Mathematical Problems in Engineering

Volume 2015, Article ID 548050, 13 pages

http://dx.doi.org/10.1155/2015/548050

## A Unified Approach to Nonlinear Dynamic Inversion Control with Parameter Determination by Eigenvalue Assignment

Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan

Received 8 September 2015; Revised 13 November 2015; Accepted 16 November 2015

Academic Editor: Rongwei Guo

Copyright © 2015 Yu-Chi Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a unified approach to nonlinear dynamic inversion control algorithm with the parameters for desired dynamics determined by using an eigenvalue assignment method, which may be applied in a very straightforward and convenient way. By using this method, it is not necessary to transform the nonlinear equations into linear equations by feedback linearization before beginning control designs. The applications of this method are not limited to affine nonlinear control systems or limited to minimum phase problems if the eigenvalues of error dynamics are carefully assigned so that the desired dynamics is stable. The control design by using this method is shown to be robust to modeling uncertainties. To validate the theory, the design of a UAV control system is presented as an example. Numerical simulations show the performance of the design to be quite remarkable.

#### 1. Introduction

In the development of high-performance aircraft, control difficulties may be encountered over some parts of flight envelope. These difficulties arise from highly nonlinear aerodynamic properties [1] in some flight conditions. In order to solve these control difficulties, nonlinear controllers are required for high-performance aircraft.

Among many control methods, nonlinear dynamic inversion (NDI) is very popular and has been widely studied for flight control designs (e.g., [2, 3]). NDI-based control systems are usually divided into fast- and slow-loop control subsystems according to the multiple time-scale method [4]. In each subsystem, Lie derivatives [5] are used to transform the nonlinear equations into linear equations. Then, linear control design methods can be employed and the control inputs are obtained by converting the linear system control variables into the original coordinates. However, the control systems obtained by feedback linearization [6] may have nonminimum phase problems for affine or nonaffine nonlinear system [7] and robust issues in case of model mismatch. A typical nonminimum phase problem may be found in flight dynamics where the altitude-elevator transfer function usually has a right-half zero. The internal state control [8] is often used to overcome these nonminimum phase problems. In addition, the fuzzy logic control [9] was also applied for solving these kinds of problems. Furthermore, to overcome the robust problems, -analysis [10] and method [11] were applied. Specially, incremental NDI (INDI) [12] was used to increase the robustness to aerodynamic uncertainties by calculating the control surface deflection changes instead of giving inputs directly.

To circumvent some aforementioned robust problems, an adaptive nonlinear model inversion control [13] was introduced, in which the design concept is similar to the conventional NDI yet without linearizing the nonlinear system. The model inversion method replaces the motion rates with a P-form or PI-form desired dynamics to negate the original dynamics. The choice of parameters in the desired dynamics is based on the bandwidth of response and time scales. The effects of different types of desired dynamics on the resulted control system were discussed [14].

Although the aforementioned NDI approaches are successful in many flight control system designs [14–16] over a large part of the flight envelope, the systems of dynamics in general have to be separated into several subsystems according to the rates of response. There are many cases in which the fast rate and the slow rate might not be distinguished so clearly however. Also, although the pole assignment method had been introduced to determine the parameters in the desired dynamics, very often the control system must first be transformed into a standard feedback control form from which the standard eigenvalue assignment method can be applied.

In consideration of the aforementioned problems existing in the current literature on control designs, a unified approach to nonlinear dynamic inversion control is proposed in this paper. The equations of motion will not be necessary to be separated into fast rate and slow rate groups, nor will they be limited to an affine system. Feedback linearizations will not be required to transform the nonlinear equations into linear equations. Nonminimum phase problems are solved by eigenvalue assignments for error dynamics. An iterative method for determining the parameters of the desired dynamics from the assigned eigenvalues of error dynamics is proposed. Analysis of robustness to model uncertainties or disturbances is conducted. This method will be ready for design without simplifying the system of equations based on physical insights once the governing dynamic equations are established and the state variables to be tracked are selected. The theory is to be developed in detail in the following sections. A UAV is introduced and its control system is designed with the developed method. Numerical simulations are conducted to validate this method.

#### 2. A Unified Approach to Nonlinear Dynamic Inversion Control

##### 2.1. Nonlinear Dynamic Inversion Control

In general, dynamic equations of motion with control inputs can be expressed bywhere is the state vector, () the control vector, and the nonlinear function representing the model of dynamics with controls. By extending the concept of dynamic inversion (DI) [4], the control vector can be assumed to be computed fromwhere is a desired state vector with its rate of change being designated.

In this paper, the desired dynamics is designated as a set of stable first-order differential equations: where represents a constant matrix with independent parameters which can be chosen. Substituting (3) into (2) yieldswhich constitutes a set of algebraic equations. Since , obviously, cannot satisfy (4) if all elements of are to be designated. It means that only part of can be designated. So let be divided into two groups, say and , where contains the state variables which are to be controlled or designated and the residual ones. Both and constitute unknown variables which are to be determined from (4). To solve a set of nonlinear algebraic equations, the Newton-Raphson iteration method can be employed as follows:where and represents the iteration number.

##### 2.2. Parameter Determination

Now, a question arises whether the state vector will asymptotically follow the desired vector if is determined from (4) and substituted in (1). In order to answer it, let (1) and (4) be examined more carefully as follows.

Subtracting (4) from (1) yieldsIf is very close to , then the above equation can be linearized as follows:With the concept that the desired variables are near constant, say , (7) can be rewritten asDefining an error vector and replacing the approximate sign with the equal sign lead (8) toEquation (9) is a set of error dynamics in which the error vector will vanish eventually if all the real parts of the eigenvalues of are negative. This is possible if is chosen appropriately. It means that the state vector can approach the desired vector once approaches . Recall that contains , a state vector to be tracked.

Notice that the eigenvalue in (9) can be determined bywhere is an identity matrix. For simplicity, if is diagonal and all elements are the same, say, , then (10) can be rewritten asIn order to make bounded, can be so chosen thatfor all . Although this method is simple, the resulting may be unnecessarily large.

For more general cases, contains a set of parameters, , which may be chosen to determine the eigenvalues and eigenvectors of . Recall that (3) must be a stable model and, therefore, the simplest way of constructing is to let all its elements vanish except those at the diagonal, which are assumed to be . In fact, some off-diagonal elements can also be allowed to exist. For example, let , and . It is trivial to prove that the latter case is also a stable model.

Now, assume that the th eigenvalue and eigenvector are and , respectively. Accordingly,In general, the eigenvector can be normalized so thatHowever, if the eigenvalue is desired rather than , then it is necessary to adjust the parameters in . Assume that an increment to is required. Thenwhere and are assumed to be approximated to and , respectively, and is also normalized so thatAccordingly, the derivative of (13) with respect to results inand the derivative of (14) becomesEquations (17) and (18) can be rearranged tofrom which can be determined. Note that represents the variation of with respect to . With all , , the th revised eigenvalue should beRecall that, in this equation, represents the th eigenvalue of , where contains the parameters , (). Now since is the th desired eigenvalue, (20) in fact becomes a set of simultaneous equations from which can be solved. With being the initial guess, revisions for parameters can be made bySince are nonlinear function of , iterative computations of (9) and (19)–(21) are required in order to obtain a set of convergent . To make it clear, the iteration procedures are summarized as follows.(1)Let the desired eigenvalues be which are all distinct.(2)Guess a set of parameters .(3)Determine the eigenvalues and eigenvectors of (9). Denote the th eigenvalue and eigenvector as and , respectively.(4)With each , determine from (19).(5)Determine from (20). If is less than a preset small value, then stop; else continue with the next step.(6)Determine from (21).(7)Replace with and go to step .At this point, it must be mentioned that the iterations will converge only if the initial guesses are very close to the true answers. It is also emphasized that the desired eigenvalues must be chosen to lie on the left-half -plane and the resulting parameters in must satisfy the stable model requirements in (3).

##### 2.3. Robust Analysis

The nonlinear dynamic inversion control design developed so far is based on a nominal dynamical model. In the real world, however, there always exist some modeling uncertainties which cannot be determined in advance. In order to check if the control design is robust, let the actual dynamical model be represented byWith this assumption, now (6) can be modified towhich can be rewritten as follows:where denotes the nonlinear part of . The solution of the above equation can be represented as follows:Note that from (9), the eigenvalues of are all in the left-half -plane since has been determined with that assumption. Also, at this point, it is not unreasonable to assume that both and the modeling differences are bounded. Therefore, if the model in (3) is stable enough, the integral part in (25) will vanish along with . Accordingly, as the time gets large enough, the state variables will approach the desired values as can be found from (25) even if there are some modeling uncertainties or disturbances.

#### 3. Nonlinear Dynamic Inversion Flight Control System Design for a UAV: An Example

##### 3.1. Flight Dynamics Equations of Motion

To illustrate the theory, a design of flight control system with the method developed is presented. Before the flight control design proceeds, a set of flight dynamics equations of motion must be formulated. Note that all aerodynamic forces and moments result from the relative motions between aircraft and the air. The aircraft is assumed to have a ground velocity:where are unit vectors in the aircraft body moving frame and the unit vectors in a fixed flat earth frame. The air is assumed to have a velocity:which is also known as the wind velocity. Then, the velocity of the aircraft relative to the air can be represented byfrom which, the aircraft total speed relative to the air, the angle of attack, and the sideslip angle can be determined, respectively, by the following equations:

With the assumptions of fixed flat earth and winds being present, the motions of aircraft with six degrees of freedom can be represented by a set of nonlinear first-order differential equations as follows:where represents the position in a fixed flat earth frame, the altitude, the elements of direction cosine matrix for transferring a fixed flat earth frame to the aircraft body moving frame, the angle of attack, the sideslip angle, the heading angle, the pitch angle, the bank angle, the roll rate, the pitch rate, and the yaw rate. Also, , , and represent, respectively, three components of the total force in an aircraft body moving frame. The three force components are composed of the thrust , the lift , the drag , the side force , and the weight ( is the aircraft mass and the gravity acceleration) by the following equations:where is the angle between the thrust and the longitudinal axis. Moreover, , , and represent the roll moment, the pitch moment, and the yaw moment, respectively, about the center of gravity, and , , , and are the components of the moment-of-inertia tensor. Furthermore, the wind disturbances on , , and are, respectively, represented by , , and which are expressed as follows:whereIn (44), , , and are the three components of the force exerted by winds. At this point, it is worthy to mention that although there is no explicit term relating wind disturbances to , , and in (39)–(41), winds do have effects on , , and through which , , and are affected. Also, winds not only have explicit effects on , , and in (36)–(38), but also have implicit effects on them through , , and which obviously depend on , , and .

To validate the method developed in this paper, a UAV as shown in Figure 1 is introduced. The parameters used for analysis are listed in Table 1.