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Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion

M. A. Mabrok, A. G. Kallapur, I. R. Petersen, A. Lanzon

arXiv:1305.1079v2math.OC

TL;DR

The paper addresses the inability of existing NI theory to cover flexible structures with free body motion. It introduces a generalized NI definition and corresponding positive-feedback stability conditions, then applies them to a flexible robotic arm. The framework broadens NI analysis to systems with free body dynamics while retaining transfer-function-based conditions.

  • Problem

    Existing NI stability results do not cover flexible structures with free body motion, including systems whose transfer functions have poles at the origin.

  • Method

    The paper defines generalized NI systems for colocated force actuators, position sensors, and free body motion, then derives positive-feedback stability results for NI plants and SNI controllers.

  • Results

    The derived stability conditions are stated solely in terms of plant and controller transfer-function properties and are independent of their system order.

  • Takeaways & Limitations

    The framework supports controller design for a broader class of NI systems and is illustrated through control of a flexible robotic arm with piezoelectric actuation and sensing.

Abstract

from arXiv · show

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. The stability conditions in this paper are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented.

I. INTRODUCTION

Highly resonant flexible structures motivate NI-based robust control, but existing NI theory cannot handle free body motion. The paper generalizes the framework and stability results to address these systems and demonstrates the approach on a flexible robotic arm.

  • Motivation: Highly resonant modes can degrade stability, performance, and vibration control, motivating robust methods for flexible structures.Their effects are sensitive to environmental changes and unmodeled dynamics.
  • Motivation: Colocated force actuators with position, velocity, or acceleration measurements support robust control of flexible structures by increasing effective damping.Positive-position feedback extends this strategy to position-sensor applications and is robust to resonant-frequency uncertainty.
  • Research gap: Existing NI stability results exclude flexible systems with free body motion because their models include poles at the origin.A disk-drive reader-head example has free body motion, yet its transfer function retains the NI frequency-response property.
  • Contributions: The paper introduces a generalized NI definition for colocated force actuators and position sensors with free body motion, extending prior definitions to allow up to two origin poles.It also derives generalized positive-feedback stability conditions for an NI plant and SNI controller.
  • Contributions: The new stability conditions use only plant and controller transfer-function properties and are independent of system order, although the proofs apply only to rational systems.The results are more general than preliminary conference versions and include free body motion in some, but not all, input-output channels.

II. PRELIMINARIES AND NOTATION

The preliminaries define the state-space and feedback notation used for NI analysis. They review existing NI and SNI conditions and characterize internal stability through the closed-loop system matrix.

  • Existing NI and SNI definitions: An SNI system has no poles in the closed right half-plane and satisfies a strict positive frequency-response inequality for every positive frequency.The reviewed framework uses this existing SNI definition alongside the generalized NI definition introduced later.
  • State-space notation: An LTI system is represented by state-space matrices A, B, C, and D with square transfer matrix G(s)=C(sI−A)^−1B+D.Strict properness means D=0, equivalently G(∞)=0.
  • Existing NI and SNI definitions: The existing NI definition excludes poles at the origin and in the open right half-plane and imposes a nonnegative frequency-response condition.Any positive-frequency pole must be simple with a positive semidefinite Hermitian residue matrix.
  • Feedback notation: For positive feedback, internal stability is defined by the closed-loop system matrix being Hurwitz.The interconnection assumes I−D̄D is nonsingular and uses the stated closed-loop matrix construction.

III. MAIN RESULTS

The paper generalizes negative imaginary systems theory to include free body dynamics and develops transfer-function stability conditions for positive-feedback interconnections with strictly negative imaginary controllers.

  • Generalized NI framework: The generalized NI definition permits poles at the origin, extending the framework to flexible systems with free body dynamics.The new definition is used instead of earlier NI definitions and supports systems whose free body motion appears in selected channels.
  • Stability conditions: The coefficients G2, G1, and G0 from the Laurent expansion encode free body motion and replace the unavailable DC-gain condition in stability analysis.The DC-gain condition is undefined when G2 or G1 is nonzero, so the paper introduces constant matrices and associated decompositions.
  • Stability conditions: For plants with G2 ≠ 0, internal stability is characterized by condition (13) plus condition (14) when Nf is positive semidefinite or condition (15) when Nf is negative semidefinite.The result applies to strictly proper NI plants and SNI controllers under the stated nonsingularity assumption.
  • Special cases: When G1 = 0 and G2 ≠ 0, Corollary 1 gives analogous necessary-and-sufficient conditions using JT ¯G(0)J and the sign of N2.For positive-semidefinite N2, conditions (17) and (18) are required; for negative-semidefinite N2, conditions (17) and (19) are required.
  • Special cases: Additional theorems simplify the criteria when null-space conditions on G2 or G1 hold, requiring only the corresponding DC-gain condition.Theorem 2 treats G1 = 0 and G2 = 0, while Theorem 4 treats G2 = 0 and G1 = 0 under a null-space inclusion.

IV. CASE STUDY: CONTROL OF FLEXIBLE ROBOTIC ARM

The case study models a flexible robotic arm as an equivalent slewing beam and uses piezoelectric actuation and sensing to study its control.

  • The robotic arm is pinned to a motor and modeled using an equivalent slewing beam.
  • Two piezoelectric patches are attached to the arm, with one serving as an actuator and the other as a sensor.
  • The system has voltage and motor-torque inputs, with corresponding piezoelectric-sensor voltage and motor-hub-angle outputs.
  • Colocation of the force actuators and position sensors indicates that the system is negative imaginary.

A. Mathematical model for the robotic arm

The robotic arm is modeled with Bernoulli–Euler beam equations, piezoelectric forcing and sensing, and mixed boundary conditions converted into a state-space representation.

  • The beam model uses Bernoulli–Euler equations with actuating and sensing piezoelectric elements.
  • Young’s modulus, second moment of inertia, density, and beam area define the beam parameters, assumed uniform along its length.Uniform EI and ρA simplify the modeling procedure; thin piezoelectric films motivate neglecting laminated-area differences.
  • The model incorporates boundary conditions at the motor joint and beam tip, while neglecting tip mass and inertia.
  • The time-domain beam equation is transformed into a Laplace-domain representation with spatial derivatives and piezoelectric forcing.
  • The resulting equations form linear ordinary differential equations with mixed boundary conditions and yield a state-space model.

B. Infinite Dimensional Transfer function Model

The infinite-dimensional arm model is expressed as a transfer-function matrix and examined against the generalized NI conditions using frequency-response and residue calculations.

  • The model relates piezoelectric voltage and motor torque to collocated sensor voltage and hub angle through a transfer-function matrix.
  • Each matrix element is an infinite-dimensional transfer function represented by ratios of numerator and denominator transcendental functions.
  • For full-length piezoelectric coverage, the transfer functions are obtained by setting x1 = 0 and x2 = L and have been experimentally verified.
  • The frequency-response skew-Hermitian part is zero for all ω ≥0, supporting the generalized NI frequency-domain condition.
  • The first eleven imaginary-axis roots are computed numerically, and residue calculations assess the NI pole conditions.
  • The log minimum-eigenvalue plot indicates that the residue matrix is positive semidefinite at system poles over the frequency range of interest.

C. Approximate Finite-dimensional Transfer Function Matrix

The irrational infinite-dimensional transfer functions are approximated by a finite-dimensional rational model, whose poles and NI properties are then checked.

  • The irrational transfer functions are approximated by rational functions to enable controller design and performance simulation.
  • The approximation uses a partial fraction expansion based on selected imaginary-axis roots of D(s).
  • The first resonant mode is used for controller design, with k = 6.6667 × 10^-8.
  • The finite-dimensional model has poles p0 = 0 and p1 = 3.4.
  • The finite-dimensional model satisfies the NI frequency-response and residue conditions because its frequency response is real symmetric and its coefficient matrices are positive semidefinite.

D. Controller design

The controller design uses an SNI integral resonant controller whose parameters satisfy the NI stability conditions, then evaluates closed-loop step responses across plant models with different mode counts.

  • The plant is paired with an SNI controller satisfying Corollary 1, guaranteeing stability of their positive-feedback interconnection.
  • An integral resonant controller is selected, with Γ > 0, Φ > 0, and symmetric Δ ensuring the controller is SNI.
  • The controller designed for the n=1 model is tested on finite-dimensional plant models with n=2,3...7 to assess performance and robustness.
  • Closed-loop performance improves as the number of modeled plant modes increases, based on the position and piezoelectric sensor responses.
  • Controller parameters were chosen by trial and error for good performance on the nominal n=1 plant model.An optimization procedure is identified as an alternative for more complicated SNI controller structures.

V. CONCLUSION

The paper extends negative imaginary systems theory to include free body dynamics and derives new positive-feedback stability results. Its application to a flexible robotic arm demonstrates the framework’s use in controller design.

  • The paper presents a new NI definition that includes systems with free body dynamics.
  • New stability results are derived for positive-feedback interconnections of negative imaginary systems.
  • The results are illustrated through control of a flexible robotic arm with a piezoelectric sensor response.

VI. APPENDIX A

Appendix A develops state-space results supporting the main stability theorem and its corollaries for NI plants interconnected positively with SNI controllers.

  • The appendix presents state-space results used to prove the main stability theorem and subsequent corollaries.
  • Theorem 5 gives necessary and sufficient internal-stability conditions for NI–SNI positive-feedback interconnections under cases determined by the matrix N.
  • When N = 0, Corollary 3 reduces internal stability to condition (61) for the NI–SNI positive-feedback interconnection.
  • Corollaries 4 and 5 specialize the stability conditions to distinct state-space cases involving n2 = 0 or n2 ≠ 0 and k = 0.
  • Supporting lemmas derive transfer-function quantities and rank, observability, controllability, and positive-real properties from minimal state-space realizations.

VII. APPENDIX B

Appendix B proves that NI systems can be transformed into a block-diagonal state-space form and uses this representation to connect the appendix stability conditions with the main theorem.

  • Any NI system with the specified minimal realization can be transformed into the block-diagonal form used in the stability analysis.
  • The transformation is constructed using state-space changes based on the Jordan form of the system matrix.
  • NI systems cannot have zero-eigenvalue Jordan blocks of order three or greater, because minimality would imply a nonzero G3 term contradicting the NI definition.
  • A full-rank factorization lemma establishes the invertible transformations needed to relate corresponding state-space blocks.
  • The proof of Theorem 1 reduces equivalence between the state-space conditions and the transfer-function conditions by using the block-diagonal representation.

2 C3bB3b since N

The passage traces algebraic proof steps that transform matrix conditions and establish equivalence among theorem and corollary stability conditions. It uses range-space relations, nonsingular transformations, rank assumptions, and substitutions into previously defined matrices.

  • Theorem 5: The proof rewrites conditions (62) and (63) from Theorem 5 by substituting previously established matrix expressions.These transformations connect the theorem’s conditions to conditions (14) and (15).
  • Theorem 5: A nonsingular transformation matrix preserves the relevant condition, allowing condition (61) to be identified with condition (13).The argument explicitly relies on invertibility of ˆR.
  • Corollaries and theorem equivalences: Theorem and corollary stability conditions are shown equivalent by matching parameter assumptions and translating conditions through Lemma 1.Examples include equivalences between conditions (18), (19), (66), and (67), and between conditions (23), (24), (70), and (71).
  • Special cases: The proofs reduce special cases involving G1 = 0 or G2 = 0 to the assumptions and conditions of Corollary 3.The reductions use identities involving C3b, B3b, C2, B2, and the matrix N.
  • Corollary and theorem proofs: Range-space inclusions and full-rank assumptions yield matrix factorizations such as G0 = JQ and G0 = C2Q.These factorizations support the derivation of special-case formulas for the matrix N.
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