A Two-Time-Scale Control Scheme for Fast Systems

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.

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

Neighboring extremals in optimization and control

Sébastien Gros

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.

EPFL2007

Finite Dimensional Methods for Differential Flatness

Basile Graf

In Control System Theory, the study of continuous-time, finite dimensional, underdetermined systems of ordinary differential equations is an important topic. Classification of systems in different categories is a natural initial step to the analysis of a given control problem. Systems of equations can often be “transformed” into other “equivalent” ones. Then, a control system is associated to a set of equivalent equations. In this setting, a property of a control system can be defined as a property that has to be satisfied by some arbitrary system of equations in the set representing the control system. Assessing such a property can be a difficult task. In this thesis, we review and study a number of ways to determine whether a multi-input nonlinear system is flat, i.e. whether it is equivalent to a linear system after some dynamical extension and change of coordinates. This is a difficult as well as a well studied problem. Therefore, coming up with some altogether new approach or solution is to a certain extent illusory. A substantial part of the text is devoted to describing existing approaches and sometimes to propose either an original alternative or an original point of view. Another part of the thesis is dedicated to the study of a drastically reduced version of the problem, where more can be said in an algorithmic way. Nonlinear control systems are first modeled as the embedding of some fibered bundle to the first jet of the time-and-state-variables manifold. The exterior system, or Pfaffian system, corresponding to the ODE then arises naturally. Input prolongations are then introduced as lifts of the previously mentioned embedding. Various filtration techniques and their applications to static feedback linearization are discussed next. A full chapter is devoted to the infinite dimensional approach involving matrix differential operators. A now classical theorem, linking integrability of the basis of a differential module after application of one such an operator, and the flatness property is discussed. The relations obtained can be decomposed in an equivalent set of differentially closed equations. We state a version of the resulting theorem where the “curvature equations” are trivial. A subproblem that has attracted the attention of researchers is the question whether a given system — subject to some state constraints — is flat. In this setting, useful concepts are that of a covering of a system by another one and the accompanying result stating that a flat system can cover only a flat system. Hence, if a “large” linear system is given together with a set of nonlinear constraints, flatness of the constrained system is assessed if the unconstrained system can be shown to cover the constrained one. Starting with the classical notion of controlled invariance and a generalized notion coined dynamic controlled invariance, sufficient conditions are discussed which also involve the notion of right invertibility and the dynamic extension algorithm. Modeling of mechanical systems by free moving point masses subject to some control forces and quadratic constraints is often effective. The resulting unconstrained equations are linear and bilinear in the state and control/Lagrangian multiplier variables. We propose a relative derived flag that leads to a filtration with guaranteed integrability at each stage. This leads to a very effective sufficient condition for the flatness of the unconstrained model. The algorithm, together with the test described in the previous section, is used to show flatness of some generalized pendulum-like equations. They are also shown to specialize to the non-holonomic car equations and to the VTOL/pendulum equations when some specific parameters are suitably chosen.

EPFL2013