Finishing Up Power Systems Project

AIRCRAFT PITCH

EECE 682

Computer Control Of Dynamic System

Project Report

Boeing Aircraft- Pitch Controller Example: Dynamics, Modeling, Simulation, Analysis

Instructor:

Dr. Adel Ghandakly Dept. Electrical and Computer Engineering

California State University, Chico Submitted By:

Nasser Al Ahbabi

AIRCRAFT PITCH

BOEING AIRCRAFT- PITCH CONTROLLER

Example: Dynamics, Modeling, Simulation, Analysis

by

Nasser Al Ahbabi

California State University, Chico.

NOVEMBER 2014 Abstract Though airplane has a number of important factors, its stability and control is a key design parameter that must be met. In an airplane the stability is defined in three angles i.e. pitch, yaw, and roll. In this paper I have focused on the pitch. The system transfer was

AIRCRAFT PITCH

obtained through analyzing the various parameter involved in the pitch control. In all the designs, I considered the design parameter requirements i.e. the percentage overshoot, steady state error, settling time, and rise time of Boeing aircraft. The designs of pitch controller using various techniques have been implemented on the system transfer function. I have provided an extension of to these techniques by using MATLAB/Simulink models that plays an important role in monitoring the results of designed controllers. In addition, I have also provided a descriptive analysis of the system response to the designed controllers and their conclusions. Keywords: Aircraft, Pitch, Ackerman, Digitized PID, Diophantine, Optimal Control, Controller , Simulink and MATLAB design. CONTENTS: INTRODUCTION

� INTRODUCTION MATHEMATICAL MODEL

� BOEING AIRCRAFT: PHYSICAL SETUP AND SYSTEM EQUATIONS � TRANSFER FUNCTION AND STATE-SPACE MODEL � DESIGN REQUIREMENTS:

CONTROLLER DESIGN

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� DESIGN 1 : DIGITIZED PID � DESIGN 2 : DIRECT METHOD ( CLOSED FORM) � DESIGN 3 : DIRECT METHOD ( DIOPHANTINE) � DESIGN 4 : POLE PLACEMENT (ACKERMAN’S FORMULA) � DESIGN 5 : OPTIMAL CONTROL

CONCLUSION REFERENCES

1. INTRODUCTION Aircrafts are perfect and good examples of a Controller system. They possess unique characteristics that make their controller design a more challenging problem. On linearization of the model we can attain results with simplified controller designs. Major parameter in the design of aircrafts entails the horizontal speed, pitch control and the throttle. The throttle controls the main motor revolutions per minute; the pitch controls the magnitude of the motor thrust. There are two inputs that are independent; the longitudinal input and the lateral cyclic input. These controls An aircraft in flight is free to rotate in three dimensions: pitch, nose up or down about an axis running from wing to wing, yaw, nose left or right about an axis running up and down; and roll, rotation about an axis running from nose to tail. In this project, only one dimension is considered. These two inputs control the angles of roll and pitch. In this project, I have used MATLAB based approach for simulation and design by applying four various controller design techniques; Digitized PID, Direct Methods (Closed form and with Diophantine) and pole placement (Ackerman’s Formula) and Optimal Control. I

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used Simulink in modeling these controller designs on the system transfer function for Direct method (Diophantine) and Optimal Control. I analyzed their response as per the design requirements.

2. MATHEMATICAL MODEL 2.1 Boeing Aircraft: Physical setup and system equations The equations governing the motion of an aircraft are a very complicated set of six nonlinear coupled differential equations. However, under certain assumptions, they can be decoupled and linearized into longitudinal and lateral equations. Aircraft pitch is governed by the longitudinal dynamics. In this example we will design an autopilot that controls the pitch of an aircraft. The basic coordinate axes and forces acting on an aircraft are shown in the figure given below.

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Assuming that the aircraft is in steady-cruise at constant altitude and velocity; thus, the thrust, drag, weight and lift forces balance each other in the x- and y-directions. We will also assume that a change in pitch angle will not change the speed of the aircraft under any circumstance (unrealistic but simplifies the problem a bit). Under these assumptions, the longitudinal equations of motion for the aircraft can are:

ά=μΩσ [-(CL +CD) α + 1 q – (CW sinϒ) θ+CL] (μ-CL) q·=μΩ [[CM-η(CL +CD)] α +[CM(1-μCL)]q+ (ηCW sinϒ)δ ) 2iyy θ·=Ωq Where; α = Angle of attack. q=Pitch rate. θ=Pitch angle. δ=Elevator deflection angle. μ=ρSḉ 4m ρ= Density of air. s= Platform area of the wing. ḉ=Average chord length. m=Mass of the aircraft. Ω=2U ḉ U=Equilibrium flight speed. CT = Coefficient of thrust. CD=Coefficient of drag. CL= Coefficient of lift. CW= Coefficient of weight. CM= Coefficient of pitch moment. ϒ= Flight path angle. σ=Constant. iyy = Normalized moment of inertia. η=μσCM=Constant. In this system, the input will be the elevator deflection angle δ and the output will be the pitch angle θ of the aircraft.

2.2 Transfer function and state-space model The linearized equations governing the motion of a Boeing’s commercial aircraft are given by; dα(t) =ά=-0.313α(t) +56.7q(t) +0.232δ(t)

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dt dq(t) = q·=-0.0139α(t) – 0.426q(t) +0.0203δ(t) dt dθ(t) = θ·=56.7q(t) dt Transfer Function The transfer function of the above system is obtained by taking the Laplace transform of the above modeling equations assumimg zero initial conditions. This gives; sA(s)=-0.313A(s) +56.7Q(s) +0.232∆(s) sQ(s)=-0.0139A(s) -0.426Q(s) +0.0203∆(s) sθ(s)=56.7Q(s) The open-loop transfer function obtained by carrying about few steps of algebra on the above equations giving; P(s)=Θ(s) = 1.151s + 0.1774 ∆(s) s3 + 0.739s2 + 0.921s State Space The above equation can be written as matrices as;

AIRCRAFT PITCH

2.3 Design requirements: In this project, I chose some design criteria in which I designed a feedback controller in response to a step command of pitch angle. The actual pitch angle overshoots less than 10%, has a rise time of less than 2 seconds, a settling time of less than 10 seconds, and a steady-state error of less than 2%. For example, if the reference is 0.2 radians (11 degrees), then the pitch angle will not exceed approximately 0.22 rad, will rise from 0.02 rad to 0.18 rad within 2 seconds, will settle to within 2% of its steady-state value within 10 seconds, and will settle between 0.196 and 0.204 radians in steady-state. In summary, the design requirements are the following.

x Overshoot less than 10% x Rise time less than 2 seconds x Settling time less than 10 seconds x Steady-state error less than 2%

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3. THE CONTROLLER DESIGN

The figure above shows a simple pitch controller design block model for the Boeing Aircraft. Taking the system transfer function derived above I will be designing the controller through the following various techniques;

� Digitized PID � Direct Methods (Closed Form and with Diophantine) � Pole Placement (Ackerman’s Formula) � Optimal Control

AIRCRAFT PITCH

3.1) AIRCRAFT PITCH: THE DIGITIZED PID CONTROLLER: A proportional-integral-derivative controller (PID controller) is a control loop feedback mechanism (controller) widely used in controlling systems in industries. A PID controller calculates an error value as the difference between a measured process variable and a desired set point. The controller attempts to minimize the error by adjusting the process through use of a manipulated variable. A block diagram of a PID controller in a feedback loop is as shown below

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The closed-loop transfer function of the system with a PID controller is: X(s) = Kds2 + Kps + Ki . F(s) s3 + (10 + Kd)s2 + (20 +Kp)s + Ki The effects of each of controller parameters Kp, Kd and Ki, on a closed-loop system are summarized in the table below.

CL RESPONSE RISE TIME OVERSHOOT SETTLING TIME S-S ERROR

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After several trial and error runs, the gains Kp =350, Ki=300, and Kd= 50 provided the desired response giving the following step response;

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.2

0.4

0.6

0.8

1

1.2

1.4 Step Response

Time (seconds)

Am pl

itu de

Kp Decrease Increase Small Change Decrease Ki Decrease Increase Increase Eliminate Kd Small Change Decrease Decrease No Change

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Increasing the derivative gain Kd in a PID controller often helps reduce overshoot. Adding a derivative control reduces the oscillation in the response a sufficient amount that the other gains can be increased to reduce the settling time. C(s)= 4.4545 (1 + 0.22s + 1.1s2) ≈ 4.45 + 0.98 + 4.90s s s This transfer function is a PID compensator with Ki = 4.45, Kp = 0.98, and Kd = 4.90. The resulting closed-loop step response is shown below.

AIRCRAFT PITCH

This response meets all of the requirements except for the settle time which at 12.6 seconds is a little larger than the given requirement of 10 seconds. The proportional gain was increased in order to reduce the system’s settle time. Increasing Kp means that I may no longer achieve the minimum possible performance metric, however, i did that in order to decrease the resulting settle time. Specifically, l changed Kp so that it equals 2. The resulting PID controller is shown below.

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The PID controller response shown below meets all of the given requirements as summarized below. � Overshoot = 5% < 10% � Rise time = 1.2 seconds < 2 seconds � Settling time = 5 seconds < 10 seconds � Steady-state error = 0% < 2%

Therefore, this PID controller will provide the desired performance of the aircraft’s pitch.

AIRCRAFT PITCH

PID controller Response

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3.2) AIRCRAFT PITCH: DIRECT METHOD – CLOSED FORM: This method is a direct analytical technique for design. In order to stabilize a control system and eventually meet the given design requirements, a feedback controller is added and hence we have both the forward and feedback path controllers. The figure below illustrates the control architecture we will employ.

P(s)=Θ(s) = 1.151s + 0.1774 ∆(s) s3 + 0.739s2 + 0.921s The closed-loop transfer function for the above with the controller C(s) simply set equal to one can be generated using the MATLAB command feedback. I scaled the response to model the fact that the pitch angle reference is a 0.2 radian (11 degree) step. Running my m-file at the command line produced the plot shown below in I added the annotations for the rise time, settling time and final value.

AIRCRAFT PITCH

The steady-state error appears to be driven to zero and there is no overshoot in the response, though the rise-time and settle-time requirements are not met. I used the MATLAB commands pole and zero to reveal the poles and zeros of the closed-loop transfer function (shown on closedform.m file) Assuming the closed-loop transfer function has the form Y(s) / R(s), the output Y(s) in the Laplace domain is calculated as follows where R(s) is a step of magnitude 0.2. Y(s) = 1.151s + 0.1774 R(s) s3 + 0.739s2 + 0.921s

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Y(s) = 1.151s + 0.1774 R(s) s3 + 0.739s2 + 0.921s Y(s) = 0.2 (1.151s + 0.1774 ) s3 + 0.739s2 + 0.921s The above expression can be expressed as partial fraction as, Y(s) = 0.2 _ 0.08 81 . _ 0.1121s + 0.08071 s s + 0.08805 s2 + 0.6509s + 2.015 The inverse Laplace transform of the above expression is taken to generate the corresponding time domain expression shown below. y(t)=0.2 – 0.0881e-0.08805t – e-0.3255t(0.1121cos(1.3816t) + 0.0320 sin(1.3816t)) Using the above equation, I generated the graph below in matlab

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0 10 20 30 40 50 60 70 -0.05

0

0.05

0.1

0.15

0.2

time (sec)

pi tc

h an

gl e

(ra d)

Closed-loop Step Response

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3.3) AIRCRAFT PITCH: DIRECT METHOD ( DIOPHANTINE)

Controller design by Diophantine method is based on the Diophantine Equation. When given the paramount and key polynomials D(z) and G(z) each having an order n and a H(z) polynomial of the order (2n-1) the Diophantine equation is given by;

d(z)D(z)+g(z)G(z)=H(z)

From the mathematical modeling, the transfer function is given as;

P(s)=Θ(s) = 1.151s + 0.1774 ∆(s) s3 + 0.739s2 + 0.921s

AIRCRAFT PITCH

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3.4) AIRCRAFT PITCH: POLE-PLACEMENT ACKERMAN The continuous-time state-space model of the aircraft pitch dynamics was derived as;

Where the input is elevator deflection angle δ and the output is the aircraft pitch angle θ. For a step reference of 0.2 radians, the following are the design criteria;

� Overshoot less than 10% � Rise time less than 2 seconds � Settling time less than 10 seconds � Steady-state error less than 2%

Discrete state-space In designing a control system a sampled-data model of the plant was generated. I used MATLAB to generate this model from a continuous-time model using the c2d command. The c2d command requires three arguments: a system model, the sampling time (Ts) and the type of hold circuit. In this design I assumed a zero-order hold (zoh) circuit In choosing a sample time, it is desired that the sampling frequency be fast compared to the dynamics of the system in order that the sampled output of the system captures the system’s full behavior, that is, so that significant inter-sample behavior isn’t missed. One measure of a system’s “speed” is its closed-loop bandwidth. A good rule of thumb is that the sampling time be smaller than 1/30th of the closed-loop bandwidth frequency. Thus, to be sure of a small enough sampling time, I used a sampling time of 1/100 sec/sample. The after running commands (shown in m-file) in MATLAB, I obtained four matrices representing the sampled-data state-space model. Hence, the discrete-time state-space model is;

AIRCRAFT PITCH

Controllability: Before designing the controller I had to verify the controllability of the system. For the system to be completely state controllable, the controllability matrix must have rank n where the rank of a matrix is the number of independent rows (or columns). The controllability matrix of a discrete-time system has the same form as a continuous-time system. C=[|A|AB|A2B|. . .|An-1B|] Since the controllability matrix is 3×3, its rank must be 3. The MATLAB command rank was used to calculate the rank of a matrix. Adding the following additional commands in an m-file and running in the MATLAB command window will produce the following output. co = ctrb(sys_d); Controllability = rank(co) Controllability =

3 Therefore, the system is completely state controllable since the controllability matrix has rank 3 The schematic of a discrete full-state feedback control system is shown below, where q-1 is the delay operator (not the aircraft’s pitch rate q). NB: I assumed that D = 0(Hence not shown).

AIRCRAFT PITCH

where K = control gain matrix x = [ alpha, q, theta ]’ = state vector theta_des = reference (r) delta = theta_des – K x = control input (u) theta = output (y) Substituting the state-feedback law δ(k)=θdes(k) – K x(k) for δ(k) to the state-space equations leads to the following assuming that all of the state variables are measured; x(k+1)=(A-BK)x(k)+Bθdes (k) θ(k)=Cx(k) Linear Quadratic Regulator (LQR) method is used to find the control matrix (K). The discrete version of the same LQR method is used. This type of control technique optimally balances the system error and the control effort based on a cost specified that defines the relative importance of minimizing errors and minimizing control effort. In the case of the regulator problem, it is assumed that the reference is zero. To use this LQR method, two parameters are defined: the state-cost weighted matrix (Q) and the control weighted matrix (R). For simplicity, I chose the control weighted matrix equal to 1 (R=1), and the state-cost matrix (Q) equal to pC’C. By employing the vector C from the output equation means I only considered those states in the output in defining the cost. In this case, θ is the only state variable in the output. The weighting factor (p) was chosen by trial and error in order to modify the step response to achieve the given requirements. Since we have a single input system, R is a scalar.

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Control matrix (K) is found by employing the MATLAB command dlqr which is the discrete-time version of the lqr command. I chose a weighting factor p =50. From matlab, the values of Q and K are found as; Q = 0 0 0 0 0 0 0 0 50 K = -0.6436 168.3611 6.9555 The stair-step response is generated as shown below ;

AIRCRAFT PITCH

Examination of the above demonstrates that the rise time, overshoot, and settling time are satisfactory. However, there is a large steady-state error but this can be corrected by introducing a precompensator to scale the overall output. Adding Precompensator Unlike other design methods, the full-state feedback system does not compare the output to the reference; instead, it compares all states multiplied by the control matrix (K x) to the reference as shown below. In obtaining the desired output, the reference input is scaled so that the output does equal the reference in steady state by introducing a precompensator scaling factor, Nbar. The basic schematic of the state-feedback system with scaling factor (Nbar) is shown below.

0 1 2 3 4 5 6 7 8 9 10 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

time (sec)

pi tc

h an

gl e

(ra d)

Closed-Loop Step Response: DLQR

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The generated the stair-step response is;

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From the plot above, the Nbar factor eliminates the steady-state error and all design requirements are satisfied.

0 1 2 3 4 5 6 7 8 9 10 0

0.05

0.1

0.15

0.2

0.25

time (sec)

pi tc

h an

gl e

(ra d)

Closed-Loop Step Response: DLQR with Precompensation

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3.5) AIRCRAFT PITCH: OPTIMAL CONTROL In optimal control, the given state space model is converted to a discrete state space model. I did this using the function c2dm that requires 6 parameters ( A,B, C, D, the sampling time Ts and the state space matrix). I implemented this in Matlab using the function ‘dlqr’.

Optimal control

AIRCRAFT PITCH

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4) CONCLUSION Among various numbers of important factors, an airplane’s stability and control is major design parameter that must be met. In an airplane the stability is defined in three angles i.e. pitch, yaw, and roll. This paper only focused to control the pitch. Overshoot, rise time, steady-state error and settling time of a pitch are minimized and regulated during the designing of the controllers using different methods and techniques. MATLAB played an important role in analyzing and comparing the results of the designed controller. I successfully designed, tested, implemented and analyzed the various controller design techniques on the Pitch Control of the Boeing Aircraft.

AIRCRAFT PITCH

5) REFERENCES [1] http://wikis.controltheorypro.com/index.php?title=Aircraft_pitch_Example [2] S. Kamalasadan and A. A. Ghandakly, “Multiple fuzzy reference model adaptive controller design for pitch-rate tracking,” IEEE Transactions on Instrumentation and Measurement, vol. 56, no. 5, pp. 1797–1808, 2007. View at Publisher · View at Google Scholar · View at Scopus [3] S. Kamalasadan and A. A. Ghandakly, “A neural network parallel adaptive controller for fighter aircraft pitch-rate tracking,” IEEE Transactions on Instrumentation and Measurement, vol. 60, no. 1, pp. 258–267, 2011. View at Publisher · View at Google Scholar · View at Scopus [4] S. E. Talole, A. Ghosh, and S. B. Phadke, “Proportional navigation guidance using predictive and time delay control,” Control Engineering Practice, vol. 14, no. 12, pp. 1445–1453, 2006. View at Publisher · View at Google Scholar · View at Scopus [5] M. Pachter, P. R. Chandler, and L. Smith, “Maneuvering flight control,” Journal of Guidance, Control, and Dynamics, vol. 21, no. 3, pp. 368–374, 1998. View at Scopus