2. Notation and definitions

required to these aircraft. Additional complexity in the use of quadrotors for commercial applications, such as parcel delivery, is that users demand satisfactory trajectory following capabilities without tuning the controller's gains prior to each mission, whenever the payload is changed. Autopilots for commercial-off-the-shelf quadrotors are currently designed assuming that the vehicle's and the payload's inertial properties are known and constant in time. Moreover, it is assumed that the propulsion system is able to deliver maximum thrust whenever needed. These assumptions considerably simplify the design of control algorithms for quadrotor helicopters, but also undermine these vehicles' reliability in challenging work conditions, such as in case the propulsion system is partly damaged or the payload is not rigidly attached to the vehicle. For instance, the authors in [4] show that if the payload's mass and matrix of inertia vary in time, then autopilots for quadrotors designed using classical control techniques, such as the proportional-derivative control, are inadequate to guarantee satisfactory trajectory tracking.

78 Adaptive Robust Control Systems

In recent years, numerous authors, such as Bouadi et al. [5]; Dydek et al. [6]; Jafarnejadsani et al. [7]; Loukianov [8]; Mohammadi & Shahri [9]; Zheng et al. [10], to name a few, employed nonlinear robust control techniques, such as sliding mode control, model reference adaptive control (MRAC), adaptive sliding mode control, and L<sup>1</sup> adaptive control, to design autopilots for quadrotors that are able to account for inaccurate modeling assumptions and compensate failures in the propulsion system. These autopilots are generally designed assuming perfect knowledge of the location of the quadrotor's center of mass, supposing that the vehicle's Euler angles are small at all times, and neglecting the inertial counter-torque. Furthermore, in several cases also the aerodynamic force and the corresponding moment are omitted. Because of these simplifying assumptions, these autopilots are inadequate for aircraft performing aggressive maneuvers, flying in adverse weather conditions, and transporting payloads not rigidly connected to the vehicle's frame [11]. The vehicle's guidance system is usually delegated to avoiding obstacles detected by proximity sensors and cameras installed aboard. For details,

In the first part of this chapter, we present the equations of motion of quadrotors and analyze those properties needed to design effective nonlinear robust controls that enable output tracking. Specifically, we present the equations of motion of quadrotors without assuming a priori that the Euler angles are small and without neglecting the inertial counter-torque and the gyroscopic effect. Since the inertial counter-torque cannot be expressed as an algebraic function of the quadrotor's state and control vectors, we account for this effect as an unmatched time-varying disturbance on the vehicle's dynamics and hence, we consider the equations of motion of a quadrotor as a nonlinear time-varying dynamical system. Successively, we verify for the first time sufficient conditions for the strong accessibility of quadrotors' altitude and rotational dynamics; strong accessibility [15] is a weak form of controllability for nonlinear time-varying dynamical systems. As a result of this analysis, we show that a conservative control law for quadrotors must prevent rotations of a π/2 angle about either of the two horizontal axes of the body reference frame; otherwise, the vehicle may be uncontrollable.

In the second part of this chapter, we present a robust autopilot for quadrotors, which is based on a version of the e-modification of the MRAC architecture [16]. This autopilot is characterized by numerous unique features. For instance, we assume that the quadrotor's inertial

see the recent works by Faust et al. [12]; Gao & Shen [13]; Lin & Saripalli [14].

In this section, we establish the notation and the definitions used in this chapter. Let R denote the set of real numbers, <sup>R</sup><sup>n</sup> the set of <sup>n</sup> 1 real column vectors, and <sup>R</sup><sup>n</sup> <sup>m</sup> the set of <sup>n</sup> <sup>m</sup> real matrices. We write <sup>1</sup><sup>n</sup> for the <sup>n</sup> � <sup>n</sup> identity matrix, 0<sup>n</sup> � <sup>m</sup> for the zero <sup>n</sup> � <sup>m</sup> matrix, and <sup>A</sup><sup>T</sup> for the transpose of the matrix A ∈ R<sup>n</sup> � <sup>m</sup>. Given a = [a1, a2, a3] <sup>T</sup>∈ R<sup>3</sup> , a� ≜ 0 �a<sup>3</sup> a<sup>2</sup> a<sup>3</sup> 0 �a<sup>1</sup> �a<sup>2</sup> a<sup>1</sup> 0 2 6 4 3 7 5 denotes

the cross product operator. We write ∥ � ∥ for the Euclidean vector norm and ∥ � ∥<sup>F</sup> for the Frobenius matrix norm, that is, given <sup>B</sup><sup>∈</sup> <sup>R</sup>n� <sup>m</sup>, <sup>∥</sup>B∥<sup>F</sup> <sup>≜</sup> <sup>t</sup>r BB<sup>T</sup> � � � � <sup>1</sup> 2 . The Fréchet derivative of the continuously differentiable function <sup>V</sup>: <sup>R</sup>n! <sup>R</sup> at <sup>x</sup> is denoted by <sup>V</sup> 0 (x)≜∂V(x)/∂x.

Definition 2.1 ([18], Def. 6.8). The Lie derivative of the continuously differentiable function <sup>V</sup>: <sup>R</sup>n! <sup>R</sup> along the vector field f : <sup>R</sup>n! <sup>R</sup><sup>n</sup> is defined as

$$L\_f V(\mathbf{x}) \triangleq V'(\mathbf{x}) f(\mathbf{x}), \qquad \mathbf{x} \in \mathbb{R}^n. \tag{1}$$

external disturbances. In practice, the proposed controller guarantees uniform ultimate bound-

where <sup>x</sup>pð Þ<sup>t</sup> <sup>∈</sup> <sup>D</sup><sup>p</sup> <sup>⊆</sup> <sup>R</sup><sup>n</sup><sup>p</sup> , <sup>t</sup> <sup>≥</sup> <sup>t</sup>0, denotes the plant's trajectory, 0<sup>n</sup>p�<sup>1</sup> <sup>∈</sup> <sup>D</sup>p, <sup>u</sup>(t)<sup>∈</sup> <sup>R</sup><sup>m</sup> denotes the control input, y(t) ∈ R<sup>m</sup> denotes the measured output, ε > 0, Ap∈ Rn<sup>p</sup> � <sup>n</sup><sup>p</sup> is unknown, Bp∈ Rn<sup>p</sup> � <sup>m</sup>,

regressor vector <sup>Φ</sup> : <sup>R</sup>n<sup>P</sup>! <sup>R</sup><sup>N</sup> is Lipschitz continuous in its argument, and <sup>b</sup><sup>ξ</sup> : ½ Þ! <sup>t</sup>0; <sup>∞</sup> <sup>R</sup><sup>n</sup><sup>p</sup> is continuous in its argument and unknown. We assume that ∥bξð Þt ∥ ≤ bξmax, t ≥ t0, and Λ is such that the pair (Ap, BpΛ) is controllable and Λmin1<sup>m</sup> ≤ Λ, for some Λmin > 0. Both Λ and Θ<sup>T</sup>

x<sup>p</sup> ∈ Dp, capture the plant's matched and parametric uncertainties, such as malfunctions in the control system; the term bξð Þ� captures the plant's unmatched uncertainties, such as external

Eq. (6) models the plant sensors as linear dynamical systems, whose uncontrolled dynamics is exponentially stable and characterized by the parameter ε > 0 ([20], Ch. 2). Given the reference signal ycmd : [t0, <sup>∞</sup>)!Rm, which is continuous with its first derivative, define <sup>y</sup>cmd, <sup>2</sup>ð Þ<sup>t</sup> <sup>≜</sup> <sup>y</sup>\_

t ≥ t0, and assume that both ycmd(�) and ycmd, 2(�) are bounded, that is, kycmd(t)k ≤ ymax, 1, t ≥ t0,

The following theorem provides a robust MRAC for the nonlinear time-varying dynamical system (5) and (6) such that the measured output y(�) is able to eventually track the reference signal ycmd(�) with bounded error, that is, there exist b > 0 and c > 0 independent of t0, and for every a∈ (0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t0, such that if ∥y(t0) �

> εC<sup>p</sup> �ε1<sup>m</sup> " #

<sup>x</sup>\_pðÞ¼ <sup>t</sup> <sup>A</sup>pxpð Þþ <sup>t</sup> <sup>B</sup>p<sup>Λ</sup> u tð Þþ <sup>Θ</sup><sup>T</sup><sup>Φ</sup> <sup>x</sup>pð Þ<sup>t</sup> � � � � <sup>þ</sup> <sup>b</sup>ξð Þ<sup>t</sup> , xpð Þ¼ <sup>t</sup><sup>0</sup> <sup>x</sup>p, <sup>0</sup>, t <sup>≥</sup> <sup>t</sup>0, (5)

, Λ ∈ R<sup>m</sup> � <sup>m</sup> is diagonal, positive-definite, and unknown, Θ ∈ R<sup>N</sup> � <sup>m</sup> is unknown, the

y t \_ðÞ¼ εCpxpð Þ� t εy tð Þ, ytð Þ¼ <sup>0</sup> Cpxp,0, (6)

Robust Adaptive Output Tracking for Quadrotor Helicopters

http://dx.doi.org/10.5772/intechopen.70723

∥y tð Þ� ycmdð Þt ∥ ≤ b, t ≥ t<sup>0</sup> þ T: (7)

xp,<sup>0</sup>

, <sup>B</sup><sup>≜</sup> <sup>B</sup><sup>p</sup>

" #b<sup>ξ</sup>ð Þ<sup>t</sup> , and consider the reference dynamical model

<sup>p</sup>ð Þ<sup>t</sup> ; y tð Þ� <sup>y</sup>cmdð Þ<sup>t</sup> � �<sup>T</sup> h i<sup>T</sup>

Cpxp,<sup>0</sup> � ycmdð Þ t<sup>0</sup> " #

� �, <sup>B</sup><sup>1</sup> <sup>≜</sup> <sup>0</sup><sup>n</sup>p�<sup>m</sup>

0<sup>m</sup>�<sup>m</sup>

Φ(xp),

81

cmdð Þt ,

∈ R<sup>n</sup>, t ≥ t0,

, t ≥ t0, (8)

� �, and

�1<sup>m</sup>

Consider the nonlinear time-varying plant and the plant sensors' dynamics

edness of the output tracking error.

and kycmd, 2(t)k ≤ ymax, 2, for some ymax, 1, ymax, 2 > 0.

For the statement of this result, let <sup>n</sup><sup>≜</sup> <sup>n</sup><sup>p</sup> <sup>+</sup> <sup>m</sup> and x tð Þ<sup>≜</sup> <sup>x</sup><sup>T</sup>

x t \_ðÞ¼ Ax tðÞþ <sup>B</sup><sup>Λ</sup> u tð Þþ <sup>Θ</sup><sup>T</sup><sup>Φ</sup> <sup>x</sup>pð Þ<sup>t</sup> � � � � <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>t</sup> , xtð Þ¼ <sup>0</sup>

where x tð Þ<sup>∈</sup> <sup>D</sup> <sup>⊆</sup> <sup>R</sup><sup>n</sup>, <sup>D</sup> <sup>≜</sup> <sup>D</sup><sup>p</sup> � <sup>R</sup><sup>m</sup>, <sup>A</sup> <sup>≜</sup> <sup>A</sup><sup>p</sup> <sup>0</sup><sup>n</sup>p�<sup>m</sup>

1<sup>n</sup><sup>p</sup> 0<sup>m</sup>�n<sup>p</sup>

note that (5) and (6) are equivalent to

ξð Þt ≜B<sup>1</sup> ycmd,2ð Þþ t εycmdð Þt

h i <sup>þ</sup>

Cp∈ R<sup>m</sup> � <sup>n</sup><sup>p</sup>

disturbances.

ycmd(t0) ∥ ≤ a, then

The zeroth-order and the higher-order Lie derivatives are, respectively, defined as

$$L\_f^0 V(\mathbf{x}) \triangleq V(\mathbf{x}), \qquad L\_f^k V(\mathbf{x}) \triangleq L\_f \left( L\_f^{k-1} V(\mathbf{x}) \right), \qquad \mathbf{x} \in \mathbb{R}^n, \qquad k \ge 1. \tag{2}$$

Given the continuously differentiable functions <sup>f</sup>, <sup>g</sup> : <sup>R</sup>n! <sup>R</sup><sup>n</sup> , the Lie bracket of f(�) and g(�) is defined as

$$\operatorname{ad}\_{\mathcal{J}} \mathbf{g}(\mathbf{x}) \triangleq \frac{\partial \mathbf{g}(\mathbf{x})}{\partial \mathbf{x}} f(\mathbf{x}) - \frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} \mathbf{g}(\mathbf{x}), \qquad \mathbf{x} \in \mathbb{R}^n. \tag{3}$$

To recall the definition of uniform ultimate boundedness, consider the nonlinear time-varying dynamical system

$$\dot{\mathbf{x}}(t) = f(t, \mathbf{x}(t)), \qquad \mathbf{x}(t\_0) = \mathbf{x}\_0, \qquad t \ge t\_0. \tag{4}$$

where x(t)∈ R<sup>n</sup> , <sup>t</sup> <sup>≥</sup> <sup>t</sup>0, <sup>f</sup> : [t0, <sup>∞</sup>) � <sup>R</sup>n!R<sup>n</sup> is jointly continuous in its arguments, <sup>f</sup>(t, �) is locally Lipschitz continuous in x uniformly in t for all t in compact subsets of t ∈[t0, ∞), and 0 = f (t, 0), t ≥ t0.

Definition 2.2 ([19], Def. 4.6). The nonlinear time-varying dynamical system (4) is uniformly ultimately bounded with ultimate bound b > 0 if there exists c > 0 independent of t<sup>0</sup> and for every a ∈(0, c), there exists T = T (a, c) ≥ 0, independent of t0, such that if ∥x<sup>0</sup> ∥ ≤ a, then ∥x(t) ∥ ≤ b, t ≥ t<sup>0</sup> + T.

#### 3. Robust MRAC for output tracking

In order to enable robust output tracking, in this section we present a robust nonlinear control law that is based on the e-modification of the conventional model reference adaptive control [16]. This control law guarantees that after a finite-time transient, the plant's measured output tracks a given reference signal within some bounded error despite model uncertainties and external disturbances. In practice, the proposed controller guarantees uniform ultimate boundedness of the output tracking error.

Consider the nonlinear time-varying plant and the plant sensors' dynamics

matrices. We write <sup>1</sup><sup>n</sup> for the <sup>n</sup> � <sup>n</sup> identity matrix, 0<sup>n</sup> � <sup>m</sup> for the zero <sup>n</sup> � <sup>m</sup> matrix, and <sup>A</sup><sup>T</sup> for the

the cross product operator. We write ∥ � ∥ for the Euclidean vector norm and ∥ � ∥<sup>F</sup> for the Frobenius

Definition 2.1 ([18], Def. 6.8). The Lie derivative of the continuously differentiable function

<sup>T</sup>∈ R<sup>3</sup>

0

x t \_ðÞ¼ f tð Þ ; x tð Þ , xtð Þ¼ <sup>0</sup> x0, t ≥ t0, (4)

, <sup>t</sup> <sup>≥</sup> <sup>t</sup>0, <sup>f</sup> : [t0, <sup>∞</sup>) � <sup>R</sup>n!R<sup>n</sup> is jointly continuous in its arguments, <sup>f</sup>(t, �) is locally

2

, a� ≜

2 6 4

(x)≜∂V(x)/∂x.

ð Þ<sup>x</sup> f xð Þ, x<sup>∈</sup> <sup>R</sup><sup>n</sup>: (1)

0 �a<sup>3</sup> a<sup>2</sup> a<sup>3</sup> 0 �a<sup>1</sup> �a<sup>2</sup> a<sup>1</sup> 0

. The Fréchet derivative of the continu-

, x∈ Rn, k ≥ 1: (2)

<sup>∂</sup><sup>x</sup> g xð Þ, x <sup>∈</sup> <sup>R</sup><sup>n</sup>: (3)

, the Lie bracket of f(�) and g(�) is

3 7 5

denotes

transpose of the matrix A ∈ R<sup>n</sup> � <sup>m</sup>. Given a = [a1, a2, a3]

matrix norm, that is, given <sup>B</sup><sup>∈</sup> <sup>R</sup>n� <sup>m</sup>, <sup>∥</sup>B∥<sup>F</sup> <sup>≜</sup> <sup>t</sup>r BB<sup>T</sup> � � � � <sup>1</sup>

<sup>V</sup>: <sup>R</sup>n! <sup>R</sup> along the vector field f : <sup>R</sup>n! <sup>R</sup><sup>n</sup> is defined as

<sup>f</sup> V xð Þ <sup>≜</sup>V xð Þ, Lk

3. Robust MRAC for output tracking

Given the continuously differentiable functions <sup>f</sup>, <sup>g</sup> : <sup>R</sup>n! <sup>R</sup><sup>n</sup>

ad<sup>f</sup> g xð Þ<sup>≜</sup> <sup>∂</sup>g xð Þ

L0

80 Adaptive Robust Control Systems

defined as

dynamical system

where x(t)∈ R<sup>n</sup>

t ≥ t0.

t ≥ t<sup>0</sup> + T.

ously differentiable function <sup>V</sup>: <sup>R</sup>n! <sup>R</sup> at <sup>x</sup> is denoted by <sup>V</sup>

Lf V xð Þ ≜V<sup>0</sup>

The zeroth-order and the higher-order Lie derivatives are, respectively, defined as

<sup>f</sup> V xð Þ <sup>≜</sup>Lf <sup>L</sup><sup>k</sup>�<sup>1</sup>

<sup>∂</sup><sup>x</sup> f xð Þ� <sup>∂</sup>f xð Þ

To recall the definition of uniform ultimate boundedness, consider the nonlinear time-varying

Lipschitz continuous in x uniformly in t for all t in compact subsets of t ∈[t0, ∞), and 0 = f (t, 0),

Definition 2.2 ([19], Def. 4.6). The nonlinear time-varying dynamical system (4) is uniformly ultimately bounded with ultimate bound b > 0 if there exists c > 0 independent of t<sup>0</sup> and for every a ∈(0, c), there exists T = T (a, c) ≥ 0, independent of t0, such that if ∥x<sup>0</sup> ∥ ≤ a, then ∥x(t) ∥ ≤ b,

In order to enable robust output tracking, in this section we present a robust nonlinear control law that is based on the e-modification of the conventional model reference adaptive control [16]. This control law guarantees that after a finite-time transient, the plant's measured output tracks a given reference signal within some bounded error despite model uncertainties and

<sup>f</sup> V xð Þ � �

$$\dot{\mathbf{x}}\_{\mathsf{P}}(t) = A\_{\mathsf{P}} \mathbf{x}\_{\mathsf{P}}(t) + B\_{\mathsf{P}} \Lambda \left[ u(t) + \Theta^{\mathsf{T}} \Phi(\mathbf{x}\_{\mathsf{P}}(t)) \right] + \widehat{\xi}(t), \quad \mathbf{x}\_{\mathsf{P}}(t\_0) = \mathbf{x}\_{\mathsf{P},0}, \quad t \ge t\_0 \tag{5}$$

$$
\dot{y}(t) = \varepsilon \mathsf{C}\_{\mathsf{P}} \mathsf{x}\_{\mathsf{P}}(t) - \varepsilon y(t), \qquad y(t\_0) = \mathsf{C}\_{\mathsf{P}} \mathsf{x}\_{\mathsf{P},0}.\tag{6}
$$

where <sup>x</sup>pð Þ<sup>t</sup> <sup>∈</sup> <sup>D</sup><sup>p</sup> <sup>⊆</sup> <sup>R</sup><sup>n</sup><sup>p</sup> , <sup>t</sup> <sup>≥</sup> <sup>t</sup>0, denotes the plant's trajectory, 0<sup>n</sup>p�<sup>1</sup> <sup>∈</sup> <sup>D</sup>p, <sup>u</sup>(t)<sup>∈</sup> <sup>R</sup><sup>m</sup> denotes the control input, y(t) ∈ R<sup>m</sup> denotes the measured output, ε > 0, Ap∈ Rn<sup>p</sup> � <sup>n</sup><sup>p</sup> is unknown, Bp∈ Rn<sup>p</sup> � <sup>m</sup>, Cp∈ R<sup>m</sup> � <sup>n</sup><sup>p</sup> , Λ ∈ R<sup>m</sup> � <sup>m</sup> is diagonal, positive-definite, and unknown, Θ ∈ R<sup>N</sup> � <sup>m</sup> is unknown, the regressor vector <sup>Φ</sup> : <sup>R</sup>n<sup>P</sup>! <sup>R</sup><sup>N</sup> is Lipschitz continuous in its argument, and <sup>b</sup><sup>ξ</sup> : ½ Þ! <sup>t</sup>0; <sup>∞</sup> <sup>R</sup><sup>n</sup><sup>p</sup> is continuous in its argument and unknown. We assume that ∥bξð Þt ∥ ≤ bξmax, t ≥ t0, and Λ is such that the pair (Ap, BpΛ) is controllable and Λmin1<sup>m</sup> ≤ Λ, for some Λmin > 0. Both Λ and Θ<sup>T</sup> Φ(xp), x<sup>p</sup> ∈ Dp, capture the plant's matched and parametric uncertainties, such as malfunctions in the control system; the term bξð Þ� captures the plant's unmatched uncertainties, such as external disturbances.

Eq. (6) models the plant sensors as linear dynamical systems, whose uncontrolled dynamics is exponentially stable and characterized by the parameter ε > 0 ([20], Ch. 2). Given the reference signal ycmd : [t0, <sup>∞</sup>)!Rm, which is continuous with its first derivative, define <sup>y</sup>cmd, <sup>2</sup>ð Þ<sup>t</sup> <sup>≜</sup> <sup>y</sup>\_ cmdð Þt , t ≥ t0, and assume that both ycmd(�) and ycmd, 2(�) are bounded, that is, kycmd(t)k ≤ ymax, 1, t ≥ t0, and kycmd, 2(t)k ≤ ymax, 2, for some ymax, 1, ymax, 2 > 0.

The following theorem provides a robust MRAC for the nonlinear time-varying dynamical system (5) and (6) such that the measured output y(�) is able to eventually track the reference signal ycmd(�) with bounded error, that is, there exist b > 0 and c > 0 independent of t0, and for every a∈ (0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t0, such that if ∥y(t0) � ycmd(t0) ∥ ≤ a, then

$$\|\|y(t) - y\_{\text{cmd}}(t)\|\| \le b, \qquad t \ge t\_0 + T. \tag{7}$$

For the statement of this result, let <sup>n</sup><sup>≜</sup> <sup>n</sup><sup>p</sup> <sup>+</sup> <sup>m</sup> and x tð Þ<sup>≜</sup> <sup>x</sup><sup>T</sup> <sup>p</sup>ð Þ<sup>t</sup> ; y tð Þ� <sup>y</sup>cmdð Þ<sup>t</sup> � �<sup>T</sup> h i<sup>T</sup> ∈ R<sup>n</sup>, t ≥ t0, note that (5) and (6) are equivalent to

$$\dot{\mathbf{x}}(t) = A\mathbf{x}(t) + B\boldsymbol{\Lambda}\left[\boldsymbol{u}(t) + \boldsymbol{\Theta}^{\mathrm{T}}\boldsymbol{\Phi}(\mathbf{x}\_{\mathrm{p}}(t))\right] + \boldsymbol{\xi}(t), \qquad \mathbf{x}(t\_{0}) = \begin{bmatrix} \mathbf{x}\_{\mathrm{p},0} \\ \mathbf{C}\_{\mathrm{p}}\mathbf{x}\_{\mathrm{p},0} - y\_{\mathrm{cmd}}(t\_{0}) \end{bmatrix}, \qquad t \ge t\_{0}, \tag{8}$$

$$\begin{aligned} & \text{where} \quad \mathbf{x}(t) \in \mathcal{D} \subseteq \mathbb{R}^{n}, \quad \mathcal{D} \triangleq \mathcal{D}\_{\mathbb{P}} \times \mathbb{R}^{m}, \quad A \triangleq \begin{bmatrix} A\_{\mathbb{P}} & \mathbf{0}\_{n\_{\mathbb{P}} \times m} \\ \varepsilon \mathbf{C}\_{\mathbb{P}} & -\varepsilon \mathbf{1}\_{m} \end{bmatrix}, \quad B \triangleq \begin{bmatrix} B\_{\mathbb{P}} \\ \mathbf{0}\_{m \times m} \end{bmatrix}, \quad B\_{1} \triangleq \begin{bmatrix} \mathbf{0}\_{n\_{\mathbb{P}} \times m} \\ -\mathbf{1}\_{m} \end{bmatrix}, \quad \text{and} \\\ & \xi(t) \triangleq B\_{1} \begin{bmatrix} \mathbf{y}\_{\text{cmd},2}(t) + \varepsilon \mathbf{y}\_{\text{cmd},1}(t) \end{bmatrix} + \begin{bmatrix} \mathbf{1}\_{n\_{\mathbb{P}}} \\ \mathbf{0}\_{m \times n\_{\mathbb{P}}} \end{bmatrix}, \quad B\_{2} \triangleq \begin{bmatrix} \mathbf{0}\_{n\_{\mathbb{P}} \times m} \\ -\mathbf{1}\_{m} \end{bmatrix}, \quad \text{and} \\\ & \xi(t) \triangleq B\_{2} \begin{bmatrix} \mathbf{0}\_{n\_{\mathbb{P}} \times m} \\ -\mathbf{1}\_{m} \end{bmatrix}, \quad \text{and} \quad \xi(t) \triangleq B\_{2} \begin{bmatrix} \mathbf{0}\_{n\_{\mathbb{P}} \times m} \\ -\mathbf{1}\_{m} \end{bmatrix}. \end{aligned}$$

$$\dot{\mathbf{x}}\_{\text{nf}}(t) = A\_{\text{nf}} \mathbf{x}\_{\text{nf}}(t) + B\_{\text{nf}} y\_{\text{cmd}}(t), \qquad \mathbf{x}\_{\text{nf}}(t\_0) = \begin{bmatrix} \mathbf{x}\_{\text{p,0}} \\ \mathbf{C}\_{\text{p}} \mathbf{x}\_{\text{p,0}} - y\_{\text{cmd}}(t\_0) \end{bmatrix}, \qquad t \ge t\_0. \tag{9}$$

e t \_ðÞ¼ <sup>A</sup>refe tð Þþ <sup>B</sup>Λ ΔK<sup>T</sup>

ultimately bounded.

Next, let <sup>x</sup>refðÞ¼ <sup>t</sup> <sup>x</sup><sup>T</sup>

and Hurwitz, B<sup>1</sup> = [0<sup>m</sup> � <sup>n</sup><sup>p</sup>

then this vector is denoted by a<sup>I</sup>

<sup>x</sup> x tð Þþ <sup>Δ</sup>K<sup>T</sup>

ref,1ð Þ<sup>t</sup> ; <sup>x</sup><sup>T</sup>

ref, <sup>2</sup>ð Þt h i<sup>T</sup>

follows from the uniform ultimate boundedness of (18) that

, �1m] T

presence of the matched uncertainty bξð Þ� ([17], pp. 317-319).

4. Modeling assumptions on quadrotors' dynamics

The Z axis is chosen so that the quadrotor's weight is given by F<sup>I</sup>

and using the same arguments as in ([17], pp. 324-325), one can prove that

y tð Þ� <sup>y</sup>cmdðÞ� <sup>t</sup> <sup>x</sup>ref, <sup>2</sup>ð Þ<sup>t</sup> � � �

cmdycmdð Þ� <sup>t</sup> ΔΘ<sup>T</sup><sup>Φ</sup> <sup>x</sup>pð Þ<sup>t</sup> � � � � <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>t</sup> , etð Þ¼ <sup>0</sup> <sup>0</sup>, t <sup>≥</sup> <sup>t</sup>0,

V e \_ ð Þ ; <sup>Δ</sup>Kx;ΔKcmd; ΔΘ <sup>&</sup>lt; <sup>0</sup> ð Þ <sup>e</sup>;ΔKx;ΔKcmd;ΔΘ <sup>∈</sup> <sup>Ω</sup>, (19)

, t ≥ t0, verify (9), where xref, 1(t)∈ Rn<sup>p</sup> and xref, 2(t) ∈ Rm. It

, and ycmd(�) is bounded, it follows from (9) that xref, 2(�) is

. Alternatively, if a∈ R<sup>3</sup> is expressed in J, then no superscript is

<sup>g</sup> ¼ mQgZ, where m<sup>Q</sup> > 0 denotes

� ≤bb, t ≥ T þ t0, (20)

Robust Adaptive Output Tracking for Quadrotor Helicopters

http://dx.doi.org/10.5772/intechopen.70723

for some compact set <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup><sup>n</sup> � <sup>R</sup><sup>n</sup> � <sup>m</sup> � <sup>R</sup><sup>m</sup> � <sup>m</sup> � <sup>R</sup><sup>N</sup> � <sup>m</sup>. Thus, it follows from Theorem 4.18 of Khalil [19] that the nonlinear dynamical system given by (18) and (11)–(13) is uniformly

for some bb > 0 and T ≥ 0, which are independent of t0. Moreover, since Aref is block-diagonal

uniformly bounded ([18], 245), that is, kxref, 2(t)k ≤ b2, t ≥ t0, for some b<sup>2</sup> ≥ 0 independent of t0. Thus, it follows from (20) that (7) is verified with <sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>b</sup> <sup>þ</sup> <sup>b</sup>2. □ It is important to notice that although the matrix Ap, which characterizes the plant's uncontrolled linearized dynamics, is unknown and hence, the augmented matrix A is unknown, the structure of the matrix A<sup>p</sup> is usually known. Thus, in problems of practical interest it is generally possible to verify the matching conditions (15) and (16), although the matrix A is unknown ([17], Ch. 9).

Remark 3.1 If the adaptive gains Kbxð Þ� , Kbcmdð Þ� , and Θb ð Þ� verify (11)–(13), respectively, with σ<sup>1</sup> = σ<sup>2</sup> = σ<sup>3</sup> = 0, then the control law (10) reduces to the conventional model reference adaptive control law for the augmented dynamical system (8) ([17], p. 298). However, conventional MRAC does not guarantee uniform ultimate boundedness of the closed-loop system in the

Let I ¼ f g O; X;Y;Z denote an orthonormal reference frame fixed with the Earth and centered at some point O, and let J ¼ f g A; x tð Þ; y tð Þ; z tð Þ , t ≥ t0, denote an orthonormal reference frame fixed with the quadrotor and centered at some point A, which is arbitrarily chosen. The axes of the reference frames I and J form two orthonormal bases of R<sup>3</sup> and if a vector a∈ R<sup>3</sup> is expressed in I,

used. In this chapter, we consider the reference frame I as an inertial reference frame; quadrotors move at subsonic velocities and are usually operated at altitudes considerably lower than 10 kilometers and hence, the error induced by this modeling assumption is negligible ([21], Ch. 5).

(18)

83

where <sup>A</sup>ref <sup>¼</sup> <sup>A</sup>ref ,<sup>1</sup> <sup>0</sup><sup>n</sup>p�<sup>m</sup> 0<sup>m</sup>�n<sup>p</sup> Aref, <sup>2</sup> " #, Aref, 1∈ Rnp�np is Hurwitz, Aref, 2∈ R<sup>m</sup> � <sup>m</sup> is Hurwitz, and Bref∈ R<sup>n</sup> � <sup>m</sup> is such that the pair (Aref, Bref) is controllable.

Theorem 3.1 Consider the nonlinear time-varying dynamical system given by Eqs. (5) and (6), the augmented dynamical system (8), and the linear time-invariant reference dynamical model (9). Define e(t) ≜x(t) � xref(t), t ≥ t0, and let

$$\boldsymbol{\gamma}\boldsymbol{\gamma}(t,\mathbf{x\_{p}},\mathbf{x}) = \widehat{\mathbf{K}}\_{\mathbf{x}}^{\mathrm{T}}(t)\mathbf{x} + \widehat{\mathbf{K}}\_{\mathrm{cmd}}^{\mathrm{T}}(t)\mathbf{y}\_{\mathrm{cmd}}(t) - \widehat{\boldsymbol{\Theta}}^{\mathrm{T}}(t)\boldsymbol{\Phi}(\mathbf{x\_{p}}), \qquad \left(t,\mathbf{x\_{p}},\mathbf{x}\right) \in [t\_{0},\boldsymbol{\omega}) \times \mathcal{D}\_{\mathbf{p}} \times \mathcal{D}, \tag{10}$$

where

$$\dot{\hat{K}}\_{\mathbf{x}}(t) = -\Gamma\_{\mathbf{x}}\left[\mathbf{x}(t)\boldsymbol{\varepsilon}^{\mathrm{T}}(t)\boldsymbol{PB} + \sigma\_{\mathrm{1}}\left|\left|\boldsymbol{B}^{\mathrm{T}}\mathbf{P}\mathbf{e}(t)\right|\right|\hat{\mathbf{K}}\_{\mathbf{x}}(t)\right], \qquad \hat{\mathbf{K}}\_{\mathbf{x}}(t\_{0}) = \mathbf{0}\_{\mathbf{n}\times\mathbf{m}} \qquad t \ge t\_{0}. \tag{11}$$

$$\dot{\hat{K}}\_{\rm cmd}(t) = -\Gamma\_{\rm cmd} \left[ y\_{\rm cmd}(t) e^{\Gamma}(t) P B + \sigma\_2 ||B^{\rm T} Pe(t)|| \widehat{K}\_{\rm cmd}(t) \right], \qquad \widehat{K}\_{\rm cmd}(t\_0) = 0\_{\rm m \times m} \tag{12}$$

$$\dot{\hat{\Theta}}(t) = \Gamma\_{\Theta} \left[ \Phi(\mathbf{x}\_{\mathcal{P}}(t)) e^{\mathbf{T}}(t) P B - \sigma\_3 ||B^{\mathrm{T}} Pe(t)|| \hat{\Theta}(t) \right], \qquad \hat{\Theta}(t\_0) = \mathbf{0}\_{\mathcal{N} \times \mathbf{m}} \tag{13}$$

the learning gain matrices Γx∈ R<sup>n</sup> � <sup>n</sup> , Γcmd∈ R<sup>m</sup> � <sup>m</sup>, and ΓΘ ∈ R<sup>N</sup> � <sup>N</sup> are symmetric positivedefinite, P∈ R<sup>n</sup> � <sup>n</sup> is the symmetric positive-definite solution of the Lyapunov equation

$$0 = A\_{\text{ref}}^{\text{T}}P + PA\_{\text{ref}} + Q \tag{14}$$

Q ∈ R<sup>n</sup> � <sup>n</sup> is symmetric positive-definite, and σ1, σ2, σ<sup>3</sup> > 0. If there exists Kx∈ R<sup>n</sup> � <sup>m</sup> and Kcmd∈ R<sup>m</sup> � <sup>m</sup> such that

$$A\_{\rm ref} = A + B\Lambda K\_{\rm x'}^{\rm T} \tag{15}$$

$$B\_{\rm ref} = B \Lambda K\_{\rm cmd}^{\rm T} \tag{16}$$

then the nonlinear time-varying dynamical system (8) with u(t) = γ(t, xp(t), x(t)), t ≥ t0, is uniformly ultimately bounded and (7) is verified.

Proof: Let ΔKx ≜Kb<sup>x</sup> � Kx, ΔKcmd ≜Kbcmd � Kcmd, and ΔΘ ≜ Θb � Θ and consider the Lyapunov function candidate

$$V(\varepsilon, \Delta \mathcal{K}\_{\rm x}, \Delta \mathcal{K}\_{\rm cmd}, \Delta \Theta) = \varepsilon^{\rm T} \mathcal{P} \varepsilon + \text{tr}\left( \left[ \Delta \mathcal{K}\_{\rm x}^{\rm T} \Gamma\_{\rm x}^{\rm T} \Delta \mathcal{K}\_{\rm x} + \Delta \mathcal{K}\_{\rm cmd}^{\rm T} \Gamma\_{\rm cmd}^{\rm T} \Delta \mathcal{K}\_{\rm cmd} + \Delta \Theta^{\rm T} \Gamma\_{\Theta}^{\rm T} \Delta \Theta \right] \Lambda \right),$$

$$(\varepsilon, \Delta \mathcal{K}\_{\rm x}, \Delta \mathcal{K}\_{\rm cmd}, \Delta \Theta) \in \mathbb{R}^{\rm x} \times \mathbb{R}^{n \times m} \times \mathbb{R}^{m \times m} \times \mathbb{R}^{N \times m},\tag{17}$$

where tr(�) denotes the trace operator. The error dynamics is given by

$$\dot{\varepsilon}(t) = A\_{\text{ref}}\varepsilon(t) + B\Lambda \left[\Delta K\_{\text{x}}^{\text{T}}\mathbf{x}(t) + \Delta K\_{\text{cmd}}^{\text{T}}y\_{\text{cmd}}(t) - \Delta \Theta^{\text{T}}\Phi(\mathbf{x}\_{\text{p}}(t))\right] + \xi(t), \quad \varepsilon(t\_{0}) = 0, \qquad t \ge t\_{0} \tag{18}$$

and using the same arguments as in ([17], pp. 324-325), one can prove that

x\_refðÞ¼ t Aref xrefð Þþ t Bref ycmdð Þt , xrefð Þ¼ t<sup>0</sup>

Bref∈ R<sup>n</sup> � <sup>m</sup> is such that the pair (Aref, Bref) is controllable.

cmdð Þ<sup>t</sup> ycmdð Þ� <sup>t</sup> <sup>Θ</sup><sup>b</sup> <sup>T</sup>

<sup>T</sup>ð Þ<sup>t</sup> PB <sup>þ</sup> <sup>σ</sup><sup>1</sup> <sup>B</sup><sup>T</sup>Pe tð Þ �

h i

� �

<sup>T</sup>ð Þ<sup>t</sup> PB <sup>þ</sup> <sup>σ</sup><sup>2</sup> <sup>B</sup><sup>T</sup>Pe tð Þ �

h i

<sup>T</sup>ð Þ<sup>t</sup> PB � <sup>σ</sup><sup>3</sup> <sup>B</sup><sup>T</sup>Pe tð Þ �

definite, P∈ R<sup>n</sup> � <sup>n</sup> is the symmetric positive-definite solution of the Lyapunov equation

Q ∈ R<sup>n</sup> � <sup>n</sup> is symmetric positive-definite, and σ1, σ2, σ<sup>3</sup> > 0. If there exists Kx∈ R<sup>n</sup> � <sup>m</sup> and

<sup>A</sup>ref <sup>¼</sup> <sup>A</sup> <sup>þ</sup> <sup>B</sup>ΛK<sup>T</sup>

<sup>B</sup>ref <sup>¼</sup> <sup>B</sup>ΛK<sup>T</sup>

then the nonlinear time-varying dynamical system (8) with u(t) = γ(t, xp(t), x(t)), t ≥ t0, is uni-

Proof: Let ΔKx ≜Kb<sup>x</sup> � Kx, ΔKcmd ≜Kbcmd � Kcmd, and ΔΘ ≜ Θb � Θ and consider the Lyapunov

<sup>x</sup>ΔKx <sup>þ</sup> <sup>Δ</sup>K<sup>T</sup>

cmdΓ<sup>T</sup>

ð Þ <sup>e</sup>;ΔKx; <sup>Δ</sup>Kcmd;ΔΘ <sup>∈</sup> <sup>R</sup><sup>n</sup> � <sup>R</sup><sup>n</sup>�<sup>m</sup> � <sup>R</sup><sup>m</sup>�<sup>m</sup> � <sup>R</sup><sup>N</sup>�m, (17)

ΘΔΘ � �<sup>Λ</sup> � �,

xΓ<sup>T</sup>

<sup>0</sup> <sup>¼</sup> <sup>A</sup><sup>T</sup>

<sup>T</sup>Pe <sup>þ</sup> <sup>t</sup><sup>r</sup> <sup>Δ</sup>K<sup>T</sup>

where tr(�) denotes the trace operator. The error dynamics is given by

h i

where <sup>A</sup>ref <sup>¼</sup> <sup>A</sup>ref ,<sup>1</sup> <sup>0</sup><sup>n</sup>p�<sup>m</sup>

82 Adaptive Robust Control Systems

e(t) ≜x(t) � xref(t), t ≥ t0, and let

<sup>γ</sup> <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � � <sup>¼</sup> <sup>K</sup>b<sup>T</sup>

\_

\_

where

0<sup>m</sup>�n<sup>p</sup> Aref, <sup>2</sup> " #

<sup>x</sup> ð Þ<sup>t</sup> <sup>x</sup> <sup>þ</sup> <sup>K</sup><sup>b</sup> <sup>T</sup>

KbxðÞ¼� t Γ<sup>x</sup> x tð Þe

the learning gain matrices Γx∈ R<sup>n</sup> � <sup>n</sup>

\_

Kcmd∈ R<sup>m</sup> � <sup>m</sup> such that

function candidate

KbcmdðÞ¼� t Γcmd ycmdð Þt e

<sup>Θ</sup><sup>b</sup> ðÞ¼ <sup>t</sup> ΓΘ <sup>Φ</sup> <sup>x</sup>pð Þ<sup>t</sup> � �<sup>e</sup>

formly ultimately bounded and (7) is verified.

V eð ;ΔKx;ΔKcmd; ΔΘÞ ¼ e

xp,<sup>0</sup>

Theorem 3.1 Consider the nonlinear time-varying dynamical system given by Eqs. (5) and (6), the augmented dynamical system (8), and the linear time-invariant reference dynamical model (9). Define

ð Þt Φ x<sup>p</sup>

�bKxð Þt

� �

� �

�bKcmdð Þt

�Θb ð Þt

, Γcmd∈ R<sup>m</sup> � <sup>m</sup>, and ΓΘ ∈ R<sup>N</sup> � <sup>N</sup> are symmetric positive-

refP þ PAref þ Q, (14)

<sup>x</sup> , (15)

cmd, (16)

cmdΔKcmd <sup>þ</sup> ΔΘ<sup>T</sup>Γ<sup>T</sup>

Cpxp,<sup>0</sup> � ycmdð Þ t<sup>0</sup> " #

� �, t; <sup>x</sup>p; <sup>x</sup> � �<sup>∈</sup> ½ Þ� <sup>t</sup>0; <sup>∞</sup> <sup>D</sup><sup>p</sup> � <sup>D</sup>, (10)

, Kbxð Þ¼ t<sup>0</sup> 0<sup>n</sup>�m, t ≥ t0, (11)

, Kbcmdð Þ¼ t<sup>0</sup> 0<sup>m</sup>�m, (12)

, Θb ð Þ¼ t<sup>0</sup> 0<sup>N</sup>�m, (13)

, Aref, 1∈ Rnp�np is Hurwitz, Aref, 2∈ R<sup>m</sup> � <sup>m</sup> is Hurwitz, and

, t ≥ t0, (9)

$$
\dot{V}(\varepsilon, \Delta \mathcal{K}\_{\mathbf{x}}, \Delta \mathcal{K}\_{\mathbf{cmd}}, \Delta \Theta) < 0 \qquad (\varepsilon, \Delta \mathcal{K}\_{\mathbf{x}}, \Delta \mathcal{K}\_{\mathbf{cmd}}, \Delta \Theta) \in \Omega,\tag{19}
$$

for some compact set <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup><sup>n</sup> � <sup>R</sup><sup>n</sup> � <sup>m</sup> � <sup>R</sup><sup>m</sup> � <sup>m</sup> � <sup>R</sup><sup>N</sup> � <sup>m</sup>. Thus, it follows from Theorem 4.18 of Khalil [19] that the nonlinear dynamical system given by (18) and (11)–(13) is uniformly ultimately bounded.

Next, let <sup>x</sup>refðÞ¼ <sup>t</sup> <sup>x</sup><sup>T</sup> ref,1ð Þ<sup>t</sup> ; <sup>x</sup><sup>T</sup> ref, <sup>2</sup>ð Þt h i<sup>T</sup> , t ≥ t0, verify (9), where xref, 1(t)∈ Rn<sup>p</sup> and xref, 2(t) ∈ Rm. It follows from the uniform ultimate boundedness of (18) that

$$\left\| \left\| y(t) - y\_{\text{cmd}}(t) - \mathbf{x}\_{\text{ref},2}(t) \right\| \right\| \leq \widehat{b}, \qquad t \geq T + t\_{0\prime} \tag{20}$$

for some bb > 0 and T ≥ 0, which are independent of t0. Moreover, since Aref is block-diagonal and Hurwitz, B<sup>1</sup> = [0<sup>m</sup> � <sup>n</sup><sup>p</sup> , �1m] T , and ycmd(�) is bounded, it follows from (9) that xref, 2(�) is uniformly bounded ([18], 245), that is, kxref, 2(t)k ≤ b2, t ≥ t0, for some b<sup>2</sup> ≥ 0 independent of t0. Thus, it follows from (20) that (7) is verified with <sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>b</sup> <sup>þ</sup> <sup>b</sup>2. □

It is important to notice that although the matrix Ap, which characterizes the plant's uncontrolled linearized dynamics, is unknown and hence, the augmented matrix A is unknown, the structure of the matrix A<sup>p</sup> is usually known. Thus, in problems of practical interest it is generally possible to verify the matching conditions (15) and (16), although the matrix A is unknown ([17], Ch. 9).

Remark 3.1 If the adaptive gains Kbxð Þ� , Kbcmdð Þ� , and Θb ð Þ� verify (11)–(13), respectively, with σ<sup>1</sup> = σ<sup>2</sup> = σ<sup>3</sup> = 0, then the control law (10) reduces to the conventional model reference adaptive control law for the augmented dynamical system (8) ([17], p. 298). However, conventional MRAC does not guarantee uniform ultimate boundedness of the closed-loop system in the presence of the matched uncertainty bξð Þ� ([17], pp. 317-319).
