Control Systems/Eigenvalue Assignment for MIMO Systems

The design of control laws for MIMO systems are more extensive in comparison to SISO systems because the additional inputs () offer more options like defining the Eigenvectors or handling the activity of inputs. This also means that the feedback matrix K for a set of desired Eigenvalues of the closed-loop system is not unique. All presented methods have advantages, disadvantages and certain limitations. This means not all methods can be applied on every possible system and it is important to check which method could be applied on the own considered problem.

Parametric State Feedback edit

A simple approach to find the feedback matrix K can be derived via parametric state feedback (in German: vollständige modale Synthese). A MIMO system


with input vector


input matrix   and feedback matrix   is considered. The Eigenvalue problem of the closed-loop system


is noted as


where   denote the assigned Eigenvalues and   denote the Eigenvectors of the closed-loop system. Next, new parameter vectors   are introduced and assigned and the Eigenvalue problem is recasted as



Controller synthesis edit

1. From Equation [1] one defines the Eigenvector with


2. The new parameter vectors   are concatenated as


where the feedback matrix K can be noted as


3. Finally, the Eigenvector definition is used to hold the full description of the feedback matrix with


The parameter vectors are defined arbitrarily but have to be linear independent.

Remarks edit

  • Method works for non-quadratic B
  • Parameter vectors   can be chosen arbitrarily

Example edit

Consider the dynamical system


which is unstable due to positive Eigenvalues  . A feedback matrix K should be found to reach a stable closed-loop system with Eigenvalues  .

1. The parameter vectors are defined as   and  

2. The resulting Eigenvectors are




3. The feedback matrix is calculated with


More precise rounding leads to a feedback matrix


Singular Value Decomposition and Diagonalization edit

If the state matrix   of system


is diagonalizable, which means the number of Eigenvalues and Eigenvectors are equal, then the transform


can be used to yield


and further


Transformation matrix M contains the Eigenvectors   as


which leads to a new diagonal state matrix


consisting of Eigenvalues  , and new input


The control law for the new input   is designed as


and the closed-loop system in new coordinates is noted as


Feedback matrix   can be used to influence or shift each Eigenvalue directly.

In the last step, the new input is transformed backwards to original coordinates to yield the original feedback matrix K. The new input is defined by




From these formulas one gains the identity


and further


Therefore, the feedback matrix is found as


Requirements edit

This controller design is applicable only if the following requirements are guaranteed.

  • State matrix A is diagonalizable.
  • The number of states and inputs are equal  .
  • Input matrix   is invertible.

Example edit

Consider the dynamical system


which is unstable due to positive Eigenvalues  . The Eigenvectors are




Thus, the transformation matrix is noted as


and the state matrix in new coordinates is derived as


The desired Eigenvalues of the closed-loop system are   and  , so feedback matrix is found with




and thus one holds


Finally, the feedback matrix in original coordinates are calculated by


Sylvester Equation edit

This method is taken from the online resource

Consider the closed-loop system


with input   and closed-loop state matrix  . The desired closed-loop Eigenvalues   can be chosen real- or complex-valued as   and the matrix of the desired Eigenvalues is noted as


The closed-loop state matrix   has to be similar to   as


which means that there exists a transformation matrix   such that


holds and further



An arbitrary Matrix   is introduced and Equation [2] is separated in a Sylvester equation



and a feedback matrix formula


Algorithm edit

1. Choose an arbitrary matrix  .

2. Solve the Sylvester equation for M (numerically).

3. Calculate the feedback matrix K.

Remarks edit

  • State matrix A and the negative Eigenvalue matrix   shall not have common Eigenvalues.
  • For some choices of G the computation could fail. Then another G has to be chosen.

Example edit

Consider the dynamical system


which is unstable due to positive Eigenvalues  . The complex-valued Eigenvalues   are desired for the closed-loop system. So, the eigenvalue matrix is noted as


Matrix G is chosen as


and Sylvester equation


is noted. The Sylvester equation is solved numerically and the transformation matrix is computed as


Finally, the feedback matrix is found as