Robust Control Invariant Research Articles

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 real-time measurements of the scheduling parameters are typically available, we allow the RCI set description and the invariance-inducing controller to be scheduling parameter dependent. Thus, the considered formulation leads to parameter-dependent conditions for the set invariance, which are replaced by sufficient Linear Matrix Inequalities (LMIs) 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. Besides 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. Using numerical examples, we show that the presented iterative algorithm can generate RCI sets for large parameter variations where commonly used robust approaches fail.

On robust control invariance and robust set stabilization of mix‐valued logical control networks

AbstractThis article investigates the robust control invariance and robust set stabilization problems based on a semi‐tensor product method for a class of mix‐valued logical control networks (MVLCNs) with disturbances. First, a calculation method for the largest robust control invariant subset contained in a given set is proposed. Second, based on the robust control invariant subset, the robust set stabilization of MVLCNs is discussed, and new results are presented. Furthermore, the design algorithm of time‐optimal state feedback stabilizers via antecedence solution technique is derived. The study of an illustrative example shows the effectiveness of the obtained new results.

Tube-Based Output Feedback Robust MPC for LPV Systems With Scaled Terminal Constraint Sets.

This article provides a solution to tube-based output feedback robust model predictive control (RMPC) for discrete-time linear parameter varying (LPV) systems with bounded disturbances and noises. The proposed approach synthesizes an offline optimization problem to design a look-up table and an online tube-based output feedback RMPC with tightened constraints and scaled terminal constraint sets. In the offline optimization problem, a sequence of nested robust positively invariant (RPI) sets and robust control invariant (RCI) sets, respectively, for estimation errors and control errors is optimized and stored in the look-up table. In the online optimization problem, real-time control parameters are searched based on the bounds of time-varying estimation error sets. Considering the characteristics of the uncertain scheduling parameter in LPV systems, the online tube-based output feedback RMPC scheme adopts one-step nominal system prediction with scaled terminal constraint sets. The formulated simple and efficient online optimization problem with fewer decision variables and constraints has a lower online computational burden. Recursive feasibility of the optimization problem and robust stability of the controlled LPV system are guaranteed by ensuring that the nominal system converges to the terminal constraint set, and uncertain state trajectories are constrained within robust tubes with the center of the nominal system. A numerical example is given to verify the approach.

Computation of robust control invariant sets with predefined complexity for uncertain systems

AbstractThis paper presents an algorithm that computes polytopic robust control‐invariant (RCI) sets for rationally parameter‐dependent systems with additive disturbances. By means of novel linear matrix inequalities (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, the proposed algorithm directly computes the RCI sets without requiring control inputs to be in a specific feedback form. 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.

Computing robust control invariant sets of constrained nonlinear systems: A graph algorithm approach

This paper deals with the computation of the largest robust control invariant sets (RCISs) of constrained nonlinear systems. The proposed approach is based on casting the search for the invariant set as a graph theoretical problem. Specifically, a general class of discrete-time time-invariant nonlinear systems is considered. First, the dynamics of a nonlinear system is approximated with a directed graph. Subsequently, the condition for robust control invariance is derived and an algorithm for computing the robust control invariant set is presented. The algorithm combines the iterative subdivision technique with the robust control invariance condition to produce outer approximations of the largest robust control invariant set at each iteration. Following this, we prove convergence of the algorithm to the largest RCIS as the iterations proceed to infinity. Based on the developed algorithms, an algorithm to compute inner approximations of the RCIS is also presented. A special case of input affine and disturbance affine systems is also considered. Finally, two numerical examples are presented to demonstrate the efficacy of the proposed method.

Observer-based output feedback robust MPC via zonotopic set-membership state estimation for LPV systems with bounded disturbances and noises

For linear parameter varying (LPV) systems with bounded disturbances and noises, this paper presents a synthesis approach to observer-based output feedback robust model predictive control (RMPC) via the zonotopic set-membership state estimation. In the off-line optimization problem, the state observer sub-gains with a minimal ellipsoidal robust positively invariant (RPI) set for the zonotopic estimation error set are designed. In the on-line control optimization problem, bounds of the current and future zontopic estimation error sets are outer approximated by scaling the minimal ellipsoidal RPI set. Then, an on-line constrained optimization problem that considers bounds of the predicted zonotopic estimation error sets is solved to guarantee robust stability of the closed-loop observer system. In the on-line receding horizon optimization, the zonotopic estimation error set is refreshed based on zonotopic properties, and its outer approximated ellipsoidal set is optimized via the technique of S-procedure. By properly considering the relationship between the zonotopic estimation error set and its outer approximated ellipsoidal set, recursive feasibility of the optimization problem can be ensured. The on-line optimization problem guarantees that the nominal estimation error system and nominal closed-loop observer system converge to the origin. When bounded disturbances and noises are considered, the estimation error system and closed-loop observer system respectively bounded within time-varying RPI sets and robust control invariant (RCI) sets are steered to a neighborhood of the origin such that robust stability of the controlled system is guaranteed. A simulation example is given to illustrate the proposed method.

Full‐complexity polytopic robust control invariant sets for uncertain linear discrete‐time systems

SummaryThis paper presents an algorithm for the computation of full‐complexity polytopic robust control invariant (RCI) sets, and the corresponding linear state‐feedback control law. The proposed scheme can be applied for linear discrete‐time systems subject to additive disturbances and structured norm‐bounded or polytopic uncertainties. Output, initial condition, and performance constraints are considered. Arbitrary complexity of the invariant polytope is allowed to enable less conservative inner/outer approximations to the RCI sets whereas the RCI set is assumed to be symmetric around the origin. The nonlinearities associated with the computation of such an RCI set structure are overcome through the application of Farkas' theorem and a corollary of the elimination lemma to obtain an initial polytopic RCI set, which is guaranteed to exist under certain conditions. A Newton‐like update, which is recursively feasible, is then proposed to yield desirable large/small volume RCI sets.

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.

On Operating Envelope Protection Design for Nonlinear Discrete-time Systems

This paper is concerned with the operating envelope protection design for a broad class of nonlinear discrete-time systems based on a Robust Control Invariant (RCI) set framework. Existing techniques for a direct design of a RCI set for a general system suffer from the computational intractability connected with the complexity of a system model and model dimension. For a nonlinear system with mild assumptions on its model here we use a suitably defined linear model with additive uncertainty confined in a finite set and its corresponding maximum RCI set. Robust constraints satisfaction for the original system is ensured for a pre-defined set of disturbance trajectories. The approach is based on a superposition of a nominal and perturbed dynamics, whereas the coupling effects between respective dynamics are encompassed with the additive uncertainty. The additive uncertainty set is estimated by employing the trajectories of disturbance and a system description. The presented approach is scalable with respect to the system dimension. A simple example from the wind energy field is used to illustrate the proposed method.

On robust control invariance of Boolean control networks

This paper investigates the robust control invariance of Boolean control networks (BCNs) via the semi-tensor product of matrices. Firstly, based on an algebraic state space representation of BCNs, two necessary and sufficient conditions are presented to check whether or not a given set is a robust control invariant set under a given state feedback controller. Secondly, by defining a series of suitable sets, all possible state feedback gain matrices under which a given set is a robust control invariant set are characterized. An illustrative example is presented to demonstrate the obtained new results.

Low-Complexity Polytopic Invariant Sets for Linear Systems Subject to Norm-Bounded Uncertainty

We propose a novel algorithm to compute low-complexity polytopic robust control invariant (RCI) sets, along with the corresponding state-feedback gain, for linear discrete-time systems subject to norm-bounded uncertainty, additive disturbances and state/input constraints. Using a slack variable approach, we propose new results to transform the original nonlinear problem into a convex/LMI problem whilst introducing only minor conservatism in the formulation. Through numerical examples, we illustrate that the proposed algorithm can yield improved maximal/minimal volume RCI set approximations in comparison with the schemes given in the literature.

Plug-and-play model predictive control based on robust control invariant sets

We consider the problem of designing decentralized controllers for large-scale linear constrained systems composed by a number of interacting subsystems. As in Riverso et al. (2013b), (i) the design of local controllers requires limited transmission of information from other subsystems and (ii) the addition/removal of a subsystem triggers the design of local controllers for child subsystems only. These properties enable Plug-and-Play (PnP) operations, and we show how to perform them while preserving global stability of the origin and constraint satisfaction. We improve several aspects of the PnP design procedure proposed in Riverso et al. (2013b) and, using recent results in the computation of Robust Control Invariant (RCI) sets, we show that all critical steps in the design of a local controller can be solved through Linear Programming (LP). Finally, an application of the proposed design procedure to a large-scale mechanical system is presented.

Robust vehicle lateral stabilisation via set-based methods for uncertain piecewise affine systems

The paper presents the design of a lateral stability controller for ground vehicles based on front steering and four wheels independent braking. The control objective is to track yaw rate and lateral velocity reference signals while avoiding front and rear wheel traction force saturation. Control design is based on an approximate piecewise-affine nonlinear dynamical model of the vehicle. Vehicle longitudinal velocity and drivers steering input are modelled as measured disturbances taking values in a compact set. A time-optimal control strategy which ensures convergence into a maximal robust control invariant (RCI) set is proposed. This paper presents the uncertain model, the RCI computation, and the control algorithm. Experimental tests at high-speed on ice with aggressive driver manoeuvres show the effectiveness of the proposed scheme.

Decentralized Robust Control Invariance for a Network of Storage Devices

Robust control of networked storage devices is considered. Each storage device is modeled as a single-state, discrete-time integrator with bounded control input, and subject to additive disturbance. The values of the disturbance are unknown but are assumed to belong to bounded sets. Nodes exchange stored resource through links of limited capacity. Characterization of the maximal robust control invariant (RCI) set is provided and decentralization of the robust feasibility problem is considered. A notion of decentralized RCI set is introduced and a parametrization of a family of decentralized RCI sets is proposed. It is shown that the set of parameters for which the decentralized RCI sets are non-empty is a polyhedral set. The result offers a possibility to use convex optimization for selection of bounds on the admissible flow through the network which ensure feasibility of the proposed decentralized design.

Robust Electronic Stability Control via Set-Based Methods for Uncertain Piecewise Affine Systems

AbstractThe paper presents the design of a lateral stability controller for ground vehicles based on front steering and four wheels independent braking. The control objective is to track yaw rate and lateral velocity reference signals while avoiding front and rear wheel traction force saturation. Control design is based on an approximate piecewise–affine nonlinear dynamical model of the vehicle. Vehicle longitudinal velocity and drivers steering input are modeled as measured disturbances taking values in a compact set. A time–optimal control strategy which ensures convergence into a maximal robust control invariant (RCI) set is proposed. This paper presents the uncertain model, the RCI computation, and the control algorithm. Simulations results on a high–fidelity vehicle model show the effectiveness of the propose scheme.

Optimized robust control invariance for linear discrete-time systems: Theoretical foundations

This paper introduces the concept of optimized robust control invariance for discrete-time linear time-invariant systems subject to additive and bounded state disturbances. A novel characterization of two families of robust control invariant sets is given. The existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear-convex case and a standard linear/quadratic program when the constraints are polyhedral or polytopic. The solution of the same optimization problem yields the corresponding feedback control law that is, in general, set-valued. A procedure for selection of a point-valued, nonlinear control law is provided.

OPTIMIZED ROBUST CONTROL INVARIANT SETS FOR CONSTRAINED LINEAR DISCRETE-TIME SYSTEMS

In this paper we introduce the concept of optimized robust control invariance for a discrete-time, linear, time-invariant system subject to additive state disturbances. A novel characterization of a family of the robust control invariant sets is given. The existence of a constraint admissible member of this family can be checked by solving a single linear programming problem. The solution of the same linear programming problem yields the corresponding feedback controller.