Enhanced PID Controller for Non-Minimum Phase Second Order Plus Time Delay System
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Purushottam Patil
Abstract
A tuning method is developed for the stabilization of the non-minimum phase second order plus time delay systems. It is well known that the presence of positive zeros pose fundamental limitations on the achievable control performance. In the present method, the coefficients of corresponding powers of s, s2 and s3 in the numerator are equated to α, β and γ times those of the denominator of the closed-loop system. The method gives three simple linear equations to get the PID parameter. The optimal tuning parameters α, β and γ are estimated by minimizing the Integral Time weighted Absolute Error (ITAE) for servo problem using fminsearch MATLAB solver aimed at providing lower maximum sensitivity function and keeping in check with the stability. The performance under model uncertainty is also analysed considering perturbation in one model parameter at a time using Kharitonov’s theorem. The closed loop performance of the proposed method is compared with the methods reported in the literature. It is observed that the proposed method successfully stabilizes and improves the performance of the uncertain system under consideration. The simulation results of three case studies show that the proposed method provides enhanced performance for the set-point tracking and disturbance rejection with improved time domain specifications.
Appendix
A Kharitonov’s Theorem [28]
This theorem gives interval for the coefficient of the characteristic equation for which the system is stable by determining the stability of vortex polynomial obtained for the boundary range of the coefficients.
The closed loop characteristic equation is given by
The characteristic equation obtained by applying Taylor’s series expansion for time delay approximation for the interval system is stated as follows
Where, ai ϵ [ail, aiu],
For i = 1,2, … n., ail is the lower limit and aiu is the upper limit,
The characteristic polynomial is said to be stable only if all four khairtonov polynomial are stable. Their stability is found by applying Routh hurwitz criterion to each equation. The khairtonov’s polynomials are gives as
The coefficient polynomial (36) is stable if and only if all the four vertex polynomials (37–38) are stable. Initial values of kp, a1 and a2 are fixed and perturbation in time delay L is substituted with limits (L–ΔL)<L<(L+ΔL) in coefficients and Kharitonov’s polynomials are obtained. These polynomial’s stability is checked with Routh–Hurwitz method. In similar procedure, stability regions for kp, a1 and a2 are obtained by varying objective parameter and keeping other parameters constant.
Nomenclature
- Greek alphabets:
- α
Coefficient for s3
- β
Coefficient for s2
- γ
Coefficient for s
- τ
Process time constant
- τI
Integral time constant
- τD
Derivative time constant
- µ
Growth rate for Monod kinetics
- Abbreviations:
- PID
Proportional, Integral, Derivative controller
- NMP
Non-Minimum Phase
- FOPTD
First Order plus Time Delay
- SOPTD
Second Order plus Time Delay
- USOPTD
Unstable second order plus time delay
- IMC
Internal Model Control
- SA
Stability Analysis
- IAE
Integral of Absolute Error
- ITAE
Integral of Time weighted Absolute Error
- ISE
Integral of Squared Error
- TV
Total Variation in manipulated variable
- G
Process transfer function
- GC
Controller transfer function
- MS
Maximum Sensitivity function
- SOPTDZ
Second Order Plus Time Delay with Zeros
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Artikel in diesem Heft
- Research Articles
- A Simplified One-Dimensional Mathematical Model to Study the Transient Thermal Behavior of an Oxidation Catalyst with Both Low and High Levels of CO Concentration at the Inlet
- Control of Integrating Process with Time Delay
- Moisture Content and Oil Uptake Variations and Modeling in Deep-Fried Hamburger Slices
- Modelling of Thermodynamic Pressure – Composition – Temperature Relationships in the Systems of Metallic Hydride Forming Materials with Gaseous Hydrogen Using C++ Software
- CFD Investigation of Al2O3 Nanoparticles Effect on Heat Transfer Enhancement of Newtonian and Non-Newtonian Fluids in a Helical Coil
- Computational Fluid Dynamics Studies of Gas-Solid Flows in a Horizontal Pipe, Subjected to an Adiabatic Wall, Using a Variable Gas Properties Eulerian Model
- Enhanced PID Controller for Non-Minimum Phase Second Order Plus Time Delay System
- Fractional Order PID Controller Design for Supply Manifold Pressure Control of Proton Exchange Membrane Fuel Cell
