Feedback Control of Negative-Imaginary SystemsFlexible structures with colocated actuators and sensors

# Feedback Control of Negative-Imaginary Systems Flexible structures with colocated actuators and sensors

Ian R. Petersen and Alexander Lanzon — July 1, 2019

## I

Highly resonant dynamics can severely degrade the performance of technological systems. Structural modes in machines and robots, ground and aerospace vehicles, and precision instrumentation, such as atomic force microscopes and optical systems, can limit the ability of control systems to achieve the desired performance. Consequently, control systems must be designed to suppress the effects of these dynamics, or at least avoid exciting them beyond open-loop levels. Open-loop techniques for highly resonant systems, such as input shaping [1], as well as closed-loop techniques, such as damping augmentation [2, 3], can be used for this purpose.

Structural dynamics are often difficult to model with high precision due to sensitivity to boundary conditions as well as aging and environmental effects. Therefore, active damping augmentation to counteract the effects of external commands and disturbances must account for parametric uncertainty and unmodeled dynamics. This problem is simplified to some extent by using force actuators combined with colocated measurements of velocity, position, or acceleration, where colocated refers to the fact that the sensors and actuators have the same location and the same direction. Colocated control with velocity measurements, called negative-velocity feedback, can be used to directly increase the effective damping, thereby facilitating the design of controllers that guarantee closed-loop stability in the presence of plant parameter variations and unmodeled dynamics [4, 1]. This guaranteed stability property can be established by using results on passive systems [5, 6]. However, the theoretical properties of negative-velocity feedback are based on the idealized assumption of colocation and require the availability of velocity sensors, which may be expensive. Also, the choice of measured variable may depend on whether the desired objective is shape control or damping augmentation.

An alternative approach to negative-velocity feedback is positive-position feedback, where position sensors are used in place of velocity sensors. Although position sensors can facilitate the objective of shape control, it is less obvious how they can be used for damping augmentation. Nevertheless, it is shown in [7, 8] that a positive-position feedback controller can be designed to increase the damping of the modes of a flexible structure. Furthermore, this controller is robust against uncertainty in the modal frequencies as well as unmodeled plant dynamics. As shown in [7, 8, 9, 10], the robustness properties of positive-position feedback are similar to those of negative-velocity feedback.

The present article investigates the robustness of positive-position feedback control of flexible structures with colocated force actuators and position sensors. In particular, the theory of negative-imaginary systems [9, 10] is used to reveal the robustness properties of multi-input, multi-output (MIMO) positive-position feedback controllers and related types of controllers for flexible structures [11, 12, 1, 13, 14]. The negative-imaginary property of linear systems can be extended to nonlinear systems through the notion of counterclockwise input-output dynamics [15, 16]. It is shown in [17] for the single-input, single-output (SISO) linear case that the results of [15, 16] guarantee the stability of a positive-position feedback control system in the presence of unmodeled dynamics and parameter uncertainties that maintain the negative-imaginary property of the plant.

Positive-position feedback can be regarded as one of the last areas of classical control theory to be encompassed by modern control theory. In this article, positive-position feedback, negative-imaginary systems, and related control methodologies are brought together with the underlying systems theory.

Table I summarizes notation used in this article, while Table II lists acronyms.

## Ii Flexible Structure Modeling

In modeling an undamped flexible structure with a single actuator and a single sensor, modal analysis can be applied to the relevant partial differential equation [18], leading to the transfer function

 P(s)=∞∑i=1ϕi(s)s2+ω2i, (1)

where each is a modal frequency, the functions are first-order polynomials, and for . In the case of a structure with a force actuator and colocated velocity sensor, the form of the numerator of (1) is determined by the passive nature of the flexible structure. Since the product of the force actuator input and the velocity sensor output represents the power provided by the actuator to the structure at time , conservation of energy implies

 E(t)≤E(0)+∫t0u(τ)y(τ)dτ (2)

for all , where represents the energy stored in the system at time , and represents the initial energy stored in the system. In this case, the variables and are dual. The passivity condition (2) implies that the transfer function is positive real according to the following definition [5].

###### Definition 1

([19, 20]) The square transfer function matrix is positive real if the following conditions are satisfied:

1. All of the poles of lie in CLHP.

2. For all in ORHP,

 P(s)+P∗(s)≥0. (3)

If is positive real, then it follows that [19, 20]

 P(ȷω)+P∗(ȷω)≥0 (4)

for all such that is not a pole of . If is a SISO transfer function, then, for all such that is neither a pole nor a zero of , (4) is equivalent to the phase condition .

###### Definition 2

([19]) The nonzero square transfer function matrix is strictly positive real if there exists such that the transfer function matrix is positive real.

If is strictly positive real, then it follows [19] that all of the poles of lie in OLHP and

 P(ȷω)+P∗(ȷω)>0 (5)

for all . If P(s) is a SISO transfer function, then (5) holds for all such that is neither a pole nor a zero of if and only if the phase condition holds for all such that is neither a pole nor a zero of .

Now consider the positive-real transfer function from force actuation to velocity measurement given by

 P(s)=∞∑i=1ψ2iss2+κis+ω2i, (6)

where, for all , is the viscous damping constant associated with the th mode and . The transfer function (6) satisfies the phase condition for all . However, (6) has a zero at the origin, and thus (5) is not satisfied for . Hence, (6) is not strictly positive real.

Now consider a lightly damped flexible structure with colocated sensor and actuator pairs. Let denote the force actuator input signals, and let denote the corresponding velocity sensor output signals. The actuator and sensor in the th colocated actuator and sensor pair are dual when the product is equal to the power provided to the structure by the th actuator at time . Now, we let

 Y(s)=P(s)U(s),

where

 U(s)=⎡⎢ ⎢⎣U1(s)⋮Um(s)⎤⎥ ⎥⎦,  Y(s)=⎡⎢ ⎢⎣Y1(s)⋮Ym(s)⎤⎥ ⎥⎦.

For , and are the Laplace transforms of and , respectively, and is the transfer function matrix of the system. Then is positive real and has the form

 P(s)=∞∑i=1ss2+κis+ω2iψiψTi, (7)

where, for all , , , and is an vector. A review of positive-real and passivity theory is given in “What Is Positive-real and Passivity Theory?”

## Iii Negative-Imaginary Systems

Mechanical structures with colocated force actuators and position sensors do not yield positive-real systems because the product of force and position is not equal to the power provided by the actuator [9, 10]. In this case, the transfer function matrix from the force actuator inputs to the position sensor outputs is of the form

 P(s)=∞∑i=11s2+κis+ω2iψiψTi, (8)

where, for all , , , and is an vector. Therefore, the Hermitian-imaginary part

 IH[P(ȷω)]=−12ȷ(P(ȷω)−P∗(ȷω))

of the frequency response function matrix satisfies

 IH[P(ȷω)]=−ω∞∑i=1κi(ω2i−ω2)2+ω2κ2iψiψTi≤0 (9)

for all That is, the frequency response function matrix for the transfer function matrix (8) has negative-semidefinite Hermitian-imaginary part for all . We thus refer to the transfer function matrix in (8) as negative imaginary. A formal definition follows.

###### Definition 3

The square transfer function matrix is negative-imaginary (NI) if the following conditions are satisfied:

1. All of the poles of lie in OLHP.

2. For all ,

 ȷ[P(ȷω)−P∗(ȷω)]≥0. (10)

A linear time-invariant system is NI if its transfer function matrix is NI.

A discussion of negative-imaginary transfer functions arising in electrical circuits is given in “Applications to Electrical Circuits.”

In the SISO case, a transfer function is negative imaginary if and only if it has no poles in CRHP and its phase is in the interval at all frequencies that do not correspond to imaginary-axis poles or zeros. Consequently, the positive-frequency Nyquist plot of a SISO negative-imaginary transfer function lies below the real axis as shown in Figure 1. Hence, a negative-imaginary transfer function can be viewed as a positive-real transfer function rotated clockwise by deg in the Nyquist plane.

Velocity sensors can be used in negative-velocity feedback control, whereas position sensors can be used in positive-position feedback [1, 7, 8, 14, 13, 11, 12]. Indeed, positive-real theory and negative-imaginary theory [9, 10] achieve internal stability by a process referred to as phase stabilization, since instability is avoided by ensuring appropriate restrictions on the phase of the corresponding open-loop systems. Gain stabilization, which is based on the small-gain theorem [19], guarantees robust stability when the magnitude of the loop transfer function is less than unity at all frequencies. As in positive-real analysis, robust stability of negative-imaginary systems [9, 10] does not require the magnitude of the loop transfer function to be less than unity at all frequencies to guarantee stability. In order to present results on the robust stability of positive-position feedback and related control schemes, we now define MIMO strictly negative-imaginary systems.

###### Definition 4

The square transfer function matrix is strictly negative-imaginary (SNI) if the following conditions are satisfied:

1. All of the poles of lie in OLHP.

2. For all ,

 ȷ[P(ȷω)−P∗(ȷω)]>0. (11)

A linear time-invariant system is SNI if its transfer function matrix is SNI.

###### Lemma 1

If the transfer function matrix is NI, respectively, SNI, and the transfer function matrix is NI, then

 P(s)=P1(s)+P2(s) (12)

is NI, respectively, SNI.

{proof}

This result follows directly from Definition 3 and Definition 4.

###### Theorem 2

Consider the NI transfer function matrices and , and suppose that the positive-feedback interconnection shown in Figure 2 is internally stable. Then the corresponding closed-loop transfer function matrix

 (13)

is NI. Furthermore, if, in addition, either or is SNI, then (13) is SNI.

{proof}

The internal stability of the positive feedback interconnection shown in Figure 2 implies that is asymptotically stable. Given , , and , define

 [y1y2]=T(ȷω)[w1w2].

Letting and , it follows from the positive feedback interconnection that and . Furthermore, using the fact that and are NI, it follows that

 ȷ[w∗1w∗2][T(ȷω)−T∗(ȷω)][w1w2] = = = ȷ(u∗1y1+u∗2y2)−ȷ(y∗1u1+y∗2u2) = ȷ(u∗1M(ȷω)u1−u∗1M(ȷω)∗u1)+ȷ(u∗2N(ȷω)u2−u∗2N(ȷω)∗u2) ≥ 0.

Since , , and are arbitrary, it follows that

 ȷ[T(ȷω)−T(ȷω)∗]≥0

for all and hence, is NI. The SNI result follows using similar arguments.

###### Theorem 3

Consider the NI transfer function matrices

 M(s)=[M11(s)M12(s)M21(s)M22(s)],  N(s)=[N11(s)N12(s)N21(s)N22(s)],

and suppose that the feedback interconnection shown in Figure 3 is internally stable. Then the corresponding closed-loop transfer function matrix

 T(s) = [M11(2)+M12(s)(I−N11(s)M22(s))−1N11(s)M21(s)N21(s)(I−M22(s)N11(s))−1M21(s) M12(s)(I−N11(s)M22(s))−1N12(s)N22(s)+N21(s)(I−M22(s)N11(s))−1M22(s)N12(s)]

is NI. Furthermore, if in addition, either or is SNI, then (3) is SNI.

{proof}

The internal stability of the feedback interconnection shown in Figure 3 implies that is asymptotically stable. Given , , and , define

 [y1y2]=T(ȷω)[w1w2].

Letting

 [u1u2]=[(I−N11(s)M22(s))−1N11(s)M21(s)(I−N11(s)M22(s))−1N12(s)(I−M22(s)N11(s))−1M21(s)(I−M22(s)N11(s))−1M22(s)N12(s)][w1w2],

it follows from the feedback interconnection shown in Figure 3 that

 [y1u2]=M(ȷω)[w1u1],  [u1y2]=N(ȷω)[u2w2]. (15)

Furthermore, using (15) and the fact that and are NI, it follows that

 ȷ[w∗1w∗2][T(ȷω)−T∗(ȷω)][w1w2] = = = ȷ([w∗1u∗1]M(ȷω)[w1u1]−[w∗1u∗1]M(ȷω)∗[w1u1]) +ȷ([u∗2w∗2]N(ȷω)[u2w2]−[u∗2w∗2]N(ȷω)∗[u2w2]) ≥ 0.

Since , , and are arbitrary, it follows that

 ȷ[T(ȷω)−T(ȷω)∗]≥0

for all and hence, is NI. The SNI result follows using similar arguments.

Underlying the stability properties of positive-position feedback is the observation that the transfer function matrix of a lightly damped flexible structure with colocated force actuators and position sensors is NI. Indeed, note that all poles of

 Pi(s)=1s2+κis+ω2iψiψTi

in the transfer function matrix (8) lie in OLHP. Also, for all ,

 ȷ[Pi(ȷω)−P∗i(ȷω)]=IH(Pi(ȷω))=2κiω(ω2i−ω2)2+κ2iω2ψiψTi≥0.

Hence, it follows from Definition 3 that each is NI. Therefore, it follows from Lemma 1 that the transfer function matrix (8) is NI.

### Iii-a The Negative-Imaginary Lemma

The following theorem, which is proved in [10, 21], provides a state-space characterization of NI systems in terms of a pair of linear matrix inequalities (LMIs). This result is analogous to the positive-real lemma [20, 19], and thus is referred to as the negative-imaginary lemma.

###### Theorem 4

Consider the minimal state-space system

 ˙x=Ax+Bu, (16) y=Cx+Du, (17)

where , , , and . The system (16), (17) is NI if and only if has no eigenvalues on the imaginary axis, is symmetric, and there exists a positive-definite matrix satisfying

 AY+YAT≤0, (18)
 B+AYCT=0. (19)

In Theorem 4 it follows from the Lyapunov inequality (18), the positive definiteness of , and the assumption that has no eigenvalues on the imaginary axis that the matrix is asymptotically stable [22, Corollary 11.8.1].

###### Corollary 5

Consider the minimal state-space system (16), (17), where , , , and . The system (16), (17) is SNI if and only if the following conditions are satisfied:

1. has no eigenvalues on the imaginary axis.

2. is symmetric.

3. There exists a positive-definite matrix such that (18) and (19) are satisfied.

4. The transfer function matrix is such that has no transmission zeros on the imaginary axis except possibly at .

{proof}

Assuming conditions 1) - 3), it follows from Theorem 4 that (16), (17) is NI. Now suppose that (16), (17) is not SNI. Then using Definition 3 and Definition 4, it follows that there exist and a nonzero vector such that

 ȷu∗[M(ȷω)−M∗(ȷω)]u=0.

Thus, has a transmission zero at , which contradicts condition 4). Hence (16), (17) is SNI.

Conversely, suppose that (16), (17) is SNI. Then, (16), (17) is NI and Theorem 4 implies that conditions 1) - 3) are satisfied. Also, it follows from Definition 4 that

 ȷ[M(ȷω)−M∗(ȷω)]>0

for all . Therefore has no transmission zeros on the imaginary axis except possibly at , and thus condition 4) is satisfied.

To illustrate Theorem 4 and Corollary 5, consider the system

 ˙x=−x+u, (20) y=x (21)

with transfer function

 M(s)=1s+1. (22)

The positive-frequency Nyquist plot of (22) given in Figure 4 shows that (20), (21) is both SNI and strictly positive real.

Applying Theorem 4 with , , , and , condition (19) can be satisfied by choosing . Then, . It now follows from Theorem 4 that (20), (21) is NI. Also, note that

 M(s)−MT(−s)=1s+1−1−s+1=2ss2−1

has no zeros on the imaginary axis except at . It then follows from Corollary 5 that (20), (21) is SNI.

Now consider the transfer function

 M(s)=2s2+s+1(s2+2s+5)(s+1)(2s+1). (23)

The positive-frequency Nyquist plot of in Figure 5 shows that for all , and thus is NI. However, Figure 5 shows that there exists such that , and thus is not SNI. Now consider the minimal realization (16), (17) of (23) given by

 A=⎡⎢ ⎢ ⎢⎣−3.5−8.5−8.5−2.5100001000010⎤⎥ ⎥ ⎥⎦,  B=⎡⎢ ⎢ ⎢⎣2.5−310⎤⎥ ⎥ ⎥⎦, (24) C=[0001],  D=0. (25)

In order to construct a matrix satisfying the assumptions of Theorem 4, note that the assumptions of Theorem 4 are equivalent to the requirement that the matrix have no eigenvalues on the imaginary axis and

 [AY+YATB+AYCTBT+CYAT0]≤0, Y>0.

Using LMI software [23], we obtain

 Y=⎡⎢ ⎢ ⎢⎣100.375−36.752.53−36.7518.5−3−12.5−3103−100.2⎤⎥ ⎥ ⎥⎦>0.

Therefore Theorem 4 implies that (16), (17), (24), (25) is NI.

Now to determine whether (16), (17), (24), (25) is SNI, note that

 M(s)−MT(−s)=2s2+s+1(s2+2s+5)(s+1)(2s+1)−2s2−s+1(s2−2s+5)(−s+1)(−2s+1) =−24(s2+1)24s8+19s6+71s4−119s2+25,

has a double zero at . Consequently, (16), (17), (24), (25) is not SNI.

### Iii-B Two Strict Negative-Imaginary Lemmas

The following theorems give sufficient conditions for the SNI property.

###### Theorem 6

Consider the minimal state-space system (16), (17), where , , , and . Suppose the following conditions are satisfied:

1. All eigenvalues of are in OLHP.

2. is symmetric.

3. There exist a positive-definite matrix and positive numbers such that is not an eigenvalue of and the matrices

 ~A=[A00−αI],  ~B=[BεI],  ~C=[C−I] (26)

satisfy

 ~A~Y+~Y~AT≤0

and

 ~B+~A~Y~CT=0.

Then (16), (17) is SNI.

The proof of Theorem 6 requires the following lemma.

###### Lemma 7

Let and . Then the transfer function matrix

 M(s)=εs+αI (27)

is SNI.

{proof}

Let the transfer function matrix (27) have minimal state-space realization

 ˙x=−αx+εu, (28) y=x. (29)

Theorem 4 and Corollary 5 can be applied to (28), (29) with , , , and . Setting , it follows that and . Hence, Theorem 4 implies that (28), (29) is NI. Furthermore,

 M(s)−MT(−s) = εs+αI−ε−s+αI = 2εss2−α2I.

Thus, has no purely imaginary transmission zeros except possibly at . Hence, it follows from Corollary 5 that (28), (29) is SNI.

Proof of Theorem 6: Let be the transfer function matrix of (16), (17). Since is not a pole of , a minimal state-space realization of the transfer function matrix is

 ˙x1=Ax1+Bu, ˙x2=−αx2+εu, y=Cx1−x2+Du.

Let

 ~A=[A00−αI],  ~B=[BεI],  ~C=[C−I],  ~D=D.

Assuming conditions 1) - 3), it follows from Theorem 4 that is NI. Then Lemma 1 and Lemma 7 imply that is SNI. \QED

To illustrate Theorem 6, we consider lightly damped flexible structures with force actuators and position sensors. An integral resonant controller [13, 14] has the form

 C(s)=[sI+ΓΦ]−1Γ, (30)

where and are positive-definite matrices. In the SISO case [13], integral resonant controllers are derived by first adding a direct feedthough to a resonant system with a colocated force actuator and position sensor. Then, application of integral feedback leads to damping of the resonant poles. Combining the direct feedthrough with the integral feedback leads to a SISO controller of the form (30). In [14], this class of SISO controllers is generalized to MIMO controllers of the form (30).

Integral resonant controllers provide integral force feedback [1], which refers to control that uses position actuators, force sensors, and integral feedback. In [1], integral feedback is modified by moving the integrator pole slightly to the left in the complex plane to alleviate actuator saturation. A SISO controller transfer function of the form (30) results from this process.

###### Theorem 8

The transfer function matrix (30) with positive definite and positive definite is SNI.

Proof: Consider the minimal state-space realization of (30) given by

 ˙x=−ΓΦx+Γu, y=x.

Let and be such that is not an eigenvalue of . The corresponding matrices in (26) are

 ~A=[−ΓΦ00−αI],  ~B=[ΓεI],  ~C=[I−I],  ~D=0.

Also, let

 ~Y=[Φ−1000]+ε⎡⎢⎣(1α+1)I(1α+1)I(1α+1)II⎤⎥⎦.

Thus,

 ~B+~A~Y~CT=0. (31)

Furthermore, note that

 [Φ−1000]

is positive semidefinite, and

 ⎡⎢⎣(1α+1)I(1α+1)I(1α+1)II⎤⎥⎦

is positive definite. Hence, .

Using the definitions of and , it follows that

 ~A~Y+~Y~AT=[−Γ000]+ε⎡⎢⎣−(1α+1)(ΓΦ+ΦΓ)−(1α+1)(ΓΦ+αI)−(1α+1)(ΓΦ+αI)T−2(α+1)I⎤⎥⎦.

Furthermore, the matrix

 [Γ000]

is positive semidefinite. For every nonzero vector of the form , we have

 [0x2]T⎡⎢⎣(1α+1)(ΓΦ+ΦΓ)(1α+1)(ΓΦ+αI)(1α+1)(ΓΦ+αI)T2(α+1)I⎤⎥⎦[0x2]>0.

Hence, it follows using Finsler’s theorem (see “What Is Finsler’s Theorem?”), Lemma S2, that there exists such that

 ⎡⎢⎣(1α+1)(ΓΦ+ΦΓ)(1α+1)(ΓΦ+αI)(1α+1)(ΓΦ+αI)T2(α+1)I⎤⎥⎦+τ[Γ000]≥0

for all . Let . Consequently, choosing implies

 ~A~Y+~Y~AT≤0. (32)

Combining (31) and (32), it follows that conditions 1) - 3) of Theorem 6 are satisfied, and therefore, the transfer function (30) is SNI.

###### Theorem 9

Consider the minimal state-space system (16), (17), where , , , and . Suppose the following conditions are satisfied:

1. All of the eigenvalues of are in OLHP.

2. is symmetric.

3. There exist a positive-definite matrix and positive numbers , , and such that , are not eigenvalues of , and the matrices

 ~A=⎡⎢⎣A000−αI000−βI⎤⎥⎦,  ~B=⎡⎢⎣BεIεI⎤⎥⎦,  ~C=[C−I−I]

satisfy

 ~A~Y+~Y~AT≤0

and

 ~B+~A~Y~CT=0.

Then (16), (17) is SNI.

The proof of Theorem 6 requires the following lemmas.

###### Lemma 10

Let , , and . Then the transfer function

 M(s)=ε(s+α)(s+β) (33)

is SNI.

{proof}

The transfer function (33) has a minimal state-space realization

 ˙x=Ax+Bu, (34) y=Cx, (35)

where

 A = [−α00−β],  B=[εε],  C=[11].

Applying Theorem 4 and Corollary 5 to (34), (35), and setting

 Y=[εα00εβ]>0,

it follows that and . Hence, Theorem 4 implies that (34), (35) is NI. Furthermore, for (34), (35), is given by

 M(s)−MT(−s) = ε(s+α)(s+β)−ε(−s+α)(−s+β) = 2ε(α+β)ss2(α+β)2−(s2+αβ)2.

Since has no imaginary transmission zeros except at , it follows from Corollary 5 that (34), (35) is SNI.

###### Lemma 11

If is an SISO SNI transfer function, then the transfer function matrix is SNI.

{proof}

This result follows directly from Definition 4.

Proof of Theorem 9: Let be the transfer function matrix of (16), (17). Since neither nor is a pole of , a minimal state-space realization of is

 ˙x1=Ax1+Bu, ˙x2=−αx2+εu, ˙x3=−βx3+εu, y=Cx1−x2−x3+Du.

Let

 ~A=⎡⎢⎣A000−αI000−βI⎤⎥⎦,  ~B=⎡⎢⎣BεIεI⎤⎥⎦,  ~C=[C−I−I],  ~D=D.

Assuming conditions 1) - 3), it follows from Theorem 4 that is NI. Finally, Lemma 1, Lemma 10, and Lemma 11 imply that is SNI. \QED

## Iv Robust Stability of Negative-Imaginary Control Systems

We now present a result given by Theorem 13 below that guarantees the robustness and stability of control systems involving the positive-feedback interconnection of an NI system and an SNI system. This positive-feedback interconnection is illustrated in Figure 2. The result is analogous to the passivity theorem given in “What Is Positive-real and Passivity Theory?” concerning the negative-feedback interconnection of a positive-real system and a strictly positive-real system.

Theorem 13 guarantees the internal stability of the positive-feedback interconnection of two systems through phase stabilization, as opposed to gain stabilization in the small-gain theorem. In phase stabilization the gains of the systems can be arbitrarily large, but the phase of the loop transfer function needs to be such that the critical Nyquist point is not encircled by the Nyquist plot. In the passivity theorem given in “What Is Positive-real and Passivity Theory?”, negative feedback is used, and thus the Nyquist point is at . Then the cascade of two positive-real systems gives a loop transfer function whose phase is in the interval . Hence, the Nyquist plot excludes the negative real axis. In NI systems, positive feedback interconnection is used and thus the Nyquist point is . This alternative Nyquist point is required since an NI system has a phase lag in the interval and thus two NI systems in cascade have a phase lag in the interval . That is, the Nyquist plot excludes the positive-real axis.

The following lemma is required in order to state the result given in Theorem 13 below.

###### Lemma 12

Let be an NI transfer function matrix. Then and are symmetric, and

 M(0)−M(∞)≥0. (36)

Also, let be an SNI transfer function matrix. Then and are symmetric, and

 N(0)−N(∞)>0. (37)

If, in addition, is positive semidefinite, then is positive definite and all of the eigenvalues of the matrix are real.

{proof}

See [10].

###### Theorem 13

Consider the NI transfer function matrix and the SNI transfer function matrix , and suppose that and . Then, the positive-feedback interconnection of and is internally stable if and only if

 λ\small max(M(0)N(0))<1. (38)
{proof}

See [10].

In the MIMO case, the proof of Theorem 13 given in [10] uses Theorem 4. In the SISO case, the sufficiency part of Theorem 13 follows directly from Nyquist arguments and thus has an intuitive interpretation. For example, consider

 M(s)=1s+1, (39)

whose positive-frequency Nyquist plot is shown in Figure 4. Also consider

 N(s)=2s2+s+1(s2+2s+5)(s+1)(2s+1), (40)

whose positive-frequency Nyquist plot is shown in Figure 5. Figure 4 shows that is SNI, whereas Figure 5 shows that is NI but not SNI. The positive-frequency Nyquist plot of the corresponding loop transfer function is shown in Figure 6. Since both and have no poles in CRHP, and the Nyquist plot of does not encircle the critical point , it follows that the positive-feedback interconnection of and is internally stable. A similar Nyquist argument is mentioned in [8] as a justification for the stability of SISO positive-position feedback systems. Furthermore, a condition equivalent to (38) is required in the result of [16].

Consider and as in Theorem 13 in the SISO case. Since is SNI, it follows that for all . Furthermore, since is NI, it follows that for all such that . Hence, satisfies for all such that . Thus the Nyquist plot of can intersect the positive-real axis only at since at infinite frequency . Thus, the Nyquist plot of does not encircle the critical point if . Hence, in the SISO case, the sufficiency part of Theorem 13 follows from the Nyquist test.

A discussion on how rigid-body modes can be handled using Theorem 13 is given in “How Are Rigid-Body Modes Handled?”.

## V Negative-Imaginary Feedback Controllers

We now apply Theorem 13 to NI feedback control systems in the case where one of the blocks in the feedback connection shown in Figure 2 corresponds to the plant, while the other block corresponds to the controller. This situation is shown in Figure 7.

Since flexible structures with colocated force actuators and position sensors are typically SNI, Theorem 13 implies that NI controllers guarantee closed-loop internal stability if the dc gain condition (38) is satisfied. Indeed, many schemes considered for controlling flexible structures rely on controllers that are NI. These schemes include positive-position feedback [7, 8, 24, 1], resonant feedback control [11, 12], and integral resonant control [13, 14]. We now consider each of these control schemes in more detail.

### V-a Positive-Position Feedback

In the SISO case, a positive-position feedback controller is a controller of the form

 C(s)=M∑i=1kis2+2ζiωis+ω2i, (41)

where , , and for . Using Nyquist arguments, the SISO transfer function , where , is SNI. Consequently, it follows from Lemma 1 that (41) is SNI. Furthermore, this result can be extended to the MIMO case to show that the transfer function matrix

 C(s)=KT(s2I+Ds+Ω)−1K, (42)

where and , is SNI [9]. A MIMO positive-position feedback controller is a controller of the form (42), while a positive-position feedback system is a control system for a flexible structure with colocated force actuators and position sensors with a controller of the form (42) [7, 8, 24, 1].

The Nyquist proof of Theorem 13 justifies the use of positive-position feedback in the SISO case. That is, since the positive-position feedback controller (41) is SNI, its phase is in the interval for all . Furthermore, since the flexible structure plant is NI, its phase is in the interval for all such that is not a zero. Hence, the phase of the loop transfer function is in the interval for all such that is not a zero. This fact, together with the strict properness of the controller (41), implies that the Nyquist plot of the loop transfer function can intersect the positive-real axis at only the frequency . Thus, the Nyquist plot of the loop transfer function does not encircle the critical point if the dc value of the loop transfer function is strictly less than unity.

### V-B Resonant Control

We now consider the exactly proper SISO SNI controller

 C(s)=M∑i=1−