The constrained unitary formalism to fuzzy fault detection filter synthesis for one class of nonlinear systems, representable by continuous-time Takagi-Sugeno fuzzy models, is presented in the chapter. In particular, a way to produce the special set of matrix parameters of the fuzzy filter is proposed to obtain the desired H ∞ norm properties of the filter transfer function matrix. The significance of the treatment in relation to the systems under influence of actuator faults is analyzed in this context, and relations to corresponding setting of singular values of filters are discussed.
- multiple models
- continuous-time Takagi-Sugeno fuzzy models
- fuzzy fault detection filters
- fuzzy state observers
Since the work of Hou and Patton , there has been much interest in the design of fault residuals for linear systems that use optimization principle in transfer function matrix of fault detection filter designed to scale up fault detection punctuality and high sensitivity to faults . While retaining these features, a novel class of fault detection filters are proposed in [3, 4], preserving the unitary implementation of the fault detection filter transfer function matrix and receipting residual signal directional properties. However, the use of this methodology for Takagi-Sugeno (TS) fuzzy systems hits the boundaries of the working sectors and requires special adaptations.
Considering the properties of TS fuzzy models [5, 6], and some specifics in frequency characteristic evaluation of multiple model structures, the approach proposed in the chapter reformulates the H norm technique suitable in TS fuzzy fault detection filter design. The problem is solved via unitary modal technique when every linear TS fuzzy filter part is designed to have the same singular values of the transfer function matrix. Since working sector constraints may cause that the stable linear filter component cannot be obtained for a linear part in TS fuzzy model, to maintain H norm of the filter, the LQ modal control principle  is used for additional stabilization. Because additional stabilization aggravates directional properties of the applied linear part, in general, if additional stabilization is necessary, the residuals are only quasi-directional. It is immediately apparent that the formulated problem is related to forcing the singular values conditioned as state observer dynamics. The chosen model of the system is selected for this chapter to be sufficiently complex in illustration of all these specifics of synthesis.
Throughout the chapter, the following notations are used: and denote the transpose of the vector and the matrix , respectively; for a square matrix means that is a symmetric positive semi-definite matrix; the symbol indicates the th-order unit matrix; denotes the set of real numbers; and and refer to the set of all -dimensional real vectors and real matrices.
2. System description
The considered class of the Takagi-Sugeno dynamic systems with additive faults is described as the following:
where , , and stand for state, control input, and measurable output, respectively; is an additive fault vector; , , , , and and the matrix products and are regular matrices for all .
The variables and , bound with the sector TS model, span the -dimensional vector of premise variables:
where is the set of normalized membership function. It is supposed that the measurable premise variables, the nonlinear sectors, and the normalized membership functions are chosen in such a way that the pairs are controllable and the pairs are observable for all .
3. Basic preliminaries from linear systems
Let the state-space description of a linear continuous-time dynamic systems take the form with equivalent meanings and dimensions as they are described in Section 2. The nature of the characterization of expected solutions to the system [(5), (6)] is given by the following results.
while the th singular value of the complex matrix is the nonnegative square root of the th largest eigenvalue of , is the adjoint of , and is the largest singular value. The singular values of the transfer function matrix are evaluated on the imaginary axis, and it is assumed that the singular values are ordered such that .
To apply in design methodology, the following result from  is quoted. If and are regular matrices, then the system matrix factorization can be realized such that
If and are regular matrices, then the system matrix factorization can be realized such that
and the transform matrix takes the form
where , , and are the left orthogonal complements to .
The idea of the following condition was derived originally as an approximation in the frequency domain for the fault transfer function matrix reflecting Eqs. (5) and (6) from . Here, it is demonstrated that it can be simply adapted for fault residual filter design. A linear fault detection filter to the system [(5), (6)] is stable and unitary if for regular and a given positive scalar the square transfer function matrix of the fault detection filter satisfies the conditions
A linear fault detection filter to the system [(5), (6)] is stable and unitary if for regular and a given positive scalar the square transfer function matrix of the fault detection filter satisfies the conditions
where is the residual filter gain matrix, is the maximal singular value of , the polynomial of order is stable, and .
Proof. Considering the fault transfer function matrix of dimension as
respectively, where is given in Eq. (15).
Specifying the following matrix product , where is a real matrix, it yields
where is a prescribed positive real value. The plus sign is introduced for the purposes that come to light in the stability ensuing development of the observer system matrix.
and it is evident that is stable if is Hurwitz, denoting here that
Rewriting the set of Eq. (22) to admit a stable solution
Therefore, the observer system matrix takes the form
implies Eq. (16).
Regarding the transfer function matrix of the state error estimate as follows
then with Eq. (29), it is
Thus, defining the fault detection filter transfer function matrix as , then
and is the vector of residual signals, then based on the following observer structure
the autonomous observer error equation is
where is the observer state, is the estimated system output, and is the observer gain matrix; the fault detection filter (37), (39) is stable and unitary if for given positive scalar and the Hurwitz matrix the conditions (15) and (16) are satisfied.
Practically, with understanding Eq. (30), the observer sensor subsystem for the fault detection filter can be designed as follows:
and, consequently, it yields
Another option is to design the observer sensor subsystem so that .
With existence of the system parameter transformation, the above structures really mean that the subset of transformed state variables whose dynamics is explicitly affected by the additive fault and the second one, whose dynamics is not affected explicitly by the fault , exists. It is important to note the fact that the eigenvalues of and of are the same whenever is related to as for any invertible . But this does not mean that if eigenvalues of the matrix are stable then eigenvalues of the matrix are also stable. Thus, as well as for a stable system, it can lead to an unstable matrix , and any additional stabilization is required.
It is important to note the fact that the eigenvalues of and of are the same whenever is related to as for any invertible . But this does not mean that if eigenvalues of the matrix are stable then eigenvalues of the matrix are also stable. Thus, as well as for a stable system, it can lead to an unstable matrix , and any additional stabilization is required.
To apply the above results, it is necessary to be able to design fault residual filter if an unstable results such that be stable without loss of unitarity.
is a solution of the continuous-time algebraic Riccati equation (CARE)
where the matrix is null matrix and and are positive definite symmetric matrices.
Then, is given as
It is in that form that is able to be exploit for specific properties of the problem in TS fuzzy fault detection filter design.
In view of the above, these results hold for continuous-time linear systems, and, in principle, Theorem 1 gives a practical method to design unitary fault residual filters for the given linear system. Similar results are obtained for unitary TS fuzzy fault detection filter design in the following section.
4. TS fuzzy fault detection filters
Using the same set of membership functions, the fuzzy fault detection filter is built on the TS fuzzy observer
where is the observer state vector, is the estimated system output vector, and and are the sets of the observer gain matrices. Additionally, the output vector of the residual TS fuzzy filter is defined as
where , . Evidently, has to be a regular matrix for all .
Formally, the following result can be simply derived. A TS fuzzy fault detection filter to the system [(1), (2)] is stable and unitary if for the set of regular matrices and , and a given positive scalar every square transfer function matrix of the fault detection filter satisfies for all the conditions
A TS fuzzy fault detection filter to the system [(1), (2)] is stable and unitary if for the set of regular matrices and , and a given positive scalar every square transfer function matrix of the fault detection filter satisfies for all the conditions
is the residual filter gain matrix, is the maximal singular value of , the polynomial of order is stable, and and are left orthogonal complements to the fault input matrix .
Proof. Because every sub-model in Eq. (47) is described by linear equations, Eqs. (15) and (16) imply directly the conditions (56) and (57), and Eq. (58) is given by Eq. (11). This concludes the proof. In practice, an additive fault typically enters through a matrix that does not depend on the sectoral boundaries defining the TS model. In this case, the synthesis is substantially simplified because is a constant matrix, and so it yields
In practice, an additive fault typically enters through a matrix that does not depend on the sectoral boundaries defining the TS model. In this case, the synthesis is substantially simplified because is a constant matrix, and so it yields
Since, independently on , the condition (52) is satisfied (), all sub-filter transfer function matrices have the same Hnorm, i.e.,
Moreover, considering that , then
That is, the Hnorm of the transfer function matrix of such defined TS fuzzy fault detection filter is independent on the system working point. Of course, this cannot be said about the dynamics of the time response of the sub-filter components.
Moreover, implies that all residual components of TS fuzzy fault detection filter have the same directional properties, which ensure unitary properties of the filter. Sectoral boundaries may cause a matrix to be such, when transformed using that will not be Hurwitz matrix. Because the transfer function matrix of the corresponding filter linear component in this case is unstable, maintaining the unitary property requires changes in the signs of the unstable eigenvalues of the associated . Applying the duality principle and inserting the additive observer gain component obtained as a solution of the Riccati equation (45) for , according to the scheme given in Lemma 2, the observer gain matrix is changed as
Sectoral boundaries may cause a matrix to be such, when transformed using that will not be Hurwitz matrix. Because the transfer function matrix of the corresponding filter linear component in this case is unstable, maintaining the unitary property requires changes in the signs of the unstable eigenvalues of the associated .
Applying the duality principle and inserting the additive observer gain component obtained as a solution of the Riccati equation (45) for , according to the scheme given in Lemma 2, the observer gain matrix is changed as
This additive stabilization results that the consequential characteristic polynomial, taking also the form
is stable since is now stable.
The price for such an additional stabilization is that if signs are changing in eigenvalues of to obtain the stable , also eigenvalues of change their signs and the resulting matrix will not be diagonal. According to Eq. (8), this does not result in a change in Hnorm, but such filter component will arrive at the unitary directional residual properties.
5. Illustrative example
where the measured output variables are water levels in tanks and the incoming flows are considered as the inputs variables ; the bounds of the state and input variables are
are positive scalars and is the sign function.
The model parameters of the system are considered as:
Minimizing the number of premise variables and excluding switching modes in controller work, the premise variables are chosen as follows
Computed from the input variable bounds, the sector bounds of the premise variables imply the numbering:
which is used in the system state matrix construction
and prescribed, moreover, that the matrix is given in such a way that the product is the identity matrix. This regularizes the residual design conditions if and are diagonal matrices.
The sector functions are trapezoidal, and the membership functions are constructed as product of three sector functions with the same ordering as .
The set of real scalars, , , and , is interactively optimized under limitations that all couples and are controllable and observable for the given set of indices , where
Consequently, the TS model matrix parameters are
Since the orthogonal complement to a square matrix does not exist, three fault detection filters can be considered for single actuator fault detection. To illustrate the design procedure, the TS fuzzy fault detection filter for the pair (, ) is considered, i.e.,
with the derived parameters
Note that in this case all with index higher than 4 lead to an unstable structure of and the resulting observer matrices need to be additionally stabilized, applying the principle given in Lemma 2.
Applying Eq. (56), the following structure of for the initial matrix is computed:
and implies that the associated TS fuzzy fault detection filter linear component can be designed directly.
where the eigenvalue spectrum of and the steady-state value of the TS fuzzy fault detection filter transfer function matrix are
respectively. It is evident that all diagonal elements of take the value . The same structure of is obtained solving with for .
Analogously, designing for the matrix , it can be seen that
Since , evidently, the associated TS fuzzy fault detection filter linear component with the unitary transfer function matrix has to be stabilized additively.
Solving also for , then
It is evident that matrix is not Hurwitz and has to be additively stabilized.
Thus, defining the weighting matrices of appropriate dimensions as
and solving the dual LQ control problem to change the sign of unstable eigenvalue of using the MATLAB function , then
It can be easily verified that
while, evidently, is not diagonal and the eigenvalues of are .
Note that the same structure of is obtained solving with the system matrices and when additional stabilization is required. Evidently, elements of this set of TS fuzzy residual filter linear components are stable, non-unitary, and without directional residual properties. Nevertheless, these properties guarantee the same singular values of the linear transfer function matrix components; as follows the result of Definition 1, the TS fuzzy residual filter will have all the singular values the same. To document this, the singular value plot of the TS fuzzy fault detection filter, as well as of all its linear parts, is equal to that presented in Figure 1. With respect to the structure of the matrices and , the comparable results are obtainable for the matrix pairs , and (, ).
The rest of gain matrices of the stable TS fault detection filter is as follows:
Since the matrices of the TS fuzzy system are not Hurwitz, the system in simulations is stabilized using the local-state feedback control laws, acting in the forced modes. Adapting the method presented in  to design the control law parameters, the local controller parameters are computed as
while is the vector of the desired steady-state system outputs.
To display simulations in the MATLAB and Simulink environment, the forced mode control is established with local controller parameter given as above for the system initial conditions and . Fault detection filter is constructed on the couple (, ) and the set of matrices and .
As the results, Figure 2 shows the TS fuzzy system output responses, illustrating their asymptotic convergence to the steady states, and Figure 3 presents the TS fuzzy fault detection filter response, reflecting a steplike 90% gain loss of the second actuator at the time instant . These examples illustrate the power that can be invoked through the prescribed H norm properties.
It can verify that TS fuzzy fault detection filters created for the couple pairs (, ) and (, ) have similar properties as that defined for the couple (, ). The difference is, for example, that in the occurrence of a single fault of the second actuator the responses of TS fuzzy fault detection filter defined for the couple (, ) naturally do not have directional properties, since the second column of is not included in its construction.
As can be seen from the solution, the sector functions defined in this way cannot create a unitary TS fuzzy fault detection filter, but the obtained orthogonal properties of the residual signals are sufficient to detect and isolate actuator faults.
6. Concluding remarks
The problem of designing the TS fuzzy fault detection filters for highly nonlinear mechanical systems representable by the TS fuzzy model is considered, to achieve the desired filter H norm property in all working point belonging to the assigned work sectors. The proposed method exploits features offered in TS fuzzy system models to design TS fuzzy fault detection filters. The rules and formulation are developed to generate residual signals with quasi-directional properties and to make the TS filter transfer function matrix with prescribed H norm properties. By a convenient choose of the sector functions, this purpose is reached using a relative small number of membership functions. If unitary definition for TS fuzzy fault detection filters is satisfied, the design methodology provides new opportunities for fault detection and isolation rules in fault tolerant nonlinear control systems, their analysis, and optimization.
The work presented in this chapter was supported by VEGA, the Grant Agency of the Ministry of Education, and the Academy of Sciences of Slovak Republic, under Grant No. 1/0608/17. This support is very gratefully acknowledged.