Interim Mode Controller Overview

• Interim mode controller is a temporary control strategy designed to stabilize system dynamics during mode transitions or when full feedback is unavailable.
• It employs methodologies such as LP scheduling, LMI-based synthesis, and event-triggered or state-triggered rules to ensure safety and bounded performance.
• This approach is applied in fields like aerospace, robotics, and smart grids to optimize resource use while meeting strict safety and operational constraints.

An Interim Mode Controller is a control system component or architecture designed to maintain stability, safety, or performance during temporary intervals when a system transitions between distinct operating modes or when full state feedback, actuation, or nominal control is momentarily unavailable. This concept appears in a range of contexts including hybrid and switched systems, robotics, aerospace, networked control, and smart grid applications. Interim mode controllers are distinguished by their use of event-driven or state-dependent switching laws, optimality criteria over finite or infinite horizons, or resource-aware scheduling, while often subject to safety constraints and cost minimization objectives.

1. Conceptual Foundations

In the context of hybrid dynamical and switched systems, an interim mode controller operates during intervals when the system is not governed by its nominal or full-mission controller—this could be due to actuator faults, loss of feedback, mode transitions, or resource constraints. It is both a "bridge" and a stabilizer, designed to keep the system state within prescribed safety or performance envelopes while awaiting recovery, reconfiguration, or the reactivation of the primary control law.

Several research threads formalize this notion:

• In Markov jump linear systems with periodic time-varying dynamics, the interim controller is synthesized to guarantee mean-square or almost-sure stability during a fault or switch-induced regime, using Lyapunov relaxations that permit temporary non-monotonicity in the value function (Shrivastava et al., 17 Sep 2024).
• In path following with intermittent state feedback, the interim controller is constructed as part of a switched systems architecture, alternating between observer-based stabilization (when feedback is available) and predictor-based propagation (when feedback is lost), with dwell time constraints explicitly guaranteeing boundedness and stability (Chen et al., 2018).
• In optimal scheduling for linear-rate multi-mode systems, the safe controller is formulated as a sequence of actions (modes and time delays) guaranteed to keep the state within a safe set—effectively serving as an interim law for state maintenance across mode transitions or resource availability changes (Wojtczak, 2013).

2. Core Mechanisms and Theoretical Guarantees

A common thread in interim mode controller design is the mathematical guarantee of bounded state evolution despite transient loss of information, control authority, or full system actuation. The methodologies can be categorized as follows:

Switched or Hybrid System Formulation:

• The system is modeled as a set of subsystems, with dynamic switching between modes (e.g., "feedback available" and "feedback denied"). The switched model allows formal Lyapunov or invariance analysis across modes (Chen et al., 2018).

Frequency Vector and Safe Set Characterization:

• Safety is verified via the existence of an implementable frequency vector $f = (f(m))_{m \in M}$ such that the weighted average derivative at the safe set boundaries does not drive the system out of bounds:

$$F_i(f, l_i) = \sum_m f(m) (b^m_i - a^m_i l_i) \ge 0, \quad F_i(f, u_i) = \sum_m f(m) (b^m_i - a^m_i u_i) \le 0$$

for all variables $i$ (Wojtczak, 2013).

Event-Triggered or State-Triggered Control Laws:

• Control inputs are updated only when a certificate function (Lyapunov $V(x)$, barrier $h(x)$, or an error measure) crosses a threshold, forming periods of controller "inactivity" and "activity." For instance, the controller is turned off once $L_f V(x, e)$ reaches a bound, and turned back on once $V(x)$ nears a performance specification $S(t)$ (Ong et al., 2022).

Optimization and LMI-Based Synthesis:

• The interim controller is the solution to a convex optimization problem, such as an LMI for quadratic cost minimization or region of attraction maximization, subject to relaxed Lyapunov decrease constraints (Shrivastava et al., 17 Sep 2024).

The significance of these frameworks lies in providing sufficient and/or necessary conditions for the existence of a controller that guarantees stability, safety, or bounded cost while the system is in an interim or non-nominal regime.

3. Algorithmic Architectures and Implementation

Several algorithmic strategies have emerged for constructing interim mode controllers:

Approach	Key Steps	Notable Application
Safe Scheduling (LP)	Solve LP for implementable frequency vector; periodic controller cycles through modes using calculated dwell times	HVAC zone control (Wojtczak, 2013)
Switched System	Define observer-based and predictor-based subsystems; enforce min/max dwell time; switching trajectory design	Path following for UAV/quadcopter (Chen et al., 2018)
LMI-Based Synthesis	Formulate mode-dependent state/output controllers with periodic Lyapunov constraints; use convex optimization	Fault-tolerant networked control (Shrivastava et al., 17 Sep 2024)
Event/State Triggered	Use certificate functions for controller on/off triggers; design performance specs (e.g., exponential decay) to bound off intervals	Spacecraft orbit control (Ong et al., 2022), constrained attitude tracking (Lei et al., 2023)

In each case, the design proceeds via (i) model abstraction and system partitioning, (ii) controller law synthesis (either by LP, convex programming, or event-trigger rule), and (iii) simulation-based or analytic validation of safety and performance properties.

For example, the safe scheduling controller for a Linear-Rate Multi-Mode System is built as follows:

1. Define all feasible modes and system parameters.
2. Solve the LP for frequency vectors $f$ as per the safe set boundary derivative conditions.
3. Remove non-implementable modes by checking equilibrium matching.
4. Calculate minimum dwell time $s$ and assign delays $t_k = f(m_k) \cdot s$ in the periodic schedule (Wojtczak, 2013).

4. Resource, Actuation, and Performance Trade-offs

Interim mode controllers are often engineered to optimize resource utilization, such as actuator usage, energy consumption, or computational effort, while maintaining system constraints:

• In spacecraft orbit stabilization, the intermittent controller fires thrusters only when necessary, as determined by a Lyapunov function and a time-varying performance threshold $S(t)$. This reduces fuel consumption while ensuring the state remains within mission-specification bounds (Ong et al., 2022).
• In spacecraft attitude tracking with actuator saturation limits, a composite event-triggered controller adaptively activates actuation to ensure ultimate boundedness, significantly reducing the number of actuation events compared to periodic controllers, yet maintaining high tracking accuracy (Lei et al., 2023).
• In HVAC systems, an optimally scheduled interim controller reduces the peak and average energy consumption by 23% and 17%, respectively, relative to heuristic (lazy) controllers, by selecting minimal mode sets and optimal dwell times (Wojtczak, 2013).

This trade-off between performance (e.g., asymptotic stability, safety, energy efficiency) and resource expenditure is formalized by the inclusion of cost functions—such as peak cost, average cost, or their weighted sum—in controller synthesis, and by explicit thresholds on how long the system can remain in an "interim" or resource-conserving mode.

5. Applications Across Domains

Interim mode controllers have been instantiated in a variety of engineering systems:

• HVAC and Building Energy Management: Controllers ensure zone temperatures remain within comfort intervals while minimizing cost and reducing actuator cycling (Wojtczak, 2013).
• Spacecraft Attitude and Orbit Control: Event-triggered and composite-triggered intermittent controllers are applied to minimize thruster firings while maintaining (set) stability or safety despite actuation constraints (Ong et al., 2022, Lei et al., 2023).
• Robotic Manipulation: Reference spreading and interim blending modes enable dual-arm robots to manage planned and unplanned simultaneous impacts without input spikes (Steen et al., 2023).
• Autonomous Vehicles: Switched systems approaches to intermittent state feedback enable safe path following even during periods of lost feedback, with guarantees provided by Lyapunov-based dwell time constraints (Chen et al., 2018).
• Fault-Tolerant Control in Networked Systems: Robust LMI-based interim controllers bridge to safe and stable operation during actuator failures or abrupt mode changes (Shrivastava et al., 17 Sep 2024).

Each application demands domain-specific adaptations (e.g., compensating for actuator saturation, handling multi-modal dynamics, or integrating resource-aware certificates), but the unifying element is maintaining safety or regulated performance during temporary periods when full nominal control is unavailable.

6. Limitations, Open Problems, and Directions

Several limitations and research challenges persist:

• Transient Overshoots: In some practical instances, such as tiltrotor transition control using gain-scheduled PID, simulations reveal thrust discontinuities or performance degradations at the switching interface—suggesting the need for smoother blending laws or finer gain scheduling (Kirst et al., 11 Nov 2024).
• Modeling Assumptions: The validity of simplified assumptions—such as uniform rotor tilt or omission of tilt rate effects—must be carefully assessed, as real systems may exhibit more complex or coupled behavior, limiting the efficacy of interim controllers based on such models (Kirst et al., 11 Nov 2024).
• Trade-offs in Stability Guarantees: Relaxed Lyapunov criteria (mean decrease over a period rather than at each step) enlarge the region of attraction for interim controllers but could theoretically allow transient excursions that challenge constraint satisfaction if not sharply designed (Shrivastava et al., 17 Sep 2024).
• Adaptive and Learning Integration: The integration of learning-based or adaptive mechanisms—e.g., data-driven local mode identification, reference-free LMPC, or kernel-based system identification—remains an active area, with ongoing work focused on robustly handling unknown or uncertain interim modes under data limitations (Kopp et al., 8 Jul 2024).
• Implementation Complexity and Verification: Recursive or patching-based Lyapunov analyses, optimization with exponentially many possible mode combinations, and event-triggered scheduling can introduce computational or verification challenges that may inhibit real-time adoption in some systems.

7. Summary Table: Interim Mode Controller Frameworks

Paper / Framework	System Type	Interim Mode Realization	Stability/Safety Guarantee	Resource/Performance Aspect
(Wojtczak, 2013)	Linear-rate multi-mode	Frequency vector, LP-based scheduling	Safe set invariance	Min peak/avg cost, polynomial time
(Chen et al., 2018)	Switched nonlinear	Observer/predictor pairing, dwell time rules	Lyapunov-based, composite error	Path following with feedback loss
(Ong et al., 2022, Lei et al., 2023)	Spacecraft/Attitude	Event/composite trigger, Lyapunov/barrier function	UUB or forward invariance	Min actuator firing, constrained
(Shrivastava et al., 17 Sep 2024)	Markov jump/periodic	LMI-based controller, periodic Lyapunov decrease	Mean-square, region of attraction	Cost and region maximization
(Kopp et al., 8 Jul 2024)	Data-driven multi-modal	Kernelized local model selection, safe set construction	LMPC, safe set, terminal const.	Data efficiency, adaptation
(Steen et al., 2023)	Dual-arm robot/impacts	Reference spreading, interim blending	Mass-spring-damper-like, input-to-state	Impact robustness, continuous input

• (Wojtczak, 2013) "Optimal Scheduling for Linear-Rate Multi-Mode Systems"
• (Chen et al., 2018) "A Switched Systems Approach to Path Following with Intermittent State Feedback"
• (Ong et al., 2022) "Stability and Safety through Event-Triggered Intermittent Control with Application to Spacecraft Orbit Stabilization"
• (Lei et al., 2023) "Composite Triggered Intermittent Control for Constrained Spacecraft Attitude Tracking"
• (Kopp et al., 8 Jul 2024) "Data-Driven Multi-Modal Learning Model Predictive Control"
• (Shrivastava et al., 17 Sep 2024) "Robust Controller Synthesis under Markovian Mode Switching with Periodic LTV Dynamics"
• (Steen et al., 2023) "Quadratic Programming-based Reference Spreading Control for Dual-Arm Robotic Manipulation with Planned Simultaneous Impacts"
• (Kirst et al., 11 Nov 2024) "Mode transition control of large-size tiltrotor aircraft"

These sources collectively establish the theoretical underpinning, algorithmic structures, performance-securing strategies, and practical instantiations that define the modern concept of the interim mode controller in technical systems.