Table 4 Chronological summary of robust control

Robust control

Sun [78]

1. Current.

N. P.

N. P.

The method is applied to the oxygen protection of FC

Experimental

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H-infinity control

Fardadi [79]

1. Air or fuel flow rate;

2. Temperature

Controller based on Standard H infinity or L-2 gain method.

N. P.

Reduce the space temperature change under significant load disturbance

A physical based dynamic model of a single co-flow SOFC repeat cell

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H-infinity control

Allag [81]

1. Current;

2. Voltage

Dynamic feedback law:

\(v\left(s\right)=-{K}_{\infty }\left(s\right)e\left(s\right)\)

\(e(t)={V}_{\mathrm{uc},t}-{V}_{\mathrm{uc}}\)

\({K}_{\infty }\): controller transfer function;

\({V}_{\mathrm{uc},t}={S}_{t}{V}_{\mathrm{max}}\): target ultracapacitor voltage.

Improve load tracking capability;

Prevent fuel shortage

Experimental

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H-infinity control

Huo [80]

1. Air or fuel flow rate

N. P.

N. P.

Reduce voltage oscillation and deviation and keep fuel utilization unchanged

Experimental model of 100 kW cross-flow SOFC system

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SMC

Dötschel [82]

1. Temperature

Corresponding variable-structure control law:

\(\left[ v \right]: = - \frac{{a\left( {\vartheta _{{{\text{FC}}}} \left( t \right),\left[ {\text{p}} \right],\left[ d \right]} \right)}}{{b\left( {\vartheta _{{{\text{FC}}}} \left( t \right),\left[ {\text{p}} \right],\left[ d \right]} \right)}}\)

-\(\frac{\widetilde{\eta }\cdot \mathrm{sign}({\vartheta }_{\mathrm{FC}}\left(t\right)-{\vartheta }_{\mathrm{FC},d})}{b\left({\vartheta }_{\mathrm{FC}}\left(t\right),\left[\mathrm{p}\right],\left[d\right]\right)}\),

v(t): control variable;

\({\vartheta }_{\mathrm{FC},d}\): desired trajectory;

[P]: interval parameter vector;

d: point-valued disturbance;

\(\widetilde{\eta }>0\).

Improve system stability

Experimental

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Rauh [83]

1. Air or fuel flow rate;

2. Temperature

Interval-based control law:

\([{v}_{\mathrm{CG},d}]:=\frac{-\widetilde{a}(x,[\mathrm{p}],[d])+{\xi }_{1,\mathrm{d}}^{(\delta )}-\sum_{r=0}^{\delta -2}{\alpha }_{r}\cdot {\widetilde{\xi }}_{1}^{(r+1)}-\widetilde{\eta }\cdot \mathrm{sign}\{s\}}{\widetilde{b}(x,[\mathrm{p}])}\)

[p] and [d]: interval parameters;

\({\xi }_{1,\mathrm{d}}^{(\delta )}\): time derivatives.

Improve system stability

Experimental

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Wu [57]

1. Air or fuel flow rate;

2. Air excess ratio;

3. Temperature

\(\dot{\delta }={B}^{-1}\left[-A+{v}_{\mathrm{ftc}}+{v}_{\mathrm{sup}}\right]\)

\({W}_{1}=6.8\times {10}^{-4}\frac{I}{{u}_{\mathrm{f},\mathrm{ref}}}\)

\({W}_{2}=1.9\times {10}^{-3}I{\lambda }_{{\mathrm{O}}_{2},\mathrm{ref}}\)

\(\delta\): opening ratio of the bypass valve;

\({W}_{1}\): inlet fuel flow;

\({W}_{2}\): inlet air flow.

Ensure the safe operation of the system; Improve system operation efficiency

A SOFC model with parameter uncertainty

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Am [84]

1. Voltage

N.P.

N.P.

Strong dynamic performance

Experimental

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