SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Bahar Taskesen
bahir.tasksen@epfl.ch
Dan A. Iancu
daniancu@stanford.edu
Cagli Koçyğit
University of Luxembourg
cagil.kocyigit@uni.lu
daniel.kuhn@epfl.ch
Linear-Quadratic-Gaussian (LQG) control is a fundamental control paradigm that has been studied and applied in various fields such as engineering, computer science, economics, and neuroscience. It involves controlling a system with linear dynamics and imperfect observations, subject to additive noise, with the goal of minimizing a quadratic cost function depending on the state and control variables. In this work, we consider a generalization of the discrete-time, finite-horizon LQG problem, where the noise distributions are unknown and belong to Wasserstein ambiguity sets centered at nominal (Gaussian) distributions. The objective is to minimize a worst-case cost across all distributions in the ambiguity set, including non-Gaussian distributions. Despite the added complexity, we prove that a control policy that is linear in the observations is optimal, as in the classic LQG problem. We propose a numerical solution method that efficiently characterizes this optimal control policy. Our method uses the Frank-Wolfe algorithm to identify the least-favorable distributions within the Wasserstein ambiguity sets and computes the controller's optimal policy using Kalman filter estimation under these distributions.
The Linear Quadratic Regulator (LQR) is a classic control problem that has served as a building block for numerous applications in engineering and computer science [3, 12], economics [29], or neuroscience [47]. It involves controlling a system with linear dynamics and imperfect observations affected by additive noise, with the goal of minimizing a quadratic state and control cost. Under the assumption that noise terms are independent and normally distributed (a case referred to as Linear-Quadratic-Gaussian, or LQG), it is well known that the optimal control policy depends linearly on the observations and can be obtained efficiently by using the Kalman filtering procedure and dynamic programming [8].
Motivated by practical settings where noise distributions may not be readily available or may not be Gaussian, this paper considers a discrete-time, finite-horizon generalization of the LQG setting where noise distributions are unknown and are chosen adversarially from ambiguity sets characterized by a Wasserstein distance and centered around nominal (Gaussian) distributions.
We show that, even under distributional ambiguity, the optimal control policy remains linear in the system's observations. Our proof is novel and does not rely on traditional recursive dynamic programming arguments. Instead, we re-parametrize the control policy in terms of the purified state observations and we derive an upper bound for the resulting minimax formulation by relaxing the ambiguity set (from a Wasserstein ball into a Gelbrich ball) while simultaneously restricting the controller to linear dependencies. We then use convex duality to prove that this upper bound matches
a lower bound obtained by restricting the ambiguity set in the dual of the minimax formulation. This implies the optimality of linear output feedback controllers, thus generalizing the classic results to a distributionally robust setting.
We also find that the worst-case distribution is actually Gaussian, which leads to a very efficient algorithm for finding optimal controllers. Specifically, we propose an algorithm based on the Frank-Wolfe first-order method that at every step solves sub-problems corresponding to classic LQG control problems, using Kalman filtering and dynamic programming. We show that this algorithm enjoys a sublinear convergence rate and is susceptible to parallelization. Lastly, we implement the algorithm leveraging PyTorch's automatic differentiation module and we find that it yields uniformly lower runtimes than a direct method (based on solving semidefinite programs) across all problem horizons.
1.1. Literature Review
This paper is related to the ample literature in control theory and engineering aimed at designing controllers that are robust to noise. The classic LQR/LQG theory, developed in the 1960s, examined linear dynamical systems in either time or frequency domain, seeking to minimize a combination of quadratic state and control costs (in time-domain) or the  $\mathcal{H}_2$  norm of the system's transfer function (in frequency domain). Motivated by findings that LQG controllers do not provide the guaranteed robust stability properties of LQR controllers [15], much effort has been devoted subsequently to designing controllers that are robust to worst-case perturbations, typically evaluated in terms of the  $\mathcal{H}_{\infty}$  norm of the system's transfer function (see, e.g., [16, 53] for a comprehensive review of  $\mathcal{H}_{\infty}$  and  $\mathcal{H}_2$  controllers). Because  $\mathcal{H}_{\infty}$  controllers tend to be overly conservative [32], various approaches have been proposed to balance the performance of nominal and robust controllers, e.g., by combining  $\mathcal{H}_2$  and  $\mathcal{H}_{\infty}$  approaches [7, 17]. A parallel stream of literature has considered risk-sensitive control [51], which minimizes an entropic risk measure instead of the expected quadratic cost. Although risk-sensitive control has a distributionally robust flavor (as the entropic risk measure is equivalent to a distributionally robust quadratic objective penalized via Kullback-Leibler divergence), risk-sensitive control models do not admit a distributionally robust formulation because the entropic risk measure is convex, but not coherent [22]. In contrast, our distributionally robust model provides a direct interpretation of the exact set of noise distributions against which the controller provides safeguards, and leads to a computationally tractable framework for finding the optimal controller.
In this sense, our work is more directly related to the literature on distributionally robust control, which seeks controllers that minimize expected costs under worst-case noise distributions [11, 33, 34, 41, 50, 52]. Closest to our work are [28, 33]. [33] proves the optimality of linear state-feedback control policies for a related minimax LQR model with a Wasserstein distance but with perfect state observations. With perfect observations, the optimal policies in the classic LQR formulation are independent of the noise distribution and are thus inherently already robust, so considering imperfect observations is what makes the problem significantly more challenging in our case. [28] studies a minimax formulation based on the Wasserstein distance with both state and observation noise but without any control policy, and focuses solely on the problem of estimating the states. Several papers have considered robust formulations with imperfect observations but for constrained systems [5, 6, 34], which are more challenging; the common approach is to restrict attention to linear feedback policies for computational tractability, and without proving their optimality.
Also related is the recent literature stream on distributionally robust optimization using the Wasserstein distance [36]. Within this stream, the closest work is [38, 44], which consider the problem of minimax mean-squared-error estimation when ambiguity is modeled with a Wasserstein distance from a nominal Gaussian distribution. Our proof builds on some ideas from these papers (e.g., relying on the Gelbrich distance in the construction of the upper bound), which it combines with ideas from control theory on purified output-feedback to obtain the overall construction. Also related is [2], which studies multistage distributionally robust problems with ambiguity sets given by a nested Wasserstein distance for stochastic processes and identifies computationally tractable cases. For a broader overview of developments related to optimal transport and Wasserstein distance with an emphasis on computational tractability and applications in machine learning, we refer to [42].
Finally, our paper is also related to literature that documents the optimality of linear/affine policies in (distributionally) robust dynamic optimization models. [10, 30] prove optimality for one-dimensional linear systems affected by additive noise and with perfect state observations, but with general (convex) state and/or control costs, [27, 49] provide computationally tractable approaches to quantifying the
suboptimality of affine controllers in finite or infinite-horizon settings, and [9, 21, 25] characterize the performance of affine policies in two-stage (distributionally) robust dynamic models.
Notation. All random objects are defined on a probability space  $(\Omega, \mathcal{F}, \mathbb{P})$ . Thus, the distribution of any random vector  $\xi : \Omega \to \mathbb{R}^d$  is given by the pushforward distribution  $\mathbb{P}_{\xi} = \mathbb{P} \circ \xi^{-1}$  of  $\mathbb{P}$  with respect to  $\xi$ . The expectation under  $\mathbb{P}$  is denoted by  $\mathbb{E}_{\mathbb{P}}[\cdot]$ . For any  $t \in \mathbb{Z}_+$ , we set  $[t] = \{0, \dots, t\}$ .
2. Problem Definition
We consider a discrete-time linear dynamical system
$$
x _ {t + 1} = A _ {t} x _ {t} + B _ {t} u _ {t} + w _ {t} \quad \forall t \in [ T - 1 ] \tag {1}
$$
with states  $x_{t} \in \mathbb{R}^{n}$ , control inputs  $u_{t} \in \mathbb{R}^{m}$ , process noise  $w_{t} \in \mathbb{R}^{n}$  and system matrices  $A_{t} \in \mathbb{R}^{n \times n}$  and  $B_{t} \in \mathbb{R}^{n \times m}$ . The controller only has access to imperfect state measurements
$$
y _ {t} = C _ {t} x _ {t} + v _ {t} \quad \forall t \in [ T - 1 ] \tag {2}
$$
corrupted by observation noise  $v_{t} \in \mathbb{R}^{p}$ , where  $C_t \in \mathbb{R}^{p \times n}$  and usually  $p \leq n$  (so that observing  $y_{t}$  would not allow reconstructing  $x_{t}$  even if there were no observation noise). The control inputs  $u_{t}$  are causal, i.e., depend on the past observations  $y_{0}, \ldots, y_{t}$  but not on the future observations  $y_{t+1}, \ldots, y_{T-1}$ . More precisely, the set of feasible control inputs  $\mathcal{U}_{y}$  is the set of random vectors  $(u_{0}, u_{1}, \ldots, u_{T-1})$  where for every  $t$  there exists a measurable control policy  $\varphi_{t}: \mathbb{R}^{p(t+1)} \to \mathbb{R}^{m}$  such that  $u_{t} = \varphi_{t}(y_{0}, \ldots, y_{t})$ . Controlling the system generates costs that depend quadratically on the states and the controls:
$$
J = \sum_ {t = 0} ^ {T - 1} \left(x _ {t} ^ {\top} Q _ {t} x _ {t} + u _ {t} ^ {\top} R _ {t} u _ {t}\right) + x _ {T} ^ {\top} Q _ {T} x _ {T}, \tag {3}
$$
where  $Q_{t} \in \mathbb{S}_{+}^{n}$  and  $R_{t} \in \mathbb{S}_{++}^{m}$  represent the state and input cost matrices, respectively. The exogenous random vectors  $x_{0}, \{w_{t}\}_{t=0}^{T-1}$  and  $\{v_{t}\}_{t=0}^{T-1}$  are mutually independent and follow probability distributions given by  $\mathbb{P}_{x_{0}}, \{\mathbb{P}_{w_{t}}\}_{t=0}^{T-1}$ , and  $\{\mathbb{P}_{v_{t}}\}_{t=0}^{T-1}$ , respectively. As the control inputs are causal, the system equations (2) imply that  $x_{t}, u_{t}$  and  $y_{t}$  can be expressed as measurable functions of the exogenous uncertainties  $x_{0}$  as well as  $w_{s}$  and  $v_{s}, s \in [t]$ , for every  $t$ . From now on we may thus assume without loss of generality that  $\Omega = \mathbb{R}^n \times \mathbb{R}^{n \times T} \times \mathbb{R}^{p \times T}$  is the space of realizations of the exogenous uncertainties,  $\mathcal{F}$  is the Borel  $\sigma$ -algebra on  $\Omega$  and  $\mathbb{P} = \mathbb{P}_{x_{0}} \otimes (\otimes_{t=0}^{T-1}\mathbb{P}_{w_{t}}) \otimes (\otimes_{t=0}^{T}\mathbb{P}_{v_{t}})$ , where  $\mathbb{P}_1 \otimes \mathbb{P}_2$  denotes the independent coupling of the distributions  $\mathbb{P}_1$  and  $\mathbb{P}_2$ .
In this context, the classic LQG model assumes that  $\mathbb{P}$  is known and Gaussian, and seeks  $u\in \mathcal{U}_y$  that minimizes  $\mathbb{E}_{\mathbb{P}}[J]$ . Appendix §A reviews the standard approach for computing optimal control inputs by estimating states through Kalman filtering techniques and using dynamic programming.
In contrast, we assume that  $\mathbb{P}$  is only known to belong to an ambiguity set  $\mathcal{W}$ , and we formulate a distributionally robust LQG problem that seeks  $u \in \mathcal{U}_y$  to minimize the worst-case expected cost:
$$
\max  _ {\mathbb {P} \in \mathcal {W}} \mathbb {E} _ {\mathbb {P}} \left[ \sum_ {t = 0} ^ {T - 1} \left(x _ {t} ^ {\top} Q _ {t} x _ {t} + u _ {t} ^ {\top} R _ {t} u _ {t}\right) + x _ {T} ^ {\top} Q _ {T} x _ {T} \right]. \tag {4}
$$
We construct the ambiguity set  $\mathcal{W}$  as a ball based on the Wasserstein distance. Specifically, we assume that a nominal Gaussian distribution  $\hat{\mathbb{P}} = \hat{\mathbb{P}}_{x_0} \otimes (\otimes_{t=0}^{T-1} \hat{\mathbb{P}}_{w_t}) \otimes (\otimes_{t=0}^T \hat{\mathbb{P}}_{v_t})$  is available so that  $\hat{\mathbb{P}}_{x_0} = \mathcal{N}(0, \hat{X}_0)$ ,  $\hat{\mathbb{P}}_{w_t} = \mathcal{N}(0, \hat{W}_t)$ , and  $\hat{\mathbb{P}}_{v_t} = \mathcal{N}(0, \hat{V}_t)$  for all  $t \in [T-1]$ , and  $\mathcal{W}$  is given by:
$$
\mathcal {W} = \mathcal {W} _ {x _ {0}} \otimes \left(\otimes_ {t = 0} ^ {T - 1} \mathcal {W} _ {w _ {t}}\right) \otimes \left(\otimes_ {t = 0} ^ {T - 1} \mathcal {W} _ {v _ {t}}\right),
$$
$$
\mathcal {W} _ {x _ {0}} = \left\{\mathbb {P} _ {x _ {0}} \in \mathcal {P} \left(\mathbb {R} ^ {n}\right): \mathrm {W} \left(\hat {\mathbb {P}} _ {x _ {0}}, \mathbb {P} _ {x _ {0}}\right) \leq \rho_ {x _ {0}}, \mathbb {E} _ {\mathbb {P} _ {x _ {0}}} [ x _ {0} ] = 0 \right\}
$$
$$
\mathcal {W} _ {w _ {t}} = \{\mathbb {P} _ {w _ {t}} \in \mathcal {P} (\mathbb {R} ^ {n}): \mathbb {W} (\hat {\mathbb {P}} _ {w _ {t}}, \mathbb {P} _ {w _ {t}}) \leq \rho_ {w _ {t}}, \mathbb {E} _ {\mathbb {P} _ {w _ {t}}} [ w _ {t} ] = 0 \}
$$
$$
\mathcal {W} _ {v _ {t}} = \{\mathbb {P} _ {v _ {t}} \in \mathcal {P} (\mathbb {R} ^ {m}): \mathbb {W} (\hat {\mathbb {P}} _ {v _ {t}}, \mathbb {P} _ {v _ {t}}) \leq \rho_ {v _ {t}}, \mathbb {E} _ {\mathbb {P} _ {v _ {t}}} [ v _ {t} ] = 0 \},
$$
and  $\mathbb{W}$  is the 2-Wasserstein distance. Thus, by construction, all exogenous random variables  $x_0, w_0, \ldots, w_{T-1}, v_0, \ldots, v_{T-1}$  are independent under every distribution in  $\mathcal{W}$ .
Definition 1 (2-Wasserstein distance). The 2-Wasserstein distance between two distributions  $\mathbb{P}_1$  and  $\mathbb{P}_2$  on  $\mathbb{R}^d$  with finite second moments is given by
$$
\mathbb {W} (\mathbb {P} _ {1}, \mathbb {P} _ {2}) = \left(\inf  _ {\pi \in \Pi (\mathbb {P} _ {1}, \mathbb {P} _ {2})} \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \| \xi_ {1} - \xi_ {2} \| _ {2} ^ {2} \pi (\mathrm {d} \xi_ {1}, \mathrm {d} \xi_ {2})\right) ^ {\frac {1}{2}},
$$
where  $\Pi (\mathbb{P}_1,\mathbb{P}_2)$  denotes the set of all couplings, that is, all joint distributions of the random variables  $\xi_{1}$  and  $\xi_{2}$  with marginal distributions  $\mathbb{P}_1$  and  $\mathbb{P}_2$ , respectively.
Our model strictly generalizes the classic LQG setting, which can be recovered by choosing  $\rho_{x_0} = \rho_{w_t} = \rho_{v_t} = 0$ . The parameters  $\rho$  thus allow quantifying the uncertainty about the nominal model and building robustness to mis-specification. We emphasize that the Wasserstein ambiguity set  $\mathcal{W}$  contains many non-Gaussian distributions and it is not readily obvious that the worst-case distribution in (4) is in fact Gaussian. However, the set  $\mathcal{W}$  is also non-convex, as it contains only distributions under which the exogenous uncertainties are independent, which makes the distributionally robust LQG problem potentially difficult to solve.
3. Nash Equilibrium and Optimality of Linear Output Feedback Controllers
We henceforth view the distributionally robust LQG problem as a zero-sum game between the controller, who chooses causal control inputs, and nature, who chooses a distribution  $\mathbb{P} \in \mathcal{W}$ . In this section we show that this game admits a Nash equilibrium, where nature's Nash strategy is a Gaussian distribution  $\mathbb{P}^{\star} \in \mathcal{W}$  and the controller's Nash strategy is a linear output feedback policy based on the Kalman filter evaluated under  $\mathbb{P}^{\star}$ .
Purified Observations. Before outlining our proof strategy, we first simplify the problem formulation by re-parametrizing the control inputs in a more convenient form (following [5, 6, 27]). Note that the control inputs in the LQG formulation are subject to cyclic dependencies, as  $u_{t}$  depends on  $y_{t}$ , while  $y_{t}$  depends on  $x_{t}$  through (2), and  $x_{t}$  depends again on  $u_{t}$  through (1), etc. Because these dependencies make the problem hard to analyze, it is preferable to instead consider the controls as functions of a new set of so-called purified observations instead of the actual observations  $y_{t}$ .
Specifically, we first introduce a fictitious noise-free system
$$
\hat {x} _ {t + 1} = A _ {t} \hat {x} _ {t} + B _ {t} u _ {t} \quad \forall t \in [ T - 1 ] \quad \text {a n d} \quad \hat {y} _ {t} = C _ {t} \hat {x} _ {t} \quad \forall t \in [ T - 1 ]
$$
with states  $\hat{x}_t\in \mathbb{R}^n$  and outputs  $\hat{y}_t\in \mathbb{R}^p$ , which is initialized by  $\hat{x}_0 = 0$  and controlled by the same inputs  $u_{t}$  as the original system (2). We then define the purified observation at time  $t$  as  $\eta_t = y_t - \hat{y}_t$  and we use  $\eta = (\eta_0,\dots ,\eta_{T - 1})$  to denote the trajectory of all purified observations.
As the inputs  $u_{t}$  are causal, the controller can compute the fictitious state  $\hat{x}_t$  and output  $\hat{y}_t$  from the observations  $y_0,\dots ,y_t$ . Thus,  $\eta_t$  is representable as a function of  $y_0,\ldots ,y_t$ . Conversely, one can show by induction that  $y_{t}$  can also be represented as a function of  $\eta_0,\dots ,\eta_t$ . Moreover, any measurable function of  $y_0,\dots ,y_t$  can be expressed as a measurable function of  $\eta_0,\dots ,\eta_t$  and vice-versa [27, Proposition II.1]. So if we define  $\mathcal{U}_{\eta}$  as the set of all control inputs  $(u_0,u_1,\dots ,u_{T - 1})$  so that  $u_{t} = \psi_{t}(\eta_{0},\dots ,\eta_{t})$  for some measurable function  $\psi_t:\mathbb{R}^{p(t + 1)}\to \mathbb{R}^m$  for every  $t\in [T - 1]$ , the above reasoning implies that  $\mathcal{U}_{\eta} = \mathcal{U}_{y}$ .
In view of this, we can rewrite the distributionally robust LQG problem equivalently as
$$
\begin{array}{l} p ^ {\star} = \left\{ \begin{array}{l l} \min  _ {x, u, y} & \max  _ {\mathbb {P} \in \mathcal {W}} \mathbb {E} _ {\mathbb {P}} \left[ u ^ {\top} R u + x ^ {\top} Q x \right] \\ \text {s . t .} & u \in \mathcal {U} _ {y},   x = H u + G w,   y = C x + v \end{array} \right. \\ = \left\{ \begin{array}{l l} \min  _ {x, u} & \max  _ {\mathbb {P} \in \mathcal {W}} \mathbb {E} _ {\mathbb {P}} \left[ u ^ {\top} R u + x ^ {\top} Q x \right] \\ \text {s . t .} & u \in \mathcal {U} _ {\eta}, \quad x = H u + G w, \end{array} \right. \tag {5} \\ \end{array}
$$
where  $x = (x_0, \ldots, x_T)$ ,  $u = (u_0, \ldots, u_{T-1})$ ,  $y = (y_0, \ldots, y_{T-1})$ ,  $w = (x_0, w_0, \ldots, w_{T-1})$ ,  $v = (v_0, \ldots, v_{T-1})$ ,  $\eta = (\eta_0, \ldots, \eta_{T-1})$ , and  $R, Q, H, G$  and  $C$  are suitable block matrices
<sup>1</sup>Our assumption that noise terms are zero-mean is consistent with the standard LQG model [8]. Requiring  $\mathbb{E}_{\mathbb{P}_{x_0}}[x_0] = 0$  is assumed for clarity and without loss of generality.
(see Appendix §B for their precise definitions). The latter reformulation involving the purified observations  $\eta$  is useful because these are independent of the inputs. Indeed, by recursively combining the equations of the original and the noise-free systems, one can show that  $\eta = Dw + v$  for some block triangular matrix  $D$  (see Appendix §B for its construction). This shows that the purified observations depend (linearly) on the exogenous uncertainties but not on the control inputs. Hence, the cyclic dependencies complicating the original system are eliminated in (5).
Subsequently, we also study the dual of (5), defined as
$$
d ^ {\star} = \left\{ \begin{array}{c c} \max  _ {\mathbb {P} \in \mathcal {W}} & \min  _ {x, u} \mathbb {E} _ {\mathbb {P}} \left[ u ^ {\top} R u + x ^ {\top} Q x \right] \\ \text {s . t .} & u \in \mathcal {U} _ {\eta}, x = H u + G w. \end{array} \right. \tag {6}
$$
The classic minimax inequality implies that  $p^{\star} \geq d^{\star}$ . If we can prove that  $p^{\star} = d^{\star}$ , that (5) has a solution  $u^{\star}$  and that (6) has a solution  $\mathbb{P}^{\star}$ , then  $(u^{\star},\mathbb{P}^{\star})$  must be a Nash equilibrium of the zero-sum game at hand [43, Theorem 2]. However, because  $\mathcal{U}_{\eta}$  is an infinite-dimensional function space and  $\mathcal{W}$  is an infinite-dimensional, non-convex set of non-parametric distributions, the existence of a Nash equilibrium (in pure strategies) is not at all evident. Instead, our proof strategy will rely on constructing an upper bound for  $p^{\star}$  and a lower bound for  $d^{\star}$ , and showing that these match.
Upper Bound for  $p^{\star}$ . We obtain an upper bound for  $p^{\star}$  by suitably enlarging the ambiguity set  $\mathcal{W}$  and restricting the controllers  $u_{t}$  to linear dependencies. We enlarge  $\mathcal{W}$  by ignoring all information about the distributions in  $\mathcal{W}$  except for their covariance matrices, and by replacing the Wasserstein distance with the Gelbrich distance. To that end, we first define the Gelbrich distance on the space of covariance matrices.
Definition 2 (Gelbrich distance). The Gelbrich distance between the two covariance matrices  $\Sigma_1, \Sigma_2 \in \mathbb{S}_+^d$  is given by
$$
\mathbb {G} (\Sigma_ {1}, \Sigma_ {2}) = \sqrt {\mathrm {T r} \left(\Sigma_ {1} + \Sigma_ {2} - 2 \left(\Sigma_ {2} ^ {\frac {1}{2}} \Sigma_ {1} \Sigma_ {2} ^ {\frac {1}{2}}\right) ^ {\frac {1}{2}}\right)}.
$$
We are interested in the Gelbrich distance because of its close connection to the 2-Wasserstein distance. Indeed, it is known that the 2-Wasserstein distance between two distributions with zero means is bounded below by the Gelbrich distance between the respective covariance matrices.
Proposition 3.1 (Gelbrich bound [24, Theorem 2.1]). For any two distributions  $\mathbb{P}_1$  and  $\mathbb{P}_2$  on  $\mathbb{R}^d$  with zero means and covariance matrices  $\Sigma_1, \Sigma_2 \in \mathbb{S}_+^d$ , respectively, we have  $\mathbb{W}(\mathbb{P}_1, \mathbb{P}_2) \geq \mathbb{G}(\Sigma_1, \Sigma_2)$ .
Recalling that  $\hat{X}_0$ ,  $\hat{W}_t$  and  $\hat{V}_t$  respectively denote the covariance matrices for  $x_0, w_t$  and  $v_t$  under the nominal distribution  $\hat{\mathbb{P}}$ , we can then define the following Gelbrich ambiguity set for the exogenous uncertainties:
$$
\mathcal {G} = \mathcal {G} _ {x _ {0}} \otimes \left(\otimes_ {t = 0} ^ {T - 1} \mathcal {G} _ {w _ {t}}\right) \otimes \left(\otimes_ {t = 0} ^ {T - 1} \mathcal {G} _ {v _ {t}}\right),
$$
$$
\mathcal {G} _ {x _ {0}} = \left\{\mathbb {P} _ {x _ {0}} \in \mathcal {P} \left(\mathbb {R} ^ {n}\right): \mathbb {E} _ {\mathbb {P} _ {x _ {0}}} [ x _ {0} ] = 0, \mathbb {E} _ {\mathbb {P}} [ x _ {0} x _ {0} ^ {\top} ] = X _ {0}, \mathbb {G} (X _ {0}, \hat {X} _ {0}) \leq \rho_ {x _ {0}} \right\}
$$
$$
\mathcal {G} _ {w _ {t}} = \left\{\mathbb {P} _ {w _ {t}} \in \mathcal {P} (\mathbb {R} ^ {n}): \mathbb {E} _ {\mathbb {P} _ {w _ {t}}} [ w _ {t} ] = 0, \mathbb {E} _ {\mathbb {P}} [ w _ {t} w _ {t} ^ {\top} ] = W _ {t}, \mathbb {G} (W _ {t}, \hat {W} _ {t}) \leq \rho_ {w _ {t}} \right\}
$$
$$
\mathcal {G} _ {v _ {t}} = \{\mathbb {P} _ {v _ {t}} \in \mathcal {P} (\mathbb {R} ^ {m}): \mathbb {E} _ {\mathbb {P} _ {v _ {t}}} [ v _ {t} ] = 0, \mathbb {E} _ {\mathbb {P}} [ v _ {t} v _ {t} ^ {\top} ] = V _ {t}, \mathbb {G} (V _ {t}, \hat {V} _ {t}) \leq \rho_ {v _ {t}} \}.
$$