PWAG

Piece-Wise Affine Generic function object

Description
Syntax
Properties
Methods
Private methods
Acknowledgements

Description

The pwag object describes a Piece-Wise Affine Generic function, i.e. a PWA function defined on a domain partitioned into irregular regions (polytopes). The generic j-th polytope $P_j$ is represented by an inequality in the form $H_j x \leq K_j$ and, for all $x$ in $P_j$ the function assumes the value $y = F_j x + G_j$ . The pwag object is derived from pwaFunction object.

Syntax

object = pwag()

Builds an empty pwag object.

object = pwag(regions,[D])

Builds a pwag object given the edges of each polytope and the matrices representing the affine function defined over each polytope. regions is an array of structures (one for each polytope) with the following fields: H, K, F, G (see description above). D is a matrix representing the domain of the function, in the form $\left[ \begin{array}{cccc} x_{min}^1 & x_{min}^2 & \ldots & x_{min}^{nx}\\ x_{max}^1 & x_{max}^2 & \ldots & x_{max}^{nx} \end{array} \right]$ . Optional argument.

object = pwag(HybridToolboxController,[D])

Builds an array of pwag objects starting from an MPC controller of class lincon or expcon (see Hybrid Toolbox). If the object is of class lincon (implicit MPC) it is converted into an expcon object (explicit MPC). The expcon object contains control functions for several (N) time istants (receding horizon), for each of these functions a pwag object is created. If soft constraints are enabled in expcon object, the last entry of the pwag array is the slack variable epsilon used to soften the constraints. D is the domain of the function (see above). Optional argument.

object = pwag(MPTController,[D])

Builds an array of pwag objects starting from an explicit MPC controller of class mptctrl (see MPT toolbox). The mptctrl object contains control functions for several (N) time istants (receding horizon), for each of these functions a pwag object is created. D is the domain of the function (see above). Optional argument.

Properties

nr - number of regions of the domain.
edges - structure with fields H, K. H and K are the matrices defining the edges of the regions, in the form $H_j x \leq K_j$ . H and K are "big" matrices containing all the edges without repetitions (if two polytopes have a common edge, it is not repeated). The edges of a particular polytope are accessed with addresses to these matrices.
functions - structure with fields F,G. F and G are the matrices defining the coefficients of the affine functions on each polytope in the form . As for edges, F and G are "big" matrices containing all the coefficients without repetitions (if two polytopes have the same function, it is not repeated). The coefficients of a particular polytope are accessed with addresses to these matrices.
regions - array of structures defining the polytopes of the partition. Each element of regions has the following fields: "Iedges": array whose k-th row contains, in the first column, the index of the row in matrices H and K corresponding to the k-th edge of the current region and, in the second column, a flag ('1' or '-1') indicating the sign of the inequality. '1' means $H_{jk} x \leq K_{jk}$ , '-1' means $H_{jk} x \geq K_{jk}$ , i.e. $\textrm{--}H_{jk} x \leq \textrm{--}K_{jk}$ . "Ifunctions": array which contains the indexes of the rows in matrices F and G corresponding to the nf affine functions defined over the current region.
tree - binary search tree associated to the domain partition, for the computation of the pwag function and for its digital implementation.

Methods

computeTree - computes the binary search tree associated to the domain partition.
disp - displays some information about the pwag object.
eval - evaluates the pwag function.
findRegion - finds the polytope containing a given point.
generateSimulinkModel - generates a Simulink model for the simulation of the closed-loop system
getCircuitPerformances - gets the performances of the circuits implementing the pwag function
getClosedLoop - gets the pwag function defining the closed loop system
getEdges - gets the edges of a region.
getFunctions - gets the coefficients of the affine function defined over a region.
getNumberOfEdges - gets the number of different edges of the domain partition.
getNumberOfRegions - gets the number of polytopes of the domain partition.
getRegions - gets information about the requested polytope(s).
plot - plots the pwag function
plotTree - plots the binary search tree associated to the domain partition.
synthesize - generates the VHDL files describing the digital circuit which implements the pwag function.
vertices - gets the vertices of the polytopes of the domain partition.

Private methods

evalmex - Evaluates the pwag function
generateMemory - Generates the memory matrix in binary form for the circuit implementation
treeExplore - Retrieves information from the binary search tree
treeStepPlot - Finds the polytope containing a given point.
quantizeData -Quantize the coefficients of the PWAG function
writeFSM - Writes the VHDL code implementing block FSM in pwag circuit
writeInterface - Writes the VHDL code implementing block pwag_ser_interface or block pwag_par_interface in pwag circuit
writeMemory - Writes the VHDL code implementing block Memory in pwag circuit
writePackage - Writes the VHDL code implementing block pwag_ser_package or block pwag_par_package in pwag circuit
writeTest - Writes the VHDL code implementing block test_pwag_ser_interface or block test_pwag_par_interface in pwag circuit

Acknowledgements

Contributors:

Alberto Oliveri (alberto.oliveri@unige.it)