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Modulating function based algebraic observer coupled with stable output predictor for LTV and sampled-data systems

Matti Noack, Ibrahima N'Doye, Johann Reger, Taous-Meriem Laleg-Kirati

Abstract

This paper proposes an algebraic observer-based modulating function approach for linear time-variant systems and a class of nonlinear systems with discrete measurements. The underlying idea lies in constructing an observability transformation that infers some properties of the modulating function approach for designing such algebraic observers. First, we investigate the algebraic observer design for linear time-variant systems under an observable canonical form for continuous-time measurements. Then, we provide the convergence of the observation error in an L2-gain stability sense. Next, we develop an exponentially stable sampled-data observer which relies on the design of the algebraic observer and an output predictor to achieve state estimation from available measurements and under small inter-sampling periods. Using a trajectory-based approach, we prove the convergence of the observation error within a convergence rate that can be adjusted through the fixed time-horizon length of the modulating function and the upper bound of the sampling period. Furthermore, robustness of the sampled-data algebraic observer, which yields input-to-state stability, is inherited by the modulating kernel and the closed-loop output predictor design. Finally, we discuss the implementation procedure of the MF-based observer realization, demonstrate the applicability of the algebraic observer, and illustrate its performance through two examples given by linear time-invariant and linear time-variant systems with nonlinear input-output injection terms.

Modulating function based algebraic observer coupled with stable output predictor for LTV and sampled-data systems

Abstract

This paper proposes an algebraic observer-based modulating function approach for linear time-variant systems and a class of nonlinear systems with discrete measurements. The underlying idea lies in constructing an observability transformation that infers some properties of the modulating function approach for designing such algebraic observers. First, we investigate the algebraic observer design for linear time-variant systems under an observable canonical form for continuous-time measurements. Then, we provide the convergence of the observation error in an L2-gain stability sense. Next, we develop an exponentially stable sampled-data observer which relies on the design of the algebraic observer and an output predictor to achieve state estimation from available measurements and under small inter-sampling periods. Using a trajectory-based approach, we prove the convergence of the observation error within a convergence rate that can be adjusted through the fixed time-horizon length of the modulating function and the upper bound of the sampling period. Furthermore, robustness of the sampled-data algebraic observer, which yields input-to-state stability, is inherited by the modulating kernel and the closed-loop output predictor design. Finally, we discuss the implementation procedure of the MF-based observer realization, demonstrate the applicability of the algebraic observer, and illustrate its performance through two examples given by linear time-invariant and linear time-variant systems with nonlinear input-output injection terms.
Paper Structure (10 sections, 3 theorems, 32 equations, 9 figures, 1 table)

This paper contains 10 sections, 3 theorems, 32 equations, 9 figures, 1 table.

Table of Contents

  1. Algebraic Observer Design
  2. Linear Time-variant Problem Setup
  3. Modulating Function Method
  4. MF-based Observer Algorithm
  5. Sampled-data Estimation via Output Predictor Closed-loop
  6. Implementation
  7. MF-based Observer Realization
  8. LTI Simulation Example with Input-Output Injection and Measurement Noise
  9. Time-variant Pendulum Dynamics with Horizontal Disturbance
  10. Conclusion

Key Result

Lemma 1

For a given LMF $\varphi:[0,T]\rightarrow\mathbb{R}^n$ of order $n$, applying the modulation operator ${\mathcal{M}}$ from eq:modop to eq:inoutop leads to with adjoint operators Therein, the collection of differentiation signal vectors $Y=[y,\dot{y},\ldots,y^{(n-1)}]^\top,$$U_j=[u_j,\dot{u}_j,\ldots,u_j^{(n-1)}]^\top$ and $D_l=[d_l,\dot{d}_l,\ldots,d_l^{(n-1)}]^\top$ are separated via the bounda

Figures (9)

  • Figure 1: UMF kernel $\varphi(\tau)=[\varphi_1,\varphi_2]^\top$ of order 2 and its derivative $\dot{\varphi}$ as defined in Equation \ref{['eq:umfcond']} of Definition \ref{['def:mf']}.
  • Figure 2: Block diagram of implementation scheme for observer \ref{['eq:sdobs']}.
  • Figure 3: Output signal $y$, noisy measurement $\tilde{y}$ and sampled data $\tilde{y}(t_i)$ (time axis zoom right).
  • Figure 4: Comparison of different output errors $e_y=y-\hat{y}$ for the scenarios of observer \ref{['eq:sdobs']}, the approach MazencMN20 and for just considering sampled data $\tilde{y}(t_i)$ in the noisy case (y-axis zoom right).
  • Figure 5: State estimation result $\hat{x}(t)$ using different algorithms.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Remark 1: Observability canonical form
  • Definition 1: Left & unitary MFs
  • Lemma 1: Adjoint equation & boundary values
  • Theorem 1: MF-based observer
  • Remark 2: Transformation of observable system
  • Theorem 2: Sampled-data state estimation
  • Remark 3: Kernel design strategies