Controllability, observability and fractional linear-quadratic problem for fractional linear systems with conformable fractional derivatives and some applications

• Published in: International journal of dynamics and control Vol. 11; no. 1; pp. 214 - 228
• Main Authors: Sadek, Lakhlifa, Abouzaid, Bouchra, Sadek, El Mostafa, Alaoui, Hamad Talibi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2023; Springer Nature B.V
• Subjects: Complexity; Continuous time systems; Control; Control and Systems Theory; Controllability; Differential equations; Dynamical Systems; Engineering; Linear systems; Mathematical analysis; Observability (systems); Optimal control; Riccati equation; Vibration; Continuous-time CF-DS; Conformable fractional differential Lyapunov equation (CF-DLE); Conformable fractional differential Riccati matrix equation (CF-DRE); Fractional calculus; Fractional linear quadratic (FLQ)
• ISSN: 2195-268X; 2195-2698
• DOI: 10.1007/s40435-022-00977-7

In the present paper, we investigate the controllability, observability and fractional linear-quadratic problem of a non-homogeneous continuous-time fractional dynamical system (CF-DS) using the conformable fractional derivatives (CFD). We show that the controllability is equivalent to a controllability matrix that has a full rank. We also show a relationship between controllability and fractional differential Lyapunov equation. We give some theorems for observability of continuous-time fractional dynamical system. Moreover, we found a relationship between the solution of conformable fractional differential Riccati matrix equation and the solution of another conformable fractional linear system. We also find an optimal control that minimizes a functional cost under a conformable fractional system CF-DS by using the solution of fractional differential Riccati equation. Finally, we offer some applications to illustrate the effectiveness of our results.