# Multi Range Multivariable Linear Controller Biology

The purpose of this report is to describe the application of a Multi-range Multivariable Linear Controller for a highly nonlinear UAV. The controller design approach is based on obtaining the sinusoidal-input describing function (SIDF) models of the nonlinear plant. The plant consists of a servo, a UAV, and a dead zone. Plant is the physical dynamic system which is to be controlled. It is then followed by linear system identification. Once a linear approximation to the describing function model of the UAV is obtained, a linear multivariable controller which in this case two lead-lag compensators is designed to achieve a robust nonlinear closed-loop system. The programming software used in this project is Matlab and the procedure is applied to control the bank angle of a UAV. The results were compared with one other design that is reported in the open literature. It is shown that the performance of the developed procedure has not been so good to assure absolute stability.

Unmanned Aerial Vehicles (UAVs) are advantageous over piloted counterparts in terms of maneuverability, low human risk, low cost and light weight, and is an important development area in the aerospace industry for the 21st century. In a military setting, they have been used in a reconnaissance and intelligence-gathering role since the 1950s, and more challenging roles are envisioned, including combat missions. UAVs are unmanned but are usually controlled by ground operators. Typically they do not operate autonomously and their inability to provide the level of safety of a manned plane keeps them from flying freely in the commercial airspace.

Ultimately, the target is to employ the UAV autonomously is such applications as survey work to assist in construction projects, aerial photography to assist in disaster mitigation, continuous weather monitoring, and crop monitoring. Smart unmanned aircraft will require significant deployment of advanced control system throughout the airframe.

There are a few works on UAV control in the open literature. In [1] they concentrated on missiles for large angles of attack. In [2] a fuzzy neural network control system was developed for UAV control. In [3] a hybrid neural adaptive controller was used for a surface-to-air missile. In that work, state-space UAV model for Linear-Quadratic regulator design is utilized. In [4] a smooth nonlinear model of an agile missile is used. In that work they used two-neural-network structure to optimize the controller using a Hamiltonian formulation. In [5], smooth actuator dynamics were included.It used a partial linearization method and designed a controller considering actuator dynamics.

A singular perturbation method to linearize equations of motion of the UAV was used in [6] which then followed by a suitable proposed control scheme. In that work, UAV dynamics are smooth, and a linear second order model is used for the actuator dynamics. In [7] a standard short-period linear equation was used to describe the motion of the missile which is then followed by application of robust techniques and a linear parameter varying controller is designed. A smooth nonlinear model of a UAV is used in [8] and a controller using quasi-linear-parameter-varying polynomial eigenstructure assignment is designed. In [9], a smooth nonlinear model for a UAV is utilized, and a longitudinal autopilot using a new nonlinear control technique is designed. An almost linear model of the UAV is utilized in [10] and a robust adaptive control scheme is applied to autopilot design for feedback linearized UAVs.

The project presented herein is about designing a Multi-range Multivariable Linear Controller for Bank-angle Control of a highly nonlinear UAV. The proposed model for the UAV is a single-input multiple-output (SIMO) system (see Fig 1). The UAV mathematical model includes discontinuous nonlinear terms, the dead zone which reduces limit cycling to eliminate small commands that pass through the system, and only a few approaches are available that deal with discontinuous nonlinear systems.

In this work, a multi-range linear controller design procedure (which is based on several describing function model of the UAV) is utilized, and a multivariable controller is designed. The results are compared with that reported in A. Nassirharand, and S.R. Mousavi Firdeh (2009) and Bushra (2009).

The nonlinear system under study may be represented in the following standard state variable format:

(1)

(2)

where x is the vector of state variables, f is a vector of nonlinear functions, u is the scalar input, y is the scalar output, g is a vector of nonlinear functions, and t is the time variable. The describing function models are generated by using a numerical approach to obtain the gain and phase of the system as a function of amplitude and frequency. In general, it is not possible to obtain a closed-form expression for the describing function model of a plant given by equations (1) and (2). Closed-form expressions for the describing function models are only available for simple nonlinear functions (Gelb and van der Velde, 1968).

The problem statement follows. Given the model for bank angle control of a UAV, design a multi-range multivariable linear controller. With reference to Figure 1, the model of the UAV follows;

(3)

(4)

(5)

(6)

Then, design a multi-range multivariable linear controller. The desired performance measures are no overshoot with a settling time of less than or equal to 2 seconds.

Figure 1.1: Schematic Block Diagram of the Closed Loop UAV Control

**1.1**

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