In this chapter, modified techniques for fault estimation in linear dynamic systems are proposed, which give the possibility to simultaneously estimate the system state as well as slowly varying faults. Using the continuous-time adaptive observer form, the considered faults are assumed to be additive, thereby the principles can be applied for a broader class of fault signals. Enhanced algorithms using H∞ approach are provided to verify stability of the observers, giving algorithms with improved performance of fault estimation. Exploiting the procedure for transforming the model with additive faults into an extended form, the proposed technique allows to obtain fault estimates that can be used for fault compensation in the fault tolerant control scheme. Analyzing the ambit of performances given on the mixed H2/H∞ design of the fault tolerant control, the joint design conditions are formulated as a minimization problem subject to convex constraints expressed by a system of linear matrix inequalities. Applied enhanced design conditions increase estimation rapidity also in noise environment and formulate a general framework for fault estimation using augmented or adaptive observer structures and active fault tolerant control in linear dynamic systems.
- linear dynamic systems
- additive fault estimation
- fault tolerant control design
- enhanced bounded real lemma
- linear matrix inequalities
- H∞ norm
- H2/H∞ control strategy
A model-based fault tolerant control (FTC) can be realized as control-laws set dependent, exploiting fault detection and isolation decision to reconfigure the control structure or as fault estimation dependent, preferring fault compensation within robust control framework. While integration of FTC with the fault localization decision technique requires a selection of optimal residual thresholds as well as a robust and stable reconfiguration mechanism , the fault estimation-dependent FTC structures eliminate a threshold subjectivism and integrate FTC and estimation problems into one robust optimization task . The realization is conditioned by observers, which performs the state reconstruction from the available signals.
The approach, in which faults estimates are used in a control structure to compensate the effects of acting faults, is adopted in modern FTC techniques [3, 4]. FTC with fault estimation for linear systems subject to bounded actuator or sensor faults, are proposed in . The observer structures are in the Luenberger form  or realized as unknown input fault observers . To guarantee the desired time response, a linear matrix inequality (LMI) based regional pole placement design strategy is proposed in  but such formulation introduces additive LMIs, which increase conservatism of the solutions. To minimize the set of LMIs of the circle regional pole placement is used; a modified approach in LMI construction is proposed in Ref. .
To estimate the actuator faults for the linear time-invariant systems without external disturbance the principles based on adaptive observers are frequently used, which make the estimation of the actuator faults by integrating the system output errors . First introduced in Ref. , this principle was applied also for descriptor systems , linear systems with time delays , system with nonlinear dynamics , and a class of nonlinear systems described by Takagi-Sugeno fuzzy models [14, 15]. Some generalizations can be found in .
The H2-norm is one of the most important characteristics of linear time-invariant control systems and so the problems concerning H2, as well as H∞, control have been studied by many authors (see, e.g. [17–20] and the references therein). Adding H2 objective to H∞ control design, a mixed H2/H∞ control problem was formulated in Ref. , with the goal to minimize H2 norm subject to the constraint on H∞ norm of the system transfer function. Such integrated design strategy corresponds to the optimization of the design parameters to satisfy desired specifications and to optimize the performance of the closed-loop system. Because of the importance of the control systems with these properties, considerable attention was dedicated to mixed H2/H∞ closed-loop performance criterion in design [22, 23] as well as to formulate the LMI-based computational technique [24, 25] to solve them or to exploit multiobjective algorithms for nonlinear, nonsmooth optimization in this design task [26, 27].
To guarantee suitable dynamics, new LMI conditions are proposed in the chapter for designing the fault observers as well as FTCs. Comparing with Ref. , the extended approach to the -stability introduced in Ref.  is used to minimize the number of LMIs in mixed H2/H∞ formulation of the FTC design and the eigenvalue circle clustering in fault observer design. In addition, different from Ref. , PD fault observer terms are comprehended through the enhanced descriptor approach , and a new design criterion is constructed in terms of LMIs. Since extended Lyapunov functions are exploited, the proposed approach offers the same degree of conservatism as the standard formulations [2, 31] but the H∞ conditions are regularized under acting of H2 constraint. Over and above, the -stability approach supports adjusting the fault estimator characteristics according to the fault frequency band.
The content and scope of the chapter are as follows. Placed after the introduction presented in Section 1, the basic preliminaries are given in Section 2. Section 3 reviews the definition and results concerning the adaptive fault observer design for continuous-time linear systems, Section 4 details the observer dynamic analysis and derives new results when using the -stability circle criterion and Section 5 recasts the extended design conditions in the framework of LMIs based on structured matrix parameters. Then, in response to fault compensation principle for such type of fault observers, Section 6 derives the design conditions for the fault tolerant control structures, reflecting the joined H2/H∞ control idea. The relevance of the proposed approach is illustrated by a numerical example in Section 7 and Section 8 draws some concluding remarks.
2. Basic preliminaries
In order to analyze whether a linear MIMO system is stable under defined quadratic constraints, the basic properties can be summarized by the following LMI forms.
Considering linear MIMO systems
where , , and are vectors of the system state, input, and output variables, respectively, is the unknown disturbance vector, is the system dynamic matrix, is the disturbance input matrix, and , are the system input and output matrices, then the system transfer functions matrices are
where is an unitary matrix and the complex number is the transform variable (Laplace variable) of the Laplace transform .
To characterize the system properties the following lemmas can be used.
3. Proportional adaptive fault observers
To characterize the role of constraints in the proposed methodology and ease of understanding the presented approach, the theorems’ proofs are restated in a condensed form in this section and also for theorems already being presented by the authors, e.g., in Refs. [38–40].
Despite different definitions, the best description for the formulation of the problem is based on the common state-space description of the linear dynamic multiinput, multioutput (MIMO) systems in the presence of unknown faults of the form
where , , and are vectors of the system, input, and output variables, respectively, is the unknown fault vector, is the system dynamics matrix, is the fault input matrix, and and are the system input and output matrices, ,
and the couple (
Limiting to the time-invariant system (16) and (17) to estimate the faults and the system states simultaneously, as well as focusing on slowly varying additive faults, the adaptive fault observer is considered in the following form 
where , , and are estimates of the system states vector, the output variables vector, and the fault vector, respectively, and is the observer gain matrix.
where is the gain matrix and , is a learning weight matrix that has to be set interactively in the design step.
In order to express unexpectedly changing faults as a function of the system and observer outputs and to apply the adaptive estimation principle, it is considered that the fault vector is piecewise constant, differentiable, and bounded, i.e., , the upper bound norm is known, and the value of is set to zero vector until a fault occurs. This assumption, in general, implies that the time derivative of can be considered as
These assumptions have to be taking into account by designing the matrix parameters of the observers to ensure asymptotic convergence of the estimation errors, Eqs. (21) and (22). The task is to design the matrix
3.1. Design conditions
If single faults influence the system through different input vectors (columns of the matrix
where the observer system matrix is
Since is linear with respect to the system parameters, it is possible to consider the Lyapunov function candidate in the following form
It is clear that the requirement
can be satisfied when Eq. (25) is satisfied.
Using the above given condition (37), the resulting formula for takes the form
and the LMI, defining the observer stability condition, is presented as
Introducing the notation
3.2. Enhanced design conditions
The observer stability analysis could be carried out generally under the assumption (29), i.e., using the forced differential equation of the form
It is evident now that acts on the state error dynamics as an unknown disturbance and, evidently, this differential equation is so not autonomous after a fault occurrence. Reflecting this fact, the enhanced approach is proposed to decouple Lyapunov matrix
where , , , and
Since Eq. (41) implies
it is possible to define the following condition based on the equality (51)
where is a symmetric positive definite matrix and is a positive scalar.
If the following requirement is introduced
Relying on Eq. (55), it is possible to write the observer stability condition as
where the following notations
Introducing the substitution
Considering that and using the Schur complement property, then the inequality (65) can be rewritten as
Since the first matrix element in the second row of Eq. (66) is zero matrix if
Thus, applying the Schur complement property, it can be written as
4. Observer dynamics with eigenvalues clustering in D-stability circle
Generalizing the approach covering decoupling of Lyapunov matrix from the observer system matrix parameters by using a slack matrix, with a good exposition of the given theorems, the observer eigenvalues placement in a circular -stability region is proposed to enable wide adaptation to faults dynamics.
Then, the time derivative of is
Generalizing the equation (75), the following condition can be set
It can be easily stated using Eq. (76) that
so, completing to square the elements in Eq. (83), it is immediate that
and using twice the Schur complement property, Eq. (85) can be rewritten as
Then, using the Schur complement property, the inequality (91) can be rewritten as
Since the second matrix element in Eq. (92) is zero matrix if
and so Eq. (93) implies the linear matrix inequality
Note, due to two integral quadratic constraints, setting the circle parameters to define -stabile region is relatively easy only for systems with single input and single output.
5. Extended design conditions
Since Eq. (98) can rewritten as follows
introducing the notations
where , , , , then it follows
and is the generalized fault observer error.
It is necessary to note that, in general, the elements of the positive definite symmetric matrix
where is a positive definite matrix. Computing the first time derivative of Eq. (109), it yields
which can be restated, using Eq. (102), as
By the Lyapunov stability theory, the asymptotic stability can be achieved if
respectively. It is evident that the matrix product is bilinear with respect to the LMI variables and . To facilitate the stability analysis, it can be written as
6. Joint design strategy for FTC
It is assumed that the systems (16) and (17) are controllable, full state feedback control, combining with additive fault compensation from , is applied and an integral component part is added to eliminate steady tracking error. In this structure, the control law takes the form
where is the reference output signal and , . Considering that in the fault-free regime
and Eq. (120) follows directly
, , and , then
where the closed-loop system matrix of the expanded system is
In order to design the system with reference attenuations
, , .
When the above conditions hold, the control law gain is
and so using the Schur complement property and the notation
Analogously, replacing in Eq. (7), the couple by the pair , the objective of H2 control is now to minimize the constraint .
Introducing the inequality
Note, to obtain a feasible block structure of LMIs, the Schur complement property has to be used to rearrange Eq. (137) to obtain Eq. (133) while the dual Schur complement property is applied to modify Eq. (139) to obtain Eq. (134).
, , and .
Defining the transform matrix
and premultiplying the left side and postmultiplying the ride side of Eq. (144) by , it yields
and with the notation
It is now easy to formulate a joint approach for integrated design of FTC, where is considered as in Eq. (118).
Note, the introduced H2H∞ control maximizes the H2 norm over all state-feedback gains while the H∞ norm constraint is optimized. The set of LMIs (152)–(154) is generally well conditioned and feasible and, since is a convergent matrix, it follows that the state of the closed-loop system converges uniformly to the desired value.
The main reason for the use of D-stability principle in the fault observer design is to adapt the fault observer dynamics to the dynamics of the fault tolerant control structure and the expected dynamics of faults. But the joint FTC design may not be linked to this principle.
7. Illustrative example
To test the effectiveness and performance of the proposed estimators, the computations are carried out using the Matlab/Simulink environment and additional toolboxes, while the observer and controller design is performed by the linear matrix inequalities formulation using the functions of SeDuMi package . The evaluation is performed in a standard condition, where the model to design the observer and the model for evaluation are the same and the simulations are performed according to the presented configuration of inputs and outputs.
Solving Eqs. (70)–(72), the fault observer design problem is solved as feasible where, with the prescribed stability region parameters
where is the observer system matrix eigenvalues spectrum. Using the same optional parameters (if necessary), there are obtained the observer gains for the design conditions introduced in Theorems 1–3 and 5–6, respectively, while
Using an extended approach presented in Theorems 7 and 8, the effect of the learning weight on the dynamic performance of the adaptive fault observer is analyzed. Setting the weight
Separated simulations of fault estimation observer outputs are realized for system under the force mode control, with the control law given as
Since separation principle holds and (
and the signal gain matrix
To evaluate the validity of the proposed compensation control scheme, weighted sinusoidal fault signals are considered. Since a weighted sinusoidal fault is suitable for evaluating the tracking performance and the robustness of the control scheme because it reflects more than slow changes in the fault magnitude, the faults in simulations are generated using the scenario
where it is adjusted .
Then, with the desired system output vector, the initial system condition and the external disturbance are chosen as follows
the faults estimates, obtained using the conditions from Theorems 1 to 6, are plotted in Figures 1–6. In all cases, the learning weight is set iteratively as
From these figures, it can be seen that fault estimation errors fast enough converge using an adaptive fault observer. Further, the extended approach with a prescribed circle -stability region is also effective in suppressing the disturbance noise effect on fault estimates.
Then, the set of control law matrix parameters is
while the eigenvalue spectrum of the closed-loop system matrix is
It is easy to see that the closed-loop system eigenvalues of the extended system strictly reflect the integral part of the control law that is, the set of inequalities (132)–(135) can be directly applied.
Based on these matrices, the closed-loop system matrix eigenvalues and the controller parameter (118) can be written out as
Finally, the design conditions are designed in such a way that the upper bounds of
while the controller matrix parameters are
and the spectrum of the closed-loop system matrix eigenvalues is
Considering the same fault generation method as above, but with , then for the desired system output vector, the initial system condition and the external disturbance chosen are as follows
the output variable responses of the closed-loop system, obtained using the conditions from Proposition 2 and Theorem 9, are shown in Figures 9 and 10 and are stable. To the structures (141), (142), and (152)–(155), the fault estimation is designed by Eq. (116).
Summarizing the obtained simulation results it can be concluded that the adaptive fault estimators, designed by the standard estimation algorithm, has the worst properties (Figure 1) that are not significantly improved even though the conditions of synthesis are enhanced by a symmetric learning weight matrix
The efficiency of the proposed algorithm to compensate the effect of an additive fault on the system output variables can be also observed. Figures 9 and 10 show that the proposed H2/H∞ method increases control robustness due to the joint mixed LMI optimization that guarantees system stability as well as the sufficient precision of compensation for a given class of slowly warring faults. Since the additive fault profile does not satisfy strictly the condition (22), its estimated time profile do not perfectly cover the actual values of the fault and where the variation of the amplitudes of
8. Concluding remarks
In this chapter, a modified approach for designing the adaptive fault observers is presented, and the -stability circle principle into fault observer design to outperform the two-stage known design approach in the fault observer dynamics adaptation is addressed. The design conditions are established as feasible problem, accomplishing under given quadratic constraints. Taking into consideration the slack updating effect, to cope with realistic operating conditions, the fault observer dynamics may be in the first case shifted to a stability region by exploiting the value of the tuning parameter. Integrated with the fault tolerant structures, H2 and H∞ norm-based analysis is carried out for compensated FTC structure to conclude about convergence of the fault compensation errors, and to derive the FTC design conditions. Using the LMI technique, the exploited mixed H2H∞ control design is possible to regularize the potential marginal feasibility of H∞-norm-based conditions. Presented illustrative example confirms the effectiveness of the proposed design alternative to construct the control structure with sufficient approximation of given class slowly warring faults and compensation of their impact on the system output variables.
The work presented in this chapter was supported by VEGA, the Grant Agency of the Ministry of Education and the Academy of Science of Slovak Republic, under Grant No. 1/0348/14. This support is very gratefully acknowledged.
Ding, S.X. (2013). Model-Based Fault Diagnosis Techniques. Design Schemes, Algorithms, and Tools, Springer-Verlag, Berlin. ISBN 978-1-4471-4798-5.
Lan, J. and Patton, R.J. (2015). Integrated design of robust fault estimation and fault-tolerant control for linear systems. Proceedings of the 54th IEEE Conference on Decision and Control CDC 2015, Osaka, Japan, pp. 5105–5110. ISBN 978-1-4799-7886-1.
Filasová, A., Gontkovič, D. and Krokavec, D. (2013). Observer-based fault estimation for linear systems with distributed time delay. Archives of Control Sciences, Vol. 23, No. 2, pp. 169–186. ISSN 1230-2384.
Wang, H. and Yang, G.H. (2009). Integrated fault detection and control for LPV systems. International Journal of Robust and Nonlinear Control, Vol. 19, No. 3, pp. 341–363. ISSN 1049-8923.
Tan, D. and Patton, R.J. (2015). Integrated fault estimation and fault tolerant control. A joint design. IFAC-PapersOnLine, Vol.48, No. 21, pp. 517–522. ISSN 1474-6670.
Oucief, N., Tadjine, M. and Labiod, S. (2016). Adaptive observer-based fault estimation for a class of Lipschitz nonlinear systems. Archives of Control Sciences, Vol. 26, No. 2, pp. 245–259. ISSN 1230-2384.
Shi, F. and Patton, R.J. (2014). Simultaneous state and fault estimation for descriptor systems using an augmented PD observer. IFAC Proceedings Volumes, Vol. 47, No. 3, pp. 8006–8011.
Shi, F. and Patton, R.J. (2015). Fault estimation and active fault tolerant control for linear parameter varying descriptor systems. International Journal of Robust and Nonlinear Control, Vol. 25, No. 5, pp. 689–706. ISSN 1049-8923.
Krokavec, D., Filasová, A. and Liššinský. P. (2016). Conditions with D-stability circle area in design of observer-based fault estimation. Applied Mathematical Sciences, Vol. 10, No. 35, pp. 1705–1717. ISSN 1312-885X.
Zhang, F., Liu, G. and Fang, L. (2009). Actuator fault estimation based on adaptive H∞ observer technique. Proceedings of the 2009 IEEE International Conference on Mechatronics and Automation ICMA 2009, Changchun, China, pp. 352–357. ISBN 978-1-4244-2692-8.
Wang, H. and Daley, S. (1996). Actuator fault diagnosis. An adaptive observer-based technique. IEEE Transactions on Automatic Control, Vol. 41, No. 7, pp. 1073–1078. ISSN 0018-9286.
Zhang, K. and Jiang, B. (2008). Analysis and design of adaptive fault estimation for time-varying delay systems. Proceedings of the 27th Chinese Control Conference, Kunming, China, pp. 38–42. ISBN 978-7-900719-70-6.
Ding, X. and Frank, P. (1993). An adaptive observer-based fault detection scheme for nonlinear dynamics systems. Proceedings of the 12th IFAC World Congres, Sydney, Australia, pp. 307–310.
Khedher, A., Benothman, K., Benrejeb, M. and Maquin, D. (2010). Adaptive observer for fault estimation in nonlinear systems described by a Takagi-Sugeno model. Proceedings of the 18th Mediterranean Conference on Control and Automation MED’10, Marrakech, Morocco, pp. 261–266. ISBN: 978-1-4244-8092-0.
Krokavec, D. and Filasová, A. (2012). A reduced-order TS fuzzy observer scheme with application to actuator faults reconstruction. Mathematical Problems in Engineering, Vol. 2012, Article ID 951953, 25 p. ISSN 1024-123X.
Zhang, K., Jiang, B. and Shi, P. (2013). Observer-Based Fault Estimation and Accomodation for Dynamic Systems, Springer-Verlag, Berlin. ISBN 978-3-642-33985-1.
Francis, B.A. (1987). A Course in H ∞ Control Theory, Springer-Verlag, Berlin. ISBN 0-387-17069-3.
Doyle, J.C., Glover, K., Khargonekar, P.P. and Francis, B.A. (1989). State-space solutions to standard H2 and H∞ control problems. IEEE Transactions on Automatic Control, Vol. 34, No. 8, pp. 831–847. ISSN 0018-9286.
Rotea, M.A. and Khargonekar, P.P. (1991). H2-optimal control with an H∞ constraint. The state feedback case. Automatica, Vol. 27, No. 2, pp. 307–316. ISSN 0005-1098.
Orlov, Y.V. and Aguilar, L.T. (2014). Advanced H ∞ Control. Towards Nonsmooth Theory and Applications, Springer Science, New York.
Khargonekar, P.P. and Rotea, M.A. (1991). Mixed H2/H∞ control. A convex optimization approach. IEEE Transaction on Automatic Control, Vol. 36, No. 7, pp. 824–831. ISSN 0018-9286.
Geromel, J.C., Peres, P.L.D. and Souza, S.R. (1992). Mixed H2/H∞ control for continuous time linear systems. Proceedings of the 31st Conference on Decision and Control, Tuscon, USA, pp. 3717–3722. ISBN 0-78030-872-7.
Bambang, R., Shimemura, E. and Uchjida, K. (1993). Mixed H2/H∞ control with pole placement. State feedback case. Proceedings of the 1993 American Control Conference, San Francisco, USA, pp. 2777–2779. ISBN 0-78030-860-3.
Scherer, C. (1995). Mixed H2/H∞ control. Trends in Control. A European Perspective, Isidori, A. (Ed.), Springer-Verlag, Berlin, pp. 173–216. TSBN 978-1-4471-3063-5.
Meisami-Azad, M., Mohammadpour, J. and Grigoriadis, K.M. (2009). Upper bound mixed H2/H∞ control and integrated design for collocated structural systems. Proceedings of the 2009 American Control Conference, St. Louis, USA, pp. 4563–4568. ISBN 978-1-4244-4524-0.
Goncalves, E.N., Palhares, R.M. and Takahashi, R.H.C. (2004). Multiobjective optimization applied to robust H2/H∞ state-feedback control synthesis. Proceedings of the 2004 American Control Conference, Boston, USA, pp. 4619–4624. ISBN 0-78038-335-4.
Nonami, K. and Sivrioglu, S. (1996). Active vibration control using LMI-based mixed H2/H∞ state and output feedback control with nonlinearity. Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, pp. 161–166. ISBN 978-0-7803-3590-5.
Krokavec, D. and Filasová, A. (2015). LMI constraints on system eigenvalues placement in dynamic output control design. Proceedings of the 2015 IEEE International Conference on Control Applications CCA 2015, Part of 2015 IEEE Multiconference on Systems and Control MSC 2015, Sydney, Australia, pp. 1749–1754. ISBN 978-1-4799-7786-4.
Zhang, K., Jiang, B. and Cocquempot, V. (2008). Adaptive observer-based fast fault estimation. International Journal of Control, Automation and Systems, Vol. 6, pp. 320–326. ISSN 1598-6446.
Filasová, A., Krokavec, D. and Serbák, V. (2015). Application of descriptor approaches in design of PD observer-based actuator fault estimation. Archives of Control Sciences, Vol. 25, No. 1, pp. 51–64. ISSN 1230-2384.
Fan, J., Zhang, Y. and Zheng. Z. (2011). Integrated adaptive fault diagnosis and state-feedback control for systems with constant actuator faults and control input constraints. Proceedings ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2011, Washington, USA, pp. 1–8. ISBN 978-0-7918-5480-8.
Debnath, L. and Bhatta, D. (2007). Integral Transforms and Their Applications, Chapman & Hall, Boca Raton. ISBN 1-58488-575-0.
Skelton, R.E., Iwasaki, T. and Grigoriadis, K.M. (1998). A Unified Algebraic Approach to Control Design, Taylor & Francis, London. ISBN 0-7484-0592-5.
Doyle, J.C, Francis, B.A. and Tannenbaum, A.R. (2009). Feedback Control Theory, Dover Publications, Mineola. ISBN 978-0-4864-6933-1.
Boyd. D. and Balakrishnan, V. (1990). A regularity result for the singular values of a transfer matrix and a qudratically convergent algorithm for computing its L∞-norm. Systems & Control Letters, Vol. 15, No. 1, pp. 1–7. ISSN 0167-6911.
VanAntwerp, J.G. and Braatz, R.D. (2000). A tutorial on linear and bilinear matrix inequalities. Journal of Process Control, Vol. 10, No. 4, pp. 363–385. ISSN 0959-1524.
Haddad, W.M and Chellaboina, V. (2008). Nonlinear Dynamical Systems and Control. A Lyapunov-Based Approach, Princeton University Press, Princeton. ISBN 978-0-6911-3329-4.
Filasová, A. and Krokavec, D. (2011). Design principles of active robust fault tolerant control systems. Robust Control. Theory and Applications, A. Bartoszewicz (Ed.), InTech, Rijeca, pp. 309–338. ISBN 978-953-307-229-6.
Krokavec, D., Filasová, A., Liššinský. P. and Serbák, V. (2015). Design of PD observer based fault estimator using a descriptor approach. Proceedings of the 26th International Workshop on Principles of Diagnosis DX-15, Paris, France, pp. 235–240.
Krokavec, D. and Filasová, A. (2016). On enhanced mixed H2/ H∞ design conditions for control of linear time-invariant systems. Proceedings of the 17th International Carpathian Control Conference ICCC 2016, Tatranská Lomnica, Slovakia, pp. 384–389. ISBN 978-1-4673-8605-0.
Patton, R.J. and Klinkhieo, S. (2009). Actuator fault estimation and compensation based on an augmented state observer approach. Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, China, pp. 8482–8487. ISBN 978-1-4244-3871-6.
Zhang, Y.M. and Jiang, J. (2008). Bibliographical review on reconfigurable fault-tolerant control systems. Annual Reviews in Control, Vol. 32, pp. 229–252, ISSN 1367-5788.
Kautsky, J., Nichols, N.K. and Van Dooren, P. (1985). Robust pole assignment in linear state feedback. International Journal of Control, Vol. 41, No. 5, pp. 1129–1155. ISSN 0020-7179.
Peaucelle, D., Henrion, D., Labit, Y. and Taitz, K. (2002). User’s Guide for SeDuMi Interface 1.04, LAAS-CNRS, Toulouse.
Wang, Q.G. (2003). Decoupling Control, Springer-Verlag, Berlin. ISBN 3-540-44128-X.