Introduction to MPC (Lasse Peters)
Autonomous Driving Scenario
Indroduction: The control task
given
reference plan : —
state estimate : 나의 속도, 위치, 다른 차의 속도
find
which control inputs achieve the plan : steering angle, pedal, …
remind: reactive Control
Longitudinal
control via PID
Lateral
control via Stanley
Limitations of Reactive Control
- Non-trivial for more complex systems
- control gains must be tuned manually
- separation into longitudinal and lateral controllers ignores coupling
- no handling of constraints such as obstacles
- ignores future decisions
Optimal Control : Model
Model of the system dynamics:
predicts the evolution of the state for a given sequence of inputs.
x_t+1 = f(x_t, u_t)
Control inputs (핸들 각도) -> dynamics -> state trajectory (자동차 경로)Model Example : Discrete 2D Bicycle
Optimal Control: Objective
- Objective
Assigns a cost to the trajectory - Example: deviation from a reference
Optimal Control: Constraints
차량이 피해야 할 방애물을 수식화 하거나
차량의 최대, 최소 가속도를 수식화 할 수 있다.
Control as Optimization Problem
In summary
Model Predictive Control (MPC)
Receding horizon control
- Start from the current state
- Find controls for a limited preview into the future
- Apply only the first input, then re-plan
MPC Design: Prediction Model Trade-off in choice of model family
- Trade-off in choice of model family: Model accuracy vs. complexity
- Data-driven approach
- Collectdata of the real system behavior
- Optimize the parameters of the model to match the behavior of the real system
Limitation: Interaction
MPC only models other agents indirectly via the dynamics
– other agents treated as dynamic obstacles with constant velocity
Challenge: Other agents also plan -> Decisions are coupled
Outlook: Dynamic Games
Ingredients of a dynamic game
- joint dynamics
이전 time 의 모든 사람의 입력이 들어가서 다음 x 예측 - Individual costs
입력으로 들어가는건 자신의 상태, 자신의 입력, 다른 사람들의 입력(not i, 세번째 항) - Solution: Nash Equilibrium
- No player can unilaterally improve
어떤 사람도 자기 혼자서 (다른 사람은 가만히 두고) 자신의 cost 를 더 낮출 수 없다.
여러 명의 사람이 각자 자신의 목적함수를 최적화하면서 동시에 움직이고, 아무도 자기 혼자 행동을 바꿔서 더 이득을 볼 수 없는 상태에 도달하면 nash equilibrium
Dynamic Game Example: Tag
2D point-mass dynamics
Objectives P1: minimize distance to P2, P2: Maximize distance to P1
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