1. 8. Stability, controllability and observability

2. 8.1 Stability For continuous-time system, the state space model is
The corresponding transfer function is
where det(sI-A) is a polynomial which forms the denominator of the system transfer function. That means the roots of det(sI-A)=0 (eigenvalues of A) are the poles of G(s).

3. 8.1 Stability For discrete-time system, the state space model is
The corresponding transfer function is
where det(zI-) is a polynomial which forms the denominator of the system transfer function. That means the roots of det(zI-)=0 (eigenvalues of  ) are the poles of G(z).

4. 8.1 Stability
Therefore, we can tell the stability of a system by calculating the eigenvalues of A or .
We can easily tell the stability of a system by inspecting its plant matrix A in continuous-time system or  in discrete-time system if A or  are in Diagonal or Jordan canonical form.

5. 8.1 Stability
Liapunov stability analysis plays an important role in the stability analysis of control systems described by state space equations. From the classical theory of mechanics, we know that a vibratory system is stable if its total energy is continually decreasing until an equilibrium state is reached. Liapunov stability is based on a generalization of this fact.

6. 8.1 Stability
Liapunov stability: If a system has an asymptotically stable equilibrium state, then the stored energy of the system displaced within a domain of attraction decays with increasing time until it finally assumes its minimum value at the equilibrium state (Details for Liapunov stability and theorems are given in section 5.6 on page in our textbook).

7. 8.1 Stability
Liapunov theorem on stability: Suppose a system is described by x’=f(x,t) where f(0,t)=0 for all t. If there exists a scalar function V(x,t) having continuous first partial derivatives and satisfying the conditions
V(x,t) is positive define (V(x,t)>0) .
V’(x,t) is negative semi-definite (V’(x,t)<0).
Then the equilibrium state at the origin is stable.

8. 8.1 Stability
Liapunov stability of Linear system: Consider the system described by x’=Ax, where x is a state vector and A is an nonsingular matrix. A necessary and sufficient condition for the equilibrium state x=0 to be asymptotically stable is that, given any positive definite Hermitian (or any positive defined real asymmetric) matrix Q, there exists a positive definite Hermitian (or a positive definite real asymmetric) matrix P such that ATP+PA=-Q.

9. 8.1 Stability
Liapunov stability of Linear system: Consider the system described by x(k+1)=x(k), where x is a state vector and  is an nonsingular matrix. A necessary and sufficient condition for the equilibrium state x=0 to be asymptotically stable is that, given any positive definite Hermitian (or any positive defined real asymmetric) matrix Q, there exists a positive definite Hermitian (or a positive definite real asymmetric) matrix P such that TP -P =-Q.

10. 8.2 Controllability
Controllability: A control system is said to be completely state controllable if it is possible to transfer the system from any arbitrary initial state to any desired state in a finite time period. That is, a control system is controllable if every state variable can be controlled in a finite time period by some unconstrained control signal. If any state variable is independent of the control signal, then it is impossible to control this state variable and therefore the system is uncontrollable.

11. 8.2.1 Controllability for a discrete-time control system
Consider the discrete-time system defined by
We assume that u(k) is a constant for the sampling period (ZOH). If given the initial state X(0), can we find a control signal to drive the system to a desired state X(k)? In other words, is this system controllable?
Using the definition just given, we shall now derive the condition for complete state controllability.

12. 8.2.1 Controllability for a discrete-time control system
Since the solution of the above equation is

13. 8.2.1 Controllability for a discrete-time control system
That is

14. 8.2.1 Controllability for a discrete-time control system
As  is an k1 matrix, we find that each term of the matrix is an k1 matrix or column vector. That is the matrix
is an kk matrix. If its determinant is not zero, we can get the inverse of this matrix. Then we have

15. 8.2.1 Controllability for a discrete-time control system
That means that if we chose the input as the above, we can transfer the system from any initial state X(0) to any arbitrary state X(k) in at most k sampling periods.
That is the system is completely controllable if and only if the inverse of controllability matrix
is available. Or the rank of the controllability matrix is k.

16. 8.2.1 Controllability for a discrete-time control system
Why do we concern about the controllability of a system?

17. 8.2.2 Examples for controllability
Example 1: Considering a system defined by
is this system controllable?
First, write the state equation for the above system

18. 8.2.2 Examples for controllability
That is
Next, find the controllability matrix

19. 8.2.2 Examples for controllability
Finally, determine the rank of controllability matrix
The rank is 2. Therefore, the system is controllable.

20. 8.2.2 Examples for controllability
Exercise 1: Given the following system
is it controllable?

21. 8.3 Observability
Observability: A control system is said to be observable if every initial state X(0) can be determined from the observation of Y(k) over a finite number of sampling periods. The system, therefore, is completely observable if every transition of the state eventually affects every element of the output vector.
In the state feedback scheme, we require the feedback of all state variables. In practical, however, some of the state variables are not accessible for direct measurement. Then it becomes necessary to estimate the un-measurable variables in order to implement the state variable feedback scheme.

22. 8.3.1 Observability for a discrete-time control system
Consider the discrete-time system defined by
we assume that u(k) is a constant for the sampling period (ZOH). If given the input signal u(k) and the output y(k), can we track back to the initial state X(0)? Or, is this system observable?

23. 8.3.1 Observability for a discrete-time control system
Since the solution of the above equation is
And y(k) is

24. 8.3.1 Observability for a discrete-time control system
Let k=0, 1, 2…n-1, we have

25. 8.3.1 Observability for a discrete-time control system
Rewrite all the above equations in matrix equation, we have

26. 8.3.1 Observability for a discrete-time control system
As C is an 1n matrix, we find that each term of the following matrix is an 1n matrix or row vector. That is the matrix is an nn matrix. If its determinant is not zero, we can get the inverse of the observability matrix.

27. 8.3.1 Observability for a discrete-time control system
Then we have

28. 8.3.1 Observability for a discrete-time control system
This means that if we knew the input and the output in n sampling periods, we can track back to the system initial state.
That is the system is completely observable if and only if the inverse of observability matrix is available. Or the rank of the observability matrix is n.

29. 8.3.1 Observability for a discrete-time control system
Why do we concern about the observability of a system?

30. 8.3.2 Examples for observability
Example 2: Given the following system
Determine its observability?

31. 8.3.2 Examples for observability
Build up the observability matrix
Find the determinant of observability matrix

32. 8.3.2 Examples for observability
Exercise 2: Given the following system
determine its observability?

33. 8.4 Controllability and observability for continuous system
For a continuous system
We can draw the similar conclusions as the discrete system

34. 8.4.1 Controllability of continuous system
A system is completely controllable if and only if the inverse of controllability matrix is available. Or the rank of the controllability matrix is n.

35. 8.4.2 Observability of continuous system
A system is completely observable if and only if the inverse of observability matrix is available. Or the rank of the observability matrix is n.

36. 8.5 Effects of discretization on controllability and observability
When a continuous-time system with complex poles is discretized, the introduction of sampling may impair the controllability and observability of the resulting discretized system. That is, pole-zero cancellation may take place in passing from the continuous-time case to the discrete-time case. Thus, the discretized system may lose controllability and observability.

37. 8.5 Effects of discretization on controllability and observability
Example 3: Consider the following continuous-time system:
Determine its controllability and observability. If the system is discretized, is the discretized system still controllable and observable?

38. 8.5 Effects of discretization on controllability and observability
This system is controllable and observable since
and

39. 8.5 Effects of discretization on controllability and observability
The discrete-time system can be obtained by

40. 8.5 Effects of discretization on controllability and observability
The discrete-time system obtained by discretization is

41. 8.5 Effects of discretization on controllability and observability
The controllability matrix
The observability matrix
If , the system will lose its controllability and observability.

42. 8.5 Effects of discretization on controllability and observability
Exercise 3: Consider the system given by following transfer function
Represent it as controllable canonical form and observable canonical form in state equations, and determine its controllability and observability.

43. Tutorials Exercise 1& 2: Given the following system
determine its controllability and observability?

44. Tutorials
Solution:

45. Tutorials
Exercise 3: Consider the system given by following transfer function
Represent it as controllable canonical form and observable canonical form in state equations, and determine its controllability and observability.

46. Tutorials
Solution:

47. Tutorials
Controllability matrix & Observability matrix

48. Tutorials
The apparent difference in the controllability and observability of the same system is caused by the fact that the original system has a pole-zero cancellation in the transfer function. If a pole-zero cancellation occurs in the transfer function, then the controllability and the observability vary, depending on how the state variables are chosen.