, Volume 51, Issue 4, pp 859–875 | Cite as

Multivariable optimal control of an industrial nonlinear boiler–turbine unit

  • Hamed Moradi
  • Gholamreza Vossoughi


Performance control of a boiler–turbine unit is of great importance due to demands for the economical operations of power plants and environmental awareness. In this paper, an optimal control strategy is designed to achieve the desired performance of a boiler–turbine unit. A multivariable nonlinear model of a utility boiler–turbine unit is considered. By manipulation of valves position for the fuel, steam and feed-water flows; output variables including the drum pressure, electric output and fluid density (and consequently drum water level) are controlled. Performance measure of the problem is defined such that the control efforts are minimized while the tracking objectives are obtained. In development of the optimal control strategy, the “variation of extremal” approach is used as an effective tool to handle the nonlinear uncertain problems. Tracking performance of the system is investigated and compared for three cases; tracking from a specific operating point to another ‘near’, ‘far’ and ‘farther’ operating point (depending on the distance between the operating points, the qualitative phrases ‘near’, ‘far’ and farther’ are used). According to the results obtained, more control efforts are required for the tracking of farther operating points (generally). Also, it is investigated that the designed optimal controller guarantees the robust performance of the system in the presence of model parametric uncertainties.


Boiler–turbine unit Nonlinear model Multivariable optimal control Tracking Parametric uncertainty 

List of symbols

\( {\bar{\mathbf{a}}} \)

Nonlinear function in state space


Steam quality

\( H_{\infty } \)

H-infinity robust technique


Hamiltonian function


Identity matrix


Performance measure (objective function)


i-th Component of P (i = 1, 2, 3)


Co-state vector

\( {\tilde{\mathbf{P}}}_{{\mathbf{x}}} \)

State influence matrix

\( {\tilde{\mathbf{P}}}_{{\mathbf{p}}} \)

Co-state influence matrix


Evaporation rate (kg/s)


i-th Component of Q (i = 1, 2, 3)


A real symmetric positive semi-definite matrix


i-th Component of R (i = 1, 2, 3)


A real symmetric positive definite matrix


Time (s)


Initial time


Final time


Valve position of fuel flow


Valve position of steam control


Valve position of feed-water flow

\( u_{i}^{*} \)

Optimal control signals in terms of state and co-state variables (i = 1, 2, 3)


Control input vector


i-th State variable


State vector


Desired state vector


Drum pressure (kg f/cm2) (y 1 = x 1)


Electric output (MW) (y 2 = x 2)


Fluid density (kg/m3) (y 3 = x 3)


Drum water level (m)

\( y_{i}^{(j)} \)

Output y i at the operating point # j


Parameter taking a small value

\( \psi_{i} \)

A nonlinear function (i = 1, 2)



The authors acknowledge the ‘National Elite Foundation of Iran’ for supporting this research.

Compliance with ethical standards

Conflict of interest

There is no conflict of interest.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Centre of Excellence in Design, Robotics and Automation, School of Mechanical EngineeringSharif University of TechnologyTehranIran

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