Safe Time-Varying Optimization based on Gaussian Processes with Spatio-Temporal Kernel

TL;DR

Evaluation on a realistic case study with gas compressors confirms that TVSafeOpt ensures safety when solving time-varying optimization problems with unknown reward and safety functions, and shows that TVSafeOpt compares favorably against SafeOpt on synthetic data, both regarding safety and optimality.

Abstract

Ensuring safety is a key aspect in sequential decision making problems, such as robotics or process control. The complexity of the underlying systems often makes finding the optimal decision challenging, especially when the safety-critical system is time-varying. Overcoming the problem of optimizing an unknown time-varying reward subject to unknown time-varying safety constraints, we propose TVSafeOpt, a new algorithm built on Bayesian optimization with a spatio-temporal kernel. The algorithm is capable of safely tracking a time-varying safe region without the need for explicit change detection. Optimality guarantees are also provided for the algorithm when the optimization problem becomes stationary. We show that TVSafeOpt compares favorably against SafeOpt on synthetic data, both regarding safety and optimality. Evaluation on a realistic case study with gas compressors confirms that TVSafeOpt ensures safety when solving time-varying optimization problems with unknown reward and safety functions.

Figures (5)

• Figure 1: Comparison of safe sets computed by TVS AFEO PT (top row) and S AFEO PT (bottom row) at $t=30, \;t=100, \text{and } t=170$. Because TVS AFEO PT takes the possible changes in time into consideration, the safe sets computed by TVS AFEO PT are contained in the ground truth safe regions while those computed by S AFEO PT have multiple violations.
• Figure 2: Comparison between TVS AFEO PT, S AFEO PT, and approximate optimization on the gas compressor case study. (a): The cardinality of the safe sets, (b): The ratio between the number of unsafe decisions in the safe sets and the cardinality of the safe sets, (c): The ratio between the number of safe decisions in the safe sets and the cardinality of the ground truth safe regions. TVS AFEO PT robustly shrinks its safe sets based on its observations and thus maintains much less violations in its safe sets than S AFEO PT and approximate optimization. It achieves this benefit at the cost of covering less of the ground truth safe region.
• Figure 3: Reward function values found by TVS AFEO PT and S AFEO PT, compared with the optimal reward function values, in the synthetic example (left), and in the compressor case study (right). In the synthetic example, TVS AFEO PT finds better reward function values than S AFEO PT. The reward function value found by TVS AFEO PT is close to the optimum when the reward function changes slowly, which supports the theoretical result. In the compressor case study, TVS AFEO PT finds lower reward function values, since it is focused on finding robust solutions adapting to significant changes of the reward function.
• Figure 4: Ground truth (solid) and linear approximation (dashed) of the operating area from compressor maps, adapted from Optimum_Noersteboe2008Influence_Zagorowska2019. For a given compressor head at time $t$ (dotted horizontal line for $H_t=120000$ J kg$^{-1}$), the mass flow $m_{it}$ through the $i$-th compressor is required to be between minimum speed (red) and surge (blue) lines, and maximum speed (violet) and choke (yellow) lines
• Figure 5: Visualization of demand (a), compressor head (b), and degradation for the compressors (c) changing with time.

Theorems & Definitions (23)

• Lemma 2.4
• proof
• Theorem 2.5
• Theorem 2.6
• Lemma B.1
• proof : Proof by induction
• proof : Proof of \ref{['thm:safety_tvsafeopt']}
• Lemma C.1
• proof
• Corollary C.2