Help Request: Time Varying LQR Solution

[acxz]

It would be great to have a an example which solves systems with time-varying matrices A, B for LQR.

I know I can just call the lqr method at every timestep with different A and B matrices, but is that the same?
We use the following, but doesn't that assume that A and B are the same during the backward pass when solving the differential equation concerning the cost to go matrix X?

ControlSystems.jl/src/matrix_comps.jl

Lines 3 to 4 in 311013e

	Compute 'X', the solution to the continuous-time algebraic Riccati equation,
	defined as A'X + XA - (XB)R^-1(B'X) + Q = 0, where R is non-singular.

i.e. Solving the cost to go matrix X backwards in time with varying A_t and B_t matrices, instead of constant A and B matrices.

Maybe I am just not understanding properly LQR in the time-varying case. Any help or guidance would be appreciated. Ideally having an example in the docs would close this issue. Thank you!

[olof3]

The toolbox only supports the case of infinite horizon and constant system dynamics, which can be solved as an algebraic Riccati equation.

Time-varying LQR is more involved. In continuous time it is necessary to solve a differential Riccati equation. One reference for further reading on this would be Liberzon's book on optimal control, chapter 6.

I think it would be nice to have this functionality in the toolbox at some point.

[acxz]

Thanks for your comments, something related and interesting that I just came across is "State-dependent Ricacati Equations" (https://ieeexplore.ieee.org/document/609663).

But yeah the big change would be having an implementation that solves the differential Riccati Equation.

[baggepinnen]

There is a generalization of this implemented in
https://github.com/baggepinnen/DifferentialDynamicProgramming.jl
I believe The iLQG algorithm will be what you are looking for in case the system is linear and time varying.