Piecewise Sliding Mode Decoupling Fault Tolerant Control System

Proposed method in the present paper could deal with fault tolerant control system by using the so called decentralized control theory with decoupling fashion sliding mode control, dealing with subsystems instead of whole system, and also in this paper we present a new design scenario to implement this method and design the controller


INTRODUCTION
The Fault tolerant control (FTC)could be a passive one or an active one [1,2].Nonlinear control scheme is one of the most widely used active(FTC)methods, because several nonlinear control methods such as sliding mode controller, feedback linearization, etc., could handle it Reconfigurable control is a critical technology [3,4] with its objectives to detect the fault and recover the functionality of the faulty system as same as that of the nominal system. Such control ideas have been implemented on a variety of military and commercial applications in last two decades to accommodate fault.
The idea to use sliding mode control for reconfiguration purposes come from the fact that this method alleviates the problems caused by uncertain or changing system dynamics or parameters. This is the case when a fault occurs in a system component Sliding mode control commutates in order to force the systems motions to act on a desired surface ( sliding surface). Sliding regimes are unaffected by perturbations satisfying the well-known matching conditions [5] The choice of the surface is related to some stabilization problem: the shape of the surface is selected a priori, leading to a set of parameters that are to be computed (adjusted) in order to obtain the desired dynamics [6].

II. PROBLEM STATMENT
Typical description for the system uncertainty caused by system faults [9] can be represented with the control input, and ( ) d t disturbance assumed to be bounded as ( ) A control law u can be designed to make the second order system (2) arrive at our control goal. However, for nonlinear MIMO models, the system dynamic is generally not in a canonical form. Rather, it has a form shown below: nonlinear functions, 1 u , 2 u are the control inputs, and 1 ( ) d t , 2 ( ) d t are external disturbances. The disturbances are assumed to be bounded as (3), one can design 1 u and 2 u respectively, as subsystem in (3). However in complex systems, where it may be presented as (3) but hard to make a subsystem control on it, could go through the idea of decoupled method to design a control u to govern the whole system.

IV. DESIGN DECOUPLED SKIDING MODE CONTROLLER
Suppose, it is possible to design a control that constrains the motion of the system to the manifold, The best approximation u of a continuous control law that would achieve 1 0 s = is thus The decoupled SMC input is to be chosen as follows for a Lyapunov function candidate And its derivative gives ( ) we can now guarantee that (10) verified. Remark 1: -as in (6), (7) and (11) respectively we could get the control law base on the second sliding surface 2 s , however, to well pose the decentralized method we select just one sliding surface 1 s , since the system is divided in two subsystem, we control the first one and take the other as feeding information for the first one, both sliding surface may be connected via z , since they are analogical and one could be augmented by the term ( 1 c z − ) as in (4), c and 2 c has strong influence on the behavior in the transient state of the system. Appropriate choice of sliding factor is necessary for achieving favorable transient response.
The control input is SMC of subsystem chosen, since in SMC theory it is assumed that 1 u u = to control the entire system, the boundary of 1 x can be assured with 0 1 u z ≺ ≺ and u z z ≤ ,means that absolute value 1 x is always bounded. Here u z is the upper limit of z therefore, z is a decaying signal, since u z is less than one.
The control action is accomplished as follows: The main object of (4) is to make 1 x and 2 x equal to zero according to the sliding-mode control theory. But when 2 0 s ≠ then 0 z ≠ in (4). This causes (6) to apply an input such that z is decreased. When z is decreased, 2 s will be decreased too.

A. piecewise sliding surface
Taking the sliding surface presented in the (4) could be written as follow: Where 1 c and Z are parameters of sliding surface S that could be written in subscript fashion such: as seen in remark1 where 0 1 We take u z z ≤ (16) The eq.(16) shows that the sliding mode is not unique, and i G will take different value even by fixing i C Still need to specify regions where the sliding occurs following the partition that we may choose.

B. Controller design
The concern now is that the state 1 1 ( , ) x x can slide along the piecewise sliding mode In way to use PWL tools elaborated in [7], equation in (15) and (16) used as piecewise sliding surface partition. then seek matched operation points with partitions, where sliding mode happen, then we select some of those point to linearize the system in way to get good approximation model of the system, and then design the controller with techniques explained in this paragraph, following the flowchart.
We write the equation out with the new defined states and by Substituting (20) into (19) we get: We can now write this in state space form Where D is zeros matrix.  The objective is to maintain the level of the tank 2 at a desired height and keep the changes of the level within a certain limit in the presence of process faults, thus, the system has one output, i.e. the liquid level in tank 2.
The fault considered is the leakage of tank 2 occurring in the draining valve, 2 V which can be modeled by an abrupt change in the flow resistance 2 R .and the dynamic relationships between the height of the levels and the inlet and outlet flow rates as follows:

A. Detection sliding mode
The choice of i G is piecewise heuristic under the condition given in (16).and the LMI feasibility problem computation to get So far we have obtained the mathematical model of the whole tank system depicted previously in the region where sliding mode occur following the algorithm described in section 5.In fact, the drainage of tank 1, tank 2 and tank 3 are assumed to be zero under normal operation, i.e. outlet valves are supposed closed. In this case.