1 Introduction

Complementary Sliding Mode Control (CSMC) by designing a pair of mutually compensating sliding surfaces to modify the Sliding Mode Control (SMC), the phase trajectory converges to the equilibrium point along the intersection of the pair of sliding surfaces^[1]. Compared with SMC, the steady-state error is reduced by a half^[2-3]. However, the control law of CSMC is still using the sign function to ensure the system robustness, it is inevitable to result in the high frequency chattering. The use of the saturation function to replace the sign function is an effective method to weaken the chattering, and the control precision of the system is also be ensured^[4-6]. However, the introduction of the saturation function adds a boundary layer parameter, which needs artificial settings^[7]. The boundary layer parameter being too large will reduce the system steady state accuracy while being too small will aggravate the chattering^[8]. Therefore, the boundary layer parameter setting is very important to ensure the performance of CSMC system^[9]. A fuzzy function is used to design an approximate saturation function for SMC to regulate the speed of the induction motor. The optimization of the boundary layer parameter can be adjusted by the fuzzy rules according to the uncertainty of the system ^[10]. A modification of the SMC is proposed to improve the speed accuracy with the saturation function boundary layer parameter designed by fuzzy rules^[11]. The nonlinear control design tool of MATLAB toolbox is used to optimize the boundary layer parameter of the saturation function of SMC, and the direct thrust control of linear motor is implemented^[12]. In this method, the four parameters: switching gain, sliding mode surface coefficient, convergence law coefficient and boundary layer parameters, are optimized simultaneously. However, this algorithm has too high computational complexity.

In the CSMC system, the joint action saturation function and its switching gain suppresses the high frequency chattering^[13-15]. Here, an optimization method of boundary layer parameter of the saturation function is proposed for CSMC strategy. Firstly, the complementary sliding surfaces are constructed by generalized and complementary sliding surfaces. Then the relationship between the boundary layer parameter and the switching gain is obtained by using the Taylor series to expand speed error. The boundary layer parameter optimization is solved for the saturation function. Moreover, the CSMC speed regulation is applied in the Permanent Magnet Synchronous Motor (PMSM) platform. Experiments show that the CSMC strategy with the optimization of the saturation function boundary layer does not require artificial settings and the optimal speed performance can be obtained at the same time.

2 CSMC Control System 2.1 PMSM Mathematical Model

Ignoring the effect of magnetic saturation, harmonic back electromotive force, hysteresis and eddy current loss, the mathematical model of the mechanical motion of the surface-type PMSM in the d-q-o synchronous rotating coordinate system is,

$ J\dot w + Bw = {T_e} - {T_L} = {K_f}{i_q} - {T_L} $

(1)

where w is the rotor mechanical angular speed, J is the moment inertia, B is the viscous damping coefficient, T_e is the motor electromagnetic torque, T_L is the load torque, K_f is the electromagnetic torque coefficient, i_q is the q-phase current in the d-q-o coordinate system.

In consideration of the parameters perturbation, friction coefficient and other uncertain factors, Eq.(1) can be described as,

$ \dot w = {A_n}w + {B_n}{i_{\rm{q}}} + H $

(2)

where A_n=-B_o/J_o, B_n=K_fo/J_o, lump uncertainty H=$\left({ - \Delta Bw - \Delta J\dot w + \Delta {K_{\rm{f}}}{i_{\rm{q}}} - {T_{\rm{L}}}} \right)/{J_{\rm{o}}}$, the symbol "o" for B_o, J_o and K_fo mean the nominal parameters, the symbol "Δ" represents the bias value of the parameters J_o, B_o, K_fo. Assume that H is a lump uncertainty with a bounded:|H|≤D, D is a positive constant.

2.2 CSMC Speed Regulation

CSMC is applied to a speed control for PMSM, the definition of tracking error e is,

$ \mathit{\boldsymbol{e}} = {\mathit{\boldsymbol{w}}^*} - \mathit{\boldsymbol{w}} $

(3)

where w^* is the given rotor angular speed.

The generalized sliding surface S_g is defined as^[1],

$ {S_{\rm{g}}} = \left( {\frac{{\rm{d}}}{{{\rm{d}}t}} + \lambda } \right)\int_0^t {e\left( \tau \right){\rm{d}}\tau } $

(4)

where the coefficient λ>0.

Derivative of Eq.(4) is,

$ {{\dot S}_{\rm{g}}} = \dot e + \lambda e = {{\dot w}^*} - {A_n}w - {B_n}{i_{\rm{q}}} - H + \lambda e $

(5)

The complementary sliding surface S_c is,

$ {S_{\rm{c}}} = \left( {\frac{{\rm{d}}}{{{\rm{d}}t}} - \lambda } \right)\int_0^t {e\left( \tau \right){\rm{d}}\tau } $

(6)

The relationship between these two sliding surfaces is^[1],

$ {{\dot S}_{\rm{g}}} = \lambda \left( {{S_{\rm{g}}} + {S_{\rm{c}}}} \right) + {{\dot S}_{\rm{c}}} $

(7)

According to Eq.(2) of the system, the tracking error converges to the boundary layer within a finite time. The robust control law of CSMC is designed as,

$ i_{\rm{q}}^* = {i_{{\rm{eq}}}} + {i_{\rm{r}}} $

(8)

where i_eq is an equivalent control component, i_r is a robust control component.

$ {i_{{\rm{eq}}}} = \frac{1}{{{B_n}}}\left[ {{{\dot w}^*} - {A_n}w + \lambda \left( {e + {S_{\rm{g}}}} \right)} \right] $

(9)

$ {i_{\rm{r}}} = \frac{1}{{{B_n}}}\left[ {\rho \cdot {\rm{sat}}\left( {\frac{{{S_{\rm{g}}} + {S_{\rm{c}}}}}{\phi }} \right)} \right] $

(10)

where ρ is a gain parameter, ρ>0, ϕ is a saturation function boundary layer thickness parameter, ϕ> 0. The sat(·) is a saturation function as follows:

$ {\rm{sat}}\left( {\frac{{{S_{\rm{g}}} + {S_{\rm{c}}}}}{\phi }} \right) = \left\{ {\begin{array}{*{20}{l}} {{\rm{sgn}}\left( {{S_{\rm{g}}} + {S_{\rm{c}}}} \right),}&{\left| {{S_{\rm{g}}} + {S_{\rm{c}}}} \right| > \phi }\\ {\frac{{{S_{\rm{g}}} + {S_{\rm{c}}}}}{\phi },}&{\left| {{S_{\rm{g}}} + {S_{\rm{c}}}} \right| \le \phi } \end{array}} \right. $

(11)

where sgn means the sign of the (S_g+S_c).

CSMC system stability proof.

Define the Lyapunov function is:

$ V = \frac{1}{2}\left( {S_{\rm{g}}^2 + S_{\rm{c}}^2} \right) $

Derivative of Lyapunov function with Eq.(4), Eq.(7)-(8),

$ \begin{array}{l} \dot V = {S_{\rm{g}}}{{\dot S}_{\rm{g}}} + {S_{\rm{c}}}{{\dot S}_{\rm{c}}} = \\ \;\;\;\;\;\;\left( {{S_{\rm{g}}} + {S_{\rm{c}}}} \right)\left( {{{\dot S}_{\rm{g}}} - \lambda {S_{\rm{c}}}} \right) = \\ \;\;\;\;\;\;\left( {{S_{\rm{g}}} + {S_{\rm{c}}}} \right)\left[ {{w^*} - \left( {{A_n}w + {B_n}{i_{\rm{q}}} + H} \right) + } \right.\\ \;\;\;\;\;\;\left. {\lambda e - \lambda {S_{\rm{c}}}} \right] = - \lambda {\left( {{S_g} + {S_{\rm{c}}}} \right)^2} - \\ \;\;\;\;\;\;H\left( {{S_{\rm{g}}} + {S_{\rm{c}}}} \right) + \left( {{S_{\rm{g}}} + {S_{\rm{c}}}} \right)\\ \;\;\;\;\;\;\;\left[ { - \rho \cdot {\rm{sat}}\left( {\frac{{{S_{\rm{g}}} + {S_{\rm{c}}}}}{\phi }} \right)} \right] \le - \lambda {\left( {{S_{\rm{g}}} + {S_{\rm{c}}}} \right)^2} + \\ \;\;\;\;\;\;\;\left| {{S_{\rm{g}}} + {S_{\rm{c}}}} \right|\left| H \right| + \left( {{S_{\rm{g}}} + {S_{\rm{c}}}} \right) \cdot \\ \;\;\;\;\;\;\;\left[ { - \rho \cdot {\rm{sat}}\left( {\frac{{{S_{\rm{g}}} + {S_{\rm{c}}}}}{\phi }} \right)} \right] \end{array} $

When ρ≥ D ≥|H|, and |S_g + S_c|>ϕ,

$ \dot V = - \lambda {\left( {{S_{\rm{g}}} + {S_c}} \right)^2} + \left| {{S_{\rm{g}}} + {S_c}} \right|\left( {\left| H \right| - \rho } \right) < 0 $

That indicates the CSMC system satisfies a large scale asymptotic stability. Therefore, the sliding modulus |S_g + S_c| will converge to the boundary layer of the saturation function in a finite time, that means |S_g + S_c|≤ ϕ From Eq.(4) and Eq.(6), it has,

$ \left| {{S_{\rm{g}}} + {S_{\rm{c}}}} \right| = 2\left| e \right| \le \phi $

(12)

Then the boundary of system tracking error e is

$ \left| e \right| \le \frac{\phi }{2} $

(13)

Eq.(13) indicates that the steady-state error of the system will enter into the boundary layer thickness of the saturation function when the CSMC system enters the sliding mode. Apparently, the saturation function replaces the sign function, adding a boundary layer parameter ϕ and the ϕ value is usually artificially setting. Moreover, ϕ has an important influence on the CSMC system performance. If the parameter ϕ value is too small, resulting in always |S_g + S_c|> ϕ the state trajectory cannot enter into the sliding mode and always stay at the boundary of the saturation function in turn, which result in the high frequency chattering. If the parameter ϕ value is too large, the CSMC system keeps in the thickness range of the boundary layer, which will lead to the increase of the system steady state error and reduce the robustness. So the optimization of the parameter ϕ is very necessary and important.

2.3 Parameter ϕ Optimization

In the CSMC system, the boundary layer parameter ϕ of the saturation function is optimized. Derivative of Eq.(3) with Eq.(2) is,

$ \dot e = {{\dot w}^*} - \dot w = {{\dot w}^*} - \left( {{A_n}w + {B_n}{i_{\rm{q}}} + H} \right) $

In CSMC system, i_q is the control variable with Eq.(9),

$ \begin{array}{*{20}{c}} {\dot e = - \lambda \left( {e + {S_{\rm{g}}}} \right) - \rho \cdot {\rm{sat}}\left( {\frac{{{S_{\rm{g}}} + {S_{\rm{c}}}}}{\phi }} \right) - H = }\\ {\psi (t) - \rho \cdot {\rm{sat}}\left( {\frac{{{S_{\rm{g}}} + {S_{\rm{c}}}}}{\phi }} \right)} \end{array} $

(14)

where ψ(t)=-H-λ(e+S_g) is a bounded function with the time variable t. It means system uncertainties, and is assumed to be |ψ(t)|≤ψ_max, ψ_max is the maximum value. During the control proceeding of the CSMC system, it is assumed that always ρ≥ψ_max, Eq.(14) can be described as follows. When (S_g + S_c)>ϕ, the trajectory of CSMC is on the reach stage, which means system error e $\gg $ϕ /2. Since Eq.(11), Eq.(14) is $\dot{e}=\psi(t)-\rho < 0$. Therefore, CSMC system error e keeps decreasing until -ϕ < (S_g + S_c) < ϕ. When (S_g + S_c) < -ϕ, the trajectory of CSMC is also on the reach stage, which means system error $e \ll-\phi / 2$. Since Eq.(11), Eq.(14) is $\dot{e}=\psi(t)+\rho>0$. Therefore, CSMC system error e keeps increasing until -ϕ < (S_g+S_c) < ϕ. When -ϕ < (S_g+S_c) < ϕ, the sign of Eq.(14) is no way to assign. But this stage belongs to the sliding mode. So, CSMC system should enter into the sliding mode and it always has Eq.(12) as:

$ \left| {{S_g}(t) + {S_c}(t)} \right| = \left| {2e(t)} \right| \le \phi $

(15)

The sampling time of the control system is assumed to be T. When t=t₁, the sliding modulus has |S_g(t₁)+S_c(t₁)|=|2e(t₁)| and the sliding modulus |S_g(t₁+T)+S_c(t₁+T)|=|2e(t₁+T)| for the next sampling time. The Taylor series is used to expand e(t₁+T) to obtain,

$ \begin{array}{l} e\left( {{t_1} + T} \right)\sim e\left( {{t_1}} \right) + \dot e\left( {{t_1}} \right) \cdot T = e\left( {{t_1}} \right) + \\ \left[ {\psi \left( {{t_1}} \right) - \rho \cdot {\rm{sat}}\left( {\frac{{{S_g}\left( {{t_1}} \right) + {S_c}\left( {{t_1}} \right)}}{\phi }} \right)} \right] \cdot T \end{array} $

(16)

Since |ψ(t)|≤ρ, so

$ \left| {\psi \left( {{t_1}} \right) - \rho \cdot {\rm{sat}}\left( {\frac{{{S_{\rm{g}}}\left( {{t_1}} \right) + {S_{\rm{c}}}\left( {{t_1}} \right)}}{\phi }} \right)} \right| \le 2\rho $

(17)

When ρ≥ψ_max,|S_g(t₁)+S_c(t₁)|>ϕ, $\dot{e}$ sign is the opposite of e sign, which means that the value of |S_g(t)+S_c(t)| keeps decreasing,|S_g(t₁+ T)+S_c(t₁+T)| approaches to ϕ. Since the extreme case as |S_g(t₁+T)+S_c(t₁+T)|=0, Eq. (15) can be expressed by Eq.(16) - (17),

$ \left| {{S_{\rm{g}}}\left( {{t_1} + T} \right) + {S_{\rm{c}}}\left( {{t_1} + T} \right)} \right| \le 4\rho T $

(18)

Let the |S_g(t₁+T)+S_c(t₁+T)|=|2e(t₁+T)|≤ϕ and maximize the accuracy of the system. The thickness parameter ϕ of the saturation function boundary layer is optimized as,

$ \phi = 4\rho T $

(19)

Eq.(19) shows that the saturation function boundary layer ϕ in the CSMC method can be optimized to 4 ρT. It means the ϕ value is designed by the switching gain ρ and the sampling time of CSMC system, and does not need artificially setting.

3 PMSM Experiments

PMSM experimental servo platform is shown in Fig. 1, composed of four components: PMSM+magnetic brake, DSP control subsystem, power inverter and computer. PMSM model: 60CB020C, rated voltage: AC220V, rated power: 200 W, rated current: 1.27 A, rated speed: 3 000 r/min, pole pairs: n_p=4, rated torque: 0.64 N·m, stator resistance: R_o=13 Ω, inductance: L_qo=L_do=L_o= 0.032H, flux linkage coefficient: Ψ_f =0.119 Wb, moment of inertia J_o=0.000 15 kg·m², sliding friction coefficient B_o=0.000 1 N·m·s/rad. Magnetic brake model: CZ02, rated torque: 2 N·m, rated current: 0.5 A, slip power: 0.15 kW. Fig. 2 is the structure of PMSM system by CSMC with the magnetic field oriented control strategy. The speed loop by using CSMC and current loop by PI controller are cascade control. The speed loop sampling frequency is 1 kHz. The control parameters are: λ=8, ρ=15, from the Eq. (19) to get ϕ=0.03.

In the experiments, the added load is 0.35 N·m, the speed reference input is: w^*=40π rad/s when 0 < t < 1 s; w^*=60π rad/s when 1 s < t < 3 s; w^*=40π rad/s when 3 s < t < 4 s. Fig. 3 shows the system speed error e with different ϕ values. Fig. 4 shows i_q^* obtained by the CSMC method with the different ϕ values. Fig. 3 and Fig. 4 show that ϕ=0.003, i_q^* chatters seriously since this ϕ value is smaller than the uncertainties. The state trajectory of CSMC switches on the boundary ϕ in turn. This time, the system speed error e≤±2 rad/s; When ϕ=0.3, there is basically no high frequency chattering in i_q^*, since this ϕ value is larger than the uncertainties. The state trajectory is always stay on inside the boundary. The system speed error becomes a little larger, and reaches -3.0 rad/s; And the optimization value ϕ=0.03, i_q^* chattering is relatively small, but the system has a smallest stable state speed error, e≤±1.5 rad/s. Both figures indict that the larger the ϕ value is, the smaller the high frequency chattering is. The proposed method to optimize the parameter ϕ of the saturation function boundary layer does not need artificial setting. This optimization method for CSMC can obtain a compromise between the high frequency chattering of the system and ensure the system steady-state error.

4 Conclusions

A parameter optimization method of the boundary layer of the saturation function is proposed to improve CSMC strategy. The boundary layer parameter is optimized as the expression of the switching gain and the sampling time, so as to avoid setting the boundary layer parameter artificially and improve the practicability of CSMC. The CSMC method is applied to the speed loop regulation in PMSM. The experimental results show that the boundary layer parameter optimization for the saturation function is effective for CSMC and can obtain a compromise between the high frequency chattering of the system and the system steady state error.