Controlled contractive sets for low-complexity constrained control

Abstract

Explicit constrained control is relatively simple when a controlled contractive set is available. However, the complexity of the explicit controller will depend on the complexity of the controlled contractive set. The ability to design a low complexity controlled contractive set is therefore desirable. Most methods for finding controlled contractive sets either assume the use of a constant linear state feedback, or is based on reachable set computations. In the first case, the assumption of a constant linear state feedback is restrictive (as MPC controllers for linear systems are typically piecewise affine), and in the second case the complexity of the controlled contractive set may be very high. Initial results on the construction of low complexity controlled contractive sets without assuming linear state feedback were reported at the IFAC World Congress in 2014. The present paper addresses shortcomings in the previous results, including the ability to handle identical modes in series (corresponding to a non-diagonalizable A matrix) and oscillatory modes. The paper suggests a method to find a flexible complexity controlled contractive set.

Aperiodic Sampled-Data Control via Explicit Transmission Mapping: A Set-Invariance Approach

“…Although such condition does not hold in general, it is still shown, under Assumptions 1 and 2, that the algorithm converges in the sense that for every λ < λ < 1, there exists a finite j ∈ N + such that the set Ω j is λ-contractive (see Theorem 3.2 in [12]). Several other algorithms have been recently proposed, see e.g., [16], [17] and see also [18] for a detailed convergence analysis. The following lemma illustrates the existence of a (non-quadratic) Lyapunov function in a given λ-contractive set:…”

Low complexity constrained control using higher degree Lyapunov functions

“…In [5] and [7], a controller based on (2) with polytopic controlled contractive sets S = {x k |F x k ≤ 1} were studied based on a piecewise linear function…”

“…Furthermore, the contractive set obtained in [5] is of fixed complexity, which does not allow trading off the complexity against the size of the contractive set. An optimization based technique has been proposed in [7] which allows the trading off complexity versus the size of the set. A solution to the optimization problem in [7] not only reduces the on-line computational complexity of the resulting constrained control, but also ensures significant reduction in the memory required to store the explicit solutions.…”

“…An optimization based technique has been proposed in [7] which allows the trading off complexity versus the size of the set. A solution to the optimization problem in [7] not only reduces the on-line computational complexity of the resulting constrained control, but also ensures significant reduction in the memory required to store the explicit solutions. However the method explained in [7] is highly non-convex, which makes it difficult to use for finding sufficiently large contractive sets for higher dimensional systems.…”

“…A solution to the optimization problem in [7] not only reduces the on-line computational complexity of the resulting constrained control, but also ensures significant reduction in the memory required to store the explicit solutions. However the method explained in [7] is highly non-convex, which makes it difficult to use for finding sufficiently large contractive sets for higher dimensional systems. Alternatively, ellipsoidal contractive sets with corresponding linear control laws can be computed, but the measure of these sets is limited by the linear structure of the control law and the inherent conservatism of the corresponding quadratic Lyapunov function.…”