Input Matrix Optimization for Desired Reachable Set Warping of Linear Systems

1. Problem Statement

The paper addresses a fundamental challenge in control design: shaping the reachable set of a dynamical system to satisfy specific performance or safety requirements.

• Context: In many engineering systems, the system dynamics ($A$) and control constraints ($U$) are fixed, but the input matrix ($B$) remains a design variable (e.g., actuator placement or gain selection).
• Objective: Given a linear system $\dot{X} = AX + Bu$ and a specific direction of interest $d$, find the optimal input matrix $B^*$ within a compact admissible set $\mathcal{B}$ that maximizes the expansion (or minimizes it for safety) of the reachable set along $d$.
• Metric: The authors define a Directional Growth Metric, $G_d(B) = d^\top (X_{d,B} - c_0)$, where $X_{d,B}$ is the boundary point of the reachable set in direction $d$, and $c_0$ is the zero-input trajectory endpoint. Maximizing $G_d(B)$ corresponds to maximizing the reachability in direction $d$.

2. Methodology

The approach relies on Pontryagin's Maximum Principle (PMP) to link the geometry of the reachable set boundary to optimal control theory.

A. Reachable Set Boundary Computation

The boundary of the reachable set $\partial \mathcal{R}(T, X_0)$ is determined by solving time-optimal control problems.

• Using PMP, the boundary trajectories are generated by integrating the system dynamics and costate equations backward/forward with a terminal costate condition $P(T) = d$.
• The optimal control $u^*(t)$ at any instant maximizes the Hamiltonian $H = P(t)^\top (AX + Bu)$.

B. Theoretical Reduction (The Main Result)

The core contribution is a theorem stating that under specific structural assumptions, the complex joint optimization over $B$ and the control input $u(t)$ reduces to a finite number of linear optimization problems.

Assumptions:

1. The system matrix $A$ has real eigenvalues.
2. The direction of interest $d$ is an eigenvector of $A^\top$.

Derivation Steps:

1. Costate Structure: Under the assumptions, the costate $P(t)$ remains collinear with $d$ for all time $t \in [0, T]$. Specifically, $P(t) = \lambda(t)d$, where $\lambda(t)$ is a scalar function.
2. Hamiltonian Maximization: Since $P(t)$ is proportional to $d$, maximizing the Hamiltonian $H$ with respect to $u$ becomes equivalent to maximizing $d^\top B u$.
3. Vertex Property: Since the control set $U$ is a convex polytope with vertices $\{u_i\}_{i=1}^N$, the optimal control $u^*$ at any time must be one of these vertices.
4. Decoupling: The problem reduces to finding a single matrix $B$ that maximizes the Hamiltonian at $t=0$ (since $H$ is constant along the trajectory). This leads to a finite set of candidates:
   • For each vertex $u_i$ of $U$, solve a linear optimization: $B_i = \arg\max_{B \in \mathcal{B}} P_0^\top B u_i$, where $P_0 = e^{A^\top T}d$.
   • Select the best pair $(B_i, u_i)$ that yields the global maximum.

Algorithm: The proposed Algorithm 1 solves $N$ linear optimization problems (one per vertex of $U$) and selects the best $B^*$ among the resulting candidates.

C. Handling Relaxed Assumptions

The paper investigates cases where the assumptions are violated:

• Complex Eigenvalues: If $A$ has complex eigenvalues, $P(t)$ rotates, causing the system to expand in multiple directions simultaneously. The algorithm still provides a "directionally informed" solution, though not globally optimal for a single direction.
• Non-Eigenvector Direction: If $d$ is not an eigenvector, the strict proportionality of $P(t)$ to $d$ breaks. However, the terminal condition $P(T)=d$ ensures the solution remains biased toward the desired direction.

3. Key Contributions

1. Theoretical Reduction: Proved that for linear systems satisfying specific spectral properties, the infinite-dimensional optimization of a control trajectory and the matrix $B$ reduces to a finite set of linear optimizations.
2. Constructive Algorithm: Provided Algorithm 1, which offers a computationally efficient method to find the optimal $B^*$ without iterative nonlinear search or simulation-based heuristics.
3. Directional Warping Framework: Formalized the concept of "warping" the reachable set, distinguishing between expanding performance capabilities and contracting safety boundaries.
4. Heuristic Extension: Demonstrated that the method remains effective even when theoretical assumptions (real eigenvalues, eigenvector alignment) are relaxed, offering a robust heuristic for general linear systems.

4. Results and Validation

The authors validated the approach on two distinct systems:

• Case 1: ADMIRE Fighter Jet Model (Linearized)

  • System: 3-state (roll, pitch, yaw rates) with 4 control inputs.
  • Properties: $A$ has real eigenvalues; the target direction $d$ (e.g., roll rate) is an eigenvector of $A^\top$.
  • Outcome: The algorithm successfully expanded the reachable set along the roll axis (increasing maneuverability) and contracted it (for safety) by switching between $\arg\max$ and $\arg\min$. The results matched the theoretical guarantees of Theorem 1.

• Case 2: Damped Oscillator

  • System: 2-state system with complex eigenvalues (violating Assumption 1).
  • Outcome: As predicted by the heuristic analysis, the reachable set expanded in multiple directions simultaneously rather than strictly along $d$. However, the expansion was still significant in the desired direction, demonstrating the method's utility even outside the strict theorem regime.

5. Significance and Future Directions

• Significance: This work bridges the gap between control synthesis (designing controllers) and system design (designing the plant/actuators). It provides a rigorous mathematical tool to optimize hardware configuration (input matrix) to guarantee specific reachability properties, which is crucial for safety-critical systems (e.g., preventing entry into unsafe regions) and performance-critical systems (e.g., maximizing maneuverability).
• Future Work:
  • Extending the optimization to the system matrix $A$ (joint optimization of $A$ and $B$).
  • Adapting the framework to nonlinear systems, where costate dynamics lack closed-form solutions, potentially requiring numerical optimal control or learning-based methods.

In summary, the paper presents a novel, computationally efficient method to "warp" the reachable set of linear systems by optimizing the input matrix, offering both rigorous guarantees under specific conditions and robust heuristics for general cases.