set invariance

F. Stoican, Hovd, M., and Olaru, S., “Explicit invariant approximation of the mRPI set for LTI dynamics with zonotopic disturbances”, in 52nd IEEE Conference on Decision and Control, Florence, Italy, 2013, p. 3237-3242.Abstract
In this paper we provide a RPI over-approximation of the mRPI set associated for linear dynamics with zonotopic disturbances.We prove that the RPI construction converges toward the mRPI set and its conservatism diminishes monotonically with respect to the complexity of the representation (a “tightness” coefficient is calculated a priori). The results are tested in illustrative examples.
Fault tolerant control based on set-theoretic methods
F. Stoican, “Fault tolerant control based on set-theoretic methods”, École Superieure d'Électricite (SUPELEC) , 2011. Online archiveAbstract
The scope of the thesis is the analysis and design of fault tolerant control (FTC) schemes through the use of set-theoretic methods. In the framework of multisensor schemes, the faults appearance and the modalities to accurately detect them are investigated as well as the design of control laws which assure the closed-loop stability. By using invariant/contractive sets to describe the residual signals, a fault detection and isolation (FDI) mechanism with reduced computational demands is implemented based on set-separation. A dual mechanism, implemented by a recovery block, which certificates previously fault-affected sensors is also studied. From a broader theoretical perspective, we point to the conditions which allow the inclusion of FDI objectives in the control law design. This leads to static feedback gains synthesis by means of numerically attractive optimization problems. Depending on the parameters selected for tuning, is shown that the FTC design can be completed by a reference governor or a predictive control scheme which adapts the state trajectory and the feedback control action in order to assure FDI. When necessary, the specific issues originated by the use of set-theoretic methods are detailed and various improvements are proposed towards: invariant set construction, mixed integer programming (MIP), stability for switched systems (dwell-time notions).
F. Stoican, Olaru, S., and Bitsoris, G., “A fault detection scheme based on controlled invariant sets for multisensor systems”, in Proceedings of the 2010 Conference on Control and Fault Tolerant Systems, Nice, France, 2010, p. 468–473.Abstract
The present paper uses set theoretic methods for the design of a fault tolerant control scheme in the case of a multisensor application. The basic principle is the separation of invariant sets for the estimations of the state and tracking error under healthy and faulty functioning. The fault scenario is based on abrupt changes of the observation equations. The main contribution is the introduction of controlled invariant sets in the fault detection mechanism. The control action is chosen so that the closed loop invariance is assured for a candidate region which accounts for the bounds on the exogenous signals (additive disturbances, noise and reference/set-points).
F. Stoican, Olaru, S., Seron, M. M., and De Doná, J. A., “Reference governor for tracking with fault detection capabilities”, in Proceedings of the 2010 Conference on Control and Fault Tolerant Systems, Nice, France, 2010, p. 546–551.Abstract
This paper presents a fault tolerant multisensor strategy for feedback control of a class of nonlinear systems upon a geometrical approach. A key point to ensure fault tolerance is the separation between healthy and faulty closed-loop behavior. Here we achieve this through set theoretic operations upon sets describing the healthy/faulty behavior of the system. The results rely both on an appropriate choice for the exogenous signals and on fixed point conditions for a nonlinear mapping which describes the gap between the nonlinear system and a linearized model in the functioning interval. A reference governor is employed such that, under a receding horizon technique, only feasible exogenous signals are provided to the system.
F. Stoican, Olaru, S., Seron, M. M., and De Doná, J. A., “A fault tolerant control scheme based on sensor switching and dwell time”, in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, Georgia, USA, 2010, p. 756–761.Abstract
The present paper deals with a switching control scheme for a plant with multiple estimator-controller-actuator pairs. The scheme has to deal with specific problems originated by the switching between the different feedback loops and accommodate to faults in the observation channels (sensors outputs). The main contribution is a fault tolerant switching scheme with stability guarantees assured by a pre-imposed dwell-time. The detection and the fault tolerance capabilities are assured through set separation for the residual signals corresponding to healthy and faulty functioning. Another contribution of the paper resides in a recovery technique for faulty sensors which makes use of a virtual sensor whose estimation, based on an optimization procedure, minimizes recovery time.
F. Stoican, Olaru, S., De Doná, J. A., and Seron, M. M., “Zonotopic ultimate bounds for linear systems with bounded disturbances”, in Proceedings of the 18th IFAC World Congress, Milano, Italy, 2011, p. 9224–9229.Abstract
This paper deals with robust invariant sets construction for discrete-time linear timeinvariant dynamics. The case of a zonotopic disturbance set is analysed in detail by exploiting the properties of these geometrical structures. A constructive method is provided for diminishing the conservatism of ultimate bound invariant sets. It is shown that the resulting zonotopic set is related to the minimal robust positively invariant set in the sense that their boundaries have common points.
H. N. Nguyen, Olaru, S., and Stoican, F., “On maximal robustly positively invariant sets”, in Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics, Noordwijkerhout, Netherlands, 2011, p. 300-305.Abstract
This paper addresses the problem of computing maximal robustly positively invariant sets for discrete-time linear time-invariant systems with disturbance inputs. It is assumed that the disturbance is unknown, additive, but bounded. The main contribution is the determination of bound of the number of steps in the iterative construction of the maximal invariant sets.
S. Olaru, De Doná, J. A., Seron, M. M., and Stoican, F., “Positive invariant sets for fault tolerant multisensor control schemes”, International Journal of Control, vol. 83, no. 12, p. 2622–2640, 2010.Abstract
This article deals with fault tolerant multisensor control schemes for systems with linear dynamics. Positive invariance is a common analysis and control design tool for systems affected by bounded constraints and disturbances. This article revisits the construction of \epsilon-approximations of minimal robust positive invariant sets for linear systems upon contractive set-iterations. The cases of switching between different sets of disturbances and the inclusion of a predefined region of the state space are treated in detail. All these results are used in multisensor control schemes which have to deal with specific problems originated by the switching between different estimators and by the presence of faults in some of the sensors. The construction of positive invariant sets for different operating regimes provides, in this context, effective fault detection information. Within the same framework, global stability of the switching strategies can be assured if the invariant sets topology allows the exclusive selection of estimates obtained from healthy sensors.
F. Stoican, Olaru, S., and Bitsoris, G., “Controlled invariance-based fault detection for multisensory control systems”, IET Control Theory & Applications Journal, vol. 7, no. 4, p. 606 – 611, 2012.Abstract
In this study, set theoretic methods are used to design a fault-tolerant scheme for a multisensor control application. The basic principle is the separation of the invariant sets for the estimations of the state and tracking error under healthy and faulty functioning. The fault scenario assumes abrupt changes of the observation equations. The main contribution of this paper is the introduction of controlled invariant sets in the fault detection mechanism. The control action is chosen in order to guarantee the closed-loop positive invariance of a candidate region when the exogenous signals (additive disturbances, noise and reference/set-points) are bounded.
F. Stoican, Olaru, S., Seron, M. M., and De Doná, J. A., “Reference governor design for tracking problems with fault detection guarantees”, Journal of Process Control, vol. 22, no. 5, p. 829–836, 2012.Abstract
The present paper deals with the reference tracking problem for processes with linear dynamics and multisensor information subject to abrupt sensor faults. A key point for fault tolerance will be the separation between healthy and faulty closed-loop behavior upon a set-characterization approach. This is achieved through set theoretic operations involving the healthy/faulty behavior of residual signals related to the system dynamics. As a main contribution, a reference governor scheme is designed using a receding horizon technique. It is shown that fault detection guarantees can be achieved by appropriate adjusting of the governor's delay/prediction window under mild assumptions on the fault scenario.
Set-Theoretic Fault Tolerant Control in Multisensor Systems
F. Stoican and Olaru, S., Set-Theoretic Fault Tolerant Control in Multisensor Systems, Engineering & Materials Science. London: Wiley - ISTE, 2013, p. 176. Buy onlineAbstract
Fault-tolerant control theory is a well-studied topic but the use of the sets in detection, isolation and/or reconfiguration is rather tangential. The authors of this book propose a systematic analysis of the set-theoretic elements and devise approaches which exploit advanced elements within the field. The main idea is to translate fault detection and isolation conditions into those conditions involving sets. Furthermore, these are to be computed efficiently using positive invariance and reachability notions. Constraints imposed by exact fault control are used to define feasible references (which impose persistent excitation and, thus, non-convex feasible sets). Particular attention is given to the reciprocal influences between fault detection and isolation on the one hand, and control reconfiguration on the other.