# Redundancy Analysis and Problem Formulation

A necessary condition for the existence of an FTC is that the system has to possess sufficient actuator redundancies to accommodate the failed control surface. In essence, FTCS design is how to manage existing redundancies in such a way that the integrity of the entire system is maintained. Therefore, the concept of redundancy is introduced in this context front.

## Redundancy Analysis

In order to meet the requirements for safety and reliability in the event of failures, the existence of redundancies is the key. Under normal conditions, the advantages of having redundant actuators may not be that obvious; however, they will play an ultimate role in counteracting failures. For clarification, some definitions for redundancy are provided herein. The relationships between the system inputs and outputs, known as functional controllability [85], are utilized to analyze the actuator redundancies.

Definition 3.1. *[85] A system is said to be functionally controllable, if for any given suitable output vector у (defined fort > 0), there exists an input vector и (defined for t > 0) for which this output vector can be achieved from the zero initial condition .*

Remark 3.1. *This definition, in fact, guarantees the independent control of the system outputs by the system inputs. In order to satisfy the functional controllability, the number of independent inputs must be greater than or equal to the number of outputs being controlled.*

Definition 3.2. *[27] For a system described as Eq. (3.7), it is said to have (m — p) degrees of actuator redundancy if the pair* (*A.bi) is completely controllable* V* (1 < *i < m), where bi is the ith column of B, and the number of independent control inputs is more than that of the system outputs being controlled.*

In a modern aircraft, for instance, several elevons are configured on the wings to provide necessary control functions for pitch and roll motions. However, if malfunctions occur in some elevons, the FTC is able to use the remaining redundant actuators to counteract the failures. Hence, the elevons constitute a set of redundant actuators.

## Problem Statement

After the development of a fault aircraft model and the completion of redundancy analysis, the design problem is to find the corresponding state/output feedback FTC such that, for all *p* defined in Chapter 3.2.3,

- 1) The closed-loop system is stable;
- 2) The desired output
**S,y***(t)*tracks the reference signal r*(t)*with zero steady-state error, that is, lim e*(t)*= 0, where e*(t)*= r*(t) —***S**_{r}y(f),S_{r}€

£->-og

R'xp is a known constant matrix which is used to define specific outputs for tracking control purposes. The reference input can be defined as r *(t) = Elicit).*

The overall structure of the proposed FTCS is shown in Fig. 3.2. A feedback strategy is selected for reconfigurable control system design. Only the pre-designed FTC laws against control surface impairments are investigated in this chapter. It is highlighted in the area enclosed by the dashed lines.

An integral control action is used to eliminate the steady-state tracking error. When the state feedback controller is designed, *a,/3,p,q,r* are used for feedback. In the case of the static output feedback, only the outputs *a,/3,p *are fed back for closed-loop control. In order to obtain a tracking controller with state/output feedback plus the tracking error integral, the augmented tracking system is introduced as

Define the augmented state vector x„ *(t) =* *(f* e (*t*) *dij* , *x ^{T} (t)* , the disturbance vector

*uj*(

_{a}*t*) = [r

^{r}

*(t) ,ш*(t

^{т}*)]*and the augmented measured out-

^{T},put vector у* _{a} (t) =* (

*f e*(

*t) dt'j , у (t) .*Then, Eq. (3.10) can be rewritten

as

where

From Eqs. (3.8) and (3.12), the augmented system can be presented in a polytopic form as follows:

FIGURE 3.2: Overall structure of proposed FTCS.

where

Choose the regulated output *z _{a} (t)* , defined by

Based on the redundancy analysis and the problem formulation, the problem of FTCS design will be addressed next. It should be mentioned that the dynamics of an aircraft depend not only on dynamic pressure, altitude, and airspeed, but also on the available area of control surface. During a normal flight, gain scheduled controllers are used to deal with different aerodynamics due to changes in dynamic pressure, altitude, and airspeed, while flying through various flight regimes. However, these flight regimes related aerodynamic changes should not be confused with those induced by loss (partial loss) of control surfaces. Regardless what regimes the aircraft is flying in, a loss of control surface area has to be accommodated through fault-tolerant flight control systems, which are the main work in this chapter. The loss in control surface area can be estimated by a parameter identification method

[86]. Subsequently, the reconfigurable control strategy can be designed based on the models of the post-fault aircraft (with the estimated surface damage) to compensate for the failures. If a transition in flight regimes has also occurred, the gain scheduled controller will become the base controller for the reconfigurable controller design. For the sake of simplicity, we have not considered the regime changes in the current chapter, because the use of gain scheduled controllers is a common practice in all flight control systems.