Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sampled-Data Controller Synthesis using Dissipative Linear Periodic Jump-Flow Systems with Design Applications

L. M. Spin, M. C. F. Donkers

Abstract

In this paper, we will propose linear-matrix-inequality-based techniques for the design of sampled-data controllers that render the closed-loop system dissipative with respect to \textcolor{blue}{quadratic supply functions}, which includes passivity and an upper-bound on the system's $\mathcal{H}_\infty$-norm as a special case. To arrive at these results, we model the sampled-data control system as a linear periodic jump-flow system, study dissipativity in terms of differential linear matrix inequalities (DLMIs) and then convert these DLMIs into a single linear matrix inequality. We will present three applications of these synthesis techniques: 1) passivity-based controller synthesis, as found in teleoperations, 2) input-output-response matching of a continuous-time filter with a discrete-time filter (by minimizing the $\mathcal{H}_{\infty}$-norm of a generalized plant) and 3) a sampled-data controller redesign problem, where the objective is to find the best sampled-data controller, in the $\mathcal{H}_{\infty}$-norm sense, for a given continuous-time controller. We will show that synthesising sampled-data controllers leads to better closed-loop system behaviour than using a Tustin discretization of a continuous-time controller.

Sampled-Data Controller Synthesis using Dissipative Linear Periodic Jump-Flow Systems with Design Applications

Abstract

In this paper, we will propose linear-matrix-inequality-based techniques for the design of sampled-data controllers that render the closed-loop system dissipative with respect to \textcolor{blue}{quadratic supply functions}, which includes passivity and an upper-bound on the system's -norm as a special case. To arrive at these results, we model the sampled-data control system as a linear periodic jump-flow system, study dissipativity in terms of differential linear matrix inequalities (DLMIs) and then convert these DLMIs into a single linear matrix inequality. We will present three applications of these synthesis techniques: 1) passivity-based controller synthesis, as found in teleoperations, 2) input-output-response matching of a continuous-time filter with a discrete-time filter (by minimizing the -norm of a generalized plant) and 3) a sampled-data controller redesign problem, where the objective is to find the best sampled-data controller, in the -norm sense, for a given continuous-time controller. We will show that synthesising sampled-data controllers leads to better closed-loop system behaviour than using a Tustin discretization of a continuous-time controller.
Paper Structure (11 sections, 3 theorems, 50 equations, 6 figures)

This paper contains 11 sections, 3 theorems, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Generalized Plant Formulation
  3. Stability & Dissipativity of SDCSs
  4. Quadratic Storage and Supply Functions
  5. Controller Synthesis
  6. Special Cases: Passivity and the $\mathcal{H}_\infty$ Norm
  7. Applications to Control Design
  8. Application to teleoperations
  9. Application to Filter Matching
  10. Application to Controller Matching
  11. Conclusion

Key Result

Lemma 3.1

Consider the LPJF system eq:ClosedLoopSD and supply functions eq:QRSsupply that satisfy $R_\mathrm{c} \preceq 0$ and $R_\mathrm{d}\preceq0$. Assume there exists a bounded matrix function $P(\tau)\succ0$ that is differentiable on the domain $\tau \in (0, h]$ that satisfies where one of the inequalities holds strictly. Then, the LPJF system eq:ClosedLoopSD is dissipative with respect to supply funct

Figures (6)

  • Figure 1: The sampled-data control system schematic.
  • Figure 2: Passivity-based control of teleoperations scheme
  • Figure 3: Block diagram of the filter-matching problem.
  • Figure 4: Comparison of discretization techniques.
  • Figure 5: Filter-matching technique for redesign of CT filters for SDCS synthesis.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Definition 3.1
  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Definition 3.2