Predictive Control of Quad-Rotor Delivering Unknown Time-Varying Payloads Based upon Extended State Observer

1. Introduction

Although the aforementioned methods have achieved satisfied control performances, they have drawbacks or the application is based on some unrealistic assumptions. For example, the control schemes based on the PID and LQR methods cannot guarantee system robustness within whole flight envelop. The MRAC method is applicable to slow time-varying system, but detailed known model information is needed. The control scheme based on SMC is insensitive to uncertainties and can stabilize the system globally. However, the prerequisite on achieving good system robustness against uncertainties is that the accurate upper bound (UB) of amplitude of the uncertainties is available. Actually, the accurate UB may not be obtained easily. Hence, an overestimation of the UB is required to determine the switching gain, resulting in high-frequency of both switching of the control input and chattering around sliding mode surface. This possibly degrades control performance and negatively affects actuator. The conventional BS method can only deal with constant or slowly changing uncertainties.

2. System Modeling and Problem Formation

The relationship between the quad-rotor and the payload is depicted in Figure 1.

In Figure 1, $\left\{{O}_{B},{X}_{B},{Y}_{B},{Z}_{B}\right\}$ represents the body frame, where ${O}_{B}$ coincides with the mass center of the aircraft. ${O}_{B}{X}_{B}{Z}_{B}$ and ${O}_{B}{Y}_{B}{Z}_{B}$ are the aircraft symmetrical planes. The distance between ${O}_{B}$ and the projection points of each rotor center on ${O}_{B}{X}_{B}{Z}_{B}$ plane is given by l. The orientation of the aircraft is described by Euler angles $\Theta ={\left[\varphi ,\theta ,\psi \right]}^{\text{T}}$ . The inertial tensor of the aircraft with respect to the body frame is denoted as $J=diag\left({I}_{x},{I}_{y},{I}_{z}\right)$ . ${T}_{1},{T}_{2},{T}_{3}$ and ${T}_{4}$ are thrusts from four rotors. ${O}_{P}^{*}$ is the projection point of ${O}_{P}$ on plane ${O}_{B}{X}_{B}{Y}_{B}$ with coordinate $\left({x}_{0},{y}_{0}\right)$ . m and ${m}_{0}$ are quad-rotor mass and payload mass, respectively. ${l}_{x}$ , ${l}_{y}$ and ${l}_{z}$ are geometrical parameters of the payload. The inertial tensor of the payload with respect to the body frame is given by:

${J}_{p}=\left[\begin{array}{ccc}\Delta {I}_{x}& \Delta {I}_{xy}& \Delta {I}_{xz}\\ \Delta {I}_{xy}& \Delta {I}_{x}& \Delta {I}_{yz}\\ \Delta {I}_{xz}& \Delta {I}_{yz}& \Delta {I}_{x}\end{array}\right]$ (1)

Remark 1: ${J}_{p}$ is an unknown matrix which is not only relative to the shape, dimensions and mass of the payload, but also relative to ${x}_{0}$ and ${y}_{0}$ (see Figure 1).

Table 1 gives the detailed physical parameters of the quad-rotor [18] used in this paper.

2.1. System Modeling

During stable flight, the roll and pitch angles of the quad-rotor are very close to zero. Thus, the kinematic model as well as Euler angle (EA) control system can be built as:

$\stackrel{˙}{\Theta }=\Omega$ (2)

According to Figure 1, the roll, pitch and yaw torques M in frame {B} can be expressed as:

$M=\left[\begin{array}{c}l\left(-{T}_{1}+{T}_{2}+{T}_{3}-{T}_{4}\right)\\ l\left(-{T}_{1}-{T}_{2}+{T}_{3}+{T}_{4}\right)\\ {k}_{c}\left(-{T}_{1}+{T}_{2}-{T}_{3}+{T}_{4}\right)\end{array}\right]$ (3)

Table 1. Physical parameters of the quad-rotor.

Denote:

$\left\{\begin{array}{l}{\tau }_{\varphi }=-{T}_{1}+{T}_{2}+{T}_{3}-{T}_{4}\\ {\tau }_{\theta }=-{T}_{1}-{T}_{2}+{T}_{3}+{T}_{4}\\ {\tau }_{\psi }={k}_{c}\left(-{T}_{1}+{T}_{2}-{T}_{3}+{T}_{4}\right)\end{array}$ (4)

where ${\tau }_{\varphi }$ , ${\tau }_{\theta }$ and ${\tau }_{\psi }$ are virtual inputs that need to be designed.

The dynamic model as well as body rate (BR) control system can be established as:

$\left(J+{J}_{p}\right)\stackrel{˙}{\Omega }=-\Omega ×\left(J+{J}_{p}\right)\Omega +\Delta M+M$ (5)

where, $J=diag\left({I}_{x},{I}_{y},{I}_{z}\right)$ ; $\Delta M={\left[{m}_{0}g\cdot {y}_{0},{m}_{0}g\cdot {x}_{0},0\right]}^{\text{T}}$ is a torque disturbance vector induced by the payload.

By recalling formulas (3) and (4), formula (5) can be written as:

$\begin{array}{l}\stackrel{˙}{\Omega }={\left(J+{J}_{p}\right)}^{-1}\left[-\Omega ×\left(J+{J}_{p}\right)\Omega +\Delta M\right]+\left[{\left(J+{J}_{p}\right)}^{-1}-{J}^{-1}\right]\cdot M+{J}^{-1}\cdot M\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\underset{{F}_{a}={F}_{a}\left(\Omega ;\Delta J,{m}_{0},{x}_{0},{y}_{0}\right)}{\underset{︸}{{\left(J+{J}_{p}\right)}^{-1}\left[-\Omega ×\left(J+{J}_{p}\right)\Omega +\Delta M\right]+\left[{\left(J+{J}_{p}\right)}^{-1}-{J}^{-1}\right]\cdot M}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{B}{\underset{︸}{diag\left(\frac{l}{{I}_{x}},\frac{l}{{I}_{y}},\frac{l}{{I}_{z}}\right)}}\cdot \underset{\Gamma }{\underset{︸}{{\left[{\tau }_{\varphi },{\tau }_{\theta },{\tau }_{\psi }\right]}^{\text{T}}}}\end{array}$ (6)

Let ${F}_{a}={\left[{f}_{p},{f}_{q},{f}_{r}\right]}^{\text{T}}$ , ${b}_{p}=l/{I}_{x}$ , ${b}_{q}=l/{I}_{y}$ , ${b}_{r}=1/{I}_{z}$ , extending formula (6) yields:

$\left\{\begin{array}{l}\stackrel{˙}{p}={f}_{p}+{b}_{p}{\tau }_{\varphi }\\ \stackrel{˙}{q}={f}_{q}+{b}_{q}{\tau }_{\theta }\\ \stackrel{˙}{r}={f}_{r}+{b}_{r}{\tau }_{\psi }\end{array}$ (7)

$\left\{\begin{array}{l}\stackrel{˙}{\Theta }=\Omega \\ \stackrel{˙}{\Omega }={F}_{a}+B\cdot \Gamma \end{array}$ (8)

2.2. Problem Formation

The problems need to be addressed in this paper are:

1) Use the ESO to estimate the nonlinear terms ${f}_{p}$ , ${f}_{q}$ and ${f}_{r}$ for feedback compensation, such that the attitude system robustness against influences from the unknown payloads can be enhanced.

3. Control Scheme Design

In this section, the ESO is used to estimate the unknown disturbance terms ${f}_{p}$ , ${f}_{q}$ and ${f}_{r}$ for feedback compensation, firstly. Then a type of predictive controller targeting MIMO system is designed for the compensated system.

Denote ${\Theta }_{d}={\left[{\varphi }_{d},{\theta }_{d},{\psi }_{d}\right]}^{\text{T}}$ as the reference Euler angles, ${\Omega }_{d}={\left[{p}_{d},{q}_{d},{r}_{d}\right]}^{\text{T}}$ as the desired body rates and ${\stackrel{^}{F}}_{a}={\left[{\stackrel{^}{f}}_{p},{\stackrel{^}{f}}_{q},{\stackrel{^}{f}}_{r}\right]}^{\text{T}}$ as the estimation of ${F}_{a}={\left[{f}_{p},{f}_{q},{f}_{r}\right]}^{\text{T}}$ . The control scheme is shown as Figure 2.

3.1. Disturbance Observation

The ESOs for observing the unknown disturbance terms ${f}_{p}$ , ${f}_{q}$ and ${f}_{r}$ are designed respectively as:

$\begin{array}{l}\left\{\begin{array}{l}{e}_{p}={z}_{p1}-p\\ {\stackrel{˙}{z}}_{p1}={z}_{p2}+{b}_{p}{\tau }_{\varphi }-{\beta }_{p1}{e}_{p}\\ {\stackrel{˙}{z}}_{p2}=-{\beta }_{p2}{e}_{p}\end{array}\\ \left\{\begin{array}{l}{e}_{q}={z}_{q1}-q\\ {\stackrel{˙}{z}}_{q1}={z}_{q2}+{b}_{q}{\tau }_{\theta }-{\beta }_{q1}{e}_{q}\\ {\stackrel{˙}{z}}_{q2}=-{\beta }_{q2}{e}_{q}\end{array}\\ \left\{\begin{array}{l}{e}_{r}={z}_{r1}-r\\ {\stackrel{˙}{z}}_{r1}={z}_{r2}+{b}_{r}{\tau }_{\psi }-{\beta }_{r1}{e}_{r}\\ {\stackrel{˙}{z}}_{r2}=-{\beta }_{r2}{e}_{r}\end{array}\end{array}$ (9)

where, ${z}_{p1}$ , ${z}_{q1}$ and ${z}_{r1}$ track p, q and r, respectively. ${z}_{p2}$ , ${z}_{q2}$ and ${z}_{r2}$

Figure 2. Sketch of the attitude control scheme.

are estimations of ${f}_{p}$ , ${f}_{q}$ and ${f}_{r}$ , respectively. That is ${\stackrel{^}{F}}_{a}={\left[{\stackrel{^}{f}}_{p},{\stackrel{^}{f}}_{q},{\stackrel{^}{f}}_{r}\right]}^{\text{T}}={\left[{z}_{p2},{z}_{q2},{z}_{r2}\right]}^{\text{T}}$ . ${\beta }_{i1}$ and ${\beta }_{i2}$ ( $i=p,q,r$ ) are gains which satisfied follow relationship [22] :

${a}_{i}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{i1}=2{a}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{i2}={a}_{i}^{2}$ (10)

Values of the parameters used in following simulation are given as: ${a}_{p}={a}_{q}={a}_{r}=100$ .

3.2. Stability Analysis

From formula (7), it is easy to find that the control object has following state space formation:

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{1}={x}_{2}+bu\\ {\stackrel{˙}{x}}_{1}=\stackrel{˙}{f}\end{array}$ (11)

Where, u is the input signal. ESO of system shown in formula (11) can be written as:

$\left\{\begin{array}{l}{e}_{1}={z}_{1}-{x}_{1}\\ {\stackrel{˙}{z}}_{1}={z}_{2}+bu-{\beta }_{1}{e}_{1}\\ {\stackrel{˙}{z}}_{2}=-{\beta }_{2}{e}_{1}\end{array}$ (12)

Denote: ${e}_{2}={z}_{2}-{x}_{2}$ . Then subtracting formula (11) from formula (12) yields:

$\left\{\begin{array}{l}{\stackrel{˙}{e}}_{1}=-{\beta }_{1}{e}_{1}+{e}_{2}\\ {\stackrel{˙}{e}}_{2}=-{\beta }_{2}{e}_{1}-\stackrel{˙}{f}\end{array}$ (13)

By denoting $E={\left[{e}_{1},{e}_{2}\right]}^{\text{T}}$ , formula (13) can be written as:

$\stackrel{˙}{E}=A\cdot E-B\cdot \stackrel{˙}{f}$ (14)

where, $A=\left[\begin{array}{cc}-2a& 1\\ -{a}^{2}& 0\end{array}\right]$ , $B=\left[\begin{array}{c}0\\ 1\end{array}\right]$ when formula (10) is considered.

Theorem: Assuming $\stackrel{˙}{f}$ is bounded with $|\stackrel{˙}{f}|\le {d}_{1}$ , then there exist a positive constant ${\epsilon }_{i}$ such that $|{e}_{i}|\le {\epsilon }_{i}$ , $i=1,2$ .

The solution of formula (14) is:

$E\left(t\right)={e}^{At}E\left(0\right)+{\int }_{0}^{t}{e}^{A\left(t-\tau \right)}B\left(-\stackrel{˙}{f}\right)\text{d}\tau$ (15)

Then it has:

$\begin{array}{c}|S|=|{\int }_{0}^{t}{e}^{A\left(t-\tau \right)}B\stackrel{˙}{f}\text{ }\text{d}\tau |\le {\int }_{0}^{t}|{e}^{A\left(t-\tau \right)}B\stackrel{˙}{f}|\text{d}\tau \\ \le {d}_{1}{\int }_{0}^{t}|{e}^{A\left(t-\tau \right)}B|\text{d}\tau \le {d}_{1}|{A}^{-1}{e}^{At}B-{A}^{-1}B|\\ \le {d}_{1}\left(|{A}^{-1}{e}^{At}B|+|{A}^{-1}B|\right)\end{array}$ (16)

The state transition matrix ${e}^{At}$ has the solution as:

${e}^{At}=\left[\begin{array}{cc}{m}_{1}\left(t\right)& {m}_{2}\left(t\right)\\ {m}_{3}\left(t\right)& {m}_{4}\left(t\right)\end{array}\right]=\left[\begin{array}{cc}\left(1-at\right){e}^{-at}& t{e}^{-at}\\ -{a}^{2}t{e}^{-at}& \left(1+at\right){e}^{-at}\end{array}\right]$ (17)

It is easy to find that ${m}_{i}\left(t\right),\left(i=1,2,3,4\right)$ are bounded, which is assumed to be $0\le |{m}_{i}\left(t\right)|\le {d}_{2}$ . Thus, it has:

$|S|\text{\hspace{0.17em}}\le {d}_{1}\left(|\left[\begin{array}{c}-\frac{1}{{a}^{2}}{m}_{4}\\ {m}_{2}-\frac{2}{a}{m}_{4}\end{array}\right]|+|\left[\begin{array}{c}-\frac{1}{{a}^{2}}\\ -\frac{2}{a}\end{array}\right]|\right)\le {d}_{1}\left(\left[\begin{array}{c}\frac{1}{{a}^{2}}{d}_{2}\\ \left(1+\frac{2}{a}\right){d}_{2}\end{array}\right]+|\left[\begin{array}{c}-\frac{1}{{a}^{2}}\\ -\frac{2}{a}\end{array}\right]|\right)\le \left[\begin{array}{c}{d}_{31}\\ {d}_{32}\end{array}\right]$ (18)

Finally, it has:

$\begin{array}{c}|E\left(t\right)|=|{e}^{At}E\left(0\right)+{\int }_{0}^{t}{e}^{A\left(t-\tau \right)}B\stackrel{˙}{f}\text{ }\text{d}\tau |\\ \le |{e}^{At}E\left(0\right)|+|S|\\ \le \left[\begin{array}{c}|{e}_{1}\left(0\right){m}_{1}\left(t\right)|+|{e}_{2}\left(0\right){m}_{2}\left(t\right)|\\ |{e}_{1}\left(0\right){m}_{3}\left(t\right)|+|{e}_{2}\left(0\right){m}_{4}\left(t\right)|\end{array}\right]+\left[\begin{array}{c}{d}_{31}\\ {d}_{32}\end{array}\right]\le \left[\begin{array}{c}{\epsilon }_{1}\\ {\epsilon }_{2}\end{array}\right]\end{array}$ (19)

The theorem is proved.

3.3. Controller Design

By using feedback compensation, the system shown in formula (8) is transformed into:

$\left\{\begin{array}{l}\stackrel{˙}{\Theta }=\Omega \\ \stackrel{˙}{\Omega }=B\cdot {\Gamma }_{0}\end{array}$ (20)

where, B has been defined in formula (6). ${\Gamma }_{0}$ is the control inputs including the parts compensating the disturbance terms ${F}_{a}$ .

It is clear that the system in formula (20) is formed by two three-input-three-output subsystems. They can be expressed by one system shown as:

$\left\{\begin{array}{l}\stackrel{˙}{X}=M\cdot U\\ Y=X\end{array}$ (21)

where, $X\in {R}^{m}$ , $Y\in {R}^{m}$ , $U\in {R}^{m}$ and $M\in {R}^{m×m}$ is full rank.

Using a sampling period T to discretize the system shown in formula (21) yields:

$Y\left(k+1\right)=Y\left(k\right)+T\cdot M\cdot U\left(k\right)$ (22)

It is assumed that within the predictive horizon, the input signal is unchanged:

$U\left(k+i\right)=U\left(k\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\ge 1$ (23)

Recalling formula (23) and applying recursion method to the system given in formula (22) yields:

$\left\{\begin{array}{l}Y\left(k+1\right)=Y\left(k\right)+T\cdot M\cdot U\left(k\right)\\ Y\left(k+2\right)=Y\left(k+1\right)+T\cdot M\cdot U\left(k+1\right)=Y\left(k\right)+2T\cdot M\cdot U\left(k\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ Y\left(k+n\right)=Y\left(k+n-1\right)+T\cdot M\cdot U\left(k+n-1\right)=Y\left(k\right)+nT\cdot M\cdot U\left(k\right)\end{array}$ (24)

where, n represents the length of the predictive horizon.

Selecting a cost function yields the following minimization problem:

$\underset{U\left(k\right)}{\mathrm{min}}J\left(k\right)=\frac{1}{2}{\left[{Y}_{d}\left(k+n\right)-Y\left(k+n\right)\right]}^{\text{T}}\cdot \left[{Y}_{d}\left(k+n\right)-Y\left(k+n\right)\right]$

where ${Y}_{d}\left(k+n\right)={\left[{y}_{1d}\left(k+n\right),\cdots ,{y}_{md}\left(k+n\right)\right]}^{\text{T}}$ is the predictive reference signal which is given.

By taking partial derivative of $J\left(k\right)$ with respect to $U\left(k\right)$ and let $\partial J\left(k\right)/\partial U\left(k\right)=0$ , the predictive control law is derived as:

$U\left(k\right)={\left(nT\cdot {M}^{\text{T}}M\right)}^{-1}{M}^{\text{T}}\left[{Y}_{d}\left(k+n\right)-Y\left(k\right)\right]$ (25)

Thus, the predictive controller for the Euler angle control system is:

${\Omega }_{d}\left(k\right)=\frac{{\Theta }_{d}\left(k+{n}_{1}\right)-\Theta \left(k\right)}{{n}_{1}T}$ (26)

The predictive controller for the body rate control system is:

$\left\{\begin{array}{l}{\Gamma }_{0}\left(k\right)={\left({n}_{2}T\cdot {B}^{\text{T}}B\right)}^{-1}{B}^{\text{T}}\left[{\Omega }_{d}\left(k+{n}_{2}\right)-\Omega \left(k\right)\right]\\ \Gamma \left(k\right)={\Gamma }_{0}\left(k\right)-{B}^{-1}\cdot {\stackrel{^}{F}}_{a}\left(k\right)\end{array}$ (27)

Values of the parameters used in following simulation are given as: ${n}_{1}=50$ , ${n}_{2}=20$ .

4. Numerical Validation

In this section, the application scenario that the quad-rotor loads and drops unknown time-varying payloads is simulated. Comparison between the developed scheme and the commonly used approaches, such as the SMC and cascade PID (CPID), is carried out to validate the superiority of the former.

The initial conditions are given as:

${\left(\varphi ,\theta ,\psi |p,q,r\right)}_{0}^{\text{T}}={\left(0,0,0|0,0,0\right)}^{\text{T}}$ (28)

The reference signals (unit: rad) are given as:

${\Theta }_{d}={\left[0.2,0.2,0.2\right]}^{\text{T}}$ (29)

Three types of payloads are delivered by the quad-rotor in different time periods. Payload mass ${m}_{0}$ (unit: m), relative position $\left({x}_{0},{y}_{0}\right)$ (unit: m) and the inertial tensor ${J}_{0}$ (unit: kg∙m2) are given as:

P1: $\left({x}_{0},{y}_{0}\right)=\left(0.1,0.1\right)$ , ${m}_{0}=1$ , ${J}_{p}=\left[\begin{array}{ccc}0.014& -0.01& 0.005\\ -0.01& 0.014& 0.005\\ 0.005& 0.005& 0.022\end{array}\right]$ ;

P2: $\left({x}_{0},{y}_{0}\right)=\left(-0.15,0.08\right)$ , ${m}_{0}=0.8$ , ${J}_{p}=\left[\begin{array}{ccc}0.007& 0.01& -0.005\\ 0.01& 0.02& 0.003\\ -0.005& 0.003& 0.024\end{array}\right]$ ;

P3: $\left({x}_{0},{y}_{0}\right)=\left(-0.18,-0.14\right)$ , ${m}_{0}=1.2$ , ${J}_{p}=\left[\begin{array}{ccc}0.03& -0.03& -0.013\\ -0.03& 0.046& -0.01\\ -0.013& -0.01& 0.064\end{array}\right]$ .

Remark 2: The computer aided design (CAD) software model CATIA is used to build the 3-D model of the payloads. Then, by giving density of the payload, values of ${m}_{0}$ and ${J}_{p}$ can be measured.

Remark 3: Though values of $\left({x}_{0},{y}_{0}\right)$ for the three used payloads are slightly different, they are in different quadrant of the plane ${O}_{B}{X}_{B}{Y}_{B}$ . Thus, the perturbation torques from different directions induced by the payloads are generated and also simulated, such that we can make this application as practical as we can.

Simulation results are illustrated as Figures 4-9.

Conclusions are drawn as:

1) Figures 4-6 reveal that the developed control scheme is superior to the one based on CPID. Although the SMC-based scheme can achieve the same control performance with the developed scheme (see Figures 4-6), Figure 8 shows chattering phenomenon of the inputs of the SMC approach, which may damage the rotors of the quad-rotor. The superiority relies on the existence of the ESO which can estimate the disturbances in a highly accurate manner (see Figure 7) for compensation without the availability of the amplitude UB of the disturbances.

2) From Figure 8, Figure 9 and three enlarged figures in Figures 4-6, it can be seen that the developed predictive controller can degrade influences from sudden changes, no surging occurs on the input signals, and fluctuation on both the output signals and the body rates is very small.

5. Conclusions

This paper develops a control scheme with anti-disturbance capability and predictive function to realize the attitude control of quad-rotor for delivering unknown time-varying payloads. The conclusions are drawn as:

Figure 4. Roll angle responses.

Figure 5. Pitch angle responses.

Figure 6. Yaw angle responses.

Figure 7. Estimation of ${F}_{a}$ .

Figure 8. Control inputs.

Figure 9. Body rates.

1) The extended state observer can estimate the uncertainties in an accurate manner, significantly enhancing system robustness. The developed predictive controller can degrade influences caused by the sudden change from sudden loading/dropping of payload.

2) Simulation results show that, the developed control scheme is significantly superior to the one based on sliding model control and cascade proportional-integral-derivative, which are commonly used in flight control of quad-rotors.

Acknowledgements

This publication was supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Cite this paper: Wang, Y. (2019) Predictive Control of Quad-Rotor Delivering Unknown Time-Varying Payloads Based upon Extended State Observer. Advances in Aerospace Science and Technology, 4, 29-41. doi: 10.4236/aast.2019.42003.
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