Stability Analysis of Distributed Vehicle Control System

1
Stability Analysis of Distributed Vehicle Control
System

• ECE598SM Class Project
• Kyoung-Dae Kim

2
Distributed Vehicle Control System (DVCS)

• Feature
• Vehicle control system
• Control components are distributed over network
• Interaction via periodic data exchange
• Entities
• Vehicle
• Supervisor component
• Controller component
• Sensor component

3
Hybrid System Model of DVCS

• Discrete interaction
• Periods Ts, Tc, Tv
• Input/Output actions
• Internal variable can change at input action
• R/C Sensor
• R/C car with continuous vehicle dynamics
• Sensor as an output action with Tv
• Constraints

4
Vehicle Model

• Bicycle type kinematic model
• Front wheel with steering
• Local origin at real wheel
• Two inputs driving, steering
• Nonholonomic constraints
• Model in cartesian coordinate

5

• Coordinate transform
• Not necessary but simplify the controller design
• (2, 4) nonholonomic system can be put in chained
  form by means of coordinate change with

6
Controller Design

• Reference trajectory
• In chanied form coordinate
• Desired State
• Desired input

7

• Errors
• Error dynamics
• Linearization w.r.t. xd

8

• State feedback controller
• Closed-loop error dynamics
• Eigenvalues at
• If we set
• Eigenvalues at

9

• Refrence trajectory
• Controller gains
• Eigenvalues at
• -5
• three -a ud1 with a 5
• Initial condition

10
Controller in DVCS

• Controller in DVCS
• Utilize the characteristics of DVCS
• Controller knows only the sampled point of the
  trajectory
• Trajectory estimation
• Two target positions
• Estimate the original trajectory as a line
  segment

11

• For each line segment
• Desired state
• Desired inputs
• Closed-loop error dynamics

12

• Refrence trajectory
• Original
• update(xd) with Ts
• 500ms, 1500ms
• Controller
• Gains for poles at
• -5, three -a ud1 with a 5
• Control(u) with Tc 10ms
• Initial condition

13
Stability Analysis

• Goal
• Check the stability of DVCS
• How does the discrete actions and constraints
  affect to the stability?
• Errors in DVCS
• Tracking error due to performance of control law
• Error caused by discrete interactions
• Variables
• Continuous variables
• Discrete variables

14

• feedback(x) action
• update(xd) action
• Discrete error

15

• If P-controller
• Closed-loop error dynamics

16

• Stability of LTI System
• Exp. stable ? Acl is Hurwitz
• BIBO stable ? all poles of H(s) are stable
• Z.I.R of the closed-loop error systems
• Acl is Hurwitz ? Exp. Stable
• Z.S.R of the closed-loop error systems
• Poles of G(s) Eigenvalues of Acl ? BIBO Stable

17

• Steady state error in (x, y) position
• Error bound

18
Conclusion

• Bicycle kinematic model of a vehicle
• Linearized state feedback controller
• Reduction from LTV to LTI system
• Utilize the characteritics of DVCS for reduction
• Proved BIBO stability of DVCS
• Derived error bound with parameters
• Periods of discrete actions
• Constraints