Observer-Based Realization of Control Systems
Daizhan Cheng Changxi Li Xiao Zhang Zhengping Ji






Observer-based Realization of Control Systems
    Daizhan Cheng, Changxi Li, Xiao Zhang, Zhengping Ji
	This work is supported partly by the National Natural Science Foundation of China (NSFC) under Grants 62073315 and 62350037.
	Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences,
		Beijing 100190; and Research Center of Semi-tensor Product of Matrices: Theory and Applications, Liaocheng Univ. Liaocheng,  P. R. China (e-mail: dcheng@iss.ac.cn).

    April 29, 2024
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Lebesgue-type of dynamic control systems and dimension-keeping semi-tensor product (DK-STP) of matrices are introduced. Using bridge matrices, the DK-STP is used to construct  approximated observer-based realization (OR) of linear control systems, as  Lebesgue-type control systems, are proposed. A necessary and sufficient condition for the OR-system to have exactly same observer dynamics is obtained. When the exact OR-system does not exist, the extended OR-system, which contains observers of the original system as part of its state variables, is presented. Moreover,  the (minimum) feedback (extended) OR-system is also constructed, and its relationship with Kalman's minimum realization is revealed. Finally, the technique developed for linear control systems has been extended
to affine nonlinear control systems. The purpose of OR-system is to provide a new technique to deal with large scale complex systems.



  Cross-dimensional projection, DK-STP,  state-observer (SO-) system, (approximate, exact, extended, feedback) OR system, linear (nonlinear) control systems.







§ INTRODUCTION


Since Kauffman proposed to use Boolean network to formulate the genetic regulatory network <cit.>, the Kalman's state space approach, which is widely used for dynamic control systems,  has also been used to represent Boolean (control) networks <cit.>. It has also been used for many other finite-valued systems, such as finite games <cit.>.

Consider a finite-valued control system. Assume there are n states (nodes or players) and m inputs, then the state evolutional equation is described as

    [ X_1(t+1)=f_1(X_1(t),⋯,X_n(t),U_1(t),⋯,U_m(t)),
    
    X_2(t+1)=f_2(X_1(t),⋯,X_n(t),U_1(t),⋯,U_m(t)),
        ⋮
    
    X_n(t+1)=f_n(X_1(t),⋯,X_n(t),U_1(t),⋯,U_m(t));;                                                                                                             Y_j(t)=h_j(X_1(t),⋯,X_n(t)),   j∈ [1,p],;                                                                                                                                                      ]

where X_i(t)∈ D_k.

Later on, by using STP, an alternative approach has been proposed, and approved to be powerful in analyzing and control design of finite-valued systems. It can be briefly described as follows:  Identifying i∼_̣k^i, i∈ [1,k] and
expressing X_i∼ x_i∈_k,  (<ref>) can be expressed as <cit.>

    [ x(t+1)=Lu(t)x(t),
    
    y(t)=Hx(t), ]

where x(t)=⋉_i=1^nx_i(t),u(t)=⋉_j=1^nu_j(t),  y(t)=⋉_s=1^py_i(t).

Comparing (<ref>) and (<ref>), it is easy to find that they are essentially different.
If we assume they are describing a finite game, then (<ref>) is a difference on state-variables while  (<ref>) is a difference on profiles, or on the functions of state-variables. We refer to Figure <ref> for this.



According to the differentiated (difference) objects, we call (<ref>) the Riemann-type system, and (<ref>) the Lebesgue-type system, (mimic to the Riemann and Lebesgue integrals). The Lebesgue-type representation of finite-valued systems has been shown to be very powerful in the system analysis and control design <cit.>.

Now consider the Kalman state space description of discrete-time linear control system is

    x(t+1)=Ax(t)+Bu(t),
    
    y(t)=Hx(t).

Its corresponding continuous-time system is

    ẋ(t)=Ax(t)+Bu(t),
    
    y(t)=Hx(t).

And (affine) nonlinear control system is:

    ẋ(t)=f(x(t))+_i=1^mg_i(x(t))u_i(t),
    
    y_j(t)=h_j(x(t)).

It is obvious that they are of the Riemann-type systems.
A natural question is: What are their corresponding Lebesgue-type systems? And are they useful? The purpose of this paper is to explore the Lebesgue-type realization of classical control systems. As the differentiated objects,  the functions are commonly considered as observers. Hence, the Lebesgue-type representation of (<ref>) or (<ref>) is also called the OR-system.

To investigate the OR-system of (<ref>) or (<ref>), some preparations are necessary. First. since the systems considered may be dimension-varying, the state spaces are considered as subsets of
^∞=∪_n=1^∞^n.
The vector space structure and the topology of ^∞ need to be established, and the cross-dimensional projection is necessary <cit.>.

Secondly, the semi-tensor product (STP) of matrices is a generalization of the conventional matrix product, which extends the conventional matrix product to two matrices of arbitrary dimensions. Fortunately, it keeps all main properties of   conventional matrix product remaining available. It was firstly proposed in 2001, more than two decades ago <cit.>. Since then, it has received various applications in different aspects, including Boolean networks <cit.>, finite games <cit.>, dimension-varying systems <cit.>, engineering problems <cit.>, finite automata <cit.>, coding <cit.>, etc.

Recently, a new STP, called the DK-STP, is proposed in <cit.>. DK-STP makes the set of m× n matrices, denoted by M_m× n, a ring (or an algebra). Using this ring structure, the analytic functions of non-square matrices can be properly defined.

Thirdly, the approximated OR-system of control systems (<ref>) and (<ref>) can be constructed by using cross-dimensional projection or DK-STP.  Staring from system (<ref>), we have its discrete-time SO-system as

    y(t+1)=HAx(t)+HBu(t):=Mx(t)+Nu(t).

Then the OR-system can be obtained by replacing x(t) by y(t) as

    y(t+1)=M y(t)+Nu(t).

Similarly, for continuous time case we have SO-system as

    ẏ(t)=HAx(t)+HBu(t):=Mx(t)+Nu(t).

And its OR-system can be obtained  as

    ẏ(t)=M y(t)+Nu(t).

Figure <ref> shows the motivation for proposing OR-system.



Assume we are facing a large or even huge complex system. Formulating it by classical state space description is difficult. In practice, we may only be interested in its some particular properties, which can be described by a set of observers, say, y^s. Then investigating y^s related OR-system is enough for our purpose. This approach has been investigated in <cit.> for finite-valued systems. This paper aimed to extend this technique to continuous time continuous state space control systems.

Fourthly, we consider when the OR-system is the exact one. That is, it represents precisely the observer dynamics of the original system.
A necessary and sufficient condition is obtained. When  the exact OR-system does not exist, the extended OR-system is proposed, which has an enlarged observers, containing the original observers as part of new observers. So it has enlarged size, which contains the exact information of original observers.

Finally, the minimum feedback OR-system is considered, which is an exact realization with smallest size. Its comparison with Kalman's minimum realization is also discussed.

The rest of this paper is organized as follows: Section 2 provides all preliminaries for our approach, including: (i) topological and vector space structure of mix-dimensional Euclidian space; (ii) best approximated linear (control) systems; (iii) DK-STP of matrices.
In Section 3 the weighted DK-STP is introduced first. Then we  show that under the DK-STP, the matrix function of square matrices can be extended to the ring of non-square matrices. The DK-STP based dynamic (control) systems are proposed and their solutions are obtained in Section 4. In Section 5 using bridge matrices, the DK-STP is used to construct the approximate OR-systems. As an example, a singular system is revealed to have a natural OR-system structure. And  necessary and sufficient condition for a linear system to have an exact  OR-system is presented. In Section 6 the OR-system of linear control systems is investigated. The technique for constructing state-feedback exact OR-system and state-feedback extended OR-system is presented in detail. Section 7 extends the results obtained for linear control systems to affine nonlinear systems. Section 8 is a brief conclusion.  The appendix A gives a list of symbols, and the appendix B is a detailed description for calculating the largest (A,B)-invariant subspace contained in the kernel of observers.







§ MATHEMATICAL PRELIMINARIES




 §.§ Topology and Vector Space Structure on ^∞


Consider

    V=^∞:=⋃_n=1^∞^n,

this subsection shows how to convert it into a topological (pseudo-) vector space <cit.>.

 Let x∈^p, y∈^q, and t=(p,q) be the least common multiple of p and q. Then

    x± y:=(x⊗_t/p) ±(y⊗_t/q)∈^t.


With this addition (subtraction) and the conventional scalar product, ^∞ becomes a pseudo-vector space <cit.>.

Next, to pose a topology on ^∞, we first define an inner product on it. Then the norm and  the distance can also be defined.

 Let x∈^p, y∈^q and t=(p,q) . Then


  (i) the inner product of x and y is defined as follows:

    <x,y>_ V:=1/t<x⊗_t/p,y⊗_t/q>;


  (ii) the norm is defined as

    x_ V:= √(<x,x>_ V) ;


  (iii) the distance is defined as

    d_ V(x,y):=x-y_ V.




The topology on ^∞, denoted by T_d, is  determined by the distance d_ V.




  (i) Let x, y∈^∞. x, y are said to be equivalent, denoted by x y, if there exist _i and _j such that

    x⊗_i=y⊗_j.



  (ii) Equivalently, x y, if and only if, d_ V(x,y)=0.

  (iii) ^∞ with classical scalar multiplication and addition (subtraction) defined by (<ref>) satisfies all requirements of vector space except that the “zero"  is not unique. All 0_n=(0,⋯,0)^T∈^n, which are equivalent, are zero. It follows that the inverse of x is
{y |  y -x}. That is why ^∞ is called a pseudo-vector space.

  (iv) Denote by

    x̅:={y | y x},

and define rx̅:=rx, and

    x̅±y̅:=x± y,

then the quotient space

    Ω:=^∞/

is a vector space.

  (v) Under the quotient topology Ω is a Housdorff space. Moreover, set

    <x̅,y̅>_ V:=<x,y>_ V.

Then Ω is an inner product space, but it is not a Hilbert space.



Next, we consider the projection from ^m to ^n.

   Let ξ∈^m. The projection of ξ on ^n, denoted by π^m_n(ξ), is defined as

    π^m_n(ξ):=_x∈_nξ - x_ V.



The projection can be expressed by matrix product form as follows.

 

    x_0:=π^m_n(ξ)=Π^m_nξ∈^n,  ξ∈^m,

where (t=(m,n))
 
    Π^m_n=1/t/n( I_n⊗^T_t/n)  ( I_m⊗_t/m) ,

Moreover, x_0 (ξ- x_0 ) (see Figure <ref>).


We refer to <cit.> for related topics about topological spaces; and to  <cit.> for some related knowledge on functional analysis.







































































 §.§ From Classical STP to DK-STP


As mentioned in Introduction, the (classical) STP was proposed over two decades ago, which is defined as follows:

 <cit.>  Let A∈ M_m× n and B∈ M_p× q, t=(n,p).
Their STP is defined as

    A⋉ B:=(A⊗ I_t/n)(B⊗ I_t/p)∈ M_mt/n× qt/p.



Up to now, the above (classical) STP has received various applications. Unfortunately, it can not be used for a linear mapping over ^∞, because the (classical) STP of a matrix with a vector is, in general, not a vector. Then a matrix-vector STP and a vector-vector STP have been defined <cit.>. Now, a natural question is: Is it possible to define an STP, which can be used for matrix-matrix product, as a compound of two linear mappings;  matrix-vector product, as a linear mapping; and vector-vector product, as an inner product, simultaneously, just as conventional matrix product, and meanwhile it can be used for factor matrices (including vectors) with arbitrary  dimensions? The answer is “yes" and the DK-STP is such one.

DK-STP is a newly proposed STP <cit.>.  All the concepts/results about DK-STP in this subsection come from <cit.>.

 Let A∈ M_m× n and B∈ M_p× q, t=(n,p). The DK-STP of A and B, denoted by A B∈ M_m× q, is defined as follows.

    A B:=(A⊗^T_t/n)(B⊗_t/p).



    Let A∈ M_m× n and B∈ M_p× q, t=(n,p). Then

    A B:=AΨ_n× pB,

where

    Ψ_n× p=(I_n⊗^T_t/n)(I_p⊗_t/p)∈ M_n× p

is called a bridge matrix of dimension n× p.


Similarly to classical STP, DK-STP satisfies the following fundamental requirements for STP.




  (i) When n=p, A B=AB.


  (ii) (Associativity)

    A (B C)=(A B) C.



  (iii) (Distributivity) Assume A,B∈ M_m× n and C is of arbitrary dimension. Then

    [ (A± B) C=A C± B C,; C (A± B)=C A± C B.;                    ]





The following proposition provides some characteristic properties of DK-STP.






  (i) Let A, B∈ M_p× q. Then A B∈ M_p× q.


  (ii) ( M_m× n,+, ) is a ring, which (a) is non-commutative, (b) without identity.


  (iii) Consider M_m× n as a vector space over , then ( M_m× n, ) is an algebra over .


  (iv) Define Lie bracket as

    [A,B]_:=A B-B A,   A,B∈ M_m× n,

then ( M_m× n, ) is a Lie algebra.



We refer to any text book of abstract algebra for concepts and fundamental properties of ring and algebra, e.g., <cit.>;
to <cit.> for Lie algebra.



§ ANALYTIC FUNCTIONS OF NON-SQUARE MATRICES




 §.§ Weighted and Ψ-Based DK-STPs


 A set of vectors is called a weight, denoted by

    ξ={ξ_n∈^n |  n=1,2,⋯},

if


  (i)

    ξ_1=1;


  (ii) ξ_i=(ξ_i^1,⋯,ξ_i^i)^T≠ 0, ξ_i^j≥ 0, i≥ 2, j∈ [1,i].




 Let ξ and η be two weights,  A∈ M_m× n, B∈ M_p× q, and t=(n,p). Then the weighted DK-STP with respect to ξ and η is defined as  follows.

    A^ξ_η  B:=(A⊗ξ^T_t/n)(B⊗η_t/p).



   Let ξ and η be two weights,  A∈ M_m× n, B∈ M_p× q, and t=(n,p). Then the weighted DK-STP with respect to ξ and η can be calculated by

    A^ξ_η B:=AΨ^ξ_η B,

where the corresponding bridge matrix

    Ψ^ξ_η=(I_n⊗ξ^T_t/n)(I_p⊗η_t/p)∈ M_n× p.



 It is easy to verify that Propositions <ref> and <ref> are also true for the  weighed DK-STP.


Observing that if we choose ξ={ξ_n=1/n_n | n=1,2,⋯} and  η={η_n=_n | n=1,2,⋯}, than

    [Ψ_p]_n× m:=(Ψ^ξ_η)_n× m=Π^m_n,

where Π^m_n is the projection matrix, defined in (<ref>). This bridge matrix is called the projecting bridge matrix.

Eq. (<ref>) is surprising, because the projection and the DK-STP are two different concepts, which have been introduced independently from two quite different topics. But Eq. (<ref>) joints them. In the sequel, one will see that this connection is physically meaningful.

If we ignore the original geometric meaning of DK-STP, we can formally define more DK-STP by define a set of bridge matrixes <cit.>.





  * 
    Ψ:={Ψ_m,n∈ M_m× n | m,n=1,2,⋯}

is a set of bridge matrices, if the following conditions are satisfied


  (i)

    (Ψ_m,n)=min(m,n);   m,n=1,2,⋯.


  (ii)

    Ψ_n,n=I_n,    n=1,2,⋯.



  * Let Ψ be given. The corresponding DK-SPT, denoted by _Ψ, can be defined as follows: Let A∈ M_m× n and B∈ M_p× q. Then the corresponding DK-STP is defined by

    A_Ψ B:=A Ψ_n× p B.




2mm

Assumption 1:  Hereafter for statement ease, unless elsewhere stated we assume

    Ψ=Ψ_p;  =_Ψ_p,

where Ψ_p is defined by (<ref>).



 §.§ Analytic Functions over Extended Rings


Since ( M_m× n, +, ) is a ring without identity, to express analytic functions of  A∈ M_m× n, we need an artificial identity.

 Define I_m× n as an identity element, satisfying


  (i)

    A I_m× n=I_m× n A=A,  ∀ A∈ M_m× n.


  (ii) For any x∈^∞,

    I_m× n x:=I_m x,    x∈^∞.








  (i) I_m× n is not a matrix, that is, it has no matrix expression.

  (ii) Eq. (<ref>) defines the interaction of I_m× n with any element in  ring M_m× n.


  (iii) Eq. (<ref>) defines its action on ^∞.



When the action of I_m× n on ^∞ is considered, it can be imagined as a matrix of dimension m× n. Then we can have some “reasonable" relations, which are considered as the consequence of its definition.


Using (<ref>), we have the following equations for the action of I_m× n:


  (i) Assume x∈^p, then

    [                 I_m× n x=I_m x; =I_mΨ_m× px=Ψ_m× px,     x∈^p. ]


  (ii) Similarly to A∈ M_m× n in (<ref>), we are reasonable to define that
 
    I_m× n x=I_m× nΨ_n× px,     x∈^p.

Comparing it with (<ref>), we have

    I_m× nΨ_n× p=Ψ_m× p.


  (iii)   Assume x∈^n, similarly to (<ref>), we have

    I_m× n x=Ψ_m× nx,     x∈^n.






    [                        M_m× n                            :=         M_m× n⋃{rI_m× n | r∈};                                                           = {rI_m× n+A |  r∈, A∈ M_m× n}. ]




( M_m× n, +,) is a ring with identity I_m× n, where

    [           (r_1I_m× n+A)+(r_2I_m× n+B):=; (r_1+r_2)I_m× n+(A+B),
        A,B∈ M_m× n; ]

and

    [       (r_1I_m× n+A) (r_2I_m× n+B); :=(r_1r_2)I_m× n+(r_1B+r_2A+A B). ]






  (i) ( M_m× n, +) is a vector space and its dimension is mn+1.

  (ii) Eq. (<ref>) means the action I_m× n  on ^∞ depends on the definition of . Precisely speaking, it depends on the corresponding bridge matrix Ψ, which determines . To be specific, we may replace
    I_m× n by I_m× n^Ψ.




Let A∈ M_m× n. Define

    A^<k>:=
    
    I_m× n,   k=0,
    A⋯ A_k,   k≥ 1.


 Let p(x)=c_nx^n+c_n-1x^n-1+⋯+c_1x+c_0 be a polynomial with c_i∈, i∈ [0,n]. A∈ M_m× n. Define the polynomial p of A as

    [                     p<A>                       :=  c_nA^<n>+c_n-1A^<n-1>+⋯;                                                   +c_1A+c_0I_m× n∈ M_m× n. ]



Using above expression and the Taylor series expansions of analytic functions, the analytic functions of non-square matrices can be defined.

 Given A∈ M_m× n.



  (i)

    e^<A>:=I_m× n+A+1/2!A^<2>+1/3!A^<3>+⋯.


  (ii)

    cos<A>:=I_m× n-1/2!A^<2>+1/4!A^<4>+⋯.


  (iii)

    sin<A>:=A-1/3!A^<3>+1/5!A^<5>+⋯.


  (iv)

    ln<I_m× n+A>=A-1/2A^<2>+1/3A^<3>+⋯.


  (v)

    [          (I_m× n+A)^<>                      =    I+  A+ (-1)/2!A^<2>;                                               + (-1) (-2)/3!A^<3>+⋯. ]


  (vi)

    cosh<A>=I_m× n+1/2!A^<2>+1/4!A^<4>+⋯.


  (vii)

    sinh<A>=A+1/3!A^<3>+1/5!A^<5>+⋯.




All other analytic functions of non-square A can be defined similarly using their Taylor series expansions.

To see when the analytic functions are properly defined, we need to know when the Taylor series expansions converge. To this end, the norm of a matrix needs to be defined.

 Let A∈ M_m× n. The DK-norm of A, denoted by A_, is defined as

    A_:=sup_0≠ x∈^mA x_ V/x_ V.



According to Definition <ref>, it is easy to calculate that

    A_=√(σ_max((AΨ_n× m)^T(AΨ_n× m))).


The convergence of Taylor series expansions of analytic functions of non-square matrices can be verified by using (<ref>).



§ DK-STP BASED DYNAMIC SYSTEM




 §.§ Action of M on ^∞


Denote the set of all matrices by

    M:=⋃_m=1^∞⋃_n=1^∞ M_m× n.


 Let A∈ M and x∈^∞. The action of  M on ^∞ , denoted by π_A:^∞^∞, is defined as

    π_A(x):=A x,   x∈^∞.



 Recall Assumption 1,
assume A ∈ M_m× n, x∈^p, and t=(n,p), then

    π_A(x):=AΨ_n× p x=1/t/nA(I_n⊗^T_t/n)(I_p⊗_t/p)x.



π_A has the following properties.

 Let A,B∈ M and x∈^∞. Then (semi-group property)

    (A(B  x)=(A B) x.



<cit.>

Let A∈ M_m× n.


  (i) ^m is an A-invariant subspace. That is,

    A x∈^m,   x∈^m.



  (ii) There exists a unique Π_A∈ M_m× m, such that

    A x=Π_Ax,  ∀ x∈^m.



  (iii)

    Π_A=AΨ_n× m.




The following result, called the generalized  Cayley-Hamilton theorem,
 is interesting.

<cit.>
 Let p(x)=x^m+c_m-1x^m-1+⋯+c_1x+c_0 be the characteristic function of Π_A, then

    A^<m+1>+p_r-1A^<m>+⋯+p_0A=0.



 In Theorem <ref> if n<m then the  characteristic function of Π_A^T can be used (to reduce the degree of the characteristic function).




 §.§ Quasi-Dynamic System over M_m× n


The following definition is based on <cit.> and <cit.>.




  (i) Let G=(G,*) be a monoid (a semi-group with identity), X be a set. ( G,X) is called a semi-group (briefly, S-) system, if there exists a mapping π:G× X X, denoted simply by (g,x)↦ gx, satisfying


  (a)

    g_1(g_2x)=(g_1*g_2)x,   g_1,g_2∈ G, ∀ x ∈ X.


  (b) Let e∈ G be the identity, then

    e(x)=x,  ∀ x∈ X.


If only (a) holds, (G,X) is called a quasi-S-system.


  (ii) Assume X is a topological space, and (G,X) is an S-system (quasi-S-) system.  (G,X) is called a (quasi-) dynamic system, if for each g∈ G the π_g:X X, defined by

    π_g: x↦ gx,   x∈ X,

is continuous.



Consider the action of monoid ( M_m× n,)  on ^∞. The discrete-time S-system can be constructed as

    y(t+1)=A(t) y(t),    A(t)∈ M_m× n, y(0)∈^∞.

Similarly, the continuous-time S-system can be constructed as

    ẏ(t)=A(t) y(t),    A(t)∈ M_m× n,  y(0)∈^∞.


It is ready to verify that both (<ref>) and (<ref>) are quasi-dynamic systems. Particularly, if the state space is restricted on ^m, they become dynamic systems.

In this paper we are only concentrated on constant systems, that is, we assume the following assumption is true.

2mm

Assumption 2: A(t)=A.

2mm

Consider system (<ref>), a direct computation shows that

    [                  y(1)=A y_0:=y_1∈^m.; y(2)=A y(1)=A^<2> y_0=A^<1>Ψ_n× my_1; y(3)=A y(2)=A^<3> y_0=A^<2>Ψ_n× my_1;                                    ⋯;                                      ]

It is obvious that the trajectory of (<ref>) is

    [                      y(t)                         =        A^<t> y_0,   t≥ 1,;                                                   = (A Ψ_n× m)^t-1 y_1, t≥ 2,;                           ]

where y_1=y(1).




  (i) (<ref>) is a quasi-dynamic system, because the identity I_m× n∈ M_m× n satisfies

    I_m× nx=I_m x=Ψ_m× nx≠ x.


  (ii) Assume  y_0∈^m,
Then (<ref>) becomes

    y(t+1)=Π_A y(t),   y(0)=y_0.

(<ref>) is a  standard difference equation over ^m.



Next, we calculate the trajectory of (<ref>).  We claim that the trajectory  is

    y(t)=e^<At> y_0.


To prove the claim,
using Taylor series expansion, (<ref>) can also be expressed  as

    [                             y(t)                                =    (I_m× n+tA+t^2/2!A Ψ_n× mA+ .;                                                                   .t^3/3!A Ψ_n× mAΨ_n× mA
    +⋯) y_0. ]

Differentiating it yields

    [                                       ẏ(t)                                          = (A+t A Ψ_n× mA+ t^2/2!  A Ψ_n× mA Ψ_n× mA.;                                                                                          .t^3/3!A Ψ_n× mAΨ_n× mAΨ_n× mA
    +⋯) y_0. ]

By definition of I_m× n we have

    A=A I_m× n=AΨ_n× mI_m× n.

Plugging it into (<ref>) yields that

    ẏ(t)=A y(t),

which completes the proof of the claim.
We then conclude that

Eq.  (<ref>) is the solution of (<ref>).


Next, we consider the corresponding control systems. For the  discrete time case, consider

    [                  y(t+1)=A y(t)+Bu(t),; y(0)=x_0∈^n,   A∈ M_m× n,  B∈ M_m× r. ]


A straightforward computation shows that

    [                                 y(1)                                    =                         A y_0+Bu(0),;                               y(t+1)                                    = Π_Ay(t)+Bu(t),   t≥ 1, y_1:=y(1)∈^m.;                                      ]

When t≥ 1 the second part of  (<ref>) is a “normal" discrete-time linear control system. The corresponding control problems can be solved using classical linear control theory.

Consider the continuous time case, we have

    [             ẏ(t)=A y(t)+Bu(t),; y(0)∈^∞, A∈ M_m× n, B∈ M_m× r. ]



A straightforward computation verifies that

    y(t)=e^<At> y_0+∫_0^te^<A(t-τ)> Bu(τ)dτ.


An alternative approach is: assume y_0∈^q.  Then it is easy to verify that

    [                                 y_0_+                                    := y(0_+)=lim_t 0_+e^<A^t>y_0=I_m× n y_0;                                                                                                       =Ψ_m× qy_0. ]

Then the system can be converted to

    [             ẏ(t)                = Π_A y(t)+ Bu(t),;                       y(0)=y_0_+. ]

Here  (<ref>) is a “normal" linear control system.



§ OR-SYSTEMS OF LINEAR (CONTROL) SYSTEMS




 §.§ Approximate OR-System


Recall linear control  system (<ref>) or (<ref>). Its  corresponding SO-system is (<ref>) or (<ref>).  It is clear that (<ref>) or (<ref>) is not a well posed dynamic (control) system.

The transition processes of  input-output system (<ref>) (or (<ref>)) and its corresponding SO-system are depicted in Figure <ref>.

2mm



2mm

Observing Figure <ref>, one sees easily that to make the SO-system a properly defined dynamic system we need a bridge
(mapping) from ^p to ^n. The bridged SO-system is called the OR-system. We give a precise definition as follows:




  (i) The OR-system of (<ref>) is defined by

    y(t+1)=M y(t)+Nu(t).


  (ii) The OR-system of (<ref>) is defined by

    ẏ(t)=M y(t)+Nu(t).




To save space hereafter we consider continuous time systems only. It is not difficult to extend all the following arguments from continuous time case to discrete time case.

In Section 4 we have discussed how to solve an OR-system, as long as the  is well defined. It is clear from aforementioned discussions that =_Ψ. That is,  is completely determined by its bridge matrix, which is exactly the “bridge mapping" required in Figure <ref> B.



  * Projection-Based :


A natural way to choose the bridge is

    Ψ=Ψ_p,

where the Ψ_p is defined by (<ref>). This is very natural, because the “bridge" defined by Ψ_p coincides with the project π^p_n:^p^n. This fact reveals the physical meaning of the relationship between projection and DK-STP.

Then we have the following result.
 Consider an SO-system

    ẏ(t)=Mẋ(t)+Nu(t),   y(0)=y_0∈^∞,

where y(t)∈^p, x(t)∈^n. Using the  project bridge matrix (<ref>), its OR-system is

    ẏ(t)=MΠ^p_n y(t)+Nu(t),   y(0_+)=I_p y_0.



We give an example to depict the procedure.


Consider the following SO-system.

    [           ẏ(t)=Mx(t)+Nu(t),; y(0)=(1,2,0,-2,-1,-1)^T∈^6, ]

where

    M=[ -1 -2  3  2 -3; -3  3 -3  2 -3;    ];  
    N=[ 4; 0;   ].

Note that

    Π^2_5=[ 0.4   0; 0.4   0; 0.2 0.2;   0 0.4;   0 0.4;     ],

we have the OR-system as

    [               ẏ(t)                  =      M y(t)+Nu(t),;                                     = MΨ_5× 2y(t)+Nu(t),;                                    :=       Ly(t)+Nu(t), ]

where

    L=MΠ^2_5=[ -0.6  0.2; -0.6   -1 ],

Moreover,

    Π^6_2=[ 1 1 1 0 0 0; 0 0 0 1 1 1 ].

Then,

    y(0_+)=Π^6_2y_0=[  3; -2 ].





  * Least-Square :


In the above project-based approach, we need only the SO-system. Now assume we have more information. Precisely speaking, we know the state-observer mapping

    y(t)=Hx(t),

such as  in Eq. (<ref>) or (<ref>). Then we can look for a “best" linear mapping as the “bridge".
Precisely speaking, we look for a linear mapping

    x(t)=Ξ y(t)

as the bridge.
Then we have

    x(t)=Ξ H x(t).

Since x(t) could be any vectors, (<ref>) leads to an algebraic equation

    I_n=Ξ H.


It is obvious that the least square solution of (<ref>) is

    Ξ= H^+,

where H^+ is the  the Moore-Penrose inverse of H <cit.>.
The physical meaning of (<ref>) is obvious. Roughly speaking, we use a pseudo-inverse of the  mapping of H:x(t) y(t), i.e., Ξ, as the inverse mapping Ξ:y(t) x(t). This is intuitively reasonable.

Particularly, if H is of full row rank,
then we have

    Ξ=H^+=H^T(HH^T)^-1.


Then we give the following definition.

  Define

    Ψ_+=Ψ_n× p:=Ξ=H^+.

The corresponding DK-STP :=_Ψ_+ is called the pseudo-inverse based DK-STP, denoted by
_+. This DK-STP is available on M_n× p.


Then we have the following result.
 Consider a control system

    [                                 ẋ(t)                                    = Ax (t)+Bu(t),   x(t),   x(0)=x_0∈^n,;                                 y(t)                                    =                    Hx(t),   y(t)∈^p. ]

Assume H is of the full row rank, using pseudo-inverse bridge matrix (<ref>), its OR-system is

    ẏ(t)=HAΨ_+y(t)+HBu(t),   y(0_+)=I_p x_0.





 §.§ Singular System


Singular systems have a natural OR structure, which can be described as follows <cit.>. A singular system consists of two parts: differential part and algebraic part. Assume it can be described as

    Eẋ(t)=Ax(t)+Bu(t),
    
    Fx(t)=0,

where E∈ M_r× n, F∈ M_(n-r)× n.

Assume

    H:=[ E; F ],


    y(t):=Ex(t),   z(t):=Fx(t).

The SO-system of (<ref>) can be expressed as

    ẏ(t)=Ax(t)+Bu(t).

Then we try to find Ψ such that (<ref>) can be expressed into OR-system as

    ẏ(t)=A_Ψ y(t)+Bu(t).


In fact, we have

    [ y(t); z(t) ]
    =[ E; F ]x(t)=Hx(t).

Setting z(t)= 0, then we the least square solution:

    x(t)≈ H^+
    [ y(t);    0 ]:=Ψ_+y(t).

Then we have approximated OR-system of (<ref>) as

    ẏ(t)≈ AΨ_+ y(t)+Bu(t).

When H is invertible, (<ref>) becomes an exact OR-system.

 Eq. (<ref>) is a simplified form of singular systems. In general, a singular linear control system is expressed as

    Wẋ(t)=Ax(t)+Bu(t),    x(t)∈^n,

where (W)=r<n. If we decompose it into two parts (differential part and algebraic part), then we have

    Eẋ(t)=Ãx(t)+B̃u(t), 
    
    Fx(t)+Du(t)=0.

Then the above approach is still applicable with a mild change that (<ref>) is replaced by

    x(t)≈ H^+[   y(t); -Du(t) ]:=Ψ_+y(t)+D̃u(t).



Next, we give a numerical example to depict singular control systems.



Consider the singular system (<ref>) with


  (i)

    E=[  1  0  1  0;  0  1  0 -1 ];  
    F=[  1  0 -1  1;  2  1  0  0 ].

Set

    H =
    [ E; F ]=
    [  1  0  1  0;  0  1  0 -1;  1  0 -1  1;  2  1  0  0 ].

Choosing a basis of ((H)) as

    B=
    [  0  1  0;  1  0 -1;  0 -1  1;  1  0  0 ].

It leads to H=BC, where

    C=
    [ 2 1 0 0; 1 0 1 0; 2 0 0 1;   ].



Then, a straightforward computation shows that

    [                                                                                                  H^+=C^T(B^THC^T)^-1B^T; =[  0.225 -0.075  0.125  0.275;   -0.2    0.4      0    0.2;  0.525 -0.125 -0.375 -0.025;   0.05  -0.35   0.25  -0.05 ] ]

The approximate OR-system (<ref>) is obtained with

    Ψ_+=[  0.225 -0.075;   -0.2    0.4;  0.525 -0.125;   0.05  -0.35 ].



  (ii) E is the same as in (<ref>) and

    F=[  1  0 -1  1;  1  1  0  0 ].

Then the corresponding H is invertible. It hollows that (<ref>)  becomes an exact OR-system with

    Ψ=
    [  1  1; -1 -1;  0 -1; -1 -2 ]
































































 §.§ Exact OR-System


In previous subsection A we construct the OR-systems by using Ψ_p and Ψ_+ respectively. As aforementioned that an obvious advantage is: when the original system is of large scale, its OR-system, concerning some particular properties of the original system, may have much smaller size. In subsection B it was shown that the OR-system might be exact. Now a critical question is: does  the dynamics of observers presented by the OR-system coincide with the output dynamics of the original system? This subsection will answer this question.

Consider ^n. It is well known from linear algebra <cit.> that all the linear functions

    f(x):=hx,   h^T, x∈^n,

form an n-dimensional vector space, called the dual space of ^n, and denoted by ^n_*.

Since the linear function f(x) in (<ref>) is uniquely determined by the row vector h,  we simply call h∈^n_* a linear function.

Let H∈ M_p× n. The subspace

    H_*:=((H))⊂^n_*,

is called a dual subspace.

 Let A∈ M_n× n. A dual subspace H_*⊂^n_* is  A-invariant, if

    H_*A⊂ H_*



The following fact is well known from linear algebra.

 Let H_*=((H)) be a dual subspace, where H∈ M_p× n.
H_* is  A-invariant, if and only if, there exists
Ξ∈ M_p× p such that

    HA=Ξ H.



Using Lemma <ref>, we have the following result.

 Consider system (<ref>). There exists an OR-system, which represents the exact dynamics of the observers of system (<ref>), if and only if, H_* is A-invariant.


Proof. (Sufficiency) Assume (<ref>) holds. Then the SO-system of (<ref>) becomes

    [           ẏ(t)              =  HAx(t)+HBu(t);                             = Ξ y(t)+HBu(t). ]

Here (<ref>) is an OR-system. That is, the observers of (<ref>) satisfies this OR-system.

(Necessity) Assume  there exists a =_Ψ such that

    [              ẏ(t)                 =     HAx(t)+HBu(t);                                   =    HA y(t)+HBu(t);                                   = HAΨ Hx(t)+HBu(t). ]

Then we must have

    HAΨ H=HA.

Setting Ξ:=HAΨ leads to (<ref>).





 Suppose H_* is A-invariant. Then by setting

    [ Ξ=HAH^+,;   Ψ=H^+, ]

the corresponding OR-system is exact. Particularly, when H has full row rank, we have

    [ Ξ=HAH^T(HH^T)^-1,;   Ψ=H^T(HH^T)^-1. ]



Proof. We prove it by assuming  H has full row rank. (The general case is exactly the same.) That is,

    H^+=H^T(HH^T)^-1.


It is easy to verify that H_* is A-invariant, if and only if, Ξ=HAH^T(HH^T)^-1 verifies
(<ref>). From the proof of Proposition <ref> one sees that Ψ satisfies

    HAΨ=Ξ=HAH^T(HH^T)^-1.

An obvious solution of (<ref>) is Ψ=H^T(HH^T)^-1.


 Assume H_* is dual A-invariant, the only choice of Ξ is the one  in (<ref>), (or (<ref>) when H has full row rank).
But the solution for Ψ may not be unique. In fact, we have

    Ψ=H^++G

with G∈ M_n× p and G in kernel of HA is also a proper solution. But different choices do not affect the corresponding OR-system.



Consider a control system

    [ ẋ(t)=Ax(t)+Bu(t),;       y(t)=Cx(t), ]

where

    A=[   0  -2   1  -6  -9;  -1  -3   4 -11 -13;   4   1  -1  10  12;   2   1  -2   7   7;  -1   0   1  -2   0;     ];  
    B=[  2;  0;  1;  1; -1;    ];


    C=[  1 -1  1 -2  0; -1  0  0 -1 -2;    ].

Set

    Ξ=CAC^T(CC^T)^-1=
    [  0 -1; -1 -1 ].

Then it is verified that

    CA=Ξ C.

Hence C_*=((C)) is A-invariant. Then we have exact OR-system as

    [                                 ẏ(t)                                    =                        Ξ y(t)+CBu(t);                                                                         = [  0 -1; -1 -1 ]y(t)+[  1; -1 ]u(t). ]






§ EXTENDED OR-SYSTEM


This section considers when the output dual subspace H_* is not A-invariant, how to construct exact OR-system. The basic idea is to enlarge  the output dual subspace to meet the invariant requirement.



 §.§ A-invariant Extended OR-System


 Let H_*⊂^n_*. The smallest dual subspace,  which is A-invariant and contains H_*, is called the A-invariant closure of H_*, and denoted by H_*^(A).





  (i) Let H^i_*, i=1,2 be A-invariant dual subspaces. Then H_*^1∩ H_*^2 is also A-invariant. So H_*^(A) exists. Because

    H_*^(A)=⋂{ S |  S⊃ H_* }.


  (ii) H_*^(A) can be calculated by an iteration as


  (1)

    V_*^0:= H_*.


  (2)

    V_*^k+1= V_*^k+ V_*^kA,   k≥ 0.


  (3)
When the increasing sequence reach k^* such that
V_*^k^*+1= V_*^k^*, then

    H_*^(A)= V_*^k^*.





Since H_*^(A) is A-invariant, we can use it to construct an OR-system.


 Consider system (<ref>). Replacing its observers by H_*^(A), then the OR-system is called its extended OR-system.


We use a numerical example to show how to construct an extended OR-system.


Consider a system Σ characterized by (<ref>) with

    [ A=[  0  1  0  0  0  0;  0  0  1  0  0  0;  1  0 -1  1 -2  1;  0  0  0  0  0  0;  0  0  0  0  1  0;  0  0  0  0  0  0;    ];  
    B=[ 0; 0; 1; 0; 0; 0;   ];;                                                                                                                                         C=[1,0,0,0,1,0]. ]


Let C_*=((C)). Using the algorithm in Remark <ref>, we have

    C_*^(A)=((H)),

where

    H=[  1  0  0  0  1  0;  0  1  0  0  1  0;  0  0  1  0  1  0;  1  0 -1  1 -1  1; -1  1  1 -1  1 -1;    ].

Define

    z=
    [ z_1 z_2 z_3 z_4 z_5 ]^T:=
    Hx.

Then we have the OR-system with respect to C_*^(A) as

    [          ż(t)             = HAx(t)+HBu(t);                          := Ãz(t)+B̃u(t),;          y(t)             =       z_1(t). ]

where

    Ã=
    [  0  1  0  0  0;  0  0  1  0  0;  0  0  0  1  0;  0  0  0  0  1;  0 -1  1  1  0;    ];
    B̃=[  0;  0;  1; -1;  1 ].








 §.§ Feedback Extended OR-System



 Consider system (<ref>), a dual space H_*⊂^n_* is said to be (A,B)-invariant dual subspace, if there exists a state feedback control

    u(t)=Fx(t),

such that H_* is A+BF invariant.


The following results are immediate consequences of the definition.

  H_* is (A,B)-invariant dual subspace, if and only if, there exist F and Ξ, such that

    H(A+BF)=Ξ H.



 Consider system (<ref>). Using bridge matrix Ψ and feedback control

    u(t)=Fx(t)+v(t),

we have approximated OR-system

    [                 ẏ(t)                    =    H(A+BF)x(t) +v(t);                                         ≈ H(A+BF) Ψ y(t)+v(t). ]


If H_* is (A,B)-invariant dual subspace, the approximated OR-system becomes an exact system

    ẏ(t)=   Ξ  y(t)+v(t).



Next, consider how to verify if a dual subspace H_* is (A,B)-invariant.

 H_*⊂^n_* is (A,B)-invariant dual subspace, if and only if,

    H_*^⊥:={x∈^n |  h(x)=0,∀ h∈ H_*}

is (A,B)-invariant.


Proof. Assume H_* is (A,B)-invariant,

    [ H_*(A+BF) H_*^⊥=((H))(A+BF) H_*^⊥;    = ((Ξ H)) H_*^⊥⊂((H)) H_*^⊥=0. ]

That is

    (A+BF) H_*^⊥⊂ H_*^⊥,

which shows H_*^⊥ is (A,B)-invariant.

Conversely, assume H_*^⊥ is (A,B)-invariant,
then (A+BF) H_*^⊥⊂ H_*^⊥. Hence, we also have

    H_*(A+BF) H_*^⊥=0.

That is,

    H_*(A+BF)⊂{ H_*^⊥}^⊥= H_*.

which means H_* is (A,B)-invariant.


 It is well known that a subspace V⊂^n is (A,B)-invariant subspace, if and only if,

    A V= V+ B,

where B=((B)).
Hence, it is a straightforward verification for (A,B)-invariant subspace. Then Proposition <ref> provides a proper way to verify whether a dual subspace is (A,B)-invariant.


Next, we consider the general case, where H_* is not (A,B)-invariant. To get a precise OR-system, we try to find a smallest Σ_*⊂^n_* such that H_*⊂Σ_* and Σ_* is (A,B)-invariant.

 The smallest subspace of ^n_* which contains H_* and is  (A,B)-invariant, is called the (A,B)-invariant closure of H_*, denoted by H_*^(A,B).


 We will show the existence and uniqueness of  (A,B)-invariant closure of H_* by constructing it.

Consider system (<ref>), there is a standard algorithm  to calculate the largest (A,B)-invariant subspace contained in H_*^⊥ <cit.>.
Then we have the following result.

 Consider system (<ref>). Let W⊂^n be the  largest (A,B)-invariant subspace contained in H_*^⊥. Then

    H_*^(A,B)=W^⊥.

 

 Proof. First, since
 
    W⊂ H_*^⊥,

Then

    W^⊥⊃ H_*.

Since W is (A,B)-invariant subspace,  W^⊥ is (A,B)-invariant dual subspace.

Second, if V_*⊃ H_* is the smallest (A,B)-invariant dual subspace, then V_*^⊥ is the largest (A,B)-invariant subspace contained in H_*^⊥. Hence
V_*^⊥=W. That is,

    V_*=W^⊥.



 Consider system (<ref>). The OR-system constructed by using H_*^(A,B)  as the observers is called the feedback OR-system of (<ref>).


 The feedback OR-system of (<ref>) is an exact system, which contains full dynamic evolutionary precess of original observers.


Proof. Since H_*^(A,B) is (A,B)-invariant, the feedback OR-system is exact. Since the original observers are part of the state variables of the   feedback OR-system, their dynamic process is completely covered by the feedback OR-system.



Since the largest (A,B)-invariant subspace contained in the  kernel of H (i.e., H_*^⊥) is unique<cit.>, the  feedback OR-system of (<ref>) on H_*^(A,B) is also unique. It can be considered as a “minimum" exact realization of (<ref>). Then it is interesting to compare it with classical realization of (<ref>) <cit.>:

 Consider the system (<ref>). There exists a coordinate transformation z=z(x), such that (<ref>) can be transformed into the following form, (which is then called the Kalman canonical form):

    [ ż(t)=Ãz(t)+B̃u(t),;       y(t)=C̃z(t), ]

where z(t)=(z^1(t),z^2(t),z^3(t),z^4(t))^T,

    Ã=[ A_11    0 A_13    0; A_21 A_22 A_23 A_24;    0    0 A_33    0;    0    0 A_43 A_44 ];B̃=[ B_1; B_2;   0;   0 ];


    C̃=(C_1,0,C_3,0),

with
(A_11, B_1) controllable and (C_1, A_11) observable. Moreover,

    [ ż^1(t)=A_11z^1(t)+B_1u(t),;   y(t)=[C_1,0,C_3,0] z(t), ]

is a minimum realization.


To compare the minimum realization (<ref>) of (<ref>) with its H_*^(A,B)-based
feedback OR-system, we consider a particular system.


Recall Example <ref>.



The largest (A,B)-invariant contained in the kernel of C_* is (Refer to Appendix 2 for the calculation.)

    V={[ 0; 0; 0; 1; 0; 0 ],
     [ 0; 0; 0; 0; 0; 1 ],
     [  1;  1;  1;  0; -1;  0 ]},

and under the state feedback

    u(t)=Fx(t),

with

    F=(-1,0,2,-1,2,-1)

we have

    (A+BF) V⊂ V.

It follows that  the (A,B)-invariant closure of C_* is

    C_*^(A,B)=((V^⊥)),

where

    V^⊥=[  1  0  0  0  1  0;  1 -1  0  0  0  0;  1  0 -1  0  0  0;    ].


Finally, the
feedback OR-system of (<ref>) with respect to  C_*^(A,B) can be obtained as

    [                                                                                   ẇ(t)                                                                                      =                                                                       V^⊥ẋ(t)+V^⊥Bv(t);                                                                                                                                                                             =                                                                         V^⊥Ax(t)+V^⊥Bv;                                                                                                                                                                             = [  0  1  0  0  1  0;  0  1 -1  0  0  0;  0  1 -1  0  0  0;    ]x(t)+[  0; -1;  0 ]v(t);                                                                                                                                                                             =                           [  0  1  0;  0 -1  1;  0 -1  1;    ]w(t)+[  0; -1;  0 ]v(t),;                                                                                   y(t)                                                                                      =                                                                                w_1(t). ]








  (i) It is easily seen that  (<ref>)-(<ref>) is under the Kalman canonical form. Hence, a minimum realization is

    [                                                   ż(t)                                                      = [  0  1  0;  0  0  1;  1  0 -1 ]z(t)+
    [ 0; 0; 1 ]u(t),;                                                   y(t)                                                      =                                           [1,0,0]z(t). ]

Though this is a minimum realization, which keeps the input-output relationship of the original system unchanged, it does not keep the output functions unchanged. So this minimum realization can not be used for realizing the dynamics of certain chosen functions as what the exact SO-systems can do.


  (ii) Compare Example <ref> with
 Example <ref>, one sees easily that the feedback extended  OR-system has less dimension than the state extended exact OR-system. This is in general true. Because the feedback ones use largest (A,B)-invariant subspace to construct its perpendicular dual subspace, which is the largest feedback invariant dual subspace. The A-invariant dual subspace can be considered as a special (A,B)-invariant dual subspace, which uses feedback control u(t)=Fx(t) with F=0. Hence, it is in general smaller than the largest one.


  (iii) The feedback extended OR-system may simultaneously solve some other control problems. For example,  the disturbance decoupling problem. Consider a linear control system

    [                    ẋ(t)                       = Ax(t)+Bu(t)+_k=1^sξ(t),;                    y(t)                       =                  Cx(t). ]

where ξ_k(t), k∈ [1,s] are disturbances. The disturbance decoupling problem (DDP) is solvable, if and only if, ξ_k∈ V, where V is the largest (A,B)-invariant subspace contained in kernel of Cx <cit.>.

Now if the DDP is solvable, then we claim that all the disturbances will be swept  out from the feedback extended OR-system. This is because

    Cξ_k=0,

and then

    C(A+BF)ξ_k⊂ C(A+BF) V⊂  C V=0.

Continuing this process, one sees easily that

    C_*^(A,B)⊂ V^⊥.

Hence all the disturbances are within  V_*^⊥. The claim follows.

Similarly, if the feedback solves the stabilization problem, i.e., A+BF is a Hurwitz matrix, then the    feedback extended exact OR-system is also asymptotically stable. In fact, the feedback extended OR-system is the smallest subsystem, involving observers.


  (iv) The arguments for continuous time control systems about  extended and feedback extended OR-systems are also applicable for discrete time control systems.





§ OR-SYSTEMS OF NONLINEAR CONTROL SYSTEMS


The above arguments on OR-systems can be extended to nonlinear control systems.  Consider system (<ref>), where x(t)∈ M, where M is an n-dimensional manifold. Then (<ref>) can be considered as an expression over a coordinate chart W with coordinates x(t)∈ W⊂^n and 0∈ W. Hereafter, we assume x(t)∈ W⊂ M. In case that W=^n, x(t) becomes a global coordinate frame.

For statement ease, we assume 0∈ W is a regular point, that is, all the distributions and co-distributions involved below are of constant dimension locally.

Then f(x),g_i(x)∈ V(W) are smooth vector fields. h_j(x)∈ C^∞(W) are smooth functions. We recall some well known knowledge for nonlinear control systems first.

 <cit.>


  (i) Let ⊂ T(W) be a distribution.  is nonsingular if ((x))=, ∀ x∈ W.  is involutive, if for any two vector fields f(x),g(x)∈ V(W),

    [f(x),g(x)]∈.


  (ii) Consider a co-vector field ξ(x)∈ V^*(W). ξ(x) is said to be exact, if there exists an h(x)∈ C^∞(W) such that

    ξ(x)=dh(x).

Denote the set of co-distributions by _*⊂ T^*(W).



 <cit.>


  (i) Let (x)⊂ T(W) be a distribution. The smallest involutive distribution contains (x) is called the involutive closure of (x), denoted by (x).

  (ii)  (x)⊂ T(W) is said to be (f,g)-invariant, if

    _f((x))⊂(x)+ G,

where G=(g_i(x) |  i∈ [1,m]).



Recall system (<ref>), we assume

2mm

Assumption A3:

    h_j(0)=0,   j∈ [1,p].


 Assumption A3 is not essential. It is used to ensure the one-one correspondence between a function h and its differential dh. This is a mimic to linear case.


It is easy to get the SO-system of system (<ref>) as

    ẏ(t)=L_f h(x(t))+_i=1^mL_g_ih(x(t))u_i(t).


Motivated by linear case, it is easy to see that if there exists  a smooth mapping

    ψ:^m W,

then an approximate OR-system can be obtained as

    ẏ=L_fh(ψ(y(t)))+_i=1^mL_g_ih(ψ(y(t)))u_i(t).


Of course, the mappings used for linear case are also the candidates of ψ, say, (i) using projection, we have x(t)=Π^p_n y(t); (ii) using pseudo-inverse, we have x(t)=[dh(x)]^+.

Next, we consider when there exists ψ such that the corresponding OR-system (<ref>) is exact.




  (i)
A co-distribution H_*⊂ T^*(W) is called an exact co-distribution, it there exist h_j(x), j∈ [1,p], such that

    H_*={dh_j(x)   j∈ [1,p]}.

Denote the set of exact co-distributions by E_*⊂_*.

  (ii) A co-distribution  H_*⊂ T^*(W) is said to be f-invariant, if there exists a smooth matrix Ξ∈ M_p× p, such that

    L_f(dH)=Ξ dH.



Note that

    L_f(dh)=dL_f(h),

which leads to

    L_f(dH)∈ E_*.


Recall system (<ref>). Denote by

    H=[ h_1(x(t));         ⋮; h_p(x(t)) ]

Then it is clear that

    H_*=((dH))

is exact.

By definition, the following result is obvious.

 If the exact co-distribution H_* defined by (<ref>) is f and g_i, i∈ [1,m] invariant, then there exists an exact OR-system of (<ref>).


Proof. Assume H_* is f and g_i, i∈ [1,m] invariant, then there exist Ξ_i∈ M_p× p, i∈[0,p], such that

    [               L_f(dH)=Ξ_0 dH,; L_g_i(dH)=Ξ_i dH,   i∈ [1,m]. ]


Note that Ξ_i are smooth matrices, using Assumption  A3, we have

    [               L_f(H)=Ξ_0 H,; L_g_i(H)=Ξ_i H,   i∈ [1,m]. ]


Then the exact OR-system is

    ẏ=Ξ_0y+_i=1^mΞ_iu_i.



Next, we consider the extended OR-system.
Assume H_* is not f and g_i, i∈ [1,m] invariant. Then we need the extended OR-system. To this end, we calculate the smallest exact co-distribution containing H_*, called its {f,g}-closure, and denoted by H^{f,g}_*. We need the following algorithm:

    [                                H^0_*:={dh_j | j∈ [1,p]},; H^k+1_*:= H^k_*+L_f( H^k_*)+_i=1^mL_g_i( H^k_*),   k≥ 0.;                                                          ]

Then it is clear that when H^k^*+1_*= H^k^*_*

    H^{f,g}_*= H^k^*_*.


From the above construction, the following result is obvious.

 Using H^k* to replace H, the obtained exact OR-system is the (smallest) extended OR-system
of the original system (<ref>).


Finally, we consider the feedback OR-system of (<ref>).

 A distribution ⊂ T^*(W) is (f,g)-invariant, if

    _f()⊂+ G,

where G=(g_1,⋯,g_m).


For verifying (f,g)-invariance,  we need the following lemma, which is called Quike lemma <cit.>.


Assume ⊂ T^*(W) is (f,g)-invariant, then there exist a neighborhood 0∈ U⊂ W and a smooth mapping
_i(x)∈ C^∞(U), i∈ [1,m], such that  is f+_i=1^mg_i_i invariant.


Because of this lemma, hereafter we consider only local case, unless  is global f+_i=1^mg_i_i invariant.

 If  ^m⊂ T^*(W) is the largest (f,g)-invariant distribution contained in (H), then  ^m is involutive,


Proof. Assume _m is the largest (f,g)-invariant distribution contained in (H), and X,Y∈_m. Then

    [X,Y](h)=L_XL_Y(h)-L_YL_X(h)=0,  ∀ h∈(H).

That is, [X,Y]∈_m.




 Let  be an f-invariant distribution. Denote by

    ^⊥:=(dξ | ξ∈ C^∞(U), L_X(ξ)=0, ∀ X∈).

Then
^⊥ is f-invariant.


Proof. Let ξ∈^⊥ and v∈. Since  is f-invariant,

    [f,v]:=v'∈.

Hence

    [f,v](ξ)=0,  ∀ξ∈^⊥.

Now

    [f,v](ξ)=L_fL_v(ξ)-L_vL_f(ξ)=L_v(L_f(ξ))=0,   ∀ξ∈^⊥.

That is

    L_f(ξ)∈^⊥,   ∀ξ∈^⊥,

which means ^⊥ is f-invariant.

Using Lemma <ref> and a similar argument as in linear case, we have the following result:

  Let _m be the largest (f,g)-invariant distribution contained in ( H_*). Then
_m^⊥ is the smallest (f,g)-invariant exact co-distribution contains H_*.


The algorithm for the largest (f,g)-invariant distribution contained in (H) is well known <cit.>. Then  _m^⊥ is also straightforward computable.

Then the feedback exact OR-system can be constructed as follows:

 Using _m^⊥ to replace H, the obtained exact OR-system is the (smallest) feedback (extended) OR-system
of the original system (<ref>).




§ CONCLUSION


This paper provides a technique to construct OR-system. First, using cross-dimensional projection and DK-STP, two kinds approximatied OR-systems are proposed respectively. Then we consider when we are able to construct an exact OR-system, which has  exactly the same dynamics of observers as the original system.
When it is not able to construct exact OR-system, we propose to construct the extended OR-system, which has the original system's observers as part of its state variables. Then the (minimum) state feedback (extended) OR-system is constructed.
Finally, all the above arguments for linear control systems have been extended to affine nonlinear control systems.

The purpose of this paper is to provide a method to deal with large complex systems.
 When a large or huge scale complex system, such as the universe, the nature, the internet, or the neural network of brain, etc., is considered, the model of overall state space is too complicated to be analyzed or even modeled. Then we may be concentrated on certain observed phenomenon, which forms the OR-system of the original system. Through investigating various OR-systems of the original system, we are able to understand or even manipulate the original complex system.

 It is a common opinion that physical theory is only a mathematical model of observations. For instance, the famous physician Hawking has said that “I don't demand that a theory correspond to reality because I don't know what reality is .... I take the positivity viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds the reality. All the one can ask is that its predictions should be agreement with observations.... All I'd concerned with is that the theory should predict the results of measurements."<cit.> That might be considered as the motivation of this work.



§ APPENDIX




 §.§  Notations:





  * ^n: n dimensional real Euclidian space.


  * ^n_*: dual space of ^n.


  * M_m× n: set of m× n matrices.


  * M_m× n: extended ring  of m× n matrices by adding identity.


  * [a,b]: the set of integers {a,a+1,⋯,b}, where a≤ b.


  * (n,p): the least common multiple of n and p.


  * _̣n^i: the i-th column of the identity matrix I_n.


  * _n:={_̣n^i| i=1,⋯,n}.


  * _k:=(1,⋯,1_k)^T.


  * L_m× n: the set of logical matrices, i.e.
L_m× n={ A∈ M_m× n | (A)⊂_m}.


  * (A): the set of columns of matrix A, and _i(A) is the i-th column of A.


  * (A): the set of rows of matrix A, and _i(A) is the i-th row of A.


  * (·): subspace or dual subspace generated by ·.


  * ⊥: perpendicular relation between x∈^n and h∈^n_*.


  * ⋉: semi-tensor product of matrices.


  * : dimension-keeping semi-tensor product of matrices.


  * V_*^(A): A-invariant closure of the dual subspace V_*⊂^n_*.


  * V_*^(A,B): (A,B)-invariant closure of the dual subspace V_*⊂^n_*.


  * T(M): tangent space of manifold M.


  * T*(M): cotangent space of manifold M.


  * V(M): set of smooth vector fields on M


  * V^*(M): set of smooth covector fields on M.


  * ⊂ T(M): distribution on M.


  * _*⊂ T^*(M): co-distribution on M.


  * E_*⊂ T^*(M): exact co-distribution on M.


  * _f(g)=[f,g]: the Lie bracket of f,g∈ V(M).





 §.§ Calculation of (A,B)-invariant subspace (for Example  <ref>)


First, we need the algorithm(<cit.> page 95):




  (i)
Set V_0=X.

  (ii)

    V_k=X∩ A^-1( B+ V_k-1),   k≥ 1.


  (iii) If V_k^*+1= V_k^*, stop.

Then the largest (A,B)-invariant subspace contained in X  is V_k^*.


Note that in the above algorithm

    A^-1( S):={x∈^n | Ax∈ S}.


To calculate  A^-1( S), we propose the following lemma.




    A^-1( S)=[A^T  S^⊥]^⊥,

where A=((A)).


Proof.
Let x∈ A^T S^⊥. Then there exists y∈ S^⊥, which means s^Ty=0, ∀ s∈ S, such that x=A^Ty. Let z∈ A^-1( S), i.e., Az=s'∈ S. So, z^Tx=z^TA^Ty=(s')^Ty=0. That is, x∈ ( A^-1 S)^⊥. Hence,

    (A^T S^⊥)^⊥⊃ A^-1( S).


Conversely, assume x∈ (A^T S^⊥)^⊥, then x^TA^Ty=0, ∀ y∈ S^⊥. That is,
Ax∈ ( S^⊥)^⊥= S. Then x∈ A^-1( S). It follows that

    (A^T S^⊥)^⊥⊂ A^-1( S).



Using algorithm <ref>, we set

    V_0:= C^⊥=((V_0)),

where

    V_0=[ -1  0  0  0  0;  0  1  0  0  0;  0  0  1  0  0;  0  0  0  1  0;  1  0  0  0  0;  0  0  0  0  1;    ]


    S_0= B+ V_0= V_0.


    S^⊥_0=(([ 1; 0; 0; 0; 1; 0 ]) ).


    A^T S^⊥_0=(([ 0; 1; 0; 0; 1; 0 ]) ).


    [                                                                                  A^-1( S_0)=(A^T S_0^⊥)^⊥; =(([  1  0  0  0  0;  0  0  0  1  0;  0  1  0  0  0;  0  0  1  0  0;  0  0  0 -1  0;  0  0  0  0  1 ]) ). ]


    [                                                                    V_1= V_0⋂ A^-1( S_0); =(([  1  0  0  0;  1  0  0  0;  0  1  0  0;  0  0  1  0; -1  0  0  0;  0  0  0  1 ]) ). ]


    S_1= B+ V_1= V_1.


    S^⊥_1=(([  1  1; -1  1;  0  0;  0  0;  0  2;  0  0 ]) ).


    A^T S^⊥_1=(([  0  0;  1  1; -1  1;  0  0;  0  2;  0  0 ]) ).


    [                                                                A^-1( S_1)=(A^T S_1^⊥)^⊥; =(([  1  0  0  0;  0  0  0  1;  0  0  0  1;  0  1  0  0;  0  0  0 -1;  0  0  1  0 ]) ). ]


    [                                                  V_2= V_1⋂ A^-1( S_1); =(([  0  0  1;  0  0  1;  0  0  1;  1  0  0;  0  0 -1;  0  1  0 ]) ). ]


    S_2= B+ V_2.

It is easy to see that

    S_2= S_1.

It follows that

    V_3= V_2,

which is the largest (A,B)-invariant subspace in C_*^⊥.



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