StabilityEquivalence  

1. The Roesser models of the form
   
   (R) x^h(i+1,j) = A11*x^h(i,j) + A12*x^v(i,j) + B1*u(i,j),
         x^v(i,j+1) = A21*x^h(i,j) + A22*x^v(i,j) + B2*u(i,j),
   
   where the Aij's and Bi's (i,j=1,2) are matrices with constant entries;
   
   
   
2. The (generalized) Fornasini-Marchesini models of the form
   
   (FM) x(i+1,j+1) = F1*x(i+1,j)+F2*x(i,j+1)+F3*x(i,j)+G1*u(i+1,j)+G2*u(i,j+1)+G3*u(i,j),
   
   where the Fi's and Gi's (i=1,2,3) are matrices with constant entries.
   

1. The equivalence problem in the sense of algebraic analysis: namely, can we transform any (R) model into an equivalent (FM) model and vice versa?
   
   
2. How can we use these equivalence transformations to study structural stability and structural stabilization issues for these classes of systems?
   

1. Check the structural stability of linear 2D discrete Roesser and generalized Fornasini-Marchesini models using the IsStable package,
   
2. Transform (if possible) a Roesser model into an equivalent generalized Fornasini-Marchesini model and vice versa.
   

1. FornasiniToRoesser2D
   
   Input: the constant matrices Fi's and Gi's defining the (FM) and the dimension dx of the state variable x of the (FM) model
        
   Output: the matrices A = (Aij)i,j and B=(B1^T B2^T)^T defining an equivalent (R) model and the respective dimensions dh(=dx) and
   dv(=dx+du) of the horizontal and vertical state variables of the (R) model
   
   
2. RoesserToFornasini2D
   
   Input: the matrices A = (Aij)i,j and B defining the (R) model and the respective dimensions dh and dv of the horizontal and vertical
   state variables of the (R) model
         
   Output: the matrices Fi's and Gi's defining an equivalent (FM) model or "Case not handled by this procedure" (see [C2015])
   
   
3. TestStructuralStabilityRoesser2D
   
   Input: the matrix A of the (R) model and the dimensions dh and dv of the horizontal and vertical state vector
         
   Output: true or false
   
   
4. TestStructuralStabilityFornasini2D
   
   Input: the matrices F1, F2 and F3 defining the (FM) model and the dimension dx of the state variable
   
   Output: true or false
   
   
5. FornasiniToOperatorMatrix2D
   
   Input: the constant matrices Fi's and Gi's defining the (FM) and the dimension dx of the state variable of the (FM) model
   
   Output: the matrix R of operators (sigma1 and sigma2) definig the associated linear system Ry=0
   
   
6. RoesserToOperatorMatrix2D
   
   Input: the matrices A = (Aij)i,j and B defining the (R) model and the respective dimensions dh and dv of the horizontal and vertical
   state variables of the (R) model
   
   Output: the matrix R of operators (sigma1 and sigma2) definig the associated linear system Ry=0
   
   
7. TestStructuralStabilityAutonomousLinearSystem2D
   
   Input: the matrix of operators R defining a square autonomous linear system
   
   Output: true or false