The Scientific and Technical Achievements of PROACTHIS
The PROACTHIS team has made significant steps in the first period in advancing Projection-Based Control (PBC) across a series of well-defined work packages (WPs). These achievements span mathematical formalization, digital implementation, stability analysis tools and advanced design frameworks.
Highlights:
WP1: Foundations and Digital Implementation of PBC
Formalizing PBC: One of the initial breakthroughs was addressing the fundamental question of well-posedness -- ensuring that the interconnected system of PBC and controlled plant have well-defined solution trajectories. By representing PBC as extensions of projected dynamical systems, originating from economical sciences, PROACTHIS provided a rigorous framework for formalizing these systems.
Digital Implementations: Modern engineering demands that PBC systems work seamlessly in digital environments. The team developed discrete-time versions of projection-based controllers, including Hybrid Integrator-Gain Systems (HIGS). These digital versions preserve the performance benefits of their continuous-time counterparts, ensuring robust and high-performant operation in real-world applications.
Connecting PBC with Control-Barrier Functions (CBFs): An exciting discovery was linking PBC with CBFs, widely used to design safety filters in control systems. This connection enables potentially smooth implementations of PBC, which have discontinuous dynamics themselves. Smooth CBF-based controllers could enable higher levels of performance and robustness.
WP2: Advanced Design Frameworks for PBC
Lyapunov-Based Stability Tools: To ensure stability, the team developed advanced tools leveraging Lyapunov functions. They utilized piecewise quadratic and piecewise affine functions, supported by linear matrix inequalities (LMIs) and linear programming (LP) tools, to analyze stability and performance properties efficiently.
Frequency-Domain Stability Analysis: The team introduced innovative frequency-domain conditions to evaluate the stability of nonlinear and split-path controllers. These tools enable engineers to assess stability directly from the plant’s frequency-response function, a method widely used in industry.
Incremental Stability Tools: New methods for incremental stability analysis were developed for PBC systems. The results ensure unique steady-state responses to periodic inputs, enabling accurate performance assessment in a “nonlinear Bode plot” style, providing more insights in the performance than scalar metrics such as L_p gains for nonlinear control systems.
WP3: Multi-Element PBC and Nonlinear Plants
Multi-Element PBC Systems: Initial results to the stability analysis of controllers with multiple PBC elements were obtained, laying the groundwork for managing complex, multi-input multi-ouput engineering systems.
Dissipative Properties: The team demonstrated that the inclusion of a simple projection elements around a nonlinear controller can immediately induce desirable dissipative properties. This can guarantees stabilization of plants that satisfy dissipativity or passivity properties, thereby expanding the applicability of PBC to nonlinear systems.
WP4: Data-Driven Design and Online Learning
PROACTHIS has made critical contributions to data-driven control design, including:
• A frequency-domain version of Willems’ fundamental lemma.
• Data-based performance analysis tools for Linear Time-Invariant (LTI) and PBC controllers.
• Frequency-based data-driven model predictive control (MPC) techniques.
These tools enable engineers to design PBC systems using easy-to-obtain frequency-based data of the plant, without the need for explicit parametric (state-space) models.
WP5: Practical Tools for Engineers
To make these advancements accessible, PROACTHIS is developing a user-friendly numerical toolbox.
