Robust state and Fault Estimation observer

Consider a state space representation of a fault-free and disturbance-free (nominal) linear time invariant system:

(4-1)

Where is the state vector, is the output vector, is the input control signal vector, , , , are all known appropriate dimension matrices, the matrix pair is assumed controllable and the pair is assumed to be observerable.

When the nominal system Eq. (4-1) is affected by actuator faults and external disturbance simultaneously, the original system is now described as:

(4-2)

where and are the actuator faults and exogenous disturbance vectors, respectively. _{, are known real constant matrices. It is very}

important to know that for solving robust FDI problems, a mathematical representation for expression of modelling uncertainty is required. Patton and Chen (Patton and Chen, 1992, 1993; Patton, Chen and Zhang, 1992; Chen, 1995) provide several methodologies to represent modelling uncertainties in structured format from various sources, as additive disturbances with an estimated distribution matrix (Chen and Patton, 1999). It can be concluded that from a mathematical point of view, the expression of the modelling uncertainty has the same effect in the system as the disturbance. As a result, a general description for system uncertainty and disturbance is expressed in a form of one distribution matrix multiplying the disturbance or uncertainty vector, i.e. , such as in Eq. (4-2) is no loss of generality. This Section is concerned with the robust fault estimator design. Hence, the following Assumptions can be made:

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Assumption 4.2: The norms of , and its first derivative of are bounded

such that:

, for all . , , are known positive

constants, namely, , , , , .

A full-order state observer for system Eq. (4-2) using output information can be designed as follows:

(4-3)

(4-4)

where is a nonlinear design function whose design is given under Theorem 4.1. The error dynamic system between the plant and the observer is then represented as:

(4-5)

(4-6)

where and . The fault estimator system can be stated mathematically as:

(4-7) Where is calculated by , is the learning rate, is the fault estimator gain matrix to be designed. Meanwhile, it should be noted that an estimate of the actuator fault can easily be derived by taking the integral of both sides of Eq. (4-7), so that .

The idea behind Eq. (4-5) is easy to grasp, if has an effect on reducing the influence from the exogenous disturbance for states and fault estimates, then the error dynamics of Eq. (4-5) becomes robust. Theorem 4.1 is established following a motivation by the observer designed by Zhu and Cen (2010). It is noted that in the work

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of (Zhu and Cen, 2010) an adaptive and robust full-order observer was constructed, the robustness is shown by the special design for the term , similar to the discontinuous control component used in sliding mode control law (Edwards and Spurgeon, 1998). However, Zhu and Cen design the observer for the fault-free case which is not the concept described here since the current work is based on the use of the term .

Theorem 4.1: Under Assumptions 4.1-4.2, if there exist symmetric positive definite

matrices _{and matrices and , such}

that the following conditions hold:

, (4-8)

(4-9)

(4-10)

where . The robust full-order observer determined by Eq. (4-3), (4-4), (4-7) has a non-linear function the term of the state estimates and output measurements, is as follows: (4-11)

where is a design matrix, is the inverse of the matrix , is a small positive constant to ensure when the time goes to infinity the state estimation and fault estimation converge asymptotically to the actual state and the actuator fault , respectively. The term is the norm of the integral , namely the norm of the fault estimate .

Proof:

Consider a Lyapunov function candidate:

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The derivative of along with the error dynamic system (4-5), (4-6) is:

(4-13) By Lemma 3.1 it is easy to show that:

(4-14)

where is a symmetric positive matrix and is a positive constant chosen appropriately by the designer. According to Eq. (4-14) and substituting Eq. (4-7) into Eq. (4-13), then Eq. (4-13) becomes:

(4-15)

where denotes the largest eigenvalue of the matrix defined in the space . In order to obtain the appropriate quadratic form to prove the Lyapunov stability of the system of Eqs. (4-5) and (4-6), a subtle mathematical transformation can be made by adding a positive term to the inequality (4-15) to change its structure, then the inequality (4-15) becomes:

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_(4-16)

To maintain the same evaluation, on subtracting the term from inequality (4-16), then inequality (4-16) becomes:

_(4-17)

Since from Eq. (4-9), then the terms and

sum to zero, hence:

(4-18)

As (Assumption 4.2), and by defining a vector as _,

Eq.(4-18) can now be rewritten as:

(4-19)

where

.

Also noting that

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(4-20)

Then Eq. (4-20) is transformed to:

(4-21)

Replace by the right hand side of Eq. (4-11) into Eq. (4-21), and the following result is obtained:

(4-22)

That is because, on the basis of Eq. (4-8),

where , and . So that which means that

_{converges asymptotically to zero. On the other hand, the state and the}

fault estimates track the trajectories of the plant states and actuator faults, respectively.

Q.E.D.

Remark 4.1: For a single input system , the matrix replaces during the

observer design. For multi-input systems, faults may occur in several actuators, at this time the matrix is a linear subspace of the matrix .

Remark 4.2: When no fault occurs, i.e. , the proposed robust observer by Eq. (4-

3), (4-4), (4-7) and (4-11) with removed fault estimation term , is robust in the sense of rejecting exogenous disturbances. This is consistent with the work of Zhu and Cen (2010). When a fault occurs, as the matrix is different from the matrix , and their columns are linearly independent respectively, the term only has the effect of rejecting the disturbances rather than rejecting the actuator fault signal. The consequence of this is that the system output is only perturbed by the actuator fault , Hence, the output error does not approach zero asymptotically. The proposed observer Eqs. (4-3), (4-4), (4-7) and (4-11) can thus serve as a fault detection observer.

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The logical fault/no-fault classification is given as follows by checking the system output residual :

where is a threshold designed for fault detection, i.e. to avoid false alarms. Meanwhile, when the actuator fault weighting matrix is unknown, the weighting matrix in the observer structure can be taken by directly using the input matrix . As the fault estimate shows the fault magnitude and identifies the location of the fault, it can also be considered as a robust fault isolation observer.

Remark 4.3: From a theoretical point of view, when , has no

significance where it’s denominator is equal to zero. For this case the observer dynamics are the same as the plant dynamics. Under this situation, no further action is required. To take care of this condition the nonlinear law is always chosen as: (4-23)

where is a small positive constant. Hence Eq. (4-23) can be used instead of Eq. (4-11) when . It is also important to point out that the inequality is nonlinear, and it is very difficult to find solutions satisfying the nonlinear inequality (due to lack of convexity). However, by setting , this inequality can be transformed into the form of Eq. (4-8), which then leads to an LMI problem.

Although inequality (4-8) can be solved efficiently using the MatLab LMI toolbox, difficulties arise when solving the inequality (4-8), Eqs. (4-9) and (4-10) simultaneously. It turns out that a simultaneous solution of inequality (4-8), Eq. (4-9) and (4-10) cannot be guaranteed. However, this problem can be converted into the following optimization

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problem (Zhang, Jiang and Cocquempot, 2008; Corless and Tu, 1998), for which the solution is more straightforward.

This procedure is summarised as follows. Solve the following two LMIs, and thereby minimize subject to inequality (4-8):

(4-24)

(4-25)

The matrices , , , can be determined by solving inequalities (4-8), (4-24) and (4-25) simultaneously. The final observer gain matrix is then given by _{, where}

_{is the inverse of the S.P.D. matrix .}