Multi-Objective Switched MPC

• Multi-objective switched MPC is a control strategy that integrates real-time optimization over conflicting objectives with discrete mode switching for enhanced system performance.
• The framework employs multiple prediction models and supervisory logic to solve parallel optimal control problems, ensuring stability and robustness under switching.
• Applications in robotics, epidemiological control, aircraft, and automotive systems demonstrate practical trade-offs between computational complexity and performance.

Multi-objective switched model predictive control (MO-SMPC) is a class of control strategies that integrate model predictive control (MPC) with both discrete switching between system modes and the simultaneous handling of multiple, potentially conflicting, performance objectives. MO-SMPC frameworks enable real-time optimization over a finite prediction horizon, dynamically adapting system behavior by selecting between multiple predictive controllers (or objective parameterizations), each corresponding to a distinct mode or set of objectives, subject to constraints on state, input, and logical switching. This approach is motivated by the need to achieve complex, multi-faceted performance (e.g., efficiency, safety, economic cost, or task-specific ergonomics) in applications such as process control, robotics, transportation, and epidemiological management.

1. Mathematical Frameworks for Multi-objective Switched MPC

The typical mathematical structure of MO-SMPC involves multiple prediction models or cost scalings, a supervisory switching policy, and dynamic constraints. Consider $M$ discrete-time prediction models:

$$x^i(k+1) = f_i(x^i(k), u^i(k)) \qquad i = 1,\dots, M,$$

with the plant state $\xi(t_k)$ and subsystems' states $x^i(k) = T^i(\xi(t_k))$ at sample $t_k$ (Niepötter et al., 15 Nov 2025). A supervisor selects the active controller $\sigma(k) \in \{1,\ldots,M\}$ at each step $k$, and applies $u(k) = u^{\sigma(k)}(k)$ to the plant.

Each model is associated with its own multi-objective cost, typically of the form:

$$J^i = \sum_{j=k}^{k+N-1} \ell_i\big(x^i(j|k), u^i(j|k)\big) + V_{f,i}\big(x^i(k+N|k)\big),$$

where $\ell_i$ may encode quadratic tracking, economic costs, or problem-specific scalarizations of multiple objectives (Niepötter et al., 15 Nov 2025, Peitz et al., 2016).

Switching may also refer to discrete system logic (e.g., SIR epidemic models (Sereno et al., 2021)), mode-specific objectives in robotics (Haninger et al., 2021), or to explicit, Pareto-driven motion primitive selection (Peitz et al., 2016). Constraints are imposed on states, inputs, and logic variables, and switching logic may itself be subject to feasibility, continuity, or stability requirements.

2. Multi-objective Formulation and Scalarization Approaches

The multi-objective nature of MO-SMPC is realized via separate, weighted costs or Pareto front computation. Standard scalarization approaches include quadratic weightings for different objectives, e.g.,

$$J = \sum_{j=0}^{N-1} \big[Q (S_j - S^*)^2 + R_u u_j^2\big] + P|S_N - S^*|,$$

as in the epidemic management context, where $Q$ penalizes deviation from herd immunity, $R_u$ penalizes intervention burden, and $P$ enforces final susceptible constraints (Sereno et al., 2021). In electric vehicle control (Peitz et al., 2016), explicit offline computation of Pareto fronts is performed over objectives such as energy and time, with real-time switching between trade-offs based on a preference parameter $\rho$.

Another approach leverages mode-specific cost functions $c_n(\cdot)$ corresponding to different task or ergonomic objectives (as in collaborative robots (Haninger et al., 2021)), aggregating the belief-weighted costs for probabilistic mode identification:

$$J_E = \sum_{\tau=t}^{t+H-1} \sum_{n=1}^N b_t[n]\, c_n(\cdot).$$

Risk-sensitive variants employ an exponential-of-quadratic cost aggregation.

3. Switching Logic, Supervisory Architecture, and Feasibility Guarantees

A defining feature of MO-SMPC is the presence of a supervisory scheme that determines when to switch between controllers or cost scalings. In the general switched MPC paradigm (Niepötter et al., 15 Nov 2025), at each time $k$, all candidate controllers solve a "composed" optimal control problem that enforces both optimality for the candidate's own model and compatibility with the restrictor models (other controllers). This "Leader–Restrictor" scheme is critical to guarantee recursive feasibility under arbitrary switching. The supervisor evaluates high-level criteria (e.g., total energy deviation, task completion, safety metrics), selects a controller that satisfies Lyapunov-like dissipativity constraints (see below), and applies its control action.

For systems with logical mode transitions (e.g., epidemic SIR models (Sereno et al., 2021), HRI tasks (Haninger et al., 2021)), switching may be governed by system thresholds or Bayesian inference. In the explicit MPC approach (Peitz et al., 2016), the switch amounts to a lookup of the motion primitive from an offline library corresponding to the current scenario and a preference weighting.

4. Stability, Robustness, and Theoretical Properties

Stability and robustness analysis in MO-SMPC leverages switched-system theory and multiple Lyapunov functions. Under the nominal (disturbance-free) regime, Lyapunov-based switching constraints are enforced:

$$\hat V_i\bigl(x^i(\tilde t^i_{2r+2})\bigr) - \hat V_i\bigl(x^i(\tilde t^i_{2r})\bigr) \leq - W_i\bigl(x^i(\tilde t^i_{2r})\bigr),$$

where $W_i$ is a positive-definite function and $\tilde t^i_{2r}$ is the $r$-th activation time of controller $i$ (Niepötter et al., 15 Nov 2025). This yields asymptotic stability for the closed loop.

In the presence of disturbances, robust MO-SMPC modifies each controller's OCP with a soft initial-state constraint and a penalty, so that a robust Lyapunov function $\overline V_i$ satisfies an input-to-state stability (ISS) dissipation inequality:

$$\overline V_i\bigl(x^i(\tilde t^i_{2r+2})\bigr) - \overline V_i\bigl(x^i(\tilde t^i_{2r})\bigr)\leq -W_i\bigl(x^i(\tilde t^i_{2r})\bigr) + \gamma_i(\|\nu(\tilde t^i_{2r+2})\|),$$

which implies ISS of the switched closed-loop (Niepötter et al., 15 Nov 2025). Recursive feasibility of the control law under arbitrary switching is guaranteed by the invariant terminal set conditions across all controllers.

5. Solution Algorithms and Real-time Implementation

Solving the MO-SMPC problem typically requires parallel (or simultaneous) solution of multiple optimal control problems, each corresponding to a mode, cost scalarization, or objective weighting. Approaches include:

• Parallel Solution: All $M$ candidate OCPs are solved at each time step. In the "composed OCP" of (Niepötter et al., 15 Nov 2025), each controller's problem incorporates restrictor constraints from the other models. Solvers such as CasADi with qpoases or BONMIN are employed for quadratic or mixed-integer nonlinear programming (Sereno et al., 2021, Niepötter et al., 15 Nov 2025).
• Explicit MPC: Offline computation of Pareto fronts and storage of scenario-based primitives enable sub-millisecond online lookup and switching (Peitz et al., 2016).
• Mode-Inference and Gaussian Processes: In physical human-robot interaction, Gaussian processes are used to infer the current discrete mode (e.g., task phase, human intent), updating a belief vector that informs the cost weighting and the model used for MPC optimization. GPs are sparsified for real-time evaluation, and warm-starting of the nonlinear solver is employed to achieve $10$–$15$ Hz control (Haninger et al., 2021).
• Mixed-Integer Programming for Switched SIR Models: In epidemic control applications, the finite set of control actions leads to a mixed-integer nonlinear programming problem solved via branch-and-bound strategies (Sereno et al., 2021).

6. Applications and Performance Trade-offs

MO-SMPC has been demonstrated across a range of domains:

• Epidemiological Management: Switched nonlinear MPC for SIR models enables simultaneous minimization of epidemic final size, social-distancing burden, and compliance with healthcare capacity constraints. Simulations show that more stringent peak infection constraints lead to longer intervention durations but avoid subsequent epidemic waves (Sereno et al., 2021).
• Aircraft Control: Application to a fixed-wing aircraft with switching between high-state-tracking and high-input-penalization MPCs achieves superior nominal and robust energy performance relative to single-MPC baselines (Niepötter et al., 15 Nov 2025).
• Automotive Systems: Offline/online explicit multi-objective MPCs for electric vehicles yield real-time capable controllers that can switch smoothly between fast and energy-saving objectives, closely matching dynamic programming benchmarks at a fraction of the computational cost (Peitz et al., 2016).
• Physical Human-Robot Interaction: Switched MPC with mode-specific costs and GP-based mode inference provides robust, flexible optimization of multiple objectives (task performance, ergonomic criteria) in collaborative robotics scenarios under mode and intent uncertainty (Haninger et al., 2021).

Trade-offs inherent in MO-SMPC design include the computational burden of solving parallel OCPs, the conservativeness of terminal set design (especially with increasing number of modes), and the complexity of supervisor design for stability and constraint satisfaction. Explicit MPCs and online learning-based adaptation are active areas for reducing these burdens.

7. Limitations, Extensions, and Ongoing Research

Current challenges and avenues for extension in MO-SMPC include:

• Terminal Set Computation: Requirement that all terminal constraint sets be invariant under all possible controllers is conservative and computationally intensive as $M$ grows (Niepötter et al., 15 Nov 2025).
• Computational Complexity: Parallel solution of M OCPs per step is demanding in high-dimensional or fast systems.
• Robustness: Methods such as tube-based MPC and soft-constraint formulations are proposed to improve robustness bounds.
• Learning-based Extensions: Integration of data-driven models, learning-based objectives, and adaptive scenario reduction in explicit MPCs is ongoing (Peitz et al., 2016, Niepötter et al., 15 Nov 2025).
• Distributed and Networked Systems: Extension to distributed architectures and networked plants is a topic for further work (Niepötter et al., 15 Nov 2025).
• Soft Switching: Convex blending or soft switching between controller outputs can mitigate discontinuities at controller transitions (Niepötter et al., 15 Nov 2025).

Ongoing research is focused on reducing conservativeness, scaling to high-dimensional and multi-agent systems, and exploiting structure in application domains to maintain real-time feasibility and theoretical guarantees.