Feedback Control of NegativeImaginary Systems
Flexible structures with colocated actuators and sensors
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 openloop levels. Openloop techniques for highly resonant systems, such as input shaping [1], as well as closedloop 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 negativevelocity feedback, can be used to directly increase the effective damping, thereby facilitating the design of controllers that guarantee closedloop 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 negativevelocity 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 negativevelocity feedback is positiveposition 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 positiveposition 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 positiveposition feedback are similar to those of negativevelocity feedback.
The present article investigates the robustness of positiveposition feedback control of flexible structures with colocated force actuators and position sensors. In particular, the theory of negativeimaginary systems [9, 10] is used to reveal the robustness properties of multiinput, multioutput (MIMO) positiveposition feedback controllers and related types of controllers for flexible structures [11, 12, 1, 13, 14]. The negativeimaginary property of linear systems can be extended to nonlinear systems through the notion of counterclockwise inputoutput dynamics [15, 16]. It is shown in [17] for the singleinput, singleoutput (SISO) linear case that the results of [15, 16] guarantee the stability of a positiveposition feedback control system in the presence of unmodeled dynamics and parameter uncertainties that maintain the negativeimaginary property of the plant.
Positiveposition feedback can be regarded as one of the last areas of classical control theory to be encompassed by modern control theory. In this article, positiveposition feedback, negativeimaginary systems, and related control methodologies are brought together with the underlying systems theory.
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
(1) 
where each is a modal frequency, the functions are firstorder 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
(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
If is positive real, then it follows that [19, 20]
(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
(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 positivereal transfer function from force actuation to velocity measurement given by
(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
where
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
(7) 
where, for all , , , and is an vector. A review of positivereal and passivity theory is given in “What Is Positivereal and Passivity Theory?”
Iii NegativeImaginary Systems
Mechanical structures with colocated force actuators and position sensors do not yield positivereal 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
(8) 
where, for all , , , and is an vector. Therefore, the Hermitianimaginary part
of the frequency response function matrix satisfies
(9) 
for all That is, the frequency response function matrix for the transfer function matrix (8) has negativesemidefinite Hermitianimaginary 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 negativeimaginary (NI) if the following conditions are satisfied:

All of the poles of lie in OLHP.

For all ,
(10)
A linear timeinvariant system is NI if its transfer function matrix is NI.
A discussion of negativeimaginary 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 imaginaryaxis poles or zeros. Consequently, the positivefrequency Nyquist plot of a SISO negativeimaginary transfer function lies below the real axis as shown in Figure 1. Hence, a negativeimaginary transfer function can be viewed as a positivereal transfer function rotated clockwise by deg in the Nyquist plane.
Velocity sensors can be used in negativevelocity feedback control, whereas position sensors can be used in positiveposition feedback [1, 7, 8, 14, 13, 11, 12]. Indeed, positivereal theory and negativeimaginary 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 openloop systems. Gain stabilization, which is based on the smallgain theorem [19], guarantees robust stability when the magnitude of the loop transfer function is less than unity at all frequencies. As in positivereal analysis, robust stability of negativeimaginary 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 positiveposition feedback and related control schemes, we now define MIMO strictly negativeimaginary systems.
Definition 4
The square transfer function matrix is strictly negativeimaginary (SNI) if the following conditions are satisfied:

All of the poles of lie in OLHP.

For all ,
(11)
A linear timeinvariant 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
(12) 
is NI, respectively, SNI.
Theorem 2
The internal stability of the positive feedback interconnection shown in Figure 2 implies that is asymptotically stable. Given , , and , define
Letting and , it follows from the positive feedback interconnection that and . Furthermore, using the fact that and are NI, it follows that
Since , , and are arbitrary, it follows that
for all and hence, is NI. The SNI result follows using similar arguments.
Theorem 3
The internal stability of the feedback interconnection shown in Figure 3 implies that is asymptotically stable. Given , , and , define
Letting
it follows from the feedback interconnection shown in Figure 3 that
(15) 
Furthermore, using (15) and the fact that and are NI, it follows that
Since , , and are arbitrary, it follows that
for all and hence, is NI. The SNI result follows using similar arguments.
Underlying the stability properties of positiveposition 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
in the transfer function matrix (8) lie in OLHP. Also, for all ,
Hence, it follows from Definition 3 that each is NI. Therefore, it follows from Lemma 1 that the transfer function matrix (8) is NI.
Iiia The NegativeImaginary Lemma
The following theorem, which is proved in [10, 21], provides a statespace characterization of NI systems in terms of a pair of linear matrix inequalities (LMIs). This result is analogous to the positivereal lemma [20, 19], and thus is referred to as the negativeimaginary lemma.
Theorem 4
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 statespace system (16), (17), where , , , and . The system (16), (17) is SNI if and only if the following conditions are satisfied:

has no eigenvalues on the imaginary axis.

is symmetric.

The transfer function matrix is such that has no transmission zeros on the imaginary axis except possibly at .
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
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
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
(20)  
(21) 
with transfer function
(22) 
The positivefrequency 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
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
(23) 
The positivefrequency 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
(24)  
(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
Using LMI software [23], we obtain
Therefore Theorem 4 implies that (16), (17), (24), (25) is NI.
IiiB Two Strict NegativeImaginary Lemmas
The following theorems give sufficient conditions for the SNI property.
Theorem 6
The proof of Theorem 6 requires the following lemma.
Lemma 7
Let and . Then the transfer function matrix
(27) 
is SNI.
Let the transfer function matrix (27) have minimal statespace realization
(28)  
(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,
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 statespace realization of the transfer function matrix is
Let
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
(30) 
where and are positivedefinite 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 statespace realization of (30) given by
Let and be such that is not an eigenvalue of . The corresponding matrices in (26) are
Also, let
Thus,
(31) 
Furthermore, note that
is positive semidefinite, and
is positive definite. Hence, .
Using the definitions of and , it follows that
Furthermore, the matrix
is positive semidefinite. For every nonzero vector of the form , we have
Hence, it follows using Finsler’s theorem (see “What Is Finsler’s Theorem?”), Lemma S2, that there exists such that
for all . Let . Consequently, choosing implies
(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 statespace system (16), (17), where , , , and . Suppose the following conditions are satisfied:

All of the eigenvalues of are in OLHP.

is symmetric.

There exist a positivedefinite matrix and positive numbers , , and such that , are not eigenvalues of , and the matrices
satisfy
and
The proof of Theorem 6 requires the following lemmas.
Lemma 10
Let , , and . Then the transfer function
(33) 
is SNI.
The transfer function (33) has a minimal statespace realization
(34)  
(35) 
where
Applying Theorem 4 and Corollary 5 to (34), (35), and setting
it follows that and . Hence, Theorem 4 implies that (34), (35) is NI. Furthermore, for (34), (35), is given by
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.
This result follows directly from Definition 4.
Iv Robust Stability of NegativeImaginary Control Systems
We now present a result given by Theorem 13 below that guarantees the robustness and stability of control systems involving the positivefeedback interconnection of an NI system and an SNI system. This positivefeedback interconnection is illustrated in Figure 2. The result is analogous to the passivity theorem given in “What Is Positivereal and Passivity Theory?” concerning the negativefeedback interconnection of a positivereal system and a strictly positivereal system.
Theorem 13 guarantees the internal stability of the positivefeedback interconnection of two systems through phase stabilization, as opposed to gain stabilization in the smallgain 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 Positivereal and Passivity Theory?”, negative feedback is used, and thus the Nyquist point is at . Then the cascade of two positivereal 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 positivereal 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
(36) 
Also, let be an SNI transfer function matrix. Then and are symmetric, and
(37) 
If, in addition, is positive semidefinite, then is positive definite and all of the eigenvalues of the matrix are real.
See [10].
Theorem 13
Consider the NI transfer function matrix and the SNI transfer function matrix , and suppose that and . Then, the positivefeedback interconnection of and is internally stable if and only if
(38) 
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
(39) 
whose positivefrequency Nyquist plot is shown in Figure 4. Also consider
(40) 
whose positivefrequency Nyquist plot is shown in Figure 5. Figure 4 shows that is SNI, whereas Figure 5 shows that is NI but not SNI. The positivefrequency 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 positivefeedback interconnection of and is internally stable. A similar Nyquist argument is mentioned in [8] as a justification for the stability of SISO positiveposition 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 positivereal 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 rigidbody modes can be handled using Theorem 13 is given in “How Are RigidBody Modes Handled?”.
V NegativeImaginary 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 closedloop 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 positiveposition 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.
Va PositivePosition Feedback
In the SISO case, a positiveposition feedback controller is a controller of the form
(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
(42) 
where and , is SNI [9]. A MIMO positiveposition feedback controller is a controller of the form (42), while a positiveposition 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 positiveposition feedback in the SISO case. That is, since the positiveposition 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 positivereal 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.
VB Resonant Control
We now consider the exactly proper SISO SNI controller