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Stability robustness of a feedback interconnection of systems with negative imaginary frequency response
A. Lanzon, I. R. Petersen
TL;DR
The paper addresses internal stability of feedback interconnections involving LTI MIMO systems with negative imaginary frequency response, a property relevant to lightly damped structures. It derives a necessary-and-sufficient DC loop-gain condition, characterizes such systems in state space, and demonstrates the result’s robustness application. The central result is that internal stability is guaranteed precisely when the DC loop gain, measured in the paper’s specified sense, is less than unity.
Problem
The paper addresses how to establish internal stability for feedback interconnections of LTI MIMO systems with negative imaginary frequency response using limited information about the systems.
Method
The paper derives a mathematical stability theorem and corollary, extends the analysis to MIMO systems, and provides a complete state-space characterization.
Results
Internal stability is characterized by the DC loop gain, measured in a precise sense, being less than unity.
Takeaways & Limitations
The result supports robust stability analysis for uncertain spillover dynamics and may assist future synthesis of systems with negative imaginary frequency response.
Abstract
from arXiv · showhide
A necessary and sufficient condition, expressed simply as the DC loop gain (ie the loop gain at zero frequency) being less than unity, is given in this paper to guarantee the internal stability of a feedback interconnection of Linear Time-Invariant (LTI) Multiple-Input Multiple-Output (MIMO) systems with negative imaginary frequency response. Systems with negative imaginary frequency response arise for example when considering transfer functions from force actuators to co-located position sensors, and are commonly important in for example lightly damped structures. The key result presented here has similar application to the small-gain theorem, which refers to the stability of feedback interconnections of contractive gain systems, and the passivity theorem (or more precisely the positive real theorem in the LTI case), which refers to the stability of feedback interconnections of positive real systems. A complete state-space characterisation of systems with negative imaginary frequency response is also given in this paper and also an example that demonstrates the application of the key result is provided.
Notation
The paper establishes notation for stable real-rational transfer-function matrices and real or complex matrices, along with eigenvalue, matrix-operation, and transfer-function-adjoint conventions.
- R∞ denotes real-rational stable transfer-function matrices of dimension n × n.
- R^n×n and C^n×n denote real and complex matrices, respectively, of dimension n × n.
- λ_i(A) denotes the i-th eigenvalue, while λ(A) denotes the maximum eigenvalue when all eigenvalues are real.
- ℜ(a) and ℑ(a) denote real and imaginary parts, while A^T, A* and M~(s) denote matrix transpose, conjugate transpose, and transfer-function adjoint.
I. INTRODUCTION
The introduction motivates negative-imaginary analysis for feedback stability, especially in lightly damped structures with spillover dynamics. It frames the paper’s contribution as a necessary-and-sufficient MIMO DC loop-gain condition, plus a state-space characterization and robustness application.
- I. INTRODUCTION: The paper seeks limited-information stability conditions for positive feedback interconnections of LTI MIMO systems with negative imaginary frequency response.
- I. INTRODUCTION: Force-actuator to co-located position-sensor transfer functions typically have negative imaginary frequency response in lightly damped structures.
- I. INTRODUCTION: Finite-mode plant models create spillover unmodeled dynamics from lightly damped modes omitted during control-system design.
- I. INTRODUCTION: Standard positive-real analysis is unhelpful for these higher-relative-degree spillover dynamics, while small-gain analysis is typically conservative for highly resonant systems.
- I. INTRODUCTION: A controller ensuring the disturbance-to-input closed-loop transfer has negative imaginary response and DC gain below 1/(Σ_{i=h+1}^H k_i) robustly stabilizes all stated spillover dynamics.
- I. INTRODUCTION: The paper formalizes the robustness theorem, extends the necessary-and-sufficient DC loop-gain condition to MIMO systems, and gives a complete state-space characterization.
- I. INTRODUCTION: Replacing the systems with sM(s) and −N(s)/s does not directly enable positive-real analysis because stability, properness, and conditional-stability differences remain.
II. SOME TECHNICAL RESULTS
The section develops technical tools for stable LTI systems with negative imaginary frequency response, including set definitions, state-space characterizations, and matrix properties. These results establish relationships between DC and infinite-frequency gains and support later stability proofs.
- System classes: The paper defines classes C and Cs for stable systems whose negative imaginary frequency response satisfies non-strict or strict inequalities.The strict class Cs is a subset of C.
- State-space characterization: Membership in Cs requires an additional check that the relevant transmission zeros lie outside the open frequency region (0, ∞).The strict frequency-domain inequality implies nonsingularity of the associated frequency-response matrix.
- State-space characterization: A complete state-space characterization of C requires a Hurwitz state matrix, symmetric feedthrough, and a positive-definite matrix satisfying AY + YA* ≤ 0 and B = −AYC*.This characterization also provides a test for membership in C.
- Gain properties: R(0) − R(∞) ≥ 0 for R(s) ∈ C, and R(0) − R(∞) > 0 for R(s) ∈ Cs.Thus, negative imaginary frequency response constrains the DC and infinite-frequency gains.
- Closure and matrix properties: The class C is closed under addition with another C system, while adding a C system to a Cs system preserves strict membership in Cs.A matrix lemma also establishes that unity is excluded from the spectrum of AB under the stated non-strict and strict negative imaginary conditions.
III. THE MAIN RESULT
The main result gives a necessary and sufficient internal-stability test for a positive feedback interconnection of systems in C and Cs. Under stated infinite-frequency assumptions, stability is determined by whether the DC loop gain is less than unity.
- Theorem 5: Theorem 5 states that M(s) ∈ C and N(s) ∈ Cs yield internal stability exactly when the DC loop gain is less than unity.The theorem also assumes M(∞)N(∞) = 0 and N(∞) ≥ 0.
- Proof strategy: The proof uses minimal realizations, positive-definite state-space matrices, and a matrix result excluding imaginary-axis singularities of I − M(jω)N(jω).Internal stability is linked to the Hurwitz property of the closed-loop realization.
- Scope conditions: The theorem requires assumptions on the systems’ gains at infinite frequency.These assumptions are explicitly noted as part of the necessary-and-sufficient result.
- Theorem 5: The internal-stability condition is equivalently expressed as N(0)^−1 − M(0) > 0.This follows from the state-space proof’s reduction of the relevant matrix inequality.
- Corollary 6: A weaker corollary restates the result in a form analogous to small-gain and passivity theorems and provides robust-stability statements for classes of perturbations.The corollary includes conditions on perturbation and nominal-system feedthrough terms.
IV. ILLUSTRATIVE EXAMPLE
The example models an uncertain two-mass lightly damped mechanical plant, separates its nominal and uncertain dynamics, and applies the proposed DC loop-gain condition to characterize robust stability.
- The plant consists of two unit masses with known wall springs and dampers, uncertain coupling stiffness k and damping α, force inputs, and displacement measurements.
- The uncertain plant is decomposed as P_Δ(s) = P(s) + Δ(s), with P(s) known and Δ(s) representing the uncertain remainder.
- The controller C(s) defines M(s), the transfer matrix from w to z, allowing the closed loop to be rearranged for robust-stability analysis.
- Robust stability for all uncertainties in the class follows from the condition λ(Δ(0)M(0)) < 1.
- For any α > 0, the controller robustly stabilizes the physical system if and only if k > 0.75.
V. CONCLUSIONS
The paper generalizes negative-imaginary stability analysis from SISO to MIMO LTI systems and identifies a precise DC loop-gain criterion. It also provides a state-space characterization while noting extensions needed for synthesis and nonlinear or time-varying systems.
- The key stability result generalizes from SISO negative-imaginary systems to MIMO LTI systems.
- Internal stability is characterized necessarily and sufficiently by a precisely defined DC loop gain being less than unity.
- The paper gives a complete state-space characterization of MIMO LTI systems with negative imaginary frequency response.
- Future work includes controller synthesis for classes C or C_s and extensions to nonlinear or time-varying systems.