Coordinates transformation for MIMO systems

NelinSys - a program tool for analysis and synthesis of nonlinear control systems

Coordinates transformation for MIMO systems

Block description

Block performs a nonlinear transformation of state variables (state coordinates) according to the exact linearization procedure. It is used together with the Linearizing control law block in order to ensure a transformation of the nonlinear state-space equations of a MIMO system

State-space description of a nonlinear system

to the linear and controllable form:
State-space description of a linear system

Following notations are used: x = (x1,x2,...,xN)^T denotes the states vector of the nonlinear system, u = (u1,u2,...,uP)^T and y = (y1,y2,...,yP)^T denote the system input and output vector, respectively. Vector q = (q1,q2,...,qNr)^T is the states vector and v = (v1,v2,...,vP)^T the input vector of the linear system resulting from the transformation. The transformation formula can be specified either by a symbolic expression or by an identifier of a symbolic variable defined in MATLAB workspace - see detailed description of block parameters below. The formula can be computed by the ExaktMimo program application included in the NelinSys tool.

Block parameters

System order (number of states, N): Positive integer specifying the number of state variables of the nonlinear system i.e. the length of the states vector x.
Relative degrees of subsystems (r1+r2+...+rK <= n): Vector of positive integer elements whose sum (so-called total relative degree of the system) is lower than or equal to system order. The total relative degree specifies the number of state variables of the linear system resulting from the transformation i.e. the length of the q vector. Subsystems of the linear system are independent on each other, their number is determined by the length of the [r1 r2 ... rK] vector and their orders by the elements of the vector.
Transformation formula - symbolic expression: Symbolic expression defining the transformation function (transformation matrix) q = Q(x); the convention for writing mathematical operations is the same as the one used by MATLAB's Symbolic Math Toolbox. State variables have to be denoted as x1, x2, ..., xN. If the transformation formula is not specified by a symbolic expression but by an identifier of a variable (see below), it is necessary to leave this field blank.
Transformation formula - variable identifier: Identifier of a symbolic variable (a class sym object) containing the expression for q = Q(x). If the transformation formula is not specified by an identifier of a variable but by a symbolic expression (see previous parameter), it is necessary to set this field to 0 (zero).

Usage example

See demo Exact linearization of a MIMO system