1 數位控制（十）

2 Continuous time SS equations

3 Discretization of continuous time SS equations

4 Liapunov Stability Analysis An equilibrium state x of the system is said to be stable in the sense of Liapunov, if corresponding to each S(e), there is an S(d) such that trajectory starting in S(d) do not leave S(e) as t increase indefinite

5 Pole placement and observer design Controllability: to transfer the system from any arbitrary initial state to any desired state. Observability: every initial state x(0) can be determined from the observation of y(kT). The controllability is the basis for the pole placement problem. The concept of observability play an important role for the design of state observer. Pole placement design technique feedback all state variables so that all poles of the closed-loop system are placed at designed location.

6 Open-loop control system Closed-loop control system with u(k)=-Kx(k)

7 Pole placement techniques Assuming all state variables are available for feedback. To design the state observer that estimates all state variables that are required for feedback.

8 Simple Pole Placement

9 Lower order system

10 Pole placement basis Completely state controllable: the desired closed-loop poles can be selected. All the state variables are feedback to place the poles. In practical measurement of all state variables may not be possible. Hence, not all state variables will be available for feedback. Completely output observable: make all state variables be observable or feedback.

11 Controllability matrix

12 Rank of a matrix A matrix A is called of rank n if the maximum number of linear independent rows (or columns) is n. Important Properties The rank is invariant under the interchange of two rows (or columns), or addition, or multiplication. For a n by n matrix A, for rank A=n imply det(A) is not equal to zero. For a n by n matrix A, rank A*=rank A, or rank A T =rank A.

13 Complete State Controllability

14 For multiple eigenvectors (Jordan form) The system is completely state controllable if and only if No two Jordan blocks are associated with the same eigenvalues, The element of any row of the transformed H that corresponds to the last row of each Jordan block are not all zero, The elements of each row of the transformed H that correspond to distinct eigenvalues are not all zero,

15 Example- completely state controllable

16 Example- not completely state controllable

17 Complete state controllability in the z plane No cancellation in the pulse transfer function. If cancellation occurs, the system cannot be controlled in the direction of the canceled mode.

18 Complete Output Controllability

19 Complete Output Controllability (w/ D)