Resilient Control Using Tube Model Predictive Control: Handling Bounded Disturbances and Attacks

This paper presents a resilient control law based on the tube Model Predictive Control (MPC) framework, designed to effectively handle bounded disturbances and attacks that fall below a predetermined threshold. This method distinguishes itself from conventional MPC by introducing an optimal initial state, denoted as $x_0^$, which is strategically chosen within a range $Z$ surrounding the real state $x$. This approach allows for robust control in the presence of uncertainties and malicious actions. The set of possible $x_0$ values forms a tubular-shaped region, hence the nomenclature 'tube' MPC.The tube-based MPC problem can be mathematically formulated as follows:/begin{equation}/n /begin{aligned}/n/Upsilon_N^(x_k) &= //operatorname*{min}{{x}0,/boldsymbol{u}} //mathcal{J}N(x_0,/overline{/boldsymbol{x}}{k},/boldsymbol{u})///n//mathrm{s.t.} /quad &/overline{x}{k+i} /in (/mathcal{X}/ominus Z), ///n&u{k+i} /in (/mathcal{U}/ominus KZ), ///n&/overline{x}{k+i+1} = f(/overline{x}{k+i}, u_{k+i}), /quad i /in /mathcal{N}{[0,N-1]},///n&/overline{x}{k+N} /in X_f /subset (/mathbb{X}/ominus /mathbb{Z}),///n&x_k /in (/overline{x}k/oplus /mathbb{Z}),/n/end{aligned}/n/end{equation}In this formulation, $/Upsilon_N$ represents the optimal value function within the optimization problem. $x_k$ refers to the measured system state acquired from sensors, and $/overline{x}k$ denotes the states in the nominal system, which represents a hypothetical system within the controller. The constraint settings can be found in /cite{Mayne2005}.The cost function used in this context is defined as:/[/mathcal{J}N = /sum{i=0}^{N-1} L(x{k+i},u{k+i}) + F(x_{k+N})/]Our proposed control scheme utilizes an online approach to solve this optimal control problem at each time instant, provided that an over-threshold FDI attack is not detected. This strategy ensures continuous adaptation to changing system conditions and mitigates the impact of potential attacks.