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Digital Control Systems Controllability&Observability

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

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

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

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

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

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

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

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

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

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CONTROLLABILITY Complete Output Controllability

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CONTROLLABILITY Complete Output Controllability

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CONTROLLABILITY Complete Output Controllability

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CONTROLLABILITY Controllability from the origin : controllability : reachability

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OBSERVABILITY

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Complete Observability of Discrete-Time Systems

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OBSERVABILITY Complete Observability of Discrete-Time Systems

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OBSERVABILITY Complete Observability of Discrete-Time Systems Observability matrix

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OBSERVABILITY Complete Observability of Discrete-Time Systems

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OBSERVABILITY Complete Observability of Discrete-Time Systems Example:

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OBSERVABILITY Complete Observability of Discrete-Time Systems Example:

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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

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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

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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.

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

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

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OBSERVABILITY Principle of Duality

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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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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