Robust Control for Systems with Bounded-Sensor Faults

Abstract

In this study an observer-based novel design of robust control system with an estimate scheme of sensor states to accommodate extended bounded-sensor faults is proposed. The sensor faults are, in general, modeled as polytopic bounds in robust control framework and are usually given as a priori assumption. But, in practice, the sensors that are subject to faults are especially vulnerable to variant conditions, such as high temperature, and humidity. Thus, their faults may fall outside the presumed polytopic bounds easily. An estimate scheme of sensor states integrated into the observerbased control system captures the sensor faults outside the presumed region. The notion of quadratic stability stabilizes the overall system and guarantees a robust performance measure of an output error signal in the presence of a set of extended admissible sensor faults. A numerical example shows the effectiveness of the proposed approaches.

1. Introduction

The problems of designing fault-tolerant control systems have attracted considerable attention. Much efforts have gone to advancing practical usage of fault-tolerant systems within the avionics industry, see [1–4], for example. These high-risk applications require fault safe operation; that is, the system can withstand tolerable faults without effects on the system operations. Theoretical results and application examples being developed in the literature are, in general, divided into two broad spectrums: active [1, 5–8] and passive [2–4, 9–18]. In the active approach, reconfigurable mechanism has been designed in the event of bounded faults. Due to flexible capacity of the mechanism, the controller in such systems is not in a fixed form. This increases the complexity of the control design. Such fault-tolerant control systems use reconfigurable structure either online or predefined control law. [5] designed a pseud-inverse technique that maintained the closed-loop system as much as possible to the original system by reconfiguring the control law, once the faults encountered. [6] used eigenstructure assignment that was capable of recovering the nominal system after faults occurred. The controller was insensitive to the variation of devices [7, 8], have used direct adaptive control schemes to update control parameters and account for the case where the sensing devices were stuck at some fixed values. The advantages of active approaches are the use of reconfigurable or adaptive structure of online controller.

In contrast, the passive fault-tolerant control is to exploit the inherent redundancy of the system components or to use the remaining functions of the component to design a fixed compensator so as to achieve a tolerable system performance in the presence of component faults. The designed fixed controller guarantees satisfactory system performance not merely during normal operations, but under variant fault conditions. [9] tolerated faults by using linear quadratic state-feedback control for a prescribed set of actuators. A given performance bound was guaranteed. The controller of [10] based on robust pole region assignment method guaranteed stability and maintained an acceptable performance in the event of actuator faults. Taken as bounded measurement noises, sensor faults are estimated by fault estimation techniques in [15]. The proposed estimator accomplished the observation of system states and measurement noises simultaneously. [16] designed a fault estimation scheme for sensor with bounded time derivative of faults. Based on linear matrix inequality techniques, [2–4, 11, 12, 17, 18] have designed robust controllers for sensor or actuator faults in the prescribed bounds with disturbance attenuation and satisfactory performance. In addition, [17, 18] treated sensor fault as a state of the system, which implies the possible boundedness of the faults. Notably, the aforementioned passive approaches have focused on the faults in a fixed bound and given as a priori information. To increase flexibility of passive approach, the advantages of active approach being adaptive are absorbed into passive design in this study.

This paper deals with extended bounded-sensor-faults, in which sensor faults may fall outside the presumed bounds in time varying or nonlinear manners. A passive form of observer-based controller with an integrated estimate scheme of sensor states captures the phenomena of extended bounded-sensor-faults. A basic idea of control design to extended faults of sensors in an observer-based control system relies on the plant states being correctly estimated under the corrupted measurement signals. To reach this goal, [11] designed a novel observer with an estimate scheme of sensor states in which the asymptotic stability of the system was restricted to against bounded and piece-wise constant faults. This result, however, is somewhat unrealistic. The study of this paper adopts the observer with estimate scheme idea as [11] but relaxes the piece-wise constant restriction to allow the sensor faults to be time-varying and/or nonlinear with bounds. Moreover, these presumed faulty sensors are allowed not to be in the presumed bounds. The observer with estimate scheme identifies these faulty conditions and computes the correct states for feedback. The notion of quadratic stability not only ensures the asymptotic stability of the system but also establishes ℒ₂-gain measure for the robust performance of a controlled error signal in the presence of a set of extended admissible sensor faults.

This paper is organized as follows. In Section 2, the system including observer structure and fault models of bounded piece-wise constant function is formulated. To proceed with Section 2, Section 3 gives the preliminary results developed in [11]. Section 4 presents the main results, which deal with the robust performance against extended bounded-sensor faults and is divided into three subsections including system reformulation, quadratic stability and Linear matrix inequality (LMI) characterizations. Section 5 includes the synthesis of observer and control gains in terms of LMIs. Section 6 demonstrates effectiveness of proposed method by a numerical example. Last section, Section 7, concludes the overall results shown in the paper.

2. Problem Formulation

Consider a linear time-invariant dynamical system with sensor faults 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐵₁𝑑(𝑡),𝑥(0)=𝑥₀,𝑦(𝑡)(𝑡),𝑦=𝐶𝑥_𝑠(𝑡)=diag𝑦(𝑡)𝜙(𝑡)=diag𝜙(𝑡)𝑦(𝑡), (2.1) where 𝑥(𝑡)∈ℜ^𝑛 is the state vectors, and 𝑢(𝑡)∈ℜ^𝑚 is the control signal of actuator. 𝑦(𝑡)∈ℜ^𝑙 is the output of the system, and 𝑦_𝑠(𝑡)∈ℜ^𝑙 is the true measured output of sensor. 𝑑(𝑡)∈ℜ^𝑑 is the disturbance. The representation of diag[𝑦(𝑡)], when the vector 𝑦(𝑡) is a vector with 𝑙 components, is a square matrix of dimension 𝑙 with the elements of 𝑦(𝑡) on the diagonal. The sensor function, 𝜙(𝑡)∈ℜ^𝑙, is to represent the remaining function of the associated sensor. For example, if a sensor 𝜙_𝑘(𝑡)=0.8, in which 𝜙_𝑘(𝑡) represents the remaining function of 𝑘th sensor in the vector 𝜙(𝑡), then we say the sensor is 80% functioning.

Now, consider a state observer with control law of the following form: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐿𝑦_𝑠(𝑡)𝜙(𝑡(𝑡,𝑢(𝑡)(𝑡),−diag)𝐶𝑥)=𝐾𝑥 (2.2) where the vectors 𝑥∈ℜ^𝑛 are the state of observer, which is an estimate of 𝑥(𝑡). Similarly, 𝜙(𝑡), an estimate of 𝜙(𝑡), which will be shown later in the sequel, is useful to observe the state 𝑥(𝑡) in the presence of sensor faults. 𝐿 and 𝐾 are the observer gain and control gain, respectively, to be designed such that the control objectives are achieved. Notice that the following expressions are interchangeable when later deriving the formulas, diag[𝐶𝑥(𝑡)]𝜙(𝑡)=diag[𝜙(𝑡)](𝐶𝑥(𝑡)) and diag[𝐶𝑥(𝑡)]𝜙(𝑡)=diag[𝜙(𝑡)](𝐶𝑥(𝑡)).

3. Preliminaries

The following assumptions are used for demonstrating the asymptotic stability based on Lyapunov method shown in the Theorem 3.1, which will be then relaxed while ℒ₂-gain robust performance is pursued in the next section. (1)𝜙_𝑘(𝑡)∈[0,1], (2)lim_Δ𝑡→0(Δ𝜙_𝑘(𝑡_𝑖)/Δ𝑡)=0, where Δ𝜙_𝑘(𝑡_𝑖)=Δ𝜙_𝑘(𝑡_𝑖+Δ𝑡)−Δ𝜙_𝑘(𝑡_𝑖), except at some time instants that 𝜙_𝑘(𝑡) jump toward zero.

The assumptions addressed above have the following interpretations: 𝜙_𝑘(𝑡)=0 means the sensor fails. 𝜙_𝑘(𝑡)=1 means the sensor works properly. A fault sensor will be such that 0<𝜙_𝑘(𝑡)<1. Thus, it is a bounded sensor fault. The lim_Δ𝑡→0(Δ𝜙_𝑘(𝑡_𝑖)/Δ𝑡)=0 and 𝜙_𝑘(𝑡) jumping toward zero mean that the sensor fault not only is a piecewise constant process but also indicate that 𝜙_𝑘(𝑡) is a bounded above function. The analysis of asymptotic stability of the closed-loop system against bounded and piecewise constant sensor faults is first studied in [11], which is revealed in the following theorem and is stated for the completeness. We define the following sets: 𝚽=diag𝜙∣𝜙=𝜙^𝑇₁𝜙^𝑇₂⋯𝜙^𝑇_𝑙^𝑇,𝜙_𝑘∈ℜ,𝜙_𝑘∈𝜙_𝑘,𝜙_𝑘, (3.1) and the vertex set of Φ is defined as 𝓑𝚽=diag𝜙∣𝜙=𝜙^𝑇₁⋯𝜙^𝑇_𝑙^𝑇,𝜙_𝑘=𝜙_𝑘or𝜙_𝑘=𝜙_𝑘, (3.2) Notice that it is easy to see that there are 2^𝑙 vertices in 𝓑Φ to represent the possible faults in a known convex set.

Theorem 3.1. Assumptions 1 and 2 hold. Consider the system (2.1) and (2.2) for the case 𝐵₁=0 and diag[𝜙]∈𝓑Φ.
If the following exist:(1)the matrices 𝑄 and 𝐿 satisfying 𝑄=𝑄^𝑇Ξ>0,₁(𝑄)<0, (3.3) where Ξ₁(𝑄)=𝐴−𝐿diag𝜙𝐶^𝑇𝑄+𝑄𝐴−𝐿diag𝜙𝐶, (3.4)(2)the matrices 𝑃 and 𝐾 satisfying 𝑃=𝑃^𝑇Ξ>0,₂(𝑃)<0, (3.5) where Ξ₂(𝑃)=(𝐴+𝐵𝐾)^𝑇𝑃+𝑃(𝐴+𝐵𝐾), (3.6)(3)for a given matrix 𝑆>0, the matrices 𝑊>0 and Γ satisfying Γ=Γ^𝑇>0,𝑊+𝑊^𝑇−𝑆>0, (3.7)
then the closed-loop systeṁ𝜙𝐶𝑥̇=𝐴𝑥+𝐵𝐾𝑥,𝑥=𝐴+𝐵𝐾−𝐿diag𝑥+𝐿𝑦_𝑠̇Γ[𝐿,𝜙=diag𝐶𝑥]^𝑇[]𝐿𝑄𝑥for𝜙∈𝐷−Γ𝑊𝜙+Γdiag𝐶𝑥^𝑇𝑄𝑥for𝜙∈̇𝜙𝐷,𝑥=𝐴𝑥−𝐿diag𝐶𝑥+𝐿𝑦_𝑠 (3.8a)(3.8b)(3.8c)(3.8d)is asymptotically stable for 𝜙∈𝐷, where 𝐷 is as follows: 𝐷=𝜙∣𝜙²≤𝜆𝑙𝜌,𝐷=𝜙∣𝜙²>𝜆𝑙𝜌. (3.9) The state error is defined as 𝑥=𝑥−𝑥. The parameters, 𝜆 and 𝜌, are the maximum and minimum eigenvalues of 𝑊^𝑇𝑆⁻¹𝑊 and 𝑊+𝑊^𝑇−𝑆, respectively, for some positive definite symmetric matrices 𝑊 and 𝑆 and 𝑊+𝑊^𝑇−𝑆>0.

Proof. Refer to [11] for the proof.

Remark 3.2. The overall closed-loop system (3.8a)–(3.8d) decomposed into input-output structured block diagram in Figure 1, where the error filter (3.8d), the adaptive law (3.8c), and the state observer (3.8b) form the augmented observer. Without sensor faults, the block diagram Figure 1 is simplified to Figure 2 where 𝜙 and 𝜙 are both identity matrices, and the block of adaptive law can be removed from Figure 1.

Figure 1 

The input-output structured block diagram of the closed-loop system (3.8a)–(3.8d).

Figure 2 

An ordinary observer-based control system.

4. Robust Performance

In the last section the asymptotic stability, based on Lyapunov method, has been shown for the system with bounded sensor faults under piecewise constant assumption. Now, a set of extended sensor faults relax the previous restrictions to not only admit bounded time varying and/or nonlinear sensor function in the vertex set, 𝓑Φ, but also let the true sensor faults, which may fall outside the presumed bound, be norm bound. A closed-loop autonomous system will be reformulated in the following subsection.

4.1. System Reformulation

Consider closed-loop system (3.8a)–(3.8d), which can be rewritten as 𝑥̇_𝑐=𝐴_𝑐𝑥_𝑐+𝐵_𝑐𝑤,𝑥_𝑐(0)∈ℬ,𝑞=𝐶_𝑞𝑥_𝑐,𝑒=𝐶_𝑒𝑥_𝑐,𝑤=diag𝜙𝑞, (4.1) where 𝑥_𝑐=𝑥𝜙𝑥,𝐵_𝑐=00𝐶−𝐿_𝑞,=𝐶𝐶0,𝐶_𝑒=𝐴𝐶00,_𝑐=𝜙Γ[]𝐿𝐴−𝐿diag𝐶00𝐵𝐾𝐴+𝐵𝐾0diag𝐶𝑥^𝑇𝑋₁0−Γ𝑊, (4.2)(4.3)(4.4)diag[𝜙]∈𝓑Φ, and diag[𝜙]=diag(𝜙₁,…,𝜙_𝑙). ℬ⊂ℜ^𝑛 defines a domain that contains the origin. The matrices 𝐾, 𝐿, Γ, and 𝑋₁ are to be determined in the sequel. The signal, 𝑤, is a bounded exogenous signal that comprises 𝜙 and 𝐶𝑥, that is, 𝑤=diag[𝜙]𝐶𝑥, where 𝜙 is the difference of estimated sensor function, 𝜙, and true sensor function, 𝜙, that is 𝜙=𝜙−𝜙. The estimation error 𝜙(𝑡) is assumed to belong to the following diagonal norm-bound set: 𝚫≜𝜙_𝑖(𝑡)∣𝜙_𝑖(𝑡)≤1,𝑖=1,…,𝑙, (4.5) which defines the extended sensor faults.

Remark 4.1. We have noticed that if the true sensor function falls within the presumed polytopic bound, 𝓑Φ, then the diagonal norm-bound set, which is confined within Δ, is partially (or maybe completely) overlaid with the 𝓑Φ. However, if the true sensor function does not fall into the set 𝓑Φ, then it is possible that 𝓑Φ intersects Δ partially or their intersection is an empty set.

Remark 4.2. It is worth noting that, under the above assumption, the signal 𝑤 is such that 𝑤^𝑇𝑤=𝑥^𝑇𝐶^𝑇diag𝜙^𝑇diag𝜙𝐶𝑥≤𝑥^𝑇𝐶^𝑇𝐶𝑥, (4.6) which for asymptotically stable systems can be bounded above and will be demonstrated in the sequel.

4.2. Robust Performance Measure

This subsection defines a robust performance measure and states an important theorem on which the robust performance is established. We assume that not all state information is available and is concerned with designing a fixed structure observer-based controller to stabilize the system (4.1) satisfying a given ℒ₂-gain constraint in response to all admissible sensor faults. The following notion of stabilizability for the system (4.1) with robust ℒ₂-gain measure is defined.

Definition 4.3. Let the constant 𝛾>0 be given. The system (4.1) is said to be stabilizable with a robust ℒ₂-gain measure 𝛾 if there exists a fixed control law 𝑢=𝐾𝑥 and observer gain 𝐿 such that for any admissible sensor faults the following conditions are satisfied.(1)The system is uniformly asymptotically stable.(2)Subject to the assumption of zero initial condition, the controlled output 𝑒 satisfies ‖𝑒‖₂‖𝑤‖₂≤𝛾. (4.7)Here, the quadratic stability with an ℒ₂-gain measure was introduced [19]. This concept is a generalization of that of quadratic stabilization to handle ℒ₂-gain measure constraint on exogenous attenuation. To this end, the characterizations of robust performance based on quadratic stability will be given in terms of matrix inequalities, where if LMIs can be found, then the computations by finite dimensional convex programming are efficient.

Theorem 4.4. Consider the closed-loop system (4.1), and the following statement holds: the closed-loop system is said to be quadratically stable with a robust ℒ₂-gain measure 𝛾 from input 𝑤 to output 𝑒 if there exists 𝑋>0 and 𝜆≥0 such that Π<0, (4.8) where 𝐴Π=^𝑇𝑐𝑋+𝑋𝐴_𝑐+𝐶^𝑇_𝑒𝐶_𝑒+𝜆𝐶^𝑇_𝑞𝐶_𝑞𝑋𝐵_𝑐𝐵^𝑇_𝑐𝑋−𝛾²𝐼. (4.9)

Proof. Let quadratic Lyapunov function be 𝑉(𝑥_𝑐)=𝑥^𝑇_𝑐𝑋𝑥^𝑇_𝑐, with 𝑋>0 such that ∀𝑥_𝑐(),𝑑,and𝑤satisfying4.1𝑉𝑥𝑑𝑡^𝑐+𝑒^𝑇𝑒−𝛾²₁𝑤^𝑇𝑤<0, (4.10)(4.11) and constraint 𝑤^𝑇𝑤≤𝑥^𝑇_𝑐𝐶^𝑇_𝑞𝐶_𝑞𝑥_𝑐. (4.12) Then, it follows from the S procedure that the equivalent condition of (4.11) is the existence of 𝜆≥0 satisfying 𝑑𝑉𝑥𝑑𝑡^𝑐+𝑒^𝑇𝑒−𝛾²₁𝑤^𝑇𝑤+𝜆𝑥^𝑇_𝑐𝐶^𝑇_𝑞𝐶_𝑞𝑥_𝑐−𝑤^𝑇𝑤≤0, (4.13) which can be equivalently written as 𝑑𝑉𝑥𝑑𝑡^𝑐+𝜆𝑥^𝑇_𝑐𝐶^𝑇_𝑞𝐶_𝑞𝑥_𝑐+𝑒^𝑇𝑒−𝛾²𝑤^𝑇𝑤≤0 (4.14) for 𝛾²=𝛾²₁+𝜆. Then the ℒ₂-gain of the (4.1) is less than 𝛾. To show this, we integrate (4.14) from 0 to 𝑇, with the initial condition 𝑥_𝑐(0)=0, to get 𝑉𝑥_𝑐(𝑇)+𝜆^𝑇₀𝑥^𝑇_𝑐𝐶^𝑇_𝑞𝐶_𝑞𝑥_𝑐𝑑𝑡+^𝑇₀𝑒^𝑇𝑒−𝛾²𝑤^𝑇𝑤𝑑𝑡≤0. (4.15) Since 𝑉(𝑥_𝑐(𝑇))+𝜆∫^𝑇₀𝑥^𝑇_𝑐𝐶^𝑇_𝑞𝐶_𝑞𝑥_𝑐𝑑𝑡≥0, this implies that ‖𝑒‖₂‖𝑤‖₂≤𝛾. (4.16) The inequality (4.8) and definition (4.9) are obtained by substituting (4.1) into (4.14). Without loss of generality, we will adopt only strict inequality in this paper. This completes the proof.

4.3. Matrix Inequality Characterizations

The following lemma is to show that the energy of the estimated output signals by observer can be limited by some matrix inequalities, which provide an upper bound of the exogenous signal, 𝑤.

Lemma 4.5. Given that 𝑥̇_𝑐=𝐴_𝑐𝑥_𝑐,𝑥_𝑐(0)∈ℬ,𝑦_𝑖=𝐶_𝑞,𝑖𝑥_𝑐,ℬ=𝑥_𝑐∣𝑥^𝑇_𝑐𝑋𝑥_𝑐≤𝜈,𝑋>0,𝜈>0, (4.17) if there exist 𝑋>0 and 𝜃_𝑖>0 satisfying 𝐴^𝑇_𝑐𝑋+𝑋𝐴_𝑐Υ<0,_𝑖>0,𝑖=1,…,𝑙, (4.18) then the following statements are equivalent:(1)𝑦²_𝑖<(𝜃_𝑖/𝜈), (2)Θ>0, where Υ_𝑖=𝑋𝐶^𝑇_𝑞,𝑖𝐶_𝑞,𝑖𝜃_𝑖𝜈𝐼[])[]𝜃,Θ=diag𝐶𝑥(−𝜈𝐼diag𝐶𝑥+Θ,Θ=diag₁,…,𝜃_𝑙. (4.19)(4.20)

Proof. ℬ is an invariant ellipsoid. Let 𝑉(𝑥_𝑐)=𝑥^𝑇_𝑐𝑋𝑥_𝑐. Since 𝑉̇𝑥_𝑐=𝑥^𝑇_𝑐𝐴^𝑇_𝑐𝑋+𝑋𝐴_𝑐𝑥_𝑐<0 (4.21) then we have 𝑉𝑥_𝑐(𝑡)≤𝑉𝑥_𝑐(0)≤𝜈. (4.22) Hence, 𝑥_𝑐(𝑡)∈ℬ,forall𝑡≥0 and 𝑦²_𝑖≤max_{𝑥𝑐∈ℬ}𝑥^𝑇_𝑐𝐶^𝑇_𝑞,𝑖𝐶_𝑞,𝑖𝑥_𝑐. (4.23) Let 𝜃_𝑖=max_{𝑥𝑐∈ℬ}𝑥^𝑇_𝑐𝐶^𝑇_𝑞,𝑖𝐶_𝑞,𝑖𝑥_𝑐. We have 𝑋𝐶^𝑇_𝑞,𝑖𝐶_𝑞,𝑖𝜃_𝑖𝜈𝐼>0,⟺𝐶^𝑇_𝑞,𝑖𝐶_𝑞,𝑖<𝜃_𝑖𝜈𝑋,∀𝑖. (4.24) From (4.23) and (4.24), we have 𝑦²_𝑖≤max_{𝑥𝑐∈ℬ}𝑥^𝑇_𝑐𝐶^𝑇_𝑞,𝑖𝐶_𝑞,𝑖𝑥_𝑐<𝜃_𝑖𝜈,∀𝑖. (4.25) This completes the proof of (1). The equivalence of (1) and (2) is straightforward.

Before stating the main theorem for the robust ℒ₂-gain measure 𝛾 of the closed-loop system (4.1), which ensures the robust performance of the original system (2.1) and (2.2) against sensor faults, the following matrices are defined: Π₁=Ξ₂𝑋₂+𝜆𝐶^𝑇𝐶𝑋₂𝐵𝐾+𝜆𝐶^𝑇()𝐶0𝐵𝐾^𝑇𝑋₂+𝜆𝐶^𝑇𝐶(⋆)−𝑋₁𝐿0−𝐿^𝑇𝑋₁−𝛾²𝐼,Π₂=−(Γ𝑊)^𝑇𝑋₃−𝑋₃(Γ𝑊)+2𝜅𝑋₃𝑋₃ΓΓ^𝑇𝑋₃,−𝜈𝐼 (4.26) where (⋆)=Ξ₁𝑋₁+(𝜆+1)𝐶^𝑇𝐶+𝑋₁𝐿Θ𝐿^𝑇𝑋₁. (4.27)

Theorem 4.6. Let the 𝛾>0, 𝜅>0, and 𝜈>0 be given. The closed-loop system (4.1) with the admissible bounded sensor faults is said to be quadratically stable with a robust ℒ₂-gain measure 𝛾, if one lets the matrix 𝑋 be in the set 𝜒𝜒=blockdiag𝑋₁,𝑋₂,𝑋₃∣𝑋_𝑖=𝑋^𝑇_𝑖>0, (4.28) there exist 𝐾, 𝐿, Γ>0, Θ>0, and 𝑋∈𝜒 such that Π₁<0, Π₂<0, and Υ_𝑖>0,𝑖=1,…,𝑙.

Proof. We consider the signals 𝑒(⋅) in response to the signals 𝑤(⋅) with the zero initial states, and a (candidate) quadratic Lyapunov function 𝑉(𝑥_𝑐)≜𝑥^𝑇_𝑐𝑋𝑥_𝑐. For any nonzero vectors, 𝜍₁ and 𝜍₂ are defined as 𝜍₁=𝑥𝑤𝑥,𝜍₂=𝜙diag[𝐶𝑥]𝐿^𝑇𝑋₁𝑥. (4.29) We have 𝜛=𝜍^𝑇₁Π₁𝜍₁+𝜍^𝑇₂Π₂𝜍₂=𝑉̇𝑥_𝑐+𝜆𝑞^𝑇𝑞+𝑒^𝑇𝑒−𝛾²𝑤^𝑇𝑤+𝑥^𝑇𝑋₁𝐿Θ𝐿^𝑇𝑋₁𝑥+𝜙^𝑇2𝜅𝑋₃𝜙, (4.30) where Θ is defined in (4.20). Let 𝜁=(^𝑥𝑐_𝑤), thus 𝜛=𝜁^𝑇Π𝜁+𝑥^𝑇𝑋₁𝐿Θ𝐿^𝑇𝑋₁𝑥+𝜙^𝑇2𝜅𝑋₃𝜙, (4.31) where Π is defined by (4.9). If the matrix inequalities, Π₁<0, Π₂<0, and Υ_𝑖>0, are satisfied, we have 𝜛<0, or equivalently, 𝜁^𝑇Π𝜁<−𝑥^𝑇𝑋₁𝐿Θ𝐿^𝑇𝑋₁𝑥−𝜙^𝑇2𝜅𝑋₃𝜙<0, (4.32) for nonzero 𝑥 and 𝜙, which implies that Π<0. Hence, by Theorem 4.4, the closed-loop system is quadratically stable with a robust ℒ₂-gain measure 𝛾 is ensured. This completes the proof.

Remark 4.7. It is highlighted that the 𝜅 is defined to be the decay rate of the estimated sensor function 𝜙 shown in (3.8a)–(3.8d) such that lim_𝑡→∞𝑒^𝜅𝑡‖𝜙‖=0, when 𝜙∈𝐷.

Remark 4.8. The ideas of using 𝜒 to set the block diagonal matrix are from the separation principle, where the observer and control gain are designed independently [20], and the lower triangular matrix structure of 𝐴_𝑐, where the diagonal elements are the major concerns of the design [21]. It is shown later in the next section that the computation of Lyapunov matrix for controller (𝑋₂) and observer (𝑋₁) is independently accomplished. Hence, in contrast to the general conservative issue of LMI due to the use of a single Lyapunov matrix for all design, the implementation of the proposed block diagonal matrix 𝜒 obtains less conservativeness. To demonstrate the fact, the robust ℒ₂-gain measure in the numerical example shows that the indexed number 𝛾 is much less than the number shown in [12].

Before presenting the synthesis results in the next section, a useful and important lemma will be stated for clarity.

Lemma 4.9 (Elimination Lemma see [19]). Given ℋ=ℋ^𝑇∈ℜ^𝑛×𝑛, 𝒱∈ℜ^𝑛×𝑚, and 𝒰∈ℜ^𝑛×𝑝 with 𝑅𝑎𝑛𝑘(𝒱)<𝑛 and 𝑅𝑎𝑛𝑘(𝒰^𝑇)<𝑛, there exists a matrix 𝐾 such that ℋ+𝒱𝐾𝒰^𝑇+𝒰𝐾^𝑇𝒱^𝑇<0 (4.33) if and only if 𝒱^𝑇_⟂ℋ𝒱_⟂<0,𝒰^𝑇_⟂ℋ𝒰_⟂<0, (4.34) where 𝒱_⟂ and 𝒰_⟂ are orthogonal complement of 𝒱 and 𝒰, respectively; that is, 𝒱^𝑇_⟂𝒱=0 and (𝒱_⟂𝒱) are of maximum rank.

5. Control and Observer Gain Synthesis

In this section according to the analyzed results shown in the last section, the observer gain, 𝐿, and control gain, 𝐾, will be synthesized. The general LMI synthesis problem involves sets of the form 𝑋∈𝜒 and a list of matrices 𝐴,𝐵,𝐶,𝐿,𝐾,Γ, and 𝑊 and scalars 𝜈, 𝜅, and 𝜃_𝑖. We will conclude the quadratic stability with a robust ℒ₂-gain measure 𝛾 control problem in an convex optimization fashion. We will also specify the details in the following, and the results will be concluded in an algorithm of computation.

Assume diag[𝜙]∈𝓑Φ and 𝜙_𝑖∈Δ,forall𝑖. Given prespecified matrices 𝐴,𝐵,𝐶,𝑊>0 and scalars 𝜈>0 and 𝜅>0. If, according to Theorem 4.6, there exist matrices 𝑋∈𝜒,𝐿,𝐾,Γ, Θ>0 and 𝛾² such that Π₁<0, Π₂<0, and Υ_𝑖>0,forall𝑖, are satisfied, then the closed-loop system is quadratically stable with a robust ℒ₂-gain measure 𝛾. Let the matrix Π₁ be decomposed into Π₁=Π_{1_1}+Π_{1_2}<0, (5.1) where Π_{1_1}=Ξ₂𝑋₂𝑋₂()𝐵𝐾0𝐵𝐾^𝑇𝑋₂(⋆⋆)−𝑋₁𝐿0−𝐿^𝑇𝑋₁−𝛾²𝐼Π_{1_2},𝐶=𝜆𝐶^𝑇𝑇0()𝐶𝐶0,⋆⋆=Ξ₁𝑋₁+𝐶^𝑇𝐶+𝑋₁𝐿Θ𝐿^𝑇𝑋₁. (5.2)(5.3)(5.4) Thus, we have 𝜍^𝑇₁Π₁𝜍₁=𝜍^𝑇₁Π_{1_1}𝜍₁+𝜍^𝑇₁Π_{1_2}𝜍₁. (5.5) It is noted that 𝜍^𝑇₁Π_{1_2}𝜍₁=𝜆𝑞^𝑇𝑞≥0 (5.6) for 𝜆≥0. Next, by S procedure, the requirement of 𝜍^𝑇₁Π₁𝜍₁<0 is equivalent to 𝜍^𝑇₁Π_{1_1}𝜍₁<0, (5.7) which implies that Π_{1_1}<0. Hence, by the well-known Schur complement, the matrix Π_{1_1}<0 can be rewritten as Ξ₂𝑋₂𝑋₂()𝐵𝐾00𝐵𝐾^𝑇𝑋₂Ξ₁𝑋₁+𝐶^𝑇𝐶−𝑋₁𝐿𝑋₁𝐿0−𝐿^𝑇𝑋₁−𝛾²𝐼00𝐿^𝑇𝑋₁0−Θ⁻¹<0. (5.8) The inequality (5.8) can be again rewritten as ℋ+𝒱𝐾𝒰^𝑇+𝒰𝐾𝒱^𝑇<0, (5.9) where 𝐴ℋ=^𝑇𝑋₂+𝑋₂𝐴0000Ξ₁𝑋₁+𝐶^𝑇𝐶−𝑋₁𝐿𝑋₁𝐿0−𝐿^𝑇𝑋₁−𝛾²𝐼00𝐿^𝑇𝑋₁0−Θ⁻¹𝑋,𝒱=²𝐵000𝐼𝐼00,𝒰=. (5.10)(5.11) Next, the orthogonal complement of 𝒱 and 𝒰 is given by 𝒱_⟂=𝑋₂⁻¹𝐵_⟂0000𝐼0000𝐼0000𝐼,𝒰_⟂=𝐼00−𝐼000𝐼000𝐼, (5.12) In which 𝐵_⟂ is defined as the orthogonal complement of 𝐵 and is such that 𝐵^𝑇_⟂𝐵=0 and [𝐵_⟂𝐵] is of maximum rank. Similarly, 𝒱 and 𝒰 are defined as orthogonal complement of 𝑉 and 𝑈, respectively.

It is followed by using the well-known Elimination Lemma stated in Lemma 4.9 that the matrix variable, 𝐾, can be eliminated from the inequality (5.9), which is equivalent to the following two inequalities: 𝒱^𝑇_⟂ℋ𝒱_⟂<0,𝒰^𝑇_⟂ℋ𝒰_⟂<0. (5.13) Substituting (5.10) and (5.12) into (5.13), we have 𝒱^𝑇_⟂ℋ𝒱_⟂=𝐵^𝑇_⟂𝑋₂⁻¹𝐴^𝑇+𝐴𝑋₂⁻¹𝐵_⟂0000Ξ₁𝑋₁+𝐶^𝑇𝐶−𝑋₁𝐿𝑋₁𝐿0−𝐿^𝑇𝑋₁−𝛾²𝐼00𝐿^𝑇𝑋₁0−Θ⁻¹<0, (5.14) which is equivalent to 𝐵^𝑇_⟂𝑋₂⁻¹𝐴^𝑇+𝐴𝑋₂⁻¹𝐵_⟂Ξ<0,₁𝑋₁+𝐶^𝑇𝐶−𝑋₁𝐿𝑋₁𝐿−𝐿^𝑇𝑋₁−𝛾²𝐿𝐼0_𝑇𝑋₁0−Θ⁻¹𝒰<0,_𝑇⟂ℋ𝒰_⟂=𝐴^𝑇𝑋₂+𝑋₂𝐴+Ξ₁𝑋₁+𝐶^𝑇𝐶𝑋₁𝐿−𝑋₁𝐿𝐿^𝑇𝑋₁−𝛾²𝐼0−𝐿^𝑇𝑋₁0−Θ⁻¹<0. (5.15)(5.16)(5.17) It is noted that (5.15), (5.16), and (5.17) cannot be solved simultaneously using LMI Toolbox of Matlab due to its nonconvexity in matrix variable, 𝑋₂. We, therefore, propose that 𝑋₂⁻¹ of (5.15) be solved first. It is easy to find 𝑋₂ using 𝑋₂⁻¹. If we let 𝐿=𝑋₁𝐿, then (5.16) and (5.17) are LMIs in variables, 𝑋₁, 𝐿, 𝛾², and Θ, which can be solved simultaneously by the LMI Toolbox of Matlab. The rest LMIs considered are Π₂<0 for a prespecified decaying rate, 𝜅>0, 𝜈>0, and matrix 𝑊>0, we have −Γ𝑊−𝑊^𝑇Γ^𝑇+2𝜅𝑋₃ΓΓ^𝑇−𝜈𝐼<0,𝑋₃>0, (5.18) where Γ=𝑋₃Γ. And lastly for Υ_𝑖>0 using Schur complement, we equivalently have 𝜈𝑋−𝜃_𝑖𝐶^𝑇_𝑞,𝑖𝐶_𝑞,𝑖>0,𝑖=1,…,𝑙, (5.19) where Θ=diag(𝜃₁,…,𝜃_𝑙).

Remark 5.1. Now, we can summarize the step of computation.(1)Find feasible solutions of 𝑋₂⁻¹>0 and thus 𝑋₂ where LMI (5.15) is satisfied.(2)Use the computed matrices 𝑋₂ found in step (1), the robust ℒ₂-gain performance problem is placed as the following optimization problem: minimize𝛾²(),(),()()𝑋subjectto5.165.175.18,and5.19₁>0,𝑋₃>0,Θ>0. (5.20)(3)Reconstruct control gain, 𝐾, and the matrices found in step (1) and (2) are substituted into the inequality (5.9), which is LMI on one matrix variable, 𝐾 and can be solved by LMI Toolbox of Mtalab.

6. Numerical Example

This example adopted from [12] will be used to illustrate the proposed design with the following parameters: 001𝐴=0100016−1−4,𝐵=,𝐶=100010. (6.1) We will test three cases to verify the superiority of the designed observer-based controller. First, case no. 1: one sensor fault where the conditions for simulation are similar to those in [12], in which the sensor 𝑦₁ is perfectly normal and sensor 𝑦₂ is subject to faults with 50% reduction in signal strength. That is, by Theorem 3.1, 𝜙₁ is equal to 1, and 𝜙₂ is allowed to be time varying or nonlinear function and varies between 0 and 0.5, that is, 0≤𝜙₂≤0.5. In the simulation, we let 𝜙₂=0.25+0.1sin(10𝑡) to examine the design. Using Matlab LMI control toolbox and following the step of computation in Remark 5.1, the matrix, 𝑋₂, is computed as follows:𝑋₂=1.58291.69590.96101.69593.81581.69590.96101.69591.5829. (6.2)

Then we solve optimization problem proposed by (5.20). We find that 𝑋₁=10⁷,,9.3574−7.9408−1.4166−7.94088.8721−0.9313−1.4166−0.93132.3478,𝐿=20.00100.026320.00100.026320.00100.0263𝐾=−7.0000−1.70001.3000 (6.3) and the optimal value of 𝛾² is 56.0603. The control gain 𝐾 places the eigenvalues of matrix 𝐴+𝐵𝐾 at {−1.00,−0.85±𝑗0.5268}. It is of interest that the eigenvalues of 𝐴−𝐿(diag[𝜙])𝐶 are placed around {−19,−2,−3} for all 𝜙₂=0.25+0.1sin(10𝑡). It is highlighted that 𝐿(diag[𝜙])𝐶 intends to closely map the fault sensor signal into its null space. This claim can be verified by viewing 𝐿diag𝜙𝐶=20.0010𝜙₁0.0263𝜙₂020.0010𝜙₁0.0263𝜙₂020.0010𝜙₁0.0263𝜙₂0, (6.4) where the effectiveness of 𝜙₂ on the eigenvalues of 𝐴−𝐿(diag[𝜙])𝐶 has been greatly reduced. Figures 3 and 4 show the complete simulation results. In Figure 3, the observed states, 𝑥, converge to the plant states, 𝑥, eventually. In Figure 4, it is noted that the estimate sensor function 𝜙₁ approaches very closely to its true value. However, for fast variation of sensor function 𝜙₂, the estimated sensor function, 𝜙₂, does not follow it but stay at certain value as 𝑡→∞. We must highlight that the estimate scheme of sensor function may not be able to approach the true sensor function, see [22] for detail, if lacking of persistent excitation, but to keep the the difference of estimated signal and true sensor function within a bound. It is seen from Figure 9 that 𝜙_𝑖∈Δ,𝑖=1,2. In Figure 4, the control input, 𝑢, is also shown. It is worth noting that the smallest possible value of 𝛾² obtained by [12] was 416.9764 (𝛾=20.42). Compare with 50.0603, the number of [12] is much larger. The reasons for that are not hard to conduct; the adaptive law and estimate scheme of faults capture the rate of change of faults, while [12] assumed only bounds of all faults.

Figure 3 

(Case no. 1): The comparison of the plant states, 𝑥, and observed states, 𝑥, of one sensor fault for the true sensor function 𝜙2∈𝓑Φ.

Figure 4 

(Case no. 1): True sensor functions, 𝜙1=1 and 𝜙2=0.25+0.1sin(10𝑡), and estimated sensor functions, 𝜙1 and 𝜙2, for 𝜙2∈𝓑Φ. The control input, 𝑢, is also shown.

The second simulation, case #2, uses the condition, which is similar to the previous case, where true sensor function 𝜙₁=1 and the true sensor function 𝜙₂ are not in the designed interval, that is, 𝜙₂∉𝓑Φ, which makes 𝜙₂>0.5 or 𝜙₂<0. In the simulation, we let true sensor function, 𝜙₂=1.5+0.1sin(10𝑡). The simulation results can actually be predicted and are similar to the previous case since the computed observer gain 𝐿 closely maps the second sensor signal into its null space. It is easy to predict that the states, 𝑥 and 𝑥, will be similar to Figure 1 and is shown in Figure 5. In Figure 6, the true sensor functions, 𝜙₁ and 𝜙₂, estimated sensor functions, 𝜙₁ and 𝜙₂, and control signals, 𝑢, are depicted. It is shown that although the estimated signal, 𝜙₂, is within the diagonal norm bound, Δ, which is shown in Figure 6, its results are deteriorated by the poor guess of its initial states.

Figure 5 

(Case no. 2): The comparison of the plant states, 𝑥, and observed states, 𝑥, of one sensor fault for the true sensor function 𝜙2∉𝓑Φ.

Figure 6 

(Case no. 2): True sensor functions, 𝜙1=1 and 𝜙2=1.5+0.1sin(10𝑡), and estimated sensor functions, 𝜙1 and 𝜙2, for 𝜙2∉𝓑Φ. The control input, 𝑢, is also shown.

The third simulation, that is, (case no. 3), is to show two sensor faults. We allow the true sensor function, 𝜙₁, shown in Figure 8, to be varying in the pulse form between 0.4 and 0.7 for the first 0.8 second and then 𝜙₁=0.6+0.03sin(10𝑡),𝑡≥0.8, while keeping 𝜙₂=0.25+0.1sin(10𝑡) for 𝑡≥0. To compute 𝐾 and 𝐿, we assume the polytopic bound with 0.4≤𝜙₁≤1 and 0≤𝜙₂≤0.5. We surprisingly found that 𝐾=(^{−6.9980−1.69651.3033}), which is close to the one sensor fault case, and 𝐿^𝑇=^{199.9852199.9852199.9852}_{0.03330.03330.0333}. Under two sensor fault cases, we have raised the 𝛾²=222.5654. It is not surprising to have the result since the signals for feedback are extremely weak. We have shown the comparison of the true states and observed states in Figure 7, where the large ripples of the observed states are produced due to the pulses of the true sensor function in the first 0.8 second but soon died out. In Figure 8, the control input, the true sensor signals, and estimate sensor signals are depicted. We recognize again from Figure 9 that although we assume that 𝜙_𝑖∈Δ,𝑖=1,2, it really depends on the good guess of initial states on the estimate scheme due to viewing the fact that 𝜙₂ maintains at its initial states. This, however, can be understood that the associated sensor function is not subject to persistent excitation [22] since it is mapped closely to its null space. Notably, for the third case, the method in [12] failed.

Figure 7 

(Case no. 3): The comparison of the plant states, 𝑥, and observed states, 𝑥, of two sensor fault for the true sensor function 𝜙𝑖∈𝓑Φ,𝑖=1,2.

Figure 8 

(Case no. 3): The true sensor functions, 𝜙1= pulses shown in the upper figure for 𝑡≤0.8 and 𝜙1=0.6+0.03sin(10𝑡),𝑡≥0.8 and 𝜙2=0.25+0.1sin(10𝑡), and the estimated sensor functions, 𝜙1 and 𝜙2. The control input, 𝑢, is also shown.

(a)

(b)

(c)

Figure 9 

Solid-diamond line represents |𝜙1|, and solid-star line represents |𝜙2|. (a), (b), and (c) are associated with simulation cases 1, 2, and 3, respectively.

7. Conclusion

This paper has developed an observer-based robust control system with an estimate scheme of sensor states against extended bounded-sensor faults. In this design, the control system not only can deal with the sensor fault in a prescribed polytopic bound but also can endure the faults outside the bound. Based on the notion of quadratic stability with a robust ℒ₂-gain measure 𝛾, sufficient conditions for the solvability of the robust control problem have been obtained, and a complete solution was given in terms of LMIs. The numerical example shows the effectiveness of the designed method.