State-dependent Control Constraints Research Articles

Run-Time Efficiency of Bilinear Model Predictive Control Using Variational Methods, With Applications to Hydronic Cooling

Effective real-time execution of nonlinear model predictive control (NMPC) on embedded systems is significantly dependent on the controller formulation. This paper studies the effect of model structure and cost functions on the computation time of scalar bilinear NMPC using variational methods, with hydronic cooling applications. Two algebraically equivalent nonlinear model structures common in literature are primarily considered: a linear state equation with state-dependent control constraints and a bilinear state equation with time-invariant rectangular control constraints. Additionally, the effects of three cost function formulations are also considered: minimum-time, quadratic regulation, and efficient state constraints. High-fidelity computer simulations, hardware-in-the-loop testing, and experiments on a bench-scale hydronic cooling system are used to study sources of computational complexity, rates of convergence, initialization techniques, and overall effectiveness of the different models and costs. These results suggest that NMPC with bilinear state equations, minimizing pump power and a one-sided quadratic state cost, converges sufficiently fast and reliably. This presents an attractive alternative to the traditionally constrained linear quadratic regulator-based NMPC on embedded systems.

Convex Optimal Control Problems with Smooth Hamiltonians

Optimal control problems with convex costs, for which Hamiltonians have Lipschitz continuous gradients, are considered. Examples of such problems, including extensions of the linear-quadratic regulator with hard and possibly state-dependent control constraints, and piecewise linear-quadratic penalties are given. Lipschitz continuous differentiability and strong convexity of the terminal cost are shown to be inherited by the value function, leading to Lipschitz continuity of the optimal feedback. With no regularity assumptions on the limiting problem, epi-convergence of costs, which can be equivalently described by pointwise convergence of Hamiltonians, is shown to guarantee epi-convergence of value functions. Resulting schemes of approximating any concave-convex Hamiltonian by continuously differentiable ones are displayed. Auxiliary results about existence and stability of saddle points of quadratic functions over polyhedral sets are also proved. Tools used are based on duality theory of convex and saddle functions.

Optimality Conditions for Control Problems Involving State Dependent Control Constraints

Optimality conditions in the form of a weak nonsmooth maximum principle for control problems with state-dependent control equality constraints are derived. They apply to problems with possibly nonsmooth data and with pointwise set constaints on some components of the control variable. An example illustrates the distinction between the conditions reported in here and some weak maximum principles previously derived. Notably, it is also shown that the necessary conditions obtained are sufficient conditions for "convex normal" problems.

On the weak maximum principle for optimal control problems with state dependent control constraints

On the weak maximum principle for optimal control problems with state dependent control constraints

Structure of the time-optimal control law for multiple arms handling a common object along specified paths

The structure of the time-optimal control law for multiple robot arms cooperatively moving a common object along a specified path with the control torque constraints is addressed. The overall mechanical system is modeled by considering the arms as closed kinematic chains using the Lagrange formulation. This results in a reduced-order dynamic model of the multi-arm system. By parameterizing the resulting dynamic model along a given path, the original higher order optimal control problem with state constraints (path constraints) is transformed into a problem of a double integrator system with state-dependent control constraints that are determined by a linear programming approach. It is then shown that the number of saturated actuators on any finite time interval along the optimal trajectory is (1+3(D-1)(m-1)), where D=2 in 2-dimensional Cartesian space, D=3 in 3-dimensional space, and m is the number of robot arms in the system. The theoretical result is validated by a numerical example.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A Survey of Viability Theory

Some theorems of viability theory which are relevant to nonlinear control problems with state constraints and state-dependent control constraints are motivated and surveyed. They all deal with viable solutions to nonlinear control problems, i.e., solutions satisfying at each instant given state constraints of a general and diverse nature. Some classical results on controlled invariance of smooth nonlinear systems are adopted to the nonsmooth case, including inequality constraints bearing on the state and state-dependent constraints on the controls. For instance, existence of a viability kernel of a closed set (corresponding to the largest controlled invariant manifold) is provided under general conditions, even when the zero-dynamics algorithm does not converge. The concepts of slow and heavy viable solutions are introduced, providing concrete ways of regulating viable solutions, by closed-loop feedbacks and closed-loop dynamical feedbacks. Viability theorems also allow the extension of Lyapunov’s second method to nonsmooth observation functions and the construction of “best” Lyapunov functions. As an application, “fuzzy differential inclusion” is presented. Proofs and complements can be found in [Viability Theory, to appear, 1991]. They rely on properties of differential inclusion (see [Differential Inclusions, Springer-Verlag, Berlin, New York, 1984]) and set-valued analysis, (see [Set-Valued Analysis, Birkhäuser, Basel, 1990]).

Pareto optimality for nonlinear infinite dimensional control systems

In this note we establish the existence of Pareto optimal solutions for nonlinear, infinite dimensional control systems with state dependent control constraints and an integral criterion taking values in a separable, reflexive Banach lattice. An example is also presented in detail. Our result extends earlier ones obtained by Cesari and Suryanarayana.

Infinite Horizon Optimal Control Problems for Semilinear Evolution Equations

In this paper we establish the existence of optimal solutions over an infinite planning horizon, of control systems governed by semilinear evolution equations and having time and state dependent control constraints. In particular, we show that every asymptotically minimizing sequence has a subsequence converging in some sense to the desired optimal pair

Optimal Operations Planning in a Large Hydro-Thermal Power System

In this paper a realistic model of seasonal production planning in the Swedish State Power Board (SSPB) System is described. This model properly accounts for the nonlinearities introduced by variable head losses and state dependent control constraints in the hydro system. The seasonal hydro-thermal problem exhibits certain complex features that prevent the use of existing nonlinear programming theory for its solution. The paper presents the development of a method of solution which recast the problem into a network flow formulation and employs network flow algorithms. This paper presents the latest developments in the area of network flow modelling for operations planning at the SSPB. For implementation of this scheme, to facilitate its use by operational engineers an information system is set up which will enable efficient handling of data bases and computational routines.

On local convergence of an algorithm for optimal control

This paper presents an algorithm for the solution of optimal control problems with state-dependent control constraints. Under several assumptions that usually hold, some properties of the optimal control are derived. Furthermore, it is proven that the sequence of controls generated by the algorithm converges locally to the optimal control under an additional condition.