Safe Adaptive Control of Parabolic PDE-ODE Cascades

Technical Summary: Safe Adaptive Control of Parabolic PDE-ODE Cascades

Problem Formulation
This paper addresses the control of a class of cascaded systems consisting of a parabolic partial differential equation (PDE) and an ordinary differential equation (ODE). The plant is modeled by a linear ODE in controllable canonical form coupled with a parabolic PDE featuring diffusion and reaction terms. A critical challenge addressed is the presence of parametric uncertainties in both the PDE subsystem (specifically the reaction coefficient $\lambda$) and the ODE subsystem (specifically the input gain $b$).

The control objective is twofold:

1. Stabilization: Achieve convergence of all plant states (both the PDE state $u(x,t)$ and the ODE state $Y(t)$) to zero.
2. Safety: Enforce state constraints defined by a barrier function $h(y_1(t), t) \geq 0$, where $y_1$ is the first state of the ODE. The controller must ensure that if the system starts in a safe set, it remains safe; if it starts in an unsafe set, it is driven back into the safe set within a user-prescribed finite time.

Methodology
The proposed solution integrates Adaptive Control Barrier Functions (aCBFs) with a Backstepping design and a Batch Least-Squares Identification (BaLSI) scheme.

1. Nominal Safe Control Design:

   • Coordinate Transformation: The ODE is transformed into a chain of integrators using a linear transformation matrix $T_z$. A high-relative-degree CBF framework is then applied to the transformed ODE states ($z_i$) to construct a sequence of barrier functions $h_i$.
   • Handling Initial Unsafe States: A time-varying auxiliary function $\sigma(t)$ is introduced. If the initial state is unsafe ($h \leq 0$), $\sigma(t)$ creates a temporary barrier that guides the state back to the safe region within a preassigned time $t_a$.
   • PDE Backstepping: A backstepping transformation is employed to map the original unstable PDE-ODE cascade into a target system. This target system consists of a stable ODE (governed by the barrier dynamics) and a stable heat equation with a damping term.
   • Control Law: The boundary control input $U(t)$ is designed to cancel the destabilizing terms and inject a decaying signal $\delta(t)$ to ensure the safety conditions are met.

2. Adaptive Extension:

   • Parameter Identification: Since $\lambda$ and $b$ are unknown, the nominal controller is replaced by an adaptive version using parameter estimates $\hat{\lambda}$ and $\hat{b}$.
   • BaLSI Scheme: A triggered batch least-squares identifier is utilized. Unlike continuous adaptation, this scheme updates parameters at discrete time instants $t_i$. By integrating the system dynamics over specific intervals, a linear regression problem is formed.
   • Finite-Time Exact Identification: A key theoretical result is that under the proposed triggering mechanism and excitation conditions, the parameter estimates converge to the exact true values in finite time (specifically at the first update instant $t_1$). This eliminates the need for persistent excitation conditions over infinite time and removes parameter estimation errors after $t_1$.

3. Stability and Safety Analysis:

   • A Lyapunov function is constructed to prove the exponential convergence of the target system states to zero.
   • The safety analysis demonstrates that the high-relative-degree CBF conditions are satisfied. The design ensures that the barrier function remains non-negative for all time if initialized safely, or becomes non-negative within the prescribed time $t_a$ if initialized unsafely.

Key Contributions
The paper claims three primary contributions:

1. Rigorous Safety Guarantees for Adaptive PDE Control: Unlike existing adaptive boundary control methods for parabolic PDEs which focus primarily on stabilization, this work provides rigorous safety guarantees under parametric uncertainties.
2. Generalization of Safe Backstepping: While previous work (e.g., [12]) addressed safe control for Stefan models (a specific parabolic PDE), this work extends the framework to a broader class of problems involving in-domain instabilities, coupled ODEs, and general safety constraints.
3. First Safe Adaptive Control for Parabolic PDE-ODE Cascades: The paper distinguishes itself from recent safe adaptive control results for hyperbolic PDE-ODE cascades (e.g., [42]) by addressing the fundamentally different dynamics of parabolic systems. To the authors' knowledge, this is the first result establishing safe adaptive control for parabolic PDEs.

Results
Numerical simulations were conducted on a second-order ODE coupled with a parabolic PDE. The simulations validated the theoretical claims:

• Stabilization: All states ($u(x,t)$ and $Y(t)$) converged to zero.
• Safety:
  • In the initially safe case, the output $y_1(t)$ remained within the safe region throughout the transient.
  • In the initially unsafe case, the controller successfully drove the output back into the safe region within the prescribed time $t_a$ (e.g., 1 second).
• Parameter Identification: The BaLSI scheme achieved exact identification of the unknown parameters $\lambda$ and $b$ at the first triggering time, after which the adaptive controller behaved identically to the nominal (known-parameter) controller.
• Robustness to Initial Conditions: The controller performed effectively regardless of whether the system started in a safe or unsafe region, and for different definitions of the safe barrier function (linear and exponential decay).

Significance and Claims
The paper positions this work as a significant step forward in the control of distributed parameter systems where safety is paramount. It bridges the gap between safety-critical control (CBFs) and adaptive control for infinite-dimensional systems with uncertainties. The authors emphasize that the combination of finite-time exact parameter identification with high-relative-degree CBFs allows for the removal of conservatism often associated with adaptive safety control, ensuring that safety is not compromised by parameter estimation errors once the identification phase is complete. The work lays the groundwork for applying these methods to practical physical systems, such as those mentioned in the introduction (fluid flow, heat transfer, chemical processes), though specific hardware implementation is reserved for future work.