Antoine Aspeel

Antoine Aspeel

I am a PhD in Control and Optimization. I am currently doing a postoc at L2S (Laboratoire des Signaux et Systèmes), part of CentraleSupélec, Paris-Saclay.

PhD Thesis — Optimal Sampling for State Estimation and Control

I started working on my Ph.D. thesis in 2017 with Prof. Raphaël Jungers and Prof. Benoit Macq. My Ph.D. thesis was motivated by an applied problem in proton therapy, a cancer treatment method that targets tumors with a proton beamline. The success of this approach relies on precise knowledge of the tumor’s position, which becomes particularly challenging for mobile tumors, such as lung tumors, that move with the patient’s breathing. One approach to address this challenge is to use X-ray images of the patient’s chest to measure the tumor’s position. However, X-ray imaging exposes both the tumor and healthy tissue to radiation, limiting the number of images that can be safely taken.

My thesis addresses the problem of determining the optimal timing for X-ray acquisitions, within a limited budget, to maximize the accuracy of the tumor tracking.

From a more abstract perspective, this problem can be formulated as optimizing the measurement times to obtain the best state estimation of a dynamical system. I studied this problem under different assumptions: linear vs. nonlinear dynamics; expected vs. worst-case error.

Despite the practical motivation behind my thesis, my research has been primarily (though not exclusively) theoretical, with a strong focus on its potential for real-world, safety-critical applications.

Linear and Gaussian dynamics: Optimal intermittent Kalman filter

In a first phase of my thesis, I focused on the case of a linear system subject to Gaussian process and measurement noises, fitting the Kalman filter formalism. My goal is to select the measurement times in order to minimize the cumulated expected mean squared error.

This research led to the following publication:

The linear system we are considering can be written

\[\begin{aligned} x_{t+1} &= Ax_t + w_t \\ y_t &= \begin{cases} Cx_t + v_t & \text{if } \sigma_t = 1 \\ \emptyset & \text{else.} \end{cases} \end{aligned}\]

where $x_t$ is the state, $y_t$ is the measurement, $w_t$ and $v_t$ are process and measurement noises. They are independent and follow a Gaussian distribution. The quantity $\sigma_t\in$ {0,1} is a binary variable describing when a measurement is acquired. This dynamics is considered over a finite time horizon $t=0,\dots,T$, and a budget of at most $N$ measurements is allowed, i.e., $ \sum_{t=0}^T\sigma_t\leq N$.

In this research, my goal is to design the measurement times $(\sigma_t)_{t=0}^T$ in order to minimize the cumulated expected mean squared estimation error

\[\sum_{t=0}^T \mathbb{E}[\|x_t-\hat{x}_t\|^2_2],\]

where

\[\hat{x}_t=\mathbb{E}[x_t| y_0,\dots,y_t]\]

is the minimum mean squared error estimate. For a given sequence of measurement times $(\sigma_t)_{t=0}^T$, the estimate $\hat{x}_t$ is given by the Kalman filter (the correction step of the Kalman filter is skipped when $\sigma_t=0$ and no measurement is available).

Time- vs. self-triggered measurements: In order to make the estimation error as small as possible, the decision $\sigma_t$ of measuring at time $t$ should depend on all the available information at that time, including the previous measurements $y_0,\dots,y_{t-1}$. Formally, this can be written

\[\sigma_t=\pi_t(\sigma_0,\dots,\sigma_{t-1};y_0,\dots,y_{t-1}),\]

where $\pi_t$ is a policy function. In that case, the measurement-times selection mechanism is called self-triggered.

In contrast, if the measurement times $\sigma_t$ are selected a priori, independently of the measurements, the measurement-times selection is called time-triggered. In general, time-triggered mechanisms are easier to implement, but self-triggered mechanisms give better performance.

Designing self-triggered mechanisms poses the following computational challenges:

\[(\pi_t)_{t=0}^T\]

that decide if a measurement is necessary at time $t$ based on the previously acquired measurements $y_0,\dots,y_{t-1}$. Since functions are infinite dimensional objects, this makes the problem infinite-dimensional and therefore seemingly intractable.
Solution: A key finding of this part of my research is that the optimal policies do not depend on the measurements. Consequently, one can optimize directly over the measurement times $(\sigma_t)_{t=0}^T$ making the problem finite-dimensional. In other words, we showed that the optimal self-triggered and time-triggered mechanisms are equivalent. In addition, the measurement times can be selected offline (i.e., before the beginning of the tracking at time $t=0$) without loss of optimality.

Using a duality argument, our method can be applied to control a linear system subject to an actuation budget while minimizing a LQR cost.

Application to tumor tracking and industrial collaboration

Throughout my Ph.D., I had the privilege of collaborating with IBA (Ion Beam Applications), the world-leading company in proton therapy. This partnership allowed me to bridge the gap between theory and practice by applying the methods I developed to tumor tracking using real medical data.

Description
Proposed clinical workflow for the treatment of lung tumors. Stars represent X-ray acquisition times.

Using real patient data, we demonstrated that optimizing X-ray acquisition times with our intermittent Kalman filter significantly improves tumor tracking, thereby enhancing the effectiveness of the treatment.

This more applied contribution resulted in the following medical publication:

Nonlinear and non-Gaussian dynamics: Optimal intermittent particle filter

In a second phase of my Ph.D., I considered the case of nonlinear dynamical systems subject to non-Gaussian noise. My work resulted in the following publications:

In that case, the optimal policy for the measurement times is not independent of the measurements, i.e., time- and self-triggered measurement mechanisms give different solutions. Therefore, we developed a time-triggered and a self-triggered method to optimize the measurement times of a particle filter.

Time-triggered particle filter

We started by considering the problem of finding the best time-triggered measurement times, i.e., the measurement times $(\sigma_t)_{t=0}^T$ must be selected offline and are not allowed to depend on the measurements $y_0,\dots,y_T$.

Description
Representation of our method to estimate the cumulated mean squared error cost for given measurement times \( \sigma_{0:T} \). The outputs generated by the block "Monte Carlo" are drawn according to the system dynamics via noise sampling. The particle filter computes an estimate \( \hat{x}^k(t) \) of \( x^k(t) \) from previous available measurements. These estimates are compared with the true \( x^k(t) \) to compute the cumulated mean squared error.

Self-triggered particle filter

To design a self-triggered particle filter, we used a receding horizon approach. First, all measurement times are selected offline using our time-triggered particle filter. Then, only the first measurement is acquired, and the remaining measurement times are re-optimized by taking into account the already received measurements. This process is repeated until the measurement budget $N$ is reached.

Overall, both of our methods have their merits: the time-triggered particle filter is easier to implement because it avoids online computations, while the self-triggered particle filter offers provably better performance. In addition, both methods outperform filtering with regularly spaced measurements (see figure below).

Histogram comparison of filtering methods
Histograms of the gain \( G \coloneqq \log_{10}\left( \dfrac{\text{MSE with regular measurements}} {\text{MSE with optimized measurements}} \right) \) for our time- and self-triggered methods. A gain \(G=+1\) indicates that the method performs 10 times better than filtering with regularly spaced measurement times. We observe that both methods outperform regularly spaced measurements, while the self-triggered method achieves the best performance.

Beyond mean squared error: Optimal sampling for worst-case control

This part of my research was published in

We considered a linear control system subject to bounded disturbances together with a linear controller with memory, i.e.,

\[u_t = \sum_{\tau \leq t} K_{(t,\tau)} y_\tau,\]

where the gains $K_{(t,\tau)}$ are design variables. Our objective was to co-design:

while satisfying two types of constraints:

Safety

The safety constraint is not convex with respect to the controller gains $K_{(t,\tau)}$. However, using the Youla parameterization allows one to rewrite the constraint linearly in terms of the Youla parameter (assuming safety and noise sets are polytopes).

Budget

The budget constraints can be written as

\[\sum_{t=0}^T \sigma_t^{\text{meas./control}} \leq N^{\text{meas./control}}.\]

For the controller structure above, the budget constraints on sensors and actuators can be handled using linear indicator constraints on the controller gains:

\[\begin{aligned} \sigma_\tau^{\text{meas.}} = 0 &\Rightarrow \left( K_{(t,\tau)} = 0 \text{ for all } t \right), \\ \sigma_t^{\text{control}} = 0 &\Rightarrow \left( K_{(t,\tau)} = 0 \text{ for all } \tau \right). \end{aligned}\]

The first constraint ensures that if a measurement is unavailable at time $\tau$, it cannot be used by the controller. The second constraint guarantees that $u_t = 0$ whenever $\sigma_t^{\text{control}} = 0$.