Design of Fractional Order Controllers using Triangular Strip Operational Matrices

To tackle plant uncertainty issues, many controller design methods are developed. The convenient one among these methods is designing a robust fractional P I λ D μ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429504433/34c6c151-5e4c-4144-a0c3-a3f653703557/content/inline8_1.tif"/> controller. This chapter proposes a simple tuning technique aimed to produce a robust noninteger order PID controller exhibiting iso-damping property during the reparameterization of a plant. The required robustness property is achieved by allowing the fractional PID control system to imitate the dynamics of a reference system with Bode’s ideal transfer function in its forward path. The objective of designing robust controller by tracking the dynamics of reference control system is defined mathematically as an H ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429504433/34c6c151-5e4c-4144-a0c3-a3f653703557/content/inline8_2.tif"/> -optimal control problem. Fractional differential systems are transformed into algebraic equations by the use of triangular strip operational matrices. The H ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429504433/34c6c151-5e4c-4144-a0c3-a3f653703557/content/inline8_3.tif"/> -optimal control problem is then changed to an ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429504433/34c6c151-5e4c-4144-a0c3-a3f653703557/content/inline8_4.tif"/> -norm minimization of a parameter ( K C , K I , K d , λ , μ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429504433/34c6c151-5e4c-4144-a0c3-a3f653703557/content/inline8_5.tif"/> ) varying square matrix. Global optimization techniques, Luus-Jaakola direct searches, and particle swarm optimizations are employed to find the optimum values of fractional PID controller parameters. The proposed method of control system design is implemented in heating furnace temperature control, automatic voltage regulator system and some integer and fractional order process models. Fractional PI^λ, fractional PD^µ, PI^λD^µD^µ2, fractional PID with fractional order filter, and the series form of fractional PID controller are designed as optimal controllers using the triangular strip operational matrix–based control design method. The performance of the proposed fractional order controller tuning technique is found to be better than the performance of some fractional order controller tuning methodologies reported in the literature. Triangular strip operational matrices proposed from the perspective of mathematics (for the solution of fractional differential and partial differential equation) finds its elegant application in the proposed method of control system design.