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Digital control of negative imaginary systems: a discrete-time hybrid integrator-gain system approach

Kanghong Shi, Ian R. Petersen

Abstract

A hybrid integrator-gain system (HIGS) is a control element that switches between an integrator and a gain, which overcomes some inherent limitations of linear controllers. In this paper, we consider using discrete-time HIGS controllers for the digital control of negative imaginary (NI) systems. We show that the discrete-time HIGS themselves are step-advanced negative imaginary systems. For a minimal linear NI system, there always exists a HIGS controller that can asymptotically stablize it. An illustrative example is provided, where we use the proposed HIGS control method to stabilize a discrete-time mass-spring system.

Digital control of negative imaginary systems: a discrete-time hybrid integrator-gain system approach

Abstract

A hybrid integrator-gain system (HIGS) is a control element that switches between an integrator and a gain, which overcomes some inherent limitations of linear controllers. In this paper, we consider using discrete-time HIGS controllers for the digital control of negative imaginary (NI) systems. We show that the discrete-time HIGS themselves are step-advanced negative imaginary systems. For a minimal linear NI system, there always exists a HIGS controller that can asymptotically stablize it. An illustrative example is provided, where we use the proposed HIGS control method to stabilize a discrete-time mass-spring system.
Paper Structure (9 sections, 3 theorems, 59 equations, 3 figures)

This paper contains 9 sections, 3 theorems, 59 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Discrete-time NI systems
  4. Discrete-time hybrid integrator-gain systems
  5. MAIN RESULTS
  6. SANI property of the HIGS
  7. Stability for the interconnection of a linear NI system and a HIGS
  8. EXAMPLE
  9. CONCLUSION AND FUTURE WORK

Key Result

Lemma 1

shi2023discrete Suppose the linear system (eq:G(z)) satisfies $\det(I-A)\neq 0$. Then the system (eq:G(z)) is NI with a positive definite quadratic storage function satisfying (eq:NNI ineq) if and only if there exists a real matrix $P=P^T>0$ such that

Figures (3)

  • Figure 1: Closed-loop interconnection of the system (\ref{['eq:G(z)']}) with the transfer matrix $G(z)$ and the HIGS $\mathcal{H}$ given in (\ref{['eq:DT-HIGS']}).
  • Figure 2: A mass-spring system with masses $m_1 = 0.04kg$, $m_2 = 0.02kg$ and spring constants $k_1=2N/m$ and $k_2 = 1N/m$. $x_a$ and $x_c$ denote the displacement of the masses $m_1$ and $m_2$, respectively. A force input $u$ is applied on the mass $m_2$.
  • Figure 3: State trajectories of the plant (\ref{['eq:example plant']}) and the HIGS (\ref{['eq:DT-HIGS']}), which are interconnected in positive feedback. Starting from nonzero initial conditions, all the state variables converge to zero. The closed-loop system is asymptotically stable, which is consistent with our expectation according to Theorem \ref{['theorem:stability']}.

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof