Projection-based Control: A Novel Paradigm for High-performance Systems

Project Information

PROACTHIS

Grant agreement ID: 101055384

DOI

10.3030/101055384

EC signature date 13 June 2022

Start date 1 November 2022

End date 31 October 2028

Funded under

• European Research Council (ERC)

Total cost

€ 2 498 516,00

EU contribution

€ 2 498 516,00

Coordinated by

TECHNISCHE UNIVERSITEIT EINDHOVEN
Netherlands

Periodic Reporting for period 1 - PROACTHIS (Projection-based Control: A Novel Paradigm for High-performance Systems)

Reporting period: 2022-11-01 to 2025-04-30

Summary of the context and overall objectives of the project

An indispensable technology:

Control systems have been crucial for many technological and scientific innovations. Today control systems are everywhere around us to regulate and optimize the operation of many devices and processes that are essential to our daily lives. However, major societal trends are currently pushing the performance requirements (accuracy, throughput and resilience) for future control systems to unparalleled levels, while the engineered systems surrounding us are increasingly hard to control due to growing complexity and uncertainty. This calls for new controller designs that can reach unprecedented levels of performance and can be synthesized through easy-to-obtain data-based system information and/or learn the optimal way of controlling the system through online observation of data.

Current control designs and their limitations:

The current control systems literature focusses on
(i) linear control designs for linear systems, or
(ii) on nonlinear control design for general nonlinear systems.
Although linear controllers for linear systems (i) can be tuned using effective time- and frequency-domain tools, the control structures lack the flexibility to optimally deal with dynamics varying over time/location and situation-dependent requirements. Moreover, linear control is subject to fundamental performance limitations and always requires design compromises. Approach (ii) uses nonlinear control structures that can have the desired flexibility and design freedom. However, general nonlinear controllers do not directly admit non-conservative design frameworks. Moreover, even if the plant is linear, yet the controller nonlinear, additional structure is present in the problem that cannot be exploited in these designs. So, although in principle nonlinear control structures can outperform linear controllers, even for linear plants, the absence of a powerful design frameworks render it hard to realize this for real systems.

Proposed solution:

In PROACTHIS, I aim to bridge this gap by creating a new control paradigm based on projection operators. By introducing projectors in control loops, specific signals are kept in well-chosen constraint sets inducing direct performance-enhancing benefits. I foresee that the mathematical structure of these projection-based controllers enables fundamental properties that were instrumental in the success of linear control and will be key to obtain effective design frameworks for this new class of controllers. Successfully developing this new system theory will pave the way towards cutting-edge control methodologies addressing the needs of future engineering systems.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Why These Achievements Matter?

Engineering systems today face increasing complexity and demands for reliability, adaptability, and high levels of performance. The scientific breakthroughs of PROACTHIS aim to provide a strong foundation for addressing these challenges. By unifying theoretical rigor with practical tools, PBC offers a potentially transformative approach to controlling modern systems, paving the way for innovations in mechatronic systems, robotics, and beyond.

Looking Ahead

As PROACTHIS progresses, it will continue to refine its theories, expand its tools, and demonstrate the real-world impact of Projection-Based Control. Especially, building a comprehensive and strong design toolkit, aligned with control engineering practice, and providing proof-of-concepts of the key control structures and their design flow on real-life demonstrators, will be instrumental to ensure broad uptake by industry.

High-level overview of goal PROACTHIS

This picture shows a nonlinear Bode plot for a projection-based closed loop

A unique periodic steady-state response to a sinusoidal input to an incrementally stable PBC loop