# reduceTemplate

parameterized order reduction of the parameterized reachable set

## Contents

- Syntax
- Description
- Input Arguments
- Output Arguments
- See Also
- References

## Syntax

[R_next,mapsParamReduce] = reduceTemplate(obj,R_abs,R_err,candTermReg,candP,flagReduceFeasible)

## Description

To limit the number of optimization variables and constraints, the order of the parameterized reachable set has to be reduced. However, standard techniques for order reduction of (polynomial) zonotopes would drop the dependence of the reduced generators on the optimization variables. This function returns the reduced reachable set as well as maps for computing the edge lengths of the reducing set (box enclosing the reduced generators) as functions of the optimization variables. The reduced generators are enclosed by a box as in [2, Sec. 4]. If we want to ensure recursive feasibility (see [1, Sec. III-D], the number of reduced generators is chosen automatically so that the reduced reachable set R_next is contained in the candidate terminal region.

## Input Arguments

obj | instance of the class computeTermRegNonlinSysPolyApproach
R_abs | parameterized reachable set of the abstraction (class: polyZonotope)
R_err | approximation of the reachable set of abstraction errors (class: polyZonotope)
candTermReg | candidate terminal region (class: polyZonotope)
candP | candidate controller parameters (double vector)
flagReduceFeasible | boolean indicating whether recursive feasiblity shall be enforced for the reduced reachable set or not (logical)

## Output Arguments

R_next | reduced approximation of the reachable set (class: polyZonotope)
mapsParamReduce | struct storing the maps for computing the lengths of the reducing set's generators
.dim | dimension (=number of parameterized; reducing generators)
.c | dummy value to setup the function handles for optimization
.G | (polynomial) maps for computing the scaling factors of the reducing generators (cell array of polyZonotopes)
.E | exponent matrix (ids in; obj.id.termReg)
.idv | ids coresponding to .E
.isConstGen | ith element is true if .G{i} is independent of obj.id.var_scalar, false otherwise
.id | ids of reducing generators with variable length
.initScales | current length of the reducing generators

## See Also

-

## References

[1] L. Schäfer and M. Althoff, "Computing Robust Control Invariant Sets of Nonlinear Systems Using Polynomial Controller Synthesis," American Control Conference, 2024, pp. 4162-4169. [2] A. Girard. "Reachability of uncertain linear systems using zonotopes", Hybrid Systems: Computation and Control, 2005, pp. 291–305.

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