The use of differential geometry methods for linearization of nonlinear electrical circuits with multiple inputs and multiple outputs (MIMOs)

Electrical circuits with concentrated parameters or electro-mechanical systems can be described by a finite number of mutually coupled ordinary differential equations and algebraic relations. Most mathematical models of physical systems are complex. They are characterized by high-order and numerous nonlinearities. A well-known nonlinear model with multiple inputs and multiple outputs which describe its dynamics in the state space can be represented by the following equations:where: f and g_i are smooth vector fields determined on a manifold M = Rⁿ, called state space, h is a smooth mapping specified for the state space M in p-dimensional output space, \( {\mathbf{R}}^{p} ,{\mathbf{h}} :\quad M \to {\mathbf{R}}^{p} \), u(t)—input vector, y(t)—output vector.

The analysis of nonlinear systems (1), especially in dynamic states, is a very difficult task of circuit theory. In most cases, there is no analytical solution of the problem (which is frequently sought) and the information about the current flow and voltage distribution can be obtained using the methods of numerical integration. As shown in Gear [1], Butcher [2], Najm [3], although they are universal and applicable to any number of differential equations, they generate a numerical solution (compiled in the form of tables, graphs, etc.). Therefore, in the search for analytical solutions the transformation of nonlinear description into a linear one by linearization (ensuring local balance of the system dynamics) is very useful in solving practical problems relating to the behaviour of nonlinear circuits.

The methods of differential geometry provide a tool for linearization and decoupling, and decomposition of a system of Eq. (1) to a linear form. The application of geometric approach to solve nonlinear problems initiated by Brockett [4] was used in control theory with observability and controllability of the systems taken into account [5, 6]. The methods of differential geometry allowed the development of efficient techniques for the analysis and synthesis of such systems. The series of publications by Byrnes and Isidori [7], Celikovsky and Nijmeijer [8], Jakubczyk and Respondek [9] which considered the problem of nonlinear transformation of a linear system by changing the coordinates (local diffeomorphism) and using feedback [2, 10, 11] considerably contributed to the development of the techniques mentioned. Furthermore, the problems of transformation of nonlinear systems to linear forms have been widely studied in the literature by Isidori [12, 13], Nijmeijer and van der Shaft [14] Bodson [15], Su [16], and Hunt et al. [17].

As Eq. (1) describes, a nonlinear system in local coordinates x in Rⁿ [where x = (x₁, x₂, …, x_n) is a local coordinate system on a smooth manifold M = X_s ⊃ x], one normally considers a transformation of state variables [12‐14, 18‐20] whose operation can be presented as follows:As a result of the transformation, the state vector in new coordinates assumes the following form:where: S(x) = [S₁(x), S₂(x), …, S_n(x)]^T is the mapping defined on the open set of Rⁿ space with Rⁿ values.

The transformation of this kind is called diffeomorphism [9, 13]. The necessary and sufficient conditions for the existence of S(x) transformation of nonlinear system into linear one are given in [14]. Transformation S(x) is difficult to find—especially for multidimensional systems so the systems which cannot be globally linearized and are transformed only into quasi-linear systems can be further linearized by the feedback. In this case, a combination of linearization by transformation of state variables and input transformation u(t) using a feedback is applied [11, 21‐25]. After transformation involving the change in coordinates and the introduction of the feedback, the state vector in a new coordinate system can be illustrated as follows:

The cited publications indicate that although geometric methods are mainly applied in control theory, they can be also used in other areas, e.g. in the theory of electrical circuits. Such attempts have already been made (for example [26‐30]), but they were too few and not exhaustive enough. It appears that geometrical methods can easily find other applications. Since the presented issues are still valid, we attempt to use geometric methods in the theory of electrical circuits. It seems that the results presented can easily find other applications. As the considered issues are still valid, we attempt to use geometrical methods in theory of electrical circuits. Some significant developments have been made in robust feedback linearization in [31‐33], but most of them are applicable for single-input nonlinear systems or when parametric uncertainties exist. The fact that the there are many more publications analysing SISO systems than those dealing with MIMO systems prompted the authors to use the methods of differential geometry for linearization of a class of nonlinear electrical circuits with multiple inputs (power sources) and multiple outputs.

The paper is organized as follows: Section 2 presents the elements of Lie algebra, used in the construction of a new base of state space. The basic definitions and theorem concerning the conditions to be met by nonlinear system to perform the linearizing procedures are included. Section 3 deals with the construction of linearizing transformation. The effective transformation and digital simulation of mathematical model of nonlinear electrical circuit MIMO showing that linearization is correct are given in Sects. 4 and 5. Conclusions and comments are presented in Sect. 6. The paper ends with a list of references.

Some basic definitions and concepts of differential geometry are presented. Attempts have been made to discuss them in a simplified and compact form. More detailed information can be found in the references [34‐36].

In the analysis of nonlinear systems, the Lie derivative deserves particular attention. It is the operation involving real-valued function h and vector field f defined on manifold M of \( {\mathbf{R}}^{n} \) space. The result of the operation is a smooth real-valued function defined for each x from M set.

where: \( \otimes \)—stands for gradient; \( \circ \)—stands for scalar product.

For example:

If g is another vector field defined on the same manifold \( M = {\mathbf{R}}^{n} \) then a derivative of scalar function along the vector field g is defined as follows:

We consider a general case of nonlinear system described by Eq. (1) with m power sources (inputs) and m outputs. The output vector of y(t) system is related to u(t) vector of power sources through state variables and nonlinear equation of state. The task of linearization consists in finding a new vector v(t) of power source (1) such that each m output block depends only on one power source, that is, has the outputs decoupled from the inputs. The terms of decoupling outputs from inputs are given by the definition:

The necessary condition for the invariance of outputs is given by the following theorem:

Hence, the sufficient condition for decoupling outputs from inputs:for each \( k \ge 0 \) and \( {\varvec{\uptau}}_{1} ,{\varvec{\uptau}}_{2} , \ldots ,{\varvec{\uptau}}_{k} \in \{ {\mathbf{f}},g_{1} , \ldots ,g_{m} \} . \)

The starting point to determine diffeomorphism S(x) transforming the system (1) into linear system is the definition of a relative degree of the system, sometimes called the characteristic number [12‐14].

If: \( L_{{\mathbf{g}}} L_{{\mathbf{f}}}^{k} h_{j} ({\mathbf{x}}) = \left( {L_{{g_{1} }} L_{{\mathbf{f}}}^{k} h_{j} ({\mathbf{x}}), \ldots ,L_{{g_{m} }} L_{{\mathbf{f}}}^{k} h_{j} ({\mathbf{x}})} \right) = 0 \)\( \forall \) k ≥ 0 and x \( \forall \) U, it is assumed that \( r_{j} = \infty \).

It should be emphasized that each number r_i is associated with the ith output of the h_i system. It is also worth noting that by differentiating the y_i output with respect to time r_i, we obtain:i.e. r_i is the number that tells us 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.

It should be pointed out that if the system has outputs decoupled from inputs then for \( {\mathbf{x}} \in {\mathbf{R}}^{n} \) the set \( L_{{{\mathbf{g}}_{i} }} L_{{\mathbf{f}}}^{{r_{i} - 1}} h_{i} ({\mathbf{x}}) \ne 0,\,1 \le i \le m \) contains V. If a nonlinear system does not have outputs decoupled from inputs, a regular static feedback can be used to change it into a system with outputs decoupled from inputs. As a result, we obtain Eq. (7).

where: u = (u₁, …, u_m)^T and α: \( M = {\mathbf{R}}^{m} \) and β: \( M = {\mathbf{R}}^{{m{\text{x}}m}} \) are smooth mappings such that matrix \( \beta ({\mathbf{x}}) \) is non-singular for any x, and v = (v₁, …, v_m) is a new input vector.

Therefore, in order to solve the problem of linearization of the system with multiple inputs and multiple outputs with initial state x_o one should find a regular static feedback defined in such neighbourhood V of x_o point that each y_i output is affected by only one v_i input, \( 1 \le i \le m \).

The feedback in this case has the form:where: E(x)—decoupling matrix and vector b(x) are given by the relations:

The proof of the theorem is presented in [14].

The method used to determine the transformation S(x) linearizing a nonlinear system (1) and coordinates z(t) of the linearized system is presented in the next section.

Let us consider the following nonlinear system (1) with m inputs and m outputs:

Let outputs be divided into m separable blocks. Let the system have a relative degree r₁, …, r_m at x_o, and let the rank of decoupling matrix E(x_o) equal m. Then r₁ + ··· + r_m ≤ n and for \( 1 \le i \le m \) the transformation of coordinates is written as:If r = r₁+ ··· + r_m is less than n it is always possible to find such n – r of functions \( S_{r + 1} ({\mathbf{x}}),\; \ldots \;,\;S_{n} ({\mathbf{x}}) \) that the mapping:has a Jacobi matrix which is non-singular at x_o and therefore determines local coordinates of transformation in the neighbourhood of x_o.

To determine a relative degree of a multidimensional system, we differentiate output “j” according to the relation:

If \( L_{{{\mathbf{g}}_{i} }} h_{j} ({\mathbf{x}}) = 0 \) differentiation should be continued for each i until for some natural r in the ith step, we obtain: \( L_{{{\mathbf{g}}_{i} }} L_{{\mathbf{f}}}^{{r_{j} - 1}} h_{j} ({\mathbf{x}}) \ne 0. \)

We differentiate outputs according to a general relation:

Thus if \( L_{{{\mathbf{g}}_{i} }} L_{{\mathbf{f}}}^{{r_{j} - 1}} h_{j} ({\mathbf{x}}) \ne 0 \), the output equation takes the following form:where: E(x) is a decoupling matrix of m × m dimension described by the relation (13).

As a result, we obtain a relative degree of the system r = r₁+ ··· + r_p, for \( 1 \le p \le m \).

If \( \det \,{\mathbf{E}}({\mathbf{x}}) \ne 0 \), then E(x) matrix is non-singular and we can determine the feedback linearizing the system. The feedback is described by the following relation (12):

To determine the transformation of state variables S(x), we use Eq. (15), which can be written in the following general relation \( S_{n}^{i} ({\mathbf{x}}) = L_{{\mathbf{f}}}^{k - 1} h_{i} ({\mathbf{x}}) \). Hence, the state variables are given by the system of equations:

Their derivatives are determined as follows:

Thus, new dynamics of a linearized system are described by an m set of the following equations:

The outputs of the system are given by relation (19).

Equations \( \dot{z}_{r + 1}^{i} = q_{r + 1} ,\; \ldots ,\;\dot{z}_{n}^{i} = q_{n} \) in (23) are determined according to the following equations:

The solution of the considered problem is defined locally in the state space for x close to x_o, in which the decoupling matrix E(x) is non-singular. It should be noted that non-singularity of the matrix is also a necessary condition for a solution to exist.

We consider an indefinite state in an electrical circuit, comprising two power sources and two nonlinear elements, as shown in Fig. 1.

We assume that the initial state of the circuit is zero (zero initial conditions), and that at t = 0 the switches W₁ and W₂ are closed simultaneously. Nonlinear current–voltage characteristics of the nonlinear resistive element are described by the second-degree polynomial of the following form:where: b is a coefficient of \( \left( {\Omega {\text{A}}^{ - 1} } \right) \) dimension.

A nonlinear coil (without losses) is described by the following relation:where: \( L(i) = (a \cdot i)^{ - 1} \) and a is a coefficient of \( \left( {\Omega {\text{A}}^{ - 1} } \right) \) dimension.

We assume that \( i_{1} = x_{1} ,\;i_{2} = x_{2} ,\;i_{4} = x_{3} ,\;u_{C} = x_{4} \) are state variables and the considered system has the form:

We order variables, adopt the notations: \( 1/L_{2} = k;\;R/L_{2} = c; \) \( b/L_{2} = d;R/L_{1} = w,1/C = l \), and employ the expression for inductance of a nonlinear coil \( L(i_{1} ) = L(x_{1} ) = (a \cdot x_{1} )^{ - 1} \) to obtain a model system of equations:for which the corresponding vectors \( {\mathbf{f}} ({\mathbf{x}}),{\mathbf{g}}_{1} ({\mathbf{x}}),{\mathbf{g}}_{2} ({\mathbf{x}}) \), \( \left( {{\mathbf{f}},{\mathbf{g}} \in {\mathbf{R}}^{4} ,n = 4} \right) \) have the form:

The equations of response (output) are as follows:

Hence, the output functions have the form:\( h_{1} ({\mathbf{x}}) = x_{3} \) and \( h_{2} ({\mathbf{x}}) = x_{4} \).

In the considered circuit, the outputs y(t) are related to the power supply vector e(t) through state variables and nonlinear equation of state. To solve the problem, we have to find a new power source defined by a regular, static feedback. The first step of linearization is to determine a relative degree of the system according to definition 2. The considered system has two inputs and two outputs. From Eq. (30), we can directly determine differentials of functions h₁(x) and h₂(x):

Calculating the Lie derivatives of function h(x) along g(x) field for h₁(x) function, we get:

Similarly, for function h₂(x) we obtain:

As the first derivatives equal zero, we calculate the derivatives of higher orders.

For h₁(x) function along g₁(x) field, we obtain:

Similarly, for g₂(x) we get:

Thus, the relative degree of the subsystem related to function h₁(x) is r₁ = 2.

For the h₂(x) function, we have:

This means that for the h₂(x) function, a relative degree of the subsystem is also r₂ = 2.

Finally, we obtain that the relative (vector) degree of the considered system is {r_1, r₂} = {2, 2} for each point x = (x₁, x₂, x₃, x₄) at which \( x_{1} \ne 0 \).

Next, we define a decoupling matrix E(x) (13) which has the following form:

The rank of E(x) equals 2.

Since \( \det {\mathbf{E}}({\mathbf{x}}) = - k \cdot w \cdot l \cdot a \cdot x_{1} \ne 0 \), it follows that matrix E(x) is non-singular and it is possible to determine feedback linearizing input–output system:where: v = [v₁, v₂]^T is a new power source vector.

We can now determine new coordinates of the considered system according to (21):

To find a normal form of the system, we have to determine b(x) vector and E⁻¹(x) matrix from Eq. (32).Since:we have to calculate corresponding Lie derivatives of h(x) function along the vector field f(x):

and

Now, the normal form of the linearized system from (22) is written as:

The expression for linearizing feedback (32) can be written as:

Thus, after transformations we get:

For the assumed feedback, the system defined by (23) is described by the following equations:

Also from the structure of power source, it follows that:

In matrix notation, the linearized system has the form:

As a result of linearization, we obtain a decomposed and decoupled linear model of the coordinate system z(t). Each state variable is defined by the differential equation with the right side, represented by the first-degree polynomial of the simplest form. To sum up, we have determined linear form of state equations for the considered system in which the ith output y_i depends only on the ith power input v_i, for i = 1, 2.

In order to verify the results obtained a numerical solution of nonlinear equation of state, given by Eqs. (28) and (30), and its linearized model (38, 39) was conducted.

Numerical solution of nonlinear equations of state was carried out for zero initial conditions x(0) = 0, parameters a = b = k = 1, c = 5, d = 10, w = 2, and for inputs e₁(t) = e₂(t) = 1(t).

Figure 2 presents time characteristics of nonlinear system outputs y₁(t) and y₂(t).

To illustrate correct performance of the transformation linearizing a nonlinear system, we compare the obtained simulation results with the results of simulations conducted for Eqs. (38) and (39) linearized by transformation onto original space x(t) = S⁻¹(z(t)). For the purpose of comparison, we conduct the following calculations:

Since: \( z_{1} ({\mathbf{x}}) = x_{3} ;\;\;z_{2} ({\mathbf{x}}) = w \cdot x_{2} - w \cdot x_{3} ;\;\;z_{3} ({\mathbf{x}}) = x_{4} ;\;\;z_{4} ({\mathbf{x}}) = l \cdot x_{2} - l \cdot x_{1} \), then determining the variables of x state we get:

New power sources v₁ i v₂ are determined from the relation:

Initial conditions were calculated as follows:

Figure 3 presents time characteristics of outputs y₁(t) and y₂(t) as shown for a nonlinear system.

As S(x) is an algebraic transformation of numerical solutions (time characteristics) of nonlinear systems state equations (Fig. 2), and the corresponding solutions of linearized systems (Fig. 3) overlap. This confirms that the derived mathematical models of linear systems subjected to linearization are correct.

The use of geometrical methods for nonlinear mathematical models makes it possible to obtain simple models of linearized systems. Thanks to that we can analyse linearized models using methods known from the theory of linear systems and then transfer the results to a nonlinear system (1) by means of inverse transformation S⁻¹(z). The transformation of nonlinear description into linear one by linearization (ensuring local balance of the system dynamics) is very useful in solving practical problems relating to the behaviour of nonlinear objects. It not only allows a simpler analysis of such systems but also makes it possible to avoid problems associated with the nonlinearity of the system.

Furthermore, it is worth noting that the numbers r₁+ ··· + r_p are called the characteristic numbers and each characteristic number r_i is associated with jth output of h_i system and tells us how many times y_i must be differentiated to receive power supply u_i explicitly. If a relative degree of the system \( r < n \) then such n – r of \( S_{r + 1} ({\mathbf{x}}),\; \ldots \;,\;S_{n} ({\mathbf{x}}) \) coordinates must be defined that the Jacobian of the mapping (16) is of full rank at x_o point. Then, making an additional assumption that \( L_{{\mathbf{g}}} S_{i} ({\mathbf{x}}) = 0 \) we obtain additional equation of the system dynamics \( \dot{z}_{i} = L_{{\mathbf{f}}}^{{}} S_{i} (x(t)) \).

Nonlinear system of multiple inputs and multiple outputs can be decomposed by a static feedback only if E(x) matrix is non-singular. Non-singularity of the matrix is thus a necessary condition for a solution to the problem. However, when choosing different output functions of the considered system, it is a frequent case that when we look for a regular feedback, a decoupling matrix is singular. In this case, dynamic feedback [12‐14] is used to solve the problem.