3.2.3. Design of Perturbation Observer

When the system outputs *y*2,3 are available, two second-order POs are designed for the estimations of states and perturbation for the subsystems

$$S2: \begin{cases} \dot{\sharp}\_{21} &=& \sharp\_{21} + l\_{21}(z\_{21} - \sharp\_{21}) + B\_2(0)u\_2 \\ \dot{\sharp}\_{22} &=& l\_{22}(z\_{21} - \sharp\_{21}), \end{cases} \tag{38}$$

$$S3: \begin{cases} \dot{\mathfrak{z}}\_{31} &=& \mathfrak{z}\_{31} + l\_{31}(\mathfrak{z}\_{31} - \mathfrak{z}\_{31}) + B\_3(0)u\_3 \\ \dot{\mathfrak{z}}\_{32} &=& l\_{32}(\mathfrak{z}\_{31} - \mathfrak{z}\_{31}), \end{cases} \tag{39}$$

where *z*ˆ21, *z*ˆ22, *z*ˆ31, and *z*ˆ32 are the estimations of *z*21, *z*22, *z*31 and *z*32, respectively, and *l*21, *l*22, *l*31 and *l*32 are gains of the observers. They are designed similarly to Equation (20).

**Remark 1.** *It should be mentioned that during the design procedure, i used in POs Equations (38) and (39) are required to be some relatively small positive constants only, and the performance of POs is not very sensitive to the observer gains, which are determined based on the upper bound of the derivative of perturbation.*

### 3.2.4. Design of Nonlinear Adaptive Controller

The estimated perturbations are used for compensating the real perturbation, and control laws of subsystems *S*2 and *S*3 can be obtained as follows:

$$
\begin{bmatrix} u\_2 \\ u\_3 \end{bmatrix} = B2(0)^{-1} \left[ \begin{bmatrix} -\mathfrak{A}\_{22} \\ -\mathfrak{A}\_{32} \end{bmatrix} + \begin{bmatrix} v\_2 \\ v\_3 \end{bmatrix} \right] \tag{40}
$$

where *<sup>v</sup>*2,3 is defined as

$$\begin{cases} v\_{2} &=& k\_{21}(z\_{21\text{r}} - \sharp\_{21}) + \sharp\_{21\text{r}} \\\ v\_{3} &=& k\_{31}(z\_{31\text{r}} - \sharp\_{31}) + \sharp\_{31\text{r}} \end{cases} \tag{41}$$

The final control law represented by currents and inductances, are expressed as follows:

$$\begin{cases} \boldsymbol{\mu\_{2}} = \boldsymbol{L\_{\rm md0}} [\boldsymbol{k\_{21}} (\boldsymbol{i\_{mdr}} - \boldsymbol{i\_{md}}) + \boldsymbol{i\_{mdr}} - \boldsymbol{\Psi\_{2}}] \\ \boldsymbol{\mu\_{3}} = \boldsymbol{L\_{\rm mq0}} [\boldsymbol{k\_{31}} (\boldsymbol{i\_{mqr}} - \boldsymbol{i\_{mq}}) + \boldsymbol{i\_{mqr}} - \boldsymbol{\Psi\_{3}}] \end{cases} \tag{42}$$

Please note that only the nominal values of *L*md0, *<sup>L</sup>*mq0, and measurements of *i*md and *<sup>i</sup>*mq are required in the NAC design.

To clearly illustrate its principle, Figure 4 shows the block diagram of the NAC.

**Figure 4.** Block diagram of nonlinear adaptive controller.

The following assumptions are made in [19,21,36–39].

**Assumption 1.** *Input gain B(x) and its derivative are bounded by* 0 < *M*1 ≤ *<sup>B</sup>*(*x*) ≤ *M*2 *,* |*B*˙(*x*)| ≤ *M*3*, where Mi, i* = 1, 2, 3 *are finite constants [for convenience we assume that <sup>B</sup>*(*x*) > 0*]. B*(0) *is chosen to satisfy:* |*B*(*x*)/*B*(0) − 1| ≤ *θ* < 1*, where θ is a positive constant. The control u is assumed to be bounded but big enough for the purpose of perturbation cancellation.*

**Assumption 2.** *The perturbation* <sup>Ψ</sup>*i*(*<sup>x</sup>*, *t*) *and its derivative* Ψ˙ *<sup>i</sup>*(*<sup>x</sup>*, *t*) *are locally Lipschitz in their arguments and bounded over the domain of interest.*

### 3.2.5. Stability Analysis of Closed-Loop System

This subsection analyzes the stability of the closed-loop system equipped with the NAC designed in the previous section.

At first, both the estimation error system and the tracking error system are obtained. On one hand, by defining estimation errors *ε*21 = *z*21 − *z*ˆ21, *ε*22 = *z*22 − *z*ˆ22, *ε*31 = *z*31 − *z*ˆ31, *ε*32 = *z*32 − *z*ˆ32, subtracting Equation (38) from Equation (37) and subtracting Equation (39) from Equation (37), the following estimation error system yields:

$$
\dot{\varepsilon}\_i = A\_i \varepsilon\_i + \eta\_i \tag{43}
$$

where

$$\begin{aligned} \varepsilon\_{i} &= \begin{bmatrix} \varepsilon\_{21} \\ \varepsilon\_{22} \\ \varepsilon\_{31} \\ \varepsilon\_{32} \end{bmatrix}, \qquad \eta\_{i} = \begin{bmatrix} 0 \\ \Psi\_{2} \\ 0 \\ \Psi\_{3} \end{bmatrix}, \\\ A\_{i} &= \begin{bmatrix} -l\_{21} & 1 & 0 & 0 \\ -l\_{22} & 0 & 0 & 0 \\ 0 & 0 & -l\_{31} & 1 \\ 0 & 0 & -l\_{32} & 0 \end{bmatrix} \end{aligned} \tag{44}$$

On the other hand, define the tracking errors as *e*21 = *y*2r − *z*21 and *e*31 = *y*3r − *z*31. It follows from Equations (24), (26), (36), (40) and (41) that

$$
\begin{bmatrix}
\dot{\varepsilon}\_{21} \\
\dot{\varepsilon}\_{31}
\end{bmatrix} = -\begin{bmatrix}
k\_{21}(\varepsilon\_{21} + \varepsilon\_{21}) + \varepsilon\_{22} \\
k\_{31}(\varepsilon\_{31} + \varepsilon\_{31}) + \varepsilon\_{32}
\end{bmatrix} \tag{45}
$$

Thus, the tracking error system can be summarized as

$$
\dot{e}\_i = M\_i e\_i + \theta\_i \tag{46}
$$

where

$$x\_i = \begin{bmatrix} \ ^{\mathcal{C}\_{21}} \\ \ ^{\mathcal{C}\_{22}} \end{bmatrix}, \qquad \theta\_i = \begin{bmatrix} \ -\:^{\mathcal{S}\_{11}} \\ \ -\:^{\mathcal{S}\_{22}} \end{bmatrix},$$

$$M\_i = \begin{bmatrix} & -k\_{21} & 0 \\ & 0 & -k\_{31} \end{bmatrix} \tag{47}$$

with *ξ*1 = *ε*22 + *k*21*ε*21 and *ξ*2 = *ε*32 + *k*31*ε*31 being the lumped estimation error.

The stability analysis of the closed-loop control system is transformed into globally uniformly ultimately bounded summarized.

**Theorem 1.** *Consider the PMSM system Equation (24) equipped the proposed NAC Equation (42) with two POs Equations (38) and (39). If the real perturbation* <sup>Ψ</sup>*i*(*<sup>x</sup>*, *t*) *defined in Equation (36) satisfies*

$$\|\Psi\_i(\mathbf{x}, t)\| \le \gamma\_1 \tag{48}$$

*then both the estimation error system Equation (43) and the tracking error system Equation (46) are, i.e.,*

$$\|\|\varepsilon\_i(t)\|\| \le 2\gamma\_1 \|\|P\_1\|\|\_\prime \|\varepsilon\_i(t)\|\| \le 4\gamma\_1 \|\|K\_i\|\| \|\|P\_1\|\| \|P\_2\|\|\_\prime \forall t \ge T \tag{49}$$

*where Pi, i* = 1, 2 *are respectively the feasible solutions of Riccati equations A*T*i P*1 + *P*1*Ai* = −*I and <sup>M</sup>*T*i P*2 + *P*2*Mi* = −*I; and Ki is a constant related to k*11, *k*21 *and k*22*.*

**Proof.** For the estimation error system Equation (43), consider the following Lyapunov function:

$$V\_{i1}(\varepsilon\_i) = \varepsilon\_i^{\Gamma} P\_1 \varepsilon\_i \tag{50}$$

The high gains of POs Equations (38) and (39) are determined by requiring Equation (21) holds, which means *Ai* is Hurwitz. One can find a feasible positive definite solution, *P*1, of Riccati equation *A*T*i P*1 + *P*1*Ai* = <sup>−</sup>*I*. Calculating the derivative of *Vi*1(*<sup>ε</sup>i*) along the solution of system Equation (43) and using Equation (48) to yield

$$\begin{array}{rcl}\dot{\mathcal{V}}\_{i1}(\varepsilon\_{i}) &=& \varepsilon\_{i}^{\mathrm{T}}(A\_{i}^{\mathrm{T}}P\_{1} + P\_{1}A\_{i})\varepsilon\_{i} + \eta\_{i}^{\mathrm{T}}P\_{1}\varepsilon\_{i} + \varepsilon\_{i}^{\mathrm{T}}P\_{1}\eta\_{i} \\ &\leq& -||\varepsilon\_{i}||^{2} + 2||\varepsilon\_{i}|| \cdot ||\eta\_{i}|| \cdot ||P\_{1}|| \\ &\leq& -||\varepsilon\_{i}|| \left(||\varepsilon\_{i}|| - 2\gamma\_{1}||P\_{1}||\right) \end{array} \tag{51}$$

Then *V* ˙ *i*1(*<sup>ε</sup>i*) ≤ 0 when *<sup>ε</sup>i* ≥ <sup>2</sup>*γ*1 *<sup>P</sup>*1 . Thus, there exists *T*1 > 0, which can lead to

$$\|\|\varepsilon\_{i}(t)\|\| \le \gamma\_{2} = 2\gamma\_{1} \|\|P\_{1}\|\|, \forall t \ge T\_{1} \tag{52}$$

For tracking error system Equation (46), one can find that *<sup>ϑ</sup>i* ≤ *Ki γ*2 with *Ki* based on *<sup>ε</sup>i*(*t*) ≤ *γ*2. Consider the Lyapunov function *Vi*2(*ei*) = *e*T*i P*2*ei*. Similarly, one can prove that there exists an instant, *T*1, the following holds

$$\|e\_i(t)\| \le 2\|K\_l\|\|\gamma\_2\|P\_2\| \le 4\gamma\_1\|K\_l\|\|\|P\_1\|\|\|P\_2\|\|, \forall t \ge T\_1 \tag{53}$$

Using Equations (52) and (53) and setting *T* = max{*<sup>T</sup>*1, *T*¯1} lead to Equation (49).

Moreover, if <sup>Ψ</sup>*i*(*<sup>x</sup>*, *t*) and Ψ˙ *<sup>i</sup>*(*<sup>x</sup>*, *t*) are locally Lipschitz in their arguments, it will guarantee the exponential convergence of the observation error [19] and closed-loop tracking error into

$$\lim\_{t \to \infty} \varepsilon\_i(t) = 0 \quad \text{and} \quad \lim\_{t \to \infty} \varepsilon\_i(t) = 0 \tag{54}$$

After the states *i*d and *i*q and their derivatives are stable that controlled by NAC. The parameter variation is considered in the error system in Equations (43) and (46), and the error system is proved as converged to zero in Equation (54). This guarantees that the estimated perturbations track the extended states defined in Equation (36), which includes the uncertainties affected by the parameter variations and disturbances, and compensates for the control input in Equation (40). Then the linearized subsystems in Equation (37) are independent of the parameters and disturbances.

**Remark 2.** *The perturbation and its derivative are assumed to locally bounded as described in Assumption 2. The existence of these bounds can be shown in the following analysis. The perturbation and its derivative can be represented as*

Ψ2 = *<sup>F</sup>*2(*x*) + *<sup>B</sup>*2(*x*)−*B*2(0) *<sup>B</sup>*2(*x*) [*k*21(*<sup>z</sup>*21*r* − *<sup>z</sup>*21) + *z*22 − *z*ˆ22] = *<sup>F</sup>*2(*x*) + *<sup>B</sup>*2(*x*)−*B*2(0) *<sup>B</sup>*2(*x*) (*k*21*e*21 + *<sup>ε</sup>*22) Ψ ˙ 2 = *F* ˙ 2(*x*) + *<sup>B</sup>*2(*x*)−*B*2(0) *<sup>B</sup>*2(0) (−Ψ˙ 2 + *k*21*<sup>e</sup>*˙21 − *ε*˙22) = *F* ˙ 2(*x*) + *<sup>B</sup>*2(*x*)−*B*2(0) *<sup>B</sup>*2(0) (*k*21*<sup>e</sup>*˙21 + *l*22*ε*21) Ψ3 = *<sup>F</sup>*3(*x*) + *<sup>B</sup>*3(*x*)−*B*3(0) *<sup>B</sup>*3(*x*) [*k*31(*<sup>z</sup>*31*r* − *<sup>z</sup>*31) + *z*32 − *z*ˆ32] = *<sup>F</sup>*3(*x*) + *<sup>B</sup>*3(*x*)−*B*3(0) *<sup>B</sup>*3(*x*) (*k*31*e*31 + *<sup>ε</sup>*32) Ψ ˙ 3 = *F* ˙ 3(*x*) + *<sup>B</sup>*3(*x*)−*B*3(0) *<sup>B</sup>*3(0) (−Ψ˙ 3 + *k*31*<sup>e</sup>*˙31 − *ε*˙32) = *F* ˙ 3(*x*) + *<sup>B</sup>*3(*x*)−*B*3(0) *<sup>B</sup>*3(0) (*k*31*<sup>e</sup>*˙31 + *l*32*ε*31)

*Considering Assumption 1, we have*

$$\begin{array}{rcl} \mid \mid \Psi\_{2} \mid & \leq & \frac{1}{1-\theta\_{2}} \mid F\_{2}(\mathbf{x}) \mid + \frac{\theta\_{2}}{1+\theta\_{2}} (\mid \mid \: \varepsilon\_{21} \mid \mid + \mid \: \varepsilon\_{22} \mid \mid \\ \mid \mid \mid \Psi\_{2} \mid & \leq & \mid \: \mathbb{P}\_{2}(\mathbf{x}) \mid + \mid \: B\_{2}(\mathbf{x}) \mid \mid \: \mu\_{2} \mid + \theta\_{2} (\mid \mid \: k\_{21} \mid \mid \mid \: \: l\_{21} \mid \mid + \\ \mid \mid \Psi\_{3} \mid & \leq & \frac{1}{1-\theta\_{3}} \mid F\_{3}(\mathbf{x}) \mid + \frac{\theta\_{3}}{1+\theta\_{3}} (\mid \: k\_{31} \mid \mid \: \: l\_{31} \mid \mid + \mid \: \varepsilon\_{32} \mid \mid \\ \mid \mid \Psi\_{3} \mid & \leq & \mid \: \mathbb{P}\_{3}(\mathbf{x}) \mid + \mid \: B\_{3}(\mathbf{x}) \mid \mid \: u\_{3} \mid + \theta\_{3} (\mid \: k\_{31} \mid \mid \: l\_{32} \mid \: \: \: l\_{32} \mid \: \: \: l\_{31} \mid \: \: \: \end{array}$$

*From the above equations, with consideration of the perturbation assumed as a smooth function of time, it can be concluded that the bound of perturbation and its derivative exist.*

*As a result, with both the Assumptions 1 and 2, the effectiveness of such perturbation observer-based control can be guaranteed.*
