Carlos Eduardo Trabuco Dórea

(A,B)-invariant polyhedral sets of linear discrete-time systems

The problem of confining the trajectory of a linear discrete-time system in a given polyhedral domain is addressed through the concept of (A, B)-invariance. First, an explicit characterization of (A, B)-invariance of convex polyhedra is proposed. Such characterization amounts to necessary and sufficient conditions in the form of linear matrix relations and presents two major advantages compared to the ones found in the literature: it applies to any convex polyhedron and does not require the computation of vertices. Such advantages are felt particularly in the computation of the supremal (A, B)-invariant set included in a given polyhedron, for which a numerical method is proposed. The problem of computing a control law which forces the system trajectories to evolve inside an (A, B)-invariant polyhedron is treated as well. Finally, the (A, B)-invariance relations are generalized to persistently disturbed systems.

Design and Implementation of Model-Predictive Control With Friction Compensation on an Omnidirectional Mobile Robot

This paper presents and discusses the implementation results of a model-predictive control (MPC) scheme with friction compensation applied to trajectory following of an omnidirectional three-wheeled robot. A cascade structure is used with an inverse kinematics block to generate the velocity references given to the predictive controller. Part of the control effort is used to compensate for the effects of static friction, allowing the use of efficient algorithms for linear MPC with constraints. Experimental results show that the proposed strategy is efficient in compensating for frictional effects as well as for tracking predefined trajectories.

(A, B)-Invariance Conditions of Polyhedral Domains for Continuous-Time Systems

This paper provides an algebraic characterization of the (A, B)-invariance property of polyhedral sets with respect to linear continuous-time systems. The family of control laws which is investigated is the set of continuous and Lipschitz functions. Some particular conditions of existence of linear state feedback laws are also presented.

Design of L-Q regulators for state constrained continuous-time systems

A new methodology to the design of LQ regulators for continuous-time systems subject to linear state constraints is proposed, consisting of two parts. In the first one, the positive invariance of the polyhedron defined by the constraints in the state space is imposed, guaranteeing thereby that the constraints will not be violated. Furthermore, the admissible constrained controllers are parameterized via the determination of the elements which are fixed in the state feedback matrix. This parameterization enables in a second part the L-Q regulator to be obtained from the solution of a parameter optimization problem subject to linear constraints, for which it is proposed a specialized feasible directions method. >

On (A, B)-invariance of polyhedral domains for discrete-time systems

The problem of confining the trajectory of a discrete-time linear system in a given polyhedral set is addressed using the positive invariance approach. The property of (A, B)-invariance of polyhedral domains is introduced. It is then geometrically and analytically characterized. In particular, this property is verified by the maximal admissible set included in a given polyhedral domain. An important class of (A, B)-invariant polyhedral domains admitting a linear state feedback invariant control law is exhibited. The results are extended to systems subject to additive disturbances.

On invariant polyhedra of continuous-time systems subject to additive disturbances

This paper presents new necessary and sufficient algebraic conditions on the existence of positively D-invariant polyhedra of continuous-time linear systems subject to additive disturbances. In particular, for a convex unbounded polyhedron containing the origin in its interior, it is also shown that the positive D-invariance conditions can be split into two lower-dimensional sets of algebraic relations: the first corresponds to disturbance decoupling conditions and the second to positive D-invariance conditions for bounded polyhedra of a reduced-order system. The stability of the overall system is discussed as well. By exploring the results obtained, an LP approach is proposed for solution of a state-constrained regulator problem in the presence of additive disturbances.

Design of Observers With Error Limitation in Discrete-Time Descriptor Systems: A Case Study of a Hydraulic Tank System

A novel descriptor observer structure is proposed in the context of discrete-time descriptor linear systems. On the basis of this new structure, the characterization of conditioned-invariant polyhedral sets with the purpose of limiting the estimation error is carried out. Limitation of the estimation error can be achieved by the computation of an as small as possible conditioned-invariant polyhedron, that contains the set of possible initial errors, and by a suitable output injection law. The approach is applied to an experimental tanks platform, for which a descriptor model is identified and validated. The obtained results confirm the merits of the proposed observer for applications in this kind of systems.

Output-feedback controlled-invariant polyhedra for constrained linear systems

A design method is proposed for output-feedback control of linear systems subject to state and control constraints, additive disturbances and measurement noise. First, necessary and sufficient conditions for a polyhedral set to be controlled-invariant under output-feedback are presented, which can be checked by the solution of a set of Linear Programming problems. Then, a dynamic output-feedback compensator structure is proposed to guarantee constraint satisfaction, through the construction of an output-feedback controlled-invariant set, from a pair composed by a conditioned-invariant and a controlled-invariant polyhedron. Based on the available measurements and on the state of the compensator, a suitable control sequence can be computed to enforce the state constraints. Differently from other approaches, the proposed method does not rely on pre-computed linear controllers or observers. Therefore, it is likely to provide a solution for larger amplitudes of disturbances and noise, and larger uncertainties on the initial state, as illustrated by numerical examples.

An iterative algorithm for constrained MPC with stability of bilinear systems

This paper presents a new algorithm for model predictive control (MPC) of constrained bilinear systems using iterative compensation of the prediction error and invariant sets for constraints satisfaction and stability guarantee. In order to improve the performance of the controller, which holds prediction as its essence, an iterative process is proposed with the objective of reducing the prediction errors due to the use of a quasi-linear approximation of the bilinear model. A study of the conditions under which the prediction error converges to zero is also provided. An important outcome of this property is that feasibility and effective state constraints satisfaction along the state trajectory can be achieved. For stability guarantee, a controlled-invariant set is computed and used as terminal constraint. Then, if the initial state is admissible, the state trajectory is assured to converge to this terminal set without violating the constraints. Once inside this region, a local controller can be used to drive the state to the operation point. Numerical examples illustrate the effectiveness of the proposed algorithm regarding convergence, constraints satisfaction and stability.

Set-Invariant Estimators for Multiple-Output Discrete-Time Systems

A new technique for the design of full-order state observers with limitation of the estimation error has been recently proposed, based on the concept of set-invariance. However, such a technique was limited to single-output linear systems. In this paper, the design of set-invariant estimators is extended to multiple-output linear discrete-time systems subject to bounded disturbances. Based on multiparametric linear programming concepts, necessary and sufficient conditions are established under which a given polyhedral set defined on the estimation error space is invariant, in the sense that the error trajectory can be kept in this set by means of a suitable output injection. Moreover, three algorithms are proposed for the computation of an invariant polyhedron which bounds the trajectory of the estimation error. Important issues such as the computation of the output injection law and the convergence of the proposed algorithms are discussed as well. Numerical examples illustrate the proposed technique