Controllability of a system of coupled harmonic oscillators

Abstract

In general, it is worthwhile to understand and control the dynamics of an existing system whose output behaves somewhat closer to the desired output rather than developing a new system which tracks the desired output, since it is beneficial for industries in many aspects; low cost, less time, etc. In this research, we control the output of a couple harmonic oscillator which has been extensively used in many Engineering Models by mainly focusing on two types of control techniques, namely source term controlling and initial condition controlling. Numerical results using MATLAB validates that controlled system output tracks the desired output for these two types of controlling. Consider the governing equations: 𝑚𝑥̈1 = − 𝑚𝑔 𝑙 𝑥1 + 𝑘(𝑥2 − 𝑥1) ,𝑚𝑥̈2 = −𝑚𝑔 𝑙 𝑥2 + 𝑘(𝑥1 − 𝑥2) 𝑥1(0) = 𝛼,𝑥2(0) = 𝛽,𝑥̇1(0) = 𝛾,𝑥̇2(0) = 𝜇.

(i) Controlling by the source term: Let the desired outputs 𝑥1 and 𝑥2 be given by 𝑥1 = 𝑟 sin(𝑝𝑡) + 𝑤 sin(𝑞𝑡),𝑥2 = −𝑟 sin(𝑝𝑡) + 𝑤 sin(𝑞𝑡) where r, w, p, and q are parameters. Then controlled system for the source term is given by 𝑚𝑥̈1 = − 𝑚𝑔 𝑙 𝑥1 + 𝑘(𝑥2 − 𝑥1) + 𝛼(𝑡), 𝑚𝑥̈2 = −𝑚𝑔 𝑙 𝑥2 + 𝑘(𝑥1 − 𝑥2) + 𝛽(𝑡) and the source terms are 𝛼(𝑡) = 𝐴 sin(𝑝𝑡) + 𝐵 sin(𝑞𝑡) and 𝛽(𝑡) = −𝐴 sin(𝑝𝑡) + 𝐵 sin(𝑞𝑡), where 𝐴 = −𝑟𝑚𝑝2 − 𝑠𝑟 + 𝑘𝑟, 𝐵 = −𝑤𝑚𝑞2 − 𝑘𝑤 − 𝑠𝑤 and 𝑠 = −(𝑚𝑔 𝑙 + 𝑘) for 𝑥1(0) = 0, 𝑥2(0) = 0,𝑥̇1(0) = 𝛾,𝑥̇2(0) = 𝜇.

(ii) Controlling by the initial condition: Let the desired output 𝑥1 and 𝑥2 be given by 𝑥1 = 𝑎𝑒√𝜆+𝜙 𝑡 + 𝑏𝑒−√𝜆+𝜙 𝑡 + 𝑐𝑒√𝜆−𝜙 𝑡 + 𝑑𝑒−√𝜆−𝜙 𝑡, 𝑥2 = 𝑎𝑒√𝜆+𝜙 𝑡 + 𝑏𝑒−√𝜆+𝜙 𝑡 − 𝑐𝑒√𝜆−𝜙 𝑡 − 𝑑𝑒−√𝜆−𝜙 𝑡 , where 𝜆 = −(𝑔 𝑙 + 𝑘 𝑚 ) and 𝜙 = 𝑘 𝑚 . By considering the system (1) the controlled system for initial conditions is given by 𝑚𝑥̈1 = −𝑚𝑔 𝑙 𝑥1 + 𝑘(𝑥2 − 𝑥1) ,𝑚𝑥̈2 = −𝑚𝑔 𝑙 𝑥2 + 𝑘(𝑥1 − 𝑥2) with initial conditions: 𝑥1(0) = 𝛼 + (𝑎 + 𝑏 − 𝑐 − 𝑑) − (𝐴 + 𝐵 − 𝐶 − 𝐷),𝑥2(0) = 𝛽 + (𝑎 + 𝑏 + 𝑐 + 𝑑) − (𝐴 + 𝐵 + 𝐶 + 𝐷),𝑥̇1(0) = 𝛾 + ( (𝑎−𝑏)−(𝐴−𝐵) √𝜆+𝜙 + (𝐶−𝐷)−(𝑐−𝑑) √𝜆−𝜙 ),𝑥̇2(0) = 𝜇 + ( (𝑎−𝑏)−(𝐴−𝐵) √𝜆+𝜙 + (𝑐−𝑑)−(𝐶−𝐷) √𝜆−𝜙 ) where 𝐴 = 1 4 (𝛼 + 𝛽 + 𝛾 √𝜆+𝜙 + 𝜇 √𝜆+𝜙 ),𝐵 = 1 4 (𝛼 + 𝛽 − 𝛾 √𝜆+𝜙 − 𝜇 √𝜆+𝜙 ), 𝐶 = −1 4 (𝛼 − 𝛽 + 𝛾 √𝜆−𝜙 − 𝜇 √𝜆−𝜙 ) ,𝐷 = − 1 4 (𝛼 − 𝛽 − 𝛾 √𝜆−𝜙 − 𝜇 √𝜆−𝜙 ).