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We discuss boundary control of a wave equation with a non-linear anti-damping boundary condition. We design structured finite-dimensional $H_\infty$-output feedback controllers which stabilize the infinite dimensional system exponentially in closed loop. The method is applied to control torsional vibrations in drilling systems with the goal to avoid slip-stick., Comment: 28 pages, 11 figures, 2 tables

International audience; Proportional-integral-derivative (PID) structured controller is the most popular class of industrial control but still could not be appropriately exploited in gain-scheduling control systems. To gain the practicability and tractability of gain-scheduling control systems, this paper addresses the (Formula presented.) gain-scheduling PID control. The design of such a controller is based on parameterized bilinear matrix inequalities, which are then solved via a bilinear matrix inequality optimization problem of nonconvex optimization. Several computational procedures are developed for its computation. The merit of the developed algorithms is shown through the benchmark examples.

International audience; We present a frequency domain based H ∞-control strategy to solve boundary control problems for systems governed by parabolic or hyperbolic partial differential equation, where controllers are constrained to be physically implementable and of simple structure suited for practical applications. The efficiency of our technique is demonstrated by controlling a reaction-diffusion equation with input delay, and a wave equation with boundary anti-damping.

The Kreiss constant $K(A)$ of a stable matrix $A$ conveys information about the transient behavior of system trajectories in response to initial conditions. We present an efficient way to compute the Kreiss constant $K(A)$, and we show how feedback can be employed to make the Kreiss constant $K(A_{cl})$ in closed loop significantly smaller. This is expected to reduce transients in the closed loop trajectories. The proposed approached is compared to potential competing techniques., 18 pages, 7 figures

International audience; The theory of interval observers requires the studied system to be cooperative. In the case of non-cooperative systems, the literature proposes to compute a state-coordinate change such that the dynamics is cooperative in the new coordinates. In this paper, a new numerical method is introduced to compute this state-coordinate change. It is based on the reformulation of the problem into a multiple dynamic linear systems stabilization problem. This problem is then solved using an existing nonsmooth technique. Computations and simulations are performed on two examples inspired by the literature.; La conception d’observateurs par intervalles requiert la coopérativité du système dynamique considéré. La littérature propose un certain nombre de méthodes pour concevoir des observateurs par intervalles sur des systèmes coopératifs et non coopératifs. Dans le deuxième cas, un changement de coordonnées est utilisé afin d’obtenir une dynamique coopérative dans la nouvelle base. Dans cet article, une nouvelle méthode permettant la détermination numérique de cette transformation est introduite. Elle repose sur une reformulation du problème mathématique de détermination d’une matrice de transformation en un problème de stabilisation d’un ensemble de modèles dynamiques linéaires. Le problème équivalent est alors résolu en faisant appel à une technique d’optimisation non lisse. L’approche est testée sur deux exemples extraits de la littérature.

Summary A novel sequential semi-definite programming method is developed for optimization subject to rank constraints on matrix-valued nonlinear functions of matrix decision variables, which arise in reduced-order linear parameter varying-linear fractional transformational control synthesis. The global convergence of the method is easily proven without any step size control. An intensive simulation shows the clear advantage of the proposed method over the state-of-the-art nonlinear matrix inequality solvers. Copyright © 2017 John Wiley & Sons, Ltd.

Summary This work deals with computing the worst-case stability and the worst-case H∞ performance of linear time-invariant systems subject to mixed real-parametric and complex-dynamic uncertainties in a compact parameter set. Our novel algorithmic approach is tailored to the properties of the nonsmooth worst-case functions associated with stability and performance, and this leads to a fast and reliable optimization method, which finds good lower bounds of μ. We justify our approach theoretically by proving a local convergence certificate. Because computing μ is known to be NP-hard, our technique should be used in tandem with a classical μ upper bound to assess global optimality. Extensive testing indicates that the technique is practically attractive. Copyright © 2016 John Wiley & Sons, Ltd.

Some applications in control require the state vector of a system to be expressed in the appropriate coordinates so as to satisfy to some mathematical properties which constrain the studied system dynamics. This is the case with the theory of linear interval observers which are trivial to implement on cooperative systems, a rather limited class of control systems. The available literature shows how to enforce this limiting cooperativity condition for any considered system through a state-coordinate transformation. This study proposes an overview of the existing numerical techniques to determine such a transformation. It is shown that in spite of being practical, these techniques have some limitations. Consequently, a reformulation of the problem is proposed so as to apply non-smooth control design techniques. A solution is obtained in both the continuous- and discrete-time frameworks. Interestingly, the new method allows to formulate additional control constraints. Simulations are performed on three examples.

International audience; We present new approaches to designing structured controllers, which are robust in the presence of mixed real parametric and complex dynamic uncertainty. As the synthesis of such controllers is inherently NP-hard, we discuss inner and outer relaxation techniques, which make this problem amenable to computations. Our relaxation methods are positively evaluated and compared on the basis of a rich test set and for a challenging missile pilot design problem.

International audience; Optimal power flow (OPF) over power transmission networks poses challenging large-scale nonlinear optimization problems , which involve a large number of quadratic equality and indefinite quadratic inequality constraints. These compu-tationally intractable constraints are often expressed by linear constraints plus matrix additional rank-one constraints on the outer products of the voltage vectors. The existing convex relaxation technique, which drops the difficult rank-one constraints for tractable computation, cannot yield even a feasible point. We address these computationally difficult problems by an iterative procedure, which generates a sequence of improved points that converge to a rank-one solution. Each iteration calls a semi-definite program. Intensive simulations for the OPF problems over networks with a few thousands of buses are provided to demonstrate the efficiency of our approach. The suboptimal values of the OPF problems found by our computational procedure turn out to be the global optimal value with computational tolerance less than 0.01%.

We use a non-smooth trust-region method for $H_\infty$-control of infinite-dimensional systems. Our method applies in particular to distributed and boundary control of partial differential equations. It is computationally attractive as it avoids the use of system reduction or identification. For illustration the method is applied to control a reaction-convection-diffusion system, a Van de Vusse reactor, and to a cavity flow control problem., Set-Valued and Variational Analysis, 2018

We propose a bundle trust-region algorithm to minimize locally Lipschitz functions which are potentially nonsmooth and nonconvex. We prove global convergence of our method and show by way of an example that the classical convergence argument in trust-region methods based on the Cauchy point fails in the nonsmooth setting. Our method is tested experimentally on three problems in automatic control.

Computing the worst-case spectral abscissa of a system with uncertain parameters allows to decide whether it is robustly stable in a given parameter range. Since this problem is NP-hard, we use a heuristic local optimization method based on a bundle trust-region strategy to compute good lower bounds. Then we employ branch-and-bound to certify the global maximum. A specific frequency sweeping technique is used to accelerate the global optimization.

International audience; For linear systems, the control law design is often performed so that the resulting closed loop meets specific frequency-domain requirements. However, in many cases, it may be observed that the obtained controller does not enforce time-domain requirements amongst which the objective of keeping a scalar output variable in a given interval. In this article, a transformation is proposed to convert prescribed bounds on an output variable into time-varying saturations on the synthesised linear scalar control law. This transformation uses some well-chosen time-varying coefficients so that the resulting time-varying saturation bounds do not overlap in the presence of disturbances. Using an anti-windup approach, it is obtained that the origin of the resulting closed loop is globally asymptotically stable and that the constrained output variable satisfies the time-domain constraints in the presence of an unknown finite-energy-bounded disturbance. An application to a linear ball and beam model is presented.

International audience; We discuss a branch and bound algorithm for global optimization of NP-hard problems related to robust stability. This includes computing the distance to instability of a system with uncertain parameters, computing the minimum stability degree of a system over a given set of uncertain parameters, and computing the worst case H_\infty norm over a given parameter range. The success of our method hinges (1) on the use of an efficient local optimization technique to compute lower bounds fast and reliably, (2) a method with reduced conservatism to compute upper bounds, and (3) the way these elements are favorably combined in the algorithm.

International audience; A new μ synthesis technique is presented that allows the design of controllers satisfying prescribed structural constraints as well as robust against mixed uncertainties. Such features are rarely jointly present in existing methods. The proposed approach relies on a numerically efficient non-smooth optimisation algorithm to obtain locally optimal solutions. The validity of the new synthesis technique is corroborated by a challenging case study in which the objective is to design a structured controller for the uncertain flexible model of a rocket in upper atmosphere flight.