Bin Meng·Yun-Bo Zhao·Jing-Jing Mu

Abstract The existence of error when compressing nonlinear functions into the coefficients of the characteristic model is known to be a key issue in existing characteristic modeling approaches,which is solved in this work by an error-free compression method.We first define a key concept of relevant states with corresponding compressing methods into their coefficients,where the coefficients are continuous and bounded and the compression is error-free.Then,we give the conditions for decoupling characteristic modeling for MIMO systems,and sequentially,we establish characteristic models for nonlinear systems with minimum phase and relative order two as well as the flexible spacecrafts,realizing the equivalence in the characteristic model theory.Finally,we explicitly explain the reasons for normalization in the characteristic model theory.

Keywords Characteristic modeling·Relevant states·Error-free compression·Flexible spacecraft·Normalization

1 Introduction

The characteristic model theory founded by Academician Wu Hongxin in the 1980s [1–3] has already witnessed many of its successful stories in the aerospace and industry fields,e.g.,the reentry lift control of the Shenzhou spacecraft [4–6],the rendezvous and docking control of the Shenzhou spacecraft and Tiangong 1 [7,8],the skip reentry control of the Chang’e-5 [9],the electrolytic aluminum control [10],just to name a few.The characteristic model theory consists of three ingredients,namely,characteristic modeling,parameter identification and all-coefficient adaptive control.In the characteristic modeling stage,the dynamics of the controlled systems is transformed to establish the characteristic model.Then,in the parameter identification stage,the projection gradient method or the projection least square method are used to identify the bounded coefficients,which is key to the success of the characteristic model.Finally,the control law is designed using the so-called all-coefficient adaptive control approach,examples of which include maintenancetracking control,golden-section adaptive control,logic integral control,logic differential control,etc.[4].

As can be understood,characteristic modeling is the first step of and key to the characteristic model theory,which has been studied extensively in the recent decades.For linear systems,the problem has been solved by proving that general linear time-invariant systems can be transformed to the second-order linear time-varying difference equations with bounded coefficients under certain conditions [11,12].For second-order affine nonlinear systems the second-order characteristic model has also been given by introducing nonlinear time scale [13].The cyclic demonstration problem is solved by a state-dependent identification projection region and a novel adaptive control method [14,15].Also,one assumption on characteristic modeling is that the compressed functions should be zero for zero system state,since otherwise the modeling errors will be infinity under certain conditions,which can be solved by the translation transformation method [16].

We notice that key to characteristic modeling is the errorfree compression of nonlinear functions into the coefficients of a characteristic model,but unfortunately error is always present in all existing methods [4].On the other hand,the reasons of the so-called“normalization”phenomenon need also be explained,where the bounds of the output coefficients of the characteristic model for different controlled systems,systems different time scales,linear or nonlinear,are all the same.Motived by the above challenges,in the present work,we

– Define a key concept of“relevant states”for nonlinear functions,which ensures the equivalence in the compression process.

– Establish the necessary and sufficient conditions for the first time,under which MIMO systems can be transformed into a decoupled characteristic model.

– Establish the characteristic models of nonlinear systems with minimum phase and relative order two as well as the flexible spacecraft,realizing the equivalence in the characteristic model theory.

– Address the normalization problem in the characteristic model theory.

In what follows,we first formulate the problem of interest in Sect.2,then present the main results in Sect.3,and finally concludes the paper in Sect.4.

2 Problem formulation and preliminaries

We first formulate the considered problems with preliminaries on characteristic modeling with error-free compression.

2.1 Problem formulation

In [19],it was proved that the dynamics of a flexible spacecraft can be transformed into a standard form of input–output linearization with minimum phase and relative order two.With this in mind,we may consider the following affine nonlinear system:

wherex1∈?n,x2∈?nandη∈?pare the system states,u∈?nis the system input,f∈?n,g∈?n×nandq∈?pare the smooth differential functions,f(0,0,0)=0 andq(0,0,0)=0.

For the system in (1),we make the following assumption.

Assumption 1(a) The system in (1) is a minimum-phase system;(b) the derivatives of functionsfandqwith their arguments are bounded;and (c)gis nonsingular.

Remark 1Assumption 1 is necessary for the global stability of the characteristic model based adaptive control.In fact,adaptive control can be rewritten as an adaptive PID control law with bounded coefficients [4],and Assumption 1 is one of the necessary conditions for the global stability of PID control in [17].

An advantage of characteristic model theory is that the coefficients of the characteristic model have determined bounds,ensuring transient stability of the closed-loop systems.The bounds are determined by adjusting the sampling period according to how fast the system dynamics can be.This system property can be measured by the eigenvalues for linear time-invariant (LTI) systems,and by the following time scale introduced in [18] for nonlinear systems.

Definition 1Define the time scale for the system in (1) as follows:

The characteristic model is of the form of linear timevarying difference equations with bounded coefficients,with its particular focus on second-order model [4,11–16],given as follows:

whereyCanduCare the output and input of the characteristic model in (2),with their dimensions being equal to those of the controlled system,anda1,a2andbare matrices with appropriate dimensions.The bounds of the output coeffi-cients are given by

and the bound of the input coefficientb(k) is given according to the specific physical properties of the input matrix of the controlled systems.In the above formulas,Tscaleis the time scale of the controlled systems,andTis the sampling period.

Whena1anda2are diagonal matrices,(2) is said to be the decoupled characteristic model.

2.2 Preliminaries

The following lemma gives an important property of the time scale.

Lemma 1Assume that Assumption1holds.For the following systems,

ProofLet the derivative variable of (5) bet,and transform(5) into the system with time scale 1,as follows:

which then means that (6) holds.The other case can be proved similarly.

Remark 2When degenerated to linear systems,Lemma 1 holds as well.It can be known from linear system theory thatf1andf2are the product and sum of eigenvalues,respectively.Eq.(6) holds from the definition of minimum time constant of linear system theory.Error-free compression of nonlinear functions into the coefficients of state variables is key to characteristic modeling.We define the following relevant states which is useful in the error-free compression.

Definition 2For a functionh(s1,s2,…,sn),if

thens1,s2,…,smare said to be a group of relevant states,ands1,s2,…,smare relevant.

Remark 3Relevant states exist in general.In fact,all states are relevant for a system with zero equilibrium according to Definition 2.From nonlinear system theory,the nonzero equilibrium can always be transferred to zero.

Remark 4There may exist multiple groups of relevant states for a system.For example,the following function

has three groups of relevant statess1;s2;s1ands2.In this present work,we need to find out the group of relevant states with the fewest elements to build the characteristic model.

For a functionh(s1,s2,…,sn) with all states being relevant,define fors1

For a functionh(s1,s2,…,sn) with relevant states beingsi,i=1,2,…,m,m

It is then easy to see that

The above results are summarized in the following lemma.

Lemma 2Consider a function h(s1,s2,…,sn)with bounded.If and only if the relevant states of h are si,i=1,…,m,m≤n,h can be compressed into the coefficients of the relevant states si without error,as shown in(9)where the coefficients are shown in(7)and(8)for m=n,and in(12)with the coefficients(10)and(11)for m

3 Main results

This section investigates the characteristic modeling problem with error-free compression for the nonlinear systems(1) and flexible spacecraft,and the normalization phenomena in the characteristic model theory.The characteristic modeling problem for the external dynamics of (1) is first considered.

3.1 Characteristic modeling for second-order affine nonlinear systems

Consider the external dynamics of (1),

wherex1i∈?,x2i∈?,ui∈?,fi∈?,gi∈?1×n,i=1,2,…,n.By Definition 1,f(0,0)=0 means there exist relevant states forfi,i=1,2,…,n,which can be expressed as

wherekjandljbelong to the set {1,2,…,n},j=1,2,…,i.By Lemma 2,designing the continuous bounded coefficients with respect to the relevant states (15) yields

Rewriting (18) in its matrix form and then taking Euclidean discretization yields

wherea1(k)∈?n×nanda2(k)∈?n×nsatisfy (3) and (4) by Lemma 1,and

It is easy to see that by further taking the output of the characteristic model asyC=x1,(19) has the form of the characteristic model in (2),i.e.,we have established the characteristic model (2) for (13).

We proceed to consider the problem of decoupling characteristic modeling,which diminishes the number of coefficient identification,hence simplifying the control design.

Lemma 2 implies that iffi,i=1,2,…,n,satisfy the following conditions:

Apparently,a1(k) anda2(k) are diagonal,which means that(24) is decoupled.It is easy to see that by further taking the output and input of the characteristic model asyC=x1anduC=u,(23) has the form of the characteristic model (2);that is,we have established the decoupled characteristic model(2) for (13),wherea1(k)∈?n×nanda2(k)∈?n×nare diagonal matrices and satisfy (3) and (4),andb(k) is with the form of (20).Eq.(21) is necessary and sufficient for establishing the decoupled characteristic model,which is said to be the decoupling condition of characteristic model.

We summarize the above derivations into the following two theorems.

Theorem 1If the nonlinear system in(13)satisfies Assumption1,then there exist relevant states for the nonlinear function f in(13),and f can be compressed into the coefficients of the relevant states without error with continuously bounded coefficients(16),as shown in(17).Furthermore,(13)can be transformed into the second-order characteristic model(2)with the coefficients satisfying(3),(4),and(20).

Theorem 2If the nonlinear system in(13)satisfies Assumption1and the decoupling condition(21),then for i=1 ,2,… ,n,fi only has two relevant states,x1i and x2i,and f can be compressed into the coefficients of the relevant states without error with continuously bounded coefficients,as shown in(22).Furthermore,(13)can be transformed into the secondorder decoupled characteristic model(2)with the coeffi-cients satisfying(3),(4),and(20),and a1and a2diagonal.

Remark 5The decoupling conditions (21) for establishing the decoupled characteristic model are given for the first time in this work.

Remark 6Equations (17) and (22) show that the compression is error-free,ensuring the equivalence in the characteristic model theory.

3.2 Characteristic modeling for the nonlinear systems in (1)

This section gives a characteristic modeling method with error-free compression for the higher-order nonlinear systems in (1),realizing the equivalence in the characteristic model theory.

Similar to (14),let

whereqi∈?,ηi∈?,i=1,2,…,p.In the following,we establish the characteristic model of (1).First,compress the nonlinear functionfandqof (1) into the coefficients of the relevant states.Similar to the deduction procedure in Section 3.1,forf,there exist relevant states,

Therefore,(1) can be rewritten in matrix form by substituting (25) and (26) into it,

whereF1,F2,Fη,Q1,Q2,andQηare matrices with appropriate dimensions.We can see that the nonlinear functionsfandqin (1) are compressed into the coefficients of the relevant states without error by comparing (1) and (27).

We further deal with the internal statesηas follows.By matrix theory,we can find the general solution ofηfrom the second equation of (27),

Substituting the third equation of (27) into (29) cancelsand then substituting (28) into the resultant equation cancelsη. This results in the third-order differential equation ofx1with the cancellation of the internal statesηand

Taking Euler discretization to (30), the third-order characteristic model can be established by further takingyC=x1anduC=u,

In characteristic model theory, the second-order one is of special significance since the intelligent adaptive control methods based on it have derived successful and widely applications [4]. Therefore, we further establish the secondorder characteristic model for (1) by introducing the online estimation methods. It follows from the first and second equations in (27) that

By designing online estimators, for example, the extended state observer (ESO), to estimate the internal statesFηη, and denoting the estimated states as(34) can be represented as follows:

Furthermore, by taking Euler discretization to (35), the second-order characteristic model (2) can be established where

We summarize the above deductions into the following theorem.

Theorem 3If the nonlinear systems in (1) satisfy Assumption 1, then (1) can be transformed into the third-order characteristic model in (31) with the coefficients satisfying (32) and(33); furthermore, if the internal states are estimated and the intermediate controlis designed as in (36), then (1) can be transformed into the second-order characteristic model(2) with the input uC =and the coefficients satisfying (3),(4), and (37).

3.3 Characteristic modeling for flexible spacecraft

This section proposes the characteristic modeling method with error-free compression for flexible spacecraft, realizing the equivalence in the characteristic model theory.

Consider the following flexible spacecraft attitude dynamics (1-3-2 Euler rotation sequence),

φ,θ,ψare the roll,the pitch,and the yaw attitudes;wx,wy,wzare the roll,the pitch and the yaw angular velocities,respectively;

represents the cross-product operator matrix;Isis the inertia matrix of spacecraft,Tsis the external torque vector acting on the spacecraft;andηL∈?landηR∈?lare the mode coordinate matrices,ξL >0 andξR >0 are the mode damping coefficients,wL >0 andwR >0 are the mode frequencies,FsL∈?3×landFsR∈?3×lare the coupled matrices between flexible and rigid bodies with the subscriptLandRbeing the left and right solar array,respectively.

Here,we consider the case where 0 ≤ψ <90?(forψ≥90?,the spacecraft model in terms of quaternion parameterization is needed).Through simple computation,C(x1)is nonsingular for 0 ≤ψ <90?.Let the main body inertial matrix be

Assumption 2Rsis nonsingular.

The following properties are given in [19].

Lemma 3The relative order of the flexible spacecraft dynamics(38)is(2,2,2),and its zero dynamics are exponentially stable.

Using differential homeomorphic transformation,(38) is transformed into the following input-output linearization form,

By comparison,we can see that (39) is already in the form of (27),meaning that it is not necessary to compress the nonlinear functions for the flexible spacecraft (38),which consequently means that Assumption 1 is unnecessary.From Theorem 3,we can obtain the characteristic modeling results directly.

Proposition 1If the flexible spacecraft in(38)satisfies Assumption2,then(38)can be transformed into the thirdorder characteristic model in(31)with the coefficients satisfying(32)and(33);furthermore,if the internal statesa(z)η are estimated asand the intermediate controlisdesigned as follows:

then(38)can be transformed into the second-order charac-teristic model(2)with the inputand the coefficientssatisfying(3),(4),and(37).

3.4 Normalization essence

This section provides insights on“normalization”in the characteristic model theory through both linear and nonlinear systems.“normalization”means that the bounds of the output coefficients of the characteristic model for different controlled systems,systems different time scales,linear or nonlinear,are all the same.

3.4.1 LTI systems

Consider the following controlled systems with different poles:

whereλi <0 andkiare real numbers fori=1,2,…,n.It follows from computer control theory that the pulse transfer function for (40) is

andG(0) is the static gain.

In the characteristic model theory,the static gain can be transformed to 1 through the input-output transformation [20,21].Thus,without loss of generality,we may assumeG(0)=1 .Inspection of (41) and (42) shows that for different systems,only the eigenvalueλiand the sampling periodTare different,but they all appear in the form of multiplication with same orders.Defining the minimum time constant [4],

In the characteristic model theory,we generally choose the sampling periodTaccording toTscale[4],

Using (43) and (44),we can obtain the same bounds for all coefficients in (41) and (42) for linear different systems (40),which implies the realization of the normalization for the linear system in (40).

3.4.2 Nonlinear systems

The reasons for the normalization for the nonlinear system in (1) can be seen from (3),(4),and (32).By choosing the sampling period according to (44),the bounds of the coefficients of the characteristic model in (2) and (31) for different controlled systems are equal,which implies the realization of normalization for the nonlinear systems in(1).

In summary,the essence of the characteristic model theory is to choose the sampling period according to the change pace of controlled systems,and further taking advantage of the structural features of the characteristic model to realize the normalization.For both linear and nonlinear systems,the output coefficient bounds of their same-order characteristic model are the same.

4 Conclusion

The characteristic modeling problem with error-free compression for nonlinear systems is investigated.A key concept of the relevant states is defined with its corresponding compression method,where the coefficients are continuous and bounded and the compression is error-free.The conditions given for decoupling characteristic modeling for MIMO systems provide bases,based on which the establishment of the characteristic models for the nonlinear systems with minimum phase and relative order two and the flexible spacecraft realizes the equivalence in the characteristic model theory.Lastly,reasons for normalization in the characteristic model theory are given.The work contributes fundamentally to the characteristic model theory.

AcknowledgementsThis work was supported by the National Key R&D Program of China (Grant Nos.2018YFA0703800 and 2018AAA0100800),the Science and Technology on Space Intelligent Control Laboratory Foundation of China (Grant No.ZDSYS-2018-04)and the National Natural Science Foundation of China (Grant Nos.U20B2054 and 51805025).