A Virtual Actuator and Sensor Approach for Fault ...

A Virtual Actuator and Sensor Approach for Fault Tolerant Control of LPV Systems Damiano Rotondoa,∗, Fatiha Nejjaria , Vicenç Puiga a Advanced

Control Systems (SAC) Research Group, Department of Automatic Control (ESAII), Technical University of Catalonia (UPC), Rambla de Sant Nebridi 10, 08222 - Terrassa (Spain)

Abstract In this paper, a Fault Tolerant Control (FTC) strategy using virtual actuators and sensors for Linear Parameter Varying (LPV) systems is proposed. The main idea of this FTC method, initially developed for LTI systems, is to reconfigure the control loop such that the nominal controller could still be used without need of retuning it. The plant with the faulty actuator/sensor is modified adding the virtual actuator/sensor block that masks the actuator/sensor fault. The suggested technique is an active FTC strategy that reconfigures the virtual actuator/sensor on-line taking into account faults and operating point changes. The stability of the reconfigured control loop is guaranteed if the faulty plant is stabilizable/detectable. The LPV virtual actuator/sensor is designed using polytopic LPV techniques and Linear Matrix Inequalities (LMIs). A two-tank system simulator is used to assess the performance of the proposed method. In particular, it is shown that the application of the proposed technique results in an improvement, in terms of performance, with respect to the LTI counterpart. Keywords: Fault Tolerant Control, Linear Parameter Varying, Virtual Actuator, Virtual Sensor, Linear Matrix Inequality

1. Introduction In control systems, actuators and sensors may be affected by partial or complete loss of effectiveness faults, as well as offsets and stucks during operation. These faults may degrade the system performance with respect to the nominal unfaulty one or, in the worst-case, end up with loss of stability. Fault Tolerant Control (FTC) allows to maintain the system performance close to the desirable one and preserve stability conditions in the presence of faults [1]. The accommodation capability of a control system depends on many factors such as the severity of the failure, the robustness of the nominal system and the presence of mechanisms that introduce redundancy in actuators and/or sensors. From the point of view of control strategies, the literature considers ∗ Corresponding

author Email addresses: [email protected] (Damiano Rotondo), [email protected] (Fatiha Nejjari), [email protected] (Vicenç Puig)

two main groups of techniques: the active and the passive (see [2] for a review). The passive FTC techniques are control laws that take into account the fault appearance as a system perturbation. Thus, within certain margins, the control law has inherent fault tolerant capabilities, allowing the system to cope with the fault presence. On the other hand, the active FTC techniques consist in adapting the control law using the information given by the Fault Detection and Isolation (FDI) block [1]. With this information, some automatic adjustments in the control loop are done after the fault appearance trying to satisfy the control objectives with minimum performance degradation. Most of the FTC methods have been proposed for Linear Time Invariant (LTI) systems. However, in recent years, the interest in Linear Parameter Varying (LPV) systems has grown due to the possibility of using such a methodology to deal with non-linear systems. The LPV techniques belong to the wider class of gain-scheduling techniques, that have consolidated as an efficient answer to analysis and synthesis problems for nonlinear systems. LPV systems were introduced by Shamma [3] to distinguish such systems from LTI and Linear Time Varying (LTV) [4]. Since then, the LPV paradigm has become a standard formalism in systems and control, for analysis, controller synthesis and even system identification. This class of system is important because gainscheduling of nonlinear systems can be performed according to the LPV paradigm, where the non-linearity is embedded in the varying parameters that depend on some endogenous signals, e.g. some system states (in this case, the system is referred to as quasi-LPV, to make a further distinction with respect to pure LPV systems, where the varying parameters only depend on exogenous signals). The LPV theory is mainly used for designing controllers for non-faulty systems, but recently it has also been used for faulty systems. For example, [5] presented a method for detecting and identifying actuator faults in nonlinear systems using LPV models and a Kalman filter that estimates the augmented states, which are directly related to faults. [6] developed an FTC strategy for polytopic LPV systems using a Static Output Feedback (SOF) in the presence of multiple actuator faults. In [7], an integrated FDI/FTC strategy based on Unknown Input Observers (UIOs) and Virtual Actuators (VAs) for LPV systems is presented. On the other hand, [8] proposed a combined use of LPV fault estimation and LPV fault compensation to meet the performance requirements in the presence of faults. In recent years, the fault-hiding paradigm has been proposed as an active strategy to obtain fault tolerance [9]. In this paradigm, the Controller Reconfiguration (CR) unit reconfigures the faulty plant instead of the controller/observer. The nominal controller is kept in the loop by inserting a reconfiguration block between the faulty plant and the nominal controller/observer when a fault occurs. The reconfiguration block is chosen so as to hide the fault from the controller point of view, allowing it to see the same plant as before the fault. In the case of sensor faults, the reconfiguration block is named virtual sensor, while in the case of actuator faults, it is named virtual actuator (see Figure 1). Initially developed for LTI systems [10], virtual sensors and actuators have been successfully extended to piecewise affine [11], Hammerstein-Wiener [12] and TakagiSugeno [13] systems. The stability of the reconfigured control loop is guaranteed if the plant is detectable/stabilizable [14]. In this paper, the virtual actuator/sensor technique is extended to LPV systems. Moreover, the two techniques are merged in a formulation that allows handling partial and complete losses of effectiveness, as well as additive faults, in one or more actuators 2

Figure 1: Virtual Actuator/Sensor FTC Scheme

and sensors at the same time. The LPV virtual actuator/sensor is designed using polytopic LPV techniques and pole placement via Linear Matrix Inequality (LMI) regions [15]. This approach requires to approximate the LPV system in a polytopic way guaranteeing the desired specifications at the expense of introducing some conservatism. As a benefit, controller design can be reduced to solve a convex optimization problem, which can be solved using solvers that are very efficient nowadays. An application of the proposed method to a two-tank system simulator is shown. The results, presented both in a graphical form and in a quantitative way, show that the FTC strategy is able to recover the original performance under all combination of faults. The main and new contributions of this work are the following: • the extension of the virtual actuator/sensor technique to LPV systems; • the merging of virtual actuators and virtual sensors in a unique scheme, allowing to cope with multiple faults affecting actuators and sensors at the same time; • the validity of the principle of separation, such that the controller, the observer, the virtual sensor and the virtual actuator can be designed separately; • the combined use of fault estimation, achieved writing the faulty equations in a regressor form in such a way that any parameter estimation technique can be used, and fault compensation for FTC within an LPV framework. The paper is organized as follows: in Section 2, the LPV faulty plant is presented, and the faults are defined. Section 3 presents the mathematical formulation of the LPV virtual actuator and sensor. In Section 4, the principle of separation for the overall control loop, including the nominal controller and the state observer, is stated, and the design using LMI pole placement is discussed. Finally, Section 5 describes the application example, a two-tank system, and some conclusions are given in Section 6. 2. Actuator and Sensor Fault Definition for LPV Systems Let us consider an LPV system in state-space form including actuator and sensor faults as follows: x f (k + 1) = A (ϑk ) x f (k) + B f (ϑk , φk ) u f (k) + fu (k) (1) 3

y f (k) = C f (ϑk , γk ) x f (k) + fy (k) (k) ∈ Rnx

(2)

(k) ∈ Rnu

where x f represents the state vector, u f denotes the control inputs and y f (k) ∈ Rny are the sensor outputs. A(ϑk ) ∈ Rn x ×nx , B f (ϑk , φk ) ∈ Rnx ×nu and C f (ϑk , γk ) ∈ Rny ×nx are time-varying matrices scheduled by ϑk := ϑ(k) ∈ Rnϑ , the vector of timevarying parameters that changes with the operating point. This vector is scheduled by some measured system variables pk := p(k) that can be estimated using some known function ϑk = f (pk ), named scheduling function. It is assumed that ϑk is available in real time and varies within a polytope Θ. fu (k) ∈ Rnu denotes the additive actuator faults, while φk := φ(k) ∈ Rnu denotes the multiplicative actuator faults, embedded in the input matrix B f (ϑk , φk ), as follows: B f (ϑk , φk ) = B (ϑk ) diag φ1 (k), . . . , φnu (k) , 0 ≤ φi (k) ≤ 1 (3) where B (ϑk ) denotes the faultless input matrix. φi represents the effectiveness of the ith actuator, such that the extreme values φi = 0 and φi = 1 represent a total failure of the ith actuator and the healthy ith actuator, respectively. In the same way, fy (k) ∈ Rny denotes the additive sensor faults, while γk := γ(k) ∈ Rny denotes the multiplicative sensor faults, embedded in the output matrix C f (ϑk , γk ) as follows: C f (ϑk , γk ) = diag γ1 (k), . . . , γny (k) C (ϑk ) 0 ≤ γi (k) ≤ 1 (4) where C (ϑk ) denotes the faultless output matrix. γi represents the effectiveness of the ith output sensor, such that the extreme values γi = 0 and γi = 1 represent a total failure of the ith sensor and the healthy ith sensor, respectively. 3. LPV Virtual Actuator/Sensor In this paper, the virtual actuator and sensor concepts [1] are extended to nonlinear systems that can be approximated by an LPV model. The main idea of this FTC method is to reconfigure the faulty plant such that the nominal controller could still be used without need of retuning. The plant with the faulty actuator/sensor is modified adding the virtual actuator/sensor block that masks the fault and allows the controller to see the same plant as before the fault. The overall scheme also includes an LPV state observer and an LPV nominal controller. When the actuator fault appears, the LPV virtual actuator reconstructs the system input vector u f (k) from the nominal output of the controller uc (k), taking into account the fault occurrence. Similarly, when the sensor fault appears, the LPV virtual sensor reconstructs the system output vector yc (k) from the faulty output vector y f (k). The faulty plant and the LPV virtual actuator/sensor are called the reconfigured LPV plant, which is connected to the nominal LPV controller. If the reconfigured LPV plant behaves like the nominal plant, the loop consisting of the reconfigured plant and the LPV controller behaves like the nominal closed-loop system. 3.1. LPV Virtual Actuator The structure of the LPV virtual actuator depends on whether or not the following rank condition is satisfied: rank B f (ϑk , φk ) = rank (B (ϑk )) , 0 ∀ϑk ∈ Θ (5) 4

The satisfaction of the rank condition (5) means that the presence of faults can be tolerated through a simple redistribution of the control inputs. This case happens either when all the faults are partial or when the effect of the lost actuators on the system can be expressed as a linear combination of the remaining actuators. In this case, the reconfiguration structure can be expressed as: u f (k) = Nva (ϑk , φk )uc (k) − fu (k)

(6)

where uc (k) is the controller output and the matrix Nva (ϑk , φk ) is given by: Nva (ϑk , φk ) = B f (ϑk , φk )† B(ϑk )

(7)

where B f (ϑk , φk )† is the pseudo-inverse of B f (ϑk , φk ). On the other hand, in cases where (5) is not satisfied, a redistribution of the control inputs is not enough, and the fault tolerance is achieved by adding, and properly using, a dynamical part to the virtual actuator, through the introduction of the virtual actuator state xva (k) and the corresponding state equation. Each of these cases is described by a particular value of the matrix B∗ (ϑk ), obtained as: B∗ (ϑk ) = B f (ϑk , φk ) Nva (ϑk , φk )

(8)

Notice that the matrix B∗ (ϑk ) does not depend on φk because the matrix Nva (ϑk , φk ) eliminates the effects of actuator partial faults1 . In such cases, the reconfiguration structure is expressed by: u f (k) = Nva (ϑk , φk ) (uc (k) − Mva (ϑk )xva (k)) − fu (k)

(9)

where Mva (ϑk )∈Rnu ×nx is the LPV virtual actuator gain matrix while the virtual actuator state xva is calculated as: xva (k+1) = A(ϑk ) + B∗ (ϑk )Mva (ϑk ) xva (k) + B(ϑk ) − B∗ (ϑk ) uc (k)

(10)

Remark 1. By comparing (6) with (9) and (10), it can be seen that the static version of the virtual actuator is a particular case of the dynamic version. In fact, in cases where (5) holds, B∗ (ϑk ) calculated as in (8) would be equal to B (ϑk ), such that uc (k) would have no effect on the virtual actuator state (10). Hence, by considering that xva (0) = 0, it would follow that xva (k) = 0 ∀k and (9) simplifies to (6). 3.2. Actuator Fault Estimation In order to apply the proposed strategy, a fault estimation is needed. In [17], a procedure for estimating the multiplicative and additive faults in the actuators has been presented. The fault estimation is formulated as a parameter estimation problem in such a way that any parameter estimation algorithm, such as least squares, could be used. In general, least squares algorithms can be formulated either in block or in recursive 1A

mathematical proof of this fact relies on the properties of the pseudo-inverse [16].

5

on-line form [18]. Once the equation is put in regressor form, the recursive formulation [19] and the block formulation [20] are interchangeable. The procedure presented in [17] started from the hypothesis that it was possible to find a state of the system that is directly influenced by the faulty actuator. This strong assumption was motivated by the fact that the procedure described in [17] was an ad hoc procedure, developed for the considered case study. In this paper, the procedure is extended to deal with systems that have multiple inputs on a state. Let us consider the state equation subject to faulty actuators, as in (1), that can be rewritten as follows: xi (k + 1) =

nx X

ai j (ϑk ) x j (k) +

j=1

nu X

φl (k)bil (ϑk ) (ul (k) + ful (k)) i = 1, . . . , n x

(11)

l=1

By taking into account the linear combination of all the state equations in (11) using coefficients κi , the following is obtained: nx X

κi xi (k + 1) =

nx nx X X

κi ai j (ϑk ) x j (k)+

κi φl (k)bil (ϑk ) (ul (k) + ful (k))

(12)

i=1 l=1

i=1 j=1

i=1

nu nx X X

that can be brought into the regression form: za (k) = ΦTa (k)µa (k)

(13)

by considering: za (k) =

nx X

κi xi (k) −

"

nx P

κi ai j (ϑk−1 ) x j (k − 1)

κi bi1 (ϑk−1 )u1 (k − 1)

...

i=1

nx P

κi bi j (ϑk−1 )u j (k − 1) #T nx P κi binu (ϑk−1 ) i=1

nx P

κi bi j (ϑk−1 ) . . .

h

(15)

i=1

i=1

µa (k) =

(14)

i=1 j=1

i=1

Φa (k) =

nx nx X X

φ1 (k − 1) ν1 (k − 1) ν j (k − 1) . . .

...

φnu (k − 1)

φ j (k − 1) νnu (k − 1)

ν j (k − 1) = φ j (k − 1) fu j (k − 1)

iT

(16) (17)

Then, by defining the following cost function, under the assumption that µa (k) is constant over the time window N: Va (µa , k, N) =

k i2 1 X 1h za (k) − ΦTa (k)µa N t=k−N+1 2

(18)

the least squares problem consists in finding: µˆ aLS (k) = arg min Va (µa , k, N) 6

(19)

where µˆ aLS (k) is the estimation of µa at time sample k. Once the least squares problem is solved2 , and the estimations φˆ 1 (k), . . ., νˆ nu (k) have been obtained, the additive fault estimations can be obtained as: fû j (k) = νˆ j (k)/φˆ j (k)

(20)

Notice that in cases where the state measurements are not available, the states estimated by a state observer should be used instead and the changes xi → xî and x j → xˆ j are introduced in (14)3 . 3.3. LPV Virtual Sensor Also in this case, the structure of the LPV virtual sensor depends on whether or not the following rank condition is satisfied: rank C f (ϑk , γk ) = rank (C (ϑk )) , 0 ∀ϑk ∈ Θ (21) The satisfaction of the rank condition (21) means that the complete system output can be directly reconstructed from y f (k). This case happens either when all the faults are partial or when the information coming from the lost sensors can be obtained as a linear combination of the outputs of the remaining sensors. In this case, the reconfiguration structure consists of a static block: yc (k) = Nvs (ϑk , γk ) y f (k) + C f (ϑk , γk ) xva (k) − fy (k) (22) where yc (k) is the same, or approximately the same, output as the nominal plant. The matrix Nvs (ϑk , γk ) is given by: Nvs (ϑk , γk ) = C (ϑk )C f (ϑk , γk )†

(23)

where C f (ϑk , γk )† is the pseudo-inverse of C f (ϑk , γk ). On the other hand, in cases where (21) is not satisfied, the information coming from the non-lost sensors is not enough, and the knowledge about the system dynamics should be used too. This is done through the introduction of a dynamical part to the virtual sensor, with a state xvs (k) and the corresponding state equation. Each of these cases is described by a particular value of the matrix C ∗ (ϑk ), obtained as: C ∗ (ϑk ) = Nvs (ϑk , γk )C f (ϑk , γk )

(24)

Notice that the matrix C ∗ (ϑk ) does not depend on γk because the matrix Nvs (ϑk , γk ) eliminates the effect of partial sensor faults1 . In such cases, the reconfiguration structure is expressed by: yc (k) = Nvs (ϑk , γk ) y f (k) + C f (ϑk , γk )xva (k) − fy (k) + C(ϑk ) − C ∗ (ϑk ) xvs (k) (25)

2 It

is assumed that the data are rich enough from the identifiability point of view. that there is a coupling effect between the fault estimation and the state observer due to the FTC correction. Such an effect can be neglected as long as the state observer is slower than the fault estimation. This can be achieved by choosing properly the design parameters of the least squares algorithm. 3 Notice

7

where the virtual sensor state xvs is calculated as: xvs (k + 1) = (A(ϑk ) + Mvs (ϑk )C ∗(ϑk )) xvs (k) + B (ϑk ) uc (k) −Mvs (ϑk )Nvs (ϑk , γk ) y f (k) + C f (ϑk , γk )xva (k) − fy (k)

(26)

being Mvs (ϑk ) ∈ Rnx ×ny the LPV virtual sensor gain matrix. Remark 2. By comparing (22) with (25) and (26), it can be seen that the static virtual sensor is a particular case of the dynamic virtual sensor. In fact, in cases where (21) holds, C ∗ (ϑk ) calculated as in (24) would be equal to C(ϑk ), such that the virtual sensor state xvs (k) would have no effect on yc (k) in (25). 3.4. Sensor Fault Estimation In order to apply the proposed strategy, a fault estimation is needed. In [21], a procedure for estimating the multiplicative and additive faults has been presented in the sensor fault case. As in the case of actuator fault estimation, the sensor fault estimation is formulated as a parameter estimation problem in such a way that any parameter estimation algorithms could be used. The LPV faulty output equation (2) takes into account the multiplicative sensor faults in the matrix C f (ϑk , γk ) and the additive sensor faults in the term fy (k) such that for the ith sensor: nx X cil (ϑk )xl (k) + fyi (k) (27) y f i (k) = γi (k) l=1

n o where y f i is the output of the sensor when a fault γi , fyi occurs and ci (ϑk ) is the ith row of matrix C(ϑk ). This equation can be brought into the regressor form: ith

z s (k) = ΦTs (k)µ s (k)

(28)

by considering: z s (k) = y f i (k) " Φ s (k) =

nx P

(29) #T

cil (ϑk )xl (k)

1

γi (k)

iT

(30)

l=1

µ s (k) =

h

fyi (k)

(31)

Since the measurements are corrupted by the faults, it is not possible to use them for solving the fault estimation problem, and the states estimated by an observer must be considered3 , through the use of the following cost function, with the assumption that µ s is constant over the time window N: V s (µ s , k, N) =

k i 1 X 1h ˆ Ts (k)µ s 2 z s (k) − Φ N t=k−N+1 2

(32)

ˆ s (k) is defined as: where Φ " ˆ s (k) = Φ

nx P

#T cil (ϑk ) xˆl (k)

l=1

8

1

(33)

being xˆl (k) the estimation of the lth state provided by the observer. Finally, the estimations γˆ i (k) and fˆyi (k), embedded in µˆ LS s (k), are obtained solving the following least squares problem2 : µˆ LS s (k) = arg min V s (µ s , k, N)

(34)

4. FTC Strategy using Virtual Actuators/Sensors 4.1. LPV Controller and Observer The LPV system (1)-(2) is controlled by a state feedback controller with tracking reference input as proposed in [22]. The feedback control law can be expressed as follows: uc (k) = ur (k) + K(ϑk )( xˆ f (k) − xr (k)) (35) where the state reference xr (k) and the feedforward control action ur (k) correspond to an equilibrium point for the reference r(k). The matrix K(ϑk ) ∈ Rnu ×nx is the LPV controller gain. The estimated state xˆ f (k) is provided by the following LPV state observer: xˆ f (k+1) = A(ϑk ) xˆ f (k) + B(ϑk )uc (k) + L(ϑk ) C(ϑk ) xˆ f (k) − yc (k) (36) where L(ϑk ) ∈ Rnx ×ny is the LPV state observer gain matrix. 4.2. Design using LMI Pole Placement An LMI approach for the design by imposing pole placement constraints is described in [15]. The main motivation for seeking pole clustering in specific regions of the complex plane is that, by constraining the eigenvalues to lie in a prescribed region, a satisfactory transient response can be ensured. A subset D of the complex plane is called an LMI region if there exist a symmetric matrix α = [αkl ] ∈ Rm×m and a matrix β = βkl ∈ Rm×m such that: D = {z ∈ C : fD (z) < 0}

(37)

fD (z) := α + zβ + z¯βT = αkl + βkl z + βlk z¯ 16k,l6m

(38)

with: Then, an LTI system is said to be D-stable if it is stable and if all its poles lie in D [15, 23]. Following [24] and with a little abuse of language, the poles of an LPV system are defined as the set of all the poles of the LTI systems obtained by freezing ϑ(k) to all its possible values ϑ ∈ Θ. It has been reported that, despite the idea of poles, as introduced, does not have a strict mathematical interpretation in the LPV case, it has a strict connection with the dynamical behavior of the system, justifying, from the engineering point of view, the abuse of language. In particular, in this paper, the quadratic D-stability for LPV systems is considered, as follows:

9

Definition 1. An LPV system x(k + 1) = A(ϑk )x(k) is quadratically D-stable, with D an LMI region defined as in (37)-(38), if there exists a symmetric positive definite matrix X = X T > 0 such that ∀ϑk ∈ Θ: !   −X A(ϑk )X    0 such that εZ − W is positive definite. In fact, Z has some minimum singular value σZ such that σZ > 0, and W has some maximum singular value σW . Also, for any non-zero vector v: vT Zv ≥ kvk2 σZ vT Wv ≤ kvk2 σW So vT (εZ − W) v ≥ kvk2 (εσZ − σW ) and kvk2 (εσZ − σW ) > 0 whenever εσZ > σW . Hence, from the definition of positive definite matrix results that εZ − W is positive definite. This completes the proof. A consequence of Theorem 2 is that, under the assumptions that: (a) the LPV controller K(ϑk ) is quadratically DK -stable; (b) the LPV observer L(ϑk ) is quadratically DL -stable; (c) the LPV virtual actuator Mva (ϑk ) is quadratically Dva -stable; and (d) the LPV virtual sensor Mvs (ϑk ) is quadratically Dvs -stable, being DK , DL , Dva and Dvs LMI regions, the overall system (47) is quadratically stable. Hence, the controller, the observer, the virtual actuator and the virtual sensor can be designed separately. Remark 4. The obtained results are valid, from a theoretical perspective, as long as the scheduling variable is perfectly measured and not affected by the faults. In cases where this assumption were not true, some additional steps should be performed in order to guarantee the overall system stability and the convergence of the estimation errors to zero. It is worth highlighting that some recent research has dealt with such an issue within the context of state estimation and observer-based control of LPV/TakagiSugeno systems [28, 29, 30, 31]. The incorporation of such theoretical results in the proposed FTC scheme will be addressed in future research. 15

Figure 3: Two-tank system

Table 1: Parameters of the two-tank system

Parameter A c12 c2

Value and Unit 1.54 · 10−2 m.2 6 · 10−4 m5/2 s . 13 · 10−4 m5/2 s

Description Area of both tanks Flow constant of the interconnecting pipe Flow constant of the outlet pipe

5. Application Example: Two-Tank System 5.1. Description of the Two-Tank System The two-tank system (see Figure 3) consists of two liquid tanks that can be filled with two identical, independent pumps that deliver the liquid flows Q1 and Q2 . The tanks are interconnected to each other through the pipe Q12 , while the outflow from the system is located at the right tank and provides a flow QN2 to the consumer. The liquid levels h1 and h2 are used as state variables in the model. The area of the cylindric tanks is denoted by A and the flow constants of the interconnecting pipe and the outflow pipe by c12 and c2 , respectively (see Table 1 for the parameters values used in this example). The non-linear model of the two-tank system is obtained from the mass balance and the Torricelli’s law, resulting in a hybrid model because it includes discontinuous functions. Under the assumption that the system is operating with h1 (t) > h2 (t), the model simplifies to: dh1 1 = (Q1 (t) − Q12 (t)) (56) dt A dh2 1 = (Q2 (t) + Q12 (t) − QN2 (t)) (57) dt A p Q12 (t) = c12 h1 (t) − h2 (t) (58) p QN2 (t) = c2 h2 (t) (59) that can be reshaped into the following quasi-LPV form: "

dh1 dt dh2 dt

#

" =

−ϑ1 (p(t)) 0 ϑ1 (p(t)) −ϑ2 (p(t))

#"

16

h1 (t) h2 (t)

# " b11 + 0

0 b22

#"

u1 (t) u2 (t)

# (60)

where u1 (t) = Q1 (t) is the first pump flow and u2 (t) = Q2 (t) is the second pump flow, while ϑ(p(t)) = [ ϑ1 (p(t)) ϑ2 (p(t)) ]T is the vector of varying parameters scheduled by p(t) = [ h1 (t) h2 (t) ]T . More precisely: √ Q12 (t) c12 h1 (t) − h2 (t) = (61) ϑ1 (p(t)) = Ah1 (t) A h1 (t) √ QN2 (t) c2 h2 (t) ϑ2 (p(t)) = = (62) Ah2 (t) A h2 (t) 1 b11 = b22 = (63) A A discrete-time quasi-LPV model can be obtained using Euler approximation with a sample time T s to allow a digital implementation of the overall scheme: "

h1 (k + 1) h2 (k + 1)

#

" =

1 − ϑ1 (p(k)) T s ϑ1 (p(k)) T s

0 1 − ϑ2 (p(k)) T s

#"

h1 (k) h2 (k)

# " b11 T s + 0

0 b22 T s

#"

# u1 (k) u2 (k) (64)

A polytopic model is obtained considering that the system operates with h1 ∈ [0.6 m, 1.8 m] and h2 ∈ [0.1 m, 0.3 m] and T s = 0.1 s, resulting in the following four vertex matrices: " # " # 0.9954 0 0.9973 0 A1 = A2 = 0.0046 0.9725 0.0027 0.9725 " A3 =

0.9954 0.0046

0 0.9841

#

" A4 =

0.9973 0.0027

0 0.9841

#

The nominal input matrix is given by the following: " # 6.4935 0 B= 0 6.4935

(65)

In the following, it is assumed that Q12 , QN2 , h1 and h2 are measured with continuous valued sensors. The measurements of the liquid levels h1 (t) and h2 (t) are assumed to be noise free and not affected by faults, and are used to schedule the quasi-LPV system. On the other hand, the measurements of the flows Q12 (t) and QN2 (t) are noisy, can be affected by faults, and are used to close the loop of the control system. Hence, the nominal output matrix is given by: " # ϑ1 (p(k)) 0 C (ϑ(p(k))) = (66) 0 ϑ2 (p(k)) such that its polytopic representation is given by: " # " 0.0459 0 0.0265 C1 = C2 = 0 0.2755 0 " # " 0.0459 0 0.0265 C3 = C4 = 0 0.1591 0 17

# 0 0.2755 # 0 0.1591

(67)

5.2. FTC Loop Design In this application example, the disk of radius r and center (−q, 0) will be used as LMI region. Given: • the nominal controller gain K(ϑ(k)), designed to be DK -stable with DK disk of radius rK and center (−qK , 0), • and the nominal state observer gain L(ϑ(k)), designed to be DL -stable with DL disk of radius rL and center (−qL , 0), the FTC design consists of: 1 (ϑ(k)), for the case of complete loss • designing the first virtual actuator gain Mva ∗ of the first actuator (fault matrix B1 ), to be D1va -stable with D1va disk of radius 1 and center (−q1 , 0), rva va 2 (ϑ(k)), for the case of complete • designing the second virtual actuator gain Mva ∗ loss of the second actuator (fault matrix B2 ), to be D2va -stable with D2va disk of 2 and center (−q2 , 0), radius rva va 1 (ϑ(k)), for the case of complete loss • designing the first virtual sensor gain Mvs ∗ of the first sensor (fault matrix C1 (ϑ (p(k)))), to be D1vs -stable with D1vs disk of 1 and center (−q1 , 0), radius rvs vs 2 (ϑ(k)), for the case of complete loss • designing the second virtual sensor gain Mvs ∗ of the second sensor (fault matrix C2 (ϑ (p(k)))), to be D2vs -stable with D2vs disk 2 and center (−q2 , 0), of radius rvs vs

where the faulty matrices B∗1 , B∗2 , C1∗ (ϑ (p(k))) and C2∗ (ϑ (p(k))) are: " B∗1 = " C1∗ (ϑ (p(k)))

=

0 0

0 0

0 6.4935

#

0 ϑ2 (p(k))

#

" B∗2 =

6.4935 0

0 0 "

C2∗ (ϑ (p(k)))

=

#

ϑ1 (p(k)) 0 0 0

#

and the LMI region specifications are chosen as: rK = 0.01 rL = 0.01 qK = −0.985 qL = −0.96

1 = r 2 = 0.01 rva va 1 qva = q2va = −0.985

1 = r 2 = 0.02 rvs vs 1 qvs = q2vs = −0.97

A solution to the system of LMIs (40)-(41) applied to this case has been found by using the YALMIP toolbox [32]. Figures 4-9 show the pole placement results for the controller, the observer, the first virtual actuator, the second virtual actuator, the first virtual sensor and the second virtual sensor, respectively. The results obtained with the proposed LPV technique are compared with the ones given by a design performed on the LTI system obtained from (64) at the frozen values corresponding to h∗1 = 1.2 m and h∗2 = 0.2 m. 18

Controller 0.02 LMI region LTI, at the design point LTI, at the vertex points LPV, at the vertex points

0.015 0.01

Imag(z)

0.005 0 −0.005 −0.01 −0.015 −0.02 0.96

0.97

0.98 Real(z)

0.99

1

Figure 4: Pole placement results with comparison between LTI and LPV techniques: Controller.

Observer 0.02 LMI region LTI, at the design point LTI, at the vertex points LPV, at the vertex points

0.015 0.01

Imag(z)

0.005 0 −0.005 −0.01 −0.015 −0.02 0.94

0.95

0.96 Real(z)

0.97

0.98

Figure 5: Pole placement results with comparison between LTI and LPV techniques: Observer.

19

Virtual Actuator 1 0.02 LMI region LTI, at the design point LTI, at the vertex points LPV, at the vertex points

0.015 0.01

Imag(z)

0.005 0 −0.005 −0.01 −0.015 −0.02 0.96

0.97

0.98 Real(z)

0.99

1

Figure 6: Pole placement results with comparison between LTI and LPV techniques: First virtual actuator.

Virtual Actuator 2 0.02 LMI region LTI, at the design point LTI, at the vertex points LPV, at the vertex points

0.015 0.01

Imag(z)

0.005 0 −0.005 −0.01 −0.015 −0.02 0.96

0.97

0.98 Real(z)

0.99

1

Figure 7: Pole placement results with comparison between LTI and LPV techniques: Second virtual actuator.

20

Virtual Sensor 1 0.03 LMI region LTI, at the design point LTI, at the vertex points LPV, at the vertex points

0.02

Imag(z)

0.01

0

−0.01

−0.02

−0.03 0.94

0.95

0.96

0.97 Real(z)

0.98

0.99

1

Figure 8: Pole placement results with comparison between LTI and LPV technique: First virtual sensor.

Virtual Sensor 2 0.03 LMI region LTI, at the design point LTI, at the vertex points LPV, at the vertex points

0.02

Imag(z)

0.01

0

−0.01

−0.02

−0.03 0.94

0.95

0.96

0.97 Real(z)

0.98

0.99

1

Figure 9: Pole placement results with comparison between LTI and LPV technique: Second virtual sensor.

21

Table 2: Computation of the measure of performance J

Controller Observer Virtual actuator 1 Virtual actuator 2 Virtual sensor 1 Virtual sensor 2

LTI 0 0.0263 0.0055 0.0044 4.3186 · 10−4 0

LPV 0 0 0.0055 0 0 0

If the whole operating range is taken into account analyzing the poles at the vertex points given by the matrices A1 , A2 , A3 and A4 , it can be seen that the LTI technique fails in guaranteeing that the poles are in the desired LMI region in the cases of the observer, the two virtual actuators, and the first virtual sensor. On the other hand, the LPV technique fails in placing the poles in the desired region of the complex plane only in the case of the first virtual actuator. However, since the first virtual actuator is stable, in case of loss of the first virtual actuator, the loop will maintain its stability, even though with a big loss of performance due to the influence of the first virtual actuator poles. The following measure is introduced in order to quantify the performance of the LPV technique with respect to the LTI technique: J=

Nσ X

! q 2 2 max 0, (Re(σi ) + q) + Im(σi ) − r

(68)

i=1

where Nσ is the number of poles and Re(σi ) and Im(σi ) indicate the real and the imaginary part of the ith pole. The measure of performance J is an index of how much the poles are far from the desired LMI region. It has been calculated for the controller, the observer, the two virtual actuator and the two virtual sensors, in both the LTI and the LPV cases, and the results have been resumed in Table 2. As expected, the proposed LPV technique shows an improvement of performance with respect to the LTI case. 5.3. Fault scenarios Hereafter, the fault scenarios are described. All the scenarios share the following characteristics: • the length of the experiment is 200s, • the desired flow QN2 to the consumer is defined as follows: ( 6 · 10−4 m3 /s, t ≤ 100s QN2 = 5 · 10−4 m3 /s, t > 100s • at time t = 0 s the system and the state observer initial conditions are the following: h i h i x0 = 1.7m 0.3m xˆ0 = 1.6m 0.25m 22

• even though, according to the theory, the dynamical part of the virtual actuator/sensor (10)/(26) should be activated only in case of complete loss of an actuator/sensor, a sharp hysteresis activation criterion is used to cope with fault estimation delays and inaccuracies. That is, a virtual actuator/sensor is activated when the corresponding fault estimation goes below the lower threshold value fˆ = 0.2 and is deactivated when the corresponding fault estimation returns above the upper threshold value fˆ = 0.3, • the flow sensor outputs are affected by a normally distributed noise with zero mean and standard deviation equal to 5% of the measurement value. The desired flow QN2 can be directly converted to a second tank liquid level reference: !2 q QN2 re f re f (69) c2 h2 = QN2 ⇒ h2 = c2 that leads to: re f

(

h2 =

t ≤ 100s t > 100s

0.2130 m, 0.1479 m,

re f

In order to apply the control law (35), the first tank liquid level reference h1 ff ff must be defined and the feedforward actions u1 and u2 have to be calculated. While re f in the nominal non-faulty case the only restriction is that h1 ∈ [0.6 m, 1.8 m], this is no longer true in case of complete loss of an actuator, as the number of possible equilibrium points of (56)-(57) reduces to one. In this paper, only a complete loss of the second pump will be considered. In this case, the only equilibrium point for a given re f second tank liquid level reference h2 is: 2 + c2 c 2 re f 12 re f h1 = h2 (70) c212 Since this paper does not deal with the problem of online reference/feedforward reconfiguration, the following assumption is made: re f

• the first tank liquid level reference h1 is calculated as (70) both when the second ff pump is available for control and when the first pump is lost. In this way, u2 = 0 and the fault will affect only the state feedback contribution to the total pump flow u2 . This leads to: re f

h1 =

(

t ≤ 100s t > 100s

1.2129 m, 0.8422 m, ff

and the first pump feedforward action results u1 = QN2 . Hereafter, the fault scenarios are defined. Each fault scenario is defined by the octuple [γ1 , γ2 , φ1 , φ2 , fy1 , fy2 , fu1 , fu2 ] where γi is the loss of effectiveness of the ith sensor, φi is the loss of effectiveness of the ith actuator, fyi is the ith sensor offset and fui is the ith actuator offset. 23

Fault Scenario 1: Actuator Faults In this fault scenario, partial, offset and total faults in the actuators are considered as follows:   [1, 1, 1, 1, 0, 0, 0, 0] t ≤ 40s      [1, 1, 0.5, 1, 0, 0, 0, 0] 40s < t ≤ 80s  [γ1 , γ2 , φ1 , φ2 , fy1 , fy2 , fu1 , fu2 ] =    [1, 1, 0.5, 1, 0, 0, 2 · 10−4 , 0] 80s < t ≤ 120s     [1, 1, 0.5, 0, 0, 0, 2 · 10−4 , 0] 120s < t ≤ 200s (71) This means that the system is working in nominal non-faulty conditions until time t = 40s. At time t = 40s, the first actuator effectiveness changes to φ1 = 0.5. At time t = 80s there appears an additive fault in the first pump fu1 = 2 · 10−4 . Finally, at time t = 120s the second pump breaks, that is, φ2 = 0. Fault Scenario 2: Sensor Faults In this fault scenario, partial and total faults in the sensors are considered as follows:   [1, 1, 1, 1, 0, 0, 0, 0] t ≤ 40s      40s < t ≤ 80s  [1, 0.5, 1, 1, 0, 0, 0, 0] [γ1 , γ2 , φ1 , φ2 , fy1 , fy2 , fu1 , fu2 ] =  (72)   [0.7, 0.5, 1, 1, 0, 0, 0, 0] 80s < t ≤ 120s     [0, 0.5, 1, 1, 0, 0, 0, 0] 120s < t ≤ 200s In this case, the system works without faults until time t = 40s. At time t = 40s, the second sensor is affected by a loss of effectiveness γ2 = 0.5. At time t = 80s, the first sensor effectiveness changes to γ1 = 0.7. Finally, at time t = 120s the first sensor is lost completely, that is, γ1 = 0. Fault Scenario 3: Actuator and Sensor Faults This fault scenario includes simultaneously faults in the actuators and sensors as follows:  [1, 1, 1, 1, 0, 0, 0, 0] t ≤ 40s     −4 , 0]   [1, 1, 1, 1, 0, 0, 2 · 10 40s < t ≤ 80s    −4 , 0] [1, 0.5, 1, 1, 0, 0, 2 · 10 80s < t ≤ 120s [γ1 , γ2 , φ1 , φ2 , fy1 , fy2 , fu1 , fu2 ] =     −4 , 0] 120s < t ≤ 160s  [1, 0.5, 1, 0, 0, 0, 2 · 10     [0, 0.5, 1, 0, 0, 0, 2 · 10−4 , 0] 160s < t ≤ 200s (73) The system works without faults until time t = 40s. At time t = 40s, a fault appears in the first actuator, which is affected by an additive fault fu1 = 2 · 10−4 . At time t = 80s, the second sensor experiments a loss of effectiveness γ2 = 0.5. Finally, at time t = 120s there is a complete loss of the second actuator φ2 = 0, followed by a complete loss of the first sensor γ1 = 0 at time t = 160s. 5.4. Simulation Results Fault estimation results, as well as liquid level responses obtained using the proposed method and a simulator of the nonlinear system (56)-(57), are given for each fault scenario. Figure 10 shows the fault estimation results in fault scenario 1. The elements of the sextuple [γ1 , γ2 , φ1 , φ2 , fu1 , fu2 ] are estimated with a delay that depends on the choice 24

of the LS parameters (e.g. the window size in block LS or the forgetting factor in recursive LS)7 . Figure 11 presents the first and second tank liquid level time responses in the fault scenario 1. The fault occurrence drives the system off the desired liquid level if no fault tolerance strategy is used. The proposed FTC approach proves to recover the nominal system performance almost perfectly. The small difference between the nominal response and the one given by the system with FTC is due to the fault estimation error and the effect of the poles introduced by the dynamical part of the virtual actuator.

γ1

1 0.5 0

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0 −4 x 10

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0 −4 x 10

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100 time [s]

120

140

160

180

200

γ2

1 0.5 0

φ1

1 0.5 0

fu1

5 0 −5

φ2

1 0.5 0

fu2

5 0 −5

Figure 10: Fault scenario 1: Fault Estimation

Figure 12 shows the fault estimation results in fault scenario 2. The sextuple 7 Notice that fault estimation results do not include an estimation of f and f as the fault estimation y1 y2 method fails in distinguishing between multiplicative and additive sensor faults. Hence, an assumption about the type of fault to be identified must be done a priori. [33] points out that additive faults can be considered as external signals acting on the system. External signals do not cause any changes in the system dynamics and, more specifically, they are not able to change the stability of the closed-loop system. For this reason, multiplicative sensor fault estimation has been preferred to the additive one.

25

First tank liquid level h1 [m] Second tank liquid level h2 [m]

1.8 reference nominal with FTC without FTC

1.6 1.4 1.2 1 0.8 0.6

0

50

100

150

200

0

50

100 Time [s]

150

200

0.3 0.25 0.2 0.15 0.1

Figure 11: Fault scenario 1: Tank liquid levels

[γ1 , γ2 , φ1 , φ2 , fu1 , fu2 ] is estimated almost perfectly. Figure 13 presents the first and second tank liquid level time responses in the fault scenario 2. The fault occurrence produces system instability if no fault tolerance strategy is used. The proposed FTC approach proves to recover the nominal system stability and performance almost perfectly. The nominal response and the one with FTC differ only because of the fault estimation error and the effect of the poles introduced by the dynamical part of the virtual sensor. Figure 14 shows the fault estimation results in fault scenario 3. The sextuple [γ1 , γ2 , φ1 , φ2 , fu1 , fu2 ] is well estimated, as in the previous cases. Figure 15 presents the first and second tank liquid level time responses in the fault scenario 3. The faults cause a loss of performance of the system and offset appearance between the real liquid level and the reference. Moreover, at time t = 160s the loss of a sensor and the consequent break of the control loop bring the system to the instability. The proposed FTC strategy shows the capacity of recovering the nominal system performance in all cases, avoiding the loss of stability due to the sensor loss, and the system state eventually converges to the desired one. A further comparison of the results is shown in Table 3, where the sums of the

26

γ1

1 0.5 0

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0 −4 x 10

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0 −4 x 10

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100 time [s]

120

140

160

180

200

γ2

1 0.5 0

φ1

1 0.5 0

fu1

5 0 −5

φ2

1 0.5 0

fu2

5 0 −5


square errors ΣS E1 and ΣS E2 , defined as follows: ΣS E1 =

L·T Xs

re f 2 T s h1 (k) − h1 (k)

(74)

re f 2 T s h2 (k) − h2 (k)

(75)

k=1

ΣS E2 =

L·T Xs k=1

where L is the simulation length, is resumed for the nominal case and for the different fault scenarios. From the analysis of the numerical results, it can be seen that the proposed FTC strategy is able to reduce the sum of the square errors in all the cases.

27



1.6 1.4 1.2 1 0.8 0.6

0

50

100

150

200

0

50

100 Time [s]

150

200

0.3 0.25 0.2 0.15 0.1


6. Conclusions This paper has proposed an FTC strategy using LPV virtual actuators and sensors for non-linear systems that can be approximated by an LPV model. Virtual actuators and sensors are defined in such a way that they can cope with different types of faults and can both be activated at the same time. The faults considered are of two types: multiplicative and additive, allowing to handle partial loss of effectiveness, offsets and complete loss of one or more actuators/sensors. This FTC method adapts the faulty plant to the nominal LPV controller instead of adapting the LPV controller to the faulty plant. In this way, the faulty plant together with the LPV virtual actuator/sensor block allows the LPV controller to see the same plant as before the fault. It has been shown that the LPV controller can stabilize the faulty plant without having to redesign it at fault time, as long as the faulty plant is stabilizable/detectable, and is able to maintain the faulty system performance close to the nominal one. The elements of the FTC loop include the LPV controller, the LPV state observer, the LPV virtual actuator and the LPV virtual sensor. All these elements are designed using polytopic LPV techniques and pole placement in LMI regions. This formulation allows a versatile, straightforward and systematic characterization of the specifications. A key contribution of the proposed approach is the blending of the virtual sensor 28

γ1

1 0.5 0

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0 −4 x 10

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0 −4 x 10

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100 time [s]

120

140

160

180

200

γ2

1 0.5 0

φ1

1 0.5 0

fu1

5 0 −5

φ2

1 0.5 0

fu2

5 0 −5


Table 3: Computation of the sums of the square errors ΣS E1 and ΣS E2 .

nominal without FTC, fault scenario 1 with FTC, fault scenario 1 without FTC, fault scenario 2 with FTC, fault scenario 2 without FTC, fault scenario 3 with FTC, fault scenario 3

29

ΣS E1 1.4629 5.8513 1.4004 332.6650 1.4554 85.9445 1.3695

ΣS E2 0.0872 0.1288 0.0806 10.8165 0.0856 1.2299 0.0790



1.6 1.4 1.2 1 0.8 0.6

0

50

100

150

200

0

50

100 Time [s]

150

200

0.3 0.25 0.2 0.15 0.1


30

technique and the virtual actuator technique in a unique scheme, that is able to provide fault tolerance against a wide class of faults, such that the control scheme is able to cope in an automatic and efficient way against simultaneous actuators and sensor faults. Another important contribution presented in the paper consists in showing that the separation principle is valid for the proposed scheme and, as a consequence, the different elements of the fault tolerant control loop can be designed separately to achieve stability, as well as the desired performance, expressed in terms of pole location constraints, under both nominal and faulty situations. The proposed approach has been applied to an illustrative application, a two-tank system simulator. It has been shown that the proposed approach can recover the system performance and avoid the loss of stability due to faults. On the other hand, it has been demonstrated that the application of the proposed technique results in an improvement of the performance with respect to the LTI counterpart. The application of the proposed approach requires a fault estimation to be available. In this paper, a least-squares approach has been used, giving satisfactory results in most cases. However, it becomes clear that the proposed approach cannot recover the desired performance when the fault estimation is not perfect (due to the intrinsic delays of on-line least-squares methods, the modeling uncertainty or the coupling effects between fault estimations). Hence, due to the importance of a good fault estimation for a satisfactory application of the proposed approach, future research will aim to add robustness both to the fault estimation method and the virtual actuator/sensor block. On the other hand, since the theoretical results obtained in this paper are valid only as long as the scheduling variable is perfectly measured and non-faulty, further research will aim at analyzing and taking into account the effects of noise and faults in the sensors measuring the scheduling variables. 7. Acknowledgements This work has been funded by the Spanish MINECO through the project CYCYT SHERECS (ref. DPI2011-26243), by the European Commission through contract iSense (ref. FP7-ICT-2009-6-270428), by UPC through the grant FPI-UPC E-01104 and by AGAUR through the contract FI-DGR 2013 (ref. 2013FIB00218). The authors would like to thank the referees for their valuable comments and suggestions, which helped to improve the paper.

31

References References [1] M. Blanke, M. Kinnaert, J. Lunze, M. Staroswiecki, Diagnosis and fault-tolerant control, Springer Berlin Heidelberg, 2006. [2] Y. Zhang, J. Jiang, Bibliographical review on reconfigurable fault-tolerant control systems, Annual Reviews in Control 32 (2008) 229–252. [3] J. S. Shamma, Analysis and design of gain scheduled control systems, Ph.D. thesis, Massachussets Institute of Technology, Department of Mechanical Engineering, 1988. [4] J. S. Shamma, An overview of LPV systems, in: J. Mohammadpour, C. Scherer (Eds.), Control of linear parameter varying systems with applications, Springer, 2012, pp. 3–26. [5] R. Hallouzi, V. Verdult, R. Babuska, M. Verhaegen, Fault detection and identification of actuator faults using linear parameter varying models, in: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005. [6] M. Rodrigues, D. Theilliol, S. Aberkane, D. Sauter, Fault tolerant control design for polytopic LPV systems, International Journal of Applied Mathematics and Computer Science 17 (2007) 27–37. [7] S. Montes de Oca, V. Puig, M. Witczak, L. Dziekan, Fault-tolerant control strategy for actuator faults using LPV techniques: Application to a two degree of freedom helicopter, International Journal of Applied Mathematics and Computer Science 22 (2012) 161–171. [8] R. Patton, S. Klinkhieo, LPV fault estimation and FTC of a two-link manipulator, in: Proceedings of the 2010 American Control Conference, Baltimore, MD, USA, 2010, pp. 4647–4652. [9] T. Steffen, Control reconfiguration of dynamical systems: Linear approaches and structural tests, volume 230 of Lecture Notes in Control and Information Sciences, Springer, 2005. [10] J. Lunze, D. Rowe-Serrano, T. Steffen, Control reconfiguration demonstrated at a two-degrees-of-freedom helicopter model, in: Proceedings of the European Control Conference (ECC), Cambridge, UK, 2003. [11] J. Richter, W. Heemels, N. van de Wouw, J. Lunze, Reconfigurable control of piecewise affine systems with actuator and sensor faults: Stability and tracking, Automatica 47 (2011) 678–691. [12] J. Richter, Reconfigurable control of nonlinear dynamical systems - A faulthiding approach, volume 408 of Lecture Notes in Control and Information Sciences, Springer, 2011. 32

[13] L. Dziekan, M. Witczak, J. Korbicz, Active fault-tolerant control design for Takagi-Sugeno fuzzy systems, Bulletin of the Polish Academy of Sciences, Technical Sciences 59 (2011) 93–102. [14] J. Lunze, T. Steffen, Control reconfiguration after actuator failures using disturbance decoupling methods, IEEE Transactions on Automatic Control 51 (2006) 1590–1601. [15] M. Chilali, P. Gahinet, H∞ design with pole placement constraints: An LMI approach, IEEE Transactions on Automatic Control 41 (1996) 358–367. [16] S. Barnett, Matrices: Methods and applications, Oxford Applied Mathematics and Computing Science Series, 1990. [17] D. Rotondo, F. Nejjari, V. Puig, Fault estimation and virtual actuator FTC approach for LPV systems, in: Proceedings of the 8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS), 2012, pp. 824–829. [18] J. Jiang, Y. Zhang, A revisit to block and recursive least squares for parameter estimation, Computers and Electrical Engineering 30 (2004) 403–416. [19] L. Ljung, System identification: Theory for the user, Prentice Hall, 1987. [20] J. Jiang, Y. Zhang, A novel variable-length sliding window blockwise leastsquares algorithm for on-line estimation of time-varying parameters, International Journal of Adaptive Control and Signal Processing 18 (2004) 505–521. [21] S. Montes de Oca, D. Rotondo, F. Nejjari, V. Puig, Fault estimation and virtual Sensor FTC approach for LPV systems, in: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, 2011, pp. 2251 – 2256. [22] G. Franklin, J. Powell, M. Workman, Digital control of dynamic systems, 3rd. ed., Addison-Wesley, 1997. [23] S. Gutman, E. Jury, A general theory for matrix root clustering in subregions of the complex plane, IEEE Transactions on Automatic Control AC-26 (1981) 853–863. [24] A. Ghersin, R. Sanchez-Peña, LPV control of a 6-DOF vehicle, IEEE Transactions on Control Systems Technology 10 (2002) 883–887. [25] X.-D. Sun, I. Postlethwaite, Affine LPV modeling and its use in gain-scheduled helicopter control, in: Proceedings of the UKACC International Conference on Control, Swansea, Wales, UK, 1998, pp. 1504–1509. [26] M. Chadli, D. Maquin, J. Ragot, Observer-based controller for Takagi-Sugeno models, in: Proceedings of the 2002 IEEE International Conference on Systems, Man and Cybernetics, 2002.

33

¨ [27] J. Schur, Uber Potenzreihen, die im Innern des Einheitskreises beschränkt sind, Journal für die reine und angewandte Mathematik 147 (1917) 205–232. [28] J. Yoneyama, H∞ filtering for fuzzy systems with immeasurable premise variables: An uncertain system approach, Fuzzy Sets and Systems 160 (2009) 1738– 1748. [29] D. Ichalal, B. Marx, J. Ragot, D. Maquin, Fault diagnosis for Takagi-Sugeno nonlinear systems, in: Proceedings of the IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS), Barcelona, Spain, 2009. [30] D. Ichalal, B. Marx, J. Ragot, D. Maquin, State estimation of Takagi-Sugeno systems with unmeasurable premise variables, IET Control Theory and Applications 4 (2010) 897–908. [31] W. Heemels, J. Daafouz, G. Millerioux, Observer-based control of discrete-time LPV systems with uncertain parameters, IEEE Transactions on Automatic Control 15 (2010) 2130–2135. [32] J. Löfberg, YALMIP : A Toolbox for Modeling and Optimization in MATLAB, in: Proceedings of the CACSD Conference, Taipei, Taiwan, 2004, pp. 284 – 289. [33] H. Niemann, J. Stoustrup, An architecture for fault tolerant controllers, International Journal of Control 78 (2005) 1091–1110.

34