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This note extends tube-based model predictive control to the control of nonlinear systems with unmodeled dynamics. The problem of obtaining robustness against unstructured uncertainty is converted into the easier problem of achieving robustness against an additional bounded disturbance while satisfying an additional output constraint. The output constraint depends on the maximum l∞ gain of the unstructured dynamics.

We propose in this note a method for computing the solution to the infinite horizon continuous-time constrained linear quadratic regulator. The method is based on two main ingredients: a multigrid method for placing a finite number of time intervals, and a piece-wise linear parameterization of the input within the intervals. The input values at the decision-time points and slopes within the time intervals are computed via quadratic programs (QPs). The grids are gradually refined to efficiently improve the accuracy of the solution, and the required matrices and vectors for all QPs are computed offline and stored to improve the online efficiency. Two examples are presented to show the main characteristics of the proposed method.

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.

State estimator design for a nonlinear discrete-time system is a challenging problem, further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances. One strategy for constrained state estimation is to employ online optimization using a moving horizon approximation. We propose a general theory for constrained moving horizon estimation. Sufficient conditions for asymptotic and bounded stability are established. We apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation. Examples are used to illustrate the benefits of constrained state estimation. Our framework is deterministic.

Practical difficulties involved in implementing stabilizing model predictive control laws for nonlinear systems are well known. Stabilizing formulations of the method normally rely on the assumption that global and exact solutions of nonconvex, nonlinear optimization problems are possible in limited computational time. In the paper, we first establish conditions under which suboptimal model predictive control (MPC) controllers are stabilizing; the conditions are mild holding out the hope that many existing controllers remain stabilizing even if optimality is lost. Second, we present and analyze two suboptimal MPC schemes that are guaranteed to be stabilizing, provided an initial feasible solution is available and for which the computational requirements are more reasonable.

Min-max feedback formulations of model predictive control are discussed, both in the fixed and variable horizon contexts. The control schemes the authors discuss introduce, in the control optimization, the notion that feedback is present in the receding-horizon implementation of the control. This leads to improved performance, compared to standard model predictive control, and resolves the feasibility difficulties that arise with the min-max techniques that are documented in the literature. The stabilizing properties of the methods are discussed as well as some practical implementation details.

In this paper two topics are explored. A new approach to the problem of obtaining an estimate of the state of a nonlinear system is proposed. The moving horizon observer produces an estimate of the state of the nonlinear system at time t either by minimizing, or approximately minimizing, a cost function over the preceding interval (horizon) [t-T,t]; as t advances, so does the horizon. Convergence of the estimator is established under the assumption that the corresponding global optimization problem can be (approximately) solved and a uniform reconstructability condition is satisfied; the latter condition is automatically satisfied for linear observable systems. The utility of the estimator for receding horizon control is explored. In particular, stability of a composite moving horizon system, comprising a moving horizon regulator and a moving horizon observer, is established. >

We present a method for the construction of a robust dual-mode, receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error. The controller is dual-mode. In a neighborhood of the origin, the control action is generated by a linear feedback controller designed for the linearized system. Outside this neighborhood, receding horizon control is employed. Existing receding horizon controllers for nonlinear, continuous time systems, which are guaranteed to stabilize the nonlinear system to which they are applied, require the exact solution, at every instant, of an optimal control problem with terminal equality constraints. These requirements are considerably relaxed in the dual-mode receding horizon controller presented in this paper. Stability is achieved by imposing a terminal inequality, rather than an equality, constraint. Only approximate minimization is required. A variable time horizon is permitted. Robustness is achieved by employing conservative state and stability constraint sets, thereby permitting a margin of error. The resultant dual-mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear, constrained systems. >

The commenter points out earlier work on the problem addressed in the above-titled paper by D.Q. Mayne and H. Michalska (ibid., vol.35, no.7, p.814-824, July 1990). In their reply, the authors acknowledge that they overlooked the earlier paper and point out differences in their treatment of the subject. >

This paper presents an algorithm for minimizing an objective function subject to conventional inequality constraints as well as to inequality constraints of the functional type: \max_{\omega \in \Omega} \phi(z,\omega) \leq 0 , where Ω is a closed interval in R , and z \in R^{n} is the parameter vector to be optimized. The algorithm is motivated by a standard earthquake engineering problem and the problem of designing linear multivariable systems. The stability condition (Nyquist criterion) and disturbance suppression condition for such systems are easily expressed as a functional inequality constraint.

Suppose that, owing to manufacturing error, the actual values of a set of parameters differ from the chosen nominal values; specifically the actual parameter vector may be anywhere in a tolerance region whose center is the nominal parameter vector. The problem considered in this paper is the determination of a set of nominal parameters so that, for every possible set of actual parameters in the tolerance region, all the design specifications can be met by tuning. A fully implementable algorithm, based on the cut-map algorithms of Eaves and Zangwill, for solving this problem is presented. The algorithm extends an earlier algorithm for the pure tolerance problem (no tuning).

We present an algorithm for solving singular value inequalities over a continuum of frequencies. The algorithm is in two parts: a master algorithm which constructs an infinite sequence of finite sets of inequalities, and a nondifferentiable optimization subalgorithm which solves these finite sets of inequalities.

This paper describes an exact penalty function algorithm for solving control problems with state, control, and terminal constraints and establishes its convergence properties. A convex optimal control problem is defined whose solution yields a search direction which satisfies the control constraints and reduces a first-order estimate of the exact penalty function. Step length is determined using an Armijo-like procedure. An adaptive procedure for adjusting the penalty parameter completes the algorithm.

This paper presents a new approach to on-line control system tuning, based on worst-case design using semi-infinite optimlzation, together with a plant uncertainty identification scheme which this approach requires.

An integrated approach to the design of practical adaptive control algorithms is presented. Many existing ideas are brought together, and the effect of various design parameters available to a user is explored. The theory is extended by showing how the problem of stabilizability of the estimated model can be overcome by running parallel estimators. It is shown how asymptotic tracking of deterministic set points can be achieved in the presence of unmodeled dynamics. >

This paper presents a survey of the basic aspects involved in the design of linear multivariable control systems via semi-infinite optimization. Specific topics treated are a) database and simulation requirements, b) techniques for the transcription of design specifications into semi-infinite inequalities, and c) semi-infinite optimization algorithms for control system design.

This paper describes an interactive, optimization-based multivariable control system design package which is currently under development at the University of California, Berkeley. The package will combine a number of subroutines from the Imperial College Multivariable Design System and the Kingston Polytechnic SLICE library with DELIGHT, the University of California, Berkeley, general purpose, interactive, optimization-based CAD system.

This paper shows how the design of feedback controllers for nonlinear systems may be formulated as an optimization problem with infinite dimensional constraints for which known algorithms may be employed. An important aspect is a method for reducing the time interval, required to ensure stability, to a finite value.

We present an algorithm based on Newton's method and a systematic enlargement of a feasible region for solving finitely, systems of nonlinear inequalities. The method depends crucially on the superlinear rate of convergence of Newton's method.

A minimum principle is obtained for discrete-time stochastic systems described by the stochastic difference equation x_{k+1} = A_{k}x_{k} + \phi_{k}(u_{k})+w_{k} where \{w_{k}, k = 0, ... ,N - \} is la sequence of independent random vector variables. The control action u k is constrained to belong to a compact set U k , and the set \phi_{k}(U_{k}), k = 0,..., N - 1 is convex. The system is open-loop.

Most minimal realization procedures utilize the Hankel matrix or properties of the controllability matrix. Recently a useful algorithm for obtaining minimal realizations has been developed by Rosenbrock using the system theory developed by him. This note shows that the algorithm can be simply developed using the properties of similarity transformation and matrix reduction procedures.

A basic algorithm for the calculation of pseudoinverses is presented and verified. Two published algorithms are shown to be special cases of the basic algorithm.