A feedback control system modelled by the differential equation * + ax + kx = 0 is known to be asymptotically stable, for k > 0, a > 0. Set up the state-space form of the equation and show that V(x₁, x₂) = kx² + (x₂ + ax₁)², x₁ = x, x₂ = x is a suitable Lyapunov function for verifying this.
A feedback control system modelled by the differential equation * + ax + kx = 0 is known to be asymptotically stable, for k > 0, a > 0. Set up the state-space form of the equation and show that V(x₁, x₂) = kx² + (x₂ + ax₁)², x₁ = x, x₂ = x is a suitable Lyapunov function for verifying this.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![1.11.1
A feedback control system modelled by the
differential equation
x + ax + kx = 0
is known to be asymptotically stable, for k > 0,
a> 0. Set up the state-space form of the equation
and show that
V(x₁, x₂) = kx² + (x₂ + ax₁)²₂ x₁ = x₂ x₂ = x
is a suitable Lyapunov function for verifying
this.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75fdb37c-adc4-4f09-9ed1-9117a89a9487%2F2e60e50e-b1de-4e1f-b522-6f94f7a7a48b%2Fzvrnn66_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.11.1
A feedback control system modelled by the
differential equation
x + ax + kx = 0
is known to be asymptotically stable, for k > 0,
a> 0. Set up the state-space form of the equation
and show that
V(x₁, x₂) = kx² + (x₂ + ax₁)²₂ x₁ = x₂ x₂ = x
is a suitable Lyapunov function for verifying
this.
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