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

Ian R. Petersen, Alexander Lanzon

arXiv:1401.7745v1eess.SYmath.OC

TL;DR

The paper addresses robust control of flexible structures whose dynamics are uncertain and whose colocated force actuators pair naturally with position sensors. It surveys negative-imaginary theory, including its definitions, feedback results, and application to flexible-structure models. The surveyed results show that stable feedback interconnections preserve NI structure and become SNI when one component is SNI.

  • Problem

    Flexible-structure models are sensitive to boundary conditions, aging, environmental effects, and unmodeled dynamics, while force–position systems are not positive real.

  • Method

    The paper surveys negative-imaginary systems, their phase-based stability theory, feedback interconnection results, and flexible-structure applications.

  • Results

    Stable feedback interconnections of NI transfer matrices are NI, and become SNI when either component is SNI; colocated force–position flexible-structure models are NI.

  • Takeaways & Limitations

    Negative-imaginary theory provides a framework for positive-position feedback and robust stability of flexible structures with colocated force actuators and position sensors.

  • Takeaways & Limitations

    The modeled transfer function is not strictly positive real because it has a zero at the origin.

Abstract

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This paper presents a survey of recent results on the theory of negative imaginary systems. This theory can be applied to the robust control of large flexible structures with colocated force actuators and position sensors.

II. FLEXIBLE STRUCTURE MODELING

The paper models flexible structures using transfer functions derived from modal analysis and connects colocated force–velocity pairs to positive-real behavior. It also shows why the resulting transfer function is not strictly positive real when it has a zero at the origin.

  • Modal analysis of the governing partial differential equation yields a transfer function for an undamped flexible structure with one actuator and one sensor.
  • Colocated force actuators and velocity sensors are dual variables because their product equals the power supplied to the structure.
  • Positive-real transfer functions satisfy pole and Hermitian-part conditions, with a corresponding SISO phase characterization.
  • A transfer function is strictly positive real only under a shifted positive-real condition, which places all poles in the open left half-plane.
  • A zero at the origin prevents transfer function (6) from satisfying the strict positive-real condition at zero frequency.
  • The resulting transfer-function matrix is positive real when the actuator and sensor pairs are colocated and dual.

III. NEGATIVE-IMAGINARY SYSTEMS

Negative-imaginary theory characterizes colocated force–position flexible-structure models and analyzes their feedback interconnections. Under internal stability, the theory preserves NI structure and becomes SNI when one component is SNI.

  • Force–position transfer functions are not positive real because force times position does not equal actuator power.
  • A transfer matrix is negative imaginary when its poles lie in the open left half-plane and its Hermitian-imaginary part is positive semidefinite for ω ≥ 0.
  • For SISO systems, negative-imaginary behavior corresponds to phase in [−π, 0] at frequencies excluding imaginary-axis poles or zeros.
  • Negative-imaginary theory supports positive-position feedback and robust stability without requiring loop-transfer magnitude below unity at every frequency.
  • The NI class is closed under the stated feedback constructions, including the positive-feedback interconnection when it is internally stable.
  • If either component in the internally stable positive-feedback interconnection is SNI, the resulting closed-loop transfer matrix is SNI.
  • The analogous feedback interconnection in Theorem 3 likewise preserves NI and becomes SNI when either component is SNI.
  • The lightly damped flexible-structure transfer matrix with colocated force actuators and position sensors is NI because its modal components are NI and the class is closed under their combination.

A. The Negative-Imaginary Lemma

The negative-imaginary lemma characterizes NI systems through state-space conditions analogous to the positive-real lemma. Its corollary adds a transmission-zero condition to characterize strict negative-imaginary systems.

  • A. The Negative-Imaginary Lemma: Theorem 4 characterizes a minimal state-space system as NI if A has no imaginary-axis eigenvalues, D is symmetric, and a positive-definite Y satisfies the stated conditions.The characterization uses Lyapunov- and coupling-type matrix conditions.
  • A. The Negative-Imaginary Lemma: The Lyapunov inequality and positive definiteness of Y imply that A is asymptotically stable.
  • A. The Negative-Imaginary Lemma: Corollary 5 characterizes SNI systems by adding the requirement that M(s)−M^T(−s) have no imaginary-axis transmission zeros except possibly at s = 0.The first three conditions are inherited from the NI characterization.
  • A. The Negative-Imaginary Lemma: The examples distinguish NI from SNI: one transfer function is both SNI and strictly positive real, while another is NI but not SNI.The latter has a positive-frequency point where its Hermitian-imaginary part is zero.
  • A. The Negative-Imaginary Lemma: The state-space characterization is illustrated by an example that satisfies Theorem 4 and is NI but fails SNI because it has a double zero at s = .

B. Two Strict Negative-Imaginary Lemmas

Two strict negative-imaginary lemmas provide sufficient constructions for SNI transfer functions. They establish SNI for modified systems and for integral resonant controllers used with flexible structures.

  • B. Two Strict Negative-Imaginary Lemmas: Theorem 6 gives sufficient state-space conditions under which a minimal system is SNI, including stable A, symmetric D, and auxiliary positive-definite matrix conditions.
  • B. Two Strict Negative-Imaginary Lemmas: Theorem 6 is established by shifting a transfer function with an auxiliary stable state and then applying the NI characterization and transmission-zero condition.
  • B. Two Strict Negative-Imaginary Lemmas: Theorem 8 proves that the integral resonant controller is SNI when Γ and Φ are positive definite.Integral resonant controllers combine direct feedthrough with integral feedback to damp resonant poles.
  • B. Two Strict Negative-Imaginary Lemmas: Theorem 9 supplies another sufficient construction for SNI systems using two auxiliary parameters α and β with α ≠ β and corresponding matrix conditions.

IV. ROBUST STABILITY OF NEGATIVE-IMAGINARY CONTROL SYSTEMS

The robust-stability result treats positive feedback between an NI system and an SNI system through phase conditions and dc-gain inequalities. In the SISO case, Nyquist arguments provide an intuitive sufficiency proof.

  • IV. ROBUST STABILITY OF NEGATIVE-IMAGINARY CONTROL SYSTEMS: The theorem provides phase stabilization rather than small-gain stabilization: arbitrarily large gains are permitted when the loop Nyquist plot avoids encircling the critical point.
  • IV. ROBUST STABILITY OF NEGATIVE-IMAGINARY CONTROL SYSTEMS: Theorem 13 states that positive feedback between NI M(s) and SNI N(s) is internally stable under M(∞)N(∞) = 0, N(∞) ≥ 0, and the stated dc-gain condition.
  • IV. ROBUST STABILITY OF NEGATIVE-IMAGINARY CONTROL SYSTEMS: The theorem's MIMO proof uses Theorem 4, whereas its SISO sufficiency proof follows directly from Nyquist arguments.
  • IV. ROBUST STABILITY OF NEGATIVE-IMAGINARY CONTROL SYSTEMS: In the SISO case, NI and SNI phase ranges place the loop phase in (−2π, 0), so the Nyquist plot cannot encircle the critical point under the theorem's conditions.
  • IV. ROBUST STABILITY OF NEGATIVE-IMAGINARY CONTROL SYSTEMS: Rigid-body modes require a separate treatment because the standard NI and SNI definitions exclude poles at the origin.A position-feedback inner loop can convert rigid-body modes into vibrational modes before applying the stability result.

V. NEGATIVE-IMAGINARY FEEDBACK CONTROLLERS

The paper applies the NI stability theorem to feedback control of flexible structures with colocated force actuators and position sensors. This links several existing controller families to a dc-gain stability condition.

  • V. NEGATIVE-IMAGINARY FEEDBACK CONTROLLERS: The plant-controller feedback configuration is obtained by treating one block as the flexible-structure plant and the other as the controller.
  • V. NEGATIVE-IMAGINARY FEEDBACK CONTROLLERS: Because colocated flexible structures are typically SNI, NI controllers guarantee closed-loop internal stability when the dc-gain condition is satisfied.
  • V. NEGATIVE-IMAGINARY FEEDBACK CONTROLLERS: The controller schemes covered include positive-position feedback, resonant feedback control, and integral resonant control.

A. Positive-Position Feedback

Positive-position feedback uses controllers that are strictly negative imaginary to stabilize flexible structures with colocated force actuators and position sensors. Nyquist arguments connect the controller and plant phase restrictions to internal stability.

  • Positive-position feedback: A SISO positive-position feedback controller has resonant terms with positive frequencies, damping coefficients, and gains.The parameters satisfy ω_i > 0, ζ_i > 0, and k_i > 0.
  • Positive-position feedback: The resulting SISO controller is SNI, and this construction extends to MIMO controllers with positive-definite matrices.The MIMO form uses D > 0 and Ω > 0.
  • Stability argument: The flexible-structure plant is NI, while the positive-position feedback controller is SNI.The controller phase lies in (−π, 0), and the plant phase lies in [−π, 0] over the relevant positive frequencies.
  • Stability argument: The positive-feedback interconnection is justified by restricting the loop phase and applying a Nyquist stability argument.The combined loop phase lies in (−2π, 0) where the imaginary-axis frequency is not a zero.

B. Resonant Control

Resonant control develops SNI controllers for flexible structures and shows how equivalent implementations can use acceleration or velocity sensing. The constructions extend from SISO to MIMO systems.

  • SISO resonant controllers: An exactly proper SISO SNI controller can be implemented as positive-position feedback using an acceleration sensor.The controller is represented as C(s) = −s^2 C̃(s), where C̃(s) is a SISO positive-position feedback controller.
  • SISO resonant controllers: The same controller can alternatively be implemented as positive-real feedback using a velocity sensor.The velocity-sensor implementation preserves the SNI property through the controller decomposition described in the paper.
  • SISO resonant controllers: A second SISO controller construction is SNI because its first term has zero imaginary part and its second term is SNI.The result follows by applying the paper’s closure lemma to the decomposed controller.
  • MIMO resonant control: The SNI controller constructions extend to MIMO resonant control systems with vector parameters.Controllers of the resulting forms are used for flexible structures with colocated force actuators and position sensors.

C. Integral Resonant Control

Integral resonant control uses SNI controller forms and a dc-gain condition to establish internal stability and tune damping in flexible structures. The same negative-imaginary framework also supports robust stability under structured uncertainty.

  • Integral resonant control: Integral resonant control uses MIMO controllers with positive-definite matrices Γ and Φ, and these controllers are SNI.Applied to flexible structures with force actuators and position sensors, this is also called integral force control.
  • Integral resonant control: 1.5498×10^-4 is the plant dc value, and choosing Φ = 1.8597×10^-4 satisfies the dc-gain condition.The parameter Γ is then selected from the root locus to maximize damping of the first resonant mode at Γ = 9.6584×10^5.
  • Integral resonant control: The closed-loop frequency response illustrates damping of the resonant modes produced by the integral resonant feedback controller.The comparison uses the open-loop plant response and the closed-loop command-to-sensor response.
  • Robust state-feedback synthesis: For structured uncertainty, an NI nominal closed loop, an SNI uncertainty, and the dc-gain condition guarantee robust stability.The state-feedback synthesis assumes full state measurements and represents the uncertain system through equations (50)–(52).
  • Robust state-feedback synthesis: The LMI conditions construct a state-feedback law u = MY^-1x that yields an NI closed-loop transfer function and robust stability.The proof establishes the NI property and verifies the assumptions of Theorem 13.
  • Robust state-feedback synthesis: The robust-stability proof verifies the dc and high-frequency conditions required by Theorem 13.It uses Gcl(∞) = 0, Δ(∞) ≥ 0, and λmax(Δ(0)Gcl(0)) < 1.

E. An LMI State-Feedback Synthesis Example

An LMI state-feedback example replaces an uncertain flexible-structure transfer function with a unity-gain nominal model and treats the difference as SNI uncertainty. The resulting controller is checked through frequency-response and dc-gain conditions.

  • Example setup: The flexible structure has colocated force actuation and position measurement, and its transfer function G(s) is assumed to be SNI.The modeling goal is state-feedback control robust against unmodeled flexible dynamics.
  • Example setup: The method replaces G(s) by a unity gain and models the error Δ(s) = G(s) − 1 as SNI uncertainty.This embeds the flexible-structure mismatch in the uncertain-system framework of Theorem 14.
  • LMI synthesis: ε = 10^-6 is used to solve the LMIs and obtain matrices Y and M, from which the state-feedback gain K is constructed.The numerical matrices and resulting gain are reported in the example.
  • Verification: The closed-loop transfer function is SNI for all ω > 0, and its dc magnitude is less than unity.With SNI uncertainty satisfying |Δ(0)| ≤ 1, Theorem 13 implies internal stability.
  • Verification: The approach remains robust when the flexible structure’s dc value is not known exactly.If G(0) is exactly unity, the dc uncertainty is zero and the associated LMI condition is unnecessary; the example instead allows uncertainty in G(0).

VI. CONCLUSION

The article connects negative-imaginary systems with classical control and passive-system theory, establishing robust stability tools for positive-feedback interconnections and applications to flexible structures and electrical circuits.

  • Negative-imaginary systems are described using classical control ideas and connected to positive-real and passive systems.
  • If one system is negative imaginary and the other strictly negative imaginary, internal stability of their positive-feedback interconnection is characterized by a dc loop gain below unity.
  • The theory provides a framework for robust stability analysis of lightly damped flexible structures with unmodeled dynamics.
  • Passivity theory guarantees internal stability for the negative-feedback interconnection of a positive-real and a strictly positive-real transfer function matrix.
  • Colocated flexible-structure actuators and sensors have counterparts in passive RLC circuits, whose source power is represented by uT(t)y(t).

X. SIDEBAR 3

The sidebar summarizes Finsler’s theorem and explains why rigid-body modes require special treatment in passivity and negative-imaginary control approaches.

  • WHAT IS FINSLER’S THEOREM?: Finsler’s theorem states that a positive semidefinite matrix can be added with a sufficiently large nonnegative multiplier to preserve positivity under a kernel condition.
  • WHAT IS FINSLER’S THEOREM?: The proof applies Finsler’s theorem after establishing xTNx = α2 > 0 for nonzero α.
  • HOW ARE RIGID-BODY MODES HANDLED?: Rigid-body modes have zero natural frequency and cannot be asymptotically stabilized by output feedback when they are unobservable and not asymptotically stable.
  • HOW ARE RIGID-BODY MODES HANDLED?: A position-feedback inner loop can convert rigid-body modes into vibrational modes, making the corresponding position states observable from velocity outputs.
  • HOW ARE RIGID-BODY MODES HANDLED?: NI and SNI definitions exclude poles at the origin, so their associated theorems cannot directly handle rigid-body modes.
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