Performance metrics

IQClab offers the possibility to consider various performance metrics in the IQC-analysis:

  • Induced L_2-gain performance (H_\infty-norm in case of LTI uncertainties)
  • H_2 performance (deterministic signal-based interpretation)
  • Generalized H_2 performance (energy-to-peak performance)
  • Passivity performance

In addition, it is possible to consider various performance metrics related to the computation of invariant sets:

  • Invariant sets in the state space under the assumption that the external disturbances are bounded in energy (e2x)
  • Invariant sets in the performance output under the assumption that the external disturbances are bounded in energy (e2z)
  • Energy-to-peak gain performance (e2p)
  • Non-zero initial condition to peak gain performance (x2p)

If no performance metric is specified, a robust stability analysis will be carried out.

Performance metricDescription
Induced L_2-gainIf selecting the induced L_2-gain performance (H_\infty-norm in case of LTI uncertainties) as performance metric, then the analysis tools perform a robustness analysis, while the induced L_2-gain from all specified performance inputs to all specified performance outputs is minimized. If feasible, this yields a guaranteed upper-bound, \gamma, on the worst-case induced L_2-gain performance for all modelled uncertainties \Delta\in\mathbf{\Delta}:

    \[\sup_{\Delta\in\mathbf{\Delta}}\|\Delta\star M\|<\gamma\]


Note: See Section 6.1 of [1] for the details on the mathematical derivation of the corresponding performance multiplier class.
H_2Similarly, if selecting the H_2-norm as performance metric, then the analysis tools perform a robustness analysis, while the the H_2-norm from all specified performance inputs to all specified performance outputs is minimized. If feasible, this yields a guaranteed upper-bound, \gamma, on the worst-case the H_2-norm performance for all modelled uncertainties \Delta\in\mathbf{\Delta}:

    \[\sup_{\Delta\in\mathbf{\Delta}}\|\Delta\star M\|_2<\gamma\]


With T=\Delta\star M, this norm is defined by

    \[\|T\|_2=\sqrt{\dfrac{1}{2\pi}\mathrm{trace}\int_{-\infty}^{\infty}T(\omega) T(\omega)^*d\omega.\]


Notes:
– This performance metric corresponds to the deterministic signal-based interpretation.
– If we let M=\left(\begin{array}{cc}M_{11}& M_{12}\\ M_{21}& M_{22} \end{array}\right) with M_{11} being dimension compatible with \Delta, then we require M_{12}(\infty)=0 and M_{22}(\infty)=0.

-See Section 6.4 of [1] for the details on the mathematical derivation of the corresponding performance multiplier class.
Generalized H_2 Similar to the H_2 performance objective it is also possible to consider the generalized H_2 metric (also called the energy-to-peak gain). If feasible, this yields a guaranteed upper-bound, \gamma, on the worst-case generalized H_2 performance for all modelled uncertainties \Delta\in\mathbf{\Delta}:

    \[\sup_{\Delta\in\mathbf{\Delta}}\|\Delta\star M\|_{2,\infty}<\gamma\]


With T=\Delta\star M, this norm is defined by

    \[\|T\|_{2,\infty}=\sqrt{\dfrac{1}{2\pi}\lambda_\mathrm{max}\int_{-\infty}^{\infty}T(\omega) T(\omega)^*d\omega,\]


where \lambda_\mathrm{max}(\cdot) denotes the maximum eigenvalue.
PassivityNext to the previous performance metrics, it is also possible to verify if the uncertain system P(\Delta)=\Delta\star M is (strictly) input passive for all uncertainties \Delta\in\mathbf{\Delta}. This means that P(\Delta)+ P(\Delta)^*\prec0 for all \Delta\in\mathbf{\Delta}.

Note: See Section 6.2 of [1] for the details on the mathematical derivation of the corresponding performance multiplier class.
e2xThis option allows to compute invariant sets in the state-space under the assumption that the external disturbances are bounded in energy. The option is abbreviated as e2x. If feasible, this guarantees that the internal state is confined to the hyper ellipsoidal region

    \[x(t)\in\left\{x\in\mathbb{R}^n: x^TH^{-1}x\leq\alpha^2\right\},\]

for \|w\|\leq\alpha.
e2zSimilarly, this option allows to compute invariant sets for the output z, again under the assumption that the external disturbances w are bounded in energy. The option is abbreviated as e2z. If feasible, this guarantees that the output z is confined to the hyper ellipsoidal region

    \[z(t)\in\left\{z\in\mathbb{R}^n: z^TH^{-1}z\leq\alpha^2\right\},\]

for \|w\|\leq\alpha.
e2pThis option, which is similar to the Generalized H_2 performance metric, allows to compute bounds on the individual components of z_j, j\in\{1,\cdots,k\}. For M_{21}(\infty)=M_{22}(\infty)=0 being strictly proper, this option yields the peak gains \gamma_j on the performance output channels z_j, j\in\{1,\cdots,k\} such that

    \[|z_j(t)|\leq\sqrt{\gamma_j}\alpha\ \ \forall t\geq0,\ j\in\{1,\cdots,k\}.\]

x2pThis option facilitates the computation of bounds on the peak gain of the performance output z for a given non-zero initial condition x_0\neq0.
With M_{21}(\infty)=0 being strictly proper, this option computes the peak gain \gamma on the performance output channel z such that

    \[\|z(t)\|=\|C_2x(t)\|\leq\gamma\|x_0\|\ \ \forall t\geq0\]

by minimizing \gamma.
Robust stabilityFinally, it is also possible to verify if the uncertain system \Delta\star M is stable for all \Delta\in\mathbf{\Delta}. This just means that all performance channels are omitted in the analysis.

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