Volume 11
Issue 12
IEEE/CAA Journal of Automatica Sinica
| Citation: | X. Xu and G.-R. Duan, “High-order fully actuated system models for strict-feedback systems with increasing dimensions,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 12, pp. 2451–2462, Dec. 2024. doi: 10.1109/JAS.2024.124599
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High-Order Fully Actuated System Models for Strict-Feedback Systems W
ith Increasing Dimensions
STRICT-FEEDBACK systems (SFSs) refer to dynamic systems consisting of multiple subsystems in a triangular structure. Control of SFSs has been widely studied for many years with a quantity of interesting results obtained, to name a few, [1]–[5]. A typical SFS is given as follows:
| {˙x1=f1(x1)+G1(x1)x2˙x2=f2(x1,x2)+G2(x1,x2)x3⋮˙xn=fn(x1,x2,…,xn)+Gn(x1,x2,…,xn)u | (1) |
where $ x_i\in\mathbb{R}^{k},\; i = 1,2,\ldots,n, $ are the state vectors, $ u\in\mathbb{R}^k $ is the control input, $ f_i:\mathbb{R}^{ik}\rightarrow\mathbb{R}^k $ and $ G_i:\mathbb{R}^{ik}\rightarrow\mathbb{R}^{k\times k} $ are sufficiently smooth functions. One of the most commonly-used method for handling control problems of such a system is the well-known backstepping method.
The backstepping technique, pioneered in the 1990s by Petar V. Kokotovick, Miroslav Krstic, and others, originally aimed at crafting stabilizing controls for SFSs [6], [7]. Its fundamental concept involves initiating the design process at the first subsystem already stabilized and then iteratively deriving new controllers to progressively stabilize subsequent subsystems until achieving control over the final external aspect. While this method assumes uniform dimensions for subsystems (i.e., $ x_i,\; i = 1,\ldots,n $), practical systems often feature nonuniform or differing dimensions. Examples include cascades of over-actuated [8] or under-actuated systems [9]. Considering SFSs with nonuniform dimensions enables the development of more general theories, encompassing uniform cases as special instances. However, designing controllers for SFSs with nonuniform dimensions via the traditional backstepping approach proves challenging, if not infeasible. Therefore, addressing the more general scenario-where subsystems possess nonuniform dimensions-requires innovative approaches.
Fortunately, there are effective approaches available to address control problems in the context of high-order fully actuated systems (HOFAS). HOFAS, also referred to as fully actuated system (FAS) approaches, were originally introduced by G. Duan in a series of papers [10]–[12]. The HOFAS approach offers a distinct perspective for handling control problems by transforming a system into a HOFAS model, which allows for the direct derivation of an exponentially stabilizing controller. Duan has demonstrated that nearly all controllable systems, regardless of linearity or nonlinearity and time-invariance or time-varying nature, can be transformed into HOFASs. The high-order FAS theory has opened up new avenues in control theory and has attracted significant research attention from scholars who have obtained numerous results in this field [13]–[22].
In [11], it is shown that first-order, second-order and high-order SFSs can all be equivalently transformed into HOFAS models. Although only SFSs with uniform dimension are considered in [11], the HOFAS approach shows its potential in handling control problems of various complicated SFSs. It is thus natural to apply HOFAS approaches to more complicated types of SFSs, one of which is the type of SFSs with increasing dimensions, which motivates this work.
This paper investigates the control of SFSs with expanding dimensions, introducing HOFAS models tailored for such systems to facilitate the direct derivation of exponentially stabilizing controllers. To our best knowledge, this work is among the first few studies concerning about control of SFSs with increasing dimensions. The contributions of this study can be delineated into two main areas.
Firstly, we present HOFAS models specifically designed for SFSs with increasing dimensions, building upon prior work wherein various SFSs were transformed into FAS models [11]. While previous research [11] predominantly addressed scenarios where subsystem dimensions remained constant, we extend these findings to encompass more intricate cases, namely SFSs with expanding dimensions. A pivotal technique employed involves the decomposition of variables based on these increasing dimensions. Additionally, whereas existing literature typically focuses on first-order SFSs, we broaden our scope to include second- and higher-order SFSs, enhancing the generality of our approach.
Secondly, we tackle the control problem associated with SFSs exhibiting increasing dimensions. Leveraging the FAS models tailored for such systems, we derive stabilized controllers directly. A notable advantage of our controllers over those designed using backstepping methods lies in the resulting closed-loop systems, which exhibit linear time-invariance. This feature underscores the efficacy of HOFAS approaches in addressing control challenges posed by more complex SFSs.
The rest of this work is organized as follows. In Section II, notations are introduced. In Sections III, HOFAS models are proposed for two types of first-order SFSs with increasing dimensions. The matrices $ G_i,\; i = 1,2,\ldots,n $ are first assumed to be constant and then assumed to be matrix functions. Second-and high-order SFSs with increasing dimensions are considered in Section IV. Conclusions are drawn in Section VI.
Throughout this paper, the following notations are used. The symbol $ I_k $ represents $ k\times k $ identity matrix. $ 0 $ represents zero matrix with appropriate dimension. The symbol $ \triangleq $ means “denoted as” in this work. For two integers $ i,j, $ where $ j\geq i $, the notation $ i\sim j $ represents the series of integers $ i, i+1, \ldots, j $. For a function $ x\in\mathbb{R}^n $, $ x^{(i)} $ denotes its i-th order derivative. Moreover, the following symbol is used:
| x(i∼j)=(x(i)x(i+1)⋮x(j)). | (2) |
For three integers $ i,j,k, $ where $ j\geq i, $ and for $ x_{pk}\in\mathbb{R}^n, \; p = i\sim j $, the following symbols are used:
| xi∼j,k=(xikxi+1,k⋮xjk) | (3) |
| x(0∼l(p))pk|p=i∼j=(x(0∼l(i))ikx(0∼l(i+1))i+1,k⋮x(0∼l(j))jk) | (4) |
where $ l:\mathbb{Z}^+\rightarrow\mathbb{Z}^+ $ is an arbitrary function mapping from positive integers into positive integers. In addition, one has the following definition:
| x(0∼l(p))pk|p=i∼j,k=1∼2=(x(0∼l(p))p1|p=i∼jx(0∼l(p))p2|p=i∼j). | (5) |
In this subsection, we consider first-order SFSs with increasing dimensions. Consider the following SFS:
| {˙x1=f1(x1)+G1x2˙x2=f2(x1,x2)+G2x3⋮˙xn=fn(x1,x2…,xn)+Gnu, | (6) |
where $ x_i\in\mathbb{R}^{k_i},\; i = 1,2,\ldots,n $, $ u\in\mathbb{R}^m $, $ G_i\in\mathbb{R}^{k_i\times k_{i+1}},\; i = 1, 2,\ldots, n-1, $ and $ G_n\in\mathbb{R}^{k_n\times m} $. In this work, it is assumed that $ k_1\leq k_2\leq \cdots\leq k_n\leq m. $ Hence, the dimensions of subsystems are increasing. Moreover, the following assumption is needed for constant matrices $ G_i,\; i = 1,2,\ldots,n. $
Assumption 1: The matrices $ G_i,\; i = 1,2,\ldots,n, $ are all of full-row rank, i.e., $ {\rm rank}(G_i) = k_i,\; i = 1,2,\ldots,n $.
Assumption 1 ensures that there exist nonsingular matrices $ Q_i, \; i = 1,2,\ldots,n $ and $ G_i, \; i = 1,2,\ldots,n $ such that
| GiQi=[ˉGi0]. | (7) |
Based on this fact, let us define $ x_{i+1,1}\in\mathbb{R}^{k_i} $, $ x_{i+1,2}\in\mathbb{R}^{k_{i+1}-k_i} $, $ f_{i1}\in\mathbb{R}^{k_i} $ and $ f_{i2}\in\mathbb{R}^{k_{i+1}-k_i} $ by
| Q−1ixi+1=(xi+1,1xi+1,2) | (8) |
| Q−1ifi+1(x1,x2,…,xi+1)=(fi+1,1(x1,x2∼i,1,x2∼i,2)fi+1,2(x1,x2∼i,1,x2∼i,2)),i=1,2,…,n−1. | (9) |
For notation convenience, we also define that $ x_{12} = x_1. $ Meanwhile, define $ u_1\in\mathbb{R}^{k_n} $ and $ u_2\in\mathbb{R}^{m-k_n} $ that
| Q−1nu=(u1u2). | (10) |
For the first subsystem, it first follows from the definition of $ x_{12} $ and $ x_{21} $ that:
| ˙x12=f1(x1)+[ˉGi0]Q−11x2=f1(x12)+ˉG1x21. | (11) |
Then, for the second subsystem, it can be obtained that
| [˙x21˙x22]=Q−11(f2(x1,x2)+[ˉG20]Q−12x3=[f21(x12,x21,x22)f22(x12,x21,x22)]+ˉG2x31. | (12) |
After an n-step induction, system (6) can be then transformed into
| {˙x12=f1(x12)+ˉG1x21˙x21=f21(x21,x1∼2,2)+[Ik10]Q−11ˉG2x31˙x22=f22(x21,x1∼2,2)+[0Ik2−k1]Q−11ˉG2x31⋮˙xn1=fn1(x2∼n,1,x1∼n,2)+[Ikn0]Q−1n−1ˉGnu1˙xn2=fn2(x2∼n,1,x1∼n,2)+[0Ikn−kn−1]Q−1n−1ˉGnu1. | (13) |
We use the newly obtained system (13) to derive a HOFAS model for the first-order SFS (6). In particular, the following theorem can be obtained.
Theorem 1: Let Assumption 1 be satisfied, then under the following transformation:
| xi1=ˉG−11∼i−1(x(i−p)p2|p=1∼i−1−Li−1(x(0∼i−1−p)p2|p=1∼i−1)),i=2,…,n | (14) |
the SFS (13) can be equivalently transformed into the following HOFAS model:
| x(n+1−p)p2|p=1∼n=Ln(x(0∼n−p)p2|p=1∼n)+ˉG1∼nu1 | (15) |
where $ L_i(\cdot) $ and $ \bar{G}_{1\sim i} $ are given iteratively by
| Li+1(x(0∼i+1−p)p2|p=1∼i+1)=(˙Li+ˉG1∼ifi+1,1(x2∼i+1,1,x1∼i+1,2)fi+1,2(x2∼i+1,1,x1∼i+1,2)) | (16) |
| ˉG1∼i+1=[ˉG1∼i00Iki+1−ki]Q−1iˉGi+1,i=1,2,…,n−1. | (17) |
with $ L_1(\cdot) = f_1(x_{12}) $ and $ \bar{G}_{1\sim 1} = \bar{G}_1 $.
Proof: We prove the theorem via the n-step induction.
Step 1: For the first subsystem, we already have that
| ˙x12=f1(x12)+ˉG1x21=L1(x12)+ˉG1∼1x21. | (18) |
Step i ($ 2\leq i\leq n-1) $: Assume that
| x(i−p)p2|p=1∼i−1=Li−1(x(0∼i−1−p)p2|p=1∼i−1)+ˉG1∼i−1xi1 | (19) |
then we prove that
| x(i+1−p)p2|p=1∼i=Li(x(0∼i−p)p2|p=1∼i)+ˉG1∼ixi+1,1. | (20) |
It follows from (19) that:
| xi1=;ˉG−11∼i−1(x(i−p)p2|p=1∼i−1−Li−1(x(0∼i−1−p)p2|p=1∼i−1)). | (21) |
We also recall the following equations from system (13):
| ˙xi1=fi1(x2∼i,1,x1∼i,2)+[Iki0]Q−1i−1ˉGixi+1,1˙xi2=fi2(x2∼i,1,x1∼i,2)+[0Iki−ki−1]Q−1i−1ˉGixi+1,1. | (22) |
Next, taking the derivatives of both sides of the (19) and substituting first equation of (22) into the result, one has that
| x(i+1−p)p2|p=1∼i−1=˙Li−1+ˉG1∼i−1˙xi1=˙Li−1+ˉG1∼i−1(fi1(x2∼i,1,x1∼i,2) +[Iki0]Q−1i−1ˉGixi+1,1)=˙Li−1+ˉG1∼i−1fi1(x2∼i,1,x1∼i,2) +[ˉG1∼i−10]Q−1i−1ˉGixi+1,1. | (23) |
By combining (23) with the last equation of (22), it can be obtained that
| x(i+1−p)p2|p=1∼i=Li(x(0∼i−p)p2|p=1∼i)+ˉG1∼ixi+1,1 | (24) |
where
| Li= (˙Li−1+ˉG1∼i−1fi1(x2∼i,1,x1∼i,2)fi2(x2∼i,1,x1∼i,2)) | (25) |
| ˉG1∼i=[ˉG1∼i−100Iki−ki−1]Q−1i−1ˉGi. | (26) |
Step n: Based on the previous $ n-1 $ steps, we can obtain that
| xi1=ˉG−11∼i−1(x(i−p)p2|p=1∼i−1 −Li−1(x(0∼i−1−p)p2|p=1∼i−1)),i=2,…,n−1 | (27) |
and
| x(n−p)p2|p=1∼n−1=Ln−1(x(0∼n−1−p)p2|p=1∼n−1) +ˉG1∼n−1xn1 | (28) |
where $ L_i(\cdot) $ and $ \bar{G}_{1\sim i} $ are given iteratively by
| Li+1(x(0∼i+1−p)p2|p=1∼i+1)=(˙Li+ˉG1∼ifi+1,1(x2∼i+1,1,x1∼i+1,2)fi+1,2(x2∼i+1,1,x1∼i+1,2)) | (29) |
| ˉG1∼i+1=[ˉG1∼i00Iki+1−ki]Q−1iˉGi+1,i=1,2,…,n−1. | (30) |
We recall the following equations from (13):
| ˙xn1=fn1(x2∼n,1,x1∼n,2)+[Ikn0]Q−1n−1ˉGnu1 | (31) |
| ˙xn2=fn2(x2∼n,1,x1∼n,2)+[0Ikn−kn−1]Q−1n−1ˉGnu1. | (32) |
With the similar discussions as that in previous steps, it can be thus obtained that
| x(n+1−p)p2|p=1∼n=Ln(x(0∼n−p)p2|p=1∼n)+ˉG1∼nu1 | (33) |
where
| Ln=(˙Ln−1+ˉG1∼n−1fn1(x2∼n,1,x1∼n,2)fn2(x2∼n,1,x1∼n,2)) | (34) |
| ˉG1∼n=[ˉG1∼n−100Ikn−kn−1]Q−1n−1ˉGn. | (35) |
It follows from the definition of $ \bar{G}_{1\sim n} $ that $ \bar{G}_{1\sim n} $ is nonsingular. Therefore, the system (33) is fully-actuated.
Remark 1: It is noted that in the HOFAS model, only part of the controller u, i.e., $ u_1 $, is used while the rest part $ u_2 $ can be removed. This is mainly because that the system (6) is “over-actuated” when the dimension of controller is greater than the state. Due to this fact, we only need part of the controller to convert the system (6) into a FAS. In fact, most of practical systems are not over-actuated, i.e., $ u_2 $ does not exist in those systems. However, there do exist “over-actuated” practical systems. For example, some real industrial systems are equipped with redundant actuators, such as spacecraft [23], ships [24], and fusion systems [25], for higher reliability and greater flexibility. These practical systems are over-actuated.
As induced in [11], based on the FAS model (15), the following exponentially stabilizing controller can be directly given:
| u1=ˉG−11∼n(−Ln+Ax(0∼n−p)p2|p=1∼n) | (36) |
where $ A\in\mathbb{R}^{\sum\limits_{i = 1}^{n}(n+1-i)k_i\times\sum\limits_{i = 1}^{n}(n+1-i)k_i} $, which yields the following linear time-invariant closed-loop system:
| x(n+1−p)p2|p=1∼n=Ax(0∼n−p)p2|p=1∼n. | (37) |
In particular, if A is chosen as a block diagonal triangular matrix in the following form:
| [A0∼n−1,10⋯00A0∼n−2,2⋯0⋮⋮⋱⋮00⋯A0n] | (38) |
where
| A0∼n−k,k=[A0kA1k⋯An−k,k],k=1,2,…,n | (39) |
with $ A_{pi},\; i = 1,2,\ldots,n, \; 0\leq p\leq n-i, $ being matrices with appropriate dimensions. In this case, the closed-loop system can be written as
| {x(n)12=n−1∑p=0Ap1x(p)12x(n−1)22=n−2∑p=0Ap2x(p)22⋮x(1)n2=A0nxn2 | (40) |
Remark 2: As is pointed out in [11], the FAS approach has two distinctive advantages compared with backstepping method. First, the FAS approach can always produce a linear time-invariant closed-loop system. Second, the FAS approach can avoid the “differential explosion” problem, which makes the method of backstepping generally not applicable to a SFS with more than 3 or 4 subsystems.
In the previous subsection, we only assume that $ G_i,\; i = 1,2,\ldots,n $, are constant matrices. In this subsection, we consider the more general case of $ G_i,\; i = 1,2,\ldots,n $, being matrix functions, which is described by the following dynamics:
| {˙x1=f1(x1)+G1(x1)x2˙x2=f2(x1,x2)+G2(x1,x2)x3⋮˙xn=fn(x1,x2,…,xn)+Gn(x1,x2,…,xn)u | (41) |
where $ x_i\in\mathbb{R}^{k_i},\; i = 1,2,\ldots,n $, $ u\in\mathbb{R}^m $. Moreover, $ f_i $ and $ G_i $ are sufficiently smooth functions. In this work, it is assumed that $ k_1\leq k_2\leq \cdots\leq k_n\leq m. $ The following assumption is needed for matrix functions $ G_i(\cdot),\; i = 1,2,\ldots,n. $
Assumption 2: For any $ x_{1\sim i}\in\mathbb{R}^{k_1+\cdots+k_i} $, the matrix function $ G_i(x_{1\sim i}) $ is of full-row rank, i.e., $ {\rm rank}(G_i) = k_i,\; i = 1, 2,\ldots,n $.
It follows from Assumption 2 that there exist sufficiently smooth functions $ Q_i: \mathbb{R}^{k_1+\cdots+k_i}\rightarrow\mathbb{R}^{k_{i+1}\times k_{i+1}} $ and $ \bar{G}_i:\mathbb{R}^{k_1+\cdots+k_i} \rightarrow \mathbb{R}^{k_i\times k_{i}} $such that:
| Gi(⋅)Qi(⋅)=[ˉGi(⋅)0] | (42) |
where $ Q_i(\cdot) $ and $ \bar{G}_i(\cdot) $ are nonsingular for any $ x_{1\sim i}\in\mathbb{R}^{k_1+\cdots+k_i} $. Define a differential homeomorphism from $ x = \{x_1^T,x_2^T,\ldots, x_n^T\}^T $to $ \bar{x} = \{\bar{x}_1^T,\bar{x}_2^T,\ldots,\bar{x}_n^T\}^T $ as follows:
| ˉx1= x1ˉx2= Q−11(x1)x2ˉx3= Q−12(x1,x2)x3⋮ˉxn= Q−1n−1(x1,x2,…,xn−1)xn. | (43) |
Then the inverse mapping from $ \bar{x} $ to x can be written as follows:
| x1= ˉx1x2= Q1(ˉx1)ˉx2≜ˉQ1(ˉx1)ˉx2x3= Q2(ˉx1,ˉQ1(ˉx1)ˉx2)ˉx3≜ˉQ2(ˉx1,ˉx2)ˉx3⋮xn= Qn−1(ˉx1,ˉQ1(¯x1)ˉx2,…,ˉQn−2(ˉx1,…,ˉxn−2)ˉxn−1)ˉxn≜ ˉQn−1(ˉx1,…,ˉxn−1)ˉxn. | (44) |
Furthermore, define $ \bar{x}_{i1} $, $ \bar{x}_{i2} $ and $ \bar{u}_1 $ as follows:
| {ˉxi1=[Iki−10]ˉxi,i=2,…,nˉxi2=[0Iki−ki−1]ˉxi,i=2,…,nˉu1=[0Ikn]ˉQ−1nu. | (45) |
Meanwhile, for notation convenience, we also define that $ \bar{x}_{12} = \bar{x}_1. $ It then follows from (43) and (44) that the following equivalent form of system (41) can be obtained:
| {˙ˉx12=ˉf1(ˉx12)+ˉG1(ˉx12)ˉx21˙ˉx21=ˉf21(ˉx21,ˉx1∼2,2)+[Ik10]ˉQ−11ˉG2(ˉx21,ˉx1∼2,2)ˉx3˙ˉx22=ˉf22(ˉx21,ˉx1∼2,2)+[0Ik2−k1]ˉQ−11ˉG2(ˉx21,ˉx1∼2,2)ˉx3⋮˙ˉxn1=ˉfn1(ˉx2∼n,1,ˉx1∼n,2)+[Ikn−10]ˉQ−1n−1ˉGn(ˉx2∼n,1,ˉx1∼n,2)ˉu1˙ˉxn2=ˉfn2(ˉx2∼n,1,ˉx1∼n,2)+[0Ikn−kn−1]ˉQ−1n−1ˉGn(ˉx2∼n,1,ˉx1∼n,2)ˉu1 | (46) |
where
| ˉf1(ˉx12)=f1(ˉx12) | (47) |
and for $ i = 2,\ldots, n, $
| ˉfi1(ˉx2∼i,1,ˉx1∼i,2)=[Iki−10]ˉQ−1i−1 ×(−˙ˉQi−1ˉxi+fi(ˉx1,ˉQ1ˉx2,…,ˉQi−1ˉxi))ˉfi2(ˉx2∼i,1,ˉx1∼i,2)=[0Iki−ki−1]ˉQ−1i−1 ×(−˙ˉQi−1ˉxi+fi(ˉx1,ˉQ1ˉx2,…,ˉQi−1ˉxi)). | (48) |
Based on the equivalent system (46), the following HOFAS model can be derived.
Theorem 2: Let Assumption 2 be satisfied, then under the following transformation:
| ˉxi1=ˉG−11∼i−1(ˉx(i−p)p2|p=1∼i−1−Li−1(ˉx(0∼i−1−p)p2|p=1∼i−1)),i=2,…,n | (49) |
the SFS (46) can be equivalently transformed into the following HOFAS model:
| ˉx(0∼n+1−p)p2|p=1∼n=Ln(ˉx(0∼n−p)p2|p=1∼n)+ˉG1∼n(ˉx(0∼n−p)p2|p=1∼n)ˉu1 | (50) |
where $ L_i(\cdot) $ and $ \bar{G}_{1\sim i}(\cdot) $ are given iteratively by
| Li+1(ˉx(0∼i+1−p)p2|p=1∼i+1)=(˙Li+˙ˉG1∼iˉxi+1,1+ˉG1∼iˉfi+1,1ˉfi+1,2) | (51) |
| ˉG1∼i+1(ˉx(0∼i+1−p)p2|p=1∼i+1)=[ˉG1∼i00Iki+1−ki]ˉQ−1iˉGi+1,i=1,2,…,n−1 | (52) |
with $ L_1 = \bar{f}_1(\bar{x}_{12}) $ and $ \bar{G}_{1\sim 1} = \bar{G}_1(\bar{x}_{12}) $.
Proof: We can also prove the theorem via the n-step induction.
Step 1: For the first subsystem, we already have
| ˙ˉx12= ˉf1(ˉx12)+ˉG1(ˉx12)ˉx21= L1(ˉx12)+ˉG1∼1(ˉx12)ˉx21. | (53) |
Step i ($ 2\leq i\leq n-1) $: Assume that
| ˉx(i−p)p2|p=1∼i−1=Li−1(ˉx(0∼i−1−p)p2|p=1∼i−1)+ˉG1∼i−1(ˉx(0∼i−1−p)p2|p=1∼i−1)ˉxi1 | (54) |
then we prove that
| ˉx(i+1−p)p2|p=1∼i=Li(ˉx(0∼i−p)p2|p=1∼i)+ˉG1∼i(ˉx(0∼i−p)p2|p=1∼i)ˉxi+1,1. | (55) |
It follows from (54) that:
| ˉxi1=ˉG−11∼i−1(⋅)(ˉx(i−p)p2|p=1∼i−1−Li−1(ˉx(0∼i−1−p)p2|p=1∼i−1)). | (56) |
We also recall the following equations from system (46):
| ˙ˉxi1= ˉfi1(ˉx2∼i,1,ˉx1∼i,2) +[Iki0]ˉQ−1i−1ˉGi(ˉx2∼i,1,ˉx1∼i,2)ˉxi+1,1˙ˉxi2= ˉfi2(ˉx2∼i,1,ˉx1∼i,2) +[0Iki−ki−1]ˉQ−1i−1ˉGi(ˉx2∼i,1,ˉx1∼i,2)ˉxi+1,1. | (57) |
Next, taking the derivatives of both sides of the (54) and substituting first equation of (57) into the result, one has that
| ˉx(i+1−p)p2|p=1∼i−1= ˙Li−1+˙ˉG1∼i−1ˉxi1+ˉG1∼i−1˙ˉxi1= ˙Li−1+˙ˉG1∼i−1ˉxi1 +ˉG1∼i−1(ˉfi1+[Iki0]ˉQ−1i−1ˉGiˉxi+1,1)= ˙Li−1+˙ˉG1∼i−1ˉxi1+ˉG1∼i−1ˉfi1 +[ˉG1∼i−10]ˉQ−1i−1ˉGiˉxi+1,1. | (58) |
By combining (58) with the last equation of (57), it can be obtained that
| ˉx(i+1−p)p2|p=1∼i= Li(ˉx(0∼i−p)p2|p=1∼i) +ˉG1∼i(ˉx(0∼i−p)p2|p=1∼i)ˉxi+1,1 | (59) |
where
| Li=(˙Li−1+˙ˉG1∼i−1ˉxi1+ˉG1∼i−1ˉfi1ˉfi2)ˉG1∼i=[ˉG1∼i−100Iki−ki−1]ˉQ−1i−1ˉGi. | (60) |
Step n: Based on the previous $ n-1 $ steps, we can obtain that
| ˉxi1= ˉG−11∼i−1(ˉx(i−p)p2|p=1∼i−1 −Li−1(ˉx(0∼i−1−p)p2|p=1∼i−1)),i=2,…,n−1 | (61) |
and
| ˉx(n−p)p2|p=1∼n−1= Ln−1(ˉx(0∼n−1−p)p2|p=1∼n−1) +ˉG1∼n−1(ˉx(0∼n−1−p)p2|p=1∼n−1)ˉxn1 | (62) |
where $ L_i(\cdot) $ and $ \bar{G}_{1\sim i}(\cdot) $ are given iteratively by
| Li+1(ˉx(0∼i+1−p)p2|p=1∼i+1)=(˙Li+˙ˉG1∼iˉxi+1,1+ˉG1∼iˉfi+1,1ˉfi+1,2) | (63) |
| ˉG1∼i+1(ˉx(0∼i+1−p)p2|p=1∼i+1)=[ˉG1∼i00Iki+1−ki]ˉQ−1iˉGi+1,i=1,2,…,n−1. | (64) |
We recall the following equations from (46):
| ˙ˉxn1= ˉfn1(ˉx2∼n,1,ˉx1∼n,2) +[Ikn0]ˉQ−1n−1ˉGn(ˉx2∼n,1,ˉx1∼n,2)ˉu1˙ˉxn2= ˉfn2(ˉx2∼n,1,ˉx1∼n,2) +[0Ikn−kn−1]ˉQ−1n−1ˉGn(ˉx2∼n,1,ˉx1∼n,2)ˉu1. | (65) |
With the similar discussions as that in previous steps, it can be thus obtained that
| ˉx(0∼n+1−p)p2|p=1∼n= Ln(ˉx(0∼n−p)p2|p=1∼n) +ˉG1∼n(ˉx(0∼n−p)p2|p=1∼n)ˉu1 | (66) |
where
| Ln=(˙Ln−1+˙ˉG1∼n−1ˉxn1+ˉG1∼n−1ˉfn1fn2)ˉG1∼n=[ˉG1∼n−100Ikn−kn−1]ˉQ−1n−1ˉGn. | (67) |
It follows from the definition of $ \bar{G}_{1\sim n} $ that $ \bar{G}_{1\sim n} $ is nonsingular for any $ \left(\bar{x}_{p2}^{(0\sim n-p)}\Big|_{p = 1\sim n}\right)\in\mathbb{R}^{\sum\limits_{i = 1}^n(n-i)k_i} $. Therefore, the system (66) is a FAS.
In [11], the author proposes more generalized SFSs of both second-order and high-order. Meanwhile, both second-order and high-order SFSs are converted into FAS models. In this subsection, we further extend the results in [11] to second-order and high-order SFSs with increasing dimensions. Proofs of all theorems in this part are omitted as they can be easily established by applying the induction method introduced in Section III together with the results in [11].
In this subsection, the following second-order SFS with increasing dimensions is first discussed:
| {¨x1=f1(x(0∼1)1)+G1x2¨x2=f2(x(0∼2)1,x(0∼1)2)+G2x3⋮¨xn−1=fn(x(0∼2)i|i=1∼n−2,x(0∼1)n−1)+Gn−1xn¨xn=fn(x(0∼2)i|i=1∼n−1,x(0∼1)n)+Gnu | (68) |
where $ x_i\in\mathbb{R}^{k_i},\; i = 1,2,\ldots,n $, $ u\in\mathbb{R}^m $, $ G_i\in\mathbb{R}^{k_i\times k_{i+1}}, \; i = 1,2,\ldots, n-1, $ and $ G_n\in\mathbb{R}^{k_n\times m} $. We adopt the similar definition for $ x_{i+1,1}\in\mathbb{R}^{k_i} $,$ x_{i+1,2}\in\mathbb{R}^{k_{i+1}-k_i} $, $ f_{i1}\in\mathbb{R}^{k_i} $, $ u_1\in\mathbb{R}^{k_n} $ and $ u_2\in\mathbb{R}^{m-k_n} $ as (8)−(10). Then system (68) can be transformed into the following equivalent form:
| {¨x12=f1(x(0∼1)1)+ˉG1x21¨x21=f21(x(0∼1)21,x(0∼1)1∼2,2)+[Ik10]Q−11ˉG2x31¨x22=f22(x(0∼1)21,x(0∼1)1∼2,2)+[0Ik2−k1]Q−11ˉG2x31⋮¨xn1=fn1(x(0∼2)12,x(0∼2)ij|i=2∼n−1,j=1∼2,x(0∼1)nj|j=1∼2)+[Ikn0]Q−1n−1ˉGnu1¨xn2=fn2(x(0∼2)12,x(0∼2)ij|i=2∼n−1,j=1∼2,x(0∼1)nj|j=1∼2)+[0Ikn−kn−1]Q−1n−1ˉGnu1. | (69) |
Based on the system (69), we can derive the HOFAS model for the second-order SFSs with increasing dimensions. In particular, we present the following theorem.
Theorem 3: Let Assumption 1 be satisfied, then under the following transformation:
| xi1=ˉG−11∼i−1(x(2i−2p)p2|p=1∼i−1−Li−1(x(0∼2i−1−2p)p2|p=1∼i−1)),i=2,…,n | (70) |
the SFS (69) can be equivalently transformed into the following HOFAS model:
| x(2n+2−2p)p2|p=1∼n=Ln(x(0∼2n+1−2p)p2|p=1∼n)+ˉG1∼nu1 | (71) |
where $ L_i(\cdot) $ and $ G_{1\sim i} $ are given iteratively by
| Li+1(⋅)=(¨Li+ˉG1∼ifi+1,1fi+1,2) | (72) |
| ˉG1∼i+1=[ˉG1∼i00Iki+1−ki]Q−1iˉGi+1,i=1,2,…,n−1 | (73) |
with $ L_1 = f_1\left(x_{12}^{(0\sim 1)}\right) $ and $ \bar{G}_{1\sim 1} = \bar{G}_1 $.
Next, similar to the first-order SFSs, the more general case where $ G_i,\; i = 1,2,\ldots,n, $ are assumed to be matrix functions can be also considered. In particular, the following second-order SFS is discussed:
| {¨x1=f1(x(0∼1)1)+G1(x(0∼1)1)x2¨x2=f2(x(0∼2)1,x(0∼1)2)+G2(x(0∼2)1,x(0∼1)2)x3⋮¨xn−1=fn(x(0∼2)i|i=1∼n−2,x(0∼1)n−1)+Gn−1(x(0∼2)i|i=1∼n−2,x(0∼1)n−1)xn¨xn=fn(x(0∼2)i|i=1∼n−1,x(0∼1)n)+Gn(x(0∼2)i|i=1∼n−1,x(0∼1)n)u | (74) |
where $ x_i\in\mathbb{R}^{k_i},\; i = 1,2,\ldots,n $, $ u\in\mathbb{R}^m $. Moreover, $ f_i $ and $ G_i $ are sufficiently smooth functions. In this work, it is assumed that $ k_1\leq k_2\leq \cdots\leq k_n\leq m. $ The following assumption is needed for matrix functions $ G_i,\; i = 1,2,\ldots,n. $
Assumption 3: For any $\left(x_p^{(0\sim 2)}\Big|_{p = 1\sim i-1},x_{i}^{(0\sim 1)}\right)\in $ $ \mathbb{R}^{3k_1+\cdots+3k_{i-1}+2k_i} $, the matrix function $ G_i(\cdot) $ is of full-row rank, i.e., $ {\rm rank}(G_i) = k_i, \; i = 1,2,\ldots,n $.
It follows from Assumption 3 that there exist differentiable functions $ Q_i: \mathbb{R}^{3k_1+\cdots+3k_{i-1}+2k_i}\rightarrow\mathbb{R}^{k_{i+1}\times k_{i+1}} $ and $\bar{G}_i: $ $ \mathbb{R}^{3k_1+\cdots+3k_{i-1}+2k_i}\rightarrow\mathbb{R}^{k_i\times k_{i}} $ such that:
| Gi(⋅)Qi(⋅)=[ˉGi(⋅)0] | (75) |
where $ Q_i $ and $ \bar{G}_i $ are of full rank. Define the mapping from $ x = \{x_1^T,x_2^T,\ldots,x_n^T\}^T $ to $ \bar{x} = \{\bar{x}_1^T,\bar{x}_2^T,\ldots,\bar{x}_n^T\}^T $ as follows:
| ˉx1= x1ˉx2= Q−11(x(0∼1)1)x2ˉx3= Q−12(x(0∼2)1,x(0∼1)2)x3⋮ˉxn= Q−1n−1(x(0∼2)i|i=1∼n−1,x(0∼1)n)xn. | (76) |
Then the inverse mapping from $ \bar{x} $ to x can be written as follows:
| x1= ˉx1x2= Q1(ˉx(0∼1)1)ˉx2≜ˉQ1(ˉx(0∼1)1)ˉx2x3= Q2(ˉx(0∼2)1,(ˉQ1(ˉx(0∼1)1)ˉx2)(0∼1))ˉx3 ≜ ˉQ2(ˉx(0∼2)1,ˉx(0∼1)2)ˉx3⋮xn= Qn−1((ˉQi−1ˉxi)(0∼2)|i=1∼n−2,(ˉQn−2ˉxn−1)(0∼1))ˉxn≜ ˉQn−1(ˉx(0∼n−i)i|i=1∼n−1)ˉxn. | (77) |
Define $ \bar{x}_{i1} $, $ \bar{x}_{i2} $ and $ \bar{u}_1 $ as follows:
| {ˉxi1=[Iki−10]ˉxi,i=2,…,nˉxi2=[0Iki−ki−1]ˉxi,i=2,…,nˉu1=[0Ikn]ˉQ−1nu. | (78) |
Meanwhile, for notation convenience, we also define that $ \bar{x}_{12} = \bar{x}_1. $ It then follows from (76) and (77) that system (74) can be equivalently transformed into the following form:
| {¨ˉx1=ˉf1(ˉx(0∼1)12)+ˉG1(ˉx(0∼1)12)ˉx21¨ˉx21=ˉf21(ˉx(0∼2)12,ˉx(0∼1)21,ˉx(0∼1)22)+[Ik10]ˉQ−11ˉG2(ˉx(0∼2)12,ˉx(0∼1)21,ˉx(0∼1)22)ˉx3¨ˉx22=ˉf22(ˉx(0∼2)12,ˉx(0∼1)21,ˉx(0∼1)22)+[0Ik2−k1]ˉQ−11ˉG2(ˉx(0∼2)12,ˉx(0∼1)21,ˉx(0∼1)22)ˉx3⋮¨ˉxn1=ˉfn1(ˉx(0∼n+1)12,ˉx(0∼n−i+2)ij|i=2∼n−1,j=1∼2,ˉx(0∼1)nj|j=1∼2)+[Ikn−10]ˉQ−1n−1×ˉGn(ˉx(0∼n+1)12,ˉx(0∼n−i+2)ij|i=2∼n−1,j=1∼2,ˉx(0∼1)nj|j=1∼2)ˉu1¨ˉxn2=ˉfn2(ˉx(0∼n+1)12,ˉx(0∼n−i+2)ij|i=2∼n−1,j=1∼2,ˉx(0∼1)nj|j=1∼2)+[0Ikn−kn−1]ˉQ−1n−1×ˉGn(ˉx(0∼n+1)12,ˉx(0∼n−i+2)ij|i=2∼n−1,j=1∼2,ˉx(0∼1)nj|j=1∼2)ˉu1 | (79) |
where
| ˉf1(ˉx(0∼1)12)=f1(ˉx(0∼1)12) | (80) |
and for $ i = 2,\ldots,n, $
| ˉfi1(⋅)=[Iki−10]ˉQ−1i−1(−¨ˉQi−1ˉxi−2˙ˉQi−1˙ˉxi+fi)ˉfi2(⋅)=[0Iki−ki−1]ˉQ−1i−1(−¨ˉQi−1ˉxi−2˙ˉQi−1˙ˉxi+fi). | (81) |
The FAS model for (79) can be illustrated via the following theorem.
Theorem 4: Let Assumption 4 be satisfied, then under the following transformation:
| ˉxi1=ˉG−11∼i−1(ˉx(2i−2p)p2|p=1∼i−1−Li−1(ˉx(0∼2i−1−2p)p2|p=1∼i−1)),i=2,…,n | (82) |
the strict-feedback system (79) can be equivalently transformed into the following HOFAS model:
| ˉx(2i+2−2p)p2|p=1∼i=Ln(ˉx(0∼2n+1−2p)p2|p=1∼n)+ˉG1∼n(ˉx(0∼2n+1−2p)p2|p=1∼n)ˉu1 | (83) |
where $ L_i(\cdot) $ and $ \bar{G}_{1\sim i}(\cdot) $ are give iteratively by
| Li+1=(¨Li+¨ˉG1∼iˉxi+1,1+2˙ˉG1∼i˙ˉxi+1,1+ˉG1∼iˉfi+1,1ˉfi+1,2) | (84) |
| ˉG1∼i+1=[ˉG1∼i00Iki+1−ki]ˉQ−1iˉGi+1,i=1,2,…,n−1 | (85) |
with $ L_1 = \bar{f}_1\left(\bar{x}_{12}^{(0\sim 1)}\right) $ and $ \bar{G}_{1\sim 1} = \bar{G}_1 $.
In this subsection, a generalized form of SFSs, where each subsystem is high-order, is discussed. The high-order SFSs are firstly considered by Duan in [11]. However, [11] only consider SFSs with uniform dimension. While the dimensions of all subsystems are assumed to be increasing in this work. In particular, we consider the following system:
| {x(m1)1=f1(x(0∼m1−1)1)+G1x2x(m2)2=f2(x(0∼m1)1,x(0∼m2−1)2)+G2x2⋮x(mn−1)n−1=fn(x(0∼mi)i|i=1∼n−2,x(0∼mn−1−1)n−1)+Gn−1xnx(mn)n=fn(x(0∼mi)i|i=1∼n−1,x(0∼mn−1)n)+Gnu | (86) |
where $ x_i\in\mathbb{R}^{k_i},\; i = 1,2,\ldots,n $, $ u\in\mathbb{R}^m $, $ G_i\in\mathbb{R}^{k_i\times k_{i+1}}, \; i = 1, 2,\ldots, n-1, $ and $ G_n\in\mathbb{R}^{k_n\times m} $. We adopt the similar definition for $ x_{i+1,1}\in\mathbb{R}^{k_i} $,$ x_{i+1,2}\in\mathbb{R}^{k_{i+1}-k_i} $, $ f_{i1}\in\mathbb{R}^{k_i} $, $ u_1\in\mathbb{R}^{k_n} $ and $ u_2\in \mathbb{R}^{m-k_n} $ as (8)−(10). Then system (86) can be transformed into the following equivalent form:
| {x(m1)12=f1(x(0∼m1−1)12)+ˉG1x21x(m2)21=f21(x(0∼m1)12,x(0∼m2−1)21,x(0∼m2−1)22)+[Ik10]Q−11ˉG2x31x(m2)22=f22(x(0∼m1)12,x(0∼m2−1)21,x(0∼m2−1)22)+[0Ik2−k1]Q−11ˉG2x31⋮x(mn)n1=fn1(x(0∼m1)12,x(0∼mi)ij|i=2∼n−1,j=1∼2,x(0∼mn−1)nj|j=1∼2)+[Ikn0]Q−1n−1ˉGnu1x(mn)n2=fn2(x(0∼m1)12,x(0∼mi)ij|i=2∼n−1,j=1∼2,x(0∼mn−1)nj|j=1∼2)+[0Ikn−kn−1]Q−1n−1ˉGnu1. | (87) |
For notation convenience, we denote that for $ j\geq i\geq 1 $,
| qi∼j=mi+mi+1+⋯+mj. | (88) |
Based on the equivalent form (87), the HOFAS model for high-order SFS (86) can be obtained, which is given by the following theorem.
Theorem 5: Let Assumption 1 be satisfied, then under the following transformation:
| xi1=ˉG−11∼i−1(x(qp∼i−1)p2|p=1∼i−1−Li−1(x(0∼qp∼i−1−1)p2|p=1∼i−1)),i=2,…,n | (89) |
the SFS (87) can be equivalently transformed into the following HOFA model:
| x_{p2}^{(q_{p\sim n})}\Big|_{p = 1\sim n} = L_n\left(x_{p2}^{(0\sim q_{p\sim n}-1)}\Big|_{p = 1\sim n}\right)+\bar{G}_{1\sim n}u_1 | (90) |
where $ L_i(\cdot) $ and $ \bar{G}_{1\sim i} $ are given iteratively by
| L_{i+1} = \left(\begin{array}{c} {L}_i^{(m_{i+1})}+\bar{G}_{1\sim i}f_{i+1,1} \\ f_{i+1,2}\end{array}\right) | (91) |
| \bar{G}_{1\sim i+1} = \left[\begin{array}{cc}\bar{G}_{1\sim i}&0\\0&I_{k_{i+1}-k_i}\end{array}\right]Q_i^{-1}\bar{G}_{i+1}, | (92) |
| i = 1,2,\ldots,n-1 | (93) |
with $ L_1 = f_1\left(x_{12}^{(0\sim m_1-1)}\right) $ and $ \bar{G}_{1\sim 1} = \bar{G}_1 $.
Similar to the first-order and second-order SFSs, we can also consider the following more general high-order SFS with $ G_i,\; i = 1,2,\ldots,n $, being matrix functions:
| \left\{ \begin{aligned} &{x}_1^{(m_1)} = f_1\left(x_1^{(0\sim m_1-1)}\right)+G_1\left(x_1^{(0\sim m_1-1)}\right)x_2 \\& {x}_2^{(m_2)} = f_2\left(x_1^{(0\sim m_1)},x_2^{(0\sim m_2-1)}\right)\\& \;\;\;\; \; \; \; \; \; \; \; \; +G_2\left(x_1^{(0\sim m_1)},x_2^{(0\sim m_2-1)}\right)x_2 \\&\;\;\; \; \; \; \; \; \; \; \; \; \vdots \\& {x}_{n-1}^{(m_{n-1})} = f_n\left(x_i^{(0\sim m_i)}\Big|_{i = 1\sim n-2},x_{n-1}^{(0\sim m_{n-1}-1)}\right)\\& \; \; \; \; \;\;\;\; \; \; \;\;\;\; \; +G_{n-1}\left(x_i^{(0\sim m_i)}\Big|_{i = 1\sim n-2},x_{n-1}^{(0\sim m_{n-1}-1)}\right)x_n\\& {x}_n^{(m_n)} = f_n\left(x_i^{(0\sim m_i)}\Big|_{i = 1\sim n-1},x_{n}^{(0\sim m_{n}-1)}\right)\\& \; \; \; \; \; \;\;\;\; \;\;\; \; \; +G_n\left(x_i^{(0\sim m_i)}\Big|_{i = 1\sim n-1},x_{n}^{(0\sim m_{n}-1)}\right)u\end{aligned}\right. | (94) |
where $ x_i\in\mathbb{R}^{k_i},\; i = 1,2,\ldots,n, $ and $ u\in\mathbb{R}^m $. Moreover, $ f_i(\cdot) $ and $ G_i(\cdot) $ are sufficiently smooth functions. In this work, we assume that $ k_1\leq k_2\leq \cdots\leq k_n\leq m. $ The following assumption is needed for system (94).
Assumption 5: For any $ \left(x_p^{(0\sim m_p)}\Big|_{p = 1\sim i-1},x_{i}^{(0\sim m_{i}-1)}\right)\in $ $\mathbb{R}^{\sum\limits_{p = 1}^{i-1}(m_p+1)k_p+m_ik_i} $, the matrix function $ G_i $ is of full-row rank, i.e., $ {\rm rank}(G_i) = k_i,\; i = 1,2,\ldots,n $.
It follows from Assumption 5 that there exist sufficiently smooth functions $ Q_i(\cdot) $ and $ \bar{G}_i(\cdot) $ for $ i = 1,2,\ldots,n $ such that:
| G_i(\cdot)Q_i(\cdot) = \left[\begin{array}{cc} \bar{G}_i(\cdot) & 0 \end{array}\right] | (95) |
where $ Q_i(\cdot) $ and $ \bar{G}_i(\cdot) $ are nonsingular. Then we define $ \bar{x} = \{\bar{x}_1^T,\bar{x}_2^T,\ldots,\bar{x}_n^T\}^T $ as follows:
| \begin{split} \bar{x}_1 = \ &x_1\\ \bar{x}_2 = \ &Q_1^{-1}\left(x_1^{(0\sim m_1-1)}\right)x_2\\ \bar{x}_3 = \ &Q_2^{-1}\left( x_1^{(0\sim m_1)},x_2^{(0\sim m_2-1)}\right)x_3\\ \vdots\\ \bar{x}_n = \ &Q_{n-1}^{-1} \left(x_i^{(0\sim m_i)}\Big|_{i = 1\sim n-1},x_{n}^{(0\sim m_{n}-1)}\right)x_n. \end{split} | (96) |
And the inverse mapping from $ \bar{x} $ to x can be written as follows:
| \begin{split} x_1 = \ &\bar{x}_1\\ x_2 = \ &Q_1\left(\bar{x}_1^{(0\sim m_1-1)}\right)\bar{x}_2\triangleq \bar{Q}_1\left(\bar{x}_1^{(0\sim m_1-1)}\right)\bar{x}_2\\ x_3 = \ &Q_2\left(\bar{x}_1^{(0\sim m_1)},\left(\bar{Q}_1\left(\bar{x}_1^{(0\sim m_1-1)}\right)\bar{x}_2\right)^{(0\sim m_2-1)}\right)\bar{x}_3\\ \ \triangleq\;& \bar{Q}_2\left(\bar{x}_1^{(0\sim m_1+m_2-2)},\bar{x}_2^{(0\sim m_2-1)}\right)\bar{x}_3\\ \vdots\\ x_n = \ &Q_{n-1}\bigg(\left(\bar{Q}_{i-1}\bar{x}_i\right)^{(0\sim m_i)}\Big|_{i = 1\sim n-2} \end{split} |
| \begin{split} \ &\left(\bar{Q}_{n-2}\bar{x}_{n-1}\right)^{(0\sim m_{n-1}-1)}\bigg)\bar{x}_n\\ \triangleq \ &\bar{Q}_{n-1}\left(\bar{x}_i^{(0\sim q_{i\sim n-1}-n+i)}\Big|_{i = 1\sim n-1}\right)\bar{x}_n. \end{split} | (97) |
Furthermore, define $ \bar{x}_{i1} $, $ \bar{x}_{i2} $ and $ \bar{u}_1 $ as follows:
| \left\{\begin{aligned} &\bar{x}_{i1} = \left[\begin{array}{cc}I_{k_{i-1}}&0\end{array}\right]\bar{x}_i,\; &&i = 2,\ldots,n\\ &\bar{x}_{i2} = \left[\begin{array}{cc}0&I_{k_i-k_{i-1}}\end{array}\right]\bar{x}_i,\; &&i = 2,\ldots,n\\ &\bar{u}_1 = \left[\begin{array}{cc}0&I_{k_n}\end{array}\right]\bar{Q}_n^{-1}u.\end{aligned}\right. | (98) |
Meanwhile, for notation convenience, we also define that $ \bar{x}_{12} = \bar{x}_1. $ It then follows from (96) and (97) that the following equivalent form of system (94) can be obtained:
| \left\{ \begin{aligned} \bar{x}_{12}^{(m_1)} = \;&\bar{f}_1\left(\bar{x}_{12}^{(0\sim m_1-1)}\right)+ \bar{G}_1\left(\bar{x}_1^{(0\sim m_1-1)}\right)\bar{x}_{21} \\ \bar{x}_{21}^{(m_2)} =\;& \bar{f}_{21}\left(\bar{x}_{12}^{(0\sim m_1+m_2-1)},\bar{x}_{21}^{(0\sim m_2-1)},\bar{x}_{22}^{(0\sim m_2-1)}\right)\\ & +\left[\begin{array}{cc}I_{k_1}&0\end{array}\right]\bar{Q}_1^{-1} \\ & \times\bar{G}_2\left(\bar{x}_{12}^{(0\sim m_1+m_2-1)},\bar{x}_{21}^{(0\sim m_2-1)},\bar{x}_{22}^{(0\sim m_2-1)}\right)\bar{x}_3 \\ \bar{x}_{22}^{(m_2)} =\;& \bar{f}_{22}\left(\bar{x}_{12}^{(0\sim m_1+m_2-1)},\bar{x}_{21}^{(0\sim m_2-1)},\bar{x}_{22}^{(0\sim m_2-1)}\right) \\ & +\left[\begin{array}{cc}0&I_{k_2-k_1}\end{array}\right]\bar{Q}_1^{-1}\\ & \times\bar{G}_2\left(\bar{x}_{12}^{(0\sim m_1+m_2-1)},\bar{x}_{21}^{(0\sim m_2-1)},\bar{x}_{22}^{(0\sim m_2-1)}\right)\bar{x}_3\\& \vdots \\ \bar{x}_{n1}^{(m_n)} =\; & \bar{f}_{n1}\bigg(\bar{x}_{12}^{(0\sim q_{1\sim n}-n+1)}, \\& \bar{x}_{ij}^{(0\sim q_{i\sim n}-n+i)}\Big|_{i = 2\sim n-1,j = 1\sim2},\bar{x}_{nj}^{(0\sim m_n-1)}\Big|_{j = 1\sim2}\bigg) \\& +\left[\begin{array}{cc}I_{k_{n-1}}&0\end{array}\right]\bar{Q}^{-1}_{n-1}\bar{G}_n\bigg(\bar{x}_{12}^{(0\sim q_{1\sim n}-n+1)}, \\& \bar{x}_{ij}^{(0\sim q_{i\sim n}-n+i)}\Big|_{i = 2\sim n-1,j = 1\sim2},\bar{x}_{nj}^{(0\sim m_n-1)}\Big|_{j = 1\sim2}\bigg)\bar{u}_1, \\& \bar{x}_{n2}^{(m_n)} = \bar{f}_{n2}\bigg(\bar{x}_{12}^{(0\sim q_{1\sim n}-n+1)}, \\ &\bar{x}_{ij}^{(0\sim q_{i\sim n}-n+i)}\Big|_{i = 2\sim n-1,j = 1\sim2},\bar{x}_{nj}^{(0\sim m_n-1)}\Big|_{j = 1\sim2}\bigg) \\ &+\left[\begin{array}{cc}0&I_{k_n-k_{n-1}}\end{array}\right]\bar{Q}^{-1}_{n-1}\bar{G}_n\bigg(\bar{x}_{12}^{(0\sim q_{1\sim n}-n+1)}, \\& \bar{x}_{ij}^{(0\sim q_{i\sim n}-n+i)}\Big|_{i = 2\sim n-1,j = 1\sim2},\bar{x}_{nj}^{(0\sim m_n-1)}\Big|_{j = 1\sim2}\bigg)\bar{u}_1\end{aligned}\right. | (99) |
where
| \bar{f}_1 = f_1 | (100) |
and for $ i = 2,\ldots,n $
| \bar{f}_{i1} = \left[\begin{array}{cc}I_{k_{i-1}}&0\end{array}\right]\bar{Q}^{-1}_{i-1}\left(-\sum\limits_{k = 1}^{m_i}C_{m_i}^{k}\bar{Q}^{(k)}_{i-1}\bar{x}_i^{(m_i-k)}+f_i\right) | (101) |
| \bar{f}_{i2} = \left[\begin{array}{cc}0&I_{k_i-k_{i-1}}\end{array}\right]\bar{Q}^{-1}_{i-1}\left(-\sum\limits_{k = 1}^{m_i}C_{m_i}^{k}\bar{Q}^{(k)}_{i-1}\bar{x}_i^{(m_i-k)}+f_i\right) | (102) |
with C being the binomial coefficient defined by
| C_{m_i}^k = \frac{m_i!}{(m_i-k)!k!}. | (103) |
Based on the equivalent system (99), the following result can be obtained for system (94).
Theorem 6: Let Assumption 5 be satisfied, then under the following transformation:
| \begin{split} \bar{x}_{i1} = &\bar{G}_{1\sim i-1}^{-1}\bigg( \bar{x}_{p2}^{(q_{p\sim i-1})}\Big|_{p = 1\sim i-1}\\ &-L_{i-1}\left(\bar{x}_{p2}^{(0\sim q_{p\sim i-1}-1)}\Big|_{p = 1\sim i-1}\right)\bigg), \; i = 2,\ldots, n \end{split} | (104) |
the SFS (99) can be equivalently transformed into the following HOFAS model:
| \bar{x}_{p2}^{(q_{p\sim n})}\Big|_{p = 1\sim n} = L_n+\bar{G}_{1\sim n}u_1 | (105) |
where $ L_i(\cdot) $ and $ \bar{G}_{1\sim i}(\cdot) $ are given iteratively by
| L_{i+1} = \left(\begin{array}{c} {L}_i^{(m_{i+1})}+\displaystyle \sum\limits_{k = 1}^{m_{i+1}}C_{m_{i+1}}^{k}\bar{G}^{(k)}_{1\sim i}\bar{x}_{i1}^{(m_{i+1}-k)}+\bar{G}_{1\sim i}f_{i+1,1} \\ f_{i+1,2}\end{array}\right) | (106) |
| \bar{G}_{1\sim i+1} = \left[\begin{array}{cc}\bar{G}_{1\sim i}&0\\0&I_{k_{i+1}-k_i}\end{array}\right]Q_i^{-1}\bar{G}_{i+1}, | (107) |
| i = 1,2,\ldots,n-1 | (108) |
with $ L_1 = \bar{f}_1\left(\bar{x}_{12}^{(0\sim m_1-1)}\right) $ and $ \bar{G}_{1\sim 1} = \bar{G}_1\left(\bar{x}_1^{(0\sim m_1-1)}\right) $.
Remark 3: Different from (43), the mapping (96) is not a differential homeomorphism from x to $ \bar{x} $ as it is not 1-1. However, it can be verified that if the system (105) is stabilized under controller $ u_1 $, then the system (94) is also stabilized. Therefore, the HOFAS model (105) is applicable for control of system (94).
In this section, we provide two numerical examples to illustrate the effectiveness of our results.
Consider the following first-order system:
| \qquad\qquad\qquad \dot{x}_1 = 0.1x_1+\left[\begin{array}{cc} 1&1 \end{array}\right]x_2 | (109a) |
| \qquad\qquad\qquad \dot{x}_2 = x_1x_2+\left[\begin{array}{ccc} 0&1&1\\1&2&2 \end{array}\right]u. | (109b) |
It can be seen that system (109) is a first-order SFS with increasing dimensions. By using the method in this paper, we can obtain the following FAS model of system (109):
| \ddot{x}_{12} = 0.1\dot{x}_{12}+x_{12}x_{21}+\left[\begin{array}{cc}0&1\end{array}\right]u_1\setcounter{equation}{110} | (110) |
| \dot{x}_{22} = x_{12}x_{22}+\left[\begin{array}{cc}1&2\end{array}\right]u_1. | (111) |
Then, we can design the following controller:
| u_1 = \left[\begin{array}{cc}-2&1\\1&0\end{array}\right]\left[\begin{array}{c}-2.1\dot{x}_{12}-x_{12}-x_{12}x_{21}\\-x_{12}x_{22}-x_{22}\end{array}\right]. | (112) |
The response is then shown by Fig. 1. The effectiveness of our controller can be thus illustrated.
Consider the following second-order system:
| \qquad\qquad \ddot{x}_1 =-0.1\dot{x}_1x_1+\left[\begin{array}{cc} 2&3 \end{array}\right]x_2 | (113a) |
| \qquad\qquad\ddot{x}_2 = \dot{x}_2+\left[\begin{array}{ccc} 0&1&2\\3&2&1 \end{array}\right]u. | (113b) |
It can be seen that system (113) is a second-order SFS with increasing dimensions. By using the method in this paper, we can obtain the following FAS model of system (113):
| {x}^{(4)}_{12} = (-0.1\dot{x}_{12}x_{12})^{(2)}+\dot{x_{21}}+\left[\begin{array}{cc}0&1\end{array}\right]u_1\setcounter{equation}{114} | (114) |
| {x}^{(2)}_{22} = \dot{x}_{22}+\left[\begin{array}{cc}3&2\end{array}\right]u_1. | (115) |
Then we can design the following controller:
| \begin{split} u_1 =\;& \left[\begin{array}{cc}-\dfrac{2}{3}& \dfrac{1}{3} \\1&0\end{array}\right]\\ &\times\left[\begin{array}{c}(0.1\dot{x}_{12}x_{12})^{(2)}-4x^{(3)}_{12}-6x^{(2)}_{12}-4x^{(1)}_{12}-4x_{12}-\dot{x_{21}}\\-3\dot{x}_{22}-x_{22}\end{array}\right]. \end{split} | (116) |
The response is then shown by Fig. 2. The effectiveness of our controller can be thus illustrated.
In this paper, we have provided HOFAS models for SFSs with increasing and increasing dimensions. First-order, second-order and high-order cases have been all considered. We first considered SFSs where $ G_i, \; i = 1,2,\ldots,n, $ are all constant matrices and then extended the results to more general cases where $ G_i,\; i = 1,2,\ldots,n, $ could be matrix functions. A coordinate transformation has been introduced to obtain the HOFAS model. Based on the HOFAS models, the exponentially stabilizing controllers have been directly given. In fact, there is no obstacle for us to extend the results to discrete-time SFSs with increasing dimensions and SFSs with state delays and increasing dimensions. Related proofs can be established by combining the methods of this paper and the existing works. Our future work may concentrate on providing HOFAS models for SFSs with decreasing dimensions.
| [1] |
M. Krstic and M. Bement, “Nonovershooting control of strict-feedback nonlinear systems,” IEEE Trans. Automatic Control, vol. 51, no. 12, pp. 1938–1943, 2006. doi: 10.1109/TAC.2006.886518
|
| [2] |
Z. Pan and T. Basar, “Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems,” IEEE Trans. Automatic Control, vol. 43, no. 8, pp. 1066–1083, 1998. doi: 10.1109/9.704978
|
| [3] |
N. Bekiaris-Liberis and M. Krstic, “Stabilization of linear strict-feedback systems with delayed integrators,” Automatica, vol. 46, no. 11, pp. 1902–1910, 2010. doi: 10.1016/j.automatica.2010.07.008
|
| [4] |
W. Li, X. Yao, and M. Krstic, “Adaptive-gain observer-based stabilization of stochastic strict-feedback systems with sensor uncertainty,” Automatica, vol. 120, p. 109112, 2020. doi: 10.1016/j.automatica.2020.109112
|
| [5] |
Y. Li, Y. Fan, K. Li, W. Liu, and S. Tong, “Adaptive optimized backstepping control-based rl algorithm for stochastic nonlinear systems with state constraints and its application,” IEEE Trans. Cybernetics, vol. 52, no. 10, pp. 10542–10555, 2021.
|
| [6] |
P. V. Kokotovic, “The joy of feedback: Nonlinear and adaptive,” IEEE Control Systems Magazine, vol. 12, no. 3, pp. 7–17, 1992. doi: 10.1109/37.165507
|
| [7] |
M. Krstic, P. V. Kokotovic, and I. Kanellakopoulos, Nonlinear and Adaptive Control Design. John Wiley & Sons, Inc., 1995.
|
| [8] |
G. Meunier, B. Boulet, and M. Nahon, “Control of an overactuated cable-driven parallel mechanism for a radio telescope application,” IEEE Trans. Control Systems Technology, vol. 17, no. 5, pp. 1043–1054, 2009. doi: 10.1109/TCST.2008.2004812
|
| [9] |
R. Olfati-Saber, “Cascade normal forms for underactuated mechanical systems,” in Proc. 39th IEEE Conf. Decision and Control (Cat. No. 00CH37187), vol. 3, 2000, pp. 2162–2167.
|
| [10] |
G. Duan, “High-order fully actuated system approaches: Part I. Models and basic procedure,” Int. Journal of Systems Science, vol. 52, no. 2, pp. 422–435, 2021. doi: 10.1080/00207721.2020.1829167
|
| [11] |
G. Duan, “High-order fully actuated system approaches: Part II. Generalized strict-feedback systems,” Int. Journal of Systems Science, vol. 52, no. 3, pp. 437–454, 2021. doi: 10.1080/00207721.2020.1829168
|
| [12] |
G. Duan, “Fully actuated system approaches for continuous-time delay systems: Part 1. Systems with state delays only,” Science China Information Sciences, vol. 66, article 112201, 2023.
|
| [13] |
D.-W. Zhang, G.-P. Liu, and L. Cao, “Coordinated control of high-order fully actuated multiagent systems and its application: A predictive control strategy,” IEEE/ASME Trans. Mechatronics, vol. 27, no. 6, pp. 4362–4372, 2022. doi: 10.1109/TMECH.2022.3156587
|
| [14] |
Y. Yu, G.-P. Liu, Y. Huang, and J. M. Guerrero, “Coordinated predictive secondary control for dc microgrids based on high-order fully actuated system approaches,” IEEE Trans. Smart Grid, vol. 15, no. 1, pp. 19–33, 2024.
|
| [15] |
R. Meng, C. Hua, K. Li, and P. Ning, “Adaptive event-triggered control for uncertain high-order fully actuated system,” IEEE Trans. Circuits and Systems II: Express Briefs, vol. 69, no. 11, pp. 4438–4442, 2022.
|
| [16] |
R. Dong, C. Hua, K. Li, and R. Meng, “Adaptive fault-tolerant control for high-order fully actuated system with full-state constraints,” Journal of the Franklin Institute, vol. 360, no. 12, pp. 8062–8074, 2023.
|
| [17] |
M. Cai, X. He, and D. Zhou, “Fault-tolerant tracking control for nonlinear observer-extended high-order fully-actuated systems,” Journal of the Franklin Institute, vol. 360, no. 1, pp. 136–153, 2023. doi: 10.1016/j.jfranklin.2022.11.025
|
| [18] |
D. Gu and S. Wang, “A high-order fully actuated system approach for a class of nonlinear systems,” Journal of Systems Science and Complexity, vol. 35, no. 2, pp. 714–730, 2022. doi: 10.1007/s11424-022-2041-4
|
| [19] |
M. Cai, X. He, and D. Zhou, “Low-power fault-tolerant control for nonideal high-order fully actuated systems,” IEEE Trans. Systems, Man, and Cybernetics: Systems, vol. 53, no. 8, pp. 4875–4887, 2023.
|
| [20] |
D.-W. Zhang, G.-P. Liu, and L. Cao, “Proportional integral predictive control of high-order fully actuated networked multiagent systems with communication delays,” IEEE Trans. Systems, Man, and Cybernetics: Systems, vol. 53, no. 2, pp. 801–812, 2022.
|
| [21] |
L. Zhang, L. Zhu, and C. Hua, “Practical prescribed time control based on high-order fully actuated system approach for strong interconnected nonlinear systems,” Nonlinear Dynamics, vol. 110, no. 4, pp. 3535–3545, 2022. doi: 10.1007/s11071-022-07820-w
|
| [22] |
A.-G. Wu, J. Zhang, and Y. Ji, “A fully actuated system approach for stabilization of discrete-time multiple-input nonlinear systems with distinct input delays,” Journal of Systems Science and Complexity, vol. 35, no. 2, pp. 670–687, 2022. doi: 10.1007/s11424-022-2046-z
|
| [23] |
Q. Hu, B. Li, B. Xiao, and Y. Zhang, Control Allocation for Spacecraft Under Actuator Faults. Springer, 2021.
|
| [24] |
A. Witkowska and R. Śmierzchalski, “Adaptive dynamic control allocation for dynamic positioning of marine vessel based on backstepping method and sequential quadratic programming,” Ocean Engineering, vol. 163, pp. 570–582, 2018. doi: 10.1016/j.oceaneng.2018.05.061
|
| [25] |
D. Khimani and M. Patil, “Fault-tolerant sliding mode control for uncertain over-actuated systems,” IFAC Proceedings Volumes, vol. 47, no. 1, pp. 748–753, 2014. doi: 10.3182/20140313-3-IN-3024.00155
|
Figures(2)
X. Xu and G.-R. Duan, “High-order fully actuated system models for strict-feedback systems with increasing dimensions,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 12, pp. 2451–2462, Dec. 2024. doi: 10.1109/JAS.2024.124599
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