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Journal of Control Science and Engineering
Volume 2011 (2011), Article ID 926712, 10 pages
http://dx.doi.org/10.1155/2011/926712
Research Article

Generalized Quadratic Linearization of Machine Models

1Department of Electrical and Electronics Engineering Department, Hindustan of Science & Technology, Chennai 603103, India
2Department of Electrical and Electronics Engineering, SSN College of Engineering Kalavakkam 603110, India

Received 10 May 2011; Revised 17 August 2011; Accepted 26 August 2011

Academic Editor: Derong Liu

Copyright © 2011 Parvathy Ayalur Krishnamoorthy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [13] 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 [68] 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 [68] 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] ̇𝑥=𝐴𝑥+𝐵𝑢+𝑓(2)(𝑥)+𝑓(3)(𝑥)++𝑓(𝑝)(𝑥)++𝑔(1)(𝑥)𝑢+𝑔(2)(𝑥)𝑢++𝑔(𝑝1)(𝑥)𝑢+,(1) where (𝐴,𝐵) are in Brunovsky normal form. That is, let 𝑘={𝑘1,𝑘2,,𝑘𝑟} be a sequence of integers such that 𝑘1𝑘2𝑘𝑟 such that 𝑘1+𝑘2++𝑘𝑟=𝑛𝐴𝐴=𝑘1000𝐴𝑘2000𝐴𝑘𝑟,𝑏𝐵=𝑘100000𝑏𝑘200000𝑏𝑘𝑟,00(2) where matrix 𝐴 is partitioned into 𝑟2 blocks while matrix 𝐵 is partitioned into 𝜇𝑟 blocks with 𝑟𝜇. Each block 𝐴𝑘𝑖 or 𝑏𝑘𝑖 is of the form 𝐴𝑘𝑖=0100001000010000,𝑏𝑘𝑖=0001.(3)𝐴𝑘𝑖 is of order 𝑘𝑖×𝑘𝑖, 𝑏𝑘𝑖 is of order 𝑘𝑖×1, and 𝐵 is 𝑛×𝜇 matrix. 𝑥=[𝑥1,𝑥2,,𝑥𝑛]𝑇 and 𝑢=[𝑢1,𝑢2,,𝑢𝜇]𝑇 are the vector input such that 𝜇<𝑛. 𝑓(𝑝)(𝑥) and 𝑔(𝑝1)(𝑥) are vector polynomials of order 𝑝 and (𝑝1), respectively, 𝑝=2,3,.

In order to linearize the system, change of coordinate and state feedback [6] of the following form is considered: ,𝐼𝑦=𝑥+𝜙(𝑥)𝑢=𝜇+𝛽(𝑥)𝑣+𝛼(𝑥),(4) where 𝜙(𝑥)=𝑗=2,3,4,𝜙(𝑗)(𝑥),𝛼(𝑥)=𝑗=2,3,4,𝛼(𝑗)(𝑥),𝛽(𝑥)=𝑗=2,3,4,𝛽(𝑗1)(𝑥),(5) and 𝐼𝜇 is the identity matrix of order 𝜇. Applying the transformations (4), (1) is reduced to ̇𝑦=𝐴𝑦+𝐵𝑣(6) provided the following equations, called homological equations [6], are satisfied for 𝑚2: 𝐴𝜙(𝑚)(𝑥)+𝐵𝛼(𝑚)(𝑥)+𝑓(𝑚)(𝑥)+𝜕𝜙(𝑚)(𝑥)𝜕𝑥𝐴𝑥=0,(7)𝐵𝛽(𝑚1)(𝑥)𝑣+𝜕𝜙(𝑚)(𝑥)𝜕𝑥𝐵𝑣+𝑔(𝑚1)(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(𝑚𝑖)(𝑥),𝑖=0,1,2,(𝑚2), and 𝜙(𝑚𝑗)(𝑥),𝑗=1,2,(𝑚2),𝑚>2. 𝑔(𝑚1)(𝑥)=𝑔(1)(𝑥); 𝑚=2, and 𝑔(𝑚1)(𝑥) is expressed in terms of 𝑔(𝑚𝑖)(𝑥),𝑖=1,2,(𝑚1), and 𝜙(𝑚𝑗)(𝑥),𝑗=1,2,(𝑚2),𝑚>2.

3. Quadratic Linearization

Consider the specialized case of system (1) containing only quadratic terms in 𝑥 as in ̇𝑥=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, 𝑦=𝑥+𝜙(2)𝐼(𝑥),𝑢=𝜇+𝛽(1)(𝑥)𝑣+𝛼(2)(𝑥),(10)

(9) is reduced to ̇𝑦=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)𝜕𝑥𝐴𝑥=0,(12)𝐵𝛽(1)(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)𝜙(𝑥)=𝑘(2)1𝜙(𝑥)𝑘(2)2(𝜙𝑥)𝑘(2)𝑟(𝑥),(14)𝑥𝑥=𝑘1𝑥𝑘2𝑥𝑘𝑟,(15)𝑓(2)𝑓(𝑥)=𝑘(2)1𝑓(𝑥)𝑘(2)2(𝑓𝑥)𝑘(2)𝑟(𝑥),(16)𝛼(2)𝛼(𝑥)=1(2)𝛼(𝑥)2(2)(𝛼𝑥)𝜇(2)(𝑥),(17) where putting 𝑝𝑖=𝑖𝑘𝑖,𝑖=1,2,𝑟,𝜙𝑘(2)𝑖𝜙(𝑥)=𝑝(2)𝑖1+1𝜙(𝑥)𝑝(2)𝑖1+𝑡𝜙(𝑥)𝑝(2)𝑖1𝜙(𝑥)𝑝(2)𝑖(𝑥),(18)𝑥𝑘𝑖=𝑥𝑝𝑖1+1𝑥𝑝𝑖1+𝑡𝑥𝑝𝑖1𝑥𝑝𝑖,(19)𝑓𝑘(2)𝑖𝑓(𝑥)=𝑝(2)𝑖1+1𝑓(𝑥)𝑝(2)𝑖1+𝑡𝑓(𝑥)𝑝𝑖1𝑓(𝑥)𝑝(2)𝑖(𝑥),(20)𝜕𝜙(2)(𝑥)=𝜙𝜕𝑥𝑘(2)1,𝑘1𝜙𝑘(2)1,𝑘2𝜙𝑘(2)1,𝑘𝑟𝜙𝑘(2)2,𝑘1𝜙𝑘(2)2,𝑘2𝜙𝑘(2)2,𝑘𝑟𝜙𝑘(2)𝑟,𝑘1𝜙𝑘(2)𝑟,𝑘2𝜙𝑘(2)𝑟,𝑘𝑟,(21) where 𝜙𝑘(2)𝑖,𝑘𝑗=𝜕𝜙𝑝(2)𝑖1+1(𝑥)𝜕𝑥𝑝𝑗1+1𝜕𝜙𝑝(2)𝑖1+1(𝑥)𝜕𝑥𝑝𝑗𝜕𝜙𝑝(2)𝑖1+𝑡(𝑥)𝜕𝑥𝑝𝑗1+1𝜕𝜙𝑝(2)𝑖1+𝑡(𝑥)𝜕𝑥𝑝𝑗𝜕𝜙𝑝(2)𝑖(𝑥)𝜕𝑥𝑝𝑗1+1𝜕𝜙𝑝(2)𝑖(𝑥)𝜕𝑥𝑝𝑗𝛽,𝑖,𝑗=1,2,𝑟,(1)𝛽(𝑥)=1(1)𝛽(𝑥)2(1)𝛽(𝑥)𝜇(1),(𝑥)(22) where 𝛽𝑖(1)𝛽(𝑥)=(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 𝐴𝑘𝑖𝜙𝑘(2)𝑖(𝑥)+𝑏𝑘𝑖𝛼𝑖(2)(𝑥)+𝑓𝑘(2)𝑖(𝑥)+𝑟𝑗=1𝜙𝑘𝑖,𝑘𝑗𝐴𝑘𝑗𝑥𝑘𝑗=0,𝑖=1,2,,𝑟.(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 𝑏𝐵=𝑘1000𝑏𝑘2000𝑏𝑘𝑟.(25)

We now put (13) in partitioned form. Using (2) and (21)–(23), the 𝑖th block row of (13) can be written as 𝑏𝑘𝑖𝛽𝑖(1)𝜙(𝑥)𝑣+𝑘𝑖,𝑘1𝑏𝑘1𝜙𝑘𝑖,𝑘2𝑏𝑘2𝜙𝑘𝑖,𝑘𝑟𝑏𝑘𝑟𝑣=0𝑣,𝑖=1,2,,𝑟.(26)

Since (26) is to be an identity in 𝑣, we can drop 𝑣 and rewrite (26) as 𝛽000000000000(1)𝑖,1(𝑥)𝛽(1)𝑖,2(𝑥)𝛽(1)𝑖,𝑟(+𝑥)𝜕𝜙𝑝(2)𝑖1+1(𝑥)𝜕𝑥𝑝1𝜕𝜙𝑝(2)𝑖1+1(𝑥)𝜕𝑥𝑝𝑟𝜕𝜙𝑝(2)𝑖1+𝑡(𝑥)𝜕𝑥𝑝1𝜕𝜙𝑝(2)𝑖1+𝑡(𝑥)𝜕𝑥𝑝𝑟𝜕𝜙𝑝(2)𝑖(𝑥)𝜕𝑥𝑝1𝜕𝜙𝑝(2)𝑖(𝑥)𝜕𝑥𝑝𝑟=0,𝑖=1,2,,𝑟.(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 𝜕𝜙𝑝(2)𝑖1+𝑡(𝑥)𝜕𝑥𝑝𝑞𝑘=0,𝑞=1,2,,𝑟,𝑡=1,2,,𝑖,1𝑖=1,2,,𝑟.(28)

4. Generalized Quadratic Linearization

Theorem 4.1. Consider the system ̇𝑥=𝐴𝑥+𝐵𝑢+𝑓(2)(𝑥),(29) where 𝑥=[𝑥1,𝑥2,,𝑥𝑛]𝑇 and 𝑢=[𝑢1,𝑢2,,𝑢𝑟]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 𝑓𝑘(2)𝑖(00𝑓𝑥)=𝑝(2)𝑖1𝑥𝑓𝑝(2)𝑖(𝑥),𝑘𝑖𝑓>2,𝑝(2)𝑖1𝑥𝑓𝑝(2)𝑖(𝑥),𝑘𝑖𝑓=2,𝑝(2)𝑖(𝑥),𝑘𝑖=1,𝑖=1,2,𝑟,(30)𝑓𝑘(2)𝑖(𝑥) is of order 𝑘𝑖×1 and 𝑥 is given by the r-tuple 𝑥=(𝑥𝑝1,𝑥𝑝2,,𝑥𝑝𝑟). Then the transformation (4), where 𝜙(𝑥)=𝜙(2)(𝑥), is given by (14), where 𝜙𝑘(2)𝑖𝜙(𝑥)=𝑝(2)𝑖1+1𝜙(𝑥)𝑝(2)𝑖1𝜙(𝑥)𝑝(2)𝑖=00𝑓(𝑥)𝑝(2)𝑖1(𝑥),𝑘𝑖𝜙2,𝑝(2)𝑖(𝑥);𝜙𝑝(2)𝑖𝑥arbitrary,𝑘𝑖=1,𝑖=1,2,,𝑟.(31)𝛼(𝑥)=𝛼(2)(𝑥) given by (17), where 𝛼𝑖(2)(𝑥)=𝑓𝑝(2)𝑖𝛽(𝑥),𝑖=1,2,,𝑟,(32)(𝑚1)(𝑥)=(1)𝑚1𝐵𝑇𝜕𝜙(2)(𝑥)𝐵𝜕𝑥𝑚1,𝑚2,(33) reduces the system (29) to ̇𝑦=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: 000𝑓0100001000010000𝑝(2)𝑖1𝑥0001𝑓𝑝(2)𝑖00𝑓(𝑥)+𝑝(2)𝑖1𝑥𝑓𝑝(2)𝑖+(𝑥)𝑟𝑗=10000𝜕𝑓𝑝(2)𝑖1𝑥𝜕𝑥𝑝𝑗1+1𝜕𝑓𝑝(2)𝑖1𝑥𝜕𝑥𝑝𝑗𝑥𝑝𝑗1+2𝑥𝑝𝑗1+3𝑥𝑝𝑗0=0,𝑖=1,2,,𝑟.(35) Note that 𝑓𝑝(2)𝑖1(𝑥) is a function of 𝑥𝑝𝑗,𝑗=1,2,,𝑟, only. Hence, 𝜕𝑓𝑝(2)𝑖1(𝑥)/(𝜕𝑥𝑝𝑗1+𝑡)=0,𝑡=1,2,,(𝑘𝑖1). Equation (35) then reduces to the following after simplification: 00𝑓𝑝(2)𝑖1𝑥0000𝑓𝑝(2)𝑖+00𝑓(𝑥)𝑝(2)𝑖1𝑥𝑓𝑝(2)𝑖+(𝑥)𝑟𝑗=100000𝜕𝑓𝑝(2)𝑖1𝑥𝜕𝑥𝑝𝑗𝑥𝑝𝑗1+2𝑥𝑝𝑗1+3𝑥𝑝𝑗0=0,𝑖=1,2,,𝑟.(36) Clearly (36) is an identity. Hence (24) is satisfied for 𝑘𝑖2. For the case 𝑘𝑖=1, it is easy to verify that (24) is satisfied as 𝜙𝑝(2)𝑖(𝑥)is a function of 𝑥𝑝𝑗,𝑗=1,2,,𝑟, only.
Equation (28) is a necessary condition for (13) to hold, and it follows easily from (31), where 𝜙𝑝(2)𝑖1+𝑡(𝑥)=0,𝑡=1,2,,(𝑘𝑖1),𝑖=1,2,,𝑟. To satisfy (13), note that, from (25), it follows that 𝐵𝑇𝐵=𝐼𝑟.(37) Premultiply (13) by 𝐵𝑇 and rearrange to yield 𝛽(1)(𝑥)=𝐵𝑇𝜕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,𝑚1, through algebraic computations as 𝑓(𝑚)(𝑥)=𝜕𝜙(𝑚1)(𝑥)𝑓𝜕𝑥(2)(𝑥)+𝑚1𝑗=2𝜕𝜙(𝑗)(𝑥)𝜕𝑥𝐵𝛼(𝑚𝑗+1)𝑔(𝑥),𝑚3,(39)(𝑚1)(𝑥)=𝑚1𝑗=2𝜕𝜙(𝑗)(𝑥)𝜕𝑥𝐵𝛽(𝑚𝑗)(𝑥),𝑚>2.(40)
For 𝑚=3, 𝑓(3)(𝑥)=𝜕𝜙(2)(𝑥)𝑓𝜕𝑥(2)(𝑥)+𝐵𝛼(2)(𝑥).(41) Partitioning 𝑓(3)(𝑥) similar to 𝑓(2)(𝑥) in (16) and considering the 𝑖th block row, 𝑓𝑘𝑖(3)(𝑥)=𝑟𝑗=1𝜙𝑘𝑖,𝑘𝑗𝑓𝑘(2)𝑗(𝑥)+𝑏𝑘𝑗𝛼𝑗(2)=(𝑥)𝑟𝑗=1𝜕𝜙𝑝(2)𝑖1+1(𝑥)𝜕𝑥𝑝𝑗1+1𝜕𝜙𝑝(2)𝑖1+1(𝑥)𝜕𝑥𝑝𝑗𝜕𝜙𝑝(2)𝑖1+𝑡(𝑥)𝜕𝑥𝑝𝑗1+1𝜕𝜙𝑝(2)𝑖1+𝑡(𝑥)𝜕𝑥𝑝𝑗𝜕𝜙𝑝(2)𝑖(𝑥)𝜕𝑥𝑝𝑗1+1𝜕𝜙𝑝(2)𝑖(𝑥)𝜕𝑥𝑝𝑗×000𝑓𝑝(2)𝑗1𝑥𝑓𝑝(2)𝑗(𝑥)+𝛼𝑗(2)(𝑥),𝑖=1,2,,𝑟.(42) Using (31) and (32) and that 𝜙𝑝(2)𝑖(𝑥) being a function of 𝑥𝑝𝑗,𝑗=1,2,,𝑟, only as above, (42) can be reduced to 𝑓𝑘𝑖(3)(𝑥)=𝑟𝑗=100000𝜕𝑓𝑝(2)𝑖1𝑥𝜕𝑥𝑝𝑗00𝑓𝑝(2)𝑗1𝑥0=0,𝑖=1,2,,𝑟.(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)𝑓𝜕𝑥(2)(𝑥)+𝜕𝜙(2)(𝑥)𝜕𝑥𝐵𝛼(3)+(𝑥)𝜕𝜙(3)(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)=𝐵𝑇(𝜕𝜙(2)(𝑥)/𝜕𝑥)𝐵𝛽(1)(𝑥)=(𝐵𝑇(𝜕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(𝑚1)(𝑥) as given in (33) is easily derived so as to satisfy (8), where 𝑔(𝑚1)(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 ̇𝑥=𝐴𝑥+𝐵𝑢+𝑓(2)(𝑥)+𝑔(1)(𝑥)𝑢 can be brought to the form ̇𝑥=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]: ̇𝑥=𝐴𝑥+𝐵𝑢+𝑓(2)(𝑥),(46) where 𝑥=[𝑥1,𝑥2,𝑥3,𝑥4,𝑥5]𝑇=[𝑖𝑠ds,𝑖𝑠qs,𝜓𝑠dr,𝜓𝑠qr,𝜔𝑟]𝑇,𝑢=[𝑢1,𝑢2,𝑢3,𝑢4]𝑇=[𝑣𝑠ds,𝑣𝑠qs,𝜓,𝜓𝑠dr], where 𝜓=𝑖𝑠qs𝜓𝑠dr𝑖𝑠ds𝜓𝑠qr, where 𝑣𝑠qs,𝑣𝑠ds,𝑖𝑠qs, and 𝑖𝑠ds represent the quadrature and direct axis voltages and currents, respectively, and 𝜓𝑠dr,𝜓𝑠qr represent direct and quadrature axis fluxes, respectively. 𝜓𝑠dr,𝜓𝑠qr represent the estimated direct and quadrature flux, respectively, and 𝜔𝑟 represents the angular velocity of the rotor. 𝑖𝑠qs and 𝑖𝑠ds represent the reference values of quadrature and direct axis currents, respectively. The superscript refers to variables in the stationary frame: 𝐿𝐴=2𝑚𝑅𝑟+𝐿2𝑟𝑅𝑠𝜎𝐿𝑠𝐿2𝑟0𝐿𝑚𝑅𝑟𝜎𝐿𝑠𝐿2𝑟𝐿0002𝑚𝑅𝑟+𝐿2𝑟𝑅𝑠𝜎𝐿𝑠𝐿2𝑟0𝐿𝑚𝑅𝑟𝜎𝐿𝑠𝐿2𝑟0𝐿𝑚𝑅𝑟𝐿𝑟0𝐿0000𝑚𝑅𝑟𝐿𝑟𝑅0𝑟𝐿𝑟0,100000𝐵=𝜎𝐿𝑠01000𝜎𝐿𝑠𝑅00000𝑟𝐿𝑟0000003𝑃2𝐿𝑚8𝐽𝐿𝑟0,𝑓(2)(𝐿𝑥)=𝑚𝜔𝑟𝜓𝑠qr𝜎𝐿𝑠𝐿𝑟𝐿𝑚𝜔𝑟𝜓𝑠dr𝜎𝐿𝑠𝐿𝑟𝜔𝑟𝜓𝑠qr𝜔𝑟𝜓𝑠dr0=𝑐1𝑥4𝑥5𝑐1𝑥3𝑥5𝑥4𝑥5𝑥3𝑥50,(47) where 𝐿𝑚,𝐿𝑟, and 𝐿𝑠 represent the magnetizing, rotor, and stator inductances, respectively. 𝑅𝑟 and 𝑅𝑠 represent the rotor and stator resistances, respectively. 𝐽represents the system moment of inertia, and 𝑃 represents the number of poles. 𝜎=1𝐿2𝑚/𝐿𝑠𝐿𝑟 and 𝑐1=𝐿𝑚/𝜎𝐿𝑠𝐿𝑟. 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: ̇𝑥=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 ,𝑓𝐴=0100000000000000000000000,𝐵=00001000010000100001(2)𝑓(𝑥)=1(2)𝑓(𝑥)2(2)𝑓(𝑥)3(2)𝑓(𝑥)4(2)(𝑓𝑥)5(2),(𝑥)(49) where 𝑓1(2)(𝑥)=𝑓1,35𝑥3𝑥5+𝑓1,45𝑥4𝑥5, 𝑓2(2)(𝑥)=𝑓2,35𝑥3𝑥5+𝑓2,45𝑥4𝑥5, 𝑓3(2)(𝑥)=𝑓3,35𝑥3𝑥5+𝑓3,45𝑥4𝑥5, 𝑓4(2)(𝑥)=𝑓4,35𝑥3𝑥5+𝑓4,45𝑥4𝑥5, 𝑓5(2)(𝑥)=𝑓5,35𝑥3𝑥5+𝑓5,45𝑥4𝑥5.

Remark 5.1. 𝑓(2)(𝑥) can be partitioned using (16) as 𝑓(2)𝑓(𝑥)=𝑘(2)1𝑓(𝑥)𝑘(2)2(𝑓𝑥)𝑘(2)3𝑓(𝑥)𝑘(2)4(𝑥),(50) where 𝑓𝑘(2)1𝑓(𝑥)=1(2)𝑓(𝑥)2(2)(=𝑓𝑥)1,35𝑥3𝑥5+𝑓1,45𝑥4𝑥5𝑓2,35𝑥3𝑥5+𝑓2,45𝑥4𝑥5,𝑓𝑘(2)2𝑓(𝑥)=3(2)=𝑓(𝑥)3,35𝑥3𝑥5+𝑓3,45𝑥4𝑥5,𝑓𝑘(2)3(𝑓𝑥)=4(2)=𝑓(𝑥)4,35𝑥3𝑥5+𝑓4,45𝑥4𝑥5,𝑓𝑘(2)4𝑓(𝑥)=5(2)=𝑓(𝑥)5,35𝑥3𝑥5+𝑓5,45𝑥4𝑥5.(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 𝐼𝑦=𝑥+𝜙(𝑥),𝑢=4+𝛽(𝑥)𝑣+𝛼(𝑥),(52) where 𝜙(𝑥)=𝜙(2)(0𝑓𝑥)=1(2)𝑥𝜙3(2)𝑥𝜙4(2)𝑥𝜙5(2)𝑥,𝛼(𝑥)=𝛼(2)(𝑥)=𝑓2(2)(𝑥)𝑓3(2)(𝑥)𝑓4(2)(𝑥)𝑓5(2)(,𝛽𝑥)(𝑚1)(𝑥)=(1)𝑚1𝐵𝑇𝜕𝜙(2)(𝑥)𝐵𝜕𝑥𝑚1,(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 ̇𝑦=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 aṡ𝑥=𝐴𝑥+𝐵𝑢+𝑓(2)(𝑥);(55)𝑥=[𝑥1,𝑥2,𝑥3]𝑡=[𝑖qs,𝑖ds,𝜔𝑒]𝑡 and 𝑢=[𝑢1,𝑢2]𝑡=[𝑣qs,𝑣ds]𝑡 where 𝑣qs,𝑣ds,𝑖qs, and 𝑖ds represent the quadrature and direct axes voltages and currents, respectively, and 𝜔𝑒 is stator angular velocity: 𝑅𝐴=𝑠𝐿qs𝐼0𝑓𝐿dm𝐿qs𝑅0𝑠𝐿ds03𝑃2𝐿dm𝐼𝑓,18𝐽00𝐵=𝐿qs001𝐿ds,𝑓00(2)𝐿(𝑥)=ds𝐿qs𝑥2𝑥3𝐿qs𝐿ds𝑥1𝑥3𝐿ds𝐿qs𝑥1𝑥2.(56)𝐼𝑓 is the field current equivalent to the permanent magnet. 𝐿ds,𝐿qs, and 𝐿dm are the direct, quadrature, and magnetizing inductances, respectively. 𝑅𝑠 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:̇𝑥=𝐴𝑥+𝐵𝑢+𝑓(2)(𝑥),(57) where 𝐴 and 𝐵 given by (2), where 𝑟=𝜇=2 and 𝑘1=2,𝑘2=1, are ,𝑓𝐴=010000000,𝐵=001001(2)𝑓(𝑥)=1(2)𝑓(𝑥)2(2)𝑓(𝑥)3(2)=𝐶(𝑥)1𝑥2𝑥3𝐶2𝑥3𝑥1𝐶3𝑥2𝑥1,(58) with 𝐶1,𝐶2,𝐶3 being constants.

Remark 5.3. 𝑓(2)(𝑥) can be partitioned using (16) as 𝑓(2)𝑓(𝑥)=𝑘(2)1𝑓(𝑥)𝑘(2)2(𝑥),(59) where 𝑓𝑘(2)1𝑓(𝑥)=1(2)𝑓(𝑥)2(2)(=𝐶𝑥)1𝑥2𝑥3𝐶2𝑥3𝑥1,𝑓𝑘(2)2𝑓(𝑥)=3(2)=𝐶(𝑥)3𝑥2𝑥1.(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 𝐼𝑦=𝑥+𝜙(𝑥),𝑢=2+𝛽(𝑥)𝑣+𝛼(𝑥),(61) where 𝜙(𝑥)=𝜙(2)0𝑓(𝑥)=1(2)𝑥𝜙3(2)𝑥,𝛼(𝑥)=𝛼(2)(𝑥)=𝑓2(2)(𝑥)𝑓3(2),𝛽(𝑥)(𝑚1)(𝑥)=(1)𝑚1𝐵𝑇𝜕𝜙(2)(𝑥)𝐵𝜕𝑥𝑚1,(62) where 𝜙3(2)(𝑥) is completely arbitrary and 𝑥=(𝑥2,𝑥3). The system then reduces to ̇𝑦=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 𝑅𝑠=1.2 ohms, 𝐿qs=12.5mH, 𝐿ds= 5.7 mH, 𝜔𝑒= 3500 rpm, 𝑃=4, 𝐽=0.0018 kgm2, and 𝜆af= 123 mWb turns. The mathematical model of the machine in the normal form is given by ̇𝑥=𝐴𝑥+𝐵𝑢+𝑓(2)(𝑥),(64) where 𝐴 and 𝐵 are as given in (57) and 𝑓(2)𝐶(𝑥)=1𝑥2𝑥3𝐶2𝑥3𝑥1𝐶3𝑥2𝑥1,(65) where 𝐶1=0.165(𝑒3),𝐶2=186.96,and𝐶3=5.754386(𝑒6). By applying Corollary 5.4 and choosing 𝜙(𝑥)=𝜙(2)0𝐶(𝑥)=1𝑥2𝑥30=00.0165(𝑒3)𝑥2𝑥30,𝛼(𝑥)=𝛼(2)𝛼(𝑥)=1(2)𝛼(𝑥)2(2)=(𝑥)186.96𝑥3𝑥15.754386(𝑒6)𝑥2𝑥1,𝛽(1)(𝑥)=𝐶1𝑥3𝐶1𝑥2=000.0165(𝑒3)𝑥30.0165(𝑒3)𝑥2,𝛽00(𝑚1)(𝑥)=(1)𝑚1𝐵𝑇𝜕𝜙(2)(𝑥)𝐵𝜕𝑥𝑚1.(66)

It is seen by verification that the system (64) becomes ̇𝑦=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,𝛽𝑚1(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.

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