Chapter 2 Evolutionary Learning of Fuzzy-Neural Networks Using a
2.4 Conclusions
This chapter proposed a RSA, which can be successfully applied in the fuzzy-neural network to search for the optimal parameters, in spite of a vast number of adjustable parameters of the fuzzy neural networks. Two examples for RSA verify that it has outstanding efficacy in learning and approximation.
Chapter 3
Indirect RSA On-Line Tuning of Fuzzy-Neural Networks for Uncertain Nonlinear Systems
In this chapter, an RSA indirect adaptive fuzzy-neural controller (RIAFC) for uncertain nonlinear systems is 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 RIAFC. To illustrate the feasibility and applicability of the proposed method, two examples of nonlinear systems controlled by the RIAFC are demonstrated.
3.1 Problem Formulation
In this section, we describe the control problem for a class of nonlinear systems, and then design the controller.
3.1.1 The Design of Certainty Equivalent Controller
Here we consider another nth-order nonlinear system of the form
1
and x=[ ,x x1 2,K,xn]T
ym
= − δ =[ , ,e e& K ( ) h s
is the state vector of the system, which is assumed to be available for measurement. We assume that f and g are uncertain continuous functions, and g is, without loss of generality, a strictly positive function. In [49], these systems are in normal form and have a relative degree n. Our objective is to design an indirect adaptive fuzzy-neural controller so that the system output y follows a given bounded reference signal . First, let
, and such that all roots
of a polynomial are in the open left half-plane. If the functions f and g are known, then the optimal control law is
ym However, since f and g are uncertain, the optimal control law (3-18) cannot be obtained. To solve this problem, we use the fuzzy-neural systems as approximators to approximate the uncertain continuous functions, f and g.
First, we replace f and g in (3-2) by fuzzy-neural networks , i.e., ˆ ( |f x wf) and ˆ ( |g x wg), respectively. Based on certainty equivalent controller [64], the resulting control law is
( ) ˆ
Substituting (3-4) in (3-1) and after some manipulations, we obtain the error equation
where
there exists a unique positive definite symmetric n Λc sI−Λc =s n +k1s(n−1)+L+kn
)
| (
(|
×n matrix P, which satisfies the Lyapunov equation:
T which means we require that , when is greater than a large constant
Vδ
≤0
V&δ Vδ
>0
V . Next, we assume the following.
Assumption 3.1: We can determine functions fU( ),x gU( )x and
Based on Assumption 3.1, equation (3-9) can be modified as
1 | | ( ) ˆ( |
Therefore, according to (3-10), we define a cost function for the RSA as
Φ = −δTPb f x wc ˆ( | f)−δTPb g x w uc( ( |ˆ g) c (3-11) in order to on-line tune weightings, a solution with the smallest cost function denotes the optimal solution. So, a better crystal can be obtained according to (3-11).
Substituting (3-12) into (3-1), the error equation becomes
ˆ ˆ ˆ Based on Assumption 3.1 and (3-14), we choose the supervisory control us as
(3-15) Equation (3-16) ensures the bounded stability of the RIAFC for the nonlinear system in (3-1).
Remark 3.1 : The concept of the supervisory control is added into our design mainly because the system states may go into the unsafe region if the RSA operations can not simultaneously generate the appropriate weightings.
To safely control the uncertain nonlinear systems, the supervisory controller must be turned on when the system states go into the unsafe region.
3.2 Description of Reduce Simulated Annealing Algorithm for On-line Controllers
For the purpose of speeding up the computation of the simulated annealing operation, the mechanism of the reduced simulated annealing algorithm has two simplified parts: (1) one perturbation in every temperature, and (2) perform compact perturbation mechanism and special cooling schedule on the solution by the cost function. The details are discussed in the following.
First, the adjustable parameters of the p-th output of the fuzzy-neural networks are as follow.
wp = ⎣ ⎦⎡wTp⎤ (3-17) where denotes a set of weighting factors in the interval , . Each element represents the adjustable parameter of the fuzzy-neural
wp Dp = −[ d dp, p]
p 0 d >
networks. Note that the solution is accepted by the cost function, that is, the optimum solution denotes the best solution of cost function.
Next, to instantaneously evaluate the stability of the closed-loop system, define the p-th cost function as
Φ = −δTPb fc ˆp( |x wf)−δTPb gcˆp( |x w ug) c (3-18) where is the estimation of the unknown dynamic and
is the estimation of the unknown dynamic A state with smallest cost function denotes the optimal solution. The detail explanation of the cost function is given later.
ˆ ( |p )
Then, according to the cost function, perturbation mechanism operators are performed. The operation procedure of the perturbation mechanism is as follows. The acceptance probability function for the new solution vector is defined as
_ _
where denotes a probability in the interval [0,1], represents Boltzmann constant and is temperature.
r Kb
T
And then, we design cooling schedule as follow:
2
The block diagram of RIAFC is given as follow:
yd Sim ulated annealing adaptive
backstepping
determ ined in the interval
Fig. 3-1 The block diagram of RIAFC
The design algorithm of RIAFC is as follows:
Step 1: Construct fuzzy-neural networks for ˆ ( |f x wf) and ˆ ( |g x wg), including fuzzy sets for x(t), the weighting vectors w and f w . g Step 2: Adjust the weighting vectors by using the RSA approach with the
cost function (3-11).
Step 3: Compute ˆ ( |f x wf) and ˆ ( |g x wg). Then, obtain the control law (3-4).
3.3 Simulation Examples of the RIAFC
This section presents the simulation results of the proposed on-line RIAFC 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 3-1: Consider the three-order nonlinear system described as
1
Thus, the RIAFC is suitable to control the system. The adjustable parameters, wf of f x x xˆ ( , , )1 2 3 are in the intervals D1=[-2,2] α =0.01, β =20, σ =10 respectively. The reference signal is given as y td( )=sin( )t in the following simulations. The initial states are set as (0)x =[0.3,1, 0.5]. The membership functions for xi, i=1,2 are given as
To apply the RIAFC to the system, the bounds fU should be obtained:
1
The simulation results are shown in Figs. 3-2, 3-3, 3-4 the RIAFC can control
the uncertain nonlinear systems to follow the desired trajectories very well. In Fig. 3-3, the tracking error reaches a bounded error (Vδ ≤ = ). Therefore, V 1 the tracking performance is very good as shown in Fig. 3-2, in which is the reference trajectory and
yd
x1 is the system output. As shown in Fig. 3-4, the chattering effect of the control input (uc + ) almost disappears after 2 us seconds, respectively. In 2 seconds, the RSA searches the neighborhood for the optimal parameters of the RIAFC.
Fig. 3-2. The system output y(t) and bounded reference y td( )
Fig. 3-3. The tracking error e
Fig. 3-4. The control input u(t)
Example 3-2: Consider the dynamic equations of the inverted pendulum system as [18]
1 2 where g=9.8 meter/sec is the acceleration due to gravity, is the mass of the cart, is the half-length of the pole, m is the mass of the pole and u is the control input. In this example, we assume =1 kg, m=0.1 kg and l=0.5 meter.
Thus, the RIAFC is suitable to control the system. The adjustable parameters,
2
D =[1,2], respectively. The reference signal is given as 2 in the following simulations. The initial states are set as
( 0.1*
y t)m = sin( )t (0) [ , 0 x = 60π ]
. The membership functions are the same as Example1.
To apply the RIAFC to the system, the bounds f ,U gU, and gL should
The design parameters are set as k1 = and 1 k2 =2, Q=diag(10,10) and 0.01
V = . Then, solve (3-8) and obtain 15 5
5 5
P ⎡ ⎤
= ⎢ ⎥
⎣ ⎦. As shown in Figs. 3-5, 3-6, 3-7 the RIAFC can control the inverted pendulum to follow the desired trajectories very well. In Fig. 3-6, the position error reaches a bounded error (Vδ ≤ =V 0.01). Therefore, the tracking performance is very good as shown in Fig. 3-5, where ym is the reference trajectory and x1 is the system output.
Fig. 3-5. The system output y(t) and bounded reference y tm( )
Fig. 3-6. The tracking error e
Fig. 3-7. The control input u(t)
3.4 Conclusions
In this chapter, an RSA indirect adaptive fuzzy-neural controller (RIAFC) 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 RIAFC with the supervisory controller guarantees the bounded stability of the closed-loop system. The simulation results show that the RSA-based adaptive fuzzy-neural controller performs on-line tracking successfully.
Chapter 4
Backstepping Adaptive Control of Uncertain Nonlinear Systems Using RSA On-Line Tuning of
Fuzzy-Neural Networks
In this chapter, an RSA backstepping adaptive fuzzy-neural controller (RBAFC) for uncertain nonlinear systems is 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 RBAFC. To illustrate the feasibility and applicability of the proposed method, two examples of nonlinear systems controlled by the RBAFC are demonstrated.
4.1 Problem Formulation
In this section, we describe the control problem for a class of nonlinear systems, and then design the backstepping controller.
4.1.1 The Design of Backstepping Controller
Here we consider another nth-order nonlinear system of the form
Where f and g are unknown smooth continuous functions, u∈R is the
system input, and x=[ ,x x1 2Lxn]T ∈Rn is the state vector. The control objective is to design the backstepping controller for system (4-1) such that all the signals in the closed-loop are uniformly stable and the statex1can track a bound reference signalymarbitrarily closely.
1 1 m
Next, the detail design procedure of the backstepping controller is described as follows.
Step 1) Define the tracking error as
(4-2) Then, the derivative of can be expressed as
& (4-3) Define the virtual control as
(4-4) using (4-4) and the fact that
z2
, equation (4-3) can be rewritten as
(4-5) Step 2) The derivative of can be expressed as
m) where c2 >0 is a design parameter. Moreover, define the error state as
2
3 3
z =x −α . Then, by using (4-6) and the fact that x&2 = +z3 α2,
equation (4-7) can be rewritten as Step 5) Consider the Lyapunov function
2
4.1.2 On-line Learning of Fuzzy-Neural Backstepping Control Using RSAOA
In practical applications, since f and g are uncertain, the optimal control law (4-13) cannot be obtained. To solve this problem, we use the fuzzy-neural systems as approximators to approximate the uncertain continuous functions f and g. Substituting (4-17) in (4-1) and after some manipulations, we obtain the error equation
In order to confine the state xi to a bounded value, Vδ must be bounded, which means we require that V&δ ≤0, when Vδ is greater than a large constant V > . Next, we assume the following. 0
Assumption 4.1: We can determine functions fU( )x , and such
Based on Assumption 4.1, equation (4-20) can be modified as
2 Therefore, according to (4-20), we define a cost function for the RSA as
0 in order to on-line tune weightings. Note that a state with the smallest cost
function denotes the optimal solution.
By incorporating a control term us into uc, the control law becomes u=uc + (4-22) us where us is supervisory control. The supervisory control us is turned on when the error function Vδ is greater than a positive constant V . If Vδ ≤ , then the supervisory control V us is turned off. That is, if the system tends to be unstable (Vδ >V), then us forces Vδ ≤ . Substituting (4-22) V into (4-1), the error equation becomes
ˆ 1 ˆ ˆ
( ) ( | ) ( ) ( | ) ( | )
n n n n c c
z& = f x − f x w −c z −z − +g x u −g x w u −g x w us (4-23) Using (4-23) and (4-19), we have
2 Based on Assumption 4.1 and (4-24), we choose the supervisory control us as
Equation (4-26) ensures the asymptotical stability of the RBAFC for the nonlinear system in (4-1).
The block diagram of RBAFC is given as here (for example in two-order nonlinear system)
ddt
ddt yd
c1 Simulated annealing adaptive
backstepping
Fig. 4-1 The block diagram of RBAFC
The design algorithm of RBAFC is as follows:
Step 1: Construct the fuzzy-neural networks for ˆ ( | )f x w and , including fuzzy sets for x(t), and the weighting vectors
ˆ ( | ) g x w w ,f w . g Step 2: Adjust the weighting vectors by using the RSA approach with the
cost function (4-21).
Step 3: Compute f x wˆ ( | ) and according to output of the fuzzy-neural network. Then, obtain the control law (4-17).
ˆ ( | ) g x w
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,
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,