Designing systems that are easy to control is important for smooth operation. However, the “ease of control” depends on the objective and application, and is not easy to evaluate in a general framework. Since many control problems have been formulated mathematically in control theory, defining “ease of control” within this framework is expected to provide a general measure for analysis and design. In classical control, input-output properties and disturbance responses are evaluated through the frequency responses of transfer and sensitivity functions [1]. For nonlinear systems, application-dependent measures are also used, such as manipulability ellipsoids in robotics [2] and design indices for drones considering fault tolerance and hovering performance [3].

In systems driven by exogenous inputs, it is important to design not only stability and tracking performance but also the response to such inputs. In particular, it is important to specify the directions in the state space along which the system is easy or difficult to move. This is particularly important in human-interactive systems, where operability and stability must be balanced. Impedance control [4] and admittance control [5] shape the relation between external forces and motion, but mainly prescribe input-output relations or local responses, rather than directional reachability over the entire state space.

The controllability Gramian [6, 7, 8, 9] is suitable for this purpose because it characterizes reachability in terms of input energy. By shaping the controllability Gramian, one can quantitatively design the apparent controllability from exogenous inputs and quantify both ease and difficulty of motion in each state direction. This idea is expected to be applicable to systems such as guidance robots for visually impaired users [10, 11] and manipulators with direct teaching [12, 13]. The authors have previously proposed Gramian shaping for such systems from the user’s perspective [14]. In the Gramian shaping, a desired Gramian is specified, and the controllability Gramian from the exogenous input is shaped by an internal feedback input so as to approach the desired one. Through the anisotropy of the Gramian, this approach enables quantitative design of ease and difficulty of control in each state direction.

To make such reachability design viable, the distance between the system Gramian and the desired Gramian should yield a tractable optimization problem to admit a clear interpretation. In the previous Gramian shaping method [14], this discrepancy was measured by the affine-invariant Riemannian (AIR) distance [15]. However, AIR-based Gramian shaping leads to a nonconvex optimization problem, so global optimality is not guaranteed and efficient algorithms are not readily available. As a result, robustness and computational efficiency are difficult to ensure. Moreover, the AIR distance has no clear physical interpretation, making it difficult to relate Gramian shaping to existing control frameworks.

To overcome these limitations, this paper proposes a new Gramian shaping method based on the Bures–Wasserstein distance (BW distance). The BW distance is defined on symmetric positive-semidefinite matrices [16] and coincides with the $2$-Wasserstein distance between Gaussian distributions with the same mean [17]. Thanks to this property, it has been used in graph generation [18], statistical theory based on BW barycenters [19, 20, 21], and geodesic analysis [22].

The main contributions of this paper are summarized as follows.

1. 1.

   We introduce the BW distance into controllability Gramian shaping from exogenous inputs to make the optimization problem convex on the set of symmetric positive definite matrices. Thus, BW-based Gramian shaping can be formulated as a convex optimization problem.

2. 2.

   Using the linear matrix inequality (LMI) representation of the BW distance, we derive a semidefinite programming (SDP) formulation of Gramian shaping, which allows various additional LMI constraints.

3. 3.

   We clarify the relation between the proposed method and $H_{2}$ control, and show that $H_{2}$ control can be interpreted as a special case of BW-based Gramian shaping.

This paper is organized as follows. Section II reviews the controllability Gramian, introduces the Gramian for exogenous inputs, and summarizes the relation between realizable Gramians and feedback gains. Section III defines the proposed method, namely, BW-based Gramian shaping and shows that it can be formulated as a convex problem. Section IV presents an SDP-based solution using an LMI representation of the BW distance. Section V discusses the relation between the proposed method and $H_{2}$ control, showing that the proposed framework provides a link between classical optimal control and optimal transport. Section VI verifies the effectiveness of the proposed method through numerical examples.

Notation: Vectors are denoted by lowercase bold letters such as $\bm{a}$, and matrices by uppercase bold letters such as $\bm{A}$. $\bm{I}_{n}$ and $\bm{O}_{n}$ denote the $n\times n$ identity matrix and zero matrix, respectively. $\bm{O}_{n\times m}$ denotes the $n\times m$ zero matrix, and $\bm{0}_{n}$ denotes the $n$-dimensional zero vector. $\bm{A}^{\top}$ and $\bm{A}^{H}$ denote the transpose and Hermitian transpose of $\bm{A}$, respectively. $\bm{A}^{\dagger}$ denotes the Moore–Penrose pseudoinverse of $\bm{A}$. $\mathbb{R}^{n}$ and $\mathbb{R}^{n\times m}$ denote the set of $n$-dimensional real vectors and the set of $n\times m$ real matrices, respectively. $\mathbb{R}_{+}$ denotes the set of nonnegative real numbers. $\mathbb{S}_{+}^{n}$ denotes the set of $n\times n$ symmetric positive-semidefinite matrices, and $\mathbb{S}_{++}^{n}$ denotes the set of $n\times n$ symmetric positive definite matrices. $\mathcal{O}(n)$ and $\mathcal{SO}(n)$ denote the orthogonal group and the special orthogonal group in dimension $n$, respectively. A diagonal matrix with diagonal entries $a_{1},a_{2},\ldots,a_{n}$ is denoted by $\mathrm{diag}\left(a_{1},a_{2},\ldots,a_{n}\right)$. A block diagonal matrix formed by the matrices $\bm{A}_{1},\bm{A}_{2},\ldots,\bm{A}_{n}$ is denoted by $\mathrm{blkdiag}\left(\bm{A}_{1},\bm{A}_{2},\ldots,\bm{A}_{n}\right)$. For $\bm{A}\in\mathbb{S}_{++}^{n}$, there exists a spectral decomposition $\bm{A}=\bm{U}\bm{\Lambda}\bm{U}^{\top}$, where $\bm{\Lambda}=\mathrm{diag}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})$, $\lambda_{i}>0$ for all $i\in\{1,2,\ldots,n\}$, and $\bm{U}\in\mathcal{O}(n)$. For a real number $\alpha$, the matrix power of a symmetric positive definite matrix is defined by $\bm{A}^{\alpha}=\bm{U}\bm{\Lambda}^{\alpha}\bm{U}^{\top}$, where $\bm{\Lambda}^{\alpha}=\mathrm{diag}(\lambda_{1}^{\alpha},\lambda_{2}^{\alpha},\ldots,\lambda_{n}^{\alpha})$.

In this section, we first review the standard controllability Gramian for stable systems. We then introduce the controllability Gramian considered in this paper for systems driven by exogenous inputs.

Consider the following linear system with state $\bm{x}\in\mathbb{R}^{n}$ and input $\bm{u}\in\mathbb{R}^{m}$:

where $\bm{A}\in\mathbb{R}^{n\times n}$ and $\bm{B}\in\mathbb{R}^{n\times m}$.

This section explains the main idea of this paper, namely, Gramian shaping based on the BW distance. Gramian shaping is a method for deforming the controllability Gramian into a desired form by exploiting the geometry of symmetric positive definite matrices [14]. This enables the design of both ease and difficulty of control for each state when the system is operated externally.

In Gramian shaping, the input is designed so that the system approaches an ideal Gramian $\bm{P}_{d}$ representing the desired controllability. The ideal Gramian may be defined from a reachable set based on input energy, or designed with reference to an ideal system. Gramian shaping is achieved by minimizing the discrepancy between the controllability Gramian (6) and the ideal Gramian $\bm{P}_{d}$. In the previous study, this discrepancy between two Gramians was measured by the affine-invariant Riemannian (AIR) distance in accordance with the geometry of symmetric positive definite matrices [14]. This made it possible to perform optimization while preserving positive definiteness. However, optimization based on the AIR distance does not admit a guaranteed global optimum, and its physical interpretation is difficult. To overcome these difficulties, this paper employs the BW distance, which is a different distance function.

The BW distance satisfies the axioms of a metric on $\mathbb{S}_{+}^{n}$ and $\mathbb{S}_{++}^{n}$. It is known to coincide with the $2$-Wasserstein distance between Gaussian distributions having the same mean.

The BW distance enables us to measure the difference between two Gramians. Our final goal is to design a feedback gain for Gramian shaping. Using Lemmas 1 and 2, however, this problem can be separated into realizable Gramian design and gain derivation. The first step is formulated as follows.

Constraint (9b) imposes positive definiteness of the Gramian, and constraint (9c) characterizes the set of Gramians realizable by the system (see Lemma 1 in Appendix). Thus, the controllability Gramian closest to the desired Gramian can be obtained. The second step is to obtain a corresponding feedback gain $\bm{K}^{\ast}$ from $\bm{P}^{\ast}$ using Lemma 2.

We now prove that Problem 1 is a strictly convex optimization problem. We prove that the objective function of Problem 1 is strictly convex.

This section explains how to solve the Gramian shaping problem numerically. As a naive approach, one can first solve Problem 1 directly and then determine the gain via Lemma 2. On the other hand, by exploiting the properties of the BW distance, we can obtain a more efficient optimization formulation.

We first reconsider Problem 1 in a form that explicitly includes gain determination.

This section discusses the relation between the proposed Gramian shaping framework and existing control theory. In particular, we focus on its connection to $H_{2}$ control.

In this section, the exogenous input $\bm{v}\in\mathbb{R}^{r}$ in (5) is assumed to be zero-mean Gaussian white noise. That is,

where $\delta(t):\mathbb{R}\to\mathbb{R}$ denotes the Dirac delta function. Then the state $\bm{x}(t)$ becomes a stochastic process, and if the closed-loop matrix $\bm{A}+\bm{B}\bm{K}$ is Hurwitz stable, the stationary covariance

exists. Moreover, $\bm{P}_{w}$ is the unique positive-semidefinite solution of the Lyapunov equation

and is expressed as

Therefore, when $\bm{v}$ is Gaussian white noise, the state covariance coincides mathematically with the controllability Gramian in Definition 2:

We now consider $H_{2}$ control for the system (4). $H_{2}$ control is based on the $H_{2}$ norm of the transfer function $\bm{G}(s)$ from the disturbance $\bm{v}$ to the state $\bm{x}$.

$H_{2}$ control is defined as the optimization problem of finding the gain $\bm{K}$ that minimizes the $H_{2}$ norm of $\bm{G}(s)$.

The $H_{2}$ control (30) can be solved as an SDP. To this end, the $H_{2}$ norm is converted into a time-domain expression via Parseval’s theorem.

Hence, $H_{2}$ control can be written as the following optimization problem.

Note that also in Problem 3, in order to compute $\bm{K}^{\ast}=\bm{E}^{\ast}\bm{P}_{w}^{\ast^{-1}}$ stably in numerical implementation, one may impose the constraint $\bm{P}_{w}\succeq\epsilon\bm{I}_{n}$. This is a numerical regularization and should be distinguished from the theoretical problem formulation.

The correspondence between the $H_{2}$ control problem in Problem 3 and the Gramian shaping problem in Problem 2 can be verified directly from the closed-form expression (8) of the BW distance. If we set $\bm{P}_{d}=\alpha\bm{I}_{n}$ and let $\alpha\to 0$, the cross term vanishes and

holds. In this case, the objective function of Problem 2 reduces to $\mathrm{tr}(\bm{P})$, and the constraints are identical. Hence, it coincides with the $H_{2}$ state-feedback design problem in Problem 3. Therefore, in this limit, BW-based Gramian shaping can be regarded as a generalization that contains $H_{2}$ control as a special case. It can be interpreted as a framework that generalizes minimization of the scalar measure $\mathrm{tr}(\bm{P}_{w})$, which measures the magnitude of the covariance (or controllability Gramian), to minimization of the distance between the covariance matrix and a target matrix. Furthermore, from a probabilistic viewpoint, under Gaussian white disturbance, the stationary distribution of the closed-loop system (5) is $\mathcal{N}(\bm{0},\bm{P}_{w})$, and the BW distance coincides with the $2$-Wasserstein distance between zero-mean Gaussian distributions. Thus, $H_{2}$ control can be reinterpreted as a design that drives the stationary closed-loop distribution $\mathcal{N}(\bm{0},\bm{P}_{w})$ toward the degenerate Gaussian distribution $\mathcal{N}(\bm{0},\bm{O}_{n})$ centered at the origin in the sense of optimal transport. This viewpoint reveals, through Gramian shaping, that classical $H_{2}$ optimal control implicitly performs an optimal transport of state distributions.

This section verifies the proposed method through numerical examples.

This paper proposed a BW-based method for shaping the controllability Gramian from exogenous inputs to the state into a desired form. We showed that the objective function is strictly convex and formulated the problem as an SDP using an LMI representation of the BW distance. This also enables the incorporation of additional LMI constraints. The proposed framework also clarified that $H_{2}$ control is a special case of BW-based Gramian shaping under Gaussian white noise. Numerical examples demonstrated anisotropic controllability design for a guidance robot, verified the ability to impose additional LMI constraints, and confirmed through a simple example that the proposed method approaches $H_{2}$ control in the corresponding limit. These results show the effectiveness of the proposed method for externally operated systems.