http://orcid.org/0000-0003-4525-4656
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ABSTRACT
Dynamic optimisation, with a particular focus on optimal control and nonzero-sum differential games, is considered. For nonlinear systems solutions sought via the dynamic programming strategy are inevitably characterised by partial differential equations (PDEs) which are often difficult to solve. A detailed overview of a control design framework which enables the systematic construction of approximate solutions for optimal control problems and differential games without requiring the explicit solution of any PDE is provided along with a novel design of a nonlinear control gain aimed at improving the ‘level of approximation’ achieved. Multi-agent systems are considered as a possible application of the theory.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Mario Sassano http://orcid.org/0000-0003-4525-4656
Thulasi Mylvaganam http://orcid.org/0000-0003-1993-4607
Alessandro Astolfi http://orcid.org/0000-0002-4331-454X
Notes
1. To further substantiate item (ii) of Definition 2.1, such condition, for instance in the case of optimal control problems dealt with in Section 3, is equivalent to the requirement that is the solution of the Algebraic Riccati Equation (ARE) associated to a LQR problem.
2. The interested reader is referred, for instance, to the literature concerning SDRE for optimal control problems reviewed, for instance, in Çimen (Citation2008).
3. In Section 4 we consider nonzero-sum differential games for which condition (ii) is equivalent to the condition that ,
, are the solution of a system of coupled algebraic Riccati equations (see, e.g. Basar & Olsder, Citation1999; Starr & Ho, Citation1969b)
4. Existence of such functions is guaranteed by the property that and
for
, respectively, while continuity is ensured by Hadamard's Lemma, see Nestruev (Citation2002).
5. The requirement that q is positive definite can be replaced by positive semidefiniteness and zero-state detectability. Namely, Assumption 3.2 can be replaced by the properties that q is positive semidefinite and that the system (Equation17(17) ) with output
is zero-state detectable, i.e.
and
for all
imply
.
6. While it is not made explicitly clear by the notation, note that the cost depends on the dynamics of ξ, i.e. on the function α, as well as on the function β.
7. Note that Theorem 3.2 is stated under the same conditions as those in Theorem 3.1, hence the conclusions of Theorem 3.1 hold, including the existence and the role of the constant .
8. The requirement regarding positive definiteness can be replaced by positive semidefinitess and zero-state detectability, similarly to what has been done in the consideration of the optimal control problem. Namely, Assumption 4.2 can be replaced by the assumptions that is positive semidefinite and that the system (Equation30
(30) ) with output
is zero-state detectable.
9. While we focus on the most common solution concept, namely the Nash equilibrium, the results can be applied to different solution concepts (see, e.g. Mylvaganam & Astolfi, Citation2014b for Stackelberg solutions).
10. Note that (as in the case of the optimal control problem) the costs ,
, depend on the dynamics of ξ, i.e. on the function α, as well as on the functions
,
.
11. Given an ellipse , the function
can be computed by straightforward computations, yielding
, with
and
, where a and b denote the major and minor semiaxis of the ellipse, respectively, and
is the rotation of the major semiaxis relative to
.
12. A path is defined as a continuously differentiable locus of points of connecting two points.
Additional information
Funding
Notes on contributors
Mario Sassano
Mario Sassano was born in Rome, Italy, in 1985. He received the B.S degree in Automation Systems Engineering and the M.S degree in Systems and Control Engineering from the University of Rome “La Sapienza”, Italy, in 2006 and 2008, respectively. In 2012 he was awarded a Ph.D. degree by Imperial College London, UK, where he had been a Research Assistant in the Department of Electrical and Electronic Engineering (2009–2012). Currently he is an Assistant Professor at the University of Rome “Tor Vergata”, Italy. His research interests are focussed on nonlinear observer design, optimal control and differential game theory with applications to mechatronical systems and output regulation for hybrid systems. He is Associate Editor of the IEEE CSS Conference Editorial Board and of the EUCA Conference Editorial Board.
Thulasi Mylvaganam
Thulasi Mylvaganam was born in Bergen, Norway, in 1988. She received the M.Eng degree in Electrical and Electronic Engineering from Imperial College London, UK, in 2010. She completed her Ph.D. in the Department of Electrical and Electronic Engineering, Imperial College London, UK, where she was a Research Associate from 2014–2016. From 2016–2017 she was a Research Fellow at the Department of Aeronautics, Imperial College London, UK, where she has been a Lecturer (Assistant Professor) since 2017. Her current research interests include nonlinear control design, differential game theory and distributed control with applications to multi-agent systems. She is Associate Editor of the IEEE CSS Conference Editorial Board.
Alessandro Astolfi
Alessandro Astolfi was born in Rome, Italy, in 1967. He graduated in electrical engineering from the University of Rome in 1991. In 1992 he joined ETH-Zurich where he obtained a M.Sc. in Information Theory in 1995 and the Ph.D. degree with Medal of Honor in 1995 with a thesis on discontinuous stabilisation of nonholonomic systems. In 1996 he was awarded a Ph.D. from the University of Rome “La Sapienza” for his work on nonlinear robust control. Since 1996 he has been with the Electrical and Electronic Engineering Department of Imperial College London, London (UK), where he is currently Professor of Nonlinear Control Theory and Head of the Control and Power Group. From 1998 to 2003 he was also an Associate Professor at the Dept. of Electronics and Information of the Politecnico of Milano. Since 2005 he has also been a Professor at Dipartimento di Ingegneria Civile e Ingegneria Informatica, University of Rome Tor Vergata. He has been a visiting lecturer in “Nonlinear Control” in several universities, including ETH-Zurich (1995–1996); Terza University of Rome (1996); Rice University, Houston (1999); Kepler University, Linz (2000); SUPELEC, Paris (2001), Northeastern University (2013).
His research interests are focussed on mathematical control theory and control applications, with special emphasis for the problems of discontinuous stabilisation, robust and adaptive control, observer design and model reduction. He is the author of more than 150 journal papers, of 30 book chapters and of over 240 papers in refereed conference proceedings. He is the author (with D. Karagiannis and R. Ortega) of the monograph “Nonlinear and Adaptive Control with Applications” (Springer-Verlag).
He is the recipient of the IEEE CSS A. Ruberti Young Researcher Prize (2007), the IEEE RAS Googol Best New Application Paper Award (2009), the IEEE CSS George S. Axelby Outstanding Paper Award (2012), the Automatica Best Paper Award (2017). He is a “Distinguished Member” of the IEEE CSS, IEEE Fellow and IFAC Fellow.
He served as Associate Editor for Automatica, Systems and Control Letters, the IEEE Trans. on Automatic Control, the International Journal of Control, the European Journal of Control and the Journal of the Franklin Institute; as Area Editor for the Int. J. of Adaptive Control and Signal Processing; as Senior Editor for the IEEE Trans. on Automatic Control; and as Editor-in-Chief for the European Journal of Control. He is currently Editor-in-Chief of the IEEE Trans. on Automatic Control. He served as Chair of the IEEE CSS Conference Editorial Board (2010–2017) and in the IPC of several international conferences. He was a Member of the IEEE Fellow Committee (2016).