^{1,2,3}

^{2}

^{1}

^{2}

^{3}

This paper develops the optimal fault-tolerant guaranteed cost control scheme for a batch process with actuator failures. Based on an equivalent two-dimensional Fornasini-Marchsini (2D-FM) model description of a batch process, the relevant concepts of the fault-tolerant guaranteed cost control are introduced. The robust iterative learning reliable guaranteed cost controller (ILRGCC), which includes a robust extended feedback control for ensuring the performances over time and an iterative learning control (ILC) for improving the tracking performance from cycle to cycle, is formulated such that it cannot only guarantee the closed-loop convergency along both the time and the cycle directions but also satisfy both the

Though studies on batch process control can be dated back to 1930s [

Meanwhile, the demand for productivity leads increasingly for the chemical plant to operate under challenging conditions, which consequently exposes the possibility of system failures. A chemical process typically has a large number of measurements and actuators. If a failure is not controlled promptly with a proper corrective action, it will degrade the process performance and even result in safety problems for the plant and personnel. Therefore, from safety as well as performance point of view, it is interesting in studying the problem of fault-tolerant control (FTC) for the system with actuator failures. Reliable control is a popular FTC method [

To exploit the repetitive nature of batch processes, iterative learning control (ILC) has been used widely. In [

In fact, there are two major issues in the robust controller design. The first is concerned with the robust stability of the uncertain closed-loop system, and the other is the robust performance. Note that the latter is more important since when controlling a system dependent on uncertain parameters, it is always desirable to design a control system which is not only stable but also guarantees an adequate level of performance. Since the so-called guaranteed cost control approach first introduced by Chang and Peng [

In this paper, according to actuator failures, the authors use the 2D theory to model, analyze, and design robust iterative learning fault-tolerant control (ILFTC) system. We first propose a 2D controller, which essentially consists of two types of controls: one is robust feedback control with extended information, using the real-time information to ensure the robust control performances along time, and the other is ILC, improving the tracking performance from cycle to cycle. Then we establish a 2D-FM model serving as a process model for the proposed design. Thus, sufficient conditions for the existence of the robust optimal ILRGCC are constructed. Further, a convex optimization problem is introduced to find the controller which minimizes the upper bound on the cost function. In addition, to solve the nonrepeatable perturbation, the control law will satisfy the

Throughout this paper, the following notations are used:

Consider process

For control input

Denote

The control objective is to design a fault-tolerant guaranteed cost control law such that the output of the process tracks a given trajectory,

For process

Design

On the other hand, for the ILC scheme, the use of more learning information may lead to a better control performance. An advantage of using the design framework proposed in this paper is that the learning information used by the ILC law can be flexibly extended, along the time and/or the cycle. In order to achieve steady-state tracking error along the time direction to be of fast convergence, here assume that the general model of the feedback/feed-forward control to be extended is described by the following linear dynamical model:

Thereby, an equivalent 2D system description of the previous batch process, combination of the extended model

Model

For 2D system

Associated with 2D system (

Introduce the following definitions to establish a procedure for the design of updating law

Denote

Assume that the

Assume that the

For a given scalar

the resulting closed-loop system (

with the zero initial condition, the controlled output

in the case when

The 2D closed-loop system

for any boundary conditions, any admissible actuator faults satisfy (

Similar results can be obtained for

The 2D closed-loop system

for any

The 2D closed-loop system

for any

In this section, we will design a reliable updating law

For any matrices

Assume that

Consider the 2D system (

Design the following quadratic Lyapunov function:

Since the inequality (

In addition, according to the definitions of

From Definition

Obviously, the system (

This completes the proof of

The robust

To solve the

Similar to Theorem

The closed-loop system (

Schematic diagram of the structure of a closed-loop system.

From the definitions of

In order to obtain the controller

It can be seen from Theorem

Injection molding, which is a batch process, mainly consists of three phases: filling, packing, and cooling [

The initial state satisfies condition (

Choose the weighting matrices

In this simulation case, we first consider repetitive parameter perturbations, that is,

(a) The guaranteed cost

(a) Tracking performances in Case

In this case, according to nonrepeatable perturbations

By an LMI framework, the optimal fault-tolerant guaranteed cost control problem via a robust and

This work is supported in part by NSFC/RGC joint Research Scheme (under Project no. N-HKUST639/09), NSFC (through Project nos. 60931160440, 61104058), and Guangzhou Scientific and Technological Project (through Project no. 11F11140010).