## 1. Introduction

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 [1], the fault estimation-dependent FTC structures eliminate a threshold subjectivism and integrate FTC and estimation problems into one robust optimization task [2]. 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 [5]. The observer structures are in the Luenberger form [6] or realized as unknown input fault observers [7]. To guarantee the desired time response, a linear matrix inequality (LMI) based regional pole placement design strategy is proposed in [8] 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. [9].

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 [10]. First introduced in Ref. [11], this principle was applied also for descriptor systems [7], linear systems with time delays [12], system with nonlinear dynamics [13], and a class of nonlinear systems described by Takagi-Sugeno fuzzy models [14, 15]. Some generalizations can be found in [16].

The H_{2}-norm is one of the most important characteristics of linear time-invariant control systems and so the problems concerning H_{2}, as well as H_{∞}, control have been studied by many authors (see, e.g. [17–20] and the references therein). Adding H_{2} objective to H_{∞} control design, a mixed H_{2}/H_{∞} control problem was formulated in Ref. [21], with the goal to minimize H_{2} 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 H_{2}/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. [5], the extended approach to the _{2}/H_{∞} formulation of the FTC design and the eigenvalue circle clustering in fault observer design. In addition, different from Ref. [29], PD fault observer terms are comprehended through the enhanced descriptor approach [30], 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 H_{2} constraint. Over and above, the

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 _{2}/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

where

To characterize the system properties the following lemmas can be used.

**Lemma 1** (*Lyapunov inequality*) [33] *The matrix* *is Hurwitz if there exists a symmetric positive definite matrix* *such that*

**Lemma 2** [34] *The matrix* *is Hurwitz and* *if there exists a symmetric positive definite matrix* *and a positive scalar* *such that*

*where* *is H*_{2} *norm of the transfer function matrix*

**Lemma 3** (*Bounded real lemma*) [35] *The matrix A is Hurwitz and *

*where* *are identity matrices and* *is H*_{∞} *norm of the disturbance transfer function matrix*

*Hereafter, * denotes the symmetric item in a symmetric matrix*.

**Lemma 4** [28] *The matrix* **A** *is* *stable Hurwitz if for given positive scalars* *there exists a symmetric positive definite matrix* *such that*

*while the eigenvalues of* *A**are clustered in the circle with the origin* *and radius ϱ within the complex plane S*.

**Lemma 5** (*Schur complement*) [36] *Let O be a real matrix, and N (*

**M**

*) be a positive definite symmetric matrix of appropriate dimension, then the following inequalities are equivalent*

**Lemma 6** (*Krasovskii lemma*) [37] *The autonomous system (1) is asymptotically stable if for a given symmetric positive semidefinite matrix* *there exists a symmetric positive definite matrix* *such that*

*where L is the weight matrix of an integral quadratic constraint interposed on the state vector q*(

*t*).

## 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 the couple (** A,C**) is observable.

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 [41]

where

The observer (19) and (20) is combined with the fault estimation updating law of the form [42]

where

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.,

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 ** J** in such a way that the observer dynamics matrix

### 3.1. Design conditions

If single faults influence the system through different input vectors (columns of the matrix ** F**), it is possible to avoid designing the estimators with the tuning matrix parameter

**> 0 and formulate the design task through the set of LMIs and a linear matrix equality.**

*G***Theorem 1** *The adaptive fault observer (19) and (20) is stable if there exists a symmetric positive definite matrix* *and matrices* *such that*

*When the above conditions hold, the observer gain matrix is given by*

*and the adaptive fault estimation algorithm is*

*where*

*and* *is a symmetric positive definite matrix which values are set interactive in design*.

*Proof*. From the system models (16) and (17) and the observer models (19) and (20), it can be obtained that

where the observer system matrix is

Since

where ** P, G** are real, symmetric, and positive definite matrices. Then, the time derivative of

where

(33) |

Inserting Eq. (21) into Eq. (34) leads to

and substituting Eq. (35) with Eq. (30) into Eq. (33), the following inequality is obtained

It is clear that the requirement

can be satisfied when Eq. (25) is satisfied.

Using the above given condition (37), the resulting formula for

and the LMI, defining the observer stability condition, is presented as

Introducing the notation

it is possible to express Eq. (39) as Eq. (24). This concludes the proof.

### 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

while

It is evident now that ** P** from the system matrices

**by introducing a slack matrix**

*A, C***in the observer stability condition, as well as to decouple the tuning parameter**

*Q**δ*from the matrix

**in the learning rate setting and using**

*G**δ*to tune the observer dynamic properties. Since the design principle for unknown input observer cannot be used, the impact of faults on observer dynamics is moreover minimized with respect to the H

_{∞}norm of the transfer functions matrix of

*G*_{f}(

*s*), while a reduction in the fault amplitude estimate is easily countervailing using the matrix

**. In this sense the enhanced design conditions can be formulated in the following way.**

*G***Theorem 2** *The adaptive fault observer (19) and (20) is stable if for a given positive* *there exist symmetric positive definite matrices* *matrices* *and a positive scalar* *such that*

*When the above conditions are affirmative the estimator gain matrix is given by the relation*

*Proof*. Using Krasovskii lemma, the Lyapunov function candidate can be considered as

where *γ* is an upper bound of *H*_{∞} norm of the transfer function matrix

(49) |

If it is assumed that Eqs. (34) and (35) hold, then the substitution of Eq. (35) into Eq. (49) leads to

(50) |

Since Eq. (41) implies

it is possible to define the following condition based on the equality (51)

where

Then, adding Eq. (52) and its transposition to Eq. (50), the following has to be satisfied

(53) |

If the following requirement is introduced

it is obvious that Eq. (54) can be satisfied when Eq. (46) is satisfied. Thus, the condition (54) allows to write Eq. (53) as follows

(55) |

Relying on Eq. (55), it is possible to write the observer stability condition as

where the following notations

are exploited.

Introducing the substitution

and using the Schur complement property with respect to the item

**Theorem 3** *The adaptive fault observer (19) and (20) is stable if there exists a symmetric positive definite matrix* *matrices* *and a positive scalar* *such that*

*When the above conditions are affirmative the estimator gain matrix is given by the relation*

*Proof*. Premultiplying the left side and postmultiplying the right side of Eq. (57) by the transformation matrix

gives

Considering that

Since the first matrix element in the second row of Eq. (66) is zero matrix if *δ* = 0 and considering that nonzero component unit of the last matrix element in this raw is certainly positive semidefinite, it can claim that

Thus, applying the Schur complement property, it can be written as

respectively. With the notation (59) then Eq. (69) gives Eqs. (61). This concludes the proof.

Comparing with Lemma 3, it can be seen that Eqs. (60)–(62) is an extended form of the bounded real lemma (BRL) structure, applicable in the design of proportional adaptive fault observers.

## 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

**Theorem 4** *The adaptive fault observer (19) and (20) is* *-stable if for given positive scalars* *there exist symmetric positive definite matrices* *matrices* *and a positive scalar* *such that*

*When the above conditions are affirmative the estimator gain matrix can be computed as*

*and the adaptive fault estimation algorithm is given by (27)*.

*Proof*. Choosing the Lyapunov function candidate as

(74) |

where *H*_{∞} norm of the transfer function matrix (43) and where the generalized observer differential equation takes the form [28]

while, with *a* > 0, *ϱ* > 0 such that *ϱ* < *a*, the matrices *A*_{cr}, *F*_{rr} are given as

Then, the time derivative of

(77) |

Assuming that, with respect to Eqs. (34) and (35), the inequality (50) holds, then Eq. (77) gives

(78) |

Generalizing the equation (75), the following condition can be set

where

(80) |

From Eq. (80), using the notation (58), the following stability condition can be obtained

where

It can be easily stated using Eq. (76) that

so, completing to square the elements in Eq. (83), it is immediate that

Substituting Eqs. (76) and (84) in Eq. (82) gives

and using twice the Schur complement property, Eq. (85) can be rewritten as

(86) |

Thus, for *A*_{e} from Eq. (30) and with the notation (59) then Eq. (86) implies Eq. (71). This concludes the proof.

**Theorem 5** (*Enhanced BRL*) *The adaptive fault observer* (*19*) *and* (*20*) *is* *stable if for given positive scalars* *there exist a symmetric positive definite matrix* *matrices* *and a positive scalar* *such that*

*When the above conditions are affirmative the estimator gain matrix can be computed by Eq*. (73).

*Proof*. Considering that in Eq. (86) ** P** =

**, then premultiplying the left side and postmultiplying the right side of Eq. (86) by the transformation matrix**

*Q*gives

(91) |

Then, using the Schur complement property, the inequality (91) can be rewritten as

(92) |

Since the second matrix element in Eq. (92) is zero matrix if *δ* = 0 and nonzero components of the elements in the second raw are positive semidefinite, it can claim that

and so Eq. (93) implies the linear matrix inequality

Thus, using Eq. (59) then Eq. (94) implies Eq. (88). This concludes the proof.

**Theorem 6** *The adaptive fault observer (19) and (20) is* *stable if for given positive scalars* *a > ϱ, there exist a symmetric positive definite matrix* *matrices* *such that*

*When the above conditions are affirmative the observer gain matrix can be computed by Eq*. (73).

*Proof*. Considering only conditions implying from fault-free autonomous system (equivalent to ** F** =

**0,**=

*C***0**), then Eq. (88) implies directly Eq. (96). This concludes the proof.

Note, due to two integral quadratic constraints, setting the circle parameters to define

## 5. Extended design conditions

In order to be able to formulate the fault observer equations incorporating the symmetric, positive definite learning weight matrix ** G**, Eqs. (21), (29), and (30) can be rewritten compositely as

Since Eq. (98) can rewritten as follows

introducing the notations

where

where

and

It is necessary to note that, in general, the elements of the positive definite symmetric matrix ** G** are unknown in advance, and have to be interactive set to adapt the observer error to the amplitude of the estimated faults. Of course, even this formulation does not mean the elimination of the matrix equality from the design conditions, because the matrix structure of

**Theorem 7**. *The adaptive fault observer (19) and (20) is stable if for a given symmetric, positive definite matrix* *there exist symmetric positive definite matrix* *and matrices* *such that*

*where* *take the structures*

*When the above conditions hold, the observer gain matrix is given by*

*Proof*. Given

where

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

Thus, Eqs. (113) and (115) imply Eqs. (105) and (106). This concludes the proof.

**Theorem 8** *The adaptive fault observer (19) and (20) is* *stable if for a given symmetric, positive definite matrix* *and positive scalars* *if there exist a symmetric positive definite matrix* *and matrices* *such that*

*where *.

*Proof*. Theorem 8, constructed as a generalization of the results giving stability conditions for adaptive fault observers, implies directly from Theorems 1 and 6. This concludes the proof.

## 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

where

and Eq. (120) follows directly

the systems (16) and (17), the fault estimation equation (21) and (121) can be expanded as

where *I*_{m} is the identity matrix of given dimension. Using the notations (118), (119), and

and applying the feedback control law (117) to the state space system in Eqs. (126) and (127), the expanded closed loop system becomes

where the closed-loop system matrix of the expanded system is

In order to design the system with reference attenuations *γ*_{2} and *γ*_{∞}, respectively, in the following is considered the transfer function matrix

**Proposition 1** *(H*_{2} *control synthesis) The state feedback control (117) to the system (126) and (127) exists and* *if for a given symmetric, positive definite matrix* *there exist symmetric positive definite matrices* *a matrix* *and a positive scalar* *such that*

*where*

When the above conditions hold, the control law gain is

*Proof*. Replacing in the inequality (6), the couple

and so using the Schur complement property and the notation

Analogously, replacing in Eq. (7), the couple _{2} control is now to minimize the constraint

Introducing the inequality

with a new matrix variable

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).

**Proposition 2** *(H*_{∞} *control synthesis) The state feedback control (117) to the systems (126) and (127) exists and* *if for a given symmetric, positive definite matrix* *there exist a symmetric positive definite matrix* *a matrix* *and a positive scalar* *such that*

*where*

*When the above conditions hold, the control law gain is*

*Proof*. Replacing in Eq. (9) the set of matrix parameters

Defining the transform matrix

and premultiplying the left side and postmultiplying the ride side of Eq. (144) by

Substituting Eq. (130) modifies the linear matrix inequality (146) as follows

and with the notation

Eq. (147) implies Eq. (141). This concludes the proof.

It is now easy to formulate a joint approach for integrated design of FTC, where

**Theorem 9** *The state feedback control (117) to the systems (126) and (127) exists and* *if for given symmetric, positive definite matrix* *and positive scalars* *there exist symmetric positive definite matrices* *and a positive scalars* *such that*

*where are* *as in Eq. (107)*, *as in Eq. (142), and* *as in Eq. (119)*.

*When the above conditions hold*

*Proof*. Prescribing a unique solution of

then Eqs. (132)–(134) and (140) and (141) in the joint sense imply Eqs. (152)–(154).

The design conditions are complemented by the inequalities (150) and (151), the same as Eq. (116). This concludes the proof.

Note, the introduced H_{2}H_{∞} control maximizes the H_{2} norm over all state-feedback gains _{∞} norm constraint is optimized. The set of LMIs (152)–(154) is generally well conditioned and feasible and, since

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 illustrate the proposed method, a system whose dynamics is described by Eqs. (16) and (17) is considered with the matrix parameters [43]

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 [44]. 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 *a* = 7, *ϱ* = 5 and the tuning parameter *δ* = 2, the resulted matrix parameters are

where

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 ** G** = 7.5 and using the optional factors as above, the resulted fault observer parameters are

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 (** A, B**) is controllable, the eigenvalues of the closed-loop system matrix

*A*_{c}=

**−**

*A***can be placed arbitrarily. Using the MATLAB function**

*BK**place.m*, the gain matrix

**is chosen that**

*K*

*A*_{c}has the eigenvalues {−1, −2, −3, −4}, i.e.,

and the signal gain matrix **W**_{w} is computed using the static decoupling principle as [45]

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

(159) |

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 ** G** = 7.5. Simulations results obtained under the same simulation conditions, but realized by applying Theorems 7 and 8 with the prescribed weight

**= 7.5, are given in**

*G***Figures 7**and

**8**.

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

Considering in the following an unforced system (126) and (127) and solving the set of linear matrix inequalities (132)–(135) to design FTC system parameters, the solution is obtained as follows

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.

When solving the design conditions (140) and (141) for the equations of the unforced systems (126) and (127), the result is the set of matrix variables

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 *H*_{2} and *H*_{∞} norm of the system transfer function are incorporated and the parameters of the feedback controllers (117) and (118) are computed from the following set of matrix variables satisfying Eqs. (152)–(155)

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

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 ** G** (

**Figure 7**). Somewhat better results can be achieved when the synthesis conditions incorporate the H

_{∞}norm of the fault transfer function (

**Figure 3**), even if they are combined with the use of an untying slack matrix

**(**

*Q***Figures 2**and

**5**). The best obtained results in accuracy and noise robustness are with the design conditions combining LMIs with constraints implying from

**Figures 4**,

**6**, and

**8**).

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 H_{2}/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 ** f**(

*t*) exceed its upper limit, there can be seen small fluctuations in compensation.

## 8. Concluding remarks

In this chapter, a modified approach for designing the adaptive fault observers is presented, and the _{2} 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 H_{2}H_{∞} 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.