Fault Tolerant Control of an Omnidirectional Robot ...

Fault Tolerant Control of an Omnidirectional Robot using a Switched Takagi-Sugeno Approach Damiano Rotondo, IEEE Student Member, Fatiha Nejjari and Vicenc¸ Puig Abstract— In this paper, a fault tolerant control (FTC) strategy for an omnidirectional robot using a switching TakagiSugeno (TS) approach is proposed. The overall solution relies on adapting the controller in order to keep the stability and some desired performances. It is shown that the design can be performed using polytopic techniques and linear matrix inequalities (LMIs). Real results obtained with a four-wheeled omnidirectional mobile robot are used to demonstrate the effectiveness of the proposed approach.

I. I NTRODUCTION The objective of a fault tolerant control (FTC) system [1]– [3] is to maintain desirable closed-loop performance, or with an acceptable degradation, and preserve stability conditions in the presence of component and/or instrument faults. The accommodation capability of a control system depends on many factors such as the fault severity, the robustness of the nominal system and the presence of redundancy in the sensors and/or the actuators. In particular, the compensation for actuator faults causing severe performance deterioration of control systems has been an important and challenging research problem [4]. FTC techniques are currently a hot topic of research, including their applications to the field of wheeled mobile robots [5]. In fact, when the mobile robots are intended to be used in hazardous environments or for long-time operation, it is needed to increase their robustness against possible failures [6]. Omnidirectional mobile robots are gaining popularity due to their enhanced mobility with respect to traditional robots [7]. In fact, they have the relevant characteristic that they can continue operating with three wheels in case some malfunctioning in the remaining wheel has been detected [8]. This makes them good platforms for testing techniques that provide fault tolerance against actuator faults. The Takagi-Sugeno (TS) framework has also been deeply studied in the automatic control literature [9], [10]. Introduced in [11], TS systems provide an effective way of representing nonlinear systems with the aid of fuzzy sets, fuzzy rules and a set of local linear models. The overall model of the system is obtained by merging the local models through fuzzy membership functions. The TS theory is This work has been funded by the Spanish MINECO through the project CYCYT SHERECS (ref. DPI2011-26243), by AGAUR through the contracts FI-DGR 2013 (ref. 2013FIB00218) and FI-DGR 2014 (ref. 2014FI B1 00172) and by the DGR of Generalitat de Catalunya (SAC group ref. 2014/SGR/374). Damiano Rotondo, Fatiha Nejjari and Vicenc¸ Puig are with Advanced Control Systems Group (SAC), Universitat Polit`ecnica de Catalunya (UPC), TR11, Rambla de Sant Nebridi, 10, 08222 Terrassa, Spain. Vicenc¸ Puig is also with Institut de Robotica i Informatica Industrial (IRI), UPC-CSIC, Carrer de Llorens i Artigas, 4-6, 08028 Barcelona, Spain.

e-mail: damiano(dot)rotondo(at)yahoo(dot)it

mainly used for designing controllers for non-faulty systems, but it has also been used for active FTC [12]–[14]. Recently, switched fuzzy systems have been introduced for dealing with complicated real systems, such as multiple nonlinear systems and switched nonlinear hybrid systems [15]–[20]. In this paper, a solution to the trajectory tracking problem in the inertial fixed coordinate system is proposed for a fourwheeled omnidirectional robot. This solution relies on the use of a reference model that describes the desired trajectory, an idea that is well-established in the LTI framework [21], and has been recently extended to cope with the control of TS systems [22], [23]. The TS controller design is performed solving a system of Linear Matrix Inequalities (LMIs), a problem for which efficient solvers are available [24], [25]. In particular, it is shown that, since there could not exist a solution within the standard TS framework, the switched TS framework is considered. By including the estimations of the actuator faults in the reference model, it is possible to obtain fault tolerance by considering the fault estimations as additional premise variables that schedule the TS error model. The effectiveness of the proposed approach is shown through experimental results obtained with a real testbed. The paper is organized as follows. Section II presents the model reference FTC approach using switched TS techniques. Section III describes the application example. Section IV presents the experimental results. Finally, the main conclusions and the possible future work are summarized in Section V. II. M ODEL R EFERENCE FTC USING S WITCHED TS T ECHNIQUES A. Switched TS System, Reference Model and Control Law Let us consider a nonlinear model described by a switched TS model, which uses sets of local models merged together using fuzzy IF-THEN rules [9], and a switching variable σ that can take a finite number of values, as follows: IF σ = 1 T HEN IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (1) (1) T HEN xi (k + 1) = Ai x(k) + Bi u(k) i = 1, . . . , N .. . ELSEIF σ = s T HEN IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (s) (s) T HEN xi (k + 1) = Ai x(k) + Bi u(k) i = 1, . . . , N .. . ELSEIF σ = S T HEN IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (S) (S) T HEN xi (k + 1) = Ai x(k) + Bi u(k) i = 1, . . . , N

(1)

where Mi j denote the fuzzy sets and N is the number of model rules; x(k) ∈ Rnx is the state vector, u(k) ∈ Rnu (s) (s) is the input vector, while Ai and Bi are matrices of appropriate dimensions. Finally, ϑ1 (k), . . . , ϑ p (k) are premise variables that can be functions of the state variables, external disturbances and/or time. Each linear consequent equation (s) (s) represented by Ai x(k) + Bi u(k) is called a subsystem. Given a value of σ = s and a pair (x(k), u(k)), the state of the switched TS system can easily be inferred by:   N (s) (s) x(k + 1) = ∑ ρi (ϑ (k)) Ai x(k) + Bi u(k)

(2)

i=1

where ϑ (k) = [ϑ1 (k), . . . , ϑ p (k)] is the vector containing the premise variables, and ρi (ϑ (k)) are defined as follows: ρi (ϑ (k)) =

wi (ϑ (k)) N

(8)

such that the control action is inferred as the weighted mean: (s)

∆u(k) = ∑ ρi (ϑ (k))Ki e(k)

(9)

i=1

∑ wi (ϑ (k))

B. Design using Quadratic Stability

(4)

For the synthesis of the switched TS controller, the following switched TS reference model is considered: IF σ = 1 T HEN : IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (1) (1) T HEN xre f ,i (k + 1) = Ai xre f (k) + Bi ure f (k) i = 1, . . . , N .. . IF σ = s T HEN : IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (s) (s) T HEN xre f ,i (k + 1) = Ai xre f (k) + Bi ure f (k) i = 1, . . . , N .. . IF σ = S T HEN : IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (S) (S) T HEN xre f ,i (k + 1) = Ai xre f (k) + Bi ure f (k) i = 1, . . . , N

IF σ = 1 T HEN IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (1) T HEN ∆ui (k) = Ki e(k) i = 1, . . . , N .. . IF σ = s T HEN IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (s) T HEN ∆ui (k) = Ki e(k) i = 1, . . . , N .. . IF σ = S T HEN : IF ϑ1 (k) is Mi1 AND . . . AND ϑ p (k) is Mip (S) T HEN ∆ui (k) = Ki e(k) i = 1, . . . , N

N

(3)

i=1

where ρi (ϑ (k)) is such that:   N ∑ ρi (ϑ (k)) = 1  i=1 ρi (ϑ (k)) ≥ 0, i = 1, . . . , N

The error system (7) is controlled through a switched TS error-feedback controller, defined as follows:

(5)

where xre f (k) ∈ Rnx is the reference state vector and ure f (k) ∈ Rnu is the reference input vector. The reference model gives the trajectory to be followed by the real
system. Given a value σ = s and a pair xre f (k), ure f (k) , the reference state can be inferred as:   N (s) (s) xre f (k + 1) = ∑ ρi (ϑ (k)) Ai xre f (k) + Bi ure f (k) (6) i=1

In this paper, both stability and pole clustering are analyzed within the quadratic Lyapunov framework [16], where the specifications are assured by the use of a single quadratic Lyapunov function. Despite the introduction of conservativeness with respect to other existing approaches, where the Lyapunov function is allowed to be parameter-varying, the quadratic approach has undeniable advantages in terms of computational complexity. In particular, the switched TS error system (7) with the error-feedback control law (9) is quadratically stable if and T > 0 and K (s) such that: only if there exist XS = XS i     (s) (s) (s) −XS Ai + B j Ki XS  0 and Ki such that:       (s) (s) (s) (s) (s) (s) T 0 and through the change of variables (s) (s) Γi , Ki X:



(s)

−X

   (s) (s) (s) T Ai X + B j Γi

(s) (s)

Ai X + B j Γi −X


(s) B f ,i φˆ (k) due to the fault using fuzzy sets, in such a way that the switched TS error model can be reshaped as:

 