Determining Initial Conditions from Output Measurements in Control Engineering, Lecture notes of Control Systems

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Control Engineering

Ramkrishna Pasumarthy

Department of Electrical Engineering, Indian Institute of Technology Madras

Module 12

Lecture 5

Observability

Motivation: Output Feedback

Consider the continuous time LTI system x^ ˙ = Ax + Bu y = Cx + Du

, x ∈ Rn, u ∈ Rm, y ∈ Rp^ (1)

Recall that if the pair (A, B) is stabilizable, then ∃ a state feedback control law u = −Kx (2)

that asymptotically stabilizes the system (1) for which A − BK is a stability matrix. Assume that the matrix C in (1) is invertible. Then x can be determined from y and u by solving the output equation: x(t) = C−^1

y(t) − Du(t)

and the feedback control law in (2) can be implemented suitably.

Motivation: Output Feedback

When the number of outputs are strictly less than the number of states then the output matrix C is not invertible and hence x cannot be reconstructed. Then the question arises

‘If only the output y can be measured where the number of outputs is strictly smaller than the states then will it be possible to implement the control law in (2)?’

Unobservable and

Unconstructible Subspaces

Introduction

When the number of outputs is strictly smaller than the number of states, instantaneous reconstruction of the state from the input u(t) and output y(t) is not possible.

Would it be possible to reconstruct the state from u(t), y(t) over an interval [t 0 , t 1 ]?

The usual tools to define the system characteristics for state estimation are, namely, observability and constructability.

▶ Observability refers to determining x(t 0 ) from the future inputs, u(t), and outputs, y(t), where t ∈ [t 0 , t 1 ]. ▶ Constructability refers to determining x(t 1 ) from the past inputs, u(t), and outputs, y(t) where t ∈ [t 0 , t 1 ].

Unobservable Subspace

Observability of the system (3) indicates determining the intial condition of the states, x 0 from the output equation (4) i.e. the conditions are to be determined for which the equation

ˇy(t) = C(t)Φ (t, t 0 ) x 0 , ∀ t ∈ [t 0 , t 1 ] (5)

can be solved without any prior knowledge of the initial condition x 0 ∈ Rn, where

ˇy(t) = y(t) −

∫ (^) t

t 0

C(t)Φ (t, τ ) B (τ ) u (τ ) dτ − D(t)u(t), ∀ t ∈ [t 0 , t 1 ]

Unobservable Subspace

Definition 8.1. Given two times t 1 > t 0 ≥ 0, the unobservable subspace on [t 0 , t 1 ] U O[t 0 , t 1 ] consists of all states x 0 ∈ Rn^ for which

C(t)Φ (t, t 0 ) x 0 = 0 , ∀ t ∈ [t 0 , t 1 ]

Definition 8.1. Given two times t 1 > t 0 ≥ 0, the system (3) is observable if its unobservable subspace contains only the zero vector i.e.

U O[t 0 , t 1 ] = 0

Overview

Summary: Module 12 Lecture 5 ▶ Observability

Contents: Module 12 Lecture 6 ▶ Observability Test