Energy Saving
Content
Energy-Saving with the Hidden Reference tracking method
ENERGY-SAVING WITH THE HIDDEN REFERENCE TRACKING METHOD
HENRY A. MENDIBURU DÍAZ Electronic E ngineer, Master in Control and Automatization [email protected]
Abstract: This paper presents a new method, the hidden reference , which reduces the signal control level when it is applied to systems that require to track step references. Two practical examples are provided. In the first one, the task is to control the position of a DC motor with a full-state feedback controller. The second example applies the hidden reference method to a self-guided vehicle controlled by optimal control, thus showing that this method eliminates the chattering in the helm s signal control. Copyright © 2004 IFAC.
Keywords: energy, motor, simulation, reference, step, tracking, trajectories, vehicle.
1. INTRODUCTION The behavior of a system s response to a control action depends on the capacity of the equipment that emits the signal control. This capacity may refer to physical variables such as voltage, current, power, angle, force, and so on, but in order to generalize, these variables will be known plainly as energy . It is clear that if a system is required to follow a step reference, a quick and exact response will request higher energy consumption. However, this unnecessary energy demand can be reduced by replacing the step reference with a special trajectory from the initial point to the final point. This trajectory can be described as a straight line or a function, such as a sine, a cosine, an exponential function, and the like, thus converting the step reference problem into a trajectory tracking problem. The key is not to follow this new trajectory but to arrive to the destination with an acceptable behavior and within a reasonably time. This paper suggests a function based on the own system behavior. This function will allow the system to achieve the desired trajectory and to reduce energy consumption not only in picks but also in the cumulative total.
time. At the same time, the reference describes the natural route that the system would follow. The key of this method is the following function:
x(t ) x(t )
Xd Xd
Td (t ) Td (t ) Xd
Xd
x(t )
XH (t )
(1)
Where: - x(t), is the state to be controlled - XH(t), is the value of the state X given by the simulation process - Xd, is the desired value of the state X - Td(t), is the desired trajectory - , , design parameters (values between 0 and 1) As shown, this function has two stages. First, when x(t) is less than or equal to certain percentage ( ) of the desired value, the desired trajectory follows the values given by the simulation process, but at the same time a certain percentage ( ) of the current error between the real state and the desired value is added to the signal control. Second, when x(t) is greater than certain percentage of the desired value, the desired trajectory is equal to the desired value. The simulated controller s gains or weights that make it possible to generate the desired trajectory could or could not be similar to those of the real controller. Nevertheless, to allow for comparability, this paper assumes similar values. Those weights are expected to search for the best possible response (either amped, sub-damped, or critically-damped). Furthermore, it is crucial to highlight that this response must not be over-damped; otherwise, expected results could not be obtained.
2. THE HIDDEN REFERENCE METHOD To obtain a trajectory from an initial point X(to) to a final point X(tf), the mathematical model of the system is simulated by performing iterations with computer software. Each iteration yields a new point that becomes part of the desired trajectory, thus making up a reference that varies in each instant of
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
1
Ms. Ing. Henry A. Mendiburu Díaz
3. APPLYING THE HIDDEN REFERENCE METHOD
V(t)
Ra
La
,
i(t) J
3.1. Position Control of a DC Motor
b
In this section our goal is to take the angular position of a DC motor from an initial point to a final reference. To perform this task, a continuous current motor of 40v and 100w was chosen (Fig.1). The motor is subject to the following technical specifications: - Moment of inertia of the rotor (J): 4.326x10-5 Kg.m2/s2 - Motor friction / damping ratio (b): 0.1 N.m/s - Electromotive force constant (Ka): 0.05665 N.m/Amp - Motor armature constant (Kt): 0.0573 - Electric resistance (Ra): 1.1 ohm - Electric inductance (La): 0.0001 H - The rotor and shaft are assumed to be rigid The mathematical model of the motor is subject to the equations (2) and (3):
V (t ) di (t ) Ra i (t ) La dt Ka (t )
Fig.1 Scheme of the DC motor It is necessary to point out that if in any of the three cases: straight line, sine, or cosine, a greater time were used for the reference before setting it constant, a greater settling time would have been obtained. This would have not been feasible, since we wanted to compare similar answers among all cases. A full-state feedback controller with gains [0.0003, 0.3390, -1.3267] was used. The results are shown in the following figures:
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
(a)
(2)
10
Signal Control (Voltage)
J
(t ) b
(t )
Kt i (t )
(3)
volts
8 6
Where V(t) is the source voltage (volts), i(t) is the armature current (Amp), (t ) is the angular position (rad), (t ) is the angular speed (rad/sec), and (t ) is the angular acceleration (rad/sec2). The state-space form of the system is given by the equation (4):
0 0 i 0 1 b J Ka La 0 Kt J Ra La i 0 0 1 La V
(b)
4 2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Fig. 2: Response to a STEP FUNCTION reference. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
(4)
(a)
First, the system was simulated in order to reach a step reference (Fig.2), and then to follow different desired trajectories. The first function used was a straight line with gradient conditioned to a final time of 0.1 sec., after this point in time, the trajectory becomes similar to the desired angular position (Fig.3). The second function was a variant of the cosine function: (-cos + 1),(Fig.4). The third function was a wave sine (Fig.5). As well as the previous one, this function uses a frequency of 2.5 Hz, and it is substituted by the desired angular position when it reaches its maximum amplitude (at /2). Finally the hidden trajectory is used with parameters =0.35 y =0.96 (Fig.6). The simulation considered an initial condition of [0.01,0,0], and a desired angular position of 1 rad., with a sampling interval of 0.005 sec.
Signal Control (Voltage) 10 8 6 volts 4 2 0 0
(b)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 time(sec)
Fig. 3: Response to a STRAIGHT LINE FUNCTION reference. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
2
Energy-Saving with the Hidden Reference tracking method
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
Table 1: Summary of results: DC Motor (*) Vmax 8.25 7.86 9.97 8.01 5.42 Pmax 60.24 54.58 54.11 56.64 25.91 Pacc 1977 1862 1897 1900 1388 Ts 0.62 0.67 0.66 0.68 0.67
(a)
Signal Control (Voltage) 10 8 6 volts
Step Line Sine Cosine Hidden
(b)
4 2 0 0
(*) Vmax maximum voltage (volt) Pmax maximum power (watts) Pacc accumulated power (watts) Ts settling time (sec.)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 time(sec)
Fig. 4: Response to a COSINE FUNCTION reference. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
3.2. Location control of a self-guided vehicle In this section the objective is to eliminate the oscillatory behavior or chattering in signal control. This is achieved by applying the proposed method called hidden reference . The time required to arrive to the destination can be reduced by modifying the signal control. When the value of the actual position is smaller than the value of the desired position, the function (t) is added to the signal control. This function (5) reduces the settling time of the system.
(t ) YH (t ) Y (t ) Y (t ) YH (t )
(a)
Signal Control (Voltage) 10 8 6 volts 4 2 0 0
(5)
(b)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 time(sec)
Fig. 5: Response to a SINE FUNCTION reference. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
Where: - YH(t), is the simulated position (m) - Y(t), is the real position (m) - (t), is the control function (radians) Suppose that a vehicle with longitude (L) 0.5 m that moves with a constant speed (v) of 3 m /s is located in the XY plane and goes toward a desired position in the Y axis. The vehicle s helm has a maximum angle of 30º (Fig. 7). Y
(a)
Signal Control (Voltage) 10 8 6 volts 4 2 0 0
(b)
X
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 time(sec)
Fig. 6: Response to a HIDDEN REFERENCE FUNCTION. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
Fig. 7: Scheme of a self-guided vehicle
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
3
Ms. Ing. Henry A. Mendiburu Díaz
The mathematical model of the system is given in the equations (6), (7), and (8):
x v cos( ) y v sin( ) v tan( ) L
The second case (Fig.9) shows the response and the signal control of the system using the hidden reference method, with parameters =0.15 y =0.9. The third case (Fig.10) shows the response and the signal control of the system using a straight-line function with gradient conditioned to a final time of 7 sec.
Position X vs Y Y (m) 40 30 20 10 10 Y (m) 40 30 20 10 0 degree 20 0 -20 0 5 10 time (sec) 15
(6) (7) (8)
Using input-state linearization, the following control law is obtained:
arctan L v sin( )
2
(a)
K1 ( y
yd )
K 2 v sen( )
(9)
14
18 Position Y(t)
22
26
X (m) 30
Where: - x, is the displacement in the X axis (m) - y, is the displacement in the Y axis (m) - yd, is the desired position in the Y axis (m) - , is the inclination angle of the vehicle (rad) - , is the helm s angle (rad) - K1, K2, are the parameters (gains) of control The system was simulated considering the initial condition [10,10] with =45º, looking for a desired position of 40m in the Y axis with =0º. The optimal controller gives gains: K1 = 1.35, K2 = 2.2, the sampling interval is 0.05 sec. The following graphs show the results of the simulations. The first case (Fig.8) shows the response and the signal control of the system using optimal control (step reference).
Position X vs Y Y (m) 40 30
(b)
5 Helm´s Angle d(t)
10
time (sec) 15
(c)
Fig. 9: Simulation using HIDDEN REFERENCE . (a) Position in the XY plane (solid), Case-1 (dashed). (b) Position Y(t) (solid), Reference (dashed). (c) Helm s angle vs time.
Position X vs Y Y (m) 40 30
(a)
20 10 10 Y (m) 40 30 20 10 0 degree 20 0 -20 0 5 10 time (sec) 15
(a)
20 10 10 Y (m) 40 30 20
14
18 Position Y(t)
22
26
X (m) 30
14
18 Position Y(t)
22
26
X (m) 30
(b)
10 0 degree 20 0 -20 0 5 10
(b)
5 Helm´s Angle d(t)
10
time (sec)15
5 Helm´s Angle d(t)
10
time (sec) 15
(c)
(c)
time (sec) 15
Fig. 8: Simulation using OPTIMAL CONTROL. (a) Position in the XY plane (solid), Reference (dashed). (b) Position Y(t) (solid), Reference (dashed). (c) Helm s angle vs time.
Fig. 10: Simulation using STRAIGHT LINE FUNCTION. (a) Position in XY plane (solid), Case-1 (dashed). (b) Position Y(t) (solid), Reference (dashed). (c) Helm s angle vs time.
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
4
Energy-Saving with the Hidden Reference tracking method
As it can be seen, the second and the third cases do not reach the response speed of the first case. Therefore, the fourth case (Fig.11) shows with the response and the signal control of the system using optimal control, but achieving a similar response to the second and third cases in order to allow the reader to compare the results, (K2=2.56).
Position X vs Y Y (m) 40 30
defined trajectory, but to move from an initial point to a final point, with a similar behavior to the response of the simulated system. - A reduction of energy consumption is achieved by substituting the maximum energy picks for low amplitude and constant amplitude curves. - The simulation of the system must not provide an over-damped response.
(a)
20 10 10 Y (m) 40 30
- This method is useful only when it is required to follow step references.
14
18 Position Y(t)
22
26
X (m) 30
5. REFERENCES Burg, T.C., Dawson, D.M., and J. Hu. (1994). Velocity tracking control for a separately excited DC motor without velocity measurements. American Control Conference pp. 1051-1056. Frank, P.M. (1990). Fault diagnosis in dynamic systems using analitical and knowledge based redundancy: a survey and some new results. Automatica 26(3), 459 474. Galler, D. and A. Kusko (1983). Control means for minimization of losses in AC and DC motor drives. IEEE transactions on industry applications IA-19(4), 561-570. Preuss, H.P. (1982). Perfect steady-state tracking and disturbance rejection by constant state feedback. International Journal of Control, 35(1), 75-93.
(b)
20 10 0 degree 20 0 -20 0 5 10 time (sec) 15
5 Helm´s Angle d(t)
10
time (sec)15
(c)
Fig. 11: Second simulation using OPTIMAL CONTROL. (a) Position in XY plane (solid), Reference (dashed). (b) Position Y(t) (solid), Reference (dashed). (c) Helm s angle vs time. Table 2: Summary of results: Self-guided Vehicle (*) Y(x) 17.8 22.0 22.7 22.0 Y(t) 11.6 11.5 12.5 12.8
Case 1 Case 2 Case 3 Case 4
(*) Y(x) trajectory of Y in function at X axis (settling position in the XY plane, meters). Y(t) trajectory of Y in function at time (settling time, seconds).
4. CONCLUSIONS - The hidden reference method uses a computersimulated system to give insight about the system's response to a known stimulus. - The method involves the conversion of a step reference problem into a trajectory tracking problem, in which the key is not to move along a
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
5
ENERGY-SAVING WITH THE HIDDEN REFERENCE TRACKING METHOD
HENRY A. MENDIBURU DÍAZ Electronic E ngineer, Master in Control and Automatization [email protected]
Abstract: This paper presents a new method, the hidden reference , which reduces the signal control level when it is applied to systems that require to track step references. Two practical examples are provided. In the first one, the task is to control the position of a DC motor with a full-state feedback controller. The second example applies the hidden reference method to a self-guided vehicle controlled by optimal control, thus showing that this method eliminates the chattering in the helm s signal control. Copyright © 2004 IFAC.
Keywords: energy, motor, simulation, reference, step, tracking, trajectories, vehicle.
1. INTRODUCTION The behavior of a system s response to a control action depends on the capacity of the equipment that emits the signal control. This capacity may refer to physical variables such as voltage, current, power, angle, force, and so on, but in order to generalize, these variables will be known plainly as energy . It is clear that if a system is required to follow a step reference, a quick and exact response will request higher energy consumption. However, this unnecessary energy demand can be reduced by replacing the step reference with a special trajectory from the initial point to the final point. This trajectory can be described as a straight line or a function, such as a sine, a cosine, an exponential function, and the like, thus converting the step reference problem into a trajectory tracking problem. The key is not to follow this new trajectory but to arrive to the destination with an acceptable behavior and within a reasonably time. This paper suggests a function based on the own system behavior. This function will allow the system to achieve the desired trajectory and to reduce energy consumption not only in picks but also in the cumulative total.
time. At the same time, the reference describes the natural route that the system would follow. The key of this method is the following function:
x(t ) x(t )
Xd Xd
Td (t ) Td (t ) Xd
Xd
x(t )
XH (t )
(1)
Where: - x(t), is the state to be controlled - XH(t), is the value of the state X given by the simulation process - Xd, is the desired value of the state X - Td(t), is the desired trajectory - , , design parameters (values between 0 and 1) As shown, this function has two stages. First, when x(t) is less than or equal to certain percentage ( ) of the desired value, the desired trajectory follows the values given by the simulation process, but at the same time a certain percentage ( ) of the current error between the real state and the desired value is added to the signal control. Second, when x(t) is greater than certain percentage of the desired value, the desired trajectory is equal to the desired value. The simulated controller s gains or weights that make it possible to generate the desired trajectory could or could not be similar to those of the real controller. Nevertheless, to allow for comparability, this paper assumes similar values. Those weights are expected to search for the best possible response (either amped, sub-damped, or critically-damped). Furthermore, it is crucial to highlight that this response must not be over-damped; otherwise, expected results could not be obtained.
2. THE HIDDEN REFERENCE METHOD To obtain a trajectory from an initial point X(to) to a final point X(tf), the mathematical model of the system is simulated by performing iterations with computer software. Each iteration yields a new point that becomes part of the desired trajectory, thus making up a reference that varies in each instant of
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
1
Ms. Ing. Henry A. Mendiburu Díaz
3. APPLYING THE HIDDEN REFERENCE METHOD
V(t)
Ra
La
,
i(t) J
3.1. Position Control of a DC Motor
b
In this section our goal is to take the angular position of a DC motor from an initial point to a final reference. To perform this task, a continuous current motor of 40v and 100w was chosen (Fig.1). The motor is subject to the following technical specifications: - Moment of inertia of the rotor (J): 4.326x10-5 Kg.m2/s2 - Motor friction / damping ratio (b): 0.1 N.m/s - Electromotive force constant (Ka): 0.05665 N.m/Amp - Motor armature constant (Kt): 0.0573 - Electric resistance (Ra): 1.1 ohm - Electric inductance (La): 0.0001 H - The rotor and shaft are assumed to be rigid The mathematical model of the motor is subject to the equations (2) and (3):
V (t ) di (t ) Ra i (t ) La dt Ka (t )
Fig.1 Scheme of the DC motor It is necessary to point out that if in any of the three cases: straight line, sine, or cosine, a greater time were used for the reference before setting it constant, a greater settling time would have been obtained. This would have not been feasible, since we wanted to compare similar answers among all cases. A full-state feedback controller with gains [0.0003, 0.3390, -1.3267] was used. The results are shown in the following figures:
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
(a)
(2)
10
Signal Control (Voltage)
J
(t ) b
(t )
Kt i (t )
(3)
volts
8 6
Where V(t) is the source voltage (volts), i(t) is the armature current (Amp), (t ) is the angular position (rad), (t ) is the angular speed (rad/sec), and (t ) is the angular acceleration (rad/sec2). The state-space form of the system is given by the equation (4):
0 0 i 0 1 b J Ka La 0 Kt J Ra La i 0 0 1 La V
(b)
4 2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Fig. 2: Response to a STEP FUNCTION reference. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
(4)
(a)
First, the system was simulated in order to reach a step reference (Fig.2), and then to follow different desired trajectories. The first function used was a straight line with gradient conditioned to a final time of 0.1 sec., after this point in time, the trajectory becomes similar to the desired angular position (Fig.3). The second function was a variant of the cosine function: (-cos + 1),(Fig.4). The third function was a wave sine (Fig.5). As well as the previous one, this function uses a frequency of 2.5 Hz, and it is substituted by the desired angular position when it reaches its maximum amplitude (at /2). Finally the hidden trajectory is used with parameters =0.35 y =0.96 (Fig.6). The simulation considered an initial condition of [0.01,0,0], and a desired angular position of 1 rad., with a sampling interval of 0.005 sec.
Signal Control (Voltage) 10 8 6 volts 4 2 0 0
(b)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 time(sec)
Fig. 3: Response to a STRAIGHT LINE FUNCTION reference. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
2
Energy-Saving with the Hidden Reference tracking method
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
Table 1: Summary of results: DC Motor (*) Vmax 8.25 7.86 9.97 8.01 5.42 Pmax 60.24 54.58 54.11 56.64 25.91 Pacc 1977 1862 1897 1900 1388 Ts 0.62 0.67 0.66 0.68 0.67
(a)
Signal Control (Voltage) 10 8 6 volts
Step Line Sine Cosine Hidden
(b)
4 2 0 0
(*) Vmax maximum voltage (volt) Pmax maximum power (watts) Pacc accumulated power (watts) Ts settling time (sec.)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 time(sec)
Fig. 4: Response to a COSINE FUNCTION reference. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
3.2. Location control of a self-guided vehicle In this section the objective is to eliminate the oscillatory behavior or chattering in signal control. This is achieved by applying the proposed method called hidden reference . The time required to arrive to the destination can be reduced by modifying the signal control. When the value of the actual position is smaller than the value of the desired position, the function (t) is added to the signal control. This function (5) reduces the settling time of the system.
(t ) YH (t ) Y (t ) Y (t ) YH (t )
(a)
Signal Control (Voltage) 10 8 6 volts 4 2 0 0
(5)
(b)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 time(sec)
Fig. 5: Response to a SINE FUNCTION reference. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
Angular Position 1 0.8 rad 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(sec)
Where: - YH(t), is the simulated position (m) - Y(t), is the real position (m) - (t), is the control function (radians) Suppose that a vehicle with longitude (L) 0.5 m that moves with a constant speed (v) of 3 m /s is located in the XY plane and goes toward a desired position in the Y axis. The vehicle s helm has a maximum angle of 30º (Fig. 7). Y
(a)
Signal Control (Voltage) 10 8 6 volts 4 2 0 0
(b)
X
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 time(sec)
Fig. 6: Response to a HIDDEN REFERENCE FUNCTION. (a) Angular position (solid), Desired trajectory (dashed). (b) Signal control or armature voltage.
Fig. 7: Scheme of a self-guided vehicle
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
3
Ms. Ing. Henry A. Mendiburu Díaz
The mathematical model of the system is given in the equations (6), (7), and (8):
x v cos( ) y v sin( ) v tan( ) L
The second case (Fig.9) shows the response and the signal control of the system using the hidden reference method, with parameters =0.15 y =0.9. The third case (Fig.10) shows the response and the signal control of the system using a straight-line function with gradient conditioned to a final time of 7 sec.
Position X vs Y Y (m) 40 30 20 10 10 Y (m) 40 30 20 10 0 degree 20 0 -20 0 5 10 time (sec) 15
(6) (7) (8)
Using input-state linearization, the following control law is obtained:
arctan L v sin( )
2
(a)
K1 ( y
yd )
K 2 v sen( )
(9)
14
18 Position Y(t)
22
26
X (m) 30
Where: - x, is the displacement in the X axis (m) - y, is the displacement in the Y axis (m) - yd, is the desired position in the Y axis (m) - , is the inclination angle of the vehicle (rad) - , is the helm s angle (rad) - K1, K2, are the parameters (gains) of control The system was simulated considering the initial condition [10,10] with =45º, looking for a desired position of 40m in the Y axis with =0º. The optimal controller gives gains: K1 = 1.35, K2 = 2.2, the sampling interval is 0.05 sec. The following graphs show the results of the simulations. The first case (Fig.8) shows the response and the signal control of the system using optimal control (step reference).
Position X vs Y Y (m) 40 30
(b)
5 Helm´s Angle d(t)
10
time (sec) 15
(c)
Fig. 9: Simulation using HIDDEN REFERENCE . (a) Position in the XY plane (solid), Case-1 (dashed). (b) Position Y(t) (solid), Reference (dashed). (c) Helm s angle vs time.
Position X vs Y Y (m) 40 30
(a)
20 10 10 Y (m) 40 30 20 10 0 degree 20 0 -20 0 5 10 time (sec) 15
(a)
20 10 10 Y (m) 40 30 20
14
18 Position Y(t)
22
26
X (m) 30
14
18 Position Y(t)
22
26
X (m) 30
(b)
10 0 degree 20 0 -20 0 5 10
(b)
5 Helm´s Angle d(t)
10
time (sec)15
5 Helm´s Angle d(t)
10
time (sec) 15
(c)
(c)
time (sec) 15
Fig. 8: Simulation using OPTIMAL CONTROL. (a) Position in the XY plane (solid), Reference (dashed). (b) Position Y(t) (solid), Reference (dashed). (c) Helm s angle vs time.
Fig. 10: Simulation using STRAIGHT LINE FUNCTION. (a) Position in XY plane (solid), Case-1 (dashed). (b) Position Y(t) (solid), Reference (dashed). (c) Helm s angle vs time.
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
4
Energy-Saving with the Hidden Reference tracking method
As it can be seen, the second and the third cases do not reach the response speed of the first case. Therefore, the fourth case (Fig.11) shows with the response and the signal control of the system using optimal control, but achieving a similar response to the second and third cases in order to allow the reader to compare the results, (K2=2.56).
Position X vs Y Y (m) 40 30
defined trajectory, but to move from an initial point to a final point, with a similar behavior to the response of the simulated system. - A reduction of energy consumption is achieved by substituting the maximum energy picks for low amplitude and constant amplitude curves. - The simulation of the system must not provide an over-damped response.
(a)
20 10 10 Y (m) 40 30
- This method is useful only when it is required to follow step references.
14
18 Position Y(t)
22
26
X (m) 30
5. REFERENCES Burg, T.C., Dawson, D.M., and J. Hu. (1994). Velocity tracking control for a separately excited DC motor without velocity measurements. American Control Conference pp. 1051-1056. Frank, P.M. (1990). Fault diagnosis in dynamic systems using analitical and knowledge based redundancy: a survey and some new results. Automatica 26(3), 459 474. Galler, D. and A. Kusko (1983). Control means for minimization of losses in AC and DC motor drives. IEEE transactions on industry applications IA-19(4), 561-570. Preuss, H.P. (1982). Perfect steady-state tracking and disturbance rejection by constant state feedback. International Journal of Control, 35(1), 75-93.
(b)
20 10 0 degree 20 0 -20 0 5 10 time (sec) 15
5 Helm´s Angle d(t)
10
time (sec)15
(c)
Fig. 11: Second simulation using OPTIMAL CONTROL. (a) Position in XY plane (solid), Reference (dashed). (b) Position Y(t) (solid), Reference (dashed). (c) Helm s angle vs time. Table 2: Summary of results: Self-guided Vehicle (*) Y(x) 17.8 22.0 22.7 22.0 Y(t) 11.6 11.5 12.5 12.8
Case 1 Case 2 Case 3 Case 4
(*) Y(x) trajectory of Y in function at X axis (settling position in the XY plane, meters). Y(t) trajectory of Y in function at time (settling time, seconds).
4. CONCLUSIONS - The hidden reference method uses a computersimulated system to give insight about the system's response to a known stimulus. - The method involves the conversion of a step reference problem into a trajectory tracking problem, in which the key is not to move along a
XI Latin-American Congress of Automatic Control , IFAC, Cuba, May/2004
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