Control Systems/Eigenvalue Assignment for MIMO Systems

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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 | edit source]

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 synthesis[edit | edit source]

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.

Example[edit | edit source]

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 Diagonalization[edit | edit source]

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

Requirements[edit | edit source]

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 | edit source]

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 Equation[edit | edit source]

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

Algorithm[edit | edit source]

1. Choose an arbitrary matrix .

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

3. Calculate the feedback matrix K.

Remarks[edit | edit source]

  • 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 | edit source]

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


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