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Hybrid integrator-gain system based integral resonant controllers for negative imaginary systems

Kanghong Shi, Ian R. Petersen

Abstract

We introduce a hybrid control system called a hybrid integrator-gain system (HIGS) based integral resonant controller (IRC) to stabilize negative imaginary (NI) systems. A HIGS-based IRC has a similar structure to an IRC, with the integrator replaced by a HIGS. We show that a HIGS-based IRC is an NI system. Also, for a SISO NI system with a minimal realization, we show there exists a HIGS-based IRC such that their closed-loop interconnection is asymptotically stable. Also, we propose a proportional-integral-double-integral resonant controller and a HIGS-based proportional-integral-double-integral resonant controller, and we show that both of them can be applied to asymptotically stabilize an NI system. An example is provided to illustrate the proposed results.

Hybrid integrator-gain system based integral resonant controllers for negative imaginary systems

Abstract

We introduce a hybrid control system called a hybrid integrator-gain system (HIGS) based integral resonant controller (IRC) to stabilize negative imaginary (NI) systems. A HIGS-based IRC has a similar structure to an IRC, with the integrator replaced by a HIGS. We show that a HIGS-based IRC is an NI system. Also, for a SISO NI system with a minimal realization, we show there exists a HIGS-based IRC such that their closed-loop interconnection is asymptotically stable. Also, we propose a proportional-integral-double-integral resonant controller and a HIGS-based proportional-integral-double-integral resonant controller, and we show that both of them can be applied to asymptotically stabilize an NI system. An example is provided to illustrate the proposed results.
Paper Structure (13 sections, 9 theorems, 71 equations, 9 figures)

This paper contains 13 sections, 9 theorems, 71 equations, 9 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Negative imaginary systems
  4. Integral resonant control
  5. Hybrid integrator-gain systems
  6. HIGS-based IRC for NI systems
  7. HIGS-based IRC
  8. NI property of HIGS-based IRC
  9. Stability for the interconnection of an NI system and a HIGS-based IRC
  10. Proportional-integral-double-integral resonant control
  11. HIGS-based $\text{PII}^2\text{RC}$
  12. Example
  13. Conclusion

Key Result

Lemma 1

xiong2010negative Let $(A,B,C,D)$ be a minimal state-space realization of an $p\times p$ real-rational proper transfer function matrix $G(s)$ where $A\in \mathbb R^{n\times n}$, $B\in \mathbb R^{n\times p}$, $C\in \mathbb R^{p\times n}$, $D\in \mathbb R^{p\times p}$. Then $G(s)$ is NI if and only if

Figures (9)

  • Figure 1: Closed-loop interconnection of an integrator $C(s)=\frac{\Gamma}{s}$ and $G(s)+D$.
  • Figure 2: Closed-loop interconnection of an IRC and a plant. This is equivalent to the closed-loop system in Fig. \ref{['fig:CT_IRC']}.
  • Figure 3: Closed-loop interconnection of a HIGS $\mathcal{H}$ and $G(s)+D$.
  • Figure 4: Closed-loop interconnection of a HIGS-based IRC and a plant. It is equivalent to the closed-loop system in Fig. \ref{['fig:HIGS-based IRC']}.
  • Figure 5: Closed-loop interconnection of a HIGS-based IRC and a plant.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Definition 1: NI systems
  • Lemma 1: NI lemma
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • ...and 7 more