## BFFPSV18

`ReachabilityAnalysis.BFFPSV18`

— Type`BFFPSV18{N, ST, AM, IDX, BLK, RBLK, CBLK} <: AbstractContinuousPost`

Implementation of the reachability method for linear systems using block decompositions.

**Fields**

`δ`

– step-size of the discretization`approx_model`

– (optional, default:`Forward`

) approximation model; see`Notes`

below for possible options`vars`

– vector with the variables of interest`block_indices`

– vector of integers to index each block that contains a variable of interest`row_blocks`

– vector of integer vectors to index variables associated to blocks of interest`column_blocks`

– vector of integer vectors to index variables in the partition`lazy_initial_set`

– (optional, default:`false`

) if`true`

, use a lazy decomposition of the initial states after discretization`lazy_input`

– (optional, default:`false`

) if`true`

, use a lazy decomposition of the input set after discretization`sparse`

– (optional, default:`false`

) if`true`

, assume that the state transition matrix is sparse`view`

– (optional, default:`false`

) if`true`

, use implementation that uses arrays views

matrix is sparse

See the `Examples`

section below for some concrete examples of these options.

**Notes**

This algorithm solves the set-based recurrence equation $X_{k+1} = ΦX_k ⊕ V_k$ by using block decompositions. The algorithm was introduced in [BFFPSV18].

Comments about some fields:

`N`

– number type of the step-size, e.g.`Float64`

`ST`

– set representation used; this is either a concrete LazySet subtype, eg.`Interval{Float64}`

, or a tuple of concrete LazySet subtypes that is commensurate with the partition

The default approximation model is:

`Forward(sih=:concrete, exp=:base, setops=:lazy)`

TODO:

- clarify assumption about contiguous blocks

**Examples**

**References**

This algorithm is essentially an extension of the method in [BFFPSV18]. Blocks can have different dimensions and the set representation can be different for each block.

For a general introduction we refer to the dissertation [SCHI18].

Regarding the approximation model, by default we use an adaptation of the method presented in [FRE11].