Synchronization and Motion Control Topics

1. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Topics in Synchronization and Motion Control Ehsan Peymani Norwegian University of Science and Technology 1 August, 2013 Trondheim, Norway Ehsan Peymani Topics in Synchronization and Motion Control

2. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Outline 1 Background 2 Synchronization in the Presence of External Disturbances 3 Speed-varying Path-following for Marine Craft 4 Motion Control using Analytical Mechanics 5 Concluding Remarks and Future Work Ehsan Peymani Topics in Synchronization and Motion Control

4. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Background In January 2009, I was invited to join CeSOS at NTNU. I started my PhD studies in August 2009 after completing my MSc in control engineering in Iran. I worked on “Motion Control using Analytical Mechanics” from October 2009 to December 2011. Professor Ali Saberi invited me to join his research group at WSU for 2012 . During my visit at WSU Almost synchronization (Synchronization in the presence of external disturbances) Speed-varying path-following I came back Norway in January 2013, and started to write my Thesis. Ehsan Peymani Topics in Synchronization and Motion Control

8. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Outline 1 Background 2 Synchronization in the Presence of External Disturbances Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents 3 Speed-varying Path-following for Marine Craft 4 Motion Control using Analytical Mechanics 5 Concluding Remarks and Future Work Ehsan Peymani Topics in Synchronization and Motion Control

9. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Synchronization in the Presence of External Disturbances 1 What kind of multi-agent systems is considered in this thesis? 2 What is the problem of synchronization? 3 How can we obtain an accurate synchronization when agents are subject to external disturbances? Ehsan Peymani Topics in Synchronization and Motion Control

10. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Multi-agent Systems A multi-agent system is referred to a network of N multi-input multi-output systems described by LTI models as Agent i : ˙xi = Aixi + Biui + Giwi yi = Cixi i ∈ S {1, 2, · · · , N} xi ∈ Rni is the state ui ∈ Rmi is the input yi ∈ Rp is the output wi ∈ Rωi is the external disturbances where limT→∞ 1 2T T −T wT i widt < ∞ Ehsan Peymani Topics in Synchronization and Motion Control

11. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Multi-agent Systems Homogeneous Multi-agent Systems if all agents are identical. Agent i : ˙xi = Axi + Bui + Gwi yi = Cxi Heterogeneous Multi-agent Systems if agents are non-identical. Ehsan Peymani Topics in Synchronization and Motion Control

12. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Multi-agent Systems – ‘Network Measurements’ Agents are allowed to exchange information according to the network’s communication topology which is described by a weighted directed graph L, with no self loops, where each node corresponds to an agent. 2446 H.F. Grip et al. / Automatica 48 (2012) 2444–2453 1 2 3 4 9876 5 10 2 1.5 2.4 2 1.3 3 2.5 1.7 1.4 2.7 1.5 1 1 0.8 Fig. 1. The depicted digraph contains multiple directed spanning trees, rooted at nodes 2, 3, 4, 8, and 9. One of these, with root node 2, is illustrated by bold arrows. where ηj ∈ Rp is a variable produced internally by agent j as part of the controller. This variable will be specified as we proceed with to zero, where the d  ˙xi ˙xK  =  Ai 0 0 AK   ei =  Ci −CK   xi xK The system (2) is i available to agent i ing ei converge to ze output-regulation m 2000). But alas, the o To achieve our objeEhsan Peymani Topics in Synchronization and Motion Control

13. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Multi-agent Systems – ‘Network Measurements’ Agents are allowed to exchange information according to the network’s communication topology which is described by a weighted directed graph L, with no self loops, where each node corresponds to an agent. 2446 H.F. Grip et al. / Automatica 48 (2012) 2444–2453 1 2 3 4 9876 5 10 2 1.5 2.4 2 1.3 3 2.5 1.7 1.4 2.7 1.5 1 1 0.8 Fig. 1. The depicted digraph contains multiple directed spanning trees, rooted at nodes 2, 3, 4, 8, and 9. One of these, with root node 2, is illustrated by bold arrows. where ηj ∈ Rp is a variable produced internally by agent j as part of the controller. This variable will be specified as we proceed with to zero, where the d  ˙xi ˙xK  =  Ai 0 0 AK   ei =  Ci −CK   xi xK The system (2) is i available to agent i ing ei converge to ze output-regulation m 2000). But alas, the o To achieve our objeEhsan Peymani Topics in Synchronization and Motion Control

14. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Multi-agent Systems – ‘Network Measurements’ The communication graph is associated with the weighted adjacency matrix AL = [aij] aij ≥ 0 is the weight of the edge from node j to node i. the Laplacian matrix L = [lij] lii = n j=1,j=i aij; lij = −aij for i = j. The sum of the elements of each row of the Laplacian matrix is zero. Ehsan Peymani Topics in Synchronization and Motion Control

15. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Multi-agent Systems – ‘Network Measurements’ Network Measurements: the quantities which are transmitted over the network. 1 Multi-agent systems with partial state coupling: ζi = N j=1 aij(yi − yj) ⇐⇒ ζi = N j=1 lijyj 2 Additional information: ˆζi = N j=1 aij(ηi − ηj) ⇐⇒ ˆζi = N j=1 lijηj where ηj ∈ Rp depends on the state of protocol j, and will be speciﬁed later. Ehsan Peymani Topics in Synchronization and Motion Control

18. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Multi-agent Systems – ‘Local Measurements’ Local Measurements: the quantities which are measured locally. 1 A network of introspective agents: ym,i = Cm,ixi Agent i has partial knowledge of its own state. 2 A network of non-introspective agents, no self-measurements are available. Ehsan Peymani Topics in Synchronization and Motion Control

21. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Multi-agent Systems – Summary Agent i : ˙xi = Aixi + Biui + Giwi yi = Cixi ζi = N j=1 aij(yi − yj) ⇐⇒ ζi = N j=1 lijyj ˆζi = N j=1 aij(ηi − ηj) ⇐⇒ ˆζi = N j=1 lijηj introspective agents ym,i = Cm,ixi non-introspective agents No local information. Ehsan Peymani Topics in Synchronization and Motion Control

22. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Synchronization Synchronization is to reach an agreement on a certain quantity of interest, which is based on the state of agents. State Synchronization lim t→∞ (xi − xj) = 0 ∀i ∈ {1, 2, · · · , N} Output Synchronization lim t→∞ (yi − yj) = 0 ∀i ∈ {1, 2, · · · , N} Ehsan Peymani Topics in Synchronization and Motion Control

23. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Synchronization – Applications synchronization of coupled oscillators distributed robotics such as formation ﬂying, ﬂocking and swarming distributed sensor fusion in mobile sensor networks distributed and parallel computing power networks quantum networks networked economics biological synchronization Ehsan Peymani Topics in Synchronization and Motion Control

24. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Synchronization – Literature Review Much of attention has been given to networks of agents in the absence of external disturbances. Synchronization in the presence of external disturbances: Articles analyzing the disturbance attenuation properties of distributed protocols without proposing any speciﬁc design that achieves synchronization with a desired degree of accuracy. Articles ensuring that the H∞ norm of the transfer function from disturbance to either the output of individual agents or a linear combination of synchronization errors is less than a given γ > 0 with LMI-based methods. No discussion on the solvability of the problem for the given γ. Limited to identical agents. Unreasonable choice of transfer function. Ehsan Peymani Topics in Synchronization and Motion Control

25. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Synchronization in the presence of External Disturbances This thesis intends to present a systematic approach to design synchronization protocols for homogeneous/heterogeneous networks of introspective/non-introspective, general linear agents, coupled with partial state information so that :::::::::::::: synchronization::: is :::::::: achieved:::: with:::: any::::::: desired::::::: degree::: of :::::::: accuracy. This concept is called “almost synchronization”. Ehsan Peymani Topics in Synchronization and Motion Control

26. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents H∞ Almost Synchronization In this thesis: The measure of the accuracy of synchronization is given by the H∞ norm of the transfer function from disturbances to the synchronization errors. The problem is called “H∞ almost synchronization”. Ehsan Peymani Topics in Synchronization and Motion Control

27. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents H∞ Almost Synchronization Deﬁne: w col {wi}, i ∈ S = {1, 2, · · · , N} ei,j yi − yj ∀i, j ∈ S, i > j The column vector of all mutual disagreements is denoted e. Consider: e = Twe(s)w. Ehsan Peymani Topics in Synchronization and Motion Control

28. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents H∞ Almost Synchronization Definition: H∞ almost synchronization Consider a multi-agent system with a communication topology L. Given a set of network graphs G and any γ > 0, the “H∞ almost synchronization” problem is to find, if possible, a linear time-invariant dynamic protocol such that, for any L ∈ G, the closed-loop transfer function from w to e satisfies Twe(s) ∞ < γ. Ehsan Peymani Topics in Synchronization and Motion Control

29. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents H∞ Almost Synchronization Question 1 Considering networks of homogeneous/heterogeneous agents, when and how H∞ almost synchronization is solvable? 1 Heterogeneous Networks of Introspective Agents 2 Homogeneous Networks of Non-introspective Agents Ehsan Peymani Topics in Synchronization and Motion Control

30. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents H∞ Almost Synchronization Question 1 Considering networks of homogeneous/heterogeneous agents, when and how H∞ almost synchronization is solvable? 1 Heterogeneous Networks of Introspective Agents 2 Homogeneous Networks of Non-introspective Agents Ehsan Peymani Topics in Synchronization and Motion Control

31. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents Networks of non-identical agents: each agent is partially aware of its own state. Agent i : ˙xi = Aixi + Biui + Giwi yi = Cixi ζi = N j=1 aij(yi − yj) ˆζi = N j=1 aij(ηi − ηj) ym,i = Cm,ixi Design procedure: Step i) Homogenization Step ii) Protocol development Ehsan Peymani Topics in Synchronization and Motion Control

32. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents Design procedure: Step 1) Homogenization Local dynamic compensators are designed in order to make all agents imitate the desired identical dynamics. Agent  ,   AgentLocal Feedback  ,   ˙xi = Axi + B(Mui + Rxi) + Ed,iwi yi = Cxi Ehsan Peymani Topics in Synchronization and Motion Control

33. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents Design procedure: Step 1) Homogenization Local dynamic compensators are designed in order to make all agents imitate the desired identical dynamics. ˙xi = Axi + B(Mui + Rxi) + Ed,iwi yi = Cxi AgentLocal Feedback  ,   Homogenization process includes 1 Squaring-down compensators; 2 Rank-equalizing compensators; 3 Dynamic feedback law. Ehsan Peymani Topics in Synchronization and Motion Control

34. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents Design procedure: Step 2) Protocol Development According to the homogenized network, for i ∈ S, construct the observer-based protocol ˙ˆxi = A ˆxi + B(Mui + Rˆxi) − −1 K(ζi − ˆζi) ui = −nq M−1 FSˆxi The protocol development is based on the time-scale structure assignment technique rooted in singular perturbation methodology. Ehsan Peymani Topics in Synchronization and Motion Control

35. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents Design procedure: Step 2) Protocol Development According to the homogenized network, for i ∈ S, construct the observer-based protocol ˙ˆxi = A ˆxi + B(Mui + Rˆxi) − −1 K(ζi − ˆζi) ui = −nq M−1 FSˆxi The protocol development is based on the time-scale structure assignment technique rooted in singular perturbation methodology. Ehsan Peymani Topics in Synchronization and Motion Control

36. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents Theorem The H∞ almost synchronization problem is solvable if 1 every agent satisﬁes: 1 (Ai, Bi, Ci) is right-invertible; 2 (Ai, Bi) is stabilizable and (Ai, Ci) is detectable; 3 (Ai, Cm,i) is detectable. 2 the communication graph is a member of the set Gβ, deﬁned as For given β > 0 and integer N0 ≥ 1, Gβ is the set of directed graphs composed of N nodes where N ≤ N0 such that every L ∈ Gβ has a directed spanning tree and the eigenvalues of its Laplacian, denoted λ1, · · · , λN , satisfy Re{λi} > β for λi = 0. Ehsan Peymani Topics in Synchronization and Motion Control

37. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents Theorem Under the stated conditions, there exists a family of linear time-invariant dynamic protocols, parameterized in terms of a tuning parameter ∈ (0, 1], of the form ˙χi = Ai( )χi + Bi( ) col {ζi, ˆζi, ym,i} ¯ui = Ci( )χi + Di( ) col {ζi, ˆζi, ym,i} where χi ∈ Rqi and i ∈ S such that (i) given β > 0, ∃ ∗ 1 ∈ (0, 1] such that, ∀ ∈ (0, ∗ 1], in the absence of disturbance, output synchronization is achieved; i.e. ei,j = yi − yj → 0, as t → ∞. (ii) given γ > 0, ∃ ∗ 2 ∈ (0, ∗ 1] such that, ∀ ∈ (0, ∗ 2], the closed-loop transfer function from w to e satisﬁes Twe(s) ∞ < γ. Ehsan Peymani Topics in Synchronization and Motion Control

38. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents 1 The notion of “H∞ almost regulated synchronization”: to suppress the eﬀect of disturbance on the regulation errors. 2 The notion of “H∞ almost formation”: to synchronize agents while agents maintain their relative outputs as desired. Ehsan Peymani Topics in Synchronization and Motion Control

39. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Heterogeneous Networks of Introspective Agents 1 The notion of “H∞ almost regulated synchronization”: to suppress the eﬀect of disturbance on the regulation errors. 2 The notion of “H∞ almost formation”: to synchronize agents while agents maintain their relative outputs as desired. Ehsan Peymani Topics in Synchronization and Motion Control

40. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Homogeneous Networks of Non-introspective Agents Networks of identical agents: No self-measurements are available. Agent i : ˙xi = Axi + Bui + Gwi yi = Cxi ζi = N j=1 aij(yi − yj) ˆζi = N j=1 aij(ηi − ηj) Design procedure: • contains 8 steps which requires the computation of the Special Coordinate Basis for (A, B, C) and (A, G, C). Ehsan Peymani Topics in Synchronization and Motion Control

41. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Homogeneous Networks of Non-introspective Agents The Special Coordinate Basis is a structural decomposition of linear time-invariant systems which reveals the systems finite and infinite zero structures and the invertibility properties. The Special Coordinate Basis decomposes the system into four separate but interconnected subsystems, namely • Invariant zero dynamics; • Non-right-invertibility dynamics; • Non-left-invertibility dynamics; • Infinite zero dynamics. In the case of introspective agents, agents are shaped into a chain of integrators of one specific order. • How are these two projects different from each other? Ehsan Peymani Topics in Synchronization and Motion Control

42. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Homogeneous Networks of Non-introspective Agents The Special Coordinate Basis is a structural decomposition of linear time-invariant systems which reveals the systems finite and infinite zero structures and the invertibility properties. The Special Coordinate Basis decomposes the system into four separate but interconnected subsystems, namely • Invariant zero dynamics; • Non-right-invertibility dynamics; • Non-left-invertibility dynamics; • Infinite zero dynamics. In the case of introspective agents, agents are shaped into a chain of integrators of one specific order. • How are these two projects different from each other? Ehsan Peymani Topics in Synchronization and Motion Control

43. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Homogeneous Networks of Non-introspective Agents The problem was solved using a novel multiple time-scale structure assignment technique. The necessary and suﬃcient conditions for the solvability of the problem were given in terms of the concepts from the geometric control theory. It was shown that if the problem of H∞ almost disturbance decoupling is solvable for a single agent, the problem of H∞ almost synchronization is solvable for a network of identical agents with any communication topology whose graph is a member of a speciﬁc family of network graphs (Gβ). Ehsan Peymani Topics in Synchronization and Motion Control

46. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Closing Remarks + The following problems are introduced and solved 1 H∞ almost synchronization 2 H∞ almost regulated synchronization 3 H∞ almost formation + The proposed protocols are parameterized in terms of a tuning parameter : 1 The structure of the controllers does not depend on the value of ; 2 The design procedure is not iterative, but one-shot design. 3 The value of can be chosen online using an optimization algorithm; 4 The order of the protocol is ﬁxed for any desired degree of accuracy. 5 The solvability of the problem is discussed. Ehsan Peymani Topics in Synchronization and Motion Control

47. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Multi-agent Systems & Synchronization H∞ Almost Synchronization Heterogeneous Networks of Introspective Agents Homogeneous Networks of Non-introspective Agents Closing Remarks + The following problems are introduced and solved 1 H∞ almost synchronization 2 H∞ almost regulated synchronization 3 H∞ almost formation + The proposed protocols are parameterized in terms of a tuning parameter : 1 The structure of the controllers does not depend on the value of ; 2 The design procedure is not iterative, but one-shot design. 3 The value of can be chosen online using an optimization algorithm; 4 The order of the protocol is ﬁxed for any desired degree of accuracy. 5 The solvability of the problem is discussed. Ehsan Peymani Topics in Synchronization and Motion Control

48. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Outline 1 Background 2 Synchronization in the Presence of External Disturbances 3 Speed-varying Path-following for Marine Craft Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares 4 Motion Control using Analytical Mechanics 5 Concluding Remarks and Future Work Ehsan Peymani Topics in Synchronization and Motion Control

49. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Path-maneuvering Scenario To force a marine craft to reach to a geometric path, and follow it with a desired speed proﬁle. ( ) ( ) ( ) ( ) ( ) ( ) Ehsan Peymani Topics in Synchronization and Motion Control

50. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Path-maneuvering Scenario The general methodology is to decompose path-maneuvering into two tasks: 1 the geometric task of converging to the path; 2 the dynamic task of assigning speed to the marine craft. ( ) ( ) ( ) ( ) ( ) ( ) Ehsan Peymani Topics in Synchronization and Motion Control

51. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Path-maneuvering Scenario The geometric task takes precedence over the dynamic task. The underlying assumption is that these two tasks are independent. Therefore, path-maneuvering controllers are composed of two decoupled controllers: speed controller that uses speed information for feedback; heading controller that uses geometric information and heading information for generating feedback commands. § ¦ ¤ ¥ The speed is not controlled according to the geometric information. The geometric information should inﬂuence the speed control loop. Ehsan Peymani Topics in Synchronization and Motion Control

54. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Speed-varying Path-maneuvering Scenario Question 2 How to incorporate the geometric information in the speed control loop such that the stability of the closed-loop system is preserved and the path-maneuvering objectives are achieved? Two methods are proposed: 1 A nonlinear approach based on backstepping 2 An approach based on the method of least squares Ehsan Peymani Topics in Synchronization and Motion Control

55. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Speed-varying Path-maneuvering Scenario Question 2 How to incorporate the geometric information in the speed control loop such that the stability of the closed-loop system is preserved and the path-maneuvering objectives are achieved? Two methods are proposed: 1 A nonlinear approach based on backstepping 2 An approach based on the method of least squares Ehsan Peymani Topics in Synchronization and Motion Control

56. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Nonlinear Approach based on Backstepping Given a straight-line path P and a desired speed proﬁle ud(t), the essence of the method is to force the underactuated marine craft to move with the desired speed ud(t) + f(e). How to ﬁnd f(e) appropriately? lime(t)→0 f(e) = 0. f(e) should be chosen in a way that it helps the path-following task happen better, in some sense. Ehsan Peymani Topics in Synchronization and Motion Control

57. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Nonlinear Approach based on Backstepping Given a straight-line path P and a desired speed proﬁle ud(t), the essence of the method is to force the underactuated marine craft to move with the desired speed ud(t) + f(e). How to ﬁnd f(e) appropriately? lime(t)→0 f(e) = 0. f(e) should be chosen in a way that it helps the path-following task happen better, in some sense. Ehsan Peymani Topics in Synchronization and Motion Control

58. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Approach based on the Method of Least Squares The design procedure is as follows: 1 Find the desired accelerations to make the geometric error exponentially stable at the origin. ρ2(γ)T ˙ν = σe 2 Find the desired accelerations to achieve the heading and the speed objectives. ˙ν = σz 3 The accelerations that achieve all the objectives H(γ) ˙ν = b(γ, φ) H(γ) = I3 ρ2(γ)T , b(γ, φ) = σz σe , φ [e, ˙e, z0, zT ]T Ehsan Peymani Topics in Synchronization and Motion Control

61. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Approach based on the Method of Least Squares ˙ν can be found such that H(γ) ˙ν − b(γ, φ) 2 is minimized. ˙ν = (H(γ)T H(γ))−1 H(γ)T b(γ, φ) The control force is given by τp b = Mb(H(γ)T H(γ))−1 H(γ)T b(γ, φ) + Cb(ν)ν + Db(ν)ν Ehsan Peymani Topics in Synchronization and Motion Control

62. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Introduction & Motivation A Nonlinear Approach using Backstepping An approach based on the method of Least Squares Closing Remarks Backstepping Least Squares The control law depends on e The control law depends on e and ˙e e inﬂuences the control laws e and ˙e appear through the function f(e). linearly in the control signals. Ehsan Peymani Topics in Synchronization and Motion Control

63. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Motivation & Objective Motion Control Methodology Outline 1 Background 2 Synchronization in the Presence of External Disturbances 3 Speed-varying Path-following for Marine Craft 4 Motion Control using Analytical Mechanics Motivation & Objective Motion Control Methodology 5 Concluding Remarks and Future Work Ehsan Peymani Topics in Synchronization and Motion Control

64. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Motivation & Objective Motion Control Methodology Motivation & Objective Question 3 How would the control law look like if Nature was the control engineer? How may one derive a motion controller for a mechanical system based on the principles of analytical mechanics? Ehsan Peymani Topics in Synchronization and Motion Control

65. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Motivation & Objective Motion Control Methodology Motivation & Objective Question 3 How would the control law look like if Nature was the control engineer? How may one derive a motion controller for a mechanical system based on the principles of analytical mechanics? Ehsan Peymani Topics in Synchronization and Motion Control

66. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Motivation & Objective Motion Control Methodology Control Methodology - The motion control problem of a mechanical system can be viewed as the problem of modeling of a constrained multi-body system. The control objectives are portrayed as the constraints conﬁning the motion of the bodies of the mechanical system. - The intention is to utilize tools from analytical mechanics to derive the equation of motion, in which the impact of constraints on motion is characterized as additional forces acting on the system. The force of constraints are applied to the system by virtue of the actuators. Ehsan Peymani Topics in Synchronization and Motion Control

67. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Motivation & Objective Motion Control Methodology Control Methodology - The motion control problem of a mechanical system can be viewed as the problem of modeling of a constrained multi-body system. The control objectives are portrayed as the constraints conﬁning the motion of the bodies of the mechanical system. - The intention is to utilize tools from analytical mechanics to derive the equation of motion, in which the impact of constraints on motion is characterized as additional forces acting on the system. The force of constraints are applied to the system by virtue of the actuators. Ehsan Peymani Topics in Synchronization and Motion Control

68. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Motivation & Objective Motion Control Methodology Control Methodology Two techniques to deal with constrained mechanical systems: The method of Lagrange multipliers This method is well known for holonomic constraints: h(q, t) = 0 This method is limited to nonholonomic constraints that are linear in velocity variables: h(q, ˙q, t) = A(q, t) ˙q + B(q, t) = 0 The method of Kalaba based on Gauss’s principle This method can be applied to any constraint that can be represented as A(q, ˙q, t)¨q + B(q, ˙q, t) = 0 Ehsan Peymani Topics in Synchronization and Motion Control

71. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Motivation & Objective Motion Control Methodology Closing Remarks Nature would exert additional forces on the system in order to cause the mechanical system to exactly satisfy the constraints. The magnitude of the additional forces is minimized according to an instantaneous cost function. Ehsan Peymani Topics in Synchronization and Motion Control

73. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Conclusion 1 Synchronization under external disturbances H∞ almost synchronization H∞ almost regulated synchronization H∞ almost formation 2 Speed-varying path-maneuvering for marine craft a method based on backstepping a method based on least squares 3 Motion control using analytical mechanics a framework for motion control using the Lagrange multiplier method review the method of Kalaba based on Gauss’s principle Ehsan Peymani Topics in Synchronization and Motion Control

77. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Future Work 1 Synchronization under external disturbances find the infimum of the H∞-norm of the transfer function, called γ∗ , for a homogeneous/heterogeneous network of non-introspective agents. 2 Speed-varying path-maneuvering for marine craft to solve the formation problem using the method of least squares. 3 Motion control using analytical mechanics to include unilateral constraints in the framework. Unilateral constraints can be used to define a proximity of convergence in formation problems. Ehsan Peymani Topics in Synchronization and Motion Control

81. Background Synchronization in the Presence of External Disturbances Speed-varying Path-following for Marine Craft Motion Control using Analytical Mechanics Concluding Remarks and Future Work Thank You for Your Attention! Ehsan Peymani Topics in Synchronization and Motion Control

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