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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.

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 [

Poincare [

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 [

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 [

To summarise the rest of the paper, in Section

Consider a multiple-input control affine system of the form [

In order to linearize the system, change of coordinate and state feedback [

Consider the specialized case of system (

(

Applying quadratic linearization to system (

We next simplify the homological equations (

Let

Using (

We now put (

Since (

Consider the system

We first show that the quadratic linearization of (

Equation (

As a result of quadratic linearization (i.e., satisfying (

For

For

Noting that from (

Proceeding this way, one can show that

The result of Theorem

A specialized result of Theorem

The squirrel cage induction motor model can be derived as below [

The system (

The result follows directly by applying the result of Theorem

The PM machine model [

The system (

The result follows directly by applying the result of Theorem

The parameters of an actual machine are obtained from [^{2}, and

It is seen by verification that the system (

Verification of quadratic linearization of PMSM through simulations is given in an earlier paper by the authors [

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 (

Finally, as per the result of Theorem