MAPS: Mode-Aware Probabilistic Scheduling

• MAPS is an adaptive control framework that leverages an IMM estimator to infer discrete friction modes without explicit measurements.
• It integrates LPV control with Bayesian mode inference to interpolate control gains smoothly for brushed DC motors under varying friction conditions.
• Validated via HILS on a QUBE-Servo 2, MAPS shows enhanced state estimation accuracy and improved transient performance over fixed-gain methods.

Mode-Aware Probabilistic Scheduling (MAPS) is an adaptive control framework that couples an Interacting Multiple Model (IMM) estimator with a Linear Parameter-Varying (LPV) control law and uses updated mode probabilities directly as interpolation weights for online gain synthesis. In the formulation introduced for a brushed DC motor with varying friction, MAPS treats friction-induced operating conditions as discrete modes, performs Bayesian mode inference in real time, and schedules controller gains probabilistically rather than through a measured deterministic scheduling variable. The resulting architecture is intended for friction-aware adaptive control in uncertain, time-varying environments and was validated in a Hardware-in-the-Loop Simulation (HILS) setting on a QUBE-Servo 2 platform (Kim et al., 16 Sep 2025).

1. Defining mechanism

The defining feature of MAPS is the replacement of traditional deterministic LPV scheduling weights with posterior mode probabilities generated online by an IMM estimator. In the DC motor formulation, the controller does not rely on explicit identification of the viscous friction coefficient at runtime. Instead, it infers the probability of each friction mode and forms a convex combination of precomputed vertex gains:

$K_k = \sum_{i=1}^M \mu_k^{(i)} K^{[i]}.$

For regulation, the control law is

$u_k = -K_k x_k,$

and for tracking, which was used in the experiments,

$$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$

The sign convention was chosen to match the hardware controller (Kim et al., 16 Sep 2025).

This construction tightly couples estimation and control. Classical LPV scheduling uses a measured or computed parameter $\rho_k$ and associated weights $\xi_i(\rho_k)$, whereas MAPS uses $\mu_k^{(i)}$ inferred from the IMM. The distinction is operationally important: the framework adapts to latent, unmeasured changes such as friction variation without explicit friction model identification. It also differs from hard switching among mode-specific controllers, because gains are blended continuously rather than selected by winner-take-all logic. This suggests a design emphasis on smooth adaptation during mode transitions rather than abrupt controller replacement.

2. System model and LPV representation

The reference application is a brushed DC motor with armature voltage input $v$ and states angular position $\theta$, angular velocity $\omega$, and armature current $i$. The coupled electrical and mechanical dynamics are

$u_k = -K_k x_k,$0

where $u_k = -K_k x_k,$1, $u_k = -K_k x_k,$2, $u_k = -K_k x_k,$3, $u_k = -K_k x_k,$4, and $u_k = -K_k x_k,$5 are motor parameters and $u_k = -K_k x_k,$6 is the viscous friction coefficient. With

$u_k = -K_k x_k,$7

the continuous-time state-space model is

$u_k = -K_k x_k,$8

with

$u_k = -K_k x_k,$9

Variations in $$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$0, for example due to temperature or load, induce distinct operating modes. The MAPS implementation uses two representative friction modes identified experimentally as $$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$1 and $$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$2 (Kim et al., 16 Sep 2025).

After forward-Euler discretization with sampling time $$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$3,

$$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$4

where

$$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$5

The scheduling parameter $$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$6 represents the viscous friction coefficient at time $$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$7. In polytopic LPV form,

$$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$8

with vertices $$e_k = x_{\mathrm{ref},k} - x_k,\qquad u_k = K_k e_k.$$9 and $\rho_k$0 for $\rho_k$1 (Kim et al., 16 Sep 2025).

For each mode $\rho_k$2, the process and measurement noises are zero-mean Gaussian:

$\rho_k$3

In the HILS validation, identical covariances were used for both modes:

$\rho_k$4

The two-mode construction is minimal rather than exhaustive. A plausible implication is that MAPS depends on whether the chosen vertex set captures the dominant friction variation relevant to control performance.

3. IMM estimation and probabilistic scheduling of gains

The estimator layer uses an IMM with $\rho_k$5 modes corresponding to $\rho_k$6 and $\rho_k$7. Mode transitions are modeled by a Markov matrix

$\rho_k$8

In the experiments, the self-transition probabilities were set to $\rho_k$9 and the switching probabilities to $\xi_i(\rho_k)$0, so $\xi_i(\rho_k)$1 and $\xi_i(\rho_k)$2 (Kim et al., 16 Sep 2025).

Given previous mode probabilities $\xi_i(\rho_k)$3, the IMM forms one-step-ahead predicted mode probabilities

$\xi_i(\rho_k)$4

and conditional mixing weights

$\xi_i(\rho_k)$5

These define the mixed initial state and covariance for each mode-conditioned Kalman filter. For each mode $\xi_i(\rho_k)$6, prediction and update are carried out with the model $\xi_i(\rho_k)$7, using innovation $\xi_i(\rho_k)$8, innovation covariance $\xi_i(\rho_k)$9, and gain $\mu_k^{(i)}$0. The corresponding likelihood is

$\mu_k^{(i)}$1

where $\mu_k^{(i)}$2 (Kim et al., 16 Sep 2025).

Posterior mode probabilities are then computed by Bayesian normalization:

$\mu_k^{(i)}$3

The fused IMM state estimate is

$\mu_k^{(i)}$4

with a corresponding covariance fusion formula that includes the within-mode covariances and the spread of mode-conditioned estimates. MAPS uses these updated $\mu_k^{(i)}$5 immediately as convex weights for control gain interpolation (Kim et al., 16 Sep 2025).

A common misunderstanding is to treat MAPS as an estimator of a continuous friction parameter. The formulation in fact operates through discrete friction modes and posterior mode probabilities. Another common misunderstanding is to equate probabilistic scheduling with mode switching. MAPS explicitly contrasts its convex blending with hard mode switching and attributes to the latter the risk of chattering and transient performance degradation during mode transitions.

4. Gain synthesis, constraints, and stability guarantees

The controller gains at the LPV vertices are synthesized by discrete-time LQR on the vertex models $\mu_k^{(i)}$6 with infinite-horizon cost

$\mu_k^{(i)}$7

where $\mu_k^{(i)}$8 and $\mu_k^{(i)}$9. The discrete algebraic Riccati equation yields $v$0, and the vertex gain is

$v$1

In the HILS implementation,

$v$2

Voltage is limited by the QUBE-Servo 2 hardware, reference amplitudes were chosen in $v$3 V, and saturation is handled by clamping the computed $v$4. No integral action is present, so anti-windup is not needed (Kim et al., 16 Sep 2025).

The stability analysis is framed through a common Lyapunov argument. The lemma stated for the vertex-closed loops is: if there exists $v$5 such that for each vertex $v$6,

$v$7

then any convex combination

$v$8

is quadratically stable for all admissible $v$9 on the probability simplex. The associated theorem states that with $\theta$0 generated by the IMM and $\theta$1, the closed loop

$\theta$2

is quadratically stable under the lemma’s conditions. The paper further states that, under bounded parameter estimation error and Lipschitz continuity of $\theta$3 and $\theta$4, MAPS preserves exponential stability when $\theta$5 and $\theta$6 are sufficiently small (Kim et al., 16 Sep 2025).

These guarantees are structurally important because they formalize the status of probability-weighted interpolation as a closed-loop design mechanism rather than a heuristic. The stability claim is not attached to a specific friction identification procedure; it is attached to convexity of the closed loop and the existence of a common Lyapunov certificate across the vertices.

5. Algorithmic workflow, HILS results, and implementation

The per-sample workflow is specified as: measure $\theta$7; perform IMM interaction to compute $\theta$8, $\theta$9, and mixed $\omega$0; run mode-conditioned Kalman prediction and update; evaluate likelihoods $\omega$1 and posterior probabilities $\omega$2; fuse the state estimate; interpolate the gain; compute $\omega$3 and $\omega$4; apply saturation; and send $\omega$5 to the plant (Kim et al., 16 Sep 2025).

The HILS platform consists of a QUBE-Servo 2 DC motor with optical encoder and amplifier, operated at a sampling rate of $\omega$6 Hz with $\omega$7 ms. Signals recorded were $\omega$8, $\omega$9, $i$0, and $i$1. Friction was emulated through a minimum-friction condition with no load and a maximum-friction condition with manual external load. The computational platform was MATLAB/Simulink Real-Time, and the reported per-step latency at $i$2 Hz was negligible (Kim et al., 16 Sep 2025).

State-estimation accuracy in HILS was reported through RMSE as follows.

State	MAPS RMSE	KF RMSE
$i$3 [rad]	0.0014	0.0038
$i$4 [rad/s]	0.5725	0.9825
$i$5 [A]	0.0223	0.0523

These correspond to an approximately $i$6 reduction for $i$7, approximately $i$8 for $i$9, and approximately $u_k = -K_k x_k,$00 for $u_k = -K_k x_k,$01 relative to the KF baseline (Kim et al., 16 Sep 2025).

Tracking performance was compared between MAPS-gain LQR and fixed-gain LQR. For a step input with no load, the two methods showed similar performance, with fixed-gain slightly lower IAE by approximately $u_k = -K_k x_k,$02. For a sine input with no load, MAPS improved RMSE to $u_k = -K_k x_k,$03 versus $u_k = -K_k x_k,$04 and IAE to $u_k = -K_k x_k,$05 versus $u_k = -K_k x_k,$06. For a step input with load, MAPS reduced IAE to $u_k = -K_k x_k,$07 versus $u_k = -K_k x_k,$08 and MAE to $u_k = -K_k x_k,$09 versus $u_k = -K_k x_k,$10. For a sine input with load, MAPS reduced RMSE to $u_k = -K_k x_k,$11 versus $u_k = -K_k x_k,$12, MAE to $u_k = -K_k x_k,$13 versus $u_k = -K_k x_k,$14, and IAE to $u_k = -K_k x_k,$15 versus $u_k = -K_k x_k,$16. Transient responses showed reduced overshoot and faster settling under friction changes with MAPS-gain LQR (Kim et al., 16 Sep 2025).

Implementation details reported for tuning and feasibility include $u_k = -K_k x_k,$17, raw IMM probabilities without optional smoothing, and complexity $u_k = -K_k x_k,$18 per step for KF factorizations. With $u_k = -K_k x_k,$19 and $u_k = -K_k x_k,$20, computations were described as trivial, and the framework ran comfortably at $u_k = -K_k x_k,$21 Hz with low latency. Practical conditions include observability and controllability of both vertex models, sensitivity of adaptation speed to the choice of $u_k = -K_k x_k,$22, and the usual estimator trade-off that larger $u_k = -K_k x_k,$23 increases adaptability but may amplify noise while larger $u_k = -K_k x_k,$24 prioritizes the model over measurements (Kim et al., 16 Sep 2025).

6. Generalization, limitations, and related uses of probabilistic scheduling

Within the LPV-adaptive-control setting, MAPS is presented as applicable to systems with mode-dependent dynamics, including varying damping in mechanical systems, aerodynamic regimes in flight control, tire-road friction in vehicles, and other hybrid or LPV plants. The stated prerequisites are a set of representative vertex models and stabilizing gains for each. The reported limitations are equally specific: limited vertex coverage may reduce performance if the true dynamics fall outside the polytopic hull; rapid, frequent switching can challenge the IMM and cause lag in $u_k = -K_k x_k,$25; poorly tuned $u_k = -K_k x_k,$26 or $u_k = -K_k x_k,$27 can degrade estimation; and weakly observable modes can produce unreliable probabilities, in which case additional sensing or richer models may be needed (Kim et al., 16 Sep 2025).

The term “probabilistic scheduling” also appears in distinct but related control and decision-making contexts. In networked control systems, a probabilistic scheduling algorithm was proposed for plants sharing a communication network of limited capacity. That framework models each plant as a two-mode Markovian jump linear system with connected/closed-loop and disconnected/open-loop modes, constructs disjoint subsets of plants of cardinality equal to the network capacity, assigns stationary probabilities to those subsets, and derives necessary and sufficient LMIs for mean-square stability of the resulting homogeneous Markov chains (Singh et al., 2022). The paper explicitly notes that it does not use the term MAPS, but its design is described as intrinsically mode-aware through MJLS characterization and probabilistic scheduling of network access.

A further development appears in fault-aware communications scheduling for robotic fleets, where an IMM-MPC framework maintains a probabilistic belief over discrete fault modes with time-inhomogeneous dynamics and optimizes a two-term objective coupling acquisition value with information gain. In that setting, mode-aware probabilistic scheduling addresses observational aliasing, such as failed contacts that cannot distinguish a lethal hardware fault from a benign communications outage under standard actions. The framework uses belief updates over discrete modes,

$u_k = -K_k x_k,$28

followed by Bayesian updating under action-conditioned observation likelihoods, and was reported to recover $u_k = -K_k x_k,$29 of lethal-fault spacecraft versus $u_k = -K_k x_k,$30 for binary-MPC in a rideshare scenario and $u_k = -K_k x_k,$31 versus $u_k = -K_k x_k,$32 in a megaconstellation scenario, across $u_k = -K_k x_k,$33 randomized trials (Schreiber et al., 9 May 2026).

These related uses clarify a terminological point. In LPV-based adaptive control, MAPS denotes probabilistic gain scheduling from inferred operating modes; in networked control, probabilistic scheduling refers to stochastic allocation of communication access; in fleet communications, it denotes receding-horizon action scheduling under mode uncertainty. The shared structure is mode awareness, probabilistic inference or probabilistic allocation, and a control or scheduling law that depends explicitly on those probabilities.