Consider the given nonlinear plant described by the nth-order differential equation shown in Equation (1) or Equation (2).

where p(x) and q(x) are unknown plant functions, u ∈ R and y ∈ R are an input and an output of the plant respectively, and is a state vector of the plant. Then, the unknown functions are identified by the indirect control algorithm given as Equation (3).

where β(x) is adaptation parameter vector and ξ(x) is fuzzy basis function vector defined as Equation (4).

The control objective of the indirect adaptive control is to determine the control input u = u(x | β) in order to satisfy Equation (5) based on fuzzy logic.

In this case, an adaptive law for adjusting the parameter β can be achieved under the assumption that the given system should be globally stable and all variables should be uniformly bounded.

If the functions p(x) and q(x) are known, the control law given as Equation (6) results in Equation (7) which implies the main control objective Equation (5).

Unfortunately, the problem is that p(x) and q(x) are unknown, and so they must be identified using adaptive fuzzy control algorithm.

If unknown plant dynamics p(x) and q(x) are substituted by respectively, the resultant control law can be given as Equation (8).

By applying Equation (8) to Equation (2), the error differential equation can be obtained as shown in Equation (9).

And Equation (9) can be transformed into error state equation shown as Equation (10).

where A_c and b_c are given as Equation (11) as followings and they can be considered as system matrix and input matrix for the derived error dynamic system given as Equation (10).

Since the control law Equation (8) was derived under the assumption that the output error should be bounded, A_c is a stable matrix, that is the roots of det(sI - A_c) = 0 are stable, there exists a positive definite matrix P which satisfy Equation (12).

If a Lyapunov function candidate V_e is chosen as Equation (13) which is positive definite except that e ≠ 0, the time derivative V_e can be derived as Equation (13).

Thus, if the time derivative is less than or equal to 0, that is the control law Equation (8) is very sufficient to control the given unknown plant. By the way, it is very difficult to find the control u_c which forces the last term of Equation (14) less than 0 such that V_e ≤ 0.

In order to control the given unknown plant by the indirect adaptive control, it is necessary to find the control inputuwhich satisfy This problem can be solved by appending another control termsuto u_c. It is called a supervisory control.

Substituting Equation (15) into Equation (2), the new error equation can be derived as Equation (16).

In this case, the time derivative of the Lyapunov function candidate is derived as Equation (17).

If it is assumed that p^u, q^u and q_L are the plant function bounds such that and , the supervisory control u_s which satisfy can be obtained as Equation (18).

It is necessary to derive the adequate adaptive law to find the adaption parameters β_p and β_q of the fuzzy logic system.

This problem can begin by defining the minimum approximation error ω shown in Equation (19).

where and are defined so as to satisfy Equation (20).

If are selected as fuzzy logic system according to Equations (3) and (4), the error equation given as Equation (16) can be rewritten as Equation (21).

where , and ξ(x) is the fuzzy basis function given as Equation (4).

In order to derive the adaptive laws, consider the Lyapunov function candidate defined as Equation (22), where ν₁ and ν₂ are positive constants.

The time derivative of V along the trajectory given as Equation (21) is derived as Equation (23).

Therefore, the adaptive laws for parameters pq and ββcan be obtained as Equation (24) such that

The structure of the indirect adaptive fuzzy controller is classified into the 1st and the 2nd-type whether the membership functions for calculating fuzzy basis functions ξ(x) given as Equation (4) are fixed or not.

Figure 4 shows the structure of the 1st-type indirect adaptive controller. Since it uses the fixed membership functions, it has a low freedom in comprising the fuzzy logic system, it needs a large number of rules inevitably. In that aspect, it wastes much time to execute all rules for computing fuzzy basis functions. On the other hand, the adaptive laws which adjust the adaptation parameters are very simple.

Figure 4: 
The structure of the 1st-type indirect adaptive fuzzy controller.

Figure 5 shows the structure of the 2nd-type indirect adaptive controller. Fuzzy basis functions are composed and calculated using the adjustable membership functions in the IF parts of the fuzzy rules. Thus fuzzy logic functions are decided using rules smaller than those of 1st-type controller. Whereas, the adaptive laws which adjust the adaptation parameters are very complicated and the adaptive laws for adjusting membership functions must be required. For this reason, the pure 2nd-type indirect adaptive fuzzy controller is not preferable to the real-time control applications.

Figure 5: 
The structure of the 2nd-type indirect adaptive fuzzy controller.

In section 3.6, the structures, advantages, and disadvantages for the 1st-type and 2nd-type indirect adaptive fuzzy controller were investigated. The 1st-type has an advantage in the type of fuzzy basis functions in the viewpoint that all basis functions are fixed. And also it has another advantage in the adaptive laws because the adaptive laws are simple. On the other hand, the 2nd-type has an advantage that it uses much smaller fuzzy rules than 1st-type because it tuned initial adaptation parameters and initial membership functions.

In this section, a new 2nd/1st-type indirect adaptive fuzzy control structure is suggested. Figure 6 presents the suggested control structure. Its control algorithm is composed by adopting the advantages of 1st-type and initial adaptation parameters and initial membership functions indirect adaptive fuzzy control structure. As meaning of the name, initial fuzzy basis functions are tuned in a way that initial adaptation parameters and initial membership functions are decided by 2nd-type in order to reduce the number of fuzzy rules. . It results in the same number of rules as those of 2nd-type structure. After then, all fuzzy basis functions are fixed during running adaption process as if it were the 1st-type case. It means that the adaptive laws are as simple as the case of 1st-type structure.

Figure 6: 
The structure of the 2nd/1st-type indirect adaptive fuzzy controller suggested in this paper.

The purpose to suggest the 2nd/1st-type indirect adaptive fuzzy control structure is not only to improve the control performance but also to implement a real-time control system in spite of using low-price CPU.

This chapter deals with an attitude control for the stabilizing system implemented with two DOF, as shown in Figure 2.

The plant dynamic for the roll plant of the implemented stabilizing system can be modeled as a highly nonlinear 2nd-order differential equation with parametric uncertainties given as Equation (25). The plant dynamic for the pitch plant can also be modeled as the same nonlinear 2nd-order differential equation as Equation (25). The only difference between the roll and pitch plant is the magnitudes of the least upper bounds for the uncertainties a₁(t) and a₂(t) in Equation (25).

Unfortunately, the exact magnitude of the each least upper bound for the two plants cannot be known.

The limits of the states x₁ and x₁ and the limit of the control input uwere prescribed as Equation (26) and the least upper bounds for uncertainties a₁(t) and a₂(t) were supposed as Equation (27).

In this case, plant constraint limits are calculated as Equation (28).

As referred in section 4.1, the dynamic characteristic of the given stabilizing system with two DOF is highly nonlinear and its model uncertainties caused by the structural complexity are not known.

Figure 7: The control structure of the 2nd/1st-type indirect adaptive fuzzy control system for each roll and pitch plant.

Figure 7: 
The control structure of the 2nd/1st-type indirect adaptive fuzzy control system for each roll and pitch plant.

For the control of the attitude of this system, the indirect adaptive fuzzy control, named 2nd/1st-type indirect adaptive fuzzy control algorithm developed in chapter 3, was applied to the roll and pitch plant, respectively. The controller structure of each plant including hardware and software are the same. In the design process of each control system, it was assumed that the plant dynamics having been modeled in section 4.1should be unknown.

Figure 7 represents a block diagram of 2nd/1st-type indirect adaptive fuzzy control structure for each roll and pitch plant. And variables and functions described in the Figure 7 are explained as followings:

θ_g : integrated value of the gyro sensor output

θ_m : motor output angle measured by potentiometer

θ_e : angle error signal

u_c : certainty equivalent control

u_s : supervisory control

u = u_c = u_s : control input for the plant

: identified plant functions

β_p(0), β_q(0) : initial adaption parameters

β_p, β_q : adjustable parameters by adaptive laws

ξ_p(x), ξ_q(x) : fuzzy basis functions

Two axis gyro sensors were used to measure the attitude of the platform motion, that is roll rate and pitch rate, of the stabilizing system. Therefore, the roll and pitch angle could be obtained by integrating the gyro sensor outputs. Two potentiometers were equipped on the rotating shafts of the roll gimbal and pitch gimbal for measuring the plant outputs. A digital controller was implemented based on CPU in view of hardware and software, in order to generate the control input for each roll and pitch plant. Naturally, the 2nd/1st-type indirect adaptive fuzzy control algorithm was implemented in CPU. By fully considering the computation time of the control algorithm, the sampling time for real time control could be selected as 0.05 second.

In order to identify the plant functions and in adaptive laws, the fuzzy basis functions ξ_p(x) and ξ_q(x) must be computed. To do this, fuzzy rules must be defined based on fuzzy membership functions.

Figure 8 shows a lookup table for generating fuzzy rules necessary to the roll plant control system design. Figure 9 shows membership functions for states x₁ and x₂ of the roll plant.

Figure 8: 
A lookup table for and of the roll plant

Figure 9 shows membership functions for states x₁ and x₂ of the roll plant, which are used in fuzzy rules.

Figure 9: 
Fuzzy membership functions for states x₁ and x₂ of the roll plant.

By using Figures 8 and 9, the corresponding fuzzy rules for the roll plant can be arranged as followings:

In the similar manner, a lookup table and membership functions must be defined for generating fuzzy rules necessary to the pitch plant control system design. Figures 10 and 11 show them, respectively.

Figure 10: 
A lookup table for and of the pitch plant.

Figure 11: 
Fuzzy membership functions for states x₁ and x₂ of the pitch plant.

By using Figures 8 and 9, the corresponding fuzzy rules for the pitch plant can be arranged as followings:

In order to test the possibility of the suggested control system, several experiments were executed using the stabilizing system shown in Figure 2. As a method to generate the platform motion of the stabilizing system, the platform on which the gyro sensors are installed was swung with the angle range about -30^o ≤ θ_g ≤ 30^o by using hands for each roll and pitch plant. The angular velocity of the gyro platform was measured about and the swing frequency was about 1/6Hz, which was similar to the rolling and pitching motion of ships.

Figure 12 shows the experimental results of the simple feedback system for the roll plant without controller. The integrated gyro sensor output is exerted on the feedback system as a reference signal and thus the potentiometer output is just the output of the system.

Figure 12: 
Angular position responses of the simple feedback system for the roll plant with time varying platform motion.

The upper figure in Figure 12 is to compare the integrated gyro sensor output with the potentiometer output. The lower figure in Figure 12 shows the output error between the integrated gyro sensor output and the potentiometer output. From the figure, it is known that the transient response of the simple feedback system for the roll plant reveals a considerable overshoot with the maximal error about ±7^o.

Figure 13 shows the experimental results of the 2nd/1st-type indirect adaptive fuzzy control system for the roll plant. From the figures, it is known that the transient response of the suggested control system for the roll plant was improved in view of an overshoot. Although the transient response revealed a poor response similar to that of the simple feedback system during transient time, it was improved within the maximal error ±2.0^o as time was passed to steady state.

Figure 13: 
Angular position responses of the suggested control system for the roll plant with time varying platform motion.

Figure 14 shows the experimental results of the simple feedback system for the pitch plant without controller. The upper figure in Figure 14 is to compare the platform motion as a reference input with the feedback system output for pitch plant. The lower figure in Figure 14 shows the output error between the platform motion and the system output. From the figure, it is known that the transient response of the simple feedback system for the pitch plant reveals a overshoot with the maximal error with the range -1.0^o to 6.0^o. A distinguished feature of the error response is that the error was biased on positive side. The reason for this bias is just the influence of the mass unbalance caused by the geometrical complexity of the pitch plant.

Figure 14: 
Angular position responses of the simple feedback system for the pitch plant with time varying platform motion.

Figure 15 shows the experimental results of the 2nd/1st-type indirect adaptive fuzzy control system for the pitch plant.

Figure 15: 
Angular position responses of the suggested control system for the pitch plant with time varying platform motion.

From the figures, it is known that the transient response of the suggested control system was improved well in view of the overshoot and the output error. The steady-state response was also improved as time was passed within the maximal error ±2.5^o and also the bias trend was disappeared.

In the conclusion, the possibility of the indirect adaptive fuzzy control system suggested in this paper for the stabilizing system, was convinced. However, the control performance in terms of the overshoot and the output error in steady-state could not be highly improved. It is noted that the only six fuzzy rules were used to compute the fuzzy basis functions. If the more numbers of fuzzy rules are used, the better performance of the fuzzy control system can be achieved.