The tool DC-11, by computing the eigenvalues, controllability matrix and observability matrix of a given system, tells the status of stability, controllability, observability and stabilizability of the system.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

## Tutorial

Example: Properties of State Representation

The input in the tool is

After clicking "Run", the results will be shown in another page.

Eigenvalues

The eigenvalues of the system are

λ1 = -1.41421356

λ2 = 1.41421356

Stability

The system is unstable.

Controllability

Controllability Matrix C(A,B) =

0.0   1.0   ;

1.0   1.0   ;

Rank [C(A,B)] = 2

The system is controllable.

Modal Controllability

Rank [A-λ1*I   B] = 2 (Full Rank. Subspace about λ1 is controllable)

Rank [A-λ2*I   B] = 2 (Full Rank. Subspace about λ2 is controllable)

Observability

Observability Matrix O(A,C) =

1.0   0.0   ;

-1.0   1.0   ;

Rank [O(A,C)] = 2

The system is observable.

Modal Observability

Rank [AT1*I   CT] = 2 (Full Rank. Subspace about λ1 is observable)

Rank [AT2*I   CT] = 2 (Full Rank. Subspace about λ2 is observable)

Stabilizability

The system is stabilizable with a state feedback.

## Subject Review

Consider a linear time-invariant (LTI) dynamic system in state representation

where x is the vector of state variables, u the vector of inputs, and y the vector of outputs. For a dynamic system with n state variables, p inputs and q outputs, the matrices and vectors in the previous equations are of the following dimensions:

For convenience of discussion, denote the LTI system described in Eqs. (1) and (2) by S.

### System Eigenvalues

The eigenvalue problem of system S is that of the matrix A in the state equation (1):

where λ is an eigenvalue, and u an associated eigenvector. The eigenvalues are the roots of the characteristic equation

which is an nth-order polynomial of λ with n roots.

### Stability Criteria

A stable LTI system is a dynamic system with a bounded response to a bounded input. Assume that the input vector u in Eq. (1) is bounded. The following three stability definitions about system S are in order:

S is marginally stable if its output is bounded, namely, ||y(t)|| is bounded for any t > 0;

S is asymptotically stable if it is marginally stable and ||y(t)||→0 as t→∞; and

S is unstable if ||y(t)|| is unbounded.

In the above definitions, ||*|| is a vector norm. For a LTI dynamic system, marginal stability is the same as Lyapunov stability.

Denote the eigenvalues of matrix A by λ1, λ2, ..., λn. The following three stability criteria are related to the eigenvalues.

Criterion 1. System S is asymptotically stable if and only if all the eigenvalues have negative real parts; namely,

Criterion 2. System S is marginally stable if and only if the following two conditions hold true:

(a) All the eigenvalues have non-positive real parts, Re(λk)≤0 for k=1,2,...,n; and

(b) For any eigenvalue λ with zero real part and multiplicity m, there are exactly m linearly independent eigenvectors determined from (λI-A)uj=0, j=1,2,...,m.

Criterion 3. System S is unstable if at least one eigenvalue has positive real part or if for any eigenvalue λ with zero real part and multiplicity m, less than m linearly independent eigenvectors can be determined from (λI-A)uj=0.

The number r of linearly independent eigenvectors associated with an eigenvalue λ of an n-by-n matrix A with multiplicity m is determined by

where rank(λI-A)≥n-m. As can be seen, only rank(λI-A)=n-m warrants the exactly m linearly independent eigenvectors.

Note: An eigenvalue that has negative real part shall be called a stable eigenvalue. An eigenvalue that has positive real part shall be called an unstable eigenvalue.

Example: Consider the following four systems with eigenvalues.

System 1: eigenvalues -1, -2±j3

System 2: eigenvalues -1, ±j3

System 3: eigenvalues 1, -2±j3

System 4: eigenvalues -1, j3, j3, -j3, -j3

By the above stability criteria, System 1 is asymptotically stable, System 2 is marginally stable, and System 3 and 4 are unstable.

The above stability criteria require the information on system eigenvalues. There are certain methods that detect stability status without having to know system eigenvalues, including Routh-Hurwitz stability criterion, Nyquist stability criterion, and Lyapunov stability criterion. These methods are not covered in this tool.

### Controllability

Definition: System S or the pair (A, B) is said to be controllable if for any initial state x(0)=x0 and any final state x(tf)=xf, there exists an input u(t) that transfers x0 to xf over a finite time interval. Otherwise, S is said to be uncontrollable.

Theorem 1 (Controllability condition): System S or the pair (A, B) is controllable if and only if the n-by-np controllability matrix

has rank n (full row rank).

Note: If S is a single-input system, the controllability matrix is an n-by-n matrix and the controllability condition becomes

Theorem 2 (Modal controllability condition): System S or the pair (A, B) is controllable if and only if the n-by-(n+p) matrix

has rank n (full row rank) at every eigenvalue λ of matrix A.

Note: An eigenvalue that does not satisfy the condition of Theorem 2 shall be called an uncontrollable eigenvalues.

The condition of Theorem 1 is equivalent to that of Theorem 2. The condition of Theorem 1 does not require information of system eigenvalues. However, the condition of Theorem 2 tells which eigenvalues render the system uncontrollable.

### Observability

Definition: System S or the pair (A, C) is said to be observable if for any unknown initial state x(0)=x0 there exists a finite time t1>0 such that the knowledge of the input and output over 0≤y≤t1 is sufficient to determine uniquely the initial state x0. Otherwise, S is said to be unobservable.

Theorem 3 (Observability condition): System S or the pair (A, C) is observable if and only if the nq-by-n observability matrix

has rank n (full column rank).

Note: If S is a single-output system, the observability matrix is an n-by-n matrix and the observability condition becomes

Theorem 4 (Modal observability condition): System S or the pair (A, C) is observable if and only if the (n+q)-by-n matrix

has rank n (full column rank) at every eigenvalue λ of matrix A.

Note: An eigenvalue that does not satisfy the condition of Theorem 4 shall be called an unobservable eigenvalues.

### Stabilizability

A LTI system is stabilizable if a state feedback can be applied to render all the eigenvalues of the closed-loop system stable. However, state feedback is only applicable to controllable eigenvalues, and does not have effect on uncontrollable eigenvalues. Thus, an unstable system S is stabilizable if one of the following two conditions is met:

(a) S is controllable; or

(b) S is uncontrollable but every uncontrollable eigenvalue is a stable one.

Example: Consider a dynamic system with matrices

The eigenvalues are

According to Theorem 2, λ1 and λ3 are controllable, but λ2 is uncontrollable. However, λ2 is stable because it has negative real part. Thus, the dynamic system is stabilizable.

### References

1. C.T. Chen, 1999, Linear System Theory and Design, 3rd Edition, Oxford University Press.

2. P.E. Sarachik, 1985, Principles of Linear Systems, Cambridge University Press, 1997.

3. B. Friedland, 1985, Control System Design: An Introduction to State Space Methods, McGraw-Hill, Inc.