Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844Vol. III (2008), Suppl. issue: Proceedings of ICCCC 2008, pp. 337-342

Intelligent Autopilot Control Design for a 2-DOF Helicopter Model

Saeed Jafarzadeh, Rooholah Mirheidari, Mohammad Reza Jahed Motlagh, Mojtaba Barkhordari

Abstract: In this paper, we introduce a new intelligent control approach called Brain Emo-tional Learning Based Intelligent Controller (BELBIC). BELBIC is a controller based onemotional model of human brain which has been introduced not for so long. This controllerhas been applied to a nonlinear model of a helicopter. Feedback linearization method hasalso been applied to the system, and the performance of two controllers has been comparedas an intelligent and a classical control method. An Input to State linearization method withsome changes has been used to control the system. The performance of the controllers hasbeen justified by the simulation.Keywords: BELBIC, Reward, Feedback Linearization, Input to State Linearization.

1 Introduction

Most of the existing results for helicopter control have been based on the linearization model or through severallinearization techniques [1, 2]. The linearized models of the helicopter have some unmodeled dynamics that makethe proposed controls unreliable when applied to original nonlinear models. Using nonlinear techniques givesbetter response and improves the performance of controlled system.

A very thorough survey of linear techniques for helicopter control has been given by Garrad and Low [1].Miniature helicopter control problems have also been discussed by Furuta et al. [3] and Kientz et al. [2]. The workof Pallett et al. [4] has served as the basis for our understanding of the helicopter model. Some other nonlinearcontrol methods have been applied to the model which has been considered in this paper. Sliding mode Control[5] and robust control method [6] are some examples of these works.

In this paper we apply two methods to the system and compare their performance. First method is a newintelligent approach called Brain Emotional Learning Based Intelligent Controller (BELBIC). This technique isbased on physiological structure of brain in excited situations and can be used in control engineering problems.Recently, there is rising tend to intelligent controllers and BELBIC is not an exception [7, 8, 9]. This controllerhas a certain structure, but it can be changed to achieve the control objectives. There are both continuous anddiscrete time BELBIC controllers [9, 10]. We use the continuous one to compare with another continuous timecontrol method (feedback linearization). To design the BEBIC controller, we should choose the appropriate rewardfunction according to physical aspects of the control problem, and tune the training coefficients of the controller toachieve desired control objectives. Second method is feedback linearization which is a classical control method.An input to state linearization has been applied to the system for this method. There has been applied a feedbacklinearization method to a model of a helicopter [11], but it is different from the model which we use here. Therefore,we design a controller by feedback linearization method for the helicopter system.

There is a complicated fifth order nonlinear state space model for the system. The system has two inputs andtwo outputs. The control objective is to track the desired set points of the outputs. The sections of the paper areas follows. Section 2 describes the fifth order model of the helicopter. Section 3 contains the theoretical resultsin BELBIC controller design and its tuning. Theoretical aspect and controller design for feedback linearizationmethod has been brought in the section 4. Section 5 shows the simulation results and finally section 6 is conclu-sions.

2 Helicopter Model

Consider the nonlinear helicopter model as follows [6]:

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338 Saeed Jafarzadeh, Rooholah Mirheidari, Mohammad Reza Jahed Motlagh, Mojtaba Barkhordari

x1 = x2

x2 = a0 +a1x2 +a2x22 +

(a3 +a4x4−

√a5 +a6x4

)x2

3

x3 = a7 +a8x3 +(a9 sinx4 +a10)x23 +u1

x4 = x5

x5 = a11 +a12x4 +a13x23 sinx4 +a14x5 +u2

x = [h h ω θ θ ]T and u = [u1 u2]T = [k1uth− k2uθ ]T

In these equations h is the height of helicopter above ground and is measured in meter. ω is the rotationalspeed of the rotor blades and is measured in radian per second. θ is the collective pitch angle of rotor bladesand is measured in radian. uth and uθ are the input to the throttle and the input to collective servomechanismsrespectively.

3 Feedback LinearizationFeedback linearization has been used successfully to address some practical control problems. These include

the control of helicopter, high performance aircraft, industrial robots, and biomedical devices [12]. The centralidea of the approach is to algebraically transform a nonlinear system dynamics into a (fully or partly) linear one, sothat linear control techniques can be applied. This differs entirely from conventional linearization in that feedbacklinearization is achieved by exact state transformation and feedback, rather than by linear approximations of thedynamics [12]. The idea of feedback linearization, i.e., of canceling the nonlinearities and imposing a desiredlinear dynamics, can be simply applied to a class of nonlinear systems described by the so-called companion form,or controllability canonical form. A system is said to be in companion form if its dynamics is represented by

xn = f (x)+b(x)u

where u is the scalar control input, x is the scalar output of interest, X = [x, x, ...,x(n−1)]T is the state vector, andf (x),b(x) are nonlinear functions of the states. This form is unique in the fact that, although derivatives of appear inthis equation, no derivative of the input u is present. Note that, in the state space representation, the above equationcan be written

ddt

x1. . .

xn−1xn

=

x2. . .xn

f (x)+b(x)u

(1)

From systems which can be expressed in the controllability canonical form, using the control input (assumingb to be non-zero)

u =1b[v− f ]

We can cancel the nonlinearities and obtain the simple input-output relation (multiple integrator form) x(n) = v.Thus, the control law v =−k0x−k1x− ...−kn−1x(n−1) with the chosen so that the polynomial pn +kn−1 pn−1 + ...+k0 has all its roots strictly in the left half complex plane, leads to the exponentially stable dynamics xn +kn−1xn−1 +...+ k0x = 0 which implies that x(t)→ 0. For tasks involving the tracking of a desired output xd(t), the controllaw v = x(n)

d − k0e− k1e− ...− kn−1e(n−1) (where e(t) = x(t)− xd(t) is the tracking error) leads to exponentiallyconvergent tracking. For the case of helicopter, we should find a state transformation to rewrite the equations inthe form of (1). We use the input to state linearization method [12].

There is a input to state linearization method for SISO systems in [12]. We apply this method with somechanges to the helicopter model which is a MIMO system. Suppose that the vector z(x) = [z1(x) ... zn(x)]T is thenew state vector, called linearized state vector. To obtain this vector, we should make the following equations

∇z1adif g = 0 i = 0, ...,n−2 and ∇z1adn−1

f g 6= 0 (2)

Intelligent Autopilot Control Design for a 2-DOF Helicopter Model 339

where is the Lie Bracket of f and g , and is a third vector field defined by ad f g = ∇g f −∇ f g .According to these equations, we can find z1(x). The linearized vector z(t) is

z(x) =[z1 L f z1 . . . Ln−1

f z1]T (3)

where L f z1 is the Lie Derivative of f and z1 , and defined by L f z1 = ∇z1 f .The outputs of the system are x1,x4 . We separate the states of the system into two sections (

[x1 x2 x3 |x4 x5

]T

), and use the first part to control the first output by the first control input, and the same for the second part.We start to find z1(x) for the first set of states of the helicopter model. In this part of the model n = 3, so we

have the following equations

∂ z1

∂ zi= 0 f or i = 2,3 ,

∂ z1

∂ zi6= 0

From these equations, we define z1 = x1. We use this method to linearize half of the system. It means that welinearized first three equations of the model by the first control input, and the last two equations by the second one.Thus, we obtain the states according to the (3)

z2 = L f z1 = X2

Z3 = L2f z1 = a0 +a1x2 +a2x2

2 +(a3 +a4x4−√

a5 +a6x4)x23

According to [12], we have

u1 = α(x)+β (x)v, α(x) =− Lnf z1

LgLn−1f z1

,β (x) =1

LgLn−1f z1

For the first three states, we have n = 3 . The first control input has been chosen so that the characteristicequation of the first output becomes (s + 30)(S + 3)2 = 0. For the last two states, we define the second controlinput to cancel the nonlinearity of the fifth equation. The characteristic equation of the second output is (s+1)2 = 0.

4 BELBIC Controller

Figure 1: Computational model of emotional learning in the Amygdala

BELBIC is an abbreviation for Brain Emotional Learning Based Intelligent Controller. Motivated by the suc-cess in functional modeling of emotions in control engineering applications, a structural model based on the limbicsystem of mammalian brain, for decision making and control engineering applications has been developed. Thecomputational model of emotional learning in the amygdala, based on Moren and Balkenius model, is depictedin Figure 2 [13, 14]. The main parts that are responsible for performing the learning algorithms are orbitofrontalcortex and amygdala.

BELBIC controller has some sensory inputs. One of the designer’s tasks is to determine the sensory inputs.This controller has two states for each sensory input. One of these two is amygdala’s output and another is theoutput of orbitofrontal cortex. Therefore, the number of sensory inputs has a key role in BELBIC controller.Usually the sensory inputs are rich signals [10].

Consider the i-th sensory input as si . Then we have amygdala and orbitofrontal cortex outputs

Ai = sivi

340 Saeed Jafarzadeh, Rooholah Mirheidari, Mohammad Reza Jahed Motlagh, Mojtaba Barkhordari

Oi = siwi

v,w are two states for the related sensory input. These states will be updated by the following equations

4vi = α.si.max(0,rew−∑Ai), 4wi = β .si.(rew−∑Ai−∑Oi−max(Si))

where α and β are training coefficients. We have a function named Reward. This function has a great role inBELBIC controllers. Reward is like its name. The controller strives to increase this reward. Therefore, the designermust define a reward function that has its maximum values in the most desired regions. This reward function couldbe frequency domain function or a normal mathematic function.

Amygdala acts as an actuator and orbitofrontal cortex acts as a preventer. Therefore the control effort ofBELBIC controller is

u = ∑Ai−∑Oi

BELBIC is a controller that has only one output. Therefore for systems with more than one control inputs wemust use one BELBIC controller for each control input. As it can be seen there are several tuning parameters foreach sensory input. The general algorithm for tuning these parameters is trial and error.

We use continuous form of BELBIC in this paper. In continuous form, the BELBIC states are updated with nota discrete relation but a continuous one. These continuous relations are

vi = α.si.(rew−Ai), wi = β .si(rew+ si +Oi−Ai)

To control the helicopter in take-off and landing problem, we have two references to be tracked. One of themis desired height and the other one is the desired value of collective pitch angle of rotor blades. We must design aBELBIC controller for each control input of helicopter. For throttle we have one sensory input and it is height’serror.

s1 = hd −h

The reward function for this BELBIC controller is as figure 2.

Figure 2: Reward Function

The reward function’s parameters are positive real numbers. As it could be seen, the BELBIC controllerreceives the maximum reward when the sensory input is zero. According to the fact that the sensory input isan error signal, the BELBIC controller tries to vanish the sensory input and it means tracking. With this rewardfunction we have reward and punishment together for BELBIC controller. There are some areas that reward takesnegative values. It could help the controllers to sense whole domain of sensory inputs.

The training coefficients and reward function’s parameters for the first BELBIC controller are as follows:

α = 5,β = 34,k1 = 1000,k2 = 100

The BELBIC controller for servo mechanical control has a sensory input and it is the error of collective pitchangle of rotor blades.

si = θd −θ

The BELBIC controller for the collective servomechanism control input is like the one for throttle controland with the same reward function but with different parameters. The training coefficients for the first BELBICcontroller are as follows

Intelligent Autopilot Control Design for a 2-DOF Helicopter Model 341

α = 3, β = 1.8, k1 = 1200, k2 = 80

5 Simulation ResultsTo see the performance of the two mentioned controllers, we have simulated the controlled system in Simulink.

The height of the helicopter should track a desired path. The second output has a set point too. A sinusoidal pathhas been considered for the height, and second output kept constant. It can be seen from these simulations thatthe tracking performance of BELBIC controller for the height is better than Feedback linearization controller.The simulation results of controller system by BELBIC controller and feedback linearization controller have beendemonstrated in the figures 3, and figure 4 demonstrates the control inputs of the system.

Figure 3: : Height (left) and Collective rotor blade angle (right) of helicopter (Solid: set point, Dashed: feedbacklinearization, Dotted: BELBIC)

Figure 4: First (left) and second (right) control input of helicopter (Solid: feedback linearization, Dotted: BELBIC)

6 Summary and ConclusionsIn this paper a BELBIC controller and a feedback linearization technique applied to a nonlinear model of a

helicopter to attain path tracking of a given path for the height of the helicopter. The system has five states andtwo control inputs. A continuous time BELBIC controller has been applied to the system, and tuned for the bestperformance. An input to state linearization has been used to control the system too. In this method the states ofthe system have been separated into two parts, and each part has been controlled by one of the control inputs. Thetransient response of the BELBIC controller is better than the case with feedback linearization controller, but inthe sense of steady state the performance of both controllers is good. However, there is an important disadvantagefor this controller that is stability guarantee. The BELBIC is a very suitable method to solve control engineeringproblems, and it can be applied to a large variety of the linear and nonlinear systems.

342 Saeed Jafarzadeh, Rooholah Mirheidari, Mohammad Reza Jahed Motlagh, Mojtaba Barkhordari

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[8] C. Lucas, S. Moghimi, “Applying BELBIC (Brain Emotion Learning Based Intelligent Controller) to an Autolanding System”, Conference: WSEAS AIKED’03, 2003.

[9] Tutunchi Ali Ghasem, “Optimal BELBIC control with application to Autopilot control”, Master’s Thesis,University of Tehran, 2006.

[10] C. Lucas, D. Shahmirzadi, N. Sheikholeslami, “Introducing BELBIC: Brain Emotional Learning Based In-telligent Controller”, International Journal of Intelligent Automation and Soft Computing, Vol. 10, No. 1, pp.11-22, 2004.

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[13] J. Moren, C. Balkenius, “A Computational Model of Emotional Conditioning in the Brain”, in Proc. workshopon Grounding Emotions in Adaptive Systems, Zurich, 1998.

[14] J. Moren, C. Balkenius, “A Computational Model of Emotional Learning in The Amygdala: From animalsto animals”, in Proc. 6th International conference on the simulation of adaptive behavior, Cambridge, Mass.,The MIT Press, 2000.

Saeed Jafarzadeh, Rooholah Mirheidari, Mojtaba BarkhordariIran University of Science and Technology

Department of Electrical EngineeringAddress: Daneshgah Street, Hengam Street, Resalat Sq., Tehran, Iran

E-mail: sjafarzadeh, rmirheidari@ee.iust.ac.ir, mbarkhordary@iust.ac.ir

Mohammad Reza Jahed MotlaghIran University of Science and Technology

Department of Computer EngineeringAddress: Daneshgah Street, Hengam Street, Resalat Sq., Tehran, Iran

E-mail: jahedmr@iust.ac.ir