Digital Control Systems Controllability&Observability.

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Presentation transcript:

Digital Control Systems Controllability&Observability

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

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

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

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

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

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

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

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

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

CONTROLLABILITY Complete Output Controllability

CONTROLLABILITY Complete Output Controllability

CONTROLLABILITY Complete Output Controllability

CONTROLLABILITY Controllability from the origin : controllability : reachability

OBSERVABILITY

Complete Observability of Discrete-Time Systems

OBSERVABILITY Complete Observability of Discrete-Time Systems

OBSERVABILITY Complete Observability of Discrete-Time Systems Observability matrix

OBSERVABILITY Complete Observability of Discrete-Time Systems

OBSERVABILITY Complete Observability of Discrete-Time Systems Example:

OBSERVABILITY Complete Observability of Discrete-Time Systems Example:

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

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

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.

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

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

OBSERVABILITY Principle of Duality

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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