Partial observer canonical form for multi-output nonlinear forced system: a new method

PurposeThe purpose of this paper is to investigate the design method of partial observer canonical form (POCF), which is one of the important research tools for industrial plants.Design/methodology/approachMotivated by the two-steps method proposed in Xu et al. (2020), this paper extends this method to the case of Multi-Input Multi-Output (MIMO) nonlinear system. It decomposes the original system into two subsystems by observable decomposition theorem first and then transforms the observable subsystem into OCF. Furthermore, the necessary and sufficient conditions for the existing of POCF are proved.FindingsThe proposed method has a wide range of applications including completely observable nonlinear system, noncompletely observable nonlinear system, autonomous nonlinear system and forced nonlinear system. Besides, comparing to the existing results (Saadi et al., 2016), the method requires less verified conditions.Originality/valueThe new method concerning design POCF has better plants compatibility and less validation conditions.


Introduction
The development of modern manufacturing industry guided by intelligent manufacturing is inseparable from the basic manufacturing equipment and integrated manufacturing system. For example, complex sensor networks are widely used in power grid, transportation system and industrial objects (Estrin et al., 1999;Akyildiz et al., 2002;Deng et al., 2015Deng et al., , 2017Hao et al., 2012;Li and Tong, 2016); and permanent magnet synchronous motor (PMSM) technology is widely used in modern power electronics technology, microchip technology and advanced control theory (Bae et al., 2001;Altaey and Kulaksiz, 2017;Schoonhoven and Uddin, 2016). However, the control system in the industrial plants, whether it is the sensor network or the servo system represented by the PMSM, is inseparable from the accurate measurement and estimation of the state in the system. Generally speaking, the actual industrial plants, especially the major equipment used in basic manufacturing, such as Tunnel Boring Machine (TBM) (Yang et al., 2019;Li et al., 2010;Zhao et al., 2015) and PMSM, often have strong At the end of the introduction, some notations of this paper should be declared. We denote the symbol colff 1 ; Á Á Á ; f n g as a matrix ½f T 1 ; Á Á Á ; f T n T , where f represents a matrix or a map with single output. Symbol ½$ m3n is a matrix with row m and column n. L f hðxÞ is Lee derivative of function hðxÞ along to vector field f ðxÞ. h$; $i is the inner product between two vector fields. A Lie bracket about two vector fields f and g is ½f ; g. We use * in a matrix to represent the nonzero and not important entry. And we use * in the subscript of a diffeomorphism to represent the Jacobian determinant of this diffeomorphism. For example, Φ Ã ¼ vΦðxÞ=vx T .

Problem formulation and preliminaries
Taking into account a multi-output nonlinear system as _ x ¼ f ðxÞ þ gðx; uÞ; (1) y ¼ hðxÞ: where x ∈ R n , u ∈ R m , y ∈ R p stands for state variables, control inputs and measurement output respectively, f ðxÞ and g i ðxÞ are smooth vector fields with dimension n, and hðxÞ is a p dimension smooth vector value function which is written as hðxÞ ¼ ðh 1 ðxÞ; h 2 ðxÞ; Á Á Á ; h p ðxÞÞ T . Following the definition of Krener and Respondek (1985), we define a group of codistributions Then the codimension at some point x of E is defined as With this description, one can state the definition of the output relative degree concerning the kth output of system (1), (2) in follows (Krener and Respondek, 1985): where cardf$g represents the number of elements of a set. One may get r 1 ≥ r 2 ≥ Á Á Á ≥ r p ≥ 1 by arranging the order of output appropriately. In order to design an observer for (1), (2) with linear error dynamics, we wander a diffeomorphism such that it can transform this system into observer canonical form (OCF) where A; C are the block diagonal matrices denoted as A ¼ blockdiagfA 1 ; Á Á Á ; A p g, C ¼ blockdiagfC 1 Á Á Á ; C p g, and for all 1 ≤ i ≤ p; with A i and C i satisfying Both Krener and Respondek (1985) and Xia and Gao (1989) proposed the sufficient and necessary conditions for transforming (1), (2) into OCF. Before introducing the OCF conditions, some definitions and notations should be introduced. They will be used Partial observer canonical form throughout the rest of this paper. We firstly give some codistributions defined in Xia and Gao (1989) for i ¼ 1; 2; Á Á Á ; p, Note that there exists a nonzero orthogonal distribution Δ corresponding to codistribution Δ ⊥ when dimfΔ ⊥ g < n. In addition, Δ and Δ ⊥ satisfies hω; H i ¼ 0 for arbitrary ω ∈ Δ ⊥ and H ∈ Δ. Then some linear equations concerning y i ¼ h i ðxÞ are introduced as: where δ i;j is Kronecker symbol. For the sake of statement, the linear equations in the form of (10), (11) are denoted as a symbol Leðf ; h i Þ. Without loss of generality, we assume τ i ∈ R n be the solution of Leðf ; h i Þ. Let θ i;1 ¼ τ i ; i ¼ 1; 2; Á Á Á ; p and further let θ i;j ¼ ½θ i;j−1 ; f ; 2 ≤ j ≤ r i . For the unification of symbols, the basic vector fields of Δ are set as θ rþ1 ; Á Á Á ; θ n with r ¼ P p i¼1 r i , i.e. Δ ¼ spanfθ rþ1 ; Á Á Á ; θ n g. Note that θ i;j ∉ Δ for all 1 ≤ i ≤ p; 1 ≤ j ≤ r i owing to linear equations (10), (11) and property hdL jþ1 Boutat and Busawon, 2011).
It is worth to be pointed out that τ i is not the unique solution of Leðf ; h i Þ. In fact, it includes two degrees of freedom. One is because the number of equations in (10), (11) is less than r. The other one is because of r < n.
We denote the solution space of Leðf ; h i Þ at point x as S 0 i ðxÞ and further denote S 0 i as the solution distribution which is produced by letting solution space move along to manifold (See the sketch map of the relationship between distribution and tangent space in Figure 1). We can also denote the tangent space at x of distribution Δ as ΔðxÞ. Then it is obvious that there is a subspace S i ðxÞ of solution space satisfying S i ðxÞ ⊂ ΔðxÞ ⊥ for arbitrary x and all 1 ≤ i ≤ p, where ΔðxÞ ⊥ represents the orthogonal complement space of Δ. S i ðxÞ is called special solution space of linear equations Leðf ; h i Þ. One may also denote a distribution S i as a special solution distribution corresponding to S i ðxÞ. By choosing a solution in S i ðxÞ and letting it move along to manifold, the vector field τ i ∈ S i proposed above could be obtained.
Moreover, there is τ i þ H ∈ S 0 i for arbitrary H ∈ Δ. In other words, S i ≡ S 0 i mod Δ. Figure 2 shows the relationship between S 0 i , S i and Δ in one of the most special forms. Lemma 1. Considering a nonlinear system (1), (2) and a point in state space x 0 . Then there exists a neighborhood U 2 containing x 0 and a diffeomorphism Φ 2 defined on U 2 such that the underlying system can be transformed into (5), (6) if and only if (1) The dimension of distribution Δ ⊥ is n, However, it is difficult for a multi-output system whose observable relative degree r ¼ P p i¼1 r i < n to satisfy Lemma 1. Thus, one hopes to find a partial observer for multioutput nonlinear which takes the form of: _ ξ 1 ¼f 1 ðξ 1 ; ξ 2 Þ þg 1 ðξ 1 ; ξ 2 ; uÞ; (12) where A i ; C i are in the form of (7), ξ 1 ; ξ 2 are states that transformed by POCF diffeomorphism with ξ 1 being the unobservable states and ξ 2 ¼ colfξ 21 ; Á Á Á ; ξ 2r g being the observable states. So POCF (12)- (14) can be used in the situation when P p i¼1 r i ¼ r < n. As the statement of introduction, there are a lot of articles about partial observers, but only a few about them concerning the partial observers with error linearization. Moreover, to the knowledge of authors, there is few or no result about POCF of multi-output nonlinear system with input. Consequently, this paper focuses on deducing the sufficient and necessary conditions of the existence of POCF for system (1)-(2) and gives a way to calculate the corresponding diffeomorphism from the original system to POCF with a new method. Remark 1. Conditions (1)-(3) of Lemma 1 are the sufficient and necessary conditions proposed in Xia and Gao (1989) that are used for transforming the autonomous system into OCF _ z ¼ Az þ γ 1 ðzÞ. These conditions can also be used for forced system in which the solutions of linear equations (10), (11) are the vector fields with connection to u. Furthermore, there are only ir i þ r iþ1 þ Á Á Á þ r p < n equations in linear equations (10), (11) so almost all of τ i , the solution of the linear equations, are not unique, except τ 1 .

The first freedom degree
The second freedom degree DeterminisƟc component Relationship between S 0 i , S i and Δ Partial observer canonical form Xia and Gao (1989) treat this kind of linear equations as an improvement because it is easier for system to satisfies communicating conditions (3). (2) in Lemma 1 proposed in (Xia and Gao, 1989) take the place of the requirement of Krener and Respondek (1985) that the system must be in special observable form. See the structure of Δ ⊥ and Δ ⊥ i in Figure 3. The red box is the codistribution Δ ⊥ , the blue box represents the vector fields spanning Δ ⊥ i . Blue area is the basic vector fields of Δ ⊥ belonging to Δ ⊥ i , we denote this area as Δ ⊥ b . And the yellow area named Δ ⊥ s represents the vector fields of Δ ⊥ i but not the basic vector fields in Δ ⊥ . Noting that Δ ⊥ i is contained in Δ ⊥ because of the restriction of dimfΔ ⊥ g. Therefore, conditions (2) actually require that all of the vector fields in Δ ⊥ s can be represented by the basic vector fields in Δ ⊥ b .

Main result
At the beginning of this section, we first introduce the basic idea of the new method concerning POCF. Firstly, calculate a diffeomorphism Φ 1 defined on a neighborhood U 1 of x 0 such that the system transformed by Φ 1 can be divided into two subsystems including observable subsystem and unobservable subsystem. Then one can design Φ 2 by using condition in Lemma 1 to transform the observable subsystem into OCF. We thus conclude Φ ¼ Φ 2 +Φ 1 . The main result of this paper is to deduce the migration results of OCF conditions under observable decomposition. Noticing that the observability decomposition theorem proposed by Isidori (1989) can be easily generalized to general nonlinear systems. Hence, we have the following lemma.
Lemma 2. A distribution Q is called a maximum invariant distribution if it is contained in spanfdh 1 ; Á Á Á ; dh p g ⊥ and invariant under f and g. Suppose a point x 0 and its neighborhood U 1 and further suppose Q is involutive and nonsingular on U 1 .
where f and g are vector fields satisfying f ¼ colf Proof. According to Frobenius theorem and Lemma in (Li, 2014, Lemma 7.1), we can obtain a diffeomorphism z ¼ Φ 1 ðxÞ by the proof process of Frobenius theorem if nonsingular involutive distribution Q is invariant under f. Then f can be transformed into an upper triangular form by Φ 1 ðxÞ. Furthermore, the first n − r terms of covector field dh in the new coordinate will be zero. Thus, we get the observable decomposition form (15)- (17). ▪ Next, some basic properties of codistribution Δ ⊥ will be given. And some of their proof is omitted because they are direct generalizations of (Tami et al., 2016) 's Lemma.
Furthermore, according to the definition of Q ⊥ , Q ⊥ is also invariant under g k ; k ¼ 1; Á Á Á ; m. Then Lemma 1 yields Q ¼ Δ is invariant under g k ; k ¼ 1; Á Á Á ; m. Bearing in mind that θ rþ1 ; Á Á Á ; θ n are the basic vector fields of Δ, one thus has ½θ i ; g k ∈ Δ for r þ 1 ≤ i ≤ n and 1 ≤ k ≤ m.
ð⇐Þ The proof of sufficiency can be completed by inverse deducing the above steps.
Theorem 2. Considering a nonlinear system (1), (2), and assume U is a neighborhood around a point x 0 in state space. If a distribution Δ. corresponding to this system is nonsingular in U with dimension n − r. Then there exists a diffeomorphism Φ ¼ Φ 2 +Φ 1 defined on U such that the original system (1), (2) can be transformed into POCF (12)-(14) if and only if (4) ½θ i;k ; g ∈ Δ; 1 ≤ k ≤ r i − 1 for arbitrary 1 ≤ i ≤ p.
Proof. Since Δ is nonsingular on the neighborhood U around x 0 , there exists a coordinate transformation Φ 1 such that the original system can be transformed into (15), (16). Then this proof will be finished in the following five steps.
(1) Given the following two groups of codistribution for all i ¼ 1; 2; Á Á Á ; p, Moreover, divide the corresponding regions of the above codistributions according to the definition in Figure 3. It is supposed to prove that condition dimfΔ It is known that for any k > r j − 1; j > i, the covector field dL k f h j ∈ Δ ⊥ s has no connection with the covector field in Δ ⊥ nΔ ⊥ i . Since the diffeomorphism does not change the independence of the vector fields and the covector fields, we conclude that the following covector fields dL k f h j ∈ Δ s ; k > r j − 1; j > i have no connection with covector fields Next, we will prove that L k f h j ¼ L k f 2 h j for arbitrary 1 ≤ k ≤ r i − 1 and all of them are only related to z 2 . This assertion will be proved by mathematical induction. Set k ¼ 1, it is obviously that h j are only related to z 2 and hence we have v h j =vz T 1 ¼ 0. It follows with Noticing that both h j and f 2 are only related to z 2 , so L f 2 h j is also only related to z 2 . Assume this assertion is fulfilled for all k ≤ r i − 2, then set k ¼ r i − 1 and one can deduce that It is apparent to show that L r i −1 f 2 h j is only related to z 2 owing to vL r i −2 Thereby, it can be deduced for arbitrary family of smooth functions (2) It can be directly deduced by the definition of observable relative degree that dimfΔ ⊥ g ¼ dimf D ⊥ g ¼ r.
(3) Considering linear equations Leð f ; h i Þ and Leð f 2 ; h i Þ, where 1 ≤ i ≤ p. Assume What need to be proved in this step is that there exists a family of proper solutions ϑ i;1 ∈ S i ðf ; h i Þ and θ i;1 ≡ ϑ i;1 mod Δ such that ½θ i;k ; θ j;l ∈ Δ for all 1 ≤ i; j ≤ p and 1 ≤ k ≤ r i ; 1 ≤ l ≤ r j if and only if ½θ i;k ;θ j;l ¼ 0.
It is known according to Lemma 3 that Δ is invariant under f ; g. Bearing in mind Therefore, ½θ i;j ; g ∈ Δ if and only if ½θ i;j ; g 2 ¼ 0.
▪ It is noted that Theorem 1 is very similar to Lemma 1 since a distribution can be spanned by a vector field 0. Thus, we can deduce a corollary from Theorem 1 which can be used for almost all smooth affine nonlinear system with observable relative degree 0 < r ≤ n. Moreover, this conclusion can be used to design POCF whether it is a single output system or a multi-output system. Corollary 1. Consider a nonlinear system (1), (2). Suppose U is a neighborhood around arbitrary point x 0 in state space. If codistribution Δ ⊥ is nonsingular on U, then there is a diffeomorphism ξ ¼ ΦðxÞ defined on U such that system (12)-(14) can be transformed into (5), (6) if and only if (2) ½θ i;k ; θ j;l ∈ Δ are satisfied for all τ i ∈ S 0 There is no doubt that it is amount to Theorem 1 when r < n. In addition, if r ¼ n, then dimfΔ ⊥ g ¼ n. It indicates Δ ¼ spanf0g. So, conditions (2) in Corollary 1 is equivalent to ½θ i;k ; θ j;l ¼ 0. Thus, this corollary degenerates into Lemma 1 when r ¼ n. In the situation of p ¼ 1, i.e. system (1), (2) is a single-output system. This corollary can deal with the same problem as what Tami et al. (2016) does. But the conditions in this conclusion are weaker than that in (Tami et al., 2016). If the system is autonomous, then this corollary degenerates into the problem discussed in Saadi et al. (2016). However, one may notice that the conditions in Saadi et al. (2016) are redundant comparing to Corollary 1.

Conclusion
To investigate the design method of POCF for a class of MIMO nonlinear system, this paper, motivated by the two steps methods of Single-output nonlinear system (Xu et al., 2020b), has generated the corresponding new methods for MIMO system. In this regard, the underlying system is divided into observable subsystem and unobservable subsystem first and then the former is transformed as OCF. Furthermore, a corollary at the end of this paper has been developed as a uniform theorem for the existing of POCF for a large class of nonlinear system, such as single-output system, multi-output system, observable system and noncompletely observable system that is considered in this paper and so on.