Nonminimum-Phase
Zeros
MUCH TO DO ABOUT NOTHING

United States, there is no floorn the popular literature
0, and negative floor numbers
there is a certain fascinaare rarely used. Despite thetion with the concept ofhuman reluctance to admit zero
zero [1]–[3]. While today
as an authentic number, it is as
the inconspicuous 0 is taken
difficult to imagine mathematics
for granted, the situation wastoday without zero as it is to imagdifferent in the distant past. For
ine technology without the wheel
example, the Romans had no symand axle.
bol for 0, a fact memorialized by the
Although the number zero is welljump from 1 B.C. to 1 A.D., a convention
known, the system-theoretic concept of ainstituted in 531 A.D. [4, p. 91]. In contrast,
system zero is virtually unknown outside of
the Mayans had a symbol for zero, and the firstday of each Mayan month was day zero [3, p. 18]. The dynamics and control theory. The purpose of this article
modern zero of mathematics slowly earned its member- is to illuminate the critical role of system zeros in conship in the club of numbers through Indian mathematics, trol-system performance for the benefit of a wide audience both inside and outside thealthough this acceptance wascontrol systems community.
achieved only through a tortuous
JESSE B. HOAGG andprocess that spanned centuries [3].
DENNIS S. BERNSTEIN
A conceptual impediment to the
POLES AND ZEROS
acceptance of zero is the difficulty in
Setting aside the notion of zero for theunderstanding the ratio 1/0. Presumably, this ratio is moment, the idea of a pole is one of the most fundamental
infinity or ∞, a much more challenging concept. That 0 concepts in system theory. Consider the continuous-timeand ∞ are close cousins casts suspicion on zero as a valid single-input, single-output (SISO) system
number. Even in modern times, the zero appears begrudgingly on your telephone keypad after the 9. In Europe, theẋ(t ) = Ax(t ) + Bu(t ),ground floor in a building is routinely labeled 0, and thusIthe meaning of floor −1 is unambiguous, whereas, in they(t ) = Cx(t ),1066-033X/07/$25.00©2007IEEE

JUNE 2007 « IEEE CONTROL SYSTEMS MAGAZINE 45

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

with scalar-valued input u(t ) and scalar-valued output
y(t ). The n × n dynamics matrix A represents the dynamics of this system, while the n × 1 column vector B represents the effect of the actuator, and the 1 × n row vector C
represents the response of the sensor. Together, (A, B, C)
defines the input-output dynamics of the system. Taking
Laplace transforms yields
 ∞y(t )e−st dt = C(sI − A)−1 x(0) + G(s)û(s),ŷ(s) 0where C(sI − A)−1 x(0) is the initial condition response of thesystem and the transfer function G is given by
G(s) = C(sI − A)−1 B.

Response to Input et

0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

0

1

2

3

4

5 6
Time

7

8

9

10

FIGURE 1 Bounded response of a nonminimum-phase system to an
unbounded input. The response of the transfer function
G(s) = (s − 1)/(s + 1)2 to the unbounded input e t is bounded andconverges to zero due to the open-right-half-plane zero z = 1.

1.2
1
0.8

y (t )

0.6
0.4
0.2
0
−0.2
−0.4

0

1

2

3

4

5 6
Time

7

8

9

10

FIGURE 2 “Wrong” initial direction with direction reversal. For thesystem G(s) = −(s − 1)/(s + 1)2 , which has one positive zero, thestep response initially moves in the “wrong” direction. Although
the operator might be tempted to abandon the control strategy,
the system eventually reverses course and reaches the desired
steady state. In this case, patience is a virtue.

To examine the poles and zeros of G, write G as
G(s) =

N(s)
,
D(s)

where the denominator polynomial is D(s)  det(sI − A),and the numerator polynomial isN(s)  C adj(sI − A)B.
Here, adj(sI − A) is the adjugate of sI − A [5, p. 20], [6, p. 42].
The roots of N are the zeros of G, while the roots of D are
the poles of G. This discussion assumes that N and D haveno common roots, which is the case if and only if (A, B, C)
is a controllable and observable triple. In this case, the npoles of G are precisely the eigenvalues of A.
The poles of G determine whether G is stable or unstable, as well as the decay rate and oscillation frequencies of
the initial condition response. The poles of G do notdepend on either the input matrix B or the output matrix
C. In contrast, the zeros of G are determined by the
dynamics matrix A as well as B and C. Hence, the zeros ofG depend on the physical placement of the sensors andactuators relative to the underlying dynamics. The conceptof a zero distinguishes control theory from dynamical systems theory.
In this article, the term transfer function refers to aproper ratio of polynomials. The transfer function G isstrictly proper if and only if G(∞) = 0, whereas G is exactly proper if and only if G(∞) = 0. For results that are
restricted to either strictly proper or exactly proper transfer
functions, the distinction is made explicitly.

THE BLOCKING EFFECT OF A ZERO
What exactly do zeros do, and how do zeros relate to thenumber zero? A zero of a transfer function is a root of thenumerator polynomial of the transfer function, and thus isa real or complex number. When the transfer function is
asymptotically stable, that is, when all of the roots of thedenominator polynomial are in the open left half plane,
each zero has a specific effect on the asymptotic responseof the transfer function for certain inputs.
To illustrate the effect of a zero on the response of an
asymptotically stable transfer function G, consider a step
input, so that the response of G approaches a steady statevalue. If, moreover, the number 0 is a zero of the transfer
function G, that is, G(0) = 0, then the steady state responseof G is zero, that is, the dc gain of the system is zero.
Next, suppose that the input to G is sinusoidal, that is,harmonic, with frequency ω . Then, the asymptotic
response is also harmonic with the same frequency ofoscillation; this response is the harmonic steady-state
response. If, moreover, the imaginary number  ω is a zeroof the transfer function G, that is, G( ω) = 0, then theamplitude of the harmonic steady-state response is zero,and thus the response converges to zero.

46 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2007

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

Finally, suppose that the input to G is the unboundedsignal et, in which case one expects the response of the system to be unbounded as well. If, however, the number 1 is azero of G, that is, G(1) = 0, then the response of the system
is not only bounded but converges to zero (see Figure 1).
In general, each zero blocks a specific input signal
multiplied by an arbitrary constant. In the case of a nonminimum-phase zero, that is, an open-right-half-plane zero,
the blocked signal is unbounded. The above observations
follow from the final value theorem (after all unstable
poles of the input are canceled by nonminimum-phasezeros of the system), and, since the system is assumed to
be asymptotically stable, are also valid for all nonzero initial conditions of all stabilizable and detectable statespace realizations.

master. The driver initially moves in one direction, forexample, to the right, and later reverses direction, moving
to the left. To see this, consider the four-wheeled car model(1)ẏ = v sin θ,vθ̇ = tan φ,


(2)
(3)

given in [10] and [11], where (x, y) is the position of thecenter of the rear wheels, θ is the angle between the car’s
longitudinal axis and the x-axis, v is the translational velocity of the rear wheels,  is the distance between the front
and rear wheels, and the control input φ is the steering

INITIAL UNDERSHOOT DUE TO ANODD NUMBER OF POSITIVE ZEROS

0.2
0y (t )

−0.2
−0.4
−0.6
−0.8
−1
0

1

2

3

4

5 6
Time

7

8

9 10

FIGURE 3 “Correct” initial direction with direction reversal. For the system G(s) = (s − 1)/(s + 1)2 , which has one positive zero, the step
response initially moves in the “correct” direction. However, after
more time passes, the system reverses and moves in the “wrong”
direction. In this case, initial optimism proves to be unfounded.

1.2
1
0.8
0.6

y(t)

The effect of positive (that is, real open-right-half-plane)zeros is evident in the step response of a system. In particular, Figure 2 shows a step response that departs in the
nonasymptotic direction; this phenomenon, which isequivalent to initial error growth, is called initial
undershoot. Note that initial undershoot is defined only fora step response whose initial and asymptotic values aredifferent. A classical result, proved in [7]–[9], states thatthe step response of an asymptotically stable, strictly proper transfer function exhibits initial undershoot if and only
if the system has an odd number of positive zeros.
Undershoot can have significant implications in practice.
Suppose, for example, that an economic plan is implemented
to boost the economy. In particular, suppose that a central
bank implements a step decrease in short-term interest rates,
which produces the undesirable effect of initially decreasing
gross domestic product, as shown in Figure 2. Supporters of
the plan might decide to abandon the plan before its ultimate
effect is known, whereas critics of the plan might feel vindicated in having opposed the plan. In reality, however, theinput-output dynamics of the economy might have an odd
number of positive zeros, in which case the appropriate
action is to wait for the system to reverse direction.
On the other hand, as illustrated in Figure 3, it might be
the case that the economy initially moves in the “correct”
direction, which would suggest that the plan is appropriate. However, the system in Figure 3 eventually reverses
course and converges to a negative value, revealing that theplan was inappropriate. To complicate matters even more,
the step response of a system with multiple positive zeros
can exhibit multiple direction reversals. For example, thestep response of a system with two positive zeros, as illustrated in Figure 4, initially moves in the “correct” direction,
reverses course to move in the “wrong” direction, and thenreverses course yet again to move in the “correct” direction.
An everyday example of positive zeros arises when driving a car backwards, a skill that every young driver mustẋ = v cos θ,

0.4
0.2
0
−0.2
−0.4

0

1

2

3

4

5 6
Time

7

8

9 10

FIGURE 4 “Correct” initial direction with multiple direction reversals.
For the system G(s) = (s − 1)2 /(s + 1)3 , which has two positive
zeros, the step response initially moves in the “correct” direction,
reverses course to move in the “wrong” direction and then reverses
course again to move in the “correct” direction. In this case,patience and fortitude are called for.

JUNE 2007 « IEEE CONTROL SYSTEMS MAGAZINE 47

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

angle measured from the car’s longitudinal axis. The fourwheeled car model contains the nonholonomic constraintẋ sin θ − ẏ cos θ = 0
and thus has an uncontrollable linearization about the zeroequilibrium. However, if we control the translational acceleration of the rear wheels, that is,v̇ = u,(4)

where u is a control input, then linearizing (1)–(4) about
the equilibrium (x0 , y0 , θ0 , v0 ) yields
δ̇x = cos(θ0 )δv − v0 sin(θ0 )δθ,
δ̇y = sin(θ0 )δv + v0 cos(θ0 )δθ,v0δ̇θ = φ,δ̇v = u,which is controllable for all constant nonzero v0 .
Now assume that the car has a constant nonzero speedv0 and φ = 0, and assume that the car’s longitudinal axis isinitially parallel to the x-axis, that is, θ0 = 0. Furthermore,
the output of the system is the y-axis position of the centerof the front wheels, given by yout = y +  sin θ , which has
the linearization yout = δy + δθ . Then the transfer function
from the steering angle control input φ to the lateral position yout isG(s) = v0

s + v0 /.
s2

If the car is driving in reverse, then G has exactly one
positive zero, namely, z = −v0 /. Thus, the lateral response
to a step input in the steering angle exhibits initial undershoot. A similar result appears in [12, p. 158]. The same
effect occurs in some aircraft as can be seen from the step

0.8

y(t )

0.6
0.4
0.2
0
−0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Time (s)FIGURE 5 Nonmonotonic step response. The step response to thetransfer function G(s) = (s2 − 10s + 27)/(s + 3)3 , which has
nonreal nonminimum-phase zeros but no positive zeros, has two
zero crossings.

response of the elevator deflection to pitch angle [13, p. 494].
These examples suggest that one source of nonminimum
phase zeros is noncolocation, that is, physical separation, ofsensing and actuation [14]–[20].

ZERO CROSSINGS DUE TO POSITIVE ZEROS
The result of [7]–[9] implies that the step response for astrictly proper transfer function having an even number of
positive zeros does not exhibit initial undershoot. Nevertheless, the step response shown in Figure 4 for a systemwith two positive zeros exhibits two direction reversals
and two zero crossings, where the term zero crossing refers
to the situation in which a signal passes through the valueof zero. In fact, we now show that if an asymptotically stable transfer function possesses at least one positive zero,then the step response of the system undergoes at least one
zero crossing.
Let z denote a positive zero of the asymptotically stabletransfer function G. The Laplace transform ŷ(s) of the output y(t ) for a unit step input is given byŷ(s) = G(s)(1/s).
Setting s = z yields ŷ(z) = G(z)(1/z). Since G(z) = 0, it follows that ŷ(z) = 0, and thus ∞e−zt y(t )dt = 0.
(5)0

Since e−zt is positive on [0, ∞), it follows that y(t ) mustcross zero on (0, ∞). In addition, (5) implies that the weighted “negative area” and “positive area” associated with y(t )are exactly equal. Note that (5) depends on z but does notdepend on either the poles or the remaining zeros of G.
Whereas it follows from [7]–[9] that the step response ofa strictly proper G with an odd number of positive zeros
has initial undershoot and thus at least one zero crossing, it
follows from (5) that at least one zero crossing occurs if G isproper and has at least one positive zero. As shown in Figure 4, the step response must possess at least two direction
reversals if G has a nonzero even number of positive zeros.
Figures 3 and 4 suggest that the number of zero crossings is equal to the number of positive zeros. In fact, a statement that the number of zero crossings is equal to thenumber of positive zeros is given in [21, p. 174] and attributed to [22], which in turn attributes the result to [23].
However, a statement of this result does not appear in [23].
Furthermore, in the second edition of [24, p. 184], the result
is attributed to [25] rather than [22]. However, the result
given in [25] is more restrictive than the statement in [24],since [25] states that the number of zero crossings is equal
to the number of positive zeros for strictly proper transfer
functions with only real poles and zeros. In fact, the statement in [24] is incorrect for transfer functions with complex
zeros. For example, the step response of a system with nopositive zeros but two complex nonminimum-phase zeros

48 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2007

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

UNDERSHOOT, OVERSHOOT,
AND ZERO CROSSINGS IN SERVO SYSTEMS
We now specialize the results given thus far to servo systems. Feedback stabilization of an unstable plant unavoidably gives rise to nonminimum-phase zeros. To see this,

1
0.9
0.8
0.7
0.6

y(t )

can exhibit two zero crossings, as shown in Figure 5.
Numerical testing suggests that the number of zero crossings in the step response is greater than or equal to thenumber of positive zeros. A proof of this conjecture is open.
For a different system with two nonreal nonminimumphase zeros, Figure 6 shows a nonmonotonic step responsewith no zero crossings. Furthermore, for yet another system with two nonreal nonminimum-phase zeros, Figure 7
shows a monotonic step response. Hence, the presence ofnonminimum-phase zeros does not guarantee the existence of either zero crossings or direction reversals. However, it is shown in [26] that, for asymptotically stable,
strictly proper systems with only real poles and real zeros,
the number of extrema in the step response (not includingt = 0) is greater than or equal to the number of zeros to theright of the rightmost pole.

0.4
0.3
0.2
0.1
0

OVERSHOOT DUE TO POSITIVE ZEROSIn addition to initial undershoot and zero crossings, thestep response of an asymptotically stable transfer function
can exhibit overshoot, that is, assume values both greaterthan and less than the asymptotic value of the step
response. In fact, the step response of an asymptoticallystable transfer function G exhibits overshoot if G(s) − G(0)has at least one positive zero. To see this, let z be a positive
zero of G(s) − G(0). Applying to G(s) − G(0) the same
steps used to derive (5), it follows that (see also [27, pp.
213–214]) ∞e−zt[y(t ) − y(∞)]dt = 0.
(6)

0

1

2

3

4

5 6
Time

7

8

9 10

FIGURE 6 Nonmonotonic step response. The step response to thetransfer function G(s) = (2s2 − s + 1)/(s + 1)3 , which has two nonreal nonminimum-phase zeros, has two direction reversals, andthus is not monotonic. However, the step response does not exhibit
any zero crossings.

1
0.9

y(t)

0

Consequently, y(t ) − y(∞) must change sign on [0, ∞),and thus y(t ) overshoots its steady-state valuey(∞) = lim t→∞ y(t ). This behavior depends on the positive zero z of G(s) − G(0) but does not depend on any otherdetails of G.
As a special case, consider a system whose step
response converges to zero, that is, y(∞) = 0, which arises
in control systems with integral action. In this case, G
exhibits overshoot if G has at least one positive zero.
Table 1 summarizes the results given above on initial
undershoot, zero crossings, and overshoot in the step
response of an asymptotically stable transfer function G.

0.5

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

1

2

3

4 5 6
Time (s)

7

8

9 10

FIGURE 7 Monotonic step response. The step response to the transfer function G(s) = (s2 − s + 4)/(s + 1)3 , which has two nonreal
nonminimum-phase zeros, is monotonic.

TABLE 1 Initial undershoot, zero crossing, and overshoot in the step response of an asymptotically stable transfer function G.
Results for exactly proper transfer functions are needed to analyze the sensitivity transfer function S, which arises in servo
systems and which is not addressed by the classical theory.

Strictly Proper

Exactly Proper

Initial UndershootIf and only if G has an
odd number of positive zeros

Zero Crossing
If G has at least
one positive zero

OvershootIf G(s ) − G(0) has atleast one positive zero

If and only if G(s ) − G(∞) has an
odd number of positive zeros

If G has at least
one positive zero

If G(s ) − G(0) has atleast one positive zero

JUNE 2007 « IEEE CONTROL SYSTEMS MAGAZINE 49

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

e
r

C

u

G

y

−

FIGURE 8 Standard servo problem with loop transfer function
L(s) = C(s)G(s). The zeros of the sensitivity transfer function
S = 1/(1 + L) from command r to error e = r − y are the poles ofL(s).

consider the servo problem in Figure 8, where
L(s) = C(s)G(s) = N(s)/D(s) represents the loop transfer
function, that is, the plant G cascaded with a proper feedback controller C under the assumption that L(s) is strictly
proper. Then the asymptotically stable closed-loop transfer
function from the reference signal r to the error signal e isgiven by the sensitivity transfer function S(s) = 1/(1 + L(s)),
that is, ê(s) = S(s)r̂(s). Since
S(s) =

D(s),N(s) + D(s)(7)

it follows that the zeros of S are precisely the poles of L.
Therefore, if either the plant or the controller is unstable and no unstable pole/zero cancellation occurs (seethe discussion below), so that L is also unstable, thenthe corresponding sensitivity transfer function S has
nonminimum-phase zeros. These nonminimum-phasezeros tend to enhance performance by decreasing themagnitude of the sensitivity function (relative to thesensitivity transfer function that would be obtained if
these zeros were not present) thereby reducing thetransmission of specific signals.

Initial Undershoot
To analyze initial undershoot in the step response of a
servo system, one might be tempted to apply the result of[7]–[9] to the sensitivity transfer function S and conclude
that the step response of S exhibits initial undershoot if
and only if S has an odd number of positive zeros (orequivalently, L has an odd number of positive poles).
However, the result of [7]–[9] does not apply because thesensitivity transfer function S is not strictly proper. Nevertheless, “Initial Undershoot Revisited” extends the resultof [7]–[9] to include exactly proper transfer functions.

Initial Undershoot RevisitedInitial undershoot describes the qualitative behavior of the step
response of a transfer function. A single-input, single-output
asymptotically stable transfer function G(s) exhibits initial undershoot if its step response initially moves in the direction that is
opposite to the direction of the asymptotic value. Initial undershootis thus equivalent to initial error growth. In [7]–[9], it is shown that astrictly proper transfer function exhibits initial undershoot if and only
if the transfer function has an odd number of positive zeros.

8
7

To analyze the step response of a servo system, however, it isnecessary to extend the definition and classification of initial
undershoot to include exactly proper transfer functions, that is,transfer functions whose numerator and denominator polynomials
have the same degree.
Let G be an asymptotically stable transfer function with relative
degree d ≥ 0, where d = 0 denotes the exactly proper case. Lety(t ) be the step response of G. The step response has the initialvalue y(0+ ) = G(∞) = lims→∞ G(s) and the asymptotic valuey(∞) = limt→∞ y(t ) = G(0). Then y(t ) exhibits initial undershoot ify (ρ) (0+ )[y(∞) − y(0+ )] < 0,

6

(s1)

y(t )

5
4

where ρ  min(d, 1) and, by the initial value theorem, y(ρ) (0+ ) lims→∞ s ρ [G(s) − G(∞)]. Note that y(t ) can exhibit initial under-

3
2

shoot only if the initial value differs from the final value, that is,y(0+ ) = y(∞). In the strictly proper case, d ≥ 1, and thus ρ = d
and y(0+ ) = 0. Hence (S1) becomes

1
0
−1

0

5

10

15

Time (s)FIGURE S1 The step response of G1 (s) = (s − 2)2 (s + 2)/(s + 1)3 ,which exhibits initial undershoot due to an odd number of positive
zeros in G1 (s) − G1 (∞). In particular, G1 (s) − G1 (∞) has one
negative zero at −2.0748 and one positive zero at 0.6748. Note thatG1 exhibits initial undershoot even though G1 itself has an evennumber of positive zeros.

y (d ) (0+ )y(∞) < 0,which is the condition given in [7]–[9].
For example, consider the step response of the exactly propertransfer function
G1 (s) =(s − 2)2 (s + 2),
(s + 1)350 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2007

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

Specifically, it is shown (with G replaced by S) that thestep response of S with L strictly proper exhibits initial
undershoot if and only if S(s) − S(∞) = S(s) − 1 has an
odd number of positive zeros. In terms of the error signal,
initial undershoot means that, after a step servo commandis introduced, the error initially grows and thus attains alarger value than the step difference due to the servo command.
Now, define the complementary sensitivity transfer function T(s) = 1 − S(s) = L(s)/(1 + L(s)) from r to y of theclosed-loop system, that is, ŷ(s) = T(s)r̂(s), given byT(s) =

N(s).
N(s) + D(s)(8)

Thus, the step response of S exhibits initial undershoot if
and only if T has an odd number of positive zeros. Furthermore, since L and T have the same zeros, the step
response of S exhibits initial undershoot if and only if L
has an odd number of positive zeros.
Note that T is strictly proper since C and thus L are
strictly proper. Therefore, the step response of T exhibits
initial undershoot if and only if T (and thus L) has an odd
number of positive zeros.

which exhibits initial undershoot, as shown in Figure S1, eventhough G1 has an even number of positive zeros. On the other
hand, the step response of the exactly proper transfer function
G2 (s) =(s − 2)(s + 2)2,(s + 1)3

does not exhibit initial undershoot, as shown in Figure S2, eventhough G2 has an odd number of positive zeros. Thus, the result of[7]–[9] is not valid for exactly proper transfer functions.

Zero Crossings
A similar result can be used to show that the error e (t )changes sign and thus has at least one zero crossing when
L has at least one positive pole p. To see this, note that p isalso a zero of S. Since S(p) = 0, applying to S the same
steps used to derive (5), which is a valid procedure despite
the fact that S is not strictly proper, yields
 ∞e−pte(t ) = 0.
0Thus, e(t ) must cross zero.
On the other hand, y(t ) has at least one zero crossing
if L has at least one positive zero z. To see this, note that,since z is a zero of L, it follows that z is also a zero of T,
and thus (5) is satisfied, which implies that y(t ) mustcross zero.

OvershootThe step response of a servo system can exhibit overshoot,that is, assume values both greater than and less than theasymptotic value. It follows from (6) that the output y(t ) ofa servo system with a step command r(t ) exhibits overshootif T(s) − T(0) has at least one positive zero. Furthermore,

undershoot because G1 (s) − G1 (∞) = (−5s2 − 7s + 7)/(s + 1)3
has exactly one positive zero, whereas the step response of
G2 (s) = (s − 2)(s + 2)2 /(s + 1)3 does not exhibit initial undershoot because G2 (s) − G2 (∞) = (−s2 − 7s − 9)/(s + 1)3 has no
positive zeros.
Note that if G is strictly proper, then G(∞) = 0, and the proposition specializes to the result presented in [7]–[9].

10PROPOSITION
The step response y(t ) exhibits initial undershoot if and only if
G(s) − G(∞) has an odd number of positive zeros.

Now, it follows immediately from the proposition that the step
response of G1 (s) = (s − 2)2 (s + 2)/(s + 1)3 exhibits initial

−2

y(t )

PROOF
L e t H(s) = G(s) − G(∞) = βN(s)/D(s) , w h e r e N a n d D a r emonic polynomials, β is a real number, and H has relative
degree ρ. Thus, y (ρ) (0+ ) = lims→∞ s ρ H(s) = β . Next, note thaty(∞) − y(0+ ) = G(0) − G(∞) = H(0) = βN(0)/D(0). Thus, (S1)
is satisfied, that is, y(t ) exhibits initial undershoot, if and only if
β 2 N(0)/D(0) < 0. Since D is Hurwitz, it follows that D(0) is positive, and thus y(t ) exhibits initial undershoot if and only if N(0) is
negative. Note that N(0) is the product of the negatives of the rootsof N, and thus y(t ) exhibits initial undershoot if and only if N has an
odd number of positive roots.

−1
−3
−4
−5
−6
−7
−8
0

5

10

15

Time (s)FIGURE
S2
The
step
responseof
G2 (s) = (s − 2)(s + 2)2 /(s + 1)3 , which does not exhibit initial
undershoot because G2 (s) − G2 (∞) has an even number of positive zeros. In particular, G2 (s) − G2 (∞) has two negative zeros
at −5.3028 and −1.6972 and no positive zeros. Note that G2
does not exhibit initial undershoot even though G2 itself has an
odd number of positive zeros.

JUNE 2007 « IEEE CONTROL SYSTEMS MAGAZINE 51

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

A conceptual impediment to the acceptance of zero is thedifficulty in understanding the ratio 1/0.
applying to S the same steps used to derive (6), which again
is a valid procedure despite the fact that S is not strictly
proper, it follows that the error e(t ) to a step command r(t )exhibits overshoot if S(s) − S(0) has at least one positive
zero. Note that e(t ) = r(t ) − y(t ), and thus e(t ) exhibits
overshoot if and only if y(t ) exhibits overshoot. Therefore,S(s) − S(0) has at least one positive zero if and only if
T(s) − T(0) has at least one positive zero.
As a special case, consider a system whose error e(t )converges to zero, that is, S(0) = 0. In particular, e(t ) converges to zero if the controller C has integral action. In thiscase, e(t ) exhibits overshoot if L has at least one positivepole. To see this, note that it follows from (7) that a positive pole p of L is also a positive zero of the sensitivity
S(s) = S(s) − S(0), which implies that e(t ) overshoots its
steady-state value, namely, zero. Consequently, y(t ) also
overshoots its steady-state value, namely, the value of thestep command.
Table 2 summarizes the results on initial undershoot,zero crossings, and overshoot for the servo system shownin Figure 8, where u(t ) is the unit step input, y(t ) is theoutput, and e(t ) is the error.

INVERTED PENDULUM ON A CARTTo demonstrate the above results, try balancing a longstick in the palm of your hand, so that you (that is, youreye, brain, and arm) are stabilizing an unstable system.
You can think of your arm as the force actuator and yourarm as the “cart,” which provides a base for the stick. It
turns out that the linearized transfer function from theforce applied by your hand to the position of your hand
has both a positive zero and a positive pole. Specifically,
this transfer function is given by [28, p. 86]
G(s) =

1 (s − z)(s + z),
M s2 (s − p)(s + p)

where p and z are the positive pole and positive zero,
respectively, given by

p=

g
mg
+
,

M


z=

g
,


where m is the mass of the stick,  is the length of the stick,
M is the mass of the cart, and g is the acceleration due togravity. The servo control system that your eye, brain, andarm implement is shown in Figure 8. Note that p > z, thatis, the pole is to the right of the zero.
Although the controller implemented by the brain is
unknown, we can draw some conclusions about properties
that any stabilizing linear time-invariant controller mustpossess. Since the plant has an odd number of real poles(specifically, one pole at p) to the right of the positive zeroz, it follows from the parity interlacing principle [28, p. 80],[29], [30] that the plant cannot be stabilized by a stable controller. In fact, the controller must have an odd number of
positive poles to the right of the zero z in order to prevent a
closed-loop pole from being attracted to z. Although it is achallenging exercise to use root locus to construct a stabilizing controller, LQG can easily be used.
For a rough test of these conditions, command yourself
to move the “cart” several inches to the left. You readily
observe that a typical controller implemented by yourbrain causes your hand to move first to the right before
moving to the left. This kind of response is also manifested
by bicycle countersteering as discussed in “Bicycle Countersteering Revisited.” The cart position y(t ) and the error
e(t ) both exhibit initial undershoot, in both cases due tothe positive zero z in T, which arises from the positiveplant zero z = g/ l, assuming that your brain does not
introduce an odd number of additional positive zeros
through C. However, with regard to initial undershoot,LQG behaves differently from the typical human brain. In

TABLE 2 Initial undershoot, zero crossing, and overshoot in the servo system shown in Figure 8, where u (t ) is the unit step
input, y (t ) is the output, and e (t ) is the error. As shown in the text,the conditions for overshoot of y (t ) and e (t )are equivalent.

y (t )e (t )

Initial UndershootIf and only if L has an odd
number of positive zeros

Zero Crossing
If L has at least
one positive zero

OvershootIf T(s) − T(0) has atleast one positive zero

If and only if L has an odd
number of positive zeros

If L has at least
one positive pole

If S(s) − S(0) has atleast one positive zero

52 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2007

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

particular, LQG can be used to design a controller withexactly one positive zero. Thus, L (and T) has two positive
zeros, and the cart position y(t ) does not exhibit initial
undershoot. For example, let M = m = 0.1 kg, l = 1 m,
g = 9.8 m/s2 , and consider the LQG controller
C(s) =−5085s3 − 21355s2 + 5820s + 3099
s4 + 45.84s3 + 981.1s2 − 39678s − 132520,(9)which has two negative zeros at −4.4273 and −0.2744 andone positive zero at 0.5018. Consequently, the step
response of the cart position y(t ) does not exhibit initial
undershoot, as shown in Figure 9. In fact, numerical experiments suggest that LQG always yields cart controllers
with an odd number of positive zeros.
In addition to initial undershoot, the positive zero z of
L implies that the cart position y(t ) has at least one zerocrossing, whereas the positive pole p of L implies that thestep response error e(t ) has at least one zero crossing.
Furthermore, in bringing the “cart” to its new position,
we observe overshoot, which is due to the positive plantpole p. This example is discussed extensively in [31].

ROBUSTNESS AND PERFORMANCE LIMITATIONSOF NONMINIMUM-PHASE ZEROSIn addition to initial undershoot and direction reversals inthe step response of a system, nonminimum-phase zeros
limit closed-loop performance. This effect can be seen bynoting that the poles of the closed-loop system are a “mixture” of the plant poles and zeros; the classical root locus
method tells us how this mixture plays itself out as theloop gain increases. In particular, as the loop gain is
increased, poles move toward zeros, and thus destabiliza-

8
6
4
2

y(t)

0
−2
−4
−6
−8
−10
−12

0

0.5

1

1.5

2
2.5
Time (s)

3

3.5

4

tion inevitably occurs when the loop transfer function has
nonminimum-phase zeros. Hence, feedback control systems have limited gain margin when the loop transfer
function has nonminimum-phase zeros, and thus limited
gain margin implies a limitation on the robustness of theclosed-loop system. A similar limitation on gain margin
occurs when proportional feedback is used and the looptransfer function has relative degree greater than 2 [14].
However, controllers can be constructed to have infinite
upward gain margin when the loop transfer function isminimum phase [32], [33].
Nonminimum-phase zeros in the loop transfer function
also limit bandwidth. To see this, it follows from asymptotic LQG theory [34, p. 369], [35] that nonminimum-phasezeros in the transfer functions from the plant disturbance

Bicycle Countersteering RevisitedT

he countersteering response in riding a bicycle is an example of initial undershoot. The constant-speed linearization ofan open-loop bicycle is unstable with a positive pole, and thepositive open-loop pole becomes a positive zero of the sensitivity transfer function. Thus, the step response of the sensitivity
transfer function has at least one zero crossing and exhibits
overshoot. In addition, the typical rider’s controller results in aloop transfer function with an odd number of positive zeros, andthus the rider-stabilized bicycle exhibits a nonminimum-phasecountersteering response, which limits maneuverability. Specifically, in turning the bicycle to the left, the rider commands a lefthand step; however, in response to this step command, thebicycle typically first turns to the right before turning to the left.
This initial undershoot behavior is discussed in [S1] and [S2].
A bicycle rider might, however, be able to use an alternative
controller that results in a nonzero even number of positive
zeros in the loop transfer function. In this case, in response to aleft-turn step command, the bicycle turns to the left, quickly
turns back to the right, and turns to the left again to complete
the left turn without initial undershoot. Figure 9 illustrates thistype of response for an LQG-controlled inverted pendulum on acart. However, as in Figure 9, the rider experiences delayedcountersteering since the bike must eventually turn back toward
the right and cross zero (see Table 2) before completing theleft-hand turn. As long as the relevant transfer function possesses at least one positive zero, a zero crossing cannot be avoided. Numerical demonstration of these properties can be basedon models given in [S3].
[S1] K.J. Astrom, R.E. Klein, and A. Lennartsson, “Bicycle dynamics andcontrol,” IEEE Control Syst. Mag., vol. 25, pp. 26–47, Aug. 2005.
[S2] D.G. Wilson, Bicycling Science, 3rd ed. Cambridge, MA: MIT Press,

FIGURE 9 The step response of the cart position with an LQG
controller. Beginning at 0, the step response initially moves in
the correct direction, overshoots the asymptotic value 1, moves
in the wrong direction past 0, and finally reverses to reach thedesired position.

2004.
[S3] D.L.M. Limebeer and R.S. Sharp, “Bicycles, Motorcycles, and Models,” IEEE Control Syst. Mag., vol. 26, pp. 34–61, Oct. 2006.

JUNE 2007 « IEEE CONTROL SYSTEMS MAGAZINE 53

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

An everyday example of positive zeros
arises when driving a car backwards.

to measurement and from the control to performance variable limit bandwidth in the sense that their mirror images
are the asymptotic locations of the closed-loop poles underhigh gain. A related phenomenon is the waterbed effect,
which concerns the effect of nonminimum phase zeros on
the peak of the sensitivity transfer function [28, p. 98].
Because poles are attracted to zeros, nonminimumphase zeros limit the use of high-gain feedback. Consequently, open-right-half-plane zeros limit the achievable
performance of fixed-gain controllers [36]–[42] as well as
adaptive controllers [43].

ZERO CANCELLATION
AND HIDDEN UNSTABLE POLES
Mathematically, a zero can cancel a pole when a pair oftransfer functions are cascaded. This property corresponds
to nothing more than the fact that the stable transfer function G(s) = (s + 1)/(s + 1) is mathematically indistinguishable from the constant transfer function G(s) = 1. Likewise,
the unstable transfer function G(s) = (s − 1)/(s − 1) is also
mathematically indistinguishable from G(s) = 1. However,unstable pole-zero cancellation is not an allowable operation in practical plant/controller cascade for the simple
reason that an arbitrarily small discrepancy between thezero at 1 and the pole at 1 results in instability.
However, even if there is no discrepancy between an
unstable pole and an unstable zero so that mathematicalcancellation occurs, the cascaded system generally has an
unbounded internal signal. To see why, consider theclosed-loop system in Figure 10, whose loop transfer function involves an unstable pole-zero cancellation. For the
servo control system shown in Figure 10, the error is givenby ê = ((s + 1)/(s + 2))r̂, which seems to indicate stability.
However, the transfer function from r to u is given byû = ((s + 1)/((s − 1)(s + 2)))r̂, which is unstable. In fact,
this transfer function exposes an otherwise hidden instability in the system. To determine the stability of a systemrepresented in terms of transfer functions, it is thus neces-

r

e
−

1
s −1

u

s −1
s +1

y

FIGURE 10 Servo feedback system. With a bounded reference signal
r , the error e = [(s + 1)/(s + 2)]r is bounded. However, the control
input u is generally unbounded due to the instability of the controller.
This system possesses a hidden unstable pole-zero cancellation.

sary to examine all transfer functions as discussed in [44, p.
123] and [45].
To demonstrate the effect of a hidden instability, weagain consider Figure 10. If we assume that r is thenonzero initial condition response of a linear timeinvariant system, then the control signal u is unbounded.
Alternatively, if r(t ) ≡ 0 but the time-domain realization of
the transfer function 1/(s − 1) has a nonzero initial condition, then the control signal u is still unbounded. Exposing
the hidden instability due to unstable pole-zero cancellation is equivalent to recognizing the presence of thisunbounded response. Hence, even when the cancellation isperfect, a nonminimum-phase controller zero cannot be
used to cancel an unstable plant pole, and an unstable controller pole cannot be used to cancel a nonminimum-phase
plant zero.

BLOCKING AND TRANSMISSIONZEROS IN MIMO SYSTEMS
While everything we have said so far applies to SISO systems, the effect of zeros on system behavior and achievable
performance is analogous but more complex in MIMO
(multiple-input, multiple-output) systems. For treatmentsof MIMO zeros, see [46]–[57].
For a nonzero l × m transfer function G, two types ofzeros are of interest. A blocking zero z ∈ C of G has theproperty that G(z) = 0. Hence, z ∈ C is a blocking zero of
G if and only if z ∈ C is a zero of every scalar entry of G.
Hence, the blocking zeros of a MIMO transfer function caneasily be determined.
The second type of zeros for a MIMO transfer function
G are the transmission zeros. To characterize transmission
zeros, it is useful to consider the Smith-McMillan form of a
MIMO transfer function [6, p. 140], [42, p. 80]. This result
states that every square or rectangular transfer function
can be transformed by means of unimodular matrices U1and U2 to a transfer function with nonzero entries appearing only on its main diagonal. (A unimodular matrix has
polynomial entries and a constant, nonzero determinant.)
The Smith-McMillan form is given by


 p1


U1 GU2 = 


q1

..

.

pr
qr



,

0(l−r)×(m−r)

where p1 , . . . , pr and q1 , . . . , qr are monic polynomials (thatis, their leading coefficients are unity), pi and qi have nocommon roots, pi is a factor of pi+1 , and qi+1 is a factor of qi.
Consequently, the roots of pr include all of the roots of
p1 , . . . , pr−1 , while the roots of q1 include all of the roots of
q2 , . . . , qr . The normal rank of G(s) is r. If p1 (z) = 0 thenG(z) = 0, and thus the roots of p1 are the blocking zeros of
G. Furthermore, note that at least one entry of the Smith-

54 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2007

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

McMillan form of G(z) is zero for every complex number zthat is a root of one of the polynomials pi . Thus, the roots ofpr are the transmission zeros of G. The analysis of transmission zeros is slightly complicated due to the fact that, asshown by the second example below, a transmission zerocan also be a pole.
For example, consider the 2 × 2 transfer functionG(s) =

1
s+1
1
s+3

1 
s+2 ,
1
s+1

1

2
2 (3s + 13s + 12)
1
2 + 13s + 14)
(3s
2



×

−2s − 4
1

3
3



1
(s+1)(s+2)(s+3)

0



s+5/3
s+1

0


s+3
.
− 12

Note that the first and third matrices are unimodular,
while the second matrix is in Smith-McMillan form. Thus,z = −5/3 is a transmission zero of G.
As another example, considerG(s) =1(s+1)21(s+1)(s+2)

a pole when a pair of transfer
functions are cascaded.
R(s) 

which has no blocking zeros and normal rank 2. To determine the transmission zeros of G, consider the factorization
G(s) =

Mathematically, a zero can cancel1(s+1)(s+2).
s+3(s+2)2G(s) = U1 (s)1(s+1)2 (s+2)20
0
U2 (s),
s+2

where U1 , U2 are the unimodular matrices
U1 (s) =(s + 2)(s3 + 4s2 + 5s + 1)

1
(s + 1)(s3 + 5s2 + 8s + 3) 1



and

U2 (s) =−(s + 2) (s + 1)(s2 + 3s + 1).
1−s(s + 2)

Hence, the McMillan degree of G is 2, the poles of G are −1and −2, the transmission zero of G is −2, and G has noblocking zeros. Note that −2 is both a pole and a transmission zero of G. Note also that, although G is strictly proper,
the Smith-McMillan form of G is improper.
Transmission zeros are usually computed by using astate-space method that involves a minimal realization ofG(s). For the SISO transfer function G(s), it is useful to notethe identity [6, p. 520]
C adj(sI − A)B = − det R(s),where R(s) is the Rosenbrock system matrix defined bysI − A B.
C
0Consequently, the complex number z is a zero of the SISO
transfer function G(s) = C(sI − A)−1 B if and only if
det R(z) = 0.
Now, suppose that G is an l × m transfer function, witha minimal realization (A, B, C), where B is an n × m matrix
and C is a l × n matrix. Then, the Rosenbrock system
matrix R has size (n + l ) × (n + m) and thus is not necessarily square. Now, z ∈ C is an invariant zero of (A, B, C) if
the rank of R(z) is less than the normal rank of R. Furthermore, it is shown in [23, p. 111] that the transmission zerosof G are exactly the invariant zeros of (A, B, C). Note thatin the case of full-state measurement, that is, C = I, therank of R(s) is n + rank B for all values of s. Hence, in thiscase, G has no transmission zeros.
By writing R(s) as


Then,



R(s) =




−A B
−I 0
−s
,
C 0
0 0

it can be seen that the invariant zeros of (A, B, C) are thegeneralized eigenvalues of the matrix pencil [58]


 

−A B
−I 0
,
.
C 0
0 0

Consequently, while the computation of poles is an eigenvalue problem, the computation of zeros is a generalized
eigenvalue problem, which is more difficult [59], [60]. Notethat there is no assumption that R(s) is square. We also
note that the first n rows of R(s) are C(s)  [sI − A B ],which is the controllability pencil, while the first n columnssI − Aof R(s) are O(s) , which is the observability pencil.
C
The PBH tests for controllability and observability arebased on C(s) and O(s), respectively.

NONMINIMUM-PHASE ZEROS IN
DISCRETE-TIME SYSTEMSThe above discussion is confined to continuous-time systems. For discrete-time systems, nonminimum-phasezeros are zeros that lie outside the unit disk. Such zeros
may or may not cause initial undershoot in the step
response [61]. In addition, the root-locus rules for discretetime systems are identical to the rules for continuous-timesystems. However, unlike continuous-time systems,

JUNE 2007 « IEEE CONTROL SYSTEMS MAGAZINE 55

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

The step response of a system with two positive zeros initially moves inthe “correct” direction, reverses course to move in the “wrong” direction,and then reverses course yet again to move in the “correct” direction.

which can have infinite gain margin for a loop transfer
function with relative degree less than or equal to two,
discrete-time systems with relative degree one or greater
have finite gain margins.
Most discrete-time systems arise as sampled continuoustime systems. In this regard it is important to note that sampled minimum-phase transfer functions are often
nonminimum phase [62], [63], [64, p. 65]. In particular, sufficiently fast sampling of a continuous-time system withrelative degree greater than two gives rise to nonminimumphase zeros [65]. Techniques for addressing nonminimumphase sampled-data systems are given in [61] and [66]–[70].
It is clear from root locus that nonminimum-phasezeros in discrete time impose limitations on robustness
and performance [71], [72]. Furthermore, as in continuous
time, discrete-time nonminimum-phase zeros prevent theuse of plant-inversion-based controllers. This limitation isapparent in adaptive control, where many methods are
restricted to minimum-phase plants [73]–[76].

CONCLUSIONS
Zeros are a fundamental aspect of systems and control theory; however, the causes and effects of zeros are more subtle than those of poles. In particular, positive zeros cancause initial undershoot (initial error growth), zero crossings, and overshoot in the step response of a system,
whereas nonminimum-phase zeros limit bandwidth. Bothof these aspects have real-world implications in many
applications. Nonminimum-phase zeros exacerbate thetradeoff between the robustness and achievable performance of a feedback control system.
From a control-theoretic point of view, a nonminimumphase zero in the loop transfer function L is arguably theworst feature a system can possess. Every feedback synthesis methodology must accept limitations due to the presence of open-right-half-plane zeros, and the mark of agood analysis tool is the ability to capture the performance
limitations arising from nonminimum-phase zeros.
While the effects of open-right-half-plane poles are evident to every student of control, the lurking dangers andlimitations of open-right-half-plane zeros are more subtle
and thus more insidious. As control practitioners, we may
despise open-right-half-plane zeros because of the difficulties they entail. However, those of us who develop control
techniques relish the challenge that open-right-half-planezeros present in our unique field of endeavor.

ACKNOWLEDGMENTS
We wish to thank Bela Palancz for providing the SmithMcMillan forms of the MIMO transfer functions, Suhail
Akhtar for suggesting Figure 1, Davor Hrovat for helpful
discussions on the nonminimum-phase behavior of rearwheel steering, John Hauser for showing us that the nonholonomic unicycle has a controllable and observablelinearization under nonzero speed, Dan Davison for helpful discussions, Carl Knospe for extensive discussions,
Mario Santillo for helpful suggestions, and several anonymous reviewers for detailed advice and suggestions.

REFERENCES[1] J.D. Barrow, The Book of Nothing. London: Jonathan Cape, 2000.
[2] R. Kaplan, The Nothing That Is: A Natural History of Zero. London: Oxford
Univ. Press, 2000.
[3] C. Seife, Zero: The Biography of a Dangerous Idea. New York: Viking, 2000.
[4] D.E. Duncan, Calendar: Humanity's Epic Struggle to Determine a True andAccurate Year. New York: Avon, 1998.
[5] R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985.
[6] D.S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton, NJ: Princeton Univ. Press, 2005.
[7] T. Norimatsu and M. Ito, “On the zero non-regular control system,” J.
Inst. Elect. Eng. Japan, vol. 81, pp. 566–575, 1961.
[8] T. Mita and H. Yoshida, “Undershooting phenomenon and its control in
linear multivariable servomechanisms,” IEEE Trans. Automat. Contr., vol. 26,
pp. 402–407, 1981.
[9] M. Vidyasagar, “On undershoot and nonminimum phase zeros,” IEEE
Trans. Automat. Contr., vol. 31, p. 440, 1986.
[10] G. Oriolo, A. De Luca, and M. Vendittelli, “WMR control via dynamic
feedback linearization: design, implementation, and experimental
validation,” IEEE Trans. Contr. Syst. Technol., vol. 10, pp. 835–852, 2002.
[11] E. Yang, D. Au, T. Mita, and H. Hu. “Nonlinear tracking control of a carlike mobile robot via dynamic feedback linearization,” Proc. Contr. 2004, U.
of Bath, UK, ID-218, 2004.
[12] D. Karnopp, Vehicle Stability. New York: Marcel Dekker, 2004.
[13] R.F. Stengel, Flight Dynamics. Princeton, NJ: Princeton Univ. Press, 2004.
[14] D.S. Bernstein, “What makes some control problems hard?,” IEEE Contr.
Syst. Mag., vol. 22, pp. 8–19, August 2002.
[15] J. Hong and D.S. Bernstein, “Bode integral constraints, colocation, andspillover in active noise and vibration control,” IEEE Trans. Contr. Syst. Technol., vol. 6, pp. 111–120, 1998.
[16] E.H. Maslen, “Positive real zeros in flexible beams,” Shock and Vibration,
vol. 2, pp. 429–435, 1995.
[17] D.K. Miu, Mechatronics: Electromechanics and Contromechanics. New York:
Springer-Verlag, 1993.
[18] J. Chandrasekar, J.B. Hoagg, and D.S. Bernstein, “On the zeros ofasymptotically stable serially connected structures,” in Proc. Conf. Decision
Control, Paradise Island, The Bahamas, Dec. 2004, pp. 2638–2643.
[19] J.B. Hoagg and D.S. Bernstein, “On the zeros, initial undershoot, andrelative degree in lumped-mass structures,” in Proc. American Control Conf.,
Minneapolis, MN, June 2006, pp. 394–399.
[20] J.B. Hoagg, J. Chandrasekar, and D.S. Bernstein, “On the zeros, initial
undershoot, and relative degree of collinear, lumped-parameter structures,”
ASME J. Dyn. Syst. Meas. Contr., submitted for publication.

56 IEEE CONTROL SYSTEMS MAGAZINE » JUNE 2007

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.

[21] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis
and Design. New York: Wiley, 1996.
[22] B.R. Holt and M. Morari, “Design of resilient processing plants–VI. The
effect of right-half-plane zeros on dynamic resilience,” Chem. Eng. Sci., vol. 40,
pp. 59–74, 1985.
[23] H.H. Rosenbrock, State Space and Multivariable Theory. Camden, NJ: Nelson, 1970.
[24] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis
and Design, 2nd ed. Hoboken, NJ: Wiley, 2005.
[25] B.A. Leon de la Barra, “On undershoot in SISO systems,” IEEE Trans.
Automat. Contr., vol. 39, pp. 578–581, 1994.
[26] M. El-Khoury, O.D. Crisalle, and R. Longchamp, “Influence of zero
locations on the number of step-response extrema,” Automatica, vol. 29,
pp. 1571–1574, 1993.
[27] G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design.
Englewood Cliffs, NJ: Prentice-Hall, 2001.
[28] J.C. Doyle, B.A. Francis, and A.R. Tannenbaum, Feedback Control Theory.
New York: Macmillan, 1992.
[29] P. Dorato, Analytic Feedback System Design: An Interpolation Approach.
Pacific Grove, CA: Brooks/Cole, 2000.
[30] M. Vidyasagar, Control System Synthesis: A Factorization Approach. Cambridge, MA: MIT Press, 1985.
[31] R.H. Middleton, “Trade-offs in linear control system design,” Automatica, vol. 27, pp. 281–292, 1991.
[32] I. Mareels, “A simple selftuning controller for stably invertible systems,”
Syst. Control Lett., vol. 4, pp. 5–16, 1984.
[33] J.B. Hoagg and D.S. Bernstein, “Direct adaptive stabilization of minimum-phase systems with bounded relative degree,” in Proc. Conf. Decision
Control, Paradise Island, The Bahamas, Dec. 2004, pp. 183–188.
[34] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems. New York:
Wiley, 1972.
[35] L. Qiu and E.J. Davison, “Performance limitations of non-minimum phasesystems in the servomechanism problem,” Automatica, vol. 29, pp. 337–349, 1993.
[36] J.S. Freudenberg and D.P. Looze, “Right half plane poles and zeros anddesign tradeoffs in feedback systems,” IEEE Trans. Autom. Contr., vol. AC-30,
pp. 555–565, 1985.
[37] D.P. Looze and J.S. Freudenberg, “Limitations of feedback properties
imposed by open-loop right half plane poles,” IEEE Trans. Automat. Contr.,
vol. 36, pp. 736–739, 1991.
[38] J. Chen, “Sensitivity integral relations and design tradeoffs in linear
multivariable feedback systems,” IEEE Trans. Automat. Contr., vol. AC-40,
pp. 1700–1716, 1995.
[39] M.M. Seron, J.H. Braslavsky, and G.C. Goodwin, Fundamental Limitationsin Filtering and Control. New York: Springer-Verlag, 1997.
[40] J. Chen, L. Qiu, and Q. Toker, “Limitations on maximal tracking accuracy,” IEEE Trans. Automat. Contr., vol. 45, pp. 326–331, 2000.
[41] K. Lau, R.H. Middleton, and J.H. Braslavsky, “Undershoot and settling
time tradeoffs for nonminimum phase systems,” IEEE Trans. Automat. Contr.,
vol. 48, pp. 1389–1393, 2003.
[42] K. Zhou, “Comparison between H2 and H∞ controllers,” IEEE Trans.
Automat. Contr., vol. 37, pp. 1261–1265, 1992.
[43] K.S. Narendra and A.M. Annaswamy, Stable Adaptive Systems. Englewood Cliffs, NJ: Prentice-Hall, 1989.
[44] K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1996.
[45] S. Kahne, “Pole-zero cancellations in SISO linear feedback systems,”
IEEE Trans. Educ., vol. 33, pp. 240–243, 1990.
[46] A.G.J. MacFarlane and N. Karcanias, “Poles and zeros of linear multivariable systems: A survey of the algebraic, geometric and complex-variable
theory,” Int. J. Control, vol. 24, pp. 33–74, 1976.
[47] B. Kouvaritakis and A.G.J. MacFarlane, “Geometric approach to analysis
and synthesis of system zeros. Part 1. Square systems,” Int. J. Control, vol. 23,
pp. 149–166, 1976.
[48] B. Kouvaritakis and A.G.J. MacFarlane, “Geometric approach to analysis and synthesis of system zeros. Part 1. Non-square systems,” Int. J. Control, vol. 23, pp. 167–181, 1976.
[49] A. Emami-Naeini, “Computation of zeros of linear multivariable systems,” Automatica, vol. 18, pp. 415–430, 1982.
[50] M.K. Sain and C.B. Schrader, “Research on system zeros: A survey,” Int.
J. Control, vol. 50, pp. 1407–1433, 1989.
[51] C.B. Schrader and M.K. Sain, “The role of zeros in the performance ofmultiinput, multioutput feedback systems,” IEEE Trans. Educ., vol. 33,
pp. 244–257, 1990.

[52] B.F. Wyman and M.K. Sain, “On the zeros of a minimal realization,” Lin.
Alg. Applicat., vol. 50, pp. 621–637, 1983.
[53] B.F. Wyman, M.K. Sain, G. Conte, and M.A. Perdon, “On the zeros
and poles of a transfer function,” Lin. Alg. Applicat., vol. 122–124, pp.
123–144, 1989.
[54] B.F. Wyman, M.K. Sain, G. Conte, and M.A. Perdon, “Poles and zeros ofmatrices of rational functions,” Lin. Alg. Applicat., vol. 157, pp. 113–139, 1991.
[55] M.J. Corless and A.E. Frazho, Linear Systems and Control. New York:
Marcel Dekker, 2003.
[56] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980.
[57] J.M. Maciejowski, Multivariable Feedback Design. Reading, MA: AddisonWesley, 1989.
[58] G.W. Stewart and J. Sun, Matrix Perturbation Theory. New York: Academic, 1990.
[59] A. Emami-Naeini and P. Van Dooren, “Computation of zeros of linear
multivariable systems,” Automatica, vol. 18, pp. 415–430, 1982.
[60] J.M. Berg and H.G. Kwatny, “Unfolding the zero structure of a linear
control system,” Lin. Alg. Applicat., vol. 258, pp. 19–39, 1997.
[61] B.L. De La Barra, M. El-Khoury, and M. Fernandez, “On undershoot in
scalar discrete-time systems,” Automatica, vol. 32, pp. 255–259, 1996.
[62] Y. Fu and G.A. Dumont, “Choice of sampling to ensure minimum-phasebehavior,” IEEE Trans. Automat. Contr., vol. 34, pp. 560–563, 1989.
[63] T. Hagiwara, “Analytic study on the intrinsic zeros of sampled-data systems,” IEEE Trans. Automat. Contr., vol. 41, pp. 261–263, 1996.
[64] K.J. Astrom and B. Wittenmark, Computer-Controlled Systems, 3rd ed.
Englewood Cliffs, NJ: Prentice-Hall, 2005.
[65] K.J. Astrom, P. Hagander, and J. Sternby, “Zeros of sampled data systems,” Automatica, vol. 20, pp. 31–38, 1983.
[66] U. Holmberg, P. Myszkorowski, and R. Longchamp, “On compensationof nonminimum-phase zeros,” Automatica, vol. 31, pp. 1433–1441, 1995.
[67] J. Tokarzewski, “System zeros via the Moore-Penrose pseudoinverse
and SVD of the first nonzero Markov parameter,” IEEE Trans. Automat.
Contr., vol. 43, pp. 1285–1291, 1998.
[68] J. Tokarzewski, Finite Zeros in Discrete Time Control Systems. New York:
Springer-Verlag, 2006.
[69] D.S. Bayard, “Stable direct adaptive periodic control using only plant orderknowledge,” Int. J. Adapt. Control Signal Process., vol. 43, pp. 551–570, 1996.
[70] P.T. Kabamba, “Control of linear systems using generalized sampled-datahold functions,” IEEE Trans. Automat. Contr., vol. AC-32, pp. 772–783, 1987.
[71] L.B. Jemaa and E.J. Davison, “Performance limitations in the robust servomechanism problem for discrete-time LTI systems,” IEEE Trans. Automat.
Contr., vol. 48, pp. 1299–1311, 2003.
[72] W. Su, L. Qiu, and J. Chen, “Fundamental limit of discrete-time systems in tracking multi-tone sinusoidal signals,” Automatica, vol. 43, pp.
15–30, 2007.
[73] K.J. Astrom and B. Wittenmark, Adaptive Control, 2nd ed. Reading, MA:
Addison-Wesley, 1995.
[74] G.C. Goodwin and K.S. Sin, Adaptive Filtering, Prediction, and Control.
Englewood Cliffs, NJ: Prentice-Hall, 1984.
[75] P. Ioannou and J. Sun, Robust Adaptive Control. Englewood Cliffs, NJ:
Prentice-Hall, 1996.
[76] I.D. Landau, Adaptive Control: The Model Reference Approach. New York:
Marcel Dekker, 1979.

AUTHOR INFORMATION
Jesse B. Hoagg received his Ph.D. in aerospace engineeringfrom the University of Michigan in 2006. His professional
experience includes appointments as a National Defense Science and Engineering Graduate Fellow for the U.S. Department of Defense and as a space scholar for the Space
Vehicles Directorate of the U.S. Air Force Research Laboratory. His research interests are in adaptive control and control
of structures.
Dennis S. Bernstein is a professor in the Aerospace Engineering Department at the University of Michigan. He is theeditor-in-chief of IEEE Control Systems Magazine and author of
Matrix Mathematics: Theory, Facts, and Formulas with Application
to Linear Systems Theory.

JUNE 2007 « IEEE CONTROL SYSTEMS MAGAZINE 57

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 10,2010 at 05:58:04 UTC from IEEE Xplore. Restrictions apply.