Reduced-Order and Full-Order Interval Observers Design for Discrete-Time System

In this paper, the design of reduced-order and full-order interval observers for discrete-time system is investigated. First, the original system is transformed into a descriptor system with the ability to estimate the unknown noise of sensors. Next, a reduced-order interval observer design method for descriptor system is presented. The state matrix of the system is obtained by linear transformation, and a Sylvester equation is constructed to find the appropriate observer gain matrix. Then, an existing design method of full-order interval observer is introduced in this paper. Finally, a simulation example is given to demonstrate that the reduced-order interval observer has more accurate interval estimation results than the full-order interval observer.


I. INTRODUCTION
State feedback plays a significant role in practical system control and fault diagnosis, but some important system states can not be obtained directly due to the limitations of measurement means, so the study of state observers attracts many scholars. And due to the unavoidable noises and disturbances in the system, the researches of robust observers have become a hot topic [1]- [5], such as high order sliding mode observer [3], unknown input observer [4]. However, the design methods proposed in the above literature require noises or disturbances to meet many restrictive conditions, for example, the observer matching condition, which is very difficult to meet for practical systems. As a result, the applications of these methods are limited.
In literature [6], the validity of approximate interval estimation is proved by theoretical research, and the concept of interval observers is first proposed in [7], then interval observers become a new method to estimate states. Compared with the traditional asymptotic observers, which make point estimation of the system states, interval observers can provide the upper and lower boundaries (x + k and x − k ) of the estimation value, so that real states of the system are always in the The associate editor coordinating the review of this manuscript and approving it for publication was Wen-Long Chin . estimation interval (k ≥ 0, x k ∈ [x − k , x + k ]) [8]. Compared with the traditional asymptotic observers, the design procedures of interval observers are much more convenient [9]. In addition, the constraints of input and output disturbances is loose, only the upper and lower boundaries of the disturbances need to be known so that the interval observer can complete an accurate interval estimation. Moreover, interval estimation contains more information about system states than traditional point estimation. Therefore, as good supplements to asymptotic observers, interval observers have been widely studied by scholars [8]- [15], and have been applied in many practical systems [16]- [18].
Descriptor systems, also known as differential-algebraic systems, are represented by differential and algebraic equations. In comparison with the standard state space representation, descriptor systems can describe a wider range of practical system models. Therefore, the researches of descriptor systems have already achieved many important results. As we know, there are many researches on interval estimation of descriptor systems based on interval observers [19]- [25]. In [19], a method of interval observer design is proposed for a class of descriptor systems which have time delays. Considering uncertain nonlinear descriptor systems are of great significance in many practical applications, [20] assumes the uncertainties are bounded and manages to make interval VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ estimations for uncertain input and output. And [21] takes into account a special situation that ( A, C) may not always detectable. On account of this, a new Luenberger-like interval observer is developed for descriptor systems and achieves good results. However, all of the above methods are based on full-order interval observers, and there are few researches on the designs of reduced-order interval observers [26]- [28]. It is worth mentioning that the structure of reduced-order interval observers is simpler than that of full-order interval observers. Also, the amount of data to be processed is much less. Therefore, reduced-order interval observers do have certain advantages in the area of interval estimation. To be specific, [27] adopts a reduced-order interval observer method to estimate the states of an induction machine system which is time-variant and gets excellent interval estimation results.
Based on discussions above, a design method of reduced-order interval observer is presented, and the estimation results are compared with a full-order method. The main contributions of this paper are summarized as follows: (1) The reduced-order interval observer designed in this paper achieves accurate state interval estimations for a complex system, and the simulation results show it is better than the existing full-order interval observer design method. (2) By constructing a descriptor system, the reduced-order interval observer can estimate the unknown sensor noise effectively.
The rest of this paper is organized as follows: Some preliminaries and problem statements are formulated in Section II. Main results are given in Section III. The simulation example and a conclusion of the whole paper are presented in Section IV and Section V, respectively.
Here, we give the definitions of the symbols used in this paper. R is the set of all real numbers, C is the set of all complex numbers, and R n + is a n dimensional set with positive elements. The A > B or A < B ( A and B denote matrices or vectors ) in the paper should be considered elementwise. If the spectral radius of a matrix A ∈ R n×n is less than one, we call it a Schur matrix. And if all the elements in the matrix A ∈ R n×n are no less than zero, we call it a non-negative matrix. For a matrix A ∈ R m×n , we give a definition to A + = max{0, A} and A − = max{0, −A}, so the relation of

II. PRELIMINARIES AND PROBLEM FORMULATION
Consider a class of discrete-time systems as follows where x k ∈ R n , u k ∈ R m and y k ∈ R p are the state vector, control input vector and output vector of the system, respectively. η k ∈ R q and w k ∈ R s denote input and sensor noises. A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , D ∈ R n×q and F ∈ R p×s , which are all constant matrices. Moreover, D and F are matrices with full column rank. In addition, it is assumed that the initial state vector of the system satisfies represent the upper and lower boundaries of the state vector respectively. In addition, the system input noise η k and initial sensor noise w k are all bounded, that is to say, they meet η − ≤ η k ≤ η + and w − ≤ w 0 ≤ w + . We assume that the system dimension satisfies p ≥ q + s.
In order to estimate the sensor noise, an augmented state vector of the system is constructed. We can get the following new system state vector And the system state space representation after transforming is given as The original system (1) can be written as In this descriptor system, E,Ā ∈ R (n+s)×(n+s) ,B ∈ R (n+s)×m , C ∈ R p×(n+s) ,andD ∈ R (n+s)×q . E is singular, i.e. rank(E) < n+s. In addition, we do not need to know the upper and lower boundaries of the sensor noise in this case. It is easy to see (2) satisfies Assuming that the original system (1) is observable, the following rank condition can be obtained According to (3), there exists a row full rank matrix which makes In (5), H 1 ∈ R (n+s)×(n+s) and H 2 ∈ R (n+s)×p are the parameter matrices that need to be designed and they should satisfy the following constraint equation Remark 1: In the process of constructing the descriptor system from the original one, only the corresponding matrix transformation is used, without using any assumptions and inferences. Therefore, the constructed descriptor system is completely equivalent to the original. Because of this, system (2) is observable as we assume system (1) is. If there is an interval observer of the descriptor system (2), we can estimate the augmented state vector, and then the upper and lower boundaries (w + k and w − k ) of the sensor noise in the original system can be obtained.
Lemma 1 [22]: For the following matrix equation is an arbitrary matrix, and N † denotes the pseudoinverse of the matrix N .
Lemma 2 [9]: Assume that the state vector Lemma 3 [13]: In the discrete-time system x k+1 = Ax k + d k , x k ∈ R n and x 0 ∈ R n are the state vector and the initial state vector of the system, respectively. Assume that A ∈ R n×n is a non-negative matrix, and d k ∈ R n + for ∀k ≥ 0. If the condition x 0 ≥ 0 is satisfied, then the discrete-time system x k+1 = Ax k + d k has a non-negative solution x k ≥ 0 for ∀k ≥ 0.

A. REDUCED-ORDER INTERVAL OBSERVER DESIGN
On the basis of Lemma 1, the solution of the row full rank matrix in (5) is is an arbitrary matrix. We can get the solutions of H 1 , H 2 , respectively In (7), Using H 1 , H 2 designed in (7), if H 1 is full column rank, then the descriptor system (2) can be transformed into the following form According to (8) and constraint equation (6) we can get The descriptor system (2) becomes Let z k =x k − H 2 y k , a transformed state equation can be obtained Selecting a matrix T = C T M T T , where M ∈ R [(n+s)−p]×(n+s) is an arbitrary matrix which make T is nonsingular. We define T −1 z k = θ k ,and substitute it into (10) Write θ k as the form of θ k = θ 1,k θ 2,k , we can get I p 0 θ 1,k θ 2,k = θ 1,k , so (11) can be expressed by (12) We can regard the corresponding parameter matrices in (12) as a whole Then, write the parameter matrices in (13) as the form of block matrices The results are presented as follows Here, we define θ 1,k θ 2,k = I 0 L I ξ 1,k ξ 2,k , and L ∈ R [(n+s)−p]×p is the gain matrix of the observer, then (16) can be easily obtained In the light of (16), the relationships among ξ 1,k , ξ 2,k , θ 1,k and θ 2,k is showed as follows Substitute (17) Based on (13) and (16), θ 2,k can be replaced by θ 2,k = Lθ 1,k + ξ 2,k = LMy k + ξ 2,k , then we bring it into (18) The system after transformation is presented as below: Remark 2: Based on (19), the reduced-order interval observer can be designed directly. What we need to do is to find out the appropriate observer gain matrix, so that the observer can meet the design conditions of asymptotic stability. In the designs of traditional asymptotic observers, for example, [24] takes an LMI method, making (A 4 − LA 2 ) a Schur matrix. But when designing interval observers, it is very difficult to find the appropriate matrix L directly because (A 4 − LA 2 ) is required to be Schur and non-negative at the same time. In literature [29], it uses a time-varying transformation. However, the calculation process is very complicated and the solution is usually hard to find. Here, a simpler linear transformation method is given as follows [13].

Proof: The upper estimation error is defined asτ
0 ≥ 0 and P is a Schur and non-negative matrix.

By using Lemma 3 we getτ
for ∀k ≥ 0. Similarly, we define the lower estimation error asτ − k = τ k − τ − k ,then we havẽ and P is a Schur and non-negative matrix, using again Lemma 3,we Like the proof result we get above, for ∀k ≥ 0, τ − k ≤ τ k . In summary, the estimation results of the interval observer always satisfy τ − k ≤ τ k ≤ τ + k for ∀k ≥ 0. Theorem 1 is proved.
In order to find the appropriate observer gain matrix L and nonsingular matrix W , we construct a Sylvester equation from P = W (A 4 − LA 2 )W −1 to calculate.
First, we give the general form of Sylvester equation From P = W (A 4 − LA 2 )W −1 , we can get an equation with the same form of (22) In (23), Q ∈ R [(n+s)−p]×p is an arbitrary matrix, and we let Q = WL. When (P ⊗ I (n+s)−p + I n+s ⊗ (−A 4 ) T ) is a nonsingular matrix, we can obtain a unique solution of L and W (⊗ is Kronecker product).

B. FULL-ORDER INTERVAL OBSERVER DESIGN
In this section, we choose the design method of a Luenberger-like interval observer in [21]. Although it considers the sensor noise, it does not estimate the noise value further. As a result, it is necessary for the sensor noise to meet the limit condition of a bounded interval [w, w], i.e. the noise belongs to an already known interval. In this paper, the method of constructing the augmented state vector does not need the above condition, so it is simpler in design procedures. Next, we briefly describe the design method of a full-order observer.
On the basis of (9), we let z k =x k − H 2 y k , then the new state equation is as follows There exists a matrix V that is nonsingular, define φ k = Vz k , (24) can be written as (25) by using linear transformation (26) is the full-order interval observer of system (25) where S = V (H 1Ā − LC)V −1 is a Schur and non-negative matrix. One can refer to [21] for more information about the proof process.

IV. SIMULATION STUDIES
In this section, we use a simulation example to demonstrate the proposed reduced-order interval observer has better performances than the full-order interval observer. Example: Consider the descriptor system we discussed in (2) with parameter matrices as follows The control input with the form of piecewise function and the system input noise are 25 1 ≤ k < 20 2 + 0.7cos(0.5k) 20 ≤ k < 60 1 + sin(k) 60 ≤ k ≤ 100 η k = 0.05sin(0.5k + 1)  It is easy to see η k is in an interval with the upper and lower boundaries η + k = 0.05 and η − k = −0.05. The selected initial conditions of the system are presented next From the results, it is extremely clear to see that the reduced-order interval observer designed in this paper has more accurate estimation results of system states than the full-order one in [21].
According to (7), we can compute the solutions of H 1 and H 2     Results of the reduced-order interval observer we proposed are presented first. So, we can compute V and the observer gain matrix L by using the method in [21]. The equation can be derived, as shown at the top of this page The simulation results are shown in Fig.1-Fig.5.

V. CONCLUSION
In this paper, a design method of reduced-order interval observers is proposed for discrete-time systems. By treating sensor noise as an augmented state, the original system is transformed into an equivalent descriptor system which is not affected by the noise. An observer is designed based on the linear transformation, and the appropriate gain matrix can be easily obtained by solving Sylvester equation. Through the simulation example, the observer designed in this paper is compared with an existing full-order interval observer. The results show that the proposed method in this paper is more effective and accurate.