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Steady-State Cascade Operators and their Role in Linear Control, Estimation, and Model Reduction Problems

John W. Simpson-Porco, Daniele Astolfi, Giordano Scarciotti

TL;DR

This work introduces steady-state cascade (SSC) operators as a unifying object for problems involving cascaded linear systems, encompassing stabilization, estimation, and model reduction. By formulating primal and dual Sylvester operators and proving equivalences between controllability/observability and injectivity/surjectivity of SSC mappings, the authors derive multiple design pathways, including recursive forwarding and low-gain strategies, with explicit moment-based parameterizations linking SSC to frequency response. The paper also demonstrates how SSC-based methods yield novel observer/estimator designs and structured reduced-order models, and extends the framework to nonlinear settings with a clear agenda for future nonlinear theory. The results provide a cohesive toolkit for cascade analysis and design, enabling data-driven (moment-based) and model-reduced approaches with robust stability guarantees under mild assumptions, and establishing a foundation for extending to nonlinear and time-varying contexts.

Abstract

Certain linear matrix operators arise naturally in systems analysis and design problems involving cascade interconnections of linear time-invariant systems, including problems of stabilization, estimation, and model order reduction. We conduct here a comprehensive study of these operators and their relevant system-theoretic properties. The general theory is leveraged to delineate both known and new design methodologies for control and observation of cascades, and to characterize structural properties of reduced models. Several entirely new designs arise from this systematic categorization, including new recursive and low-gain design frameworks for observation of cascaded systems. The benefits of the results beyond the linear time-invariant setting are demonstrated through preliminary extensions for nonlinear systems, with an outlook towards the development of a similarly comprehensive nonlinear theory.

Steady-State Cascade Operators and their Role in Linear Control, Estimation, and Model Reduction Problems

TL;DR

This work introduces steady-state cascade (SSC) operators as a unifying object for problems involving cascaded linear systems, encompassing stabilization, estimation, and model reduction. By formulating primal and dual Sylvester operators and proving equivalences between controllability/observability and injectivity/surjectivity of SSC mappings, the authors derive multiple design pathways, including recursive forwarding and low-gain strategies, with explicit moment-based parameterizations linking SSC to frequency response. The paper also demonstrates how SSC-based methods yield novel observer/estimator designs and structured reduced-order models, and extends the framework to nonlinear settings with a clear agenda for future nonlinear theory. The results provide a cohesive toolkit for cascade analysis and design, enabling data-driven (moment-based) and model-reduced approaches with robust stability guarantees under mild assumptions, and establishing a foundation for extending to nonlinear and time-varying contexts.

Abstract

Certain linear matrix operators arise naturally in systems analysis and design problems involving cascade interconnections of linear time-invariant systems, including problems of stabilization, estimation, and model order reduction. We conduct here a comprehensive study of these operators and their relevant system-theoretic properties. The general theory is leveraged to delineate both known and new design methodologies for control and observation of cascades, and to characterize structural properties of reduced models. Several entirely new designs arise from this systematic categorization, including new recursive and low-gain design frameworks for observation of cascaded systems. The benefits of the results beyond the linear time-invariant setting are demonstrated through preliminary extensions for nonlinear systems, with an outlook towards the development of a similarly comprehensive nonlinear theory.
Paper Structure (36 sections, 9 theorems, 107 equations, 5 figures, 1 table)

This paper contains 36 sections, 9 theorems, 107 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Steady-State Cascade Operators
  3. Definitions and Interpretation
  4. System-Theoretic Properties of the SSC Operators
  5. Parameterization via Frequency Response and Moments
  6. Applications in Control, Estimation, and Model Reduction of Linear Systems
  7. Stabilization of the Cascade $\Sigma \rightarrow \Sigma^{\prime}$
  8. Stabilization with $\mathdutchcal{C}_{\rm d}$
  9. "Fix $G$, Design $K_{\eta}$"
  10. "Fix $K_{\eta}$, Design $G$"
  11. Low-Gain Stabilization with $\mathdutchcal{C}_{\rm d}$
  12. "Fix $G$, Design $K_{\eta}(\epsilon)$
  13. "Fix $K_{\eta}$, Design $G(\epsilon)$
  14. Low-Gain Stabilization with $\mathdutchcal{C}_{\rm p}$
  15. "Fix $G$, Design $K_{\eta}(\epsilon)$
  16. ...and 21 more sections

Key Result

Theorem 1

Suppose that $\mathrm{eig}(A) \cap \mathrm{eig}(F) = \emptyset$ and consider the operators $\mathdutchcal{C}_{\rm p}$ and $\mathdutchcal{C}_{\rm d}$ defined in Eq:GenL. The following statements are equivalent: Moreover, and the converses of SSCNewItm:6a, resp. SSCNewItm:6b, holds if $G^{\sf T}\ker(\lambda I_{\nu}-F^{\sf T}) = \mathbb{C}^p$ for all $\lambda \in \mathrm{eig}(F)$, resp. for all $\l

Figures (5)

  • Figure 1: The cascade interconnection $\Sigma^{\prime} \to \Sigma$.
  • Figure 2: The cascade interconnection $\Sigma \to \Sigma^{\prime}$.
  • Figure 3: Disturbance rejection response in four-tank system. Tuning parameters: gain $K$ computed via LQR with $Q = \mathrm{diag}(3,3,1,1)$ and $R = 0.1I$. Subsequent gains required in the designs computed via LQR with $Q = I$ and $R = \{10^6, 5\cdot 10^6, 10^4, 3\cdot10^6, 10^6,10^4\} \times I$.
  • Figure 4: Disturbance estimation in four-tank system. Tuning parameters: $L_x$ computed via LQR with $Q = \mathrm{diag}(3,3,1,1)$ and $R = I$. Subsequent gains required in the designs computed via LQR with $Q = I$ and $R = \{10^2, 10^2, 10^5\} \times I$.
  • Figure 5: Disturbance estimation for a linear system driven by a Van der Pol oscillator.

Theorems & Definitions (26)

  • Remark 1: Literature on Sylvester Equations
  • Theorem 1: SSC Operators and Right-Invertibility
  • Remark 2: Cascade Controllability
  • Remark 3: Meaning of $\boldsymbol{G^{\sf T}\ker(\lambda I_{\nu}-F^{\sf T})} = \boldsymbol{\mathbb{C}^p}$
  • proof
  • Theorem 2: SSC Operators and Left-Invertibility
  • Corollary 1: SSC Operators and Invertibility
  • Remark 4: Effect of State Feedback and Output Injection
  • Theorem 3: SSC Operators and Moments
  • proof
  • ...and 16 more