⚡ electrical engineering

Predictor-Feedback Stabilization of Linear Switched Systems with State-Dependent Switching and Input Delay

This paper proposes a predictor-feedback control design for linear switched systems with state-dependent switching and input delay, utilizing a novel exact predictor state construction and multiple Lyapunov functionals to establish uniform exponential stability.

Original authors: Andreas Katsanikakis, Nikolaos Bekiaris-Liberis, Delphine Bresch-Pietri

Published 2026-03-23

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CC BY 4.0

Original authors: Andreas Katsanikakis, Nikolaos Bekiaris-Liberis, Delphine Bresch-Pietri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are driving a car, but there's a strange twist: your steering wheel controls the car's direction, but the car doesn't react until 5 seconds later.

Now, add a second twist: the rules of the road change depending on where you are. If you are in the "city zone," you must drive slowly and turn sharply. If you are in the "highway zone," you must drive fast and stay straight. The problem is, you don't know which zone you will be in 5 seconds from now, because that depends on where you steer right now.

This is the exact nightmare scenario described in the paper by Katsanikakis, Bekiaris-Liberis, and Bresch-Pietri. They are dealing with complex machines (called Linear Switched Systems) that have two major problems:

Input Delay: The machine takes time to react to your commands.
State-Dependent Switching: The machine's internal rules change automatically based on its current position or speed.

The Problem: The "Blind" Driver

Most existing control methods are like a driver who only looks at the rearview mirror. They see where the car was and try to correct it. But if the car reacts 5 seconds late, and the road rules change based on where the car will be, looking in the rearview mirror isn't enough. You will crash because you are reacting to the past, not the future.

The Solution: The "Crystal Ball" (Predictor Feedback)

The authors invented a new way to control these systems. Think of it as giving the driver a Crystal Ball.

Instead of just looking at where the car is now, their method calculates exactly where the car will be 5 seconds from now, assuming the current plan continues.

The Magic Trick: Usually, predicting the future is impossible because you don't know which "mode" (city vs. highway) the car will switch into. But because the rules for switching are based on the car's own future position, the authors found a mathematical way to solve this "chicken and egg" problem.
The Result: They created a formula that acts like a crystal ball. It says, "If I steer this way, the car will enter the 'city zone' at 2:03 PM, so I must switch my steering logic now to prepare for that."

How They Proved It Works (The Safety Net)

In engineering, you can't just say, "It looks like it works." You have to prove it mathematically.

The Backstepping Metaphor: Imagine you are trying to balance a long pole on your hand. It's hard. But if you imagine the pole is actually a series of smaller, easier-to-balance poles stacked on top of each other, you can solve the problem step-by-step. The authors used a technique called Backstepping to break the complex, delayed system into simpler, manageable pieces.
The Multiple Energy Maps: To prove the car won't crash, they drew several "energy maps" (Lyapunov functions). Imagine a landscape with hills and valleys. The goal is to make sure the car always rolls down into a valley (stability) and never climbs a hill (instability). They proved that no matter how the car switches between "city" and "highway" modes, these maps ensure the car always stays on a path that leads to a safe stop.

The Real-World Test

They tested their "Crystal Ball" controller on a computer simulation of a two-mode system.

The Scenario: They used a switching rule inspired by communication networks (like Wi-Fi routers deciding which device gets to send data).
The Outcome: Even with a significant delay (1 second in the simulation, which is huge in control theory), the system stabilized perfectly. The car didn't crash; it smoothly navigated the changing rules and the delayed reaction.

Why This Matters

This paper is a big deal because it solves a problem that was previously thought to be very difficult or impossible to solve with simple, exact math.

Before: Engineers had to guess or use very restrictive rules to handle delays and changing modes.
Now: They have a precise "recipe" (predictor feedback) that can handle long delays and automatic rule changes, provided the rules depend on the system's state.

In a nutshell: The authors gave a "blind" driver a crystal ball and a set of safety maps, allowing them to drive a car that reacts slowly and changes its own driving rules based on where it's going, without ever crashing. This could lead to better autonomous vehicles, more stable power grids, and smarter communication networks.

1. Problem Statement

The paper addresses the control of linear switched systems subject to two simultaneous challenges:

State-Dependent Switching: The system dynamics switch between different linear modes based on the current state $X(t)$ , rather than a time-dependent signal. The state space is partitioned into disjoint regions $\{\Omega_i\}$ , where the active mode $\sigma(X)$ is determined by which region the state occupies.
Arbitrarily Long Input Delay: There is a constant, potentially long delay $D$ in the control input $U(t-D)$ .

The Core Challenge:
Existing literature typically handles either input delays in time-dependent switching systems or state delays in state-dependent systems. The combination of state-dependent switching and input delay creates a unique difficulty: the future evolution of the system (and thus the future switching mode) depends on the future state, which is unknown. This makes the construction of a standard predictor state (which requires knowing future system parameters) an implicit problem, similar to nonlinear systems with input delay.

2. Methodology

The authors propose a Predictor-Feedback Control Design that achieves complete delay compensation. The methodology consists of three main pillars:

A. Construction of the Exact Predictor State

Since the switching signal $\sigma$ depends on the state, the predictor state $P(\theta)$ (representing the future state at time $t+\theta$ ) satisfies an implicit integral equation:
$P(\theta) = X(t) + \int_{t-D}^{\theta} [A_{\sigma(P(s))}P(s) + B_{\sigma(P(s))}U(s)] ds$
Because the mode $\sigma(P(s))$ depends on the unknown future trajectory $P(s)$ , this is implicit. The authors derive two ways to handle this:

Implicit Definition: Treating the system as a nonlinear system where the switching logic is embedded in the integral.
Semi-Explicit Formula: By leveraging the fact that the system is linear within each mode, the authors derive a recursive formula (Eq. 14). This involves:
- Starting from the current state $X(t)$ .
- Propagating the trajectory forward in time using the current mode's dynamics.
- Detecting when the trajectory leaves the current region $\Omega_i$ (a switching event).
- Switching to the new mode's dynamics and continuing the propagation until the delay horizon $D$ is reached.
- This results in a piecewise exponential solution that can be computed algorithmically.

B. Control Law

The control input is designed using the computed predictor state:
$U(t) = K_{\sigma(P(t))} P(t)$
where $K_i$ are feedback gains designed for the delay-free system in each mode $i$ .

C. Stability Analysis via Backstepping and Multiple Lyapunov Functionals

To prove stability, the authors employ a Backstepping Transformation:

Transformation: They define a target system variable $W(\theta) = U(\theta) - K_{\sigma(P(\theta))}P(\theta)$ . This transforms the original delayed system into a target system where the delay term is isolated in a stable manner ( $W(t)=0$ for $t \ge 0$ ).
Multiple Lyapunov Functionals (MLF): They construct a family of Lyapunov functionals $V_i(t)$ $V_{i} (t)$ for each mode.
- The functional includes a quadratic term for the state and an integral term for the distributed delay (the $W$ variable).
- Key Condition: They impose a continuity condition on the Lyapunov matrices at the switching surfaces ( $\Omega_{i,j}$ ), ensuring $V_i(X) = V_j(X)$ when switching occurs. This prevents the Lyapunov function from jumping up at switching instants, which is crucial for stability in switched systems.
Norm Equivalence: They prove that the norms of the original system variables $(X, U)$ and the transformed variables $(X, W)$ are equivalent, allowing stability results from the target system to be mapped back to the original system.

3. Key Contributions

Novel Predictor Construction: The paper provides the first exact predictor construction for linear switched systems with state-dependent switching and input delay. It bridges the gap between linear predictor methods (usually for time-dependent switching) and nonlinear implicit predictor methods.
Semi-Explicit Implementation: While the predictor is theoretically implicit, the authors provide a computable, semi-explicit algorithm that exploits the linearity of the subsystems, making the design implementable.
Stability Proof for State-Dependent Switching with Delay: The authors establish uniform exponential stability for the closed-loop system. This is achieved by combining backstepping transformations with a novel construction of multiple Lyapunov functionals that account for both the input delay and the state-dependent switching logic.
Relaxation of Assumptions: Unlike many existing works that restrict delay length or require time-dependent switching, this method handles arbitrarily long delays and state-dependent switching rules.

4. Results

Theoretical: Theorem 3.1 proves that the closed-loop system is uniformly exponentially stable. Specifically, the state $X(t)$ and the control input energy decay exponentially to zero, regardless of the initial conditions and the length of the delay $D$ .
Simulation:
- A two-mode system was simulated with a switching rule motivated by communication networks (specifically the "Try-Once-Discard" protocol, where the mode with the highest Lyapunov value is selected).
- The simulation used a delay $D=1$ second.
- Outcome: The state trajectories converged to zero, and the control input successfully compensated for the delay. The switching signal behaved as expected, switching between modes based on the state trajectory.
- The authors compared the implicit numerical integration method (used in simulation for speed) with the semi-explicit formula, noting that both are viable depending on system complexity.

5. Significance

This work is significant because it addresses a practical control problem where delays and switching are coupled by the system state, a scenario common in:

Vehicle Control: Switching between throttle and braking based on speed/acceleration with actuator delays.
Epidemic Modeling: Implementing quarantine measures based on infection counts with policy implementation lags.
Networked Control Systems: Protocol switching based on network congestion or state, subject to communication delays.

By providing a rigorous stability guarantee and a constructive design method, this paper fills a critical gap in the literature, enabling the control of complex hybrid systems that were previously difficult to stabilize due to the combination of long delays and state-dependent logic.

Future Work

The authors plan to extend this methodology to a broader class of systems using a hybrid systems framework to relax the current restrictions on the switching signal form. They also intend to conduct a deeper analysis of the computational complexity and accuracy trade-offs between the implicit and semi-explicit predictor implementations.