Abstract
Model predictive control (MPC), sometimes referred to as the receding horizon control, is an optimization-based approach to stabilization of discrete-time control systems. It is well-known that infinite-horizon optimal control, with the Linear-Quadratic Regulator [1] as the fundamental example, can provide optimal controls that result in asymptotically stabilizing feedback [8].
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Notes
- 1.
\(\mathbb{N}\) denotes the set of non-negative integers, and we use \(\mathbb{N}_{N}:=\{ 0, 1,\ldots,N - 1,N\}\) for any \(N \in \mathbb{N}\).
- 2.
Terms admissible control sequence/s are used interchangeably with feasible control sequence/s.
- 3.
The function \(f: \mathbb{R}^{n} \rightarrow [-\infty,\infty ]\) is lower semicontinuous at \(\overline{x} \in \mathbb{R}^{n}\) ( on \(\mathbb{R}^{n}\) ) if \(f(\overline{x}) \leq \liminf _{x\rightarrow \overline{x}}f(x)\) ( \(f(\overline{x}) \leq \liminf _{x\rightarrow \overline{x}}f(x)\) for every \(\overline{x} \in \mathbb{R}^{n}\) ).
- 4.
The function \(f: \mathbb{R}^{n} \rightarrow [-\infty,\infty ]\) is upper semicontinuous at \(\overline{x} \in \mathbb{R}^{n}\) ( on \(\mathbb{R}^{n}\) ), if \(\limsup _{x\rightarrow \overline{x}}f(x) \leq f(\overline{x})\) ( \(\limsup _{x\rightarrow \overline{x}}f(x) \leq f(\overline{x})\) for every \(\overline{x} \in \mathbb{R}^{n}\) ).
- 5.
For a convex \(f: \mathbb{R}^{n} \rightarrow (-\infty,\infty ]\), \(\partial f(x) = \left \{y \in \mathbb{R}^{n}\ :\ \forall x' \in \mathbb{R}^{n},\ f(x') \geq f(x) + y^{T}(x' - x)\right \}\), and if f is differentiable at \(x \in \mathbb{R}^{n}\) then ∂f(x) reduces to ∇f(x).
- 6.
We say that x N results from u N−1 and x if, for each \(k \in \mathbb{N}_{N-1}\), x k+1 = f(x k, u k) with x 0 = x.
- 7.
A function f: [0, ∞) → [0, ∞) is called a \(\mathcal{K}_{\infty }\)-class function if it is continuous, strictly increasing, f(0) = 0, and f(x) → ∞ as x → ∞.
- 8.
In other words, for each \((x,u) \in \mathbb{R}^{n} \times \mathbb{R}^{m}\), the set F(x, u) is the set of all limits z = limi → ∞ z i with z i = f(x i, u i) and (x i, u i) → (x, u).
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Acknowledgements
R. Goebel was partially supported by the Simons Foundation Grant 315326.
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Goebel, R., Raković, S.V. (2019). Set-Valued and Lyapunov Methods for MPC. In: Raković, S., Levine, W. (eds) Handbook of Model Predictive Control. Control Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77489-3_3
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