Chapter 4 Backstepping Adaptive Control of Uncertain Nonlinear
4.2 Simulation Examples of the RBAFC
This section presents the simulation results of the proposed on-line RBAFC for a class of uncertain nonlinear systems to illustrate the stability of the
closed-loop system is guaranteed, and all signal involved are bounded.
Example 4-1: Consider the three-order nonlinear system described as
1
Our objective is to control the system state x1 to track the reference trajectory yd . Clearly, (4-27) is in the form of (4-1). Thus, the RBAFC is suitable to control the system. The adjustable parameters, w of f f x x xˆ ( , , )1 2 3 are in the intervals D =[-2,2] 1 α =0.01, β =20, σ =10, . The reference signal is given as
T0 = 1000 ( ) sin( )
y td = t in the following simulations. The initial states are set as (0)x =[0.5,1, 0.5]. The membership functions for xi, i=1,2 are given as
To apply the RBAFC to the system, the bounds fU should be obtained:
1 simulation results are shown in Figs. 4-2, 4-3, 4-4. As shown in Figs. 4-2, 4-3 the RBAFC can control the uncertain nonlinear systems to follow the desired
=
trajectories very well. In Fig. 4-3, the tracking error reaches a bounded error (Vδ ≤ =V 0.05). Therefore, the tracking performance is very good as shown in Fig. 4-2, in which yd is the reference trajectory and x1 is the system output.
As shown in Fig. 4-4, the chattering effect of the control input ( ) almost disappears after 1 seconds, respectively. In 1 second, the RSA searches the neighborhood for the optimal parameters of the RBAFC.
uc+us
Fig. 4-2. The system output y(t) and bounded reference y td( )
Fig. 4-3. The tracking error e
Fig. 4-4. The control input u(t)
Example 4-2: Consider the two-order nonlinear system described as [18]
1 2
Our objective is to control the system state to track the reference trajectory in the following simulations. The initial states are set as
m(
y t)=0.1* sin(t) (0
x ) [
60π , 0]
= − . The membership functions are the same as Example1.
To apply the RBAFC to the system, the bounds f ,U gU and gL should
be obtained: simulation results are shown in Figs. 4-5, 4-6, 4-7. As shown in Figs. 4-5, 4-6, the RBAFC can control the uncertain nonlinear systems to follow the desired trajectories very well. In Fig. 4-6, the tracking error reaches a bounded error (Vδ ≤ =V 0.02). Therefore, the tracking performance is very good as shown in Fig. 4-5, in which ym is the reference trajectory and x1 is the system output.
Fig. 4-5. The system output y(t) and bounded reference y tm( )
Fig. 4-6. The tracking error e
Fig. 4-7. The control input u(t)
4.3 Conclusions
In this chapter, an RSA backstepping adaptive fuzzy-neural controller (RBAFC) has been proposed. The free parameters of the adaptive fuzzy-neural controller can be successfully tuned on-line via the RSA approach with a special evaluation mechanism, instead of solving complicated mathematical equations. The RBAFC with the supervisory controller guarantees the bounded stability of the closed-loop system. The simulation results show that the RSA-based backstepping adaptive fuzzy-neural controller performs on-line tracking successfully.
Chapter 5
Design of Fuzzy-neural Controller Using Reduce Simulated Annealing Algorithms for MIMO
Nonlinear Systems
In this chapter, a simulated annealing adaptive fuzzy-neural control scheme is proposed for a class of multiple-input multiple-output (MIMO) nonlinear systems. The control scheme incorporates a compact simulated annealing algorithm and fuzzy-neural networks into backstepping design. The reduce simulated annealing (RSA) algorithm is used to adjust the parameters of the fuzzy-neural networks in order to instantaneously generate the appropriate control strategy. The reduce simulated annealing algorithm has a simplified procedure with an energy cost function which is used to evaluate the real-time stability of the closed-loop systems. The state represents an adjustable parameter of the fuzzy-neural networks. To illustrate the feasibility and applicability of the proposed method, an example of the double inverted pendulums controlled by the proposed method is provided.
5.1 Problem Formulation and Fuzzy-Neural Networks
In this section, we describe the control problem for a class of MIMO nonlinear systems, and then design the backstepping controller. In addition, the structure of the fuzzy-neural networks is briefly reviewed in this section.
5.1.1 Problem Formulation and Backstepping Control Design
First, consider the MIMO nonlinear systems as
1 2 input of the p-th subsystem, is a positive unknown constant,
is the state vector, and is the state vector of the p-th subsystem. Our control objective is to develop the backstepping controller so that the state trajectory
up
xp can asymptotically track a bounded command ypd.
Next, the detail design procedure of the backstepping controller under the assumption of the known system dynamics fp is described as follows.
Step 1) Define a tracking error as
track the bounded command ypd . Thus, define an error state as
2 p1 equation (5-3) can be rewritten as
(5-5)
1 2 1
p p p
z& =z −c zp1
Step 2) Differentiating zp2 can be expressed as
z&p2 =x&p2−α&p1= xp3− −( c zp1&p1+&&y )pd
Step 5) Consider the Lyapunov function as follows By differentiating (5-15) and using (5-5), (5-8), (5-11) and (5-14), we have
The state trajectory xp1 can asymptotically track the bounded command
pd. y
On the basis of the aforementioned description, the backstepping controller of the nth order nonlinear system can be summarized as the following lemma.
Lemma 5.1: Consider the nth-order nonlinear systems (5-1). Let and
. Then, the state trajectory
pi 0
c > x can asymptotically track the bounded
command ypd.
5.1.2 Description of MIMO Fuzzy-neural Networks
The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an training input data xpq, p=1, 2,L,m, , and the output data , the ith fuzzy rule has the following form:
1, 2, , p
where is a rule number, i A and ipq are the fuzzy sets. By using product inference, center-average and singleton fuzzifier, the output of the fuzzy-neural networks can be expressed as:
i is a weighting vector,
1 2
The typical fuzzy-neural networks have a total of four layers. Nodes at layer I are input nodes that represent input linguistic variables. Nodes at layer II represent the values of the membership function of total linguistic variables.
At layer III, nodes are the values of the fuzzy basis vector ξ. Each node at layer III is a fuzzy rule. The links between layer III and layer IV are full connected by the weighting factors. Layer IV is the output layer.
5.2 Development of Simulated Annealing Adaptive Fuzzy-neural Control Scheme
In this section, based on the reduced simulated annealing algorithm as mentioned above, a method for developing on-line backstepping adaptive fuzzy-neural controllers is proposed.
Since f and p are uncertain, the optimal control law (5-13) cannot be obtained. To solve this problem, the fuzzy-neural system is used to
bp
approximate the uncertain continuous function fp. First, the uncertain continuous function fp in (5-13) is replaced by fuzzy-neural networks (5-18), i.e., f x yˆp( ,&pd |wp). The resulting control law Suppose that . Then, substituting (5-20) in (5-1) and after some manipulations, we obtain the error equation
up =u instantaneously evaluate the stability of the closed-loop system. A solution with the smallest energy cost function denotes the optimal solution.
Since the system states may go into the unsafe region if the simulated annealing operations can not simultaneously generate the appropriate weightings of the fuzzy-neural networks in some time interval, the concept of the supervisory controller is incorporated into the simulated annealing adaptive backstepping fuzzy-neural controller to guarantee that the system
states are confined to the safe region. By incorporating a supervisory control term ups into upc, the control law becomes
p pc
u =u +ups (5-23) The supervisory control term is added when the function is greater than a positive limit . If
ups
V V
V
Vu ≤ u, then the supervisory control term is canceled. That is, if the system tends to enter the unsafe region ( ), then
forces the system to return to the safe region.
ups
Vu
V >
ups
Substituting (5-23) into (5-1), the error equation becomes
( 1) Suppose that . Then, substituting (5-26) into (5-25), we have
0( 0)
From (5-27), the bounded stability of the closed-loop system for the nonlinear system in (5-1) can be guaranteed.
The design algorithm of the proposed scheme is given as follows.
Step 1: Construct the fuzzy-neural networks for f x yˆp( ,&pd |wp) including
fuzzy sets for x, and the weighting vectors wp.
Step 2: Adjust the weighting vectors by using the reduced simulated annealing algorithm with the energy cost function.
Step 3: Compute fˆp( ,x y&pd |wp) according to (5-18). Then, obtain the control law (5-20).
5.3 Simulation Examples
This section presents the simulation results of the proposed method for a class of MIMO nonlinear systems, and then illustrates that the stability of the closed-loop system is guaranteed. Consider the following problem of balancing double inverted pendulums connected by a spring [43]
x&11 = x12
where xi1 is the angular position of the ith pendulum from the vertical reference, and ui is a torque input. It is assumed that both xi1 and x& are i1 available for measurement. The parameters of the double inverted pendulums are chosen as m1 = 0.5 kg m2 =0.5 kg, J1 =0.5 kg and kg,
N⋅m/rad, and m.
J2 = 0.5 2
k= r=1
Our objective is to control the system state xp1 to track the reference trajectory . It is clear that from (5-28) and (5-1), the proposed method is suitable to control the system. The design parameters of the simulated annealing algorithm are given T=1000,
ypd
Kb = 6.25×10−5. The adjustable parameters wi of f xˆ ( , , ,i 11 x12 x21 x22) are in the intervals Di =[-1,1] and α =0.01, β =20, σ =10 . The reference signals are set as
1d( ) 0.5sin( ) 0.2 cos( ) 0.1sin(2 ) . The membership functions for ) should be obtained as
1
(case 2). The simulation results are shown in Figs. 5-1-5-8. In the case 5-1, Figs. 5-1-5-4 show that the proposed control algorithm can control the MIMO uncertain nonlinear systems to follow the desired trajectories, and the tracking error do not reach the safe limit ( ). In the case 2, Figs.
5-5-5-8 show that the system outputs
c11 c12 c21 c22 signals and very well, respectively. Compared with the results of the case 1, the case 2 can achieve the better tracking performance at the
y ( )1d t y ( )2d t
expense of the chattering effect of the control input in some time intervals as shown in Fig. 5-7. The chattering effect is due to adding the supervisory control term when the tracking error exceeds the safe limit (V >Vu =0.02).
Fig. 5-1. The system output y11( )t and bounded reference y1d( )t (case1)
Fig. 5-2. The system output y21( )t and bounded reference y2d( )t (case1)
Fig. 5-3. The control inputs u1 and u2 (case1)
Fig. 5-4. The tracking errors z11( )t and z21( )t (case1)
Fig. 5-5. The system output and bounded reference (case2)
11( ) y t
1d( ) y t
Fig. 5-6. The system output and bounded reference (case2)
21( ) y t
2d( ) y t
Fig. 5-7. The control inputs u1 and u2 (case2)
Fig. 5-8. The tracking errors z11( )t and z21( )t (case2)
5.4 Conclusions
In this chapter, a simulated annealing adaptive fuzzy-neural control scheme has proposed for a class of MIMO nonlinear systems. The weighting
parameters of the fuzzy-neural controller can be successfully tuned instantaneously via the reduced simulated annealing algorithm with an energy evaluation mechanism, instead of solving complicated mathematical equations.
Using the energy evaluation mechanism can evaluate the real-time stability of the closed-loop systems in order to generate the appropriate control strategy.
The supervisory control term added into the simulated annealing adaptive backstepping fuzzy-neural networks guarantees the safeness of the MIMO closed-loop system. Moreover, the proposed design algorithm has been successfully applied to control double inverted pendulums connected by a spring. The simulation results show that the simulated annealing adaptive backstepping fuzzy-neural control scheme performs on-line tracking successfully.
Chapter 6
Design of Fuzzy-neural Controllers for DC Servo motors Using Reduced Simulated Annealing
Algorithms
In this chapter, to demonstrate the applicability of propose method on practical system, we utilize the fuzzy-neural controller for tracking control of DC servo motors. The weighting factors of the fuzzy-neural controller are tuned on-line via the RSA approach. The RSA backstepping adaptive fuzzy-neural controller discussed in Chapter 5 is used to control a DC servo motor.
6.1 Problem Description of DC Servo Motors
Consider the dynamics system of the servo motors given as Fig.6-1.
M T
θ
Load B
J
-+ vb
L R
va
M otors circuit
ia
Fig.6-1 Circuits of the servomotor
where R is the armature resistance, L is the armature inductance, θ is the angular displacement of the motor shaft, B is the friction constant, J is the armature moment of inertia of the motor, and T is the load torque.
From Fig. 6-1, the differential equation for the armature circuit is
a
a a
v Ri Ldi v
= + dt + b (6-1) where denotes the armature current. The voltage is the back electromotive force of the motor and is proportional to the speed of the motor shaft. It is given by
where is back-emf constant. For a DC servo motor, the torque generated by the motor is proportional to the armature current. That is
Kb
, then, substituting (6-3) in (6-4), and according to
(6-1), (6-2), (6-4), the representation of the state space is given as
[ ]
Let . According to (6-2), (6-4) and (6-6), equation (6-5) can be written as:( )
Our control objective is to develop the backstepping controller so that the DC servo motor system output can asymptotically track a bounded command.
6.2 Simulation Results
This section presents the simulation results of the proposed method for a class of nonlinear system, and then illustrates that the stability of the closed-loop system is guaranteed. Equation (6-7) can be rewritten as
1 2 Chapter 5 for SISO system. The control objective is to force the state x1 of the servo motor to follow the reference signal . The control law of the proposed method is
yd
u=uc +us (6-9) where uc is the backstepping adaptive fuzzy-neural controller designed as
2 2 1
T=1000, Kb=6.25 10× −5 α=0.01, β =20, σ =10. The initial states are set
To apply the RBAFC to the system, the bounds
1 2 2 1 2
( , ) ( )1 62.0919 2 U( , )
f x x x x x
= − τ ≤ x ≡ f (6-11) Two cases are simulated. In the case 1, the design parameters are set as
1 ,
c =15 c2 =5 and V =0.1 and the reference signal is set as . The simulation results are shown in Figs. 6-2, 6-3, 6-4. In the case 2, the design parameters are set as
( ) 60 0.1 y td = sin( t)
c1 =15, c2 =10 and V =0.1 and the reference signal is set as y td( )=60 100− e−t. The simulation results are shown in Figs. 6-5, 6-6, 6-7.
Fig. 6-2 The system output y t( ) and bounded reference y td( )(case1)
Fig. 6-3. The tracking error e(case1)
Fig. 6-4. The control input u(t) (case1)
Fig. 6-5 The system output y t( ) and bounded reference y td( )(case2)
Fig. 6-6. The tracking error e(case2)
Fig. 6-7. The control input u(t)(case2)
6.3 The Hardware Structure and Experimental Results
The MT22R2-24 DC servomotor system shown in Fig.6-8 was made by SME Company. The module output of this motor and the output of motor are
shown in Fig.6-9. The output voltage of the control module is 10V and the output voltage of the motor is 75V (3000RPM). The specification of MT22R2-24 is shown in Table 6-1.
Fig.6-8 The MT22R2-24 DC servomotor
The control modules output (V)
The control modules output (V)
Fig.6-9 Relationship of the control modules input and output
Table 6-1 The specifications of MT22R2-24.
ITEM SPECIFICATION UNIT
Max. Voltage(V) 120 Volts Max. Speed(RPM) 5000 RPM Armature Moment of inertia(J) 0.0006 Kg-m^2
Torque Constant(Kt) 0.23 N-m/Amp Voltage Constant(Kb) 0.23 Volts-sec/Rad Resistance(R) 3.11809 Ohm Peak Stall Torque 8.0 N-m Acceleration at Peak Torque 13300 Rad/sec^2 Mechanical Time Constant 16 Milliseconds
B 0.00203 N-m-s/rad Motor Weight 4.1 Kg
The block diagram of hardware implementation is shown in Fig. 6-10.
The hardware structure is composed of a person computer, USB I/O 24 module, microcontroller (82G516), switching DC-DC converter and DC servo motor system. The proposed method is implemented by using the person computer with 5-ms sampling interval, and the servo driver motor system is controlled by Pulse-Width Modulation (PWM) method where switch-duty ratio D∈[-1,1] is varied to adjust the output of the Buck DC-DC converter.
The control input voltage in PC needs to be transformed into the switch duty ratio D, and then the servo driver motor system is controlled by the output voltage of the converter. The duty ratio of PWM is described as
u
1 , 75
Fig.6-10 The block diagram of hardware implementation Fig.6-10 The block diagram of hardware implementation
The experimental control system block diagram is shown in Fig. 6-11.
According to initial states, design parameters and reference signals, two experimental cases are given to test the performance of the controlled systems.
The experimental control system block diagram is shown in Fig. 6-11.
According to initial states, design parameters and reference signals, two experimental cases are given to test the performance of the controlled systems.
Fig. 6-11. Control system block diagram
The membership functions, the initial states and the adjustable parameters wf of ˆf are the same as section 6-2. In the case 1, the reference signal is set
as . The design parameters are set as , .
The tracking response is shown in Fig.6-12, Fig.6-13, and Fig.6-14. In the case 2, the desired reference signal is set as . The design parameters are set as ,
( 60sin(0.1 )
y td )= t c1 =15
0e−t
2 10
c =
( ) 60 10 y td = −
1 15
c = c2 =10. The tracking response is shown in Fig.6-15, Fig.6-16, and Fig. 6-17.
Fig. 6-12 The system output x1 and bounded reference r(case1)
Fig. 6-13. The tracking error e(case1)
Fig. 6-14. The control input u(t)(case1)
Fig. 6-15 The system output x1 and bounded reference r(case2)
Fig. 6-16. The tracking error e(case2)
Fig. 6-17. The control input u(t) (case2)
6.4 Conclusions
In this chapter, the RBAFC have been used in DC servo motor through simulation and experiment. Compare with experiment results, it is obvious that performance in simulation is better than in experiment with the same conditions. That results from the temperature of motor, the signal disturbance, the damage of component or equipment, the preciseness of circuits design and so on. However, the RBAFC still achieves the desired effect in position tracking of the servo motor experiment.
Chapter 7
Summaries and Suggestions for Future Research
7.1 Summaries
In general, the universal approximators are trained via gradient-based methods, which may only find a local minimum solution during the learning process. Therefore, in Chapter 2 of this thesis, we proposed a reduced simulated annealing optimization algorithm (RSAOA) to adjust the weightings of fuzzy-neural networks. From the off-line simulation results, a suited way of learning for the fuzzy-neural network is provided.
In Chapter 3, the RSA-based indirect adaptive fuzzy-neural controller (RIAFC) for uncertain nonlinear systems has been proposed by using a reduced simulated annealing algorithm (RSA). The weighting factors of the adaptive fuzzy-neural controller are tuned on-line via the RSA approach. For the purpose of on-line tuning these parameters and evaluating the stability of the closed-loop system, a cost function is included in the RSA approach. In addition, in order to guarantee that the system states are confined to the safe region, a supervisory controller is incorporated into the proposed method.
Simulation results have shown that the proposed RSA-based indirect adaptive fuzzy-neural controller (RIAFC) scheme can rapidly learn the unknown system dynamics, and achieve favorable tracking performance.
The RSAOA-based fuzzy neural backstepping control scheme has been proposed to control a class of SISO and MIMO nonlinear systems in Chapter 4 and Chapter 5, respectively. The control scheme incorporates the backstepping design technique with a fuzzy neural network. The weighting factors of the fuzzy neural controller are tuned on-line via the RSAOA approach. The free parameters of the fuzzy-neural controller can be successfully tuned on-line via
the RSAOA approach with a special evaluation mechanism. In order to guarantee that the system states are confined to the safe region, a supervisory controller is incorporated into the RSAOA-based fuzzy neural backstepping controller. Simulation results have shown that the proposed RSAO-based fuzzy neural backstepping control scheme can rapidly learn the unknown system dynamics, and achieve favorable tracking performance.
Experimental results are shown in Chapter 6. The RSAOA-based fuzzy neural backstepping control scheme has been used to control the MT22R2-24 DC servo motor. Experimental results verify the control scheme can achieve the desired effect in tracking.
7.2 Suggestions for Future Research
In this thesis, we only discuss with the affine nonlinear systems. The future research is recommended to work toward more general nonlinear systems.
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