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Projection-based Controllers with Inherent Dissipativity Properties

Hoang Chu, S. J. A. M van den Eijnden, W. P. M. H. Heemels

Abstract

Projection-based Controllers (PBCs) are currently gaining traction in both scientific and engineering communities. In PBCs, the input-output signals of the controller are kept in sector-bounded sets by means of projection. In this paper, we will show how this projection operation can be used to induce useful passivity or general dissipativity properties on broad classes of (unprojected) nonlinear controllers that otherwise would not have these properties. The induced dissipativity properties of PBC will be exploited to guarantee asymptotic stability of negative feedback interconnections of passive nonlinear plants and suitably designed PBC, under mild conditions. Proper generalizations to so-called $(q,s,r)$-dissipativity will be presented as well. For illustrating the effectiveness of PBC control design via these passivity-based techniques, two numerical examples are provided.

Projection-based Controllers with Inherent Dissipativity Properties

Abstract

Projection-based Controllers (PBCs) are currently gaining traction in both scientific and engineering communities. In PBCs, the input-output signals of the controller are kept in sector-bounded sets by means of projection. In this paper, we will show how this projection operation can be used to induce useful passivity or general dissipativity properties on broad classes of (unprojected) nonlinear controllers that otherwise would not have these properties. The induced dissipativity properties of PBC will be exploited to guarantee asymptotic stability of negative feedback interconnections of passive nonlinear plants and suitably designed PBC, under mild conditions. Proper generalizations to so-called -dissipativity will be presented as well. For illustrating the effectiveness of PBC control design via these passivity-based techniques, two numerical examples are provided.
Paper Structure (12 sections, 1 theorem, 31 equations, 6 figures)

This paper contains 12 sections, 1 theorem, 31 equations, 6 figures.

Table of Contents

  1. Introduction
  2. System description and problem formulation
  3. System configuration and plant model
  4. Projection-based control
  5. Problem formulation
  6. Design of projection-based controllers
  7. Main results
  8. Existence of solutions
  9. Stability analysis
  10. Mass-spring-damper system
  11. TORA system
  12. Conclusion

Key Result

Theorem 1

Consider the plant eq:plant5 and suppose Assumption assumption: detectable is satisfied. Furthermore, suppose the PBC controller eq:controller5 to have a sector condition satisfying Assumption assumption: sector and the $z_2$-dynamics to be ISS with respect to $(z_1,v)$. Then the closed-loop PBC sys

Figures (6)

  • Figure 1: Negative feedback interconnection of a plant and a controller.
  • Figure 2: Comparison simple gain controller and projected controller on a mass-spring-damper system.
  • Figure 3: Projected controller input-output operating on a mass-spring-damper system.
  • Figure 4: Time evolution of the position of the mass and the angle of the pendulum of the TORA system with three controllers (C0: simple gain, C1 and C2: PBC)
  • Figure 5: The input-output pair of two PBC controllers operating on the TORA system (plotted against time and against each other)
  • ...and 1 more figures

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Remark 1
  • Definition 4.1
  • Theorem 1
  • proof