Induction motor control: homotopy continuation approach and power efficiency maximization

Induction motor control: homotopy continuation approach and power efficiency maximization

Alex Borisevich

June 21. 2012

Introduction




   The goal of this report is to present two main results obtained in research work during March-May 2012 in Laboratory of Electrical Drives at Reutlingen University. This work devoted to developing some advanced control algorithms for induction motors. Research is partially supported by Robert Bosch Center for Power Electronics (RBZ) and SEW-Eurodrive GmbH.
   In first chapter abstract problem of output setpoint tracking for affine non-linear system is considered. Presented approach combines state feedback linearization and homotopy numerical continuation in subspaces of phase space where feedback linearization fails. The method of numerical parameter continuation for solving systems of nonlinear equations is generalized to control affine non-linear dynamical systems. Application of proposed method demonstrated on the speed and rotor magnetic flux control in the three-phase asynchronous motor.
   In second chapter the problem of maximizing the efficiency of the induction motor under part-load condition considered. By measuring the difference between power consumption of the quadrature and direct supply channels in dq coordinate system a simple criterion for optimal power loss operation is obtained. The approach differs in that the calculation of developed criterion requires only one parameter of the motor, the the stator inductance. The value of the criterion used in the correction of the flux (magnetizing current) for its optimality tuning.
   The author is grateful for all help and constructive discussion to professor of Technical Faculty Dr.-Ing. Gernot Schullerus. Also author must thank for the technical and motivational support staff of Technical Faculty, namely Prof. Dr.-Ing. Jurgen Trost, Dr. Bernd Petereit, Dr. Daniel Fierro.

1

Chapter 1





                Switching strategy based on homotopy continuation for non-regular affine systems with application in induction motor control




            1.1    Motivation


   Let the affine nonlinear system with m inputs and m outputs in state space of dimension n is given:

m
x = f(x) + y^gi(x)ui, y = h(x),                     (1-1)
i=1
   where x е X C Rⁿ, y e Y C Rm, u e U C Rm, maps f : Rⁿ ^ Rⁿ, gi : Rⁿ ^ Rⁿ, h : Rⁿ ^ Rm are smooth vector fields f,g,h e C'■. Func tions f (x) a nd g(x) are considered as bounded on X.
   Systems of the form (1.1) are the most studied objects in the nonlinear control theory.
   There are several most famous control methods for systems of type (1.1) : feedback linearization [1, 2, 3], application of differential smoothness [4], Lyapunov functions and its generalizations [5], including a backstepping [6], also sliding control [7] and approximation of smooth dynamic systems by hybrid (switching) systems and hybrid control [8].
   All of these control techniques have different strengths and weaknesses, their development is currently an active area of research, and the applicability and practical implementation has been repeatedly confirmed in laboratory tests and in commercial hardware.
   Approach described below is based on the method of numerical parameter continuation for solving systems of nonlinear equations [9], which deals with parametrized combination of the original problem, and some very simple one with a known solution. The immediate motivation for the use of parameter continuation method in control problems is a series of papers [10, 11], in which described the application of these methods directly in the process of physical experiments.
   In this paper we consider the solution of the output zeroing problem for the system (1.1) with relative degrees rj > 1 that expands earlier obtained in [12] and [13] results for a case rj = 1. Further it is supposed that (1.1) it is free from zero-dynamics, i.e. n = 52”=₁ rj.
   The article consists of several parts. We briefly review the necessary facts about the method of parameter continuation and feedback linearization. Next, we represent the main result, an illustrative example of the method, as well as an example of controlling three-phase induction motor.


            1.2    Problem statement and motivation


   In this paper we consider the problem of nonlinear output regulation for affine nonlinear system. In particular, we will solve the problem of output regulation to constant setpoint (without loss of generality, regulation to 0).

2

Definition 1. Given the system of form (1.1). Problem of output regulation to zero (aka output zeroing) is Uie design of such state-feedback a)ntr= Zaw u(t) = u(x) application of which asymptotically drives the sy^t^m output to 0: lim, . ᵥ y(t) = 0.

    The output zeroing problem of affine nonlinear systems can be solved using mentioned above feedback linearization method. The main idea of the method consists in the transformation using a nonlinear feedback nonlinear system N : u(t) ^ y(t) to the linear one L : v(t) ^ y(t) with the same outputs y, but new inputs v. After that, the resulting linear system L can be controlled by means of linear control theory.
    Suppose that a control problem of N can be in principle solved, i.e. there is exists a satisfying input signal u*(t), which gives the output response y*(t). The essence of problems in feedback linearization comes from that the response y*(t) may not be in any way■ reproduced by system L which is obtained after linearization.
    The simplest specific example is the system x = u, y = h(x) = x(x² — 1) + 1, x(0) = 1 for which the problem of output zeroing y ^ 0 is needed to solve.
    If the system under consideration was a constant relative degree, the use of control v = —y after feedback linearization would give the output trajectory of y(t) = exp(—t), which is everywhere decreasing y(t) < 0.
    In this case, the nonlinearity y = h(x) has two limit points xj ₂ = ±1/V%, in which h'ₓ(x( ₂) = 0. Any trajectory y(t), that connects y(0) = 1 with y(T) = 0 passes sequentially through the points yj = h(3⁻¹/²) and y2 = h(—3⁻¹/²), and besides yj > yj. Hence, anу trajectory y(t) on the interval (0,ti) should decrease with time (Figure 1), on the interval (ti,t₂) increase, and in the interval again decrease. Such a trajectory is not reproducible using the feedback linearization.


Figure 1. Output trajectories of linearized system with constant relative degree and system with y = h(x) = x(x² — 1) + 1

    The behavior of the system in Figure 1 can be interpreted as follows: in the intervals (0, ti) and (t₂, T) the system can be linearized in the usual manner and presented in the form y = v. On the interval (ti, t₂) system behavior differs from the original, and the trajectory need to move in the opposite direction from the y = 0, which is the same as control of system y = —v. A similar situation arises in numerical methods for finding roots and optimization of functions with singularities, where in order to achieve optimum or find a root motion in the direction opposite to predicted by Newton’s method is needed. We can use the parameter A € [0,1] to indicate the motion direct ion. Increasing of parameter A > 0 corresponds to the movement of y(t) in the direction to the desired setpoint y = 0, and parameter decreases A < 0 in the opposite movement. The points of direction change A(t) = 0 correspond to overcoming the singularities of h(x). In fact, this idea is the basis of the approach proposed below.



            1.3  Background


  In this section we present known facts needed to understand the main result. Finally, we come to the conclusion that the numerical homotopy methods can be used not only for solving nonlinear equations, but also for control of nonlinear affine systems. Here and below we will always consider a setpoint tracking problem.

        1.3.1   Feedback linearization

Definition 2. MIMO nonlinear system has relative degree rj for оutput yj in 6 C Rⁿ if at least for one function gi is true


L Lf 1hj =0                                               (1.2)
    where Lf X = 'ff⁴ f (x) = P”=i '''^fx fi(x)     a Lie deriva tive of function X along a vector field f.
    It means that at least one input uₖ influences to output yj after rj integrations.
    Number r = 52 Г i rj is called as the total relative degree of system. If r = n and matrix

•Г1 - 1 7

A(x) =

/ LgᵢLf¹⁻¹hi(x)

L
 gm

\LS₁ Lf

hm (x)

L„ Lf
gm f

hi(x)

hm(x)

(1-3)

is full rank, then the original dynamical system (1.1) in 6 equivalent to system:

yjrj) = Lf hj + Xх LgiLf 1hj • Ui = B(x) + A(x) • u i=1

(1-4)

The nonlinear feedback

u = A(x) 1[v — B (x)]                                      (1-5)
   converts in subspace 6 original dynamical system (1.1) to linear:

y⁽rj ) = Vj                                          (1.6)
    Control of a nonlinear system (1.1) consists of two feedback loops, one of which implements a linearizing transformation (1.5), second one controls the system (1.6) by any known method of linear control theory.
    A significant drawback, which limits the applicability of the feedback linearization in practice is requirement of relative degree r constancy and full-rank of matrix A(x) in the whole phase space 6.


        1.3.2 Numerical continuation method

   Let it is necessary to solve system of the nonlinear equations

ф(£) = о

(1-7)

    where ф : Rm н Rm is vector-valued smooth nonlinear function.
    Lets Q C Rm is open set and C(Q) is set of continuous maps from its closure Q tо Rm. Functions F₀,F1 e C(Q) are homotopic (homotopy equivalent) if there exists a continuous mapping

H : Q x [0,1] н Rm

(1-8)

   that H (£, 0) = F₀(£) a nd H (£, 1) = F1(£) for all £ e Q. It can be shown [9] that the equation H(£, X) = 0 has solution (£, X) for all X e [0,1]. The objective of all numerical continuation methods is tracing of implicitly defined function H(£, X) = 0 foг X e [0,1].
   Lets H : D н Rm is C-^continuous function on an open set D C Rm⁺¹, and the Jacobian matrix DH(£, X) is full-rank rank DH(£, X) = m for all (£, X) e D. Then, for all (£, X) e D exists a unique vector т e Rm⁺¹ such as

DH(£, X) • т = 0, кт||₂ = 1, det (DH<’ X⁾"j > 0,

and mapping

Ф : D ■ Rm⁺¹, Ф : (£, X) н т

(1-9)



(1-Ю)

is locally Lipschitz on D.
Function (1.17) specifies the autonomous differential equation