探讨无偏移线性MPC的动态稳定性及其应用
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无偏移线性模型预测控制（Offset-free Linear Model Predictive Control, OFL-MPC）是一种先进的控制策略，其核心理念是确保在稳定状态下受控输出能够精确地达到预设的设定值。OFL-MPC的实现涉及几个关键组件：组合估计器，用于处理系统的人工干扰和实时状态估计；稳态目标计算，这在双层MPC架构中尤为重要，即控制策略分为上下两层，分别负责短期优化和长期规划；以及动态控制器，它根据系统的当前状态和预测来实时调整控制输入。
文献中对OFL-MPC的稳定性条件已有详尽研究。在理想情况下，这些条件通常建立在系统渐近稳定的假设之上，这意味着系统的误差会随着时间的推移无限接近于零。然而，双层MPC中的动态稳定性问题却相对较少被关注。动态稳定性涉及到控制器在不断变化的系统环境中如何保持稳定性能，例如，当系统受到外部扰动或者内部参数变化时，控制器能否继续保持输出的无偏移特性。
在实际应用中，确保动态稳定性至关重要，因为它直接影响到控制系统的鲁棒性和响应速度。动态稳定性分析可能包括考察控制器的抗干扰能力、系统的瞬态响应、以及在存在不确定性或非线性因素时的适应性。对于动态稳定性，研究人员可能会探讨以下几点：
1. 控制器设计：如何设计一个能够自适应地调整参数，以补偿系统动态变化的控制器，确保在任何时刻都能维持输出的准确性。
2. 耐受性分析：评估控制器在面对各种扰动（如噪声、模型不准确等）时的性能，以保证系统在实际运行中的稳定性。
3. 稳定性边界：确定在何种程度的扰动下，OFL-MPC还能保持无偏移特性，这对于系统的安全运行至关重要。
4. 滚动优化与预测更新：频繁地更新预测模型和优化策略，使得控制器能快速适应环境变化，同时保持稳定性能。
尽管渐近稳定性是OFL-MPC的基础，但深入理解并优化动态稳定性对于提升整体控制性能和实际应用的可行性具有重要意义。因此，进一步研究动态稳定性条件、改进算法和控制器设计，对于无偏移线性模型预测控制的未来发展具有深远影响。未来的研究可能需要结合实际系统模型和应用背景，以寻求更高效、鲁棒的动态控制策略。
A Discussion on Stability of Offset-free Linear Model Predictive Control
Baocang Ding, Tao Zou and Hongguang Pan
Abstract— An offset-free controller is one that drives con-
trolled outputs to their set-point values at the steady state. In
the literature, conditions for offset-free linear model predictive
control are given for combined estimator (for both the artificial
disturbance and system state), steady-state target calculation,
and dynamic controller. In the presence of steady-state target
calculation, the framework for MPC is called the double-layered
MPC. Usually, the offset-free property of the double-layered
MPC is obtained under the assumption that the system is
asymptotically stable. This paper discusses on the dynamic
stability property which has been rarely considered in the
literature of double-layered MPC.
Index Terms— Model predictive control, Offset-free control,
Stability
I. INTRODUCTION
Model predictive control (MPC) differs from other control
methods mainly in its implementation of the control moves.
Usually, MPC solves a finite-horizon optimal control prob-
lem at each sampling instant, so that the control moves for
the current time and a period of future time (say, totally N
sampling intervals) are obtained. However, only the current
control move is applied to the plant. At the next sampling
instant, the same kind of optimization is repeated with the
new measurements [12]. One is used to compare implement-
ing MPC to passing the street or playing chess which have
a similar pattern with MPC: acting while optimizing. In a
degree, for a lot of engineering problems the unique pattern
of MPC is not artificial, but inevitable [4].
Most of the MPC procedures applied in the industrial
processes lack theoretical guarantee of stability. Usually,
industrial MPC adopts a finite-horizon optimization, without
a special weighting on the output prediction at the end of
the prediction horizon. Theoretically, the regulation problem
for the nominal MPC can have guarantee of stability by
imposing a special terminal weighting on the system state
predictions and a special constraint on the terminal state
prediction [10]. The authors in [10] give a comprehensive
framework. However, [10] does not solve everything for the
stability of MPC. Many studies should be developed, perhaps
without an end. For example, in the past 10 years, the studies
on the robust MPC for regulation problem go far beyond
[10].
This work is supported by National Nature Science Foundation under
Grant 60934007, Grant 61174095 and Grant 61074059.
Baocang Ding and Hongguang Pan are with the Ministry of Education
Key Lab For Intelligent Networks and Network Security (MOE KLINNS
Lab), Department of Automation, School of Electronic and Information
Engineering, Xi’an Jiao Tong University, Xi’an 710049, P. R. China
baocang.ding@gmail.com; hongguangpan@163.com
Tao Zou is with the Department of Automation, Zhejiang University of
Technology, Hangzhou, 310032, P. R. China
tzou@zjut.edu.cn
We could say that, for the case of regulation problem
when the system state is measurable, the research on MPC
is becoming mature (see e.g. [2]–[5], [8]). For the case of
regulation problem when the system state is unmeasurable,
and there is no model parametric uncertainty, the research
on MPC is becoming mature (see e.g. [14]–[16]). For other
cases (output feedback MPC for the systems with parametric
uncertainties, tracking MPC, etc.), there are many undergo-
ing researches.
The industrial MPC adopts a more complex framework
than the synthesis approaches of MPC (a synthesis approach
of MPC is that with guaranteed stability). Its hierarchy is
shown in, e.g., [6]. In other words, the synthesis approaches
of MPC have not been sufficiently developed to include the
industrial MPC.
The goal of the steady-state target calculation is to re-
calculate the targets from the local optimizer every time
the MPC controller executes. This must be done because
disturbances entering the systems or new input information
from the operator may change the location of the optimal
steady-state. Today, the separation of the MPC algorithm into
steady-state target and dynamic control move calculations is
a common part of industrial MPC technology [6].
In the linear MPC framework, offset-free control is usually
achieved by adding step disturbance to the process model.
The most widely-used industrial MPC implementations as-
sume a constant output disturbance that can lead to sluggish
rejections of disturbances that enter the process elsewhere. In
[9], [11], some general disturbance models that accommodate
unmeasured disturbances entering through the process input,
state, or output, have been proposed.
In a more general sense, the disturbance model can incor-
porate any nonlinearity, uncertainty, and physical disturbance
(measured or unmeasured). The disturbance can be estimated
by the Kalman filter (or the usual observer). The estimated
disturbance is assumed to be step-like, i.e., unchanging in the
future, at each sampling time (MPC refreshes its solution
at each sampling time). The estimated disturbance drives
the steady-state target calculation, in order to refresh the
new target value for the control move optimization. MPC
incorporates both the steady-state target calculation and the
control move optimization is called doubled-layered MPC.
When the system state is also unmeasurable, the doubled-
layered MPC is the output tracking MPC (see e.g. [13]).
In this paper, we give some preliminary results for the
stability of double-layered MPC or output tracking MPC.
These results could be useful for incorporating the industrial
MPC into the synthesis approaches of MPC.
Notations: For any vector x and positive-definite matrix
80
978-1-4577-2072-7/12/$26.00
c
2012 IEEE
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