168 research outputs found
Stability Verification of Neural Network Controllers using Mixed-Integer Programming
We propose a framework for the stability verification of Mixed-Integer Linear
Programming (MILP) representable control policies. This framework compares a
fixed candidate policy, which admits an efficient parameterization and can be
evaluated at a low computational cost, against a fixed baseline policy, which
is known to be stable but expensive to evaluate. We provide sufficient
conditions for the closed-loop stability of the candidate policy in terms of
the worst-case approximation error with respect to the baseline policy, and we
show that these conditions can be checked by solving a Mixed-Integer Quadratic
Program (MIQP). Additionally, we demonstrate that an outer and inner
approximation of the stability region of the candidate policy can be computed
by solving an MILP. The proposed framework is sufficiently general to
accommodate a broad range of candidate policies including ReLU Neural Networks
(NNs), optimal solution maps of parametric quadratic programs, and Model
Predictive Control (MPC) policies. We also present an open-source toolbox in
Python based on the proposed framework, which allows for the easy verification
of custom NN architectures and MPC formulations. We showcase the flexibility
and reliability of our framework in the context of a DC-DC power converter case
study and investigate its computational complexity
On feasibility, stability and performance in distributed model predictive control
In distributed model predictive control (DMPC), where a centralized
optimization problem is solved in distributed fashion using dual decomposition,
it is important to keep the number of iterations in the solution algorithm,
i.e. the amount of communication between subsystems, as small as possible. At
the same time, the number of iterations must be enough to give a feasible
solution to the optimization problem and to guarantee stability of the closed
loop system. In this paper, a stopping condition to the distributed
optimization algorithm that guarantees these properties, is presented. The
stopping condition is based on two theoretical contributions. First, since the
optimization problem is solved using dual decomposition, standard techniques to
prove stability in model predictive control (MPC), i.e. with a terminal cost
and a terminal constraint set that involve all state variables, do not apply.
For the case without a terminal cost or a terminal constraint set, we present a
new method to quantify the control horizon needed to ensure stability and a
prespecified performance. Second, the stopping condition is based on a novel
adaptive constraint tightening approach. Using this adaptive constraint
tightening approach, we guarantee that a primal feasible solution to the
optimization problem is found and that closed loop stability and performance is
obtained. Numerical examples show that the number of iterations needed to
guarantee feasibility of the optimization problem, stability and a prespecified
performance of the closed-loop system can be reduced significantly using the
proposed stopping condition
Multi-parametric programming : novel theory and algorithmic developments
Imperial Users onl
Bilevel programming for analysis of low-complexity control of linear systems with constraints
In this paper we use bilevel programming to find the maximum difference between a reference controller and a low-complexity controller in terms of the infinity-norm difference of their control laws. A nominal MPC for linear systems with constraints, and a robust MPC for linear systems with bounded additive noise are considered as reference controllers. For possible low-complexity controllers we discuss partial enumeration (PE), Voronoi/closest point, triangulation, linear controller with saturation, and others. A small difference in the norm between a low-complexity controller and a robust MPC may be used to guarantee closed-loop stability of the low-complexity controller and indicate that the behaviour or performance of the low-complexity controller will be similar to that of the reference one. We further discuss how bilevel programming may be used for closed-loop analysis of model reduction
Special Bilevel Quadratic Problems for Construction of Worst-Case Feedback Control in Linear-Quadratic Optimal Control Problems under Uncertainties
Almost all mathematical models that describe processes, for instance in industry, engineering or natural sciences, contain uncertainties which arise from different sources. We have to take these uncertainties into account when solving optimal control problems for such processes. There are two popular approaches : On the one hand the so-called closed-loop feedback controls, where the nominal optimal control is updated as soon as the actual state and parameter estimates of the process are available and on the other hand robust optimization, for example worst-case optimization, where it is searched for an optimal solution that is good for all possible realizations of uncertain parameters.
For the optimal control problems of dynamic processes with unknown but bounded uncertainties we are interested in a combination of feedback controls and robust optimization. The computation of such a closed-loop worst-case feedback optimal control is rather difficult because of high dimensional complexity and it is often too expensive or too slow for complex optimal control problems, especially for solving problems in real-time. Another difficulty is that the process trajectory corresponding to the worst-case optimal control might be infeasible. That is why we suggest to solve the problems successively by dividing the time interval and determining intermediate time points, computing the feedback controls of the smaller intervals and allowing to correct controls at these fixed intermediate time points. With this approach we can guarantee that for all admissible uncertainties the terminal state lies in a given prescribed neighborhood of a given state at a given final moment. We can also guarantee that the value of the cost function does not exceed a given estimate.
In this thesis we introduce special bilevel programming problems with solutions of which we may construct the feedback controls. These bilevel problems can be solved explicitly. We present, based on these bilevel problems, efficient methods and approximations for different control policies for the combination of feedback control and robust optimization methods which can be implemented online, compare these approaches and show their application on linear-quadratic control problems
Bilevel optimization for bunching mitigation and eco-driving of electric bus lines
The problems of bus bunching mitigation and of the energy management of groups of vehicles are traditionally treated separately in the literature, and formulated in two different frameworks. The present work bridges this gap by formulating the optimal control problem of the bus line eco-driving and regularity control as a smooth, multi-objective nonlinear program. Since this nonlinear program only has few coupling variables, it is shown how it can be solved in parallel aboard each bus such that only a marginal amount of computations need to be carried out centrally. This leverages the decentralized structure of a bus line by enabling parallel computations and reducing the communication loads between the buses, which makes the problem resolution scalable in terms of the number of buses. Closed-loop control is then achieved by embedding this procedure in a model predictive control. Stochastic simulations based on real passengers and travel times data are realized for several scenarios with different levels of bunching for a line of electric buses. Our method achieves fast recoveries to regular headways as well as energy savings of up to 9.3% when compared with traditional holding or speed control baselines
Computation of Input Disturbance Sets for Constrained Output Reachability
Linear models with additive unknown-but-bounded input disturbances are
extensively used to model uncertainty in robust control systems design.
Typically, the disturbance set is either assumed to be known a priori or
estimated from data through set-membership identification. However, the problem
of computing a suitable input disturbance set in case the set of possible
output values is assigned a priori has received relatively little attention.
This problem arises in many contexts, such as in supervisory control, actuator
design, decentralized control, and others. In this paper, we propose a method
to compute input disturbance sets (and the corresponding set of states) such
that the resulting set of outputs matches as closely as possible a given set of
outputs, while additionally satisfying strict (inner or outer) inclusion
constraints. We formulate the problem as an optimization problem by relying on
the concept of robust invariance. The effectiveness of the approach is
demonstrated in numerical examples that illustrate how to solve safe reference
set and input-constraint set computation problems
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