Table 5 Chronological summary of MPC

MPC

Jurado [85]

1. Air or fuel flow rate

Cost function:

\(\mathrm{min} J({H}_{\mathrm{p}1},{H}_{\mathrm{p}2},{H}_{\mathrm{c}}, \lambda )=\sum_{j={H}_{\mathrm{p}1}}^{{H}_{\mathrm{p}2}}{(w(k+j)-\widehat{y}(k+j))}^{2}+\lambda \sum_{j=1}^{{H}_{\mathrm{c}}}\Delta {u}^{2}(k+j-1)\)

\(\widehat{y}(k+j)\): predicted system output;

\(w(k+j)\): modified setpoint;

\({H}_{\mathrm{p}1}\): minimum costing horizon;

\({H}_{\mathrm{p}2}\): maximum costing or prediction horizon;

\({H}_{\mathrm{c}}\): control horizon;

\(\lambda\): move suppression coefficient.

Meet power demand

Multivariable fuzzy Hammerstein model of SOFC

**

**

**

Sanandaji [87]

1. Air or fuel flow rate;

2. Voltage

N. P.

N. P.

Improve load tracking capability

Linear parameter variation model of SOFC system

**

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**

Kupilik [88]

1. Air or fuel flow rate

Objective function：

\(J=\mathrm{minimize }{x}_{k,N}^{^{\prime}}{P}_{\mathrm{f}}{x}_{k,N}+\sum_{i=0}^{N-1}{x}_{k,i}^{T}Q{x}_{k,i}+{u}_{k,i}^{T}R{u}_{k,i}+{\Vert {Q}_{\mathrm{Y}}({y}_{k,i}-{y}_{\mathrm{ref},i})\Vert }_{2}\)

\({\mathrm{y}}_{\mathrm{ref},i}\): reference trajectory;

\({P}_{\mathrm{f}}\), \(Q\), \(R\) and \({Q}_{Y}\): weighting matrix;

\({x}_{k,N}\): state at time k.

Meet the change of load demand; Reduce carbon deposition

Experimental model of movable 1.5 kW SOFC system

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Horalek [89], [90]

1. Air or fuel flow rate

Multilinear MPC controller:

\(J\left(k\right)=\sum_{p=1}^{N}\sum_{j=1}^{{n}_{y}}{({\lambda }_{j}^{y}({y}_{j,\mathrm{ref}}\left(k+p|k\right)-{y}_{j}\left(k+p|k\right)))}^{2}+\sum_{p=0}^{{N}_{u}-1}\sum_{j=1}^{{n}_{u}}{({\lambda }_{j}^{\Delta u}\Delta {u}_{j}\left(k+p|k\right))}^{2}\)

y: controlled variable;

\({y}_{\mathrm{ref}}\): reference;

\(\Delta u\): increments of manipulated variables;

p: Prediction horizon.

Stable output voltage

Nonlinear SOFC state model

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Miaomiao [91]

1. Current density

Performance index function:

\(J\left({U}_{N},x\left(0\right)\right)=\sum_{k=0}^{N-1}\left({x}_{k}^{^{\prime}}Q{x}_{k}+{u}_{k}^{^{\prime}}R{u}_{k}\right)+{x}_{N}^{^{\prime}}P{x}_{N}\)

x: collection of the bounded parameter;

Q: state weight;

R: input weight;

N: control domain.

Reduce computation and storage

Piece wise affine (PWA) model of SOFC

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Frenkel [92]

1. Power

Control law:

\({u}_{k}={e}_{1}^{\mathrm{T}}\cdot {\widetilde{u}}_{k}={e}_{1}^{\mathrm{T}}({\mathrm{S}}_{v}\cdot {\widetilde{y}}_{\mathrm{d},\mathrm{k}+1}-\mathrm{k}\cdot {\mathrm{x}}_{k})\)

\(\mathrm{k}\): control gain.

N. P.

Experimental

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Data-driven PC

Wang [86]

1. Air or fuel flow rate

Cost function:

\(\mathrm{min }{J}_{k}=\mathrm{min }{\left({\widehat{y}}_{k+1/L}-{r}_{k+1/L}\right)}^{T}Q\left({\widehat{y}}_{k+\frac{1}{L}}-{r}_{k+\frac{1}{L}}\right)+\Delta {u}_\frac{k}{M}^{T}{R}_{M}\Delta {u}_\frac{k}{M}+{u}_\frac{k}{M}^{T}{P}_{M}{u}_\frac{k}{M}\)

\(L\): prediction horizon;

\(M\): control horizon;

\({\widehat{y}}_{k+1/L}\): vector of prediction of

future output;

\({r}_{k+1/L}\): vector of future reference signals;

\(Q\) , \({R}_{M}\) and \({P}_{M}\): weighting matrices with block diagonal structure;

\({u}_{k/M}\): vector of future manipulated signal.

Verify the effectiveness of data-driven predictive control algorithm

N. P.

**

**

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NMPC

Huo [38]

1. Air or fuel flow rate;

2. Voltage

Cost function:

\(J=\sum_{j=1}^{{H}_{\mathrm{p}}}{[{\widehat{V}}_{\mathrm{dc}}\left(k+j\right)-{V}_{\mathrm{dcr}}(k+j)]}^{2}+\lambda \sum_{i=1}^{{H}_{\mathrm{c}}}{[{N}_{\mathrm{f}}\left(k+i\right)-{N}_{\mathrm{f}}(k+i-1)]}^{2}\)

\({\widehat{V}}_{\mathrm{dc}}\left(k+j\right)\): predicted output voltage;

\({H}_{\mathrm{c}}\): control horizon;

\({N}_{\mathrm{f}}\left(k+i\right)\): manipulated variable;

\(\lambda\): weighting factor.

Stable output voltage

Hammerstein model of SOFC

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**

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Zhang [93]

1. Air or fuel flow rate;

2. Current density

Cost function:

\(\Phi =\mathrm{min }\left\{\sum_{k=T}^{T+{N}_{\mathrm{c}}}L\left({w}_{k},{v}_{k}\right)+\frac{1}{2}V\left({w}_{T+{N}_{\mathrm{c}}}\right)+\sum_{k=T}^{T+{N}_{\mathrm{c}}}(\frac{1}{2}{\eta }_{\mathrm{r},k}{O}_{\mathrm{r}}{\eta }_{\mathrm{r},k}^{T}+{O}_{\mathrm{r}}{\eta }_{\mathrm{r},k})\right\}\)

\(\mathrm{V}\left({w}_{T+{N}_{\mathrm{c}}}\right)\): penalty term;

\(\eta\): relaxation factor;

Q_r: penalty matrix of regulator cost function.

SOFC under effective control constraints

Experimental

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Yang [94]

1. Temperature

Objective function:

\(J=\sum_{i=1}^{P}{q}_{i}{[y\left(k+i\right)-{y}_{\mathrm{m}}(k+i)]}^{2}+\sum_{j=1}^{M}{r}_{j}\Delta {u}^{2}(k+j-1)\)

\({q}_{i}\) and \({r}_{j}\): weight coefficient;

P: prediction horizon;

M: control horizon;

\(y\left(k+i\right)\): actual output;

\({y}_{\mathrm{m}}(k+i)\): model predictive output.

Effectively control temperature of SOFC stack

Improved TS fuzzy model of SOFC stack

**

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Murshed [95]

1. Voltage

Objective function:

\(\mathrm{min }J=\sum_{i=1}^{N}\left[{\Vert \widehat{x}\left(k+\frac{i}{k}\right)-{x}_{\mathrm{ref}}\Vert }_{Q}^{2}+{\Vert u\left(k+\frac{i}{k}\right)-{u}_{\mathrm{ref}}\Vert }_{R}^{2}+{\Vert \Delta u\left(k+\frac{i}{k}\right)\Vert }_{S}^{2}\right]+{W}_{\mathrm{indirect}}[\sum {\dot{n}}_{i,E}{\int }_{{T}_{\mathrm{ref}}}^{{T}_{\mathrm{AE}}}{C}_{p,i}(T)\mathrm{d}T+{\dot{n}}_{i,E}\frac{1-r}{r}{\int }_{{T}_{\mathrm{ref}}}^{{T}_{\mathrm{B}}}{C}_{p,i}(T)\mathrm{d}T]\)

Q, R, S and \({W}_{\mathrm{indirect}}\): weighting matrices;

\({T}_{\mathrm{ref}}\): minimum temperature;

\({T}_{\mathrm{AE}}\), \({T}_{\mathrm{B}}\): exit temperatures;

\({\dot{n}}_{i,E}\): flow rate.

Improve the operation efficiency of the system

Experimental

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Bhattacharyya [96]

1. Air or fuel flow rate;

2. Voltage

Optimization formula:

\(\mathrm{min}{ \Vert {W}_{1}(\widehat{y}-\xi )\Vert }_{2}+{\Vert {W}_{2}u\Vert }_{2}\)

\({W}_{1}\): weighting matrix;

\({W}_{2}\): move suppression matrix.

Meet the step change of load

Experimental

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**

Lee [97]

1. Air or fuel flow rate;

2. Voltage

Performance index:

\(J\left(k,k+1\right)\triangleq x{\left(k|k\right)}^{T}\mathcal{z}x\left(k|k\right)+u{\left(k|k\right)}^{T}\gamma u\left(k|k\right)+V\left(k+1|k\right)\)

\(V\left(k+1|k\right)\): terminal cost;

x: state vector;

u: control input;

\(\mathcal{z}\)>0, \(\gamma\)>0.

Improve load tracking capability

Sector bounded nonlinear model of SOFC system

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Wu [27]

1. Air or fuel flow rate

Control input:

\({u}_{k}\left(i\right)={u}_{k-1}\left(i\right)+{K}_{\mathrm{P}}{e}_{k-1}\left(i\right)+{K}_{\mathrm{D}}{\dot{e}}_{k-1}\left(i\right) i=1,\dots ,{N}_{\mathrm{c}}\)

\({N}_{\mathrm{c}}\): control window;

k: iteration index;

\({e}_{k-1}\left(i\right)\): tracking error;

\({\dot{e}}_{k-1}\left(i\right)\): error difference;

\({K}_{\mathrm{P}}\) and \({K}_{\mathrm{D}}\): learning gains.

Extend system life

Experimental model of SOFC system

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Awryńczuk [28]

1. Air or fuel flow rate

Cost function:

\(J\left(k\right)=\sum_{p=1}^{N}{({V}_{\mathrm{dc}}^{\mathrm{sp}}\left(k+p|k\right)-{\widehat{V}}_{\mathrm{dc}}\left(k+p|k\right))}^{2}+\lambda \sum_{p=0}^{{N}_{\mathrm{u}}-1}{(\Delta {q}_{\mathrm{f}}\left(k+p|k\right))}^{2}\)

\(\lambda\): weighting coefficient;

\({\widehat{V}}_{\mathrm{dc}}\left(k+p|k\right)\): predicted values;

\({V}_{\mathrm{dc}}^{\mathrm{sp}}\left(k+p|k\right)\): set-point values;

\({N}_{\mathrm{u}}\): control horizon.

Meet fuel use restrictions;

Good quality control

Continuous time model of SOFC system

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GPC

Deng [98]

1. Current density

Cost function:

\(J=E\left\{\sum_{j=1}^{N}{(\widehat{y}\left(k+j\right)-{y}_{\mathrm{r}}(k+j))}^{2}+\sum_{j=1}^{M}\lambda (j){(\Delta u(k+j-1))}^{2}\right\}\)

E{ }: expectation operator;

N: maximum costing horizon;

M: control horizon;

\(\lambda (j)\): control weighting sequence;

\(\widehat{y}\left(k+j\right)\): optimum ahead prediction;

\({y}_{\mathrm{r}}(k+j)\): future reference trajectory.

Improve load tracking capability; Extended stack life

Fractional order model of SOFC

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Jiang [99]

1. Air or fuel flow rate;

2. Temperature

Objective function:

\(J=E\left\{\sum_{j=1}^{{N}_{2}}{[\widehat{y}\left(k+j\right)-{y}_{\mathrm{r}}(k+j)]}^{2}+\sum_{j=1}^{{N}_{\mathrm{u}}}{\eta }_{j}{[\Delta U(k+j-1)]}^{2}\right\}\)

E{ }: expectation operator;

\({\eta }_{j}\): control weighting sequence which limits the amplitude of the control sequence;

\({N}_{2}\): maximum costing horizon;

N_u: control horizon;

\(\widehat{y}\left(k+j\right)\): predictive system output at the (k+j) th instance;

\({y}_{\mathrm{r}}(k+j)\): reference value at the (k+j) th instance.

Improve system response speed and stability; Improve anti-interference ability

TS fuzzy model of SOFC system

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Pohjoranta [32]

1. Temperature

Objective function:

\(J\left({N}_{1},{N}_{2},{N}_{3}\right)=\sum_{j={N}_{1}}^{{N}_{2}}{\Vert \widehat{y}\left(t+j|t\right)-w(t+j)\Vert }_{R}^{2}+\sum_{j=1}^{{N}_{3}}{\Vert u\left(t+j-1\right)-u(t+j-2)\Vert }_{Q}^{2}\)

\(\widehat{y}\left(t+j|t\right)\): optimum j-step-ahead prediction of the output y;

\({N}_{1}\), \({N}_{2}\): prediction horizon;

\({N}_{3}\): control horizon;

\(w(t+j)\): set point trajectory;

R and Q: positive definite weight matrices.

Extended service life

Simulink model of 10 kW SOFC system

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Jiang [100, 101]

1. Air or fuel flow rate

Objective function:

\(J=E\left\{\sum_{j=1}^{{N}_{2}}{\left[\widehat{y}\left(k+j\right)-{y}_{\mathrm{r}}(k+j)\right]}^{2}+\sum_{j=1}^{{N}_{\mathrm{u}}}{\eta }_{j}{\left[\Delta U(k+j-1)\right]}^{2}\right\}\)

E{ }: expectation operator;

\({\eta }_{j}\): control weighting sequence;

\({N}_{2}\): maximum cost horizon;

\({N}_{\mathrm{u}}\): control horizon;

\(\widehat{y}\left(k+j\right)\): predictive system output;

\({y}_{\mathrm{r}}(k+j)\): reference value.

Improve load tracking capability; Extend system life

Simulink model of 5kW SOFC system

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Boubaker [102]

1. Air or fuel flow rate

N.P.

N.P.

Improve response speed

Dynamic model of SOFC system.

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CMPC

Li [53]

1. Voltage

Performance index:

\(J\left(k\right)=\sum_{i=0}^{N-1}[{\left(x\left(k+i+\frac{1}{k}\right)-\overline{x }\right)}^{T}\mathrm{Q}\left(x\left(k+i+\frac{1}{k}\right)-\overline{x }\right)+R\Delta {q}_{\mathrm{f}}^{2}(k+i/k)]+\psi (x\left(k+N+\frac{1}{k}\right)-\overline{x })\)

N: prediction horizon;

\(x\left(k+i+\frac{1}{k}\right)\): predicted state;

\(\overline{x }\): equilibrium value of the state vector;

k: current time.

Successfully handle control and control motion constraints; Satisfactory closed-loop performance is obtained

Dynamic model of SOFC system

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Spivey [103]

1. Air or fuel flow rate;

2. Temperature;

3. Pressure

Objective function:

\(\mathrm{min} J=\frac{1}{2}{\left(y-{y}_{\mathrm{ref}}\right)}^{T}Q\left(y-{y}_{\mathrm{ref}}\right)+\frac{1}{2}\Delta {u}^{T}R\Delta u+\frac{1}{2}{\xi }^{T}V\xi\)

y: vector of controlled variables at all prediction time steps;

\({y}_{\mathrm{ref}}\): reference trajectory;

\(Q\), \(R\) and \(V\): weight matrices;

\(\Delta u\): change in manipulated variables between each control time step;

\(\xi\): slack variables.

Extended service life

A dynamic, quasi-two-dimensional model for a high-temperature tubular SOFC combined with ejector and pre-reformer models

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Fuzzy MPC

Wu [14]

1. Air or fuel flow rate

Preference index:

\(\mathrm{min }J={\Vert \frac{1}{2}(\left|Y\left(k\right)-{Y}_{\mathrm{h}}\left(k\right)\right|+\left|Y\left(k\right)-{Y}_{\mathrm{l}}\left(k\right)\right|-\left|{Y}_{\mathrm{h}}\left(k\right)-{Y}_{\mathrm{l}}\left(k\right)\right|)\Vert }_{Q}^{2}+{\Vert \Delta U(k)\Vert }_{R}^{2}+{\Vert \Delta Y(k)\Vert }_{W}^{2}\)

\({Y}_{\mathrm{h}}\left(k\right)\): higher limit sequence;

\({Y}_{\mathrm{l}}\left(k\right)\): lower limit sequence;

\(\Delta Y(k)\): increment of output prediction;

Q, R and W: weight coefficient matrix

Stable output voltage;

Stable fuel utilization

Fuzzy model with region tracking for SOFC system

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AMPC

Liu [104]

1. Voltage

Objective function:

\(\mathrm{min}J\left(x\left(t\right),U\left(t\right),{N}_{P},{N}_{C}\right)\)

\(s.t.\left\{\begin{array}{c}x\left(t+1\right)=Ax\left(t\right)+{\mathrm{B}}_{\mathrm{u}}u\left(t\right)+{\mathrm{B}}_{\mathrm{d}}d\left(t\right),\\ \left(t=t,t+1,\dots ,t+{N}_{p}-1\right); \\ y\left(t\right)=Cx\left(t\right); \\ \frac{2{K}_{r}I}{0.9}\le u\left(t\right)\le \frac{2{K}_{r}I}{0.7} \end{array} \right.\)

N.P.

Improve output voltage tracking ability and dynamic performance

Dynamic model of SOFC system

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