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

Różowicz, Sebastian; Zawadzki, Andrzej

doi:https://doi.org/10.1155/2018/9405256

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 9405256 | https://doi.org/10.1155/2018/9405256

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

Sebastian Różowicz¹and Andrzej Zawadzki¹

Academic Editor: Vyacheslav Kalashnikov

Received26 Apr 2018

Revised08 Sept 2018

Accepted28 Oct 2018

Published06 Nov 2018

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.

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 where f and are smooth vector fields defined on the manifold M = , called the state space, h is a smooth mapping defined on the state space M into p-dimensional space of outputs, and : .

The methods of differential geometry allow for the linearization of nonlinear systems (1) to the following linear form [1–3]: where z and v are new state and input vectors, w is an output vector in a linearized system, and A, B, and C 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 variables x with new variables z describing the system in a new state space [1–13]. The linearizing transformation can be written aswhere S(x) is linearizing transformation and z(t) a new state vector

After transformation the state vector is of the form:where is a mapping defined on an open set of space with values and the following properties:(i) is locally invertible; i.e., there exists the inverse mapping (that can be transformed back to the original components of state vector x) such that where is the image in the S transformation;(ii) S and 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 transformation u(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

As 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. Section 2 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 manifold M deserves particular attention. The result of the operation is a smooth real-valued function defined for each x from the M set as follows.

Definition 1. Let h mapping be a smooth scalar function of n variables, , and a vector field defined on manifold ; then the Lie derivative of scalar function in the direction of field f is a scalar function given by the formula:There exists the following recurrence formula for this derivative:The second operation concerns vector fields f and g defined on the manifold M = space. The result of the operation (defined below) is a new smooth vector field.

Definition 2. Let f and g be vector fields defined on the manifold M = . Lie brackets of the vector field are called the third vector field, defined by the following relation: where – scalar product.
For such notation of square brackets we obtain the recurrence formulaeThe direct linearization occurs for nonlinear systems for which the vector field represented by Lie brackets 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 , to the controllable and observable linear system (2).(b)System (1) satisfies (C), (O), and , for all , all , and all (where = f), when at least two of the ’s differ from .(c)System (1) satisfies (C), (O), and constant for all , all , and all (d)System (1) satisfies (C), (O), and and constant for all , all , and all (e)System (1) satisfies (C), (O), and , for all , where ’s are real numbers and constant for all , all , and all The proof of Theorem 3 and the extensive discussion of conditions are presented in [53].

In the analysed model of the nonlinear system (1) the output of y(t) is related to input u(t) through the state variables and the nonlinear equation of state. A simple and direct form of the relation between the output y(t) (or its derivatives) and the input u(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 degree r₁(x), …, r_p(x) of a smooth nonlinear system includes the smallest natural numbers such that for each :If k0 and xU, it is assumed that .
It should be emphasized that each number r_i is associated with the i-th output of the h_i system. It is also worth stressing that by differentiating the y_i output with respect to time r_itimes we obtainThat is, r_i is the number that tells how many times the system output y_i(t) should be differentiated to obtain the supply u_i in “an explicit form”, that is, the linear relation between supply u_i and output y_i.
If the characteristic numbers of the system are finite, then we can define decoupling matrix:The method of determining the coordinate (local transformation) of a linearized system is as follows.

Lemma 5. Let the system have a relative degree r₁,..., r_p at the point , and let the rank of decoupling matrix be p. Then r₁+ ... + n and for coordinate transformation is New coordinates for are defined asTheir derivatives are determined as follows:By calculating other components in the same way we obtain a general form of linearized system of The output equation is written asThe above equations represent a normal form of equations describing a nonlinear system with m inputs and p outputs, with a relative degree r₁,..., r_p at the point .
It should be noted that in (15a) coefficients are equal to matrix elements, where is replaced by , and coefficients are the elements of the vector:where is replaced by .

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 Theorem 3 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 aswhere u = and α: and β: are smooth mappings with the property that matrix is nonsingular for any x and v = (v₁,..., ) is a new supply vector.
The application of regular, static feedback to the system (1) results inwhich is of the same form as the considered nonlinear system but now with the newly defined inputs (supplies) (v₁,..., ).

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) having m inputs and m outputs described byLet the outputs of the system be divided into m disjoint blocks.

Here, the task of linearization consists in finding a new supply of the system (19) so that each of the m output 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 if there exists a neighbourhood V of point in which for each for any vector field , from the set ) and if a relative degree of the system r₁,... r_m is finite and constant on V.
In order to solve the problem of linearization for the system with multiple outputs and multiple inputs with the initial state one should find a regular, static feedback defined in the neighbourhood V of the point such that each output is affected by one input , . The feedback in this case isThe decoupling matrix E(x) and the vector b(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 Theorem 3. Here, the combination of linearization by transformation of state variables and the transformation of input function u(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 Definition 4:
Ifwe 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, for and any the relative degree of the system 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 output y we obtainwhereTherefore, if for a given point then in the neighbourhood of this point the following transformation of system input can be defined:It is clear that the relation between a new input v and output y is linear.

If, however, for any point then the differentiation of output should be repeated. HenceAnd again if then a new supply can be written as

Otherwise, the differentiation procedure should be continued up to the step where for some natural r we obtain for some .

The output differentiation is expressed by the following general relation:Finally, a new supply iswhich leads to the relation:and we obtain the linear relation between input v and output y.

Step 2. If a relative degree of the system r = n then linearizing transformation S(x) is determined by relation (12). Hence, for the system with single input and single output we get

Step 3. We determine the state vector in new coordinates .

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 supply v is obtained as follows:Hence, we get the equations of states (15a) and (15b):and output

Step 6. If the relative degree of the system r < n, then such coordinates are defined as follows: the Jacobian of the mapping (14) is full-rank at a point (with an additional assumption that ).
Thus we obtain additional equations of system dynamics:Assuming thatthe equations are

Step 7. We determine the state vector in new coordinates:

Step 8. We determine the scalar function and define regular feedback by the relation:

Step 9. The system dynamics in new coordinates can be represented by the linearized system of and the output

Example 9. We consider transient state in the electrical circuit with nonlinear resistor as shown in Figure 1.

Current-voltage characteristics of a nonlinear resistive element are described by a second-degree polynomial:where a – coefficient of dimension.

Assuming that the state variables are currents flowing through the inductances and the capacitor voltage, , the state vector becomes . The state equation modelling the considered electrical circuit is given bywhere , , and vectors , are

In the circuit described by (41) the output y(t) is related to the power supply e(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 function h(x)): and are considered.

Case 1. Suppose that we have to find current in the branch with a nonlinear resistor.
Since the output equation is(i)we determine the relative degree of the system. Directly from the above equation, we can determine the differential of the function present in the formulas for the Lie derivative of h function along the fields f and g, respectively:Calculating derivatives one by onewe obtain , so for any point x = (x₁, x₂, x₃) the relative degree of the system is r(x) = 3.(ii)the next step is to determine the S(x) transformation of state variables and new coordinates:(iii)regular, static feedback linearizing the system is of the form: Hence, a new supply is(iv)we determine normal form of linearized system for the above feedback:The system dynamics of new coordinate system can be represented by the matrix:The correct work of the algorithm is confirmed by the results obtained by the simulation of linearized system. The results are presented in Figure 2.

Figure 2 shows the numerical solutions of (48) after transformation of state variables z(t) of the linear system into variables x(t) of nonlinear system.

Case 2. Now let us find voltage on the capacitor . The output function isso the differential of the function is (i)by calculating the relative degree of the system we obtainthat for any point x = (x₁, x₂, x₃) the relative degree of the system is r (x) = 2.

Since r < n using the algorithm in accordance with Step 6 we have to determine the third component of transformation S(x) by solving the partial equation:The function, which satisfies the above equation and the condition , can be given by the formula:Thus, the new state variables are of the form:The feedback is given asAnd a vector of new supply isBy determining the normal form of the linearized system we getDynamics of the system in new coordinates is written asThe 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 of m inputs and m outputs 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 Definition 6 (as in one-dimensional case) we determine a relative degree of nonlinear system by calculating the Lie derivative from the function , along the vector fields f and g_i(x), respectively.

Note 10. To determine the relative degree of nonlinear multidimensional system we differentiate outputs:If then for each i differentiation is carried out until in the ith step for some natural r we obtain that .

To differentiate outputs we use general relation:

If , then the output equation of linearized system is written aswhere is m x m decoupling matrix defined by (11).

As a result we obtain the relative degree of the system r = r₁ + …+ r_p, for .

Step 2. We determine the decoupling matrix (11) and vector (16).

Step 3. If , then matrix E(x) is nonsingular we determine the feedback linearizing the system as

Step 4. From a general relation we determine the transformation of state variables S(x) for a multidimensional system: .

Step 5. We determine state variables using .

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, (Figure 3). We assume that the initial state of the circuit is equal to zero and at t = 0 we close switches W₁ i W₂. simultaneously. We also assume that current-voltage characteristics of nonlinear resistive element , where b is a coefficient of dimension. Nonlinear coil (without loss) is given by , where and a is a coefficient of dimension.

We assume that are state variables. We order variables and introduce notations:

and the expression for the inductance of nonlinear coil to obtain the model system of equations in which the vectors and become

The output equations are

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 function h₁(x) we obtainSimilarly, for the function h₂(x) we getAs the first derivatives are equal to zero, we calculate derivatives of consecutive rows. For the function h₁(x) we getAs a result the relative degree of subsystem r₁ = 3. From the calculations for the function h₂(x) we obtain r₂ = 2 because

Finally, we find that the relative (vector) degree of the considered system equals . As r₁ + r₂ = n the next step of the algorithm is to determine decoupling (decomposing) matrix E(x):

Since the determinant of the matrix , E(x) is nonsingular and we can determine feedback linearizing output-input system:

where v₁i v₂ are components of a new supply (output) vector.

After transformations we getNew coordinates are given byThe normal form of linearized system according to (14) is

Also, the structure of the assumed power supply implies that and

Finally, we obtain the linear form of the state equations for the system in which the ith input y_i depends only on the ith power input v_i, for i = 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 (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 Theorem 3 determines how the transformation directly linearizing the system, or the transformation with feedback should be sought.

It should be stressed that the numbers r₁ + …+ r_p are called the characteristic numbers and any characteristic number r_i is associated with j-th output of the system h_i which determines how many times y_i must be differentiated to get the power u_i explicite.

Nonlinear system with multiple inputs and multiple outputs can be decomposed by a static feedback only if the matrix E(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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2018 Sebastian Różowicz and Andrzej Zawadzki. 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.

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