Learn and Control while Switching: Guaranteed Stability and Sublinear Regret

Control allocation is a vast and richly studied field that addresses the problem of distributing control efforts over redundant actuators to achieve a specific performance. Along this line, supervisory switching control that enables operation with different subsets of actuators is a practical approach for control allocation [1, 2, 3, 4]. In this class of problems, the switching time, the switch mode (which represents a subset of actuators), or both of these parameters can be either specified as part of the general decision-making process or determined by an external unit. Three selected problem classes that illustrate the significance of switching in over-actuated systems are Fault Tolerant Control (FTC), Time-Triggered Protocol (TTP), and moving target defense (MTD). These examples differ in their approach to defining and designing these crucial variables. Designing an algorithm to efficiently switch between different actuating modes when system parameters are unknown presents a significant challenge in terms of performance. We refer to this problem as ”learn and control while switching.” In this context, it is inefficient to disregard previously acquired information about these parameters and start the learning and control process from scratch each time a mode switch occurs. Our goal is to create an algorithm that capitalizes on previously acquired knowledge when commencing a new actuation mode. More specifically, our focus lies on an online Optimism in the Face of Uncertainty (OFU) based algorithm within the framework of Linear Quadratic (LQ) settings. This class of algorithms builds a confidence ellipsoid containing the unknown system parameters, allowing us to design a feedback gain by adopting an optimistic perspective toward this defined set. A naive approach to address our proposed problem would involve the repetitive application of the OFU-based algorithm whenever a mode switch takes place. However, this method is likely to be inefficient, given that it disregards previously acquired knowledge. In this research endeavor, we introduce a novel strategy that leverages learned information. This strategy involves the projection of recently constructed ellipsoids into the mode space that the system has transitioned into. Following this projection, we establish a confidence ellipsoid for control design, incorporating both the knowledge stored within the projected ellipsoid and the online measurements gathered after initiating the actuation. The core technical challenge in constructing this ellipsoid involves confidence set construction while incorporating the already learned information stored within the projected confidence ellipsoid. In our setting, unlike [5], who use the trace norm of the system parameters for regularization in least square estimation and confidence set construction, we utilize a weighted Frobenius norm. This approach provides us with an opportunity to incorporate the projected ellipsoid into confidence ellipsoid construction. To fully incorporate this idea, we also forgo the confidence ellipsoid normalization applied in [5]. As a result, our confidence ellipsoid representation differs slightly from that of [5]. Regarding the stability guarantee in the switching setting, we need to determine a minimum average dwell time that ensures the system slows down the switching process sufficiently to guarantee boundedness of the state norm during the switching operation.

Actuator faults can cause a decay in performance (e.g., impact on energy consumption of actuator) or even more devastating incidents [6]. This kind of safety concerns is often considered within the literature on so-called Fault Tolerant Control (FTC) [7, 8], where the goal of switching is to bypass a failing actuator [9], [10]. [11].

In TTP systems [12, 13], where control must occur through communication channels whose availability varies according to a predetermined/ known a priori schedule, the goal of switching is to conform to the specifications of that protocol.

As another specific application of switching systems, we can refer to the MTD strategy in cyber-physical security of control systems. MTD, which first appeared in the computer science literature [14] is a proactive strategy using which a system can hide its dynamics from a potential attacker by randomly switching between a different subset of actuators. This strategy seems practical considering that adversarial agents have limited resources and cannot compromise all the actuators simultaneously. In [15, 16], the authors apply MTD for identifying malicious attacks on sensors and actuators. Furthermore, in [4] and [17] the authors show an effort to address this challenge for a system with known dynamics.

One significant difference between these applications is that some are instances of ”direct switching” whereas others belong to the ”indirect switching” category [18], which impacts the determination of ”when and what mode to switch to”. For example, in TTP, both the switch time and actuation mode are pre-specified and given. On the other hand, for the FTC, the switch time is simply when an anomaly is detected. For this class of problems, we propose a mechanism that picks the best actuating mode to switch to by examining the richness of learned information up to that point. As for the MTD application, given that high-frequency switches are desirable, we let the system switch as often as desired unless stability is violated. For this aim, we constrain the system to stay in a mode for some minimum average dwell time (MADT). The next actuating method is specified with a similar strategy as in the FTC case. In all cases, the algorithm guarantees closed-loop stability and that parameters are learned well-enough to ensure low 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). The former is addressed by designing a MADT while latter requires detailed insight into available RL algorithms in LQ control literature.

Several studies have recently attempted to address switched systems control under different assumptions when the system model is unknown. Authors in [19] designed data-driven control laws for switched systems with discrete-time linear systems and unknown switch signals. The stability of an unknown switched linear system is investigated probabilistically by observing only finite trajectories in [20]. Furthermore, in [21, 22] design of stabilizing controllers for a switched system is addressed by solely relying on system structure and experimental data and without an explicit plant identification.

Learning-based methods in LQ framework are divided into model-free ([23] [24, 25, 26, 27]) and model-based RL approaches. The former ignores parameter estimates and outputs sub-optimal policy by solely relying on the history of data and directly minimizing the cost function [26]. The latter category usually comes with guaranteed regret, thanks to system identification as a core step in control design. We use similar properties to obtain a guaranteed regret.

A naive approach to model-based adaptive control is a simple philosophy that estimates a system’s unknown parameters and treats them as actual parameters in optimal control design. This algorithm is known as certainty equivalence in the literature, and since it decouples the estimation problem from the control one, it may lead to strictly sub-optimal performance [28]. Model-Based algorithms, in general, can be divided into three main categories, namely ”Offline estimation and control synthesis,” ”Thompson Sampling-based,” and ”Optimism in the Face of Uncertainty (OFU)” algorithms. OFU that couples estimation and control design procedures have shown efficiency in outperforming the naive certainty equivalence algorithm. Campi and Kumar in [28] proposed a cost-based parameter estimator, which is an OFU based approach to address the optimal control problem for linear quadratic Gaussian systems with guaranteed asymptotic optimality. However, this algorithm only guarantees the convergence of average cost to that of the optimal control in limit and does not provide any bound on the measure of performance loss for the finite horizon. Abbasi-Yadkori and Szepesvari [29] proposed a learning-based algorithm to address the adaptive control design of the LQ control system in the finite horizon, with worst case regret bound of $O(\sqrt{T})$. Using $l_{2}$-regularized least square estimation, they construct high-probability confidence set around unknown parameters of the system and design an algorithm that optimistically plays concerning this set [30]. Along this line, Ibrahimi and et. al. [31] later on proposed an algorithm that achieves $O(p\sqrt{T})$ regret bound with state-space dimension of $p$. Authors in [32] proposed an OFU-based learning algorithm with mild assumptions and $O(\sqrt{T})$ regret. Furthermore, in [33] propose an OFU-based algorithm which for joint stabilization and regret minimization that leverages actuator redundancy to alleviate state explosion during initial time steps when there is low number of data.

The objective function to be minimized in the algorithm [30] is non-convex which brings about a computational hurdle. However, recently Cohen et al., in [5], through formulating the LQ control problem in a relaxed semi-definite program (SDP) that accounts for the uncertainty in estimates, proposed a computationally efficient OFU-based algorithm. Moreover, unlike the state-norm bound given by [30] which is loose, [5] guarantees strongly sequentially stabilizing policies enabling us to come ups with tight upper-bound, which ensures small MADT (appropriate for MTD application). These advantages motivate us to use the relaxed SDP framework in our analysis rather than [30].

The remainder of the paper is organized as follows: Section II provides the problem statement and then is followed by Section III which gives a brief background review, overview of projection technique, and augmentation technique. Moving on to Section V, we first provide an overview of the projection-based strategy. Then, we detail the main algorithmic steps and summarize the stability analysis, which includes MADT design and algorithm performance, particularly in terms of the regret bound. Section VI briefly discusses the extension of the proposed algorithm to the FTC and MTD type of applications. Afterwards, numerical experiment is given in Section VII. Finally, we conclude the key achievements in Section VIII. The most rigorous analysis of the algorithms (e.g., stability and regret bounds) and technical proofs are provided in Appendix A while leaving complimentary proofs to Appendix B. .

Throughout this paper, we use the following notations: $\|A\|$ denotes the operator norm i.e., $\|A\|=\max_{x,\|x\|\leq 1}\|Ax\|$. We denote trace norm by $\|\bullet\|_{*}(=\sqrt{\operatorname{Tr}(\bullet^{\top}\bullet)})$. The entry-wise dot product between matrices is denoted by $A\bullet B=\operatorname{Tr}(A^{\top}B)$. The weighted norm of a matrix $A$ with respect to $V$ is interchangeably denoted by $\operatorname{Tr}(A^{\top}VA)$ or $\|A\|_{V}^{2}$. We denote a $m$ dimensional identity matrix by $I_{m}$. We denote the ceiling function by $\lceil x\rceil$ that maps $x$ to the least integer greater than or equal to $x$.

Consider the following time-invariant discrete time LQR system

where $A_{*}\in\mathbb{R}^{n\times n}$, $B_{*}\in\mathbb{R}^{n\times m}$ are unknown and $Q\in\mathbb{R}^{n\times n}$ and $R\in\mathbb{R}^{m\times m}$ are known and respectively positive definite matrices.

Let $\mathbb{B}$ be the set of all columns, $b^{i}_{*}$ ($i\in\{1,...,m\}$) of $B_{*}$. An element of its power set $2^{\mathbb{B}}$ is a subset $\mathcal{B}_{j}$, $j\in\{1,...,2^{m}\}$ 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. With this definition in place we can express the system dynamics as follows:

or equivalently

where $\Theta^{\sigma(t)}_{*}=(A_{*},B^{\sigma(t)}_{*})^{\top}$ is controllable and $\sigma:{0}\;\cup\mathbb{N}\rightarrow\mathcal{M}$ is a given right-continuous piecewise constant switching signal where $\mathcal{M}$ denotes the set of all controllable modes $(A_{,}B^{i}_{*})$.

The associated cost with this mode is

where $R^{\sigma(t)}\in\mathbb{R}^{d_{\sigma(t)}\times d_{\sigma(t)}}$ is a block of $R$ which penalizes the control inputs of the actuators of mode $\sigma(t)\in\mathcal{M}.$

Since we are considering arbitrary, externally-determined, switching patterns (which could include switching to a particular mode at some time instant, then never switching out of it), controllability of the system in each mode is necessary to ensure controllability of the switched system. This mode-by-mode controllability assumption is also useful to enable OFU-style learning at each epoch.

The noise process satisfies the following assumption which is a standard assumption in controls community (e.g., [30], [5] and [34]).

Our goal is to design a strategy which can efficiently transfer the control authority between different subsets of actuators when the system parameters are unknown.

Relying on the past measurements, the problem is designing a sequence $\{u^{\sigma(t)}_{t}\}$ of control input such that the average expected cost

is minimized where $\sigma(t)\in\mathcal{I}$, where $\mathcal{I}=[i_{\tau_{1}}\;i_{\tau_{2}}\;...i_{\tau_{ns}}]$, represents the switch sequence. Each element in the set $\mathcal{I}$ specifies both the mode index $i_{\tau_{j}}\in\mathcal{M}$ and the time $\tau_{j}$ at which the switch to that mode occurs. In the context of TTP applications, this sequence is typically predetermined, and thus the system has access to both the switch times and the corresponding modes. In contrast, for FTC, while the switch times $\tau_{j}$ are determined by an anomaly detector, the decision regarding which mode to switch to is made by the algorithm itself. Therefore, the algorithm plays a pivotal role in selecting the next mode, denoted as $i_{\tau_{j}}\in\mathcal{M}$. When the system parameters are known, it may be relatively straightforward to specify the next mode that results in the lowest average expected cost. However, this task becomes significantly more challenging in the absence of such information. The MTD application introduces an even greater level of complexity, as the algorithm must autonomously decide both when and to which mode to switch. In this work, our primary focus is on TTP setting. In Section VI we will provide a brief overview of the general solutions for the other two settings.

As a measure of optimality loss having to do with the lack of insight into the model, the regret of a policy is defined as follows:

where $J_{*}(\Theta_{*}^{\sigma(t)},Q,R)$ is the optimal average expected cost of actuating in mode $\sigma(t)\in\{1,...,2^{m}\}$.

Note that having access to $\vartheta_{i}$ and $\nu_{i}$ for all $i$ separately is quite demanding and, hence, in practice, we only require knowledge of $\bar{\vartheta}:=\max_{i}\vartheta_{i}$ and $\bar{\nu}=\max_{i}\nu_{i}$. The assumption of having access to an initial stabilizing policy like $K_{0}$ is a common practice in the literature (as seen in [5] and [35]). Nevertheless, it is possible to develop a stabilizing policy in real-time by relying on some a priori bounds. For further details on this approach, we encourage readers to explore [34] and [33].

Developing a ”learn and control while switching” strategy requires overcoming two core challenges: performance and stability. We propose an efficient OFU-based algorithm equipped with a projection tool that can efficiently transfer the learned information from an actuating mode to others to achieve the former goal. This strategy outperforms the naive approach of repeatedly applying the standard OFU algorithm. Fast recurrent switching between the actuating modes can cause explosion of state norm over finite horizon. Hence, in TTP setting, we operate under the assumption that the switches occur at a sufficiently slow rate on average, with the time interval between two consecutive switches $\tau_{j+1}-\tau_{j}$ being longer than some minimum average dwell time $\tau_{MAD}$.

Before presenting the strategy in more detail (and, in a later section, the specifics of the algorithms), we start by reviewing the main ingredients of our approach to build the foundation for our analysis.