- Email: [email protected]
On undershoot in scalar discrete-time systems
Pergamon
Automama. Vol. 32. No. 2, pp. 255-259, 19% Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved ow5-109X/% $15.W + o.Kl
ooo5-1098(95)ool24-7
Brief Paper
On Undershoot BERNARD0
LE6N
in Scalar Discrete-time
DE LA BARRA,?
Key Words-Discrete-time
MARIO
EL-KHOURYS
systems; non-minimum-phase
Systems*
and MARIO
FERNANDEZt
zeros; step response; undershoot.
Assumption
2. G(z) satisfies Assumption non-negative real poles.
Abstrad-We
identify a class of asymptotically stable, strictly proper, non-minimum-phase, scalar discrete time systems whose step response is shown to exhibit undershoot a given number of times. The conditions governing this correspondence can be verified by inspection and have a simple expression in terms of the zero/pole locations of the underlying system. Both qualitative and quantitative aspects of the undershooting phenomenon are discussed, and a simple but non-trivial example is used to gain further insight into the main result of this paper.
1 and has only
Under Assumption 2, it is possible to anticipate that undershooting effects will be due to the number real/complex nature of the non-minimum-phase zeros, not to any oscillating effects associated with the poles. following definition characterizes a multiple undershoot response.
the and and The step
Definition 1. Let G(z) be as in Assumption
1, and let y, be the corresponding time-domain response to a positive unit step. Then it is said that yk displays type-r, undershoot, r, P 1, if there exists exactly r, different values of k. say O
1. Introduction
It is known that continuous-time systems with an odd number of real open right-half-plane zeros display initial inverse response to a step input (see e.g. Hara et al., 1986; Vidyasagar, 1986). A similar situation follows for the discrete-time case if the system has an odd number of positive real non-minimum-phase zeros (see e.g. Deodhare and Vidyasagar, 1992). In this paper we contribute towards the time-domain characterization of non-minimum-phase responses by identifying a class of discrete-time systems whose step response is shown to display undershoot a given number of times. The constitute the discrete-time results to be presented counterpart of those reported in Leon de la Barra (1994a), and cannot be derived via sampling arguments. The paper is organized as follows. Section 2 contains definitions and background material. In Section 3 we present the main result, which qualitatively connects real nonminimum-phase discrete-time zeros and undershoot in the This result constitutes an important step response. contribution towards the characterization of the qualitative behaviour of the time-domain response in terms of zero/pole locations. Quantitative features of the undershooting phenomenon are also presented in the latter section. Conclusions are given in Section 4. This paper complements previous work by the authors; see e.g. El-Khoury (1991), El-Khoury et al. (1993a, b) and Leon de la Barra (1992, 1994a, b) for a more elaborate picture of a number of related issues.
i=l,...,
Yk,KCO, (yk,-Yk,+,)~
r,.
i=l.....
(1) r,,
(2)
i=l....,r,,
(3)
and either (Y,,-Y,,-i)(yk,-Yk,+,)>O,
or 3i, 15 i 5 r,, 3j(i), j(i) 2 1 such that 1,
I=
Yk,--I = Yk,
, j(i),
(4) (5)
(yk,-yk,-,(,)~I)(yk,-Yk,+I)>O,
Note that the ‘logic’ used in Definition 1 is (l)-(2) and {(3) or (4)-(5)}. Figures 1 and 2 illustrate Definition 1 for the case K >O. It could also be of interest to know whether or not y, starts off in the opposite direction from the steady-state value K. With this in mind, the notion of initial undershoot is also introduced. Definition 2. If yk exhibits type-r, undershoot, said to display initial undershoot if Vk,
y,~sO
r, 2 1, yk is
O
(6)
3. Non-minimum-phase zeros and undershoot in the step response 3.1. Qualitative aspects of the undershooting phenomenon.
2. Preliminaries
To gain insight into the effect of non-minimum-phase zeros upon the step response, we now assume that G(z) satisfies Assumption 1, and that is given explicitly by
The following two assumptions will assist us to precisely state a number of quoted and original results.
G(z) ~
Assumption
1. G(z) is a strictly proper, asymptotically stable, non-minimum-phase, scalar discrete-time system with non-zero DC gain K.
K
mt
(z - zs)
ry(z_p,)’
G(l)=Kfo,
(7)
where the parameter K satisfies
(1 -p,)
KirI;:l
*Received 1 October 1993; revised 25 March 1994; received in final form 17 June 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. KuZera under the direction of Editor Ruth F. Curtain. Corresponding author Dr Bernard0 Leon de la Barra. Tel. +56 2 6784215/6727832; Fax +56 2 6727832/6953881; E-mail [email protected]. t Department of Electrical Engineering, University of Chile, PO Box 412-3, Santiago, Chile. $ CSEM, Maladihre 71, 2007 Neuchltel, Switzerland.
n::,
(1 -z,)
K.
Using basic zeta-transform properties in (7), it can be shown that the positive unit-step response y, associated with G(z) satisfies 0 yk =
iK
(Osk%n,(k =n,),
1).
(9)
where n,-n,-n, is the relative degree of G(z). It is important to highlight that the parameter K in (8) and (9) is 255
256
Brief Papers Yk+
Yk
n”
K
k; - 1
I
ki
I
4
.
- 1
k, -j(i)
ki + I
‘..
I
ki - 1
ki
ki + 1
I
I
1
I k; -j(i)
k
.
.
k
.
.
‘.
.
j(i)
Fig. 1. Illustration of the conditions (l)-(3)
for K > 0.
directly proportional to the distance from the poles to z = 1, and inversely proportional to the distance from the zeros to z = 1.t From the same equations, we then see that the first non-zero value of the step response can be extremely large, relative to the steady-state value K, if G(z) has (non-minimum-phase) zeros significantly ‘closer’ to z = 1 than the poles. It is also clear from (8) that K will have opposite sign to the steady-state value K if and only if G(z) has an odd number of positive real non-minimum-phase discrete-time zeros. This ‘last observation has previously appeared in El-Khourv (1991). Deodhare and Vidvasagar (1992), and Leon de Ia Barra’(1992) and leads naturally?o the foliowing fact. Fact 1. yk displays initial undershoot if and only if G(z) has an odd number of positive real non-minimum-phase discrete-time zeros. Remark 1. Note that Fact 1 remains valid when G(z) has complex-conjugate and/or negative real poles. It is also important to highlight that complex-conjugate nonminimum-phase discrete-time zeros do not have any influence on determining the initial undershoot. Also keep in mind that the previous statements in this subsection remain valid when multiple zeros and poles are allowed in G(z).
Note also that initial undershoot has been called type-A and that undershoot without initial undershoot has been called type-B undershoor (see e.g. Deodhare and Vidyasagar, 1990, 1992). To the best of the authors’ knowledge, Fact 1 constitutes the only available result that relates the qualitative nature of the undershoot in the step response to non-minimum-phase discrete-time zeros. To prepare for the main result of this subsection, we introduce the following definitions.
undershoot,
Definition 3. If G(z) satisfies Assumption 2 and has nP - r poles at the origin and r distinct non-zero poles, we define a pole bracket as the open interval (p,,pj+,) between two different and consecutive poles 0
The following theorem provides lower and upper bounds on the number of changes of sign (n) in the impulse response t Note that this point has been previously called the ‘edge of the unit circle’ (see e.g. Jury, 1955). f For a system having only stable non-negative real poles, the fastest pole is the one closest to the origin, and the slowest pole is the one closest to z = 1.
.
times
Fig. 2. Illustration of the conditions (1) (2) and (4), (5) for K > 0. (gk) of G(z). It can also be shown that n coincides with the number of extrema in the corresponding step response. Theorem 1. If G(z) satisfies Assumption 2 and has only real zeros then n is bounded as m,+m,5~5ml+mz+m3-p. Proof
See El-Khoury
(10)
(1991) or El-Khoury et al. (1993a).
Note that the same authors have shown that, for plants satisfying the conditions of the theorem, all those zeros located to the left of the fastest pole do not have any influence upon n. This is why these zeros have not been taken into account. Thus we have the following corollary. Corollary 1. If G(Z) satisfies Assumption 2, has only real zeros, and meets the so-called ‘bracketing’ condition p = m3, together with m2 5 1, then y,, will display exactly type-r, undershoot, with r, + int [$(m, + l)]. Proof:
See the Appendix.
Note that Corollary 1 was previously shown by El-Khoury (1991) for the case m, = 0. To get a better feel for Fact 1, Theorem 1 and Corollary 1, we present the following example. Example
1. Consider
the eight-order
discrete-time
system
given by &(z ~ 0.45)(z - 055)(Z - 0.9) x (z. - 2.5)(z - 3.3)(z - 5)(z - 10) G(z) + (Z ~ O.l)(Z - 0.2)(z - 0.3)(z - 0.4) x (z - 0.5)(z - 0.6)(z - 0.7)(z - 0.8) It can easily be seen
that
the
above
01)
system satisfies 1 determines that n = 5, i.e. y,, the step response of the system (ll), will display five extrema. Correspondingly, Corollary 1 predicts that r, = 2, i.e. that y, will display type-2 undershoot. Similarly, given that there is an even number of positive real non-minimum-phase zeros, Fact 1 predicts that yk will not exhibit initial undershoot. In other words, the step response will start in the right direction, changing direction and reversing its sign three times, before finally moving toward its steady-state value. In Figs 3 and 4 we have displayed the step response of the system (ll), verifying what we have anticipated. Note from Fig. 3 that yk also exhibits overshoot. This is due to the existence of a real open-unit-disk zero to the right of the slowest pole; see Leon de la Barra (1992), Leon de la Barra and Goodwin (1991) and McWilliams (1993) for a thorough discussion of a number of related issues. 3.2. Quantitative aspects of the undershooting phenomenon. Fact 1, Theorem 1 and Corollary 1 give important insight into the ‘shape’ of undershoot, but fall short of quantitative information relating to magnitudes. The Iatter feature is clearly of primary importance to a designer of
p = m3 = 2, m, = 1 and m, = 4. Theorem
257
Brief Papers
0
5
10
15
20
k
25
30
35
40
45
Fig. 3. Step response of the system (11).
feedback loops (see e.g. Franklin, 1985; Zatiriou and Morari, 1990; 1985; Moore, 1989; Moore and Bhattacharyya, Middleton, 1991). In this regard, the following rule of thumb constitutes a handy complement to the above results. Rule of Thumb 1. Let G(z) be as in Assumption 1, with unit DC gain and only one positive real non-minimum-phase discrete-time zero, located at z = b, b > 1. Let kp be the rise time of yk up to level Q, O< LY< 1,t as defined by k: + f,“f,(I/y, 2 LY Vk = I},
(12)
with a zero at z = b, b > 1, under a general two-degree-offreedom configuration, and for any nonlinear or time-varying controller. We also refer the reader to Boyd and Barratt (1991) for additional motivation on achievable design specifications. In Fig. 5 we have pictured (14) for a few locations of the positive real non-minimum-phase zero. It is evident from this figure that the closer the zero is to z = 1, the smaller is the feasibility set. Specifically, if we want to keep the_ left-hand side of the inequality (14) say below about 100 V (%), we need to have
>lW(l
K”,
and let the peak undershoot be given by v~s~p~,~~{-y~}. Under the above conditions, v is lower-bounded as (13) Derivation.
See the Appendix.
Remark 2. It is known that in a feedback context (see e.g. Middleton, 1991: Deodhare and Vidyasagar, 1992), and if the open-loop plant has a positive real non-minimum-phase discrete-time zero, the step response from the reference input to the system output must exhibit some undershoot. In fact, under the requirement of internal stability, it can easily be seen by using Rule of Thumb 1 that the closed-loop will be design specifications k: 5 Kg,, and v s V,,, achievable if and only if
mc,X
+ a/P) log b
(15)
It is clear from this inequality that to keep the undershoot small, a large rise time is required relative to the time constant$ of the positive real non-minimum-phase zero. Only one positive real non-minimum-phase zero was assumed in Rule of Thumb 1. This does not constitute an important loss of generality, in the sense that if there were several positive real non-minimum-phase zeros then the one(s) closest to z = 1 would impose the most critical bound on the rise time. It is also clear that the right-hand side of (15) is ‘proportional’ to the rise-time threshold LY.
Fig. 4. Initial behaviour of the step response of the system (II).
4. Conclusions We have derived, for a class of discrete-time systems having only real zeros and non-negative real poles, the exact number of times that the associated step response displays undershoot. Note from (10) and the proof of Corollary 1 that the whole point of the ‘bracketing’ condition p = m3, together with mz 5 1, is to make sure that both bounds on the number of changes of sign of yk coincide. For zero/pole patterns not satisfying both conditions, Theorem 1 does not enable us to exact/y determine the number of undershoots in the associated step response. Nevertheless. this result constitutes an important step towards the complete characterization of the effects of non-minimum-phase discrete-time zeros upon step response. In a feedback context, we quantitatively described the trade-off existing between achievable closed-loop peak undershoot and rise time of the corresponding reference step response. It was clearly shown how the size of the feasibility set becomes smaller when the non-minimum-phase zeros approach z = 1. The overall characterization of the effect of non-minimumphase discrete-time zeros, both real and complex, upon the step response of systems having complex-conjugate poles remains an open problem. On the other hand, it has been
t The usual choices for a are in the upper fifth of this range.
r 2 0, is given by ilog rl-‘.
(14)
It is important to stress that the trade-off embodied in this inequality holds for any LTI discrete-time open-loop plant
,._,,.,............ 4
yk
.o..-
$ The time constant of a discrete-time
zero (pole) r * re”,
258
Brief Papers 10
I
I
6 ‘b.
Y 4
n:., ‘..
1
‘,. 0,..,‘..,
b = I.05 .Q..b = I.10 +.b = I.20 .*---
4
7
10
Fig. 5. Trade-off between achievable peak undershoot and rise-time specifications for a = 0.8.
shown that for discrete-time plants having only non-negative real poles, negative real non-minimum-phase zeros do not determine extrema in the corresponding step response; see El-Khoury (1991) and Ledn de la Barra (1994b) for alternative proofs of this statement. It has also been proved that each pair of complex-conjugate non-minimum-phase zeros does not determine any extrema if there is a pair of non-negative real poles whose average is larger or equal than their real part: see Ledn de la Barra (1994b) for the proof of this latter statement. Recent related work has appeared in McWilliams and Sain (1994), and Holmberg et al. (1995).
Acknowledgemen&-The
authors are grateful to the anonymous referees for a number of suggestions that helped to improve the paper. The first author also acknowledges the support of CONICYT (Comisi6n National de Investigaci6n Cientffica y Tecnohjgica, Chile) under Grant 1950745.
References
Boyd, S. P. and C. H. Barratt Design. Limits of Performance.
(1991). Linear Controller Prentice-Hall, Englewood
Cliffs, NJ. Deodhare, G. and M. Vidyasagar (1990). Design of non-overshooting feedback control systems. In Proc. 29th IEEE Conf. on Decision and Conrrol, Honolulu, HI, Vol. 3, pp. 1827-1834. Deodhare, G. and Vidyasagar, M. (1992). Control system design via infinite linear programming. Int. J. Conrrol, 55, 1351-1380. El-Khoury, M. (1991). Influence des zCros d’une fonction de transfert sur le comportement dynamique d’un systkme lineaire et application au, rCglage polynomial. ScD thesis, Institut d’Automatique, Ecole Polytechnique FCderale de Lausanne. El-Khoury, M., 0. D. Crisalle and R. Longchamp (1993a). Discrete transfer-function zeros and step response extrema. In Preprinrs 12th IFAC World Congress, Sydney, Vol. 3, pp. 81-86. El-Khoury, M., 0. D. Crisalle and R. Longchamp (1993b). Influence of zero locations on the number of step-response extrema. Automatica, 29, 1571-1574. Franklin, G. F. (1985). Direct digital control design via pole placement techniques. In S. G. Tzafestas (Ed.), Applied Digital Control, pp. 111-148. North-Holland, Amsterdam. Hara. S.. N. Kobavashi and T. Nakamizo (1986). Design of nor&dershootihg multivariable servd systems ising dynamic compensators. Inr. .I. Control, 44,331-342.
Holmberg, U., P. Myszkorowski, Y. Piguet and R. Longchamp (1995). On compensation of nominimumphase zeros. Auromarica, 31, 1433-1441. Jury, E. I. (1955). The effect of pole and zero locations on the transient response of sampled-data systems. Trans. AIEE
Pt II: ADolic. Ind.. 74, 41-48.
Le6n de la Barr&-B. A. (1992). Zeros and their influence on time and frequency domain properties of scalar feedback systems. PhD thesis, University of Newcastle, New South Wales: also available from UMI Dissertation Services, Ann Arbor, MI. Le6n de la Barra, B. A. (1994a). On undershoot in SISO systems. IEEE Trans. Autom. Control, AC-%, 578-581. Le6n de la Barra, B. A. (1994b). Sufficient conditions for monotonic discrete time step responses. ASME .I. Dyn. Syst. Meus. Conrrol, 116, 8101814.-
Le6n de la Barra. B. A. and G. C. Goodwin (19911. ~ , Ooen . unit disk zeros of scalar discrete time transfer functions: implications for feedback design. In Proc. 30th IEEE Conf on Decision and Conrrol, Brighton, U.K., Vol. 3, p. 2672-2673. McWilliams, L. H. (1993). Qualirariue features of linear time-invariant system transienr responses: PhD dissertation, Universitv of Notre Dame. IN: also available from UMI Dissertatcon Services, Ann Arbor, MI. McWilliams, L. H. and M. K. Sain (1994). Qualitative features of discrete-time system responses. In Proc. 33rd IEEE Conf: on Decision and Control, Lake Buena Vista, FL, Vol. 1, pp. 18-23. Middleton, R. H. (1991). Trade-offs in linear control system design. Automatica, 27, 281-292. Moore, K. L. (1989). Design techniques for transient PhD dissertation, Texas A&M response control. University. Moore, K. L. and S. P. Bhattacharyya (1990). A technique for choosing zero locations for minimal overshoot. IEEE Trans. Autom. Conrrol, AC-35,577-580.
Vidyasagar, M. (1986). On undershoot and nonminimum phase zeros. IEEE Trans. Aurom. Control, AC-31,440. Zafiriou, E. and M. Morari (1985). Digital controllers for SISO systems: a review and a new algorithm. Int. J. Control, 42,855-876. Appendix Proof of Corollary 1. Let us define C?(z) A zG(z)/(z - 1). It can then be seen that m2 5 1 guarantees that fi3 = m3 + m, together with p = m3 + m2, i.e. G(z) satisfies the corresponding ‘bracketing’ condition p = fii,. Thus, noting that 6, = 0, we have from (10) that +j = m,, i.e. y* has exactly m,
Brief Papers changes of sign, and the result follows. Note also that the application of Theorem 1 to G(z) leads to n = m, + mz, i.e., g, has exactly m, + mz changes of sign. This determines that, for m2 = 1, y,, also exhibits overshoot (see Example 1). Derivation of Rule of Thumb 1. It is clear that the zeta-transform summation representation of zG(z)/(z - 1) has /zl> 1 as its domain of convergence, and is given explicitly by
zG(z) -=2 z-1
ykz
-k.
259
at z = h, we have X 0= 2 y,b-k= k=O
kp-1
G4.2)
and, after using the definition of peak undershoot, we obtain
2
Y,J-k
k=LP
= *~‘b-*(_y,)~v*E’b--*, (A.3) k=O
Now it follows from this inequality that
(A.1)
k=”
The zero at z = b, b > 1, is obviously in the domain of convergence of (A.l). Consequently, after evaluating (A.l)
W.
2 y,b-& + c y,&*, k=” k=/$
after using the fact that y* 2 a Vk 2 kp.
k=O
Automama. Vol. 32. No. 2, pp. 255-259, 19% Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved ow5-109X/% $15.W + o.Kl
ooo5-1098(95)ool24-7
Brief Paper
On Undershoot BERNARD0
LE6N
in Scalar Discrete-time
DE LA BARRA,?
Key Words-Discrete-time
MARIO
EL-KHOURYS
systems; non-minimum-phase
Systems*
and MARIO
FERNANDEZt
zeros; step response; undershoot.
Assumption
2. G(z) satisfies Assumption non-negative real poles.
Abstrad-We
identify a class of asymptotically stable, strictly proper, non-minimum-phase, scalar discrete time systems whose step response is shown to exhibit undershoot a given number of times. The conditions governing this correspondence can be verified by inspection and have a simple expression in terms of the zero/pole locations of the underlying system. Both qualitative and quantitative aspects of the undershooting phenomenon are discussed, and a simple but non-trivial example is used to gain further insight into the main result of this paper.
1 and has only
Under Assumption 2, it is possible to anticipate that undershooting effects will be due to the number real/complex nature of the non-minimum-phase zeros, not to any oscillating effects associated with the poles. following definition characterizes a multiple undershoot response.
the and and The step
Definition 1. Let G(z) be as in Assumption
1, and let y, be the corresponding time-domain response to a positive unit step. Then it is said that yk displays type-r, undershoot, r, P 1, if there exists exactly r, different values of k. say O
1. Introduction
It is known that continuous-time systems with an odd number of real open right-half-plane zeros display initial inverse response to a step input (see e.g. Hara et al., 1986; Vidyasagar, 1986). A similar situation follows for the discrete-time case if the system has an odd number of positive real non-minimum-phase zeros (see e.g. Deodhare and Vidyasagar, 1992). In this paper we contribute towards the time-domain characterization of non-minimum-phase responses by identifying a class of discrete-time systems whose step response is shown to display undershoot a given number of times. The constitute the discrete-time results to be presented counterpart of those reported in Leon de la Barra (1994a), and cannot be derived via sampling arguments. The paper is organized as follows. Section 2 contains definitions and background material. In Section 3 we present the main result, which qualitatively connects real nonminimum-phase discrete-time zeros and undershoot in the This result constitutes an important step response. contribution towards the characterization of the qualitative behaviour of the time-domain response in terms of zero/pole locations. Quantitative features of the undershooting phenomenon are also presented in the latter section. Conclusions are given in Section 4. This paper complements previous work by the authors; see e.g. El-Khoury (1991), El-Khoury et al. (1993a, b) and Leon de la Barra (1992, 1994a, b) for a more elaborate picture of a number of related issues.
i=l,...,
Yk,KCO, (yk,-Yk,+,)~
r,.
i=l.....
(1) r,,
(2)
i=l....,r,,
(3)
and either (Y,,-Y,,-i)(yk,-Yk,+,)>O,
or 3i, 15 i 5 r,, 3j(i), j(i) 2 1 such that 1,
I=
Yk,--I = Yk,
, j(i),
(4) (5)
(yk,-yk,-,(,)~I)(yk,-Yk,+I)>O,
Note that the ‘logic’ used in Definition 1 is (l)-(2) and {(3) or (4)-(5)}. Figures 1 and 2 illustrate Definition 1 for the case K >O. It could also be of interest to know whether or not y, starts off in the opposite direction from the steady-state value K. With this in mind, the notion of initial undershoot is also introduced. Definition 2. If yk exhibits type-r, undershoot, said to display initial undershoot if Vk,
y,~sO
r, 2 1, yk is
O
(6)
3. Non-minimum-phase zeros and undershoot in the step response 3.1. Qualitative aspects of the undershooting phenomenon.
2. Preliminaries
To gain insight into the effect of non-minimum-phase zeros upon the step response, we now assume that G(z) satisfies Assumption 1, and that is given explicitly by
The following two assumptions will assist us to precisely state a number of quoted and original results.
G(z) ~
Assumption
1. G(z) is a strictly proper, asymptotically stable, non-minimum-phase, scalar discrete-time system with non-zero DC gain K.
K
mt
(z - zs)
ry(z_p,)’
G(l)=Kfo,
(7)
where the parameter K satisfies
(1 -p,)
KirI;:l
*Received 1 October 1993; revised 25 March 1994; received in final form 17 June 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. KuZera under the direction of Editor Ruth F. Curtain. Corresponding author Dr Bernard0 Leon de la Barra. Tel. +56 2 6784215/6727832; Fax +56 2 6727832/6953881; E-mail [email protected]. t Department of Electrical Engineering, University of Chile, PO Box 412-3, Santiago, Chile. $ CSEM, Maladihre 71, 2007 Neuchltel, Switzerland.
n::,
(1 -z,)
K.
Using basic zeta-transform properties in (7), it can be shown that the positive unit-step response y, associated with G(z) satisfies 0 yk =
iK
(Osk%n,(k =n,),
1).
(9)
where n,-n,-n, is the relative degree of G(z). It is important to highlight that the parameter K in (8) and (9) is 255
256
Brief Papers Yk+
Yk
n”
K
k; - 1
I
ki
I
4
.
- 1
k, -j(i)
ki + I
‘..
I
ki - 1
ki
ki + 1
I
I
1
I k; -j(i)
k
.
.
k
.
.
‘.
.
j(i)
Fig. 1. Illustration of the conditions (l)-(3)
for K > 0.
directly proportional to the distance from the poles to z = 1, and inversely proportional to the distance from the zeros to z = 1.t From the same equations, we then see that the first non-zero value of the step response can be extremely large, relative to the steady-state value K, if G(z) has (non-minimum-phase) zeros significantly ‘closer’ to z = 1 than the poles. It is also clear from (8) that K will have opposite sign to the steady-state value K if and only if G(z) has an odd number of positive real non-minimum-phase discrete-time zeros. This ‘last observation has previously appeared in El-Khourv (1991). Deodhare and Vidvasagar (1992), and Leon de Ia Barra’(1992) and leads naturally?o the foliowing fact. Fact 1. yk displays initial undershoot if and only if G(z) has an odd number of positive real non-minimum-phase discrete-time zeros. Remark 1. Note that Fact 1 remains valid when G(z) has complex-conjugate and/or negative real poles. It is also important to highlight that complex-conjugate nonminimum-phase discrete-time zeros do not have any influence on determining the initial undershoot. Also keep in mind that the previous statements in this subsection remain valid when multiple zeros and poles are allowed in G(z).
Note also that initial undershoot has been called type-A and that undershoot without initial undershoot has been called type-B undershoor (see e.g. Deodhare and Vidyasagar, 1990, 1992). To the best of the authors’ knowledge, Fact 1 constitutes the only available result that relates the qualitative nature of the undershoot in the step response to non-minimum-phase discrete-time zeros. To prepare for the main result of this subsection, we introduce the following definitions.
undershoot,
Definition 3. If G(z) satisfies Assumption 2 and has nP - r poles at the origin and r distinct non-zero poles, we define a pole bracket as the open interval (p,,pj+,) between two different and consecutive poles 0
The following theorem provides lower and upper bounds on the number of changes of sign (n) in the impulse response t Note that this point has been previously called the ‘edge of the unit circle’ (see e.g. Jury, 1955). f For a system having only stable non-negative real poles, the fastest pole is the one closest to the origin, and the slowest pole is the one closest to z = 1.
.
times
Fig. 2. Illustration of the conditions (1) (2) and (4), (5) for K > 0. (gk) of G(z). It can also be shown that n coincides with the number of extrema in the corresponding step response. Theorem 1. If G(z) satisfies Assumption 2 and has only real zeros then n is bounded as m,+m,5~5ml+mz+m3-p. Proof
See El-Khoury
(10)
(1991) or El-Khoury et al. (1993a).
Note that the same authors have shown that, for plants satisfying the conditions of the theorem, all those zeros located to the left of the fastest pole do not have any influence upon n. This is why these zeros have not been taken into account. Thus we have the following corollary. Corollary 1. If G(Z) satisfies Assumption 2, has only real zeros, and meets the so-called ‘bracketing’ condition p = m3, together with m2 5 1, then y,, will display exactly type-r, undershoot, with r, + int [$(m, + l)]. Proof:
See the Appendix.
Note that Corollary 1 was previously shown by El-Khoury (1991) for the case m, = 0. To get a better feel for Fact 1, Theorem 1 and Corollary 1, we present the following example. Example
1. Consider
the eight-order
discrete-time
system
given by &(z ~ 0.45)(z - 055)(Z - 0.9) x (z. - 2.5)(z - 3.3)(z - 5)(z - 10) G(z) + (Z ~ O.l)(Z - 0.2)(z - 0.3)(z - 0.4) x (z - 0.5)(z - 0.6)(z - 0.7)(z - 0.8) It can easily be seen
that
the
above
01)
system satisfies 1 determines that n = 5, i.e. y,, the step response of the system (ll), will display five extrema. Correspondingly, Corollary 1 predicts that r, = 2, i.e. that y, will display type-2 undershoot. Similarly, given that there is an even number of positive real non-minimum-phase zeros, Fact 1 predicts that yk will not exhibit initial undershoot. In other words, the step response will start in the right direction, changing direction and reversing its sign three times, before finally moving toward its steady-state value. In Figs 3 and 4 we have displayed the step response of the system (ll), verifying what we have anticipated. Note from Fig. 3 that yk also exhibits overshoot. This is due to the existence of a real open-unit-disk zero to the right of the slowest pole; see Leon de la Barra (1992), Leon de la Barra and Goodwin (1991) and McWilliams (1993) for a thorough discussion of a number of related issues. 3.2. Quantitative aspects of the undershooting phenomenon. Fact 1, Theorem 1 and Corollary 1 give important insight into the ‘shape’ of undershoot, but fall short of quantitative information relating to magnitudes. The Iatter feature is clearly of primary importance to a designer of
p = m3 = 2, m, = 1 and m, = 4. Theorem
257
Brief Papers
0
5
10
15
20
k
25
30
35
40
45
Fig. 3. Step response of the system (11).
feedback loops (see e.g. Franklin, 1985; Zatiriou and Morari, 1990; 1985; Moore, 1989; Moore and Bhattacharyya, Middleton, 1991). In this regard, the following rule of thumb constitutes a handy complement to the above results. Rule of Thumb 1. Let G(z) be as in Assumption 1, with unit DC gain and only one positive real non-minimum-phase discrete-time zero, located at z = b, b > 1. Let kp be the rise time of yk up to level Q, O< LY< 1,t as defined by k: + f,“f,(I/y, 2 LY Vk = I},
(12)
with a zero at z = b, b > 1, under a general two-degree-offreedom configuration, and for any nonlinear or time-varying controller. We also refer the reader to Boyd and Barratt (1991) for additional motivation on achievable design specifications. In Fig. 5 we have pictured (14) for a few locations of the positive real non-minimum-phase zero. It is evident from this figure that the closer the zero is to z = 1, the smaller is the feasibility set. Specifically, if we want to keep the_ left-hand side of the inequality (14) say below about 100 V (%), we need to have
>lW(l
K”,
and let the peak undershoot be given by v~s~p~,~~{-y~}. Under the above conditions, v is lower-bounded as (13) Derivation.
See the Appendix.
Remark 2. It is known that in a feedback context (see e.g. Middleton, 1991: Deodhare and Vidyasagar, 1992), and if the open-loop plant has a positive real non-minimum-phase discrete-time zero, the step response from the reference input to the system output must exhibit some undershoot. In fact, under the requirement of internal stability, it can easily be seen by using Rule of Thumb 1 that the closed-loop will be design specifications k: 5 Kg,, and v s V,,, achievable if and only if
mc,X
+ a/P) log b
(15)
It is clear from this inequality that to keep the undershoot small, a large rise time is required relative to the time constant$ of the positive real non-minimum-phase zero. Only one positive real non-minimum-phase zero was assumed in Rule of Thumb 1. This does not constitute an important loss of generality, in the sense that if there were several positive real non-minimum-phase zeros then the one(s) closest to z = 1 would impose the most critical bound on the rise time. It is also clear that the right-hand side of (15) is ‘proportional’ to the rise-time threshold LY.
Fig. 4. Initial behaviour of the step response of the system (II).
4. Conclusions We have derived, for a class of discrete-time systems having only real zeros and non-negative real poles, the exact number of times that the associated step response displays undershoot. Note from (10) and the proof of Corollary 1 that the whole point of the ‘bracketing’ condition p = m3, together with mz 5 1, is to make sure that both bounds on the number of changes of sign of yk coincide. For zero/pole patterns not satisfying both conditions, Theorem 1 does not enable us to exact/y determine the number of undershoots in the associated step response. Nevertheless. this result constitutes an important step towards the complete characterization of the effects of non-minimum-phase discrete-time zeros upon step response. In a feedback context, we quantitatively described the trade-off existing between achievable closed-loop peak undershoot and rise time of the corresponding reference step response. It was clearly shown how the size of the feasibility set becomes smaller when the non-minimum-phase zeros approach z = 1. The overall characterization of the effect of non-minimumphase discrete-time zeros, both real and complex, upon the step response of systems having complex-conjugate poles remains an open problem. On the other hand, it has been
t The usual choices for a are in the upper fifth of this range.
r 2 0, is given by ilog rl-‘.
(14)
It is important to stress that the trade-off embodied in this inequality holds for any LTI discrete-time open-loop plant
,._,,.,............ 4
yk
.o..-
$ The time constant of a discrete-time
zero (pole) r * re”,
258
Brief Papers 10
I
I
6 ‘b.
Y 4
n:., ‘..
1
‘,. 0,..,‘..,
b = I.05 .Q..b = I.10 +.b = I.20 .*---
4
7
10
Fig. 5. Trade-off between achievable peak undershoot and rise-time specifications for a = 0.8.
shown that for discrete-time plants having only non-negative real poles, negative real non-minimum-phase zeros do not determine extrema in the corresponding step response; see El-Khoury (1991) and Ledn de la Barra (1994b) for alternative proofs of this statement. It has also been proved that each pair of complex-conjugate non-minimum-phase zeros does not determine any extrema if there is a pair of non-negative real poles whose average is larger or equal than their real part: see Ledn de la Barra (1994b) for the proof of this latter statement. Recent related work has appeared in McWilliams and Sain (1994), and Holmberg et al. (1995).
Acknowledgemen&-The
authors are grateful to the anonymous referees for a number of suggestions that helped to improve the paper. The first author also acknowledges the support of CONICYT (Comisi6n National de Investigaci6n Cientffica y Tecnohjgica, Chile) under Grant 1950745.
References
Boyd, S. P. and C. H. Barratt Design. Limits of Performance.
(1991). Linear Controller Prentice-Hall, Englewood
Cliffs, NJ. Deodhare, G. and M. Vidyasagar (1990). Design of non-overshooting feedback control systems. In Proc. 29th IEEE Conf. on Decision and Conrrol, Honolulu, HI, Vol. 3, pp. 1827-1834. Deodhare, G. and Vidyasagar, M. (1992). Control system design via infinite linear programming. Int. J. Conrrol, 55, 1351-1380. El-Khoury, M. (1991). Influence des zCros d’une fonction de transfert sur le comportement dynamique d’un systkme lineaire et application au, rCglage polynomial. ScD thesis, Institut d’Automatique, Ecole Polytechnique FCderale de Lausanne. El-Khoury, M., 0. D. Crisalle and R. Longchamp (1993a). Discrete transfer-function zeros and step response extrema. In Preprinrs 12th IFAC World Congress, Sydney, Vol. 3, pp. 81-86. El-Khoury, M., 0. D. Crisalle and R. Longchamp (1993b). Influence of zero locations on the number of step-response extrema. Automatica, 29, 1571-1574. Franklin, G. F. (1985). Direct digital control design via pole placement techniques. In S. G. Tzafestas (Ed.), Applied Digital Control, pp. 111-148. North-Holland, Amsterdam. Hara. S.. N. Kobavashi and T. Nakamizo (1986). Design of nor&dershootihg multivariable servd systems ising dynamic compensators. Inr. .I. Control, 44,331-342.
Holmberg, U., P. Myszkorowski, Y. Piguet and R. Longchamp (1995). On compensation of nominimumphase zeros. Auromarica, 31, 1433-1441. Jury, E. I. (1955). The effect of pole and zero locations on the transient response of sampled-data systems. Trans. AIEE
Pt II: ADolic. Ind.. 74, 41-48.
Le6n de la Barr&-B. A. (1992). Zeros and their influence on time and frequency domain properties of scalar feedback systems. PhD thesis, University of Newcastle, New South Wales: also available from UMI Dissertation Services, Ann Arbor, MI. Le6n de la Barra, B. A. (1994a). On undershoot in SISO systems. IEEE Trans. Autom. Control, AC-%, 578-581. Le6n de la Barra, B. A. (1994b). Sufficient conditions for monotonic discrete time step responses. ASME .I. Dyn. Syst. Meus. Conrrol, 116, 8101814.-
Le6n de la Barra. B. A. and G. C. Goodwin (19911. ~ , Ooen . unit disk zeros of scalar discrete time transfer functions: implications for feedback design. In Proc. 30th IEEE Conf on Decision and Conrrol, Brighton, U.K., Vol. 3, p. 2672-2673. McWilliams, L. H. (1993). Qualirariue features of linear time-invariant system transienr responses: PhD dissertation, Universitv of Notre Dame. IN: also available from UMI Dissertatcon Services, Ann Arbor, MI. McWilliams, L. H. and M. K. Sain (1994). Qualitative features of discrete-time system responses. In Proc. 33rd IEEE Conf: on Decision and Control, Lake Buena Vista, FL, Vol. 1, pp. 18-23. Middleton, R. H. (1991). Trade-offs in linear control system design. Automatica, 27, 281-292. Moore, K. L. (1989). Design techniques for transient PhD dissertation, Texas A&M response control. University. Moore, K. L. and S. P. Bhattacharyya (1990). A technique for choosing zero locations for minimal overshoot. IEEE Trans. Autom. Conrrol, AC-35,577-580.
Vidyasagar, M. (1986). On undershoot and nonminimum phase zeros. IEEE Trans. Aurom. Control, AC-31,440. Zafiriou, E. and M. Morari (1985). Digital controllers for SISO systems: a review and a new algorithm. Int. J. Control, 42,855-876. Appendix Proof of Corollary 1. Let us define C?(z) A zG(z)/(z - 1). It can then be seen that m2 5 1 guarantees that fi3 = m3 + m, together with p = m3 + m2, i.e. G(z) satisfies the corresponding ‘bracketing’ condition p = fii,. Thus, noting that 6, = 0, we have from (10) that +j = m,, i.e. y* has exactly m,
Brief Papers changes of sign, and the result follows. Note also that the application of Theorem 1 to G(z) leads to n = m, + mz, i.e., g, has exactly m, + mz changes of sign. This determines that, for m2 = 1, y,, also exhibits overshoot (see Example 1). Derivation of Rule of Thumb 1. It is clear that the zeta-transform summation representation of zG(z)/(z - 1) has /zl> 1 as its domain of convergence, and is given explicitly by
zG(z) -=2 z-1
ykz
-k.
259
at z = h, we have X 0= 2 y,b-k= k=O
kp-1
G4.2)
and, after using the definition of peak undershoot, we obtain
2
Y,J-k
k=LP
= *~‘b-*(_y,)~v*E’b--*, (A.3) k=O
Now it follows from this inequality that
(A.1)
k=”
The zero at z = b, b > 1, is obviously in the domain of convergence of (A.l). Consequently, after evaluating (A.l)
W.
2 y,b-& + c y,&*, k=” k=/$
after using the fact that y* 2 a Vk 2 kp.
k=O