| Publications about 'linear quadratic regulator (LQR)' |
| Articles in journal or book chapters |
| This article investigates the robustness of gradient descent algorithms under perturbations. The concept of small-disturbance input-to-state stability (ISS) for discrete-time nonlinear dynamical systems is introduced, along with its Lyapunov characterization. The conventional linear \emph{Polyak-\L{}ojasiewicz} (PL) condition is then extended to a nonlinear version, and it is shown that the gradient descent algorithm is small-disturbance ISS provided the objective function satisfies the generalized nonlinear PL condition. This small-disturbance ISS property guarantees that the gradient descent algorithm converges to a small neighborhood of the optimum under sufficiently small perturbations. As a direct application of the developed framework, we demonstrate that the LQR cost satisfies the generalized nonlinear PL condition, thereby establishing that the policy gradient algorithm for LQR is small-disturbance ISS. Additionally, other popular policy gradient algorithms, including natural policy gradient and Gauss-Newton method, are also proven to be small-disturbance ISS. |
| Motivated by the growing use of Artificial Intelligence (AI) tools in control design, this paper takes the first steps towards bridging the gap between results from Direct Gradient methods for the Linear Quadratic Regulator (LQR), and neural networks. More specifically, it looks into the case where one wants to find a Linear Feed-Forward Neural Network (LFFNN) feedback that minimizes a LQR cost. This paper starts by computing the gradient formulas for the parameters of each layer, which are used to derive a key conservation law of the system. This conservation law is then leveraged to prove boundedness and global convergence of solutions to critical points, and invariance of the set of stabilizing networks under the training dynamics. This is followed by an analysis of the case where the LFFNN has a single hidden layer. For this case, the paper proves that the training converges not only to critical points but to the optimal feedback control law for all but a set of measure-zero of the initializations. These theoretical results are followed by an extensive analysis of a simple version of the problem (the ``vector case''), proving the theoretical properties of accelerated convergence and robustness for this simpler example. Finally, the paper presents numerical evidence of faster convergence of the training of general LFFNNs when compared to traditional direct gradient methods, showing that the acceleration of the solution is observable even when the gradient is not explicitly computed but estimated from evaluations of the cost function. |
| Conference articles |
| This paper studies gradient dynamics subject to additive stochastic noise, which may arise from sources such as stochastic gradient estimation, measurement noise, or stochastic sampling errors. To analyze the robustness of such stochastic gradient systems, the concept of small-covariance noise-to-state stability (NSS) is introduced, along with a Lyapunov-based characterization. Furthermore, the classical Polyak–Lojasiewicz (PL) condition on the objective function is generalized to the $\mathcal{K}$-PL condition via comparison functions, thereby extending its applicability to a broader class of optimization problems. It is shown that the stochastic gradient dynamics exhibit small-covariance NSS if the objective function satisfies the $\mathcal{K}$-PL condition and possesses a globally Lipschitz continuous gradient. This result implies that the trajectories of stochastic gradient dynamics converge to a neighborhood of the optimum with high probability, with the size of the neighborhood determined by the noise covariance. Moreover, if the $\mathcal{K}$-PL condition is strengthened to a $\mathcal{K}_\infty$-PL condition, the dynamics are NSS; whereas if it is weakened to a general positive-definite-PL condition, the dynamics exhibit integral NSS. The results further extend to objectives without globally Lipschitz gradients through appropriate step-size tuning. The proposed framework is further applied to the robustness analysis of policy optimization for the linear quadratic regulator (LQR) and logistic regression. |
| In this work we study the convergence of gradient methods for nonconvex optimization problems -- specifically the effect of the problem formulation to the convergence behavior of the solution of a gradient flow. We show through a simple example that, surprisingly, the gradient flow solution can be exponentially or asymptotically convergent, depending on how the problem is formulated. We then deepen the analysis and show that a policy optimization strategy for the continuous-time linear quadratic regulator (LQR) (which is known to present only asymptotic convergence globally) presents almost global exponential convergence if the problem is overparameterized through a linear feed-forward neural network (LFFNN). We prove this qualitative improvement always happens for a simplified version of the LQR problem and derive explicit convergence rates for the gradient flow. Finally, we show that both the qualitative improvement and the quantitative rate gains persist in the general LQR through numerical simulations. |
| Motivated by the current interest in using Artificial intelligence (AI) tools in control design, this paper takes the first steps towards bridging results from gradient methods for solving the LQR control problem, and neural networks. More specifically, it looks into the case where one wants to find a Linear Feed-Forward Neural Network (LFFNN) that minimizes the Linear Quadratic Regulator (LQR) cost. This work develops gradient formulas that can be used to implement the training of LFFNNs to solve the LQR problem, and derives an important conservation law of the system. This conservation law is then leveraged to prove global convergence of solutions and invariance of the set of stabilizing networks under the training dynamics. These theoretical results are then followed by and extensive analysis of the simplest version of the problem (the ``scalar case'') and by numerical evidence of faster convergence of the training of general LFFNNs when compared to traditional direct gradient methods. These results not only serve as indication of the theoretical value of studying such a problem, but also of the practical value of LFFNNs as design tools for data-driven control applications. |
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