A robust reduced-order observers design approach for linear discrete periodic systems

This paper investigates the problem of designing reduced-order observers for linear discrete-time periodic (LDP) systems. In case that the linear discrete-time periodic system is observable, an algebraic equivalent system is obtained by non-singular linear transformation, and the partial states to be observed are separated simultaneously. Then the considered problem is transformed into the problem of solving a class of periodic Sylvester matrix equation and an iterative algorithm for periodic reduced-order state observers design is derived. In addition, robust consideration based on periodic reduced-order state observers for LDP systems is also conducted. At last, one numerical example is worked out to illustrate the effectiveness of the proposed approaches.


1.
Introduction. Linear discrete-time periodic systems are the simplest and most important time-varying systems. Many time-varying systems in practical engineering can be approximated as a simple linear discrete period periodic system. With the development of aerospace, networks and some high technology in recent years, linear discrete periodic systems have been paid renewed attentions in the control theory community, and have achieved many important results (see [13]- [2] and references therein). In [18], a novel lifting method was proposed that converted the linear periodic system to an augmented Linear Time-Invariant (LTI) system, and the design problem of spacecraft attitude control using magnetic torques was solved. [20] studied the local control of discrete-time periodic linear systems subject to input saturation by using the multi-step periodic invariant set approach. In addition, stabilization of periodic systems with input and output delays was investigated in [21].
State observer is an important research topic in control theory and control engineering, the purpose of observer is to estimate the state of another dynamic system. The state observer's theory stems from the Kalman filter in [4] and the Luenberger are formulated. Section 4 offers an illustrative numerical example to verify the effectiveness of the proposed method. We come to conclusion in Section 5.
Notation The superscripts "T" and " − 1" stand for matrix transposition and matrix inverse, respectively; R n denotes the n-dimensional Euclidean space; i, j represents the integer set {i, i + 1, . . . , j − 1, j}, tr(A) means the trace of matrix A and Ψ A denotes the monodromy matrix A T −1 A T −2 · · · A 0 with period T.
2. Preliminaries and problem formulation. Consider LDP systems with the following state space representation where t ∈ Z, the set of integers, x t ∈ R n , u t ∈ R r and y t ∈ R m are respectively the state vector, the input vector and the output vector, A t , B t , C t are matrices of compatible dimensions satisfying To recall standard concept and results for periodic systems, for t ∈ Z denote by where 0 m,n denotes the zero element of R m×n , and I n denotes the identity matrix of dimension n. The notions and results concerning the reachability and observability of the periodic system (1) have been introduced and analyzed in [3]. We provide them in the following.
The periodic system (1) is reachable at time t if and only if the following matrix A t − R t (λ) B t (7) has full row rank for all λ ∈ C, or equivalently, for all the eigenvalues of Φ A (t+T, t).
has full column rank for all λ ∈ C, or equivalently, for all the eigenvalues of Φ A (t + T, t).
When there exists some restrictions in practice, the state of system (1) can not be fully gotten by hardware, but the input u(t) and the output y(t) can be measured. Considering that the output y t of the system already contains part of information of the system state x t , the direct use of this part of information can construct a state observer with dimension lower than the estimated system.
Here, we conclude the problem of periodic reduced-order state observer for linear discrete-time periodic (LDP) systems as follows: Problem. Given a completely reachable and completely observable LDP system (1), find a state observer with dimension n − m, whose state variable is denoted by z t , such that the original states x t can be reconstituted by the combination of system output y t and the observer state z t .
Based on the above idea, we would like to present an non-singular linear transformation firstly, by which the partial states to be observed can be separated. Lemma 2.3. Given estimated system (1), selecting a group of (n−m)×n dimension matrix R t , t ∈ 0, T − 1, arbitrarily such that the following n × n matrix P t , t ∈ 0, T − 1, are non-singular.
where Q 1t ∈ R n×m , Q 2t ∈ R n×(n−m) . Then the following formula can be established Then we have Since algebraic equivalent systems have the same controllability and observability, (A t , C t ) is observable if and only if (A t , C t ) is observable. According to Lemma 2.2, by denoting the column rank as rank c , the observability of matrix pair (A t , C t ) at time t means

ROBUST REDUCED-ORDER OBSERVERS DESIGN FOR LDP SYSTEMS 2803
According to (11), the above equation can be rewritten as Then, we can get Rearranging the rows of this matrix gives That is to say where A and R are give by (3) and (6) with compatible matrix dimensions. Again, according to Lemma 2.2, the above equation denote that matrix pair (A 22t , A 12t ) is observable at time t. Thus the proof is accomplished.
Utilizing the linear non-singular transformation x t = P t x t provided in Lemma 2.3, we can see that the estimated system (1) is algebraically equivalent to the following system To design a periodic observer for system (1), a natural thought is to design a periodic observer for system (16) firstly, then by linear inverse transformation of state x(t), to obtain the estimation of original state x(t). From (16), we can see that for the transformed state x t , its substate x 1t is the output y t of the original system (1). So it can be used directly without reconstruction. What need to be reconstructed here is the n − m dimension substate vector x 2t of x t , so lower order periodic state observer with dimension n − m can meet requirements.
Thus, problem 2 can be degenerated into the following problem.
Problem. For completely reachable and completely observable LDP system (16), find periodic matrix L t ∈ R (n−m)×m , t ∈ 0, T − 1, such that the state observer of the substate vector x 2t can be given as follows And for any x 0 , z 0 and u t , there holds (17) is an n − m order dynamic system with the input vectors u t and y t .
3.1. Periodic reduced-order observer design. The first thing to consider is the existence condition for periodic reduced-order state observer. To do this, we would like to give the following theorem.
Theorem 3.1. For a given LDP system (16), assume that periodic matrix pair (A 22t , A 12t ) is observable and L t , t ∈ 0, T − 1, are a group of matrices that make periodic matrix pair (A 22t −L t A 12t ) stable, then system (17) constitutes an observer of the state component x 2t of the system (16).
Proof. By formula (16) we can obtain Define Formula (18) can be expressed as canonical as follows: Since (A 22t − L t A 12t ) is observable, the system (20) is the (n − m) dimensional subsystem of system (16), the full-dimensional state observer of system (20) exists and has the following form and one can configure all the eigenvalues of (A 22t − L t A 12t ) by selecting L t . Then we substitute equation (19) into system (21), From system (22) and formula (23), we can obtain Observer (17) can be obtained from equation (23) and equation (25). Thus the proof is accomplished.
According to the above theorem, the problem of reduced-order observer design can be converted into the stabilization of periodic matrix pair (A 22t , A 12t ). In the following, we would like to adopt poles assignment technique to achieve this purpose.
In the following, we will firstly present an iterative algorithm to compute periodic reduced order observer gains L(t), t ∈ 0, T − 1, which can solve Problem 2.

While
6. Let X t = X t (k). The real periodic matrix L t can be obtained as Remark 2. The main part of the algorithm does not contain nested loops, so the computational complexity of the algorithm is O(n − m).
On the convergence and correctness of the Algorithm 1, we give the following lemma, which is similar with the corresponding lemma of literature [10] and its proof is omitted here. Next, we will give the main result of this section. Theorem 3.3. Consider the completely observable periodic discrete-time linear system (1). If matrices Q t , A t , B t , C t are determined by equations (11)- (12), and matrices L t are generated from Algorithm 1, then a periodic reduced-order observer for system (1) with dimension n − m can be given by That is to say, for any x 0 , z 0 , and u t the composite system composed of system (1) and system (27) satisfies the following relationship Proof. According to Lemma 3.2, N t (k), t ∈ 0, T − 1, are convergent sequences. Denote lim k→∞ N t (k) = N t , t ∈ 0, T − 1. Then N t = 0, t ∈ 0, T − 1. Construct the following index function: By this equation, we have: Then the least squares solution (X * 0 , X * 1 , · · · , X * T −1 ) satisfies for t = 0, 1, · · · , T − 1.
Since N t = 0, we can conclude J = 0, which leads to By equation (26), it can be rewritten as premultiplying the both sides of equation (30) with X −1 t+1 gives Continuous multiplication of the above equation for t ∈ T − 1, 0, gives This means when T is an odd number, or when T is an even number. According to Algorithm 1, all the poles of Ψ F lie in the unit circle. By the above relation, matrices L t can stabilize periodic matrix pair (A 22t , A 12t ). This is to say, the periodic observer gain matrices L t generated from Algorithm 1 is a solution to Problem 2.
Once problem 2 is solved, the reconstruction of the transformed state x t can be attained. Considering Then we can get the reconstruction state x t of the system state x t as To summarize, combining equations (35) and (17), we can obtain a (n − m) dimensional state observer for system (1) with the following expression 3.2. Robust consideration. In practice, it is inevitable that there exist some uncertain disturbances in system running, which can lead to the deviation of the real system data from the nominal system data. However, the periodic observers are designed according to the nominal system. This will result in inaccuracy even instability on the observation error. Therefore, it is indispensable to design a robust observer which is insensitive to the small perturbation on system data. In this section, based on a perturbation analysis result on periodic data, we would like to provide a periodic robust observer design program.
For normalization, the problem of robust observer design for linear discrete-time periodic system (1) can be described as follows.
Problem. Consider the completely observable linear discrete-time periodic system (1), seek the periodic matrix L t ∈ R (n−m)×m , t ∈ 0, T − 1, such that the following conditions are met: 1. Observer system (36) gives an asymptotic estimation of state x t . 2. The reduced-order observer gains are as insensitive as possible to small perturbations on system matrices.
Here, we review a previous research result on perturbation analysis, which can be found in reference [11].
λ n } is the Jordan canonical form of matrix Ψ. For a real scalar ε > 0, ∆ i (ε) ∈ R n×n , i ∈ 0, T − 1, are matrix functions of ε satisfying where ∆ i ∈ R n×n , i ∈ 0, T − 1, are constant matrices. Then for any eigenvalue λ of matrix Ψ(ε) = (A(T − 1) + ∆ T −1 (ε)) (A(T − 2) + ∆ T −2 (ε)) · · · (A(0) + ∆ 0 (ε)) , the following relation holds: According to Lemma 3.4, one could take the robust performance index of problem 3.2 as On the other hand, since small gains also mean robustness in some sense, one can adopt another robustness index as Taking a tradeoff between the two index gives where 0 ≤ α ≤ 1 is a weighting factor. Combined with the iterative algorithm for reduced-order observer design, the problem of robust periodic reduced-order observer design can be converted into an optimization problem. The corresponding algorithm can be summarized in the following. and denote the optimal decision matrix by D opt t , t ∈ 0, T − 1. 3. Substituting D opt t into steps 3-5 of algorithm 1 gives optimization matrices X opt t . 4. The robust periodic reduced-order observer gain matrices can be obtained as

4.
A numerical example. Consider LDP system (1) with periodic T = 3. Its parameters are given as follows: It is known by simple verification that this system is completely reachable and completely observable. The state dimension is 3 and the output dimension is 1. Therefore, a reduce-order observer with dimension 2 can be expected. According to lemma 2.3, we can obtain the non-singular matrix P t , t ∈ 0, 1, 2, as shown below Let the parameter matrix D t , t ∈ 0, 1, 2, be in the following form: According to Algorithm 1, by choosing parameter matrices D t , we can obtain a group of gains for reduced-order observer as  To check the effectiveness of the designed observers, the observed errors by the two observers are taken into consideration. Let reference input be v(t) = 0.3[sin( π 2 + t) + cos( π 2 + t)] and assume that initial states of the system (1) and the observer (36) are respectively taken as  Figure 1. From the simulation results, we can see that both the two observers can track the system states very well, even the reduced order observer gains L rand t are generated randomly by Algorithm 1. Moreover, the error trajectories by L robu t have smaller overshoot and faster convergence speed, which can lead better transient and steady performance. Therefore, we can conclude that the robust reduced-order observer design algorithm proposed in the paper is very effective.

5.
Conclusion. This paper mainly introduces the design method of periodic reduced-order state observer for a class of LDP systems. Following the design ideas of reduce-order observer for LTI systems, by non-singular linear transformation, a part of state information can be completely represented by system output and the other state information can be reconstructed via the periodic poles assignment technique. In addition, robust periodic reduced-order observer is also considered in this paper. Correspondingly, two detailed design algorithms are provided and their effectiveness are illustrated by a simulation example.
Interesting future research topics include: (i) to study distributionally robust design for LDP systems under the uncertainties with unknown distributions; (ii) to consider probabilistic control of Markov jump linear discrete periodic systems.