Generalized Quadratic Linearization of Machine Models

Abstract

In the exact linearization of involutive nonlinear system models, the issue of singularity needs to be addressed in practical applications. The approximate linearization technique due to Krener, based on Taylor series expansion, apart from being applicable to noninvolutive systems, allows the singularity issue to be circumvented. But approximate linearization, while removing terms up to certain order, also introduces terms of higher order than those removed into the system. To overcome this problem, in the case of quadratic linearization, a new concept called “generalized quadratic linearization” is introduced in this paper, which seeks to remove quadratic terms without introducing third- and higher-order terms into the system. Also, solution of generalized quadratic linearization of a class of control affine systems is derived. Two machine models are shown to belong to this class and are reduced to only linear terms through coordinate and state feedback. The result is applicable to other machine models as well.

1. Introduction

Control of nonlinear systems is gaining increasing attention in recent years due to its technical importance and its impact on various applications as well. When nonlinearities of systems become significant, linear control techniques generally fail to produce the desired results. However, feedback linearization of nonlinear systems allows linear control methods to be applied effectively to the resultant linearized system.

In the application of exact linearization technique [1–3] to involutive systems, an issue of singularity needs to be addressed in practical situations. The singularity is caused when a function of the system state tends towards zero causing the state feedback to fail. Approximate linearization technique due to Krener [4] based on a Taylor series expansion, applicable to non-involutive systems, has the merit of allowing the problem of singularity to be overcome.

Poincare [5] derived what are known as homological equations for approximate linearization of autonomous differential systems. Kang and Krener [6–8] extended the result for the approximate linearization of control affine systems and derived what are termed generalized homological equations. Tall [9, 10] extended Kang and Krener's normal forms to multi-input systems. Devanathan [11, 12] developed necessary and sufficient conditions for quadratic linearization of control affine systems with a single input.

Using the approximate linearization technique, the problem of singularity of state feedback can be circumvented by appropriate definition of state feedback. But the approximate linearization, while removing nonlinearities up to a certain higher order, introduces nonlinearities into the system of higher order than those removed. Introduction of higher-order nonlinearities into a system which originally may not have such nonlinearities is a matter of concern.

In the case of machine models, for example, the second-order dynamic models as given in [13] seem adequate to describe the normal behaviour of the machine. Applying quadratic linearization based on approximate linearization technique to this model introduces third- and higher-order terms into the model even though the machine does not possess nonlinearities of this order during normal operation.

In this paper, a new concept called “generalized quadratic linearization” which seeks to remove the second-order nonlinearity in the model without introducing third-and higher-order nonlinearities in the process is introduced. This is in contrast to the existing quadratic linearization techniques due to Kang and Krener [6–8] and Devanathan [11, 12] which introduce third- and-higher order terms in the process of removing second-order terms. The generalized quadratic linearization, being a stronger condition than quadratic linearization, imposes additional constraints on the quadratic polynomials of the system than those imposed due to quadratic linearization alone. Sufficient conditions on the quadratic polynomials together with coordinate and state feedback transformations are derived for a class of systems for which generalized quadratic linearization is applicable. In particular, the results are shown to apply to the induction motor and permanent magnet synchronous motor models. The results can be extended to other machine models as well. Application of generalized quadratic linearization to machine models also helps to avoid the issue of singularity which is a drawback attributed to the existing exact linearization of machine models [2, 3]. This is the main contribution of the paper.

To summarise the rest of the paper, in Section 2, homological equations for arbitrary order linearization are introduced. Section 3 helps to simplify the homological equations for quadratic linearization, thus paving the way for the central result on generalized quadratic linearization presented in Section 4. In Section 5, generalized quadratic linearization is applied to two machine models including a real-world numerical example. In Section 6, the paper is concluded.

2. Background

Consider a multiple-input control affine system of the form [7, 14, 15] 𝑥̇=+𝐴⋯𝑥++𝐵𝑔𝑢+₍₁₎𝑓⁽²⁾(𝑥)+𝑓⁽³⁾(𝑥)+⋯+𝑓^(𝑝)(𝑥)(𝑥)𝑢+𝑔⁽²⁾(𝑥)𝑢+⋯+𝑔^(𝑝−1)(𝑥)𝑢+⋯,(1) where (𝐴,𝐵) are in Brunovsky normal form. That is, let 𝑘={𝑘₁,𝑘₂,…,𝑘_𝑟} be a sequence of integers such that 𝑘₁≥𝑘₂≥⋯≥𝑘_𝑟 such that 𝑘₁+𝑘₂+⋯+𝑘_𝑟=𝑛𝐴=⎡⎢⎢⎢⎢⎢⎢⎣𝐴^𝑘10𝐴0_𝑘2⋯0⋮0⋮0⋯⋯⋮𝐴⋮0_𝑘𝑟𝐵=⎡⎢⎢⎢⎢⎢⎢⎣𝑏_𝑘1⎤⎥⎥⎥⎥⎥⎥⎦,0𝑏0_𝑘2⋯00⋯0⋮0⋮0⋯⋯⋮𝑏⋮0_𝑘𝑟⋮0⋯⋮⋮00⋯0⎤⎥⎥⎥⎥⎥⎥⎦,(2) where matrix 𝐴 is partitioned into 𝑟² blocks while matrix 𝐵 is partitioned into 𝜇𝑟 blocks with 𝑟≤𝜇. Each block 𝐴_𝑘𝑖 or 𝑏_𝑘𝑖 is of the form 𝐴_𝑘𝑖=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⋮0000⋮1000⋮0100⋯⋯⋯⋯⋮⋮0010⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑏_𝑘𝑖=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⋮0001⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(3)𝐴_𝑘𝑖 is of order 𝑘_𝑖×𝑘_𝑖, 𝑏_𝑘𝑖 is of order 𝑘_𝑖×1, and 𝐵 is 𝑛×𝜇 matrix. 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇 and 𝑢=[𝑢₁,𝑢₂,…,𝑢_𝜇]^𝑇 are the vector input such that 𝜇<𝑛. 𝑓^(𝑝)(𝑥) and 𝑔^(𝑝−1)(𝑥) are vector polynomials of order 𝑝 and (𝑝−1), respectively, 𝑝=2,3,….

In order to linearize the system, change of coordinate and state feedback [6] of the following form is considered: 𝑢=𝐼𝑦_𝜇=𝑥+𝜙(𝑥),+𝛽(𝑥)𝑣+𝛼(𝑥),(4) where 𝜙(𝑥)=𝛽_{𝑗=2,3,4,…}(𝑥𝜙)^(𝑗)=(𝑥),_{𝑗=2,3,4,…}𝛼(𝑥)=_{𝑗=2,3,4,…}𝛼^(𝑗)(𝑥),𝛽^(𝑗−1)(𝑥),(5) and 𝐼_𝜇 is the identity matrix of order 𝜇. Applying the transformations (4), (1) is reduced to 𝑦̇=𝐴𝑦+𝐵𝑣(6) provided the following equations, called homological equations [6], are satisfied for 𝑚≥2: −𝐴𝜙^(𝑚)(𝑥)+𝐵𝛼^(𝑚)(𝑥)+𝑓^(𝑚)(𝑥)+𝜕𝜙^(𝑚)(𝑥)𝜕𝑥𝐴𝑥=0,(7)𝐵𝛽^(𝑚−1)(𝑥)𝑣+𝜕𝜙^(𝑚)(𝑥)𝜕𝑥𝐵𝑣+𝑔^{(𝑚−1)}(𝑥)𝑣=0,∀𝑣,(8) where 𝑓′^(𝑚)(𝑥)=𝑓^(𝑚)(𝑥) for 𝑚=2 and 𝑓′^(𝑚)(𝑥) is expressed in terms of 𝑓^{(𝑚−𝑖)}(𝑥),𝑖=0,1,2,…(𝑚−2), and 𝜙^{(𝑚−𝑗)}(𝑥),𝑗=1,2,…(𝑚−2),𝑚>2. 𝑔′^(𝑚−1)(𝑥)=𝑔⁽¹⁾(𝑥); 𝑚=2, and 𝑔′^(𝑚−1)(𝑥) is expressed in terms of 𝑔^{(𝑚−𝑖)}(𝑥),𝑖=1,2,…(𝑚−1), and 𝜙^{(𝑚−𝑗)}(𝑥),𝑗=1,2,…(𝑚−2),𝑚>2.

3. Quadratic Linearization

Consider the specialized case of system (1) containing only quadratic terms in 𝑥 as in 𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥).(9) Quadratic linearization of (9) involves specialization of the result discussed in Section 2 for 𝑚=2 and simplifying it by applying 𝑔⁽¹⁾(𝑥)=0. That is, applying the following coordinate transformation and feedback, 𝑢=𝐼_𝜇𝑦=𝑥+𝜙⁽²⁾(𝑥),+𝛽⁽¹⁾(𝑥)𝑣+𝛼⁽²⁾(𝑥),(10)

(9) is reduced to 𝑦̇=𝐴𝑦+𝐵𝑣+𝑂⁽³⁾(𝑦,𝑣),(11) where 𝑂⁽³⁾(𝑦,𝑣) corresponds to terms of order 3 or higher, provided the following equations are satisfied: −𝐴𝜙⁽²⁾(𝑥)+𝐵𝛼⁽²⁾(𝑥)+𝑓⁽²⁾(𝑥)+𝜕𝜙⁽²⁾(𝑥)𝐵𝛽₍₁₎𝜕𝑥𝐴𝑥=0,(12)(𝑥)𝑣+𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵𝑣=0,∀𝑣.(13)

Remark 3.1. Applying quadratic linearization to system (9) introduces third- and higher-order terms into the system (see (11)) even though originally the system does not possess nonlinearities of this order. To avoid this situation, we introduce a new concept of generalized quadratic linearization which seeks to remove the second-order nonlinearity in the system without introducing third- and higher-order nonlinearities in the process. Sufficient conditions for the generalized quadratic linearization for a class of systems are discussed in the next section.

We next simplify the homological equations (12) and (13) by representing them in partitioned form, using the Brunovsky normal form introduced in Section 2.

Let 𝜙⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣𝜙^𝑘(2)1𝜙(𝑥)_𝑘(2)2𝜙⋮(𝑥)_𝑘(2)𝑟(𝑥)⎤⎥⎥⎥⎥⎥⎥⎦,(14)𝑥=⎡⎢⎢⎢⎢⎢⎢⎣𝑥^𝑘1𝑥_𝑘2𝑥⋮_𝑘𝑟⎤⎥⎥⎥⎥⎥⎥⎦,(15)𝑓⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣𝑓^𝑘(2)1𝑓(𝑥)_𝑘(2)2𝑓⋮(𝑥)_𝑘(2)𝑟(𝑥)⎤⎥⎥⎥⎥⎥⎥⎦,(16)𝛼⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣𝛼¹⁽²⁾𝛼(𝑥)₂₍₂₎𝛼⋮(𝑥)_𝜇(2)(𝑥)⎤⎥⎥⎥⎥⎥⎥⎦,(17) where putting 𝑝_𝑖=∑^𝑖𝑘_𝑖,𝑖=1,2,…𝑟,𝜙^𝑘(2)_𝑖(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜙^{𝑝(2)𝑖−1+1}𝜙⋮(𝑥)_{𝑝(2)𝑖−1+𝑡}𝜙⋮(𝑥)_{𝑝(2)𝑖−1}𝜙(𝑥)_𝑝(2)𝑖(𝑥)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(18)𝑥_𝑘𝑖=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑥^{𝑝𝑖−1+1}𝑥⋮_{𝑝𝑖−1+𝑡}𝑥⋮_𝑝𝑖−1𝑥_𝑝𝑖⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(19)𝑓^𝑘(2)_𝑖(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓^{𝑝(2)𝑖−1+1}𝑓⋮(𝑥)_{𝑝(2)𝑖−1+𝑡}𝑓⋮(𝑥)_𝑝𝑖−1𝑓(𝑥)_𝑝(2)𝑖(𝑥)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(20)𝜕𝜙⁽²⁾(𝑥)𝜕𝑥=⎡⎢⎢⎢⎢⎢⎢⎣𝜙^{𝑘(2)1,𝑘1}𝜙^𝑘(2)_1,𝑘2⋯𝜙^𝑘(2)_1,𝑘𝑟𝜙^𝑘(2)_2,𝑘1𝜙^𝑘(2)_2,𝑘2⋯𝜙^𝑘(2)_2,𝑘𝑟𝜙⋮⋮⋮⋮_{𝑘(2)𝑟,𝑘1}𝜙^𝑘(2)_𝑟,𝑘2⋯𝜙^𝑘(2)_{𝑟,𝑘𝑟}⎤⎥⎥⎥⎥⎥⎥⎦,(21) where 𝜙^𝑘(2)_{𝑖,𝑘𝑗}=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜕𝜙^{𝑝(2)𝑖−1+1}(𝑥)𝜕𝑥_{𝑝𝑗−1+1}⋯𝜕𝜙^{𝑝(2)𝑖−1+1}(𝑥)𝜕𝑥_𝑝𝑗𝜕𝜙⋮_{𝑝(2)𝑖−1+𝑡}⋮⋮(𝑥)𝜕𝑥_{𝑝𝑗−1+1}⋯𝜕𝜙^{𝑝(2)𝑖−1+𝑡}(𝑥)𝜕𝑥_𝑝𝑗𝜕𝜙⋮_𝑝(2)𝑖⋮⋮(𝑥)𝜕𝑥_{𝑝𝑗−1+1}⋯𝜕𝜙^𝑝(2)𝑖(𝑥)𝜕𝑥_𝑝𝑗𝛽⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑖,𝑗=1,2,…𝑟,₍₁₎(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣𝛽¹⁽¹⁾𝛽(𝑥)₂₍₁₎𝛽⋮(𝑥)_𝜇(1)(𝑥)⎤⎥⎥⎥⎥⎥⎥⎦,(22) where 𝛽^𝑖(1)(𝑥)=𝛽^(1)𝑖,1(𝑥),𝛽^(1)𝑖,2(𝑥),…,𝛽^(1)𝑖,𝜇(𝑥),𝑖=1,2,…,𝜇.(23)

Using (2), (14)–(17), and (21), the 𝑖th block row of (12) can be written as −𝐴_𝑘𝑖𝜙^𝑘(2)_𝑖(𝑥)+𝑏_𝑘𝑖𝛼^𝑖(2)(𝑥)+𝑓^𝑘(2)_𝑖(𝑥)+^𝑟_𝑗=1𝑖=1𝜙,2,…_{𝑘𝑖,𝑘𝑗}𝐴,_𝑘𝑗𝑟𝑥._𝑘𝑗=0,(24) The second homological equation, namely, (13), can be simplified as follows. Since the effect of control input 𝑢 in (9) is dependent on the 𝐵 matrix only, without loss of generality, we can then assume that 𝜇=𝑟, because the terms in 𝐵 matrix are nonzero only for the first 𝑟 inputs. So, we can write the revised 𝐵 matrix as 𝐵=⎡⎢⎢⎢⎢⎢⎢⎣𝑏^𝑘10𝑏0_𝑘2⋯0⋮0⋮0⋯⋯⋮𝑏⋮0_𝑘𝑟⎤⎥⎥⎥⎥⎥⎥⎦.(25)

We now put (13) in partitioned form. Using (2) and (21)–(23), the 𝑖th block row of (13) can be written as 𝑏_𝑘𝑖𝛽^𝑖(1)(𝑥)𝑣+𝜙_{𝑘𝑖,𝑘1}𝑏_𝑘1𝜙_{𝑘𝑖,𝑘2}𝑏_𝑘2∀⋯𝑣,𝑖𝜙=1_{𝑘𝑖,𝑘𝑟},𝑏2,_𝑘𝑟…𝑣,=𝑟.0(26)

Since (26) is to be an identity in 𝑣, we can drop 𝑣 and rewrite (26) as ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝛽⋮⋮0000⋮⋮0000⋯⋯⋯⋯⋮⋮⋮⋮0000_(1)𝑖,1+(𝑥⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣)𝜕𝜙𝛽^(1)𝑖,2(𝑥_{𝑝(2)𝑖−1+1})⋯𝛽^(1)𝑖,𝑟(𝑥)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(𝑥)𝜕𝑥_𝑝1⋯𝜕𝜙^{𝑝(2)𝑖−1+1}(𝑥)𝜕𝑥_𝑝𝑟𝜕𝜙⋮_{𝑝(2)𝑖−1+𝑡}⋮⋮(𝑥)𝜕𝑥_𝑝1⋯𝜕𝜙^{𝑝(2)𝑖−1+𝑡}(𝑥)𝜕𝑥_𝑝𝑟𝜕𝜙⋮_𝑝(2)𝑖⋮⋮(𝑥)𝜕𝑥_𝑝1⋯𝜕𝜙^𝑝(2)𝑖(𝑥)𝜕𝑥_𝑝𝑟𝑖=1,2,…,⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝑟=.0,(27) Since 𝛽^(1)𝑖,𝑙(𝑥),𝑙=1,2,…,𝑟, are arbitrary, it is clear that a necessary condition for (13) to be satisfied is 𝜕𝜙^𝑝(2)_{𝑖−1+𝑡}(𝑥)𝜕𝑥_𝑝𝑞=0,𝑞=1,2,…,𝑟,𝑡=𝑖1=,21,,…2,…,𝑘,_𝑖𝑟.−1,(28)

4. Generalized Quadratic Linearization

Theorem 4.1. Consider the system 𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥),(29) where 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇 and 𝑢=[𝑢₁,𝑢₂,…,𝑢_𝑟]^𝑇,𝑛>𝑟. (𝐴,𝐵) are in Brunovsky normal form as given in (2), (3), and (25). 𝑓⁽²⁾(𝑥) is given by (16), where 𝑓^𝑘(2)_𝑖(𝑥)=⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩⎡⎢⎢⎢⎢⎢⎣𝑓⋮^{𝑝(2)𝑖−1}00𝑓_𝑝(2)𝑖𝑥^(𝑥)⎤⎥⎥⎥⎥⎥⎦,𝑘^𝑖𝑓>2,_{𝑝(2)𝑖−1}𝑓_𝑝(2)𝑖𝑥^(𝑥),𝑘^𝑖𝑓=2,_𝑝(2)𝑖(𝑥)𝑖,=1,2,𝑘…_𝑖=𝑟,1,(30)𝑓^𝑘(2)_𝑖(𝑥) is of order 𝑘_𝑖×1 and 𝑥^ is given by the r-tuple 𝑥^=(𝑥_𝑝1,𝑥_𝑝2,…,𝑥_𝑝𝑟). Then the transformation (4), where 𝜙(𝑥)=𝜙⁽²⁾(𝑥), is given by (14), where 𝜙^𝑘(2)_𝑖(𝑥)=⎧⎪⎪⎪⎨⎪⎪⎪⎩⎡⎢⎢⎢⎢⎣𝜙^{𝑝(2)𝑖−1+1}𝜙⋮(𝑥)_{𝑝(2)𝑖−1}𝜙(𝑥)_𝑝(2)𝑖(𝑥)⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎣𝑓⋮00^𝑝(2)_𝑖⁻¹(𝑥′)⎤⎥⎥⎥⎦,𝑘^𝑖𝜙≥2,_𝑝(2)𝑖(𝑥′);𝜙^𝑝(2)_𝑖𝑖𝑥=^1,a2rb,i…tra,r𝑟y.,𝑘_𝑖=1,(31)𝛼(𝑥)=𝛼⁽²⁾(𝑥) given by (17), where 𝛽𝛼^𝑖(2)(𝑥)=−𝑓^𝑝(2)_𝑖(𝑥),𝑖=1,2,…,𝑟,(32)_(𝑚−1)(𝑥)=(−1)^𝑚−1𝐵^𝑇𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵^𝑚−1,𝑚≥2,(33) reduces the system (29) to 𝑦̇=𝐴𝑦+𝐵𝑣.(34)

Proof. We first show that the quadratic linearization of (29) is achieved using (10), assuming (30)–(33). That is, it needs to be verified that (12) and (13) are satisfied. To verify that (12) is satisfied, it is sufficient to verify that (24) is satisfied. Substituting (30)–(32) into (24), we need to satisfy the following: −⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⋮0000⋮1000⋮0100⋯⋯⋯⋯⋮⋮0010⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓⋮_{𝑝(2)𝑖−1}000−⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⋮0001⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦𝑓_𝑝(2)𝑖𝑥^⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(𝑥)+⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓⋮_{𝑝(2)𝑖−1}00𝑓_𝑝(2)𝑖𝑥^+(𝑥)_𝑟⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑗=1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜕𝑓⋮00_{𝑝(2)𝑖−1}⋯⋯⋮⋮00𝑥^𝜕𝑥_{𝑝𝑗−1+1}⋯𝜕𝑓^{𝑝(2)𝑖−1}𝑥^𝜕𝑥_𝑝𝑗⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑥_{𝑝𝑗−1+2}𝑥_{𝑝𝑗−1+3}𝑥⋮_𝑝𝑗𝑖=1,2,…0,𝑟.⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=0,(35) Note that 𝑓^𝑝(2)_𝑖−1(𝑥′) is a function of 𝑥_𝑝𝑗,𝑗=1,2,…,𝑟, only. Hence, 𝜕𝑓^𝑝(2)_𝑖−1(𝑥′)/(𝜕𝑥_{𝑝𝑗−1+𝑡})=0,𝑡=1,2,…,(𝑘_𝑖−1). Equation (35) then reduces to the following after simplification: −⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓⋮_{𝑝(2)𝑖−1}000𝑥^⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦−⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓_𝑝(2)𝑖⋮000(𝑥)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓⋮^{𝑝(2)𝑖−1}00𝑓_𝑝(2)𝑖𝑥^+(𝑥)_𝑟⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑗=1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⋮⋮00⋯⋯⋮⋮⋮⋮00𝜕𝑓⋮⋮0_{𝑝(2)𝑖−1}𝑥^𝜕𝑥_𝑝𝑗⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑥_{𝑝𝑗−1+2}𝑥_{𝑝𝑗−1+3}𝑥⋮_𝑝𝑗0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=0,𝑖=1,2,…,𝑟.(36) Clearly (36) is an identity. Hence (24) is satisfied for 𝑘_𝑖≥2. For the case 𝑘_𝑖=1, it is easy to verify that (24) is satisfied as 𝜙^𝑝(2)_𝑖(𝑥′)is a function of 𝑥_𝑝𝑗,𝑗=1,2,…,𝑟, only.
Equation (28) is a necessary condition for (13) to hold, and it follows easily from (31), where 𝜙^𝑝(2)_{𝑖−1+𝑡}(𝑥)=0,𝑡=1,2,…,(𝑘_𝑖−1),𝑖=1,2,…,𝑟. To satisfy (13), note that, from (25), it follows that 𝐵^𝑇𝐵=𝐼_𝑟.(37) Premultiply (13) by 𝐵^𝑇 and rearrange to yield 𝛽⁽¹⁾(𝑥)=−𝐵^𝑇𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵(38) which is (33) for 𝑚=2. Substitution of (38) satisfies (13). Hence, the result holds for 𝑚=2.
As a result of quadratic linearization (i.e., satisfying (12) and (13)), higher-order terms are left in the system as per (11) even though originally (29) did not have third- and higher-order terms. These are now to be removed to effect generalized quadratic linearization. The higher-order terms are functions of 𝛼(𝑥),𝜙(𝑥), and 𝛽(𝑥) and are given for the case 𝑔^(𝑚)(𝑥)=0,𝑚≥1, through algebraic computations as 𝑓^(𝑚)(𝑥)=𝜕𝜙^(𝑚−1)(𝑥)𝜕𝑥𝑓⁽²⁾(𝑥)+^𝑚−1_𝑗=2𝜕𝜙^(𝑗)(𝑥)𝜕𝑥𝐵𝛼^{(𝑚−𝑗+1)}𝑔𝑚≥3,(𝑥),(39)_{(𝑚−1)}(𝑥)=^𝑚−1_𝑗=2𝜕𝜙^(𝑗)(𝑥)𝜕𝑥𝐵𝛽^{(𝑚−𝑗)}(𝑥),𝑚>2.(40)
For 𝑚=3, 𝑓^(3)(𝑥)=𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝑓⁽²⁾(𝑥)+𝐵𝛼⁽²⁾(𝑥).(41) Partitioning 𝑓′⁽³⁾(𝑥) similar to 𝑓⁽²⁾(𝑥) in (16) and considering the 𝑖th block row, 𝑓^𝑘_𝑖⁽³⁾(𝑥)=^𝑟_𝑗=1=𝜙_{𝑘𝑖,𝑘𝑗}𝑓^𝑘(2)_𝑗(𝑥)+𝑏_𝑘𝑗𝛼^𝑗(2)(𝑥)_𝑟_𝑗=1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜕𝜙^𝑝(2)_𝑖−1+1(𝑥)𝜕𝑥_{𝑝𝑗−1+1}⋯𝜕𝜙^{𝑝(2)𝑖−1+1}(𝑥)𝜕𝑥_𝑝𝑗𝜕𝜙⋮_{𝑝(2)𝑖−1+𝑡}⋮⋮(𝑥)𝜕𝑥_{𝑝𝑗−1+1}⋯𝜕𝜙^{𝑝(2)𝑖−1+𝑡}(𝑥)𝜕𝑥_𝑝𝑗𝜕𝜙⋮_𝑝(2)𝑖⋮⋮(𝑥)𝜕𝑥_{𝑝𝑗−1+1}⋯𝜕𝜙^𝑝(2)𝑖(𝑥)𝜕𝑥_𝑝𝑗×⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓⋮⋮_{𝑝(2)𝑗−1}000⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝑓_𝑝(2)𝑗𝑥^(𝑥)+𝛼^𝑗(2)(𝑥)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑖=1,2,…,𝑟.(42) Using (31) and (32) and that 𝜙^𝑝(2)_𝑖(𝑥) being a function of 𝑥_𝑝𝑗,𝑗=1,2,…,𝑟, only as above, (42) can be reduced to 𝑓^𝑘_𝑖⁽³⁾(𝑥)=^𝑟_𝑗=1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⋮⋮00⋯⋯⋮⋮⋮⋮00𝜕𝑓⋮⋮0_{𝑝(2)𝑖−1}𝑥^𝜕𝑥_𝑝𝑗⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓⋮_{𝑝(2)𝑗−1}00𝑖=1,2,…0𝑥,𝑟^.⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=0,(43) Hence, 𝑓′⁽³⁾(𝑥)=0. Homological equation (7), for 𝑚=3, is satisfied if we put 𝜙⁽³⁾(𝑥)=0 and 𝛼⁽³⁾(𝑥)=0.
For 𝑚=4, 𝑓^(4)(𝑥)=𝜕𝜙⁽³⁾(𝑥)𝜕𝑥𝑓⁽²⁾(𝑥)+𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵𝛼⁽³⁾+𝜕𝜙₍₃₎(𝑥)(𝑥)𝜕𝑥𝐵𝛼⁽²⁾(𝑥).(44) As 𝜙⁽³⁾(𝑥)=0 and 𝛼⁽³⁾(𝑥)=0, it follows from (44) that 𝑓′⁽⁴⁾(𝑥)=0. Hence, one can have 𝜙⁽⁴⁾(𝑥)=0and𝛼⁽⁴⁾(𝑥)=0 to satisfy (7) for 𝑚=4. Proceeding this way, it can be proved that 𝑓^(𝑚)(𝑥)=0,𝑚>3.
Noting that from (40), 𝑔′⁽²⁾(𝑥)=(𝜕𝜙⁽²⁾(𝑥)/𝜕𝑥)𝐵𝛽⁽¹⁾(𝑥) for 𝑚=3, and (8) can be written as 𝐵𝛽⁽²⁾(𝑥)+𝜕𝜙⁽³⁾(𝑥)𝜕𝑥𝐵+𝑔^(2)(𝑥)=0,(45) from which using the fact that 𝜙⁽³⁾(𝑥)=0,𝛽⁽²⁾(𝑥)=−𝐵^𝑇(𝜕𝜙⁽²⁾(𝑥)/𝜕𝑥)𝐵𝛽⁽¹⁾(𝑥)=(𝐵^𝑇(𝜕𝜙⁽²⁾(𝑥)/𝜕𝑥)𝐵)² using (38) is derived. Substitution of 𝛽⁽²⁾(𝑥) satisfies (8) for 𝑚=3.
Proceeding this way, one can show that 𝑓′^(𝑚)=0,𝑚>3, and we can choose 𝜙^(𝑚)=0and𝛼^(𝑚)=0,𝑚>3, to satisfy (7). 𝛽^(𝑚−1)(𝑥) as given in (33) is easily derived so as to satisfy (8), where 𝑔′^(𝑚−1)(𝑥) is given in (40), using the same technique as for 𝑚=2,3 successively as above, hence the result.

Remark 4.2. The result of Theorem 4.1 provides a solution to the generalized quadratic linearization of a class of systems defined by (29). Equation (29) may appear to be a strong condition. But this is needed because of the requirement that the third- and all higher-order nonlinearities, which are usually introduced into the system during the course of quadratic linearization, identically vanish. As shown in Section 5, system (29) can represent many electric machine models [13] of which induction motor and permanent magnet synchronous motor are examples. Also, temperature and flow variables in heat transfer (composition and flow variables in mass transfer) processes, for example, combine to produce quadratic nonlinearity [16] of the form (9) which is a general form of (29). Further, even if a quadratic model (9) does not correspond exactly to model (29), one can always tune the coordinate and state feedback on the lines described by the authors in [17] to bring the system to a linear normal form. It can also be pointed out that quadratic system of the form 𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥)+𝑔⁽¹⁾(𝑥)𝑢 can be brought to the form 𝑥̇=𝐴𝑥+𝐵𝑣+𝑓^(2)(𝑥) through state feedback as shown in [6]. Hence the forms (9) and (29), which are a special case of (9), are more general than they appear to be.

Remark 4.3. A specialized result of Theorem 4.1 appears in an earlier paper by the authors [18]. In [18], the concept of generalized quadratic linearization is applied to a class of control affine systems with a specific configuration of Brunovsky form for multiple inputs. A PMSM model is shown to belong to this class and the generalized quadratic linearization of PMSM is derived. In Theorem 4.1, however, we have considered a more general class of systems with no assumption on the configuration of the Brunovsky form for multiple inputs. Hence, the theoretical result of [18] is a special case of the result of Theorem 4.1.

5. Generalized Quadratic Linearization of Machine Models

5.1. Squirrel Cage Induction Motor

The squirrel cage induction motor model can be derived as below [13]: 𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥),(46) where 𝑥=[𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅]^𝑇=[𝑖^𝑠_ds,𝑖^𝑠_qs,𝜓^𝑠_dr,𝜓^𝑠_qr,𝜔_𝑟]^𝑇,𝑢=[𝑢₁,𝑢₂,𝑢₃,𝑢₄]^𝑇=[𝑣^𝑠_ds,𝑣^𝑠_qs,𝜓^,𝜓^𝑠_dr], where 𝜓^=𝑖^𝑠∗_qs𝜓^𝑠_dr−𝑖^𝑠∗_ds𝜓^𝑠_qr, where 𝑣^𝑠_qs,𝑣^𝑠_ds,𝑖^𝑠_qs, and 𝑖^𝑠_ds represent the quadrature and direct axis voltages and currents, respectively, and 𝜓^𝑠_dr,𝜓^𝑠_qr represent direct and quadrature axis fluxes, respectively. 𝜓^𝑠_dr,𝜓^𝑠_qr represent the estimated direct and quadrature flux, respectively, and 𝜔_𝑟 represents the angular velocity of the rotor. 𝑖^𝑠∗_qs and 𝑖^𝑠∗_ds represent the reference values of quadrature and direct axis currents, respectively. The superscript refers to variables in the stationary frame: 𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝐿^2𝑚𝑅_𝑟+𝐿^2𝑟𝑅_𝑠𝜎𝐿_𝑠𝐿^2𝑟0𝐿^𝑚𝑅_𝑟𝜎𝐿_𝑠𝐿^2𝑟0−𝐿_2𝑚00𝑅_𝑟+𝐿^2𝑟𝑅_𝑠𝜎𝐿_𝑠𝐿^2𝑟0𝐿^𝑚𝑅_𝑟𝜎𝐿_𝑠𝐿^2𝑟𝐿_𝑚0𝑅_𝑟𝐿_𝑟0𝐿0000_𝑚𝑅_𝑟𝐿_𝑟0−𝑅^𝑟𝐿_𝑟𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1000000⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝜎𝐿_𝑠01000𝜎𝐿_𝑠0000−0𝑅_𝑟𝐿_𝑟0000−3𝑃0₂0𝐿_𝑚8𝐽𝐿_𝑟𝑓₍₂₎0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐿^𝑚𝜔_𝑟𝜓^𝑠_qr𝜎𝐿_𝑠𝐿_𝑟−𝐿^𝑚𝜔_𝑟𝜓^𝑠_dr𝜎𝐿_𝑠𝐿_𝑟−𝜔_𝑟𝜓^𝑠_qr𝜔_𝑟𝜓^𝑠_dr0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑐¹𝑥₄𝑥₅−𝑐₁𝑥₃𝑥₅−𝑥₄𝑥₅𝑥₃𝑥₅0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(47) where 𝐿_𝑚,𝐿_𝑟, and 𝐿_𝑠 represent the magnetizing, rotor, and stator inductances, respectively. 𝑅_𝑟 and 𝑅_𝑠 represent the rotor and stator resistances, respectively. 𝐽represents the system moment of inertia, and 𝑃 represents the number of poles. 𝜎=1−𝐿^2𝑚/𝐿_𝑠𝐿_𝑟 and 𝑐₁=𝐿_𝑚/𝜎𝐿_𝑠𝐿_𝑟. The model (46) can be reduced to normal form in a standard way using linear transformation of state variables [19] and linear state feedback as follows, where 𝑥,𝑢, 𝐴,𝐵 and 𝑓⁽²⁾(𝑥) are retained for simplicity of notation: 𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥),(48) where 𝐴 and 𝐵 given by (2), where 𝑟=𝜇=4 and 𝑘₁=2,𝑘₂=𝑘₃=𝑘₄=1, are 𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣0000010000𝑓000000000000000⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣01000001000001000001₍₂₎⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓¹⁽²⁾𝑓(𝑥)₂₍₂₎𝑓(𝑥)₃₍₂₎𝑓(𝑥)₄₍₂₎𝑓(𝑥)₅₍₂₎(𝑥)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(49) where 𝑓¹⁽²⁾(𝑥)=𝑓_1,35𝑥₃𝑥₅+𝑓_1,45𝑥₄𝑥₅, 𝑓²⁽²⁾(𝑥)=𝑓_2,35𝑥₃𝑥₅+𝑓_2,45𝑥₄𝑥₅, 𝑓³⁽²⁾(𝑥)=𝑓_3,35𝑥₃𝑥₅+𝑓_3,45𝑥₄𝑥₅, 𝑓⁴⁽²⁾(𝑥)=𝑓_4,35𝑥₃𝑥₅+𝑓_4,45𝑥₄𝑥₅, 𝑓⁵⁽²⁾(𝑥)=𝑓_5,35𝑥₃𝑥₅+𝑓_5,45𝑥₄𝑥₅.

Remark 5.1. 𝑓⁽²⁾(𝑥) can be partitioned using (16) as 𝑓⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣𝑓^𝑘(2)1𝑓(𝑥)_𝑘(2)2𝑓(𝑥)_𝑘(2)3𝑓(𝑥)_𝑘(2)4(𝑥)⎤⎥⎥⎥⎥⎥⎥⎦,(50) where 𝑓^𝑘(2)₁(𝑥)=⎡⎢⎢⎣𝑓¹⁽²⁾𝑓(𝑥)₂₍₂₎(𝑥)⎤⎥⎥⎦=⎡⎢⎢⎣𝑓^1,35𝑥₃𝑥₅+𝑓_1,45𝑥₄𝑥₅𝑓_2,35𝑥₃𝑥₅+𝑓_2,45𝑥₄𝑥₅𝑓_𝑘(2)2⎤⎥⎥⎦,𝑓_𝑘(2)3(𝑥)=𝑓³⁽²⁾(𝑥)=𝑓_3,35𝑥₃𝑥₅+𝑓_3,45𝑥₄𝑥₅,𝑓_𝑘(2)4(𝑥)=𝑓⁴⁽²⁾(𝑥)=𝑓_4,35𝑥₃𝑥₅+𝑓_4,45𝑥₄𝑥₅,(𝑥)=𝑓⁵⁽²⁾(𝑥)=𝑓_5,35𝑥₃𝑥₅+𝑓_5,45𝑥₄𝑥₅.(51) As 𝑓¹⁽²⁾(𝑥) is a function of 𝑥₃,𝑥₄, and 𝑥₅ only and hence a function of 𝑥^=(𝑥₂,𝑥₃,𝑥₄,𝑥₅), one can write 𝑓¹⁽²⁾(𝑥)=𝑓¹⁽²⁾(𝑥′). The systems given by (48) and (49) are then in line with conditions of Theorem 4.1 as given in (30). Hence, Theorem 4.1 can be applied to system (48) to obtain the solution of generalized quadratic linearization which is given in Corollary 5.2.

Corollary 5.2. The system (48) can be quadratic linearized in the generalized sense using the transformation 𝑢=𝐼𝑦₄=𝑥+𝜙(𝑥),+𝛽(𝑥)𝑣+𝛼(𝑥),(52) where 𝜙(𝑥)=𝜙⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑓¹⁽²⁾0𝜙₃₍₂₎𝑥^𝜙₄₍₂₎𝑥^𝜙₅₍₂₎𝑥^𝛼(𝑥)=𝛼₍₂₎𝑥^⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(𝑥)=⎡⎢⎢⎢⎢⎢⎢⎣−𝑓²⁽²⁾−𝑓₃₍₂₎(𝑥)−𝑓₄₍₂₎(𝑥)−𝑓₅₍₂₎(𝑥)𝛽(𝑥)_(𝑚−1)⎤⎥⎥⎥⎥⎥⎥⎦,(𝑥)=(−1)^𝑚−1𝐵^𝑇𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵^𝑚−1,(53) where 𝜙³⁽²⁾(𝑥^),𝜙⁴⁽²⁾(𝑥^), and 𝜙⁵⁽²⁾(𝑥′) are completely arbitrary, and 𝑥^=(𝑥₂,𝑥₃,𝑥₄,𝑥₅). The system reduces to 𝑦̇=𝐴𝑦+𝐵𝑣.(54)

Proof. The result follows directly by applying the result of Theorem 4.1 to (48) where 𝑛=5,𝑟=4,𝜇=4,𝑘₁=2, and 𝑘₂=𝑘₃=𝑘₄=1;𝑝₁=2,𝑝₂=3,𝑝₃=4, and 𝑝₄=5.

5.2. Permanent Magnet Synchronous Motor

The PM machine model [13] can be expressed as𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥);(55)𝑥=[𝑥₁,𝑥₂,𝑥₃]^𝑡=[𝑖_qs,𝑖_ds,𝜔_𝑒]^𝑡 and 𝑢=[𝑢₁,𝑢₂]^𝑡=[𝑣_qs,𝑣_ds]^𝑡 where 𝑣_qs,𝑣_ds,𝑖_qs, and 𝑖_ds represent the quadrature and direct axes voltages and currents, respectively, and 𝜔_𝑒 is stator angular velocity: 𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−𝑅^𝑠𝐿_qs0−𝐼^𝑓𝐿_dm𝐿_qs0−𝑅^𝑠𝐿_ds3𝑃₂0𝐿_dm𝐼^𝑓𝐵8𝐽=⎡⎢⎢⎢⎢⎢⎣100⎤⎥⎥⎥⎥⎥⎥⎦,𝐿_qs001𝐿_ds𝑓00₍₂₎⎤⎥⎥⎥⎥⎥⎦,(𝑥)=⎡⎢⎢⎢⎢⎢⎣−𝐿^ds𝐿_qs𝑥₂𝑥₃𝐿_qs𝐿_ds𝑥₁𝑥₃𝐿_ds−𝐿_qs𝑥₁𝑥₂⎤⎥⎥⎥⎥⎥⎦.(56)𝐼^𝑓 is the field current equivalent to the permanent magnet. 𝐿_ds,𝐿_qs, and 𝐿_dm are the direct, quadrature, and magnetizing inductances, respectively. 𝑅_𝑠 is the stator resistance. 𝑃 is the number of poles, and 𝐽 is the system moment of inertia. The model (55) can be reduced to the following normal form in a standard way using linear transformation of state variables [19] and linear state feedback, where 𝑥, 𝑢, 𝐴, 𝐵, and 𝑓⁽²⁾(𝑥) are retained for simplicity of notation:𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥),(57) where 𝐴 and 𝐵 given by (2), where 𝑟=𝜇=2 and 𝑘₁=2,𝑘₂=1, are 𝑓𝐴=⎡⎢⎢⎢⎢⎣000100000⎤⎥⎥⎥⎥⎦,𝐵=⎡⎢⎢⎢⎢⎣010001₍₂₎⎤⎥⎥⎥⎥⎦,(𝑥)=⎡⎢⎢⎢⎢⎣𝑓¹⁽²⁾𝑓(𝑥)₂₍₂₎𝑓(𝑥)₃₍₂₎(𝑥)⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝐶¹𝑥₂𝑥₃𝐶₂𝑥₃𝑥₁𝐶₃𝑥₂𝑥₁⎤⎥⎥⎥⎥⎦,(58) with 𝐶₁,𝐶₂,𝐶₃ being constants.

Remark 5.3. 𝑓⁽²⁾(𝑥) can be partitioned using (16) as 𝑓⁽²⁾(𝑥)=⎡⎢⎢⎣𝑓^𝑘(2)1𝑓(𝑥)_𝑘(2)2(𝑥)⎤⎥⎥⎦,(59) where 𝑓^𝑘(2)₁(𝑥)=⎡⎢⎢⎣𝑓¹⁽²⁾𝑓(𝑥)₂₍₂₎(𝑥)⎤⎥⎥⎦=⎡⎢⎢⎣𝐶¹𝑥₂𝑥₃𝐶₂𝑥₃𝑥₁𝑓_𝑘(2)2⎤⎥⎥⎦,(𝑥)=𝑓³⁽²⁾(𝑥)=𝐶₃𝑥₂𝑥₁.(60) Note that 𝑓¹⁽²⁾(𝑥) is a function of 𝑥′=(𝑥₂,𝑥₃) and hence 𝑓¹⁽²⁾(𝑥)=𝑓¹⁽²⁾(𝑥′). The system given by (57) is in line with conditions of Theorem 4.1 as given in (30). Hence, Theorem 4.1 can be applied to system (57) to obtain the solution of generalized quadratic linearization which is given in Corollary 5.4.

Corollary 5.4. The system (57) can be quadratic linearized in the generalized sense using the transformation 𝑢=𝐼𝑦₂=𝑥+𝜙(𝑥),+𝛽(𝑥)𝑣+𝛼(𝑥),(61) where 𝜙(𝑥)=𝜙⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎣𝑓𝜙¹⁽²⁾₃₍₂₎0𝑥^𝛼(𝑥)=𝛼₍₂₎𝑥^⎤⎥⎥⎥⎥⎦,(𝑥)=⎡⎢⎢⎣−𝑓²⁽²⁾−𝑓₃₍₂₎(𝑥)𝛽(𝑥)_(𝑚−1)⎤⎥⎥⎦,(𝑥)=(−1)^𝑚−1𝐵^𝑇𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵^𝑚−1,(62) where 𝜙³⁽²⁾(𝑥′) is completely arbitrary and 𝑥′=(𝑥₂,𝑥₃). The system then reduces to 𝑦̇=𝐴𝑦+𝐵𝑣.(63)

Proof. The result follows directly by applying the result of Theorem 4.1 to (57) where 𝑛=3,𝑟=2, and 𝜇=2;𝑘₁=2,𝑘₂=1;𝑝₁=2;𝑝₂=3.

5.3. Real-World Example of PMSM

The parameters of an actual machine are obtained from [20] as 𝑅_𝑠=1.2 ohms, 𝐿_qs=12.5mH, 𝐿_ds= 5.7 mH, 𝜔_𝑒= 3500 rpm, 𝑃=4, 𝐽=0.0018 kgm², and 𝜆_af= 123 mWb turns. The mathematical model of the machine in the normal form is given by 𝑥̇=𝐴𝑥+𝐵𝑢+𝑓⁽²⁾(𝑥),(64) where 𝐴 and 𝐵 are as given in (57) and 𝑓⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎣𝐶¹𝑥₂𝑥₃𝐶₂𝑥₃𝑥₁𝐶₃𝑥₂𝑥₁⎤⎥⎥⎥⎥⎦,(65) where 𝐶₁=−0.165(𝑒−3),𝐶₂=−186.96,and𝐶₃=5.754386(𝑒6). By applying Corollary 5.4 and choosing 𝜙(𝑥)=𝜙⁽²⁾(𝑥)=⎡⎢⎢⎢⎢⎣𝐶₁𝑥0₂𝑥₃0⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣−0.0165(𝑒0−3)𝑥²𝑥₃𝛼(𝑥)=𝛼₍₂₎0⎤⎥⎥⎥⎥⎦,(𝑥)=⎡⎢⎢⎣𝛼¹⁽²⁾𝛼(𝑥)₂₍₂₎(𝑥)⎤⎥⎥⎦=⎡⎢⎢⎣186.96𝑥³𝑥₁−5.754386(𝑒6)𝑥₂𝑥₁𝛽₍₁₎⎤⎥⎥⎦,(𝑥)=⎡⎢⎢⎣−𝐶¹𝑥₃−𝐶₁𝑥₂=⎡⎢⎢⎣0.00165(𝑒−03)𝑥⎤⎥⎥⎦₃0.0165(𝑒−3)𝑥₂𝛽00_(𝑚−1)⎤⎥⎥⎦,(𝑥)=(−1)^𝑚−1𝐵^𝑇𝜕𝜙⁽²⁾(𝑥)𝜕𝑥𝐵^𝑚−1.(66)

It is seen by verification that the system (64) becomes 𝑦̇=𝐴𝑦+𝐵𝑣(67) through transformation (61), given 𝜙(𝑥),𝛼(𝑥), and 𝛽(𝑥) as above. The above system is completely linearized with no higher-order terms left behind.

5.4. Simulation Results

Verification of quadratic linearization of PMSM through simulations is given in an earlier paper by the authors [18]. The simulation results show that the closed-loop dynamic response of PMSM, before linearization, exhibits a behaviour that corresponds to varied responses for variations in reference and load conditions. However, it is verified that the quadratic linearized PMSM system gives a uniform closed-loop response for different reference and load conditions, as is expected of a linear system.

6. Conclusion

In this paper, linearization of machine models, which are predominantly quadratic in nature, is considered. Since the machine models only exhibit higher-order nonlinearities such as core loss, stray loss, and saturation, under extreme conditions, quadratic models can be used to represent the machines during normal operation.

The existing exact feedback linearization techniques introduce singularities in the system. This may result in the the faliure of linearization and the corresponding control technique, during the course of operation of the machine. Application of Poincare's approximate linearization technique due to Kang and Krener can remove the quadratic nonlinearity through a homogeneous transformation. This technique does not suffer from singularity issues but, being an approximate method, introduces higher-order terms in the process of quadratic linearization. Hence, in this paper, the concept of generalized quadratic linearization technique is introduced, wherein the higher-order terms introduced during the process of quadratic linearization are removed even as the quadratic term is removed.

A solution to the problem of generalized quadratic linearization is given for a class of control affine systems. Induction motor and PM motor models involving quadratic nonlinearity are considered and are quadratically linearised in the generalized sense. The proposed method can also be extended to wound rotor and synchronous machine models as well.

The generalized quadratic linearization technique proposed in this paper can also be considered as the approximate feedback linearization equivalent of exact linearization of a class of systems of the form (29). Since the solution of the generalized quadratic linearization involves removing third- and higher-order nonlinearities, the result of the paper can also been seen as a contribution towards a special case of the open problem on arbitrary order linearization [21] of control affine systems.

Finally, as per the result of Theorem 4.1, 𝛽(𝑥)=∑_𝑚=2,3,…𝛽^𝑚−1(𝑥) whose convergence needs to be established. Also, Theorem 4.1 of Section 4 deals only with sufficient conditions for generalized quadratic linearization. The question arises as to what are the necessary conditions for (9) to be generalized quadratically linearizable in general: this is our future work.