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Digital Control Systems Controllability&Observability.

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Presentation on theme: "Digital Control Systems Controllability&Observability."— Presentation transcript:

1 Digital Control Systems Controllability&Observability

2 CONTROLLABILITY Complete State Controllability for a Linear Time Invariant Discrete-Time Control System

3 CONTROLLABILITY Complete State Controllability for a Linear Time Invariant Discrete-Time Control System Controllability matrix

4 CONTROLLABILITY Complete State Controllability for a Linear Time Invariant Discrete-Time Control System

5 CONTROLLABILITY Complete State Controllability for a Linear Time Invariant Discrete-Time Control System Condition for complete state controllability

6 CONTROLLABILITY Complete State Controllability for a Linear Time Invariant Discrete-Time Control System Condition for complete state controllability

7 CONTROLLABILITY Complete State Controllability for a Linear Time Invariant Discrete-Time Control System Condition for complete state controllability Example:

8 CONTROLLABILITY Complete State Controllability for a Linear Time Invariant Discrete-Time Control System Condition for complete state controllability Example:

9 CONTROLLABILITY Determination of Control Sequence to Bring the Initial State to a Desired State

10 CONTROLLABILITY Condition for Complete State Controllability in the z-Plane Example:

11 CONTROLLABILITY Complete Output Controllability

12 CONTROLLABILITY Complete Output Controllability

13 CONTROLLABILITY Complete Output Controllability

14 CONTROLLABILITY Controllability from the origin : controllability : reachability

15 OBSERVABILITY

16

17 Complete Observability of Discrete-Time Systems

18 OBSERVABILITY Complete Observability of Discrete-Time Systems

19 OBSERVABILITY Complete Observability of Discrete-Time Systems Observability matrix

20 OBSERVABILITY Complete Observability of Discrete-Time Systems

21 OBSERVABILITY Complete Observability of Discrete-Time Systems Example:

22 OBSERVABILITY Complete Observability of Discrete-Time Systems Example:

23 OBSERVABILITY Popov-Belevitch-Hautus (PBH) Tests for Controllability/Observability S D LTI is observable iff S D LTI is constructible iff S D LTI is controllable/reachable/controllable from the origin iff S D LTI is controllable to zero iff

24 OBSERVABILITY Popov-Belevitch-Hautus (PBH) Tests for Controllability/Observability S D LTI is observable iff S D LTI is constructible iff S D LTI is controllable/reachable/controllable from the origin iff S D LTI is controllable to zero iff

25 OBSERVABILITY Condition for Complete Observability in the z-Plane Example: Since, det ( ), rank ( ) is less than 3. Note: A square matrix A n×n is non-singular only if its rank is equal to n.

26 OBSERVABILITY Condition for Complete Observability in the z-Plane Example: Since, det ( )=0, rank ( ) is less than 3.

27 OBSERVABILITY Principle of Duality S1:S1: S2:S2:

28 OBSERVABILITY Principle of Duality

29 OBSERVABILITY Principle of Duality S 1 is completely state controllabe S 2 is completely observable. S 1 is completely observable S 2 is completely state controllable.

30 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Transforming State-Space Equations Into Canonical forms:

31 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Transforming State-Space Equations Into Canonical forms:

32 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Transforming State-Space Equations Into Canonical forms:

33 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Transforming State-Space Equations Into Canonical forms:

34 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Transforming State-Space Equations Into Canonical forms:

35 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Transforming State-Space Equations Into Canonical forms:

36 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Invariance Property of the Rank Conditions for the Controllability Matrix and Observability Matrix

37 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Kalman’s Controllability Decomposition Kalman Decomposition:

38 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Kalman Decomposition Kalman Decomposition:

39 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Kalman’s Controllability Decomposition

40 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Kalman’s Controllability Decomposition Partition the transformed into

41 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Kalman’s Controllability Decomposition Example: x(k+1)= x(k) + u(k)

42 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Kalman’s Observability Decomposition

43 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Kalman’s Controllability and Observability Decomposition

44 USEFUL TRANSFORMATION IN STATE-SPACE ANALYSIS AND DESIGN Kalman’s Controllability and Observability Decomposition

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