Stabilizing Dynamical Systems via Policy Gradient Methods

Stabilizing an unknown control system is one of the most fundamental problems in control systems engineering. A wide variety of tasks - from maintaining a dynamical system around a desired equilibrium point, to tracking a reference signal (e.g a pilot’s input to a plane) - can be recast in terms of stability. More generally, synthesizing an initial stabilizing controller is often a necessary first step towards solving more complex tasks, such as adaptive or robust control design (Sontag, 1999, 2009).

In this work, we consider the problem of finding a stabilizing controller for an unknown dynamical system via direct policy search methods. We introduce a simple procedure based off policy gradients which provably stabilizes a dynamical system around an equilibrium point. Our algorithm only requires access to a simulator which can return rollouts of the system under different control policies, and can efficiently stabilize both linear and smooth, nonlinear systems.

Relative to model-based approaches, model-free procedures, such as policy gradients, have two key advantages: they are conceptually simple to implement, and they are easily adaptable; that is, the same method can be applied in a wide variety of domains without much regard to the intricacies of the underlying dynamics. Due to their simplicity and flexibility, direct policy search methods have become increasingly popular amongst practitioners, especially in settings with complex, nonlinear dynamics which may be challenging to model. In particular, they have served as the main workhorse for recent breakthroughs in reinforcement learning and control (Silver et al., 2016; Mnih et al., 2015; Andrychowicz et al., 2020).

Despite their popularity amongst practitioners, model-free approaches for continuous control have only recently started to receive attention from the theory community (Fazel et al., 2018; Kakade et al., 2020; Tu and Recht, 2019). While these analyses have begun to map out the computational and statistical tradeoffs that emerge in choosing between model-based and model-free approaches, they all share a common assumption: that the unknown dynamical system in question is stable, or that an initial stabilizing controller is known. As such, they do not address the perhaps more basic question, how do we arrive at a stabilizing controller in the first place?

We now present our main results establishing how our algorithm, discount annealing, provably stabilizes linear dynamical systems via a reduction to direct policy search methods. We begin with the following preliminaries on the Linear Quadratic Regulator (LQR).

To simplify our presentation, we adopt the shorthand $J_{\mathrm{lin}}(K\mid\gamma):=J_{\mathrm{lin}}(K\mid\gamma,A,B)$ in cases where the system dynamics $(A,B)$ are understood from context. Furthermore, we assume that $(A,B)$ is stabilizable and that $\lambda_{\min}(Q),\lambda_{\min}(R)\geq 1$. It is a well-known fact that $K_{\star,\gamma}:=\operatorname*{arg\,min}_{K}J_{\mathrm{lin}}(K\mid\gamma,A,B)$ achieves the minimum LQR cost over all possible control laws. We begin our analysis with the observation that the discounted LQR problem is equivalent to the undiscounted LQR problem with damped dynamics matrices.²²2This lemma is folklore within the controls community, see e.g. Lamperski (2020).

From this equivalence, it follows from basic facts about LQR that a controller $K$ satisfies $J_{\mathrm{lin}}(0\mid\gamma,A,B)<\infty$ if and only if $\sqrt{\gamma}(A+BK)$ is stable. Consequently, for $\gamma<\rho(A)^{-2}$, the zero controller is stabilizing and one can solve the discounted LQR problem via direct policy search initialized at $K$ = 0 (Fazel et al., 2018). At this point, one may wonder whether the solution to this highly discounted problem yields a controller which stabilizes the undiscounted system. If this were true, running policy gradients (defined in Eq. 2.1) to convergence, on a single discounted LQR problem, would suffice to find a stabilizing controller.

Unfortunately, the following proposition shows that this is not the case.

We now describe the discount annealing procedure for linear systems (Figure 1), which provably recovers a stabilizing controller $K$. For simplicity, we present the algorithm assuming access to noisy, bounded cost and gradient evaluations which satisfy the following definition. Employing standard arguments from (Fazel et al., 2018; Flaxman et al., 2005), we illustrate how these evaluations can be efficiently implemented using polynomially many samples drawn from a simulator in Appendix C.

The procedure leverages the equivalence (Lemma 2.1) between discounted costs and damped dynamics for LQR, and the consequence that the zero controller is stabilizing if we choose $\gamma_{0}$ sufficiently small. Hence, for this discount factor, we may apply policy gradients initialized at the zero controller in order to recover a controller $K_{1}$ which is near-optimal for the $\gamma_{0}$ discounted objective.

Our key insight is that, due to known stability margins for LQR controllers, $K_{1}$ is stabilizing for the $\gamma_{1}$ discounted dynamics for some discount factor $\gamma_{1}>(1+c)\gamma_{0}$, where $c$ is a small constant that has a uniform lower bound. Therefore, $K_{1}$ has finite cost on the $\gamma_{1}$ discounted problem, so that we may again use policy gradients initialized at $K_{1}$ to compute a near-optimal controller $K_{2}$ for this larger discount factor. By iterating, we have that $\gamma_{t}~{}\geq~{}(1+c)^{t}\gamma_{0}$ and can increase the discount factor up to 1, yielding a near-optimal stabilizing controller for the undiscounted LQR objective.

The rate at which we can increase the discount factors $\gamma_{t}$ depends on certain properties of the (unknown) dynamical system. Therefore, we opt for binary search to compute the desired $\gamma$ in the absence of system knowledge. This yields the following guarantee, which we state in terms of properties of the matrix $P_{\star}$, the optimal value function for the undiscounted LQR problem, which satisfies $\min_{K}J_{\mathrm{lin}}(K\mid 1)=\mathrm{tr}\left[P_{\star}\right]$ (see Appendix A for further details).

We remark that since $\varepsilon$ need only be polynomially small in the relevant problem parameters, each call to $\varepsilon\text{-}\mathtt{Grad}$ and $\varepsilon\text{-}\mathtt{Eval}$ can be carried out using only polynomially many samples from a simulator which returns finite horizon system trajectories under various control policies. We make this claim formal in Appendix C.

We now extend the guarantees of the discount annealing algorithm to smooth, nonlinear systems. Whereas our study of linear systems explicitly leveraged the equivalence of discounted costs and damped dynamics, our analysis for nonlinear systems requires access to system rollouts under damped dynamics, since the previous equivalence between discounting and damping breaks down in nonlinear settings.

More specifically, in this section, we assume access to a simulator which given a controller $K$, returns trajectories generated according to $\mathbf{x}_{t+1}=\sqrt{\gamma}G_{\mathrm{nl}}(\mathbf{x}_{t},K\mathbf{x}_{t})$ for any damping factor $\gamma\in(0,1]$, where $G_{\mathrm{nl}}$ is the transition operator for the nonlinear system. While such trajectories may be infeasible to generate on a physical system, we believe these are reasonable to consider when dynamics are represented using software simulators, as is often the case in practice (Lewis et al., 2003; Peng et al., 2018).

The discount annealing algorithm for nonlinear systems is almost identical to the algorithm for linear systems. It again works by repeatedly solving a series of quadratic cost objectives on the nonlinear dynamics as defined below, and progressively increasing the damping factor $\gamma$.

The normalization by $d_{x}/r^{2}$ above is chosen so that the nonlinear objective coincides with the LQR objective when $G_{\mathrm{nl}}$ is in fact linear. Relative to the linear case, the only algorithmic difference for nonlinear systems is that we introduce an extra parameter $r$ which determines the radius for the initial state distribution. As established in Theorem 2, this parameter must be chosen small enough to ensure that discount annealing succeeds. Our analysis pertains to dynamics which satisfy the following smoothness definition.

For simplicity, we assume $\beta_{\mathrm{nl}}~{}\geq~{}1$ and $r_{\mathrm{nl}}~{}\leq~{}1$. Using Assumption 1, we can apply Taylor’s theorem to rewrite $G_{\mathrm{nl}}$ as its Jacobian linearization around the equilibrium point, plus a nonlinear remainder term.

Rather than trying to directly understand the behavior of stabilization procedures on the nonlinear system, the key insight of our nonlinear analysis is that we can reason about the performance of a state-feedback controller on the nonlinear system via its behavior on the system’s Jacobian linearization. In particular, the following lemma establishes how any controller which achieves finite discounted LQR cost for the Jacobian linearization is guaranteed to be exponentially stabilizing on the damped nonlinear system for initial states that are small enough. Throughout the remainder of this section, we define $J_{\mathrm{lin}}(\cdot\mid\gamma):=J_{\mathrm{lin}}(\cdot\mid\gamma,A_{\mathrm{jac}},B_{\mathrm{jac}})$ as the LQR objective from Definition 2.1 where $(A,B)=(A_{\mathrm{jac}},B_{\mathrm{jac}})$.

The second main building block of our nonlinear analysis is the observation that if the dynamics are locally smooth around the equilibrium point, then by Lemma 3.1, decreasing the radius $r$ of the initial state distribution $\mathbf{x}_{0}\sim r\cdot\mathcal{S}^{d_{x}-1}$ reduces the magnitude of the nonlinear remainder term $f_{\mathrm{nl}}$. Hence, the nonlinear system smoothly approximates its Jacobian linearization. More precisely, we establish that the difference in gradients and costs between $J_{\mathrm{nl}}(K\mid\gamma,r)$ and $J_{\mathrm{lin}}(K\mid\gamma)$ decrease linearly with the radius $r$.

Lastly, because policy gradients on linear dynamical systems is robust to inexact gradient queries, we show that for $r$ sufficiently small, running policy gradients on $J_{\mathrm{nl}}$ converges to a controller which has performance close to the optimal controller for the LQR problem with dynamic matrices $(A_{\mathrm{jac}},B_{\mathrm{jac}})$. As noted previously, we can then use Lemma 3.2 to translate the performance of the optimal LQR controller for the Jacobian linearization to an exponential stability guarantee for the nonlinear dynamics. Using these insights, we establish the following theorem regarding discount annealing for nonlinear dynamics.

We note that while our theorem only guarantees that the controller is stabilizing around a polynomially small neighborhood of the equilibrium, in experiments, we find that the resulting controller successfully stabilizes the dynamics for a wide range of initial conditions. Relative to the case of linear systems where we leveraged the monotonicity of the LQR cost to search for discount factors using binary search, this monotonicity breaks down in the case of nonlinear systems and we instead analyze a random search algorithm to simplify the analysis.

In this section, we evaluate the ability of the discount annealing algorithm to stabilize a simulated nonlinear system. Specifically, we consider the familiar cart-pole, with $d_{x}=4$ (positions and velocities of the cart and pole), and $d_{u}=1$ (horizontal force applied to the cart). The goal is to stabilize the system with the pole in the unstable ‘upright’ equilibrium position. For further details, including the precise dynamics, see Section D.1. The system was simulated in discrete-time with a simple forward Euler discretization, i.e., $\mathbf{x}_{t+1}=\mathbf{x}_{t}+{T_{s}}\dot{\mathbf{x}}_{t}$, where $\dot{\mathbf{x}}_{t}$ is given by the continuous time dynamics, and ${T_{s}}=0.05$ (20Hz). Simulations were carried out in PyTorch (Paszke et al., 2019) and run on a single GPU.

Setup. The discounted annealing algorithm of Figure 1 was implemented as follows. In place of the true infinite horizon discounted cost $J_{\mathrm{nl}}(K\mid\gamma,r)$ in Eq. C.4 we use a finite horizon, finite sample Monte Carlo approximation as described in Appendix C,

Here, $J_{\mathrm{nl}}^{(H)}(K\mid\gamma,\mathbf{x})=\sum_{j=0}^{H-1}\mathbf{x}_{t}^{\top}Q\mathbf{x}_{t}+\mathbf{u}_{t}^{\top}R\mathbf{u}_{t}$, is the length $H$, finite horizon cost of a controller $K$ in which the states evolve according to the $\sqrt{\gamma}$ damped dynamics from Eq. C.5 and $\mathbf{u}_{t}=K\mathbf{x}_{t}$. We used $N=5000$ and $H=1000$ in our experiments. For the cost function, we used $Q={T_{s}}\cdot I$ and $R={T_{s}}$. We compute unbiased approximations of the gradients using automatic differentiation on the finite horizon objective $J_{\mathrm{nl}}^{(H)}$.

Instead of using SGD updates for policy gradients, we use Adam (Kingma and Ba, 2014) with a learning rate of $\eta=0.01/r$. Furthermore, we replace the policy gradient termination criteria in Step 2 (Eq. 2.2) by instead halting after a fixed number $(M=200)$ of gradient descent steps. We wish to emphasize that the hyperparameters $(N,H,\eta,M)$ were not optimized for performance. In particular, for $r=0.1$, we found that as few as $M=40$ iterations of policy gradient and horizons as short as $H=400$ were sufficient. Finally, we used an initial discount factor $\gamma_{0}=0.9\cdot\|A_{\mathrm{jac}}\|_{2}^{-2}$, where $A_{\mathrm{jac}}$ denotes the linearization of the (discrete-time) cart-pole about the vertical equilibrium.

Results. We now proceed to discuss the performance of the algorithm, focusing on three main properties of interest: i) the number of iterations of discount annealing required to find a stabilizing controller (that is, increase $\gamma_{t}$ to 1), ii) the maximum radius $r$ of the ball of initial conditions $\mathbf{x}_{0}\sim r\cdot\mathcal{S}^{d_{x}-1}$ for which discount annealing succeeds at stabilizing the system, and iii) the radius ${r_{\text{roa}}}$ of the largest ball contained within the region of attraction (ROA) for the policy returned by discount annealing. Although the true ROA (the set of all initial conditions such that the closed-loop system converges asymptotically to the equilibrium point) is not necessarily shaped like a ball (as the system is more sensitive to perturbations in the position and velocity of the pole than the cart), we use the term region of attraction radius to refer to the radius of the largest ball contained in the ROA.

Concerning (i), discount annealing reliably returned a stabilizing policy in less than 9 iterations. Specifically, over 5 independent trials for each initial radius $r\in\{0.1,0.3,0.5,0.7\}$ (giving 20 independent trials, in total) the algorithm never required more than 9 iterations to return a stabilizing policy.

Concerning (ii), discount annealing reliably stabilized the system for $r~{}\leq~{}0.7$. For $r\approx 0.75$, we observed trials in which the state of the damped system ($\gamma<1$) diverged to infinity. For such a rollout, the gradient of the cost is not well-defined, and policy gradient is unable to improve the policy, which prevents discount annealing from finding a stabilizing policy.

Concerning (iii), in Table 1 we report the final radius ${r_{\text{roa}}}$ for the region of attraction of the final controller returned by discount annealing as a function of the training radius $r$. We make the following observations. Foremost, the policy returned by discount annealing extends the radius of the ROA beyond the radius used during training, i.e. ${r_{\text{roa}}}>r$. Moreover, for each $r>.1$, the ${r_{\text{roa}}}$ achieved by discount annealing is greater than the ${r_{\text{roa}}}=0.703$ achieved by the exact optimal LQR controller and the ${r_{\text{roa}}}=0.506$ achieved by the exact optimal ${\mathcal{H}_{\infty}}$ controller for the system’s Jacobian linearization (see Table 1). (The ${\mathcal{H}_{\infty}}$ optimal controller mitigates the effect of worst-case additive state disturbances on the cost; cf. Section D.2 for details).

One may hypothesize that this is due to the fact that discount annealing directly operates on the true nonlinear dynamics whereas the other baselines (LQR and $\mathcal{H}_{\infty}$ control), find the optimal controller for an idealized linearization of the dynamics. Indeed, there is evidence to support this hypothesis. In Figure 3 presented in Appendix D, we plot the error $\|K_{\text{pg}}^{\star}(\gamma_{t})-K_{\text{lin}}^{\star}(\gamma_{t})\|_{F}$ between the policy $K_{\text{pg}}^{\star}(\gamma_{t})$ returned by policy gradients, and the optimal LQR policy $K_{\text{lin}}^{\star}(\gamma_{t})$ for the (damped) linearized system, as a function of the discount factor $\gamma_{t}$ used in each iteration of the discount annealing algorithm. For small training radius, such as $r=0.05$, $K_{\text{pg}}^{\star}(\gamma_{t})\approx K_{\text{lin}}^{\star}(\gamma_{t})$ for all $\gamma_{t}$. However, for larger radii (i.e $r=0.7$), we see that $\|K_{\text{pg}}^{\star}(\gamma_{t})-K_{\text{lin}}^{\star}(\gamma_{t})\|_{F}$ steadily increases as $\gamma_{t}$ increases.

That is, as discount annealing increases the discount factor $\gamma$ and the closed-loop trajectories explore regions of the state space where the dynamics are increasingly nonlinear, $K_{\text{pg}}^{\star}$ begins to diverge from $K_{\text{lin}}^{\star}$. Moreover, at the conclusion of discount annealing $K_{\text{pg}}^{\star}(1)$ achieves a lower cost, namely [15.2, 15.4] vs [16.5, 16.8] (here $[a,b]$ denotes [$\min$, $\max$] over 5 trials) and larger ${r_{\text{roa}}}$, namely [0.769, 0.777] vs [0.702, 0.703], than $K_{\text{lin}}^{\star}(1)$, suggesting that the method has indeed adapted to the nonlinearity of the system. Similar observations as to the behavior of controllers fine tuned via policy gradient methods are predicted by the theoretical results from Qu et al. (2020).

This works illustrates how one can provably stabilize a broad class of dynamical systems via a simple model-free procedure based off policy gradients. In line with the simplicity and flexibility that have made model-free methods so popular in practice, our algorithm works under relatively weak assumptions and with little knowledge of the underlying dynamics. Furthermore, we solve an open problem from previous work (Fazel et al., 2018) and take a step towards placing model-free methods on more solid theoretical footing. We believe that our results raise a number of interesting questions and directions for future work.

In particular, our theoretical analysis states that discount annealing returns a controller whose stability properties are similar to those of the optimal LQR controller for the system’s Jacobian linearization. We were therefore quite surprised when in experiments, the resulting controller had a significantly better radius of attraction than the exact optimal LQR and $\mathcal{H}_{\infty}$ controllers for the linearization of the dynamics. It is an interesting and important direction for future work to gain a better understanding of exactly when and how model-free procedures are adaptive to the nonlinearities of the system and improve upon these model-based baselines. Furthermore, for our analysis of nonlinear systems, we require access to damped system trajectories. It would be valuable to understand whether this is indeed necessary or whether our analysis could be extended to work without access to damped trajectories.

As a final note, in this work we reduce the problem of stabilizing dynamical systems to running policy gradients on a discounted LQR objective. This choice of reducing to LQR was in part made for simplicity to leverage previous analyses. However, it is possible that overall performance could be improved if rather than reducing to LQR, we instead attempted to run a model-free method that directly tries to optimize a robust control objective (which explicitly deals with uncertainty in the system dynamics). We believe that understanding these tradeoffs in objectives and their relevant sample complexities is an interesting avenue for future inquiry.

We would like to thank Peter Bartlett for many helpful conversations and comments throughout the course of this project, and Russ Tedrake for support with the numerical experiments. JCP is supported by an NSF Graduate Research Fellowship. MS is supported by an Open Philanthropy Fellowship grant. JU is supported by the National Science Foundation, Award No. EFMA-1830901, and the Department of the Navy, Office of Naval Research, Award No. N00014-18-1-2210.