**5. ISBP approach**

( ( )) ( ( )) ( ) 0 [ ] *<sup>T</sup>*

.

, *Y* ∈*R <sup>m</sup>*0×(*n*<sup>+</sup> *<sup>p</sup>*)

, and a positive scale *η* satisfying

0

*I*



represent the *i*th row of matrices *X* , *Y* and *K* = *XP*, *G* =*YP*. Then, the saturation

−1

1

*u*


*Q*


1

. Note that S is defined as the safety

. The proof can be found in [13].

*x*


*Q*

h

, *Z* ∈*R <sup>p</sup>*×*m*<sup>0</sup>

, diagonal positive‐

 j*Kx k T Kx k Gx k* - £

Based on this lemma, saturation compensator can be designed in the following theorem.

**Theorem 2:** Given LQR parameter matrices *Qu*, *Qx*, if there exist a symmetric positive‐definite

, ,,,, inf *W XY Z SWr*

0, 1,..., \* *X Y i i I m*

> 1 2

*D WD I W*

\* \*\* 0 0 <sup>0</sup> \* \*\*\* <sup>000</sup>

\* 200 000 \*\* 0 000

*T T TT T TT T T*

é ù - ê ú - --

*W Y WA C X W*

\* \*\*\* \* 0 0 \* \*\*\* \* \* 0

ë û -

compensator will be *Ec* <sup>=</sup>*ZS* <sup>−</sup><sup>1</sup> and the closed‐loop postfault system will be stable if *x*(*k*) inside

According to Theorem 2, the safety region S is an invariant set which means that if initial states and steady states of a system are inside the set, the state trajectory from initial states to steady states will also be inside the set. In this way, a postfault system with a fault‐tolerant controller can be seen as a new closed‐loop system with the initial state *x*(*kf* ), where *kf* is instant when the fault is detected Clearly, a postfault system will be safe if initial state *x*(*kf* ) is inside the

On the other hand, the related admissible set of reference *Ω*ref is also achieved by Theorem 2 and it has a closed relationship with the controller. In the next section, the controller design

\* \*\*\* \* \* \*

*W* é ù - ³ = ê ú ë û

j

, matrices *X* ∈*R <sup>m</sup>*0×(*n*<sup>+</sup> *<sup>p</sup>*)

1

0 0

h


the stability domain S={*x*(*k*)| *x <sup>T</sup>* (*k*)*Px*(*k*)≤1} with *P* =*W* <sup>−</sup><sup>1</sup>

region in this chapter. *Ωref* is also achieved with *Qr* =*Wr*

and the reference design will be composed together.

*r*

*S SB Z R*

, *Wr* <sup>∈</sup>*<sup>R</sup> <sup>p</sup>*<sup>×</sup> *<sup>p</sup>*

is verified for any positive‐definite matrix *T* ∈*R <sup>m</sup>*0×*m*<sup>0</sup>

matrix *W* ∈*R* (*n*<sup>+</sup> *<sup>p</sup>*)×(*n*<sup>+</sup> *<sup>p</sup>*)

where *Xi*

safety region S.

, *Yi*

definite matrices *S* ∈*R <sup>m</sup>*0×*m*<sup>0</sup>

122 Recent Progress in Some Aircraft Technologies

This section proposes an ISBP approach that can plan a feasible trajectory based on the desired path nodes under both external environment constraints and their own dynamic limits [14]. In order to consider environment constraints and dynamic constraints at the same time, invariant set is used as a bridge to connect the two kinds of constraints because the invariant set is calculated based on the Lyapunov function that is linked to dynamic model, and on the other hand, the invariant set has certain geometrical shapes, such as ellipsoid, so that it can easily be represented in work space with obstacles.

Note that, in this section, we have assumed that the heading of UH is kept to be 0, pitch angle and roll angle of UH are kept small so that the position of UH in the world coordinate system can be considered as the integration of velocities in the body coordinate system.
