friction-inertia-estimation.md

Field	Value
Title	Mechanical Parameters Identification — Friction and Inertia
Type	theory
Status	approved
Version	1.0.0
Component	service-mechanical-ident
Date	2025-01-01

Overview

Accurate knowledge of the mechanical plant parameters — rotor moment of inertia $J$, viscous friction coefficient $B$, and Coulomb (static) friction torque $\tau_0$ — is required to design an optimal speed or position loop, to implement feed-forward compensation, and to estimate system health over time.

This identification procedure uses Recursive Least Squares (RLS) to fit the parameters of Newton's rotational equation of motion to real-time measurements of torque (estimated from $i_q$) and speed/ acceleration (derived from the encoder). RLS operates online in every control cycle, converging to the true parameters as the motor is excited with varying speed profiles.

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Prerequisites

Symbol	Meaning	Unit
$J$	Rotor moment of inertia	kg·m²
$B$	Viscous friction coefficient	N·m·s/rad
$\tau_0$	Coulomb friction (static offset)	N·m
$\tau_e$	Electrical (motor) torque = $\frac{3}{2}p\psi_f i_q$	N·m
$\tau_{load}$	External load torque (treated as disturbance)	N·m
$\omega$	Rotor mechanical angular velocity	rad/s
$\alpha$	Rotor angular acceleration = $\dot\omega$	rad/s²
$\mathbf{\theta}$	Parameter vector $[J,\ B,\ \tau_0]^T$	mixed
$\mathbf{\phi}$	Regressor vector $[\alpha,\ \omega,\ 1]^T$	mixed
$\lambda$	RLS forgetting factor	—
$\mathbf{P}$	Error covariance matrix (3×3)	—
$\mathbf{K}$	Kalman gain vector (3×1)	—

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Mathematical Foundation

1. Mechanical Equation of Motion

The PMSM rotor dynamics are described by Newton's second law for rotation. Assuming viscous friction and a constant Coulomb friction offset:

$$\tau_e = J\dot\omega + B\omega + \tau_0 + \tau_{load}$$

Ignoring unmeasured load torque (treated as noise by the estimator), this becomes a linear regression:

$$\tau_e = \mathbf{\phi}^T \mathbf{\theta}$$

where:

$$\mathbf{\phi} = \begin{pmatrix}\dot\omega \\ \omega \\ 1\end{pmatrix}, \qquad \mathbf{\theta} = \begin{pmatrix}J \\ B \\ \tau_0\end{pmatrix}$$

The regression form makes the parameter vector $\mathbf{\theta}$ directly estimable from measured input-output data $(\mathbf{\phi}, \tau_e)$ using least squares.

2. Batch Least Squares (Background)

If $N$ measurements are collected:

$$\mathbf{y} = \mathbf{\Phi}\,\mathbf{\theta} + \mathbf{e}$$

where $\mathbf{y} \in \mathbb{R}^N$ is the torque sequence, $\mathbf{\Phi} \in \mathbb{R}^{N \times 3}$ is the regressor matrix, and $\mathbf{e}$ is noise. The ordinary least-squares solution is:

$$\hat{\mathbf{\theta}} = (\mathbf{\Phi}^T\mathbf{\Phi})^{-1}\mathbf{\Phi}^T\mathbf{y}$$

This requires storing all $N$ measurements and inverting a 3×3 matrix. RLS computes the same result recursively, processing one sample at a time with $O(n^2)$ cost where $n=3$ is the number of parameters.

3. Recursive Least Squares — Derivation

Define the weighted least-squares cost with exponential forgetting factor $\lambda \in (0, 1]$:

$$J_N = \sum_{k=1}^{N} \lambda^{N-k}\!\left(\tau_e[k] - \mathbf{\phi}[k]^T\hat{\mathbf{\theta}}\right)^2$$

The exponential weight $\lambda^{N-k}$ discounts older measurements, giving the estimator a finite effective memory window of $\approx 1/(1-\lambda)$ samples.

Applying the matrix inversion lemma (Woodbury identity) to the recursive update of $\mathbf{P}_N = (\sum_{k=1}^N \lambda^{N-k}\mathbf{\phi}[k]\mathbf{\phi}[k]^T)^{-1}$ yields:

$$\boxed{ \mathbf{K}[n] = \frac{\mathbf{P}[n-1]\,\mathbf{\phi}[n]}{\lambda + \mathbf{\phi}[n]^T\mathbf{P}[n-1]\,\mathbf{\phi}[n]} }$$ $$\boxed{ \hat{\mathbf{\theta}}[n] = \hat{\mathbf{\theta}}[n-1] + \mathbf{K}[n]\!\left(\tau_e[n] - \mathbf{\phi}[n]^T\hat{\mathbf{\theta}}[n-1]\right) }$$ $$\boxed{ \mathbf{P}[n] = \frac{1}{\lambda}\!\left(\mathbf{I} - \mathbf{K}[n]\,\mathbf{\phi}[n]^T\right)\mathbf{P}[n-1] }$$

The scalar $e[n] = \tau_e[n] - \mathbf{\phi}[n]^T\hat{\mathbf{\theta}}[n-1]$ is called the innovation (prediction error). The gain $\mathbf{K}$ weights how much to move the estimate in response to the innovation.

Forgetting Factor

For $\lambda = 1$ (no forgetting) the RLS is equivalent to batch LS over the entire history. For $\lambda < 1$ older data is exponentially discounted. The effective memory window is:

$$N_{eff} = \frac{1}{1 - \lambda}$$

With $\lambda = 0.998$: $N_{eff} = 500$ samples. This allows the estimator to track slow parameter variations (e.g. bearing wear increasing $B$) without diverging from the current plant.

Trade-off: smaller $\lambda$ → faster tracking, higher noise sensitivity. At $\lambda = 0.998$ and 1 kHz update rate, the effective window is 500 ms — appropriate for slowly varying mechanical parameters.

4. Torque and Kinematics Estimation

Torque: The motor torque is estimated from the q-axis current:

$$\tau_e = \frac{3}{2}\,p\,\psi_f\,i_q = K_T\, i_q$$

The torque constant $K_T = \frac{3}{2}p\psi_f$ must be known from the electrical identification stage or from the motor datasheet.

Speed: Measured directly from the encoder as:

$$\omega[n] = \frac{\theta_m[n] - \theta_m[n-1]}{T_s}$$

Acceleration: Estimated by finite difference of speed:

$$\dot\omega[n] = \frac{\omega[n] - \omega[n-1]}{T_s}$$

The double finite difference amplifies measurement noise. A low-pass filter (e.g. moving average) on $\omega$ before differentiation reduces this effect at the cost of bandwidth.

5. Convergence Criteria

The algorithm has converged to reliable estimates when:

1. Innovation is small: $|e[n]| = |\tau_e[n] - \mathbf{\phi}[n]^T\hat{\mathbf{\theta}}[n]| < \epsilon_e$

   This indicates the model fits the data well (low residual error).

2. Gain trace is small: $\mathrm{tr}(\mathbf{K}[n]) < \epsilon_K$

   This indicates the covariance has collapsed (the estimator is no longer updating significantly), i.e. the estimate has stabilised.

The two thresholds in use: $\epsilon_e = 10^{-4}$ and $\epsilon_K = 10^{-2}$.

The gain trace criterion is a proxy for $\mathrm{tr}(\mathbf{P}[n])$ shrinking, which proves that $\hat{\mathbf{\theta}}[n]$ has converged to a minimum-variance estimate given the data.

6. Observability Requirement

RLS identifies $J$, $B$, and $\tau_0$ simultaneously only if the regressor matrix $\mathbf{\Phi}$ is persistently exciting of order 3 — i.e. the columns $[\dot\omega, \omega, 1]$ are linearly independent over the observation window. In practice this requires:

• $\dot\omega \neq 0$: the motor must accelerate and decelerate (constant-speed operation cannot identify $J$).
• $\omega$ must vary over a sufficient range to separate $B\omega$ from $\tau_0$.
• Both positive and negative torques should appear if possible (else $\tau_0$ is poorly identified).

A typical identification profile: triangular speed reference — ramp up, hold briefly, ramp down — repeated $\ge 2$ times.

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Block Diagrams

RLS Signal Flow

graph TD
    ENC[Encoder θ_m] --> SPD[Speed ω\nfinite difference]
    SPD --> ACC[Acceleration α\nfinite difference]
    IQ[Current i_q] --> TRQ[Torque τ_e\n= K_T · i_q]

    ACC --> REG[Regressor φ\n= [α, ω, 1]]
    SPD --> REG

    REG --> KG[Gain K\n= P·φ / λ + φᵀPφ]
    TRQ --> INN[Innovation e\n= τ_e − φᵀθ̂]
    KG --> UPD[θ̂ update\nθ̂ += K·e]
    INN --> UPD
    REG --> COV[P update\nP = 1/λ·I−Kφᵀ·P]
    KG --> COV

    UPD --> OUT_J[Ĵ = θ̂[1]]
    UPD --> OUT_B[B̂ = θ̂[2]]
    UPD --> OUT_T[τ̂₀ = θ̂[3]]

    COV --> CONV{Convergence\ncheck}
    UPD --> CONV
    CONV -->|tr K < ε_K\nAND &#124;e&#124; < ε_e| DONE[Converged]

Effective Memory Window (Forgetting Factor)

Sample weight w[k] = λ^(N-k)

w │
1.0│ ●
   │  ●
   │    ●
0.5│       ●
   │          ●
   │               ●
   │                    ●
0.0└─────────────────────────────→ k (samples into past)
   N   N-50  N-100  N-200  N-500
            ↑
       N_eff = 500 samples at λ=0.998
       (data older than ~500 samples has weight < 0.37)

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Numerical Properties

Property	Value / Condition
Parameter vector size	3: $[J,\ B,\ \tau_0]^T$
Covariance matrix	$3\times 3$ symmetric positive definite
Forgetting factor	$\lambda = 0.998$
Effective window	$N_{eff} = 1/(1-\lambda) = 500$ samples
Convergence check $\epsilon_e$	$10^{-4}$ (innovation threshold)
Convergence check $\epsilon_K$	$10^{-2}$ (gain trace threshold)
RLS per-step cost	$O(n^2) = O(9)$ — 9 multiply-add ops for $n=3$
Numerical stability	$\mathbf{P}$ can become indefinite; use regularisation if $\lambda < 1$ for long runs
Initialisation	$\mathbf{P}_0 = \alpha_0 \mathbf{I}$, $\hat{\mathbf{\theta}}_0 = \mathbf{0}$

Sensitivity Analysis

Source of Error	Effect
$K_T$ error	All parameters scaled proportionally ($\tau_e$ is biased)
Encoder quantisation	Noise on $\alpha$; use long averaging window
Constant-speed operation	$J$ unidentifiable ($\dot\omega = 0$ makes regressor rank-deficient)
Very short identification time	$\mathbf{P}$ not collapsed; noisy estimates
$\lambda$ too large	Slow convergence to parameter changes
$\lambda$ too small	Noisy, unstable estimates at steady state

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Worked Example

Motor: $J = 2 \times 10^{-4}\ \text{kg·m}^2$, $B = 5 \times 10^{-4}\ \text{N·m·s/rad}$, $\tau_0 = 0.01\ \text{N·m}$, $K_T = 0.3\ \text{N·m/A}$, $f_s = 1\ \text{kHz}$.

Regressor at one sample with $\dot\omega = 50\ \text{rad/s}^2$, $\omega = 100\ \text{rad/s}$, $i_q = 2\ \text{A}$:

$$\mathbf{\phi} = \begin{pmatrix}50\\ 100\\ 1\end{pmatrix}, \quad \tau_e = K_T\, i_q = 0.3 \times 2 = 0.6\ \text{N·m}$$

True output:

$$\mathbf{\phi}^T\mathbf{\theta}_{true} = 50 \times 2\times10^{-4} + 100 \times 5\times10^{-4} + 0.01 = 0.01 + 0.05 + 0.01 = 0.07\ \text{N·m}$$

The difference 0.6 − 0.07 = 0.53 N·m is the load torque $\tau_{load}$, which becomes the innovation residual from the estimator perspective — it will be treated as noise and averaged out over $N_{eff}$ samples.

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Limitations & Assumptions

• Assumes: $\tau_{load}$ is zero mean over the effective window (i.e. no sustained external load that would be attributed to $\tau_0$ or $J$).
• Assumes: Motor torque constant $K_T$ is known a priori (from electrical identification).
• Assumes: The mechanical system is linear in $J$, $B$, $\tau_0$ — nonlinear effects (e.g. position-dependent cogging, Stribeck friction) are not modelled.
• Does not handle: Rapidly varying $J$ (e.g. moving load). The $N_{eff} = 500$ sample window means changes faster than 500 ms update time are tracked with lag.
• Does not handle: Identification in the presence of external vibration or impact loads.

References

1. Ljung, L. — System Identification: Theory for the User, 2nd ed., Prentice Hall, 1999.
2. Åström, K.J. & Wittenmark, B. — Adaptive Control, 2nd ed., Addison-Wesley, 1995.
3. Kim, J.-H. et al. — "Online Parameter Identification of PMSM Using Recursive Least-Squares Algorithm", IEEE International Electric Machines and Drives Conference, 2007.