Nonlinear Port-Hamiltonian Systems ConFlex Network Meeting, July 3, 2020 Arjan van der Schaft in collaboration with Bernhard Maschke, Romeo Ortega, · · · Bernoulli Institute for Mathematics, CS & AI Jan C. Willems Center for Systems and Control University of Groningen, the Netherlands Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 1 / 100
Most of the talk can be found in AvdS, Dimitri Jeltsema: Port-Hamiltonian Systems Theory: An Introductory Overview, 2014 pdf available from my home page: www.math.rug.nl/˜arjan and in Chapters 6, 7 of AvdS: L 2 -Gain and Passivity Techniques in Nonlinear Control, 3rd ed. 2017 Further background: Modeling and Control of Complex Physical Systems; the Port-Hamiltonian Approach, GeoPleX consortium, Springer, 2009 Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 2 / 100
Introduction • Port-Hamiltonian systems theory as systematic framework for multi-physics systems: modeling for control • Is based on viewing energy and power as ’lingua franca’ between different physical domains • Combines classical Hamiltonian dynamics with network structure, including energy-dissipation and interaction with environment • Unifies lumped-parameter and distributed-parameter physical systems • Bridges the gap between modeling and control. • Identification of underlying physical structure in the mathematical model provides powerful tools for analysis, simulation and control Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 3 / 100
Outline 1 Port-Hamiltonian systems 2 Port-Hamiltonian formulation of network dynamics 3 Properties of port-Hamiltonian systems 4 Distributed-parameter port-Hamiltonian systems 5 Including thermodynamics ? 6 Passivity-based control of port-Hamiltonian systems 7 IDA Passivity-based control 8 New control paradigms emerging Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 4 / 100
The basic picture e S e R storage dissipation D f S f R e P f P Figure: Port-Hamiltonian system Every physical system that is modeled in this way defines a port-Hamiltonian system. For control purposes ’any’ physical system can be modeled this way. Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 5 / 100
Port-based modeling is based on viewing the physical system as interconnection of ideal basic elements, linked by energy flow. Linking done via conjugate vector pairs of flow variables f ∈ R k and effort variables e ∈ R k , with product e T f equal to power. In some cases (e.g., 3D mechanical systems) f ∈ F (e.g., linear space of twists) and e ∈ E = F ∗ (e.g., wrenches), with product defined by pairing. Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 6 / 100
Basic elements: • (1) Energy-storing elements ˙ = − f x ∂ H e = ∂ x ( x ) , H energy function and hence d dt H = e T f . • (2) Energy-dissipating elements: e T f ≤ 0 R ( f , e ) = 0 , • (3) Energy-routing elements: - generalized transformers: � f 1 � � e 1 � f 1 = Mf 2 , e 2 = − M T e 1 f = , e = , f 2 e 2 - generalized gyrators: J = − J T f = Je , Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 7 / 100
• (4) Ideal interconnection and constraint equations: e 1 = e 2 = · · · = e k , f 1 + f 2 + · · · + f k = 0 or f 1 = f 2 = · · · = f k , e 1 + e 2 + · · · + e k = 0 f = 0 , or e = 0 (3) and (4) share the following two properties: Power-conservation e T f = e 1 f 1 + e 2 f 2 + · · · + e k f k = 0 , and k linear and independent equations. Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 8 / 100
From energy-routing elements and interconnection equations to Dirac structures This means energy-routing elements and interconnection and constraint equations have following two properties in common. Described by linear equations: f , e ∈ R k Ff + Ee = 0 , satisfying e T f = e 1 f 1 + e 2 f 2 + · · · + e k f k = 0 and � � rank = k F E Energy-routing elements (3) and interconnection and constraint equations (4) are grouped into one geometric object: the linear space of flow and effort variables satisfying all equations, called Dirac structure. Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 9 / 100
Definition of Dirac structures Definition A (constant) Dirac structure is a subspace (typically F = R k = E ) D ⊂ F × E such that (i) e T f = 0 for all ( f , e ) ∈ D , (ii) dim D = dim F . Example: for any skew-symmetric map J : E → F its graph { ( f , e ) ∈ F × E | f = Je } is Dirac structure. Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 10 / 100
Alternative definition; e.g., for infinite-dimensional case Symmetrization of power e T f leads to indefinite bilinear form ≪ , ≫ on F × E : e T a f b + e T ≪ ( f a , e a ) , ( f b , e b ) ≫ := b f a , ( f a , e a ) , ( f b , e b ) ∈ F × E Definition A (constant) Dirac structure is subspace D ⊂ F × E such that D = D ⊥ ⊥ , where ⊥ ⊥ denotes orthogonal companion with respect to ≪ , ≫ . Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 11 / 100
Coordinate-free definition of pH systems e S e R storage dissipation D f S f R e P f P Start from the Dirac structure, defined as subspace of space of all flows f = ( f S , f R , f P ) and all efforts e = ( e S , e R , e P ) Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 12 / 100
Constitutive relations: ’Close’ the energy-storing ports of D by relations ∂ H − ˙ x = f S , ∂ x ( x ) = e S and the energy-dissipating ports by R ( f R , e R ) = 0 This leads to the port-Hamiltonian system x ( t ) , f R ( t ) , f P ( t ) , ∂ H ( − ˙ ∂ x ( x ( t )) , e R ( t ) , e P ( t )) ∈ D t ∈ R R ( f R ( t ) , e R ( t )) = 0 N.B.: in general in differential-algebraic equations (DAE) format. Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 13 / 100
Example (The ubiquitous mass-spring system) Two energy-storage elements: 2 kq 2 (potential energy) • Spring Hamiltonian H s ( q ) = 1 q ˙ = − f s = velocity dH s e s = dq ( q ) = kq = force 2 m p 2 (kinetic energy) 1 • Mass Hamiltonian H m ( p ) = p ˙ = − f m = force dp ( p ) = p dH m = = velocity e m m Note the slight difference with ’classical’ mechanical modeling, where one starts from identifying q as the position of mass, defining the velocity ˙ q and momentum p = m ˙ q . Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 14 / 100
Example (Mass-spring system cont’d) Dirac structure linking flows f s , f m , F and efforts e s , e m , v : f s = − e m = − v , f m = e s − F Power-conserving since f s e s + f m e m + vF = 0. Yields pH system � � � � � ∂ H � � � ∂ q ( q , p ) q ˙ 0 1 0 = + F ∂ H p ˙ − 1 0 ∂ p ( q , p ) 1 � ∂ H � ∂ q ( q , p ) � � v = 0 1 ∂ H ∂ p ( q , p ) with H ( q , p ) = H s ( q ) + H m ( p ) Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 15 / 100
Example (Magnetically levitated ball) ∂ H ∂ q ( q , p , φ ) q ˙ 0 1 0 0 I = ∂ H = ∂ H V , ˙ − 1 0 0 ∂ p ( q , p , φ ) + 0 ∂ϕ ( q , p , φ ) p ϕ ˙ 0 0 − R 1 ∂ H ∂ϕ ( q , p , φ ) Coupling electrical/mechanical domain via Hamiltonian H ( q , p , φ ) H ( q , p , ϕ ) = mgq + p 2 ϕ 2 2 m + 2 L ( q ) Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 16 / 100
Example (Synchronous machine) I 3 0 31 0 31 ˙ ∂ H ψ s − R s 0 3 0 31 0 31 ∂ψ s 1 V s ˙ ∂ H 0 3 − R r 0 31 0 31 ψ r 0 3 0 0 31 ∂ψ r = + V f 0 ∂ H 0 13 0 13 − d − 1 p ˙ ∂ p τ 0 13 0 1 ˙ ∂ H 0 13 0 13 1 0 θ 0 13 0 0 ∂θ ∂ H ∂ψ s 0 3 0 31 0 31 I s I 3 ∂ H � � ∂ψ r I f = 0 13 1 0 0 0 0 , R s > 0 , R f > 0 , d > 0 ∂ H ω 0 13 0 13 1 0 ∂ p ∂ H ∂θ � ψ s � � � H ( ψ s , ψ r , p , θ ) = 1 + 1 L − 1 ( θ ) 2 J p 2 ψ T ψ T s r 2 ψ r Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 17 / 100
Example (DC motor) R L + ω V J K τ I b _ 6 interconnected subsystems: • 2 energy-storing elements: inductor L with state ϕ (flux), and rotational inertia J with state p (angular momentum); • 2 energy-dissipating elements: resistor R and friction b ; • gyrator K ; • voltage source V . Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian systems 18 / 100
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