State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system

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State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system Jorge Aurelio Brizuela Mendoza a,n, Carlos Manuel Astorga Zaragoza b, Arturo Zavala Río c,1, Leo Pattalochi d, Francisco Canales Abarca e,2 a

Centro Nacional de Investigación y Desarrollo Tecnológico (CENIDET), Interior Internado Palmira S/N, Col. Palmira, Cuernavaca, Morelos 62490, Mexico Centro Nacional de Investigación y Desarrollo Tecnológico (CENIDET), Mexico c Instituto Potosino de Investigación Científica y Tecnológica (IPICYT), Camino a la Presa San José 2055, Col. Lomas 4 sección, 78126. San Luis Potosí, S.L.P., Mexico d Centro Nacional de Investigación y Desarrollo Tecnológico (CENIDET)/École Supérieure des, Sciences et Technologies de L'Ingénieur de Nancy (ESSTIN), Mexico e ABB Corporate Research, Brown Boveri Strasse 6, 5400 Baden, Switzerland b

art ic l e i nf o

a b s t r a c t

Article history: Received 1 May 2015 Received in revised form 23 November 2015 Accepted 27 November 2015 This paper was recommended for publication by Dr. Steven Ding

This paper deals with an observer design for Linear Parameter Varying (LPV) systems with high-order time-varying parameter dependency. The proposed design, considered as the main contribution of this paper, corresponds to an observer for the estimation of the actuator fault and the system state, considering measurement noise at the system outputs. The observer gains are computed by considering the extension of linear systems theory to polynomial LPV systems, in such a way that the observer reaches the characteristics of LPV systems. As a result, the actuator fault estimation is ready to be used in a Fault Tolerant Control scheme, where the estimated state with reduced noise should be used to generate the control law. The effectiveness of the proposed methodology has been tested using a riderless bicycle model with dependency on the translational velocity v, where the control objective corresponds to the system stabilization towards the upright position despite the variation of v along the closed-loop system trajectories. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: LPV systems Time-varying parameter Actuator fault detection Actuator fault estimation State estimation

1. Introduction The Linear-Parameter-Varying (LPV) modeling has represented, over the last few decades, a simple way to approach nonlinear dynamics. This kind of systems belong to the general class of Linear-Time-Varying (LTV) systems [1–3], allowing the approximation of complex systems based on a set of parameters whose value may change along the system trajectories [4]. A LPV system is considered as a parameter-dependent system, in which the

n

Corresponding author. Tel.: þ 52 7773627795. E-mail addresses: [email protected] (J.A. Brizuela Mendoza), [email protected] (C.M. Astorga Zaragoza), [email protected] (A. Zavala Río), [email protected] (L. Pattalochi), [email protected] (F. Canales Abarca). URLS: http://www.cenidet.edu.mx/ (C.M. Astorga Zaragoza), http://www.ipicyt.edu.mx (A. Zavala Río), https://www.esstin.univ-lorraine.fr/ (L. Pattalochi), http://new.abb.com/ch (F. Canales Abarca). 1 Tel.: þ52 444 342000. 2 Tel.: þ41 58 588 33 00.

control design is defined in terms of causal functions of the timevarying parameter — current control values cannot depend on the future values of the time-varying parameter — in contrast with LTV systems design. The distinction from Linear Time Invariant (LTI) systems is, with the previously commented in mind, clear: the LPV systems are nonstationary [5]. In addressing the LPV representation, several formulations exist: polytopic, Linear Fractional Transformation (LFT) and LPV affine. The polytopic LPV systems have been broadly studied because they provide a suitable representation computed as the combination of linear models — that approach the system behavior at a finite number of operating points — visualized as the vertices of a polytope [6,7]. The LFT representation of LPV systems involves the separation between the varying and the non-varying part of the model [8]. Finally, the affine formulation considers an infinite number of equilibrium points. Within the last case, there is an additional representation which involves dependency on the time-varying parameter on multiple degree: the polynomial LPV systems [9,10]. Such a particular formulation of LPV systems presents a difficult consideration: the stability analysis leads to problems related to Linear

http://dx.doi.org/10.1016/j.isatra.2015.11.026 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

2

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Matrix Inequalities with dependency on the time-varying parameter, named parameterized Linear Matrix Inequalities (PLMI). In other words, the main difficulty with polynomial LPV systems corresponds to the solution of a PLMI and, consequently, a less studied case. On the other hand, the continued growth of the control systems requires maintain an acceptable level of reliability, in which case the Fault Tolerant Control (FTC) [11] becomes relevant. In this direction, an important issue concerns to the estimation of the fault affecting the system [12–14]. The reason comes intuitively: if the fault magnitude is known, then it is possible to take corrective actions to ensure the control objectives. Such corrective actions may involve the on-line modification of the control gains by considering the fault estimation or the addition of virtual actuators and/or sensors [15]. This can be viewed within a special FTC case: Active Fault Tolerant Control. As a result, the importance of fault estimation lies on the possibility to provide information about the fault to be used in the FTC scheme. Further, the previous information can be also taken as fault indicator, isolation and identification, issue related to the Fault Diagnosis and Isolation (FDI). In terms of control system design, moreover, it is highly desirable the computation of a control law that contributes to a smooth and unforced operation of the actuator of the system, which could hardly be accomplished in the presence of noise at the system outputs. Consequently, the contribution of this paper refers the design of an observer in charge of the estimation of actuator fault and system state, reducing the effect of the noise presented at the system outputs. The previous is achieved by maintaining the stability characteristics of LPV systems. The LPV systems stability, in contrast with the LTI systems stability, is related to the fact that the Lyapunov analysis leads to an expression in which the maximum rate for the time-varying parameter — involved in the system dynamics — is taken into account. Thus, the proposed observer design maintains its stability considering the maximum rate of the time-varying parameter, as a result of the stability conditions for LPV systems. In the model of the considered application, the rate of the time-varying parameter corresponds to the translational acceleration of the bicycle. The proposed observer finds potential applications in FTC schemes due to the fault estimation. The estimated state, on the other hand, could be used to generate a control signal with less noise corruption. In the LPV fault reconstruction literature, [16] designs an observer using the sliding mode methodology for fault estimation with application to a Boeing 747-100 LPV model, through an affine LPV representation. In [17], the fault estimation and its compensation into the closed-loop system are addressed. An affine LPV model of a two-link manipulator is considered, in which the control objectives are maintained through the proposed methodology. Ref. [18] deals with a state observer for affine LPV systems, with a computed solution as a linear combination between the timevarying parameters and their boundaries. Furthermore, for a winding machine system, a polytopic LPV sensor fault detection filter has been developed in [19]. Although the works in these references are applied to LPV systems, the actuator fault estimation remains insufficiently explored in the polynomial LPV systems framework. The previous matter is viewed as the main motivation for the results presented throughout this paper. Consequently, the design of the observer is based on the extension of the LTI robust methodologies to polynomial LPV systems, considering the stability characteristics for this kind of systems. The proposed observer provides the system state and actuator fault estimation, using a polynomial LPV model of a riderless bicycle, affected by an actuator fault and measurement noise. The proposed observer is used to build the control law, by means of the estimated state with less noise than the one presented at the system output. The fault estimation, meanwhile, can be used within FDI or FTC approaches. The paper structure is as follows: Section 2 presents the definition

of polynomial LPV systems, as well as their controllability and observability conditions. Section 3 addresses the riderless bicycle model and its dynamical analysis. Section 4 provides the preliminaries on the stabilization of the riderless bicycle polynomial LPV model. Section 5 presents the design of the state and actuator fault estimation observer, besides the controller design. It addresses the observer and control gain computation at the end. Section 6 shows the simulation results considering different types of actuator faults. Finally, a brief discussion is presented in Section 7 and conclusions in Section 8.

2. Polynomial LPV systems The present section addresses the definition of polynomial LPV systems along with their structural properties. Consider the following dynamical system: x_ ¼ Aðζ ðtÞÞx þ Bðζ ðtÞÞu y ¼ Cðζ ðtÞÞx

ð1Þ

where x A Rn , u A Rp and y A Rs correspond to the state, input and output (vector) variables, respectively. Aðζ ðtÞÞ, Bðζ ðtÞÞ and Cðζ ðtÞÞ represent parameter-dependent matrices of compatible dimensions. ζ ðtÞ A Rm is the time-varying parameter vector. Normally, ζ ðtÞ is considered measurable and bounded, with bounded time ratio ζ ðtÞ, i.e. ( ν; ν 4 0 such that J ζ ðtÞ J r ν and J ζ_ ðtÞ J r μ with μ 40. If Aðζ ðtÞÞ, Bðζ ðtÞÞ and/or Cðζ ðtÞÞ adopt the form

χ ðζ Þ ¼ χ 0 þ

k X m X

χ f½ði  1Þm þ jg ζ ðtÞij

ð2Þ

i¼1j¼1

for some k Z1, where χl, l ¼ 0; …; km, are matrices of compatible dimensions (depending on the referred matrix Aðζ ðtÞÞ, Bðζ ðtÞÞ or Cðζ ðtÞÞ), then the state model (1) corresponds to a polynomial LPV system. As an example (from now on, the argument of ζ will be dropped for the sake of simplicity), consider ζ A R2 , i.e. two timevarying parameters ζ1 and ζ2 (i.e. m ¼ 2). With k ¼ 2, Eq. (2) corresponds to

χ ðζ Þ ¼ χ 0 þ χ 1 ζ 1 þ χ 2 ζ 2 þ χ 3 ζ 21 þ χ 4 ζ 22

ð3Þ

As a result, the system dependency in the time-varying parameter vector will appear in polynomial form. In order to show the structural properties for this kind of systems and, according to [20], the controllability condition for polynomial LPV systems can be seen as the extension of this criterion applied to LTI systems. The polynomial LPV system (1), considering the representation (2), is controllable if rank½Bðζ Þ Aðζ ÞBðζ Þ…Aðζ Þn  1 Bðζ Þ ¼ n

8 Jζ J rν

ð4Þ

In analogous way, a polynomial LPV system will be observable if

2

6 6 rank6 6 4

Cðζ Þ

Cðζ ÞAðζ Þ ⋮ Cðζ ÞAðζ Þn  1

3 7 7 7¼n 7 5

8 Jζ J rν

ð5Þ

3. The riderless bicycle model and its dynamical analysis The riderless bicycle can be formulated in terms of a general frame which includes a rigid body, a front frame composed by the handle, and the tires. The general representation for this system is given as [21] Q q€ þ vW q_ þ ðgE0 þ v2 E1 Þq ¼ f

ð6Þ

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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where q A R2 is the angular position vector containing the roll angle ϕ and the handlebar angle δ, i.e. q ¼ ½ϕ δT , and f A R2 is the input force vector composed by the torques applied to the general frame T ϕ and handlebar T δ , i.e. f ¼ ½T ϕ T δ T , as shown in Fig. 1. In Eq. (6), Q and W are the mass and damping constant matrices, respectively; E0 and E1 are stiffness matrices, and g is the gravity acceleration. The referred matrices, Q, W, E0 and E1, are related to the physical properties of the prototype and the effects of its interaction with the environment. The motion equation (6) has been validated for small variations around ϕ and δ, considering the bicycle at upright position [21]. h i _ δ_ T , the translaConsidering the state vector as x ¼ ϕ δ ϕ tional velocity of the bicycle as the time-varying parameter, and taking the torque applied to the handlebar Tδ as a unique input _ , the state space repreforce along with the outputs δ and ϕ sentation for (6) can be expressed as x_ ¼ AðvÞx þ Bu y ¼ Cx

ð7Þ

where [9] 2

0 6 0 6 AðvÞ ¼ 6 4 13:67 4:857

0

1

0

0

0:225  1:319v

 0:164v

2

10:81  1:125v2

3 0  6 7 0 0 6 7 B¼6 7C¼ 4  0:339 5 0

3:621v

0

3

7 7 7  0:552v 5

0

0

0

1

0

The controllability and observability criteria for the LPV system of the riderless bicycle can be computed using (4) and (5). Thus, for the controllability condition, we follow the next procedure (where j  j is used to denote the determinant of a matrix): P c ðvÞ ¼ j B AðvÞB AðvÞ2 B AðvÞ3 Bj P c ðvÞ ¼  5806:5321v4 þ19463:1274v2 22:3115



ð9Þ

By getting the positive real roots of Pc(v) the following translational velocity set, where the controllability is lost, has been found: v ¼ f0:03386; 1:8305g

ð10Þ

The observability condition, meanwhile, can be computed from the transpose of Eq. (5) since rank½Q  ¼ rank½Q T  (where Q is a given matrix). Let us point out that Eq. (5) will result in a rectangular matrix by considering the defined in Eq. (8). As a result, the transpose of (5), referred from now on as Λo ðvÞ, is 2 0 0 0 13:67 4:857 6 1 0 0 0:225  1:319v2 10:81  1:125v2 6 Λo ðvÞ ¼ 6 40 1 0  0:164v 3:621v 0

2

1

3.1. Controllability and observability conditions

1

 2:388v

3

0

1

 0:552v

 4:922v

37:9v

0:837v3  6v 13:67  1:971v2

 2:089v3  24:996 4:857 9:240v2

0:089v2 þ 0:225

2:578v2 þ10:81

 2:388v 187:961  26:520v2

3

7 2:5v4 17:758v2 þ5:51 7 7 7 0:648v3 6:350v 5 1:711v3  14:087v

ð8Þ

ð11Þ In order to compute rank½Λo ðvÞ ¼ rank½Λo ðvÞ , since it corresponds to a rectangular matrix which is depending on the timevarying parameter v, the procedure begins with a non-zero determinant operation considering a matrix of n  1 dimension. Once this has been solved, the procedure continues by adding the remaining rows and columns in order to obtain square matrices of n dimension. For each one of those square matrices, the determinant should be computed (the determinant operation will result in an equation involving the time-varying parameter v). Thus, for all v such that rank½Λo ðvÞ an, the LPV model is not observable. From Eq. (11), the considered square matrix of ðn  1Þ  ðn  1Þ dimension corresponds to an identity matrix formed by the rows 2  4 and columns 1  3. By following the mentioned procedure, incorporating the row 1 and column 4: 2 3 0 0 0 13:67 " # 6 1 0 0 0:225 1:319v2 7 013 13:67 7 Λo1 ¼ 6 ð12Þ 6 7¼ V1 I3 40 1 0 5 0:164v T

7:457 It can be seen that the matrix A(v) is depending on the translational velocity with general representation AðvÞ ¼ A0 þ A1 v þ A2 v2 , which is consistent with Eq. (2). Fig. 2 shows the open-loop eigenvalues of A(v) as functions of the time-varying parameter v. It is worth to note that the open-loop criterion, since LPV systems are related to LTV systems, is not a sufficient condition for determine the stability of the system.

δ δ

Fig. 1. Front view of the bicycle and roll angle ϕ. Top view of the bicycle: handlebar angle δ and forces f.

0

1

0:552v

where I3 is the identity matrix A R33 and V 1 A R31 with j Λo1 j ¼  13:67. Incorporating the row 1 and column 5: 2 3 0 0 0 4:857 " # 6 1 0 0 10:81 1:125v2 7 013 4:5857 7 Λo2 ¼ 6 ¼ ð13Þ 6 7 V2 I3 40 1 0 5 3:621v 0

Fig. 2. (a) Open-loop eigenvalues; (b) real part of open-loop eigenvalues as functions of the translational velocity v.

0

0

1

2:388v

in which case j Λo2 j ¼  4:857. In analogous way, matrices Λo3, Λo4 and Λo5 were found to be " # " # 013  4:922v 013 37:9v Λo3 ¼ ; Λo4 ¼ V3 V4 I3 I3 " # 2 013 187:961  26:520v Λo5 ¼ ð14Þ I3 V5

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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4

By computing the respective determinants of these last three matrices, it can be concluded that the riderless bicycle LPV model is observable 8 v, since there is not a coincident value for v in the solutions. As a result, the controllability and observability analysis allow to define the following translational velocity interval for the bicycle, for practical purposes:

Φ≔½0:5; 1:7

ð15Þ

where the translational velocity is expressed in m/s. The defined as Φ corresponds to the range of v where the controllability and observability conditions are not lost.

4. Preliminaries on the stabilization of a riderless bicycle Consider the system (7) with measurement noise z and an actuator fault expressed in additive form fa. The control objective is to maintain the vertical position of the bicycle by using, as control input, the torque applied to the handlebar axis T δ (despite a varying translational velocity v): x_ ¼ AðvÞx þBu þBf f a y ¼ Cx þ Dz

ð16Þ

where the matrix Bf considers the columns of B associated with the actuator under fault condition and fa the fault magnitude vector. D and z represent the noise matrix and its magnitude, which are affecting the outputs of the system. Let us point out the significance of the fault modeled as in (16). According to [23, section 2.3.1], an actuator fault corresponds to a variation of the global control input applied to the system. This can be characterized through the columns of B and their fault magnitude (expressed in matrix form), which include the fault functions associated with each actuator or input signal to the system. In order to estimate the state x and the magnitude of the fault fa, an observer with the following structure is adopted [24–27]: ^ x^_ ¼ AðvÞx^ þ Bu þ Bf f^a þK 0 ðvÞðy  yÞ _ ^ f^ a ¼ L0 ðvÞðy  yÞ y^ ¼ C x^

ð17Þ

^ f^ a where K 0 ðvÞ and L0 ðvÞ are the time-varying observer gains; x, and y^ represent the estimates of the state, fault and output, respectively. The objective is to design the observer (17) in order to ^ by provide the fault estimation f^ a as well as the system state x, minimizing the measurement noise effects presented at the measured outputs. Consider the control law: u ¼ T δ ¼ kðvÞx^

ð18Þ

where k(v) corresponds to the control gains (with time-varying parameter dependency) and x^ the estimated state. Consider the dynamics of the state estimation error ee ¼ x  x^ along with estimation error for the actuator fault ef ¼ f a  f^a : ^ e_ e ¼ AðvÞx þ Bu þ Bf f a ðAðvÞx^ þBu þBf f^a þ K 0 ðvÞðy yÞÞ ¼ ðAðvÞ  K 0 ðvÞCÞee þ Bf ef K 0 ðvÞDz ^ ¼ f_a  L0 ðvÞCee  L0 ðvÞDz e_ f ¼ f_a  L0 ðvÞðy  yÞ

ð19Þ

The control law (18), in terms of the state estimation error ee, is u ¼ T δ ¼ kðvÞx^ ¼ kðvÞðx  ee Þ

ð20Þ

consequently, the closed-loop system can be computed as x_ ¼ ðAðvÞ þ BkðvÞÞx  BkðvÞee þ Bf f a

ð21Þ

As a result, the estimation errors ee, ef and the closed-loop system can be represented in matrix form as follows: 32 3 2 _ 3 2 x x AðvÞ þ BkðvÞ  BkðvÞ 0 76 ee 7 6 e_ e 7 6 0 AðvÞ K ðvÞC B ¼ 4 5 4 5 0 f 54 e_ f

0 2

Bf 6 þ4 0 0

 L0 ðvÞC 0 32 3 fa 0 0 6 7  K 0 ðvÞD 0 7 54 z 5 f_  L ðvÞD I 0

ef ð22Þ

a

Recalling, the linear separation principle addresses the dynamic conditions to be fulfilled in such a way that the observer and control design can be performed independently. To explain the previous, let us point out that, from Eq. (22), if a suitable k0 ðvÞ is computed, the state estimation error ee will be ee -0 as t-1. As a result, the term _ will disappear and, involved in the closed-loop system  BkðvÞ (in x) finally, the control design can be performed from the remaining term AðvÞ þ BkðvÞ in Eq. (21). The observer, in turn, will be able to provide the fault estimation f^a by designing L0 ðvÞ such that ef -0 as t-1. The main issue in (22) corresponds to how to deal with the effects due to the measurement noise z and the fault fa, within the estimation error dynamics and the closed-loop system. Then, the observer design should consider a procedure capable to guarantee the observer stability despite the fault and measurement noise magnitude. As a consequence, the conditions fa ¼0 and f a a 0 should be taken into account in the observer design, addressed in the present article by extending the procedure existing for LTI systems to polynomial LPV systems. The main contribution of the present work consists in guarantee the observer stability despite the fault occurrence and measurement noise.

5. The riderless bicycle stabilization system 5.1. Observer design: stability To begin with the observer design, consider the error estimation dynamics in Eq. (19) represented in matrix form as follows: #" # " #" # " # " ee e_ e z AðvÞ  K 0 ðvÞC Bf  K 0 ðvÞD 0 ¼ þ ð23Þ ef e_ f f_ a  L0 ðvÞC 0  L0 ðvÞD I h iT  T Defining the vectors Δ ¼ ee ef and Ψ ¼ z f_a , Eq. (23) can _ ¼ AðvÞ ~ Δ þ B~ Ψ with be rewritten in compact form as Δ " # " # AðvÞ  K 0 ðvÞC Bf  K 0 ðvÞD 0 ~ AðvÞ ¼ ; B~ ¼ ð24Þ L0 ðvÞC 0  L0 ðvÞD I To ensure the state and fault estimation error convergence, a T Lyapunov function VðΔ; vÞ ¼ Δ PðvÞΔ has been considered, where P (v) denotes a symmetric positive definite matrix (let us point out the dependence of the Lyapunov function on the time-varying parameter v). In order to guarantee Δ-0 as t-1, it is necessary to have V_ ðΔ; vÞ o 0 [22], 8 Δ A U  Rn (U is a subset of Rn , since the model is linear) 8 v A Φ. Recall that Φ has been defined as the translational velocity set where the model does not have loss of controllability and observability conditions (see end of Section 3.1). As a result, the problem for state and fault estimation is reduced to find two parameter-dependent observer gains K 0 ðvÞ and L0 ðvÞ such that the asymptotic convergence of Δ is accomplished if Ψ ¼ 0 and the error vector Δ ultimately bounded if Ψ a 0. Both conditions can be formulated in mathematical terms as

Δ-0 as t-1 8 t if Ψ ¼ 0 J Δ J Q Δ r ðςðvÞÞ2 J Ψ J Q Ψ 8 t if Ψ a 0

ð25Þ

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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where Q Δ and Q Ψ are positive-defined matrices, J Δ J Q Δ ¼ Δ Q Δ Δ, T J Ψ J Q Ψ ¼ Ψ Q Ψ Ψ and ςðvÞ is the time-varying attenuation level. The expressions in (25) are fulfilled if [24] T

T T V_ ðΔ; vÞ þ Δ Q Δ Δ  ðςðvÞÞ2 Ψ Q Ψ Ψ o 0

ð26Þ

which in turn, corresponds to d dt

T

     ∂P ðvÞ ΔT P ðvÞ A ðvÞΔ þ B Ψ þ ΔT v_ Δ þ ð A ðvÞΔ þ B Ψ ÞT P ðvÞΔ ∂v T

for some Q ðvÞ ¼ Q ðvÞ 4 0. Since the inverse of a symmetric and positive-definite matrix is also symmetric and positive-definite, the pre-multiplication and post-multiplication operation in (35) by Q ðvÞ  1 gives [22] ð27Þ

In inequality (27), it is important to clarify the significance of T the Δ v_ ∂PðvÞ ∂v Δ term. In LPV framework, the stability analysis considers the time-varying parameter rate, which, in this particular case, corresponds to the bicycle translational acceleration. This is the main characteristic of the LPV systems stability. Going on with the observer design, Eq. (27) can be represented as 3" # " #T 2 ~ þ AðvÞ ~ T PðvÞ þ v_ ∂PðvÞ þ Q PðvÞB~ Δ Δ 4 PðvÞAðvÞ Δ ∂v 5 o0 T Ψ Ψ  ðςðvÞÞ2 Q Ψ B~ PðvÞ ð28Þ As a result, the convergence of the estimation error will be accomplished if the parameterized LMI (PLMI): 2 3 ~ ~ T PðvÞ þ v_ ∂PðvÞ þQ PðvÞB~ PðvÞAðvÞ þ AðvÞ Δ ∂v 4 5o0 ð29Þ T ðςðvÞÞ2 Q Ψ B~ PðvÞ is solvable with PðvÞ ¼ PðvÞT 4 0. In order to solve (29), a modification must be performed: #   " K 0 ðvÞ AðvÞ Bf ~ ~ ½C 0  AðvÞ ¼ A 1 ðvÞ  K s ðvÞC 1 ¼ ð30Þ L0 ðvÞ 0 0 " B~ ¼  K s ðvÞB~ 1 þ C 2 ¼ 

K 0 ðvÞ L0 ðvÞ

#

 ½D 0 þ

0

0

0

I

ð35Þ

T

T 

T

under the consideration that the solution of (32) guarantees the convergence of the estimation error ee, allows the design of the control gain k(v) such that x-0 when t-1 from the term AðvÞ þ BkðvÞ in Eq. (34). For this to be achieved, the controller design considers the Lyapunov function Vðx; vÞ ¼ xT Q ðvÞx whose derivative is [22] ∂Q ðvÞ o0 V_ ðx; vÞ ¼ Q ðvÞðAðvÞ þ BkðvÞÞ þ ðAðvÞ þBkðvÞÞT Q ðvÞ þ v_ ∂v

ΔT P ðvÞΔ_ þ ΔT ðP ðvÞÞΔ þ Δ_ PðvÞΔ þ ΔT Q Δ Δ−ðςðvÞÞ2 Ψ T Q Ψ Ψ o 0

þ Δ Q Δ Δ−ðςðvÞÞ2 Ψ Q Ψ Ψ o0 Δ A ðvÞP ðvÞΔ   T T T ∂P ðvÞ þ Δ P ðvÞ B Ψ þ Δ v_ Δ þ ΔT A ðvÞT P ðvÞΔ þ Ψ T B P ðvÞΔ ∂v T T þ Δ Q Δ Δ−ðςðvÞÞ2 Ψ Q Ψ Ψ o0

5

 ð31Þ

where I is the identity matrix of appropriate dimensions. Such a modification is made in order to eliminate the bilinear condition PðvÞK 0 ðvÞ and PðvÞL0 ðvÞ. Further, by defining K fo ðvÞ ¼ PðvÞK s ðvÞ and MðvÞ ¼ ðςðvÞÞ2 , the following is proposed.

ðAðvÞ þ BkðvÞÞQ ðvÞ−1 þ Q ðvÞ−1 ðAðvÞ þ BkðvÞÞT þ v_ Q ðvÞ−1

∂Q ðvÞQ ðvÞ−1 o0 ∂v

AðvÞQ ðvÞ−1 þ BkðvÞQ ðvÞ−1 þ Q ðvÞ−1 AðvÞT þ Q ðvÞ−1 kðvÞT BT " # ∂Q ðvÞ ∂Q ðvÞ−1 Q ðvÞ−1 þ Q ðvÞ þ v_ Q ðvÞ−1 o0 ∂v ∂v

ð36Þ

let us point out that " # ∂Q ðvÞ ∂Q ðvÞ  1 ∂Q ðvÞ _ ðvÞ  1 _ ðvÞ  1 Q ðvÞ  1 þ Q ðvÞ Q ðvÞ  1 vQ ¼ vQ ∂v ∂v ∂v ∂Q ðvÞ  1 ∂Q ðvÞ ∂Q ðvÞ  1 _ ðvÞ  1 Q ðvÞ  1 ¼  v_ ¼ 0 ) vQ ∂v ∂v ∂v By combining (36) and (37) V_ ðx; vÞ o 0 corresponds to

þ v_

ð37Þ

AðvÞQ ðvÞ  1 þ BkðvÞQ ðvÞ  1 þ Q ðvÞ  1 AðvÞT þ Q ðvÞ  1 kðvÞT BT  v_

∂Q ðvÞ  1 o0 ∂v

ð38Þ

Finally, by defining XðvÞ ¼ Q ðvÞ  1 and YðvÞ ¼ kðvÞXðvÞ in order to eliminate the bilinear condition kðvÞXðvÞ, the following PLMI is obtained: AðvÞXðvÞ þ BYðvÞ þ XðvÞT AðvÞT þ YðvÞT BT  v_

∂XðvÞ o0 ∂v

ð39Þ

As a result, the asymptotic stability is guaranteed if the PLMI (39) is solvable with XðvÞ ¼ XðvÞT 4 0. The final solution for the control gain is computed by kðvÞ ¼ YðvÞXðvÞ  1

ð40Þ

5.3. Observer gain and control gain computation

is feasible with κ ðvÞ ¼ PðvÞA~ 1 ðvÞ  K fo ðvÞC 1 þ A~ 1 ðvÞT PðvÞ C T1 K fo ðvÞT þ v_ ∂PðvÞ PðvÞ ¼ffi PðvÞT 4 0, ½K 0 ðvÞj ; L0 ðvÞT ¼ K s ðvÞ ¼ PðvÞ  1 K fo ðvÞ and ∂v , p ffiffiffiffiffiffiffiffiffiffi ςðvÞ ¼ MðvÞ.

The PLMI restrictions (32) and (39) can be seen as a set of LMIs with time dependency due to the translational velocity v. The methods to solve this kind of problems are the discretization method [22], the relaxation method [28] and the sum of Squares method [29]. In this paper the discretization method is used. This method is based on the discretization for the time-varying parameter range using a difference approximation for the derivative term. Once the discrete problem has been solved, an interpolation method is used to generate the continuous solution. With the previously commented in mind, the discrete PLMI for the control gains corresponds to

5.2. Controller design

AðjhÞXðjhÞ þ BYðjhÞ þ XðjhÞT AðjhÞT þ YðjhÞBT 7 v_

Proposition 1. The observer (17) will provide the fault and state estimation in the sense that ee -0 as t-1 if Ψ ¼ 0, and bounded by a factor ςðvÞ when Ψ a 0, if 2 3 κ ðvÞ þQ Δ  K fo ðvÞB~ 1 þ PðvÞC 2 4 5 o0 ð32Þ T  B~ 1 K fo ðvÞT þ C T2 PðvÞ  MðvÞQ Ψ

The following control law is proposed: h i u ¼ kðvÞx^ ¼ kϕ ðvÞ kδ ðvÞ kϕ_ ðvÞ kδ_ ðvÞ x^

XðjhÞ ¼ XðjhÞT 4 0

Yðjhþ hÞ  YðjhÞ o0 h ð41Þ

where k(v) is the control gain and x^ is provided by the observer given in (17). The closed-loop system in fault-free case (i.e. fa¼0 in Eq. (16)):

and for Proposition 1 (see Eq. (32)), related to the proposed observer: 2 3 κ ðjhÞ þ Q Δ  K fo ðjhÞB~ 1 þ PðjhÞC 2 4 5o0 ð42Þ T  B~ 1 K fo ðjhÞT þ C T2 PðjhÞ MðjhÞQ Ψ

x_ ¼ ðAðvÞ þ BkðvÞÞx  BkðvÞee

where

ð33Þ

ð34Þ

κ ðjhÞ ¼ PðjhÞA~ 1 ðjhÞ  K fo ðjhÞC 1 þ A~ 1 ðjhÞT PðjhÞ  C T1 K fo ðjhÞT

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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6

Fig. 3. (a) Observation error eigenvalues; (b) real part of eigenvalues as functions of the translational velocity v; (c) time-varying attenuation level.

7 v_ Pðjh þ hÞh  PðjhÞ, PðjhÞ ¼ PðjhÞT 4 0, ½K 0 ðjhÞ L0 ðjhÞT ¼ K s ðjhÞ ¼ PðjhÞ  1 pffiffiffiffiffiffiffiffiffiffiffiffi K fo ðjhÞ and ςðjhÞ ¼ MðjhÞ. In order to improve the results, a LMI region for the controller and observer has been considered [30]. A LMI region corresponds to a procedure which allows, in terms of LMIs, the closed-loop eigenvalues allocation within a specific subset of the complex plane. In this context, the closed-loop eigenvalues, as a result of the control gains, will be allocated to the left side of βc (where βc defines the subset of the complex plane where the closed-loop eigenvalues will remain or the LMI region) by solving AðjhÞXðjhÞ þBYðjhÞ þ XðjhÞT AðjhÞT þ YðjhÞBT þ 2XðjhÞβc o0 T

XðjhÞ ¼ XðjhÞ 4 0

function: K s ðvÞ ¼ Z F3 v3 þ Z F2 v2 þ Z F1 v þ Z F0 2

6  3:589 6 6 Z F3 ¼ 6 6  2:327 6 4  17:585  14:965 2

ð43Þ

The LMI region for the observer, on the other hand, will be defined by the parameter βo (βo defines the LMI region in analogous way that the LMI region for the closed-loop eigenvalues) and the following PLMI: 2 3 κ~ ðjhÞ þQ Δ þ 2βo PðjhÞ  K fo ðjhÞB~ 1 þ PðjhÞC 2 4 5 o0 ð44Þ T  B~ 1 K fo ðjhÞT þC T2 PðjhÞ  MðjhÞQ Ψ þ 2β o PðjhÞ where κ~ ðjhÞ ¼ PðjhÞA~ 1 ðjhÞ  K fo ðjhÞC 1 þ A~ 1 ðjhÞT PðjhÞ  C T1 K fo ðjhÞT . In the inequalities (41)–(44), j ¼ 1; 2; …; N is the gridding timevarying parameter space and h 40 the step width. Let us point out the significance of N and h. N is considered as the number of points in which the time-varying parameter range will be divided, whereas h corresponds to the interval contained within two consecutive points. In Eqs. (41)–(44), the term v_ is the maximum derivative value related to the translational velocity that the observer and controller will be able to cope, which, in the presented context, corresponds to the acceleration of the vehicle. The final solution for the controller and observer will be computed by solving (41)–(43) and Eqs. (42)–(44) simultaneously, respectively.

6. Simulation results 6.1. Control and observer gains: solutions The PLMI set (41)–(44) was solved using h ¼ 0:02, N ¼ 60, and v_ ¼ 0:05 with YALMIP Toolbox [31] and MATLAB. In Eqs. (42)–(44), Q Δ ¼ I  10  3 , Q Ψ ¼ 750I and βo ¼ 6 were considered. The final observer gains (considering the interpolation of the discrete solutions) are presented in the following 3rd grade polynomial

 8:899

22:240

6  7:723 6 6 Z F1 ¼ 6 6  41:298 6 4  799:597  437:676

4:481

ð45Þ 2

3

2:486 7 7 7 1:911 7 7; 7  10:529 5  17:049

6  5:491 6 6 Z F2 ¼ 6 6 20:876 6 4 20:132 20:385 2

3

7:315

7 7 7 7; 7 7  217:092 5  242:273 0:558 11:004

11:555

9:530

3

3:120 7 7 7 8:127 7 7 7 135:237 5 149:398

38:508

6 137:35127 6 6 Z F0 ¼ 6 6 7:818 6 4 2282:818 1338:391

28:576

ð46Þ

3

0:280 7 7 7 26:794 7 7 7 13:078 5 3:758 ð47Þ

For the control gains, the LMI region considered was defined by

βo ¼ 1. The solution, by solving (41) and (43) simultaneously, interpolated to 4th grade polynomial functions, was found to be 2

 79:0188 6 3:6333 6 kðvÞT ¼ 6 4  22:1225 0:3839

283:6894 10:3396 79:8812 3:1946

 234:4882  1:5758 68:4279 8:6688

165:4273 22:6184 41:8261 10:3002

2 3 3 v4 262:0917 6 3 7 6v 7 7 15:8127 7 76 v2 7 76 7 5 70:3465 6 6 7 4 v 5 3:9232 1

ð48Þ The eigenvalues corresponding to the estimation error matrix (30) are displayed in Fig. 3. The real and imaginary parts of the observer eigenvalues are presented in Fig. 3a, the real part of the eigenvalues as functions of the translational velocity v in Fig. 3b and the time-varying attenuation level ςðjhÞ for Eq. (25) in Fig. 3c. Finally, from Fig. 4, the error convergence of the state and fault estimation can be concluded, as a result of the estimation error eigenvalues allocation. Fig. 5 shows the controller gains as functions of the translational velocity v. The asterisks correspond to the discrete values computed by solving Eqs. (41) and (43). The continuous line, on the other hand, was obtained by using Eq. (48). The closed-loop eigenvalues for AðvÞ þ BkðvÞ in Eq. (34) are presented in Figs. 6 and 7, concluding with the system stabilization 8 v A Φ despite the variation of the translational velocity v.

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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6.2. Non-faulty case closed-loop simulation Once the observer and controller gains have been computed, the simulation of the system (16) by using the control law (33) and the estimated x^ from (17) can be carried out. Gaussian noises z1 with mean value 0 and variance 3  10  3 and z2 with mean 0 and

Fig. 4. Observation error eigenvalues as functions of the translational velocity v.

7

variance 2  10  3 and D ¼ 0:2I for (16) were considered. The initial conditions applied were x ¼ ½0 0:1  0:04 0T and x^ ¼ ½0 0:075  0:02 0T . Fig. 8 shows the response of the controlled system with a translational velocity from 1.7 to 1.35 m/s (see Fig. 8d). Figs. 8a and b present the real and the estimated outputs yδ and yϕ_ , showing the effectiveness of the proposed design. It can be seen that it is able to compute the state estimation with less noise than the presented at the system outputs. The control law, the estimated roll angle ϕ and the velocity of the handlebar δ_ can be seen in Figs. 8c, e and f, respectively. Let us point out that the LPV model of the riderless bicycle is valid for small variations around ϕ and δ, which should be consistent with the closed-loop state magnitude. By taking the previously mentioned into account, the measurement noise at the system outputs presents a magnitude of approximately 7 0:04 rad or 7 2:291. This represents a considerable measurement noise with respect to the LPV model characteristics. Despite this fact, the proposed design provides the state estimation in order to build the control law with less noise, which in turn will contribute to the smooth operation of the actuator of the system.

Fig. 5. Discrete and interpolated controller gains: (a) kϕ ; (b) kδ ; (c) kϕ_ ; (d) kδ_ .

Fig. 6. (a) Closed-loop eigenvalues; (b) real part of closed-loop eigenvalues as functions of the translational velocity v.

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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6.3. Closed-loop simulation with BIAS actuator fault This section shows the performance of the closed-loop system by considering a BIAS [23] actuator fault in the system (16). From now on, the numerical value for Bf has been taken as the B matrix in Eq. (8), since it is related to the system input. For this

Fig. 7. Closed-loop eigenvalues as functions of the translational velocity v.

simulation, similar initial conditions were considered as in the previous case. Figs. 9 and 10 present the results of a BIAS faulty case fa ¼ 0.25 at t ¼ 2:5 s. The real and reconstructed outputs for the system are depicted in Fig. 9a, whence one can see the fault occurrence and its implication in the closed-loop system. Fig. 9a presents a deviation from the equilibrium point in yδ (upright position) caused by the applied BIAS fault magnitude; where the referred output finishes at 1 rad equivalent to a 5.73°. The translational velocity variation, finally, is shown in Fig. 9d. Finally, let us point out that v_ ¼ j 1:701:35 7 j ¼ 0:05 (see Fig. 9d), according to the considered value used to compute the control and observer gains. The control law T δ , in turn, is depicted in Fig. 9c. The estimated and the real fault are presented in Fig. 10. It is possible to see that the fault estimation converges correctly in approximately t¼0.5 s. Under the consideration that the translational velocity is changing its value continuously, the actuator fault estimation error ef ¼ f a  f^ a stay close to 0, once the fault estimation has been estimated correctly. The state estimation error, meanwhile, is depicted in Fig. 10c. The observer stability is maintained despite on the fault occurrence.

Fig. 8. (a) Real and estimated output yδ ; (b) real and estimated output yϕ_ ; (c) control law T δ ; (d) translational velocity v; (e) roll angle estimated ϕ; (f) velocity of the _ handlebar estimated δ.

Fig. 9. (a) Real and estimated output yδ ; (b) real and estimated output yϕ_ ; (c) control law Tδ; (d) translational velocity variation.

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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Fig. 10. (a) Real and actuator fault estimation; (b) fault estimation error ef; (c) state estimation error ee.

Fig. 11. (a) Real and estimated output yδ ; (b) real and estimated output yϕ_ ; (c) control law Tδ; (d) translational velocity variation.

Fig. 12. (a) Real and actuator fault estimation; (b) fault estimation error ef; (c) state estimation error ee.

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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Fig. 13. (a) Real and estimated output yδ ; (b) real and estimated output yϕ_ ; (c) control law Tδ; (d) translational velocity variation.

Fig. 14. (a) Real and actuator fault estimation; (b) fault estimation error ef.

6.4. Closed-loop simulation with a loss of effectiveness in the actuator

estimation. The actuator fault estimation ef is depicted in Fig. 12b. Finally, the state estimation error can be seen in Fig. 12c.

In this simulation, a loss of effectiveness [23] in the actuator with magnitude 45%, i.e. f a ¼  0:45 in t ¼ 4:5 was applied. A translational velocity variation from 1 to 1.6 m/s has been adopted, by considering the same initial conditions in x and x^ mentioned in Section 6.2. For this test, an increasing translational velocity was considered, in order to evaluate the proposed stabilization system design. The results obtained in this case lead to interpretations analog to those perceived in the previous faulty case. It is possible to see, from Fig. 11a, that the fault condition generates a strong inference in the closed-loop system presented in yδ output, due to the fault magnitude. In Fig. 11c the control law, after the actuator fault, presents a variation from the equilibrium, caused by the fault magnitude. Fig. 12a shows the applied actuator fault and its correctly

6.5. Closed-loop simulation with time-varying actuator fault As a final test, a time-varying actuator fault was considered by using f a ¼ 0:25 sin 54t for 2:5 o t o 8 s. in Eq. (16). For this test, with respect to the previous, Gaussian noises z1 with mean value 0 and variance 6  10  3 and z2 with mean 0 and variance 4  10  3 for (16) were considered. The most difficult task of the proposed design consists in guaranteeing the convergence of ee despite on the fault magnitude. With the previously mentioned in mind, the proposed methodology must be evaluated under a time-varying actuator fault, besides its operation by considering an increment on the noise presented at the system outputs. Figs. 13a and b show the real and estimated outputs yδ and yϕ_ , respectively. The control law can be seen in Fig. 13c and, finally, Fig. 13d presents the translational velocity. The real and estimated faults, on the other

Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i

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hand, are depicted in Fig. 14a, while the actuator fault estimation error ef ¼ f a  f^a is shown in Fig. 14b. As in previous tests, the state estimation error is presented. From Fig. 14b, a small variation from 0 for the actuator estimation error ef can be seen. This is because there is no an additional condition imposed in the observer design to cope with this particular type of faults. In other words, the classical linear system theory stipulates that there is an error in the case of tracking references using integer actions. Despite this fact, the proposed design is able to maintain its stability in spite of any actuator fault type.

11

corroborated through simulations considering BIAS, loss of effectiveness and time-varying faults.

Acknowledgements To the Consejo Nacional de Ciencia y Tecnología (CONACYT) and the Instituto Potosino de Investigación Científica y Tecnológica (IPICYT), in Mexico.

References 7. Discussion As mentioned in previous sections, the proposed methodology can be used in a fault tolerant system scheme. In order to design the FTC system, the observability and controllability conditions for the failed system should be accomplished. The previous corresponds to an important issue to be taken into account before the FTC design. In order to highlight the importance of these requirements, let us point out that the information about the fault (due to the fact that the observer is providing its estimation at all time) will be used to build the mathematical representation of the failed system. This faulty representation is the reference for the FTC design. In this direction, since the actuator fault is related to the input matrix B, it may change the controllability condition for the faulty system. In other words, the control law will not generate the system convergence in the case of the lack of controllability condition and, consequently, the FTC scheme cannot be designed or even considered. As previously commented, the actuator fault is directly related to the controllability condition for the faulty system, however, it does not change the observability condition. As a result, the estimation error for the observer will be bounded regarding the fault occurrence in the system. This can be seen from Proposition I, specially by the ςðvÞ factor. Thus, in a faulty condition for the system, the proposed design will provide the estimation of the fault and state, which, in turn, as long as the faulty system remains controllable, will generate the possibility of the FTC design.

8. Conclusions The results presented in this paper correspond to the design of an observer which is simultaneously in charge on the actuator fault and state estimation, applied to a riderless bicycle polynomial LPV model. Taking the translational velocity of the bicycle as a time-varying parameter along the system trajectories, the observer is able to estimate the state and the actuator fault despite a BIAS, loss of effectiveness or time-varying fault occurrence. As an additional contribution, the estimation comes with less noise than the actual system output measurements and, as a result from the test carried out, the correct performance for the proposed observer can be concluded. It is important to point out that the proposed design gives rise to a versatile observer, allowing to generate a stabilizing control signal with a less noisy estimated state (with respect to noise level acting on the system outputs). Moreover, the actuator fault estimation can be used for FTC purposes, by using it within an active fault tolerant control scheme. For example, a fault tolerant control based on the on-line reconfiguration of the control gains, or a virtual sensors/actuator based FTC. The proposed design can be used as fault diagnosis approach, in terms of a fault occurrence indicator, isolation and identification, due to the fact that it is continuously providing the fault estimation. Finally, the efficiency and usefulness of the observer have been

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Please cite this article as: Brizuela Mendoza JA, et al. State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.026i