# Design of Reduced-Order Multiple Observers for Uncertain Systems with Unknown Inputs

Authors: Mihai Lungu
Journal: Complexity
 (2018)

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## Abstract

The paper presents the design of a new reduced-order multiple observer for the estimation of the state associated with Takagi-Sugeno systems with unknown inputs, this being only the second reduced-order multiple observer ever designed. The design of reduced-order multiple observers which can achieve the finite-time state reconstruction for nonlinear systems described by multiple models is a niche area problem; the author of this paper continuing his work started with the introduction of the reduced-order multiple observer concept. The new multiple observer is a combination of a typical reduced-order observer for linear-time invariant multivariable systems and a full-order multiple observer for Takagi-Sugeno systems. The sufficient stability conditions of the observer are derived via the Lyapunov theory and its robustness is improved by means of a novel and efficient method which cancels the negative effect of the uncertainties appearing in the system. To validate the suggested design algorithm, the steps of the design procedure have been summarized and software implemented for the concrete case of a light aircraft lateral-directional motion.


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## Body

## 1. Introduction

In many real-world applications, there are difficulties in obtaining the measurement of the state variables describing the functioning of a system; sometimes, this is even impossible because of the physical constraints and/or economical restrictions; the usage of observers instead of sensors is a solution largely adopted in order to avoid these problems. The state reconstruction is also affected by some unmeasurable or inaccessible inputs (denoted as unknown inputs), which can be at the origin of biased estimations if these unknown inputs are not correctly taken into consideration [1–3]. The estimation of the states and unknown inputs (noises, measurement uncertainties, faults of sensors or actuators, etc.) for a physical system is needed in order to conceive a control strategy able to minimize the negative effects of the disturbances [4, 5]. There are differences between the observers designed for linear or nonlinear systems, the design process being more difficult in the second case; to overcome this drawback, viable solutions can be the model order reduction [6, 7] or the usage of the linearization method to obtain a linear system, because this technique (detailed in [8]) allows the transformation of any nonlinear system into the so-calledmultiple model—sum of linear models, each of them characterizing the system in a specific operating regime. The principle of the multiple model approach is based on the reduction of the system’s complexity by the decomposition of its operating space in a finite number of operating zones [9]; one obtains local models which are linear, affine, and time invariant systems due to the presence of some linearization constants. The relative contribution of each submodel is quantified by a weighting function. Initially, the multiple observers have been used for failure detection [10, 11] and switching control [12], where the multiple observers are designed for candidate plant models. In the research area of multiple models’ estimation, the aim of the paper [13] was the state estimation for perturbed output nonlinear systems; a Lyapunov based sliding mode multiple observer is presented, the proof of the observer stability being ensured by a suitable choice of estimation gains—solutions of linear matrix inequalities (LMIs). In [14], the authors focus on the state estimation of a nonlinear system described by a Takagi-Sugeno (T-S) multiple model having unknown inputs and outputs; the proposed approach consists of a mathematical transformation which enables the “conversion” of the unknown outputs into “pseudo” unknown inputs—eliminated through the designed multiple observer. In [15], a joint state and parameter estimation observer, proposed for T-S systems whose matrices depend on unknown parameters, is applied to a system model for a heat-exchanger-zones combination and the implementation challenges; the unknown parameters are rewritten using the sector nonlinearity approach through weighting functions and sector extreme values; also, an L2 formulation is used to handle the situation arising out of the difference between the actual and estimated weighting functions. Almost all papers dealing with the design of multiple observers assume that the weighting functions depend on measurable variables, few works being devoted to the case of nonmeasurable decision variables [3]; the paper [16] proposes a method which reduces the conservatism of the precedent works by reducing the number of linear matrix inequalities to be solved and relaxing the conditions for this method to be applied. Another multiple observer has been designed in [8]; it uses the interpolation principle for local observers and the calculation of the multiple observer gains is used to calculate the gains of the local observers. The output feedback controller’s design for Takagi-Sugeno fuzzy system with time-varying state-delay has been investigated in [17]. Also, a great amount of research works has been carried out on application of T-S fuzzy systems for networked nonlinear systems and fuzzy stochastic systems [18].In all works from literature dealing with the design of multiple observers for the estimation of states and unknown inputs, the designed observers arefull-order multiple observers (FOMOs); until now, there is only one paper in the literature (paper [19]) dealing with the design of areduced-order multiple observer(ROMO) for Takagi-Sugeno systems with unknown inputs; the observer belongs to the author of this paper who introduced for the first time the concept of reduced-order multiple observer, the design of such observers being, therefore, a niche research theme. The main advantage of the reduced-order multiple observers is related to the decrease of the system sensors’ number: some of the state variables are measured, while the others are estimated [19]; moreover, all the other advantages of general reduced-order observers can be considered advantages of ROMOs with respect to FOMOs. The main drawbacks of the ROMO from [19] are (1) the large number of existence conditions; (2) the usage of pole placement method because it can lead to the undesired phenomena called theeigenvalues’ sensitivity; (3) lack of solutions for robustness’ improvement in the case of uncertainties in the system; (4) high convergence time for the estimation error. The ROMO designed in this paper has smaller number of existence conditions, does not use the pole placement technique, and includes an innovative algorithm to increase its robustness with respect to uncertainties; moreover, it will be shown that the problem of ROMO’s design for T-S unknown input systems can be reduced to a standard one (the unknown inputs do not interfere in the equations of the observer).The observer designed in this paper (the second ROMO ever obtained) is a mixture of the approaches presented in [4, 20, 21], improved by a rigorous method to increase its robustness. In [20], the necessary and sufficient conditions are presented for the existence and the design of an unknown input functional observer for linear-time invariant (LTI) multivariable systems, subjected to unknown inputs, the existence of the observer being verified by computing a nullspace of a known matrix and testing some matrices’ rank conditions. In [21], a full-order observer without unknown inputs reconstruction is suggested in order to achieve the finite-time reconstruction of the state vector for a class of unknown inputs LTI systems, the observer design approach being based on an innovative method to obtain a full-order observer with no existence conditions. The paper [9] presents the design methodology of a FOMO associated with Takagi-Sugeno descriptor systems, a method to improve the observer robustness being also provided. In the present paper, the approaches from these three papers are partially used and additional and innovative new elements within the design approach are brought: the new form of the reduced-order observer, a new method for solving matrix equation systems with more unknowns than equations, an innovative method to improve the observer’s robustness with respect to uncertainties, and a “while” loop based method to increase the new ROMO’s generality and applicability, as well as to decrease its existence conditions’ number.The paper is organized as follows: the structure of a general multiple model is presented in Section2; the design methodology of the new reduced-order multiple observer is provided in Section 3; in the next section, the new ROMO’s robustness is improved by means of robustness terms introduced in the dynamics of the observer; a summarization of all the design approach steps is achieved in the fifth section, while in Section 6 the new observer is validated by means of numerical simulations for the concrete case of a light aircraft lateral-directional motion during landing; a brief comparison between the only two existing ROMOs (the one designed in this paper and the one from [19]) as well as between the new ROMO and other FOMOs from literature is also achieved. Finally, some conclusions are shared in Section 7.


## 2. Structure of the Takagi-Sugeno Multiple Model

Depending on the connections existing between its local models, two types of multiple models can be distinguished:(1) multiple models having homogeneous submodels (the local models have the same state) and (2) multiple models having heterogeneous submodels (the local models have not the same state); the first variant is more commonly used, being introduced by Takagi and Sugeno in the fuzzy modeling [22] and by Johansen and Foss in the modeling using the concept of multiple model [23].A large class of nonlinear systems can be represented by the well-known T-S multiple models due to their simplicity; using the interpolation between some linear submodels and their associated activation functions, any nonlinear system can be described by means of the T-S representation. Any nonlinear system can be transformed into a multiple model by using many approaches, the linearization method being the most used one; using the resulting interpolation of the linear models, a multiple model general form is obtained [9]:(1)x˙t=∑i=1MμiξtAixt+Biut+Dvt+di,yt=Cxt,where xt∈Rn is the state vector, ut∈Rm the known input vector, vt∈Rq the unknown input vector, and yt∈Rr the measurable output vector, and C∈Rr×n, D∈Rn×q and, for the local model “i”, Ai∈Rn×n, Bi∈Rn×m, and di∈Rn are considered known and constant matrices. The activation (weighting) functions μiξt, i=1,M¯, depending on the so-called decision vector ξt, have the following properties [24]: ∑i=1Mμiξt=1; 0≤μiξt≤1, ∀i=1,M¯; M is the number of local models depending on the precision of modeling, the complexity of the nonlinear system to be approximated by using a multiple model, and the structure of the activation functions. The matrices Ai, Bi, C, D, and di can be obtained by the direct linearization of the nonlinear model around several operating points, or by means of an identification procedure [9].Notations. Throughout the paper, the following notations are used: XT denotes the transpose of the matrix X, X>0 means that X is a symmetric positive definite matrix, and · represents the Euclidean norm for vectors and the spectral norm for matrices, while X∗ is the right inverse of the full row rank matrix X, defined by the equation: X∗=XTXXT-1 with the property: XX∗=I.


## 3. Design of the New Reduced-Order Multiple Observer

### 3.1. Problem Statement

The new reduced-order multiple observer is designed in this section for the case of T-S nonlinear systems with unknown inputs; after that, its robustness is improved by adding robustness terms in the observer’s equations considering the case of uncertain multiple models. The direct and simple method to design the observer for T-S systems with unknown inputs reduces the design procedure of reduced-order observers with unknown inputs to the design problem of multiple observers. The new observer should estimate any state for Takagi-Sugeno multiple models with unknown inputs; the problem addressed in this paper is the design of ap-order reduced-order multiple observer which generates the required p state vector of the form:(2)zt=Lxt,where L∈Rp×n is a full row rank matrix, while zt∈Rp is the vector containing a part of the system’s state variables or combinations of these variables. The matrix L has been chosen of the following form:(3)L=T+FC,where T∈Rp×n, F∈Rp×r; the full row rank matrix L is judiciously chosen and the matrices T and F have to be determined.Without loss of generality, the next assumptions are considered in the paper: (A1)rankC=r, rankD=q; (A2) the pairs C,Ai are observable; (A3) p>r>q; (A4) n=p+r. For matrix C in (1) a particular form is chosen: C=0r×n-rC2, with C2∈Rr×r being full rank matrix [25, 26]. This is not restrictive as long as the matrix C is full row rank (condition (A1)), because there will always be an orthogonal transformation such that the output equation of the system is yt=0C2xt. It will be proved later that the only existence condition of ROMO is rankCD=rankD=q.Considerz^t being the estimation of zt; the obtaining of this vector is achieved by using a p order ROMO which has been obtained by modifying the form of the reduced-order functional observer from [20] and adapting it to the case of Takagi-Sugeno multiple models in [9]. The new ROMO can estimate the components of the state vector xt by letting the matrix L comprise those rows of identity matrix In that correspond to the state variables to be estimated; having in mind that L is a full row rank matrix, the estimated state vector x^t will be obtained using (2) and the right inverse of the matrix L; that is, x^t=L∗z^t=LTLLT-1z^t.The reduced-order multiple observer for estimating the state vector of Takagi-Sugeno systems is chosen as follows:(4)w˙t=∑i=1MμiξtNiwt+Hiut+Jiyt+Gi,z^t=wt+Fyt,with wt∈Rp; to fully know the equations of the new ROMO, the matrices Ni∈Rp×p, Ji∈Rp×r, Hi∈Rp×m, Gi∈Rp×1, T∈Rp×n, and F∈Rp×r should be determined such that z^t converges to zt.


### 3.2. Global Convergence Conditions of the Reduced-Order Multiple Observer

For the determination of the new ROMO’s unknown matrices, one considers the error vector of the observeret=zt-z^t. It is easy to prove that the dynamics of the new reduced-order multiple observer is(5)e˙t=∑i=1MμiξtNie+∑i=1MμiξTAi-NiT-JiCxt+TBi-Hiut+TDvt+Tdi-Gi.The convergence of the ROMO is synthesized into the following theorem.Theorem 1.
Consider the Takagi-Sugeno multiple model (1) with the assumptions (A1)–(A4). et→0 as t→∞ for any x(0), w(0), v(t), and u(t) if and only if the matrix N~=∑i=1MμiξNi is Hurwitz σN~∈C- and(6)TAi-NiT-JiC=0,TBi-Ni=0,TD=0,Tdi-Gi=0.Proof.
​
Sufficiency. If conditions (6) are satisfied, the error dynamics becomes e˙t=N~et; further, one gets et=e0·exp(N~t)=e0·exp(N~t) and, regardless of the value of e0, if N~ is Hurwitz, limt→∞⁡et=0 results for any x(0), w(0), v(t), and u(t).
Necessity. To proof the necessity of the theorem, the following property of propositions is used: Proposition P→ propositions P1  Λ  P2  Λ  P3  Λ  P4⇔ΓP1  V ΓP2  V ΓP3  V ΓP4→ΓP (Λ = and, V = or, Γ = non). If N~ is not Hurwitz σN~∉C-, even for vt=0, ut=0, x0=0 and ∀e0≠0⇒e˙0=N~e0≠0⇒limt→∞⁡et≠0; if TD≠0, TBi-Ni≠0, or Tdi-Gi≠0, then there exist u(t) and v(t) to make limt→∞⁡et≠0; also, if TAi-NiT-JiC≠0, because one supposed that there are no states which tend to zero for any values of the vectors u(t) and v(t), xt≠0 results and the error dynamics is e˙t=N~et+TAi-NiT-JiCxt︸≠0≠N~et⇒limt→∞⁡et≠0.To determine other existence conditions of ROMO and to calculate the unknown matrices from its equations, one considers the Lyapunov functionVe=eTPe, where P is a symmetrical and positive-defined matrix; using the dynamics of observer’s error, one obtains the derivative of the function Ve as follows:(7)V˙e=∑i=1MμiξeTNiTP+PNie.The convergence of the reduced-order multiple observer V˙<0 is achieved if the nonlinear matrix inequality(8)NiTP+PNi<0is satisfied (constraint of the observer) for any i=1,M¯. Having in mind that, in the design algorithm of the ROMO (Section 5), the determination of the matrices Ni will be achieved before solving the matrix inequalities (8), these will become linear matrix inequalities and will be easily solved by using the LMI theory.To conclude, the new reduced-order multiple observer’s design problem has been reduced now to the finding of the matricesNi,Ji,Hi,Gi,T, and P that satisfy the constraints:(9)TAi-NiT-JiC=0,TBi-Hi=0,TD=0,Tdi-Gi=0,σN~∈C-,NiTP+PNi<0;this problem is equivalent to the standard problem of observers’ design when all the inputs are known.


### 3.3. Design Approach

The matricesNi, T, Ji, Ai, D are partitioned as(10)Ni=N1iN2iN3iN4i,T=T1T2T3T4,Ji=J1iJ2i,Ai=A1iA2iA3iA4i,D=D1D2,T3=T31T32,with(11)Ni∈Rp×p,N1i∈Rr×r,N2i∈Rr×p-r,N3i∈Rp-r×r,N4i∈Rp-r×p-r,T∈Rp×n,T1∈Rr×n-r,T2∈Rr×r,T3∈Rp-r×n-r,T4∈Rp-r×r,T31∈Rp-r×n-p,T32∈Rp-r×p-r,Ji∈Rp×r,J1i∈Rr×r,J2i∈Rp-r×r,D∈Rn×q,D1∈Rp×q,D2∈Rr×q,Ai∈Rn×n,A1i∈Rn-r×n-r,A2i∈Rn-r×r,A3i∈Rr×n-r,A4i∈Rr×r.The design of the new observer is concentrated into the following innovative theorem.Theorem 2.
Consider the Takagi-Sugeno multiple model (1); using assumptions (A1)–(A4) and the condition rankCD=q, the reduced-order multiple observer (4) is convergent if, for any i=1,M¯, there exists a symmetrical and positive-defined matrix P verifying the matrix inequality (8), if the matrix N~=∑i=1MμiξNi is Hurwitz, and the matrices T, Gi, Hi, F, Ni, Ji have the forms:(12)T=Ip×n-D1CD+C,Gi=Tdi,Hi=TBi,F=L-Ip×n+D1CD+C0n-r×rIrC2-1,Ni=N1iA~1i·0n-p×p-rIp-r·T32-1N3iA~3i·0n-p×p-rIp-r·T32-1,Ji=A~2i-A~1i·0n-p×p-rIp-r·T32-1·T4C2-1A~4i-A~3i·0n-p×p-rIp-r·T32-1·T4C2-1,where the matrices N1i and N3i have been randomly chosen, and(13)A~1i=T1A1i+T2A3i-N1iT1,A~2i=T1A2i+T2A4i-N1iT2,A~3i=T3A1i+T4A3i-N3iT1,A~4i=T3A2i+T4A4i-N3iT2.Proof.
The proof of Theorem2 mainly involves the demonstration of the expressions (12), issue which is equivalent to the solving of the first four equations (9). The first equation to solve is the third equation (9), that is, TD=0; to determine the matrix T from this equation, one chooses the matrix T of the following form: T=Ip×n+EC, with Ip×n being the identity matrix with p lines and n columns and E∈Rp×r being an unknown matrix to be calculated. Using C=0r×n-rC2 and partitioning the known matrix D as in (10) and (11), one successively gets(14)Ip×n+ECD=0⟺D1+ECD=0⟺E=-D1CD+,where CD+ is the generalized pseudoinverse of CD, given by [27, 28]: CD+=CDTCD-1CDT. The generalized pseudoinverse of CD can be obtained if and only if CD is full column rank, namely, rankCD=q; till now, this is the only existence condition of the reduced-order multiple observer. With (14), one obtains the first expression (12), that is, T=Ip×n-D1CD+C.
After the determination of the matrixT, by means of the second equation (9), Hi=TBi, i=1,M¯, results, and, by using the fourth equation (9), one gets Gi=Tdi, i=1,M¯. The matrix L is judiciously chosen by the designer of the observer having in mind the states which are directly measured by means of the system sensors (the forms of the matrices C and C2 give this information) and the states that are not measured and should be estimated. The determination of the matrix F is achieved now by using (3) and the obtained expression of the matrix T; thus, from (3) one obtains FC=L-T⇔F0r×n-rC2=L-T. By right multiplying of this equation with 0n-r×rIr, taking into account the fact that rankC=rankC2=r (C2 is a full rank matrix), one successively obtains F=L-T·0n-r×rIr·C2-1=L-Ip×n+D1CD+C·0n-r×rIr·C2-1. This way, the first four expressions (12) have been demonstrated.
The only unknown matrices from (4) of the ROMO are now Ni and Ji; these are the solutions of the first equation (9) and subject of two constraints: (1)  σN~∈C-; (2) for any i=1,M¯, there exists a matrix P>0 which verifies the linear matrix inequality (8). One can use the method from [21] where the solving of an equation with two unknown matrices has been done in the design approach of a full-order observer for linear-time invariant systems with unknown inputs or the method from [25], where the matrices Ni are calculated as(15)Ni=TAi-KiC,where Ki are obtained with the pole placement technique such that σNi∈C-. Replacing (15) into the equation to be solved, that is, TAi-NiT-JiC=0, and taking into account the equations T=Ip×n+EC, E=-D1CD+, and T=Ip×n-D1CD+C, the following is obtained:(16)Ji=-TAiE+KiIp×n+EC⟺Ji=TAiD1CD++KiIp×n-D1CD+C;to choose stable eigenvalues for the matrices Ni, the pair TAi,C should be observable [25]. If the pair TAi,C is not observable, the calculation of matrices Ki is made such that the observer is asymptotically stable if and only if TAi,C is detectable. In [25], the equivalence of this condition with other two (see the next theorem) is proved.
Theorem  3 (see [25]).If conditionsrankD=rankCD=qandrankT=n-qhold, the following 3 conditions are equivalent:(1)
the pairTAi,Cis observable or at least detectable;(2)
ranksT-TAiC=n, ∀s∈C, Res≥0;(3)
ranksIn-AiDC0=n+q, ∀s∈C, Res≥0.
The main disadvantage of the approach presented in [25] is related to the choice of eigenvalues for the matrices Ni; the choice of the eigenvalues for the matrix Ni has a great influence on the observer’s poles and convergence speed. The pole placement technique is easy to implement in the case of observers’ design, but it has some disadvantages: (1) the technique is difficult to be used in the case of systems with big order or for poorly controlled systems; (2) if one chooses fast poles for the observer, the advantage is that the observer estimation error decays rapidly, but the disadvantage is that the system needs perfect sensors and/or noise free environment; (3) if one chooses slow poles for the observer, the advantage is that the system is less sensitive to process disturbances and measurement noise, but the disadvantage is that the observer estimation error decays slowly [21].
The disadvantages of the pole placement technique lead to the conclusion that the approach from [21] is better than the one presented in [25]; therefore, one chooses to adapt the technique from [21] to the present case—multiple observer design for T-S multiple models with unknown inputs. This step firstly involves the partitioning of the matrices Ni, T, Ji, Ai as in (10) and (11); substituting (10) into the first equation (9), one obtains(17)N1iT1+N2iT3-T1A1i-T2A3i=0r×n-r,N1iT2+N2iT4+J1iC2-T1A2i-T2A4i=0r×r,N3iT1+N4iT3-T3A1i-T4A3i=0p-r×n-r,N3iT2+N4iT4+J2iC2-T3A2i-T4A4i=0p-r×r.
The previous matrix equation system has 6 unknown variables (the matricesN1i, N2i, N3i, N4i, J1i, J2i) and only 4 equations. The system can be solved if only the number of unknown variables is equal to or less than the number of equations. Therefore, to transform the equation system (17) into a compatible determined system, two of the six matrices are considered random matrices. One chooses N1i∈Rr×r and N3i∈Rp-r×r randomly, but the approach also works well if, instead of these matrices, any other two matrices are chosen randomly. The other two matrices, together with J1i and J2i, are now the 4 unknown variables of system (17) which becomes compatible determined, its solving being not a difficult task. Using the notations (13), system (17) gets the form:(18)N2iT3=A~1i,N2iT4+J1iC2=A~2i,N4iT3=A~3i,N4iT4+J2iC2=A~4i.By using the partitioning of matrix T3 as in (10) and (11), the solving of the first equation (18) is equivalent with N2iT31T32=A~1i, which, by right multiplying with 0n-p×p-rIp-r, leads to N2i=A~1i·0n-p×p-rIp-r·T32-1; in the same way, one solves the third equation (18) and gets N4i=A~3i·0n-p×p-rIp-r·T32-1; one supposed that T32 is a full rank matrix. The matrices J1i and J2i are determined from the second and the fourth equations of system (18); having again in mind that C2 is a full rank matrix, the following is obtained:(19)J1i=A~2i-N2iT4C2-1=A~2i-A~1i·0n-p×p-rIp-r·T32-1·T4C2-1,J2i=A~4i-N4iT4C2-1=A~4i-A~3i·0n-p×p-rIp-r·T32-1·T4C2-1;the last two expressions (12) have been demonstrated.
The determined matricesNi should satisfy two constraints: (C1) σN~∈C-; (C2) for any i=1,M¯, there exists a matrix P>0 which verifies the matrix inequality (8). Using the matrices Ni, i=1,M¯, the matrix N~ and its eigenvalues are obtained; also, the M linear matrix inequalities (8) are solved; if the matrices Ni, i=1,M¯, assure the stability of the matrix N~ (all its eigenvalues are placed in the left-hand side of the complex plane) and the existence of a matrix P>0 such that, for any i=1,M¯, the LMIs (8) are satisfied, the matrices Ni have been obtained properly, the dynamics of the observer error has a homogeneous form and, according to Theorem 1, et→0 as t→∞ for any x(0), w(0), v(t), and u(t). Otherwise, other random matrices N1i and N3i are chosen and system (18) is again solved until the constraints (C1) and (C2) are fulfilled. In the new algorithm software implementation, the fulfillment of the two constraints is made in a “while” loop. To conclude, one managed to avoid the pole placement technique; thus, no condition regarding the observability or detectability of the pair TAi,C is needed in our design approach and, therefore, the only existence condition of the observer is rankCD=rankD=q. As a consequence, the introduced innovation element (the “while” loop) increases the generality and convergence of the new ROMO. The theorem’s proof is now complete.Remark 4.
It is well known that the information provided by the sensors of a system is in connection with the output matrix of the system (C) and vice versa; if one chooses to measure the last r states from the state vector x, the matrix C has the form C=0r×n-rIr; in this case, the only existence condition of the reduced-order multiple observer is always satisfied and, therefore, the new algorithm has no existence conditions and only 4 assumptions have to be made. Thus, another important advantage of the new ROMO is the lack of a priori restrictions on the class of Takagi-Sugeno multiple models that can be considered.The block diagram associated with the ensemble multiple model—reduced-order multiple observer—is presented in Figure1.Figure 1
Block diagram of the reduced-order multiple observer.


## 3.1. Problem Statement

The new reduced-order multiple observer is designed in this section for the case of T-S nonlinear systems with unknown inputs; after that, its robustness is improved by adding robustness terms in the observer’s equations considering the case of uncertain multiple models. The direct and simple method to design the observer for T-S systems with unknown inputs reduces the design procedure of reduced-order observers with unknown inputs to the design problem of multiple observers. The new observer should estimate any state for Takagi-Sugeno multiple models with unknown inputs; the problem addressed in this paper is the design of ap-order reduced-order multiple observer which generates the required p state vector of the form:(2)zt=Lxt,where L∈Rp×n is a full row rank matrix, while zt∈Rp is the vector containing a part of the system’s state variables or combinations of these variables. The matrix L has been chosen of the following form:(3)L=T+FC,where T∈Rp×n, F∈Rp×r; the full row rank matrix L is judiciously chosen and the matrices T and F have to be determined.Without loss of generality, the next assumptions are considered in the paper: (A1)rankC=r, rankD=q; (A2) the pairs C,Ai are observable; (A3) p>r>q; (A4) n=p+r. For matrix C in (1) a particular form is chosen: C=0r×n-rC2, with C2∈Rr×r being full rank matrix [25, 26]. This is not restrictive as long as the matrix C is full row rank (condition (A1)), because there will always be an orthogonal transformation such that the output equation of the system is yt=0C2xt. It will be proved later that the only existence condition of ROMO is rankCD=rankD=q.Considerz^t being the estimation of zt; the obtaining of this vector is achieved by using a p order ROMO which has been obtained by modifying the form of the reduced-order functional observer from [20] and adapting it to the case of Takagi-Sugeno multiple models in [9]. The new ROMO can estimate the components of the state vector xt by letting the matrix L comprise those rows of identity matrix In that correspond to the state variables to be estimated; having in mind that L is a full row rank matrix, the estimated state vector x^t will be obtained using (2) and the right inverse of the matrix L; that is, x^t=L∗z^t=LTLLT-1z^t.The reduced-order multiple observer for estimating the state vector of Takagi-Sugeno systems is chosen as follows:(4)w˙t=∑i=1MμiξtNiwt+Hiut+Jiyt+Gi,z^t=wt+Fyt,with wt∈Rp; to fully know the equations of the new ROMO, the matrices Ni∈Rp×p, Ji∈Rp×r, Hi∈Rp×m, Gi∈Rp×1, T∈Rp×n, and F∈Rp×r should be determined such that z^t converges to zt.


## 3.2. Global Convergence Conditions of the Reduced-Order Multiple Observer

For the determination of the new ROMO’s unknown matrices, one considers the error vector of the observeret=zt-z^t. It is easy to prove that the dynamics of the new reduced-order multiple observer is(5)e˙t=∑i=1MμiξtNie+∑i=1MμiξTAi-NiT-JiCxt+TBi-Hiut+TDvt+Tdi-Gi.The convergence of the ROMO is synthesized into the following theorem.Theorem 1.
Consider the Takagi-Sugeno multiple model (1) with the assumptions (A1)–(A4). et→0 as t→∞ for any x(0), w(0), v(t), and u(t) if and only if the matrix N~=∑i=1MμiξNi is Hurwitz σN~∈C- and(6)TAi-NiT-JiC=0,TBi-Ni=0,TD=0,Tdi-Gi=0.Proof.
​
Sufficiency. If conditions (6) are satisfied, the error dynamics becomes e˙t=N~et; further, one gets et=e0·exp(N~t)=e0·exp(N~t) and, regardless of the value of e0, if N~ is Hurwitz, limt→∞⁡et=0 results for any x(0), w(0), v(t), and u(t).
Necessity. To proof the necessity of the theorem, the following property of propositions is used: Proposition P→ propositions P1  Λ  P2  Λ  P3  Λ  P4⇔ΓP1  V ΓP2  V ΓP3  V ΓP4→ΓP (Λ = and, V = or, Γ = non). If N~ is not Hurwitz σN~∉C-, even for vt=0, ut=0, x0=0 and ∀e0≠0⇒e˙0=N~e0≠0⇒limt→∞⁡et≠0; if TD≠0, TBi-Ni≠0, or Tdi-Gi≠0, then there exist u(t) and v(t) to make limt→∞⁡et≠0; also, if TAi-NiT-JiC≠0, because one supposed that there are no states which tend to zero for any values of the vectors u(t) and v(t), xt≠0 results and the error dynamics is e˙t=N~et+TAi-NiT-JiCxt︸≠0≠N~et⇒limt→∞⁡et≠0.To determine other existence conditions of ROMO and to calculate the unknown matrices from its equations, one considers the Lyapunov functionVe=eTPe, where P is a symmetrical and positive-defined matrix; using the dynamics of observer’s error, one obtains the derivative of the function Ve as follows:(7)V˙e=∑i=1MμiξeTNiTP+PNie.The convergence of the reduced-order multiple observer V˙<0 is achieved if the nonlinear matrix inequality(8)NiTP+PNi<0is satisfied (constraint of the observer) for any i=1,M¯. Having in mind that, in the design algorithm of the ROMO (Section 5), the determination of the matrices Ni will be achieved before solving the matrix inequalities (8), these will become linear matrix inequalities and will be easily solved by using the LMI theory.To conclude, the new reduced-order multiple observer’s design problem has been reduced now to the finding of the matricesNi,Ji,Hi,Gi,T, and P that satisfy the constraints:(9)TAi-NiT-JiC=0,TBi-Hi=0,TD=0,Tdi-Gi=0,σN~∈C-,NiTP+PNi<0;this problem is equivalent to the standard problem of observers’ design when all the inputs are known.


## 3.3. Design Approach

The matricesNi, T, Ji, Ai, D are partitioned as(10)Ni=N1iN2iN3iN4i,T=T1T2T3T4,Ji=J1iJ2i,Ai=A1iA2iA3iA4i,D=D1D2,T3=T31T32,with(11)Ni∈Rp×p,N1i∈Rr×r,N2i∈Rr×p-r,N3i∈Rp-r×r,N4i∈Rp-r×p-r,T∈Rp×n,T1∈Rr×n-r,T2∈Rr×r,T3∈Rp-r×n-r,T4∈Rp-r×r,T31∈Rp-r×n-p,T32∈Rp-r×p-r,Ji∈Rp×r,J1i∈Rr×r,J2i∈Rp-r×r,D∈Rn×q,D1∈Rp×q,D2∈Rr×q,Ai∈Rn×n,A1i∈Rn-r×n-r,A2i∈Rn-r×r,A3i∈Rr×n-r,A4i∈Rr×r.The design of the new observer is concentrated into the following innovative theorem.Theorem 2.
Consider the Takagi-Sugeno multiple model (1); using assumptions (A1)–(A4) and the condition rankCD=q, the reduced-order multiple observer (4) is convergent if, for any i=1,M¯, there exists a symmetrical and positive-defined matrix P verifying the matrix inequality (8), if the matrix N~=∑i=1MμiξNi is Hurwitz, and the matrices T, Gi, Hi, F, Ni, Ji have the forms:(12)T=Ip×n-D1CD+C,Gi=Tdi,Hi=TBi,F=L-Ip×n+D1CD+C0n-r×rIrC2-1,Ni=N1iA~1i·0n-p×p-rIp-r·T32-1N3iA~3i·0n-p×p-rIp-r·T32-1,Ji=A~2i-A~1i·0n-p×p-rIp-r·T32-1·T4C2-1A~4i-A~3i·0n-p×p-rIp-r·T32-1·T4C2-1,where the matrices N1i and N3i have been randomly chosen, and(13)A~1i=T1A1i+T2A3i-N1iT1,A~2i=T1A2i+T2A4i-N1iT2,A~3i=T3A1i+T4A3i-N3iT1,A~4i=T3A2i+T4A4i-N3iT2.Proof.
The proof of Theorem2 mainly involves the demonstration of the expressions (12), issue which is equivalent to the solving of the first four equations (9). The first equation to solve is the third equation (9), that is, TD=0; to determine the matrix T from this equation, one chooses the matrix T of the following form: T=Ip×n+EC, with Ip×n being the identity matrix with p lines and n columns and E∈Rp×r being an unknown matrix to be calculated. Using C=0r×n-rC2 and partitioning the known matrix D as in (10) and (11), one successively gets(14)Ip×n+ECD=0⟺D1+ECD=0⟺E=-D1CD+,where CD+ is the generalized pseudoinverse of CD, given by [27, 28]: CD+=CDTCD-1CDT. The generalized pseudoinverse of CD can be obtained if and only if CD is full column rank, namely, rankCD=q; till now, this is the only existence condition of the reduced-order multiple observer. With (14), one obtains the first expression (12), that is, T=Ip×n-D1CD+C.
After the determination of the matrixT, by means of the second equation (9), Hi=TBi, i=1,M¯, results, and, by using the fourth equation (9), one gets Gi=Tdi, i=1,M¯. The matrix L is judiciously chosen by the designer of the observer having in mind the states which are directly measured by means of the system sensors (the forms of the matrices C and C2 give this information) and the states that are not measured and should be estimated. The determination of the matrix F is achieved now by using (3) and the obtained expression of the matrix T; thus, from (3) one obtains FC=L-T⇔F0r×n-rC2=L-T. By right multiplying of this equation with 0n-r×rIr, taking into account the fact that rankC=rankC2=r (C2 is a full rank matrix), one successively obtains F=L-T·0n-r×rIr·C2-1=L-Ip×n+D1CD+C·0n-r×rIr·C2-1. This way, the first four expressions (12) have been demonstrated.
The only unknown matrices from (4) of the ROMO are now Ni and Ji; these are the solutions of the first equation (9) and subject of two constraints: (1)  σN~∈C-; (2) for any i=1,M¯, there exists a matrix P>0 which verifies the linear matrix inequality (8). One can use the method from [21] where the solving of an equation with two unknown matrices has been done in the design approach of a full-order observer for linear-time invariant systems with unknown inputs or the method from [25], where the matrices Ni are calculated as(15)Ni=TAi-KiC,where Ki are obtained with the pole placement technique such that σNi∈C-. Replacing (15) into the equation to be solved, that is, TAi-NiT-JiC=0, and taking into account the equations T=Ip×n+EC, E=-D1CD+, and T=Ip×n-D1CD+C, the following is obtained:(16)Ji=-TAiE+KiIp×n+EC⟺Ji=TAiD1CD++KiIp×n-D1CD+C;to choose stable eigenvalues for the matrices Ni, the pair TAi,C should be observable [25]. If the pair TAi,C is not observable, the calculation of matrices Ki is made such that the observer is asymptotically stable if and only if TAi,C is detectable. In [25], the equivalence of this condition with other two (see the next theorem) is proved.
Theorem  3 (see [25]).If conditionsrankD=rankCD=qandrankT=n-qhold, the following 3 conditions are equivalent:(1)
the pairTAi,Cis observable or at least detectable;(2)
ranksT-TAiC=n, ∀s∈C, Res≥0;(3)
ranksIn-AiDC0=n+q, ∀s∈C, Res≥0.
The main disadvantage of the approach presented in [25] is related to the choice of eigenvalues for the matrices Ni; the choice of the eigenvalues for the matrix Ni has a great influence on the observer’s poles and convergence speed. The pole placement technique is easy to implement in the case of observers’ design, but it has some disadvantages: (1) the technique is difficult to be used in the case of systems with big order or for poorly controlled systems; (2) if one chooses fast poles for the observer, the advantage is that the observer estimation error decays rapidly, but the disadvantage is that the system needs perfect sensors and/or noise free environment; (3) if one chooses slow poles for the observer, the advantage is that the system is less sensitive to process disturbances and measurement noise, but the disadvantage is that the observer estimation error decays slowly [21].
The disadvantages of the pole placement technique lead to the conclusion that the approach from [21] is better than the one presented in [25]; therefore, one chooses to adapt the technique from [21] to the present case—multiple observer design for T-S multiple models with unknown inputs. This step firstly involves the partitioning of the matrices Ni, T, Ji, Ai as in (10) and (11); substituting (10) into the first equation (9), one obtains(17)N1iT1+N2iT3-T1A1i-T2A3i=0r×n-r,N1iT2+N2iT4+J1iC2-T1A2i-T2A4i=0r×r,N3iT1+N4iT3-T3A1i-T4A3i=0p-r×n-r,N3iT2+N4iT4+J2iC2-T3A2i-T4A4i=0p-r×r.
The previous matrix equation system has 6 unknown variables (the matricesN1i, N2i, N3i, N4i, J1i, J2i) and only 4 equations. The system can be solved if only the number of unknown variables is equal to or less than the number of equations. Therefore, to transform the equation system (17) into a compatible determined system, two of the six matrices are considered random matrices. One chooses N1i∈Rr×r and N3i∈Rp-r×r randomly, but the approach also works well if, instead of these matrices, any other two matrices are chosen randomly. The other two matrices, together with J1i and J2i, are now the 4 unknown variables of system (17) which becomes compatible determined, its solving being not a difficult task. Using the notations (13), system (17) gets the form:(18)N2iT3=A~1i,N2iT4+J1iC2=A~2i,N4iT3=A~3i,N4iT4+J2iC2=A~4i.By using the partitioning of matrix T3 as in (10) and (11), the solving of the first equation (18) is equivalent with N2iT31T32=A~1i, which, by right multiplying with 0n-p×p-rIp-r, leads to N2i=A~1i·0n-p×p-rIp-r·T32-1; in the same way, one solves the third equation (18) and gets N4i=A~3i·0n-p×p-rIp-r·T32-1; one supposed that T32 is a full rank matrix. The matrices J1i and J2i are determined from the second and the fourth equations of system (18); having again in mind that C2 is a full rank matrix, the following is obtained:(19)J1i=A~2i-N2iT4C2-1=A~2i-A~1i·0n-p×p-rIp-r·T32-1·T4C2-1,J2i=A~4i-N4iT4C2-1=A~4i-A~3i·0n-p×p-rIp-r·T32-1·T4C2-1;the last two expressions (12) have been demonstrated.
The determined matricesNi should satisfy two constraints: (C1) σN~∈C-; (C2) for any i=1,M¯, there exists a matrix P>0 which verifies the matrix inequality (8). Using the matrices Ni, i=1,M¯, the matrix N~ and its eigenvalues are obtained; also, the M linear matrix inequalities (8) are solved; if the matrices Ni, i=1,M¯, assure the stability of the matrix N~ (all its eigenvalues are placed in the left-hand side of the complex plane) and the existence of a matrix P>0 such that, for any i=1,M¯, the LMIs (8) are satisfied, the matrices Ni have been obtained properly, the dynamics of the observer error has a homogeneous form and, according to Theorem 1, et→0 as t→∞ for any x(0), w(0), v(t), and u(t). Otherwise, other random matrices N1i and N3i are chosen and system (18) is again solved until the constraints (C1) and (C2) are fulfilled. In the new algorithm software implementation, the fulfillment of the two constraints is made in a “while” loop. To conclude, one managed to avoid the pole placement technique; thus, no condition regarding the observability or detectability of the pair TAi,C is needed in our design approach and, therefore, the only existence condition of the observer is rankCD=rankD=q. As a consequence, the introduced innovation element (the “while” loop) increases the generality and convergence of the new ROMO. The theorem’s proof is now complete.Remark 4.
It is well known that the information provided by the sensors of a system is in connection with the output matrix of the system (C) and vice versa; if one chooses to measure the last r states from the state vector x, the matrix C has the form C=0r×n-rIr; in this case, the only existence condition of the reduced-order multiple observer is always satisfied and, therefore, the new algorithm has no existence conditions and only 4 assumptions have to be made. Thus, another important advantage of the new ROMO is the lack of a priori restrictions on the class of Takagi-Sugeno multiple models that can be considered.The block diagram associated with the ensemble multiple model—reduced-order multiple observer—is presented in Figure1.Figure 1
Block diagram of the reduced-order multiple observer.


## 4. Improvement of the Observer’s Robustness

The matricesAi from (1) may be not completely known; in this case, some small unknown variances ΔAi can appear, the multiple model being affected by uncertainties (modeling unknown parameters). Therefore, the improvement of the ROMO’s robustness is a problem to solve from the unknown small variances’ point of view such that the negative effect of the unknown matrices on the system state estimation is minimized, if not canceled. The apparition of the uncertain matrices ΔAi in the dynamics of the system is the negative consequence of faults, calculation errors, modeling errors, or small variations of the system during the linearization process [29]. If the uncertain matrices are taken into consideration, the multiple model, initially described by (1), becomes now(20)x˙t=∑i=1MμiξtAi+ΔAixt+Biut+Dvt+di,yt=Cxt;if ΔAi=0, (1) are obtained; otherwise, the effect of the unknown matrices ΔAi cannot be completely eliminated but only minimized. To overcome the drawback related to the presence of the uncertain matrices (considered unknown terms) and to improve the robustness of the ROMO with respect to these uncertainties, the robustness (compensation) terms γit,i=1,M¯, are added in the equations of the observer; its new form is(21)w˙t=∑i=1MμiξtNiwt+Hiut+Jiyt+Gi+γi,z^t=wt+Fyt.The determination of the terms γit is achieved by using the Lyapunov theory; this approach represents another innovative element of the paper. The expression of the robustness terms is presented within the following theorem.Theorem 5.
For anyi=1,M¯, if there exist a matrix P->0 and the positive scalars β1, β2, satisfying the constraint(22)NiTP-+P-Ni+β1T~iA+β1β2-1δ~iAIP-P--β1-1I<0,the state estimation error of the observer (21) converges to zero if(23)γit=β11+β2δ~iAz^tTz^t2·P~rtrTtrt,rt≠0,0,rt=0,where rt=yt-y^t, P~=P--1LC∗+T, and the time variant matrices T~iA=L∗TTΔAiTTΔAiL∗ are bounded, that is, T~iA<δ~iA.Proof.
To deduce the expression of the robustness termsγit,i=1,M¯, one calculates again the dynamics of the ROMO’s error e˙t=z˙t-z^˙(t)=Tx˙t-w˙(t) and, by using the previous equations and wt=Txt-et, the following is obtained:(24)e˙t=∑i=1MμiξNiet-TiAxt-γit,where TiA=T·ΔAi; to obtain equation (24), the fulfillment of the constraints from Theorem 2 has been considered. For the ease of notations, in the below presented approach, the time dependence (t) is omitted.
Using the Lyapunov functionVe=eTP-e, with P- being symmetrical and positive-defined matrix and the dynamics of observer’s error being (24), one obtains the derivative of the function Ve as follows:(25)V˙=e˙TPe+eTPe˙=∑i=1MμiξeTNiTP-+P-Nie+∑i=1MμiξxTTiATP-e+eTP-TiAx-2γiTP-e;the equation γiTP-e+eTP-γi=2γiTP-e has been used.
During the proof of this theorem, the following inequality is used [8]:(26)XTY+YTX≤βXTX+β-1YTY,for any matrices X and Y and positive scalars β.
Using inequality (26), written for the set: Σ1:X=TiAx,Y=P-e,β=β1, the following is obtained: xTTiATP-e+eTP-TiAx≤β1xTT-iAx+β1-1eTP-2e, with T-iA=TiAT·TiA. By using these, (25) becomes(27)V˙≤∑i=1MμiξeTNiTP-+P-Ni+β1-1P-2e+∑i=1Mμiξβ1xTT-iAx-2γiTP-e.
Using the expression of the errore=z-z^=Lx-z^⇔x=L∗e+z^, the term β1xTT-iAx can be expressed as follows:(28)β1xTT-iAx=β1z^TT~iAz^+β1eTT~iAe+β1z^TT~iAe+eTT~iAz^,with T~iA=L∗TTΔAiTTΔAiL∗=LTLLT-1TTΔAiTTΔAiLTLLT-1. Also, considering that the matrices T~iA are bounded, that is, T~iA<δ~iA, using again inequality (26) for the set Σ2:X=z^,Y=e,β=β2, inequality (27) gets the form:(29)V˙≤∑i=1MμiξeTNiTP-+P-Ni+β1-1P-2+β1T~iA+β1β2-1δ~iAIe+∑i=1Mμiξβ11+β2δ~iAz^Tz^-2γiTP-e.Imposing β11+β2δ~iAz^Tz^-2γiTP-e=0, (29) becomes V˙≤∑i=1MμiξeTNiTP-+P-Ni+β1-1P-2+β1T~iA+β1β2-1δ~iAIe; the convergence of the ROMO for uncertain Takagi-Sugeno systems with unknown inputs is achieved if, for any i=1,M¯, there exists a symmetrical and positive-defined matrix P- which verifies the nonlinear matrix inequality:(30)NiTP-+P-Ni+β1-1P-2+β1T~iA+β1β2-1δ~iAI<0,which, by using the Schur lemma, is equivalent to the linear matrix inequality (22).
The obtaining of (23) is achieved by means of the relationship β11+β2δ~iAz^Tz^=2γiTP-e; from this, one gets γi=β11+β2δ~iAz^Tz^/2e+P--1T=β11+β2δ~iAz^Tz^/2·e+P--1T. Using the equations (1)  P-=P-T; (2)  e=Lx-x^=LC∗y-y^=LC∗r, with r=y-y^ (residue of the ROMO); (3)  Z+T=ZTZ-1ZTT=ZTTZTZ-1T=ZTTZTZ-1=ZT∗, for any column vector Z; (4)  r+T=rT∗=rrTr-1=r/rTr, one calculates e+P--1T as follows:(31)e+P--1T=P--1e+T=P--1LC∗r+T=P--1r+LC∗+T=P--1LC∗+TrrTr=P~rrTr,where P~=P--1LC∗+T=P--1LCTCCT-1+T.
With this, the expression of the robustness terms becomes(32)γi=β11+β2δ~iAz^Tz^2·P~·rrTr.
A problem which can appear in the ROMO’s implementation process is related to the case when, althoughr→0, the robustness terms γi are not bounded. For this problem, the solution from [8] can be borrowed: γi are null if r<ε, where ε is a positive constant having small values, chosen by the observer designer. In this case, the observer estimation error does not converge asymptotically to zero but very close according to the values of ε. Thus, the expression of the robustness terms (23) has been deduced and the proof of the theorem is now complete.


## 5. Design Algorithm for Software Implementation

The ROMO’s design procedure, presented in detail in Sections3 and 4, can be summarized now into the following algorithm.Step 1.
One introduces the matrices/vectorsAi,Bi,di,i=1,M¯, and the constant matrices C, D; r=rankC=rankC2 and p=n-r (n is the number of states) are calculated; one checks the fulfillment of the ROMO’s assumptions ((A1)–(A4)) and existence condition: rankCD=q; the expressions of the decision vector ξ(t) and activation functions μiξt,i=1,M¯, are established.Step 2.
One judiciously chooses the full row rank matrixL.Step 3.
The matricesNi,T,Ji,Ai,D are partitioned with (10) and (11); the matrix D1 is obtained by using (11), while the matrices T,Gi,Hi, and F are calculated by means of first four equations (12).Step 4.
One randomly chooses the matricesN1i∈Rr×r and N3i∈Rp-r×r,i=1,M¯, and the matrices (13) are calculated. Using these matrices and the last two equations (12), the last four unknown matrices N2i,N4i,J1i,J2i are obtained.Step 5.
The matricesNi and Ji, i=1,M¯, are built, and the matrix N~=∑i=1MμiξNi and its eigenvalues are calculated. For any i=1,M¯, the LMIs (8) are solved with respect to P; if the constraints (C1) and (C2) are satisfied, the matrices Ni and Ji have been obtained properly; otherwise one returns to Step 4 and repeats Steps 4 and 5 until the two conditions are fulfilled.Step 6.
By means of the calculated matrices, one designs the observer described by (4)—the case without uncertainties; the time histories of the states and estimated states are obtained; using these vectors, the residual r is calculated. If the condition r<ε is not fulfilled, the positive scalars β1, β2 are chosen and the bounds δ~iA are calculated; the LMIs (22) are solved with respect to matrix P-; then, one calculates the matrix P~ and the robustness terms γit, i=1,M¯, by means of (23); these additional terms are introduced in the equations of the observer for the case of dynamics with uncertainties. For the validation of the reduced-order multiple observer, the dynamics associated with the multiple model ((1): case without uncertainties or (20): case with uncertainties) is also used.


## 6. Numerical Simulation Results

### 6.1. Numerical Simulation Setup

To validate the new reduced-order observer, any dynamics meeting the assumptions and the existence condition can be chosen; in this paper, the new ROMO is software implemented and validated in Matlab for the case of a light aircraft lateral-directional motion; the aircraft dynamics, withM=2, has the form (1) where the following matrices have been considered [24]:(33)A1=-0.30-339.810-5.400.1-8.33.75000-28.60.370-0.6400-9.50010.0100.01000010.00100.001000-0.0100-100000.010-0.0010-5,A2=-0.280-339.810-5.200.1-7.63.95000-27.60.340-0.5900-9.60010.0100.01000010.00200.001000-0.0200-9.90000.020-0.0010-4.95,B1=000000000-0.012100010,B2=000000000-0.0099.8009.8,d1=0.2·1111111,d2=0.1·1111111,C=C1C2=03×4I3;D=0.500.51020T.The state of the system is x=VyωxωzφψδrδaT, with Vy being aircraft lateral velocity, ωx the roll angular rate, ωz the yaw angular rate, φ the roll angle of aircraft, ψ the yaw angle of aircraft, δr the rudder deflection, and δa the ailerons’ deflection; the two components of the system input vector u=δrcδacT are the commands of the rudder and ailerons, respectively [24].Aircraft flight is often influenced by disturbances like wind shears, atmospheric turbulences, or errors of the sensors. From aircraft dynamics’ point of view, these represent unknown inputs; an observer for systems with unknown inputs may estimate these unknown inputs, but, more important, it must estimate the system states with very small errors [4]; therefore, in this paper, the unknown input vector v(t) has been randomly chosen. The decision and the activation functions have been chosen of the following forms: ξt=ut, μ1ξt=0.41-tanh⁡ξt, μ2ξt=1-μ1ξt, where the input has the form u=-K-x^, with the gain matrix K- determined by using the ALGLX optimal algorithm borrowed from [24]; K- can be also calculated by means of the pole placement technique or other methods.


### 6.2. Results and Discussion

For the above presented matrices, one obtainsn=7, m=2, q=1, and p=4. The observer validation is performed for both the dynamics without uncertainties ΔAi=0 and with uncertainties ΔAi≠0. The solving of the linear matrix inequalities (8) and (22) is performed using the Matlab LMI tool. From the forms of the matrices C and C2, one can remark that the last three states of the system ψ,δr,δa are measured; within Step 2 of the new algorithm, one judiciously chooses the full row rank matrix L; since this matrix gives information regarding the estimated states, one chooses L=Ip0p×n-p; thus, the first p=4 states are estimated, while the last 3 are measured. Having in mind the form of the matrix C, one gets yt=0r×n-rIrxt=0r×n-rIrx1tx2t=x1t; in other words, x1(t) contains the measurable states, while zt=Ip0p×n-pxt=Ip0p×n-px1tx2t=x2t contains the states to be estimated (the last r states of the system). Thus, ROMO estimates x2(t), providing x^2t; the estimate of the state vector x(t) is obtained by concatenation of the vectors x1(t) and x^2t; that is, x^t=x1tx^2t=ytz^t.For the first simulation case (without uncertainties), within Steps3, 4, and 5, one obtains(34)T=10000-0.250010000000100-0.25000010-0.50,G1=0.0150.020.0150.01,G2=0.00750.010.00750.005,H1=-2.5000-2.50-50,H2=-2.45000-2.450-4.90,F=00.25000000.25000.50,N1=diag-5;-10;-15;-20,N2=diag-7.5;-15;-22.5;-30,J1=0-4.27000-29.760-11.0300.01-4.80,J2=0-4.26000-29.770-11.0400.01-4.790;the solutions of the two LMIs (8) are P1=diag0.195;0.105;0.06;0.04, P2=diag0.072;0.035;0.023;0.017.For increasing the robustness of the ROMO, one uses the new technique involving the adding ofM robustness terms in the observer’s equations; thus, for case 2 (dynamics with uncertainties), one has chosen ΔAi as a percent of 1% from their initial values (Ai); also, one has chosen ε=0.01 and different values for the scalars β1 and β2  β1=β2=0.8;1.6;2.4;3.2 in order to analyze the influence of these constants on the observer’s convergence time and convergence errors. Solving the two linear matrix inequalities (22) for β1=β2=0.8, the following is obtained:(35)P-1=10-2·13.5300006.65-0.0100-0.014.66-0.0700-0.073.27,P-2=10-2·8.8200004.34-0.0100-0.013.05-0.0500-0.052.15.The new architecture from Figure1 is software implemented in Matlab, Figure 2(a); the Matlab/Simulink subsystem “subsystem vectors gama_1, gama_2” (subsystem for the calculation of the vectors γit,i=1,M¯) is presented in Figure 2(b).Figure 2
Matlab/Simulink model for the new reduced-order multiple observer.
(a) (b)For case1  ΔAi=0, Figure 3 presents the time histories of the estimation errors, that is, p=4 components of the vector et=zt-z^t; from Figure 3 one remarks the canceling of the estimation errors, hence the effectiveness of the developed ROMO scheme. This is equivalent to the achievement of the state reconstruction (the convergences z^t→zt and x^t→xt). The convergence speed of the ROMO (less than 2 seconds) represents an excellent convergence speed in the research area of the T-S multiple models.Figure 3
The new observer’s estimation errors; case1: dynamics without uncertainties.
(a) (b) (c) (d)For case2  ΔAi≠0, Figure 4 presents again the time histories of the estimation errors’ four components for different values of the scalars β1 and β2. One remarks the following: (1) the convergence time is approximately the same as in case of dynamics without uncertainties due to the robustness’ improvement approach; (2) the increase of the scalars β1 and β2 does not modify the convergence time of ROMO; (3) the increase of the scalars β1 and β2 leads to the increase of the overshoots’ absolute values for Figure 4(a) (from 7.1 m/s for β1=β2=0.8 to 10.35 m/s for β1=β2=3.2) and to the decrease of the overshoots’ absolute values for Figure 4(b) (from 20.05 deg/s for β1=β2=0.8 to 18.35 deg/s for β1=β2=3.2).Figure 4
The new observer’s estimation errors; case2: dynamics with uncertainties.
(a) (b) (c) (d)Remark 6.
One can make a brief comparison between the new designed reduced-order multiple observer and the only existing ROMO in the literature—the one designed in [19]. The first drawback of the observer from [19] is related to its existence conditions’ number; thus, the observer from [19] has 4 existence conditions, while the one designing this paper has only one existence condition. The decrease of the existence conditions’ number is due to the “while” loop which improves the generality character of the ROMO. The second main disadvantage of the reduced-order multiple observer from [19] is related to the usage of pole placement (eigenstructure assignment) method, the disadvantages of this technique being presented in the third section of the paper; it is possible that the choice of the observer’s poles leads to the unwanted situation when limt→∞⁡et≠0, this phenomena being called theeigenvalues’ sensitivity; no robustness’ improvement method is included in [19] for this possible problem. The third disadvantage of the observer from [19] with respect to the new ROMO from this paper is related to the lack of an analysis regarding the influence of uncertainties on the observer’s performances. From the dynamic characteristics’ point of view, the comparison between the 2 convergent reduced-order observers for T-S multiple models affected by unknown inputs cannot be done because different T-S multiple models have been used in the two papers; this is because two of the present ROMO’s assumptions ((A3) and (A4)) do not allow the usage of the Takagi-Sugeno multiple model from [19]. However, it has been shown that the new observer’s convergence speed is an excellent one in the research area of the T-S multiple models. Taking into account all the above conclusions, one can remark the superiority of the new designed reduced-order multiple observer from existence conditions and robustness improvement point of view.Remark 7.
One can also make a brief comparison between the new designed ROMO and other existing FOMOs from the literature. Comparing to the state estimation error obtained by a method without taking into account the uncertainties (method proposed in [27]), one can conclude that the new proposed method is more robust against modeling uncertainties and unknown inputs. Another strong point of the new designed ROMO is the class of systems for which the state estimation is possible; the advantages of the used multiple models are related to the ability to extend the tools of the linear theory to nonlinear systems, to the possibility of reducing the complexity of nonlinear systems by constructing linear models that will be aggregated by using weighting functions, and to the fact that these aggregating functions possess the convexity property. Another advantage of the new ROMO with respect to other papers ([1, 2]) dealing with the design of FOMOs for T-S multiple models is related to the existence conditions which are less restrictive. Also, the new design procedure is effective because (1) it addresses the unknown inputs and the uncertainties decoupling problems in the same framework (and not as two independent problems); (2) it can be carried out using commercially available software, such as Matlab; (3) it provides all the degrees of design freedom which can be further utilized to achieve some additional system specifications; (4) the simulation results demonstrate superior convergence of the proposed ROMO compared to those in [1, 2, 19].


## 6.1. Numerical Simulation Setup

To validate the new reduced-order observer, any dynamics meeting the assumptions and the existence condition can be chosen; in this paper, the new ROMO is software implemented and validated in Matlab for the case of a light aircraft lateral-directional motion; the aircraft dynamics, withM=2, has the form (1) where the following matrices have been considered [24]:(33)A1=-0.30-339.810-5.400.1-8.33.75000-28.60.370-0.6400-9.50010.0100.01000010.00100.001000-0.0100-100000.010-0.0010-5,A2=-0.280-339.810-5.200.1-7.63.95000-27.60.340-0.5900-9.60010.0100.01000010.00200.001000-0.0200-9.90000.020-0.0010-4.95,B1=000000000-0.012100010,B2=000000000-0.0099.8009.8,d1=0.2·1111111,d2=0.1·1111111,C=C1C2=03×4I3;D=0.500.51020T.The state of the system is x=VyωxωzφψδrδaT, with Vy being aircraft lateral velocity, ωx the roll angular rate, ωz the yaw angular rate, φ the roll angle of aircraft, ψ the yaw angle of aircraft, δr the rudder deflection, and δa the ailerons’ deflection; the two components of the system input vector u=δrcδacT are the commands of the rudder and ailerons, respectively [24].Aircraft flight is often influenced by disturbances like wind shears, atmospheric turbulences, or errors of the sensors. From aircraft dynamics’ point of view, these represent unknown inputs; an observer for systems with unknown inputs may estimate these unknown inputs, but, more important, it must estimate the system states with very small errors [4]; therefore, in this paper, the unknown input vector v(t) has been randomly chosen. The decision and the activation functions have been chosen of the following forms: ξt=ut, μ1ξt=0.41-tanh⁡ξt, μ2ξt=1-μ1ξt, where the input has the form u=-K-x^, with the gain matrix K- determined by using the ALGLX optimal algorithm borrowed from [24]; K- can be also calculated by means of the pole placement technique or other methods.


## 6.2. Results and Discussion

For the above presented matrices, one obtainsn=7, m=2, q=1, and p=4. The observer validation is performed for both the dynamics without uncertainties ΔAi=0 and with uncertainties ΔAi≠0. The solving of the linear matrix inequalities (8) and (22) is performed using the Matlab LMI tool. From the forms of the matrices C and C2, one can remark that the last three states of the system ψ,δr,δa are measured; within Step 2 of the new algorithm, one judiciously chooses the full row rank matrix L; since this matrix gives information regarding the estimated states, one chooses L=Ip0p×n-p; thus, the first p=4 states are estimated, while the last 3 are measured. Having in mind the form of the matrix C, one gets yt=0r×n-rIrxt=0r×n-rIrx1tx2t=x1t; in other words, x1(t) contains the measurable states, while zt=Ip0p×n-pxt=Ip0p×n-px1tx2t=x2t contains the states to be estimated (the last r states of the system). Thus, ROMO estimates x2(t), providing x^2t; the estimate of the state vector x(t) is obtained by concatenation of the vectors x1(t) and x^2t; that is, x^t=x1tx^2t=ytz^t.For the first simulation case (without uncertainties), within Steps3, 4, and 5, one obtains(34)T=10000-0.250010000000100-0.25000010-0.50,G1=0.0150.020.0150.01,G2=0.00750.010.00750.005,H1=-2.5000-2.50-50,H2=-2.45000-2.450-4.90,F=00.25000000.25000.50,N1=diag-5;-10;-15;-20,N2=diag-7.5;-15;-22.5;-30,J1=0-4.27000-29.760-11.0300.01-4.80,J2=0-4.26000-29.770-11.0400.01-4.790;the solutions of the two LMIs (8) are P1=diag0.195;0.105;0.06;0.04, P2=diag0.072;0.035;0.023;0.017.For increasing the robustness of the ROMO, one uses the new technique involving the adding ofM robustness terms in the observer’s equations; thus, for case 2 (dynamics with uncertainties), one has chosen ΔAi as a percent of 1% from their initial values (Ai); also, one has chosen ε=0.01 and different values for the scalars β1 and β2  β1=β2=0.8;1.6;2.4;3.2 in order to analyze the influence of these constants on the observer’s convergence time and convergence errors. Solving the two linear matrix inequalities (22) for β1=β2=0.8, the following is obtained:(35)P-1=10-2·13.5300006.65-0.0100-0.014.66-0.0700-0.073.27,P-2=10-2·8.8200004.34-0.0100-0.013.05-0.0500-0.052.15.The new architecture from Figure1 is software implemented in Matlab, Figure 2(a); the Matlab/Simulink subsystem “subsystem vectors gama_1, gama_2” (subsystem for the calculation of the vectors γit,i=1,M¯) is presented in Figure 2(b).Figure 2
Matlab/Simulink model for the new reduced-order multiple observer.
(a) (b)For case1  ΔAi=0, Figure 3 presents the time histories of the estimation errors, that is, p=4 components of the vector et=zt-z^t; from Figure 3 one remarks the canceling of the estimation errors, hence the effectiveness of the developed ROMO scheme. This is equivalent to the achievement of the state reconstruction (the convergences z^t→zt and x^t→xt). The convergence speed of the ROMO (less than 2 seconds) represents an excellent convergence speed in the research area of the T-S multiple models.Figure 3
The new observer’s estimation errors; case1: dynamics without uncertainties.
(a) (b) (c) (d)For case2  ΔAi≠0, Figure 4 presents again the time histories of the estimation errors’ four components for different values of the scalars β1 and β2. One remarks the following: (1) the convergence time is approximately the same as in case of dynamics without uncertainties due to the robustness’ improvement approach; (2) the increase of the scalars β1 and β2 does not modify the convergence time of ROMO; (3) the increase of the scalars β1 and β2 leads to the increase of the overshoots’ absolute values for Figure 4(a) (from 7.1 m/s for β1=β2=0.8 to 10.35 m/s for β1=β2=3.2) and to the decrease of the overshoots’ absolute values for Figure 4(b) (from 20.05 deg/s for β1=β2=0.8 to 18.35 deg/s for β1=β2=3.2).Figure 4
The new observer’s estimation errors; case2: dynamics with uncertainties.
(a) (b) (c) (d)Remark 6.
One can make a brief comparison between the new designed reduced-order multiple observer and the only existing ROMO in the literature—the one designed in [19]. The first drawback of the observer from [19] is related to its existence conditions’ number; thus, the observer from [19] has 4 existence conditions, while the one designing this paper has only one existence condition. The decrease of the existence conditions’ number is due to the “while” loop which improves the generality character of the ROMO. The second main disadvantage of the reduced-order multiple observer from [19] is related to the usage of pole placement (eigenstructure assignment) method, the disadvantages of this technique being presented in the third section of the paper; it is possible that the choice of the observer’s poles leads to the unwanted situation when limt→∞⁡et≠0, this phenomena being called theeigenvalues’ sensitivity; no robustness’ improvement method is included in [19] for this possible problem. The third disadvantage of the observer from [19] with respect to the new ROMO from this paper is related to the lack of an analysis regarding the influence of uncertainties on the observer’s performances. From the dynamic characteristics’ point of view, the comparison between the 2 convergent reduced-order observers for T-S multiple models affected by unknown inputs cannot be done because different T-S multiple models have been used in the two papers; this is because two of the present ROMO’s assumptions ((A3) and (A4)) do not allow the usage of the Takagi-Sugeno multiple model from [19]. However, it has been shown that the new observer’s convergence speed is an excellent one in the research area of the T-S multiple models. Taking into account all the above conclusions, one can remark the superiority of the new designed reduced-order multiple observer from existence conditions and robustness improvement point of view.Remark 7.
One can also make a brief comparison between the new designed ROMO and other existing FOMOs from the literature. Comparing to the state estimation error obtained by a method without taking into account the uncertainties (method proposed in [27]), one can conclude that the new proposed method is more robust against modeling uncertainties and unknown inputs. Another strong point of the new designed ROMO is the class of systems for which the state estimation is possible; the advantages of the used multiple models are related to the ability to extend the tools of the linear theory to nonlinear systems, to the possibility of reducing the complexity of nonlinear systems by constructing linear models that will be aggregated by using weighting functions, and to the fact that these aggregating functions possess the convexity property. Another advantage of the new ROMO with respect to other papers ([1, 2]) dealing with the design of FOMOs for T-S multiple models is related to the existence conditions which are less restrictive. Also, the new design procedure is effective because (1) it addresses the unknown inputs and the uncertainties decoupling problems in the same framework (and not as two independent problems); (2) it can be carried out using commercially available software, such as Matlab; (3) it provides all the degrees of design freedom which can be further utilized to achieve some additional system specifications; (4) the simulation results demonstrate superior convergence of the proposed ROMO compared to those in [1, 2, 19].


## 7. Conclusions

The purpose of this study was to design a new reduced-order multiple observer, combining a typical reduced-order observer for LTI systems with unknown inputs and a full-order multiple observer, to achieve the finite-time reconstruction of the system’s state associated with T-S multiple models. The main advantage of the new designed ROMO is the decrease of the necessary sensors’ number. To cancel the possible system uncertainties and to improve the observer robustness, some compensation terms have been added in the observer design approach. To ensure the canceling of the state estimation errors, one has considered quadratic Lyapunov functions; one has been shown that the problem of reduced-order multiple observers for Takagi-Sugeno multiple models with unknown inputs can be reduced to the standard case when the unknown input vector does not interfere in the observer equations. The steps of the design procedure have been software implemented to validate the new ROMO; as an example, one has considered the lateral-directional motion of a light aircraft; the errors of the observer have been canceled even in the presence of unknown inputs and uncertainties. Future works will concern the application of these methods to the construct of robust ROMOs for T-S fuzzy systems with time delays.



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*Source: 5231989-2018-05-14.xml*