Classical state feedback controller for nonlinear systems using mean value theorem: closed loop-FOC of PMSM motor application

Abstract

The problem of state feedback controllers for a class of Takagi-Sugeno (T-S) Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the differential mean value theorem (DMVT) and convex theory. The proposed design approach is based on the mean value theorem (MVT) to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients as linear parameter varying (LPV) systems. Using the Lyapunov theory, stability conditions are obtained and expressed in terms of linear matrix inequalities (LMIs). The controller gains are then obtained by solving linear matrix inequalities. The effectiveness of the proposed approach for closed loop-field oriented control (CL-FOC) of permanent magnet synchronous machine (PMSM) drives is demonstrated through an illustrative simulation for the proof of these approaches. Furthermore, an extension for controller design with parameter uncertainties and perturbation performance is discussed.

Keywords

Takagi-Sugeno (T-S) fuzzy systems / sector nonlinearity / nonlinear controller / linear matrix inequality (LMI) approach / differential mean value theorem (DMVT) / field oriented control (FOC) / linear parameter varying (LPV)

Introduction

Tremendous research activities on the problem of implementation and observers-based control for nonlinear dynamic systems show that there is a growing interest in the control theory in the last decade.

Specific performances require a careful choice of controller which demands completely the knowledge of states and inputs of the system. However, very less approaches concern the observer-based control problem for nonlinear systems.

The problem of state feedback controllers for nonlinear systems remains an open research subject up to now. Obtaining a systematic control method for nonlinear systems under linear matrix inequality (LMI) conditions poses a great challenge.

Fuzzy representation of nonlinear systems is an especially important topic [ 1− 3]. Nonlinear systems can be represented by Takagi-Sugeno (T-S) fuzzy rules with consequent parts as linear subsystems [ 4, 5]. Some works [ 6− 8] on controller and observer design were analyzed for a class of nonlinear systems using a differential mena value theorem (DMVT) and sector nonlinearity approaches.

Permanent magnet synchronous motors (PMSMs) are widely used in high performance servo applications [ 9] due to their high efficiency, high power density, and large torque to inertia ratio [ 10− 12]. However, PMSMs are highly nonlinear multivariable dynamic systems and, without speed sensors and under load and parameter perturbations, it is difficult to control their speed with high precision using the conventional control strategies [ 13, 14].

Linearization and/or high-frequency switching-based nonlinear speed control techniques, such as feedback linearization control, sliding mode control, back stepping, state dependant Riccati equation (SDRE) and TS fuzzy controls have been implemented for PMSM drives [ 15].

This paper deals with the problem of classical state feedback with a modified parallel distributed controller (MPDC) for a class of Lipschitz nonlinear systems with nonlinear output measurements.

The proposed design approach is based on the mean value theorem (MVT) to express the nonlinear error dynamics of the combined system and controller as a convex combination of known matrices with time varying coefficients as LPV systems. These kinds of problems have been solved by using MVT and sector nonlinearity to nonlinear terms in the state error controller equation [ 16, 17]. The controller gains can be designed based on the results for that class of nonlinear systems by using the Lyapunov theory [ 1, 13], so, stability conditions are obtained and expressed in terms of linear matrix inequalities. Finally, the controller gains are then obtained by solving LMIs [ 18]. The main results on controller design are given in Sections 2 and 3.

This paper focuses on speed and currents control of the PMSM motor using a nonlinear controller design technique under closed loop-field oriented control (FOC). It is based on the use of the T-S model representing the behavior of the nonlinear system. The contribution of this work concerns the T-S model with immeasurable premise variables (e.g. the state of the system). Such a model is commonly encountered when using the sector nonlinearity approach [ 1, 19]. The main contributions of this paper are merging FOC and DMVT controller design for application in PMSM motor control.

Nonlinear controller design

Problem statement

This section presents an efficient methodology for designing controllers for the class of nonlinear systems with nonlinear output measurement described by state equation,

(1) \dot{x}=f(x)+g(x)u,

and nonlinear output

(2) y=h(x),

where x\in R^n is the state vector, u\in R^p is the input vector, y\in R^m is the output measurement vector. Functions f(x):R^n\to R^n , h(x):R^n\to R^m and g(x):R^n\to R^n\times R^p are nonlinear. In addition, f(x) and h(x) are assumed to be differentiable.

It could be represented as a T-S model [ 13]

(3) \dot{x}(t)={\displaystyle \sum }_{i=\mathrm{1}}^r{\mu }_i(x(t))(A_{i}x(t)+B_{i}x(t)),

(4) y\left(t\right)={\displaystyle \sum }_{i=1}^r{\mu }_i\left(x\left(t\right)\right)C_{i}x\left(t\right).

First, the following matrices are introduced:

(5) A_0={\displaystyle \frac{1}{r}}{\displaystyle \sum }_{i=1}^r{A}_i, B_0={\displaystyle \frac{1}{r}}{\displaystyle \sum }_{i=1}^r{B}_i, {\overline{A}}_i=A_i-A_0, {\overline{ B}}_i=B_i-B_0, {\overline{C}}_i=C_i-C_0.

Then, it is easy to rewrite the above model in the Lipchitzien form as

(6) \dot{x}\left(t\right)=A_{0}x\left(t\right)+B_{0}u\left(t\right)+{\displaystyle \sum }_{i=1}^r{\mu }_i\left(x\left(t\right)\right)\left({\overline{A}}_{i}x\left(t\right)+{\overline{B}}_{i}u\left(t\right)\right),

(7) y(t)=C_{\mathrm{0}}x(t)+{\displaystyle \sum }_{i=\mathrm{1}}^r{\mu }_i(x(t))(C_{i}x(t)).

where it appears that matrices A₀ and B₀ play the role of nominal values of the system and {\bar{A}}_i and {\bar{B}}_{i } are variations around these values.

A new function can be defined as

(8) Ф(x,u)={\displaystyle \sum }_{i=\mathrm{1}}^r{\mu }_i(x(t))\left({\bar{A}}_{i}x(t)+{\bar{B}}_{i}u(t)\right),

(9) \Psi \left(x\right)={\displaystyle \sum }_{i=1}^r{\mu }_i\left(x\left(t\right)\right)\left(C_{i}x\left(t\right)\right).

Comparing Eqs. (8) and (9) with Eqs. (6) and (7), respectively, it is found

(10) \{\begin{array}{l}Ф(x,u)=(f(x)-A_{\mathrm{0}}x)+(g(x)u-B_{\mathrm{0}}u)\hfill \\ \mathrm{\Psi }(x)=h(x)-C_{\mathrm{0}}x\hfill \end{array}.

State feedback control

A structure of the modified dynamic PDC control for T-S model (Eqs. (3) and (4)) is given as follows:

The feedback command to be discussed has the form as in classical design.

(11) u=-K_{\mathrm{0}}(x(t)-x_c(t)),

where K₀ is the gain of the controller, and x_c(t) is the desired state and supposed to be a stepwise signal.

The objective is to determine the gain matrix K₀ such that nonlinear system (Eqs. (1) and (2)) becomes globally asymptotically stable under the action of linear static feedback (Eq. (11)).

Using Eqs. (5) and (11), the closed loop error state dynamic equation can be obtained as follows,

(12) \stackrel{·}{e}(t)=\stackrel{·}{x}(t)-\stackrel{·}{x_c}(t)=\stackrel{·}{x}(t)-(A_{\mathrm{0}}-B_{\mathrm{0}}{K}_{\mathrm{0}})x(t)+\mathrm{\Phi }(x,u)+B_{\mathrm{0}}{K}_{\mathrm{0}}{x}_c(t).

Note that the stability analysis of Eq. (12) cannot be directly achieved with the help of the tools developed for T-S systems. The key point of the proposed controller design is to obtain a suitable form of the state controller error in order to reuse the tools proposed for stability and relaxed stability analysis of T-S systems. In conclusion, the objective is to find the gain K₀ of Eq. (12) that stabilizes the state controller error equation.

Before going to the synthesis of the controller, the MVT and the sector nonlinear transformation have to be introduced.

MVT for bounded Jacobian systems

In this sub-section, a mathematical tool is presented, which will be used subsequently to develop the controller gains in the next section. First, the MVT for vector functions is presented [ 6, 16]. Then, the canonical basis for writing a vector function with a composition form is defined. A modified form of the MVT for vector functions is introduced.

Fisrt, n different entries of the nonlinear vector function is introduced.

\mathrm{\Phi }(w):{\mathrm{R}}^{n+p}\to {\mathrm{R}}^n , are denoted {\mathrm{\Phi }}_i(w) , which follows

(13) \mathrm{\Phi }(w)={[{\mathrm{\Phi }}_{\mathrm{1}}(w)\cdots {\mathrm{\Phi }}_n(w)]}^{\mathrm{T}},

where {\mathrm{\Phi }}_i(w):{\mathrm{R}}^{n+p}\to \mathrm{R},\mathit{i}=\cdots \mathit{n} and w=(x, u)=(w₁, w₂, …, w_n+p).

Let us denote e_n(i) the vector of {\mathrm{R}}^n with all entries being null, except the ith being equal to one, given as

(14) e_n(i)={\left[\begin{array}{lllllll}\mathrm{0}& \cdots & \mathrm{0}& \mathrm{1}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{1}& \cdots & i-\mathrm{1}& i& i+\mathrm{1}& \cdots & n\end{array}\right]}^T.

\mathrm{\Phi }(w) could be written as

(15) Ф(w)={\displaystyle \sum }_{i=\mathrm{1}}^n{e}_n(i){Ф}_i(w).

Theorem 1 [ 16] Consider {\mathrm{\Phi }}_i(w):{\mathrm{R}}^{n+p}\to \mathrm{R} . Let a, b∈R^n+p. If {\mathrm{\Phi }}_i(w) is differentiable on [a, b], there exists two constant vectors c∈R^n+p andξ(c)∈R^n+p satisfying \xi (c)\in [x^m,x^M]\left(i.e.,{\xi }_{ij}(c)\in \left[{\underset{\_}{\xi }}_{ij},{\bar{\xi }}_{ij}\right]\right) with {\xi }^M=\left(\cdots ,{\bar{\xi }}_{ij},\cdots \right) and {\xi }^m=\left(\cdots ,{\underset{\_}{\xi }}_{ij},\cdots \right) , for i = 1, …, n and j = 1, …, n + p such that

(16) {Ф}_i(a)-{Ф}_i(b)={\displaystyle \frac{\partial {Ф}_i}{\partial w_j}}(c)(a-b)={\xi }_{ij}(c)(a-b).

Applying the MVT to nonlinear functions (Eq. (16)), it is obtained for a,b∈R^n+p:

(17) Ф(a)-Ф(b)={\displaystyle \frac{\partial Ф}{\partial w}}(c)(a-b)=\left({\displaystyle \sum }_{i=\mathrm{1}}^n{\displaystyle \sum }_{j=\mathrm{1}}^{n+p}{e}_n(i)e_n^T(j){\displaystyle \frac{\partial {Ф}_i}{\partial w_j}}(c)\right)(a-b)\mathrm{.}

In the case in this paper, Eq. (18) can be obtained.

(18) (Ф(x(t),u)-Ф(x_c(t),(u_c(t)))={\displaystyle \frac{\partial Ф}{\partial w}}(c)(w-w_c)={\displaystyle \frac{\partial Ф}{\partial w}}(c)\left[\begin{array}{l}x-x_c\\ u-u_c\end{array}\right],

where c(t)∈[(x_c(t), u_c(t)), (x(t),u(t))].

For PMSM-machine control, function g(x) will be replaced by B₀ and at the nominal equilibrium point (regulation x_c(t)=0), the function Ф(x_c(t),(u_c(t))=(f(x_c(t))-A_{\mathrm{0}}{x}_c(t))+(g(x)-B_{\mathrm{0}})u_c(t)=\mathrm{0} , then Eq. (18) reduces to Ф(x(t),u(t))={\displaystyle \frac{\partial Ф}{\partial x}}(c)x(t) .

In addition, for tracking case (x_c(t)= stepwise signal), function Φ(x_c(t), u_c(t))≠0 then Eq. (18) can be reduced as Ф(x(t),u(t))=Ф(x_c(t),u_c(t))+{\displaystyle \frac{\partial Ф}{\partial x}}(c)(x(t)-x_c(t)) .

Based on the MVT, the gradient

(19) \{{\displaystyle \frac{\partial Ф}{\partial x}}\left(c\right)={\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^n{e}_n\left(i\right)e_n^T\left(j\right){\displaystyle \frac{\partial {Ф}_i}{\partial x_j}}\left(c\right).

Replacing Eq. (19) in Eq. (12), the state error equation becomes

(20) \stackrel{·}{e}=\left[A_{\mathrm{0}}-B_{\mathrm{0}}{K}_{\mathrm{0}}+{\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{e}_n(i)}}{e}_n^T(j){\displaystyle \frac{\partial {\mathrm{\Phi }}_i}{\partial x_j}}(c)\right]e+[B_{\mathrm{0}}{K}_{\mathrm{0}}{x}_c+\mathrm{\Phi }(x_c(t),u_c(t))].

Assumptions

(21) {\underline{\xi }}_{ij}\le {\displaystyle \frac{\partial {\mathrm{\Phi }}_i}{\partial x_j}}(c)={\xi }_{ij}\le {\overline{\xi }}_{ij}; {\overline{\xi }}_{ij}\ge \max \left({\displaystyle \frac{\partial {\mathrm{\Phi }}_i}{\partial x_j}}(c)\right) and {\underline{\xi }}_{ij}\le \min \left({\displaystyle \frac{\partial {\mathrm{\Phi }}_i}{\partial x_j}}(c)\right),

such that each nonlinearity {\displaystyle \frac{\partial {Ф}_i}{\partial x_j}} can be replaced using the sector nonlinearity by

(22) \{{\displaystyle \frac{\partial {Ф}_i}{\partial x_j}}(c)={\bar{\delta }}_{ij}{\bar{\xi }}_{ij}+{\underset{\_}{\delta }}_{ij}{\underset{\_}{\xi }}_{ij}.

Consequently

(23) {\displaystyle \sum }_{i=\mathrm{1}}^n{\displaystyle \sum }_{j=\mathrm{1}}^n{e}_n(i)e_n^T(j){\displaystyle \frac{\partial {Ф}_i}{\partial x_j}}(c)={\displaystyle \sum }_i^n{\displaystyle \sum }_j^n\left({\overline{\delta }}_{ij}{H}_{ij}{\overline{\xi }}_{ij}+{\underline{\delta }}_{ij}{H}_{ij}{\underline{\xi }}_{ij}\right).

Such that the weighting functions

(24) \{\begin{array}{l}{\delta }_{ij}^{\mathrm{1}}={\displaystyle \frac{{\displaystyle \frac{\partial {Ф}_i}{\partial x_j}}-{\underset{\_}{\xi }}_{ij}}{{\bar{\xi }}_{ij}-{\underset{\_}{\xi }}_{ij}}},\hfill \\ {\delta }_{ij}^{\mathrm{2}} = {\displaystyle \frac{{\bar{\xi }}_{ij}-{\displaystyle \frac{\partial {Ф}_i}{\partial x_j}}}{{\bar{\xi }}_{ij}-{\underset{\_}{\xi }}_{ij}}}.\hfill \end{array}

with {\displaystyle \sum }_{e=1}^2{\delta }_{ij}^e=1 such that \mathrm{0}\le {\delta }_{ij}^e\le \mathrm{1} .

H_ij is a zeros matrix elsewhere unless in the position indicated by the ith raw and jth column it takes one.

Replacing Eq. (23) in Eq. (20), Eq. (25) can be obtained.

(25) \stackrel{·}{e}={\displaystyle \sum _{i=\mathrm{1}}^q{h}_i(\xi )(A_{\mathrm{0}}-B_{\mathrm{0}}{K}_{\mathrm{0}}+A_i)e+[B_{\mathrm{0}}{K}_{\mathrm{0}}{x}_c+\mathrm{\Phi }(x_c(t),u_c(t))]}.

So, the final state Eq. (25) can be expressed as

(26) \stackrel{·}{e}={\displaystyle \sum _{i=\mathrm{1}}^q{h}_i(\xi )G_{i}e+[B_{\mathrm{0}}{K}_{\mathrm{0}}{x}_c+\mathrm{\Phi }(x_c(t),u_c(t))]}

The weighting functions h_i(·) are defined by the sector nonlinearity approach in T-S fuzzy [ 5, 8] systems by using the local weighting functions {\delta }_{ij}(·) defined above.

Then, the control design problem consists in finding the controller gain K₀ such that the system Eq. (26) is asymptotically stable. Therefore, use can be made of the results and relaxations techniques in Refs. [ 8, 17].

Stability studies

The stability of Eq. (26) is studied by the quadratic Lyapunov function of the first term. On the other hand, the second term acts in the feed forward and does not affect the stability of the equation. The stability is studied by the quadratic Lyapunov function [ 18, 19] with common matrix to Eq. (27).

(27) \stackrel{·}{e}={\displaystyle \sum _{i=1}^q{h}_i\left(\xi \right)[\left(A_0-B_0{K}_0+A_i\right)e+\left[B_0{K}_0{x}_c+\mathrm{\Phi }\left(x_c\left(t\right),u_c\left(t\right)\right)\right]}

(27)with

V\left(e(t)\right)=e^T(t)\mathit{Pe}(t),P=P^{\mathrm{T}}>0.

The stability is related to the derivative with respect to t so,

(28) \dot{V}(e(t))=e^T(t)((A_{\mathrm{0}}^{T}P+P{A}_{\mathrm{0}}-K_{\mathrm{0}}^{\mathrm{T}}{B}_{\mathrm{0}}^{\mathrm{T}}P-P{B}_{\mathrm{0}}{K}_{\mathrm{0}})+A_h^{T}P+P{A}_h)e(t),

where A_h={\displaystyle \sum }_{i=1}^q{h}_i(\xi )A_i with q\le {\mathrm{2}}^{n^{\mathrm{2}}} .

The stability of the state equation is ensured if the time derivative of Lyapunov Eq. (28) is negative definite, which leads to the following time dependent LMIs:

(29)

The convex sum property of the weighting functions makes it possible to obtain time independent inequalities

(30)

for i=1, …, q.

To express inequality (30) in term of LMI, the change of variables K₀P=M is used and the conditions of LMI are obtained as follows with α representing the rate of convergence. It can be pointed out that the dynamics may also present an oscillatory phenomenon. The performance of the controller can be improved by pole assignment in an LMI region.

Theorem 2 The closed-loop dynamics (27) is asymptotically stable, if there exist P=P^T>0, such that

(31)

for i=1, …,q and j=1, …,r. Moreover, if all the conditions are satisfied, the controller gain is

(32) K_{\mathrm{0}}=MP.

Augmented state feedback regulator

The purpose of introducing the presented MVT-PI-controller is to ensure zero steady-state tracking error for stepwise reference signals in the presence of disturbances or model uncertainties [ 2, 16]. Its principle is based on the well-known procedure of introducing an integral action in the forward channel. A new state variable is introduced to integrate the tracking error, and then the extended modified PDC system controller can be described as

(33) u(t)=\left[\begin{array}{ll}{K}_{\mathrm{0}}& K_I\end{array}\right]\left[\begin{array}{l}e(t)\\ e{(t)}_I\end{array}\right]=\bar{K}\overline{e(t)},

where {\dot{e}}_I(t)=x(t)-x_c(t) .

The closed-loop of the augmented system with state tracking error becomes

(34) \dot{\overline{e(t)}}={\displaystyle \sum }_{i=\mathrm{1}}^q{h}_i(\zeta )\left(\overline{G_i}\overline{e(t)}+\overline{D_i}\bar{v}(t)\right),

with \overline{G_i}=\overline{A_i}-\overline{B_i}\bar{K} ,

such that

(35) {\bar{A}}_{\mathrm{i}}=\left[\begin{array}{ll}{A}_{\mathrm{0}}\hfill & \mathrm{0}\hfill \\ I\hfill & \mathrm{0}\hfill \end{array}\right],\bar{B}=\left[\begin{array}{l}{B}_{\mathrm{0}}\hfill \\ \mathrm{0}\hfill \end{array}\right],{\bar{D}}_{\mathrm{i}}=\left[\begin{array}{ll}{A}_{\mathrm{0}}+A_i\hfill & D\hfill \\ I\hfill & \mathrm{0}\hfill \end{array}\right],\bar{v}(t)=\left[\begin{array}{l}{x}_c(t)\hfill \\ C_r(t)\hfill \end{array}\right]

with load C_r(t).

The stability theorem for augmented system (34) and the convergence of the controller can be derived by means of Lyapunov direct by choosing a quadratic function that can be solved by an LMI tool in a way similar to the case I) with minor modifications of the matrices of the augmented system.

Theorem 3 The closed-loop dynamics (34) of the augmented system is asymptotically stable, if there exists P=P^T, such that

(36)

for i=1, …,q.

Moreover, if all the conditions are satisfied, the controller gain is \bar{K}=M{P}^{-\mathrm{1}} .

Mathematic formulation of PMSM machine with closed loop FOC

Dynamic model of PMSM drive

The governing equation of an AC motor consists of two parts, electrical and mechanical systems.

Electrical governing equation

The mathematical model of the PMSM is composed of stator windings and permanent magnets mounted on the rotor surface (surface mounted PMSM). By using the theory of synchronous reference system, the voltage, current, and inductance of each phase of the PMSM are transferred to the two axes d-q axes. The electrical equations of the PM synchronous motor can be described in the rotor rotating reference frame, written in the (d-q axis) rotor flux reference frame as [ 9, 12]

(37) \begin{array}{l}{\displaystyle \frac{d}{dt}}{i}_d={\displaystyle \frac{\mathrm{1}}{L_d}}{u}_d-{\displaystyle \frac{R}{L_d}}{i}_d+{\displaystyle \frac{L_q}{L_d}} p{\omega }_r{i}_q,\hfill \\ {\displaystyle \frac{d}{dt}}{i}_q={\displaystyle \frac{\mathrm{1}}{L_q}}{u}_q-{\displaystyle \frac{R}{L_q}}{i}_q+{\displaystyle \frac{i_q}{L_q}}p{\omega }_r{i}_d-\Phi {\displaystyle \frac{p{\omega }_r}{L_q}},\hfill \\ {\displaystyle \frac{d}{dt}}{\omega }_r={\displaystyle \frac{\mathrm{1}}{J}}(T_e-F{\omega }_r-C_r).\hfill \end{array}

Mechanical governing equation

The torque that is generated by the energy conversion process is used to drive mechanical loads. Its expression is related to mechanical parameters via the fundamental law of the dynamics as

(38) T_e=\mathrm{1.5}p(\Phi i_q+(L_d-L_q)i_d{i}_q).

The motor parameters used are:

Moment of inertia J=6.36×10⁻⁴ kg·m²;

Stator resistance R_s=4.55 Ω;

Stator inductance L_d=L_q=0.0116 H;

Flux linkage established by magnets Φ=0.317 Wb;

Friction factor F=6.11×10⁻³ Nms;

Pole pair n_p=2;

Load torque C_r.

In the d-q reference frame, the three-phase PMSM can be described as the state space

(39) \{\begin{array}{l}{\dot{x}}_{\mathrm{1}}=-{\displaystyle \frac{R}{L_d}}{x}_{\mathrm{1}}+p{\displaystyle \frac{L_q}{L_d}}{x}_{\mathrm{2}}{x}_{\mathrm{3}}+{\displaystyle \frac{\mathrm{1}}{L_d}}{u}_{\mathrm{1}},\hfill \\ {\dot{x}}_{\mathrm{2}}=-p{\displaystyle \frac{L_d}{L_q}}{x}_{\mathrm{1}}{x}_{\mathrm{3}}-{\displaystyle \frac{R}{L_q}}{x}_{\mathrm{2}}-p{\displaystyle \frac{\mathrm{\Phi }}{L_q}}{u}_{\mathrm{2}},\hfill \\ {\dot{x}}_{\mathrm{3}}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}p{\displaystyle \frac{\mathrm{\Phi }}{J}}{x}_{\mathrm{2}}-{\displaystyle \frac{F}{J}}{x}_{\mathrm{3}}-{\displaystyle \frac{C_r}{J}},\hfill \end{array}

where

(40) x={[x_{\mathrm{1}} x_{\mathrm{2}} x_{\mathrm{3}}]}^{\mathrm{T}}={[i_d i_q \omega ]}^{\mathrm{T}} \mathrm{a}\mathrm{n}\mathrm{d} u={[u_{\mathrm{1}} u_{\mathrm{2}}]}^{\mathrm{T}}={[u_d u_q]}^{\mathrm{T}}.

Closed loop field oriented control (CL-FOC)

The basic principle in controlling the PMSM is based on FOC, as illustrated in Fig. 1 [ 20, 21]. This is obtained by letting the permanent magnet flux linkage to be aligned in d-axis and stator torque component vector, i_q is kept along q-axis direction. This means that the value of the courant i_d is kept zero in order to achieve the field orientation. To implement the FOC concept in closed loop, the desired states have to be determined assuring FOC control such that:

For a fixed desired speed, {\omega }_d , and assuming fixed load torque C_r , to satisfy field orientation condition which mean that the current aligned in d-axis is kept zero (x_dd=0). Then the desired value for the current x_qd aligned in q-axis would be determined from Eq. (39).

(41) \{\begin{array}{l}{x}_{qd}(t)\approx {ℒ}^{-\mathrm{1}}\left\{{\displaystyle \frac{\left(Js+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}p{\displaystyle \frac{\Phi }{J}}\right){\Omega }_d(s)-{\displaystyle \frac{\mathrm{1}}{J}}{C}_r(s)}{\mathrm{1}+t_{\mathrm{2}}s}}\right\},\hfill \\ x_{wd}(t)\approx {ℒ}^{-\mathrm{1}}\left\{{\displaystyle \frac{{\Omega }_d(s)}{\mathrm{1}+t_{\mathrm{3}}s}}\right\}.\hfill \end{array}

The terms τ₃ and τ₂ represent a chosen time constants based on practical considerations.

Controller design

After stabilizing the system, a controller is proposed in monitoring states in the form of state Eq. (26) to the desired values in assuring FOC.

First, it is assumed that the state vector is completely measured, which leads to the output,

(42) y(t)=cx(t) \mathrm{such} \mathrm{that} C=\left(\begin{array}{lll}\mathrm{1}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right).

The proposed approach is applied and the dynamic of controlled state Eq. (26) in closed loop is obtained and reduced to

(43) \dot{x}\left(t\right)={\displaystyle \sum }_{i=1}^8{h}_i\left(\xi \right)\left(A_0+A_i-B_0{K}_0\right)x\left(t\right)+B_0{K}_0{x}_c\left(t\right),

where (A_i+A₀) are equal to

(44) {\displaystyle \frac{\partial Ф}{\partial x}}(c)={\displaystyle \frac{\partial f}{\partial w}}(c)={\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{4}}{\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{4}}{e}_n^T(i)e_n(j){\displaystyle \frac{\partial f_i}{\partial w_j}}(c).

Following the proposed approach, there is

(45) {\displaystyle \frac{\partial f}{\partial x}}(c)=\left(\begin{array}{lll}-{\displaystyle \frac{R}{L_s}}& p{c}_{\mathrm{3}}& p{c}_{\mathrm{2}}\\ -p{c}_{\mathrm{3}}& -{\displaystyle \frac{R}{L_s}}& -p{c}_{\mathrm{1}}-p{\displaystyle \frac{\Phi }{L_s}}\\ \mathrm{0}& {\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}p{\displaystyle \frac{\mathrm{\Phi }}{J}}& -{\displaystyle \frac{F}{J}}\end{array}\right),

where c∈[x, x_c].

Defining c₁ and c₂ and c₃ as new premise variables, it is easy to compute a T-S representation of the Jacobian {\displaystyle \frac{\partial f}{\partial x}}(c) in the form of

(46) \{\begin{array}{l}{\displaystyle \frac{\partial f_{\mathrm{1}}}{\partial x_{\mathrm{2}}}}(c)={\xi }_{\mathrm{12}}=p{c}_{\mathrm{3}}=- {\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{1}}}}(c),\\ {\displaystyle \frac{\partial f_{\mathrm{1}}}{\partial x_{\mathrm{3}}}}(c)={\xi }_{\mathrm{13}}=p{c}_{\mathrm{2},}\\ {\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{3}}}}(c)={\xi }_{\mathrm{23}}=-p{c}_{\mathrm{1}}-p{\displaystyle \frac{\Phi }{L_s}}.\end{array}

The premise variables are bounded as

(47) \{\begin{array}{l}\mathrm{0} \mathrm{rad}/s\le \omega \le \mathrm{200} \mathrm{rad}/s\\ -\mathrm{6}A\le i_q\le \mathrm{6}A,\\ -\mathrm{6}A\le i_d\le \mathrm{6}A\end{array}

So

(48) \{\begin{array}{l}{\gamma }_{\mathrm{1}}^{\mathrm{0}}={\displaystyle \frac{{\xi }_{\mathrm{12}}-{\underline{\xi }}_{\mathrm{12}}}{{\overline{\xi }}_{\mathrm{12}}-{\underline{\xi }}_{\mathrm{12}}}},{\gamma }_{\mathrm{1}}^{\mathrm{1}}={\displaystyle \frac{{\overline{\xi }}_{\mathrm{12}}-{\xi }_{\mathrm{12}}}{{\overline{\xi }}_{\mathrm{12}}-{\underline{\xi }}_{\mathrm{12}},}}\\ {\gamma }_{\mathrm{2}}^{\mathrm{0}}={\displaystyle \frac{{\xi }_{\mathrm{13}}-{\underline{\xi }}_{\mathrm{13}}}{{\overline{\xi }}_{\mathrm{13}}-{\underline{\xi }}_{\mathrm{13}}}},{\gamma }_{\mathrm{2}}^{\mathrm{1}}={\displaystyle \frac{{\overline{\xi }}_{\mathrm{13}}-{\xi }_{\mathrm{13}}}{{\overline{\xi }}_{\mathrm{13}}-{\underline{\xi }}_{\mathrm{13}}}},\\ {\gamma }_{\mathrm{3}}^{\mathrm{0}}={\displaystyle \frac{{\xi }_{\mathrm{23}}-{\underline{\xi }}_{\mathrm{23}}}{{\overline{\xi }}_{\mathrm{23}}-{\underline{\xi }}_{\mathrm{23}}}},{\gamma }_{\mathrm{3}}^{\mathrm{1}}={\displaystyle \frac{{\overline{\xi }}_{\mathrm{23}}-{\xi }_{\mathrm{23}}}{{\overline{\xi }}_{\mathrm{23}}-{\underline{\xi }}_{\mathrm{23}}}}.\end{array}

The weighting functions are given by h_{\mathrm{1}}={\gamma }_{\mathrm{1}}^{\mathrm{1}}{\gamma }_{\mathrm{2}}^{\mathrm{1}}{\gamma }_{\mathrm{3}}^{\mathrm{1}}, h_{\mathrm{2}}={\gamma }_{\mathrm{1}}^{\mathrm{1}}{\gamma }_{\mathrm{2}}^{\mathrm{1}}{\gamma }_{\mathrm{3}}^{\mathrm{0}},\cdots , h_{\mathrm{8}}={\gamma }_{\mathrm{1}}^{\mathrm{0}}{\gamma }_{\mathrm{2}}^{\mathrm{0}}{\gamma }_{\mathrm{3}}^{\mathrm{0}} with the following matrices

(49) A_{\mathrm{1}}=\left(\begin{array}{lll}-\mathrm{392.2}& \mathrm{401}& \mathrm{13}\\ \mathrm{399}& -\mathrm{392.2}& \mathrm{67.7}\\ \mathrm{0}& \mathrm{1495.3}& \mathrm{9.6}\end{array}\right),A_{\mathrm{2}}=\left(\begin{array}{lll}-\mathrm{392.2}& \mathrm{1}& \mathrm{13}\\ -\mathrm{1}& -\mathrm{392.2}& -\mathrm{67.7}\\ \mathrm{0}& \mathrm{1495.3}& -\mathrm{9.6}\end{array}\right),A_{\mathrm{3}}=\left(\begin{array}{lll}-\mathrm{392.2}& \mathrm{401}& -\mathrm{11}\\ \mathrm{399}& \mathrm{392.2}& -\mathrm{67.7}\\ \mathrm{0}& \mathrm{1495.3}& -\mathrm{9.6}\end{array}\right),A_{\mathrm{4}}=\left(\begin{array}{lll}-\mathrm{392.2}& \mathrm{1}& -\mathrm{11}\\ -\mathrm{1}& -\mathrm{392.2}& -\mathrm{67.7}\\ \mathrm{0}& \mathrm{1495.3}& -\mathrm{9.6}\end{array}\right),A_{\mathrm{5}}=\left(\begin{array}{lll}-\mathrm{392.2}& \mathrm{401}& \mathrm{13}\\ \mathrm{399}& -\mathrm{392.2}& -\mathrm{43.7}\\ \mathrm{0}& \mathrm{1495.3}& -\mathrm{9.6}\end{array}\right),A_{\mathrm{6}}=\left(\begin{array}{lll}-\mathrm{392.2}& \mathrm{1}& \mathrm{13}\\ -\mathrm{1}& -\mathrm{392.2}& -\mathrm{43.7}\\ \mathrm{0}& \mathrm{1495.3}& -\mathrm{9.6}\end{array}\right),A_{\mathrm{7}}=\left(\begin{array}{lll}\mathrm{392.2}& \mathrm{401}& -\mathrm{11}\\ \mathrm{399}& -\mathrm{392.2}& -\mathrm{43.7}\\ \mathrm{0}& \mathrm{1495.3}& -\mathrm{9.6}\end{array}\right),A_{\mathrm{8}}=\left(\begin{array}{lll}\mathrm{392.2}& \mathrm{1}& -\mathrm{11}\\ -\mathrm{1}& -\mathrm{392.2}& -\mathrm{43.7}\\ \mathrm{0}& \mathrm{1495.3}& -\mathrm{9.6}\end{array}\right),

for the P-controller.

The stability of system Eq. (43) is guaranteed if a solution exists in the constraints of LMI in Eq. (26).

The MVT approach gives the following matrix gain K₀ guaranteeing the exponential convergence of the proposed P-controller:

(50) K_{\mathrm{0}}={\mathrm{10}}^{\mathrm{4}}\left[\begin{array}{lll}\mathrm{0.0019}& -\mathrm{0.0004}& -\mathrm{0.0013}\\ \mathrm{0.0003}& \mathrm{0.0060}& \mathrm{6.0072}\end{array}\right]

For the PI-controller, the augmented gain

(51) \bar{K}={\mathrm{10}}^{\mathrm{4}}=\left[\begin{array}{lllll}-\mathrm{0.0002}& \mathrm{0.0000}& \mathrm{0.0000}& -\mathrm{9.0003}& \mathrm{0.0367}\\ -\mathrm{0.0000}& -\mathrm{0.0002}& \mathrm{0.1166}& -\mathrm{0.0454}& -\mathrm{4.3726}\end{array}\right].

Simulation results of closed loop−FOC of PMSM machine

The proposed control design is applied to a PMSM motor under closed loop FOC in order to track the desired currents i_d, i_q and the angular rotor velocity ω_r with perturbation C_r and parameter uncertainties for the two cases (P-controller and PI-controller).

The desired trajectories of the benchmark (desired tracking states) are such that, after that the real speed is carried with a great transient to 160 rad/s and from 0 to 0.7 s and followed by a reduction of the speed to 100 rad/s at 0.7 s to 1.5 s. Then the load torque is applied between t = 1.2 s until 2 s. Finally, it is followed by a reverse process of the speed to ‒100 rad/s at 1.5 s. This first simulated result makes it possible to test the performance and the robustness of the controller MVT, with complete measured states, from high speed to the reverse speed. From 0 to 0.7 s, the speed is carried to its nominal value (160 rad/s) and remains constant. This phase is defined to test the controller behavior during a great transient speed. Then, the motor is driven to reach again a constant low speed value from 0.7 s until 1.5 s (see Fig. 2). With the P-controller, the speed and currents of the PMSM machine given in Fig. 2 converge to the real values under conditions from very high to reverse process and vice versa. A small static error (5%) occurs when the motor speed increases (between 0 s and 0.7 s) due to natural characteristics of the P-controller and an acceptable error occurs due to load torque variation (between 1.2 s and 2 s), as shown in Fig. 2 (a), (b) and (c).

The effectiveness of the proposed PI-controller with the same benchmark from transient rotor speed from zero to the nominal value 160 rad/s and the reverse transient process to ‒100 rad/s at t = 1.5 s (Fig. 3) is tested. It can be affirmed that a very small static and dynamic error occurs during the transient and permanent regime between the real and the desired rotor speed and the other states (currents), essentially I_d-current in order to assure FOC as shown in Fig. 4.

By introducing a variation on R_s and L_s in the PMSM, machine parameters and a test of a forced attenuated sinusoidal desired signal input are given in Figs. 5 (a), (b) and (c) and Fig. 6 respectively which show that the MVT-P and PI-controller perform well. The robustness under parametric variation of the stator resistance R_s and inductance L_s and the tracking of the different signals behaviors are accomplished.

Simulation tests are conducted without and with stator inductance L_s having a smooth mismatch of a maximum of 50% of the nominal value at t = 0.5 s. Using the modified PDC control law, the results of this test are shown in Fig. 5 (b) and (c).

It is clear that when considering stator inductance variation, two things arise with the CL-FOC control: a very oscillatory response at the transition of the references and applied torques, and a zero steady-state error occurred in motor speed and currents.

The general conclusion is that, in all cases, the speed and the currents track the desired values and show a robustness of the proposed scheme under parametric uncertainties and unknown load torque variations.

Conclusions

In this paper, a new nonlinear controller design technique has been investigated with nonlinear measurements, obtained by DMVT methodology, in controlling currents and speed of PMSM under CL-FOC application. The approach used is based on the MVT to express the nonlinear error dynamics as a convex combination of known matrices with time varying as LPV systems. The controller gain is then obtained by solving the LMIs. The simulated results verify the efficiency of the proposed controller scheme in terms of overshoot, speed and disturbance rejection. Switching effects of inverter and friction effects have been neglected during controller design. Simulations have been performed using the Matlab/Simulink environment.

The developed approach can enable controller design for a large class of differentiable nonlinear systems with a globally bounded and with nonlinear measurements. In the future, inverter dynamics and testing of the proposed algorithm in real experimental setup will be considered.

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