Parameter Dependent Robust Control Invariant Sets for LPV Systems with Bounded Parameter Variation Rate

Real-time measurements of the scheduling parameter of linear parameter-varying (LPV) systems enables the synthesis of robust control invariant (RCI) sets and parameter dependent controllers inducing invariance. We present a method to synthesize parameter-dependent robust control invariant (PD-RCI) sets for LPV systems with bounded parameter variation, in which invariance is induced using PD-vertex control laws. The PD-RCI sets are parameterized as configuration-constrained polytopes that admit a joint parameterization of their facets and vertices. The proposed sets and associated control laws are computed by solving a single semidefinite programing (SDP) problem. Through numerical examples, we demonstrate that the proposed method outperforms state-of-the-art methods for synthesizing PD-RCI sets, both with respect to conservativeness and computational load.Comment: 8 pages, 6 figure

Computation of Parameter Dependent Robust Invariant Sets for LPV Models with Guaranteed Performance

This paper presents an iterative algorithm to compute a Robust Control Invariant (RCI) set, along with an invariance-inducing control law, for Linear Parameter-Varying (LPV) systems. As the real-time measurements of the scheduling parameters are typically available, in the presented formulation, we allow the RCI set description along with the invariance-inducing controller to be scheduling parameter dependent. The considered formulation thus leads to parameter-dependent conditions for the set invariance, which are replaced by sufficient Linear Matrix Inequality (LMI) conditions via Polya\u27s relaxation. These LMI conditions are then combined with a novel volume maximization approach in a Semidefinite Programming (SDP) problem, which aims at computing the desirably large RCI set. In addition to ensuring invariance, it is also possible to guarantee performance within the RCI set by imposing a chosen quadratic performance level as an additional constraint in the SDP problem. The reported numerical example shows that the presented iterative algorithm can generate invariant sets which are larger than the maximal RCI sets computed without exploiting scheduling parameter information

Computation of Robust Control Invariant Sets with Predefined Complexity for Uncertain Systems

This paper presents an algorithm that computes polytopic robust control-invariant (RCI) sets for rationally parameter-dependent systems with additive disturbances. By means of novel LMI feasibility conditions for invariance along with a newly developed method for volume maximization, an iterative algorithm is proposed for the computation of RCI sets with maximized volumes. The obtained RCI sets are symmetric around the origin by construction and have a user-defined level of complexity. Unlike many similar approaches, fixed state feedback structure is not imposed. In fact, a specific control input is obtained from the LMI problem for each extreme point of the RCI set. The outcomes of the proposed algorithm can be used to construct a piecewise-affine controller based on offline computations

Computation of Parameter Dependent Robust Invariant Sets for LPV Models with Guaranteed Performance

This paper presents an iterative algorithm to compute a Robust Control Invariant (RCI) set, along with an invariance-inducing control law, for Linear Parameter-Varying (LPV) systems. As the real-time measurements of the scheduling parameters are typically available, in the presented formulation, we allow the RCI set description along with the invariance-inducing controller to be scheduling parameter dependent. The considered formulation thus leads to parameter-dependent conditions for the set invariance, which are replaced by sufficient Linear Matrix Inequality (LMI) conditions via Polya's relaxation. These LMI conditions are then combined with a novel volume maximization approach in a Semidefinite Programming (SDP) problem, which aims at computing the desirably large RCI set. In addition to ensuring invariance, it is also possible to guarantee performance within the RCI set by imposing a chosen quadratic performance level as an additional constraint in the SDP problem. The reported numerical example shows that the presented iterative algorithm can generate invariant sets which are larger than the maximal RCI sets computed without exploiting scheduling parameter information.Comment: 32 pages, 5 figure

Full-complexity characterization of control-invariant domains for systems with uncertain parameter dependence

This letter proposes an algorithm to find a robust control invariant (RCI) set of desired complexity and the associated linear, state-feedback control law. The candidate RCI set is restricted to be symmetric around the origin. The algorithm is applicable to rational parameter dependent systems with bounded additive disturbance. The system constraints are framed as simple affine inequalities whereas the invariance condition as a set of sufficient LMI conditions. The proposed iterative algorithm is guaranteed to be recursively feasible and converge to some stationary point

Control of Constrained Dynamical Systems with Performance Guarantees: With Application to Vehicle motion Control

In control engineering, models of the system are commonly used for controller design. A standard control design problem consists of steering the given system output (or states) towards a predefined reference. Such a problem can be solved by employing feedback control strategies. By utilizing the knowledge of the model, these strategies compute the control inputs that shrink the error between the system outputs and their desired references over time. Usually, the control inputs must be computed such that the system output signals are kept in a desired region, possibly due to design or safety requirements. Also, the input signals should be within the physical limits of the actuators. Depending on the constraints, their violation might result in unacceptable system failures (e.g. deadly injury in the worst case). Thus, in safety-critical applications, a controller must be robust towards the modelling uncertainties and provide a priori guarantees for constraint satisfaction. A fundamental tool in constrained control application is the robust control invariant sets (RCI). For a controlled dynamical system, if initial states belong to RCI set, control inputs always exist that keep the future state trajectories restricted within the set. Hence, RCI sets can characterize a system that never violates constraints. These sets are the primary ingredient in the synthesis of the well-known constraint control strategies like model predictive control (MPC) and interpolation-based controller (IBC). Consequently, a large body of research has been devoted to the computation of these sets. In the thesis, we will focus on the computation of RCI sets and the method to generate control inputs that keep the system trajectories within RCI set. We specifically focus on the systems which have time-varying dynamics and polytopic constraints. Depending upon the nature of the time-varying element in the system description (i.e., if they are observable or not), we propose different sets of algorithms.The first group of algorithms apply to the system with time-varying, bounded uncertainties. To systematically handle the uncertainties and reduce conservatism, we exploit various tools from the robust control literature to derive novel conditions for invariance. The obtained conditions are then combined with a newly developed method for volume maximization and minimization in a convex optimization problem to compute desirably large and small RCI sets. In addition to ensuring invariance, it is also possible to guarantee desired closed-loop performance within the RCI set. Furthermore, developed algorithms can generate RCI sets with a predefined number of hyper-planes. This feature allows us to adjust the computational complexity of MPC and IBC controller when the sets are utilized in controller synthesis. Using numerical examples, we show that the proposed algorithms can outperform (volume-wise) many state-of-the-art methods when computing RCI sets.In the other case, we assume the time-varying parameters in system description to be observable. The developed algorithm has many similar characteristics as the earlier case, but now to utilize the parameter information, the control law and the RCI set are allowed to be parameter-dependent. We have numerically shown that the presented algorithm can generate invariant sets which are larger than the maximal RCI sets computed without exploiting parameter information.Lastly, we demonstrate how we can utilize some of these algorithms to construct a computationally efficient IBC controller for the vehicle motion control. The devised IBC controller guarantees to meet safety requirements mentioned in ISO 26262 and the ride comfort requirement by design