Organizers:
Manuel Guerra
CEMAPRE/ISEG – Universidade de Lisboa
Jo˜
ao Miranda Lemos
INESC-ID/IST – Universidade de Lisboa
Nuno Bastos
Contents
PROGRAM v
INVITED TALKS AND TUTORIAL 1
CONTRIBUTED TALKS 5
POSTERS 25
SPEAKERS AND POSTER PRESENTERS 39
AUTHORS 40
PROGRAM
Monday, June 12th, 2017
9h30–10h00 Registration
10h00–10h45 Invited talk: Andrey Sarychev 10h45–11h00 Poster’s abstracts
11h00–11h30 Contributed talks:
Margarida Camarinha Margarida Ferreira 11h30–12h00 Coffee break and Posters 12h00–13h30 Contributed talks:
Ana P. Lemos-Pai˜ao Delfim F.M. Torres Fernando Fontes Jo˜ao Henriques Silv´erio Rosa Sofia Lopes 13h30–15h00 Lunch
15h00–15h45 Invited talk: Ugo Boscain 15h45–16h30 Contributed talks:
Maria do Ros´ario de Pinho Dmitri Karamzin
Miguel Oliveira 16h30–17h00 Coffee break and Posters 17h00–18h00 Contributed talks:
Fernando Lobo Pereira Nuno M. Brites
Pedro Louren¸co Jos´e Manuel Igreja 18h00–18h30 Tutorial: Manuel Guerra
INVITED TALKS AND
TUTORIAL
Simultaneous control of ensembles of nonlinear
systems
Andrey Sarychev
Dip. di Matematica e Informatica U.Dini, University of Florence, Italy [email protected]
Over the last decade there is a growing interest with regard to the control of ensembles (parameterized families) of nonlinear control systems
˙xθ = fθ(xθ, u), θ ∈ Θ ⊂ Rν, (1) by a single θ-independent control u(·). Such problem arises for example, when one seeks for a control, which may compensate a dispersion of parameters.
One of notable examples is Bloch model in NMR spectroscopy, seen as a bilinear control system in SO(3) with a parameter subject to dispersion. Partial controllability results for this model have been obtained by N.Khaneja and S.Li, who also suggested application of Campbell-Hausdorff formula for ”generating higher order Lie brackets ... which carry higher order powers of the dispersion parameters”.
An alternative problem setting amounts to finding for a control system ˙x = f (x, u) a ”simultaneous control” u(t), which (approximately) drives an ensemble of points x(θ), θ ∈ Θ, to a target z(θ). In our presentation we opt for Lp-approximate controllability:
R
Θkx(T ; θ) − z(θ)k
pdθ < p.
In a recent publication [1] with A.Agrachev and Yu.Baryshnikov we aimed at intro-ducing Lie algebraic (”geometric control”) approach to the controllability of (1).
We started with finite ensembles (finite Θ), to which Lie rank criteria of exact con-trollability can be applied after proper modification. We proved that the property of global controllability for a finite ensemble of control-linear systems is generic and, as an example, established global controllability by means of a single scalar control for a finite ensemble of rigid bodies with generic inertial parameters.
For continual ensembles (1) achieving exact controllability would require, in general, infinite-dimensional set of control parameters. Instead we fix the dimension of control and study L1-approximate controllability.
In the setting we first considered a model example of ”controlling the holonomy” for an ensemble of 2-distributions in R3, where we get necessary and sufficient for approximate
systems) on a manifold, for which we formulate sufficient approximately controllability criteria in terms of Lie algebraic span. This is a version of Rashevsky-Chow theorem for control-linear ensembles.
In what regards the ”simultaneous control” of ensembles of points x(θ) then the controllability criteria are similar in spirit to the previously mentioned results, but the formulations and the proofs differ. We establish genericity of the property of exact global controllability for finite point ensembles, and advance with a formulation of a version of Rashevsky-Chow theorem for continual point ensembles.
References
[1] A.Agrachev, Yu.Baryshnikov, A. Sarychev (2016). Ensemble Controllability by Lie Algebraic Methods, ESAIM COCV, v. 22, pp. 921-938.
EPCO 2017
Portuguese Meeting on Optimal Control
Intrinsic random walks in Riemannian geometry
via volume sampling
Ugo Boscain
CMAP-´Ecole Polytechnique, France [email protected]
We relate some constructions of stochastic analysis to differential geometry, via ran-dom walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics (solutions of the corresponding optimal control problem) and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric signifi-cance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.
Optimal control:
deterministic, stochastic, and ramifications
Manuel Guerra
CEMAPRE and ISEG – Universidade de Lisboa [email protected]
In the first part of this talk, we outline and compare some methods and issues that are of outstanding importance to optimal control in the deterministic and the stochastic cases. Then, we relate both cases to some types of control problems on semigroups, outlining both the finite and infinite-dimensional cases. We present some interesting features of such problems, hopefully pointing to interesting directions of research.
CONTRIBUTED TALKS
Variational obstacle avoidance problem on
Riemannian manifolds
Margarida Camarinha
CMUC, Department of Mathematics, University of Coimbra,Portugal [email protected]
Anthony Bloch
Department of Mathematics, University of Michigan, USA [email protected]
Leonardo Colombo
Department of Mathematics, University of Michigan, USA [email protected]
We introduce variational obstacle avoidance problems on Riemannian manifolds and derive necessary conditions for the existence of their normal extremals. The problem consists of minimizing an energy functional depending on the velocity and covariant ac-celeration, among a set of admissible curves, and also depending on a navigation function used to avoid an obstacle on the workspace, a Riemannian manifold. In particular, we study the case when the workspace is a Lie group endowed with a left-invariant Rieman-nian metric. We apply the results to the obstacle avoidance problem of a planar rigid body and a unicycle.
Optimal control problems with dynamics involving a
normal cone
M. Margarida A. Ferreira
DEEC - Systec, Faculdade de Engenharia, Universidade do Porto [email protected]
Gueorgui Smirnov Universidade do Minho [email protected] M. do Ros´ario de Pinho
DEEC - Systec, Faculdade de Engenharia, Universidade do Porto [email protected]
We consider a class of optimal control problems in which the dynamics is given by a differential inclusion involving a normal cone to a given convex set. In this way, the traditional regularity hypotheses associated to the differential inclusion are not satisfied and implicit state constraints are also present in the problem. For this class of problems, we prove existence of solution and deduce necessary conditions of optimality in the form of Pontryagin’s Maximum principle. A main role in this work is played by a sequence of ordinary differential equations that approximates the original dynamics. An illustrative example will be presented and related work on this subject will be discussed.
EPCO 2017
Portuguese Meeting on Optimal Control
A sufficient condition for linear delayed optimal
control problems
Ana P. Lemos-Pai˜ao1
CIDMA, Department of Mathematics, University of Aveiro, Portugal [email protected]
Cristiana J. Silva2
CIDMA, Department of Mathematics, University of Aveiro, Portugal [email protected]
Delfim F. M. Torres3
CIDMA, Department of Mathematics, University of Aveiro, Portugal [email protected]
This research responds to an open question by proving a sufficient optimality condition for linear problems of optimal control with delays in state and control variables. In the proof of our main result we transform a linear problem with delays to another one without. This allows us to use a known theorem that ensures a sufficient optimality condition for linear problems of optimal control without delays. Furthermore, an illustrative example is given.
References
[1] L. G¨ollmann, D. Kern, H. Maurer (2009). Optimal control problems with delays in state and control variables subject to mixed control-state constraints Optim. Control Appl. Meth. 30, no. 4, 341–365.
[2] L. G¨ollmann, H. Maurer (2014). Theory and applications of optimal control problems with multiple time-delays J. Ind. Manag. Optim. 10, no. 2, 413–441.
[3] T. Guinn (1976). Reduction of Delayed Optimal Control Problems to Nondelayed Problems J. Optim. Theory Appl. 18, no. 3, 371–377.
[4] E. B. Lee, L. Markus. (1967). Foundations of Optimal Control Theory. John Wiley & Sons, Inc. New York-London-Sydney.
[5] C. J. Silva, H. Maurer, D. F. M. Torres (2017). Optimal control of a tuberculosis model with state and control delays Math. Biosci. Eng. 14, no. 1, 321–337.
1Lemos-Pai˜ao is supported by the Portuguese Foundation for Science and Technology (FCT) under
project UID/MAT/04106/2013 (CIDMA) and by the Ph.D. fellowship PD/BD/114184/2016.
2Silva is supported by FCT through projects UID/MAT/04106/2013 (CIDMA) and
PTDC/EEI-AUT/2933/2014 (TOCCATA) and by the post-doc fellowship SFRH/BPD/72061/2010.
3Torres is supported by FCT within UID/MAT/04106/2013 (CIDMA) and
PTDC/EEI-AUT/2933/2014 (TOCCATA), funded by Project 3599 – Promover a Produ¸c˜ao Cient´ıfica e Desen-volvimento Tecnol´ogico e a Constitui¸c˜ao de Redes Tem´aticas and FEDER funds through COMPETE 2020, Programa Operacional Competitividade e Internacionaliza¸c˜ao (POCI).
Optimal control of a delayed HIV model
Delfim F. M. Torres1 University of Aveiro [email protected] Cristiana J. Silva2 University of Aveiro [email protected]We propose a model for the human immunodeficiency virus type 1 (HIV-1) infec-tion with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function, representing the efficiency of reverse tran-scriptase inhibitors, and consider the pharmacological delay associated with the control. Finally, we propose and analyze an optimal control problem with state and control de-lays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs.
References
[1] D. Rocha, C. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model, Math. Methods Appl. Sci., in press. DOI: 10.1002/mma.4207
[2] F. Rodrigues, C. J. Silva, D. F. M. Torres and H. Maurer, Optimal control of a delayed HIV model, Discrete Contin. Dyn. Syst. Ser. B, to appear.
[3] C. J. Silva, H. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng. 14 (2017), no. 1, 321–337.
[4] C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst. 35 (2015), no. 9, 4639–4663.
1This research was supported by the Portuguese Foundation for Science and Technology (FCT) within
projects UID/MAT/04106/2013 (CIDMA) and PTDC/EEI-AUT/2933/2014 (TOCCATA).
2Supported by UID/MAT/04106/2013 (CIDMA), project PTDC/EEI-AUT/2933/2014 (TOCCATA)
EPCO 2017
Portuguese Meeting on Optimal Control
Trajectories with guaranteed constraint satisfaction
in numerically-solved continuous–time optimal
control problems
Fernando A.C.C. Fontes1
SYSTEC–ISR, Universidade do Porto [email protected]
Lu´ıs Tiago Paiva
SYSTEC–ISR, ISEP–IPP, Universidade do Porto [email protected]
In this work we consider continuous–time optimal control problem and develop a numerical method that guarantees that the constraints imposed along the trajectory are in fact satisfied for all times.
The problem is relevant and non–trivial in situations where a continuous–time internal representation of the system is considered, while the optimal control problem is solved numerically using a discrete representation.
We propose a condition that when verified on a finite set of time instants (using limited computational power) can guarantee that the trajectory constraints are satisfied on an uncountable set of times. We devise an algorithm for the numerical solution of constrained nonlinear optimal control problems that combines a guaranteed constraint satisfaction strategy with an adaptive mesh refinement strategy.
1In this research, we acknowledge the support of FEDER/COMPETE/NORTE2020/POCI/FCT
funds through grants UID/EEA/00147/2013|UID/IEEA/00147/006933–SYSTEC, NORTE-01-0145-FEDER-000033–Stride, and PTDC-EEI-AUT-2933-2014|16858–TOCCATA.
A new high-order discontinuous Galerkin method for
the numerical solution of continuous and bang-bang
optimal control problems
Jo˜ao Henriques1
LAETA, IDMEC, Instituto Superior T´ecnico, Universidade de Lisboa [email protected]
Jo˜ao M. Lemos2
INESC-ID, Instituto Superior T´ecnico, Universidade de Lisboa [email protected]
Lu´ıs E¸ca3
LAETA, IDMEC, Instituto Superior T´ecnico, Universidade de Lisboa [email protected]
Lu´ıs Gato4
LAETA, IDMEC, Instituto Superior T´ecnico, Universidade de Lisboa [email protected]
Ant´onio Falc˜ao5
LAETA, IDMEC, Instituto Superior T´ecnico, Universidade de Lisboa [email protected]
Pseudo-Spectral (PS) methods are one of the most important and popular tools to compute solutions of optimal control problems (OCPs) [1, 2]. In the PS methods, the state and control trajectories are approximated in the entire domain by single polynomials where the unknown coefficients are computed at the Legendre-Gauss-Lobatto (LGL) or Chebyshev-Gauss-Lobatto (CGL) points. The capability of achieving high-order accuracy for smooth OCPs is probably the most appealing characteristic of the PS methods [3]. The high-order accuracy characteristic of the PS methods is usually lost for bang-bang OCPs due to the Gibbs phenomenon that results from the usage of smooth interpolation functions to compute non-smooth solutions [4].
The aim of the current research is to present a new high-order method that was devised to solve both continuous and bang-bang optimal control problems within the framework of the Pontryagins Maximum Principle. The computational time domain is divided in N consecutive time elements Ij, j = {1..., N }, where the numerical computation of the
states and co-states is performed using a Discontinuous Galerkin (DG) finite element time-stepping method. The finite element function space consists of piecewise Legendre polynomials. The inter-element boundary conditions are weakly enforced, hence its name of Discontinuous Galerkin.
1Supported by FCT researcher grant IF/01457/2014 and WAVEBUOY project
PTDC/MAR-TEC/0914/2014. Supported by FCT through IDMEC under LAETA PEst-OE/EME/LA0022.
2Supported by FCT through INESC-ID under UID/CEC/50021/2013 and
PTDC/EEIPRO/0426/2014.
3Supported by FCT through IDMEC under LAETA PEst-OE/EME/LA0022. 4Supported by FCT through IDMEC under LAETA PEst-OE/EME/LA0022. 5Supported by FCT through IDMEC under LAETA PEst-OE/EME/LA0022.
EPCO 2017
Portuguese Meeting on Optimal Control
The control trajectory is also approximated using the finite element function space. One of the novelties of the method is the iterative computation of the control trajectory. It is shown that the necessary conditions of optimality for the optimal problem within each time element Ij are:
• Find the optimal control function u(t) in each Ij such that
max
u(t)∈U
Z
Ij
H(t, x, u, λ)dt, (1) along the optimal trajectories for x and λ.
As such, the computation of the optimal control trajectory is recasted in the problem of finding the optimal u that maximizes (1).
The advantages of the proposed method are: the transformation of a global optimiza-tion problem in a series of optimizaoptimiza-tion problems solved independently for each finite element and a continuous polynomial solution for the state, costate and control vari-ables. Another characteristic of the DG method is the simple implementation of local mesh and polynomial refinement (hp-refinement) that reveals significant advantages when addressing OCPs with bang-bang solutions.
The contributions of the work are: i) the approximation of the Pontryagins Maximum Principle for optimal control problems within the framework of the DG finite element method; ii) maximization of the integral of the Hamiltonian function by approximating the control variables with the same basis functions of the state and adjoint variables; iii) a new iterative method for the solution of the Pontryagins conditions; and iv) a h-refinement algorithm to compute the switching instants of bang-bang OCPs.
The new method is an effective high-order alternative to the Pseudo-Spectral methods for linear and non-linear optimal control problems. Three examples of application are given: a continuous OCP, a bang-bang OCP and a real application of the method to solve a bang-bang OCP with a very large number of switching instants.
References
[1] G. Elnagar, M. A. Kazemi, M. Razzaghi (1995). The pseudospectral Legendre method for discretizing optimal control problems. IEEE Transactions on Automatic Control 40 (10) 1793-1796. doi:10.1109/9.467672.
[2] I. M. Ross, M. Karpenko (2012). A review of pseudospectral optimal con-trol: From theory to flight. Annual Reviews in Control 36 (2) 182-197. doi:10.1016/j.arcontrol.2012.09.002.
[3] E. Tohidi, S. L. Noghabi (2013). An efficient Legendre pseudospectral method for solving nonlinear quasi bang-bang optimal control problems. Journal of Applied Mathematics, Statistics and Informatics 8 (2) 73-85. doi:10.2478/v10294-012-0016-0.
[4] B. Fornberg (1996). A Practical Guide to Pseudospectral Methods. Cambridge Uni-versity Press. doi:10.1017/CBO9780511626357.
[5] M. Shamsi (2011). A modified pseudospectral scheme for accurate solution of bang-bang optimal control problems. Optimal Control Applications and Methods 32 (6) 668680. doi:10.1002/oca.967.
[6] D. G. Luenberger (1979). Introduction to dynamic systems: theory, models, and applications. J. Wiley & Sons, New York, Chichester, Brisbane.
[7] P. Lasaint, P.-A. Raviart (1974). On a finite element method for solving the neutron transport equation. In: Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), 89-123. Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York.
[8] R. Boucher, W. Kang, Q. Gong (2014). Feasibility of the Galerkin optimal control method. In: Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on, pp. 66716676. doi:10.1109/CDC.2014.7040436.
[9] G. B. Arfken, H. J. Weber, F. E. Harris (2013). Mathematical Methods for Physicists, 7th Edition. Academic Press, Boston. doi:10.1016/B978-0-12-384654-9.00030-X.
EPCO 2017
Portuguese Meeting on Optimal Control
Optimal control of the customer dynamics in a
marketing model
Silv´erio Rosa1
University of Beira Interior and Instituto de Telecomunica¸c˜oes [email protected]
C´esar Silva
University of Beira Interior [email protected]
Paulo Rebelo
University of Beira Interior [email protected]
Helena Alves
University of Beira Interior [email protected]
Pedro G. Carvalho
University of Beira Interior [email protected]
We consider a compartmental model to study the most suitable marketing strategy in a system for the evolution of the number of regular customers and referral customers in some corporation. This model, recently proposed in [1], is modified to a non-autonomous optimal control problem and the marketing policies are the control variables. The exis-tence and uniqueness of the solution, of the optimal control problem, is discussed. Some simulation is presented to validate the model.
References
[1] C´esar M. Silva, Silv´erio Rosa, Helena Alves and Pedro G. Carvalho (2016), A math-ematical model for the customer dynamics based on marketing policy, Applied Math-ematics and Computation, Volume 273, 42–53.
1This work was supported by FCT through Instituto de Telecomunica¸c˜oes (project
Multi-crop irrigation system planning via optimal
control
Sofia O. Lopes1
CMAT, Universidade do Minho [email protected] Fernando A.C.C. Fontes
SYSTEC–ISR, Universidade do Porto [email protected]
In this work we address the annual planning of irrigation systems for multiple crops. Using optimal control methods, we devise the time profile of the water introduced in a reservoir to supply different fields with different types of crops, with the criterion of minimizing the total water consumption. The solution is characterized analytically and a numerical solution is also obtained. The numerical solution is validated by confirming that it satisfies optimality conditions.
References
[1] Lopes, S. O., Fontes, F. A., Pereira, R., de Pinho, M., Gon¸calves, A. M. (2016). Optimal control applied to an irrigation planning problem. Mathematical Problems in Engineering, 2016.
[2] Fontes, F. A., Lopes, S. O. (2013). Normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints. Journal of Mathematical Analysis and Applications, 399(1), 27-37.
1In this research, we acknowledge the support of FEDER/COMPETE/NORTE2020/POCI/FCT
funds through grants UID/EEA/00147/2013|UID/IEEA/00147/006933–SYSTEC, UID/MAT/00013/ 2013–CMAT, NORTE-01-0145-FEDER-000033–Stride, and PTDC-EEI-AUT-2933-2014|16858– TOCCATA.
EPCO 2017
Portuguese Meeting on Optimal Control
Multiprocesses with state constraints
Maria do Ros´ario de Pinho1
SYSTEC, FEUP [email protected] Hasnaa Zidani
ENSTA- Paris Tech, France
We focus on optimal control for systems defined by three different continuous control systems, each one defined in three different regions of the state space, Ω1, Ω2 and Ω3.
Here Ω2 is the boundary between Ω1 and Ω3. The aim is to drive the state from an initial
position inside Ω1 to a target into Ω3 in a fixed time interval while minimizing a certain
cost. Necessary optimality conditions in the form of a Maximum Principle for similar optimal control for problems involving such systems have been studied in the literature; see, for example, [1], [4], [2]. Our problem of interest is closed related to those treated in [1] but, as in [2] we couple the different dynamic systems with state constraints. Indeed, the state dynamics changes depending on the state space and the switching between dynamics are enforced by state constraints. In contrast with [2] we do not enforce the system to live in Ω2. Indeed, the interval of time the system lives in Ω2 may be 0 or
positive. This is a feature covered in [3] for unconstrained optimal processes but not in [2]. We show through examples how this feature of our work may be of importance. We also discuss some straightforward generalizations of our work.
References
[1] G. Barles, A. Briani and R. Trelat (2016), Value Function and Optimal Trajectories for Regional Control Problems via Dynamic Programming and Pontryagin Maximum Principles, In: HAL Id: hal–01313559, May 2016.
[2] P. E. Caines, F. Clarke and R. Vinter (2006), A Maximum Principle for Hybrid Optimal Control Problems with Pathwise State Constraints, In: Proceedings of the 45th IEEE Conference on Decision and Control.
[3] F. Clarke and R. Vinter (1989), Optimal Multiprocesses, In: SIAM J. Control and Optimization, Vol. 27, No. 5, pp. 1072–1091.
[4] A. V. Dmitruck and A.M. Kaganovich (2008), The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle, In: Systems and Control Letters, Vol. 27, pp. 964–970.
1MdR de Pinho wishes to thank the support of the project PTDC/EEIAUT/ 2933/2014,
TOC-CATTA, funded by FEDER funds through COMPETE2020 - POCI and by national funds through FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia. This work was done while she was visiting ENSTA- Paris Tech and she would like to express her gratitude from the warm way she was received by Hasnaa Zidani in ENSTA.
A generalized existence theorem in impulsive control
problems
Dmitri Karamzin
dmitry [email protected] Fernando Lobo Pereira [email protected]
Some optimal control problems do not have solution in the class of classic controls. This suggests the need of a relaxation or extension of the control problem ensuring the existence of a solution in some enlarged class of controls.
This work aims at the development of an extension for optimal control problems with nonlinear control dynamics and control functions which take values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the concept of generalized controls. However it is adapted for the case of discontinuous arcs. This leads to the notion of generalized impulsive control.
EPCO 2017
Portuguese Meeting on Optimal Control
Lipschitzian regularity of solution to a problem of
calculus of variations
Miguel Oliveira Minho University nunomiguel [email protected] Georgi Smirnov Minho University [email protected]We obtain explicit estimate for Lipschitz constant of solution to a problem of calculus of variations. The approach is based on transformation of the problem into a time-optimal control problem, suggested by Gamkrelidze. This estimate is used to obtain complexity bounds for numerical optimization methods.
Model predictive control coordination for the
sustainable management of production factors
in agriculture
Fernando Lobo Pereira [email protected]
This presentation discusses a decision support system based on a Model Predictive Control (MPC) Scheme in order to conciliate sustainable short term economic returns of a set of agricultural production units that compose a given region and the its long term environmental sustainability.
The overall coordination is achieved by a decentralized, adaptive, and hierarchic struc-ture that, on the one hand, promotes the long term common good by approximating the solution to an infinite horizon optimal control problem, and, on the other hand, provides agro-chemical indicators to each one of the local farmers.
This is a very complex problem and this work is a preliminary effort in which the emphasis will be in the problem formulation in a simple context. More precisely, two production units will be modeled and an MPC based coordinator to define the produc-tion factors for each producproduc-tion unit will be considered. In this process, the maximum principle of Pontryagin will play a considerable role in formulating the dynamics taken into account by the coordinator.
EPCO 2017
Portuguese Meeting on Optimal Control
Fisheries management in random environments:
comparison of harvesting policies for the logistic
model
Nuno M. Brites1
CIMA, Instituto de Investiga¸c˜ao e Forma¸c˜ao Avan¸cada, Universidade de ´Evora [email protected]
Carlos A. Braumann2
CIMA, Instituto de Investiga¸c˜ao e Forma¸c˜ao Avan¸cada, and
Departamento de Matem´atica, Escola de Ciˆencias e Tecnologia, Universidade de ´Evora [email protected]
We model fish population growth subjected to harvesting taking into account the effect of the environmental random fluctuations on the growth dynamics. For that we use stochastic differential equations (SDE). In particular, we consider a logistic model plus random environmental perturbations for the natural growth and subtract a harvesting yield term based either on a variable or on a constant fishing effort.
There is previous work on the optimal design of the harvesting policy with the purpose of maximizing the expected accumulated profit (discounted by a depreciation rate) over a finite time horizon. We consider a profit structure which includes revenues to be proportional to the yield and costs to be quadratic on the effort per unit time. The harvesting efforts of the optimal policies vary with the randomly varying population size and such policies can, under certain conditions, even be of bang-bang type. These policies, borrowed from the financial world where data is abundant, are not applicable to harvesting since they require constant evaluation of the population size and they have frequent random changes in harvesting effort incompatible with the logistics of fishing.
Our approach, based on sustainable and applicable fishing policies with constant ef-fort, leads to sustainability of the population and to a stationary distribution of the population size and do not require evaluation of population size. We determine the con-stant harvesting effort policy that optimizes the expected sustainable profit per unit time and check what we lose profitwise when using this policy instead of the optimal inapplica-ble policy with variainapplica-ble effort. Applying Monte Carlo simulations and using population parameters based on real data, we show that our approach is almost as profitable as the first.
1Work developed at Centro de Investiga¸c˜ao em Matem´atica e Aplica¸c˜oes (CIMA), a research center
funded by Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT) under the project UID/MAT/04674/2013, and with a PhD grant from FCT with reference SFRH/BD/85096/2012.
2Work developed at CIMA, a research center funded by FCT under the project UID/MAT/04674/
Towards uncertainty optimization in active SLAM
Pedro Louren¸co1
Institute for Systems and Robotics (ISR/IST), LARSyS, Instituto Superior T´ecnico, Universidade de Lisboa, Portugal [email protected]
Pedro Batista2
Instituto Superior T´ecnico, Universidade de Lisboa, Portugal [email protected]
Paulo Oliveira3
Institute of Mechanical Engineering,
Associated Laboratory for Energy, Transports and Aeronautics, Lisbon, Portugal [email protected]
Carlos Silvestre4
Department of Electrical and Computer Engineering,
Faculty of Science and Technology of the University of Macau, China [email protected]
Simultaneous localization and mapping (SLAM) is the problem of navigating a vehicle in an unknown environment, by building a map of the area and using this map to deduce its location, without the need for a priori knowledge of location. The main paradigm in SLAM is to move to gain new knowledge and improve what is known. In the formative years of SLAM (see [1]), the question of how to move was completely separated from the estimation problem. However, in recent years, several works addressed the issue of intelligent moving in the context of SLAM, thus introducing Active SLAM (see [2] and [3]). The objective of Active SLAM is to plan ahead the motion of the vehicle in order to maximize the explored areas and minimize the uncertainty associated with the estimation. These two objectives are, in a sense, complementary: exploration involves moving in previously unvisited terrain with the objective of increasing the overall knowledge of the environment, while the latter is exploitation, i.e., it involves revisiting areas to maximize the information gain. In this work, the focus is on the exploitation part of Active SLAM, which requires that some form of exploration is already done.
Not specifically applied to Active SLAM, but still with the objective of uncertainty reduction, an interesting idea is proposed in [4]. The author recovers a traditional optimal control and estimation result [5], and, using the Pontryagin minimum principle [6], derives a control strategy for mobile sensors that influences the uncertainty evolution in a Kalman filter. Given the linear character of the Kalman filter, this is only possible if any of the
1The work of P. Louren¸co was supported by the PhD. Student Grant SFRH/BD/89337/2012 from
the Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT).
2The work of P. Batista was supported by FCT through ISR, under contract LARSyS
UID/EEA/50009/2013.
3The work of P. Oliveira was supported by the FCT through IDMEC, under contract LAETA
UID/EMS/50022/2013.
4The work of C. Silvestre was supported by the University of Macau Project MYRG2015-00126-FST,
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parameters of the filter depends, even if indirectly, on the input of another system. This work addresses the problem of optimizing the uncertainty in a simultaneous localization and mapping filter with globally exponentially stable error dynamics [7]. This is done by designing an optimization problem that weighs the final uncertainty, the average uncertainty in the horizon considered, and the cost of the control. Using the Pontryagin minimum principle and building on [5] and [4], the optimization problem is transformed into a two-point boundary value problem that encodes necessary conditions for the input that minimizes the uncertainty. A strategy is proposed to solve this problem numerically, and several particular examples are analysed in depth.
References
[1] H. Durrant-Whyte and T. Bailey, “Simultaneous Localisation and Mapping (SLAM): Part I The Essential Algorithms,” IEEE Robotics & Automation Magazine, vol. 13, no. 2, pp. 99–110, 2006.
[2] H. J. S. Feder, J. J. Leonard, and C. M. Smith, “Adaptive mobile robot navigation and mapping,” The International Journal of Robotics Research, vol. 18, no. 7, pp. 650–668, 1999.
[3] C. Leung, S. Huang, and G. Dissanayake, “Active SLAM using Model Predictive Control and Attractor based Exploration,” in Proc. of the 2006 IEEE/RSJ Interna-tional Conference on Intelligent Robots and Systems, Oct 2006, pp. 5026–5031.
[4] I. I. Hussein, “Kalman filtering with optimal sensor motion planning,” in Proc. of the 2008 American Control Conference, Seattle, Washington, USA, June 2008, pp. 3548–3553.
[5] M. Athans and E. Tse, “A direct derivation of the optimal linear filter using the maximum principle,” IEEE Transactions on Automatic Control, vol. 12, no. 6, pp. 690–698, December 1967.
[6] M. Athans and P. Falb, Optimal Control: An Introduction to the Theory and Its Applications, ser. Dover Books on Engineering Series. Dover Publications, 2006.
[7] P. Louren¸co, B. J. Guerreiro, P. Batista, P. Oliveira, and C. Silvestre, “Simultane-ous Localization and Mapping for Aerial Vehicles: a 3-D sensor-based GAS filter,” Autonomous Robots, vol. 40, pp. 881–902, Jun. 2016.
Self-organized formations in multiple-agent systems
using D-MPC
Jos´e Manuel Igreja1
ADEEEA-ISEL-IPL/INESC-ID, Lisboa, Portugal [email protected]
Jo˜ao Miranda Lemos
IST-Univ. Lisboa/INESC-ID, Lisboa, Portugal [email protected]
Self-organized or soft flock formations [1] deals, in multi-agent systems (MAS), with very flexible self-organizing groups of agents where each agent should not be in a fixed formation position, but rather is able to a find its own position in the group quickly and naturally. In this way the formation should change dynamically with the environment, the group behavior and the assigned mission. This presentation focus in coordinated behavior between agents that use implicit rules or dependencies, inducing self-organizing social behavior, while performing one mission with a common goal. Agents coordination is understood as a weaker form of cooperation where each agent plays a preset role. In MAS, (Objective) Coordination has been defined as managing dependencies in inter-agent activities [2]. In distributed control, the coordination of MAS concept refers to all the various types of control algorithms that depend on information interchange between subsystems. For this purpose, Distributed Model Predictive Control (D-MPC) [3] is used to control and coordinate agents by suitable designing of cost functions and coupling dynamical constraints. The resulting algorithms solve a sequence, in time, of multiple static non-convex optimization problems, with nonlinear coupling constraints, and apply a typical receding horizon policy for mission control. In distributed control setups, each agent moves according to the distributed control algorithm and shares information with other agents in such a way that individual and collective behavior may be related with a global outcome, while preserving the notion of soft flock formation or flexible group. The concept is validated by developing D-MPC algorithms that can be used to maneuver mechanical agents in a 2-D spacial scenario with obstacles and collision avoidance. Several examples are included, where feedback stability and Pareto optimal and Nash Equilibrium outcomes are discussed.
References
[1] C. S. Ho,Y.-S. Ong, X. Chen, A.-H. Tan (2012). FAME, Soft Flock Formation Control for Collective Behavior Studies and Rapid Games Development. Lecture Notes in Computer Science, vol 7673. Springer, Berlin, Heidelberg.
[2] Michael Schumacher(2001). Objective Coordination in Multi-Agent System Engineer-ing. Lecture Notes in Computer Science, Vol. 2039, Lecture Notes in Artificial Intel-ligence, Springer-Verlag.
1This work has been supported by FCT, Portugal, under contracts UID/CEC/50021/2013 and
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[3] J. M. Maestre, R.R Negenborn (2014). Distributed Model Predicitve Control Made Easy, Control and Automation: Science and Engineering, Vol. 69, 1st Edition, Springer.
POSTERS
On the design of nonlinear model predictive control
schemes with stability guarantees
Andrea Alessandretti1
Faculty of Engineering, University of Porto, Porto, Portugal [email protected]
A. Pedro Aguiar
Faculty of Engineering, University of Porto, Porto, Portugal [email protected]
Despite the fact that the set fo sufficient condition for the design of MPC control schemes with closed- loop stability guarantees is well established, the design tools that allow satisfying such assumptions are still very limited. This is especially true in the case where the performance index is of the economic kind, i.e., that does not simply penalize the distance between the current state and the desired one but it rather represents a more complex control objective. As a result, the design is often based using linearization tech-niques that lead to MPC controllers with a limited region of attraction and therefore reducing their range of applicability.
In this poster, we present two of design methods, both for discrete and continuous time systems, for the design of a stabilizing terminal set and terminal cost for model predictive control scheme with economic performance index. It is shown how these results can be used in the design of a nonlinear MPC with a global region of attraction even in the case of systems with constrained inputs.
1This work was partially supported by FCT R&D Unit SYSTEC -
POCI-01-0145-FEDER-006933/SYSTEC funded by ERDFCOMPETE2020FCT/MECPT2020 and project STRIDE - NORTE-01-0145-FEDER-000033, funded by ERDFNORTE 2020.
Design of minimum time trajectories for autonomous
underwater vehicles
Joana Fonseca DEEC, FEUP [email protected] An´ıbal Matos INESC-TEC, FEUP [email protected]Maria do Ros´ario de Pinho1 SYSTEC, FEUP
We determine minimum time vertical trajectories of an AUV using a dynamical model of a Mares autonomous underwater vehicle developed in [1]. The minimum time optimal control problem of interest is solved numerically using the direct method: it is first discretized and the transcribed nonlinear programming problem is then solved using IPOPT (see [2]). The solution is then partially validated using the Maximum Principle (see [4]) and it is compared to other possible trajectories to assert its optimality. The solution of our original problem is also compared to the minimum time trajectory obtained using a simplified model proposed in [3].
A remarkable finding of our work is the fact that the trajectories obtained with both models are a wobbling line joining the initial position and the target set. Here we discus this phenomenon and we propose some sub-optimal solutions imposing piece wise constant controls with a reduced number of switching times.
Additionally, we study how the minimum time for our original problem changes as a function of the target set.
References
[1] BMM Ferreira (2009), Modela¸c˜ao e controlo de ve´ıculo submarino com quatro graus de liberdade, In: Tese de doutoramento, FEUP.
[2] Andreas W¨achter and Lorenz T. Biegler (2006), On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106, 25–57.
[3] M. Chyba, N.E. Leonard and E.D. Sontag (2003), Singular trajectories in multi-input time-optimal problems. Application to controlled mechanical systems, In: Jour-nal on Dynamical and Control Systems, vol. 9 (1), pp. 7388.
[4] Richard Vinter (2000). Optimal Control. Birkh¨auser, Boston.
1MdR de Pinho wishes to thank the support of the project PTDC/EEIAUT/ 2933/2014,
TOC-CATTA, funded by FEDER funds through COMPETE2020 - POCI and by national funds through FCT - Fundacao para a Ciencia e a Tecnologia.
EPCO 2017
Portuguese Meeting on Optimal Control
Optimal control of an epidemiological viral marketing
model using indirect and direct methods
Jo˜ao N.C. Gon¸calves
Algoritmi and Department of Production and Systems, University of Minho, Portugal [email protected]
Helena Sofia Rodrigues
School of Business Studies, Polytechnic Institute of Viana do Castelo and CIDMA, Department of Mathematics, University of Aveiro, Portugal [email protected]
M. Teresa T. Monteiro
Algoritmi and Department of Production and Systems, University of Minho, Portugal [email protected]
The complexity of optimal control problems requires the use of numerical methods to compute control and optimal state trajectories for a dynamical system, aiming to optimize a given performance index [2]. Based on a real viral advertisement, the dynamics of a SIR viral marketing epidemic model with optimal control is studied and indirect and direct methods are analyzed and compared. With the main goal of maximize the diffusion of information with a low cost, an optimal control problem is formulated and studied, based on a system of ODEs proposed in [1]. The existence and uniqueness of the solution are proved. Our results suggest that low investment costs in marketing strategies fulfill the proposed objective. Moreover, numerical simulations allow to provide control intervention strategies and show that the cost of implement control policies is a crucial parameter for the spreading of marketing messages within a target population.
References
[1] A. Karnick, P. Dayama (2012). Optimal Control of Information Epidemics. In: Proc. IEEE Commun. System. Networks Conf., 17.
[2] A. Rao (2009). A survey of numerical methods for optimal control. Technical Report AAS 09-334. Dep. of Mechanical and Aerospace Engineering, University of Florida.
Banking risk as an epidemiological model:
an optimal control approach
Olena Kostylenko1
CIDMA, Department of Mathematics, University of Aveiro, Portugal [email protected]
Helena Sofia Rodrigues
CIDMA, Department of Mathematics, University of Aveiro and,
School of Business Studies, Polytechnic Institute of Viana do Castelo, Portugal [email protected]
Delfim F. M. Torres
CIDMA, Department of Mathematics, University of Aveiro, Portugal [email protected]
Abstract: The process of contagiousness spread modelling is well-known in epi-demiology. However, the application to banking market is quite recent. In this work, we present a system of ordinary differential equations, simulating data from the largest European banks. Then, an optimal control problem is formulated in order to study the impact of a possible measure of the Central Bank in the economy. The proposed approach enables qualitative specifications of contagion in banking obtainment and an adequate analysis and prognosis within the financial sector development and macroeconomic as a whole. We show that our model describes well the reality of the largest European banks. The main simulations were done using MATLAB and BOCOP optimal control solver, and the main results are taken for three distinct scenarios.
Introduction
Mathematical models of spread of epidemics are widely used in different fields of studies and directly related to everyone’s life. Our research is focused on studies of contagion in banking market using an epidemiological approach through mathematical modelling. Banks are directly at the centre of the financial system. Crisis in banking is one of the most serious type of financial crisis. Therefore, it is very important to study how the crisis spreads in banking market, to find its basic laws and methods to control it. A good understanding of their propagation mechanism makes possible to find and propose suitable policy interventions, which can most effectively reduce their contagious spread. This is the main goal of our research. Namely, we show that the application of epidemi-ological models is able to describe well the nature and character of the contagion spread
1This research was supported by the Portuguese Foundation for Science and Technology (FCT) within
project UID/MAT/04106/2013 (CIDMA).
This work is part of first author’s Ph.D., which is carried out at University of Aveiro under the Doctoral Program in Applied Mathematics MAP-PDMA, of Universities of Minho, Aveiro and Porto, and is also supported by the Ph.D. fellowship PD/BD/114188/2016.
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Portuguese Meeting on Optimal Control
and its behaviour over time. Our analysis identifies which measures to adopt and when they must be taken in order to prevent affects and serious negative effects for a particular bank, and for economy as a whole.
Methodology
The processes of contagion in economy are very similar to the disease propagation in a population (from one individual to another). This similarity allows us to consider contagion in such type of financial sector, as banking, using the same mathematical model of infection spreading as used in epidemiology. The SIR models are simulated with the MATLAB software. Since one of the main goals is to avoid the wide dissemination of a contagion, we pose an optimal control problem in Bolza form and solve it using the optimal control solver BOCOP.
Findings
The statistical data is taken, with respect to the year 2012, for the 169 largest European banks with total assets over than 10 bn Euro. The contagion and recovery propaga-tion rates used in our paper follow the statistics and empirical findings of Philippas, Koutelidakis and Leontitsis [1]. We obtained graphics for different scenarios of contagion spreading among European banks, which depend on to the location of the first initially contagious bank. It is found that the different economical situation of countries, where contagion begins, affects to the period of time needed to achieve the contagion-free equi-librium.
Conclusions
We investigated the dynamic behaviour of contagiousness in the banking sector, at the macroeconomic level, using a SIR epidemic model and optimal control. The features of contamination were shown to depend on the parameter values of the transmission and recovery rates, as well as on the country for which the process of infection begins. The scale of negative consequences for different scenarios of bank risk contagion were identified. Banks at risk from countries with greater financial influence in Europe tend to propagate more severely the banking crisis; at the same time, the recovery period is longer. An optimal control problem was proposed, in order to reduce the number of contagious banks, to prevent large-scale epidemic contagiousness, and to avoid serious financial and economic consequences.
References
[1] D. Philippas, Y. Koutelidakis, A. Leontitsis (2015). Insights into European interbank network contagion. Managerial Finance. 41, no.8, 754–772.
Electricity generation via optimal control of
underwater kite power systems
Lu´ıs Tiago Paiva1
SYSTEC–ISR, ISEP–IPP, Universidade do Porto [email protected]
Fernando A.C.C. Fontes
SYSTEC–ISR, Universidade do Porto [email protected]
In this work we address the problem of generating electricity through the control of Underwater Kite Power Systems (UKPS). This is an highly nonlinear problem for which the optimization is challenging.
We develop a 2D optimal control problem (OCP) formulation, based on a continuous– time model of the kite, to determine trajectories and controls for the kite that maximize the total energy produced in a given time interval. According to our results, the problem can be promptly solved with high level of accuracy when using our multi–level adaptive mesh refinement algorithm. The results confirm the amount of electrical power that can be generated with such device, emphasizing the advantage of its use.
1In this research, we acknowledge the support of FEDER/COMPETE/NORTE2020/POCI/FCT
funds through grants UID/EEA/00147/2013|UID/IEEA/00147/006933–SYSTEC, NORTE-01-0145-FEDER-000033–Stride, and PTDC-EEI-AUT-2933-2014|16858–TOCCATA.
EPCO 2017
Portuguese Meeting on Optimal Control
Radial functions for solving partial integro-differential
equations
Nedjem Eddine Ramdani
Department of Mathematics University of Batna 2, Algeria [email protected]
Abdelaziz Mennouni
Department of Mathematics University of Batna 2, Algeria [email protected]
In this work, we investigate a method for solving a class of partial integro differential equations over Ω an open bounded domain in Rn. The present method based on radial
functions. We prove the existence of the solution for the approximate equation, and we give a new error estimate for the numerical solutions.
References
[1] Belytschko T, Lu YY, GuL (1994) Element-free Garlerkin methods, Int J Numer Methods Eng 1994; 37:229-56.
[2] Onate E, Idelsohn S, Zienkiewicz OC, (1996) Taylor RL. A finite Point method in computational mechanics: applications to convective transport and fluid flow, Int J Numer Methods Eng 1996; 39: 3839-66.
[3] Lancaster P, Salkauskas K. (1981) Surfaces generated by moving least squares meth-ods, Math Comput 1981; 37: 141-58.
[4] Wendland H. (2005) Scattered data approximation, Cambridge, UK: Cambridge University Press; 2005.
[5] Levin D. (1998) The approximation power of moving least-squares, Math Comput 1998; 67: 1517-31.
[6] Armentano MG, Duran RG. (2001) Error estimates for moving least square approx-imations, Appl Numer Math 2001; 37: 397-416.
Optimal control of virus propagation in computers
Paulo Rebelo1
Universidade da Beira Interior, 6201 − 001 Covilh˜a. [email protected]
Silv´erio Rosa2
Universidade da Beira Interior, 6201 − 001 Covilh˜a. [email protected]
C´esar Silva3
Universidade da Beira Interior, 6201 − 001 Covilh˜a. [email protected]
The aim of this work is to consider optimal control problems for the propagation of computer virus. A non-autonomous model is presented. Two optimal control problems are considered.
References
[1] Jianguo Ren, Xiaofan Yang, Lu-Xing Yang, Yonghong Xu and Fanzhou Yang (2012). A delayed computer virus propagation model and its dynamics. Chaos, Solitons & Fractals, Nonlinear Science, and Nonequilibrium and Complex Phenomena, Chaos, Solitons & Fractals 45 (2012) 7479.
[2] Xiaofan Yang and Lu-Xing Yang (2012). Towards the Epidemiological Modeling of Computer Viruses. Hindawi Publishing Corporation, Discrete Dynamics in Nature and Society, Volume 2012, Article ID 259671, 11 pages doi:10.1155/2012/259671.
[3] Jos´e Roberto Castilho Piqueira, Betyna Fern´andez Navarro and Luiz Henrique Alves Monteiro (2005). Epidemiological Models Applied to Viruses in Computer Networks. Journal of Computer Science 1 (1): 31-34, 2005 ISSN 1549-3636.
1This work was supported by FCT through Centro de Matem´atica e Aplica¸c˜oes da UBI (CMA-UBI). 2This work was supported by FCT through Instituto de Telecomunica¸c˜oes (project
UID/EEA/50008/2013).
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Portuguese Meeting on Optimal Control
Switching-time parameterization approaches for the
unit commitment problem
Lu´ıs A.C. Roque1
LIADD-INESC-TEC, DMA, Instituto Superior de Engenharia do Porto [email protected]
Fernando A.C.C. Fontes
SYSTEC–ISR, Faculdade de Engenharia, Universidade do Porto [email protected]
Dalila B.M.M. Fontes
LIADD-INESC-TEC, Faculdade de Economia, Universidade do Porto [email protected]
The Unit Commitment Problem (UCP) is a well-known combinatorial optimization problem in power systems and the subject of intense research. The problem involves both the on/off decisions and output power levels of generating units over a given planning horizon. The objective is the minimization of the total operating costs while a set of technological and operational constraints must be satisfied.
The UCP is typically formulated as a mixed-integer optimization problem with both binary variables and real variables. However, the complexity due to integer variables makes the UCP a hard nonconvex problem to be addressed with current optimization solvers. A large variety of optimization methods have been employed to find solutions, ranging from exact methods or approaches based on them (such as dynamic programming, branch-and-bound, Lagrangian relaxation) to heuristic methods (genetic algorithms, sim-ulated annealing, particle swarm, tabu search).
Here, the problem is formulated as a switching-time parameterized optimal control problem involving only real-valued controls. Compared to the usual mixed-integer pro-gramming models in the literature, our continuous time formulation has the advantage that all decision variables are real-valued, which enables the use of more efficient optimiza-tion methods for its soluoptimiza-tion. Moreover, addioptimiza-tional benefits can result with the possibility of dealing with irregular or fast-sampled time intervals, or even continuous-time varying demand data.
1In this research, we acknowledge the support of FEDER/COMPETE/NORTE2020/POCI/FCT
funds through grants UID/EEA/00147/2013|UID/IEEA/00147/006933–SYSTEC, NORTE-01-0145-FEDER-000033–Stride, and PTDC-EEI-AUT-2933-2014|16858–TOCCATA.
Optimal control of pre-exposure prophylaxis in HIV
prevention
Cristiana J. Silva1 University of Aveiro [email protected] Delfim F. M. Torres2 University of Aveiro [email protected]Pre-exposure prophylaxis (PrEP) consists in the use of an antiretroviral medication to prevent the acquisition of HIV infection by uninfected individuals and has recently demonstrated to be highly efficacious for HIV prevention. We propose a new epidemi-ological model for HIV/AIDS transmission including PrEP. Existence, uniqueness and global stability of the disease free and endemic equilibriums are proved. An optimal con-trol problem with a mixed state concon-trol constraint is then proposed and analyzed, where the control function represents the PrEP strategy and the mixed constraint models the fact that, due to PrEP costs, epidemic context and program coverage, the number of in-dividuals under PrEP is limited at each instant of time. The objective is to determine the PrEP strategy that satisfies the mixed state control constraint and minimizes the number of individuals with pre-AIDS HIV-infection as well as the costs associated with PrEP. The optimal control problem is studied analytically. Through numerical simulations, we demonstrate that PrEP reduces HIV transmission significantly.
References
[1] C. J. Silva and D. F. M. Torres, Modeling and optimal control of HIV/AIDS pre-vention through PrEP, Discrete Contin. Dyn. Syst. Ser. S, to appear.
1Supported by UID/MAT/04106/2013 (CIDMA), project PTDC/EEI-AUT/2933/2014 (TOCCATA)
and the post-doc fellowship SFRH/BPD/72061/2010.
2This research was supported by the Portuguese Foundation for Science and Technology (FCT) within
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Portuguese Meeting on Optimal Control
Aircraft landing using dynamic image-based visual
servo control
Zhiqi Tang1
Instituto Superior T´ecnico, Universidade de Lisboa, Portugal [email protected]
Rita Cunha
Instituto Superior T´ecnico, Universidade de Lisboa, Portugal [email protected]
Tarek Hamel
I3S UNSA-CNRS, University of Nice Sophia Antipolis, France [email protected].
Carlos Silvestre
Department of Electrical and Computer Engineering, Faculty of Science and
Technology, University of Macau, on leave from IST, Universidade de Lisboa, Portugal [email protected]
The landing maneuver is still one of the most critical and dangerous flight phases, responsible for the majority of aircraft accidents [1]. In this work, we propose a vision-based controller to automatically steer a fixed-wing Unmanned Aerial Vehicle (UAV) during the first three stages of landing: alignment, glide-slope, and flare. The use of vision for automatic landing of fixed-wings UAVs has been extensively researched. For example, in [2] and [3] optical flow information to sense and control the height above the the ground. Other solutions rely on vision to estimate the 3-D position and orientation of the aircraft with respect to the runway and then apply a traditional control scheme much in the same way as with an Instrument Landing System (ILS) or a differential GPS system. To this end, a reliable computer generated model of the runway is required and extreme care in camera calibration [4, 5].
More recently, Image-based Visual Servo solutions, which use directly image measure-ments in the control laws, have also been proposed [6, 7]. This work builds on the same principles to provide a cohesive solution that tackles the first three landing phases up until touchdown. We consider a similar guidance dynamics, where the airspeed is regulated to a constant value and the virtual force input defines references to be tracked by the high-gain inner-loop control system. For the guidance control law, a proportional-derivative structure is adopted.
For the alignment and glide phases, the position error is encoded in line features, represented by the so-called bi-normalized Pl¨ucker coordinates [8]. For alignment, these are extracted from images of the side edges of the runway. For the glide phase, virtual line features are derived from the images of the front corners of the runway and the desired glide slope angle. The main novelty is the use of optical flow measurements instead of airspeed measurements for the derivative term, which is inspired by the work in [9].
1This was partially funded by the Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT) under the NETSys
Doctoral Program and the FCT Investigator Program IF/00921/2013 and by the Macao Science and Technology, Development Fund under Grant FDCT/048/2014/A1.
Using optical flow for the velocity-like term eliminates the need for including a crosswind estimator and considerably simplifies the control laws. As a second contribution, we adopt a different approach for stability present an alternative proof of stability, which relies on writing the kinematics in terms of the position error instead of the visual error and has the advantage of imposing less restrictive conditions on the tuneable gains. A technical result is also provided, which highlights the fact that the visual error is a passive function of the position error.
For the flare phase, images of the side edges of the runway are again used but this time in a modified form to provide a direct position error term, simply scaled by the height above the runway. Using again the translational optical flow, we obtain a velocity term, which is also scaled by the height above the runway. No explicit separation between the control laws for horizontal alignment and touchdown is required and the need for a crosswind estimator is also eliminated. Simulation results illustrate the effectiveness of the proposed control solution for the three phases of the landing maneuver.
References
[1] D. Foyle, A. Goodman, and B. Hooey (2003). Nasa aviation safety program con-ference on human performance modeling of approach and landing with augmented displays. National Aeronautics and Space Administration.
[2] D. B. Barber, S. R. Griffiths, T. W. McLain, and R. W. Beard (2007). Autonomous landing of miniature aerial vehicles. Journal of Aerospace Computing, Information, and Communication. 4, no. 5, 770–784.
[3] S. Thurrowgood, R. J. D. Moore, D. Soccol, M. Knight, and M. V. Srinivasan (2014). A biologically inspired, vision-based guidance system for automatic landing of a fixed-wing aircraft. Journal of Field Robotics, 31, no. 4, 699–727.
[4] M. K. Kaiser, N. Gans, and W. Dixon (2010). Vision-based estimation for guidance, navigation, and control of an aerial vehicle. IEEE Transactions on Aerospace and Electronic Systems, 46, no. 3, 1064–1077.
[5] M. Laiacker, K. Kondak, M. Schwarzbach, and T. Muskardin (2013). Vision aided automatic landing system for fixed wing uav. 2013 IEEE/RSJ International Con-ference on Intelligent Robots and Systems, 2971–2976.
[6] F. Le Bras, T. Hamel, R. Mahony, C. Barat, and J. Thadasack (2014). Approach maneuvers for autonomous landing using visual servo control. IEEE Transactions on Aerospace and Electronic Systems, 50, no. 2,1051–1065.
[7] P. Serra, R. Cunha, T. Hamel, C. Silvestre, and F. Le Bras (2015). Nonlinear image-based visual servo controller for the flare maneuver of fixed-wing aircraft using optical flow. Control Systems Technology, IEEE Transactions on, 23, no. 2, 570–583.
[8] R. Mahony and T. Hamel (2005). Image-based visual servo control of aerial robotic systems using linear image features. IEEE Transactions on Robotics, 21, no. 2, 227– 239.
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[9] L. Rosa, T. Hamel, R. Mahony, and C. Samson (2014). Optical-flow based strategies for landing vtol uavs in cluttered environments. IFAC Proceedings Volumes, 47, no. 3, 3176–3183.
Active piezoelectric vibration control for a
Timoshenko beam
Kenan Yildirim
Mus Alparslan University, Turkey [email protected]
In the present study, the vibration control problem for the Timoshenko beam modeled as an equation including the derivatives of the state variable with respect to the time variable at the fourth order is considered and solved. Time dependent optimal control voltage function is derived by means of a maximum principle. Numerical results obtained by MATLAB are given in table and graphical forms to show the validity of the piezoelec-tric vibration control of the beam. It can be concluded from numerical results that the proposed vibration control algorithm can be applied for the other Timoshenko beams, which have different structural properties, for the damping out the undesired vibrations in the structure.
EPCO 2017
Portuguese Meeting on Optimal Control
SPEAKERS AND POSTER PRESENTERS
Andrea Alessandretti, 25 Ugo Boscain, 3 Nuno M. Brites, 19 Margarida Camarinha, 5 M. Margarida A. Ferreira, 6 Joana Fonseca, 26
Fernando A.C.C. Fontes, 9
Jo˜ao N.C. Gon¸calves, 27 Manuel Guerra, 4
Jo˜ao Henriques, 10
Jos´e Manuel Igreja, 22
Dmitri Karamzin, 16 Olena Kostylenko, 28
Ana P. Lemos-Pai˜ao, 7 Fernando Lobo Pereira, 18 Sofia O. Lopes, 14
Pedro Louren¸co, 20
Miguel Oliveira, 17
Lu´ıs Tiago Paiva, 30
M. do Ros´ario de Pinho, 15
Nedjem Eddine Ramdani, 31 Paulo Rebelo, 32
Lu´ıs A.C. Roque, 33 Silv´erio Rosa, 13
Andrey Sarychev, 1 Cristiana J. Silva, 34
Zhiqi Tang, 35
Delfim F.M. Torres, 8
AUTHORS
A. Pedro Aguiar, 25 Andrea Alessandretti, 25 Helena Alves, 13 Pedro Batista, 20 Anthony Bloch, 5 Ugo Boscain, 3 Carlos A. Braumann, 19 Nuno M. Brites, 19 Margarida Camarinha, 5 Pedro G. Carvalho, 13 Leonardo Colombo, 5 Rita Cunha, 35Lu´ıs E¸ca, 10
Ant´onio Falc˜ao, 10
M. Margarida A. Ferreira, 6 Joana Fonseca, 26
Dalila B.M.M. Fontes, 33
Fernando A.C.C. Fontes, 9, 14, 30, 33
Lu´ıs Gato, 10
Jo˜ao N.C. Gon¸calves, 27 Manuel Guerra, 4
Tarek Hamel, 35 Jo˜ao Henriques, 10
Jos´e Manuel Igreja, 22
Dmitri Karamzin, 16 Olena Kostylenko, 28
Jo˜ao M. Lemos, 10, 22 Ana P. Lemos-Pai˜ao, 7
Fernando Lobo Pereira, 16, 18 Sofia O. Lopes, 14
Pedro Louren¸co, 20
Abdelaziz Mennouni, 31 M. Teresa T. Monteiro, 27
Miguel Oliveira, 17 Paulo Oliveira, 20
Lu´ıs TiagoPaiva, 9
Lu´ıs Tiago Paiva, 30
M. do Ros´ario de Pinho, 6, 15, 26
Nedjem Eddine Ramdani, 31 Paulo Rebelo, 13, 32
Helena Sofia Rodrigues, 27, 28 Lu´ıs A.C. Roque, 33
Silv´erio Rosa, 13, 32
Andrey Sarychev, 1 C´esar Silva, 13, 32 Cristiana J. Silva, 7, 8, 34 Carlos Silvestre, 20, 35 Gueorgui Smirnov, 6, 17 Zhiqi Tang, 35 Delfim F.M. Torres, 7, 8, 28, 34 Kenan Yildirim, 38 Hasnaa Zidani, 15
