High‐order sliding‐mode functional observers for multiple‐input multiple‐output (MIMO) linear time‐invariant systems with unknown inputs

For arbitrary multiple‐input multiple‐output linear time invariant systems with unknown inputs this article provides sufficient conditions to estimate linear functionals of the state variables. When the unknown input is uniformly bounded these conditions are strictly weaker than the classical conditions for functional unknown input observers, well‐known in the literature, and generalize previous results using discontinuous differentiators. Furthermore, a general methodology is proposed to design functional observers, that are able to estimate the functionals, when possible, exactly and in finite‐time or fixed‐time. Instead of using a cascade of Luenberger observers and high‐order sliding‐mode differentiators, standard in the literature for this problem, a bi‐homogeneous observer of reduced order is proposed in the article. Proofs of the convergence are provided using smooth Lyapunov functions and an academic example illustrates the behavior of the proposed observer for a system not tractable with the available methods.


INTRODUCTION
Functional observers provide an asymptotic converging estimation of a function of the states, in contrast to state observers, for which the full state is to be estimated.Functional observers are very important, since it is not always necessary to estimate the full state.For example, to implement a linear feedback control law u = Kx, it suffices to estimate the functional Kx.For fault detection and many other control tasks a similar situation appears.Beyond being simpler, it is in general also less demanding in terms of observability/detectability requirements to estimate a functional of the state than the full state, as it is well-known (for an overview see Reference 1 and the recent paper 2 ).So, for example, it is impossible to estimate the state of an undetectable system, but it is still possible to obtain some functionals of the state.For linear time invariant systems necessary and sufficient conditions to obtain functional observers are well-known, and there are several methods to calculate them. 1,2or systems with Unknown Inputs a state Unknown Input Observer (UIO) or a Functional Unknown Input Observer (FUIO) intends to estimate the states or a functional of the states, despite of the presence of the (completely) unknown inputs, respectively. 3,4(F)UIOs are very flexible, since the unknown input can describe many unmodeled phenomena and/or highly nonlinear effects and they are a key ingredient in the fault detection and isolation field. 5,6It is not surprising that the conditions under which an observer for systems with unknown inputs can be designed are much stronger than the ones for systems without unknown inputs.These conditions are very well-known and established 3 (see also References 7-10 and References 4,11-13), and several design methods have been proposed for LTI systems.
The necessary and sufficient conditions for the existence of a FUIO require, 3,4 on the one side, the dynamics decoupled from the unknown input to be detectable or observable, and, on the other side, that the states of the dynamics affected by the unknown input can be obtained without using time derivatives.For the estimation of the state this latter condition is fulfilled only if the relative degree between unknown input and the measured output is one, while the former condition is equivalent to the system having only stable invariant zeros, or no zeros at all. 3,4,13hese conditions are rather stringent, and it would be desirable to relax them.However, since they are necessary, it is impossible to overcome them, without imposing further restrictions on the system or relaxing properties of the observer.Although the detectability/observability conditions of the dynamics decoupled from the unknown input are impossible to relax, it is possible to lessen the relative degree conditions, if (ideal) differentiators are used (see References 4 and 13).Since ideal differentiators are unfeasible, we are again faced with the necessity of imposing extra requisites on the system to lower the design conditions for FUIOs.
Usual (continuous) differentiators, as for example, linear or homogeneous ones, [14][15][16][17] are able only to approximately estimate (asymptotically) the derivatives of a signal, when its highest derivative is bounded.In contrast, the High-Order Sliding Mode (HOSM) differentiator 18,19 is a discontinuous system that can estimate exactly, robustly and in finite-time an arbitrary number l > 0 of the derivatives of a signal, as far as its (l + 1)th derivative is uniformly bounded.Levant's differentiator opened the possibility of relaxing the relative degree condition in the design of UIOs, under the extra requirement that the unknown input is bounded.This has been proposed in numerous works for the estimation of the state, [20][21][22][23][24][25] and pursued for a functional of the state in References 13 and 12.However, as it is detailed in References 13 and 12, beyond the boundedness assumption, several other restrictions have to be imposed, and additional difficulties appear when using a differentiator.For example, known or unknown inputs have to be differentiable.This is partially avoided constructing an UIO consisting of a cascade of a Luenberger Observer and a HOSM differentiator, 20,22,24,25 what strongly increases the order of the observer.Using the same cascade structure, a set of sufficient conditions for the design of FUIOs are presented in References 13 and 12.The boundedness of the unknown input in Levant's differentiator has been relaxed, allowing growing unknown inputs with a known upper bound, by using variable gains (see References 26 and 27, which also study the accuracy of the explicit Euler integration).
The objective of this article is to extend our previous work in Reference 13, and the contributions are the following: (i) We relax further the conditions to construct FUIO for multiple-input multiple-output (MIMO) LTI systems.(ii) Instead of a cascade of a Luenberger observer and a differentiator, we propose discontinuous FUI observers, which reduce the order and do not require differentiation.(iii) Using the homogeneous in the bi-limit injection terms of the differentiator recently proposed in Reference 28, we are able to accelerate (when possible) the convergence of the estimation dynamics, obtaining finite-time or fixed-time convergence.(iv) A unified analysis and design framework is proposed, so that the design of state or functional observers, with linear or nonlinear, finite-time or fixed-time dynamics is possible.(v) We also extend the recent work reported in Reference 29, that considers state UI observers, not including FUIOs.The design in Reference 29 relies on a very special triangular structure for the system.We relax strongly this condition, allowing more general (nontriangular) feedback interconnected forms for the design.This requires a Lyapunov-based convergence proof, that is unnecessary in Reference 29.(vi) We propose much simpler injection terms as those proposed originally in Reference 28, and also used in Reference 29.This strongly simplifies the realization of the observer.
The rest of the article is organized as follows.In Section 2 some preliminaries are introduced.The problem formulation and an overview of the state of the article is provided in Section 3. The new Functional Unknown Input Observer proposed in this article is presented in Section 4. The proof of the main results, using smooth Lyapunov functions, is detailed in Section 5, while an illustrative academic example is given in Section 6.Finally, some conclusions are drawn in Section 7.

PRELIMINARIES
The notation used in the article is fairly standard.We recall briefly some definitions of homogeneity and homogeneity in the bi-limit.However, for precise definitions and properties, we refer the reader to References 30-32 for homogeneity of continuous or discontinuous systems, and to References 33 and 34 for homogeneity in the bi-limit of continuous or discontinuous systems, respectively.For a vector x ∈ R n , all real values  > 0, and n positive real numbers r i > 0 the dilation operator is defined as Constants r i > 0 are the weights of the coordinates x i , and r ∶= [r 1 , … , r n ] is the vector of weights.
A function V ∶ R n  → R m is said to be r-homogeneous of degree l ∈ R, or (r, l)-homogeneous for short, if for all  > 0 and for all x ∈ R n the identity V (Δ r  x) =  l V (x) holds.Consider a Filippov Differential Inclusion 30,35,36 where F (x) ∈ R n is an upper-semicontinuous nonempty compact convex set-valued function.The Differential Inclusion (1) is r-homogeneous of degree l ∈ R, or (r, l)-homogeneous for short, if the identityF(Δ r  x) =  l Δ r  F(x) holds for all  > 0 and for all x ∈ R m .For a Differential Equation ̇x = f (x) the standard definition of homogeneity 30 coincides with that of (1) if we consider the Differential Inclusion ̇x ∈ {f (x)}.Homogeneity of the Differential Inclusion (1) means that (1) is invariant under the transformation of the time and coordinates given by is said to be homogeneous in the 0-limit with associated triple (r 0 , l 0 ,  0 ), if it is approxi- mated near x = 0 by the (r 0 , l 0 )-homogeneous function  0 .It is said to be homogeneous in the ∞-limit with associated triple is said to be homogeneous in the bi-limit, or bl-homogeneous for short, if it is homogeneous in the 0-limit and homogeneous in the ∞-limit.Similar definitions apply for Differential Inclusion (1) or Differential Equations.
Consider the system (1) and assume that the origin is a strongly asymptotically stable (AS) equilibrium point.Define an open ball centered at the origin with radius r > 0 by B r = {x ∈ R n ∶ ‖x‖ < r}.The origin of system is finite-time stable (FTS) if it is AS and for every x 0 ∈ B r ⧵ {0}, any solution x (t, x 0 ) reaches x (t, x 0 ) = 0 at some finite time moment t = T (x 0 ) and remains there ∀t ≥ T (x 0 ), where Homogeneous systems (1) have important properties as for example, that local stability implies global stability and if the homogeneous degree is negative asymptotic stability implies finite time stability: 19,30,32 Assume that the origin of a Filippov Differential Inclusion (1) is strongly locally Asymptotic Stable and that it is r−homogeneous of degree l < 0; then, x = 0 is strongly globally finite-time stable and the settling time is continuous at zero and locally bounded.Stability of homogeneous systems (1) can be studied by means of homogeneous Lyapunov functions: 19,30,31,[36][37][38] Assume that the origin of a homogeneous Filippov DI (1) is strongly globally AS.Then, there exists a C ∞ homogeneous strong Lyapunov Function.
Along this article, we use the following notation.For an integer n, i = ⟨1, n⟩ means i = 1, … , n.For a real variable z ∈ R and a real number p ∈ R the symbol ⌈z⌋ p = |z| p sign(z) is the signed power p of z.According to this ⌈z⌋ 0 = sign (z), and if p is an odd number then ⌈z⌋ p = z p and |z| p = z p for any even integer p.Moreover, ⌈z⌋ p ⌈z⌋ q = |z| p+q , ⌈z⌋ p ⌈z⌋ 0 = |z| p , and ⌈z⌋ 0 |z| p = ⌈z⌋ p .

PROBLEM FORMULATION AND OVERVIEW
Consider a MIMO LTI system without feedthrough (for simplicity) given by where (t) ∈ R n , (t) ∈ R m , and (t) ∈ R p are the state, unknown input and measured output, respectively.If the unknown input  (⋅) is a Lebesgue measurable function the solutions of ( 2) are Carathéodory solutions.The objective is to reconstruct (t) ∈ R q using only measurements of (t).Without loss of generality, we assume that rank (C) = p, rank (D) = m, rank (E) = q.For simplicity in the presentation we do not consider a known input u, since it does not modify the (observability) properties and it is simple to include it in the observer design.A similar argument is valid for a feedthrough interconnection between input and output.The problem we are interested in is the following: given a triple (A, D, C) find the set of matrices E such that a convergent estimation of  can be obtained.Furthermore, design an observer performing the estimation.This is the so called functional observation problem.We are interested in relaxing the classical conditions for the existence and construction of such observers.We do not seek for a minimal order observer, but we aim to obtain observers with strong convergence properties when possible, that is, finite-time or fixed-time convergence.

Description of 𝚺 in normal or special forms
For the characterization of the required properties allowing the existence of the FUIOs, and also to perform their design, it is convenient to write the system Σ in special forms.
It is well-known (see e.g., References 4,39-41) that for system Σ, there always exist state x = T, input w = R and output y = Q regular transformations such that the transformed system is an interconnection of the following six subsystems and the output, that is, the functional of the state, to be estimated is given by The subsystems have the following properties: 1. Subsystem Σ s , given by the triple (A s , C s , D s ), is affected by the unknown input, it is strongly observable*, x s ∈ R n s , w s ∈ R m s , y s ∈ R p s , with m s = p s , so the subsystem is square, and rank D s = rank C s = m s .Furthermore, there exist an integer  ≥ 0 and constant matrices M i such that where y (i) s represents the ith time derivative of y s .2. Subsystem Σ o , given by the pair (A o , C o ), is not affected by the unknown input, it is observable, and 3. Subsystem Σ d is not affected by the unknown input, A d is Hurwitz, so that it is detectable, and x d ∈ R n d .4. Subsystem Σ u is not affected by the unknown input, the eigenvalues of A u have non negative real parts, that is, ℜ { i (A u )} ≥ 0, so that it is undetectable and x u ∈ R n u . 5. The subsystem, composed of Σ  and Σ  , is affected by the unknown input, it is not strongly observable, Recall that subsystems Σ d and Σ u correspond to the zero dynamics of the system.The union of the eigenvalues of A d and A u are the invariant zeros of the system.Those of A d are asymptotically stable, that is, they have negative real parts, while those of A u are not asymptotically stable, that is, they have zero or positive real parts.
To establish our results, we will require to express the strongly observable Σ s and the observable Σ o subsystems in more specific forms.We refer to the special coordinate basis, introduced in Reference 40, and presented in detail in Reference 41(chapter 5).It is very convenient and there is a Matlab toolbox providing for its construction.
It is classical that the observable subsystem Σ o can be written either in the observer normal form Reference 41(eqs.5.3.13-5.3.15) or the observability normal form (see e.g., Reference 42(chapter 6) for a detailed discussion about normal forms).We will use this latter one, since it requires the bl-homogeneity of the observer, while the observer normal form can be implemented with a homogeneous differentiator (as already done in Reference 29,43).The observability normal form of Σ o is given by the p o subsystems For the strongly observable subsystem Σ s we will make use of the special form given in Reference 41(eqs.5.3.19-5.3.21), which is composed of m s = p s subsystems for i = 1, … , p s , where n s = ∑ p s i=1 n s,i and A (ss,i),j , a s,i , a so,i , a sd,i , a su,i , a s,i , and a s,i are constant row vectors of appropriate dimensions.Note that n s,i is the relative degree between the unknown input w s,i and the output y s,i of the subsystem Σ s,i , and they correspond to the zeros at infinity of the system.

Classical and relaxed conditions
Under the assumptions that the unknown input w (t) is a completely arbitrary signal (as long as solutions of the system exist), for example, it may be unbounded, and that not ideal differentiators are used, necessary and sufficient conditions for the existence of State (UIO's) or Functional Unknown Input Observers (FUIO's) for system (2), are well-known (see e.g., References 3,4,13).If the full state of the system Σ has to be estimated, that is, E = I, the necessary and sufficient conditions have been termed strong* detectability by Hautus, 3 and it is equivalent to the fact that for system Σ for any unknown input w and initial state, y (t) → 0 as t → ∞ implies x (t) → 0 as t → ∞.In the special coordinates (3)-( 8) strong* detectability means that subsystems Σ u , Σ  and Σ  do not exist, that is, n u = n  = n  = 0, and that matrix C s is invertible.Moreover, the UIO has assignable dynamics if also subsystem For estimation of a functional of the state (9) (FUIO), necessary and sufficient conditions are weakened to Reference 4(theorem 2) Moreover, the convergence velocity of the FUIO dynamics can be assigned arbitrarily if, additionally, E d = 0.
Sufficiency of conditions ( 13)-( 14) is simply obtained by noting that only the observable x o and detectable x d states have to be estimated, and the part of the states x s required by E s correspond to the output y s .In that case a (reduced order) linear observer is given by References 4 and 13 The estimation dynamics of x o can be arbitrarily assigned, since (C o , A o ) is observable, while the estimation dynamics of x d is completely determined by matrix A d .From x s it is only possible to obtain the value of its output y s .
As shown in Reference 4(proof of Theorem 2), it is completely impossible to estimate the states x u , x  and x  , since subsystems Σ u , Σ  and Σ  correspond to the indistinguishable dynamics of the system Σ.And therefore it is impossible to relax condition (13) for estimation of a functional z, or the corresponding conditions n u = n  = n  = 0 for the estimation of the state.However, condition ( 14) for FUIO's or the invertibility of C s for UIO's can be relaxed, if we have perfect differentiators, as property (10) of the strong observable subsystem Σ s shows.In this case, observer (15) can be extended using ideal differentiators to Note that in (16) only derivatives of the output y s are used.However, if a known input u is present in system Σ, for the estimation of x s also derivatives of the input signal u are necessary, what is more probematic since input signals are usually less smooth than output ones.For estimation of the full state, that is, E = I, the necessary and sufficient conditions n u = n  = n  = 0 for the existence of the UIO ( 16) is called by Hautus 3 strong detectability and is equivalent to the fact that for system Σ for any unknown input w, whenever y (t) = 0 then x (t) → 0 as t → ∞.Assignability of the convergence velocity of ( 16) requires, additionally, n d = 0 for state estimation or E d = 0 for functional estimation.If system Σ satisfies n u = n  = n  = n d = 0 is termed by Hautus 3 strongly observable, and this is equivalently characterized as: for any unknown input w and initial condition, whenever y (t) = 0 for all t ≥ 0 implies that x (t) = 0 for all t ≥ 0. Necessity of the condition (13) for the existence of a FUIO or, correspondingly, strong detectability for an UIO (as e.g., ( 16)), has been shown for example, in References 3 and 4(theorem 2) and derives from the fact that it is completely impossible to estimate by any means the states x u , x  , and x  , since they are indistinguishable.
Since estimation of x s , except if C s is invertible, requires the use of ideal differentiators, it is natural to ask how to realize such an algorithm.Consider that the ideal differentiator in ( 16) is to be realized by a dynamical system with an Input-Output map xs =  (y s ).Since, as it is well-known, the differentiation operation is unbounded, that is, an arbitrary small change in the argument can result in an arbitrary large change in the image, a very good (or perfect) realization of  will not be  ∞ −Input-Output Stable (IOS), that is, an input with small  ∞ size results in an output with a small  ∞ size.For example, if one applies an arbitrary small input y s (t) = a sin(t), with small a, the output xs (t) will be an arbitrarily large signal xs (t) = a  sin(t + ), for large .Although this is a good property to take derivatives, it implies that the sensitivity to noise will be extremely high, since  will not filter out but amplify the high frequency noise.
A possibly more desirable property of the realization  is one being  ∞ −Input-Output Stable, since such a system will have less sensitivity to measurement noise: a small noise in y s will lead to a small error in the estimate xs .However, such a stable realization xs =  S (y s ) is not able to always estimate correctly x s from y s for Σ s , since for this system, in general, y s (t) → 0 when t → ∞ does not imply that x s (t) → 0 when t → ∞.
Note that y(t) → 0, but x 2 has no limit and x 3 (t) is unbounded.And an estimator, such as xs =  S (y s ), that is Input-Output stable, cannot estimate correctly the state of the system.This shows the necessity of conditions ( 13) and ( 14) for a FUIO, if an Input-Output Stable observer is to be implemented.Note that the proof of the sufficiency and necessity of ( 13) and ( 14) for the existence of a FUIO given in Reference 3 it is assumed that the FUIO is linear and time invariant.However, as explained in Reference 4 and in the previous paragraph, the necessity is valid for estimators which are input-output stable.
Since conditions ( 13) and ( 14) are necessary and sufficient, to dispose of ( 14) it is required to strengthen some other condition in the problem formulation.Here, we will assume: Assumption 1.The unknown input w is uniformly bounded.
If only exact convergence in finite time or asymptotically is desired, this condition can be relaxed to: The unknown input w is finally uniformly bounded.
The introduction of the high-order sliding mode (HOSM) differentiator, 18,19 a discontinuous algorithm able to estimate exactly, robustly and in finite-time an arbitrary number l > 0 of the derivatives of a signal, as far as its (l + 1)th derivative is uniformly bounded, opened the possibility of fully relaxing condition (14).And this has been attempted in numerous works, [20][21][22][23][24][25] and also in the special issue. 45These works consider only the estimation of the state, that is the case when E = I.The recent papers 12,13 extend these results to the FUIO.However, as it is detailed in References 12 and 13, beyond Assumption 1, several other restrictions are required to dispose of condition (14).One of the difficulties of using a differentiator is that, when a known input is present, the input signal has also to be differentiated.To avoid this 20,22,24,25 introduce an Unknown Input Observer consisting of a cascade of a Luenberger Observer and a HOSM differentiator.This highly increases the order of the observer.Moreover, boundedness of the unknown input is not sufficient, since the state x s also depends on x u , x  , and x  , which should be also required to be bounded.A set of sufficient conditions for the FUIO are presented in References 12 and 13, which relax condition (14) imposing additional conditions to (13), and using a cascade structure for the realization of the FUIO.
Our main objective in this work is to determine necessary and sufficient conditions such that x s can be estimated exactly using observers with constant gains, that is, being  ∞ −Input-Output stable, and assuming only that the unknown input w satisfies Assumption 1.In other words, under Assumption 1, which conditions have to replace (14) to be able to estimate x s exactly.
Assumption 1 permits the use of linear or homogeneous observers, [14][15][16][17] but at the cost of an approximate instead of an exact estimation of the states or functional of the states and using high-gain.
Remark 1. Assumption 1 can be relaxed if differentiators or observers that are not  ∞ −Input-Output Stable are allowed, at the cost of having unboundedly increasing sensibility to the noise of the observer.A possible implementation of a differentiator not being  ∞ −Input-Output Stable is provided by Differentiators with (unboundedly) growing gains, as those proposed in References 26,27, or also 44, where an estimator for the system (17) is constructed.However, we will restrict ourselves here to construction of FUIO's being  ∞ −input-output stable.
5. In a recent work, 29 have performed part of this program for state observers, not including FUIOs.Methodologically, they rely strongly on a very special triangular structure for system Σ, which is derived in that work and Reference 43.
Although this is a nice feature of linear systems, and it simplifies strongly the convergence analysis, we allow for a more general kind of interconnections between the subsystems in (3)-( 9).In our case the convergence proof is more general than in Reference 29, but also more involved.We think that this will allow a simpler extension to more general frameworks than the LTI one.Furthermore, a simplified realization of the output injection terms of the observer is presented in this article, compared to the ones proposed in References 29 and 28.

THE EXTENDED FUNCTIONAL UNKNOWN INPUT OBSERVER
The states x u , x  , and x  of the undetectable Σ u and of the not strongly observable Σ  , Σ  subsystems, respectively, are imposible to estimate.Thus, we aim at estimating the states x s , x o , and x d of the strongly observable Σ s , the observable Σ o and the detectable Σ d subsystems.In order to estimate these states we propose an observer, based on the bi-homogeneous differentiator proposed in Reference 28.This observer is given by the interconnection of observers for these subsystems Σ o , Σ d , and Σ s , and a bounded error estimation of the state x  of subsystem Σ  .
The full extended functional unknown input observer (EFUIO) is given by the following interconnected set of observers, The detailed form of observers Ω s and Ω o , and in particular of their correction terms Φ s and Φ o , respectively, will be given in the next Section 4.1.Since these correction terms can be discontinuous, the solutions of (18) are understood in the sense of Filippov 30,35 .
Observer (18) corresponds to a copy of the dynamics of part of the plant (3)- (8).Similar to the previous observers ( 15) and ( 16) it has a copy of the observable Σ o and the detectable Σ d subsystems.The correction term Φ o of Ω o can stabilize the observation error and, moreover, it can modify the convergence properties beyond the possibilities of the linear correction terms in (15) and (16).As it will be clarified below, Φ o can induce finite-time or fixed-time convergence.For the detectable part Σ d no correction term is available, and thus its form and velocity of convergence cannot be modified: it is exponential depending exclusively on the eigenvalues of A d .
In contrast to ( 15) and ( 16), observer (18) contains a copy of the strongly observable Σ s and of the not strongly observable but internally stable Σ  subsystem.The estimators for these two subsystems replaces the ideal differentiator contained in (16), which is not realizable.The correction term Φ s of observer Ω s can strongly modify the convergence properties, not only accelerating it and inducing finite-time and fixed-time convergence, but also, and more importantly in this case, by allowing to fully counteract the effects of bounded unknown input signals for the exact estimation of x s .The estimator Ω  has no correction term, and its dynamics is fully determined by the properties of the subsystem itself, in particular, by the Hurwitz system's matrix A  .Since subsystem Σ  has unknown inputs and x  is not observable/detectable, the estimation x cannot converge (in general) to its true value x  .However, as it will be shown, under some conditions x will provide a rough estimation of x  , that is, x − x  is (asymptotically) uniformly bounded, even in case x  is unbounded.This rough estimation of x  is used by Ω s for the exact estimation of x s .Observe that since x  is in general an unbounded signal it is necessary to implement observer Ω  .This observer is not necessary if D s A s = 0.
Note that no copies of subsystems Σ u , Σ  are used in the observer, since their states are not observable/detectable and their trajectories grow unboundedly.This explains also that the realization of the observer in original coordinates may be problematic, since it compromises the internal stability of the observer.All gains of the observer are constant, and the state estimation error dynamics is Input-to-State stable from the noise to the state estimation error.

4.1
Detailed structure of observers  s and  o In particular, for the convergence proof, it is convenient to write the observers Ω s and Ω o in the special form given by ( 12) and ( 11) with positive external gains k ,j > 0, for  ∈ {(s, i) , (o, )}, and a positive tuning gain L > 0, appropriately selected.
The output injection terms φ(s,i),j (⋅) are obtained from the functions by scaling the positive internal gains  (s,i),j > 0,  (s,i),j > 0 as Here (⋅) → (⋅) means that the left expression is replaced by the right expression in the function. > 0 a is a further positive tuning gain, that has to be appropriately selected.Functions  (s,i),j are the sum of two (signed) power functions, with powers selected as r (s,i),0,n s,i = 1, r (s,i),0,j = r (s,i),0,j+1 − which are completely defined by two parameters, d 0 and d ∞ .
Similarly, the output injection terms φ(o,),j (⋅) are obtained from the functions by scaling the positive internal gains  (o,),j > 0,  (o,),j > 0 as They are the sum of two (signed) power functions, with powers selected as which are also completely defined by the two parameters, d 0 and d ∞ , and the selected value of r o ≥ 1.They have to satisfy For convenience, relations ( 23) and ( 26) are extended to j = n s,i + 1, and j = n o, + 1, respectively.
The output injection terms ( 21) and ( 24) are much simpler than the ones proposed in Reference 28, whose complexity grows with the order.This simplifies greatly the implementation of the observer.
Remark 2. Note that the last weights for the strongly observable subsystem Σ s are set to one (23), while those for the observable one (26) can be selected different from one.This allows to have for example, discontinuous injection terms in the observer Ω s and continuous ones in the observer Ω o selecting d 0 = −1 and r o > 1.If r o = 1 then both will be discontinuous.Remark 3. We have considered in the observer ( 19)-( 20) a unique value of L and  for all subsystems.This allows a simplified proof and is sufficient to show the convergence of the observer.However, in the application it is advantageous to have different values of L and  for each subsystem.
Finally we note that the negative homogeneous terms used in the observer injections proposed here have a rather high-gain character near zero error.However, this high-gain is only local in contrast to for example, the High-Gain Observers, 14,15 for which the high-gain acts globally.

Exact and approximate observers
We consider two different kinds of EFUIO's, depending on the selection of the parameter d 0 .Proofs of the forthcoming Theorems 1 and 2 are given in Section 5.

Exact extended functional unknown input observers
Due to the presence of (bounded) unknown inputs w, if condition ( 14) is not satisfied, observers with all continuous injection terms (11) for the strongly observable subsystems Ω s,i , will be in general not able to exactly estimate the corresponding states x s,i .For this to be possible it is necessary to select d 0 = −1, so that injection terms φ(s,i),n s,i (⋅) in ( 19) are discontinuous.The main result is the following Theorem 1.Consider a LTI system Σ (2) with unknown inputs satisfying Assumption 1. Suppose that conditions (13) and in ( 3) and ( 7) are satisfied.Choose d 0 = −1 and d ∞ ≥ 0. Under these conditions, for arbitrary positive internal gains  ,j > 0,  ,j > 0, there exist positive external gains k ,j > 0 for  = {(s, i) , (o, )}, and parameters L > 0 and  > 0 sufficiently large such that the estimation error of system (18) has a globally and asymptotically stable equilibrium point if the unknown inputs are zero, that is, w s = 0, w  = 0.Moreover, for bounded unknown inputs, system ( 18) is an exact globally convergent FUIO for system Σ (2), for arbitrary matrices E s , E o , and E d .Conversely, the conditions are also necessary for the proposed observer.Furthermore, the estimation error dynamics is Input-to-State Stable with respect to the measurement noise.□ Note that for the convergence of the observer it is not required that the estimation of x  converges to the true value.Only a bounded estimation is used.
The Theorem establishes the sufficiency of the conditions ( 13) and (27).The necessity of the conditions is established under two considerations: (1) For the class of observers with constant gains considered, which are ISS with respect to measurement noise.(2) That E s is arbitrary.
As discussed previously, Assumption 1 or conditions (27) can still be weakened or relaxed.For example, using a gain-varying observer or, for some specific E s , performing a more detailed structure analysis of the plant.

Approximate extended functional unknown input observers
If only continuous injection terms (21) are used in the observer, that is, d 0 ≠ −1, then only an approximate estimation of the states x s,i can be obtained.The main result is the following Theorem 2. Consider a LTI system Σ (2) with unknown inputs satisfying Assumption 1. Suppose that conditions ( 13) and ( 27) are satisfied.Choose −1 < d 0 ≤ 0 and d ∞ ≥ 0. Under these conditions, for arbitrary positive internal gains  ,j > 0,  ,j > 0, there exist positive external gains k ,j > 0 for  = {(s, i) , (o, )}, and parameters L > 0 and  > 0 sufficiently large such that the estimation error of system (18) has a globally and asymptotically stable equilibrium point if the unknown inputs are zero, that is, w s = 0, w  = 0.Moreover, for bounded unknown inputs, system ( 18) is an approximate globally convergent FUIO for system Σ (2), for arbitrary matrices E s , E o , and E d .Furthermore, the ultimate bound on the estimation error can be reduced arbitrarily by increasing the value of , and the estimation error dynamics is Input-to-State Stable with respect to the measurement noise.□

Discussion on conditions (27)
To better understand the meaning of conditions (27) for the convergence of the observer (18)    27) are necessary and sufficient: D  A u = 0 and A  = 0. Similarly, for observer error Ξ s the inputs x  and x u are unbounded, and therefore problematic for the boundedness of e s .Conditions D s A su = 0 and D s A s = 0 from (27) are necessary and sufficient to assure that these signals do not affect the final behavior of e s .A block diagram of the error dynamics, when conditions (27) are satisfied, is presented in Figure 1.
From the remaining inputs to Ξ s , e o and e d converge to zero, while e  and w s are finally bounded.When the correction term Φ s is a continuous function, then e s can be at the best uniformly and ultimately bounded.This is the case for the approximate observers of Theorem 2. However, a discontinuous injection term Φ s is able to fully counteract the effect of bounded unknown inputs and force e s to converge to zero in finite-time.This happens when d 0 = −1 and corresponds exactly to the situation of Theorem 1.The previous discussion also shows the necessity of the conditions (27) for the convergence of the observer, since if any of them is violated one of the estimators will have an unbounded unknown input signal.

Tuning of the observers
Gain Selection: The different gains in the observers have a different role, and their selection has a natural hierarchy, so that they should be selected in the following order: 1.The internal gains  ,j > 0,  ,j > 0 can be selected freely.Their selection influences the relative weight of the low degree term and the high degree one.
2. The external gains k ,j > 0 have the objective of stabilizing the observers in the absence of interconnections and external perturbations, that is, when we consider that w = 0, a o, = 0, a s,i = 0, a so,i = 0, a sd,i = 0, a su,i = 0, a s,i = 0.And so they can be designed as for the differentiator proposed in Reference 28.
3. The tuning parameter L is selected to assure the convergence in presence of the interconnections, but not of the bounded perturbations.
For setting its value we assume that w = 0, a su,i = 0, a s,i = 0.By increasing its value beyond the minimal value to assure stability the convergence velocity will be increased.
4. The tuning parameter  is selected to assure the convergence in presence of the unknown input and the bounded perturbations caused by the bounded error estimation of x  .
Conditions ( 27) relax the classical condition ( 14) for the existence of a FUIO.Theorem 1 extends the result of theorem 6 in Reference 13.For example, in Reference 13 subsystem Σ u is required to be absent from system Σ, but not in Theorem 1.Moreover, observers Ω o and Ω s in Theorem 1 are able to converge in finite-time or in fixed-time, in contrast to their counterparts in Reference 13, which converge exponentially.Moreover, observers Ω o and Ω s do not require a differentiator in cascade, as required in Reference 13.
It is important to note that observer Ω (18) is not a simple use of a HOSM differentiator, replacing the ideal derivatives in observer ( 16), but it is an observer, with discontinuous injection terms.
In contrast to Reference 29 we consider here the more general and complex case of estimating a functional z = Ex of the state (9), and not only the full state as in Reference 29.Furthermore, the injection terms we use are much simpler than those used in Reference 29.Moreover, the proof in Reference 29 relies on a triangular structure of the system Σ in the special coordinates.So, for example, the observable subsystem Σ o has to be written in the observer form, and not in the observability form given in (11).A similar restriction has to be imposed on the strongly observable subsystem Σ s in (12).In fact, our proof is still valid if more interconnections are given.For example, if in (11) the last equation is replaced by ̇x(o,),n o, = a o, x o + a os, x s + A (os,),n o, y s , that is, when a connection from subsystem Σ s into subsystem Σ o, depends not only on the output y s but also on the state x s .The triangular structure of Reference 29 has two advantages: (i) It facilitates the proof of the convergence of the observer, since the error system is in cascade.In fact, no formal convergence proof is given in Reference 29.In our case the triangular structure is not given, and thus the proof is more involved.(ii) The triangular structure allows the use of a homogeneous observer, as for example, d 0 = d ∞ < 0. In the more general situation we consider this is not possible, and it is required that d ∞ ≥ 0. Note that if we use the triangular structure obtained in Reference 29, we can also select a homogeneous observer, that is, d 0 = d ∞ .

Convergence properties and assignment
The convergence properties of the observer (18) and its subsystems depend on the properties of the system, on one side, and on the selection of the parameters d 0 and d ∞ on the other side.We consider the behavior of the partial observers in (18).We assume in all cases that Assumption 1, conditions ( 13) and ( 27) are satisfied and that appropriate gains have been selected.

Observer Ω d
For this observer the estimation error dynamics from ( 28) is given by This system is a linear time invariant system without inputs, and this leads immediately to Lemma 1.The estimation error e d of observer Ω d in (18) has e d = 0 as a globally and exponentially stable equilibrium point.

Its convergence velocity is determined by the eigenvalues of A d and it cannot be modified. □
The proof of this results is classical.

Estimator Ω 𝜂
For this estimator the estimation error dynamics from ( 28) is given by This is a LTI system with a (bounded) unknown input w  and in cascade interconnection with Ξ d .Standard arguments lead to Lemma 2. The estimation error e  and e d of observer Ω d and estimator Ω  in ( 18) has e d = 0, e  = 0 as a globally and exponentially stable equilibrium point in the absence of unknown input w  = 0. Its convergence velocity is determined by the eigenvalues of A d and A  and it cannot be modified.For a bounded w  the trajectories of the system (Ξ d , Ξ  ) are globally and ultimately bounded.In particular, there exists a constant b ≥ 0 (depending on the bound of w  ) such that for every a > 0 there is a T(a) such that and lim t→∞ e d (t) = 0.Moreover, if lim t→∞ w  (t) = 0 then also lim t→∞ e  (t) = 0 .□ The proof of this results is also classical.

Observer Ω o
For this observer the estimation error dynamics from ( 28) is given by This is an autonomous, (in general) non linear and possibly discontinuous system, depending on the selection of d 0 , d ∞ and r o .It is shown in Section 5 the following Lemma 3. The estimation error e o of observer Ω o in (18) has e o = 0 as an equilibrium point which is 1. globally and exponentially stable if d 0 = d ∞ = 0 and r 0 ≥ 1; 2. globally and finite-time stable if d 0 < 0, d ∞ ≥ 0 and r 0 ≥ 1;

globally and fixed-time stable if d
Moreover, its convergence velocity can be assigned arbitrarily.□

Observer Ω s
For this observer the estimation error dynamics from ( 28) is given by This is an (in general) non linear and possibly discontinuous system, depending on the selection of d 0 , d ∞ , and r o , with a (bounded) unknown input w s and in cascade interconnection with Ξ d , Ξ o , and Ξ  .It is shown in Section 5 the following Lemma 4. Consider the estimation error e s of the observer Ω s .e s = 0 is an equilibrium point which is 1. globally and exponentially stable if d 0 = d ∞ = 0, r o ≥ 1, w s = 0 and w  = 0.If D s (A sd e d + A s e  ) = 0 its convergence velocity can be assigned.
2. Globally and asymptotically stable if d 0 < 0, d ∞ ≥ 0, r 0 ≥ 1, w s = 0 and w  = 0.If D s (A sd e d + A s e  ) = 0 its convergence is in finite-time, or in fixed-time if d ∞ > 0, and its velocity can be assigned.

Globally and finite-time stable if d
is uniformly bounded (for t ≥ 0) by a constant and d ∞ > 0 its convergence is in fixed-time, and its velocity can be assigned.

Special cases
We have considered so far a rather general situation for the design of the FUIO.If system Σ satisfies special conditions, then the generic observer can be reduced to more classical results.Some particular situations for state observers, that is, E = I, are given by: 1. Σ has no unknown inputs w and it is observable.In that case observer Ω o estimates the state in finite-time or in fixed-time with an assignable convergence dynamics.In case we select d 0 = d ∞ = 0, we recover a linear observer.2. Σ has no unknown inputs w and it is detectable.In that case observer Ω o together with Ω d estimate the whole state exponentially.Although the state x o can be estimated in finite-time or in fixed-time with an assignable convergence dynamics, the state x d can be only recovered exponentially, with a dynamics completely determined by the eigenvalues of A d .In case we select d 0 = d ∞ = 0, we recover a linear observer.3. Σ has unknown inputs w and it is strong* detectable, that is, only subsystems Σ o , Σ d , and Σ s exist, and y s = x s .In that case observer Ω o together with Ω d estimate the whole state exponentially.Ω s is not required.Although the state x o can be estimated in finite-time or in fixed-time with an assignable convergence dynamics, the state x d can be only recovered exponentially, with a dynamics completely determined by the eigenvalues of A d .In case we select d 0 = d ∞ = 0, we recover a linear observer.In case subsystem Σ d is absent, then an assignable convergent observer is obtained, and it can converge in finite-time or in fixed-time.4. Σ has unknown inputs w and it is strongly detectable, that is, only subsystems Σ o , Σ d , and Σ s exist, and y s ≠ x s .In that case observers Ω o , Ω s , selecting d 0 = −1, together with Ω d , estimate the whole state exponentially.The state x o can be estimated in finite-time or in fixed-time with an assignable convergence dynamics.The state x d can be only recovered exponentially, with a dynamics completely determined by the eigenvalues of A d .The state x s can be estimated in finite-time or in fixed-time with an assignable convergence dynamics in the second convergence phase, after the estimation of the state x d reaches a certain thereshold, depending on the gain .Selecting d 0 ≠ −1, only an approximate estimation of the state x s can be attained, which can be made as small error as desired by increasing .In case we select d 0 = d ∞ = 0, we recover a linear approximate observer.In case subsystem Σ d is absent, that is, system is strongly observable, then Ω o , Ω s is an assignable convergent observer, and it can converge exactly in finite-time or in fixed-time for d 0 = −1, or approximately for d 0 ≠ −1.
Obviously, in all these cases an arbitrary functional of the involved (detectable) states can be also estimated.Note that when subsystems Σ u , Σ  or Σ  are present, it is impossible to estimate the full state, and only true functional observers can be constructed, that is, E ≠ I.

OBSERVATION ERROR DYNAMICS AND LYAPUNOV FUNCTION
In this section we recall the Lyapunov function used to prove the convergence of the observer.The Lyapunov function can be used also for estimation of the convergence time and the calculation of the gains k i that render the algorithm stable, but we do not detail this here.

Observation error dynamics
Defining the observation errors as e (s,i),j ≜ x(s,i),j − x (s,i),j , i = 1, … , p s ; j = 1, … , n s,i , their dynamics satisfy where w s,i , w  , and w  are the unknown inputs, which are assumed to be bounded by a constant Δ, that is, , respectively.Note that conditions (27) imply that in (33) a su,i = 0, a s,i = 0, and in (36) A  = 0 and D  A u = 0. Taking these assumptions into account and scaling the differentiation error as we obtain ] , (38) where L n o,j −k+1  (o,j),k , and we have used the relations † (obtained easily from ( 21) and ( 24)) φ(s,i),j ( For the construction of the Lyapunov function, perform the state transformation where for j = 1, k (s,i),0 = k (o,),0 = 1, together with the time transformation  = Lt.If we denote by x ′ = dx d the derivative with respect to , the dynamics of ( 33)-( 35) become where

5.2
Lyapunov Function for the subsystem composed of the observable and strongly observable subsystems  * o, and  *

s,i
First, note that the output injection terms given by ( 24) are simpler and different form the ones used in Reference 28 for the stability proof of the differentiator.However, the proof in Reference 28 carries over to the case with injection terms given by (24), since all requirements are fulfilled.These conditions are basically that the injection terms are bl-homogeneous and that they are obtained from succesive compositions of the functions used in the construction of the Lyapunov function.This latter condition is clarified below in (45).The former condition is simple, using the bl-homogeneity of function  (o,),j and the properties of composition of bl-homgeneous functions.In this latter case functions (24) can be written as the composition of functions  (o,),l (s) negative definite.In the absence of ψw this implies the asymptotic stability of z so = 0.If d 0 < 0 this convergence is in finite-time, and if d ∞ > 0 it is in fixed-time.If ψw is ultimately bounded, and d 0 = −1, it is possible to select the gain  > 0 sufficiently large, so that V ′ so (z so ) is negative definite, and thus finite-time stability is assured.Finally, if d 0 > −1, then the input-state map ψw → z so is input-to-state stable (ISS), and given an ultimate bound for ψw the ultimate bound for z so can be set arbitrarily small by selecting  > 0 sufficiently large.
Global and asymptotic stability of the zero estimation error, e s = 0, e o = 0, e d = 0, e  = 0, in the absence of unknown inputs is simple, since all subsystems have globally and asymptotically stable equilibria.
We do not provide all the details of the proofs of Lemmata 3 and 4.However, the inequalities proved above for the Lyapunov functions lead to the conclusions of the Lemmata using standard arguments.
The proof of the ISS of the estimation error dynamics from measurement noise input, when the output y(t) is perturbed by a noise signal y(t) + (t), to the estimation error, can be obtained similarly to the procedure used in Reference 28.
The necessity of condition ( 13) has been already established, and it is related to the indistinguishability of states x u , x  and x  .Necessity of (27) for the convergence of observer ( 18) can be derived directly from the estimation error (28).If any of the conditions of ( 27) is violated, then either Ξ s or Ξ  will have an unbounded unknown input, that make (some of) their states, e s or e  , unbounded.This cannot be compensated with the correction terms with constant gains, and therefore e s will not converge to zero.Thus, some of the components of x s cannot be estimated by the observer.

EXAMPLE
Consider the MIMO LTI system, written in the special form as with a sd = a s = 1,  = 0.2 and where The states of the strong observable subsystem are x s = [x The full extended functional unknown input observer (EFUIO) is given by the following interconnected set of partial observers ] xo + a sd xd + a s x , ] xo + y s , F I G U R E 2 Simulation result for the FUIO.The graphs illustrate the evolution of the not measured states of the plant, and the estimation errors of the linear FUI observer (in the third row) and of the HOSM FUI Observer (in the last row) .It is clear that the states grow unboundedly due to the instability of the plant's dynamics, but the observer is able to estimate correctly the states x s , x o , x d , but not x  and x u .
x  can be estimated within a bounded error.
For both FUIOs all other parameters are selected identical.For the simulations the external gains have been selected as k s = , and the tuning gains  = 25 and L = 20.The unknown inputs have been set as w 1 = 12.5 sin(t) + 12.5 cos(2t), and w 2 = 12.5 sin(1.5t)+ 12.5 cos(3t).The initial conditions for the plant's states have been set to zero, that is, x i,0 = 0, and xi,0 = 1 for the observer.We have used a fixed-step explicit Euler method, with integration step  = 5 × 10 −6 .
Figure 2 shows the simulation results for the plant and the observers.The graphs in the first and the second rows are identical for both observers.The third row corresponds to the Linear FUI Observer, while the fourth row to the HOSM EFUI Observer.
1. First row: The upper three graphs present the time evolution of the unmeasured states x s2 , x s3, x o2 , that grow unboundedly, since the system is unstable.2. Second row: The three graphs in the second row show the evolution of the growing states x d , x  on the left, and x u on the right, while in the middle the estimation errors of the corresponding (open loop) observers for x d , x  is shown.The convergence of e d cannot be assigned, and it is given by the stable transmission zero at z = −3.Note that the estimation F I G U R E 3 Simulation result for the FUIO.The graphs illustrate the evolution of the not measured states of the plant, and the estimation errors of the linear FUI observer (in the third row) and of the HOSM FUI Observer (in the last row) .The initial conditions of the observer have been multiplied here by 10 4 .Note that the change in the convergence velocity of the HOSM FUIO is much smaller than that for the linear FUIO, a property obtained from the positive value of p ∞ = 0.15.
error of x  , e  , does not converge to zero, but to a neighborhood of zero, which depends on the size of the unknown input.The velocity of this convergence depends on the eigenvalues of the stable dynamics A  = −4, and cannot be modified.

Third row:
The three graphs in the third row illustrate the estimation errors of the Linear FUI Observer.Note that e s does not converge to zero, due to the effect of the unknown input.e o , however, converges asymptotically, with an assignable dynamics.This happens despite of the bounded unknown inputs.4. Fourth row: The three graphs in the fourth row illustrate the estimation errors of the HOSM EFUI observer.Here, in contrast to the linear observer, e s converges to zero in fixed-time, with an assignable convergence time.We should note here that this fixed-time is achieved only after both e d and e  have attained a neighborhood of zero.e o converges in fixed-time, with an assignable dynamics.This happens despite of the bounded unknown inputs.
For the simulations we have used the same values of L and  for both observers Ω s and Ω o .However, a better tuning can be achieved if different gain values are selected for each one.
Figure 3 shows the simulation results for the plant and the observers for much larger initial conditions of the observer, that is, xi,0 = 10 4 .
d ∶ { ̇x d = −3x d + y s + y o , =1 n o, and A (os,),j and a o, are constant row vectors of appropriate dimensions.Although the observability form has a triangular structure, we do not specify it in (11) since we do not make use of it.
it is useful to write the estimation error dynamics.Defining the estimation errors as e s = xs − x s , e o = xo − x o , e d = xd − x d , e  = x − x  , the dynamics is given by Ξ s ∶ ̇es = A s e s − K s Φ s (C s e s ) + D s (A so e o + A sd e d + A s e  − w s ) − D s ( A su x u + A s x  ) ,

)
Ξ d and Ξ o are systems without inputs.SinceA d is Hurwitz, e d converges to zero exponentially at a velocity determined by A d and it cannot be modified.Selecting the correction term K o Φ o , e o can be forced to converge to zero at an arbitrary rate, and, depending on the selection of d 0 and d ∞ , the convergence can be exponential, in finite-time or in fixed-time.Contrastingly, subsystem Ξ  , is internally stable, since A  is Hurwitz, but is driven by some inputs.e d and w  (bounded) are not problematic for the final boundedness of x  , but x  and x u are, since they grow unboundedly.So to assure final boundedness of e  two conditions from ( + a so,i e o + a sd,i e d − a su,i x u + a s,i e  − a s,i x  − w s,i , o ,  d , e  , w s,i ) = a s,i  s + a so,i  o + a sd,i  d + a s,i   − w s,i  (o,j),k + a sd,i  d + a s,i   − w s,i 1 , x 2 , x 3 ], those of the observable subsystem are x o = [x 4 , x 5 ], the detectable subsystem's state is x d = x 6 , and x u = x 7 , x  = x 8 .Matrices A s and A o , corresponding to the strong observable and the observable subsystems, are anti-Hurwitz, there are two invariant zeros at z = {−3, 2} and A  = −4 is Hurwitz.Since condition(27)of Theorem 1 is fulfilled, states {x 1 , … , x 6 } can be estimated, but not states x 7 , x 8 .And therefore any functional z formed as a linear combination of {x 1 , … , x 6 } can be also estimated, that is, with E i arbitrary for i = 1, … , 6 and E 7 = E 8 = 0. Clearly, the system is not strongly observable but neither strongly detectable nor strong* detectable.No previous result can be used to design a FUIO for this system.