006. Recommended by Associate Editor F. Bullo. This work was supported by the Natural Sciences and Engineering Research Council of Canada. The authors are with the Department of Electrical and Computer Engineering, the University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2006.878745
Fig. 1. Control configurations considered in this note. (a) General two degrees-of-freedom feedback scheme. (b) Classical unity feedback one degreeof-freedom feedback scheme. (c) Open-loop scheme. (d) Observer-based statefeedback control scheme with integral action.
a general two degrees-of-freedom (2-DOF) control configuration, as shown in Fig. 1(a); note that classical 1-DOF control [see Fig. 1(b)], open-loop control [see Fig. 1(c)], and standard control schemes based on state-space methods [such as that shown in Fig. 1(d)] are special cases of Fig. 1(a). Denote the closed-loop transfer function from R(s) to Y (s) in Fig. 1(a) by H (s). Assuming closed-loop stability, the zeros of H (s) include all the NMP zeros of P (s). Assuming that P (s) is strictly proper, then H (s) is also strictly proper. Hence, throughout this note H (s) is assumed to be a stable strictly proper NMP transfer function with input R(s) and output Y (s). We are especially concerned about the unit step response of H (s), so R(s) = 1=s throughout. The following characteristics of the step response, y (t), will be used (see Fig. 2). • The steady-state value of y (t), denoted y , is y = y (t). The final value theorem immediately gives limt y = H (0). Throughout, we assume y > 0, although similar results can be derived if y < 0 by considering the signal 0y (t) instead of y (t). • The undershoot of the step response, denoted yus , is the smallest nonnegative number such that
1
1
!1
1
1
y ( t)
0
1
yus
8 0 t
:
In this note, we say that y (t) “exhibits undershoot” if yus > 0. (Readers should beware that alternate definitions have been used. For example, in [2] the term undershoot is used in the more limited situation where there exists a T > 0 such that y (t) < 0 for 0 < t < T ). • The overshoot of y (t), denoted yos , is the smallest nonnegative number such that y ( t)
( 1+ y
yos )
8 0 t
We say that y (t) “exhibits overshoot” if yos
0018-9286/$20.00 © 2006 IEEE
:
>
0.
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1379
Fig. 3. Factoring
H (s) as in (6).
step response of H (s) exhibits undershoot. To derive the undershoot bound, first use (2) and the fact that 0y (t) yus to obtain
1
Hs y y > y > y > yt y y >
Fig. 2. Representative step response of ( ) used to show the definitions of ), overshoot ( ), and settling time steady-state value ( ), undershoot ( 0, 0, and 0. If ( ) does not ( ). For this example, = 0; similarly, if ( ) does not exhibit overshoot, exhibit undershoot, then = 0. It is assumed that 0 throughout, although similar results then can be derived if 0.
y
y
t
y
y
0:
(1)
As ts ! 0, the bound (1) tends to 1. Proof: Since z is a zero of H (s), it is true that Y (z ) = H (z )R(z ) = H (z )=z = 0. Now, the Laplace transform of y (t) is
1 Y
(s ) =
0st dt
y ( t) e
z
1
0zt dt y ( t) e
1 Y
(z ) =
y ( t) e
0zt dt = 0:
with positive
(2)
0zt dt
):
(3)
0 98 1 for 2 [ s 1) to compute :
0:98y1 e
y
t ;
t
0zt dt =
0:98y1
t
z
e
0zt
:
(4)
Combine (3) and (4) to obtain (1). In the next two sections, we investigate step response overshoot.
II. CASE WHERE H (s) HAS AT LEAST TWO REAL NMP ZEROS Theorem 1 establishes that the presence of at least one real NMP zero in H (s) is enough to guarantee undershoot, and that the undershoot grows without bound as the settling time tends to zero. The following theorem, the main result of this note, shows that a similar result holds for step response overshoot, assuming that H (s) has at least two real NMP zeros. The intuition behind the proof is that one zero leads to undershoot while the other zero introduces negative derivation action [see (7)]; the effect of the negative derivative action is to increase y (t), and the increase gets more severe as the settling time decreases. Theorem 2: Let H (s) be a stable strictly proper transfer function with real NMP zeros at s = z1 and s = z2 and with any number of additional zeros located anywhere in the complex plane. Assume y1 > 0. Then yos
s
yus e
0
0zt (1 0 e
0
which (due to the stability of H (s)) converges for all real part, including the point s = z . Hence
t
0y(t)e0zt dt
yus
Second, use the fact that y (t) that
t
The following theorem is critical in the development that follows, and is important in its own right since it shows that real NMP zeros lead to undershoot. The proof is included for completeness. Theorem 1 [3]: Let H (s) be a stable strictly proper transfer function with a real NMP zero at s = z and with any number of additional zeros located anywhere in the complex plane. Assume y1 > 0. Then, the step response of H (s) exhibits undershoot that, as a fraction of the steady-state value, is bounded by
t 0
=
1
j
0zt dt = y ( t) e
y
1
>
e
0:98 z t 1
0
1
0
z1 ts
z 1 ts
01
:
(5)
For sufficiently small ts , the bound in (5) is positive, so the step response necessarily exhibits overshoot. As ts ! 0, the bound (5) tends to 1. Proof: Begin by factoring H (s) as
0
Since y1 > 0, (2) implies that time intervals where y (t) is positive must be balanced by time intervals where y (t) is negative. Thus, the
H (s )
=
G (s )
1
0
1 z1
s
(6)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
Fig. 4. Representative plot of f (t), including a zoom-in near t = t
showing the two possible cases discussed in the text.
where G(s) is stable and strictly proper with a NMP zero at s = z2 . Let f (t) denote the step response of G(s) and, as usual, let y (t) denote the step response of H (s) (see Fig. 3). By (6)
y ( t)
=
f ( t)
0
1
Inequalities (9) and (10) imply that jy (tsf ) 0 y1 j 0:02y1 ; in other words, signal y (t) must have a longer settling time than signal f (t):
ts d
z1 dt
f ( t)
(11)
tsf :
(7) From (11) and (8), the following bound on fus results:
holds for all t > 0. The signal f (t) tends to a positive number as ! 1, namely y1 , since H (0) = G(0). Fig. 4 shows a representative plot of f (t). Since G(s) is NMP, Theorem 1 implies that f (t) exhibits undershoot: more precisely, denoting the minimum value of f (t) by 0fus , Theorem 1 states
fus
t
fus
0:98y1
01
z t
e
(8)
y (tsf )
= f (tsf )
0
= 1:02y1
1
d
Next, choose
t
:
f ( t)
=
z1 dt
0
1
d
z1 dt
f (tsf )
1 02 :
(9)
In the second case, f (tsf ) = 0:98y1 and, therefore, (d=dt)f (tsf ) and
t2
dt
f (t
3
)=
f (t2 ) t2
max y (t) 2[0;1)
y (tsf )
= f (tsf )
0
= 0:98y1
z1 dt
0
1
d
z1 dt
f (tsf )
y (t
3
0
0 98 :
y1 :
(10)
0 0
fus
=
t1
t2
t1
:
(13)
)
0
0
fus
fus
= fus
f (tsf )
f (t 1 )
3 = f (t ) +
>
d
0 0
The machinery now exists to determine a bound on the peak value of y (t). In particular, use (7), (13), the fact that f (t3 ) 0fus , the fact that ts > t2 0 t1 , and (12) to obtain
0
1
(12)
:
fus
3 = f (t ) y1 :
01
z t
0 g and define
t
f (tsf )
0:98y1 e
to be any value in the (necessarily nonempty) set t1 := maxft : f (t) = 0; 0 t < t2 g, as indicated in Fig. 4. By the mean value theorem, there exists a t3 in the interval (t1 ; t2 ) such that
f
d
where tsf is the settling time of f (t) (again, see Fig. 4). It is desirable to express the bound (8) in terms of ts (the settling time of y (t)) instead of tsf . There are two cases to consider, as shown in Fig. 4. In the first case, f (tsf ) = 1:02y1 and, therefore, (d=dt)f (tsf ) 0. Equation (7) yields
z t
d
f (t
z1
+
0
)
t1
1
z1
t2
fus
1
z1
ts
z1 ts
01
t2
fus
1
3
1
fus
+
0:98y1 e
1
z1 dt
0
t1
01 1 z 1 ts
01
:
(14)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
Fig. 5. Lower bound on the step response overshoot for Example 1. The dashed line marks where the bound becomes positive.
Finally, use (14) and the relationship max t
2[0;1)
y (t)
=
y
os
+
y1
to
obtain (5). Note that the bound in Theorem 2 is equally valid with z1 and z2 exchanged, but neither of the two arrangements of z1 and z2 yields a consistently tighter bound. Also note that, although the bound may be conservative in particular examples, it does prove that the step response of a transfer function with at least two real nonminimum phase zeros exhibits overshoot if the settling time is short enough. Example 1: Consider the plant
P (s) = (s 0 1)(s 03 2) : (s + 1)
(15)
Assume that any of the control configurations shown in Fig. 1 is used, and, for simplicity, assume that perfect steady-state step tracking is required, i.e., y1 = 1. Using z1 = 1 and z2 = 2, the bound on overshoot in (5) was computed as a function of the settling time, and is plotted in Fig. 5 as a solid curve. Note that the bound is negative, and therefore provides no meaningful information, for ts > 0:4237 s. However, for ts < 0:4237 s, the bound is positive and, therefore, there must be overshoot for every possible stabilizing linear time-invariant control scheme. As expected, the overshoot tends to infinity as ts tends to zero. To demonstrate more explicitly the overshoot trend for this example, consider an observer-based state-feedback controller with integral action, as shown in Fig. 1(d). Let x_ = Ax + Bu, y = Cx be any minimal state–space realization of P (s). To describe the controller in Fig. 1(d), introduce
A~ :=
A C
0 0
~ := and B
B 0
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Fig. 6. Closed-loop step responses for Example 1 using three different values of . The inset plot illustrates that y = 1 for each of the curves. The example clearly demonstrates overshoot associated with multiple real NMP zeros.
where the two controller gains K := [K1 K2 ] and H , are chosen ~ ) and (A 0 HC ) are all in the such that the eigenvalues of (A~ 0 BK open left-half complex plane. (The closed-loop poles are, of course, ~ ) and those of (A 0 HC ).) the union of the eigenvalues of (A~ 0 BK ~ ) are The gain K was computed so that the eigenvalues of (A~ 0 BK placed at f0 ; 02 ; 03 ; 04 g (where > 0 is a parameter), and H was computed so that the eigenvalues of (A 0 HC ) are placed at f05 ; 06 ; 07 g. Fig. 6 shows the final closed-loop step responses for three values of . In all cases, zero initial conditions are assumed. Observe that, due to the integrator in the feedback loop, y1 = 1 holds. Also note the presence of overshoot in the plots; the overshoot is much larger than the bound (5), indicating that Theorem 1 can be conservative. Finally, it is interesting that the overshoot is significantly larger in magnitude than the undershoot, implying that overshoot is, at least for some examples, a greater cost associated with multiple real NMP zeros than is undershoot. III. CASE WHERE H (s) HAS ONE REAL NMP ZERO Theorem 2 establishes that step response overshoot results from the presence of two real NMP zeros and sufficiently small settling time. It is natural to ask if the same type of result holds if H (s) has only one NMP zero. In fact, it does not. Theorem 3: Let H (s) be a stable strictly proper transfer function with exactly one NMP zero, at s = z . Then there does not exist a nontrivial lower bound (possibly dependent on ts , y1 , and z ) on the step response overshoot of H (s). Proof: Consider the special case where all the poles of H (s) are real and distinct:
H (s ) = 0 K (s 0 z ) : (s + p )
:
n
i
i=1
The controller equations are then
In (16), K > 0, z > 0, and p > p 01 > 1 1 1 > p1 > 0 with n > 1. As usual, let y(t) denote the step response of H (s). Then, y(t) exhibits the following characteristics: 1) y (0) = 0 (since H (s) is strictly proper); 2) y1 > 0 (by direct calculation of H (0)); n
z_ = y 0 r x^_ = (A 0 BK1 0 HC )^x 0 BK2 z + Hy u = 0 K1 x^ 0 K2 z
(16)
p
n
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REFERENCES
Fig. 7. Closed-loop step responses for Example 2 using three different values of . The example clearly demonstrates that there need not be overshoot if only one NMP zero is present.
3) y (t) exhibits undershoot (using Theorem 1); 4) y (t) has exactly one extrema for t > 0 (using the results in [13]). From these four properties, we conclude that y (t) does not exhibit overshoot, no matter how small ts is. Consequently, there does not exist a nontrivial lower bound on the overshoot. Theorem 3 does not imply that there is never closed-loop step response overshoot for a plant that has a single NMP zero; indeed, it is not difficult to generate counterexamples where overshoot is exhibited because of the positions of other poles and zeros. However, the theorem does establish that a single NMP zero does not necessarily lead to step response overshoot, as demonstrated by the following example. Example 2: Consider the plant
P (s) = (s 0 1)3 (s + 1) with the same “observer-based state-feedback with integral action” controller constructed in Example 1. Note that P (s) is the same as in Example 1, except one of the NMP zeros has been removed. Fig. 7 shows the final closed-loop step responses for three values of . The step responses exhibit no overshoot. IV. CONCLUSION The main contribution of this note is a proof that multiple real NMP zeros lead to step response overshoot for sufficiently small settling time. Although motivated by the tracking problem, Theorems 1 and 2 also have immediate consequences if H (s) is interpreted as the transfer function from r to u in Fig. 1(a); in this case, unstable poles of P (s) show up as NMP zeros of H (s) and, therefore, Theorems 1 and 2 imply that the control signal has undershoot and (if ts is small enough) overshoot if P (s) has at least two unstable real poles. Like almost all results in the performance limitation literature, the focus of this work has been on real NMP zeros, and both insight and formulas for the complex NMP zero case are sparse.
[1] J. S. Freudenberg and D. P. Looze, “Right half-plane poles and zeros and design tradeoffs in feedback systems,” IEEE Trans. Autom. Control, vol. AC-30, no. 6, pp. 555–565, Jun. 1985. [2] M. Vidyasagar, “On undershoot and nonminimum phase zeros,” IEEE Trans. Autom. Control, vol. AC-31, no. 5, pp. 440–440, May 1986. [3] R. H. Middleton, “Trade-offs in linear control system design,” Automatica, vol. 27, no. 2, pp. 281–292, 1991. [4] L. Qiu and E. Davison, “Performance limitations of nonminimum phase systems in the servomechanism problem,” Automatica, vol. 29, no. 2, pp. 337–349, 1993. [5] J. Chen, L. Qiu, and O. Toker, “Limitations on maximal tracking accuracy,” IEEE Trans. Autom. Control, vol. 45, no. 2, pp. 326–331, Feb. 2000. [6] K. Lau, R. Middleton, and J. Braslavsky, “Undershoot and settling time tradeoffs for nonminimum phase systems,” IEEE Trans. Autom. Control, vol. 48, no. 8, pp. 1389–1393, Aug. 2003. [7] M. M. Seron, J. H. Braslavsky, and G. C. Goodwin, Fundamental Limitations in Filtering and Control. London, U.K.: Springer-Verlag, 1997, Communications and Control Engineering. [8] J. Freudenberg, R. Middleton, and A. Stefanopoulou, “A survey of inherent design limitations,” in Proc. Amer. Control Conf., Chicago, IL, Jun. 2000, pp. 2987–3001. [9] M. Bement and S. Jayasuriya, “Use of state feedback to achieve a nonovershooting step response for a class of nonminimum phase systems,” ASME J. Dyna. Syst., Measure., Control, vol. 126, pp. 657–660, Sep. 2004. [10] S. Darbha and S. Bhattacharyya, “On the synthesis of controllers for nonovershooting step response,” IEEE Trans. Autom. Control, vol. 48, no. 5, pp. 797–800, May 2003. [11] S. Jayasuriya and J.-W. Song, “On the synthesis of compensators for nonovershooting step response,” ASME J. Dyna. Syst., Measure., Control, vol. 118, pp. 757–763, Dec. 1996. [12] Y. Kim, L. Keel, and S. Bhattacharyya, “Transient response control via characteristic ratio assignment,” IEEE Trans. Autom. Control, vol. 48, no. 12, pp. 2238–2244, Dec. 2003. [13] M. El-Khoury, O. D. Crisalle, and R. Longchamp, “Influence of zero locations on the number of step-response extrema,” Automatica, vol. 29, no. 4, pp. 1571–1574, 1993.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
On Overshoot and Nonminimum Phase Zeros James Stewart and Daniel E. Davison
Abstract—It is widely known that real nonminimum phase zeros lead to step response undershoot, and that the size of the undershoot necessarily tends to infinity as the settling time tends to zero. In this note, we show that the presence of two or more real nonminimum phase zeros can lead to step response overshoot in addition to undershoot. A lower bound on the overshoot is derived, and it is shown that the overshoot, like the undershoot, necessarily tends to infinity as the settling time tends to zero. The results are derived for single-input–single-output linear time-invariant continuous-time systems, and apply to both open-loop control and general two degree-of-freedom closed-loop control. Index Terms—Nonminimum phase zeros, overshoot, transient response, undershoot.
I. INTRODUCTION This note focuses on the control of continuous-time plants that have real nonminimum phase (NMP) zeros, i.e., zeros on the positive real axis in the complex plane. It is well established that NMP zeros limit the achievable closed-loop performance; representative works that discuss such performance limitations include [1]–[6], and good overviews of these and related results include [7] and [8]. Perhaps the most familiar characteristic of real NMP zeros is that they lead to undershoot in the step response. The purpose of this note is to show that NMP zeros can also lead to overshoot in the step response. The main result is that if the plant has two or more real NMP zeros (and any number of additional zeros located anywhere in the complex plane), then there is overshoot if the speed of response is fast enough. The conclusions of this note complement, and in no way contradict, work on the synthesis of compensators that result in a nonovershooting closed-loop step response. For example, in [9], a control strategy is designed to guarantee a nonovershooting step response, but the results assume there is at most one NMP zero (and also use state feedback, whereas the results in the present note apply to output feedback); in [10], a nonovershooting control scheme is devised that applies even if the plant has multiple NMP zeros, but the scheme relies critically on making the closed-loop response sufficiently slow; and in [11] and [12], nonovershooting controllers are considered where there are no constraints on the speed of response, but the results apply only if the plant is minimum phase. The conclusions of this note also complement other results related to step response overshoot and pole-zero locations. For example, it is well known that a single minimum-phase zero can lead to step response overshoot (e.g., see [7]); of course, a key difference between minimum-phase zeros and NMP zeros is that it is possible to cancel out the former, but not the latter, and still maintain closed-loop stability. Similarly, a real unstable plant pole leads to closed-loop step response overshoot, assuming that a one degree-of-freedom control configuration is used (e.g., see [7]). Throughout this note, let P (s) denote the known plant which has at least one NMP zero, let r denote the reference signal, and let y denote the plant output. The controller structure under investigation is Manuscript received September 21, 2005; revised February 3, 2006. Recommended by Associate Editor F. Bullo. This work was supported by the Natural Sciences and Engineering Research Council of Canada. The authors are with the Department of Electrical and Computer Engineering, the University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2006.878745
Fig. 1. Control configurations considered in this note. (a) General two degrees-of-freedom feedback scheme. (b) Classical unity feedback one degreeof-freedom feedback scheme. (c) Open-loop scheme. (d) Observer-based statefeedback control scheme with integral action.
a general two degrees-of-freedom (2-DOF) control configuration, as shown in Fig. 1(a); note that classical 1-DOF control [see Fig. 1(b)], open-loop control [see Fig. 1(c)], and standard control schemes based on state-space methods [such as that shown in Fig. 1(d)] are special cases of Fig. 1(a). Denote the closed-loop transfer function from R(s) to Y (s) in Fig. 1(a) by H (s). Assuming closed-loop stability, the zeros of H (s) include all the NMP zeros of P (s). Assuming that P (s) is strictly proper, then H (s) is also strictly proper. Hence, throughout this note H (s) is assumed to be a stable strictly proper NMP transfer function with input R(s) and output Y (s). We are especially concerned about the unit step response of H (s), so R(s) = 1=s throughout. The following characteristics of the step response, y (t), will be used (see Fig. 2). • The steady-state value of y (t), denoted y , is y = y (t). The final value theorem immediately gives limt y = H (0). Throughout, we assume y > 0, although similar results can be derived if y < 0 by considering the signal 0y (t) instead of y (t). • The undershoot of the step response, denoted yus , is the smallest nonnegative number such that
1
1
!1
1
1
y ( t)
0
1
yus
8 0 t
:
In this note, we say that y (t) “exhibits undershoot” if yus > 0. (Readers should beware that alternate definitions have been used. For example, in [2] the term undershoot is used in the more limited situation where there exists a T > 0 such that y (t) < 0 for 0 < t < T ). • The overshoot of y (t), denoted yos , is the smallest nonnegative number such that y ( t)
( 1+ y
yos )
8 0 t
We say that y (t) “exhibits overshoot” if yos
0018-9286/$20.00 © 2006 IEEE
:
>
0.
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1379
Fig. 3. Factoring
H (s) as in (6).
step response of H (s) exhibits undershoot. To derive the undershoot bound, first use (2) and the fact that 0y (t) yus to obtain
1
Hs y y > y > y > yt y y >
Fig. 2. Representative step response of ( ) used to show the definitions of ), overshoot ( ), and settling time steady-state value ( ), undershoot ( 0, 0, and 0. If ( ) does not ( ). For this example, = 0; similarly, if ( ) does not exhibit overshoot, exhibit undershoot, then = 0. It is assumed that 0 throughout, although similar results then can be derived if 0.
y
y
t
y
y
0:
(1)
As ts ! 0, the bound (1) tends to 1. Proof: Since z is a zero of H (s), it is true that Y (z ) = H (z )R(z ) = H (z )=z = 0. Now, the Laplace transform of y (t) is
1 Y
(s ) =
0st dt
y ( t) e
z
1
0zt dt y ( t) e
1 Y
(z ) =
y ( t) e
0zt dt = 0:
with positive
(2)
0zt dt
):
(3)
0 98 1 for 2 [ s 1) to compute :
0:98y1 e
y
t ;
t
0zt dt =
0:98y1
t
z
e
0zt
:
(4)
Combine (3) and (4) to obtain (1). In the next two sections, we investigate step response overshoot.
II. CASE WHERE H (s) HAS AT LEAST TWO REAL NMP ZEROS Theorem 1 establishes that the presence of at least one real NMP zero in H (s) is enough to guarantee undershoot, and that the undershoot grows without bound as the settling time tends to zero. The following theorem, the main result of this note, shows that a similar result holds for step response overshoot, assuming that H (s) has at least two real NMP zeros. The intuition behind the proof is that one zero leads to undershoot while the other zero introduces negative derivation action [see (7)]; the effect of the negative derivative action is to increase y (t), and the increase gets more severe as the settling time decreases. Theorem 2: Let H (s) be a stable strictly proper transfer function with real NMP zeros at s = z1 and s = z2 and with any number of additional zeros located anywhere in the complex plane. Assume y1 > 0. Then yos
s
yus e
0
0zt (1 0 e
0
which (due to the stability of H (s)) converges for all real part, including the point s = z . Hence
t
0y(t)e0zt dt
yus
Second, use the fact that y (t) that
t
The following theorem is critical in the development that follows, and is important in its own right since it shows that real NMP zeros lead to undershoot. The proof is included for completeness. Theorem 1 [3]: Let H (s) be a stable strictly proper transfer function with a real NMP zero at s = z and with any number of additional zeros located anywhere in the complex plane. Assume y1 > 0. Then, the step response of H (s) exhibits undershoot that, as a fraction of the steady-state value, is bounded by
t 0
=
1
j
0zt dt = y ( t) e
y
1
>
e
0:98 z t 1
0
1
0
z1 ts
z 1 ts
01
:
(5)
For sufficiently small ts , the bound in (5) is positive, so the step response necessarily exhibits overshoot. As ts ! 0, the bound (5) tends to 1. Proof: Begin by factoring H (s) as
0
Since y1 > 0, (2) implies that time intervals where y (t) is positive must be balanced by time intervals where y (t) is negative. Thus, the
H (s )
=
G (s )
1
0
1 z1
s
(6)
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Fig. 4. Representative plot of f (t), including a zoom-in near t = t
showing the two possible cases discussed in the text.
where G(s) is stable and strictly proper with a NMP zero at s = z2 . Let f (t) denote the step response of G(s) and, as usual, let y (t) denote the step response of H (s) (see Fig. 3). By (6)
y ( t)
=
f ( t)
0
1
Inequalities (9) and (10) imply that jy (tsf ) 0 y1 j 0:02y1 ; in other words, signal y (t) must have a longer settling time than signal f (t):
ts d
z1 dt
f ( t)
(11)
tsf :
(7) From (11) and (8), the following bound on fus results:
holds for all t > 0. The signal f (t) tends to a positive number as ! 1, namely y1 , since H (0) = G(0). Fig. 4 shows a representative plot of f (t). Since G(s) is NMP, Theorem 1 implies that f (t) exhibits undershoot: more precisely, denoting the minimum value of f (t) by 0fus , Theorem 1 states
fus
t
fus
0:98y1
01
z t
e
(8)
y (tsf )
= f (tsf )
0
= 1:02y1
1
d
Next, choose
t
:
f ( t)
=
z1 dt
0
1
d
z1 dt
f (tsf )
1 02 :
(9)
In the second case, f (tsf ) = 0:98y1 and, therefore, (d=dt)f (tsf ) and
t2
dt
f (t
3
)=
f (t2 ) t2
max y (t) 2[0;1)
y (tsf )
= f (tsf )
0
= 0:98y1
z1 dt
0
1
d
z1 dt
f (tsf )
y (t
3
0
0 98 :
y1 :
(10)
0 0
fus
=
t1
t2
t1
:
(13)
)
0
0
fus
fus
= fus
f (tsf )
f (t 1 )
3 = f (t ) +
>
d
0 0
The machinery now exists to determine a bound on the peak value of y (t). In particular, use (7), (13), the fact that f (t3 ) 0fus , the fact that ts > t2 0 t1 , and (12) to obtain
0
1
(12)
:
fus
3 = f (t ) y1 :
01
z t
0 g and define
t
f (tsf )
0:98y1 e
to be any value in the (necessarily nonempty) set t1 := maxft : f (t) = 0; 0 t < t2 g, as indicated in Fig. 4. By the mean value theorem, there exists a t3 in the interval (t1 ; t2 ) such that
f
d
where tsf is the settling time of f (t) (again, see Fig. 4). It is desirable to express the bound (8) in terms of ts (the settling time of y (t)) instead of tsf . There are two cases to consider, as shown in Fig. 4. In the first case, f (tsf ) = 1:02y1 and, therefore, (d=dt)f (tsf ) 0. Equation (7) yields
z t
d
f (t
z1
+
0
)
t1
1
z1
t2
fus
1
z1
ts
z1 ts
01
t2
fus
1
3
1
fus
+
0:98y1 e
1
z1 dt
0
t1
01 1 z 1 ts
01
:
(14)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
Fig. 5. Lower bound on the step response overshoot for Example 1. The dashed line marks where the bound becomes positive.
Finally, use (14) and the relationship max t
2[0;1)
y (t)
=
y
os
+
y1
to
obtain (5). Note that the bound in Theorem 2 is equally valid with z1 and z2 exchanged, but neither of the two arrangements of z1 and z2 yields a consistently tighter bound. Also note that, although the bound may be conservative in particular examples, it does prove that the step response of a transfer function with at least two real nonminimum phase zeros exhibits overshoot if the settling time is short enough. Example 1: Consider the plant
P (s) = (s 0 1)(s 03 2) : (s + 1)
(15)
Assume that any of the control configurations shown in Fig. 1 is used, and, for simplicity, assume that perfect steady-state step tracking is required, i.e., y1 = 1. Using z1 = 1 and z2 = 2, the bound on overshoot in (5) was computed as a function of the settling time, and is plotted in Fig. 5 as a solid curve. Note that the bound is negative, and therefore provides no meaningful information, for ts > 0:4237 s. However, for ts < 0:4237 s, the bound is positive and, therefore, there must be overshoot for every possible stabilizing linear time-invariant control scheme. As expected, the overshoot tends to infinity as ts tends to zero. To demonstrate more explicitly the overshoot trend for this example, consider an observer-based state-feedback controller with integral action, as shown in Fig. 1(d). Let x_ = Ax + Bu, y = Cx be any minimal state–space realization of P (s). To describe the controller in Fig. 1(d), introduce
A~ :=
A C
0 0
~ := and B
B 0
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Fig. 6. Closed-loop step responses for Example 1 using three different values of . The inset plot illustrates that y = 1 for each of the curves. The example clearly demonstrates overshoot associated with multiple real NMP zeros.
where the two controller gains K := [K1 K2 ] and H , are chosen ~ ) and (A 0 HC ) are all in the such that the eigenvalues of (A~ 0 BK open left-half complex plane. (The closed-loop poles are, of course, ~ ) and those of (A 0 HC ).) the union of the eigenvalues of (A~ 0 BK ~ ) are The gain K was computed so that the eigenvalues of (A~ 0 BK placed at f0 ; 02 ; 03 ; 04 g (where > 0 is a parameter), and H was computed so that the eigenvalues of (A 0 HC ) are placed at f05 ; 06 ; 07 g. Fig. 6 shows the final closed-loop step responses for three values of . In all cases, zero initial conditions are assumed. Observe that, due to the integrator in the feedback loop, y1 = 1 holds. Also note the presence of overshoot in the plots; the overshoot is much larger than the bound (5), indicating that Theorem 1 can be conservative. Finally, it is interesting that the overshoot is significantly larger in magnitude than the undershoot, implying that overshoot is, at least for some examples, a greater cost associated with multiple real NMP zeros than is undershoot. III. CASE WHERE H (s) HAS ONE REAL NMP ZERO Theorem 2 establishes that step response overshoot results from the presence of two real NMP zeros and sufficiently small settling time. It is natural to ask if the same type of result holds if H (s) has only one NMP zero. In fact, it does not. Theorem 3: Let H (s) be a stable strictly proper transfer function with exactly one NMP zero, at s = z . Then there does not exist a nontrivial lower bound (possibly dependent on ts , y1 , and z ) on the step response overshoot of H (s). Proof: Consider the special case where all the poles of H (s) are real and distinct:
H (s ) = 0 K (s 0 z ) : (s + p )
:
n
i
i=1
The controller equations are then
In (16), K > 0, z > 0, and p > p 01 > 1 1 1 > p1 > 0 with n > 1. As usual, let y(t) denote the step response of H (s). Then, y(t) exhibits the following characteristics: 1) y (0) = 0 (since H (s) is strictly proper); 2) y1 > 0 (by direct calculation of H (0)); n
z_ = y 0 r x^_ = (A 0 BK1 0 HC )^x 0 BK2 z + Hy u = 0 K1 x^ 0 K2 z
(16)
p
n
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REFERENCES
Fig. 7. Closed-loop step responses for Example 2 using three different values of . The example clearly demonstrates that there need not be overshoot if only one NMP zero is present.
3) y (t) exhibits undershoot (using Theorem 1); 4) y (t) has exactly one extrema for t > 0 (using the results in [13]). From these four properties, we conclude that y (t) does not exhibit overshoot, no matter how small ts is. Consequently, there does not exist a nontrivial lower bound on the overshoot. Theorem 3 does not imply that there is never closed-loop step response overshoot for a plant that has a single NMP zero; indeed, it is not difficult to generate counterexamples where overshoot is exhibited because of the positions of other poles and zeros. However, the theorem does establish that a single NMP zero does not necessarily lead to step response overshoot, as demonstrated by the following example. Example 2: Consider the plant
P (s) = (s 0 1)3 (s + 1) with the same “observer-based state-feedback with integral action” controller constructed in Example 1. Note that P (s) is the same as in Example 1, except one of the NMP zeros has been removed. Fig. 7 shows the final closed-loop step responses for three values of . The step responses exhibit no overshoot. IV. CONCLUSION The main contribution of this note is a proof that multiple real NMP zeros lead to step response overshoot for sufficiently small settling time. Although motivated by the tracking problem, Theorems 1 and 2 also have immediate consequences if H (s) is interpreted as the transfer function from r to u in Fig. 1(a); in this case, unstable poles of P (s) show up as NMP zeros of H (s) and, therefore, Theorems 1 and 2 imply that the control signal has undershoot and (if ts is small enough) overshoot if P (s) has at least two unstable real poles. Like almost all results in the performance limitation literature, the focus of this work has been on real NMP zeros, and both insight and formulas for the complex NMP zero case are sparse.
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