Antonis Papachristodoulou

Antonis Papachristodoulou, James Anderson, Giorgio Valmorbida et al.

The release of SOSTOOLS v4.00 comes as we approach the 20th anniversary of the original release of SOSTOOLS v1.00 back in April, 2002. SOSTOOLS was originally envisioned as a flexible tool for parsing and solving polynomial optimization problems, using the SOS tightening of polynomial positivity constraints, and capable of adapting to the ever-evolving fauna of applications of SOS. There are now a variety of SOS programming parsers beyond SOSTOOLS, including YALMIP, Gloptipoly, SumOfSquares, and others. We hope SOSTOOLS remains the most intuitive, robust and adaptable toolbox for SOS programming. Recent progress in Semidefinite programming has opened up new possibilities for solving large Sum of Squares programming problems, and we hope the next decade will be one where SOS methods will find wide application in different areas. In SOSTOOLS v4.00, we implement a parsing approach that reduces the computational and memory requirements of the parser below that of the SDP solver itself. We have re-developed the internal structure of our polynomial decision variables. Specifically, polynomial and SOS variable declarations made using sossosvar, sospolyvar, sosmatrixvar, etc now return a new polynomial structure, dpvar. This new polynomial structure, is documented in the enclosed dpvar guide, and isolates the scalar SDP decision variables in the SOS program from the independent variables used to construct the SOS program. As a result, the complexity of the parser scales almost linearly in the number of decision variables. As a result of these changes, almost all users will notice a significant increase in speed, with large-scaleproblems experiencing the most dramatic speedups. Parsing time is now always less than 10% of time spent in the SDP solver. Finally, SOSTOOLS now provides support for the MOSEK solver interface as well as the SeDuMi, SDPT3, CSDP, SDPNAL, SDPNAL+, and SDPA solvers.

Luca Furieri, Yang Zheng, Antonis Papachristodoulou et al.

This paper proposes a novel input-output parametrization of the set of internally stabilizing output-feedback controllers for linear time-invariant (LTI) systems. Our underlying idea is to directly treat the closed-loop transfer matrices from disturbances to input and output signals as design parameters and exploit their affine relationships. This input-output perspective is particularly effective when a doubly-coprime factorization is difficult to compute, or an initial stabilizing controller is challenging to find; most previous work requires one of these pre-computation steps. Instead, our approach can bypass such pre-computations, in the sense that a stabilizing controller is computed by directly solving a linear program (LP). Furthermore, we show that the proposed input-output parametrization allows for computing norm-optimal controllers subject to quadratically invariant (QI) constraints using convex programming.

Liqun Zhao, Konstantinos Gatsis, Antonis Papachristodoulou

Reinforcement learning (RL) has demonstrated impressive performance in various areas such as video games and robotics. However, ensuring safety and stability, which are two critical properties from a control perspective, remains a significant challenge when using RL to control real-world systems. In this paper, we first provide definitions of safety and stability for the RL system, and then combine the control barrier function (CBF) and control Lyapunov function (CLF) methods with the actor-critic method in RL to propose a Barrier-Lyapunov Actor-Critic (BLAC) framework which helps maintain the aforementioned safety and stability for the system. In this framework, CBF constraints for safety and CLF constraint for stability are constructed based on the data sampled from the replay buffer, and the augmented Lagrangian method is used to update the parameters of the RL-based controller. Furthermore, an additional backup controller is introduced in case the RL-based controller cannot provide valid control signals when safety and stability constraints cannot be satisfied simultaneously. Simulation results show that this framework yields a controller that can help the system approach the desired state and cause fewer violations of safety constraints compared to baseline algorithms.

Yang Zheng, Giovanni Fantuzzi, Antonis Papachristodoulou

Optimization over non-negative polynomials is fundamental for nonlinear systems analysis and control. We investigate the relation between three tractable relaxations for optimizing over sparse non-negative polynomials: sparse sum-of-squares (SSOS) optimization, diagonally dominant sum-of-squares (DSOS) optimization, and scaled diagonally dominant sum-of-squares (SDSOS) optimization. We prove that the set of SSOS polynomials, an inner approximation of the cone of SOS polynomials, strictly contains the spaces of sparse DSOS/SDSOS polynomials. When applicable, therefore, SSOS optimization is less conservative than its DSOS/SDSOS counterparts. Numerical results for large-scale sparse polynomial optimization problems demonstrate this fact, and also that SSOS optimization can be faster than DSOS/SDSOS methods despite requiring the solution of semidefinite programs instead of less expensive linear/second-order cone programs.

Yifan Xie, Zuxun Xiong, Julian Berberich et al.

This paper proposes a data-driven controller design method for unknown nonlinear systems based on a Koopman bilinear realization. Using Koopman operator theory, the nonlinear system can be represented as a bilinear discrete-time system with a residual error term. The residual error is proportionally bounded by the norm of the lifted state and input, while the system matrices of the bilinear model are unknown. Assuming that bounds on the residual error are available, the unknown system matrices are characterized via a set-membership representation using the collected input-state data pairs of the nonlinear system. A data-driven controller design method is proposed to ensure stability for all bilinear systems within this set-membership description and for all admissible residual errors. More specifically, we design a rational state-feedback controller that stabilizes the bilinear model with residual error and, consequently, the original nonlinear system, by solving a sum-of-squares (SOS) program. The effectiveness of the proposed approach is demonstrated through numerical examples.

Aivar Sootla, Yang Zheng, Antonis Papachristodoulou

Diagonally dominant matrices have many applications in systems and control theory. Linear dynamical systems with scaled diagonally dominant drift matrices, which include stable positive systems, allow for scalable stability analysis. For example, it is known that Lyapunov inequalities for this class of systems admit diagonal solutions. In this paper, we present an extension of scaled diagonally dominance to block partitioned matrices. We show that our definition describes matrices admitting block-diagonal solutions to Lyapunov inequalities and that these solutions can be computed using linear algebraic tools. We also show how in some cases the Lyapunov inequalities can be decoupled into a set of lower dimensional linear matrix inequalities, thus leading to improved scalability. We conclude by illustrating some advantages and limitations of our results with numerical examples.

Luca Furieri, Yang Zheng, Antonis Papachristodoulou et al.

We consider the problem of designing a stabilizing and optimal static controller with a pre-specified sparsity pattern. Since this problem is NP-hard in general, it is necessary to resort to approximation approaches. In this paper, we characterize a class of convex restrictions of this problem that are based on designing a separable quadratic Lyapunov function for the closed-loop system. This approach generalizes previous results based on optimizing over diagonal Lyapunov functions, thus allowing for improved feasibility and performance. Moreover, we suggest a simple procedure to compute favourable structures for the Lyapunov function yielding high-performance distributed controllers. Numerical examples validate our results.

Yang Zheng, Giovanni Fantuzzi, Antonis Papachristodoulou

This paper introduces a notion of decomposition and completion of sum-of-squares (SOS) matrices. We show that a subset of sparse SOS matrices with chordal sparsity patterns can be equivalently decomposed into a sum of multiple SOS matrices that are nonzero only on a principal submatrix. Also, the completion of an SOS matrix is equivalent to a set of SOS conditions on its principal submatrices and a consistency condition on the Gram representation of the principal submatrices. These results are partial extensions of chordal decomposition and completion of scalar matrices to matrices with polynomial entries. We apply the SOS decomposition result to exploit sparsity in matrix-valued SOS programs. Numerical results demonstrate the high potential of this approach for solving large-scale sparse matrix-valued SOS programs.

Liam Perreault, Idris Kempf, Kirill Sechkar et al.

Cybergenetic gene expression control in bacteria enables applications in engineering biology, drug development, and biomanufacturing. AI-based controllers offer new possibilities for real-time, single-cell-level regulation but typically require large datasets and re-training for new systems. Data-enabled Predictive Control (DeePC) offers better sample efficiency without prior modelling. We apply DeePC to a system with two inputs (optogenetic control and media concentration) and two outputs (expression of gene of interest and host growth rate). Using basis functions to address nonlinearities, we demonstrate that DeePC remains robust to parameter variations and performs among the best control strategies while using the least data.

Aivar Sootla, Yang Zheng, Antonis Papachristodoulou

In this paper, we introduce a set of block factor-width-two matrices, which is a generalisation of factor-width-two matrices and is a subset of positive semidefinite matrices. The set of block factor-width-two matrices is a proper cone and we compute a closed-form expression for its dual cone. We use these cones to build hierarchies of inner and outer approximations of the cone of positive semidefinite matrices. The main feature of these cones is that they enable a decomposition of a large semidefinite constraint into a number of smaller semidefinite constraints. As the main application of these classes of matrices, we envision large-scale semidefinite feasibility optimisation programs including sum-of-squares (SOS) programs. We present numerical examples from SOS optimisation showcasing the properties of this decomposition.

Matthew Newton, Antonis Papachristodoulou

Neural networks have shown great success in many machine learning related tasks, due to their ability to act as general function approximators. Recent work has demonstrated the effectiveness of neural networks in control systems (known as neural feedback loops), most notably by using a neural network as a controller. However, one of the big challenges of this approach is that neural networks have been shown to be sensitive to adversarial attacks. This means that, unless they are designed properly, they are not an ideal candidate for controllers due to issues with robustness and uncertainty, which are pivotal aspects of control systems. There has been initial work on robustness to both analyse and design dynamical systems with neural network controllers. However, one prominent issue with these methods is that they use existing neural network architectures tailored for traditional machine learning tasks. These structures may not be appropriate for neural network controllers and it is important to consider alternative architectures. This paper considers rational neural networks and presents novel rational activation functions, which can be used effectively in robustness problems for neural feedback loops. Rational activation functions are replaced by a general rational neural network structure, which is convex in the neural network's parameters. A method is proposed to recover a stabilising controller from a Sum of Squares feasibility test. This approach is then applied to a refined rational neural network which is more compatible with Sum of Squares programming. Numerical examples show that this method can successfully recover stabilising rational neural network controllers for neural feedback loops with non-linear plants with noise and parametric uncertainty.

Tomoki Sadatoshi, Antonis Papachristodoulou, Yutaka Hori

Moment dynamics in stochastic chemical kinetics often involve an infinite chain of coupled equations, where lower-order moments depend on higher-order ones, making them analytically intractable. Moment bounding via semidefinite programming provides guaranteed upper and lower bounds on stationary moments. However, this formulation suffers from the rapidly growing size of semidefinite constraints due to the combinatorial growth of moments with the number of molecular species. In this paper, we propose a sparsity-exploiting matrix decomposition method for semidefinite constraints in stationary moment bounding problems to reduce the computational cost of the resulting semidefinite programs. Specifically, we characterize the sparsity structure of moment equations, where each reaction involves only a subset of variables determined by its reactants, and exploit this structure to decompose the semidefinite constraints into smaller ones. We demonstrate that the resulting formulation reduces the computational cost of the optimization problem while providing practically useful bounds.

Junaid Ahmed Memon, Allan Andre Do Nascimento, Kostas Margellos et al.

This paper presents a prototyping framework for distributed control of multi-robot systems, aimed at bridging theory and practical testing of distributed optimization algorithms. Using the Single Program, Multiple Data (SPMD) paradigm, the framework emulates distributed control on a single computer, with each core running the same algorithm using local states and neighbour-to-neighbour communication. We demonstrate the framework on a four-quadrotor position-swapping task using a non-cooperative game-theoretic distributed algorithm. Computational time and trajectory data are compared across the supported dynamics levels: a point-mass model, a high-fidelity quadrotor model, and an experimental hardware testbed using Crazyflie quadcopters. The results show that the framework provides a low-cost and accessible approach for validating distributed algorithms.

Liqun Zhao, Keyan Miao, Konstantinos Gatsis et al.

Reinforcement learning (RL) excels in applications such as video games, but ensuring safety as well as the ability to achieve the specified goals remains challenging when using RL for real-world problems, such as human-aligned tasks where human safety is paramount. This paper provides safety and stability definitions for such human-aligned tasks, and then proposes an algorithm that leverages neural ordinary differential equations (NODEs) to predict human and robot movements and integrates the control barrier function (CBF) and control Lyapunov function (CLF) with the actor-critic method to help to maintain the safety and stability for human-aligned tasks. Simulation results show that the algorithm helps the controlled robot to reach the desired goal state with fewer safety violations and better sample efficiency compared to other methods in a human-aligned task.

Carl R Richardson, Declan S Jagt, Matthew M Peet et al.

It has recently been shown that the evolution of a state, described by a Partial Differential Equation (PDE), can be more conveniently represented as the evolution of the state's highest spatial derivative (the ``fundamental state''), which lies in $L_2$ and has no boundary conditions (BCs) or continuity constraints. For linear PDEs, this yields a Partial Integral Equation (PIE) parametrized by Partial Integral (PI) operators mapping the fundamental state to the PDE state. In this paper, we show that for polynomial PDEs, the dynamics of the fundamental state can instead be compactly expressed as a distributed polynomial in the fundamental state, parametrized by a new tensor algebra of PI operators acting on the tensor product of the fundamental state. We further define a SOS parametrization of the distributed polynomial and use this to construct a distributed SOS program, for testing local stability of polynomial PDEs.

Jan-Peter Calliess, Antonis Papachristodoulou, Stephen J. Roberts

This work proposes a new method for simultaneous probabilistic identification and control of an observable, fully-actuated mechanical system. Identification is achieved by conditioning stochastic process priors on observations of configurations and noisy estimates of configuration derivatives. In contrast to previous work that has used stochastic processes for identification, we leverage the structural knowledge afforded by Lagrangian mechanics and learn the drift and control input matrix functions of the control-affine system separately. We utilise feedback-linearisation to reduce, in expectation, the uncertain nonlinear control problem to one that is easy to regulate in a desired manner. Thereby, our method combines the flexibility of nonparametric Bayesian learning with epistemological guarantees on the expected closed-loop trajectory. We illustrate our method in the context of torque-actuated pendula where the dynamics are learned with a combination of normal and log-normal processes.