**4. Reconfigurable controller design**

In this section, AHC‐based antiwindup controller design method is introduced which can be achieved by solving a set of linear matrix inequalities (LMIs). At the same time, the related invariant‐set‐based safety region is also calculated. The definition of invariant set and the controller design under actuator constraints are given as follows.

#### **4.1. Invariant set**

Considering a linear system

$$\mathbf{x}(k+1) = A\mathbf{x}(k)$$

the (positively) invariant set can be given by the following definition.

**Definition 1 [10]:** The set S⊂*R <sup>n</sup>* is said invariant for the above system if for all *x*(0)∈S the solution *x*(*k*)∈S. If *x*(0)∈S implies *x*(*k*)∈S for *k* >0, then we say that S is positively invariant.

Based on Definition 1, the definition of robustly controlled invariant set for the following system is provided:

$$\begin{cases} \mathbf{x}(k+\mathbf{l}) = A\mathbf{x}(k) + B\boldsymbol{u}(k) + D\boldsymbol{o}(k) \\ \mathbf{y}(k) = C\mathbf{x}(k) \end{cases}$$

**Definition 2 [10]:** The set S⊂*R <sup>n</sup>* is said robustly controlled invariant for the above system if there exists a continuous feedback control law *u*(*k*)= *Kx*(*k*), which ensures the existence and uniqueness of the solution on *R* <sup>+</sup> such that S is positively invariant for the closed loop system.

An invariant set S can be constructed by the following theorem.

**Theorem 1 [11]:** The set S*<sup>ρ</sup>* ={*x*(*k*)∈*R <sup>n</sup>* |*V* (*x*(*k*))≤*ρ*} with *ΔV* (*x*(*k*))≤0 for all *x*(*k*)∈S*ρ* is positively invariant where *V* is a function.

For convenience, *ρ* is chosen equal to 1 and simplify S*ρ* as S={*V* (*x*(*k*))≤1}. Hence, the construc‐ tion of S finds a function *V* , which implies *ΔV* (*x*(*k*))≤0 when *V* (*x*(*k*))≤1. Obviously, if we assume *V* (*x*(*k*))≥0, *V* would be a candidate of the Lyapunov function. More specially, if the Lyapunov candidate function is in quadratic form such that *V* (*x*(*k*))= *x <sup>T</sup>* (*k*)*Px*(*k*), the invariant set will be an ellipsoid and this nature will be used later.

#### **4.2. Controller design under actuator constraints**

Consider linear discrete UH model, Eq. (2), with actuator constraints:

$$\begin{cases} \boldsymbol{\chi}\_h(k+\mathsf{l}) = \boldsymbol{A}\_h \boldsymbol{\chi}\_h(k) + \boldsymbol{B}\_h \boldsymbol{\sigma}(\boldsymbol{u}\_h(k)),\\ \boldsymbol{\chi}\_h(k) = \boldsymbol{C}\_h \boldsymbol{\chi}\_h(k) \end{cases}$$

Where *σ*(*uh* (*k*)) represents constrained control inputs such that −1≤*u*(*k*)≤1 and the saturation feature can be defined by *σ*(*ui* (*k*))=sgn(*ui* (*k*))min{1, |*ui* (*k*)|}, where sgn( ⋅ ) is a sign function and *i* =1, ..., *m*. Then, considering AHC model, the above system can be rewritten as

$$\begin{cases} \boldsymbol{\chi}\_h(k+\mathsf{l}) = \boldsymbol{A}\_h \boldsymbol{\chi}\_h(k) + \boldsymbol{B}\_{h0} \boldsymbol{\sigma}(\boldsymbol{\nu}(k)) + \boldsymbol{B}\_{h\boldsymbol{\zeta}} \overline{\boldsymbol{\mu}}\_h \\ \quad \boldsymbol{\chi}\_h(k) = \boldsymbol{C}\_h \boldsymbol{\chi}\_h(k) \end{cases}$$

Where *Bh* <sup>0</sup> = *Bh Λ* and *Bhf* = *Bh* (*I* −*Λ*). In the following discussion, assume the number of *λ<sup>i</sup>* =0 is *mf* , which implies *mf* actuators cannot respond to the control signal and *m*<sup>0</sup> =*m*−*mf* is the number of fault‐free/partial‐fault actuators that can respond to the control signal. In other words, *m*<sup>0</sup> represents the number of nonzero columns of matrix *Bh* 0. In this way, the postfault system with different actuator faults can be represented.

In order to achieve set‐point tracking offset free and compensate actuator saturation, an integrator with a saturation compensator is introduced as follows:

$$e(k+\mathbf{l}) = e(k) + T\_s \left( \text{ref} - C\_h \mathbf{x}\_h(k) \right) - T\_s E\_c \phi(\mathbf{v}(k))$$

where *e*(*k*) is the integrator state vector, *Ec* is the pending compensator matrix, *φ*(*v*(*k*))=*v*(*k*)−*σ*(*v*(*k*)) is the difference of controller outputs and actuator outputs under saturation [12], ref∈*Ω*ref={ref|ref*<sup>T</sup> Qr*ref≤1} is a set‐point reference, *Qr* is a constant matrix, and *Ts* is sampling period. Thus, a new open‐loop system with extend state vector *x*(*k*)= *xh <sup>T</sup>* (*k*) e*<sup>T</sup>* (*k*) *<sup>T</sup>* can be expressed by

$$\begin{cases} \mathbf{x}(k+1) = A\_\circ \mathbf{x}(k) + B\mathbf{v}(k) + Do(k) - (B + RE\_\circ)\mathbf{\bar{\rho}}(\mathbf{v}(k)) \\ \mathbf{y}(k) = \mathbf{Cx}(k) \end{cases} \tag{4}$$

where *ω*(*k*)= *u*¯ *<sup>T</sup>* ref*<sup>T</sup> <sup>T</sup>* ,

( 1) ( ) ( )

*x* ì += + +

*x k Ax k Bu k*

)

**Definition 2 [10]:** The set S⊂*R <sup>n</sup>* is said robustly controlled invariant for the above system if there exists a continuous feedback control law *u*(*k*)= *Kx*(*k*), which ensures the existence and

**Theorem 1 [11]:** The set S*<sup>ρ</sup>* ={*x*(*k*)∈*R <sup>n</sup>* |*V* (*x*(*k*))≤*ρ*} with *ΔV* (*x*(*k*))≤0 for all *x*(*k*)∈S*ρ* is positively

For convenience, *ρ* is chosen equal to 1 and simplify S*ρ* as S={*V* (*x*(*k*))≤1}. Hence, the construc‐ tion of S finds a function *V* , which implies *ΔV* (*x*(*k*))≤0 when *V* (*x*(*k*))≤1. Obviously, if we assume *V* (*x*(*k*))≥0, *V* would be a candidate of the Lyapunov function. More specially, if the Lyapunov candidate function is in quadratic form such that *V* (*x*(*k*))= *x <sup>T</sup>* (*k*)*Px*(*k*), the invariant

( 1) ( ) ( ( ))

Where *σ*(*uh* (*k*)) represents constrained control inputs such that −1≤*u*(*k*)≤1 and the saturation

(*k*))min{1, |*ui*

<sup>0</sup> ( 1) ( ) ( ( ))

ì += + +

*<sup>h</sup> h h <sup>h</sup> hf x k Ax k B vk*

*B u* <sup>í</sup>

Where *Bh* <sup>0</sup> = *Bh Λ* and *Bhf* = *Bh* (*I* −*Λ*). In the following discussion, assume the number of *λ<sup>i</sup>* =0 is *mf* , which implies *mf* actuators cannot respond to the control signal and *m*<sup>0</sup> =*m*−*mf* is the number of fault‐free/partial‐fault actuators that can respond to the control signal. In other words, *m*<sup>0</sup> represents the number of nonzero columns of matrix *Bh* 0. In this way, the postfault

In order to achieve set‐point tracking offset free and compensate actuator saturation, an

s

*k k*

s

(*k*)|}, where sgn( ⋅ ) is a sign function and

() () *h hh h h*

*k x kC* ì *k* =+ +

*x Ax B u*

*h hh*

*i* =1, ..., *m*. Then, considering AHC model, the above system can be rewritten as

() () *h hh*

*y k Cx k*

( )

such that S is positively invariant for the closed loop system.

*k*

*D*

w

() (

*y Ck*

*k*

í î =

An invariant set S can be constructed by the following theorem.

set will be an ellipsoid and this nature will be used later.

Consider linear discrete UH model, Eq. (2), with actuator constraints:

*y*

(*k*))=sgn(*ui*

î =

integrator with a saturation compensator is introduced as follows:

system with different actuator faults can be represented.

î =

í

**4.2. Controller design under actuator constraints**

uniqueness of the solution on *R* <sup>+</sup>

120 Recent Progress in Some Aircraft Technologies

invariant where *V* is a function.

feature can be defined by *σ*(*ui*

$$\begin{aligned} A\_s &= \begin{bmatrix} A\_h & 0 \\ -T\_s C\_h & I \end{bmatrix}, B = \begin{bmatrix} B\_{h0} \\ 0 \end{bmatrix}, D = \begin{bmatrix} D\_1 \ D\_2 \end{bmatrix} = \begin{bmatrix} B\_{h'} & 0 \\ 0 & T\_s I \end{bmatrix}, \\\ R &= \begin{bmatrix} 0 \\ T\_s I \end{bmatrix} and \ C = \begin{bmatrix} C\_h & 0 \end{bmatrix} \end{aligned}$$

Note that, if *mf* =0, *ω*(*k*)=ref, then *D* = 0 (*TsI*)*<sup>T</sup> <sup>T</sup>* will hold.

Based on linear quadratic regulator (LQR) theory, state‐feedback controller *v*(*k*)= *Kx*(*k*) can be calculated. Thus, the closed‐loop postfault system can be expressed by

$$\begin{cases} \varkappa(k+l) = A\varkappa(k) + D\alpha(k) - (B + RE\_c)\wp(\nu(k)) \\ \qquad \wp(k) = C\varkappa(k) \end{cases}$$

where *A*=(*Ae* + *BK*). In order to guarantee the closed‐loop system stability and to track offset‐ free under external input *ω*(*k*) and actuator constraints, the following theorem is proposed. As a basis, first a lemma is given which defines a set of system states related to actuator saturation.

**Lemma 1 ([12]):** Define the following polyhedral set with matrix *G* ∈*R <sup>m</sup>*0×(*n*<sup>+</sup> *<sup>p</sup>*) :

$$\mathcal{E} = \left\{ \mathbf{x}(k) \, \middle| \, \left| (K\_i - G\_i)\mathbf{x}(k) \, \middle| \, \left| \le 1, i = 1, \dots, m \right\} \right.\right\}$$

where *Ki* and *Gi* represent the *i*th row of matrices *K* and *G*. If *x*(*k*)∈ℰ, then the relation

$$\left[\boldsymbol{\varrho}(\boldsymbol{K}\boldsymbol{\mathfrak{x}}(k))^{\top}\boldsymbol{T}\Big[\boldsymbol{\varrho}(\boldsymbol{K}\boldsymbol{\mathfrak{x}}(k)) - \boldsymbol{G}\boldsymbol{\mathfrak{x}}(k)\Big] \leq 0\,\,\|\boldsymbol{S}\|$$

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

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 matrix *W* ∈*R* (*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 *η* satisfying

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

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


where *Xi* , *Yi* represent the *i*th row of matrices *X* , *Y* and *K* = *XP*, *G* =*YP*. Then, the saturation 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 the stability domain S={*x*(*k*)| *x <sup>T</sup>* (*k*)*Px*(*k*)≤1} with *P* =*W* <sup>−</sup><sup>1</sup> . Note that S is defined as the safety region in this chapter. *Ωref* is also achieved with *Qr* =*Wr* −1 . The proof can be found in [13].

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 safety region S.

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 and the reference design will be composed together.
