Ph. D. Project
Title:
Data-driven Optimal Fault-Tolerant Control
Dates:
2023/10/01 - 2026/09/30
Supervisor(s):
Other supervisor(s):
HAMELIN Frédéric (hamelin.frederic@univ-lorraine.fr)
Description:
The current power of computing devices and the availability of mathematical and algorithmic tools for processing high-dimensional data have led to a strong revival of interest in the use of data for the control of complex dynamic systems. This use of data is also strongly driven by the emergence of artificial intelligence techniques in automatic control. If the synthesis of controllers by indirect approaches, i.e., by identifying the model of the system on the basis of data and then following a second step of designing the controller from the identified model, is not new, direct methods where the synthesis of the controller is directly driven by data are emerging as an effective alternative for the control of dynamics that are difficult to model or unknown. The behavioral FTC approach initiated in our work and summarized in the monograph [JYS18a] is being rediscovered today as one of the appropriate mathematical frameworks for data-driven direct analysis and synthesis. Indeed, a recent wave of research activities driven by a result of J.C. Willems in the behavioral framework, known as the "fundamental lemma", has emerged in the last two years [MR17], [Waa+20], [DT20]. This fundamental lemma has become the backbone of these recent research efforts on data-driven control. In simple terms, this lemma states that, under appropriate assumptions, any input-output trajectory of a linear time-invariant system can be described as a linear combination of a single previously recorded trajectory [Wil+05].
More precisely, under the condition that the input signals are sufficiently rich, the so-called persistence of excitation condition, the behavior of the system over horizons of length L, i.e. the set of these trajectories over this horizon, is in the image space of a Hankel matrix constructed from these trajectories. This image space can be parameterized by an arbitrary vector of dimension compatible with this Hankel matrix. This constitutes a data-based representation of a system that allows all its signals to be generated using the above-mentioned parameter and opens up a vast field of possibilities for the direct analysis and synthesis of dynamic systems without the use of a classical model.
The image representation is closely related to the so-called "kernel" representation of our work in fault tolerant control [JYS18a]. This kernel representation describes the behavioral equations of a system as trajectories belonging to the kernel space of a polynomial matrix. It can be shown that this kernel representation is equivalent to the image representation and this equivalence allows us to extend our work in fault tolerant control with a new insight given by the fundamental lemma of Willems.
In particular, solutions to FTC problems using classical techniques based on state or external representations (LQR, model matching, H∞, MPC, etc...) become reformulable directly using the data matrix. A particular perspective on which we wish to focus our attention in this thesis project is that of data-driven FTC control via predictive control formulated using the image representation. Indeed, one of the major difficulties of fault-tolerant control based on model-based predictive control (MPC) is the need to have a real-time post-fault model for the computation of the predictions on which the calculation of the optimal commands in the faulty mode(s) is based. Very recently, the classical problem of linear control and finite-horizon quadratic cost tracking (LQR) has been reformulated using image representation and its solution implemented via the leaky horizon principle leads to data-based predictive control in the behavioral framework [CLD19], [Ber+21], [Hua+22]. Unfortunately, the proposed solutions are not transposable to fault-tolerant control for the following reason: the data of the Hankel matrix are previously recorded data in open loop. However, a faulty system goes through dynamic modes that are not the nominal mode described by the image space of this Hankel matrix of data previously recorded offline. This raises the question of how to update this Hankel matrix during a failure:
1) - How to build the data windows that form the columns of the Hankel matrix
following a system failure with regard to the requirement of a rapid accommodation to defects?
2) - how to update this Hankel matrix of data in real time?
3) - how to satisfy the requirement of the persistence of the input signal excitation in closed loop so that the image of the Hankel matrix is the set of trajectories in the faulty mode?
4) -Is it possible to formulate the predictive control optimization problem by directly integrating
this requirement of persistence of the excitation of the control input, either in the cost function, or as a constraint? This problem of the dual role of the input signal, i.e., excitation and joint control, is a dual control problem whose solution is not trivial.
The generic character of the expected results should allow their application to a wide range of systems. In particular, the validation of the results will be done on the newly built CRAN Eco-sûr platform, dedicated to the problems of control for the energy efficiency of buildings. The platform has many measurement systems that provide data on many variables related to indoor and outdoor climatic conditions of buildings for their monitoring and control.
More precisely, under the condition that the input signals are sufficiently rich, the so-called persistence of excitation condition, the behavior of the system over horizons of length L, i.e. the set of these trajectories over this horizon, is in the image space of a Hankel matrix constructed from these trajectories. This image space can be parameterized by an arbitrary vector of dimension compatible with this Hankel matrix. This constitutes a data-based representation of a system that allows all its signals to be generated using the above-mentioned parameter and opens up a vast field of possibilities for the direct analysis and synthesis of dynamic systems without the use of a classical model.
The image representation is closely related to the so-called "kernel" representation of our work in fault tolerant control [JYS18a]. This kernel representation describes the behavioral equations of a system as trajectories belonging to the kernel space of a polynomial matrix. It can be shown that this kernel representation is equivalent to the image representation and this equivalence allows us to extend our work in fault tolerant control with a new insight given by the fundamental lemma of Willems.
In particular, solutions to FTC problems using classical techniques based on state or external representations (LQR, model matching, H∞, MPC, etc...) become reformulable directly using the data matrix. A particular perspective on which we wish to focus our attention in this thesis project is that of data-driven FTC control via predictive control formulated using the image representation. Indeed, one of the major difficulties of fault-tolerant control based on model-based predictive control (MPC) is the need to have a real-time post-fault model for the computation of the predictions on which the calculation of the optimal commands in the faulty mode(s) is based. Very recently, the classical problem of linear control and finite-horizon quadratic cost tracking (LQR) has been reformulated using image representation and its solution implemented via the leaky horizon principle leads to data-based predictive control in the behavioral framework [CLD19], [Ber+21], [Hua+22]. Unfortunately, the proposed solutions are not transposable to fault-tolerant control for the following reason: the data of the Hankel matrix are previously recorded data in open loop. However, a faulty system goes through dynamic modes that are not the nominal mode described by the image space of this Hankel matrix of data previously recorded offline. This raises the question of how to update this Hankel matrix during a failure:
1) - How to build the data windows that form the columns of the Hankel matrix
following a system failure with regard to the requirement of a rapid accommodation to defects?
2) - how to update this Hankel matrix of data in real time?
3) - how to satisfy the requirement of the persistence of the input signal excitation in closed loop so that the image of the Hankel matrix is the set of trajectories in the faulty mode?
4) -Is it possible to formulate the predictive control optimization problem by directly integrating
this requirement of persistence of the excitation of the control input, either in the cost function, or as a constraint? This problem of the dual role of the input signal, i.e., excitation and joint control, is a dual control problem whose solution is not trivial.
The generic character of the expected results should allow their application to a wide range of systems. In particular, the validation of the results will be done on the newly built CRAN Eco-sûr platform, dedicated to the problems of control for the energy efficiency of buildings. The platform has many measurement systems that provide data on many variables related to indoor and outdoor climatic conditions of buildings for their monitoring and control.
Keywords:
Fault-tolerant control, data-based optimal control, Hankel matrix, data
Department(s):
| Control Identification Diagnosis |
