An internal model control design method for failure-tolerant control with multiple objectives

Abstract

The selection of control structure is a critical step in the design of a control system since it can largely affect the achievable control performance. This paper presents a general internal model control (IMC) structure with multiple degrees-of-freedom for the design of control systems with multiple objectives (i.e., reference tracking and load and output disturbance rejection), so that controllers are designed independently of each other. The optimal performance of controllers is shown to remain intact when a controller is taken off-line, for example, due to actuator and/or sensor failures. This feature circumvents the need to redesign the remaining on-line controllers for optimal failure-tolerant control. The proposed control approach is used to derive IMC control structures for multi-loop cascade and coordinated control systems. The performance of the control approach is demonstrated on a simulated thin-film drying process in continuous pharmaceutical manufacturing for several multi-loop control structures and a variety of control-loop component failures.

Introduction

The design of the control structure, which is the specification of the interconnection of measurements, exogenous inputs, and manipulated variables, greatly influences the performance achievable by a control system. In practice, control systems are commonly required to fulfill multiple objectives in terms of reference tracking and load and output disturbance rejection, which cannot be adequately specified using a single performance measure. This realization has led to the development of numerous control structures that have multiple degrees-of-freedom, where each degree-of-freedom is tasked with addressing some subset of the control objectives (Grimble, 1988, Brosilow, Markale, 1992, Limebeer, Kasenally, Perkins, 1993, Pottmann, Henson, Ogunnaike, Schwaber, 1996, Zhou, Ren, 2001, Dehghani, Lanzon, Anderson, 2006, Vilanova, Serra, Pedret, Moreno, 2006, Liu, Zhang, Gao, 2007, Vidyasagar, 2011). As failures in system components inevitably occur in practice, an important practical consideration in control structure selection is to ensure that the selected control structure and its associated controllers have graceful performance degradation during component failures (Blanke, Kinnaert, Lunze, Staroswiecki, 2003, Isermann, 2006, Zhou, Frank, 1998, Paulson 2019).

Internal model control (IMC) (Morari and Zafiriou, 1989) is a control design method developed in the 1970s–1980s with several useful features, including that it provides a convenient theoretical framework for the design of two degrees-of-freedom control systems (i.e., feedback and feedforward controller) when model uncertainty is present (Vilanova, 2007). The basic idea of IMC control design is to combine an optimal controller obtained from the nominal process model with a low-pass filter to tradeoff closed-loop performance with robustness to model uncertainties. As the IMC structure is a particular case of the Youla-Jabr-Bongiorno-Kucera (or Youla for short) parametrization of controllers that preserve closed-loop stability, the original control design problem can be replaced by simply the selection of an arbitrary parameter that appears affinely in the closed-loop transfer function. This allows the IMC control structure to ensure internal nominal stability of the closed-loop system (Morari, Zafiriou, 1989, Braatz, 1996). Furthermore, the IMC control structure allows for separate controller design for performance and robustness to alleviate the tradeoff between performance and robustness in the traditional feedback framework (Zhou and Ren, 2001).

This paper addresses the IMC control design with optimal failure tolerance. An extension of the IMC control structure is presented for failure-tolerant control with multiple objectives related to reference tracking and rejection of load and output disturbances. The proposed IMC control structure enables the design of control systems with multiple degrees-of-freedom, where a controller for each exogenous input is designed independently of the other controllers. Thus, the control structure preserves the optimal performance of the remaining controllers when any of the controllers are taken off-line (e.g., due to actuator and/or sensor failures). That is, without compromising the best achievable control performance, the proposed IMC control structure alleviates the inherent suboptimality of a classical feedback control structure in dealing with failures of one or more controllers in a control system with multiple objectives. Additionally, the proposed IMC approach to failure tolerance is distinct from robust control methods that account for potential actuator and/or sensor failures in that these methods can result in overly conservative performance when no actuator and/or sensor failures occur. This is because the control system is designed with respect to worst-case performance in robust control methods (e.g., see Zhou and Ren, 2001).

The proposed multiple degrees-of-freedom IMC controller design approach is also extended for the design of multi-loop control systems by deriving a general control structure for multi-loop cascade and coordinated control systems. The performance of the control approach is demonstrated using a thin-film drying process in continuous pharmaceutical manufacturing (Mesbah et al., 2014).

Notation and Preliminaries. Throughout the paper, a finite-dimensional process is denoted by $P\left(s\right)\in \mathcal{R}{\mathcal{H}}_{\infty },$ where s is the Laplace variable and $\mathcal{R}{\mathcal{H}}_{\infty }$ denotes the real rational subspace of ${\mathcal{H}}_{\infty }$ consisting of all proper and rational transfer matrices. The exogenous inputs are bounded signals, i.e., $r,l_m,l_u,d_m,d_u\in {\mathcal{L}}_p[0,\infty ),$ where ${\mathcal{L}}_p[0,\infty )$ encompasses all signal sequences on [0, ∞) that have finite p-norm. The real-valued function ‖ · ‖ denotes any norm defined over the linear vector space of the signals. The induced system norm | · | is defined as the supremum of an output signal norm over a norm-bounded set of an input signal (Zhou et al., 1996).

Definition 1

(Internal Stability (Morari and Zafiriou, 1989)): A linear time-invariant (LTI) closed-loop system is internally stable if transfer functions between all bounded exogenous inputs to the closed-loop system and all outputs are stable (i.e., have all poles in the open left-half plane).

Definition 2

(Robust Stability (Morari and Zafiriou, 1989)): A closed-loop system is robustly stable if the controller C ensures the internal stability of the closed-loop system for all $P\in \mathcal{P},$ where $\mathcal{P}$ is the set of uncertain processes.