Stabilization of inexact MPC schemes Markus K¨ ogel and Rolf Findeisen Abstract— In model predictive control often the underlying optimization problem is not solved exactly to meet hard real- time bounds or to save computations. This jeopardizes prop- erties of the closed loop system, such as stability, performance or recursive feasibility. We present a framework for linear system with polytopic constraints and quadratic performance criteria, which guarantees recursive feasibility and stability subject to inexact solutions. We combine the approach with simple optimization methods to obtain real-time feasibility and stability. I. I NTRODUCTION Model predictive control (MPC) is often used to stabilize and ”optimally” control constrained linear systems, c.f. [1]– [6]. By now it is well known how stability and recursive feasibility in MPC can be guaranteed, assuming that the underlying optimization problem is solved exactly, c.f. [4]– [6]. However often only inexact solutions of the required optimization problems are available, e.g. for fast systems and for control using low cost hardware. One challenge in MPC is the solution of the optimization problem, which needs to done at each time instance. By now many tailored solution approaches exist. For small systems one can use explicit MPC [7], [8], which allows to solve the optimization problem mostly offline and requires online basically only the evaluation of a look up table. The optimization problem can also be solved online. Several works consider tailored optimization algorithms to allow an efficient solution. Tailored interior-point methods are investigated in e.g. [9]–[16] and specialized active set methods in e.g. [17]. Gradient based methods or so-called first order methods are considered in [18]–[28]. Besides the advancements of the solution speed of the un- derlying optimization problem it can often not be guaranteed that the exact solution or at least a suboptimal, but feasible solution can be found within the available time. To allow a stabilizing feedback with guaranteed constraint satisfaction the inexact solution needs to be directly considered. Stability in the presence of inexact solution satisfying the equality constraints due to the dynamics, but not the state and input constraints is guaranteed in the works [20], [22] using constraint tightening. Furthermore, if a feasible solution is available, then one can guarantee, using tailored MPC setups, straightforwardly stability by employing optimization M. K¨ ogel and R. Findeisen are with the Institute for Automation Engi- neering, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany. {markus.koegel, rolf.findeisen}@ovgu.de. Partial support by the International Max Planck Research School Magde- burg and the German Research Foundation, grant FI 1505/3-1, are gratefully acknowledged. methods, which guarantee cost decrease, if possible, and maintains feasibility, see [16], [29]. We present a framework to guarantee stability subject to inexact solutions consisting of a feasibility recovery scheme and a robustified optimization problem. Compared to existing results we assume that an approximation of the exact solution is available, which satisfies the state and input constraints, but violates the equality constraints (of the dynamics). In particular, we propose a procedure to remove the mis- match in the dynamics and combine it with ideas from robust MPC, see [4], [30], to guarantee that after this recovery the constraints are still satisfied. The framework can be combined with tailored optimization methods to obtain sta- bilizing control laws with hard bounds on the computational complexity. The considered class of optimization methods is general, e.g. it includes also distributed methods, such as [27]. As an example we discuss a combination of a quadratic penalty method with Nesterov’s fast gradient method, c.f. [31], [32] and illustrate it with an example. The remainder of this paper is structured as follows. Section II states the framework and the problem. Section III contains as main result a framework to guarantee stability and constraint satisfaction in the presence of inexact solu- tions. In Section IV we combine the framework with simple optimization methods to obtain a real-time control scheme. The notation is standard. A ⊕ B, A ⊖ B denote the Minkowski sum, Minkowski difference, see [4]. For a matrix M , M> 0 means that M = M T and M is positive definite. II. PROBLEM FORMULATION This section outlines first the considered problem class and reviews MPC based on exact optimization. A. Considered problem class We consider constrained, linear, discrete time systems x(k + 1) = Ax(k)+ Bu(k), (1a) x(k) ∈ X,u(k) ∈ U, (1b) where x(k) ∈ R n is the state and u(k) ∈ R p the input. For simplicity of presentation we assume that the system is controllable. The sets X and U are compact, convex polytopes and contain the origin in their interior. B. MPC based on exact optimization A common method to stabilize the system (1) and guaran- tee constraint satisfaction is MPC. In MPC an input sequence u k and a state trajectory x k defined over a horizon N x k =  x T k|k ...x T k+N|k  T , u k =  u T k|k ...u T k+N−1|k  T , (2)