In this paper, we derive an expression for the loss of optimal performance (compared to the corresponding linear-quadratic optimal performance with the instantaneous full-state feedback) when the continuous-time finite-horizon linear-quadratic optimal controller uses the estimates of the state variables obtained via a reduced-order observer. It was shown that the loss of optimal performance value can be found by solving the differential Lyapunov equation whose dimensions are equal to dimensions of the reduced-order observer. A proton exchange membrane fuel cell example is included to demonstrate the loss of optimal performance as a function of the final time. It can be seen from the simulation results that the loss of optimal performance value can be very large. The loss of optimal performance value can be drastically reduced by using the proposed least-square formulas for the choice of the reduced-order observer initial conditions.
This paper presents a new technique for design of full-state feedback controllers for linear dynamic systems in three stages. The new technique is based on appropriate partitioning of the linear dynamic system into linear dynamic subsystems. Every controller design stage is done at the subsystem level using only information about the subsystem (reduced-order) matrices. Due to independent design in each stage, different subsystem controllers can be designed to control different subsystems. Partial subsystem level optimality and partial eigenvalue subsystem assignment can be achieved. Using different feedback controllers to control different subsystems of a system has not been present in any other known linear full-state feedback controller design technique. The new technique requires only solutions of reduced-order subsystem level algebraic equations. No additional assumptions were imposed except what is common in linear feedback control theory (the system is controllable (stabilizable)) and theory of three time-scale linear systems (the fastest subsystem state matrix is invertible)). The local full-state feedback controllers are combined to form a global full-state controller for the system under consideration. The presented results are specialized to the three time-scale linear control systems that have natural decomposition into slow, fast, and very fast subsystems, for which numerical ill conditioning is removed and solutions of the design algebraic equations are easily obtained. The proposed three-stage three time-scale feedback controller technique is demonstrated on the eighth-order model of a fuel cell model.
In this paper a full nonlinear dynamic control oriented mathematical model of Proton Exchange Membrane (PEM) fuel cell system is developed. The model is structured as a nonlinear five state space model. The derivation of each state equation is based on physics fundamental principles using thermodynamic theory of ideal gas mixtures, conservation of mass law, flow dynamics in serpentine flow channels and diffusion. The output of proposed model, stack voltage, is developed from Nernst equation that includes three main types of losses occurring in the fuel cell. The unknown parameters of the model are estimated and fitted using sets of steady state experimental data. Stack polarization curve of the proposed model is validated by using sets of data for three different values of inlet pressures. Experimental setup used to attain data is the Greenlight Innovation G60 fuel cell test station system and TP50 Fuel Cell stack.
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