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Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 563 T-S Fuzzy Tracking Control Based on H∞ Performance with Output Feedback for Pendulum-Cart System
Hanny Megawati Rosalinda1, Trihastuti Agustinah2, Khairurizal Alfathdyanto1
[email protected], [email protected]
1Politeknik Elektronika Negeri Surabaya
2Institut Teknologi Sepuluh Nopember Article Information Abstract Submitted : 28 Apr 2023
Reviewed : 29 Apr 2023 Accepted : 3 May 2023
In some practices, not all state variables are available because of limited or noisy measurements. Thus, via output feedback, an observer is used to estimate the unmeasured states. To apply linear controllers to the pendulum-cart system, the Takagi-Sugeno fuzzy model is utilized by linearizing the system in more than one operating point. The effect of disturbances on tracking performance is reduced to the prescribed attenuation level by H∞ performance. The stability of the whole closed-loop system is investigated using the Lyapunov function. Sufficient conditions are derived in terms of a set of Linear Matrix Inequality (LMI) to obtain the controller and observer gain. Simulation results show that the proposed control method can make the system track the sinusoidal reference signal, maintain stability, and attenuate the effect of disturbances to less than the prescribed attenuation level measured by L2 gain. In the implementation process, an adjustment is needed to move the observer’s pole and speed up the observer’s responses.
Keywords T-S Fuzzy tracking control, Output feedback, H∞ Performance, LMI, Pendulum-cart system
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 564 A. Introduction
Robots have been widely used in our society. It plays an important role in taking over hazardous and high-risk jobs. As robots become easier to integrate with other systems, many have used them, not only in the military field [1, 2] but also in medical [3, 4, 5] and in our daily life, such as segway. The characteristics of those mentioned robotics systems can be represented in a pendulum system [6].
For example, in [3], an unconstrained two-pendulums is used to model a humanoid gesture: a human leg. The first pendulum represents the thigh, the second is the shank and the joint is the hip and the knee. It has simple structures, yet it shares the same characteristics of instability, nonlinearity, and multivariate [7]. The process of pendulum research also investigates many control issues, so that it becomes effective in examining a new control method and theory. Thus, an inverted pendulum can be considered a fundamental benchmark that can be found in many laboratories.
Various control methods have been evaluated in inverted pendulum systems.
Generally, the strategies for handling nonlinearity are divided into 2 ways. First, a nonlinear control algorithm is directly applied [8-11]. Although it can approach the nonlinear characteristics, it has more difficult design processes because of the complexity of that system itself. The second strategy, linear controllers are used by approaching the system into a linear system [1, 3, 12-17]. Antonio-Cruz et al approximated the nonlinear pendulum and pendubot into a linear model around its operating point [12]. The nonlinearity of the system: the limit cycle, caused by dead zone and static friction, is eliminated by its controller design. Thus, the design is specifically made for the system. Hasanah et al linearized a two-wheeled balancing robot using the Jacobian matrix so that a PID controller with Particle Swarm Optimization tuning can be used [14]. Linear controllers, indeed, are easier to find in literature along with their excellence. However, linearizing a system only represents its characteristics around its operating point, not the whole range of the system.
Takagi-Sugeno (T-S) fuzzy model, through Parallel Distributed Compensation (PDC) scheme, is considered an effective way to represent the whole nonlinear characteristic with linear control algorithms by linearizing the system in some operating points [13, 15]. Every operating point on linearized plant rules is compensated by every gain on controller rules with the same fuzzy membership function. This PDC scheme can be used not only in controller gain but also in observer gain [18-20].
To deal with disturbances, many researchers have been using H∞
performance to attenuate it so that it has a very small effect on tracking performance [13, 15, 21-23]. Agustinah et al adjusted the reference model so that it can be applied to a pendulum-cart system [15]. However, when applying the control method to a real plant, it differentiates available states to generate another state. In some cases, indeed there are state variables that could not be measured or are unavailable. Measuring state can be physically difficult and costly. In this paper, we develop the control method [15] by using an output feedback scheme, instead of state feedback.
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 565 In the real plant implementation process, there is an interesting issue we have to investigate. The observer fails to estimate, unlike the simulation. This is because observer gain and controller gain are computed interdependently by the algorithm. In consequence, the eigenvalue, which determines the speed response [24], of each system (controller-observer) cannot be set. For practical control design, the speed response of an observer system must be assured to be higher than the speed response of the main system. Thus, the control system design is broken into two separate parts (controller-observer), which facilitates the design, and the stability of the whole system is still guaranteed [25-27].
This paper is organized as follows. Section 2 addresses a comprehensive explanation of the original system and its model. Section 3 presents the tracking control algorithms. The results of the simulation and implementation of the control design to the pendulum-cart system are discussed in section 4. The conclusion of these projects is given in Section 5.
B. Dynamical of The Inverted Pendulum-Cart System
The system consists of only one rigid pendulum body attached to a cart that can swing the pendulum in one rotational joint, by gravity and the moment of inertia (see Fig. 1(a)). The cart is controllable to left and right, along a horizontal track, connected to a DC motor via a belt. The objective of the proposed design is that the cart can track a sinusoidal reference signal while maintaining the pendulum around its unstable equilibrium point.
Figure1. (a). Acting forces on the system (b). Linearizing in some operating points Its mathematical model can be derived using Newton’s second law, stating translational motion horizontally (1), vertically (2), and rotational motion (3) [28].
(1)
(2)
(3) Where is the force applied to the cart (N), is the frictional force, is the mass of the pendulum (kg), is the mass of the cart (kg), is the cart position (m), is the distance from the axis of rotation to the center of mass of the pendulum-cart system (m), is the pendulum angle from the vertical axis (rad), is the normal force, = 9.8 m/s2 is the gravity constant, is the pendulum friction constant (kg·m2/s), and is the moment of inertia of the pendulum cart system to the center of mass.
Let T where x1 is cart position (m) measured from the midpoint of rail, x2 is pendulum position (rad) from the vertical axis, x3 is cart
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 566 velocity (m/s), and x4 is pendulum velocity (rad/s), the state space system can be described by:
(4)
with
dan
It can be interpreted from (4) that the system is nonlinear. Thus, linearization is needed to apply a linear controller. It can be seen in Fig. 1(b) that a nonlinear function can be represented by some linear function, linearized in more than one operating point, to capture a wider range of characteristics. It is practicable thanks to the Fuzzy T-S scheme, called parallel distributed compensation.
The system can be linearized around its equilibrium point using the Jacobian matrix [17]. Equilibrium points are solutions of the differential equation of the system; a point where the rate of change of state is equal to zero. The mathematical model of the inverted pendulum-cart system in (4) can be written as:
(5)
where
;
Then, the equilibrium points ( ) can be obtained through:
(6) The solutions of (6) are:
(7)
with k is a real number and n is an integer. The objective of the proposed control system is to track the cart while maintaining the pendulum angle at 0°, the
operating points are chosen to be ,
and . Regardless of pendulum
frictions, the linear system of (5) using the Jacobian formula is:
(8)
with
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 567
C. Tracking Control Design
Using the scheme of Parallel Distributed Compensation of the T-S Fuzzy model, the nonlinear system is represented by the combination of 3 linear models, linearized in 3 different operating points. Each of the linear models will be compensated by the 3 controllers respectively. As seen in Fig. 3, the tracking error er (t) is a difference between the reference states and system states. The system states are estimated by the observer because not all the real states can be measured. The controller, the plant, the observer, and the model reference will be augmented into one matrix to be analyzed its global stability using the Lyapunov function. The compensation gains (K) in the controller are calculated so that the system satisfies the H∞ tracking performance.
Figure2. Proposed control design 3.1. Takagi-Sugeno Fuzzy Model
This T-S Fuzzy model is mathematical model-based. Linearized models are utilized on plant rule to capture the dynamic of the nonlinear system. The premises of the plant rules are the operating points in the linearization process. That rule is expressed in (9) [15].
Plant rule i:
If
Then (9)
; i = 1, 2,..,q
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 568 where q denotes the number of fuzzy rules, j denotes the number of fuzzy sets in one rule, is the fuzzy set, denotes the premise variable (function of a measured state variable), u denotes the control input, denotes a bounded disturbance, and denotes a measurement noise. denotes the state vector, denotes the outputs of the system, A, B, C, and D are the matrices with appropriate sizes. The fuzzy plant is inferred below [15]:
(10)
where
Weighting and has some properties as follows:
The controller rule is generated by the Parallel Distributed Compensation scheme. Each linear model on plant rule will be compensated by each gain on controller rule whose fuzzy set corresponds, as follows:
Controller rule i:
If
Then ; i = 1,2,..,q (11)
where
denotes controller gains, denotes tracking errors obtained from difference values between the actual states and the reference states ( ). Not all the states are measured, then, estimated states ( ), produced by the observer, is utilized in that equation. Premises and fuzzy sets in the controller rule are precisely identic with the premises and the fuzzy set in the plant rule, and so does the observer rules.
Observer rule i:
If
Then
; i = 1, 2,.., (12)
where denotes observer gain. The overall fuzzy observer model is declared in (13) [14].
(13)
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 569 3.2. Augmented System
There is more than 1 sub-system in this overall control system design that must be stabilized. Therefore, those sub-systems are merged into one to simplify the controller design. Those systems are observer, plant, controller, and reference model. A reference model is used to assist the system track the sinusoidal reference signal, as follows:
(14) where r denotes a sinusoidal reference signal, and are system and input matrices with appropriate size so that has the same size as system states.
Observer error, denoted , is desired to be convergence at 0 if ∞. The equation of observer error is shown in (15).
(15) Substituting (10) and (13), the differential of observer error is (16).
(16)
Substituting (11) into (10), we get:
(17) Finally, using (14), the augmented matrix is obtained as follows:
where
3.3. Lyapunov Stability
Based on Lyapunov’s theory, system stability can be determined by the energy inside the system. The candidate Lyapunov function, used in this control system, is in the following form:
(19) where
(18)
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 570 The differential of that Lyapunov function is also used to determine stability. The differential of (19) is:
(20) The following closed-loop system:
(21) can be guaranteed to be stable, if there is matrix fulfilled the following requirements:
1.
2.
Requirement number 2 is obtained by substituting (21) with (20).
, then the result of the substitution is (22).
(22) 3.4. H∞ tracking performance
In this sub-section, H∞ tracking performance is used to determine the value of , , and P. The objective of this control system is to make the system stable and get the H∞ tracking performance based on tracking error as in the following equation [14]
(23) where tf is the final time, denotes an attenuation level, and Q is a weighting positive definite matrix. The inequality in (23) can be modified into (24) if the system in (18) and the initial condition system are considered.
(24) where
Equation (24) has a physical meaning that all the effects, given by (t), to tracking error will be attained to the attenuation level ( ). To satisfy (24), we must fulfill (25). Whatever (t) is, the norm of tracking error divided by the norm of (t) has to be equal to or less than the prescribed value .
(25)
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 571 Inequality in (25) can be extended by substituting , and to be (26).
(26) where
The matrix in (26) is modified by Schur Complement so that it has the LMI form as follows:
(27)
where
The desired parameters from the matrix are and . It cannot be solved directly by one step LMI numeric method. First, some of those 5 parameters must be found through particular inequality. Then, by substituting the parameters that have been found, we search for the other parameters. The inequality used for the first step is . Where and , that inequality can be expressed in the following form:
(28) Schur Complement is utilized to change that inequality into the LMI form (29).
(29)
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 572 where
Parameters and can be obtained by getting and first via LMI numerical method. The second step is substituting and into (19) and obtaining the other parameters , and . The matrix is derived already into the standard LMI form.
An attenuation level is determined as small as possible to increase H∞
tracking performance. However, there is also the appropriate positive definite matrix P to be considered. The determination of attenuation level can be expressed as follows:
s.t. and (25)
D. Results and Discussion
Using the LMI toolbox in MATLAB, the parameter , and are obtained using the algorithm in the previous section. The optimal attenuation level is found to be .. Selecting the weighting matrix
, the solutions are:
; ;
;
;
;
We have simulated the original nonlinear system in MATLAB Simulink. The proposed control design in the previous section is then applied to the system (5). It can be seen in Fig. 3 that the cart successfully follows the reference signal while maintaining the pendulum in its inverted position.
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 573 A disturbance ( ) is also added to the control signal to test the system’s robustness (see Fig. 4). Deviation occurs slightly during the disturbances, but the system is capable to bring the cart and the pendulum back on track. In numbers, it has been satisfied Eq. (18), one of the objectives of the control design. The value of L2-gain (measured, 0.0004508) is far less than the prescribed attenuation level (0.14).
Figure 3. The trajectories of system outputs with disturbances
Figure 4. Disturbances added to the control signal
One of the reasons that the control design fits the objectives is because the observer has good performance, too. As in (5), the outputs are only the state variables x1 and x2, while the others are unavailable (x3 and x4). The observer can estimate the unavailable state variables as depicted in Fig. 5. The actual and estimated values of x3 have no difference in transient and steady time, without disturbances. Whereas state variables x4 have a very small difference just on undershoot value as big as 0.003 rad/s.
The second test is done by adding noise to one of the measured states, cart position. The average value of noise is desired to be 0.001-0.002. This value is equal to 10-20% maximum value of cart position while steady. Noise is added through random numbers with mean 0 and variance 0.001 (standard deviation 0.0316).
It can be concluded that noise doesn’t affect the system too much as seen in Fig. 6. This is supported by the tracking error, measured at 0.04735. This value is close to the tracking error in the experiment without noise, in which at 0.04466.
The value of L2-gain is 0.0006026, which is still less than the attenuation level. The response of the pendulum proves that the system still can be stabilized in the origin.
The system could return to the appropriate position despite disturbances and noises because of the compensation given by the control signal. When disturbances are given, the control signal increases in the amount of disturbance
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 574 value, 0.3 N. Then, the system can push back the value of the control signal to normal value, when there are no disturbance and noise. It shows that the presence of disturbance and noise can be compensated by the value of appropriate K and L.
Figure 5. The trajectories of unmeasured states with disturbances
Figure 6. The Trajectories of system outputs with noise Table 1. Eigenvalue comparison
System with controller Observer
1 -9.97 + 2.11i; -9.97 - 2.11i; -5.86; -2.1; -2; -3 -152.86; -2.59; -2.56; -0.56 2 -8.96 + 7.63i; -8.96 - 7.63i; -4.06; -2.13; -2; -3 -152.86; -2.59; -2.56; -0.56 3 -4.49 +12.03i; -4.49 +12.03i; -2.45 + 0.27i; -2.45 + 0.27i;
-2; -3 -152.86; -2.59; -2.56; -0.56
However, not all dynamics of the system can be captured in simulation. An engineering adjustment is needed in real-time experiments in the pendulum-cart system. The system cannot be stable unless the observer calculates faster than the controller. With the generated parameter ( and ), the dominant eigenvalue of the system with controllers is smaller than the eigenvalue of the system with observers (see Table 1). It resulted in lower responses from the observer. Pole placement is, then, used to move the dominant eigenvalue far to the left. The pole is put in -37.5+27i, -37.5+27i, -38.8+2i, and -38.8+2i.
Fig. 7 shows comparisons of the fuzzy design (three linear models) to the same controller with only one linear model. It indicates that with a wider range of the model, the system can move the cart faster to the reference signal; transient response is improved.
With the same disturbances as in Fig. 4, the cart deviates but still returns to its position. As expected, the performance is not as well as in the simulation since the observer gain is replaced by the new pole placement gain. However, the
Attribution-ShareAlike 4.0 International License Vol. 12, No. 2, Ed. 2023 | page 575 disturbance is still successfully attenuated, indicated by the value of L2-gain to be 0.001 (in simulation: 0.0004508, prescribed: 0.14).
Figure 7. Comparison of fuzzy and linear control in a real-time experiment
Figure 8. The system outputs with disturbances in a real-time experiment E. Conclusion
An output feedback scheme, to improve the algorithm of T-S fuzzy tracking control design with H∞ tracking performance for the pendulum-cart system is developed in this paper. From simulation results, the proposed control method has been verified to make the system track the sinusoidal reference signal, maintain the stability, and attenuate the effect of disturbance to output performance, measured by L2-Gain, to less than the prescribed attenuation level ρ. The observer could generate unavailable states and make them converge with their actual value quickly despite disturbances. For practical control design, the eigenvalue of each sub-system (controller-observer) must be assured. A separation principle with H∞
tracking performance both in a controller and an observer can be developed for further studies.
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