A SLIDING MODE TECHNIQUE FOR RECONSTRUCTION OF UNKNOWN CONTROL INPUT IN LINEAR SYSTEMS

The unknown input estimation problem for linear dynamic systems exposed to the disturbances is studied in the paper. A novel approach to designing sliding mode observer is suggested for systems not satisfying matching, minimum phase, and detectability conditions. To apply this approach, the reduced-order model of the original system is used. Such a model allows to reduce the dimension of the sliding mode observer, besides the restrictions placed on the original system are relaxed


Introduction
This work investigates the problem of unknown input estimation in linear control systems.The unknown input can be interpreted as a function describing faults in engineering systems, and the problem is reduced to the fault diagnosis one.This problem was intensively studied for the past 30 years; see, for example, [Blanke et al.(2006); Ding(2014); Samy et al.(2011); Witczak(2014)].Different tools for solving the problem have been developed: identification, diagnostic observers, parity relations.The interesting method of identification is based on sliding mode observers (SMO), it uses features of sliding motion studied in [Utkin(1992)].
To relax the matching condition, two methods are used.The first one is based on a high-order sliding mode differentiator [Bejarano and Fridman(2010); Fridman et al.(2007); Floquet et al.(2007); Fridman et al.(2008); Yang et al.(2013)]; such a differentiator generates the derivatives of the outputs to transform the original system into a system which satisfies the matching condition.Multiple SMOs in cascade are used in the second method [Tan and Edwards(2009)].Both methods proved good results but assume that the system is a minimum phase and the estimation scheme has a complex structure.
The method suggested by [Alwi et al.(2009)] relaxes the matching condition but the unknown input estimate is corrupted by the fault derivative.The method suggested in [Rios et al.(2014)] does not ensure asymptotic convergence, as a result, the estimation errors are only bounded.The minimum phase condition is relaxed in [Bejarano et al.(2009)] for systems where the unknown inputs are at the dynamics and output, and sufficient and necessary conditions for unknown input and state reconstruction are received.These conditions are less limiting than strong detectability.The method in [Bejarano(2011)] under some restrictions solved the problem of partial unknown input estimation.In [Hmidi et al.(2020)] and [Wang et al.(2017)] the detectability condition is used instead of the minimum phase one.
Note that all cited above papers consider vector unknown inputs and vector functions describing faults.Besides, the dimension of SMOs in [Edwards et al.(2000)] and similar papers coincides with that of the original system.
The main contribution of the present paper is that SMOs are designed for systems which do not satisfy common conditions: minimum phase, matching, and detectability ones.This is explained by that SMO is designed not for the full order model of the original system but for its reduced order model.Such a model has not some special features of the full order model which prevent the possibility for SMO design.As an example, the full order model is non-detectable while the reduced order model has the detectability property.Besides, the canonical form of matrices describing such a model is used.As a result, the limitations placed on the full order model, are relaxed and the dimension of the observer decreases.
Consider a control system where x(t) ∈ R n is the state vector, u(t) ∈ R m is the vector of control, y(t) ∈ R l is the output vector, A, B, C, Z, and D are known matrices, z(t) ∈ R is the unknown bounded input, ∥z(t)∥ ≤ β, d(t) ∈ R p is the unmatched disturbance, one assumes that d(t) is an unknown bounded function.
Remark 1.The relation z(t) ∈ R corresponds to the most probable single faults.If z(t) ∈ R s , s > 1, the suggested method should be applied to each component of the vector function z(t) presenting multiple faults.
The method suggested in [Yan and Edwards(2007)] assumes that system (1) meets two conditions: 1) rank(C[Z D]) = rank([Z D]), 2) system is a minimum phase; the methods suggested in [Hmidi et al.(2020); Wang et al.(2017)] assume that (1) is detectable.In our paper, the unknown input estimation problem will be solved without these conditions.The method suggested in the paper is based not on the full order model of the original system but on its reduced order model, besides, the identification canonical form of such a model is used.
Note that another type of matching condition with respect to the disturbance was considered in [Antipov et al.(2021); Utkin et al.(2022)]: rank(B) = rank(B D).
In our paper we assume that the disturbance is unmatched.
2 Reduced order model design Let r z be minimal relative degree of y with respect to z(t).The output y * (t) corresponds to r z and the matrix R * exists satisfying the condition R * y(t) = y * (t).Assumption means that r z < ∞ and the unknown input can be identified.
Remark 2. If Im(Z) ⊆ Ker(V (n) ), the unknown input cannot be estimated because it lies in the unobservable part of system (1).If Im(Z) ̸ ⊆ Ker(V (n) ), the output y * (t) is affected by the unknown input which can be estimated.
The problem is solved on the basis of the reduced order model of (1) generally described as follows: where x * (t) ∈ R k is the vector of state A * , B * , G * , C * , D * , and Z * are matrices to be determined.One assumes that x * (t) = Φx(t) for the matrix Φ to be determined.As is known [Zhirabok et al.(2017a); Zhirabok et al.(2017b)], that matrices R * and Φ meet the conditions (3) Our purpose is to develop the method to construct the model (2) insensitive to the disturbance that allows solving the problem of unknown input estimation.
We specify the matrices A * and C * in the identification canonical form 2) is observable.Otherwise, system (2) can be transformed into the observable canonical form [Kwakernaak and Sivan(1972)], and then the observable part of this form can be transformed into (4) of less dimension.
Based on (4), one obtains from (3) the following relations: (5) Here Φ i and G * i are i-th rows of the matrices Φ and J * , i = 1, ..., k, respectively.One can show [Zhirabok et al.(2017b)] that ( 5) are transformed into the equation where Insensitivity to the disturbance in the form of condition ΦD = 0 one can take into consideration as [Zhirabok et al.(2017a); Zhirabok et al.(2017b)] where Clearly, this equation and ( 6) can be written as Solve this equation for minimal k ≥ r z and design the model ( 8) as Note that assumption Im(Z) ̸ ⊆ Ker(V (n) ) means that Z * ̸ = 0.If (7) has not solution for all k < n, the model is sensitive to the disturbance and the problem of exact unknown input estimation has no solution.In this case, the problem of approximate unknown input estimation can be solved [Zhirabok et al.(2019)].Another approach is to estimate the disturbance d(t) and then use such an estimate in (2) [Alvarez-Sanchez et al.(2013)].
Remark 4. The model ( 9) corresponds to that in [Edwards et al.(2000); Wang et al.(2017)] with stable matrix A * 2 since the original system is a minimum phase or detectable by assumption.In contrast to this, A * 2 in our approach may be unstable; stability of SMO is provided below by feedback.
3 Sliding mode observer design SMO is sought in the from ) where v(t) = sign(e 1 ), the coefficients l 1 and L 2 provide the observer stability; they exist due to the canonical form (4).
The error e(t) is described by or where A * * is stable matrix.
Remark 5.It is known that if e(t) meets the equation ė(t) = F e(t) + g(t) with stable matrix F and the function g(t) is bounded, then e(t) is bounded as well that is ∥e(t)∥ ≤ γ for some scalar γ.
Theorem.The observer (10) generates the following estimates for the function z and when Z * 2 ̸ = 0, where ε is a small positive scalar.
Proof.Note that proof is similar to that developed in [Wang et al.(2017)].Firstly one proves that ∥e(t)∥ ≤ δ for some δ.Because ∥v(t)∥ = 1 and the function z(t) is bounded, then ∥Z * z(t) + Kv(t)∥ ≤ g 0 for some scalar g 0 .It is evident from (12) and Remark 5 that the function e(t) is bounded and ∥e(t)∥ ≤ δ for some δ.
Secondly, one proves that e 1 = 0 in finite time due to choices of the observer gains; as a result, sliding motion is achieved.Introduce Lyapunov candidate function V 1 = e 2 1 and using (12) take its derivative: If k 1 satisfies then V1 < 0, and sliding motion (e 1 = ė1 = 0) happens in finite time.
Thirdly, one proves that e 2 = 0 in finite time due to choices of the observer gains, and sliding motion is achieved.Because A * * is stable matrix, there exist positive definite symmetric matrices P and W such that A T * * P + P A * * = −W.Write down the matrix P in the form P = p 1 P 2 P T 2 P 3 where P 3 is positive definite symmetric matrix.Consider Lyapunov candidate function V 2 = e T P e and take its derivative using (11): V2 = e T (A T * * P + P A * * )e − 2e T P (Z * z + Kv).
It is evident from ( 12) and e 1 = ė1 = 0 that A * 1 e 2 = k 1 v + Z * 1 z.Using k 2 = P −1 3 A T * 1 k 3 for some k 3 > 0 and e 1 = 0, we obtain If one chooses k 3 and k 1 as and then k 3 is chosen as then V2 < 0. Theorem has been proved.Based on ( 15), ( 16), and ( 17), the coefficients k 1 and k 3 should be chosen as It is evident from ( 12) that if Z * 2 = 0, the unknown input z(t) is estimated based on the first expression in ( 12) by ( 13); otherwise, one uses the second expression in (12) and obtains ( 14).

Practical example
Consider the control system (18) Equations ( 18) constitute a model of electric servoactuator where x 1 is the output rotation angle at the reducer output shaft; x 2 is the output rotation velocity at the motor output shaft; x 3 is the current through the servoactuator windings; i p is the reducing ratio of the reducer; J H is the moment of inertia of the electric servoactuator and of the rotating parts of the reducer; K ω and K M are the respective coefficients of the counter EMF and of the torque; K d is the torques of the Coulomb friction at the motor output shaft; R m and L m are the active and inductive resistances of the electric servoactuator windings, respectively; K U is the amplification factor.
One assumes that the unknown input z(t) = − R(t) Lm x 3 (t) is caused by the deviation R(t) of the active resistances from its nominal value R m .
Note that the unknown input estimation problem for this example can be solved by known methods (for example, [Edwards et al.(2000)]) but we use our approach to demonstrate its efficiency.

Illustrative example
Consider non-minimum phase and non-detectable system control system: (20) The system is described by the following matrices: It can be shown that the system is not minimum phase and detectable.Really, Rozenbrock matrix R(s) for system ( 20) is given by .
Since rank(R(s)) = 5 under s = 1, then s = 1 is invariant zero of (A, [D Z], C), therefore the system is no minimum phase.Besides, one can shown that Ker(V (3) ) = (0 0 1 0) T and the unobservable part of the system presented by x 3 is unstable, therefore the system is non-detectable.Then the methods suggested in [Tan and Edwards(2003); Wang et al.(2017)] cannot be used in this case but SMO can be designed by our method.

Conclusion
The problem of unknown input estimation for systems which do not satisfy the matching, minimum phase, and detectability conditions is studied.The suggested methods is based on the reduced order model of the original system insensitive to the disturbance.The practical and illustrative examples show the effectiveness of the proposed method.Future work is developing algorithms for unknown input and parameter estimation for nonlinear systems such that considered in [Furtat et al.(2022)] to compare the results with [Furtat and Orlov(2020)].