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Digital Control Systems Controllability&Observability
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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
OBSERVABILITY
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
SDLTI is observable iff
SDLTI is constructible iff
SDLTI is controllable/reachable/controllable from the origin iff
SDLTI is controllable to zero iff
𝑟𝑎𝑛𝑘 [𝑠𝐼−𝐺 ⋮ 𝐻 ]=𝑛 𝐺𝑟𝑎𝑛𝑘 [𝜆𝑖 𝐼−𝐺 ⋮ 𝐻 ]=𝑛
[ 𝐶𝑠𝐼 −𝐺]
[ 𝑠𝐼−𝐺 ⋮ 𝐻 ]
𝑟𝑎𝑛𝑘 [ 𝐶𝜆 𝐼−𝐺 ]=𝑛 ∀ 𝜆∈ℂ
𝑟𝑎𝑛𝑘 [ 𝑠𝐼−𝐺 ⋮ 𝐻 ]=𝑛 ∀ 𝜆∈ℂ
𝑟𝑎𝑛𝑘 [𝜆 𝐼−𝐺 ⋮ 𝐻 ]=𝑛 ∀ 𝜆∈ℂ/ {0 }
𝑟𝑎𝑛𝑘 [ 𝐶𝜆 𝐼−𝐺 ]=𝑛 ∀ 𝜆∈ℂ/ {0 }
OBSERVABILITY
Popov-Belevitch-Hautus (PBH) Tests for Controllability/Observability
SDLTI is observable iff
SDLTI is constructible iff
SDLTI is controllable/reachable/controllable from the origin iff
SDLTI is controllable to zero iff
𝑟𝑎𝑛𝑘 [𝑠𝐼−𝐺 ⋮ 𝐻 ]=𝑛 𝐺𝑟𝑎𝑛𝑘 [𝜆𝑖 𝐼−𝐺 ⋮ 𝐻 ]=𝑛
[ 𝐶𝑠𝐼 −𝐺]
[ 𝑠𝐼−𝐺 ⋮ 𝐻 ]
𝑟𝑎𝑛𝑘 [ 𝐶𝜆 𝐼−𝐺 ]=𝑛 ∀ 𝜆∈ℂ
𝑟𝑎𝑛𝑘 [ 𝑠𝐼−𝐺 ⋮ 𝐻 ]=𝑛 ∀ 𝜆∈ℂ
𝑟𝑎𝑛𝑘 [𝜆 𝐼−𝐺 ⋮ 𝐻 ]=𝑛 ∀ 𝜆∈ℂ/ {0 }
𝑟𝑎𝑛𝑘 [ 𝐶𝜆 𝐼−𝐺 ]=𝑛 ∀ 𝜆∈ℂ/ {0 }
OBSERVABILITY
Condition for Complete Observability in the z-Plane
Example:
Since, det ( ), rank ( ) is less than 3.
Note: A square matrix An×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: S2:
OBSERVABILITY
Principle of Duality
OBSERVABILITY
Principle of Duality
S1 is completely state controllabe S2 is completely observable.
S1 is completely observable S2 is completely state controllable.
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Transforming State-Space Equations Into Canonical forms:
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Transforming State-Space Equations Into Canonical forms:
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Transforming State-Space Equations Into Canonical forms:
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Transforming State-Space Equations Into Canonical forms:
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Transforming State-Space Equations Into Canonical forms:
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Transforming State-Space Equations Into Canonical forms:
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Invariance Property of the Rank Conditions for the Controllability Matrix and Observability Matrix
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Kalman’s Controllability Decomposition
Kalman Decomposition:
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Kalman Decomposition
Kalman Decomposition:
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Kalman’s Controllability Decomposition
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Kalman’s Controllability Decomposition
Partition the transformed into
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Kalman’s Controllability Decomposition
Example:
x(k+1)= x(k) + u(k)
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Kalman’s Observability Decomposition
1
1
A W AW
B W B
C CW
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Kalman’s Controllability and Observability Decomposition
USEFUL TRANSFORMATION IN STATE-SPACE ANALYSISAND DESIGN
Kalman’s Controllability and Observability Decomposition
0
0
VT
W
𝑇=[𝑉 00 𝑊 ]
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