§2-3 Observability of Linear Dynamical Equations 1. Definition of observability Observability studies the possibility of estimating the state from the output. Definition 2-6 The dynamical equation is said to be observable at t0, if there exists a finite t1>t0 such that for any state x(t0) at time t0, the knowledge of the input and the output over the time interval suffices to determine the state x(t0). Otherwise, the dynamical equation is said to be unobservable at t0.

Remark: The system is said to be unobservable if there exists a state x(t0) such that x(t0) can not be determined by and uniquely. Question How to determine the state x(t0) with and output ? Example 2-11 Consider the following system whose state transition matrix is

The solution is

Note that Pre-multiplying

known known

we have known It is easy to verify that for .

Hence,

2. Criteria for observability Theorem 2-8 Dynamical equation is observable at time t0 if and only if there exists a finitet1>t0, such that the n columns of matrix is linearly independent over [t0, t1].

2). Pre-multiplying both sides of the equation (＊) with Proof Sufficiency: 1). Consider (＊) 2). Pre-multiplying both sides of the equation (＊) with we have

3). Integrating both sides from t0 to t1, we have Form Theorem 2-1, it follows that V(t0, t1) is nonsingular if and only if the columns of C(t) (t, t0) is linearly independent over [t0, t1].

Necessity: the proof is by contradiction. Assume that the system is observable but the columns of are linearly dependent for any . Then there exists a , such that If we choose , then we have which means that x(t0) can not be determined by y.

Corollay 2-8 The dynamical equation (2-1) is observable at time t0 if and only if there exists a finite time t1>t0 such that the matrix V(t0, t1) is nonsingular, where

Theorem 2-10 Suppose that A(t) and C(t) of the state equation (A(t), B(t), C(t)) are n-1 times continuously differentiable. Then the dynamical equation is observable at t0 if there exists a finite t1>t0 such that where

5. The observability criteria for LTI systems Theorem 2-11 For the n-dimensional linear time invariant dynamical equation （2-21） the following statements are equivalent: All columns of CeAt are linearly independent on [t0, +). (2)All columns of C(sIA)1are linearly independent over C.

(3) The matrix is nonsingular for any t0 ≥0 and t > t0. (4) The n q n observability matrix

(6) For every eigenvalue of A,

§2- 4 Controllability and observability of Jordan canonical form 1. Equivalence transformation Consider Let and . Then we have where

Theorem 2-13: The controllability and observability of a linear time-invariant dynamical equation are invariant under any equivalence transformation. Proof From the Theorem 2-6, is controllable if and only if It is easy to verify that

2. Criteria for controllability and observability of the jordan-form dynamical equations Typical Jordan-canonical form matrix is as follows

Example Determine the controllability and observability of the following system 5 5 5 5 -5 -5 Using PBH rank test:

5 5 5 5 -5 -5 Using PBH rank test:

当系统矩阵有重特征值时，常常可以化为若当形，这时A、 B、C的形式如下：

Theorem 2-14 System in Jordan canonical form is controllable if and only if the rows of the following matrix are linearly independent. System in Jordan canonical form is observable if and only if the columns of are linearly independent.

proof Let Ai be an nith order block, we only need to check Because other sub blocks are full row rank, and

By using PBH test the last row and the first column are zero because Aij is of Jordan canonical form. Therefore, if the matrix formed by the last rows of are linearly independent, then

Similarly, we can prove the observability for system in a Jordan canonical form.

Example Determine the controllability and observability of the following system

Substituting in , we have which is of full row rank .

Substituting in , we have which is of full row rank .

are linearly independent. Therefore, the system is controllable.

Determine the observability of the following system:

Substituting in , we have The sub-block is of full column rank.

Substituting in , we have Because the column C121 is zero, the system is unobservable.

Example Consider the single input system It is easy to check that the system is controllable by using PBH test.