SRM Polytopic (Quasi-)LPV Model Summary for Python Verification

This document summarizes the LPV/polytopic modeling decisions we made so far, so you can implement Python code to compare: 1) the original (nonlinear/quasi-LPV) SRM dynamics, and
2) the polytopic LPV approximation constructed over excited-current operating regions.

We assume unipolar phase currents and 90° periodicity of inductance profiles.

0. Notation

Phases: (j \in {a,b,c})

Given functions (from SRM magnetic model / identification):

• (L_j(\theta)): phase inductance
• (L’_j(\theta)=\dfrac{\partial L_j(\theta)}{\partial\theta})

Motor parameters:

• (R): phase resistance
• (J_m): inertia
• (B_m): viscous friction coefficient

Inputs:

• Phase voltages (v_a,v_b,v_c)
• Disturbance torque (T_d)

1. “Reference” (Original) Quasi-Nonlinear SRM State Model (for simulation baseline)

1.1 State, input, output

State: [ x=\begin{bmatrix} i_a & i_b & i_c & \omega & \theta \end{bmatrix}^\top ] Input: [ u=\begin{bmatrix} v_a & v_b & v_c & T_d \end{bmatrix}^\top ] Output (measured currents): [ y=\begin{bmatrix} i_a & i_b & i_c \end{bmatrix}^\top,\qquad C=\begin{bmatrix} I_{3\times3} & 0_{3\times2} \end{bmatrix} ]

1.2 Dynamics (baseline model)

Electrical (per phase): [ \dot i_j = -\frac{R}{L_j(\theta)}\,i_j -\frac{L’_j(\theta)}{L_j(\theta)}\,i_j\,\omega +\frac{1}{L_j(\theta)}\,v_j ]

Mechanical: [ \dot\omega = -\frac{B_m}{J_m}\omega +\sum_{j\in{a,b,c}}\frac{1}{J_m}L’_j(\theta)\,i_j^2 -\frac{1}{J_m}T_d ]

Kinematics: [ \dot\theta=\omega ]

This model is “ground truth” for comparison.
If you use phase offsets (e.g., (L_a(\theta)=L(\theta-\phi_a))), implement them consistently in the functions (L_j(\theta)), (L’_j(\theta)).

2. Goal: Polytopic LPV Approximation with Vertex-LMI-Friendly Constant Matrices

We wanted a polytopic form where each vertex has constant (A) matrices, to improve feasibility of vertex-wise LMI conditions.

Key modeling idea (B):

Use a local affine approximation of the term (\dfrac{R}{L_j(\theta)}) so that part of it becomes a (\theta)-state coupling term (i.e., populating (A(1:3,5))).

3. Scheduling / Periodicity

3.1 90° periodicity

Use the wrapped angle: [ \bar\theta=\mathrm{mod}!\left(\theta,\frac{\pi}{2}\right)\in\left[0,\frac{\pi}{2}\right] ]

3.2 Theta grid for polytopic partition (9 points)

[ \theta_k=\frac{k-1}{8}\cdot\frac{\pi}{2},\quad k=1,\dots,9 ]

3.3 Theta weights (\zeta_k(\bar\theta)) (hat/linear interpolation)

If (\bar\theta\in[\theta_m,\theta_{m+1}]): [ \zeta_m=\frac{\theta_{m+1}-\bar\theta}{\theta_{m+1}-\theta_m},\qquad \zeta_{m+1}=\frac{\bar\theta-\theta_m}{\theta_{m+1}-\theta_m} ] [ \zeta_k=0\ (k\neq m,m+1) ] Properties: [ 0\le \zeta_k\le 1,\qquad \sum_{k=1}^9\zeta_k=1 ]

4. Current Vertices (Unipolar) and “Excited-only” Polytope

We assume: [ i_a,i_b,i_c \in [0,I_{\max}] ] and we construct current vertices from ({0,I_{\max}}) for each phase: [ \sigma=(\sigma_a,\sigma_b,\sigma_c)\in{0,1}^3 ] Vertex currents: [ i_a^{(\sigma)}=\sigma_a I_{\max},\quad i_b^{(\sigma)}=\sigma_b I_{\max},\quad i_c^{(\sigma)}=\sigma_c I_{\max} ]

4.1 Excluding the (0,0,0) vertex

We exclude (\sigma=000) to focus on excited conditions. Define: [ \mathcal S={001,010,011,100,101,110,111} ] So 7 current vertices remain.

4.2 Current barycentric weights (per phase)

For each phase: [ \mu_{j,0}(i_j)=1-\frac{i_j}{I_{\max}},\qquad \mu_{j,1}(i_j)=\frac{i_j}{I_{\max}} ] Then for each (\sigma\in{0,1}^3): [ w_\sigma(i)=\mu_{a,\sigma_a}(i_a)\,\mu_{b,\sigma_b}(i_b)\,\mu_{c,\sigma_c}(i_c) ] and (\sum_\sigma w_\sigma = 1).

Weight for excluded vertex: [ w_{000}=\mu_{a,0}\mu_{b,0}\mu_{c,0} ]

4.3 Normalized weights over excited vertices

For ((i_a,i_b,i_c)\neq(0,0,0)), define: [ \tilde w_\sigma(i)=\frac{w_\sigma(i)}{1-w_{000}(i)},\qquad \sigma\in\mathcal S ] Then: [ 0\le \tilde w_\sigma\le 1,\qquad \sum_{\sigma\in\mathcal S}\tilde w_\sigma = 1 ]

Practical note: to avoid numerical issues when currents are tiny, define an “excited region” [ i_a+i_b+i_c \ge I_{\min} > 0 ] and only use the excited-polytope model there.

5. Final Polytopic Weights (Total 9×7 = 63 vertices)

Define combined weights: [ \Xi_{k,\sigma}(x)=\zeta_k(\bar\theta)\,\tilde w_\sigma(i) \qquad (k=1,\dots,9,\ \sigma\in\mathcal S) ] Then: [ 0\le \Xi_{k,\sigma}\le 1,\qquad \sum_{k=1}^9\sum_{\sigma\in\mathcal S} \Xi_{k,\sigma}=1 ]

6. Vertex Matrix Construction: (A_{k,\sigma}), (B_k), (C)

State and input remain: [ x=\begin{bmatrix} i_a&i_b&i_c&\omega&\theta\end{bmatrix}^\top,\quad u=\begin{bmatrix}v_a&v_b&v_c&T_d\end{bmatrix}^\top ] [ C=\begin{bmatrix}I_{3\times3} & 0_{3\times2}\end{bmatrix} ]

6.1 Precompute inductance terms at theta grid points

At each theta vertex (\theta_k): [ L_{j,k}=L_j(\theta_k),\qquad L’_{j,k}=L’_j(\theta_k) ]

Define: [ g_{j,k}=\frac{L’{j,k}}{L{j,k}} ]

6.2 Local affine approximation of (f_j(\theta)=\frac{R}{L_j(\theta)})

We approximate near (\theta_k): [ f_j(\theta)\approx a_{j,k}+b_{j,k}\theta ] where: [ b_{j,k}=f’j(\theta_k)=\frac{d}{d\theta}\Big(\frac{R}{L_j(\theta)}\Big)\Big|{\theta_k} = -R\frac{L’{j,k}}{L{j,k}^2} ] [ a_{j,k}=f_j(\theta_k)-b_{j,k}\theta_k=\frac{R}{L_{j,k}}-b_{j,k}\theta_k ]

This yields: [ -\frac{R}{L_j(\theta)}i_j \approx -a_{j,k} i_j - b_{j,k} i_j\,\theta ] so we get a constant (\theta)-column at each current vertex by replacing (i_j) with (i_j^{(\sigma)}).

6.3 Vertex (A_{k,\sigma}\in\mathbb{R}^{5\times 5})

Use vertex current values: [ i_a^{(\sigma)}=\sigma_a I_{\max},\quad i_b^{(\sigma)}=\sigma_b I_{\max},\quad i_c^{(\sigma)}=\sigma_c I_{\max} ]

Nonzero entries of (A_{k,\sigma}) (state order ([i_a,i_b,i_c,\omega,\theta])):

(i) “Resistance” part (diagonal via affine approx) [ A_{k,\sigma}(1,1)=-a_{a,k},\quad A_{k,\sigma}(2,2)=-a_{b,k},\quad A_{k,\sigma}(3,3)=-a_{c,k} ]

(ii) Omega coupling in current equations (from (-\frac{L’}{L}i\omega)) [ A_{k,\sigma}(1,4)=-g_{a,k}\,i_a^{(\sigma)},\quad A_{k,\sigma}(2,4)=-g_{b,k}\,i_b^{(\sigma)},\quad A_{k,\sigma}(3,4)=-g_{c,k}\,i_c^{(\sigma)} ]

(iii) Theta-column coupling (from affine approx of (-\frac{R}{L}i)) [ A_{k,\sigma}(1,5)=-b_{a,k}\,i_a^{(\sigma)},\quad A_{k,\sigma}(2,5)=-b_{b,k}\,i_b^{(\sigma)},\quad A_{k,\sigma}(3,5)=-b_{c,k}\,i_c^{(\sigma)} ]

(iv) Mechanical torque term (using original torque model without 1/L) We use the identity (L’ i^2 = (L’i)\,i) and replace one (i) by vertex value. [ A_{k,\sigma}(4,1)=\frac{1}{J_m}L’{a,k}\,i_a^{(\sigma)},\quad A{k,\sigma}(4,2)=\frac{1}{J_m}L’{b,k}\,i_b^{(\sigma)},\quad A{k,\sigma}(4,3)=\frac{1}{J_m}L’{c,k}\,i_c^{(\sigma)} ] Friction: [ A{k,\sigma}(4,4)=-\frac{B_m}{J_m} ]

(v) Kinematics [ A_{k,\sigma}(5,4)=1 ]

All other entries are zero.

6.4 Vertex (B_k\in\mathbb{R}^{5\times4})

We keep input matrices depending only on (\theta_k): [ B_k(1,1)=\frac{1}{L_{a,k}},\quad B_k(2,2)=\frac{1}{L_{b,k}},\quad B_k(3,3)=\frac{1}{L_{c,k}},\quad B_k(4,4)=-\frac{1}{J_m} ] All other entries are zero.

6.5 Output matrix

[ C=\begin{bmatrix}I_{3\times3} & 0_{3\times2}\end{bmatrix} ]

7. Final Polytopic LPV Model (63 vertices)

Dynamics: [ \dot x = \sum_{k=1}^9\sum_{\sigma\in\mathcal S}\Xi_{k,\sigma}(x)\,A_{k,\sigma}\,x \;+\; \sum_{k=1}^9 \zeta_k(\bar\theta)\,B_k\,u ] Output: [ y=Cx ]

8. How to Use This in Python Comparison

You will likely implement two simulators:

8.1 Baseline simulator (original model)

• Use the equations in Section 1 with your exact (L_j(\theta),L’_j(\theta)).

8.2 Polytopic simulator

At each time step: 1) Wrap angle: (\bar\theta=\mathrm{mod}(\theta,\pi/2)) 2) Compute (\zeta_k(\bar\theta)) 3) Compute current weights (\mu_{j,0},\mu_{j,1}) 4) Compute unnormalized (w_\sigma), normalize to (\tilde w_\sigma) over (\sigma\in\mathcal S) 5) Compute (\Xi_{k,\sigma}=\zeta_k \tilde w_\sigma) 6) Evaluate:

• (A_\text{poly}=\sum_{k,\sigma}\Xi_{k,\sigma} A_{k,\sigma})
• (B_\text{poly}=\sum_k \zeta_k B_k) 7) Integrate: [ \dot x = A_\text{poly}x + B_\text{poly}u ]

Suggested metrics

• Trajectory error: (|x_\text{baseline}-x_\text{poly}|)
• Current RMSE per phase
• Speed and angle error over excited intervals
• Sensitivity vs. (I_{\min}) threshold (if used)

9. Implementation Notes / Caveats

1) Excluded vertex normalization: The normalization (\tilde w_\sigma) is only safe when (1-w_{000}) is not tiny. Use an excited threshold (I_{\min}) or fallback model near zero current.

2) Vertex count: 63 constant vertices = feasible for LMI but still sizable. If your switching strategy implies only 1–2 phases conduct at a time, you can optionally exclude unrealistic current vertices (e.g., 111).

3) Affine approximation accuracy: The affine approximation of (\frac{R}{L(\theta)}) around each (\theta_k) is a modeling approximation. Its mismatch is part of what you will quantify in simulation.

4) Theta periodicity: Ensure consistent use of (\bar\theta) in the polytopic weights and inductance evaluation.

10. Checklist of What You Need to Provide in Code

• Functions:
  • L_a(theta), L_b(theta), L_c(theta)
  • dL_a(theta), dL_b(theta), dL_c(theta)
• Constants: R, Jm, Bm, Imax
• Input trajectory: v_a(t), v_b(t), v_c(t), Td(t)
• Initial conditions: i_a(0), i_b(0), i_c(0), w(0), theta(0)

End of document.