**5.1. Reachable set**

Considering System (4) with controller *v*(*k*)= *Kx*(*k*), the definition of reachable set is given.

**Definition 3 [15]:** The reachable set S*r* is defined as

$$\mathcal{S}\_r = \left\{ \mathbf{y}(k) = C\mathbf{x}(k) \mid \mathbf{x}(k) \in \mathcal{S}\_n \right\}.$$

$$\mathcal{S}\_{\scriptscriptstyle{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\cdots}}}}}}}}}}}}}}}}}}}}}}}}}}}$$
}}}

where *Ω<sup>v</sup>* ={*v*(*k*)| −1≤*vi* ≤1, *i* =1, ..., *m*0}.

#### **5.2. ISBP method**

For the convenience of discussion, a 2D work space is used in the following whereas the proposed ISBP approach can be expanded in 3D condition directly. Consider a preknown environment as shown in **Figure 3**, where the initial location is *L* <sup>0</sup> and the goal location is *L* 5. Based on the preknown environment, a lot of path planning approaches can be used to find a feasible path against external environment constraints such as obstacles. Suppose this path is represented by a series of nodes such as *L* <sup>0</sup>∼ *L* <sup>5</sup>, as shown in **Figure 3**.

The target of the proposed ISBP approach is to calculate feasible controller reference inputs to move the UH from the initial location to the goal location based on the feasible path and satisfy both external environment constraints and UH's limits.

**Figure 3.** Two‐dimensional environment and feasible path.

Consider a simple example in the preknown environment as shown in **Figure 4(a)** where the initial location is *L* <sup>0</sup> and the goal location is *L* <sup>1</sup>. Suppose a feasible path is represented by two nodes *L* <sup>0</sup> and *L* <sup>1</sup> and their connection is shown in the same figure.

Clearly, the work space is state space; thus, the location in work space is equality to the state such as the initial state, which is equal to L0. Thus, an invariant set S can be constructed according to plant dynamic model (4) under controller *v*(*k*)= *Kx*(*k*), which also guarantees obstacle avoidance. Suppose the invariant set is S0 and the related reachable set is S*r*0 as shown in **Figure 4(b)**. Clearly, because of the existence of constraints such as obstacles and UH's limits, the invariant set S<sup>0</sup> may not contain the goal location *L* <sup>1</sup> as shown in **Figure 4(b)**. Closely, the goal location *L* <sup>1</sup> is also outside the reachable set S*r*0, which means *L* <sup>1</sup> is unreachable under this condition. Thus, the system states should be continued moving. Suppose the intersection of reachable set S*r*<sup>0</sup> and feasible path *<sup>L</sup>* <sup>0</sup>*<sup>L</sup>* ¯

<sup>1</sup> is *c*<sup>1</sup> as shown in **Fig‐ ure 4(b)**. In order to move state from *c*1 to *L* <sup>1</sup>, a new invariant set, whose center is *c*1, is required. Suppose the state of the new system is *x*¯(*k*)<sup>=</sup> *<sup>x</sup>*(*k*)− *xc*1, where *xc*1 is a constant vector whose position is equal to *c*1 and other elements are 0. Based on the new system, we can construct the second invariant set S1 whose center is *c*1 as shown in **Figure 4(c)**. The related reachable set S*r*<sup>1</sup> is also shown. Obviously, the goal location *L* <sup>1</sup> is inside the reachable set S*r*1 now, which means *L* <sup>1</sup> is reachable under this condition. In this way, we can choose the center of S1, *c*1, and the goal location *L* <sup>1</sup> as a sequence of the controller reference input. Finally, Figure 4(d) shows the practical trajectory that is obtained by the UH actual operation.

Self-Healing Control Framework Against Actuator Fault of Single-Rotor Unmanned Helicopters http://dx.doi.org/10.5772/62480 125

**Figure 4.** ISBP in 2D condition.

**Figure 3.** Two‐dimensional environment and feasible path.

124 Recent Progress in Some Aircraft Technologies

nodes *L* <sup>0</sup> and *L* <sup>1</sup> and their connection is shown in the same figure.

this condition. Thus, the system states should be continued moving. Suppose the intersection of reachable set S*r*<sup>0</sup> and feasible path *<sup>L</sup>* <sup>0</sup>*<sup>L</sup>* ¯

practical trajectory that is obtained by the UH actual operation.

Consider a simple example in the preknown environment as shown in **Figure 4(a)** where the initial location is *L* <sup>0</sup> and the goal location is *L* <sup>1</sup>. Suppose a feasible path is represented by two

Clearly, the work space is state space; thus, the location in work space is equality to the state such as the initial state, which is equal to L0. Thus, an invariant set S can be constructed according to plant dynamic model (4) under controller *v*(*k*)= *Kx*(*k*), which also guarantees obstacle avoidance. Suppose the invariant set is S0 and the related reachable set is S*r*0 as shown in **Figure 4(b)**. Clearly, because of the existence of constraints such as obstacles and UH's limits, the invariant set S<sup>0</sup> may not contain the goal location *L* <sup>1</sup> as shown in **Figure 4(b)**. Closely, the goal location *L* <sup>1</sup> is also outside the reachable set S*r*0, which means *L* <sup>1</sup> is unreachable under

**ure 4(b)**. In order to move state from *c*1 to *L* <sup>1</sup>, a new invariant set, whose center is *c*1, is required. Suppose the state of the new system is *x*¯(*k*)<sup>=</sup> *<sup>x</sup>*(*k*)− *xc*1, where *xc*1 is a constant vector whose position is equal to *c*1 and other elements are 0. Based on the new system, we can construct the second invariant set S1 whose center is *c*1 as shown in **Figure 4(c)**. The related reachable set S*r*<sup>1</sup> is also shown. Obviously, the goal location *L* <sup>1</sup> is inside the reachable set S*r*1 now, which means *L* <sup>1</sup> is reachable under this condition. In this way, we can choose the center of S1, *c*1, and the goal location *L* <sup>1</sup> as a sequence of the controller reference input. Finally, Figure 4(d) shows the

<sup>1</sup> is *c*<sup>1</sup> as shown in **Fig‐**

#### **5.3. Calculation of the invariant set and the reachable set**

According to Theorem 2, the invariant set S and the reference set *Ωref* are obtained. Assume the reachable set S*r* defined by Definition 3 is included in the invariant set S such that S*<sup>r</sup>* ⊂S. Because the reference is bounded such as ref∈*Ω*ref and the integrator can guarantee all references to be reachable such as *y*(*∞*)=ref, the reachable set S*<sup>r</sup>* can be denoted by <sup>S</sup>*<sup>r</sup>* ={ref|ref*<sup>T</sup> Qr*ref≤1}. Thus, the invariant set and reachable set neglecting external environ‐ ment constraints are obtained.

However, in order to guarantee controlled plant obstacle avoidance, the state trajectory should be kept outside all of spheres that represent obstacles. For achieving it, the invariant set S should not intersect with all of obstacle spheres. Suppose S0⊂*R <sup>n</sup>*<sup>+</sup> *<sup>p</sup>* is a sphere that is defined by S<sup>0</sup> ={*x*(*k*)| *<sup>x</sup> <sup>T</sup>* (*k*)*Q*0*x*(*k*)≤1}, where *Q*<sup>0</sup> =diag *<sup>q</sup>*1, ..., *qn*<sup>+</sup> *<sup>p</sup>* , *qi* =1 / *<sup>d</sup>*min <sup>2</sup> , *d*min is the distance between the center of the nearest obstacle and the current system equilibrium point.

According to the above analysis, the following theorem is proposed to calculate the maximum invariant set S and the reachable set S*r* of system (4) under both external environment constraints and UH's limits.

**Theorem 3:** Given *η* >0, LQR parameter matrices *Qu*, *Qx*, symmetric positive‐definite matri‐ ces *W*0 and *R*, if there exist a symmetric positive‐definite matrix *<sup>W</sup>* <sup>∈</sup>*<sup>R</sup>* (*n*<sup>+</sup> *<sup>p</sup>*)×(*n*<sup>+</sup> *<sup>p</sup>*) , matrices *X* ∈*R <sup>m</sup>*0×(*n*<sup>+</sup> *<sup>p</sup>*) , *Y* ∈*R <sup>m</sup>*0×(*n*<sup>+</sup> *<sup>p</sup>*) , *Z* ∈*R <sup>p</sup>*×*m*<sup>0</sup> , diagonal positive‐definite matrices *S* ∈*R <sup>m</sup>*0×*m*<sup>0</sup> , *Wr* <sup>∈</sup>*<sup>R</sup> <sup>p</sup>*<sup>×</sup> *<sup>p</sup>* , and a positive scale *λ* satisfying

$$\inf\_{\mathcal{W}, \mathcal{X}, \mathcal{Y}, \mathcal{Z}, \mathcal{S}, \mathcal{W}\_r} \mathcal{X}$$

$$
\begin{bmatrix} 1 & X\_i - Y\_i \\ \ast & W \end{bmatrix} \ge \mathbf{0}, I = 1, \dots, m\_0
$$


$$\left[\begin{array}{cc} I\_{\rho} & \mathbf{0}\_{\rho \times n} \end{array} \right]^{\mathsf{T}} W\_{r} \left[\begin{array}{cc} I\_{\rho} & \mathbf{0}\_{\rho \times n} \end{array} \right] \leq W$$

<sup>0</sup> \* *<sup>r</sup> R I W* é ù l ê ú £ ë û *W W*£ <sup>0</sup>

> l> 0

where *W*<sup>0</sup> =*Q*<sup>0</sup> −1 , then obstacle avoidance and −1≤*v*(*k*)≤1 would be guaranteed, furthermore, the ellipsoid S={*x*(*k*)| *x <sup>T</sup>* (*k*)*Px*(*k*)≤1} is an invariant set with *P* =*W* <sup>−</sup><sup>1</sup> and <sup>S</sup>*<sup>r</sup>* ={ref|ref*<sup>T</sup> Qr* ref≤1} is a reachable set with *Qr* <sup>=</sup>*Wr* −1 , which satisfies S*<sup>r</sup>* ⊂S.

Clearly, Theorem 2 is included in Theorem 3 because the invariant set is a bridge to connect environment and UH dynamic constraints. Thus, by the ISBP approach, the controller design and controller reference calculation are achieved simultaneously when the preknown work space, the desired path nodes, and the UH dynamic model are given.

The relationship between the related sets and the obstacles are shown in **Figure 5** assuming that all sets are projected to a 2D plane.

**Figure 5.** The relationship of related sets.

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

where *W*<sup>0</sup> =*Q*<sup>0</sup>

−1

<sup>S</sup>*<sup>r</sup>* ={ref|ref*<sup>T</sup> Qr* ref≤1} is a reachable set with *Qr* <sup>=</sup>*Wr*

space, the desired path nodes, and the UH dynamic model are given.

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

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

126 Recent Progress in Some Aircraft Technologies

, and a positive scale *λ* satisfying

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

1

0 0

h


*r*

*S SB Z R*

, ,,,, 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 -

 0 0 *<sup>T</sup> p pn r p pn I WI W* ´ ´ é ùé ù £ ë ûë û

> <sup>0</sup> \* *<sup>r</sup> R I W* é ù l

ê ú £ ë û

*W W*£ <sup>0</sup>

l> 0

the ellipsoid S={*x*(*k*)| *x <sup>T</sup>* (*k*)*Px*(*k*)≤1} is an invariant set with *P* =*W* <sup>−</sup><sup>1</sup> and

Clearly, Theorem 2 is included in Theorem 3 because the invariant set is a bridge to connect environment and UH dynamic constraints. Thus, by the ISBP approach, the controller design and controller reference calculation are achieved simultaneously when the preknown work

, then obstacle avoidance and −1≤*v*(*k*)≤1 would be guaranteed, furthermore,

, which satisfies S*<sup>r</sup>* ⊂S.

−1

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

*W* é ù - ³ = ê ú ë û

l

0

*I*



1

*u*


*Q*


1

*x*


*Q*

, diagonal positive‐definite matrices *S* ∈*R <sup>m</sup>*0×*m*<sup>0</sup>

,

In other words, based on the calculated invariant set S and the reachable set S*r*, a new center *c*<sup>1</sup> can be computed which is the intersection point of S*<sup>r</sup>* and the feasible path. The next step is to calculate the second invariant set and the reachable set whose center is *c*1 until the last reachable set covers the final path point.
