Mustafa Khammash

Lorenzo Fagiano, Aldo U. Zgraggen, Manfred Morari et al.

An approach to control tethered wings for airborne wind energy is proposed. A fixed length of the lines is considered, and the aim of the control system is to obtain figure-eight crosswind trajectories. The proposed technique is based on the notion of the wing's "velocity angle" and, in contrast with most existing approaches, it does not require a measurement of the wind speed or of the effective wind at the wing's location. Moreover, the proposed approach features few parameters, whose effects on the system's behavior are very intuitive, hence simplifying tuning procedures. A simplified model of the steering dynamics of the wing is derived from first-principle laws, compared with experimental data and used for the control design. The control algorithm is divided into a low-level loop for the velocity angle and a high-level guidance strategy to achieve the desired flight patterns. The robustness of the inner loop is verified analytically, and the overall control system is tested experimentally on a small-scale prototype, with varying wind conditions and using different wings.

Lorenzo Fagiano, Khahn Huynh, Bassam Bamieh et al.

A study on filtering aspects of airborne wind energy generators is presented. This class of renewable energy systems aims to convert the aerodynamic forces generated by tethered wings, flying in closed paths transverse to the wind flow, into electricity. The accurate reconstruction of the wing's position, velocity and heading is of fundamental importance for the automatic control of these kinds of systems. The difficulty of the estimation problem arises from the nonlinear dynamics, wide speed range, large accelerations and fast changes of direction that the wing experiences during operation. It is shown that the overall nonlinear system has a specific structure allowing its partitioning into sub-systems, hence leading to a series of simpler filtering problems. Different sensor setups are then considered, and the related sensor fusion algorithms are presented. The results of experimental tests carried out with a small-scale prototype and wings of different sizes are discussed. The designed filtering algorithms rely purely on kinematic laws, hence they are independent from features like wing area, aerodynamic efficiency, mass, etc. Therefore, the presented results are representative also of systems with larger size and different wing design, different number of tethers and/or rigid wings.

Corentin Briat, Mustafa Khammash

Protein mean and variance levels in a simple stochastic gene expression circuit are controlled using proportional integral feedback. It is shown that the protein mean level can be globally and robustly tracked to any desired value using a simple PI controller that satisfies explicit sufficient conditions. Controlling both the mean and variance on the other hand requires the use of an additional control input, chosen here as the mRNA degradation rate. Local robust tracking of mean and variance is proved to be achievable using multivariable PI control, provided that the reference point satisfies necessary conditions imposed by the system. Even more importantly, it is shown that there exist PI controllers that locally, robustly and simultaneously stabilize all the equilibrium points inside the admissible region. Simulation examples illustrate the results.

Corentin Briat, Mustafa Khammash

Sufficient conditions for the design of a simple class of interval observers for linear impulsive systems subject to minimum and range dwell-time constraints are obtained and formulated in terms of infinite-dimensional linear programs. The proposed approach is fully constructive in the sense that suitable observer gains can be extracted from the solution of the optimization problems and is flexible enough to be extended to include performance constraints and parametric uncertainties. In order to be solvable, the infinite-dimensional linear programs are relaxed using a method based on sum of squares which is known to be asymptotically exact in the present case. Three examples are given for illustration: the first one pertains on the interval observation of an impulsive system under a minimum dwell-time constraint, the second one is about the interval observation of an aperiodic sampled-data system and the last one is about the interval observation of a linear switched system.

Lorenzo Fagiano, Mustafa Khammash

Polynomial Chaos Expansions represent a powerful tool to simulate stochastic models of dynamical systems. Yet, deriving the expansion's coefficients for complex systems might require a significant and non-trivial manipulation of the model, or the computation of large numbers of simulation runs, rendering the approach too time consuming and impracticable for applications with more than a handful of random variables. We introduce a novel computationally tractable technique for computing the coefficients of polynomial chaos expansions. The approach exploits a regularization technique with a particular choice of weighting matrices, which allow to take into account the specific features of Polynomial Chaos expansions. The method, completely based on convex optimization, can be applied to problems with a large number of random variables and uses a modest number of Monte Carlo simulations, while avoiding model manipulations. Additional information on the stochastic process, when available, can be also incorporated in the approach by means of convex constraints. We show the effectiveness of the proposed technique in three applications in diverse fields, including the analysis of a nonlinear electric circuit, a chaotic model of organizational behavior, finally a chemical oscillator.

Corentin Briat, Mustafa Khammash

Linear Parameter-Varying (LPV) systems with piecewise differentiable parameters is a class of LPV systems for which no proper analysis conditions have been obtained so far. To fill this gap, we propose an approach based on the theory of hybrid systems. The underlying idea is to reformulate the considered LPV system as an equivalent hybrid system that will incorporate, through a suitable state augmentation, information on both the dynamics of the state of the system and the considered class of parameter trajectories. Then, using a result pertaining on the stability of hybrid systems, two stability conditions are established and shown to naturally generalize and unify the well-known quadratic and robust stability criteria together. The obtained conditions being infinite-dimensional, a relaxation approach based on sum of squares programming is used in order to obtain tractable finite-dimensional conditions. The approach is finally illustrated on two examples from the literature.

Corentin Briat, Mustafa Khammash

Delays are an important phenomenon arising in a wide variety of real world systems. They occur in biological models because of diffusion effects or as simplifying modeling elements. We propose here to consider delayed stochastic reaction networks. The difficulty here lies in the fact that the state-space of a delayed reaction network is infinite-dimensional, which makes their analysis more involved. We demonstrate here that a particular class of stochastic time-varying delays, namely those that follow a phase-type distribution, can be exactly implemented in terms of a chemical reaction network. Hence, any delay-free network can be augmented to incorporate those delays through the addition of delay-species and delay-reactions. Hence, for this class of stochastic delays, which can be used to approximate any delay distribution arbitrarily accurately, the state-space remains finite-dimensional and, therefore, standard tools developed for standard reaction network still apply. In particular, we demonstrate that for unimolecular mass-action reaction networks that the delayed stochastic reaction network is ergodic if and only if the non-delayed network is ergodic as well. Bimolecular reactions are more difficult to consider but an analogous result is also obtained. These results tell us that delays that are phase-type distributed, regardless of their distribution, are not harmful to the ergodicity property of reaction networks. We also prove that the presence of those delays adds convolution terms in the moment equation but does not change the value of the stationary means compared to the delay-free case. Finally, the control of a certain class of delayed stochastic reaction network using a delayed antithetic integral controller is considered. It is proven that this controller achieves its goal provided that the delay-free network satisfy the conditions of ergodicity and output-controllability.

Corentin Briat, Ankit Gupta, Mustafa Khammash

The antithetic integral feedback motif recently introduced in Briat, Gupta & Khammash (Cell Systems, 2017) is known to ensure robust perfect adaptation for the mean dynamics of a given molecular species involved in a complex stochastic biomolecular reaction network. However, it was observed that it also leads to a higher variance in the controlled network than that obtained when using a constitutive (i.e. open-loop) control strategy. This was interpreted as the cost of the adaptation property and may be viewed as a performance deterioration for the overall controlled network. To decrease this variance and improve the performance, we propose to combine the antithetic integral feedback motif with a negative feedback strategy. Both theoretical and numerical results are obtained. The theoretical ones are based on a tailored moment closure method allowing one to obtain approximate expressions for the stationary variance for the controlled network and predict that the variance can indeed be decreased by increasing the strength of the negative feedback. Numerical results verify the accuracy of this approximation and show that the controlled species variance can indeed be decreased, sometimes below its constitutive level. Three molecular networks are considered in order to verify the wide applicability of two types of negative feedback strategies. The main conclusion is that there is a trade-off between the speed of the settling-time of the mean trajectories and the stationary variance of the controlled species; i.e. smaller variance is associated with larger settling-time.

Corentin Briat, Mustafa Khammash

Controlling stochastic reactions networks is a challenging problem with important implications in various fields such as systems and synthetic biology. Various regulation motifs have been discovered or posited over the recent years, the most recent one being the so-called Antithetic Integral Control (AIC) motif in Briat et al. (Cell Systems, 2016). Several favorable properties for the AIC motif have been demonstrated for classes of reaction networks that satisfy certain irreducibility, ergodicity and output controllability conditions. Here we address the problem of verifying these conditions for large sets of reaction networks with fixed topology using two different approaches. The first one is quantitative and relies on the notion of interval matrices while the second one is qualitative and is based on sign properties of matrices. The obtained results lie in the same spirit as those obtained in Briat et al. (Cell Systems, 2016) where properties of reaction networks are independently characterized in terms of control theoretic concepts, linear programming conditions and graph theoretic conditions.

Corentin Briat, Ankit Gupta, Mustafa Khammash

Homeostasis is a running theme in biology. Often achieved through feedback regulation strategies, homeostasis allows living cells to control their internal environment as a means for surviving changing and unfavourable environments. While many endogenous homeostatic motifs have been studied in living cells, some other motifs may remain under-explored or even undiscovered. At the same time, known regulatory motifs have been mostly analyzed at the deterministic level, and the effect of noise on their regulatory function has received low attention. Here we lay the foundation for a regulation theory at the molecular level that explicitly takes into account the noisy nature of biochemical reactions and provides novel tools for the analysis and design of robust homeostatic circuits. Using these ideas, we propose a new regulation motif, which we refer to as {\em antithetic integral feedback, and demonstrate its effectiveness as a strategy for generically regulating a wide class of reaction networks. By combining tools from probability and control theory, we show that the proposed motif preserves the stability of the overall network, steers the population of any regulated species to a desired set point, and achieves robust perfect adaptation -- all with low prior knowledge of reaction rates. Moreover, our proposed regulatory motif can be implemented using a very small number of molecules and hence has a negligible metabolic load. Strikingly, the regulatory motif exploits stochastic noise, leading to enhanced regulation in scenarios where noise-free implementations result in dysregulation. Finally, we discuss the possible manifestation of the proposed antithetic integral feedback motif in endogenous biological circuits and its realization in synthetic circuits.

Corentin Briat, Mustafa Khammash

Ergodicity and output controllability have been shown to be fundamental concepts for the analysis and synthetic design of closed-loop stochastic reaction networks, as exemplified by the use of antithetic integral feedback controllers. In [Gupta, Briat & Khammash, PLoS Comput. Biol., 2014], some ergodicity and output controllability conditions for unimolecular and certain classes of bimolecular reaction networks were obtained and formulated through linear programs. To account for context dependence, these conditions were later extended in [Briat & Khammash, CDC, 2016] to reaction networks with uncertain rate parameters using simple and tractable, yet potentially conservative, methods. Here we develop some exact theoretical methods for verifying, in a robust setting, the original ergodicity and output controllability conditions based on algebraic and polynomial techniques. Some examples are given for illustration.

Milad Siami, Nader Motee, Gentian Buzi et al.

This paper develops some basic principles to study autocatalytic networks and exploit their structural properties in order to characterize their inherent fundamental limits and tradeoffs. In a dynamical system with autocatalytic structure, the system's output is necessary to catalyze its own production. We consider a simplified model of Glycolysis pathway as our motivating application. First, the properties of these class of pathways are investigated through a simplified two-state model, which is obtained by lumping all the intermediate reactions into a single intermediate reaction. Then, we generalize our results to autocatalytic pathways that are composed of a chain of enzymatically catalyzed intermediate reactions. We explicitly derive a hard limit on the minimum achievable $\mathcal L_2$-gain disturbance attenuation and a hard limit on its minimum required output energy. Finally, we show how these resulting hard limits lead to some fundamental tradeoffs between transient and steady-state behavior of the network and its net production.

Corentin Briat, Mustafa Khammash

While the design of optimal peak-to-peak controllers/observers for linear systems is known to be a difficult problem, this problem becomes interestingly much easier in the context of interval observers because of the positive nature of the error dynamics. Indeed, by exploiting several recent results on positive systems, we propose a novel and non-conservative approach formulated in terms of tractable finite-dimensional linear programs for designing a class of interval observers achieving minimum peak-to-peak gain. The optimal observer is notably shown to be uniform over the set of all possible mappings between observation errors and their weighted versions, which parallels a recent result on the stabilization of linear positive systems. Results pertaining on the interval observation of time-delay and discrete-time systems are then obtained as a direct application of the proposed method, emphasizing then its versatility. Several examples on the interval observation of linear and nonlinear systems are finally given for illustration.