On Normal Forms of Nonlinear Systems Affine in Control - IEEE Xplore

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On Normal Forms of Nonlinear Systems Affine in Control Xinmin Liu, Member, IEEE, and Zongli Lin, Fellow, IEEE

, and

Abstract—The nonlinear equivalences of both finite and infinite zero structures of linear systems have been well understood for single input single output systems and have found many applications in nonlinear control theory. The extensions of these notions to multiple input multiple output systems have proven to be highly sophisticated. In this paper, we propose constructive algorithms for decomposing a nonlinear system that is affine in control. These algorithms require modest assumptions on the system and apply to general multiple input multiple output systems that do not necessarily have the same number of inputs and outputs. They lead to various normal form representations and reveal the structure at infinity, the zero dynamics and the invertibility properties, all of which represent nonlinear equivalences of relevant linear system structural properties.

in a neighborhood of

Index Terms—Geometric approach, infinite zero structure, invertibility, nonlinear systems, normal forms, structural properties, zero dynamics.

where , , and is the zero dynamics. With a state feedback, this normal form reduces to the zero dynamics cascaded with a clean chain of integrators linking the input to the output. Here by clean we mean that no other signal enters the middle of the chain. Such a nice feature is extended to nonlinear systems with more than one input output pair. That is, a special class of square can be transformed invertible nonlinear system with into the zero dynamics cascaded with clean chains of integrators. To do this, the notion of vector relative degree was introhas a vector duced in [25], [26]. System (1) with at if relative degree

I. INTRODUCTION HE nonlinear analogues of linear system structural properties, such as relative degrees (or infinite zero structure), zero dynamics (or finite zero structure) and invertibility properties, have played critical roles in recent literature on the analysis and control design for nonlinear systems (see, e.g., [1]–[19] and the references therein for a sample of this literature). The normal forms that are associated with these structural properties, along with the basic tools like those reported in [20]–[24], have enabled many major breakthroughs in nonlinear control theory. Consider a multiple input multiple output (MIMO) nonlinear system of the form

T

(3) If system (1) has a relative degree , then on an appropriate set , it takes the following of coordinates in a neighborhood of normal form (see, e.g., [12])

(4)

(5) in a neighborhood of

, and

(1)

(6)

, and are the state, input and where output, respectively. Let the mappings , and be smooth containing the origin , with in an open set and . Denote and col . in (1), A single input single output system, i.e., if has a relative degree at

at If system (1) has a vector relative degree , then with an appropriate change of coordinates, it can be described by

(2)

which contains clean chains of integrators. Moreover, if the is involutive distribution spanned by the column vectors of , a set of local coordinates can be in a neighborhood of selected such that . The clean chains of integrators are called a prime form in [27] for linear systems, and the necessary and sufficient geometric conditions for the existence of prime forms for nonlinear systems is developed in [28]. There is a large body of nonlinear systems and control literature based on the form (7) (see e.g., [29]–[34], for a small sample).

Manuscript received December 22, 2008; revised October 23, 2009; accepted May 22, 2010. Date of publication June 01, 2010; date of current version February 09, 2011. Recommended by Associate Editor D. Liberzon. X. Liu is with the Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597 USA. Z. Lin is with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2010.2051634

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The conditions for the existence of a vector relative degree, (5) and (6), though similar to (2) and (3) in form, are not easy to be satisfied. For example, a simple change of coordinates in the output space could undermine property (5). In general, the vector relative degree is a rather restrictive structural property that not even all square invertible linear systems, with the freedom of choosing coordinates for the state, output and input spaces and state feedback, could possess. A square invertible in general can only be translinear system with chains of informed into the zero dynamics cascaded with tegrators, with all but one chains containing output injection terms (see [35]). That is, there are interconnections between these chains. For example, consider a linear system with

(8) The system contains two chains of integrators of lengths 1 and 3. The parameter represents an output injection term, which in turn represents the interconnections between the two chains. Such an interconnection cannot be removed through coordinate transformations and state feedback, and thus system (8) cannot be represented by two clean chains of integrators. To see this, suppose that there exist nonsingular coordinate transformations , and such that

which indicates that the system can be decoupled into two clean chains of integrators. Denote . By and , we obtain . The entry of is . So, , consequently, is singular. This is a contradiction. A major generalization of the form (7) was made in [8], [24], [36], [37], where MIMO square invertible systems are considered. In [8], with the Zero Dynamics Algorithm, a sequence of are nested submanifolds defined, and system (1) is transformed into the form

(9)

, and , , and the static state feedback , , with the matrix being smooth and nonsingular. In the algorithm, the ranks of certain matrices are assumed to be where , is given by

,

, in

constant on these nested submanifolds. With some stronger assumptions imposed in the algorithm [24], [36], i.e., the ranks of certain matrices are assumed to be constant for all (not just in these submanifolds), one can have all . Moreover, if certain vector fields commute, one can select coor. Thus, system (9) becomes dinates such that

(10) The applications of the form (10) in solving the problems of asymptotic stabilization, disturbance decoupling, tracking and regulation can be found in [24] and the references therein. The infinite zeros of a linear system can be defined either through the root locus theory or as the Smith-McMillan zeros of the transfer function at infinity [38], [39]. They can also be characterized in state-space [27], [40]. On the other hand, the structure at infinity was introduced for a certain class of nonlinear systems in [41], and was further developed for smooth systems or analytic systems in [42] and for meromorphic systems in [43]–[45]. In [42] and [5, Ch.9], formal zeros at infinity are defined in terms of a set of geometric conditions. In par, ticular, for system (1) defined on a smooth manifold a sequence of the locally controlled invariant distribuin are defined, tions , where , , and where , denotes the set of smooth vector fields on a smooth denotes the tangent bundle of . Under manifold , and the assumption of the distributions and on having constant dimensions, the formal zeros at infinity can be defined. Formal zeros at infinity play an important role in the input output decoupling problem by static state feedback, in which after possible relabeling of inputs, the control does not . But this structure information influence the output , does not show in a normal form in [5], [42]. In [11], [45], a linear-algebraic strategy is developed based on the use of vector spaces over the field of meromorphic functions. As a counterpart to the above differential-geometric approach, the algebraic approach considers system (1) with , and being meromorphic. Except for some singular points, the two approaches lead to the same results as in [46], in particular, the same notions of rank and structure at infinity. The structure at , infinity is related to a chain of subspaces , , where , and where are meromorphic functions. The struc. ture at infinity is then determined by With a generalized state space transformation, a regular generalized state feedback and a universal additive output injection, system (1) can be transformed into a canonical form, which contains time derivatives of inputs and shows the structure at infinity explicitly.

LIU AND LIN: ON NORMAL FORMS OF NONLINEAR SYSTEMS AFFINE IN CONTROL

As pointed out in [8], if all , the set of integers in (10) corresponds to the vector relative degree, which in this case, represents the infinite zero structure if the system is linear. These integers however are not related to , the infinite zero structure of linear systems when and thus cannot be defined as the nonlinear equivalence of and expected to play a similar role as infinite zeros. To see this, con: sider the following linear system

(11) which is in the form of (10) with and . However, according to [27], [35], the system is invertible with two infinite . Therefore, the integers and in the form (10) zeros do not generalize the notion of infinite zero structure of linear systems. Invertibility of linear systems was first studied in [47]–[49]. In these references, inversion algorithms and invertibility criteria are given. Invertibility of nonlinear control systems was considered in [1], [3], which generalized the structure algorithm for linear systems [48]. References [46], [50], and [51] carry out a systematic study of invertibility of general nonlinear systems that are not necessarily affine in control. The authors gave a list of equivalent conditions for the right and the left invertibility of linear systems, and examined when and how these conditions can be generalized to nonlinear systems. Based on [3], reference [16] explicitly constructs the left inverse of an affine output-input stable system. Invertibility of nonlinear systems can also be determined by using the structure algorithm in [44]–[46]. A key feature of the normal forms is that they represent a system in several interconnected subsystems. These subsystems, along with the interconnections that exist among them, lead us to a deeper insight into how control would take effect on the system, and thus to the construction of control laws that meet our design specifications. The structure of a linear system has been studied in great depth. In 1973, Morse [27] showed that, under a group of state, input and output transformations, state feedback and output injection, any matrix triple is uniquely characterized by three lists of positive integers and a list of monic polynomials. By identifying state variables in the structure algorithm in [48], Sannuti and Saberi [35] explicitly constructed state, input and output transformations that transform a general MIMO system, not necessarily square, into a so-called special coordinate basis form, which displays all structural properties of the system, including the finite and infinite zero structures and invertibility properties. Motivated by the many efforts reported in the nonlinear control literature and a complete understanding and numerous applications of the structural decomposition of linear systems, we make an attempt to study structural properties of affine nonlinear systems beyond the case of square invertible systems. For a general nonlinear system (1) in the absence of the vector relative degree assumption, we develop an algorithm, which is re-

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ferred to as the infinite zero structure algorithm and, under certain constant rank assumptions over , results in diffeomorphic state, input and output transformations and state feedback laws under which the system can be represented in normal forms. In the special case when the system is square and invertible, our normal forms take a form similar to those in [24], [36], but with an additional property that allows the normal forms to reveal the nonlinear equivalence of infinite zeros of linear systems. In addition, our development enhances the existing results in some other ways. First, fewer assumptions are required. Second, the resulting normal forms explicitly show invertibility structures and nonlinear equivalence of invariant zeros. Third, our development applies to general MIMO nonlinear systems that are not necessarily square. The infinite zero structure algorithm will also be adapted to develop normal forms that reveal system structural properties when the output is restricted to zero. The adapted algorithm will be referred to as the zero output structure algorithm. The assumptions required will also be in the form of constant ranks, but in a sequence of nested subsets, rather than the more stringent constant ranks on as required by the infinite zero structure algorithm. Our results on zero output normal forms inherit the features pertaining to the infinite zero structure algorithm and thus enhance the existing results on the zero output normal forms in similar ways as the normal forms resulting from the infinite zero structure algorithm. The remainder of this paper is organized as follows. The infinite zero structure algorithm and the resulting normal forms are presented in Section II. The zero output structure algorithm and the resulting normal forms are given in Section III. Section IV contains a few examples that illustrate the main results of the paper. A brief conclusion to the paper is drawn in Section V. For clarity in the presentation, all proofs are given in the appendices. II. NORMAL FORMS AND STRUCTURAL PROPERTIES OF NONLINEAR SYSTEMS In this section, we will find diffeomorphic state, input and output transformations and static state feedback laws under which system (1) can be represented in normal forms and discuss about the intrinsic structural properties these normal forms reveal. A. Infinite Zero Structure Algorithm Both the infinite zero structure algorithm and the algorithm in [24], [36] involve repetitive differentiations of the output and, under certain constant rank assumptions, identification of functions to serve as new state variables. What distinguish the infinite zero structure algorithm are how we select the output for further differentiations and how we identify the state variables. Initial Step. Let

,

,

and

Step . We start with row rank Assumption for

, , where the matrix has full . Suppose that the following assumption holds. : The matrix

, and there exists an

has constant rank

.

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such that the matrix for

next step and from which new state variables will be selected, , which contains and part of , but also is of full row rank and in such a way that

is of full row rank

.

Let

be such that (12)

Denote (13) The matrix

has full row rank

and

Thus, there exist unique smooth functions

such that (14) Define

B. Infinite Zeros (15)

If and , then increase by 1 and repeat the above step. Otherwise, go to Final Step. Final Step. Let

More specifically, we first identify , then define based on and , rather than on , . Such an approach will be helpful in selecting state variables that render the more informative normal forms. , which plays We note that in the algorithm of [24], [36], , is defined based on , which plays a similar role as , . a role similar to collective role of in such a way, Moreover, by choosing the function we are able to carry out the algorithm with fewer constant rank assumptions than the algorithm in [24], [36], and more importantly, allow the algorithm to be applicable to square but non-invertible systems and non-square systems. In [24], [36], the additional constant rank condition is assured of . We also device criteria for the above repetitive procedure to stop. The times the derivatives are taken on each output variable and which stopping criterion is met determine the structure at infinity and the invertibility properties, respectively.

, we have (16)

Let

We now extend the linear system notion of infinite zeros to nonlinear systems. Definition 2.2: Suppose that system (1) is regular. The infinite zeros of the system are the set of integers as identified in the infinite zero structure algorithm. Roughly speaking, each integer in the set represents a chain of integrators of length connecting an input and output pair. We will justify Definition 2.2 as an extension of the linear system notion of infinite zeros to nonlinear systems by showing that the set is invariant under diffeomorphic state, input and output transformations, and static state feedback. in Consider a diffeomorphic state transformation , we have (17)

Denote the set

. Define a set of integers as

where

(18) End. Definition 2.1: System (1) is said to be regular, if Assumption , , are satisfied. In each step of the infinite zero structure algorithm, we identify not only , which will be further differentiated in the

Following the infinite zero structure algorithm, it is easy to verify the following result. Lemma 2.1: If system (1) is regular, then system (17) is regular too. Moreover, both systems have the same infinite zeros. We also have the following result.

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Lemma 2.2: The infinite zeros of system (1) are invariant under with 1) input transformation being smooth and nonsingular; with being 2) output transformation nonsingular; with 3) static state feedback being smooth. Proof: See Appendix A. We will further justify Definition 2.2 by applying the infinite zero structure algorithm to linear systems (see Theorem 2.3), as obtained in the infinite zero the set structure algorithm coincides with the infinite zeros of this linear system as defined in [27], [35]. C. Normal Forms We will base on the infinite zero structure algorithm to derive normal forms of system (1). Denote

(19) By (14) and (15)

(20) For the notational brevity, denote as , for . We first define the new states reprewith senting the dynamics of integrators, which connect to the output the input

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is of full row rank for Assumption : The matrix . , Note that Assumption is automatically satisfied if , , in the infinite zero structure algorithm are independent of . Lemma 2.3: Suppose that system (1) is regular, and that , , , in the infinite zero structure algorithm are constant matrices. Then, the matrix is of full row rank for . Proof: See Appendix B. By the infinite zero structure algorithm, we know that is of full row rank. Note that , , are the coefficients in . is of full row rank. In what folSo, under Assumption , ’s with lows, we augment the state variables additional state variables to form a full set of state variables for the system. Similarly, we also need to augment the input ’s with advariables ’s and the output variables additional output variables ditional input variables and to form a full input vector and output vector, respectively. Note that contains states, and thus , , define chains containing a total of integrators. If , we introduce the to reorder the states such permutation matrix integrators and corresponds that each chain contains to only one input and one output, where with

identity matrix

and . Define

being the -th column of the

(25) where Note that have

. In view of (19) and (20), we

Denote

(21) Note that if a new set of coordinates

Let

for

, then

. Define

(22) (26) (27)

(23) (24) It is obvious that , and . To construct a new set of coordinates, we need the following assumption.

(28) where a diffeomorphism on

is smooth and such that ,

is is

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smooth and such that the matrix is nonsingular, and is such that the constant matrix is nonsingular. Denote

Let

be smooth functions and

.. .

.. .

..

,

.. .

.

.. .

By (21), the dynamics does not relate to the state feed. Inequality (31) follows from this fact. backs with Remark 2.1: The results of Theorem 2.1 are applicable to general MIMO systems that are not necessarily square. For square and invertible systems, the form (30) is in the same form as the one derived in [24], [36], where no vector relative degree assumption is required either. However, normal form in Theorem 2.1 possesses an extra property (31) (see Corollary 2.1 and Example 4.1). Such a property plays a key role in defining the nonlinear equivalence of the infinite zeros of linear systems. In what follows, we further simplify the normal form in Theorem 2.1. in (27) such Assumption : There exists a that the distribution spanned by the column vectors of is involutive. Theorem 2.2: Suppose that the conditions in Theorem 2.1 and Assumption are satisfied. Then there exists a set of coordinates in such that the system takes the form of Theorem 2.1 with

Define (32) (29) where

.. .

.. .

..

.

.. .

.. .

We have the following result. Theorem 2.1: Suppose that system (1) is regular, and that be as obtained Assumption holds. Let in the infinite zero structure algorithm. There exist a set of coordinates in , i.e., diffeomorphic state, input and output transformations, such that the system takes the following form:

(30)

where

,

Proof: See Appendix C. Let us apply the infinite zero structure algorithm to a linear , i.e., system (1) with , system and . It is obvious that Assumptions , , and automatically hold. . There Theorem 2.3: Consider a linear system exist nonsingular state, input and output transformations, and state feedback such that the system takes the form

, with the matrix being nonsingular for ,

and (31)

(33) does not contain dywhere the subsystem namics that is simultaneously controllable and observable, , , with being nonsingular, and , for , . The form (33) can be achieved by some additional state transformations on the linear counterpart of (30). In (33), the dynamics of depends only on and , , only depends on . It can be verified that the finite and zeros are given by the simultaneously uncontrollable and un. The infinite zeros are observable dynamics of . The system is left invertible if is absent, is absent, invertible if both and are right invertible if and are present. absent, and degenerate if both Some remarks on the infinite zero structure algorithm and normal forms are given as follows. Remark 2.2: In the infinite zero structure algorithm, there is only one constant rank assumption in each step, while in the

LIU AND LIN: ON NORMAL FORMS OF NONLINEAR SYSTEMS AFFINE IN CONTROL

constrained dynamics algorithm [5] and the zero dynamics algorithm [8], [24], [36], each step involves two constant rank assumptions (see the explain after Definition 2.1). However, in the infinite structure algorithm, to construct a new set of coordinates, Assumption is needed. Assumption automatically holds if certain matrices are constant (see Lemma 2.3). Remark 2.3: The infinite zero structure algorithm stops at when (16) is satisfied. Carrying on the algorithm furStep , for . This ther would not increase . That is, . Suppose can be seen in two cases. Case 1: such that . there exists a is a full row rank Then, by the algorithm, matrix and . This is a contradiction. Case 2: . Suppose there exists a such . Then, that . However, it can be easily verified that is a matrix with a full row rank. This is also a contradiction. Remark 2.4: As observed in [8], it is in general difficult to construct a set of coordinates such that (32) is satisfied. It entails partial differential equations. the solution of a system of , , However, in the special case that in (30) are independent of , i.e., is a constant, , the by renaming the state variable in (30) disappears under the new set of term coordinates. Remark 2.5: The variables , , , constitute all the states associated with , the structure at infinity. Note that for some with is not defined. For each , the states , , form chains of integrators, and each chain contains integrators. However, except for the smallest such that , in which form clean chains of integrators that link the transformed inputs to the transformed , for each remaining with , the equaoutputs tions governing the states represent chains of integrators injected into with the previous transformed inputs . the integrators with

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, . Suppose is the locally maxfor each , imal zero output submanifold with represents the tangent space to at . where Then, there exists a unique smooth mapping such that the vector field is tangent to . The pair is called the zero dynamic of (1). for a square invertible nonlinear The global version of system is defined in [24], [36] as a controlled invariant smooth . embedded submanifold of Here, we want to use the form (30) to derive the zero dynamics of general nonlinear system in . In particular, let in (30). It then follows from the dynamic equations that and , . Consequently, the remaining dynamics reduces to (34) Let be the smallest distribution that is invariant for (34) and contains the distribution spanned by the column vectors of , and be the smallest codistribution that is invariant for (34) and contains the codistribution spanned by the row . Note that the distribution characterizes vectors of local strong accessibility and the codistribution characterizes local observability. The subsystem (34) does not contain any dynamics that is both strong locally accessible (by ) and locally observable (through ). Otherwise, the infinite zeros . Thus by [5], [8], we have are no longer the following result. Lemma 2.5: Consider system (34). Assume that the distributions , and of (34) each has a constant dimension. Then there exist a set of coordinates such that (34) takes the form

(35)

D. Invertibility and Zero Dynamics

with

Equation (16) in the infinite zero structure algorithm indicates the invertibility property of the system. Lemma 2.4: System (1) is left invertible if , , invertible if , and right invertible if . degenerate if Equivalently, the system in Theorem 2.1 is left invertible if is absent, right invertible if is absent, invertible if both and are absent, and degenerate if both and are present. In [8], the zero dynamics of a nonlinear system is defined for be a smooth cona square invertible nonlinear system. Let is said to be locally nected submanifold of . The manifold controlled invariant at if there exist a smooth mapping such that is locally invariant under the vector field . A zero output submanifold in a neighborhood for the nonlinear system (1) is a smooth connected subof manifold , which is locally controlled invariant at and

. The decomposition (35) allows us to decompose normal form (30) into four distinct subsystems (see Example 4.2) as we can do in a linear system ([27], [35]). In a generalization to the notion of invariant zero of linear systems [35], the dynamics is referred to as the zero dynamics of system (1). has been studied in [8], [36]. The case of is In this case, and are absent from (30), and directly obtained as the zero dynamics of system (1).

and

E. Normal Forms of Square Invertible Systems We now consider the normal forms of system (1) with , i.e., a square invertible system, which has been considered in [8], [24], [36]. In this case, and do not exist and we have the following result, as a corollary to Theorems 2.1 and 2.2.

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Corollary 2.1: Suppose that a square invertible system (1) has , and Assumption holds. infinite zeros Under a new set of coordinates, the system takes the form,

(36) where

with the being nonsingular, and

matrix

III. NORMAL FORMS OF NONLINEAR SYSTEMS RELATING TO THE ZERO OUTPUT In determining the zero dynamics, a normal form representation of the nonlinear system is also given in [8], which displays structure information along one special output trajectory, the zero output. Two constant rank assumptions are made in the , . nested submanifolds Here, we will show that the infinite zero structure algorithm can be adapted for the same problem. In particular, for system (1), we will introduce Assumption , , in the , , rather than for all . nested subsets Because the nested subsets , , are related to the zero output, we refer to the resulting algorithm as the zero output structure algorithm.

(37) Zero Output Structure Algorithm If, in addition, the distribution spanned by the column vectors of is involutive for , then there exist a set of coordinates , . such that Note that the form given in Corollary 2.1 is the same as (10) except for the additional structural property (37). The equations in (36) display a triangular structure of the control inputs that enter the system. Property (37) imposes an additional structure within each chain of integrators on how control inputs enter the system. With this additional structural property, the set represents infinite zeros when the system is linear. To see the significance of property (37), we transfer system (11) into the normal form in Corollary 2.1

,

Initial Step. Let

,

and

.

Step . We start with and , where the matrix has full row rank in , with and being a . neighborhood of Assumption

: The matrix

in such that

, and there exists an

has a constant rank

(38) is the connected component of

where containing

.

Suppose that Assumption is satisfied. Let and be has full row rank as in (12)–(13). Thus, the matrix for , and by using the following state and output transformations and state feedback: Therefore, there exist smooth functions , , such that (39) where in

is a matrix valued smooth function with . Denote (40)

It is obvious that and , which coincide with the infinite zeros of this linear system (see, e.g., [27], [35]). Remark 2.6: The normal form in Corollary 2.1 can be further simplified by using the method in [24], [36]. Under the assumption that certain vector fields commute, there exist a set of coordinates such that the dynamics of in Corollary 2.1 simplifies to .

and define

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If and , then increase by 1 and repeat the above step. Otherwise, go to Final Step.

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We next consider system (1) with in the zero output structure algorithm, which has been considered in do not exist and we have [8], [24], [36]. In this case, and the following result. Corollary 3.1: Suppose that the conditions in Theorem 3.1 hold with . Then, there exist a set of local coordinates such that the system takes the form

Final Step. The same as the final step in the infinite zeros structure algorithm in Section II-A. End. Definition 3.1: The point is said to be a regular point , , in the zero of system (1) if Assumption output structure algorithm are satisfied. and are Note that in Step , the choice of the matrices not unique. Lemma 3.1: Suppose that and , , are , , and . Then different choices yielding

(45)

where , with

, , in

, and , being nonsingular, and

,

(42) (46) where is a nonsingular matrix valued smooth function, and is smooth with in . Proof: See Appendix D. The following result follows directly from Lemma 3.1. Lemma 3.2: The set , and hence the set , as identified in the zero output structure algorithm are invariant with respect to the choice of matrices and , . , and as in (22)–(24). We have the Define following crucial result. Lemma 3.3: Let be a regular point of system (1). Then, , and are of full row rank. Proof: See Appendix E. Theorem 3.1: Consider system (1). Suppose that is a be as obtained in the regular point. Let zero output structure algorithm. There exist a set of coordinates, i.e., diffeomorphic state, input and output transformations, such that the system assumes the following form:

(43)

where

, , , being nonsingular, and

, in , with

, and ,

,

The submanifold

is given as

Remark 3.1: The normal form in Corollary 3.1 is the same as the result in [8, Chapter 6], except that there is property (46) here. Remark 3.2: The zero output structure algorithm requires milder regularity assumptions than the infinite zero structure algorithm. For example, consider

By the infinite zero structure algorithm, the system is not regular, since

does not have a constant rank in a neighborhood of . is a However, by the zero output structure algorithm, regular point with and . Remark 3.3: We have similar results as in Lemmas 2.1, 2.2 and 2.5 for zero output structure algorithm. If the point of system (1) is regular, then the point of system (17) is regular too. The set of the integers as identified in the zero output structure algorithm are invariant under the state, input and output transformations, and state feedback. The zero dynamics can be computed similarly as in Lemma 2.5. IV. EXAMPLES

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Examples 4.1 and 4.2 illustrate the infinite zero structure algorithm, and Example 4.3 illustrates the zero output structure algorithm.

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Example 4.1: Consider system (1) with

The system is defined globally, i.e., infinite zero structure algorithm. Step 1.

,

. Let

. Thus, .

Step 2. . Let

,

Initial Step. Let

,

and

,

Step 1. . Let

,

. Hence, . Thus, .

,

,

, , .

, . Let

,

.

,

,

Thus,

Step 3.

,

,

,

,

.

,

and ,

. Find

We obtain

.

. .

,

,

.

,

such that

, i.e.,

, . Let

. Hence, Final Step. .

.

. Thus, ,

,

.

,

. So, ,

.

. Let

Hence,

,

,

,

Final Step. , . It is obvious that . Let

,

,

Step 2.

.

, . Thus,

, ,

Step 3. So,

. So,

, ,

,

and . We carry out the infinite zero structure algorithm as follows.

. We apply the

. . Let . Thus

.

Let

with , and . We take the following , and further transformation on . Let . Then, the system takes the following form: The form (30) is given by

The system is invertible with two infinite zeros of order 2 and 3. . Note The zero dynamic degenerates to the single point that . Example 4.2: Consider system (1) with

with . The zero dynamics is . It is also clear from the normal form above that the system has an infinite zero of order 1 and is not invertible. Example 4.3: Consider system (1) with

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It is obvious that and output structure algorithm as follows.

. We carry out the zero

.

Step 1. Let

. Let

Final Step. .

,

,

and

, we have

, .

. . Let

. Hence, , ,

. Then

Letting

,

,

Step 2. Let Thus,

Let

,

. Hence, . Thus, ,

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,

, . .

, and

The distribution spanned by the column vectors of is not involutive. Define , and . Consequently, in , the form (43) is given by

By (14) and (15)

Similarly, letting . Thus, Similarly, letting , Thus,

and

, we obtain

and

, we obtain

. . APPENDIX B PROOF OF LEMMA 2.3

, and . We want to show that all row vectors in the following list are linearly independent: Denote

.. .

By letting

, we obtain the zero dynamics

.

V. CONCLUSION We have presented constructive algorithms for decomposing an affine nonlinear system into its normal form representations. Such algorithms generalize the existing results in several ways. They require less restrictive assumptions on the system and apply to general MIMO systems that do not necessarily have the same number of inputs and outputs. The resulting normal forms reveal various nonlinear equivalences of linear system structural properties. These algorithms and the resulting normal forms are thus expected to facilitate the solution of several nonlinear control problems. Initial results have shown that these normal forms simplify the conventional backstepping design and motivate new backstepping design procedures that are able to stabilize systems on which the conventional backstepping is not applicable [52].

.. .

.. .

..

.

The rows of are linearly independent since the matrix is of full row rank. Next, we show that the rows of , and are linearly independent. To do this, consider

(47) By row operation, the right hand side of (47) can be transformed to

Considering

, we have

APPENDIX A PROOF OF LEMMA 2.2 1) and 2) are obvious from the infinite zero structure algorithm. 3) Apply the infinite zero structure algorithm to the closedloop system

Therefore, (47) is of full row rank for . Hence the row vec, , and are linearly independent. tors Similarly, the row vectors of are linearly independent.

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APPENDIX C PROOF OF THEOREM 2.2 Note that (48) are linearly independent for Thus, the column vectors of . By Frobenius’ Theorem, there exist real-valued such that the rows of functions are linearly independent and (49) Thus,

spans the kernel space of . Suppose that

. By (39), we have . is nonsingular, has full row rank , and hence its rows form a basis of the solution in . By (41), we have space of . Assume that Since

and the rows of in

form a basis of the solution space of . By (39) and (41), we

have

satisfies Hence

Considering

(49)

and , we have , . Thus, . where is formed from In view of (48) and the fact that , has full column rank, some rows of implying that and hence . Therefore, and the space spanned by the row vectors of has a dimension of . elements from Selecting to form in (26), and by (49), we have for . Consequently, with

which leads to (32). APPENDIX D PROOF OF LEMMA 3.1 We first establish the following result. Lemma 6.1: Let

We have (50) (51) where

and . Moreover, the rows of form a basis of the solution space of the in . homogeneous linear equation Proof: We carry out the proof by induction. By As, the matrix has a constant rank in sumption

Thus,

and . The matrix is of is of full row rank. full row rank, since and have and The matrices rows, respectively, and the rank of is . Thus, the rows of form a basis of the solution space in . of Now we are ready to prove Lemma 3.1. We do it by induc. According to the algorithm, tion. Consider , thus, . The rows of form a basis of the solution space of the homogeneous linear equain . Similarly, the rows of tion span the same solution space in . Therefore, , where is a nonthe matrix is smooth with in singular and smooth, and . Thus, by (41), , where in . , equation (42) are satisAssume that, for fied. That is

where in

is nonsingular and . Thus

is smooth with (52)

LIU AND LIN: ON NORMAL FORMS OF NONLINEAR SYSTEMS AFFINE IN CONTROL

with and

in

,

where

and

being nonsingular, where

.. .

.. .

.. .

By (52), we know that . We also have Thus,

..

.. .

.

Therefore,

is equivalent to

where

with

and

. It is obvious that

in

. By (50),

and in . The rows of and span the solution spaces of homogeneous linear equations and in , respectively. By (53), , and thus

By (39)

where is a matrix, and is in . Due to the structure smooth with of and , we know that in . is nonsingular in a neighborhood of , which Thus, . By (54) contains

(55) And by (41), we have

Thus, multiplying (55) to the right by and (53), we have

and using (51)

.

By the zero output structure algorithm, we know that is of full row rank. Note that , , are the coefficients in . So if is of is of full row rank. Thus, we only need to full row rank, is of full row rank. We prove it by induction. prove that . We first Recall that prove that the row vectors of , or , are linearly independent. It follows directly from the fact that has full row rank. Assume that the rows of are linearly independent. We want to prove that the rows of are linearly independent. , , and Let . Define

(54) is a nonsingular matrix valued smooth function, in . Denote

in

APPENDIX E PROOF OF LEMMA 3.3

. (53)

where and

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where

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By (41)

REFERENCES

(56) Let (57) Thus,

. Since the matrix is of full row rank in , we have (58)

in

. Thus, by (57) and (58) (59)

By (56)

where

, . Consider

, are matrix valued functions of . We have

(60) , for By (59) and (60), we have and . In conclusion, the row vectors of are linearly independent.

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Xinmin Liu (M’08) received the M.S. from Xiamen University, Xiamen, China, in 1998, the M.E. degree from the National University of Singapore in 2000, and the Ph.D. degree from the University of Virginia, Charlottesville, in 2010. His current research interests include nonlinear control, biological control, numerical analysis, and image processing.

Zongli Lin (S’89–M’90–SM’98–F’07) received the B.S. degree in mathematics and computer science from Xiamen University, Xiamen, China, in 1983, the M.Eng. degree in automatic control from the Chinese Academy of Space Technology, Beijing, China, in 1989, and the Ph.D. degree in electrical and computer engineering from Washington State University, Pullman, in 1994. He is a Professor of Electrical and Computer Engineering at University of Virginia. His current research interests include nonlinear control, robust control, and control applications. Dr. Lin was an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and the IEEE/ASME TRANSACTIONS ON MECHATRONICS. He has served on the operating committees and program committees of several conferences. He currently serves on the editorial boards of several journals and book series, including Automatica, Systems & Control Letters, Science China Information Science, and the IEEE Control Systems Magazine. He is an elected member of the Board of Governors of the IEEE Control Systems Society.