functionanalysis

Information

Syntax

Modelica_LinearSystems2.StateSpace.Analysis.analysis(ss);
   or
Modelica_LinearSystems2.StateSpace.Analysis.analysis(
  ss,
  analyseOptions=analyseOptions,
  fileName,
  systemName,
  description);

Description

This function analyzes a state space system

der(x) = A * x + B * u
    y  = C * x + D * u     (1)
    x(t=0) = x0

based on its poles, i.e. the eigenvalues, and the zeros of the system. The system will be checked for stability, controllability and observability. In the case that the system is not stable stabilizability and detectability are examined. Furthermore, stability, controllability, observability, stabilizability, and detectability are indicated for each eigenvalue.

Stability

System (1) is stable if and only if all eigenvalues of the matrix A have negative real parts. The calculation of the eigenvalues is based on the LAPACK routine dgeev.

Controllability

System (1) is said to be controllable if, starting from any initial state x₀, the system can be driven by appropriate inputs to any final state x₁ within some finite time window. Equivalent is that the eigenvalues of A-BK can arbitrarily be assigned by an appropriate choice of the matrix K.

Stabilizability

System (1) is said to be stabilizable if all the unstable eigenvalues, i.e. all s with Re(s)>=0, of A are controllable. Therefore, a controllable system is always stabilizable. An equivalent definition of stabilizability is, that a system is said to be stabilizable if there exist a matrix K such that A-BK is stable.

Observability

System (1) is said to be observable if the (arbitrary) initial state x₀ can be uniquely determined from any state x(t₁), t₁>0, from the knowledge of the input u(t) and output y(t). With other words, from the system's outputs it is possible to determine the behavior of the entire system. Equivalent is, that the eigenvalues of A-LC can be arbitrarily be assigned by an appropriate choice of matrix L. Observability is called the dual concept of controllability, since a system (A,B,C,D) is observable if the system (A^T, C^T, B^T, D^T) is controllable.

Detectability

System (1) is said to be detectable if all the unstable eigenvalues, i.e. all s with Re(s)>=0, of A are observable. Therefore, a observable system is always detectable. An equivalent definition of detectability is, that a system is said to be detectable if there exist a matrix L such that A-LC is stable. Detectability is called the dual concept of stabilizability, since a system (A,B,C,D) is detectable if the system (A^T, C^T, B^T, D^T) is stabilizable.

Algorithm to test controllability/stabilizability and observability/detectability respectively

The test of controllability and stabilizability is performed with the staircase algorithm which transforms the system (A,B,C,D) into the controller-Hessenberg form (A_H, B_H, C_H, D) with A_H is a block upper Hessenberg matrix and B_H=[B₁; 0] with triangular matrix B₁ with rank(B₁) = rank(B). In A_H=[A_c, *,0, A_nc) the eigenvalues of the matrices A_c and A_nc are the controllable eigenvalues and uncontrollable eigenvalues of A respectively. The test of observability and detectability is performed by testing the system (A^T, C^T, B^T, D^T) with respect to controllability and stabilizability.

Solution of a linear time invariant system

The solution x(t) of the initial value problem (1) consists of the homogeneous part (zero input response) x_h(t) and the inhomogeneous part x_i(t). The zero input solution is given by

xh(t) = exp(A*(t-t0))x0.

The system can also be represented as a linear combination of the modal states z,

x = Vz

i.e. the states of a similar system, with

der(z) = V-1AVz + V-1Bu

where the system matrix V^-1AV is the real Jordan form. For single real eigenvectors the system is decoupled, i.e. the solution of the modal states are denoted by

zi = exp(si t)*z0i

The behavior of the modal states is determined as the solution of a linear first order differential equation for real eigenvalues. Since this behavior is well known, the behavior of the x_i can at least roughly be estimated by means of the behavior of the most relevant modal states. Therefore, the contribution of the modal states to the states is computed as an indication of the original system behavior.

Contribution of the modal states to the states

Generally, as described above, the states of the system can be described as linear combination of modal states and, therefore, the states can be characterized to a certain extend by the modal states if the proportions of the combination are known. Hence, for each modal state z_i of the vector z the elements |v_i,j|/|v_i| of the corresponding right eigenvector v_i indicate the proportion of z_i that is contributed to the state x_j. On the other hand, the composition of xi is indicated by the elements |v_i,j|/|v_i^T|, i.e. the elements |v_i,j|/|v_i^T| of the corresponding row v_i^T of the eigenvector matrix V indicate the proportion of the state x_i that is contributed by the modal state z_j.

Example

  ss=StateSpace(
    A=[-3,2,-3,4,5,6; 0,6,7,8,9,4; 0,2,3,0,78,6; 0,1,2,2,3,3; 0,13,34,0,0,1; 0,
      0,0,-17,0,0],
    B=[1,0; 0,1; 1,0; 0,1; 1,0; 0,1],
    C=[0,0,1,0,1,0; 0,1,0,0,1,1],
    D=[0,0; 0,0],
    xNames={"x1","x2","x3","x4","x5","x6"},
    uNames={"u1","u2"}, yNames={"y1","y2"});

  String fileName="analysis.html";
  String systemName="Demonstration System";
  String description="System to demonstrate the usage of Modelica_LinearSystems2.StateSpace.Analysis.anlysis()"

algorithm
  Modelica_LinearSystems2.StateSpace.Analysis.analysis(ss, fileName=fileName, systemName=systemName, description=description)
//  gives:

System report

The system Demonstation System

der(x) = A * x + B * u
    y  = C * x + D * u

is defined by

        x1   x2   x3   x4   x5   x6            u1  u2
    x1  -3    2   -3    4    5    6         x1  1   0
    x2   0    6    7    8    9    4         x2  0   1
A = x3   0    2    3    0   78    6     B = x3  1   0
    x4   0    1    2    2    3    3         x4  0   1
    x5   0   13   34    0    0    1         x5  1   0
    x6   0    0    0  -17    0    0         x6  0   1

        x1   x2   x3   x4   x5   x6            u1  u2
C = y1   0    0    1    0    1    0     D = y1  0   0
    y2   0    1    0    0    1    1         y2  0   0

Description

System to demonstrate the usage of Modelica_LinearSystems2.StateSpace.Analysis.analysis()

Characteristics

The system
is not stable
but it is controllable
and therefore it is stabilizable
The system is not observable
but it is detectable

Eigenvalues analysis

Real eigenvalues

Conjugated complex pairs of eigenvalues

In the table above, the column contribution to states lists for each eigenvalue the states to which thecorresponding modal state contributes most. This information is based on the two largest absolute values of the corresponding right eigenvector (if the second large value is less than 5 % of the largest contribution, it is not shown).

In the next table, for each state in the column correlation to modal states, the modal states which contribute most to the coresponding state are summarized, i.e. the state is mostly composed of these modal states This information is based on the two largest absolute values of row i of the eigenvector matrix that is associated with eigenvalue i (if the second large value is less than 5 % of the largest contribution, it is not shown). This only holds if the modal states are in the same order of magnitude. Otherwise, the modal states listed in the last column might be not the most relevant one.

Invariant zeros

Inputs

Contents

Revisions