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Simultaneous compensation of input delay and state/input quantization for linear systems via switched predictor feedback

Florent Koudohode, Nikolaos Bekiaris-Liberis

Abstract

We develop a switched predictor-feedback law, which achieves global asymptotic stabilization of linear systems with input delay and with the plant and actuator states available only in (almost) quantized form. The control design relies on a quantized version of the nominal predictor-feedback law for linear systems, in which quantized measurements of the plant and actuator states enter the predictor state formula. A switching strategy is constructed to dynamically adjust the tunable parameter of the quantizer (in a piecewise constant manner), in order to initially increase the range and subsequently decrease the error of the quantizers. The key element in the proof of global asymptotic stability in the supremum norm of the actuator state is derivation of solutions' estimates combining a backstepping transformation with small-gain and input-to-state stability arguments, for addressing the error due to quantization. We extend this result to the input quantization case and illustrate our theory with a numerical example.

Simultaneous compensation of input delay and state/input quantization for linear systems via switched predictor feedback

Abstract

We develop a switched predictor-feedback law, which achieves global asymptotic stabilization of linear systems with input delay and with the plant and actuator states available only in (almost) quantized form. The control design relies on a quantized version of the nominal predictor-feedback law for linear systems, in which quantized measurements of the plant and actuator states enter the predictor state formula. A switching strategy is constructed to dynamically adjust the tunable parameter of the quantizer (in a piecewise constant manner), in order to initially increase the range and subsequently decrease the error of the quantizers. The key element in the proof of global asymptotic stability in the supremum norm of the actuator state is derivation of solutions' estimates combining a backstepping transformation with small-gain and input-to-state stability arguments, for addressing the error due to quantization. We extend this result to the input quantization case and illustrate our theory with a numerical example.
Paper Structure (9 sections, 6 theorems, 110 equations, 4 figures)

This paper contains 9 sections, 6 theorems, 110 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Control Design
  3. Linear Systems With Input Delay & State Quantization
  4. Properties of the Quantizer
  5. Predictor-Feedback Law Using Almost Quantized Measurements
  6. Stability of Switched Predictor-Feedback Controller Under State Quantization
  7. Extension to Input Quantization
  8. Simulation Results
  9. Conclusions and Future Work

Key Result

Theorem 1

Consider the closed-loop system consisting of the plant pde_representation--pde_representation1 and the switched predictor-feedback law control_quantizer--switching_parameter. Let the pair $(A, B)$ be stabilizable. If $\Delta$ and $M$ satisfy then for all $X_{0} \in \mathbb{R}^{n}$, $u_{0} \in \mathcal{C}_{lpw}([0, D], \mathbb{R})$, there exists a unique solution such that $X(t) \in AC\left(\math

Figures (4)

  • Figure 1: An approximate quantizer with $\varepsilon$-layer.
  • Figure 2: Left: The norm $|X(t)| + \|u(t)\|_{\infty}$ of the closed-loop system \ref{['pde_representation']}--\ref{['pde_representation1']}, \ref{['quantizer_matrix']}, \ref{['quantizer_simus']}, for $D = 1$, $M = 2$, $\Delta =\dfrac{M}{100}$, under the predictor-feedback law \ref{['control_quantizer']}--\ref{['switching_parameter']}, \ref{['M3']}--\ref{['T']}, with parameters $\overline{M}=0.6,$$\overline{M}_1=2,$$\Omega=0.63,$$T=2,$ and $\mu_0=1.$ The dashed line is the switching signal $\mu(t)M\overline{M}$. Right: The respective states of the closed-loop system.
  • Figure 3: Left: The norm $|X(t)| + \|u(t)\|_{\infty}$ of the closed-loop system \ref{['pde_representation']}--\ref{['pde_representation1']}, for $D = 1$, $M = 2$, $\Delta =\dfrac{M}{100}$, under the nominal feedback law $U(t)=\mu K\int_{0}^{D}e^{A(D-y)}Bq\left(\dfrac{u(y)}{\mu}\right)dy+\mu K e^{AD}q\left(\dfrac{X_1}{\mu}\right) \quad q\left(\dfrac{X_2}{\mu}\right)^T,$ for fixed $\mu = 0.1$ and $q$ defined in \ref{['quantizer_matrix']}, \ref{['quantizer_simus']}, with parameters $\overline{M}=0.6$, $\overline{M}_1 = 2$, $\Omega=0.63$, $T = 2$, and $\mu_0 = 1$. Right: The respective states of the closed-loop system.
  • Figure 4: Left: The norm $|X(t)| + \|u(t)\|_{\infty}$ of the closed-loop system \ref{['pde_representation']}--\ref{['pde_representation1']}, for $D = 1$, $M = 2$, $\Delta =\dfrac{M}{100}$, under the nominal feedback law $U=\mu K\int_{0}^{D}e^{A(D-y)}Bq\left(\dfrac{u(y)}{\mu}\right)dy+\mu K e^{AD}q\left(\dfrac{X_1}{\mu}\right) \quad q\left(\dfrac{X_2}{\mu}\right)^T,$ for fixed $\mu = 100$ and $q$ defined in \ref{['quantizer_matrix']}, \ref{['quantizer_simus']}, with parameters $\overline{M}=0.6$, $\overline{M}_1 = 2$, $\Omega=0.63$, $T = 2$, and $\mu_0 = 1$. Right: The respective states of the closed-loop system.

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof