Transactions of Nanjing University of Aeronautics and Astronautics  2018, Vol. 35 Issue (4): 693-701   PDF    
Fault Tolerant Control Scheme Design for Formation Flight Control System of Multiple Unmanned Aerial Vehicles
Qian Moshu1,2, Xiong Ke2     
1. College of Electrical Engineering and Control Science, Nanjing Tech University, Nanjing 211816, P. R. China;
2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China
Abstract: The command tracking problem of formation flight control system (FFCS) for multiple unmanned aerial vehicles (UAVs) with sensor faults is discussed. And the objective of the addressed control problem is to design a robust fault tolerant tracking controller such that, for the disturbances and sensor faults, the closed-loop system is asymptotically stable with a given disturbance attenuation level. A robust fault tolerant tracking control scheme, combining an observer with H performance, is proposed. Furthermore, it is proved that the designed controller can guarantee asymptotic stability of FFCS despite sensor faults. Finally, a simulation of two UAV formations is employed to demonstrate the effectiveness of the proposed approach.
Key words: formation control    fault tolerant control (FTC)    unmanned aerial vehicle (UAV)    robust control    
0 Introduction

An unmanned aerial vehicle (UAV) is an aircraft without a human pilot aboard. The flying of UAV, which depends on the flight control system (FCS) including actuator, sensor and so on, is controlled either autonomously by on board computers or by the remote control of a pilot[1-5]. The UAV not only has many potential military and civil applications but also has great scientific significance in academic research. One of the interesting topics is the cooperative control of UAV in formation flight, in which a group of UAVs fly in a desired graph formation[6-10]. There are reasons to believe that the efficiency of group performance is higher than that of single UAV. There have been significant researches in the area of cooperative control of multiple vehicles system in recent years. For examples, the authors mainly focused on the motion consensus and formation control of UAV swarm in Ref.[11], and a novel formation control algorithm suitable for both leaders and followers was designed, in which leaders can be influenced by feedback from their teammates. A normal PID control law was studied for the closed formation of UAVs in Ref.[12], and the bound condition of a safe distance was given. In Ref. [13], the authors respectively proposed centralized and decentralized event-triggered control laws, which achieved the the circle formation. The fault of single UAV or agent will influence the entire formation and even safe, so it is necessary to study the fault tolerant control (FTC) problem of the formation[14]. Several approaches have been exploited to solve the problem about accomplishing a certain formation for faulty UAVs. A mathematical model using dual quaternion is employed to describe the spacecraft formation. The effectiveness of the proposed control method is further demonstrated in the presence of actuator fault, disturbance and parameter uncertainty as well. Moreover, the finite-time stability of the closed-loop system is guaranteed by the adaptive feedback FTC law[15]. The authors investigate the time varying fault tolerant control formation problem for high-order multi-agent systems with actuator failure in Ref. [16]. Using the adaptive online updating strategies, the bounds of actuator failures can be unknown. Then an algorithm is proposed to determine the control parameters of the FTC, where an approach to expand the feasible formation set and the formation feasible conditions is given. A fault tolerant formation control scheme of UAV based on Kalman filter is presented, which can deal with both GPS sensor failure and wireless communication packet losses in Ref.[17]. A centralized null space based approach is introduced to compose a complex mission for swarms. Bernulli model and Extended Gilbert model are employed to value the impact of the main packet loss in wireless networks[18]. The works mentioned above perform effectively in the design of formation controller, but the sensor faults in multi-UAV formation are not studied seriously. Once the attitude, velocity or position sensor of a UAV fails, the formation flight control system (FFCS) will be influenced, which motivates us to study further[19-21].

Motivated by the above discussion, the robust fault tolerant tracking control design is discussed in this paper for FFCS of multi-UAV in which sensor faults happen to a single UAV. The dynamics model of UAV formation including sensor faults and disturbances is established. Then an observer-based feedback control approach is employed. Finally, a simulation example is given to illustrate the validity of the proposed approach.

1 Problem Statement

In this study, the typical leader and wing-man formation patterns are adopted, and a typical triangle formation of UAV is shown in Fig. 1[22].

Fig. 1 Triangle formation of UAV

The aerodynamic coupling effects are ignored for the simplicity, and then a linear state space model for the FFCS of UAV is obtained [12]

$ \left\{ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\dot {\bar x}}} = \mathit{\boldsymbol{A\bar x}} + \mathit{\boldsymbol{Bu}} + \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}{\mathit{\boldsymbol{w}}_1}}\\ {\mathit{\boldsymbol{\bar y}} = \mathit{\boldsymbol{C\bar x}}} \end{array}} \right. $ (1)

where

$ \mathit{\boldsymbol{\bar x}} = {\left[ {\begin{array}{*{20}{c}} x&{{V_W}}&y&{{\psi _W}}&z&\zeta \end{array}} \right]^{\rm{T}}} $
$ \mathit{\boldsymbol{u}} = \left[ {\begin{array}{*{20}{c}} {{V_{{W_C}}}}\\ {{\psi _{{W_C}}}}\\ {{h_{{W_C}}}} \end{array}} \right],{\mathit{\boldsymbol{w}}_1} = \left[ {\begin{array}{*{20}{c}} {{V_L}}\\ {{\psi _L}}\\ {{h_{{L_C}}}} \end{array}} \right] $
$ \mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} 0&{ - 1}&0&{ - {y_0}/{\tau _{{\psi _W}}}}&0&0\\ 0&{ - 1/{\tau _{{V_W}}}}&0&0&0&0\\ 0&0&0&{\left( {\frac{{{x_0}}}{{{\tau _{{\psi _W}}}}} - {V_{{L_0}}}} \right)}&0&0\\ 0&0&0&{ - 1/{\tau _{{\psi _W}}}}&0&0\\ 0&0&0&0&0&1\\ 0&0&0&0&{ - \frac{1}{{{\tau _{{h_a}}}{\tau _{{h_b}}}}}}&{ - \left( {\frac{1}{{{\tau _{{h_a}}}}} + \frac{1}{{{\tau _{{h_b}}}}}} \right)} \end{array}} \right] $
$ \mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} 0&{\frac{{{y_0}}}{{{\tau _{{V_W}}}}}}&0\\ {\frac{1}{{{\tau _{{V_W}}}}}}&0&0\\ 0&{\frac{{{x_0}}}{{{\tau _{{\psi _W}}}}}}&0\\ 0&{\frac{1}{{{\tau _{{\psi _W}}}}}}&0\\ 0&0&0\\ 0&0&{\frac{1}{{{\tau _{{h_a}}}{\tau _{{h_b}}}}}} \end{array}} \right],\mathit{\boldsymbol{ \boldsymbol{\varTheta} }} = \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&0&0\\ 0&{{V_{{L_0}}}}&0\\ 0&0&0\\ 0&0&0\\ 0&0&{ - \frac{1}{{{\tau _{{h_a}}}{\tau _{{h_b}}}}}} \end{array}} \right] $

where $ \mathit{\boldsymbol{\bar x, u}}, \mathit{\boldsymbol{\bar y}}$ and w1 are respectively state vector, control input, output vector and process disturbance vector. x, y and z are the separation distances between leader and follower with respect to the wing UAV. VL, VW are the leader and the follower velocities. ψL, ψW are the leader and the follower heading angles. CRm×6 is a diagonal constant matrix, and m is a given positive constant. More parameters refer to Ref.[12].

In order to meet the requirements of actual conditions, the following important aspects should be taken into consideration simultaneously.

(1) Sensor fault. When the attitude, velocity or position sensor of a UAV occurs gain fault, the fault of FFCS will occur, and the measure values of x, y, z, ψW, VW and ζ will fully or partially be inflected. Under this condition, the output can be generally characterized as[23]

$ {{\mathit{\boldsymbol{\bar y}}}_F} = \mathit{\boldsymbol{F\bar y}} $ (2)

where ${{\mathit{\boldsymbol{\bar y}}}_F} \in {{\bf{R}}^q} $ is the actual measure output of system in sensor faulty case, FRq×m denotes the sensor gain fault parameter and satisfies the following inequality constraints: $ 0 \le \mathit{\boldsymbol{\underline F}} = {\rm{diag}}\{ {\underline f _1}, {\underline f _2}\} \le \mathit{\boldsymbol{F}} = {\rm{diag}}\{ {f_1}, {f_2}\} \le \mathit{\boldsymbol{\bar F}} = {\rm{diag}}\{ {{\bar f}_1}, {{\bar f}_2}\} , {f_i}(i = 1, 2)$ are the output proportion coefficients of system in sensor faulty case.

Here matrices F0 and $ {\mathit{\boldsymbol{\tilde F}}}$ are given by

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{F}}_0} = {\rm{diag}}\left\{ {{f_{01}},{f_{02}}} \right\} = \frac{{\mathit{\boldsymbol{\underline F}} + \mathit{\boldsymbol{\bar F}}}}{2} = }\\ {{\rm{diag}}\left\{ {\frac{{{{\underline f }_1} + {{\bar f}_1}}}{2},\frac{{{{\underline f }_2} + {{\bar f}_2}}}{2}} \right\}} \end{array} $ (3)
$ \mathit{\boldsymbol{\tilde F}} = {\rm{diag}}\left\{ {{{\tilde f}_1},{{\tilde f}_2}} \right\} = \frac{{\mathit{\boldsymbol{\bar F}} - \mathit{\boldsymbol{\underline F}} }}{2} = {\rm{diag}}\left\{ {\frac{{{{\bar f}_1} - {{\underline f }_1}}}{2},\frac{{{{\bar f}_2} - {{\underline f }_2}}}{2}} \right\} $ (4)

The sensor fault model parameter F can be turned into

$ \mathit{\boldsymbol{F}} = {\mathit{\boldsymbol{F}}_0} + {\mathit{\boldsymbol{F}}_\delta } = {\mathit{\boldsymbol{F}}_0} + {\rm{diag}}\left\{ {{\delta _1},{\delta _2}} \right\} $ (5)

where $ {\delta _i} \le {{\tilde f}_i}\left( {i = 1, 2} \right)$.

(2) Input constraint. In view of the limited power of actuator, the actual control input force should be confined into a certain range, which means that

$ {\left\| {\mathit{\boldsymbol{u}}\left( t \right)} \right\|_2} \le {u_{\max }} $ (6)

where umax denotes the maximum control input.

(3) Disturbance. In order to obtain an actual dynamic performance of FFCS, the disturbance vector w, which includes process disturbance w1 and outside disturbance w2, is assumed to belong to a bounded set $\mathit{\boldsymbol{\Im}} $ defined as $\mathit{\boldsymbol{\Im}} \underline{\underline \Delta } \{ \mathit{\boldsymbol{w}} \in {\underline{\underline \Delta } ^r}:|\mathit{\boldsymbol{w|}} \le \mathit{\boldsymbol{\bar w}}\} $ for some constant non-negative values ${\mathit{\boldsymbol{\bar w}}} $, that is, $ \mathit{\boldsymbol{w}}{\rm{(}}t{\rm{)}} \in \mathit{\boldsymbol{\Im}} $ for all time instants t≥0.

According to the above mentioned sensor faults and disturbances, the FFCS model (Eq.(1)) of UAV can be described by

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{\dot {\bar x}}} = \mathit{\boldsymbol{A\bar x}} + \mathit{\boldsymbol{Bu}} + \mathit{\boldsymbol{ \boldsymbol{\varTheta} w}}\\ {{\mathit{\boldsymbol{\bar y}}}_F} = \mathit{\boldsymbol{F\bar y = FC\bar x}} \end{array} \right. $ (7)

where w is a bounded disturbance.

To design the tracking controller, a reference model is introduced for FFCS

$ {{\mathit{\boldsymbol{\dot {\bar x}}}}_r} = {\mathit{\boldsymbol{A}}_r}{{\mathit{\boldsymbol{\bar x}}}_r} + \mathit{\boldsymbol{r}} $ (8)

where ${{\mathit{\boldsymbol{\bar x}}}_r} \in {{\bf{R}}^6} $ is the reference state, Ar is a specified Hurwitz matrix and rR6 is a bounded reference input.

Since UAV is a low cost aircraft and state variables are not all measured exactly, a close-loop system tracking controller based on an observer is designed as

$ \mathit{\boldsymbol{\dot {\hat {\bar x}} }} = \mathit{\boldsymbol{A\hat {\bar x}}} + \mathit{\boldsymbol{Bu}} + \mathit{\boldsymbol{L}}\left( {{{\mathit{\boldsymbol{\bar y}}}_F} - \mathit{\boldsymbol{\hat {\bar y}}}} \right) $ (9)
$ \mathit{\boldsymbol{\hat {\bar y}}} = \mathit{\boldsymbol{C\hat {\bar x}}} $ (10)
$ \mathit{\boldsymbol{u}} = - \mathit{\boldsymbol{K}}\left( {{{\mathit{\boldsymbol{\bar x}}}_r} - \mathit{\boldsymbol{\hat {\bar x}}}} \right) $ (11)

where $ {\hat{\bar{\mathit{\boldsymbol{x}}}}}$ is the estimated value of ${\mathit{\boldsymbol{\bar x}}} $, L the gain matrix of observer to be designed, and K an unknown control gain matrix with appropriate dimension.

The attenuation of external disturbances is guaranteed considering the H performance related to the tracking error ${{\mathit{\boldsymbol{\bar x}}}_r} - \mathit{\boldsymbol{\bar x}} $ as [24]

$ \int_0^{{t_f}} {{{\left( {{{\mathit{\boldsymbol{\bar x}}}_r} - \mathit{\boldsymbol{\bar x}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{Q}}\left( {{{\mathit{\boldsymbol{\bar x}}}_r} - \mathit{\boldsymbol{\bar x}}} \right){\rm{d}}t} \le {\eta ^2}\int_0^{{t_f}} {\left( {{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{r}} + {\mathit{\boldsymbol{w}}^{\rm{T}}}\mathit{\boldsymbol{w}}} \right){\rm{d}}t} $ (12)

where Q is a given positive definite weighting matrix, tf denotes the final time, and η is a specified attenuation level. By defining the state estimated error ${\mathit{\boldsymbol{e}}_{\rm{0}}} = \mathit{\boldsymbol{\bar x}} - {\hat{\bar{\mathit{\boldsymbol{x}}}}} $, the tracking error $ {\mathit{\boldsymbol{e}}_{\rm{p}}} = \mathit{\boldsymbol{\bar x - }}{{\mathit{\boldsymbol{\bar x}}}_r}$, the state vector of the global closed-loop system $\mathit{\boldsymbol{\tilde x}} = {[{\mathit{\boldsymbol{e}}_{\rm{0}}}, {\mathit{\boldsymbol{e}}_{\rm{p}}}, {{\mathit{\boldsymbol{\bar x}}}_r}]^{\rm{T}}} $, then combining the control law (Eq.(11)), the system (Eq.(7)), and the observers (Eqs.(9), (10)), the following closed-loop system model can be obtained

$ \mathit{\boldsymbol{\dot {\tilde x}}} = \mathit{\boldsymbol{\tilde A\tilde x}} + \mathit{\boldsymbol{\tilde B\tilde \varphi }} $ (13)

where

$ \mathit{\boldsymbol{\tilde A}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{LC}}}&{\mathit{\boldsymbol{L}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}}&{\mathit{\boldsymbol{L}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}}\\ { - \mathit{\boldsymbol{BK}}}&{\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}}}&{\mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{A}}_r}}\\ 0&0&{{\mathit{\boldsymbol{A}}_r}} \end{array}} \right] $
$ \mathit{\boldsymbol{\tilde S}} = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}&0\\ \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}&{ - \mathit{\boldsymbol{I}}}\\ 0&\mathit{\boldsymbol{I}} \end{array}} \right],\mathit{\boldsymbol{\tilde \varphi }} = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{w}}\\ \mathit{\boldsymbol{r}} \end{array}} \right] $

Note that with the state vector $ {\mathit{\boldsymbol{\tilde x}}}$, in-Eq.(12) can be written with $\mathit{\boldsymbol{\tilde Q}} = {\rm{diag}}\left\{ {0, \mathit{\boldsymbol{Q}}, 0} \right\} $

$ \int_0^t {{{\mathit{\boldsymbol{\tilde x}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde Q\tilde x}}{\rm{d}}t} \le {\eta ^2}\int_0^t {{{\mathit{\boldsymbol{\tilde \varphi }}}^{\rm{T}}}\mathit{\boldsymbol{\tilde \varphi }}{\rm{d}}t} $ (14)

where $ {{\mathit{\boldsymbol{\tilde \varphi }}}^{\rm{T}}}\mathit{\boldsymbol{\tilde \varphi }} = {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{r}} + {\mathit{\boldsymbol{w}}^{\rm{T}}}\mathit{\boldsymbol{w}}$.

The objective is now to compute the gain matrices K, L to ensure the asymptotic stability of Eq.(13) thus guaranteeing Eq.(14). A straightforward result is summarized in the following theorem.

Theorem 1  If there exists a matrix $ \mathit{\boldsymbol{\tilde P}} = {{\mathit{\boldsymbol{\tilde P}}}^{\rm{T}}} > 0$ and a positive constant η such that the following linear matrix inequality (LMI) is satisfied

$ \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde P}} + \mathit{\boldsymbol{\tilde P\tilde A}} + \mathit{\boldsymbol{I}}}&{\mathit{\boldsymbol{\tilde P\tilde S}}}\\ {{{\mathit{\boldsymbol{\tilde S}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde P}}}&{ - {\eta ^2}\mathit{\boldsymbol{I}}} \end{array}} \right] < 0 $ (15)

then the asymptotic stability of the closed-loop formation control system Eq.(13) is ensured and the H tracking control performance Eq.(14) is guaranteed with an attenuation level η.

Proof  Consider the following candidate Lyapunov function

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{V}} = {{\mathit{\boldsymbol{\tilde x}}}^{\rm{{\rm T}}}}\mathit{\boldsymbol{\tilde P\tilde x}}}&{\mathit{\boldsymbol{P}} = {\mathit{\boldsymbol{P}}^{\rm{T}}} > 0} \end{array} $ (16)

The stability of the closed-loop model (Eq.(13)) is satisfied under the H performance (Eq.(14)) with the attenuation level η if

$ \mathit{\boldsymbol{\dot V}} + {{\mathit{\boldsymbol{\tilde x}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde Q\tilde x}} - {\eta ^2}{{\mathit{\boldsymbol{\tilde \varphi }}}^{\rm{T}}}\mathit{\boldsymbol{\tilde \varphi }} \le 0 $ (17)

Eq.(17) leads to

$ \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\tilde x}}}^{\rm{T}}}\left[ {{{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde P}} + \mathit{\boldsymbol{\tilde P\tilde A}} + \mathit{\boldsymbol{\tilde Q}}} \right]\mathit{\boldsymbol{\tilde x}} + {{\mathit{\boldsymbol{\tilde x}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde P\tilde S\tilde \varphi }} + }\\ {{{\mathit{\boldsymbol{\tilde \varphi }}}^{\rm{T}}}{\mathit{\boldsymbol{S}}^{\rm{T}}}\mathit{\boldsymbol{P\tilde x}} - {\eta ^2}{{\mathit{\boldsymbol{\tilde \varphi }}}^{\rm{T}}}\mathit{\boldsymbol{\tilde \varphi }} \le 0} \end{array} $ (18)

or equivalently

$ \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\tilde x}}}\\ {\mathit{\boldsymbol{\tilde \varphi }}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde P}} + \mathit{\boldsymbol{\tilde P\tilde A}} + \mathit{\boldsymbol{\tilde Q}}}&{\mathit{\boldsymbol{\tilde P\tilde S}}}\\ {{\mathit{\boldsymbol{S}}^{\rm{T}}}\mathit{\boldsymbol{P}}}&{ - {\eta ^2}\mathit{\boldsymbol{I}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\tilde x}}}\\ {\mathit{\boldsymbol{\tilde \varphi }}} \end{array}} \right] \le 0 $ (19)

The goal is now to obtain a tractable LMI problem that allows searching the gain matrices (both the control matrix K and the observer gain matrix L) to prove the closed-loop stability (finding $ \mathit{\boldsymbol{\tilde P}} > 0$ to ensure the prescribed attenuation level η).

Remark  Most of former results just consider the wireless communication or GPS fault of the multiple aircrafts formation[14, 17], but few attempts have been made towards solving the controller design problem with sensor faults. Thus, the novelty of model (Eq.(13)) with respect to existing results is that the sensor fault of FFCS is taken into consideration.

2 Fault Tolerant Control Design

In this section, the fault tolerant formation controller design problem of multiple UAVs is investigated. The design requirements mentioned above are analyzed separately, and the obtained results are utilized for the tracking controller design.

Lemma 1[25]  For real matrices X, Y and S=ST>0 with appropriate dimensions and a positive constant γ, the following inequalities hold

$ {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{Y}} + {\mathit{\boldsymbol{Y}}^{\rm{T}}}\mathit{\boldsymbol{X}} \le \gamma {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{X + }}{\gamma ^{ - 1}}{\mathit{\boldsymbol{Y}}^{\rm{T}}}\mathit{\boldsymbol{Y}} $ (20)
$ {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{Y}} + {\mathit{\boldsymbol{Y}}^{\rm{T}}}\mathit{\boldsymbol{X}} \le {\mathit{\boldsymbol{X}}^{\rm{T}}}{\mathit{\boldsymbol{S}}^{ - 1}}\mathit{\boldsymbol{X + }}{\mathit{\boldsymbol{Y}}^{\rm{T}}}\mathit{\boldsymbol{SY}} $ (21)

Lemma 2[26]  For real matrices A, B, W, Y, Z and a regular matrix Q with appropriate dimensions, one has

$ \begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{Y}} + {\mathit{\boldsymbol{B}}^{\rm{T}}}{\mathit{\boldsymbol{Q}}^{ - 1}}\mathit{\boldsymbol{B}}}&{{\mathit{\boldsymbol{W}}^{\rm{T}}}}\\ \mathit{\boldsymbol{W}}&{\mathit{\boldsymbol{Z}} + \mathit{\boldsymbol{AQ}}{\mathit{\boldsymbol{A}}^{\rm{T}}}} \end{array}} \right] < 0 \Rightarrow }\\ {\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{Y}}&{{\mathit{\boldsymbol{W}}^{\rm{T}}} + {\mathit{\boldsymbol{B}}^{\rm{T}}}{\mathit{\boldsymbol{A}}^{\rm{T}}}}\\ {\mathit{\boldsymbol{W}} + \mathit{\boldsymbol{AB}}}&\mathit{\boldsymbol{Z}} \end{array}} \right] < 0} \end{array} $ (22)

Lemma 3[27]  Let a matrix Ξ < 0, a matrix X with appropriate dimension such that XTΞX≤0, and a scalar α, the following inequality holds

$ {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varXi} X}} \le - \alpha \left( {{\mathit{\boldsymbol{X}}^{\rm{T}}} + \mathit{\boldsymbol{X}}} \right) - {\alpha ^2}{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}^{ - 1}} $

Lemmas 1-3 formulate the conditions under which the inequality meets the negative definite. Based on these propositions, the following theorem presents a controller design method via convex optimization.

Theorem 2  When the sensor fault parameter matrix F of closed-loop system (Eq.(13)) is unknown and satisfies Eqs.(3)-(5), if there exist positive symmetric matrices $ {\mathit{\boldsymbol{P}}_1} = \mathit{\boldsymbol{P}}_1^{\rm{T}} > 0, {\mathit{\boldsymbol{P}}_3} = \mathit{\boldsymbol{P}}_3^{\rm{T}} > 0,\mathit{\boldsymbol{N}} = {\mathit{\boldsymbol{N}}^{\rm{T}}} > 0, {\rm{ }}\mathit{\boldsymbol{S}} = {\mathit{\boldsymbol{S}}^{\rm{T}}} > 0$, real matrices Y and Z, a positive constant η, such that the following LMI conditions are satisfied

$ \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^1}&0&{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{13}^1}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}&0&\mathit{\boldsymbol{Z}}&0&0&\mathit{\boldsymbol{Z}}&0\\ * &{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{22}^1}&0&0&0&0&{\mathit{\boldsymbol{NQ}}}&{\mathit{\boldsymbol{ \boldsymbol{\hat \varGamma} }}_{28}^1}&0&{\mathit{\boldsymbol{N}}{\mathit{\boldsymbol{C}}^{\rm{T}}}}\\ *&* &{\mathit{\boldsymbol{ \boldsymbol{\hat \varGamma} }}_{33}^1}&0&{0.5{\mathit{\boldsymbol{P}}_3}}&0&0&0&0&0\\ *&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0&0&0&0&0\\ *&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0&0&0&0\\ *&*&*&*&* &{ - \mathit{\boldsymbol{I}}}&0&0&0&0\\ *&*&*&*&*&* &{ - 2\mathit{\boldsymbol{Q}}}&0&0&0\\ *&*&*&*&*&*&* &{\mathit{\boldsymbol{S}} - \mathit{\boldsymbol{I}}}&0&0\\ *&*&*&*&*&*&*&* &{ - \mathit{\boldsymbol{S}}{{\mathit{\boldsymbol{\tilde F}}}^{ - 2}}}&0\\ *&*&*&*&*&*&*&*&* &{ - \mathit{\boldsymbol{S}}{{\mathit{\boldsymbol{\tilde F}}}^{ - 2}}} \end{array}} \right] < 0 $ (23)

where $ \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^1 = 0.5[{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{ZC}} + {({\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{ZC}})^{\rm{T}}}], {\rm{ }}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{13}^1 = \mathit{\boldsymbol{Z}}(\mathit{\boldsymbol{I}} - {\mathit{\boldsymbol{F}}_0})C, {\rm{ }}$ $\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{22}^1 = 0.5[\mathit{\boldsymbol{AN}} + \mathit{\boldsymbol{BY}} + {\left( {\mathit{\boldsymbol{AN}} + \mathit{\boldsymbol{BY}}} \right)^{\rm{T}}}], $ $ {\rm{ }}\mathit{\boldsymbol{ \boldsymbol{\hat \varGamma} }}_{33}^1 = 0.5({\mathit{\boldsymbol{P}}_3}{\mathit{\boldsymbol{A}}_r} + \mathit{\boldsymbol{A}}_r^{\rm{T}}{\mathit{\boldsymbol{P}}_3}) + {\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{SC}}, {\rm{ }}\mathit{\boldsymbol{ \boldsymbol{\hat \varGamma} }}_{28}^1 = {\mathit{\boldsymbol{N}}^{\rm{T}}}{\mathit{\boldsymbol{C}}^{\rm{T}}}{(\mathit{\boldsymbol{I}} - {\mathit{\boldsymbol{F}}_0})^{\rm{T}}}$.

$ \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^1}&\mathit{\boldsymbol{I}}&0&0&0&0&0&0\\ * &{ - 2\mathit{\boldsymbol{N}}}&0&0&0&0&0&{{\mathit{\boldsymbol{Y}}^{\rm{T}}}}\\ *&* &{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{33}^2}&{\mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{A}}_r}}&\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}&{ - \mathit{\boldsymbol{I}}}&{{\mathit{\boldsymbol{N}}^{\rm{T}}}\mathit{\boldsymbol{Q}}}&0\\ *&*&* &{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{44}^2}&0&{0.5{\mathit{\boldsymbol{P}}_3}}&0&0\\ *&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0&0\\ *&*&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0\\ *&*&*&*&*&* &{ - 2\mathit{\boldsymbol{Q}}}&0\\ *&*&*&*&*&*&* &{ - \mathit{\boldsymbol{I}}} \end{array}} \right] < 0 $ (24)

where $\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^2 = 0.5[{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{ZC}} + {({\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{ZC}})^{\rm{T}}}], \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{33}^2 = 0.5[\mathit{\boldsymbol{AN}} + \mathit{\boldsymbol{BY}} + {\left( {\mathit{\boldsymbol{AN}} + \mathit{\boldsymbol{BY}}} \right)^{\rm{T}}}] + $ $\mathit{\boldsymbol{B}}{\mathit{\boldsymbol{B}}^{\rm{T}}}, \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{44}^2 = 0.5\cdot({\mathit{\boldsymbol{P}}_3}{\mathit{\boldsymbol{A}}_r} + \mathit{\boldsymbol{A}}_r^{\rm{T}}{\mathit{\boldsymbol{P}}_3}) $ .Then the asymptotic stability of the closed-loop formation control system (Eq.(13)) is ensured and the H tracking control performance (Eq. (14)) is guaranteed with an attenuation level η. Moreover, if a solution exists, the gains K and L are obtained using K=YN-1 and L=P-1Z.

Proof  Assuming that $ \mathit{\boldsymbol{\tilde P}} = {\rm{diag}}\{ {\mathit{\boldsymbol{P}}_1}, {\mathit{\boldsymbol{P}}_2}, {\mathit{\boldsymbol{P}}_3}\} $ and P2=P2T>0, in-Eq.(19) can be rewritten as

$ \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varPi} }}_{11}}}&{{\mathit{\boldsymbol{ \boldsymbol{\varPi} }}_{12}}}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}&0\\ * &{{\mathit{\boldsymbol{ \boldsymbol{\varPi} }}_{22}}}&{{\mathit{\boldsymbol{P}}_2}\left( {\mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{A}}_r}} \right)}&{{\mathit{\boldsymbol{P}}_2}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}&{ - {\mathit{\boldsymbol{P}}_2}}\\ *&* &{{\mathit{\boldsymbol{P}}_3}{\mathit{\boldsymbol{A}}_r} + \mathit{\boldsymbol{A}}_r^{\rm{T}}{\mathit{\boldsymbol{P}}_3}}&0&{{\mathit{\boldsymbol{P}}_3}}\\ *&*&* &{ - {\eta ^2}\mathit{\boldsymbol{I}}}&0\\ *&*&*&* &{ - {\eta ^2}\mathit{\boldsymbol{I}}} \end{array}} \right] < 0 $ (25)

where

$ \begin{array}{l} {\mathit{\boldsymbol{ \boldsymbol{\varPi} }}_{11}} = {\mathit{\boldsymbol{P}}_1}\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{LC}}} \right) + {\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{LC}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{P}}_1},{\mathit{\boldsymbol{ \boldsymbol{\varPi} }}_{12}} = {\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}}\left( {\mathit{\boldsymbol{I}} - } \right.\\ \left. \mathit{\boldsymbol{F}} \right)\mathit{\boldsymbol{C}} + {\mathit{\boldsymbol{K}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}{\mathit{\boldsymbol{P}}_2},{\mathit{\boldsymbol{ \boldsymbol{\varPi} }}_{22}} = {\mathit{\boldsymbol{P}}_2}\left( {\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}}} \right) + {\left( {\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{P}}_2} + \\ \mathit{\boldsymbol{Q}} \end{array} $

For a convenient design, in-Eq.(19) is simplified as

$ {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}^1} + {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}^2} < 0 $

where

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}^1} = }\\ {\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\bar \varGamma} }}_{11}^1}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}&0\\ * &{\mathit{\boldsymbol{ \boldsymbol{\bar \varGamma} }}_{22}^1}&0&0&0\\ *&* &{0.5\left( {{\mathit{\boldsymbol{P}}_3}{\mathit{\boldsymbol{A}}_r} + \mathit{\boldsymbol{A}}_r^{\rm{T}}{\mathit{\boldsymbol{P}}_3}} \right)}&0&{0.5{\mathit{\boldsymbol{P}}_3}}\\ *&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0\\ *&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}} \end{array}} \right]} \end{array} $
$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}^2} = }\\ {\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\bar \varGamma} }}_{11}^2}&{{\mathit{\boldsymbol{K}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}{\mathit{\boldsymbol{P}}_2}}&0&0&0\\ * &{\mathit{\boldsymbol{ \boldsymbol{\bar \varGamma} }}_{22}^2}&{{\mathit{\boldsymbol{P}}_2}\left( {\mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{A}}_r}} \right)}&{{\mathit{\boldsymbol{P}}_2}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}&{ - {\mathit{\boldsymbol{P}}_2}}\\ *&* &{0.5\left( {{\mathit{\boldsymbol{P}}_3}{\mathit{\boldsymbol{A}}_r} + \mathit{\boldsymbol{A}}_r^{\rm{T}}{\mathit{\boldsymbol{P}}_3}} \right)}&0&{0.5{\mathit{\boldsymbol{P}}_3}}\\ *&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0\\ *&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}} \end{array}} \right]} \end{array} $
$ \begin{array}{l} \mathit{\boldsymbol{ \boldsymbol{\bar \varGamma} }}_{11}^1 = 0.5\left[ {{\mathit{\boldsymbol{P}}_1}\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{LC}}} \right) + {{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{LC}}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{P}}_1}} \right],\mathit{\boldsymbol{ \boldsymbol{\bar \varGamma} }}_{22}^1 = 0.5 \cdot \\ \left[ {{\mathit{\boldsymbol{P}}_2}\left( {\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}}} \right){{\left( {\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{P}}_2} + \mathit{\boldsymbol{Q}}} \right],\mathit{\boldsymbol{ \boldsymbol{\bar \varGamma} }}_{11}^2 = 0.5\left[ {{\mathit{\boldsymbol{P}}_1}\left( {\mathit{\boldsymbol{A}} - } \right.} \right.\\ \left. {\left. {\mathit{\boldsymbol{LC}}} \right) + {{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{LC}}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{P}}_1}} \right],\mathit{\boldsymbol{ \boldsymbol{\bar \varGamma} }}_{22}^2 = 0.5\left[ {{\mathit{\boldsymbol{P}}_2}\left( {\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}} - } \right.} \right.\\ \left. {{{\left( {\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{P}}_2} + \mathit{\boldsymbol{Q}}} \right]. \end{array} $

Then in-Eq.(25) holds, if Γ1 < 0 and Γ2 < 0.

Using the well-known property in Lemma 1, the following inequality can be deduced

$ \begin{array}{l} \mathit{\boldsymbol{e}}_0^{\rm{T}}{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{e}}_p} + \mathit{\boldsymbol{e}}_0^{\rm{T}}{\mathit{\boldsymbol{C}}^{\rm{T}}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right){\mathit{\boldsymbol{L}}^{\rm{T}}}{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{e}}_0} \le \\ \mathit{\boldsymbol{e}}_0^{\rm{T}}{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}}{\left( {{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{e}}_0} + \mathit{\boldsymbol{e}}_p^{\rm{T}}{\mathit{\boldsymbol{C}}^{\rm{T}}}{\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{e}}_p} \end{array} $

Based on the above, Γ1 < 0 is equivalent to

$ \begin{array}{l} {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varGamma} }}}^1} = \\ \;\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\tilde \varGamma} }}_{11}^1}&0&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}&0\\ * &{\mathit{\boldsymbol{ \boldsymbol{\tilde \varGamma} }}_{22}^1}&0&0&0\\ *&* &{0.5\left( {{\mathit{\boldsymbol{P}}_3}{\mathit{\boldsymbol{A}}_r} + \mathit{\boldsymbol{A}}_r^{\rm{T}}{\mathit{\boldsymbol{P}}_3}} \right)}&0&{0.5{\mathit{\boldsymbol{P}}_3}}\\ *&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0\\ *&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}} \end{array}} \right] < 0 \end{array} $ (26)

where $\mathit{\boldsymbol{ \boldsymbol{\tilde \varGamma} }}_{22}^1 = 0.5[{\mathit{\boldsymbol{P}}_2}\left( {\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}}} \right) + {\left( {\mathit{\boldsymbol{A}} + \mathit{\boldsymbol{BK}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{P}}_2} + \mathit{\boldsymbol{Q}}] + {\mathit{\boldsymbol{C}}^{\rm{T}}}{\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}, $ $\mathit{\boldsymbol{ \boldsymbol{\tilde \varGamma} }}_{11}^1 = 0.5[{\mathit{\boldsymbol{P}}_1}\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{LC}}} \right) + {\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{LC}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{P}}_1}] + {\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L }}{({\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{L}})^{\rm{T}}} $.

Then, one proceeds a bijective change of variables followed by a pre-post multiply of the in-Eq.(26) by diag{I, N, I, I, I} with Z=P1L, N=P2-1 and KP2-1=KN=Y, and obtains

$ \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^1}&0&{\mathit{\boldsymbol{Z}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)\mathit{\boldsymbol{C}}}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}&0&\mathit{\boldsymbol{Z}}&0&0\\ * &{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{22}^1}&0&0&0&0&{{\mathit{\boldsymbol{N}}^{\rm{T}}}\mathit{\boldsymbol{Q}}}&{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{28}^1}\\ *&* &{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{33}^1}&0&{0.5{\mathit{\boldsymbol{P}}_3}}&0&0&0\\ *&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0&0&0\\ *&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0&0\\ *&*&*&*&* &{ - \mathit{\boldsymbol{I}}}&0&0\\ *&*&*&*&*&* &{ - 2\mathit{\boldsymbol{Q}}}&0\\ *&*&*&*&*&*&* &{ - \mathit{\boldsymbol{I}}} \end{array}} \right] < 0 $ (27)

where $\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^1 = 0.5[{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{ZC}} + {({\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{ZC}})^{\rm{T}}}], \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{22}^1 = 0.5[\mathit{\boldsymbol{AN}} + \mathit{\boldsymbol{BY}} + {\left( {\mathit{\boldsymbol{AN}} + \mathit{\boldsymbol{BY}}} \right)^{\rm{T}}}], $ $ \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{33}^1 = 0.5({\mathit{\boldsymbol{P}}_3}{\mathit{\boldsymbol{A}}_r} + \mathit{\boldsymbol{A}}_r^{\rm{T}}{\mathit{\boldsymbol{P}}_3}), \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{28}^1 = {\mathit{\boldsymbol{N}}^{\rm{T}}}{\mathit{\boldsymbol{C}}^{\rm{T}}}{\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{F}}} \right)^{\rm{T}}}$.

Similarly, Γ2 < 0 can be turned into

$ {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}^2} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^2}&\mathit{\boldsymbol{I}}&0&0&0&0&0&0\\ * &{ - 2\mathit{\boldsymbol{N}}}&0&0&0&0&0&{{\mathit{\boldsymbol{Y}}^{\rm{T}}}}\\ *&* &{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{33}^2}&{\mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{A}}_r}}&\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}&{ - \mathit{\boldsymbol{I}}}&{{\mathit{\boldsymbol{N}}^{\rm{T}}}\mathit{\boldsymbol{Q}}}&0\\ *&*&* &{0.5\left( {{\mathit{\boldsymbol{P}}_3}{\mathit{\boldsymbol{A}}_r} + \mathit{\boldsymbol{A}}_r^{\rm{T}}{\mathit{\boldsymbol{P}}_3}} \right)}&0&{0.5{\mathit{\boldsymbol{P}}_3}}&0&0\\ *&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0&0\\ *&*&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0\\ *&*&*&*&*&* &{ - 2\mathit{\boldsymbol{Q}}}&0\\ *&*&*&*&*&*&* &{ - \mathit{\boldsymbol{I}}} \end{array}} \right] < 0 $ (28)

where $ \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^2 = 0.5({\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{ZC}}) + 0.5{({\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{ZC}})^{\rm{T}}}, \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{33}^2 = 0.5[\mathit{\boldsymbol{AN}} + \mathit{\boldsymbol{BY}} + {\left( {\mathit{\boldsymbol{AN}} + \mathit{\boldsymbol{BY}}} \right)^{\rm{T}}}] + \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{B}}^{\rm{T}}}$.

Using Eqs.(3)-(5), in-Eq.(27) can be rewritten as

$ {\mathit{\boldsymbol{M}}^1} = \mathit{\boldsymbol{M}}_0^1 + \mathit{\boldsymbol{\xi }}{\mathit{\boldsymbol{F}}_\delta }{\mathit{\boldsymbol{\beta }}^{\rm{T}}} + \mathit{\boldsymbol{\beta }}{\mathit{\boldsymbol{F}}_\delta }{\mathit{\boldsymbol{\xi }}^{\rm{T}}} + \mathit{\boldsymbol{\mu }}{\mathit{\boldsymbol{F}}_\delta }{\mathit{\boldsymbol{v}}^{\rm{T}}} + \mathit{\boldsymbol{v}}{\mathit{\boldsymbol{F}}_\delta }{\mathit{\boldsymbol{\mu }}^{\rm{T}}} < 0 $ (29)
$ \mathit{\boldsymbol{M}}_0^1 = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{11}^1}&0&{\mathit{\boldsymbol{Z}}\left( {\mathit{\boldsymbol{I}} - {\mathit{\boldsymbol{F}}_0}} \right)\mathit{\boldsymbol{C}}}&{{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}&0&\mathit{\boldsymbol{Z}}&0&0\\ * &{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{22}^1}&0&0&0&0&{{\mathit{\boldsymbol{N}}^{\rm{T}}}\mathit{\boldsymbol{Q}}}&{\mathit{\boldsymbol{ \boldsymbol{\hat \varGamma} }}_{28}^1}\\ *&* &{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{33}^1}&0&{0.5{\mathit{\boldsymbol{P}}_3}}&0&0&0\\ *&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0&0&0\\ *&*&*&* &{ - 0.5{\eta ^2}\mathit{\boldsymbol{I}}}&0&0&0\\ *&*&*&*&* &{ - \mathit{\boldsymbol{I}}}&0&0\\ *&*&*&*&*&* &{ - 2\mathit{\boldsymbol{Q}}}&0\\ *&*&*&*&*&*&* &{ - \mathit{\boldsymbol{I}}} \end{array}} \right] $

where $ \mathit{\boldsymbol{\xi }} = {[ - {\mathit{\boldsymbol{Z}}^{\rm{T}}}, 0, 0, 0, 0, 0, 0, 0]^{\rm{T}}}, \mathit{\boldsymbol{\beta }} = {\left[ {0, 0, C, 0, 0, 0, 0, 0} \right]^{\rm{T}}}, $ $ \mathit{\boldsymbol{ \boldsymbol{\hat \varGamma} }}_{28}^1 = {\mathit{\boldsymbol{N}}^{\rm{T}}}{\mathit{\boldsymbol{C}}^{\rm{T}}}{(\mathit{\boldsymbol{I}} - {\mathit{\boldsymbol{F}}_0})^{\rm{T}}}, {\rm{ }}\mathit{\boldsymbol{\upsilon }} = {\left[ {0, 0, 0, 0, 0, 0, 0, \mathit{\boldsymbol{I}}} \right]^{\rm{T}}}~~{\rm{and }}~~\mathit{\boldsymbol{\mu }} = {\left[ {0, - \mathit{\boldsymbol{CN}}, 0, 0, 0, 0, 0, 0} \right]^{\rm{T}}}$.

For Lemma 1 and S=ST>0, in-Eq.(29) is equivalent to

$ {\mathit{\boldsymbol{M}}^1} \le \mathit{\boldsymbol{M}}_0^1 + \mathit{\boldsymbol{\xi }}{\mathit{\boldsymbol{S}}^{ - 1}}{{\mathit{\boldsymbol{\tilde F}}}^2}{\mathit{\boldsymbol{\xi }}^{\rm{T}}} + \mathit{\boldsymbol{\beta S}}{\mathit{\boldsymbol{\beta }}^{\rm{T}}} + \mathit{\boldsymbol{\mu }}{\mathit{\boldsymbol{S}}^{ - 1}}{{\mathit{\boldsymbol{\tilde F}}}^2}{\mathit{\boldsymbol{\mu }}^{\rm{T}}} + \mathit{\boldsymbol{vS}}{\mathit{\boldsymbol{v}}^{\rm{T}}} $

Applying Schur complement theorem on the above inequalities, the in-Eq.(23) is deduced, then Theorem 3 is proved. When sensor fault of FFCS is unknown and satisfies conditions (Eqs.(3)-(5)), the asymptotic stability of the close-loop FFCS (Eq.(13)) is ensured.

3 Simulation Results

Simulations are performed for FFCS consisting of two UAVs. The expected formation geometries are specified by xc=20 m, yc=20 m and zc=0 m, respectively. The initial state of leader UAV are set as: VL0=30 m/s, ψL0=10°, hL0=1 000 m. The initial state of follower UAV are set as: VW0=20 m/s, ψW0=20°, hW0=1 010 m. The initial formation geometries are defined as: x0=35 m, y0=35 m and z0=-10 m. The dynamic matrices A and B are valuated for the data listed as[8]: The velocity time constant τVW=5 s, the heading time constant τψW=0.75 s, the altitude time constant τha=0.3 s, τhb=3.85 s. Outside disturbance is assumed as: w2=0.02·[2sint, cos0.5t, 3sint-cost]T. Furthermore, the parameter matrix of the reference model is given by

$ {\mathit{\boldsymbol{A}}_r} = \left[ {\begin{array}{*{20}{c}} {0.5637}&{ - 1.0683}&{2.8774}&{ - 9.422}&0&0\\ { - 8.4503}&{0.2027}&0&0&{0.6296}&0\\ 0&0&{ - 2.9534}&{ - 6.8129}&0&0\\ 0&0&{ - 0.6912}&{ - 3.1775}&0&0\\ 0&{ - 1.3284}&0&0&{4.1532}&{1.0078}\\ { - 0.7359}&0&0&0&{0.6304}&{ - 6.1824} \end{array}} \right] $

The resulting design is simulated in Matlab's Simulink package by the mathworks. To verify the superiority of the control approach, it is assumed that unknown sensor faults occur in the sensor channels of x, VW and y, respectively. It is assumed that $\mathit{\boldsymbol{\bar F}} = 0.1\;{\rm{diag}}\left\{ {1, 1, 1} \right\}, \mathit{\boldsymbol{\underline F}} = 0.9\cdot{\rm{diag}}\left\{ {1, 1, 1} \right\} $.

In the following, different simulation results are given using a robust controller without FTC and an observer-based FTC scheme developed in this study, respectively. First of all, a robust tracking controller without FTC is adopted. As shown in Figs. 2, 3, the robust controller without FTC cannot guarantee the state and control input of FFCS having a satisfactory dynamic performance. For example, the evolution of x cannot asymptotically track command xc.

Fig. 2 State evolution of FFCS using controller without FTC

Fig. 3 Control input evolution of FFCS using controller without FTC

Correspondingly, according to the linear matrix inequality in Theorem 3, one can deduce the fault tolerant tracking controller matrix K and observer gain matrix L as

$ \mathit{\boldsymbol{K}} = \left[ {\begin{array}{*{20}{c}} {0.3578}&{ - 7.1436}&{ - 2.2473}&{7.1387}&{13.6481}&{14.3787}\\ {0.0487}&{ - 0.0057}&{2.2459}&{ - 9.7714}&{ - 0.0001}&{ - 0.0001}\\ { - 0.0889}&{0.4104}&{ - 0.003}&{0.1804}&{ - 13.6884}&{ - 14.4836} \end{array}} \right] $
$ \mathit{\boldsymbol{L}} = {10^3} \cdot \left[ {\begin{array}{*{20}{c}} {5.5405}&{6.1796}&{ - 9.2905}\\ {5.5182}&{6.2120}&{ - 9.3002}\\ {5.4071}&{6.0587}&{ - 9.0407}\\ {5.7957}&{6.5002}&{ - 9.8711}\\ {5.6223}&{6.3028}&{ - 9.4959}\\ {5.6187}&{6.2989}&{ - 9.4925} \end{array}} \right] $

By utilizing the robust fault tolerant controller obtained in Theorem 3, it can be seen from Fig. 4 that the unknown sensor fault can be well regulated. Both x and y asymptotically track the specified commands xc and yc within 6 s. Moreover, as shown in Fig. 5, the robust fault tolerant tracking controller developed in this paper, makes that control input responses of the close-loop FFCS have a satisfactory dynamic performance in spite of the sensor fault, which demonstrates the efficiency of the proposed approach.

Fig. 4 State evolution of FFCS using robust fault tolerant controller

Fig. 5 Control input evolution of FFCS using robust fault tolerant controller

4 Conclusions

In this study, a novel fault tolerant tracking control design approach is proposed for a class of multiple UAVs formation control systems with sensor faults using both LMI technique and Lyapunov stability approach, which guarantees that the faulty UAVs formation has the good fault tolerant capability. Firstly, a formation control model of multiple UAVs considering the disturbances and sensor faults is introduced and a sensor gain fault model is also given. Then a robust fault tolerant tracking controller is designed in unknown sensor faulty case by means of a generalized observer. Finally, the simulation results are shown to exhibit the effectiveness of designed FTC approach.

Acknowledgements

This work was supported in part by the Post Doctoral Research Foundation of Jiangsu Province (No.1701140B), the National Natural Science Foundation of China (No.61403195), and the GF Research and Development Project of the Nanjing Tech Universities (No.201709).

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