AReN: assured ReLU NN architecture for model predictive control of LTI systems
AReN: assured ReLU NN architecture for model predictive control of LTI systems
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DOI：
10.1145/3365365.3382213
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发表时间：
2019-11
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通讯作者：
James Ferlez;Yasser Shoukry
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作者：
James Ferlez;Yasser Shoukry
在本文中，我们考虑自动设计整流线性单元 (ReLU) 神经网络 (NN) 架构的问题，该架构足以为具有二次成本的 LTI 系统实现最优模型预测控制 (MPC) 策略。具体来说，我们提出了 AReN，一种生成 Assured ReLU 架构的算法。 AReN 将具有二次成本规范的 LTI 系统作为输入，并输出 ReLU NN 架构，并确保存在精确实现关联 MPC 控制器的网络权重。因此，AReN 为控制 LTI 系统的 ReLU NN 架构设计提供了新的见解：AReN 可以在训练开始之前建议适当的 NN 架构，而不是根据数据训练启发式选择的 NN 架构，或者迭代许多架构直到找到合适的架构。虽然之前的一些工作受到 ReLU NN 控制器和最优 MPC 控制器都是连续分段线性 (CPWL) 函数这一事实的启发，但利用这种相似性来设计具有正确性保证的 NN 架构仍然难以实现。 AReN 使用两个新颖的功能来实现这一目标。首先，我们重新解释了最近通过 ReLU NN 实现 CPWL 函数的结果，以表明 CPWL 函数可以通过 ReLU 架构来实现，该架构由函数中不同仿射区域的数量决定。其次，我们证明我们可以有效地过度近似最优 MPC 控制器中的仿射区域数量，而无需精确求解 MPC 问题。总之，这些结果将 MPC 问题与 ReLU NN 实现联系起来，而无需显式求解 MPC：结果是一个 NN 架构，可以保证它可以实现 MPC 控制器。我们通过数值结果展示了 AReN 在设计神经网络架构方面的有效性。
In this paper, we consider the problem of automatically designing a Rectified Linear Unit (ReLU) Neural Network (NN) architecture that is sufficient to implement the optimal Model Predictive Control (MPC) strategy for an LTI system with quadratic cost. Specifically, we propose AReN, an algorithm to generate Assured ReLU Architectures. AReN takes as input an LTI system with quadratic cost specification, and outputs a ReLU NN architecture with the assurance that there exist network weights that exactly implement the associated MPC controller. AReN thus offers new insight into the design of ReLU NN architectures for the control of LTI systems: instead of training a heuristically chosen NN architecture on data - or iterating over many architectures until a suitable one is found - AReN can suggest an adequate NN architecture before training begins. While several previous works were inspired by the fact that ReLU NN controllers and optimal MPC controllers are both Continuous, Piecewise-Linear (CPWL) functions, exploiting this similarity to design NN architectures with correctness guarantees has remained elusive. AReN achieves this using two novel features. First, we reinterpret a recent result about the implementation of CPWL functions via ReLU NNs to show that a CPWL function may be implemented by a ReLU architecture that is determined by the number of distinct affine regions in the function. Second, we show that we can efficiently over-approximate the number of affine regions in the optimal MPC controller without solving the MPC problem exactly. Together, these results connect the MPC problem to a ReLU NN implementation without explicitly solving the MPC: the result is a NN architecture that has the assurance that it can implement the MPC controller. We show through numerical results the effectiveness of AReN in designing an NN architecture.