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Multi-centralized control system design based on equivalent transfer functions using gain and phase-margin specifications for unstable TITO process

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Abstract

In this paper, a new multi centralized control system method proposed for unstable systems using gain and phase-margin specifications. Effective relative gain array based loop interactions are considered to design of off-diagonal controllers. Unstable first order plus time delay models are changed to second order plus time delay model without losing model identity to design a two degree of freedom PID diagonal controllers. Equivalent transfer function method Xiong et al. (J Process Control 17(8):665–673, 2007) is extended to unstable system. Simulation results show that; designed robust control system gives good response and control action. Performance evaluation is done by integral absolute error criteria.

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Abbreviations

EETRA:

Effective energy transmission ratio array

ERGA:

Effective relative gain array

EOTF:

Effective open-loop transfer function

ETF:

Effective transfer function

GPM:

Gain and phase margin

IAE:

Integral absolute error

ISE:

Integral square error

ISE:

Multi-centralized control system

MIMO:

Multi-input multi-output

PI:

Proportional integral

PID:

Proportional integral derivative

PTM:

Process transfer function matrix

RFA:

Relative frequency array

RGA:

Relative gain array

SISO:

Single-input single-output

SOPTD:

Second order plus time delay

TITO:

Two-input two-output

UFOPTD:

Unstable first order plus time delay

\(\Omega \) :

Critical frequency matrix

E :

Effective energy transmission ratio array

\(\otimes \) :

Hadamard multiplication

\(\Omega \) :

Critical frequency array

\(\Phi \) :

Effective relative gain array

\(\Gamma \) :

Relative frequency array

\(\Lambda \) :

Relative gain array

\(u_{i}\) :

Controller out-put signals

\(y_{i}\) :

Process response

\(g_{p,ij}(s)\) :

Transfer function

\(g_{ij}(0)\) :

Steady state gain

\(\theta _{ij}\) :

Time delay

\(a_{1,ij}\) :

Process time constant

\(a_{2,ij}\) :

Process 2nd order polynomial time constant

\(e_{i}\) :

Error signals

\(r_{i}\) :

Reference signals

\(e_{ij}\) :

Energy transmission ratio

\(\omega _{c,ij}\) :

Critical frequency

\(\omega _{u,ij}\) :

Ultimate frequency

\(g_{ij}^0 (j\omega )\) :

Steady state gain of the ETF

\(\lambda _{ij}\) :

Relative gain array element

\(\gamma _{ij}\) :

Relative frequency array element

\(\odot \) :

Hadamard division

\(\hat{g}_{pij}(s)\) :

Equivalent transfer function element

\(\hat{g}_{pij}(0)\) :

ETF steady state gain

\(\hat{\theta }_{ij}\) :

ETF time delay

\(\phi _m\) :

Phase margin

\(\omega _m\) :

Gain margin frequency

\(\omega _p\) :

Phase margin

\(s_{w}\) :

Set-point weight

\(G_{p}(s)\) :

Process transfer function

\(G_{c}(s)\) :

Controller transfer function

\(\hat{G}_{p}(0)\) :

ETF gain matrix

\(\dot{K}_{p,ij}\) :

2DoF PID gain

\(A_m\) :

Gain margin matrix

\(K_{p,ij}\) :

PID proportional gain

\(K_{1,ij}\) :

Inner-loop P controller

\(K_{i,ij}\) :

Integral gain

\(K_{d,ij}\) :

Derivative gain

2DoF:

Two degree of freedom

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  1. Indian Institute of Technology Madras, Chennai, India

    Chandra Shekar Besta

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Besta, C.S. Multi-centralized control system design based on equivalent transfer functions using gain and phase-margin specifications for unstable TITO process. Int. J. Dynam. Control 6, 817–826 (2018). https://doi.org/10.1007/s40435-017-0345-3

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  • DOI: https://doi.org/10.1007/s40435-017-0345-3

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