Table 2 Overview of the approximation techniques for uncertainty propagation with R _k the variable to which the parametric uncertainty is propagated

Rationale

Linearization of state equations around \({\bar {\boldsymbol {\theta }}}\)

Approximate distribution by a fixed number of

Approximate response of the model (at sampling

parameters (sigma points)

points) as a pth order polynomial function of θ

Uncertainty distribution

Normal

Any symmetric, unimodal distribution

Any

Equations

State equations + Sensitivity equations:

State equations for SPs:

State equations for sampling points:

\(\left \{\begin {array}{lll} \dot {\mathbf {S}}_{\mathrm {{LIN}}}(t) &=& \frac {\partial \mathbf {f}(\mathbf {x},\mathbf {u},\boldsymbol {\theta }_{\text {nom}},t)}{\partial \mathbf {x}} \mathbf {S_{\mathrm {{LIN}}}} + \frac {\partial \mathbf {f}(\mathbf {x},\mathbf {u},\boldsymbol {\theta }_{\text {nom}},t)}{\partial \boldsymbol {\theta }},\\ \mathbf {S_{\mathrm {{LIN}}}}(0) &=& \mathbf {0} \end {array} \right. \)

\(\dot {\mathbf {x}}_{i}=\mathbf {f}(\mathbf {x_{i}},\mathbf {u}, \mathbf {\boldsymbol {\pi }_{i}},t)\) with i=0,…,2n _θ

\(\dot {\mathbf {x}}_{i} = \mathbf {f}(\mathbf {x_{i}},\mathbf {u}, \mathbf {\boldsymbol {\pi }_{i}},t)\) with i=0,…,n _s −1

Total n _states

(n _θ +1)n _x

(2n _θ +1)n _x

\(\frac {(n_{\theta }+p)!}{n_{\theta }!p!}n_{x}\)

Sampling points

–

Sigma points: 2n _θ +1

Collocation points: \(\frac {(n_{\theta }+p)!}{n_{\theta }!p!}\)

Expected value of R _k

R _k

\(\frac {1}{n_{\theta }+\kappa }\left (\kappa R_{k}(\boldsymbol {\pi }_{0}) + \frac 12 \sum ^{2n_{\theta }}_{i=1}R_{k}(\boldsymbol {\pi }_{i}) \right)\)

\(a_{R_{k},0}^{(p)}\)

Variance on R _k

\(\mathbf {P_{\mathrm {{R_{k}R_{k},LIN}}}} = \frac {\partial R_{k}}{\partial \mathbf {x}}\mathbf {P_{\mathrm {{LIN}}}}\left (\frac {\partial R_{k}}{\partial \mathbf {x}}\right)^{\top }\)

\(\mathbf {P_{\mathrm {{R_{k}R_{k},SP}}}} = \frac {1}{n_{\theta }+\kappa }\left (\kappa (R_{k}(\boldsymbol {\pi }_{0}) -\bar {R_{k}})(R_{k}(\boldsymbol {\pi }_{0})-\bar {R_{k}})^{\top } \right)\)

\( {\mathbf{P}}_{{\mathrm{R}}_{\mathrm{k}}{\mathrm{R}}_{\mathrm{k}},PCE}^{(p)}=\sum_{j=1}^{L-1}{\left({a}_{R_k,j}^{(p)}\right)}^2\mathbf{E}\left[{\varPhi}_j^2\left(\theta \right)\right] \)

\(+\frac {1}{n_{\theta }+\kappa }\left (\frac 12 \sum ^{2n_{\theta }}_{i=1}(R_{k}(\boldsymbol {\pi }_{i}) -\bar {R_{k}})(R_{k}(\boldsymbol {\pi }_{i}) -\bar {R_{k}})^{\top } \right)\)

with: P _LIN=S _LIN(t)Σ S _LIN(t)^⊤

with: \(\left \{\begin {array}{lll} \mathbf {\boldsymbol {\pi }_{0}} &=& \boldsymbol {\theta }_{\text {nom}} \;, \\ \mathbf {\boldsymbol {\pi }_{i}} &=& \boldsymbol {\theta }_{\text {nom}}+\sqrt {(n_{\mathrm {\theta }}+\kappa){\boldsymbol {\Sigma }}}_{i} \;\text {with}\; i=1, \hdots, n_{\mathrm {\theta }} \;, \\ \mathbf {\boldsymbol {\pi }_{i}} &=& \boldsymbol {\theta }_{\text {nom}}-\sqrt {(n_{\mathrm {\theta }}+\kappa)\boldsymbol {\Sigma }}_{i-n_{\mathrm {\theta }}} \;\text {with}\; i=n_{\mathrm {\theta }}+1, \hdots, 2n_{\mathrm {\theta }} \;. \\ \kappa &=& 3-n_{\theta } \; \end {array}\right.\)

with: a=(Λ Λ ^⊤)⁻¹ Λ R _k,s

Optimization problem

\(\underset {\mathbf {u, x}, t_{\mathrm {f}}}{\text {min}} \quad \{J_{1}, \dots, J_{n_{J}}\}\)

\(\underset {\mathbf {u, x}, t_{\mathrm {f}}}{\text {min}} \quad \{J_{1}, \dots, J_{n_{J}}\}\)

\(\underset {\mathbf {u, x}, t_{\mathrm {f}}}{\text {min}} \quad \{J_{1}, \dots, J_{n_{J}}\}\)

\(\bar {J_{i}}_{\text {LIN}}+ \alpha _{J_{i}} \sqrt {\mathbf {P_{\mathrm {{J_{i}J_{i},SP}}}}}\)

\(\bar {J_{i}}_{\text {SP}}+ \alpha _{J_{i}} \sqrt {\mathbf {P_{\mathrm {{J_{i}J_{i},SP}}} }}\)

\(\bar {J_{i}}_{\text {PCE}}^{(p)}+ \alpha _{J_{i}} \sqrt {\mathbf {P_{\mathrm {{J_{i}J_{i},PCE}}}}^{(p)}}\)

s.t. \(\left \{ \begin {array}{lll} \dot {\mathbf {x}} &=& \mathbf {f}(\mathbf {x},\mathbf {u},\boldsymbol {\theta },t) \\ \dot {\mathbf {S}}_{\mathrm {{LIN}}}(t) &=& \frac {\partial \mathbf {f}(\mathbf {x},\mathbf {u},\boldsymbol {\theta }_{\text {nom}},t)}{\partial \mathbf {x}} \mathbf {S_{\mathrm {{LIN}}}} + \frac {\partial \mathbf {f}(\mathbf {x},\mathbf {u},\boldsymbol {\theta }_{\text {nom}},t)}{\partial \boldsymbol {\theta }},\\ \mathbf {S_{\text {LIN}}}(0) &=& \mathbf {0} \\ \mathbf {x}(0)&=&\mathbf {x}_{0} \\ 0 &\geq & \bar {c}_{\text {prob},i, \text {LIN}} \\ &&+ \alpha _{c_{\text {prob},i}} \sqrt {\mathbf {P_{\mathrm {c_{\text {prob},i},c_{\text {prob},i}, LIN}}}} \end {array} \right.\)

s.t. \(\left \{ \begin {array}{lll} \dot {\mathbf {x}}_{i}&=&\mathbf {f}(\mathbf {x_{i}},\mathbf {u}, \mathbf {\boldsymbol {\pi }_{i}},t) \text {with}\; i = 0, \hdots, 2n_{\theta } \\ \mathbf {x}(0)&=&\mathbf {x}_{0} \\ 0 &\geq & \bar {c}_{\text {prob},i, \text {SP}} \\ &&+ \alpha _{c_{\text {prob},i}} \sqrt {\mathbf {P_{\mathrm {c_{\text {prob},i},c_{\text {prob},i},SP}}}} \\ \end {array} \right.\)

s.t. \( \left\{\begin{array}{lll}{\dot{\mathbf{x}}}_i& =& \mathbf{f}\left({\mathbf{x}}_i,\mathbf{u},{\pi}_i,t\right)\\ &&\mathrm{with} i=0,\dots, {n}_s-1\\ {}\mathbf{x}(0)& =& {\mathbf{x}}_0\\ {}{{\mathbf{R}}_{\mathbf{k},\mathbf{s}}}^{(p)}& =& {\left({\varLambda}^{(p)}\right)}^{\top }{{\mathbf{a}}_{{\mathbf{R}}_{\mathbf{k}}}}^{(p)}\\ {}\mathrm{with} k=1,\dots, {n}_R\\ {}0& \ge & {\bar{c}}_{\mathrm{prob},i,\mathrm{P}\mathrm{C}\mathrm{E}}^{(p)}\\ {}+{\alpha}_{c_{\mathrm{prob},i}}\sqrt{{{\mathbf{P}}_{{\mathbf{c}}_{\mathbf{prob},\mathbf{i}},{\mathbf{c}}_{\mathbf{prob},\mathbf{i}},\boldsymbol{P}\boldsymbol{C}\boldsymbol{E}}}^{(p)}}\\ {}\mathrm{with} i=1,\dots, {n}_{c_{\mathrm{prob}}}\end{array}\right. \)