$\mathcal{H}_2/\mathcal{H}_\infty$ Optimal Control with Sparse Sensing and Actuation

Abstract

In this paper, we present novel convex optimization formulations for designing full-state and output-feedback controllers with sparse actuation that achieve user-specified $\mathcal{H}_2$ and $\mathcal{H}_\infty$ performance criteria. For output-feedback control, we extend these formulations to simultaneously design control laws with sparse actuation and sensing. The sparsity is induced through the minimization of a weighted $\ell_1$ norm, promoting the efficient use of sensors and actuators while maintaining desired closed-loop performance. The proposed methods are applied to a nonlinear structural dynamics problem, demonstrating the advantages of simultaneous optimization of the control law, sensing, and actuation architecture in realizing an efficient closed-loop system.

Figures (13)

• Figure 1: A tensegrity cantilever structure under gravity acting downwards. The bars $(b_i)$ are shown in black, and the cables $(c_j)$ are shown in red.
• Figure 2: Minimum $\|u_i(t)\|_\infty$ for a given $\mathcal{H}_2$ and $\mathcal{H}_\infty$ performance, with full-state feedback controller.
• Figure 3: State trajectories for a given $\mathcal{H}_2$ and $\mathcal{H}_\infty$ performance, with full-state feedback controller and nonlinear dynamics. The figure also shows the open-loop trajectories.
• Figure 4: Control trajectories for a given $\mathcal{H}_2$ and $\mathcal{H}_\infty$ performance, with full-state feedback controller and nonlinear dynamics.
• Figure 5: Minimum $\|u_i(t)\|_\infty$ for a given $\mathcal{H}_2$ and $\mathcal{H}_\infty$ performance, with full-state feedback controller. The channel-wise norms were subject to upper bounds $\gamma_i\leq10$ and $\gamma_i\leq81$ shown by dashed lines for $\mathcal{H}_2$ and $\mathcal{H}_\infty$ design problems, respectively.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Remark 1
• Remark 2
• Theorem 3
• Theorem 4
• Remark 3
• Definition 1
• Proposition 1
• Lemma 1