How to Use QSR.nb

QSR.nb is a Mathematica program to obtain a polynomial-in-the-state 
representation (PSR) and a quadratic-in-the-state representation (QSR) of a 
given continuous-time dynamical system. QSR.nb consists of the following four 
parts. 

1) Definition of Functions
  Several functions are defined by executing this part. A user does not need to 
modify this part. 


2) Definition of State Equation and Output Equation
  A user needs to modify only this part to define a system: 
      dx/dt = f(x) + B(x)u
          y = h(x) + D(x)u
The state vector xv and control input vector uv are defined as
         xv = Array[x,dimx]; 
         uv = Array[u,dimu];
where dimx and dimu denote the dimensions of the state vector and the control 
input vector and are given by the user. Then, xv is a list of the form {x[1], x
[2], ... , x[dimx]}. The lists fx and hx consist of elements of f(x) and h(x), 
respectively. TrBx is a list of lists, that is, each element is a list 
consisting of a column of B(x). Therefore, TrBx represents the transpose of B(x).
 TrDx is also a list of lists consisting of columns of D(x) and represents the 
transpose of D(x). 


3) Immersion into a Polynomial-in-the-State Representation
  A PSR of the system defined above is obtained by executing this part. The 
immersion alpha(x) is obtained as a variable alpha. The new state vector z = 
alpha(x) is governed by: 
      dz/dt = f(z) + B(z)u
          y = h(z) + D(z)u
New functions f(z), B(z), h(z) and D(z) are obtained as variables fz, TrBz, hz, 
and TrDz, respectively. 


4) Immersion into a Quadratic-in-the-State Representation
  A QSR of the PSR obtained above is obtained by executing this part. The 
immersion beta(z) from PSR is obtained as a variable bet, and the immersion 
gamma(x) from the original system is obtained as a variable gam. The new state 
vector w = gamma(x) ( = beta(z) ) is governed by: 
      dw/dt = f(w) + B(w)u
          y = h(w) + D(w)u
The new functions f(w), B(w), h(w) and D(w) are obtained as variables fw, TrBw, 
hw, and TrDw, respectively. 


The usage is simple. Modify only 2) Definition of State Equation and Output 
Equation, and execute all. Some warning may occur concerning spelling of symbols,
 because some symbols have similar names. Such warning may be ignored. 



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Toshiyuki Ohtsuka
Department of Computer-Controlled Mechanical Systems
Graduate School of Engineering
Osaka University
2-1 Yamadaoka
Suita, Osaka 565-0871
Japan
E-mail ohtsuka@mech.eng.osaka-u.ac.jp
http://www-newton.mech.eng.osaka-u.ac.jp/~ohtsuka/