Neighboring extremals in optimization and control

Optimization arises naturally when process performance needs improvement. This is often the case in industry because of competition – the product has to be proposed at the lowest possible cost. From the point of view of control, optimization consists in designing a control policy that best satisfies the chosen objectives. Most optimization schemes rely on a process model, which, however, is always an approximation of the real plant. Hence, the resulting optimal control policy is suboptimal for the real process. The fact that accurate models can be prohibitively expensive to build has triggered the development of a field of research known as Optimization under Uncertainty. One promising approach in this field proposes to draw a strong parallel between optimization under uncertainty and control. This approach, labeled NCO tracking, considers the Necessary Conditions of Optimality (NCO) of the optimization problem as the controlled outputs. The approach is still under development, and the present work is today's most recent contribution to this development. The problem of NCO tracking can be divided into several subproblems that have been studied separately in earlier works. Two main categories can be distinguished : (i) tracking the NCO associated with active constraints, and (ii) tracking the NCO associated with sensitivities. Research on the former category is mature. The latter problem is more difficult to solve since the sensitivity part of the NCO cannot be directly measured on the real process. The present work proposes a method to tackle these sensitivity problems based on the theory of Neighboring Extremals (NE). More precisely, NE control provides a way of calculating a first-order approximation to the sensitivity part of the NCO. This idea is developed for static and both nonsingular and singular dynamic optimization problems. The approach is illustrated via simulated examples: steady-state optimization of a continuous chemical reactor, optimal control of a semi-batch reactor, and optimal control of a steered car. Model Predictive Control (MPC) is a control scheme that can accommodate both process constraints and nonlinear process models. The repeated solution of a dynamic optimization problem provides an update of the control variables based on the current state, and therefore provides feedback. One of the major drawbacks of MPC lies in the expensive computations required to update the control policy, which often results in a low sampling frequency for the control loop. This limitation of the sampling frequency can be dramatic for fast systems and for systems exhibiting a strong dispersion between the predicted and the real state such as unstable systems. In the MPC framework, two main methods have been proposed to tackle these difficulties: (i) The use of a pre-stabilizing feedback operating in combination with the MPC scheme, and (ii) the use of robust MPC. The drawback of the former approach is that there exists no systematic way of designing such a feedback, nor is there any systematic way of analyzing the interaction between the MPC controller and this additional feedback. This work proposes to use the NE theory to design this additional feedback, and it provides a systematic way of analyzing the resulting control scheme. The approach is illustrated via the control of a simulated unstable continuous stirred-tank reactor and is applied successfully to two laboratory-scale set-ups, an inverted pendulum and a helicopter model called Toycopter. The stabilizing potential of NE control to handle fast and unstable systems is well illustrated. In the case of a strong dispersion between the state trajectories predicted by the model and the real process, robust MPC becomes infeasible. This problem can be addressed using robust MPC based on multiple input profiles, where the inherent feedback provided by MPC is explicitly taken into account, thereby increasing the size of the set of feasible inputs. The drawback of this scheme is its very high computational complexity. This work proposes to use the NE theory in the robust MPC framework as an efficient way of dealing with the feasibility issue, while limiting the computational complexity of the approach. The approach is illustrated via the control of a simulated unstable continuous stirred-tank reactor, and of an inverted pendulum.

Structure-preserving approaches and data-driven closure modeling for model order reduction

Nicolò Ripamonti

In this thesis, we propose model order reduction techniques for high-dimensional PDEs that preserve structures of the original problems and develop a closure modeling framework leveraging the Mori-Zwanzig formalism and recurrent neural networks. Since high-fidelity approximations of PDEs often result in a large number of degrees of freedom, the need for iterative evaluations for numerical optimizations and rapid feedback is computationally challenging.The first part of this thesis is devoted to conserving the high-dimensional equation's invariants, symmetries, and structures during the reduction process. Traditional reduction techniques are not guaranteed to yield stable reduced systems, even if the target problem is stable. In the context of fluid flows, the skew-symmetric structure of the problem entails the preservation of the kinetic energy of the system. By preserving the same structure at the level of the reduced model, we obtain enhanced stability, and accuracy and the reduced model acquires physical significance by preserving a surrogate of the energy of the original problem. Next, we focus on Hamiltonian systems, which, being driven by symmetry, are a source of great interest in the reduction community. It is well known that the breaking of these symmetries in the reduced model is accompanied by a blowup of the system energy and flow volume. In this thesis, geometric reduced models for Hamiltonian systems are further developed and combined with the dynamically orthogonal methods, addressing the poor reducibility in time of advection-dominated problems. The reduced solution is expressed as a linear combination of a finite number of modes and coincides with the symplectic projection of the high-fidelity Hamiltonian problem onto the tangent space of the approximating manifold. An error surrogate is used to monitor the approximation ability of the reduced model and make a change in the rank of the approximating system if necessary. The method is further developed through a combination of DEIM and DMD to reduce non-polynomial nonlinearities while preserving the symplectic structure of the problem and applied to the Vlasov-Poisson system.In the second part of the thesis, we consider several data-driven methods to address the poor accuracy in the under-resolved regime for Galerkin reduced models via a closure term. The closure term is developed systematically from the Mori-Zwanzig formalism by introducing projection operators on the spaces of resolved and unresolved scales, thus resulting in an additional memory integral term. The interaction between different scales turns out to be nonlocal in time and dominated by a high-dimensional orthogonal dynamics equation, which cannot be solved precisely and efficiently. Several classical methods in the field of statistical mechanics are used to approximate the memory term, exploiting the finiteness of the memory kernel support. We conclude this thesis by showing through numerical experiments how long short-term memory networks, i.e., machine learning structures characterized by feedback connections, represent a valid tool for approximating the additional memory term.

EPFL2022

A Two-Time-Scale Control Scheme for Fast Systems

Dominique Bonvin, Davide Buccieri, Sébastien Gros, Philippe Müllhaupt

Model predictive control (MPC) is a very effective approach to control nonlinear systems, especially when the systems are high dimensional and/or constrained. MPC formulates the problem of input trajectory generation as an optimization problem. However, due to model mismatch and disturbances, frequent re-calculation of the trajectories is typically called for. This paper proposes a two-time-scale control scheme that uses less frequent repeated trajectory generation in a slow loop and time-varying linear feedback in a faster loop. Since the fast loop reduces considerably the effect of uncertainty, trajectory generation can be done much less frequently. The problem of trajectory generation can be treated using either optimization-based MPC or flatness-based system inversion. As proposed, the method cannot handle hard constraints. Both MPC and the two-time-scale control scheme are tested via the simulation of a flying robotic structure. It is seen that the MPC scheme is too slow to be considered for real-time implementation on a fast system. In contrast, the two-time-scale control scheme is fast, effective, and robust.

2007

Methodologie d'Optimisation Dynamique et de Commande Optimale des Petites Stations d'Epuration a Boues Activees

Benoît Chachuat

The adoption of stricter effluent requirements by the European Union rises large problems for small communities having few economical and technical resources. These problems motivate the synthesis of advanced optimisation-based controllers in order to enhance the performances of small-size wastewater treatment plants. The aim of this study is to develop a methodology of dynamic optimisation and optimal control of small-size alternating aerobic-anoxic activated sludge plants. The first part deals with the improvements which can be potentially obtained through the application of dynamic optimisation techniques. Two problems are considered: the minimisation of nitrogen discharge and the reduction of the operating costs. These are constrained, non-convex and high-dimensional problems that exhibit hybrid discrete/continuous and combinatorial behaviour. In both cases, the application of the resulting optimal aeration strategies leads to large improvements of the process performances, while satisfying the discharge requirements and the operating constraints; it is also verified that the long-term implementation of the optimal control profiles guarantees durable process improvement. The embedding of the optimal aeration strategies within closed-loop controllers is dealt with in the second part of the thesis. Optimal control consists in (i) the on-line joint observation/estimation of both state variables a nd parameters and, (ii) the use of a non-linear model predictive control scheme to update the aeration profiles. Beforehand, a reduced model is derived by simplifying the general ASM~1 model and sensitivity analyses are performed to quantify the influence of unmeasured disturbances and model mismatch on the optimisation results. Numerical implementations show that the resulting closed-loop controller brings large improvements with respect to usual operating modes either in terms of nitrogen discharge or in terms of energy consumption.