# Discrete-time MPC with Prescribed Degree of Stability

## Abstract

In this chapter, we discuss discrete-time model predictive control with a prescribed degree of stability. In Section 4.2, we begin with illustrative examples to show how the prediction horizon in the discrete-time model predictive control affects the stability and numerical condition of the algorithm. In Section 4.3, we propose the use of an exponentially weighted moving horizon window in model predictive control design, which converts the numerically ill-conditioned Hessian matrix into a numerically well-conditioned Hessian in the presence of a large prediction horizon. The solution is extended to predictive control with an infinite horizon. In Section 4.4, asymptotic stability for predictive control designed using an infinite horizon is achieved through exponential data weighting and modification of the weight matrices. In Section 4.5 we show how to design a predictive control system with a prescribed degree of stability. In Section 4.6, we discuss the tuning parameters in the design of model predictive control systems. In Section 4.7, we introduce exponentially weighted constrained control. In the final section (see Section 4.8), an additional benefit when using exponential data weighting is illustrated through a case study of predictive control of a complex system. The results are established first by using the control vector *Δu* with the Laguerre pole *a* = 0, and then extended to the general case using Laguerre functions where 0 ≤ *a* < 1. The case of smaller *N* is of particular interest to us as an approximation to the optimal closed-loop performance (see Section 4.6).

## Keywords

Model Predictive Control Prediction Horizon Laguerre Function Recede Horizon Control Prescribe Degree## Preview

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