**4. Electric Medium Passenger Bus Vector Control System Based on Adaptive Robust Current Control**

Compared with the application of motors in other aspects, the motor drive system of electric medium-sized buses has higher requirements for the current following ability and control accuracy of the motor. During the operation of electric medium-sized buses, due to the complicated operating environment, it will encounter various non-linear disturbances, resulting in current and torque ripples in the system. Therefore, this paper chooses to study from the perspective of control, and uses adaptive robust control to overcome the effects of nonlinear disturbances, thereby improving the tracking accuracy of the current loop.

According to Equation (1), a voltage model containing non-ideal back-emf can be obtained as shown in Equation (21).

$$L\frac{d}{dt}i\_{\boldsymbol{q}} = \boldsymbol{u}\_{\boldsymbol{q}} - \boldsymbol{R}\dot{\boldsymbol{q}}\_{\boldsymbol{q}} - \boldsymbol{c}\_{\boldsymbol{q}} + \boldsymbol{\Delta}\_{\boldsymbol{q},} \tag{21}$$

where *eq* is the non-ideal back-EMF, and Δ*q* is the sum of the deviation voltages caused by all nonlinear disturbances.

Assume that the non-ideal back EMF model is:

$$e\_{\mathfrak{q}} = \overline{\mathcal{S}}\_{\mathfrak{k}} \mathcal{K}\_{\mathfrak{k}} \tag{22}$$

where in *Se* represents a wave function containing a fundamental wave and a 6th harmonic, *Ke* represents a back EMF coefficient matrix. Their expressions are shown in Equations (23) and (24), respectively.

$$\overline{S}\_{\varepsilon} = \frac{3}{2} a \iota\_{\varepsilon} [1 \cos(6 \,\sigma\_{\varepsilon})],\tag{23}$$

$$K\_{\mathfrak{e}} = \begin{bmatrix} K\_{\mathfrak{q}1} \ K\_{\mathfrak{q}6} \end{bmatrix}^T,\tag{24}$$

Bringing Equation (22) into Equation (21), the following equation of state can be obtained:

$$L\frac{d}{dt}i\_q = \mu\_q - Ri\_q - \overline{S}\_\varepsilon K\_\varepsilon + \Delta\_{q\prime} \tag{25}$$

To establish a standard adaptive robust control model, let . *x* = *<sup>d</sup> dt iq*, let γ<sup>T</sup> = *Se* and σ = *Ke*. Define the amount of virtual control *u* as:

$$
\overline{\boldsymbol{\mu}} = \boldsymbol{\mu}\_q - \mathrm{Ri}\_{q\prime} \tag{26}
$$

where *uq* is the actual output of the controller and *Riq* is calculated from the known amount and the feedback amount.

Substituting the above assumption and Equation (26) into Equation (25), Equation (26) is obtained as shown below.

$$L\dot{\mathbf{x}} = \overline{\mathbf{u}} + \boldsymbol{\gamma}^T \boldsymbol{\sigma} + \boldsymbol{\Delta}\_{\Psi} \tag{27}$$

The adaptive robust current controller designed in this paper makes the output of the double-disc AFPMSM model overcome the effects of non-ideal back EMF and other nonlinear disturbances. The tracking error with the expected value *xd* is as small as possible.

It can be seen from the equation of state, Equation (27) that the previously designed double-disc AFPMSM model clearly includes parameter uncertainties and nonlinear disturbances. Adaptive robust control can compensate the uncertainty in the system through the design of adaptive law, and synthesize the robust control law to overcome the influence of nonlinear disturbance, so it is suitable for the design of PMSM current controller. According to Equations (27) and (28) can be obtained.

$$
\overline{\boldsymbol{\mu}} = \boldsymbol{L}\dot{\boldsymbol{x}} - \boldsymbol{\gamma}^T \boldsymbol{\sigma} - \boldsymbol{\Delta}\_{\boldsymbol{\Psi}} \tag{28}
$$

The control law form of the adaptive robust controller is as shown in Equation (29).

$$
\overline{u} = u\_a + u\_{r\prime} \tag{29}
$$

where *ua* is the compensation term for adaptive control, which can be expressed as:

$$\mu\_d = L\dot{\mathbf{x}}\_d - \boldsymbol{\chi}^T \boldsymbol{\mathfrak{t}}\_\prime \tag{30}$$

where . *xd* is the differential of the expected value of the q-axis current and σˆ is the estimated value of the unknown parameter σ.

*u*r is a robust control term and can be expressed as:

$$
\mu\_r = \mu\_{r1} + \mu\_{r2} \tag{31}
$$

In Equation (31), *u*r1 is a linear proportional feedback term and *u*r1 = –*k*r1*z*, *k*r1 is a proportional coefficient, and *z* = *x–xd* represents a tracking error. According to the control law represented by Equations (28) and (29), the dynamic equation of the system tracking error is:

$$L\dot{z} - \mu\_{r1} = \mu\_{r2} - \left[\gamma^T \overleftarrow{\sigma} - \Delta\_q\right],\tag{32}$$

where <sup>σ</sup> is the parameter estimation bias and <sup>σ</sup> <sup>=</sup> <sup>σ</sup><sup>ˆ</sup> <sup>−</sup> <sup>σ</sup>, *<sup>u</sup>*r2 is the robust control term. According to the robust control principle, the design requirements are:

$$\begin{cases} z\mu\_{r2} \le 0\\ z\left|\mu\_{r2} - \left[\gamma^T \widetilde{\sigma} - \Delta\_{\overline{q}}\right]\right| \le \varepsilon \quad \text{'} \end{cases} \tag{33}$$

In the above equation, ε represents any positive integer, the first condition can be guaranteed to be naturally dissipated, and the second condition indicates that *ur*<sup>2</sup> can suppress nonlinear disturbance, modeling error and estimation error of adaptive parameters. There are many ways to select *ur*<sup>2</sup> that meet the conditions [19]. The most common one is:

$$\begin{cases} \begin{aligned} \mu\_{l2} &= -\frac{1}{4\epsilon}h^2 z \\ h &= \left| \gamma^T \right| |\sigma\_{\text{max}} - \sigma\_{\text{min}}| + \Delta\_{\text{qmax}} \end{aligned} \tag{34}$$

It can be known from the control law in Equation (29) and the tracking error dynamic in Equation (32) that the tracking performance of the adaptive robust controller depends on the design of the robust control term *u*r. Since the adaptive law design is synthesized by tracking error, the parameter projection method is used to modify the adaptive law [20]. Therefore, the adaptive law of adaptive robust control is expressed as:

$$
\dot{\hat{\sigma}} = \text{Proj}(\Gamma \gamma z),
\tag{35}
$$

where Γ is a diagonal adaptive law matrix, and Pr*oj*(λ) is a projection operator, which can be expressed as:

$$\text{Proj}(\lambda) = \begin{cases} 0, \text{if} \begin{cases} \text{ } \hat{\sigma} = \sigma\_{\text{min}} \text{ and } \lambda < 0\\ \text{ } \hat{\sigma} = \sigma\_{\text{max}} \text{and } \lambda > 0\\ \lambda, \text{others} \end{cases} \end{cases} \tag{36}$$

When using the control law of Equation (29) and the adaptive law of Equation (35), the adaptive robust current control block diagram is shown in Figure 7.

**Figure 7.** Adaptive robust current control block diagram.

#### **5. Simulation and Experimental Results**

#### *5.1. Drive System Simulation*

A two-disc AFPMSM simulation system is set up, and the current controller adopts adaptive robust control method and PI control, respectively. Finally, compare the current tracking performance of the two controllers. The current controller adopts the control strategy of *id* = 0, the d-axis current adopts PI controller, the q-axis current adopts adaptive robust controller, the given current *iq*\* is 150 A, and the *iq* current response waveform is shown in Figure 8. The three-phase current waveform is shown in Figure 9.

**Figure 8.** *Iq* current response waveform based on adaptive robust current controller.

**Figure 9.** Three-phase current waveform based on adaptive robust current controller.

When the given current *iq*\* is also 150 A, and the d-axis and q-axis currents all use the PI controller, the *iq* current response waveform is shown in Figure 10, and the three-phase current waveform is shown in Figure 11.

**Figure 10.** *Iq* current response waveform based on PI current controller.

It can be seen from the simulation waveform that the control system based on adaptive robust current controller has a response time of about 14 ms from 0 A to 150 A, and the current has almost no overshoot, and the current fluctuation is small at steady state. The control system based on the PI current controller has a response time of approximately 21 ms from 0 A to 150 A and a current overshoot of approximately 6%. The simulation results show that the adaptive robust current controller designed in this paper has better current control performance.

**Figure 11.** Three-phase current waveform based on PI current controller.
