## 3 Main Results
Since, in many practical cases, the matrix \(F\) is the ratio of a multi-affine or quadratic matrix function to a multi-affine or quadratic non-null polynomial with respect to the uncertain parameters and respect to bounded functions, the determination of \(G\) can be easily made by using Lemmas 2, 3 and 4, with:_  
(i): \(G=gI,\ g>0\) _in the hypothesis that_ \(F\) _is a p.d. matrix (in the case of (4) and (9));__;_
2. \(G=gJ^{T}\)_,_ \(g>0\) _where_ \(J\) _is the Jacobian matrix of a suitable coordinate transformation (e.g., the case of robots in their workspace or equipped with cameras used as sensors, in the cases of ships, drones, or satellites);_
3. \(G=F^{\dagger}=F^{T}(FF^{T})^{-1}\) _if_ \(F\) _is independent of_ \(p\) _(e.g., in the case of kinematic inversion) or, by evaluating_ \(G\) _in correspondence of the nominal value_ \(\hat{p}\) _of_ \(p\)_, in the hypothesis that the variations of_ \(p\) _are sufficiently bounded;_
4. _other more suitable matrices_ \(G\) _(see Application_ 3_)._  
**Remark 7**. : After verifying that \(f\) _satisfies the bounding condition of class \(K_{\gamma_{0}}\) (2) and determining a matrix \(G\) satisfying the positivity condition (1), since the tracking error decreases when a increases, the single parameter a can be determined via simulation or experimentally for an assigned class of references \(r\)._  
**Remark 8**. : _Inequalities used to prove Theorems 1, 2, 3 (especially to prove that when the design parameter a increases the tracking error norm is inversely proportional to \(a^{2}\)) can make very conservative the stated results. A much less conservative estimate of the tracking error norm, i.e., providing smaller values of \(\rho_{1}\) and larger values of \(\rho_{2}\), by taking into account that \(2\rho\dot{\rho}=2x^{T}\dot{P}x\) can be obtained by using the following relation:_  
\[\dot{\rho}=\chi(\rho)=\max_{x\in C_{p,\ \rho},\ p\in\wp,T\in[0,\ t_{f}]}x^{T}P(Ax +Bw)/\rho \tag{51}\]  
_that does not use any inequality. The offline use of (51) provides an error much near to the true one, as it turns out to be from Applications 3 and 4, and Examples 2 and 3 in [34]._  
The determination of the majorant system by using (51) can be easily made by using the Matlab command \"fmincon\" (which finds a constrained minimum of a function F(X) of several variables and, hence, also a constrained maximum changing the sign to F(X)) and/or by using randomized algorithms (e.g., see [22]). Note that the current problem is not the computational burden of the controller design, but its simplicity of realization and its effectiveness. **Remark 9**. : _To reduce the gains of the controller and the control signal, above all during the transient phase, it is possible to:_  
1. _smooth the trajectories with appropriate filters and suitable initial conditions;_
2. _better identify the process parameters and the disturbances, and use a compensation signal_ \(u_{c}\) _to reduce the norm of_ \(w\)_;_
3. _use a connection trajectory if the initial error is excessive_ _[_35_]__;_
4. _slow down_ \(r(t)\)_, i.e., replace the reference signal_ \(r(t)\) _by_ \(r(st)\)_, with_ \(s<1\)_._  
Moreover, consider also the following advantages (see (45) and (46), (49) and (50), Applications 1-4, Example 6 in the Appendix A and [16, 19, 34, 35, 38]):  
1. the control signals are without the chattering phenomenon, since the proposed control law does not present discontinuities and does not have high gains,
2. the proposed control laws can be also realized by using simple analogical circuits,
3. the stated theoretical results are useful to obtain other analytical and synthetic results, significant both from a theoretical and practical point of view, and to analyze complex systems with parametric and structural uncertainties.