This paper proposes a novel robust Model Predictive Control (MPC) scheme for linear discrete-time systems affected by model uncertainty described by interval matrices. The key feature of the proposed method is a bound on the uncertainty propagation along the prediction horizon which exploits a set-theoretic over-approximation of each term of the uncertain system impulse response. Such an approximation is based on matrix zonotopes and leverages the interval matrix structure of the uncertainty model. Its main advantage is that all the relevant bounds are computed offline, thus making the online computational load independent of the number of uncertain parameters. A variable-horizon MPC formulation is adopted to guarantee recursive feasibility and to ensure robust asymptotic stability of the closed-loop system. Numerical simulations demonstrate that the proposed approach is able to match the feasibility regions of the most effective state-of-the-art methods, while significantly reducing the computational burden, thereby enabling the treatment of nontrivial dimensional systems with multiple uncertain parameters.
Model Predictive Control (MPC) is widely recognized as an effective technique to address multivariable control problems with state and input constraints [1,2,3]. Robust MPC is an active research area that deals with the application of MPC to systems affected by uncertainty. The main challenges in this context is to provide theoretical guarantees on recursive feasibility, robust stability and constraint satisfaction of the MPC scheme, in the presence of disturbances and unmodeled dynamics.
The case of bounded additive disturbances has been investigated in great detail. Minimizing the worst-case performance over all possible disturbance sequences requires the solution of min-max control problems [4,5], whose computational complexity typically increases exponentially with the dimension of the state and input vectors. A more efficient, yet more conservative, approach is the so-called tube-MPC (see, e.g., [6,7,8,9]), in which robust constraint satisfaction is achieved by suitably tightening the nominal constraints to cope with the effect of disturbances on the predicted nominal trajectory.
Parametric uncertainty in the dynamic model is usually more challenging to deal with, mainly because uncertainty propagation along the prediction horizon depends on the system states and inputs. Early works consider systems with uncertain impulse responses [10]. A commonly used uncertainty description assumes that the system matrices belong to a polytopic set, thus leading to min-max optimization problems aiming at minimizing the worst-case performance with respect to all admissible models. Notably, in [4] it is shown that the MPC control law can be computed explicitly offline by using multiparametric programming, although the complexity of the solution rapidly grows even for systems of small dimension. Hence, the research has focused on finding a suitable compromise between conservatism and computational complexity. Methods based on ellipsoidal invariant sets and linear matrix inequalities have been proposed [11,12,13]. Conservatism reduction has been pursued by employing polytopic tubes and invariant sets, see, e.g., [14,15,16]. A detailed review of these approaches can be found in [3,Chapter 5].
More recently, researchers have turned their attention towards methods that employ linear time-varying state feedback controllers. These approaches are potentially advantageous for reducing conservativeness with respect to tube-based approaches, but make it more difficult to guarantee recursive feasibility of the MPC scheme. For example, in [17] uncertainty propagation is expressed in terms of norm bounds on the perturbation of the nominal predicted trajectory, that can be computed offline but must hold for any admissible input sequence. In [18], a system level synthesis approach is adopted to derive an uncertainty bound which depends on the controller parameters. Both methods employ a shrinking horizon MPC along with a backup control law to cope with the lack of recursive feasibility of the robust optimal control problem. Moreover, the number of constraints of such a problem depends on the number of vertices of the model uncertainty polytope, which typically grows exponentially with the number of uncertain parameters. Hence, such techniques appear to be suitable to problems involving low-order systems with few uncertain parameters.
In this paper, a new robust MPC scheme is presented for linear discrete-time systems, affected by model uncertainties described by interval matrices. Such uncertain models are commonly used, for example, in setmembership identification [19,20,21]. The main novelty of the proposed approach lies in the way the set-valued dynamics of the uncertainty propagation along the prediction horizon is expressed in terms of the nominal trajectory. Namely, this is done by computing a set-theoretic bound on each term of the uncertain system impulse response. These bounds exploit the interval matrix model description to propagate uncertainty in the form of matrix zonotopes. A key feature is that all the bounds can be computed offline, which makes the computational burden of the online optimal control problem independent of the number of uncertain parameters. A variable-horizon MPC approach is adopted [22,23], by including the horizon length among the optimization variables of the optimal control problem. This allows one to achieve recursive feasibility and to guarantee that the state trajectories converge in finite time to a suitable robust invariant set, designed to guarantee robust asymptotic stability of the control scheme. Numerical simulations show that the proposed technique yields feasibility domains comparable to those provided by the method introduced in [18], which was shown to outperform both tube-based approaches and the method in [17]. Remarkably, this is achieved with a much lower computational burden, which allows the proposed approach to be applied to systems of nontrivial dimension, with several unc
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