10,404 research outputs found
Stability of hybrid model predictive control
In this paper we investigate the stability of hybrid systems in closed-loop with Model Predictive
Controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability and
exponential stability. A general theory is presented which proves that Lyapunov stability is achieved for
both terminal cost and constraint set and terminal equality constraint hybrid MPC, even though the
considered Lyapunov function and the system dynamics may be discontinuous. For particular choices
of MPC criteria and constrained Piecewise Affine (PWA) systems as the prediction models we develop
novel algorithms for computing the terminal cost and the terminal constraint set. For a quadratic MPC
cost, the stabilization conditions translate into a linear matrix inequality while, for an 1-norm based
MPC cost, they are obtained as 1-norm inequalities. It is shown that by using 1-norms, the terminal
constraint set is automatically obtained as a polyhedron or a finite union of polyhedra by taking a
sublevel set of the calculated terminal cost function. New algorithms are developed for calculating
polyhedral or piecewise polyhedral positively invariant sets for PWA systems. In this manner, the on-line
optimization problem leads to a mixed integer quadratic programming problem or to a mixed integer
linear programming problem, which can be solved by standard optimization tools. Several examples
illustrate the effectiveness of the developed methodology
Approximate Dynamic Programming for Constrained Piecewise Affine Systems with Stability and Safety Guarantees
Infinite-horizon optimal control of constrained piecewise affine (PWA)
systems has been approximately addressed by hybrid model predictive control
(MPC), which, however, has computational limitations, both in offline design
and online implementation. In this paper, we consider an alternative approach
based on approximate dynamic programming (ADP), an important class of methods
in reinforcement learning. We accommodate non-convex union-of-polyhedra state
constraints and linear input constraints into ADP by designing PWA penalty
functions. PWA function approximation is used, which allows for a mixed-integer
encoding to implement ADP. The main advantage of the proposed ADP method is its
online computational efficiency. Particularly, we propose two control policies,
which lead to solving a smaller-scale mixed-integer linear program than
conventional hybrid MPC, or a single convex quadratic program, depending on
whether the policy is implicitly determined online or explicitly computed
offline. We characterize the stability and safety properties of the closed-loop
systems, as well as the sub-optimality of the proposed policies, by quantifying
the approximation errors of value functions and policies. We also develop an
offline mixed-integer linear programming-based method to certify the
reliability of the proposed method. Simulation results on an inverted pendulum
with elastic walls and on an adaptive cruise control problem validate the
control performance in terms of constraint satisfaction and CPU time
Reliably-stabilizing piecewise-affine neural network controllers
A common problem affecting neural network (NN) approximations of model
predictive control (MPC) policies is the lack of analytical tools to assess the
stability of the closed-loop system under the action of the NN-based
controller. We present a general procedure to quantify the performance of such
a controller, or to design minimum complexity NNs with rectified linear units
(ReLUs) that preserve the desirable properties of a given MPC scheme. By
quantifying the approximation error between NN-based and MPC-based
state-to-input mappings, we first establish suitable conditions involving two
key quantities, the worst-case error and the Lipschitz constant, guaranteeing
the stability of the closed-loop system. We then develop an offline,
mixed-integer optimization-based method to compute those quantities exactly.
Together these techniques provide conditions sufficient to certify the
stability and performance of a ReLU-based approximation of an MPC control law
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