Uncertainty quantification of model predictive control for nonlinear systems with parametric uncertainty using hybrid pseudo-spectral method

Figures & data

Table 1. Wiener–Askey scheme (Xiu & Karniadakis, 2002)

Distribution type	Random variable distribution	Polynomial basis	Domain
Continuous	Gaussian	Normalized Hermite Polynomials	$[-\mathrm{\infty },\mathrm{\infty }]$
	Uniform	Legendre Polynomials	$[a,b]$
	Gamma	Laguerre Polynomials	$[0,\mathrm{\infty }]$
	Beta	Jacobi Polynomials	$[a,b]$
Discrete	Poisson	Charlier Polynomials	$\{0,1,2,...\}$
	Binomial	Krawtchouk Polynomials	$\{0,1,2,...\}$
	Negative Binomial	Miexner Polynomials	$\{0,1,2,...,N\}$
	Hypergeometric	Hahn Polynomials	$\{0,1,2,...,N\}$

Table 2. Parameters of CSTR

$V_R=0.1m^3$	$R=8.314J/\left(molK\right)$
$C_{A0s}=1.0kmol/m^3$	$T_{A0s}=350K$
$\mathrm{\Delta }H=N\left(-4.78\times {10}^4,5\times {10}^2\right)J/mol$	$k_0=N\left(72\times {10}^9,5\times {10}^8\right)1/min$
$E=8.314\times {10}^{4}J/mol$	$c_p=0.239J/\left(gK\right)$
$\rho =1000kg/m^3$	$F=0.1m^3/min$
$T_{Rs}=380K$	$C_{Rs}=0.57kmol/m^3$

Figure 1. Different realizations of terminal constraints ${\mathbb{X}}_f$ for 36 collocation points

Figure 2. Effect of the parameter uncertainty on the control input for CSTR optimal control problem

Figure 3. Mean of the optimal states of the CSTR with MC and gPC uncertainty propagation

Figure 4. Variance of the optimal states of the CSTR with MC and gPC uncertainty propagation

Table 3. Computational time of gPC and MC uncertainty propagation for CSTR optimal control problem

Method	Number of samples	Computational time
gPC	36	11 min
MC	30,000	35 h

Figure 5. Effect of the parameter uncertainty on the control for optimal control problem of Example 2

Figure 6. Mean of the optimal states of Example 2 with MC and gPC uncertainty propagation

Figure 7. Variance of the optimal states of Example 2 with MC and gPC uncertainty propagation

Table 4. Computational time of gPC and MC uncertainty propagation for optimal control problem of Example 2

Method	Number of samples	Computational time
gPC	400	18 min
MC	31,500	27 h

Xiu, D., & Karniadakis, G. E. (2002). The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing, 24, 619–644. doi:10.1137/S1064827501387826

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