Return True if to_region is reachable from_region.

For details see solve_feasible.

Compute S0 \subseteq P1 from which P2 is N-reachable.

N-reachable = reachable in horizon n. The system dynamics are ssys. The closed-loop algorithm solves for one step at a time, which keeps the dimension of the polytopes down.

Parameters:

• N - The horizon length
• closed_loop (bool) - If true, take 1 step at a time. This keeps down polytope dimension and handles disturbances better.
• use_all_horizon (bool) - Used for closed loop algorithm. If true, then allow reachability also in less than N steps.
• trans_set - If specified, then force transitions to be in this set. Otherwise, P1 is used.
• P2 (Polytope or Region)
• ssys (LtiSysDyn)
• P1 (Polytope or Region)

Returns: Polytope or Region

the subset S0 of P1 from which P2 is reachable

Compute S0 \subseteq P1 from which P2 is closed-loop N-reachable.

Parameters:

• ssys - system dynamics
• N (int > 0) - horizon length
• use_all_horizon (bool) -
  • if True, then take union of S0 sets
  • Otherwise, chain S0 sets (funnel-like)
• trans_set - If provided, then intermediate steps are allowed to be in trans_set.

  Otherwise, P1 is used.

• P2 (Polytope or Region)
• P1 (Polytope or Region)

Compute the components of the polytope:

   L [x(0)' u(0)' ... u(N-1)']' <= M

which stacks the following constraints:

• x(t+1) = A x(t) + B u(t) + E d(t)
• [u(k); x(k)] \in ssys.Uset for all k

If list_P is a Polytope:

• x(0) \in list_P if list_P
• x(k) \in Pk for k= 1,2, .. N-1
• x(N) \in PN

If list_P is a list of polytopes:

• x(k) \in list_P[k] for k= 0, 1 ... N

The returned polytope describes the intersection of the polytopes for all possible

Parameters:

• ssys (LtiSysDyn) - system dynamics
• N - horizon length
• disturbance_ind - list indicating which k's that disturbance should be taken into account. Default is [1,2, ... N]
• Pk (Polytope)
• list_P (list of Polytopes or Polytope)
• PN (Polytope)

Calculate the array d_hat such that:

   d_hat = max(G*DN_extreme),

where DN_extreme are the vertices of the set D^N.

This is used to describe the polytope:

   L*x <= M - G*d_hat.

Calculating d_hat is equivalen to taking the intersection of the polytopes:

   L*x <= M - G*d_i

for every possible d_i in the set of extreme points to D^N.

Parameters:

• G - The matrix to maximize with respect to
• D - Polytope describing the disturbance set
• N - Horizon length

Returns:

d_hat: Array describing the maximum possible effect from the disturbance