Distributed Nonlinear MPC Formation Control with Limited Bandwidth

1. WeC07.6 Distributed Nonlinear MPC Formation Control with Limited Bandwidth Sami El-Ferik, Bilal A. Siddiqui (KFUPM) Frank L. Lewis (UTA) The 2013 American Control Conference, Washington, DC

2. Distributed NMPC Formation Control with Limited Bandwidth Outline • Introduction • Problem Statement • CF-DNMPC Algorithm • Stability Analysis • Simulation Results ACC’13

3. Introduction Literature Review Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Cooperative Teams of Multiple Agents • A multi-agent system (MAS) is a system composed of multiple interacting intelligent agents within an environment. • Cooperation b/w autonomous agents working in teams extend individual agents capabilities by: • Division of tasks • Division of labor • Distributed sensing • Robustness to single points of failure • Communication plays a key role in cooperation. ACC’13

4. Introduction Literature Review Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Multi-agent Formation Control • Multi-agent formation control consists of • Cohesion: attraction to neighbors • Alignment: velocity and average heading agreement • Collision Avoidance: repulsion from neighbors within minimal distance • Obstacle Avoidance: repulsion from static or dynamic foreign objects. • Depending on communication topology, Formation Control can be: • Centralized: Single hub collects data & computes ctrl action for all agents • Distributed: Ctrl & Estimation done locally; communication b/w agents • Decentralized Control is advantageous because: • Centralized ctrl is computationally expensive • Agents may be geographically distributed • Faults in communication with central hub will cause system failure • Limited throughput/bandwidth of communication channel. ACC’13

5. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Formation Control by Nonlinear MPC • Work on multi-agent formation control using MPC pioneered @ Caltech [Dunbar2001]. Need to know neighbors’ models removed recently [Dunbar2012], by communicating error trajectories. • A generalized framework for distributed NMPC for cooperative control of team of agents is proposed in [Allgöwer2011]. • [Shim2003] considered collision avoidance by repulsive potential field in NMPC & using position, velocity of other vehicles to predict trajectories. • Distributed NMPC framework to vehicles receiving information after fixed delay from their neighbors [Franco2008]. Delayed state information projected using forward forgetting-factor. Results conservative. • Cost penalties penalty with priority strategy for collision avoidance in NMPC, the neighbors’ randomly delayed information projected using linear recurrence [Chao2012]. No stability proofs. ACC’13

6. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Research Objectives • Leader-follower formation control of constrained AUVs robust to propagation delays. • Limited network throughput demands drastic reduction in packet size. • Distributed control is to be formulated, such that agents operate without knowing dynamics of other agents. • Collision avoidance to be ensured. • Stability to be guaranteed both for strong and weak network topologies. ACC’13

7. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Local Control Distributed Nonlinear MPC External Inputs • DT Nonlinear Dynamics for i=1,….N agents • Distributed FH Cost Function: • Distributed NMPC: min s.t. Local Cost Cooperation Cost Terminal Cost u Apply first element ACC’13

8. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Data Compression, Tail Approximation • Agents need to communicate planned trajectories: • To reduce packet size -----› Data compression by NN -----› for all j (neighborhood of Ai) • But, is received at Aj after delay -----› w • Using time-stamps, is estimated; using NN, traj is re-sampled w -----› Ai at time tAj at t+Δji Calculate Compress & Broadcast Delay Receive & est Resample ACC’13

9. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Collision Course Definition dij (t+Npi) j dij (t+2) dij (t+1) dij (t) i Rimin ACC’13

10. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Collision Avoidance Algorithm • Two agents Ai and Aj on collision course if, • The cost for optimization is modified with repulsion potential: • ,where • is a spatial filter strictly decreasing in ACC’13

11. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Increment time index t=t+1 Distributed NMPC Formation Control with Limited Bandwidth Modify Cost Ĵt=J t(1+Φt) Yes Collision Course? Take local sensor data for localization Target No End ACC’13

12. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Stability Analysis: Preliminaries • Nonlinear DT dynamics • Input to State Stability • ISpS ISpS x0 0 c ISS ACC’13

13. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth ISpS Lyapunov Function Ω 0 Ξ c • For and , if • , • , • , for all Then, is an ISpS Lyapunov function. Theorem 1 • If a system admits an ISpS Lyapunov function in region • then it is regional ISpS stable. ACC’13

14. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Stability of Individual Agents Xif 0 XiMPC C(ξ) • Universal Approximators: Any smooth function can be approximated by a 2-layer NN with bounded error: ---------> , • Theorem 2 (Stab. of Agents without C.A) : • Let be local ISpS Lyapunov function for Ai. If there exists a “terminal set” Xif which is RPI, where a local terminal control laws kif are applicable, then with certain Lipschitz conditions and nonlinear bounds on local, cooperative and terminal components of cost function Jit , the individual agents Ai are ISpS, with robust output admissible set XiMPC • Difference between ISS and ISpS is the ball with radius c. Since, , where αc is a K-function, ISpS converge to asymptotic stability if there is no uncertainty in received trajectory, ξ=0. ACC’13

15. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Stability of Individual Agents • Terminal Control Law: key to ensure stability is design of terminal control kif(xi). • Lemma 2:Let, local cost be cooperative cost bounded as and terminal cost Q, R, S, Qf are positive definite (Qf is symmetric), then there exists terminal control such that is stable. To find Qf and Kf ACC’13

16. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Stability of Individual Agents with C.A. • Theorem 3:Let be local ISpS Lyapunov function for Ai without collision avoidance. It can be shown that the modified cost , where is also an ISpS Lyapunov function under certain Lipschitz conditions and bounds if an optimal trajectory can be found such that average distance between increases : Thus, an Agent under the collision avoidance algorithm avoids collision and is ISpS stable. j i j i ACC’13

17. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Stability of Team of Agents • Mixed Graph: Both directed and undirected edges. • Team of agents Ai each provided with ISpS Lyapunov function • Recall, for ISPS: and • These comparison functions depend on Lipschitz constants, nonlinear bounds on cost and prediction horizon. • Design a function , such that • Then, network gain of Ai to Aj is 1 γ31 γ21 γ12 γ13=0 2 3 γ23 ACC’13

18. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Stability of Team of Agents Strongy Connected Network Weekly Connected Network Theorem 4: If each agent Ai is supplied with local ISpS Lyapunov function , and there are appropriately designed network gains , then the team of agents is ISpS stable as long as the network is at least weakly connected and following Small Gain conditions are met: ACC’13

19. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Simulation Study • Nonlinear Dynamics Constraints Delay • Cost Function: • Optimization Parameters Neural Network Np=50,Nc=30,Q=0.1diag(1,1,10,1,10,1) Layers = 1 R=0.01, Sij=0.25Q (i=2,3),S1j=0.2Sij Neurons/layer=6 ACC’13

20. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Simulation Study: Strongly Connected Data compressed by 72% ACC’13

21. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Simulation Study: Strongly Connected Constraint ACC’13

22. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Simulation Study: Strongly Connected ACC’13

23. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Simulation Study: Weekly Connected ACC’13

24. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Simulation Study: Weekly Connected Constraint ACC’13

25. Introduction Literature Rev. Problem Statement CF-DNMPC Algorithm Stability Analysis Simulations Conclusion Distributed NMPC Formation Control with Limited Bandwidth Simulation Study: Weakly Connected Data compressed by 72% ACC’13

26. Distributed NMPC Formation Control with Limited Bandwidth Thank You ACC’13