§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 nonzero state x(t0) at time t0, the knowledge of the input u[t0, t1] and the output y[t0, t1] over the time interval [t0, t1] suffices to determine the state x(t0).

Remark: The system is said to be unobservable if there exists a nonzero state x(t0) such that x(t0) can not be determined by u[t0, t1] and y[t0, t1] uniquely. Question: How can we determine the state x(t0) with input u[t0, t1] and output y[t0, t1] ? Example 2-11 Consider the following system whose state transition matrix is

For simplicity, let u=0. Then, the solution is Therefore, Since y is a scalar, it is impossible to determine x10 and x20 from the knowledge of y only. The way out is as follows.

Pre-multiplying both sides of the above equation by we obtain that Integrating the equation from t0 to t1 yields

known known

we have known It is easy to verify that for t1>t0.

Hence,

2. Criteria for observability Theorem 2-8 Dynamical equation is observable at time t0 if and only if there exists a finite t1>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) are linearly independent over [t0, t1].

Necessity: the proof is by contradiction. Assume that the system is observable but the columns of C(t)(t, t0) are linearly dependent for any t1>t0 . Then, there exists a column vector 0, such that If we choose x(t0)=, then we have which means that x(t0) can not be determined by y.

Corollary 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 2-9 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

3. Theorem of duality Consider the dynamical equations and Theorem 2-10: (1). The system (I) is controllable at t0  the system (II) is observable at t0.

(2). The system (I) is observable at t0the system (II) is controllable at t0. Proof: Let (t, t0) be the state transition matrix of system (I). Then, it can be checked that is the state transition matrix of system (II). Therefore, the system (I) is controllable at t0  t1>t0, such that the n rows of (t0, )B() are linearly independent on [t0, t1] 

That is, the n columns of B. (t) That is, the n columns of B*(t)*–1(, t0) are linearly independent on [t0, t1] the system (II) is observable at t0. Similarly, one can also prove statement (2) of the theorem.

4. 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: (1). (221) is observable for any t0 in [0, +); (2). All the columns of CeAt are linearly independent on [t0, +).

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

(5). All columns of C(sIA)1 are linearly independent over . (6) For every eigenvalue i 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 Theorem 2-6, It is easy to verify that Hence, the controllability is invariant under any equivalence transformation. The same conclusion is also true for observability.

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

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

5 5 5 5 -5 -5 PBH test:

Example. Determine the controllability and observability of the following system

Substituting 1 in [A–I, B], we have which is of full row rank .

Substituting 2 in [A–I, B], we have which is of full row rank .

The two matries are linearly independent. Therefore, the system is controllable.

Then, we determine the observability of the system:

Substituting 1 in , we have The sub-block is of full column rank.

Substituting 2 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.