Solve the problem of stochastic reachability of a target tube (a lower bound
on the maximal reach probability and an affine controller synthesis) using
chance-constrained optimization and uniform risk allocation
=============================================================================
SReachPointCcAuniform implements the chance-constrained underapproximation to
the problem of stochastic reachability of a target tube to construct an affine
controller. This technique is inspired from Algorithms 1 and 2 of
M. Vitus and C. Tomlin, "On feedback design and risk allocation in chance
constrained control", In Proc. Conf. Dec. & Ctrl., 2011.
In contrast to their original algorithm, we have a chance constraint on
the input and the state. Further, the lower bound on the reachability (state
constraint) depends on how high the input chance constraint satisfaction
probability is. Therefore, we perform two levels of bisection
--- one to maximize the probability of constraint satisfaction for the
state, and the other to meet the chance constraint on the input. However,
to save time, we check only for feasibility in the input bisection.
Subsequently, the obtained solution is discounted for input constraint
violation using Theorem 1 of
A. Vinod and M. Oishi. Affine controller synthesis for stochastic reachability
via difference of convex programming. In Proc. Conf. Dec. & Ctrl., 2019.
(submitted). https://hscl.unm.edu/affinecontrollersynthesis/
=============================================================================
[lb_stoch_reach, opt_input_vec, opt_input_gain, risk_alloc_state, ...
risk_alloc_input] = SReachPointCcAu(sys, initial_state, safety_tube, ...
options)
Inputs:
-------
sys - System description (LtvSystem/LtiSystem object)
initial_state- Initial state for which the maximal reach probability must be
evaluated (A numeric vector of dimension sys.state_dim)
safety_tube - Collection of (potentially time-varying) safe sets that
define the safe states (Tube object)
options - Collection of user-specified options for 'chance-affine-uni'
(Matlab struct created using SReachPointOptions)
Outputs:
--------
lb_stoch_reach
- Lower bound on the stochastic reachability of a target tube
problem computed using chance constraints and
difference-of-convex techniques
opt_input_vec,
opt_input_gain
- Controller U=MW+d for a concatenated input vector
U = [u_0; u_1; ...; u_{N-1}] and concatenated disturbance
vector W=[w_0; w_1; ...; w_{N-1}].
- opt_input_gain: Affine controller gain matrix of dimension
(sys.input_dim*N) x (sys.dist_dim*N)
- opt_input_vec: Open-loop controller: column vector
dimension
(sys.input_dim*N) x 1
risk_alloc_state
- Risk allocation for the state constraints
risk_alloc_input
- Risk allocation for the input constraints
See also SReachPoint.
Notes:
* See @LtiSystem/getConcatMats for more information about the notation used.
============================================================================
This function is part of the Stochastic Reachability Toolbox.
License for the use of this function is given in
https://sreachtools.github.io/license/