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Mixed Regular and Impulsive Sampled-data LQR

Jamal Daafouz, Jérôme Lohéac, Romain Postoyan

Abstract

We investigate the benefits of combining regular and impulsive inputs for the control of sampled-data linear time-invariant systems. We first observe that adding an impulsive term to a regular, zero-order-hold controller may help enlarging the set of sampling periods under which controllability is preserved by sampling. In this context, we provide a tailored Hautus-like necessary and sufficient condition under which controllability of the mixed regular, impulsive (MRI) sampled-data model is preserved. We then focus on LQR optimal control. After having presented the optimal controllers for the sampled-data LQR control in the MRI setting, we consider the scenario where an impulsive disturbance affects the dynamics and is known ahead of time. The solution to the so-called preview LQR is presented exploiting both regular and impulsive input components. Numerical examples, that include an insulin infusion benchmark, illustrate that leveraging both future disturbance information and MRI controls may lead to significant performance improvements.

Mixed Regular and Impulsive Sampled-data LQR

Abstract

We investigate the benefits of combining regular and impulsive inputs for the control of sampled-data linear time-invariant systems. We first observe that adding an impulsive term to a regular, zero-order-hold controller may help enlarging the set of sampling periods under which controllability is preserved by sampling. In this context, we provide a tailored Hautus-like necessary and sufficient condition under which controllability of the mixed regular, impulsive (MRI) sampled-data model is preserved. We then focus on LQR optimal control. After having presented the optimal controllers for the sampled-data LQR control in the MRI setting, we consider the scenario where an impulsive disturbance affects the dynamics and is known ahead of time. The solution to the so-called preview LQR is presented exploiting both regular and impulsive input components. Numerical examples, that include an insulin infusion benchmark, illustrate that leveraging both future disturbance information and MRI controls may lead to significant performance improvements.
Paper Structure (12 sections, 4 theorems, 57 equations, 4 figures)

This paper contains 12 sections, 4 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Controllability analysis
  4. Optimal MRI-LQR design
  5. Cost function
  6. Optimal design
  7. Preview in H$_2$ optimal control
  8. Controller implementation
  9. Examples
  10. Example 1 in Souza2013
  11. Optimal insulin infusion
  12. Conclusion

Key Result

Theorem 1

Suppose the pair $(A,B)$ is controllable, and let $T\in\mathbb{R}_{>0}$. The pair $(A_{d,T},)$ is controllable if and only if for every $\mu\in\{\lambda\in\sigma(A) \text{ s.t. } \exists(\ell,\gamma)\in\mathbb{Z}\times(\sigma(A)\setminus\{\lambda\}) \text{ s.t. } (\lambda-\gamma)T=2i\pi\ell\}$. $\Box$

Figures (4)

  • Figure 1: Optimal costs in Section \ref{['subsect:ex1-souza']}: regular LQR control (red); impulse LQR control (green); MRI-LQR control (blue).
  • Figure 2: Optimal MRI cost in Section \ref{['subsect:ex1-souza']}: with and without preview.
  • Figure 3: Insulin infusion example: BGL response in open-loop and with different MRI-LQR strategies together with the corresponding control inputs.
  • Figure 4: Insulin infusion example: BGL response given by the MRI-LQR strategy with and without preview ($N=2$ and $N=0$, respectively).

Theorems & Definitions (10)

  • Definition 1
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Theorem 2
  • Theorem 3
  • Remark 3