Research

Unlike linear systems, nonlinear systems are difficult to ensure stability. Port-Hamiltonian systems, a subset of nonlinear systems, are described by Hamilton’s canonical equations with added control inputs and can represent many physical systems. A key advantage of such systems is the ability to utilize the Hamiltonian function, which represents the system’s energy, as a Lyapunov function candidate in control. This approach falls under the category of control methods known as “passivity-based control.”

Our research group studies nonlinear control methods for mechanical port-Hamiltonian systems by designing artificial potential functions that depend on position and velocity. The methods enable us to ensure Lyapunov stability of the closed-loop system. Recently, we have been focusing on developing techniques to realize sliding mode control within the framework of passivity-based control by using non-differentiable potential functions.

Refueling spacecraft in space is difficult, so it is an important issue to minimize fuel consumption. Obtaining the trajectory that minimizes fuel consumption requires solving the $L^{1}$-optimal control problem, a challenge known for its complexity. The $L^{1}$-optimal control input exhibits sparsity, staying at zero for most of the control period, which allows strategies that take advantage of this feature. We are conducting research on various mathematical optimization algorithms, including algorithms for solving the $L^{1}$-optimal control problem.

Additionally, we investigate optimization algorithms with stochastic parameters to find global optimal solutions in problems characterized by multiple local optima. This approach is expected to identify global optimal solutions even for problems where deterministic methods may only find local optimal solutions.

We develop optimization algorithms for diverse applications and provide theoretical proofs of their convergence and other essential properties.

Controlling systems, such as spacecraft and aircraft, requires the identification of their mathematical models. However, building these models solely from a priori physical knowledge has its limitations. Therefore, “system identification,” a method of constructing mathematical models from data obtained by measuring system behavior, has been the subject of active research.

Our team proposes system identification methods for complex nonlinear systems using neural networks, which have the property of approximating any continuous function. The proposed architecture consists of a state estimator, $E_{\theta}$, and an output predictor, $P_{\phi}$. It is designed to reconstruct the original state-space representation. These components, $E_{\theta}$ and $P_{\phi}$, are neural networks, making the identification of nonlinear systems feasible. Additionally, the identified model can be employed in control designs that require fast online computations, including model predictive control, because the structure enables multi-step-ahead prediction in a single computation.

We are currently working on the identification of systems with complex physical phenomena that could not be modeled accurately before. We are also applying this method to real systems, such as an automatic driving vehicle.

When controlling a system, certain specifications, such as fast response and reduced input energy consumption, may be required. Control that maximizes or minimizes the objective function, which represents the degree of the achievement of target goals, is known as “optimal control.” Optimal control, especially for nonlinear systems, has been researched actively. The problem of finding optimal control inputs as time series data can be reduced to Hamilton’s canonical equations using the calculus of variations. However, applying the obtained feedforward optimal control inputs to a real system can lead to challenges like instability or undesirable performance due to uncertainties, disturbances, and modeling errors. Therefore, incorporating a feedback structure in the control system is generally required. An optimal feedback control law can be derived from a value function that is a solution to the Hamilton-Jacobi-Bellman (HJB) equation, but finding this solution is extremely difficult.

In this laboratory, we have focused on developing an optimal feedback control law for nonlinear systems by utilizing a database containing solutions to Hamilton’s canonical equations, instead of directly solving the HJB equation. The optimal control input for each instance is determined using data identified as closest to the current state in the database. Recently, we have been conducting additional research, including the modeling of value functions using neural networks. Furthermore, we are validating the effectiveness of our proposed method by applying it to real machines, such as drones.