Practical Tracking for Minimum-Phase 1-d Heat Equation with Arbitrary Uniformly Continuous Bounded Reference Signals

In this paper, we tackle the output tracking problem for a specific one-dimensional, minimum-phase heat equation with an arbitrary reference signal. This represents a long-standing and unresolved challenge, embodying a prototypical non-collocated output tracking issue for systems governed by partial differential equations. Specifically, the control input is applied at the boundary, whereas the regulated output can be positioned anywhere within the interval, making it associate with a particularly challenging ill-posed problem. Firstly, we introduce an exponential convergent tracking algorithm that utilizes tracking error feedback control for sufficiently smooth reference signals. Subsequently, we propose a practical tracking methodology that approximates the reference signal with a smooth function. Our numerical simulations corroborate the theoretical findings.