Outline ANU Workshop on Systems and Control Conservation laws and invariance principles in networked control systems Zhiyong Sun The Australian National University, Canberra, Australia 1
Content 1. Background and preliminary 2. Conservation (and decay) laws in distributed systems 3. Dimensional-invariance principles 4. Conclusions 2
Starting examples: formation control Potential function: p i : agent i s position; d k_ij : the desired distance between agent i and j. Control law: 3
Typical system equations A sample of references R Olfati-Saber, RM Murray CDC 2002; DV Dimarogonas, KH Johansson CDC 2008, ACC 2009, Automatica 2010; J Cortés Automatica 2009; L Krick, ME Broucke, BA Francis IJC 2009; F Dörfler, B Francis ECC 2009, TAC 2010; M Cao, AS Morse et al. CDC 08, CIS 2011; BDO Anderson, TH Summers, S Dasgupta, et. al. CDC 2009, NecSys 2010, AUCC 2011; KK Oh, HS Ahn ACC 2011, Automatica 2011, IJRNC 2014, Automatica 2015; YP Tian, Q Wang Automatica 2013; BDO Anderson, U Helmke SIAM-SICON 2014; X Cai, M de Queiroz IEEE CST 2014 ASME-DSMC 2014; X Chen, RW Brockett CDC 2014, ACC 2015;... 4
Background and motivation Invariance example: formation centroid The center of the mass of the formation (i.e., formation centroid position) is stationary. Questions: More invariance properties in networked systems Extensions to more general distributed control systems 5
Alternative system equations 6 Potential function: p i : agent i s position; p i - p j : relative position between agents i and j; d k_ij : the desired distance between agents i and j. Control law:
Background and motivation In this talk 7 Several invariance/conservation principles in general networked systems Conservation laws of linear/angular momentum Dimensional-invariance principles in networked systems
Content 1. Background and preliminary 2. Conservation (and decay) laws in networked control systems 3. Dimensional-invariance principles 4. Conclusions 8
Background and motivation Insights from Noether s theorem 9 Every differentiable symmetry of the action of a physical system has a corresponding conservation law.
Background and motivation Symmetry in distributed/networked systems 10 The invariance of a predefined potential function under certain group actions. Infinitesimal generator corresponding to the group action along each agent s evolution. The undirected graph topology modelling the interactions between neighbouring agents. Gradient-based controllers for a multi-agent coordination control Example: Translation invariance and rotation invariance. Gradient descent control law:
Background and motivation General rules: Conservation law I 11 Consider a potential function denoted by V(p 1, p 2,..., p n ) V is invariant under translation action, A distributed coordination system evolving as a gradient descent flow of V with an underlying undirected interaction graph. Then the linear momentum of the distributed coordination system is conserved and is zero. Conservation of linear momentum
Background and motivation General rules: Conservation law II Consider a potential function denoted by V(p 1, p 2,..., p n ) V is a function of relative vectors (i.e., p i p j with (i, j) Edge set) V is invariant under rotation action A distributed coordination system evolving as a gradient descent flow of V with an underlying undirected interaction graph. Then the angular momentum of the distributed coordination system is conserved and is zero. Conservation of angular momentum 12
Background and motivation Extensions to double-integrator models p i : agent i s position; v i : agent i s velocity; Coordination function Graph Laplacian Conservation/decay laws of linear momentum and angular momentum Conservation law: The overall linear momentum is conserved. Decay law: The angular momentum decays exponentially fast to zero. 13
Background and motivation Application example: formation steering control The linear momentum (or the evolution of formation centroid position) could be used as an anchor for the steering control. The formation angular momentum is associated with attitude control or re-orientating of the whole formation. Distributed maneuvering control for rotational and translational formations de Marina, H. G., Jayawardhana, B., & Cao, M. (2016). Distributed rotational and translational maneuvering of rigid formations and their applications. IEEE Transactions on Robotics, 32(3), 684-697. 14
Content 1. Background and preliminary 2. Conservation (and decay) laws in distributed systems 3. Dimensional-invariance principles 4. Conclusions 15
Starting examples: Collinear/coplanar issues in formation systems The sets of collinear or coplanar positions are invariant for 2- D or 3-D formation systems. If agents start in a lower dimensional space, then they will always remain in that lower dimensional space. 16
Starting examples: Collinear/coplanar issues in formation systems Conversely, if all agents start at non-collinear (non-coplanar) positions, then their positions will be non-collinear (noncoplanar) at any finite time. 17
18 Questions: Background invariance and motivation principles on space dimensions A universal principle for networked control systems? Conditions on the coupling/interaction terms?
Rank-preserving flow A(t) and B(t) can be time-varying and state-dependent. Uwe Helmke and John B. Moore, Optimization and Dynamical Systems, Chapter 5. Springer, 1994. 19
Example on formation systems: system equations The fundamental equation: A compact form: 20
Matrix differential equations The formation control system: Key point: to obtain a matrix differential equation. Denote The rank of P and Z reflects the dimension of the embedding space for all the agents positions. 21
Main result: dimension-invariance principle Matrix differential equation for P: Stress matrix Matrix differential equation for Z: Theorem: rank(p) and rank(z) are constant for all finite time, along any solution p(t). 22
Extensions of rank-preserving property Topology independent It also holds for switching topology and time-varying topology. Extensions to other networked systems It is not limited to networked formation systems (also holds for other networked control systems, e.g. linear consensus networks). 23
Networked Agent models systems: scalar coupling case (*) where each agent lives in R d, w ij is a scalar (constant or timevarying) coupling weight between agents j and i. The coupling/coefficient weights could be constant, timevarying, or state-dependent; We do not require w ij = w ji, i.e., the coupling weight could be asymmetric. Theorem: The solutions of the coupled dynamical system (*) have the dimensional-invariance principle. 24
Networked Agent models systems: scalar coupling case (*) Theorem: The solutions of the coupled dynamical system (*) have the dimensional-invariance principle. Dimensional-invariance principle Subspace-preserving principle 25
Examples: networked systems with dimensional invariance (scalar couplings) 26 A universal principle that applies for many typical networked systems.
Networked systems: matrix coupling case 27 (**) where each agent lives in R d, W ij R d d is a matrix (constant, time-varying or state-dependent) coupling weight between agents j and i. The coupled dynamical systems (**) have the dimensionalinvariance principle if and only if the coefficient and coupling matrices W ij satisfy the following condition for some matrix A R d d and scalars {b ij }.
Examples: networked systems with dimensional invariance (matrix coupling) 28 Matrix coupling case: conditions apply
Examples: networked systems living in 2D Agents positions cannot escape to a higher dimensional space, nor can they shrink to a lower dimensional space. 29
Examples: networked systems living in 3D Applications: Equilibrium analysis for networked systems; Transient behaviour on agents position evolutions 30
Content 1. Background and preliminary 2. Conservation (and decay) laws in distributed systems 3. Dimensional-invariance principles 4. Conclusions 31
Conclusions Conservation laws in distributed control systems Invariance and symmetry lead to conservations of linear/angular momentums. Applications to formation steering/maneuvering control. Dimensional invariance principles in networked systems Dimensions of spaces spanned by agents positions are invariant. Applications to transient behavior analysis and equilibrium analysis. 32
Related papers Zhiyong Sun, Uwe Helmke, and Brian D. O. Anderson. Rigid formation shape control in general dimensions: an invariance principle and open problems. CDC 15 Zhiyong Sun, and Changbin Yu. Dimensional-invariance principles in coupled dynamical systems. Submitted to IEEE Transactions on Automatic Control, in revision, 2017 Zhiyong Sun, Shaoshuai Mou, Brian D. O. Anderson, and Changbin Yu. Conservation and decay laws in distributed coordination control systems. Automatica, 87, 1-7, 2018. 33
34 Thanks Questions? E-mail: zhiyong.sun@anu.edu.au Web: https://sites.google.com/view/zhiyong-sun