Design and control of a self-adjusting outdoor landscape wall

This paper designs an outdoor landscape wall, which is equipped with an electrical motor, a sunlight intensity sensor and a rain sensor. The plants in the landscape wall will be rotated indoors when the weather is bad. To the contrary, based on signals from the sunlight intensity sensor, the plants can be rotated outdoors to get suitable sunlight. The dynamics of the electrical motor current control loop is much higher than the requirement of the position control loop. The whole control system can be divided into two subsystems: out-loop position control system and inner-loop current control system. An adaptive control strategy is proposed for out-loop position control. A nonlinear controller based on feedback linearization is developed for inner-loop current control. The two subsystems are synthesized with a first-order filter. Simulations are conducted to verify the proposed control strategy. The simulation results demonstrate that high-performance position tracking can be achieved under parameter uncertainty and disturbance.

2. According to system characteristic, the whole system is divided into two subsystems.A first-order filter is used to synthesize out-loop and inner-loop controllers.It avoids the differential explosion problem in conventional backstepping controller design procedure.
This article is organized as follow.The design of the self-adjusting outdoor landscape wall is showed in "Design of the self-adjusting outdoor landscape wall" section; the system dynamic model is established in "System dynamic model" section; the developed control strategy is presented in "Control strategy design" section; simulations are conducted in "Simulations" section; conclusions are drawn in "Conclusions" section.

Results
Design of the self-adjusting outdoor landscape wall.A self-adjusting outdoor landscape wall is designed and shown in Fig. 1.Several rows of flowerpot modules 100 are arranged in frame 1 from top to bottom.As shown in Fig. 2, flowerpot modules 100 includes a rotating shaft 2 and flowerpots 3 fixed on rotating shaft 2. The rotating shaft 2 is arranged on both sides of frame 1, and its end extends out of one side of frame 1. Flowerpot mounting supports 4 are arranged on rotating shaft 2. A flowerpot 3 is fixed on each flowerpot mounting support 4. Flowerpot 3 and flowerpot mounting support 4 are connected by bolts or clamps.The bottom of flowerpot mounting support 4 has a shaped hole.And rotating shaft 2 passes through the shaped hole to drive flowerpot mounting support 4 to rotate.In addition, the distance between each row is determined by the height of the plant and the size of flowerpot 3. The bottom of frame 1 is installed with a tilting water guide tank 12, which can guide water to the outlet.This design meets the requirements of different plant number on the landscape wall.The adjustment is flexible and easy to operate.Different number of plants can be arranged on different rows.
Frame 1 is equipped with a drive module to rotate shaft 2, which consists transmission wheel 5, transmission belt 6 and motor module.Transmission wheel 5 is fixed on the end of rotating shaft 2, which extends out the side of frame 1.The adjacent transmission wheels 5 are connected with transmission belt 6.The motor module includes motor 7, whose output end is equipped with an output wheel 8. Output wheel 8 is connected with transmission shaft 2. Transmission belt 6 connects transmission wheels 5 on rotating shaft 2 to realize synchronous turning control.The end of bottom rotating shaft 2 extending out the side of frame 1 is installed with an input wheel 9.And input wheel 9 and output wheel 8 are connected by transmission belt 6.The radius ratio of output wheel 9 and input wheel 8 is 1:5, i.e., the deceleration ratio is 5:1.The transmission is adapted to different flowerpot layout layers, which can be convenient and flexible to assembly.A signal acquisition module 10 is arranged at the top of frame 1. Signal acquisition module 10 includes a sunlight intensity sensor and a rainfall sensor.The   System dynamic model.The system dynamic model of the designed self-adjusting outdoor landscape wall can be simplified into where J is the is the lumped moment of inertia, θ is the rotational angle, B is the viscous friction coefficient, d represents the unmodeled forces and disturbances, and T is the drive torque of the motor.
The drive torque of the motor can be expressed as where C t is the torque constant and I is the current of the motor.
For the internal dynamics of the motor circuit, where R is the resistance, L is the inductance, U is the input voltage.And E a is the back electromotive force, and T be the state variables, the dynamics of the system can be rewritten in state space,

Control strategy design.
The plants in the landscape wall can be rotated according to the signals from the sunlight intensity sensor or the rainfall sensor, even remote control commands.To make the plants in the landscape wall in a better condition, a controller is designed for the landscape wall under parameter uncertainties and disturbances.Adaptive control is employed for out-loop position control to compensate the system uncertainties and disturbances.Feedback linearization is used in inner-loop current control to decide the actual control input.The controllers of out loop and inner loop are synthesized with a first-order filter.
Out-loop position control.Define the position tracking error as z 1 = x 1d − x 1 , a filtered tracking error z 2 = ż1 + k 1 z 1 , and an auxiliary signal v = ẋ1d + k 1 z 1 .x 1d and ẋ1d are the desired positon and its derivative.k 1 is a positive constant.And let z = [z 1 , z 2 ] T .From (4), it can be obtained where H is a lumped uncertainty, and is dependent on J , B , d.
The approximation of H can be expressed in the forms of approximations J , B and d, The approximation error of H is defined as In a compact expression, let H = T W and H = T W , where And define W = W − W. The control input torque is designed as And the adaptive law is designed as where Ŵ = diag(Ŵ 1 , Ŵ 2 , Ŵ 3 ) is a positive definite diagonal matrix, i = 1, 2, 3 .And Proj W i (•) is a projection function where W imax and W imin are known, i = 1, 2, 3.

Define a following Lyapunov function
The time derivative of V 1 is given by Because viscous friction coefficient of practice system B ≥ 0 , V1 ≤ 0 .Thus, the stability of the out-loop posi- tion control can be guaranteed.
Inner-loop current control.Define the position Let x 3d be the desired current.And define z 3 = x 3d − x 3 .So,

Define the control input voltage as
where k 3 is a positive constant.
Define a following Lyapunov function The time derivative of V 2 is given by Therefore, V2 ≤ 0 , the stability of the inner-loop current control can be guaranteed.
First-order filter.The controllers of out loop and inner loop are synthesized with a first-order filter to avoid the differential explosion problem in conventional backstepping controller design procedure.Figure 3 shows the first-order filter.The input of the filter is the out-loop desired torque T d , and the output is the inner-loop desired current x 3d and its derivative ẋ3d .The saturation function keeps the desired current x 3d in an appropriate range.Besides, the time constant τ should be chosen according to the properties of the selected motor.The stabilities of inner loop and out loop are guaranteed.And the dynamic requirement of the out-loop position control is not very high and is much lower than the dynamics of the inner-loop current control.The two controllers are synthesized with a first-order filter.Therefore, the stability of the whole system can be guaranteed.The proposed adaptive controller for the designed self-adjusting outdoor landscape wall is shown in Fig. 4.
(   dt , whose parameters are shown in Table 2.And the parameters cannot set bigger, because oscillation will occur.The second is nonlinear controller without parameter adaption (NC), i.e., Ŵ = 0 , the rest parameters are same as AC.The third is sliding mode controller (SMC).SMC is based on NC, and its control input torque is designed as The third is AC, whose parameters are shown in Table 3.
The quantity and scale of the plants in the landscape wall can be various, that means J, B are unknown.The tested desired trajectory is shown in Fig. 5a.Firstly, assuming d is constant, i.e., d = 0.2 Nm.The initial values of the adaptive parameters in AC are set as 0. The values of J, B of NC are set as the 0.8 times of the true values, so does SMC.The tracking errors of the four controllers are shown in Fig. 5b.Among the four controllers, AC demonstrates the smallest errors and the error will converge to zero finally.SMC obtain better control performance comparing with NC, because sliding mode term is applied.The control voltage of AC is shown in Fig. 5c.The estimated parameters of AC are shown in Fig. 5d-f.The estimations converge to their true values gradually.
Secondly, assuming d is variable, i.e., d = 0.2 sin π 10 t Nm .Same desired trajectory shown in Fig. 5a are tested.The initial values of the adaptive parameters in AC and parameters in NC and SMC are set the same as before.Figure 6a shows the tracking errors of the four controllers.AC still has the best tracking performance.The control voltage of AC is shown in Fig. 6b.The estimated parameters of AC are shown in Fig. 6c-e.The reason why the tracking errors of AC does not converge to zero and its parameter estimations cannot converge to true values is that a variable d is used in simulation.But even with a variable d , AC still shows the best tracking performance, which demonstrates the universality of AC.
Finally, assuming d is random, i.e., uniform random number of Matlab is used, and its maximum and mini- mum value is ± 0.2 Nm and sample time is 0.1 s.Same desired trajectory shown in Fig. 5a are tested.The initial values of the adaptive parameters in AC and parameters in NC are set the same as before.Figure 7a shows the tracking errors of the four controllers.AC still has the best tracking performance.The control voltage of AC is shown in Fig. 7b.The estimated parameters of AC are shown in Fig. 7c-e.Why the tracking errors of AC does not converge to zero and why its parameter estimations cannot converge to true values, the reason is same.But even with a random d , AC still shows the best tracking performance, which also demonstrates the universality of AC.

Discussion
In this paper, a new self-adjusting outdoor landscape wall is designed.The outdoor landscape wall is driven by an electrical motor and equipped with a sunlight intensity sensor and a rain sensor.The plants in the landscape wall can be rotated according to the signals from the sensors.The dynamics of the electrical motor current control loop is much higher than the requirement of the position control loop.The whole control system can be divided

Figure 1 .
Figure 1.Structure of the designed self-adjusting outdoor landscape wall.

Figure 3 .
Figure 3. Schematic diagram of the first-order filter.

Figure 4 .
Figure 4. Schematic diagram of the proposed adaptive controller.

Figure 5 .Figure 6 .Figure 7 .
Figure 5. (a) Desired trajectory; (b) tracking errors with a constant d; (c) control voltage of AC with a constant d; (d) estimation of J with a constant d; (e) estimation of B with a constant d; (f) estimation of d with a constant d.
Plants on the landscape wall can be rotated inside for protection in bad weather.2. Sunlight requirement of plants can be satisfied through keeping plants in suitable position.3. Self-adjustment can be realized according to the signals from the sunlight intensity sensor or the rainfall sensor, even remote control commands.

Table 1 .
Parameters of the motor and system.

Table 2 .
Parameters of PID.

Table 3 .
Parameters of AC and SMC.