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NonLinear Model Predictive Control
Given:
System: Cost function: where:
Calculate:
Initial state: U that minimizes J(X,U)
Non-linear MPC Robert Platt Northeastern University NonLinear Model - - PowerPoint PPT Presentation
Non-linear MPC Robert Platt Northeastern University NonLinear Model - - PowerPoint PPT Presentation
Non-linear MPC Robert Platt Northeastern University NonLinear Model Predictive Control Given: System: Cost function: where: Initial state: Calculate: U that minimizes J( X,U ) NonLinear Model Predictive Control Given: System: Cost
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SLIDE 3
NonLinear Model Predictive Control
Given:
System: Cost function: where:
Calculate:
Initial state: U that minimizes J(X,U)
SLIDE 4
NonLinear Model Predictive Control
Given:
System: Cost function: where:
Calculate:
Initial state: U that minimizes J(X,U)
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Minimize: Subject to:
NonLinear Model Predictive Control
But, this is a nonlinear constraint – so how do we solve it now?
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LTV (linear time varying) problem
Given: a nonlinear system: A quadratic cost fn A nominal trajectory: Find: a controller that works nearby nominal trajectory
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Idea: time varying linear system
Linear Time Invariant (LTI): Linear Time Variant (LTV):
Notice time time dependence – each time step is linearized differently
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Linear Time Varying LQR (LTV)
Similar first order taylor series expansion as before:
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Linear Time Varying LQR (LTV)
Similar first order taylor series expansion as before: Solution: solve LQR for this TV system: Resulting controller is linearized about nominal trajectory:
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Linear Time Varying LQR (LTV)
Similar first order taylor series expansion as before: Solution: solve LQR for this TV system: Resulting controller is linearized about nominal trajectory:
Key points: – point linearized about changes on each time step – nominal trajectory does not even need to be feasible! – you DO NEED to know what time step you’re at throughout
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Question
Think about cart-pole We have shown that you can use linearize LQR about an arbitrary trajectory. Can you just linearize about the current operating point on each time step?
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Back to nonlinear control problem...
Minimize: Subject to: LTV LQR only works if you have a nominal trajectory How do you get the nominal trajectory???
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Iterative LQR
Key observation: If: you start with a bad nominal trajectory, run LTV LQR linearized about it, and then integrate forward the resulting locally optimal policy… Then: the resulting trajectory will be better (lower cost) than the nominal trajectory you started with
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Iterative LQR
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Iterative LQR
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Iterative LQR
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Iterative LQR
Why is this not zero? When can we skip these terms?
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Iterative LQR
Why is this not zero? When can we skip these terms? How does this fix problem?
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Iterative LQR
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Iterative LQR
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Iterative LQR
iLQR works well in the MPC context – stabilization to the trajectory will handle most small deviations from nominal – can iterate the process to handle larger deviations