Joint Stabilization and Regret Minimization through Switching in Over-Actuated Systems (extended version)

The past few years have witnessed a growing interest in an online learning-based Linear Quadratic (LQ) control, in which an unknown LTI system is controlled while guaranteeing a suitable scaling of the regret (defined as the average difference between the closed-loop quadratic cost and the best achievable one, with the benefit of knowing the plant’s parameters) over a desired horizon $[0,T]$.

Table 1 summarizes the regret’s scaling achieved by several recent works in the literature (Abbasi and Szepesvári (2011); Mania et al. (2019); Cohen et al. (2019); Lale et al. (2020a)). As can be seen, the bounds scale like $\sqrt{T}$, but also include an exponential term in $m$ (the dimension of the plant’s state plus inputs) when an initial stabilizing controller is unavailable. Recently, Chen and Hazan (2021) have shown that this dependency is unavoidable in this setting, at least for an initial exploration time period, by providing a matching lower bound. On closer inspection, this undesirable dependency stems from an exponential growth of the system’s state during the initial period of learning, which eventually contributes a $\mathcal{O}(m^{m})$ term like that appearing on the last row of Table 1. Apart from negatively impacting regret, this transient state growth can also be damageable if, e.g., the linear plant to be controlled is in fact the linearization of a nonlinear process around some particular equilibrium, and the state can be driven outside the neighborhood of this equilibrium where the linearization is adequate.

A natural idea to try and partially alleviate this effect in over-actuated systems is to reduce the ambient dimension $``m"$ during the initial period by employing only a subset of all available actuators, before potentially switching to a different mode to simultaneously learn and control the plant. The goal of this paper is to show that this approach yields the desired result (a lower bound on the plant’s state and the regret than in the presence of all actuators) and to provide a rigorous proof of the achieved $\mathcal{O}((n+d_{i})^{(n+d_{i})})$ bound (where $d_{i}$ is number of actuators during initial exploration). We note that, while the idea is conceptually simple, obtaining these rigorous guarantees is not completely straightforward, and requires a revisit of the tools of Lale et al. (2020a) to overcome a potential challenge: ensuring that, at the end of the period when a strict subset of actuators is used, all entries of the B-matrix (including those corresponding to unused actuators) are sufficiently well-learned for the closed-loop to become stable and the regret to remain appropriately bounded. This can only be achieved by adding additional exploratory noise with respect to the approach of Lale et al. (2020a) and results in an additional linear term in the regret bound.

* requirement for initial stabilizing controller.

While this idea of leveraging over-actuation can generally be applied to any model-based type of online learning algorithms presenting the exponential scaling mentioned above, we focus on the special kind of methods based on “Optimism in the Face of Uncertainty (OFU)”.

OFU type of algorithms, which couple estimation and control design procedures, have shown their ability to outperform the naive certainty equivalence algorithm. Campi and Kumar (1998) propose an OFU based approach to address the optimal control problem for LQ setting with guaranteed asymptotic optimality. However, this algorithm only guarantees the convergence of average cost to that of the optimal control in the limit but does not provide any bound on the measure of performance loss in finite time. Abbasi and Szepesvári (2011) propose a learning-based algorithm to address the adaptive control design of LQ control system in finite time with worse case regret bound of $\mathcal{O}(\sqrt{T})$ with exponential dependence in dimension. Using $l_{2}$-regularized least square estimation, they construct a high-probability confidence set around unknown parameters of the system and designed an algorithm that optimistically plays with respect to this set. Along this line, many works attempt to get rid of the exponential dependence with further assumption, e.g. highly sparse dynamics (see Ibrahimi et al. (2012)) or access to a stabilizing controller (see Cohen et al. (2019)). Furthermore, Faradonbeh et al. (2020) propose an OFU-based learning algorithm with mild assumptions and $\mathcal{O}(\sqrt{T})$ regret. This class of algorithms was extended by Lale et al. (2020b, 2021) to LQG setting where there is only partial and noisy observations on the state of system. In addition, Lale et al. (2020a) propose an algorithm with more exploration for both controllable and stabilizable systems.

The remainder of the paper is organized as follows: Section 2 reviews the preliminaries, assumptions and presents the problem statement and formulation. Section 3 overviews the proposed initial exploration (IExp), stabilizing OFU (SOFUA) algorithms and discusses in detail how to choose best actuating mode for the initial exploration purpose. Furthermore, in Section 3 the performance analysis (state norm upper-bound and regret bound) is given while leaving the details of the analysis to Section 6 and technical theorems and proofs of analysis to Section 7. Numerical experiments are given in Section 4. Finally, Section 5 summarizes the paper’s key contributions.

Consider the following linear time invariant dynamics and the associated cost functional given by:

where the plant and input matrices $A_{*}\in\mathbb{R}^{n\times n}$ and $B_{*}\in\mathbb{R}^{n\times d}$ are initially unknown and have to be learned and $(A_{*},B_{*})$ is controllable. $Q_{*}\in\mathbb{R}^{n\times n}$ and $R_{*}\in\mathbb{R}^{d\times d}$ represent known and positive definite matrices. $\omega_{t+1}$ denotes process noise, satisfying the following assumption.

The problem is designing a sequence $\{u_{t}\}$ of control inputs such that the regret $\mathcal{R}_{T}$ defined by

achieves a desired specification which scales sublinearly in $T$. The term $J_{*}(\Theta_{*},Q_{*},R_{*})$ in (2) where $\Theta_{*}=(A_{*}\;B_{*})^{\top}$ denotes optimal average expected cost. For LQR setting with controllable pair $(A,\;B)$ we have $J_{*}(\Theta,Q,R)=\bar{\sigma}_{\omega}^{2}trace(P(\Theta,Q,R))$, where $P(\Theta,Q,R)$ is the unique solution of discrete algebraic riccati equation (DARE) and the average expected cost minimizing policy has feedback gain of

While the regret’s exponential dependency on system dimension appears in the long-run in Abbasi and Szepesvári (2011) the recent results of Mania et al. (2019) on the existence of a stabilizing neighborhood, makes it possible to design an algorithm that only exhibits this dependency during an initial exploration phase (see Lale et al. (2020a)).

After this period, the controller designed for any estimated value of the parameters is guaranteed to be stabilizing and the exponentially dependent term thus only appears as a constant in overall regret bound. As explained in the introduction, this suggests using only a subset of actuators during initial exploration to even further reduce the guaranteed upperbound on the state.

In the remainder of the paper, we pick the best actuating mode (i.e. subset of actuators) so as to minimize the state norm upper-bound achieved during initial exploration and characterize the needed duration of this phase for all system parameter estimates to reside in the stabilizing neighborhood. This is necessary to guarantee both closed loop stability and acceptable regret, and makes it possible to switch to the full actuation mode.

Let $\mathbb{B}$ be the set of all columns, $b^{i}_{*}$ ($i\in\{1,...,d\}$) of $B_{*}$. An element of its power set $2^{\mathbb{B}}$ is a subset $\mathcal{B}_{j}$ $j\in\{1,...,2^{d}\}$ of columns corresponding to a submatrix $B_{*}^{j}$ of $B_{*}$ and mode $j$. For simplicity, we assume that $B^{1}_{*}=B_{*}$ i.e., that the first mode contains all actuators. Given this definition we write down different actuating modes dynamics with extra exploratory noise as follows

where $\Theta^{i}_{*}=(A_{*},B^{i}_{*})^{\top}$ is controllable.

The associated cost with this mode is

where $R_{*}^{i}\in\mathbb{R}^{d_{i}\times d_{i}}$ is a block of $R_{*}$ which penalizes the control inputs of the actuators of mode $i$.

We have the following assumption on the modes which assists us in designing proposed strategy.

By slightly abusing notation, we drop the superscript label for the actuating mode 1 (e.g. $\Upsilon_{1}=\Upsilon$, $s^{1}=s$, and $\mathcal{S}_{c}^{1}=\mathcal{S}_{c}$). It is obvious that $s^{i}\leq s$ $\forall i$.

Note that the item (1) in Assumption 2 is typical in the literature of OFU-based algorithms (see e.g., Abbasi and Szepesvári (2011); Lale et al. (2020a)) while (2) in fact always holds in the sense that $\sup_{\Theta^{i}\in\mathcal{S}^{i}}\|A_{*}+B^{i}_{*}K({\Theta}^{i},Q_{*},R^{i}_{*})\|$ and $J_{*}(\Theta_{*}^{i},Q_{*},R^{i}_{*})-J_{*}(\Theta_{*},Q_{*},R_{*})$ are always bounded (see e.g., Abbasi and Szepesvári (2011); Lale et al. (2020a)). The point of (2), then, is that upper-bounds on their suprema are available which can in turn be used to bound regret explicitly. The knowledge of these bounds does not affect Algorithms 1 and 2 but their value enters Algorithm 3 for determination of the best actuating mode and the corresponding exploration duration. In that sense ”best actuating mode” should be understood as ”best given the available information”.

Boundedness of $S^{i}_{c}$’s implies boundedness of $P(\Theta^{i},Q_{*},R_{*}^{i})$ with a finite constant $D_{c}^{i}$ (see Anderson and Moore (1971)), (i.e., $D_{c}^{i}=\sup\{\left\lVert P(\Theta^{i},Q_{*},R_{*}^{i})\right\rVert\mid\Theta^{i}\in\mathcal{S}^{i}_{c}\}$). We define $D=\max_{i\in\mathcal{B}^{*}}D^{i}$. Furthermore, there exists $\kappa^{i}_{c}>1$ such that $\kappa^{i}_{c}=\sup\{\left\lVert K(\Theta^{i},Q_{*},R_{*}^{i})\right\rVert\mid\Theta^{i}\in\mathcal{S}^{i}_{c}\}$.

Recalling that the set of actuators of mode $i$ is $\mathcal{B}_{i}$, we denote its complement by $\mathcal{B}_{i}^{c}$ (i.e. $\mathcal{B}_{i}\cup\mathcal{B}_{i}^{c}=\{1,...,d\}$). Furthermore, we denote the complement of control matrix $B_{*}^{i}$ by $\bar{B}^{i}_{*}$.

If some modes fail to satisfy Assumption 2 they can simply be removed from the set $2^{\mathbb{B}}$ without affecting algorithm or the derived guarantees.

In this section, we propose an algorithm in the spirit of that first proposed by Lale et al. (2020a) which leverages actuator redundancy in the ”more exploration” step to avoid blow up in the state norm while minimizing the regret bound. We break down the strategy into two phases of initial exploration, presented by Algorithm (IExp), and optimism (Opt), given by SOFUA algorithm.

The IExp algorithm, which leverages exploratory noise, is deployed in the actuating mode $i^{*}$ for duration $T_{c}^{i^{*}}$ to reach a stabilizing neighborhood of the full-actuation mode and alleviate state explosion while minimizing regret.

Afterwards, Algorithm 2 which leverages all the actuators comes into play. This algorithm has the central confidence set, given by the Algorithm 1, as an input. The best actuating mode $i^{*}$ that guarantees minimum possible state norm upper-bound and initial exploration duration $T^{i^{*}}_{c}$ is determined by running Algorithm 3 at the subsection 3.3.

In this section, we demonstrate on a practical example how the use of our algorithms successfully alleviates state explosion during the initial exploration phase. We consider a control system with drift and control matrices to be set as follows:

We choose the cost matrices as follows:

The Algorithm 3 outputs the exploration duration $T^{i^{*}}_{c}=50s$ and best actuating mode $i^{*}$ for initial exploration with corresponding control matrix $B_{*}^{i^{*}}$ and $R_{*}^{i^{*}}$

It has graphically been shown in Abbasi-Yadkori (2013) that the optimization problem (11) is generally non-convex for $n,d>1$. Because of this fact, we decided to solve optimization problem (11) using a projected gradient descent method in Algorithm 1 and 2, with basic step

where $L^{i}=\bar{\sigma}^{2}_{\omega}+\vartheta^{2}\bar{\sigma}^{2}_{\nu}$ for $i=i^{*}$ and $L^{i}=\bar{\sigma}^{2}_{\omega}$ for $i=1$. $\nabla_{\Theta^{i}}f$ is the gradient of $f$ with respect to $\Theta^{i}$. $\mathcal{C}^{i}_{t}(\Theta^{i})$ is the confidence set, $PROJ_{g}$ is Euclidean projection on $g$ and finally $\gamma$ is the step size. Computation of gradient $\nabla_{\Theta^{i}}$ as well as formulation of projection has been explicited in Abbasi-Yadkori (2013), similar to which we choose the learning rate as follows:

We apply the gradient method for 100 iterations to solve each OFU optimization problem and apply the projection technique until the projected point lies inside the confidence ellipsoid. The inputs to the OFU algorithm are $T=10000$, $\delta=1/T$, $\lambda=1$, $\sigma_{\omega}=0.1$, $s=1$ and we repeat simulation $10$ times.

As can be seen in Fig. 1, the maximum value of the state norm (attained during the initial exploration phase) is smaller when using mode $i^{*}$ than when all actuators are in action.

Regret-bound for both cases is linear during initial exploration phase, however SOFUA guarantees $\mathcal{O}(\sqrt{T})$ regret for $T>50s$.

In this work, we proposed an OFU principle-based controller for over-actuated systems, which combines a step of ”more-exploration” (to produce a stabilizing neighborhood of the true parameters while guaranteeing a bounded state during exploration) with one of ”optimism”, which efficiently controls the system. Due to the redundancy, it is possible to further optimize the speed of convergence of the exploration phase to the stabilizing neighborhood by choosing over actuation modes, then to switch to full actuation to guarantee an $\mathcal{O}(\sqrt{T})$ regret in closed-loop, with polynomial dependency on the system dimension.

A natural extension of this work is to classes of systems in which some modes are only stabilizable. Speaking more broadly, the theme of this paper also opens the door to more applications of switching as a way to facilitate learning-based control of unknown systems, some of which are the subject of current work.