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

Controllability matrix

CONTROLLABILITY

Condition for complete state controllability

CONTROLLABILITY

Example:

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

Controllability from the origin : controllability : reachability

OBSERVABILITY

Complete Observability of Discrete-Time Systems

OBSERVABILITY

Observability matrix

OBSERVABILITY

Example:

OBSERVABILITY

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

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:

Invariance Property of the Rank Conditions for the Controllability Matrix and Observability Matrix

Kalman’s Controllability Decomposition

Kalman Decomposition:

Kalman Decomposition

Kalman Decomposition:

Partition the transformed into

Example:

x(k+1)= x(k) + u(k)

Kalman’s Observability Decomposition

A W AW

Kalman’s Controllability and Observability Decomposition

𝑇=[𝑉 00 𝑊 ]

Digital Control Systems Controllability&Observability.

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