Sasa V. Rakovic

Invariant approximations of the minimal robust positively Invariant set

This note provides results on approximating the minimal robust positively invariant (mRPI) set (also known as the 0-reachable set) of an asymptotically stable discrete-time linear time-invariant system. It is assumed that the disturbance is bounded, persistent and acts additively on the state and that the constraints on the disturbance are polyhedral. Results are given that allow for the computation of a robust positively invariant, outer approximation of the mRPI set. Conditions are also given that allow one to a priori specify the accuracy of this approximation.

Brief paper: Robust output feedback model predictive control of constrained linear systems: Time varying case

The problem of output feedback model predictive control of discrete time systems in the presence of additive but bounded state and output disturbances is considered. The overall controller consists of two components, a stable state estimator and a tube based, robustly stabilizing model predictive controller. Earlier results are extended by allowing the estimator to be time varying. The proposed robust output feedback controller requires the online solution of a standard quadratic program. The closed loop system renders a specified invariant set robustly exponentially stable.

Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances

Piecewise affine (PWA) systems are useful models for describing non-linear and hybrid systems. One of the key problems in designing controllers for these systems is the inherent computational complexity of controller synthesis and analysis. These problems are amplified in the presence of state and input constraints and additive but bounded disturbances. In this paper, we exploit set invariance and parametric programming to devise an efficient robust time optimal control scheme. Specifically, the state is driven into the maximal robust invariant set /spl Omega//sub /spl infin// in minimum time. We show how to compute /spl Omega//spl tilde//sub /spl infin// and derive conditions for finite time computation.

Stochastic Tubes in Model Predictive Control With Probabilistic Constraints

Recent developments in stochastic MPC provided guarantees of closed loop stability and satisfaction of probabilistic and hard constraints. However the required computation can be formidable for anything other than short prediction horizons. This difficulty is removed in the current paper through the use of tubes of fixed cross-section and variable scaling. A model describing the evolution of predicted tube scalings simplifies the computation of stochastic tubes; furthermore this procedure can be performed offline. The resulting MPC scheme has a low online computational load even for long prediction horizons, thus allowing for performance improvements. The approach is illustrated by numerical examples.

Reachability analysis of discrete-time systems with disturbances

This paper presents new results that allow one to compute the set of states that can be robustly steered in a finite number of steps, via state feedback control, to a given target set. The assumptions that are made in this paper are that the system is discrete-time, nonlinear and time-invariant and subject to mixed constraints on the state and input. A persistent disturbance, dependent on the current state and input, acts on the system. Existing results are not able to address state- and input-dependent disturbances and the results in this paper are, therefore, a generalization of previously published results. One of the key aims of this paper is to present results such that one can perform the relevant set computations using polyhedral algebra and computational geometry software, provided the system is piecewise affine and the constraints are polygonal. Existing methods are only applicable to piecewise affine systems that either have no control inputs or no disturbances, whereas the results in this paper remove this limitation. Some simple examples are also given that show that, even if all the relevant sets are convex and the system is linear, convexity of the set of controllable states cannot be guaranteed.

Brief paper: Explicit use of probabilistic distributions in linear predictive control

The guarantee of feasibility given feasibility at initial time is an issue that has been overlooked by many of the recent papers on stochastic model predictive control. Effective solutions have recently been proposed, but these carry considerable online computational load and a degree of conservativism. For the case that the elements of the random additive disturbance vector are independent, the current paper ensures that probabilistic constraints are met and that a quadratic stability condition is satisfied. A numerical example illustrates the efficacy of the proposed algorithm, which achieves tight satisfaction of constraints and thereby attains near-optimal performance.

Brief paper: Feedback and invariance under uncertainty via set-iterates

We examine discrete-time control systems under non-parametric disturbances. Sets which a given control feedback makes invariant under the disturbance are analyzed via lifting the feedback operation to the space of sets. Properties of being an attractor of the disturbed dynamics and being a minimal invariant set are derived from the corresponding notions of the set-dynamics, yielding, in turn, useful characterizations and error estimates for numerical algorithms which detect the minimal invariant sets. Concrete numerics for some examples of practical feedback rules are offered.

Parameterized Tube Model Predictive Control

This paper develops a parameterized tube model predictive control (MPC) synthesis method. The most relevant novel feature of our proposal is the online use of a single tractable linear program that optimizes parameterized, Minkowski decomposable, state and control tubes and an associated, fully separable, nonlinear, control policy. The induced control policy enjoys a higher degree of nonlinearity than existing tube MPC and robust MPC using disturbance affine control policy. Our proposal offers greater generality than the state of the art robust MPC methods. It is conjectured, and also established in three cases, that our proposal is equivalent, feasibility-wise, to dynamic programming (DP). It is also shown that, under natural assumptions, our method is computationally efficient while it possesses rather strong system theoretic properties.

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.

Technical communique: A logarithmic-time solution to the point location problem for parametric linear programming

The optimiser of a (multi) parametric linear program (pLP) is a piecewise affine function defined over a polyhedral subdivision of the set of feasible states. Once this affine function has been pre-calculated, the optimal solution can be computed for a particular parameter by determining the region that contains it. This is the so-called point location problem. In this paper, we show that this problem can be written as an additively weighted nearest neighbour search that can be solved in time linear in the dimension of the state space and logarithmic in the number of regions. It is well-known that linear model predictive control (MPC) problems based on linear control objectives (e.g., 1- or ~-norm) can be posed as pLPs, and on-line calculation of the control law involves the solution to the point location problem. Several orders of magnitude sampling speed improvement are demonstrated over traditional MPC and closed-form MPC schemes using the proposed scheme.