# Input-Output Transformation Using the Feedback of Nonlinear Electrical Circuits: Algorithms and Linearization Examples

Authors: Sebastian Różowicz; Andrzej Zawadzki
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper addresses the problem of nonlinear electrical circuit input-output linearization. The transformation algorithms for linearization of nonlinear system through changing coordinates (local diffeomorphism) with the use of closed feedback loop together with the conditions necessary for linearization are presented. The linearization stages and the results of numerical simulations are discussed.


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## Body

## 1. Introduction

While analyzing actual physical systems we often have to deal with nonlinear problems. Most of them can be described by a system of differential equations in the normal form of Cauchy system. This applies to the dynamics of electrical circuits with concentrated parameters, as well as mechanical or electromechanical systems. The research of such systems is reduced to mathematical analysis of the model, which in a general case is nonlinear. The theory of linear systems of differential equations in the normal form of Cauchy system is explicit with respect to the time derivatives. Therefore, the linearized-preferably linear models of nonlinear systems are sought. This allows considering the dynamic behaviour of the object as a linear one and then going back to the original system by the inverse transformation. This approach may be useful for qualitative research including issues of stability, control, and the existence of periodic solutions.In general, the nonlinear model describing the dynamics in the state space can be represented by(1)x˙=fx+∑i=1mgixui,x∈Rn,u∈Rmy=hx,y∈Rpwheref and gi are smooth vector fields defined on the manifoldM = Rn, called the state space,h is a smooth mapping defined on the state spaceM intop-dimensional space of outputs, and Rp,h:M→Rp.The methods of differential geometry allow for the linearization of nonlinear systems (1) to the following linear form [1–3]: (2)dzdt=Az+Bvy=Czwherezandv are new state and input vectors,w is an output vector in a linearized system, andA,B, andC are matrixes of proper size.Models (2) are obtained as a result of linearizing transformation, transforming the nonlinear system by changing coordinates in the state space. The change consists in the replacement of the original state variablesxwith new variableszdescribing the system in a new state space [1–13]. The linearizing transformation can be written as(3)Sx:x⇒zwhereS(x) is linearizing transformation andz(t) a new state vectorAfter transformation the state vector is of the form:(4)z=Sxwhere S(x)=S1x,S2x,…,SnxT is a mapping defined on an open set of Rn space with Rn values and the following properties:(i)
S is locally invertible; i.e., there exists the inverse mapping S-1 (that can be transformed back to the original components of state vectorx) such that(5)x=S-1zforanyzinIm⁡SwhereIm⁡S is the image in theS transformation;(ii)
S and S-1 are smooth mappings.The transformation of this kind is called a diffeomorphism [2, 3, 10, 14].The design methods of nonlinear control systems have been developed for many years. Based upon differential geometry the necessary and sufficient conditions for linearization of nonlinear control system (by changing coordinates) have been obtained [1–4, 14–17]. Those conditions, except for the planar case, turn out to be restrictive. Consequently, the natural problem of finding normal forms for nonlinearizable systems appears. The problem is, however, very complex and has been extensively studied.The systems that cannot be linearized and transform only to quasilinear systems can be further linearized by feedback. In this case, the combination of linearization by the transformation of state variables and input transformationu(t) with feedback [18–27] should be used. As a result of coordinate transformation and the introduction of feedback, the state vector in a new coordinate system can be represented as(6)z=Sx+feedbackAs differential equations are difficult to solve and many systems are not feedback linearizable, the feedback linearization has been extended in various ways (1). One of them is the method based upon the theory of singularities of vector fields and distributions discussed in [16, 28–31]. Other methods were proposed by a few authors including Gardner [32] describing the geometry of feedback equivalence [33–36], by Bonnard [37, 38] and by Jakubczyk [39, 40] using basing on the Hamiltonian formalism for optimal control problems. The concepts of partial linearization, approximate linearization, pseudolinearization, extended linearization, etc.; see, e.g., [14, 17, 22, 41–47], introduced. Numerous papers on linearization of nonlinear system with the use of closed feedback loop inspired by Kang and Krener [24] was presented for single-input systems by Tall and Respondek [48] and for multi-input systems by Tall [49]. In [50] the authors solved these two problems by defining algorithms allowing the explicit computation of the linearizing state (resp. feedback) coordinates for any nonlinear control system that is linearizable (resp., feedback linearizable).Transformation of nonlinear system into linear one by linearization (with the local equivalence of the system dynamics ensured) is very useful in solving practical problems concerning nonlinear behaviour of objects. This approach simplifies the analysis of nonlinear systems. Given the relevance of the problems the attempt has been made to develop efficient algorithms for linearization of a given certain class of nonlinear systems.This paper aims to develop linearization algorithms of input-output nonlinear systems through the transformation of state space with the use of closed-loop feedback. The use of geometric methods in the theory of electrical circuits has been presented. The linearization of nonlinear SISO systems has been discussed and extended for the linearization of MIMO systems. The elements of Lie algebra [3, 5, 6, 51, 52] have been used for the construction of linearizing transformation (6). It is worth mentioning that the linearization techniques have already proved to be very useful and are still of interest nowadays.The paper is organized as follows. Section2 provides basic definitions and theorems concerning the conditions to be met for the linearization of a nonlinear system. The elements of the Lie algebra used to construct a new base of state space are presented. The algorithms employed for the linearization of nonlinear mathematical models and the examples (with emphasis on electrical circuits) proving that the algorithms are correct are given in Section 3. Conclusions and comments are presented in Section 4, followed by the bibliography.


## 2. Preliminaries

This section outlines some basic definitions and concepts of differential geometry. The efforts were made to present them in a simplified and concise form. The indepth explanation can be found in the bibliography. In the analysis of nonlinear systems the operation including real function h and vector field f defined on the space manifoldM deserves particular attention. The result of the operation is a smooth real-valued function defined for each x from theM set as follows.Definition 1.
Leth mapping Rn→Rn be a smooth scalar function ofn variables, x=(x1,x2,…,xn)T,∈Rn, and f:Rn→Rn a vector field defined on manifold M=Rn; then the Lie derivative of scalar function h(x)=h(x1,x2,…xn) in the direction of fieldf is a scalar function given by the formula:(7)Lfh=∂h∂x∘f=∑i=1n∂h∂xifiThere exists the following recurrence formula for this derivative:7aLf0h=h-zeroderivative  fori=1,2,…7bLfnh=LfLfn-1h-nthderivative
The second operation concerns vector fieldsf andg defined on the manifoldM = Rn space. The result of the operation (defined below) is a new smooth vector field.Definition 2.
Letf andg be vector fields defined on the manifoldM = Rn. Lie brackets of the vector field are called the third vector field, defined by the following relation: (8)f,g=∂g∂x∘f-∂f∂x∘g,where ∘– scalar product.
For such notation of square brackets we obtain the recurrence formulae8aadf0g=g8badfig=f,adfi-1gThe direct linearization occurs for nonlinear systems for which the vector field represented by Lie brackets g,adf1g,adf2g,…,adfn-1g satisfies the conditions of Theorem 3. After introducing the notations of the conditions of controllability (C) and observability (O) Theorem 3 takes the following form.Theorem 3.
The following conditions are equivalent [53]: (a)
System (1) is equivalent, under a (local) diffeomorphism z=S(x), to the controllable and observable linear system (2).(b)
System (1) satisfies (C), (O), and Lgi1Lgi2⋯Lgiκhj=0, for all 1≤j≤p, all κ≥1, and all 0≤iq≤m (where g0 =f), when at least two of the iq’s differ from 0.(c)
System (1) satisfies (C), (O), and LgjLfqhi= constant for all 1≤j≤m, all 1≤i≤p, and all 0≤q≤κj+μi-1.(d)
System (1) satisfies (C), (O), and [adfqgj,adfrgi]=0,forall1≤i,j≤m,all0≤q≤κj,0≤r≤Ki and LgiLfqhi= constant for all 1≤j≤m, all 1≤i≤p, and all 0≤q≤κj-1.(e)
System (1) satisfies (C), (O), and dLfμihi=∑j=1p∑k=0μi-1CjkidLfkhj, for all 1≤i≤p, where Cjki’s are real numbers and LgiLfqhi= constant for all 1≤j≤m, all 1≤i≤p, and all 0≤q≤μi-1.
The proof of Theorem3 and the extensive discussion of conditions are presented in [53].In the analysed model of the nonlinear system (1) the output ofy(t) is related to inputu(t) through the state variables and the nonlinear equation of state. A simple and direct form of the relation between the outputy(t) (or its derivatives) and the inputu(t) should be found. The starting point to determine diffeomorphism transforming the system (1) into linear system (2) is the definition of a relative degree of the system, sometimes called the characteristic number [1–3].Definition 4.
A relative degreer1(x), …,rp(x) of a smooth nonlinear system includes the smallest natural numbers such that for each j∈1,…,p:(9a)iLgLfkhjx=Lg1Lfkhjx,…,LgmLfkhjx=0,k=0,…,rj-2,∀x∈U(9b)iiLgLfrjx-1hjx=Lg1Lfrjx-1hjx,…,LgmLfrjx-1hjx≠0,forcertainx∈UIf LgLfkhj(x)=Lg1Lfkhj(x),…,LgmLfkhj(x)=0  ∀  k≥0 andx∈U, it is assumed that rj=∞.
It should be emphasized that each numberri is associated with thei-th output of thehi system. It is also worth stressing that by differentiating theyi output with respect to timeritimes we obtain(10)yiri=Lfrihix+LgiLfrihixui,i∈1,…,pThat is,ri is the number that tells how many times the system outputyi(t) should be differentiated to obtain the supplyui in “an explicit form”, that is, the linear relation between supplyui and outputyi.
If the characteristic numbers of the system are finite, then we can define decoupling matrix:(11)Ex=Lg1Lfr1-1h1,…,LgmLfr1-1h1⋮Lg1Lfrp-1hp,…,LgmLfrp-1hpThe method of determining the coordinate (local transformation) of a linearized system is as follows.Lemma 5.
Let the system have a relative degreer1,...,rp at the point xo, and let the rank of decoupling matrix E(xo) bep. Thenr1+  ... + rp≤n and for 1≤i≤p coordinate transformation is (12)S1ix=hixS2ix=Lfhix⋮Sriix=Lfri-1hixNew coordinates for 1≤i≤p are defined as(13)z1,z2,…,zri=S1i,S2i,…,Srii=h1x,Lfh1x,…Lfri-1hpxTheir derivatives are determined as follows:(14)z˙1=ddth1x=dh1dxxx˙=dh1dxxfx=Lfh1x=z2By calculating other components in the same way we obtain a general form of linearized system of (15a)z˙1i=z2iz˙1i=z3i⋮z˙ri-1i=zriiz˙rii=biz,w+∑i=1maijz,wujzThe output equation is written as(15b)yi=z1i
The above equations represent a normal form of equations describing a nonlinear system withminputs andpoutputs, with a relative degreer1,...,rp at the point xo.
It should be noted that in (15a) coefficients aij are equal to E(x) matrix elements, where x is replaced by S-1(z), and coefficients biz,w are the elements of the vector:(16)bx=Lfr1h1x,Lfr2h2x,⋯,LfrphpxTwhere x is replaced by S-1(z).Intuitively, it seems that a set of nonlinear systems that can be transformed by coordinate change is very small. Therefore, if a nonlinear system does not meet the conditions of Theorem3 we consider the transformation of the original model into an equivalent one with simpler dynamics resulting from the transformation of the system state and feedback (6).Definition 6.
Regular state static feedback of nonlinear system (1) is defined as(17)u=αx+βxvwhereu = u1,…,umT and α: M=Rm and β: M=Rmxm are smooth mappings with the property that matrix β(x) is nonsingular for anyx andv= (v1,..., vm) is a new supply vector.
The application of regular, static feedback to the system (1) results in(18)dxtdt=fxt+∑i=1mgixαix+∑j=1m∑i=1mgixβjixvjwhich is of the same form as the considered nonlinear system but now with the newly defined inputs (supplies) (v1,..., vm).Regular, static feedback can be used to solve a much larger group of linearization than is the case of transformation of state coordinates. This allows for the linearization of the system, its decoupling, and decomposition. On the other hand, the use of feedback alters the dynamics of the original system.Let us consider the case of a nonlinear system (1) havingm inputs andm outputs described by(19)x˙=fx+∑i=1mgixuiy=hx=h1x,…,hmxLet the outputs of the system be divided intom disjoint blocks.Here, the task of linearization consists in finding a new supply of the system (19) so that each of themoutput blocks depends on one supply only; i.e., its outputs are decoupled from the inputs.Definition 7.
Nonlinear system (19) has inputs decoupled from outputs around the point xo if there exists a neighbourhoodVof xo point in which LgjLτ1…Lτkh(x)=0 for each x∈V for any vector field τ1,τ2,…,τk, from the set f,g1,…,gm) and if a relative degree of the systemr1,...rm is finite and constant onV.
In order to solve the problem of linearization for the system with multiple outputs and multiple inputs with the initial statexo one should find a regular, static feedback defined in the neighbourhoodV of the xo point such that each output yi is affected by one input vi, 1≤i≤m. The feedback in this case is(20)u1u2⋮um=E-1xv1v2⋮vm-b1xb2x⋮bmxThe decoupling matrixE(x) and the vectorb(x) are given by relations (11) and (16).


## 3. Input-Output Linearization Algorithms of Nonlinear System

The analysis of nonlinear systems, modelling the input-output and undergoing transformations linearizing the system locally includes the following tasks:(i)
linearization of systems with single input and single output—SISO;(ii)
linearization of systems with multiple input and multiple output—MIMO.The proposals of their algorithmic solutions are presented in the subsequent subsections.


### 3.1. Algorithm Linearizing Single Input and Single Output (SISO)

In most cases the transformation of nonlinear systems does not make them equivalent to linear ones. This is true for the systems that do not satisfy the conditions of Theorem3. Here, the combination of linearization by transformation of state variables and the transformation of input functionu(t) with the use of feedback (17) should be applied.The following algorithm describes linearization of a nonlinear system with one power supply and one output.Step 1.
We determine a relative degree of the system from Definition4:
If(21)LgLfihx=0for0≤i≤r-1LgLfr-1hx≠0we can define the input transformation of nonlinear system, which means that there exists a regular and static feedback and the system can be linearized.
If, however,LgLfkh(x)=0 for k≥0 and any x∈Rn the relative degree of the system r=∞ then a regular and static feedback linearizing the system does not exist.Note 8.
The definition of a relative degree of the system can be understood as follows.
By differentiating the outputywe obtain(22)y˙=∂hx∂xx˙=∂hx∂xfx+gxu=Lfhx+Lgxuwhere(23)Lfhx=∂hx∂xfx,andLghx=∂hx∂xgx.Therefore, if Lgh(x)≠0 for a given point xo∈Ωx then in the neighbourhood of this point the following transformation of system input can be defined:(24)u=1Lghxv-LfhxIt is clear that the relation between a new inputv and outputy is linear.If, however,Lgh(x)=0 for any point xo∈Ωx then the differentiation of output should be repeated. Hence(25)y¨=∂Lfhx∂xx˙=∂Lfhx∂xfx+gxu=Lf2hx+LgLfhxuAnd again if LgLfh(x)≠0 then a new supply can be written as(26)u=1LgLfhxv-Lf2hxOtherwise, the differentiation procedure should be continued up to theith step where for some naturalrwe obtain LgLfrh(x)≠0 for some xo∈Ωx.The output differentiation is expressed by the following general relation:(27)yi=∂Lfi-1hx∂xx˙=∂Lfi-1hx∂xfx+gxu=Lfihx+LgLfi-1hxuFinally, a new supply is(28)u=1LgLfr-1hxv-Lfrhxwhich leads to the relation:(29)yr=vand we obtain the linear relation between inputv and outputy.Step 2.
If a relative degree of the systemr =nthen linearizing transformationS(x) is determined by relation (12). Hence, for the system with single input and single output we get(30)S1x=hxS2x=Lfhx⋮Srx=Lfr-1hxStep 3.
We determine the state vector in new coordinatesz=S(x)=S1(x),S2(x),…,Sr(x),T.Step 4.
We determine a regular, static feedback linearizing the analyzed system using (17) and new input (supply)v.Step 5.
For one-dimensional case the system dynamics in the new coordinates and with a new supplyv is obtained as follows:(31)z˙1=∂S1∂xdxdt=∂h∂xdxdtz˙2=∂S2∂xdxdt=∂Lfh∂xdxdt⋮z˙r-1=∂Sr-1∂xdxdt=∂Lfr-2h∂xdxdtz˙r=vHence, we get the equations of states (15a) and (15b):(32a)z˙1=S2xt=Lfhxt=z2tz˙2=S3xt=Lf2hxt=z3t⋮z˙r-1=Srxt=Lfr-1hxt=zrtz˙r=vand output (32b)y=z1Step 6.
If the relative degree of the systemr <n, then such n-r coordinates Sr+1(x),…,Sn(x) are defined as follows: the Jacobian of the mapping (14) is full-rank at a point xo (with an additional assumption that LgSi(x)=0).
Thus we obtain additional equations of system dynamics:(33)z˙i=∂Si∂xfxt+gxtut=LfSixtut+LgSxtut=LfSixtAssuming that(34)qiz=LfSiS-1zr+1≤i≤nthe equations are(35)z˙i=qiztStep 7.
We determine the state vector in new coordinates:(36)z=Sx=S1x,S2x,…,Sr+1x,…,SnxTStep 8.
We determine the scalar functionLfr-1h(x) and define regular feedback by the relation:(37)u=1LgLfr-1hxv-LfrhxStep 9.
The system dynamics in new coordinates can be represented by the linearized system of(38a)z˙1=z2,z˙2=z3⋮z˙r-1=zr,z˙r=v,z˙r+1=qr+1⋮z˙n=qnand the output (38b)y=z1Example 9.
We consider transient state in the electrical circuit with nonlinear resistor as shown in Figure1.Figure 1
Nonlinear circuit of the third order.Current-voltage characteristics of a nonlinear resistive element are described by a second-degree polynomial:(39)ui=ai2wherea – coefficient of ΩA-1 dimension.Assuming that the state variables are currents flowing through the inductances and the capacitor voltage,x1=i1,x2=i2,x3=uC, the state vector becomes x(t)=i1,i1,uCT=x1(t),x2(t),x3(t)T. The state equation modelling the considered electrical circuit is given by(40)x˙1=-b·x3-k·x1-x22+b·etx˙2=-k·x1-x22+b·etx˙3=1Cx1where b=1/L1=1/L2, k=a/L1=a/L2, and vectors f(x),g(x), f,g∈R3,n=3 are(41)fx=-bx3-kx1-x22-kx1-x221Cx1,gx=bb0In the circuit described by (41) the outputy(t) is related to the power supplye(t) through state variables and nonlinear equation of state. To solve the problem we have to find a new power supply with a regular static feedback. For that purpose we use SISO algorithm for linearization.Two cases of different output signals (output functionh(x)): y=h(x)=x1-x2 and y=h(x)=x3 are considered.Case 1.
Suppose that we have to findi3(t) current in the branch with a nonlinear resistor.
Sincei3(t)=i1(t)-i2(t)=x1-x2 the output equation is(42)y=hx=x1-x2(i)
we determine the relative degree of the system.
Directly from the above equation, we can determine the differential of the functionh(x) present in the formulas for the Lie derivative ofh function along the fieldsf andg,respectively:(43)∂hx∂x=∂hx∂x1∂hx∂x2∂hx∂x3=1-10
Calculating derivatives one by one(44)Lghx=∂hx∂x∘gx=1-10bb0T=0;LgLfhx=Lg∂hx∂x∘fx=0LgLf2hx=LgLfLfhx=LgLf-bx3=Lg-bCx1=-bC00bb0=-b2C
we obtainLgLf2h(x)≠0, so for any pointx = (x1,x2,x3) the relative degree of the system isr(x) = 3.(ii)
the next step is to determine theS(x) transformation of state variables and new coordinates:(45)S1x=hx=x1-x2=z1S2x=Lfhx=-bx3=z2S3x=Lf2hx=-bCx1=z3(iii)
regular, static feedback linearizing the system is of the form:(46a)e=1LgLf2hxv-Lf3hx=-Cb2v-b2C·x3+bC·k·x1-x22
Hence, a new supply is(46b)v=b2C·x3+bC·k·x1-x22-b2C·e(iv)
we determine normal form of linearized system for the above feedback:(47)z˙1x=ddtS1x=ddthx=ddtx1-x2=x˙1-x˙2=-b·x3=z2xz˙2x=ddtS2x=ddtLfhx=ddt-b·x3=-b·x˙3=-bC·x1=z3xz˙3x=ddtS3x=ddtLf2hx=ddt-bCx˙1=b2C·x3+bCk·x1-x22-b2C·e=vx
The system dynamics of new coordinate system can be represented by the matrix:(48)z˙1z˙2z˙3=010001000z1z2z3+001v;y=100z1z2z3
The correct work of the algorithm is confirmed by the results obtained by the simulation of linearized system. The results are presented in Figure2.Figure 2
Linearized system.Figure2 shows the numerical solutions of (48) after transformation of state variablesz(t) of the linear system into variablesx(t) of nonlinear system.Case 2.
Now let us find voltage on the capacitoruc=x3. The output function is(49)y=hx=x3so the differential of the h(x) function is (50)∂hx∂x=∂hx∂x1∂hx∂x2∂hx∂x3=001(i)
by calculating the relative degree of the system we obtain(51)Lghx=∂hx∂x∘gx=001bb0T=0,LgLfhx=Lg∂hx∂x∘fx=Lg1Cx1=bCthat for any pointx = (x1,x2,x3) the relative degree of the system isr (x) = 2.Sincer <n using the algorithm in accordance with Step 6 we have to determine the third component of transformationS(x) by solving the partial equation:(52)∂S3x∂x∘gx=∂S3x∂x1∂S3x∂x2∂S3x∂x3bb0T=b·∂S3x∂x1+b·∂S3x∂x2The function, which satisfies the above equation and the condition LgSi(x)=0, can be given by the formula:(53)S3x=x1-x2Thus, the new state variables are of the form:(54)z1=S1x=hx=x3,z2=S2x=Lfhx=1Cx1,z3=S3x=x1-x2The feedback is given as(55a)e=1LgLfhxv-Lf2hx=Cbv--bCx3-kC·x1-x22And a vector of new supply is(55b)v=-bCx3-kC·x1-x22+bC·eBy determining the normal form of the linearized system we get(56)z˙1x=ddtS1x=ddthx=ddtx3=x˙3=1Cx1=z2xz˙2x=ddtS2x=ddtLfhx=ddt1Cx1=1C·x˙1=-bC·x3-kC·x1-x22+bC·e=vz˙3x=ddtS3x=ddtx1-x2=x˙1-x˙2=-b·x3=-b·z1xDynamics of the system in new coordinates is written as(57)z˙1=z2z˙2=vz˙3=-b·z1y=z1orinmatrixnotation:z˙1z˙2z˙3=010000-b00z1z2z3+010v;y=100z1z2z3The resulting system of equations is equivalent to the system described by the mathematical model considered for Case 1.


### 3.2. Linearizatrion Algorithm for Multi-Input and Multi Output Systems (MIMO)

Let us consider a nonlinear system ofm inputs andmoutputs described by (19). Linearization algorithm for MIMO system is an extension of SISO algorithm onto the case of multidimensional system and consists of following steps:Step 1.
From Definition6 (as in one-dimensional case) we determine a relative degree of nonlinear system by calculating the Lie derivative from the function hi(x), (1≤i≤m) along the vector fieldsf andgi(x), respectively.Note 10.
To determine the relative degree of nonlinear multidimensional system we differentiatejth outputs:(58)y˙j=Lfhjx+∑i=1mLgihjxuiIf Lgihj(x)=0 then for eachi differentiation is carried out until in the ith step for some naturalrwe obtain that LgiLfrj-1hj(x)≠0.To differentiate outputs we use general relation:(59)yji=Lfrjhjx+∑i=1mLgLfrj-1hjxuiIfLgiLfrj-1hj(x)≠0, then the output equation of linearized system is written as(60)yjr1yjr2⋮yjrm=Lfr1h1Lfr2h2⋮Lfrmhm+Exu1u2⋮umwhere E(x) ism xm decoupling matrix defined by (11).As a result we obtain the relative degree of the systemr =r1 + …+rp, for 1≤p≤m.Step 2.
We determine the decoupling matrix (11) and vector (16).Step 3.
Ifdet⁡E(x)≠0, then matrixE(x) is nonsingular we determine the feedback linearizing the system as u=-Ex-1∘bx+Ex-1∘vStep 4.
From a general relation we determine the transformation of state variablesS(x) for a multidimensional system: Ski(x)=Lfk-1hi(x).Step 5.
We determine state variables usingzki(x)=Lfk-1hi(x).Step 6.
We determine the state equation of linearized system and the equations of system output.Example 11.
The functionality of the algorithm is presented by electrical circuit with two nonlinear elements, (Figure3). We assume that the initial state of the circuit is equal to zero and att = 0 we close switchesW1 iW2. simultaneously. We also assume that current-voltage characteristics of nonlinear resistive element u=b·i·ln⁡i, whereb is a coefficient of ΩA-1dimension. Nonlinear coil (without loss) is given by u=L(i)(di/dt), where L(i)=(a·i)-1 anda is a coefficient of ΩA-1 dimension.Figure 3
Diagram of nonlinear circuit of the 5th order with two power sources.We assume thati1=x1,i2=x2,i4=x3,uC1=x4,uC2=x5 are state variables. We order variables and introduce notations:R 1 / L 2 = a ; R 1 / L 2 = b ; 1 / L 1 = c ; 1 / C 1 = w , 1 / C 2 = l , 1 / L 2 = k and the expression for the inductance of nonlinear coil L(i1)=L(x1)=(x1)-1 to obtain the model system of equations in which the vectors f(x),g1(x),g2(x) and f,g∈R5,n=5 become(61)fx=-x1x4-R1·x12+R1x1x2-ax1+x2-b·ln⁡x2+kx5-c·x4+c·x5w·x1+w·x3l·x2-l·x3;g1x=x10000;g2x=0k000The output equations are(62)y1=x3,y2=x5.The considered system has two outputs and two inputs. The first step of the linearization algorithm is to determine a relative degree of the system. For functionh1(x) we obtain(63)Lg1h1x=00100x10000T=0Lg2h1x=001000k000T=0Similarly, for the functionh2(x) we get(64)Lg1h2x=00001x10000T=0Lg2h2x=000010k000T=0As the first derivatives are equal to zero, we calculate derivatives of consecutive rows. For the functionh1(x) we get(65)Lg1Lfh1x=Lg1-cx4+cx5=0Lg2Lfh1x=Lg2-cx4+cx5=0Lg1Lf2h1x=-cwlc-cw-lc00x10000T=-c·w·x1≠0Lg2Lf2h1x=-cwlc-cw-lc000k000T=k·l·c≠0As a result the relative degree of subsystemr1 = 3. From the calculations for the functionh2(x) we obtainr2 = 2 because (66)Lg1Lfh2x=Lg1l·x2-l·x3=0Lg2Lfh2x=Lg2l·x2-l·x3=0l-l000k000T=k·l≠0Finally, we find that the relative (vector) degree of the considered system equalsr1,r2=3,2. Asr1 +r2 =n the next step of the algorithm is to determine decoupling (decomposing) matrixE(x): (67)Ex=c·w·x1-k·l·c0k·lSince the determinant of the matrixdet⁡E(x)≠0,E(x) is nonsingular and we can determine feedback linearizing output-input system:(68)e1e2=E-1xv1v2-b1xb2x=1k·l·c·w·x1k·lk·l·c0c·w·x1v1v2-Lf3h1xLf2h2xwherev1iv2 are components of a new supply (output) vector.After transformations we get(69)v1=b1x+a11x·e1+a12x·e2andv2=b2x+a21x·e1+a22x·e2New coordinates are given by(70)z1x=S11x=h1x=x3z2x=S21x=Lfh1x=-c·x4+c·x5z3x=S31x=Lf2h1x=c·w·x1+l·c·x2-x3l·c+w·cz4x=S12x=h2x=x5z5x=S22x=Lfh2x=l·x2-l·x3The normal form of linearized system according to (14) is(71)z˙1x=ddth1x=ddtx3=x˙3=-c·x4+c·x5=z2xz˙2x=ddtLfh1x=ddt-c·x4+c·x5=-c·x˙4+c·x˙5=-cwx1+wx3+clx2+lx3=z3xz˙3x=ddtLf2h1x=c·w·x˙1+l·c·x˙2-x˙3l·c+w·c=cw-x1x4-R1·x12+R1x1x2+x1e1+lc-ax1+x2-b·ln⁡x2+kx5+ke2+c·x4-c·x5l·c+w·c=v1z˙4x=ddth2x=ddtx5=x˙5=l·x2-l·x3=z5xz˙5x=ddtLfh2x=ddtl·x2-l·x3=l·x˙2-l·x˙3=l-ax1+x2-b·ln⁡x2+kx5+k·e2-l·-c·x4+c·x5=v2Also, the structure of the assumed power supply implies thaty1=z1 and y2=z3Finally, we obtain the linear form of the state equations for the system in which the ith inputyi depends only on the ith power inputvi, fori = 1, 2 is obtained.


## 3.1. Algorithm Linearizing Single Input and Single Output (SISO)

In most cases the transformation of nonlinear systems does not make them equivalent to linear ones. This is true for the systems that do not satisfy the conditions of Theorem3. Here, the combination of linearization by transformation of state variables and the transformation of input functionu(t) with the use of feedback (17) should be applied.The following algorithm describes linearization of a nonlinear system with one power supply and one output.Step 1.
We determine a relative degree of the system from Definition4:
If(21)LgLfihx=0for0≤i≤r-1LgLfr-1hx≠0we can define the input transformation of nonlinear system, which means that there exists a regular and static feedback and the system can be linearized.
If, however,LgLfkh(x)=0 for k≥0 and any x∈Rn the relative degree of the system r=∞ then a regular and static feedback linearizing the system does not exist.Note 8.
The definition of a relative degree of the system can be understood as follows.
By differentiating the outputywe obtain(22)y˙=∂hx∂xx˙=∂hx∂xfx+gxu=Lfhx+Lgxuwhere(23)Lfhx=∂hx∂xfx,andLghx=∂hx∂xgx.Therefore, if Lgh(x)≠0 for a given point xo∈Ωx then in the neighbourhood of this point the following transformation of system input can be defined:(24)u=1Lghxv-LfhxIt is clear that the relation between a new inputv and outputy is linear.If, however,Lgh(x)=0 for any point xo∈Ωx then the differentiation of output should be repeated. Hence(25)y¨=∂Lfhx∂xx˙=∂Lfhx∂xfx+gxu=Lf2hx+LgLfhxuAnd again if LgLfh(x)≠0 then a new supply can be written as(26)u=1LgLfhxv-Lf2hxOtherwise, the differentiation procedure should be continued up to theith step where for some naturalrwe obtain LgLfrh(x)≠0 for some xo∈Ωx.The output differentiation is expressed by the following general relation:(27)yi=∂Lfi-1hx∂xx˙=∂Lfi-1hx∂xfx+gxu=Lfihx+LgLfi-1hxuFinally, a new supply is(28)u=1LgLfr-1hxv-Lfrhxwhich leads to the relation:(29)yr=vand we obtain the linear relation between inputv and outputy.Step 2.
If a relative degree of the systemr =nthen linearizing transformationS(x) is determined by relation (12). Hence, for the system with single input and single output we get(30)S1x=hxS2x=Lfhx⋮Srx=Lfr-1hxStep 3.
We determine the state vector in new coordinatesz=S(x)=S1(x),S2(x),…,Sr(x),T.Step 4.
We determine a regular, static feedback linearizing the analyzed system using (17) and new input (supply)v.Step 5.
For one-dimensional case the system dynamics in the new coordinates and with a new supplyv is obtained as follows:(31)z˙1=∂S1∂xdxdt=∂h∂xdxdtz˙2=∂S2∂xdxdt=∂Lfh∂xdxdt⋮z˙r-1=∂Sr-1∂xdxdt=∂Lfr-2h∂xdxdtz˙r=vHence, we get the equations of states (15a) and (15b):(32a)z˙1=S2xt=Lfhxt=z2tz˙2=S3xt=Lf2hxt=z3t⋮z˙r-1=Srxt=Lfr-1hxt=zrtz˙r=vand output (32b)y=z1Step 6.
If the relative degree of the systemr <n, then such n-r coordinates Sr+1(x),…,Sn(x) are defined as follows: the Jacobian of the mapping (14) is full-rank at a point xo (with an additional assumption that LgSi(x)=0).
Thus we obtain additional equations of system dynamics:(33)z˙i=∂Si∂xfxt+gxtut=LfSixtut+LgSxtut=LfSixtAssuming that(34)qiz=LfSiS-1zr+1≤i≤nthe equations are(35)z˙i=qiztStep 7.
We determine the state vector in new coordinates:(36)z=Sx=S1x,S2x,…,Sr+1x,…,SnxTStep 8.
We determine the scalar functionLfr-1h(x) and define regular feedback by the relation:(37)u=1LgLfr-1hxv-LfrhxStep 9.
The system dynamics in new coordinates can be represented by the linearized system of(38a)z˙1=z2,z˙2=z3⋮z˙r-1=zr,z˙r=v,z˙r+1=qr+1⋮z˙n=qnand the output (38b)y=z1Example 9.
We consider transient state in the electrical circuit with nonlinear resistor as shown in Figure1.Figure 1
Nonlinear circuit of the third order.Current-voltage characteristics of a nonlinear resistive element are described by a second-degree polynomial:(39)ui=ai2wherea – coefficient of ΩA-1 dimension.Assuming that the state variables are currents flowing through the inductances and the capacitor voltage,x1=i1,x2=i2,x3=uC, the state vector becomes x(t)=i1,i1,uCT=x1(t),x2(t),x3(t)T. The state equation modelling the considered electrical circuit is given by(40)x˙1=-b·x3-k·x1-x22+b·etx˙2=-k·x1-x22+b·etx˙3=1Cx1where b=1/L1=1/L2, k=a/L1=a/L2, and vectors f(x),g(x), f,g∈R3,n=3 are(41)fx=-bx3-kx1-x22-kx1-x221Cx1,gx=bb0In the circuit described by (41) the outputy(t) is related to the power supplye(t) through state variables and nonlinear equation of state. To solve the problem we have to find a new power supply with a regular static feedback. For that purpose we use SISO algorithm for linearization.Two cases of different output signals (output functionh(x)): y=h(x)=x1-x2 and y=h(x)=x3 are considered.Case 1.
Suppose that we have to findi3(t) current in the branch with a nonlinear resistor.
Sincei3(t)=i1(t)-i2(t)=x1-x2 the output equation is(42)y=hx=x1-x2(i)
we determine the relative degree of the system.
Directly from the above equation, we can determine the differential of the functionh(x) present in the formulas for the Lie derivative ofh function along the fieldsf andg,respectively:(43)∂hx∂x=∂hx∂x1∂hx∂x2∂hx∂x3=1-10
Calculating derivatives one by one(44)Lghx=∂hx∂x∘gx=1-10bb0T=0;LgLfhx=Lg∂hx∂x∘fx=0LgLf2hx=LgLfLfhx=LgLf-bx3=Lg-bCx1=-bC00bb0=-b2C
we obtainLgLf2h(x)≠0, so for any pointx = (x1,x2,x3) the relative degree of the system isr(x) = 3.(ii)
the next step is to determine theS(x) transformation of state variables and new coordinates:(45)S1x=hx=x1-x2=z1S2x=Lfhx=-bx3=z2S3x=Lf2hx=-bCx1=z3(iii)
regular, static feedback linearizing the system is of the form:(46a)e=1LgLf2hxv-Lf3hx=-Cb2v-b2C·x3+bC·k·x1-x22
Hence, a new supply is(46b)v=b2C·x3+bC·k·x1-x22-b2C·e(iv)
we determine normal form of linearized system for the above feedback:(47)z˙1x=ddtS1x=ddthx=ddtx1-x2=x˙1-x˙2=-b·x3=z2xz˙2x=ddtS2x=ddtLfhx=ddt-b·x3=-b·x˙3=-bC·x1=z3xz˙3x=ddtS3x=ddtLf2hx=ddt-bCx˙1=b2C·x3+bCk·x1-x22-b2C·e=vx
The system dynamics of new coordinate system can be represented by the matrix:(48)z˙1z˙2z˙3=010001000z1z2z3+001v;y=100z1z2z3
The correct work of the algorithm is confirmed by the results obtained by the simulation of linearized system. The results are presented in Figure2.Figure 2
Linearized system.Figure2 shows the numerical solutions of (48) after transformation of state variablesz(t) of the linear system into variablesx(t) of nonlinear system.Case 2.
Now let us find voltage on the capacitoruc=x3. The output function is(49)y=hx=x3so the differential of the h(x) function is (50)∂hx∂x=∂hx∂x1∂hx∂x2∂hx∂x3=001(i)
by calculating the relative degree of the system we obtain(51)Lghx=∂hx∂x∘gx=001bb0T=0,LgLfhx=Lg∂hx∂x∘fx=Lg1Cx1=bCthat for any pointx = (x1,x2,x3) the relative degree of the system isr (x) = 2.Sincer <n using the algorithm in accordance with Step 6 we have to determine the third component of transformationS(x) by solving the partial equation:(52)∂S3x∂x∘gx=∂S3x∂x1∂S3x∂x2∂S3x∂x3bb0T=b·∂S3x∂x1+b·∂S3x∂x2The function, which satisfies the above equation and the condition LgSi(x)=0, can be given by the formula:(53)S3x=x1-x2Thus, the new state variables are of the form:(54)z1=S1x=hx=x3,z2=S2x=Lfhx=1Cx1,z3=S3x=x1-x2The feedback is given as(55a)e=1LgLfhxv-Lf2hx=Cbv--bCx3-kC·x1-x22And a vector of new supply is(55b)v=-bCx3-kC·x1-x22+bC·eBy determining the normal form of the linearized system we get(56)z˙1x=ddtS1x=ddthx=ddtx3=x˙3=1Cx1=z2xz˙2x=ddtS2x=ddtLfhx=ddt1Cx1=1C·x˙1=-bC·x3-kC·x1-x22+bC·e=vz˙3x=ddtS3x=ddtx1-x2=x˙1-x˙2=-b·x3=-b·z1xDynamics of the system in new coordinates is written as(57)z˙1=z2z˙2=vz˙3=-b·z1y=z1orinmatrixnotation:z˙1z˙2z˙3=010000-b00z1z2z3+010v;y=100z1z2z3The resulting system of equations is equivalent to the system described by the mathematical model considered for Case 1.


## 3.2. Linearizatrion Algorithm for Multi-Input and Multi Output Systems (MIMO)

Let us consider a nonlinear system ofm inputs andmoutputs described by (19). Linearization algorithm for MIMO system is an extension of SISO algorithm onto the case of multidimensional system and consists of following steps:Step 1.
From Definition6 (as in one-dimensional case) we determine a relative degree of nonlinear system by calculating the Lie derivative from the function hi(x), (1≤i≤m) along the vector fieldsf andgi(x), respectively.Note 10.
To determine the relative degree of nonlinear multidimensional system we differentiatejth outputs:(58)y˙j=Lfhjx+∑i=1mLgihjxuiIf Lgihj(x)=0 then for eachi differentiation is carried out until in the ith step for some naturalrwe obtain that LgiLfrj-1hj(x)≠0.To differentiate outputs we use general relation:(59)yji=Lfrjhjx+∑i=1mLgLfrj-1hjxuiIfLgiLfrj-1hj(x)≠0, then the output equation of linearized system is written as(60)yjr1yjr2⋮yjrm=Lfr1h1Lfr2h2⋮Lfrmhm+Exu1u2⋮umwhere E(x) ism xm decoupling matrix defined by (11).As a result we obtain the relative degree of the systemr =r1 + …+rp, for 1≤p≤m.Step 2.
We determine the decoupling matrix (11) and vector (16).Step 3.
Ifdet⁡E(x)≠0, then matrixE(x) is nonsingular we determine the feedback linearizing the system as u=-Ex-1∘bx+Ex-1∘vStep 4.
From a general relation we determine the transformation of state variablesS(x) for a multidimensional system: Ski(x)=Lfk-1hi(x).Step 5.
We determine state variables usingzki(x)=Lfk-1hi(x).Step 6.
We determine the state equation of linearized system and the equations of system output.Example 11.
The functionality of the algorithm is presented by electrical circuit with two nonlinear elements, (Figure3). We assume that the initial state of the circuit is equal to zero and att = 0 we close switchesW1 iW2. simultaneously. We also assume that current-voltage characteristics of nonlinear resistive element u=b·i·ln⁡i, whereb is a coefficient of ΩA-1dimension. Nonlinear coil (without loss) is given by u=L(i)(di/dt), where L(i)=(a·i)-1 anda is a coefficient of ΩA-1 dimension.Figure 3
Diagram of nonlinear circuit of the 5th order with two power sources.We assume thati1=x1,i2=x2,i4=x3,uC1=x4,uC2=x5 are state variables. We order variables and introduce notations:R 1 / L 2 = a ; R 1 / L 2 = b ; 1 / L 1 = c ; 1 / C 1 = w , 1 / C 2 = l , 1 / L 2 = k and the expression for the inductance of nonlinear coil L(i1)=L(x1)=(x1)-1 to obtain the model system of equations in which the vectors f(x),g1(x),g2(x) and f,g∈R5,n=5 become(61)fx=-x1x4-R1·x12+R1x1x2-ax1+x2-b·ln⁡x2+kx5-c·x4+c·x5w·x1+w·x3l·x2-l·x3;g1x=x10000;g2x=0k000The output equations are(62)y1=x3,y2=x5.The considered system has two outputs and two inputs. The first step of the linearization algorithm is to determine a relative degree of the system. For functionh1(x) we obtain(63)Lg1h1x=00100x10000T=0Lg2h1x=001000k000T=0Similarly, for the functionh2(x) we get(64)Lg1h2x=00001x10000T=0Lg2h2x=000010k000T=0As the first derivatives are equal to zero, we calculate derivatives of consecutive rows. For the functionh1(x) we get(65)Lg1Lfh1x=Lg1-cx4+cx5=0Lg2Lfh1x=Lg2-cx4+cx5=0Lg1Lf2h1x=-cwlc-cw-lc00x10000T=-c·w·x1≠0Lg2Lf2h1x=-cwlc-cw-lc000k000T=k·l·c≠0As a result the relative degree of subsystemr1 = 3. From the calculations for the functionh2(x) we obtainr2 = 2 because (66)Lg1Lfh2x=Lg1l·x2-l·x3=0Lg2Lfh2x=Lg2l·x2-l·x3=0l-l000k000T=k·l≠0Finally, we find that the relative (vector) degree of the considered system equalsr1,r2=3,2. Asr1 +r2 =n the next step of the algorithm is to determine decoupling (decomposing) matrixE(x): (67)Ex=c·w·x1-k·l·c0k·lSince the determinant of the matrixdet⁡E(x)≠0,E(x) is nonsingular and we can determine feedback linearizing output-input system:(68)e1e2=E-1xv1v2-b1xb2x=1k·l·c·w·x1k·lk·l·c0c·w·x1v1v2-Lf3h1xLf2h2xwherev1iv2 are components of a new supply (output) vector.After transformations we get(69)v1=b1x+a11x·e1+a12x·e2andv2=b2x+a21x·e1+a22x·e2New coordinates are given by(70)z1x=S11x=h1x=x3z2x=S21x=Lfh1x=-c·x4+c·x5z3x=S31x=Lf2h1x=c·w·x1+l·c·x2-x3l·c+w·cz4x=S12x=h2x=x5z5x=S22x=Lfh2x=l·x2-l·x3The normal form of linearized system according to (14) is(71)z˙1x=ddth1x=ddtx3=x˙3=-c·x4+c·x5=z2xz˙2x=ddtLfh1x=ddt-c·x4+c·x5=-c·x˙4+c·x˙5=-cwx1+wx3+clx2+lx3=z3xz˙3x=ddtLf2h1x=c·w·x˙1+l·c·x˙2-x˙3l·c+w·c=cw-x1x4-R1·x12+R1x1x2+x1e1+lc-ax1+x2-b·ln⁡x2+kx5+ke2+c·x4-c·x5l·c+w·c=v1z˙4x=ddth2x=ddtx5=x˙5=l·x2-l·x3=z5xz˙5x=ddtLfh2x=ddtl·x2-l·x3=l·x˙2-l·x˙3=l-ax1+x2-b·ln⁡x2+kx5+k·e2-l·-c·x4+c·x5=v2Also, the structure of the assumed power supply implies thaty1=z1 and y2=z3Finally, we obtain the linear form of the state equations for the system in which the ith inputyi depends only on the ith power inputvi, fori = 1, 2 is obtained.


## 4. Conclusion

The presented examples show that the equivalent linear models are obtained by applying the developed transformation algorithms with feedback linearization to mathematical models. Thus, the linear models can be analysed using methods known from the theory of linear systems. The obtained results can be applied to a nonlinear system (1) using the inverse transformation S-1(z). This methodology can be used to precisely analyse the spark discharge as a dynamic phenomenon in the ignition system of a motor vehicle, which allows for high-precision response [54].It is worth noting that proper fulfillment of the Theorem3 determines how the transformation directly linearizing the system, or the transformation with feedback should be sought.It should be stressed that the numbersr1 + …+rp are called the characteristic numbers and any characteristic numberri is associated with j-th output of the systemhi which determines how many timesyi must be differentiated to get the poweruiexplicite.Nonlinear system with multiple inputs and multiple outputs can be decomposed by a static feedback only if the matrixE(x) is nonsingular.Therefore, nonsingularity of this matrix is a prerequisite for a solution of the problem. However, when choosing different output functions of the test system, to find a regular feedback, it often turns out that the decoupling matrix is singular.



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