Robust Multivariable Control of a PMSM Via μ-Synthesis Under Structured Parametric Uncertainties

The robust control of nonlinear electromechanical systems in the presence of structured parametric uncertainties is a fundamental research area in modern control engineering. Among these systems, the permanent magnet synchronous motor (PMSM) plays a central role in highperformance industrial applications such as electric vehicles, drones, industrial robots, and aerospace propulsion systems, due to its excellent torque-to-weight ratio, high efficiency, and fast dynamics [1-3]. However, the performance of these systems strongly depends on the accuracy of the model used for controller design. In practice, the physical parameters of the PMSM-namely stator resistance $R_s, d$ - and $q$-axis inductances $L_d$ and $L_q$, magnet flux linkage $\phi_f$, mechanical inertia $J_m$, and viscous friction coefficient $B$ are subject to significant variations due to temperature, magnetic saturation, mechanical wear, or material aging [4-6]. These variations compromise the stability and performance of classical field-oriented control (FOC) strategies based on PI regulators, which are designed for a nominal model and exhibit limitations under severe parametric uncertainties, such as overshoot, steady-state error, and poor disturbance rejection [7-9]. To overcome these limitations, several advanced approaches have been proposed, such as adaptive control, fuzzy logic, or model predictive control [10-12]. However, these methods often lack formal guarantees of robust stability. In contrast, robust control based on the $H \infty / \mu$ framework provides a mathematically rigorous approach to designing controllers capable of maintaining performance and stability under structured uncertainties [13, 14]. Unlike standard $H \infty$ synthesis, which treats uncertainties as unstructured, $\mu$-synthesis exploits their physical nature (real, bounded, decoupled parameters), thereby reducing the conservatism of the solution and providing finer robustness guarantees [15, 16].

The field of robust PMSM control has been the subject of extensive research aimed at improving stability and performance under parametric uncertainties and exogenous disturbances. Several studies have used synthesis and $\mu$-analysis to guarantee robust stability and disturbance rejection under severe parameter variations [17, 18]. Others have explored nonlinear strategies such as backstepping control [19], finite-control-set model predictive control (FCS-MPC) [20, 21], or disturbance-observer-based adaptive control [22, 23]. Recent works have demonstrated the effectiveness of robust MIMO control for electric motor drives under parametric uncertainties, as well as the importance of frequency weighting functions for disturbance rejection. These contributions confirm the relevance of $\mu$-synthesis for multivariable electromechanical systems [24-27].

In this work, we propose a MIMO model of the PMSM with two inputs $u_d, u_q$ (axis voltages) and two outputs, $i_d, \omega_m$ (axis current and mechanical speed), developed in the (d,q) frame by exploiting FOC to ensure effective decoupling between the d and q axes. This model, although simplified, captures the dominant dynamics of the PMSM and constitutes a solid basis for the synthesis of a robust controller. Six physical parameters ($R_s, L_d, L_q, \phi_f, J_m, B$) are modeled as uncertain with ±30% variations, reflecting real operating conditions [4, 5].

The $\mu$-synthesis framework combines robust control design and $\mu$-analysis to synthesize a controller that ensures both nominal performance and robust stability in the presence of structured uncertainties. A robust controller is then obtained via a D-K iteration procedure, which alternates between an $H \infty$ synthesis step (K-step) and a scaling analysis step (D-step) to minimize $\mu$ over the entire frequency range of interest. This approach has been successfully applied to complex systems such as altitude test facilities and electromechanical drives under load disturbances, demonstrating its ability to ensure precise tracking and excellent disturbance rejection even under fast commands and strong perturbations [24-27].

A generalized MIMO plant is constructed, and a μ synthesis with diagonal weighting functions is implemented via a D-K iteration, yielding a reduced-order controller. The weighting functions are chosen to reflect performance, disturbance rejection, and control effort objectives, in accordance with the mixed-sensitivity loop-shaping structure. The controller obtained from the D-K algorithm is typically high-order. To facilitate implementation, order reduction is performed via modal approximation while preserving robust stability properties [26, 27].

The resulting controller is validated through simulation, demonstrating its effectiveness in reference tracking and disturbance rejection. The results show that the closed-loop system maintains a stable and accurate dynamic response, even under severe parametric variations, as encountered in real operating conditions. This study confirms the effectiveness of μ-synthesis for robust PMSM control in uncertain environments.

3.1 Robustness problem formulation

The design of a robust controller for the PMSM relies on the $\mu$-synthesis framework, a rigorous method to ensure both robust stability and nominal performance in the presence of structured parametric uncertainties [24, 25-27]. The nominal model $G(s)$, established in the previous section, is augmented with a structured uncertainty block $\Delta$, grouping variations in the six physical parameters: stator resistance $R_s$, inductances $L_d, L_q$, magnet flux $\phi_f$, inertia $J_m$, and viscous friction $B$. Each uncertainty is modeled as a normalized multiplicative perturbation bounded in $H \infty$ norm:

$p=p_0\left(1+\delta_p \Delta_p\right),\left|\Delta_p\right| \leq 1$

where, $p_0$ is the nominal value, $\delta_p$ the uncertainty level (e.g., 0.3 for ±30%), and $\Delta_p$ a normalized variable. This structure captures the combined effects of parametric variations while preserving system causality [26, 27].

3.2 Structure and generalized plant

The fundamental structure of $\mu$-synthesis relies on the Linear Fractional Transformation (LFT) (Figure 1), which separates the system into two parts: a nominal model $\mathrm{P}(\mathrm{s})$ and a structured uncertainty block $\Delta$. The controller $K(s)$ is interconnected via a lower LFT (LLFT), while the uncertainty block is connected via an upper LFT (ULFT) [25-27].

The generalized plant $P(s)$ is described in state-space form as:

$P_{\Delta}:\left(\begin{array}{c}x(t) \\ v(t) \\ z(t) \\ y(t)\end{array}\right)=\left(\begin{array}{cccc}A & B_d & B_w & B_u \\ C_v & D_{v d} & D_{v w} & D_{v u} \\ C_z & D_{z d} & D_{z w} & D_{z u} \\ C_y & D_{y d} & D_{y w} & D_{y u}\end{array}\right)\left(\begin{array}{c}x(t) \\ d(t) \\ w(t) \\ u(t)\end{array}\right)$     (5)

where:

• $x(t)$: system state,
• $d(t)$: exogenous disturbances (uncertainties, noise),
• $w(t)$: reference signals,
• $u(t)$: control input,
• $v(t)$: output of the uncertainty block $\Delta$,
• $z(t)$: weighted output (performance),
• $y(t)$: measurement.

The ULFT interconnection between $\Delta_p$ and $\Delta$ is defined by:

$\begin{gathered}v=P_{11} d+P_{12} w, w=P_{21} d+P_{22} w, d=\Delta v \\ \Rightarrow z=\operatorname{ULFT}\left(P_{\Delta}, \Delta\right) w\end{gathered}$     (6)

The LLFT interconnection between $P_{\Delta}$ and $K$ is defined by:

$\begin{gathered}u=P_{12} K\left(I-P_{22} K\right)^{-1} P_{21} w+P_{11} \mathrm{w} \\ \Rightarrow z=\operatorname{LLFT}\left(P_{\Delta}, \Delta\right) w\end{gathered}$     (7)

Figure 1 illustrates this standard interconnection [1].

Figure 1. Generalized plant interconnection with the controller and uncertainty blocks [26, 27]

3.3 Definition and interpretation of the structured singular value ($\mu$)

The structured singular value $\mu$, or structured singular value, is a fundamental measure of robustness of a closed-loop system in the presence of structured uncertainties [24, 25, 27]. For a closed-loop system represented by the transfer function $T_{z w}(s)= \operatorname{LLFT}\left(P_{\Delta}, K\right)(s)$, $\mu$ is defined as:

$\mu_{\Delta}\left(T_{z w}(j w)\right)=\left(\min _{\Delta \in \Delta}\left(\frac{\bar{\sigma}(\Delta)}{\operatorname{det}\left(I-T_{z w}(j w) \Delta\right)}=0\right)\right)^{-1}$     (8)

The system is robustly stable if and only if:

$\sup _{\omega \in R} \mu_{\Delta}\left(T_{z w}(j \omega)\right)<1$     (9)

This condition ensures closed-loop stability even under extreme parametric variations. An upper bound of $\mu$ is given by:

$\mu_{\Delta}\left(T_{z w}(j \omega)\right) \leq \inf _{D \in D} \bar{\sigma}\left(D(j \omega) T_{z w}(j \omega) D^{-1}(j \omega)\right)$     (10)

where, D is the set of scaling matrices compatible with the structure of D [8]. This bound is minimized during the D-K iteration.

3.4 Frequency weighting functions and MIMO structure

A generalized plant $P(s)$ is constructed using frequency weighting functions to formulate a mixed-sensitivity control problem. These weights are chosen to reflect design objectives [24, 25, 27].

• $W_1(s)$: sensitivity weighting, shaping tracking accuracy and low-frequency disturbance rejection,
• $W_2(s)$: control effort weighting, limiting actuator demand and preventing inverter saturation,
• $W_3(s)$: complementary sensitivity weighting, ensuring robustness against unmodeled dynamics (e.g., residual coupling, fast parameter variations, highfrequency noise).

These functions are defined as diagonal matrices:

$\begin{aligned} & W_1(s)=\operatorname{diag}\left(W_{1, i_d}(s), W_{1, \omega}(s)\right) \\ & W_2(s)=\operatorname{diag}\left(W_{2, i_d}(s), W_{2, \omega}(s)\right) \\ & W_3(s)=\operatorname{diag}\left(W_{3, i_d}(s), W_{3, \omega}(s)\right)\end{aligned}$     (11)

Each element is a stable, minimum-phase first-order filter, designed to jointly achieve:

• high gain at low frequencies for precise tracking and disturbance rejection,
• controlled roll-off to limit closed-loop bandwidth and avoid noise amplification,
• bounded high-frequency gain to constrain control effort.

The diagonal structure of the weights is justified by the decoupled nature of the PMSM multivariable model in the (d,q) frame. After FOC, the d and q axes are nearly independent: $i_d$ primarily controls flux, while $i_q$ controls torque. This separation allows diagonal weighting, simplifying controller synthesis while maintaining satisfactory performance [24, 26]. It also reduces computational complexity and avoids convergence issues associated with full matrices.

The tuning process is systematic and iterative: first, frequency domain constraints are enforced to guarantee robust stability and performance margins; second, time-domain validation under representative operating conditions confirms satisfactory dynamic response, disturbance rejection, and control authority. This two-stage approach ensures that the weighting functions translate high-level design requirements precision, robustness, and real time implement ability into mathematically tractable specifications for µ-synthesis.

Figure 2 shows the block diagram of the proposed experiment, where the augmented plant is fed only by reference signals. This structure simplifies analysis and synthesis by focusing all performance objectives on reference response [27].

Figure 2. The block diagram of the proposed experiment containing the augmented plant which contains as inputs the reference signals only

3.5 D-K iteration algorithm

$\mu$-synthesis cannot be solved directly because the structured singular value $\mu$ is not a convex norm. An effective solution is the D-K iteration algorithm, which alternates between two steps:

1. K-step ($H \infty$ Synthesis): With fixed scaling matrices $D(j \omega)$, solve a standard $H \infty$ problem. Find a controller $K(s)$ minimizing the norm of the scaled system $H \propto \left\|D(s) T_{z w}(s) D^{-1}(s)\right\|_{\infty}$ [13].
2. D-step ($\mu$-Analysis): With fixed controller $K(s)$, compute at each frequency $\omega$ a scaling matrix $D(j \omega)$ that minimizes the largest singular value of the scaled system: $\min _D \bar{\sigma}\left(D(j \omega) T_{z w}(j \omega) D^{-1}(j \omega)\right)$ then interpolate the $D(j \omega)$ by a stable, minimum-phase system.

The $D$ and $K$ steps are iterated until convergence, i.e., until $\mu<1$ over the entire frequency range of interest [25, 27].

This paper has presented a robust multivariable control strategy for a PMSM based on $\mu$-synthesis, specifically designed to maintain high performance under significant structured parametric uncertainties of up to ±30%. A MIMO model was developed in the (d,q) reference frame, leveraging FOC principles to decouple the machine's dynamics. The uncertainties in key parameters—stator resistance, d- and q-axis inductances, magnetic flux, moment of inertia, and viscous friction—were explicitly modeled, allowing for a systematic design process.

The controller was synthesized using the D–K iteration method, resulting in a high-order solution that was subsequently reduced to a practical order of 6 while preserving the critical robustness properties. The final controller requires only the measurement of rotor speed, simplifying the sensor requirements and the overall control architecture.

Simulation results in Simulink demonstrate the controller’s exceptional performance. It achieves precise speed tracking, maintains zero d-axis current for optimal flux regulation, and exhibits an immediate and robust response to both speed inversions and load torque disturbances. The complete absence of speed drop under load and the seamless transition during speed reversal underscore the controller’s robustness.

This work demonstrates the significant potential of $\mu$-synthesis for robust PMSM control in challenging, real-world conditions where parameter variations are inevitable. The approach provides a rigorous, model-based alternative to conventional control methods, ensuring stability and performance guarantees across the entire uncertainty domain. The successful validation of the reduced-order controller highlights its practical feasibility for industrial applications such as electric vehicles, robotics, and precision drives.

Nevertheless, the validity of the proposed approach is bounded by two key assumptions. First, it relies on an accurate machine model and effective d q axis decoupling via field-oriented control. This assumption may be challenged under extreme operating conditions, particularly when significant thermal effects alter the stator resistance and permanent magnet flux, or when unmodeled nonlinearities such as magnetic saturation and inverter dead time effects become dominant. Second, while the sixth order realization is theoretically suitable for embedded implementation, critical real time constraints, including computational load, execution latency, and minimum sampling frequency have not yet been quantified and will be rigorously assessed during experimental validation.

Future work will focus on experimental validation of the controller on a physical test bench. This will provide further evidence of its real-time performance and robustness. Additionally, the integration of this control strategy with advanced techniques like field-weakening control will be explored to extend the operational speed range of the motor.