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 FeedbackEdit

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

[1]

 

Controller synthesisEdit

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.

ExampleEdit

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

 

and

 

3. The feedback matrix is calculated with

 

More precise rounding leads to a feedback matrix

 

Singular Value Decomposition and DiagonalizationEdit

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

 

and

 

From these formulas one gains the identity

 

and further

 

Therefore, the feedback matrix is found as

 

RequirementsEdit

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.

ExampleEdit

Consider the dynamical system

 

which is unstable due to positive Eigenvalues  . The Eigenvectors are

 

and

 

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

 

and thus one holds

 

Finally, the feedback matrix in original coordinates are calculated by

 

Sylvester EquationEdit

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

[2]

 

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

[Sylvester]

 

and a feedback matrix formula

 

AlgorithmEdit

1. Choose an arbitrary matrix  .

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

3. Calculate the feedback matrix K.

RemarksEdit

  • 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.

ExampleEdit

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