Negative Imaginary Systems Theory in the Robust Control of Highly Resonant Flexible Structures

Negative Imaginary Systems Theory in the Robust Control of Highly Resonant Flexible Structures

Ian R. Petersen Research supported by the Australian Research Council (ARC) Ian R. Petersen is with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia. i.r.petersen@gmail.com
Abstract

This paper covers recent developments in the theory of negative imaginary systems and their application to the control of highly resonant flexible structures. The theory of negative imaginary systems arose out of a desire to unify a number of classical methods for the control of lightly damped structures with collocated force actuators and position sensors including positive position feedback and integral force feedback. The key result is a stability result which shows why these methods are guaranteed to yield robust closed loop stability in the face of unmodelled spillover dynamics. Related results to be presented connect the theory of negative imaginary systems to positive real systems theory and a negative imaginary lemma has been established which is analogous to the positive real lemma. The paper also presents recent controller synthesis results based on the theory of negative imaginary systems.

I Introduction

The problem of vibration control for flexible structures arises in a variety of applications such as large space structures, flexible dynamics in air vehicles, nano-positioning, the control of atomic force microscopes and the control of optical interfermetric sensing systems; e.g., see [1, 2]. A particular problem in vibration control for flexible structures is that unmodelled (spillover) dynamics can severely degrade the control system performance or lead to instability. Also, uncertainties in mode frequencies and damping can lead to similar problems of poor control system performance or instability; e.g., see [3, 4].

An approach to the control of flexible structures which addresses these robustness issues is positive-position feedback, which involves the use of collocated force actuators and position sensors [5, 6]. Roughly speaking a positive position feedback controller consists of a transfer function matrix which has the same form as a finite mode model of a flexible structure. This approach was originally proposed as an alternative to passivity based velocity feedback approaches in the control of large space structures. The original papers on positive position feedback included stability proofs based on Routh-Hurwitz and Nyquist techniques but the robustness properties were never fully quantified. The research presented in this paper aims to develop a general systems theory framework for positive position control and related control methods for flexible structures. An important special case of positive-position feedback involves the use of piezoelectric actuators and sensors. 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 unmodelled plant dynamics. We consider the theory of negative-imaginary systems to reveal the robustness properties of MIMO positive-position feedback controllers and related types of controllers for flexible structures; e.g., see [7, 8, 1, 9, 10, 11, 12, 13, 14, 15].

The negative-imaginary property of linear systems can be extended to nonlinear systems through the notion of counterclockwise input-output dynamics [16, 17, 18]. It is shown in [19] for the single-input, single-output (SISO) linear case that the results of [16, 17] guarantee the stability of a positive-position feedback control system in the presence of unmodelled dynamics and parameter uncertainties that maintain the negative-imaginary property of the plant.

I-a Flexible Structure Modeling

In modeling an undamped flexible structure with collocated force actuators and position sensors, modal analysis can be applied to the relevant partial differential equation leading to the transfer function matrix

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

of the frequency response matrix satisfies

for all That is, the frequency response matrix has a negative-semidefinite Hermitian-imaginary part for all . We thus refer to the transfer function matrix as negative imaginary. Any flexible structure with collocated force actuators and position sensors will have a negative imaginary transfer function matrix.

Ii Negative Imaginary Transfer Function Matrices

In this section, we present the formal definition of a negative imaginary transfer function matrix. We also present a number of extensions to this notion, which have appeared in the literature.

Definition 1 ([20, 21, 22, 23])

A square real-rational proper transfer function matrix is termed negative imaginary (NI) if

  1. has no poles at the origin and in ;

  2. for all except values of where is a pole of ;

  3. If , , is a pole of , it is at most a simple pole, and the residue matrix is positive semidefinite Hermitian.

In the SISO case, a transfer function is negative imaginary if and only if it has no poles in open right half plane (ORHP) or at the origin and its phase is in the interval at all frequencies. Consequently, the Nyquist plot of a SISO negative-imaginary transfer function lies below the real axis; see Figure 1.

Fig. 1: Nyquist plot of a negative-imaginary system.
Remark 1

Note that this definition has been recently extended to allow for a simple pole at the origin; [24, 25]. However, in this case, this leads to a slightly different form of the stability condition. Also, in [26], a lossless version of this definition is given along with a corresponding negative imaginary lemma.

The following lemma from [21, 23] describes the relationship between negative imaginary transfer function matrices and positive real transfer function matrices.

Lemma 1 ([21, 23])

Given a square real-rational strictly proper transfer function matrix . Then is negative imaginary if and only if

  1. has no poles at the origin;

  2. is positive real.

The following definition defines the strict version of the negative imaginary property.

Definition 2 (See [20, 21].)

A square real-rational proper transfer function matrix is termed strictly negative imaginary (SNI) if

  1. has no poles in ;

  2. for .

Remark 2

In the paper [27], the strict negative imaginary property is extended to a notion of strongly strict negative imaginary (SSNI) systems by imposing the additional conditions that

and

This extension is useful in characterizing the strict negative imaginary property using LMIs and in controller synthesis.

The following lemmas state some useful properties of negative imaginary transfer function matrices.

Lemma 2 ([21, 23])

A square real-rational proper transfer function matrix is negative imaginary if and only if and is negative imaginary.

Lemma 3 ([15])

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

(1)

is NI. Also, if in addition is SNI, then is SNI.

Lemma 4 ([15])

Consider the NI transfer function matrices and , and suppose that the positive-feedback interconnection of this transfer functions is internally stable. Then the corresponding closed-loop transfer function matrix

(2)

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

We can also can also extend the above definitions of NI systems to the finite bandwidth case.

Definition 3 ([28])

A square real-rational proper transfer function matrix is said to be finite frequency negative imaginary (FFNI) with bandwidth if it satisfies the following conditions:

  1. has no poles at the origin and in the open right-half of the complex plane;

  2. for all , where
    ;

  3. Every pole of on , if any, is simple and the corresponding residue matrix of is positive semidefinite Hermitian, where ;

  4. .

Based on the above definition, we have the following properties.

Lemma 5 ([28])

If is an FFNI transfer function matrix with bandwidth , then the following properties hold:

  1. for all , where
    .

  2. Every pole of in , if any, is simple and the corresponding residue matrix of is negative semidefinite Hermitian, where .

Note that the stability result given in [29], uses a notion similar to the FFNI property defined above along with a small gain property to establish robust stability.

Iii Some Negative Imaginary Lemmas

The following Theorem provides a state space condition for a system to be negative imaginary.

Theorem 1 (Negative Imaginary Lemma,[20, 21, 22, 23].)

Let be a minimal state-space realization of an real-rational proper transfer function matrix , where , , , . Then is negative imaginary if and only if

  1. , , and

  2. there exists a matrix , , such that

The following alternative form of the negative imaginary lemma also holds in the case that the system has a simple pole at the origin.

Theorem 2 (Modified NI Lemma,[22, 23, 24, 25].)

Let be a minimal realization of , where , , , , . Then is negative-imaginary if and only if and there exist matrices , , , and such that the following LMI is satisfied:

(3)
Remark 3

Note that a version of this negative imaginary lemma for descriptor systems is presented in [30].

The following theorems provide corresponding conditions for strict negative imaginary properties.

Theorem 3 (Weak SNI Lemma,[22, 23].)

Let be a minimal state-space realization of an real-rational proper transfer function matrix , where , , , . Then is strictly negative imaginary if and only if

  1. is Hurwitz, , ;

  2. There exists a matrix , , such that

  3. The transfer function matrix has full column rank at for any . Here . That is, for any .

The following theorem provides a purely LMI approach to test the strong SNI property.

Theorem 4

[Strong SNI Lemma, [27, 31].] Given a square transfer function matrix with a state-space realization , where , , and . Suppose has normal rank and is observable. Then, is Hurwitz and is Strongly Strict Negative Imaginary if and only if and there exists a matrix such that

(4)

The following theorem from [28] gives a finite bandwidth version of the Negative Imaginary Lemma.

Theorem 5 (FFNI Lemma [28].)

Consider a real-rational proper transfer function matrix with a minimal state-space realization . Suppose all poles of are in the closed left-half of the complex plane, and the poles on the imaginary axis, if any, are simple. Let a positive scalar be given. Also, suppose that if has eigenvalues () such that , the residue of at is given by . Then the following statements are equivalent:

  1. The transfer function matrix is FFNI with bandwidth .

  2. , , and the transfer function matrix with a minimal state-space realization is FFPR with bandwidth .

  3. , , and for all if has any eigenvalues on where . Also, there exist real symmetric matrices and such that

    (5)
    (6)
    (7)
  4. , , and for all if has any eigenvalues on where . Also, there exist real symmetric matrices and such that

    (8)
    (9)
    (10)

In addition to the above negative imaginary lemmas, which involve LMI conditions to test if a given system is negative imaginary or strictly negative imaginary, the following result uses an eigenvalue test of a corresponding Hamiltonian matrix to test if a given system is NI.

Theorem 6 ([32])

Consider a transfer function matrix with a minimal realization . Also, suppose is a Hurwitz matrix, , and . Then, is NI if and only if the following conditions are satisfied:

  1. The Hamiltonian matrix,

    has no pure imaginary eigenvalues with odd multiplicity.

  2. has only positive real eigenvalues for all , where , and is the set of frequencies listed in strictly increasing order, such that Here, where .

Remark 4

This approach has been extended in the paper [33] to consider the problem of perturbing a non-NI system model in order to obtain an NI system model.

Also, the paper [32] gives the following result for testing the NI property of SISO systems.

Assumption 1

Let , and suppose the transfer function has all of its poles and zeros in the closed left half of the complex plane excluding poles at the origin. Also, any pole on the imaginary axis is assumed to be simple.

Theorem 7 ([32])

The SISO transfer function with , and satisfying Assumption 1 is NI if and only if the following conditions are satisfied:

  1. The matrix has no eigenvalues of odd (algebraic) multiplicity on the open negative real axis.

  2. All residues of corresponding to poles on the imaginary axis are positive.

Iv Stability of Interconnections of Negative Imaginary Systems

The following theorem, which is the main stability result in the theory of negative imaginary systems, gives a necessary and sufficient condition for the stability of the positive feedback interconnection (see Figure 2) between an NI system and an SNI system.

Fig. 2: A positive feedback interconnection.
Theorem 8 ([20, 21, 22, 23])

Given a negative imaginary transfer function matrix and a strictly negative imaginary transfer function matrix that also satisfy and . Then the positive feedback interconnection is internally stable if and only if .

Remark 5

Note that in the papers [34, 35], the robustness properties obtained by this theorem are compared with the robustness properties which are obtained from the analysis approach. Also, in the papers [36, 37], this stability robustness result is extended to allow for additional norm bounded uncertain parameters.

The above theorem can also be extended to give a sufficient condition for closed loop stability in the case in which the NI system has a simple pole at the origin.

Theorem 9 ([24, 25])

Consider a strictly proper negative imaginary transfer function matrix with minimal state space realization , which may contain a simple pole at the origin. Also, consider a strictly negative imaginary transfer function matrix . Then the closed-loop positive feedback interconnection between and is internally stable if and the matrix is not singular.

V Negative-Imaginary Feedback Controllers

In this section, we present a number of different classes of commonly used negative imaginary controllers; see also [15].

V-a Positive-Position Feedback Controllers

In the SISO case, a positive-position feedback controller (e.g., see [5, 6, 38, 39, 12, 40, 41, 42, 43, 44]) is a controller of the form

(11)

where , , and for . Such controllers are always SNI.

V-B Resonant Controllers

Resonant controllers are proper but not strictly proper SISO SNI controllers of the form

(12)

where , , and for ; e.g, see [8, 45]. The controller (12) can be implemented as the positive-position feedback controller (11) using an acceleration sensor rather than a position sensor. Alternatively, (12) can be implemented as the positive-real feedback controller

where , , and for , using a velocity sensor rather than a position sensor. Such controllers are always SNI.

V-C Integral Resonant Controllers

MIMO transfer function matrices of the form

are SNI. Here, is a positive-definite matrix and is a positive-definite matrix. The use of a controller of this form when applied to a flexible structure with force actuators and position sensors is referred to as integral resonant control, or integral force control [1, 9, 46, 47, 48, 49, 10, 50, 51].

Vi State-Feedback Controller Synthesis

Consider the feedback control system in Figure 3 in the case that full state feedback is available. In this case, the negative imaginary lemma can be used to synthesize a state-feedback control law such that the resulting closed-loop system is NI. Indeed, suppose the uncertain system shown in Figure 3 is described by the state equations

(13)
(14)
(15)

where the uncertainty transfer function matrix is assumed to be SNI with and . Applying the state-feedback control law yields the closed-loop uncertain system

(16)
(17)
(18)

The corresponding nominal closed-loop transfer function matrix is

(19)
Fig. 3: A feedback control system.
Theorem 10 ([52, 15])

Consider the uncertain system (13), (14), (15) and suppose there exist matrices , , and a scalar such that

(20)
(21)
(22)

Here the parameter is chosen to be sufficiently small. Then the state-feedback control law is robustly stabilizing for the uncertain system (13), (14), (15).

Remark 6

Note that in the papers [53, 54], an alternative approach to controller synthesis is considered by connecting the negative imaginary synthesis problem to the control problem.

Vii Conclusions

We have surveyed some recent results in the area of negative imaginary systems theory. This results characterize the class of negative imaginary systems and give methods for testing if a given system is negative imaginary. These results also show that the class of NI systems yields a robust stability analysis result, which broadly speaking can be captured by saying that if one system is negative imaginary and the other system is strictly negative imaginary, then a necessary and sufficient condition for internal stability of the positive-feedback interconnection of the two systems is that the dc loop gain is less than unity. This result provides a framework for the analysis of robust stability of lightly damped flexible structures with unmodelled dynamics.

References

  • [1] A. Preumont, Vibration Control of Active Structures, 2nd ed.   Dordrecht: Kluwer, 2002.
  • [2] S. Devasia, E. Eleftheriou, and S. O. R. Moheimani, “A survey of control issues in nanopositioning,” IEEE Transactions On Control Systems Technology, vol. 15, no. 5, pp. 802–823, Sept. 2007.
  • [3] S. P. Bhat and D. K. Miu, “Precise point-to-point positioning control of flexible structures,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 112, pp. 667–674, 1990.
  • [4] P. Dang, F. L. Lewis, K. Subbarao, and H. Stephanou, “Shape control of flexible structure using potential field method,” in 17th IEEE International Conference on Control Applications, San Antonio, TX, September 2008, pp. 540–546.
  • [5] C. Goh and T. Caughey, “On the stability problem caused by finite actuator dynamics in the collocated control of large space structures,” International Journal of Control, vol. 41, no. 3, pp. 787–802, 1985.
  • [6] J. Fanson and T. Caughey, “Positive position feedback control for large space structures,” AIAA Journal, vol. 28, no. 4, pp. 717–724, 1990.
  • [7] D. Halim and S. O. R. Moheimani, “Spatial resonant control of flexible structures-application to a piezoelectric laminate beam,” IEEE Transactions on Control Systems Technology, vol. 9, no. 1, pp. 37–53, 2001.
  • [8] H. R. Pota, S. O. R. Moheimani, and M. Smith, “Resonant controllers for smart structures,” Smart Materials and Structures, vol. 11, no. 1, pp. 1–8, 2002.
  • [9] S. S. Aphale, A. J. Fleming, and S. O. R. Moheimani, “Integral resonant control of collocated smart structures,” Smart Materials & Structures, vol. 16, no. 2, pp. 439–446, Apr. 2007.
  • [10] B. Bhikkaji, S. O. R. Moheimani, and I. R. Petersen, “Multivariable integral control of resonant structures,” in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, December 2008, pp. 3743 – 3748.
  • [11] T. McKelvey and S. O. R. Moheimani, “Estimation of phase constrained MIMO transfer functions with application to flexible structures with mixed collocated and non-collocated actuators and sensors,” in Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, July 2005, pp. 219–224.
  • [12] S. Moheimani, B. Vautier, and B. Bhikkaji, “Experimental implementation of extended multivariable PPF control of an active structure,” IEEE Transactions on Control Systems Technology, vol. 14, no. 3, pp. 433–455, 2006.
  • [13] C. H. Cai and G. Hagen, “Stability analysis for a string of coupled stable subsystems with negative imaginary frequency response,” IEEE Transactions On Automatic Control, vol. 55, no. 8, pp. 1958–1963, Aug. 2010.
  • [14] I. A. Mahmood, S. O. R. Moheimani, and B. Bhikkaji, “A new scanning method for fast atomic force microscopy,” IEEE Transactions On Nanotechnology, vol. 10, no. 2, pp. 203–216, Mar. 2011.
  • [15] I. R. Petersen and A. Lanzon, “Feedback control of negative-imaginary systems,” Control Systems Magazine, vol. 30, no. 5, pp. 54 – 72, 2010.
  • [16] D. Angeli, “On systems with counterclockwise input-output dynamics,” in Proceedings of the IEEE Conference on Decision and Control, Atlantis, Bahamas, December 2004, pp. 2527–2532.
  • [17] ——, “Systems with counterclockwise input-output dynamics,” IEEE Transactions on Automatic Control, vol. 51, no. 7, pp. 1130–1143, 2006.
  • [18] C. H. Cai and G. Hagen, “Coupling of stable subsystems with counterclockwise input-output dynamics,” 2010 American Control Conference, pp. 3458–3463, 2010.
  • [19] A. K. Padthe, J. H. Oh, and D. S. Bernstein, “Counterclockwise dynamics of a rate-independent semilinear Duhem model,” in Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, December 2005, pp. 8000–8005.
  • [20] A. Lanzon and I. R. Petersen, “A modified positive-real type stability condition,” in Proceedings of the 2007 European Control Conference, Kos Greece, July 2007, pp. 3912–3918.
  • [21] ——, “Stability robustness of a feedback interconnection of systems with negative imaginary frequency response,” IEEE Transactions on Automatic Control, vol. 53, no. 4, pp. 1042–1046, 2008.
  • [22] J. Xiong, I. R. Petersen, and A. Lanzon, “Stability analysis of positive feedback interconnections of linear negative imaginary systems,” in Proceedings of the 2009 American Control Conference, St Louis, MO, June 2009, pp. 1855 – 1860.
  • [23] ——, “A negative imaginary lemma and the stability of interconnections of linear negative imaginary systems,” IEEE Transactions on Automatic Control, vol. 55, no. 10, pp. 2342 – 2347, 2010.
  • [24] M. A. Mabrok, A. G. Kallapur, I. R. Petersen, and A. Lanzon, “Stability analysis for negative imaginary feedback systems including an integrator,” in Proceedings of the 8th Asian Control Conference, Kaohsiung, Taiwan, May 2011.
  • [25] ——, “A new stability result for the feedback interconnection of negative imaginary systems with a pole at the origin,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando FL, December 2011.
  • [26] J. Xiong, I. R. Petersen, and A. Lanzon, “On lossless negative imaginary systems,” in Proceedings of the 2009 Asian Control Conference, Hong Kong, August 2009.
  • [27] A. Lanzon, Z. Song, S. Patra, and I. R. Petersen, “A strongly strict negative-imaginary lemma for non-minimal linear systems,” Communications in Information and Systems, vol. 11, no. 2, pp. 139–152, 2011.
  • [28] J. Xiong, I. R. Petersen, and A. Lanzon, “Finite frequency negative imaginary systems,” in Proceedings of the 2010 American Control Conference, Baltimore, MD, June 2010.
  • [29] S. Patra and A. Lanzon, “Stability analysis of interconnected systemswith “mixed” negative-imaginary and small-gain properties,” IEEE Transactions on Automatic Control, vol. 56, no. 6, pp. 1395–1400, 2011.
  • [30] M. Mabrok, A. G. Kallapur, I. R. Petersen, and A. Lanzon, “A negative imaginary lemma for descriptor systems,” in Proceedings of the 2011 Australian Control Conference, Melbourne, November 2011.
  • [31] Z. Song, S. Patra, A. Lanzon, and I. R. Petersen, “On state-space characterization for strict negative-imaginariness of LTI systems,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando FL, December 2011.
  • [32] M. A. Mabrok, A. G. Kallapur, I. R. Petersen, and A. Lanzon, “Spectral conditions for the negative imaginary property of transfer function matrices,” in 18th IFAC World Congress, Milan, Italy, 2011.
  • [33] M. Mabrok, I. R. Petersen, A. G. Kallapur, and A. Lanzon, “Enforcing a system model to be negative imaginary via perturbation of hamiltonian matrices,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando FL, December 2011.
  • [34] S. Engelken, A. Lanzon, and I. R. Petersen, “Mu analysis for interconnections of systems with negative imaginary frequency response,” in Proceedings of the 2009 European Control Conference, Budapest, Hungary, August 2009.
  • [35] S. Engelken, S. Patra, A. Lanzon, and I. R. Petersen, “Stability analysis of negative imaginary systems with real parametric uncertainty – the SISO case,” IET Control Theory & Applications, vol. 4, no. 11, pp. 2631–2638, 2010.
  • [36] Z. Song, A. Lanzon, S. Patra, and I. R. Petersen, “Analysis of robust performance for uncertain negative-imaginary systems using structured singular value,” in Proceedings of the 18th Mediterranean Conference on Control and Automation, Marrakech, Morocco, 2010.
  • [37] Z. Song, A. Lanzon, S. Patra, and I. Petersen, “Robust performance analysis for uncertain negative-imaginary systems,” International Journal of Robust and Nonlinear Control, vol. 22, no. 3, pp. 262–281, 2012.
  • [38] S. O. R. Moheimani, B. J. G. Vautier, and B. Bhilkkaji, “Multivariable PPF control of an active structure,” 2005 44th IEEE Conference On Decision and Control & European Control Conference, Vols 1-8, pp. 6824–6829, 2005.
  • [39] M. Ratnam, B. Bhikkaji, A. J. Fleming, and S. O. R. Moheimani, “PPF control of a piezoelectric tube scanner,” 2005 44th IEEE Conference on Decision and Control & European Control Conference, Vols 1-8, pp. 1168–1173, 2005.
  • [40] S. N. Mahmoodi and M. Ahmadian, “Active vibration control with modified positive position feedback,” Journal of Dynamic Systems Measurement and Control-transactions of the ASME, vol. 131, no. 4, p. 041002, July 2009.
  • [41] I. A. Mahmood and S. O. R. Moheimani, “Improvement of accuracy and speed of a commercial AFM using positive position feedback control,” 2009 American Control Conference, Vols 1-9, pp. 973–978, 2009.
  • [42] ——, “Making a commercial atomic force microscope more accurate and faster using positive position feedback control,” Review of Scientific Instruments, vol. 80, no. 6, p. 063705, June 2009.
  • [43] Y. K. Yong, B. Ahmed, and S. O. R. Moheimani, “Atomic force microscopy with a 12-electrode piezoelectric tube scanner,” Review of Scientific Instruments, vol. 81, no. 3, p. 033701, Mar. 2010.
  • [44] B. Ahmed and H. R. Pota, “Dynamic compensation for control of a rotary wing uav using positive position feedback,” Journal of Intelligent & Robotic Systems, vol. 61, no. 1-4, pp. 43–56, Mar. 2011.
  • [45] S. O. R. Moheimani and B. J. G. Vautier, “Resonant control of structural vibration using charge-driven piezoelectric actuators,” 2004 43rd IEEE Conference On Decision and Control (CDC), Vols 1-5, pp. 5368–5373, 2004.
  • [46] S. S. Aphale, A. J. Fleming, and S. O. R. Moheimani, “Integral control of collocated smart structures,” Active and Passive Smart Structures and Integrated Systems 2007, vol. 6525, p. 652521, 2007.
  • [47] E. Pereira, S. O. R. Moheimani, and S. S. Aphale, “Analog implementation of an integral resonant control scheme,” Smart Materials & Structures, vol. 17, no. 6, p. 067001, Dec. 2008.
  • [48] B. Bhikkaji and S. O. R. Moheimani, “Integral resonant control of a piezoelectric tube actuator for fast nanoscale positioning,” IEEE-ASME Transactions On Mechatronics, vol. 13, no. 5, pp. 530–537, Oct. 2008.
  • [49] ——, “Fast scanning using piezoelectric tube nanopositioners: A negative imaginary approach,” 2009 IEEE/ASME International Conference On Advanced Intelligent Mechatronics, Vols 1-3, pp. 274–279, 2009.
  • [50] B. Bhikkaji, S. O. R. Moheimani, and I. R. Petersen, “A negative imaginary approach to modeling and control of a collocated structure,” IEEE/ASME Transactions on Mechatronics, vol. 17, no. 4, pp. 717–727, 2012.
  • [51] E. Pereira, S. S. Aphale, V. Feliu, and S. O. R. Moheimani, “Integral resonant control for vibration damping and precise tip-positioning of a single-link flexible manipulator,” IEEE-ASME Transactions on Mechatronics, vol. 16, no. 2, pp. 232–240, Apr. 2011.
  • [52] I. R. Petersen, A. Lanzon, and Z. Song, “Stabilization of uncertain negative-imaginary systems via state-feedback control,” in Proceedings of the 2009 European Control Conference, Budapest, Hungary, August 2009.
  • [53] A. Lanzon, Z. Song, and I. R. Petersen, “Reformulating negative imaginary frequency response systems to bounded-real systems,” in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, December 2008, pp. 322 – 326.
  • [54] Z. Song, A. Lanzon, S. Patra, and I. R. Petersen, “Towards controller synthesis for systems with negative imaginary frequency response,” IEEE Transactions on Automatic Control, vol. 55, no. 6, pp. 1506–1511, 2010.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
375788
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description