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Positive Observers Revisited

David Ohlin, Anders Rantzer, Emma Tegling

Abstract

The paper shows that positive linear systems can be stabilized using positive Luenberger-type observers, contradicting previous conclusions. This is achieved by structuring the observer as monotonically converging upper and lower bounds on the state. Analysis of the closed-loop properties under linear observer feedback gives conditions that cover a larger class than previous observer designs. The results are applied to nonpositive systems by enforcing positivity of the dynamics using feedback from the upper bound observer. The setting is expanded to include stochastic noise, giving conditions for convergence in expectation using feedback from positive observers.

Positive Observers Revisited

Abstract

The paper shows that positive linear systems can be stabilized using positive Luenberger-type observers, contradicting previous conclusions. This is achieved by structuring the observer as monotonically converging upper and lower bounds on the state. Analysis of the closed-loop properties under linear observer feedback gives conditions that cover a larger class than previous observer designs. The results are applied to nonpositive systems by enforcing positivity of the dynamics using feedback from the upper bound observer. The setting is expanded to include stochastic noise, giving conditions for convergence in expectation using feedback from positive observers.
Paper Structure (10 sections, 6 theorems, 23 equations, 2 figures)

This paper contains 10 sections, 6 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Problem setup
  5. Positive observers
  6. Closing the loop
  7. Relaxed conditions
  8. Feedback positivization
  9. Stochastic noise
  10. Conclusion

Key Result

Proposition 1

(Monotonicity) Consider the system ${x(t+1) = Ax(t)}$ with ${A\ge0}$. For ${a, b\in\mathbb{R}^n_+}$ it holds that

Figures (2)

  • Figure 1: Evolution of the first state component $x_1$ of \ref{['eq:exsys']} and corresponding upper and lower estimates $\overline{x}_1$ and $\underline{x}_1$ with observer and feedback gains according to \ref{['eq:exgains']}. The component $x_1$ contains the unstable mode in \ref{['eq:exsys']}, which is stabilized by feedback from the lower estimate when the observer state approaches the true value $x$. Each element of the initial state $x(0)$ is drawn randomly from the interval $\left[0,1\right]$, with estimates initialized to ${\overline{x}(0)=\mathbf{1}}$ and ${\underline{x}(0)=\mathbf{0}}$, dictating the transient performance. The chosen gains stabilize the errors and dynamics.
  • Figure 2: Trajectory of the first state component $x_1$ and corresponding bounds for the system in Example 3. Due to the process and measurement noise, the bounds are no longer guaranteed to hold at every time step $t$. Regardless, invariance ${X\in\mathcal{X}}$ of the expected behavior is sufficient to guarantee asymptotic convergence to a bounded steady state, despite instability of the autonomous dynamics \ref{['eq:ex3sys']}.

Theorems & Definitions (18)

  • Proposition 1
  • Proposition 2
  • Definition 1
  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • ...and 8 more