Nonlinear backstepping design for the underactuated TORA system

The nonlinear feedback cascade model of the underactuated translational oscillators with rotating actuator is obtained through a collocated partial feedback linearization and a global change of coordinates. A nonlinear controller is designed with the backsteping technology, which treats the state variables as virtual control inputs to design the virtual controllers step by step. The system stability is proved with the Lyapunov stability theorem. The simulation results show the system under any initial states can be asymptotically stabilized to the origin and the controller has a good control performance.

1. Introduction

The TORA (translational oscillators with rotating actuator) is a classical underactuated mechanical system, which is composed of a non-actuated translational oscillators and an actuated pendulum. It was developed as a simplified model of a dual spin spacecraft [1]. One research motivation to control the TORA comes from the possible use to suppress the translational vibration with a rotational actuation. The other is an independent interest to control the TORA as a benchmark problem in the nonlinear control design.

The control problem on the TORA has attracted a considerable amount of attentions from many researchers. Some focus on the intelligent design methods [2-4] and some focus on the nonlinear design methods [5-11]. Among the intelligent design methods, a self-tuning fuzzy sliding mode control is proposed [2-3] and there are some fuzzy rules and parameters to be preset. A neural network controller is proposed with SPSA Algorithm [4] and a time-consuming learning process is needed. Among the nonlinear design methods, the energy-based design [5-7], the backstepping design [8-9] and sliding mode based design [10], the design processes and the obtained controllers are relatively cumbersome. A comparison study of nonlinear control techniques for the TORA is performed [5].

In this paper, the TORA system is transformed into a nonlinear feedback cascade system through a collocated partial feedback linearization and a global change of coordinates. Then a nonlinear backstepping design method is proposed. The design procedure treats the state variables as virtual control inputs to design the virtual controllers step by step, and the system stability is proved via the Lyapunov stability theorem.

2. The TORA system model

The TORA is shown schematically in Fig. 1. The oscillator consists of a cart of mass $m_{1}$ connected to a fixed wall by a linear spring of stiffness $k$ . The cart is constrained to have one-dimensional travel and its position is $q_{1}$ . The rotational actuator with the output torque is attached to the cart and it has a pendulum with the equivalent mass $m_{2}$ , the rotate radius $r$ , the moment of inertia $I$ and the rotate angle $q_{2}$ . Since the rotating torque can not drive the cart directly, it is a benchmark example of the underactuated mechanical system, which has one control input $τ$ and two configuration variables ( $q_{1}$ , $q_{2}$ ), and its Euler-Lagrange equations of motion can be obtained as:

1

\{\begin{array}{l} (m_{1} + m_{2}) {\ddot{q}}_{1} + m_{2} r c o s (q_{2}) {\ddot{q}}_{2} - m_{2} r s i n (q_{2}) {\dot{q}}_{2}^{2} + k q_{1} = 0, \\ m^{2} r c o s (q_{2}) {\ddot{q}}_{1} + (m_{2} r^{2} + I) {\ddot{q}}_{2} + m_{2} g r s i n (q_{2}) = τ . \end{array}

Fig. 1TORA system configuration

Since the partial feedback linearization technology proposed by Spong [12] can simplify the system dynamics. The following collocated partial feedback linearization is adopted:

2

τ = ((m_{2} r^{2} + I) - \frac{{(m_{2} r c o s (q_{2}))}^{2}}{m_{1} + m_{2}}) u + \frac{{(m_{2} r)}^{2} \sin (q_{2}) \cos (q_{2})}{m_{1} + m_{2}} {\dot{q}}_{1}^{2} - k m_{2} r q_{1} c o s (q_{2}) + m_{2} g r s i n (q_{2}),

and it can simplify the dynamics of the shape variable $q_{2}$ as:

{\ddot{q}}_{2} = u .

Olfati-Saber proposed a global change of coordinates [13] such as:

3

\{\begin{array}{l} x_{1} = q_{1} + \frac{m_{2} r s i n (q_{2})}{m_{1} + m_{2}}, \\ x_{2} = q_{2}, \\ x_{3} = (m_{1} + m_{2}) {\dot{q}}_{1} + m_{2} r c o s (q_{2}) {\dot{q}}_{2}, \\ x_{4} = {\dot{q}}_{2} . \end{array}

It can transform the system dynamics into a nonlinear system as:

4

\{\begin{array}{l} {\dot{x}}_{1} = \frac{x_{3}}{m_{1} + m_{2}}, \\ {\dot{x}}_{2} = x_{4}, \\ {\dot{x}}_{3} = - k x_{1} + \frac{k m_{2} r s i n x_{2}}{m_{1} + m_{2}}, \\ {\dot{x}}_{4} = u . \end{array}

Therefore, the system model of TORA can be described with a nonlinear feedback cascade model [14] as:

5

\{\begin{array}{l} {\ddot{x}}_{1} = - \frac{k x_{1}}{m_{1} + m_{2}} + \frac{k m_{2} r}{{(m_{1} + m_{2})}^{2}} s i n x_{2}, \\ {\ddot{x}}_{2} = u . \end{array}

3. The nonlinear controller design through backstepping

Since the model of TORA can be transformed into a cascade nonlinear system through the collocated partial feedback linearization (2) and the global change of coordinates (3), a nonlinear controller can be designed with the backstepping technology [14]. The design process is:

Step 1.

From the dynamic equation of state $x_{1}$ in the TORA system model (5):

6

{\ddot{x}}_{1} = - \frac{k}{m_{1} + m_{2}} x_{1} + \frac{k m_{2} r}{{(m_{1} + m_{2})}^{2}} s i n x_{2} .

It can be seen that only $x_{2}$ is in the dynamic equation (6) of state $x_{1}$ . So $x_{2}$ can be looked as the virtual control input to Equation (6) to control the system state $x_{1}$ and ${\dot{x}}_{1}$ . A reference trajectory $x_{2 r}$ is defined for $x_{2}$ to track as:

x_{2 r} = \frac{k}{m_{1} + m_{2}} x_{1} - \frac{k m_{2} r}{{(m_{1} + m_{2})}^{2}} s i n x_{2} + x_{2} - k_{1} x_{1} - k_{2} {\dot{x}}_{1},

which leads to an error $e_{2}$ defined as:

7

e_{2} = x_{2} - x_{2 r} = - \frac{k}{m_{1} + m_{2}} x_{1} + \frac{k m_{2} r}{{(m_{1} + m_{2})}^{2}} s i n x_{2} + k_{1} x_{1} + k_{2} {\dot{x}}_{1},

where, $k_{1}$ and $k_{2}$ are positive constants that are chosen such that:

{\ddot{x}}_{1} + k_{2} {\dot{x}}_{1} + k_{1} x_{1} = 0,

is asymptotically stable at $(x_{1}, {\dot{x}}_{1}) = (0,0)$ .

The dynamic Equation (6) of state $x_{1}$ can be described as:

8

[\begin{matrix} {\dot{x}}_{1} \\ {\ddot{x}}_{1} \end{matrix}] = [\begin{matrix} 0 1 \\ - k_{1} - k_{2} \end{matrix}] [\begin{matrix} x_{1} \\ {\dot{x}}_{1} \end{matrix}] + [\begin{matrix} 0 \\ 1 \end{matrix}] e_{2} .

Define $X_{1} = [\begin{matrix} x_{1} \\ {\dot{x}}_{1} \end{matrix}]$ , $A = [\begin{matrix} 0 1 \\ - k_{1} - k_{2} \end{matrix}]$ , $B = [\begin{matrix} 0 \\ 1 \end{matrix}]$ . So the Equation (8) is written as:

{\dot{X}}_{1} = A X_{1} + B e_{2} .

According to the control theory for the linear time-invariant system [15], the choose of $k_{1}$ and $k_{2}$ can implies the existence of a positive definite symmetry matrix $P$ and $Q$ so that:

A^{T} P + P A = - Q .

Consider a scalar positive definite Lyapunov function given by:

V_{1} = \frac{1}{2} (X_{1}^{T} P X_{1} + e_{2}^{2}) \geq 0 .

Its time derivative ${\dot{V}}_{1}$ is given by:

9

{\dot{V}}_{1} = - \frac{1}{2} X_{1}^{T} Q X_{1} + e_{2} ({\dot{e}}_{2} + X_{1}^{T} P B) = - \frac{1}{2} X_{1}^{T} Q X_{1} + e_{2} (- \frac{k}{m_{1} + m_{2}} {\dot{x}}_{1} + \frac{k m_{2} r {\dot{x}}_{2}}{{(m_{1} + m_{2})}^{2}} c o s x_{2} + k_{1} {\dot{x}}_{1} + k_{2} {\ddot{x}}_{1} + X_{1}^{T} P B),

We note that the variable ${\dot{x}}_{2}$ enters the right hand side of Equation (9). We now proceed to look ${\dot{x}}_{2}$ as the control variable and design a reference trajectory ${\dot{x}}_{2 r}$ for it to make the second term of right hand in Equation (9) be non-positive.

Step 2.

In step 1, the time derivative of the Lyapunov function $V_{1}$ is obtained in Equation (9). In order to make the ${\dot{V}}_{1}$ be negative definite, the state ${\dot{x}}_{2}$ is looked as the virtual control input in Equation (9). A reference trajectory ${\dot{x}}_{2 r}$ is defined as:

{\dot{x}}_{2 r} = \frac{k}{m_{1} + m_{2}} {\dot{x}}_{1} - \frac{k m_{2} r {\dot{x}}_{2}}{{(m_{1} + m_{2})}^{2}} c o s x_{2} - k_{1} {\dot{x}}_{1} - k_{2} {\ddot{x}}_{1} - X_{1}^{T} P B + {\dot{x}}_{2} - k_{3} e_{2} .

The tracking error ${\hat{e}}_{2}$ for ${\dot{x}}_{2}$ can be defined as:

10

{\hat{e}}_{2} = {\dot{x}}_{2} - {\dot{x}}_{2 r} = - \frac{k}{m_{1} + m_{2}} {\dot{x}}_{1} + \frac{k m_{2} r {\dot{x}}_{2}}{{(m_{1} + m_{2})}^{2}} c o s x_{2} + k_{1} {\dot{x}}_{1} + k_{2} {\ddot{x}}_{1} + X_{1}^{T} P B + k_{3} e_{2} .

So:

{\dot{V}}_{1} = - \frac{1}{2} X_{1}^{T} Q X_{1} - k_{3} e_{2}^{2} + e_{2} {\hat{e}}_{2} .

We modify the scalar positive Lyapunov function $V_{1} \geq 0$ as:

V_{2} = V_{1} + \frac{1}{2} {\hat{e}}_{2}^{2} \geq 0 .

Differentiating $V_{2}$ :

11

{\dot{V}}_{2} = - \frac{1}{2} X_{1}^{T} Q X_{1} - k_{3} e_{2}^{2} + {\hat{e}}_{2} (e_{2} + {\dot{\hat{e}}}_{2}) = - \frac{1}{2} X_{1}^{T} Q X_{1} - k_{3} e_{2}^{2}

+ {\hat{e}}_{2} (e_{2} - \frac{k}{m_{1} + m_{2}} {\ddot{x}}_{1} + \frac{k m_{2} r {\ddot{x}}_{2}}{{(m_{1} + m_{2})}^{2}} c o s x_{2} - \frac{k m_{2} r {\dot{x}}_{2}^{2}}{{(m_{1} + m_{2})}^{2}} s i n x_{2} + k_{1} {\ddot{x}}_{1} - \frac{k k_{2}}{m_{1} + m_{2}} {\dot{x}}_{1} + \frac{k_{2} k m_{2} r {\dot{x}}_{2}}{{(m_{1} + m_{2})}^{2}} c o s x_{2} + X_{1}^{T} P B + k_{3} {\dot{e}}_{2}) .

Step 3.

The system control variable $u = {\ddot{x}}_{2}$ enters in the right hand of the Equation (11). So, In order to make the ${\dot{V}}_{2}$ be negative definite, we can make the following equation hold:

12

e_{2} - \frac{k}{m_{1} + m_{2}} {\ddot{x}}_{1} + \frac{k m_{2} r {\ddot{x}}_{2}}{{(m_{1} + m_{2})}^{2}} c o s x_{2} - \frac{k m_{2} r {\dot{x}}_{2}^{2}}{{(m_{1} + m_{2})}^{2}} s i n x_{2} + k_{1} {\ddot{x}}_{1} - \frac{k k_{2}}{m_{1} + m_{2}} {\dot{x}}_{1} + \frac{k_{2} k m_{2} r {\dot{x}}_{2}}{{(m_{1} + m_{2})}^{2}} c o s x_{2} + X_{1}^{T} P B + k_{3} {\dot{e}}_{2} = - k_{4} {\hat{e}}_{2} .

Such that:

13

{\dot{V}}_{2} = - \frac{1}{2} X_{1}^{T} Q X_{1} - k_{3} e_{2}^{2} - k_{4} {\hat{e}}_{2}^{2} \leq 0.

Therefore, the control law can be obtained from (12) as:

14

u = - \frac{{(m_{1} + m_{2})}^{2}}{k m_{2} r c o s x_{2}} \{e_{2} - \frac{k}{m_{1} + m_{2}} {\ddot{x}}_{1} - \frac{k m_{2} r {\dot{x}}_{2}^{2}}{{(m_{1} + m_{2})}^{2}} s i n x_{2} + k_{1} {\ddot{x}}_{1} + \frac{k_{2} k m_{2} r {\dot{x}}_{2}}{{(m_{1} + m_{2})}^{2}} c o s x_{2} - \frac{k k_{2}}{m_{1} + m_{2}} {\dot{x}}_{1} + {\dot{X}}_{1}^{T} P B + k_{3} {\dot{e}}_{2} + k_{4} {\hat{e}}_{2}\} .

Theorem 1: The TORA system described by Equation (1) is asymptotically stable under the control input (2) and the nonlinear controller (14) with the feedback cascade model (5) and the global change of coordinates (3).

Proof:

The backstepping design process has proved: the time derivative of the chosen positive definite Lyapunov function $V_{2}$ is negative definite. That is to say, the three terms of the right hand in Equation (13) is asymptotically approach to 0. Since that the first term $- \frac{1}{2} {\bar{x}}^{T} Q \bar{x}$ approaches to 0, $(x_{1}, {\dot{x}}_{1})$ must asymptotically approach to (0,0). From the second term $- k_{3} e_{2}^{2}$ approaches to 0, it is known from Equation (7) that $x_{2}$ must asymptotically approach to 0. The third term $- k_{4} {\hat{e}}_{2}^{2}$ approaches to 0 implies that ${\dot{x}}_{2}$ asymptotically approach to 0. Therefore, the system states $(x_{1}, {\dot{x}}_{1}, x_{2}, {\dot{x}}_{2})$ of the TORA described by Equation (5) asymptotically approach to (0,0,0,0).

The global change of coordinates (3) is an invertible transformation, which is:

15

\{\begin{array}{l} q_{1} = x_{1} - \frac{m_{2} r s i n x_{2}}{m_{1} + m_{2}}, \\ {\dot{q}}_{1} = {\dot{x}}_{1} - \frac{m_{2} r c o s x_{2}}{m_{1} + m_{2}} {\dot{x}}_{2}, \\ q_{2} = x_{2}, \\ {\dot{q}}_{2} = {\dot{x}}_{2} . \end{array}

It can be seen from Equation (15) that $(x_{1}, {\dot{x}}_{1}, x_{2}, {\dot{x}}_{2})$ asymptotically approach to (0,0,0,0) implies that $(q_{1}, {\dot{q}}_{1}, q_{2}, {\dot{q}}_{2})$ approach to (0,0,0,0). That is to say, the control input $τ$ that is calculated with Equation (2), (14) and (4) can asymptotically stabilize the TORA system.

Remark 1: There is a singularity when $x_{2} = π / 2$ in the controller (14). The method to deal with the singularity is: a linear controller is designed through the LQR method [15] with the linearized system around $x_{2} \in (π / 2 - Δ, π / 2 + Δ)$ for the TORA system:

16

u = K x .

The controller switch between the nonlinear controller (14) and the linear controller (16) according to the system state $x_{2}$ .

Remark 2: Only three steps are needed to obtain the nonlinear controller with the backstepping technology. The first step is designed for a one-input and two-state system, the second step is designed for a one-input and on state system, and the third step is to obtain the controller. The design process and the proposed controller are simple compared with some the literature [5-6].

4. Simulation studies

In order to test the proposed nonlinear control algorithm, the following system parameters [16] are used: $m_{1} =$ 10, $m_{2} =$ 1, $k =$ 1, $r =$ 1, $I =$ 1. The parameters of the nonlinear controller are chosen as $P = [6.5 1; 1 0.3],$ $Q = [10 0; 0 1],$ $K = [\begin{matrix} –2 & –2 & 8 & 4 \end{matrix}],$ $k_{1} =$ 5, $k_{2} = 5,$ $k_{3} = 0.5$ , $k_{4} = 0.5$ .

The simulation results are shown in Fig. 2-4, where the position unit and angle unit respectively are meter and radian to calculate the $V_{2}$ . The Fig. 2 is obtained under the initial state $(x_{1}, {\dot{x}}_{1}, x_{2}, {\dot{x}}_{2}) = (100,0,0,0)$ i.e. $(q_{1}, {\dot{q}}_{1}, q_{2}, {\dot{q}}_{2}) = (100,0,0,0)$ , Fig. 3 is obtained under the initial state $(x_{1}, {\dot{x}}_{1}, x_{2}, {\dot{x}}_{2}) = (0,0,90°,0)$ i.e. $(q_{1}, {\dot{q}}_{1}, q_{2}, {\dot{q}}_{2}) = (- 90.9,0,90°,0)$ and Fig. 4 are obtained under the initial state $(x_{1}, {\dot{x}}_{1}, x_{2}, {\dot{x}}_{2}) = (100,0,90°,0)$ i.e. $(q_{1}, {\dot{q}}_{1}, q_{2}, {\dot{q}}_{2}) = (9.09, 90°,0,0)$ .

Fig. 2Simulation results under system state q1,q˙1,q2,q˙2=(100,0,0,0) i.e. x1,x˙1,x2,x˙2=(100,0,0,0)

a) Position and velocity of the cart

b) Angle and angle velocity of the pendulum

c) Control torque $τ$

d) The Lyapunov function $V_{2}$

It can be seen form the simulation results that the TORA system is asymptotically stable under any initial states with the proposed control algorithm. At the same time, it is shown that it is possible to suppress the translational vibration with a rotational actuation. On the other hand, the control performance can be improved through adjusting the parameters of the proposed controller.

From Fig. 3(d), the Lyapunov function is not a monotonically decreasing function. The reason is the controller switches between the nonlinear controller (14) and the linear controller (16). When the linear controller is in use, the Lyapunov function may be not a monotonically decreasing function. Lots of simulation experiments show that the parameters $k_{1}$ , $k_{2}$ , $k_{3}$ , $k_{4}$ respectively correspond to the system states $x_{1}$ , ${\dot{x}}_{1}$ , $x_{2}$ , ${\dot{x}}_{2}$ therefore it is easy to adjust the parameters for a improved system performance.

Fig. 3Simulation results under system state q1,q˙1,q2,q˙2=(–90.9,90°,0,0) i.e. x1,x˙1,x2,x˙2=(0,0,90°,0)

a) Position and velocity of the cart

b) Angle and angle velocity of the pendulum

c) Control torque $τ$

d) The Lyapunov function $V_{2}$

Fig. 4Simulation results under system state q1,q˙1,q2,q˙2=(9.09,90°,0,0) i.e. x1,x˙1,x2,x˙2=(100,0,90°,0)

a) Position and velocity of the cart

b) Angle and angle velocity of the pendulum

c) Control torque $τ$

d) The Lyapunov function $V_{2}$

5. Conclusions

The underactuated TORA is transformed to a nonlinear feedback cascade system with a collocated partial feedback linearization and a global change of coordinates. A nonlinear control algorithm is proposed from the backstepping technology. The design procedure treats the state variables as virtual control inputs to design the virtual controllers step by step. The first step is designed for a one-input and two-state system, the second step is designed for a one-input and one-state system, and the third step is to obtain the nonlinear controller. The system stability and the control performance are proved via the Lyapunov stability theorem or simulation results.

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About this article

Received

29 January 2013

Accepted

30 November 2013

Published

31 March 2014

Keywords

underactuated mechanical systems

TORA

backstepping

vibration control

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 61175086) and a Project of Shandong Province Higher Educational Science and Technology Program (No. J13LN25).

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.