Stabilization of Nonlinear Systems with Uncertain Equilibrium States with Application to Energy Efficient HVAC Systems

压缩机是现代建筑供暖、通风和空调（HVAC）系统的主要驱动部件。这些压缩机的性能和效率特征由它们的特性曲线给出，该曲线反映了压缩机的稳态压力上升随气体流量的变化。特性曲线的峰值不仅决定了压缩机的最大运行能力/效率，而且还标志着系统不稳定的起始点，即压缩机喘振。电涌的控制方法在文献中得到了广泛的研究。这些控制器将允许HVAC系统在其最高效率下安全运行。然而，这些喘振控制方法的一个共同限制是，它们需要准确了解压缩机的稳态特性，并且当特性曲线引入误差时，它们的有效性会迅速下降。本研究项目的目的是为这种情况和类似情况开发必要的理论工具，并使用已开发的理论来设计未知或不确定压缩机特性曲线的喘振控制器。将开发分析工具，以在开发的控制器下最大化HVAC系统的安全操作区域，并量化系统可以容忍的特性曲线中的最大不确定性水平。最后，这些结果将导致浪涌控制方法的发展，使暖通空调系统在其最高能效条件下安全连续运行，并且对可能影响系统稳态特性的外部条件变化具有鲁棒性。本课题的主要研究目标是为具有未知或不确定平衡状态的非线性系统的鲁棒镇定发展新的控制理论方法。大多数非线性系统的分析和控制方法都假定对平衡状态有精确的了解。然而，由于模型的不确定性和/或系统的混沌动力学，当涉及复杂的非线性动力学时，并不总是能够准确地获得这些信息。当非线性系统受到严格的控制约束时，不确定平衡状态带来的挑战会被放大。例如，不确定性可能在控制工作中引入静态偏移，从而限制了可用于补偿不需要的外部干扰的控制。此外，不确定的平衡状态会降低对被控对象的建模假设的准确性，从而增加开环动力学的不确定性。这些因素可以显著减少稳态的吸引域，即使达到稳定。本研究将提供理论论据，以证明文献中所考虑的控制未知稳态混沌系统的方法的有效性。此外，这些控制方法将扩展到使用鲁棒和最优控制技术的新功能。所提出的研究将为具有不确定稳态的约束系统提供新的控制方法，以及最大化相应吸引力域的优化方法。本研究的激励应用是主动控制压缩机喘振，以达到提高暖通空调系统能效的目的。将所开发的控制方法应用于具有不确定特性曲线的浪涌的主动控制将证明该项目所开发的方法的工业相关性。

Compressors are the main driving components of heating, ventilation and air conditioning (HVAC) systems in modern buildings. The performance and efficiency characteristics of these compressors are given by their characteristic curves, which map the steady-state compressor pressure rise as a function of the gas flow-rate. The peak of the characteristic curve not only determines the maximum operating capacity/efficiency of a compressor, but it also marks the inception point for a system instability known as compressor surge. Control methods for surge have been studied extensively in the literature. These controllers would allow HVAC systems to operate safely at their peak efficiency. However, a common limitation of these surge control methods is that they require exact knowledge of the compressor steady-state characteristics, and their effectiveness degrades rapidly when errors are introduced to the characteristic curve. The objective of this research project is to develop the necessary theoretical tools for this and similar situations, and to use the developed theory to design surge controllers for unknown or uncertain compressor characteristic curves. Analytical tools will be developed to maximize the safe operating region of an HVAC system under the developed controllers, and to quantify the maximum level of uncertainty in the characteristic curve that the system can tolerate. Finally, these results will lead to the development of a surge control method that allows HVAC systems to operate safely and continuously at their peak energy efficiency condition, and is robust to variations in external conditions that may affect the steady state characteristics of the system. The main research objective of this project is to develop new control-theoretic methods for the robust stabilization of nonlinear systems with unknown or uncertain equilibrium states. Most methods for the analysis and control of nonlinear systems assume precise knowledge of the equilibrium states. It is however not always possible to obtain this information accurately when complex nonlinear dynamics are involved, due to model uncertainties and/or system chaotic dynamics. The challenges introduced by uncertain equilibrium states are amplified when nonlinear systems are subjected to strict control constraints. For example, uncertainties may introduce a static offset in the control effort, limiting the control available to compensate for unwanted external disturbances. Furthermore, uncertain equilibrium states can reduce the accuracy of modeling assumptions on the controlled plant, adding further uncertainty to the open loop dynamics. These factors can significantly reduce the domain of attraction of the steady states, even if stability is achieved. This research will provide theoretical arguments to demonstrate the validity of methods considered in the literature for the control of chaotic systems with unknown steady states. Furthermore, these control methods will be extended to incorporate new capabilities using robust and optimal control techniques. The proposed research will offer novel control methods for constrained systems with uncertain steady states, and optimization methods to maximize the corresponding domains of attraction. The motivating application of this research is active control of compressor surge to achieve increased energy efficiency of HVAC systems. Application of the developed control methods to the active control of surge with uncertain characteristic curves will demonstrate the industrial relevance of the methods developed in the project.