Abstract

Internal model control (IMC) design method based on quasi-LPV (Linear Parameter Varying) system is proposed. In this method, the nonlinear model is firstly transformed to the linear model based on quasi-LPV method; then, the quadrotor nonlinear motion function is transformed to transfer function matrix based on the transformation model from the state space to the transfer function; further, IMC is designed to control the controlled object represented by transfer function matrix and realize quadrotor trajectory tracking. The performance of the controller proposed in this paper is tested by tracking for three reference trajectories with drastic changes. The simulation results indicate that the control method proposed in this paper has stronger robustness to parameters uncertainty and disturbance rejection performance.

1. Introduction

In the last few years, quadrotor helicopters have received widespread attention for their many advantages and relevant technologies have also been important research topics [1]. A quadrotor helicopter is a complex system with high nonlinearities, strong couplings, and underactuation, and it is constantly affected by aerodynamic disturbance, unmodeled dynamics, and parametric uncertainty. Quadrotor helicopter control is a challenging problem [2]. Therefore, the research for the control systems of quadrotor helicopters has been widely conducted in the automatic control field [3].

Among all the control systems, robust nonlinear controllers have good control effect compared to classic controllers. The common robust control methods include sliding mode control [4, 5] and nonlinear control [6, 7]. These methods especially sliding mode control can effectively control the quadrotor, provide good dynamic performance, and assure robust stability. Sliding mode control and its improved methods are used in wider control field except for quadrotor. Sliding mode control has been used in underactuated two-wheeled mobile robot [8], steer-by-wire systems with AC motors [9], orbital stabilization of inverted-pendulum systems [10], stochastic polynomial systems with unmeasured states [11], electric drive [12], multimachine power system [13], and motion control [14]. But these methods directly rely on quadrotor nonlinear equations to achieve the control for the quadrotor, so the design method is complex and it is difficult to be mastered by engineers. For this reason, in this paper, the nonlinear model is firstly transformed to the linear model based on quasi-LPV (Linear Parameter Varying) method; the quadrotor model is then represented by the transfer function using the transformation model from the state space to the transfer function; finally, IMC (internal model control) is designed. IMC can inhibit the time-delay and has strong robustness.

In order to analyze and design gain scheduled control, Shamma proposed the LPV system in 1988 [15]. Later, Shamma and Athans further studied and improved the LPV system [16, 17]. In the last decades, the LPV system which is independent of both LTI (Linear Time Invariant) and LTV (Linear Time Varying) systems has become a standard formalism in systems analysis and controller synthesis. In 2012, Shamma summarizes the research results for the LPV systems [18]. Quasi-LPV [19, 20] is an extension of the LPV. The LPV controller designs based on the LPV system can be found in the literature in [21, 22] and their references.

IMC (internal model control) is proposed by Garcia and Morari in 1982 [23] for analyzing the two predictive control systems MAC (model algorithmic control) [24] and DMC (dynamic matrix control) [25] and as an extension for the Smith prediction assessment which is also called Smith time-delay compensator. IMC makes the Smith time-delay compensator design simpler and has stronger robustness and interference rejection ability. IMC is widely used in process control; for example, an internal model control method is designed for controlling the adiabatic reaction temperature of autothermal reforming (ATR) reactor [26], inverted decoupling internal model control method is used to control square stable multivariable time-delay systems [27], and an improvement method of the IMC is used to control MIMO (multiple input-multiple output) first order time-delay nonsquare systems [28]. In recent years, IMC has been expanded to the robotic arm, smart car control systems, and so forth. For example, IMC is used to control hydraulically driven robotic arm [29] and smart car’s speed [30]; modified IMC schemes with fuzzy supervisor are proposed to control the speed of heavy duty vehicle (HDV) [31]. But IMC is rarely used to control quadrotor; only the literature in [32] is found. In the literature in [32], discrete-time IMC is used for quadrotor trajectory tracking control. In this paper, aiming at the characteristics of quadrotor, a controller design strategy combining quasi-LPV system and IMC is proposed to track the given trajectory, where the quasi-LPV system is only used to transform the nonlinear model to the linear model, which is convenient to get the transfer function of the quadrotor and further control quadrotor.

Quasi-LPV system is an extension of the LPV system. Correspondingly, the control method for LPV system can also be applied to quasi-LPV system. IMC design for LPV system can be found in the literature in [33, 34]. IMC in the literature in [33] is a one degree of freedom IMC which is designed to do compromise between the performance and robustness. Practically, we can use two degrees of freedom IMC as shown in this paper and the literature in [34]. In two degrees of freedom IMC, feedforward IMC only needs to consider tracking performance and feedback filter only needs to consider robustness and disturbance rejection performance. In addition, the author of the literature in [33] points out that the benefit of design presented in his paper is eliminating the need for adjusting IMC filter. But it can be seen from the literature in [33] that design weights need to be chosen for solving the LMI problem. The choice of the design weights is also a process of trial and error. Therefore, the literature in [33] eliminates the need to adjust IMC filter but at the same time increases the need to choose design weights. In the literature in [34], the author uses the generalized IMC (GIMC) based on LMI to control LPV system. The feedback controller in GIMC controller is equivalent to feedforward controller in IMC, and the conditional controller in GIMC is equivalent to feedback filter part in two degrees of freedom IMC. According to the literature in [35], the parameter of feedback filter is often set as the half of loop delay time, which has been verified to be correct in many applications. Therefore, it is not necessary to use LMI to adjust the filter parameter. In addition, the author of the literature in [34] points out that LPV system cannot be used to treat transfer functions, but it can be seen from the proof of Lemma  1 that the state matrices of LPV system can do matrix addition, multiplication, and inverse operations. Therefore, the concept of the transfer function in linear system can be generalized to LPV system and further generalized to quasi-LPV system.

The motivation of this paper is to design a controller which is simple but has strong robustness to parameters uncertainty and disturbance rejection performance. The control strategy in this paper which integrates the advantages of the quasi-LPV system and IMC can realize the above purpose. The contributions of this paper are as follows: the integration of the quasi-LPV system and IMC is firstly proposed to control quadrotor and this design method can be generalized to control other nonlinear systems; the concept of the transfer function in linear system is generalized to quasi-LPV system and the two degrees of freedom IMC in which the parameter of feedback filter is adjusted according to the literature in [35] are used to control quadrotor; It can be seen from the transfer function matrix of quadrotor that the relationships of three attitude angles are as follows: yaw angle is decoupled with roll and pitch angles, roll angle is coupled with pitch angle, and when , roll angle is decoupled with pitch angle; the virtual inputs are proposed to simplify the position loop transfer function and a new calculation method for the desired values of roll and pitch angles is proposed, which is not restricted by the yaw angle, and therefore the control performance of controller adopted in this paper is better, which can make quadrotor fly with better mobility.

The remainder of the paper is organized as follows: quadrotor helicopter model is described in Section 2. The calculation of quasi-LPV model and the establishment of the quadrotor system transfer function are given in Section 3. In Section 4, the IMC design and the calculation of the desired roll angle, pitch angle, and total thrust are presented. Simulation results are reported in Section 5. Finally, we draw some conclusions and shed light on future work in Section 6.

2. Quadrotor Model

The structure of a quadrotor helicopter is shown in Figure 1, where represents the body coordinate system and represents the ground coordinate system.

Figure 1 

The structure of a quadrotor helicopter.

Given an aircraft, the positive direction in the body coordinate frame is usually defined as the moving direction, the positive direction as the right side of the moving direction, and the positive direction as the vertical downward direction; this is also termed forward-right-downward coordinate frame [36].

Because the fly altitude of quadrotor helicopters is limited in the atmosphere, the “flat earth hypothesis” can be adopted to consider the ground coordinate frame as an inertial coordinate frame to simplify the modeling complexity. In order to facilitate navigation and way-point tracking, the axis directions of the ground coordinate frame are chosen as north-east-down navigation frame directions, namely, that the axis points to the north, the axis points to the east, and the axis is perpendicular to the ground and points to the center of the earth.

The two main mechanism-modeling methods for the quadrotor include Newton-Euler formalism and Lagrange-Euler formalism [37–39]. In this paper, quadrotor helicopter nonlinear model is obtained by the Newton-Euler formalism and the rotor dynamics is not considered during the model establishment.

The procedure of the Newton-Euler formalism modeling is as follows: firstly projecting lift forces acting on the aircraft to the ground coordinate frame; secondly analyzing the linear motion of the aircraft with Newton’s second law in the inertial coordinate system and the angular motion of the aircraft with the law of moment of momentum in the body coordinate system. Some assumptions are made in the process of quadrotor modeling as follows: quadrotor is a rigid body; the structure is symmetric; the center of gravity and the origin of body coordinate system are coincident and ground effect is ignored.

2.1. Kinematics Model

The transformation matrix between two rectangular coordinate systems is orthogonal. , , and denote rotation matrices produced by the ground coordinate frame rotating roll angle , pitch angle , and yaw angle around , , and axes, respectively, and the expressions are as follows:

The rotation matrix from the ground coordinate system to the body coordinate system is the product of formulae (1), which denote rotation around the axis followed by rotation around axis and finally followed by rotation around axis; namely,Therefore, the transformation matrix from the ground coordinate system to the body coordinate system is given byThe specific expression is given bywhere and .

The angular velocity components , , and are the projection values on the body coordinate system of rotation angular velocity which denotes the rotation from the ground coordinate system to the body coordinate system. The relationships between angular velocity components and the attitude angle change rates are shown in Figure 2.

Figure 2 

The relationships between angular velocity components and the attitude angle change rate.

The transformation matrix from to is given bywhereAround hovering position, is assumed as a unit matrix [36, 40].

2.2. Dynamic Model

The dynamics model is composed of the rotational and translational motions. The rotational motion is fully actuated, while the translational motion is underactuated. In the body coordinate system, the rotational motion equations are derived according to the law of momentum theorem and gyroscopic effect of quadrotor; they are given bywhere is the inertia matrix of quadrotor which is diagonal under the hypothesis of structure symmetry and the elements , , and are, respectively, inertia matrices of , , and axes. The last item on the left side of (7) represents the gyroscopic effect which is caused by the inertia of the rotors and relative speed , where () represents the th rotor speed. The aerodynamic force and moment produced by the th rotor are directly proportional to the square of the rotor speed. The relationships are given bywhere and are the aerodynamic force and moment constants, respectively. The moments acting on the quadrotor in the body coordinate system are given bywhere is the moment arm which represents the distance from the axis of a rotor to the center of quadrotor.

The translational motion equations are obtained in the ground coordinate system by the method of Newton’s second lawwhere is the position of quadrotor in the ground coordinate system, is the mass of quadrotor, is the acceleration of gravity, and is the total lift force acting on quadrotor in the body coordinate system; namely,

2.3. The Motion Equations of Quadrotor

Synthesizing the kinematics and dynamics models of quadrotor, the motion equations of quadrotor can be derived as follows:where , , and are the attitude angle of quadrotor; , , and are the position of quadrotor; , , , and are the control input variables, which can be, respectively, calculated by , , , and in which and are the aerodynamic force and moment constants, respectively; , , and are, respectively, inertia matrices of , , and axes. is the inertia matrix of the rotor and is rotors relative speed, where () represents the th rotor speed; is the moment arm which represents the distance from the axis of a rotor to the center of quadrotor; is the mass of quadrotor; and is the acceleration of gravity.

3. The Calculation of Quasi-LPV Model and the Establishment of the Quadrotor System Transfer Function

3.1. Brief Introduction of the Quasi-LPV Systems

Quasi-LPV is an extension of the LPV. A LPV system is a linear time-varying system whose matrices depend on a vector of time-varying parameters which are either measured in real time or estimated using some known scheduling function. The advantage of this class of systems is that it embeds the system nonlinearities in the varying parameters which make the nonlinear system become a linear system with the parameter varying. In the pure LPV system, the varying parameters only depend on exogenous signals and, in the quasi-LPV system, the varying parameters can be functions of the states, inputs or outputs. The state space model of LPV system is given bywhere the state space matrices , , , and are the function of the time-varying parameter ; is the state vector; is the system output; is the control input vector.

3.2. The Calculation of Quasi-LPV Model

Quasi-LPV method is used to transform the quadrotor nonlinear model represented by (12) into linear model in this paper. Because position loop of quadrotor is related to the attitude angle state variables, attitude loop of quadrotor is related to the first derivative of yaw angle and rotor rotating angular velocity which is input, and quasi-LPV model is used to represent the quadrotor nonlinear motion equations in this paper. Unlike LPV system, the state space matrix of quasi-LPV system is the function of time-varying states , , , and input . The state vector is chosen as . The output vector is chosen as . The control input vector is represented as . Considering the hypothesis of quadrotor structure symmetry, that is, is equal to and is the half of , the coefficient matrix of the state equation derived by quasi-LPV method is given bywhere

3.3. The Establishment of the Quadrotor System Transfer Function

Using , the transfer function matrix of quadrotor helicopter can be obtained aswhere , , , and .

It can be seen from the transfer function matrix of quadrotor that the position loop is underactuated with three outputs and only one input; the attitude loop is fully actuated; the transfer function of position loop is time-varying, which varies with the attitude angle; yaw angle is decoupled with roll angle and pitch angle; roll angle is coupled with pitch angle. The coupling transfer function of roll angle and pitch angle can be extracted and written as

Because is much smaller than and , the gyroscopic effect item can be ignored. Under the hypothesis of quadrotor structure symmetry, is equal to . The simplified coupling transfer function of roll and pitch angles can be obtained by eliminating common factor of (17) and substituting with . It is given by

In this paper, the control design is conducted only in . In this situation, (18) can be rewritten as

It can be seen from (19), that the roll angle and pitch angle are decoupled. Therefore, the attitude angle stability of quadrotor can be realized by individually designed controllers for three attitude angles. For such a simple second-order system represented by yaw angle loop in (16) and (19), the internal model controller can be used. Only two controller parameters need to be adjusted in IMC.

The position transfer function in (12) can be extracted and written as

If the virtual inputs are defined as

then the transfer function of position loop can be written as

To sum up, in six inputs and six outputs , the transfer function of quadrotor is given by

4. The IMC Design and the Calculation of the Desired Roll Angle, Pitch Angle, and Total Thrust

4.1. Quadrotor Control Structure

The quadrotor control structure adopted in this paper is shown in Figure 3. The inner and outer loops both adopt the internal model control.

Figure 3 

Quadrotor control structure.

4.2. IMC Design Principle

The general structure of IMC [35] is shown in Figure 3, in which is the controlled object, is the controlled object model, is the feedforward item of IMC, is the feedback filter, and are the output of the controlled object and controlled quantity, is the output of controlled object model, is a given value (reference trajectory), and is the external disturbance. In IMC system, is mainly used for the reliable tracking for a given input and is used to adjust the robustness and reject disturbance (see Figure 4).

Figure 4 

The general structure of IMC.

According to the different values, IMC system can be called one degree of freedom IMC system or two degrees of freedom IMC system. When , IMC system is called One degree of freedom IMC system, otherwise it is called two degrees of freedom IMC system. The basic structure of IMC which is proposed by Garcia and Morari in 1982 is shown in Figure 5. It is essentially one degree of freedom IMC system.

Figure 5 

The basic structure of IMC.

In Figure 5, is the transfer function of disturbance channel, and usually ; the inner of the dotted box is the internal structure of the whole control system and the process model is also included besides the controller in the inner of the dotted box; therefore, this control structure shown in Figure 5 is entitled internal model control. The equivalent structure of IMC can be derived by doing equivalent transformation for Figure 5, and it is shown in Figure 6. It can be seen from Figure 6 that it is a unit feedback control system.

Figure 6 

The equivalent structure of IMC.

It can be seen from Figure 6 that the relation between the feedback controller and IMC is given by

According to Figure 6, the closed-loop transfer functions under the action of control and disturbance are respectively given by

The closed-loop transfer function for one degree of freedom IMC system is derived asFrom Figure 6, feedback signal is straightforwardly derived as

If the model is accurate, that is, , (27) and (28), respectively, become (29) and (30); namely,

After introducing the controlled object model, it can be seen from (29) that the output of the controlled object only contains the product items but not division items. Therefore, IMC has good tracking performance. It can be seen from (30) that the amount of feedback becomes disturbance estimator from full feedback of output. At the moment, IMC is equivalent to a disturbance estimator, and can be designed to fully compensate the disturbance effect on output. Therefore, IMC has good disturbance rejection performance. When the model does not completely match with the controlled object, that is, , the model error exists, the amount of feedback includes some information of the model mismatch, and thus the designed can compensate the amount of model mismatch; that is to say, the IMC is robust to the model mismatch.

In conclusion, IMC actually belongs to a kind of robust control. It has good tracking performance and ability to reject disturbance and has robustness to the model mismatch. When there is no feedback filter, that is, one degree of freedom IMC system, the tracking performance, disturbance rejection performance, and robustness can be traded off only by feedforward item of IMC. When there is feedback filter, that is, two degrees of freedom IMC system, feedback signal is given as

It can be seen from (31) that the design of only needs to consider tracking performance which is the rapidity of response and the intensity of control, and robustness and disturbance rejection performance is adjusted by the feedback filter .

4.2.1. The Design Procedure of

This paper uses the modified offset method [41] for the design. The principle of the offset method is as follows: supposing the controlled object is stable, firstly use to eliminate the minimum phase section of , and then add a first-order or second-order feedforward filter to regulate the rapidity of the response, the control intensity, and robustness and disturbance rejection performance. The advantage of this method is that the design and adjustment are intuitive and simple, and it can be applied to continuous and discrete systems. But for nonminimum phase object, due to the instable zero of that cannot be eliminated, the response performance of the system cannot be guaranteed; that is to say, overshoot and negative overshoot likely appear. The design procedure of the modified offset method is as follows: firstly do decomposition for the controlled object model which is decomposed into minimum phase part and all-pass filter part , and then use the offset method to design of minimum phase part. The specific design procedure is given as follows.

Step 1 (the decomposition of the controlled object model ). can be decomposed into two items and and can be written aswhere is an all-pass filter which contains all time delays and zero in right half plane; is a transfer function with minimum phase characteristic which is stable and does not contain the prediction term.

Step 2 (IMC design). IMC is designed asIn (34), is the low pass filter; the purpose of adding a filter into the controller is to guarantee the stability and robustness of the system. The principle of choosing is making rational; that is, can be realized through the decomposition of (32) and filter choice of (34). is filter parameters; it is the only parameter in design.
In the filter design, should be large enough to ensure that is rational. The greater is, the slower the closed-loop output response is and the softer the operating variables change. In addition, is approximately proportional to the closed-loop bandwidth. Therefore, an initial estimate of the filter parameters can be got and, in practice, can be adjusted on-line as needed.

4.2.2. The Design of

It can be seen from Figure 6 and (28), when there is no model mismatch and disturbance, that the feedback signal , the original system is equal to open-loop system, and only the feedforward controller works at this time. Therefore, the feedback filter is set specifically for model mismatch and disturbance, the most common structure for is first-order low-pass filter, and it can be expressed as

Equation (35) only has one adjustable parameter; the structure is simple, but performance improvement effect is remarkable and there is a lot of mature conclusions. In the multivariable time-delay systems, is often set as the half of loop delay time [35].

4.3. IMC Design for the Attitude and Position Loops

According to the feedforward and feedback controllers of IMC design method introduced in Section 4.2, when the position and attitude controlled object vector is , the feedforward controller is given aswhere , , , , , and are, respectively, controller design parameters corresponding to the controlled variables , , , , , and loops.

4.4. The Calculation of the Desired Roll Angle, Pitch Angle, and Total Thrust

Equation (21) can be transformed as

Subtracting the expression derived by multiplying (38) with from the expression derived by multiplying (37) with , can be obtained; further, the desired value of pitch angle can be obtained as

Adding the expression derived by multiplying (37) with and the expression derived by multiplying (38) with together, (41) can be obtained as

can be derived by (41) dividing (39); further, the desired value of roll angle can be obtained as

The total thrust can be obtained by adding the both sides square of (37), (38), and (39) together; it can be written as

To sum up, after the position loop controllers , , and are got, the desired values of roll and pitch angles and the total thrust can be calculated by using (40), (42), and (43), which can complete the join of attitude loop and position loop. Finally, the control for quadrotor position and yaw angle can be realized by using (36), (40), (42), and (43) and quadrotor trajectory tracking can be further realized.

5. Simulation Results

To test the trajectory tracking performance, the constant interference rejection performance, and robustness to parameter uncertainty of the control strategy proposed by this paper, the trajectory tracking experiments for the quadrotor flight path proposed in the literature in [42, 43] were carried out and constant interference and parameter uncertainty were added in the constant interference rejection performance and robustness tests. The quadrotor parameters used in this paper are as follows:  kg, m/s²,  m, kg·m², kg·m²,  kg·m², , and . In simulation, constant wind and ±30% uncertainties of inertia parameter were, respectively, added into three kinds of trajectory to test constant interference rejection performance and robustness to parameter uncertainty of the control strategy proposed in this paper. Constant wind was introduced at different times. At s, constant wind in direction  m/s was introduced; at s, constant wind in direction  m/s was introduced; while at s, constant wind in direction  m/s was added. In this paper, the IMC parameters of position and attitude loop are the same; the parameters of feedforward controller are as follows: . The literature in [44] pointed out that the delay due to the electronic speed controller (ESC) driver is 0.2 s; therefore, the parameters of the feedback controller in position and attitude loop are all chosen as .