This paper proposes the topics of sliding mode control for nonlinear Takagi-Sugeno systems based on a state observer with application to single-link flexible joint robotic. Firstly, a state observer relying on estimated premise variables is constructed, based on which we define an integral-type switching surface function on the estimation space. Secondly, by the equivalent control method, a sliding mode dynamics with an error system is obtained. Then, an adaptive variable structure controller is constructed to make sure that the predefined switching surface will be arrived in finite-time. Furthermore, stability analysis with an _{∞} performance is analyzed for the whole closed-loop system by linear matrix inequality condition. Finally, simulation study based on the robotics is conducted to confirm the validity of the proposed observer-based fuzzy controller.

The research of nonlinear systems has always been the most popular topic in control theory and applications. Since no matter from internal and external perturbations or mechanism of dynamics, systems suffer from non-linearities is more common in practice. However, many traditional tools were good at tackling linear systems and invalid for nonlinear systems. Therefore, the issue of nonlinear control strategies comes to researchers' eyes, one of which was the Takagi-Sugeno (T-S) fuzzy approaches^{[1]}, which was viewed as a powerful mathematical method in dealing with complex nonlinear dynamics. Relying on a group of "IF-THEN" rules, the T-S fuzzy model provides accurate approximation of smooth nonlinear terms via fuzzy "blending" of local linear dynamics with the help of membership functions. Recently, fruitful research results have been proposed in literature by virtue of the T-S fuzzy approach. For example, the fault prognostics when the degradation phenomena exhibit nonlinear and time-varying dynamics by error based revolving T-S fuzzy model in^{[2]}; An input-output stabilization issue for time-delay systems via T-S fuzzy method was studied in^{[3]}; In^{[4]}, the fuzzy observer design and _{∞}^{[5-7]} and references therein.

Sliding mode control (SMC)^{[8]} has received great attention in the last decades for its powerful effectiveness in dealing with complex systems. It is already well known that the SMC has many good features, for instance, totally insensitive to the matched system disturbance, simplicity in computation and better transient performance. Therefore, it has witnessed great efforts been undertaken in the application of SMC, such as switching power converters^{[9]}; robot manipulators^{[10]}; Furuta pendulum^{[11]}; electric circuits^{[12]}, fuel cell systems^{[13]}, etc.; In general, the design of SMC contains two steps: (a) The switching surface design; (b) the sliding mode controller design. For the issue (a), it is worth noting that significant efforts have been devoted to the integral-type sliding surface design since it is acknowledged that the reaching action is no longer required in a conventional SMC approach by designing an integral switching hyperplane. Thus, the robustness can be achieved along the whole sliding surface, which motivates us to further investigate this issue in the paper. Recently, a few nice works have been reported on this issue, for instance, several crucial problems regarding the performance, modification, and improvement of integral SMC was discussed in^{[14]}; the problems of observer-based integral SMC and fault estimation for nonlinear systems was investigated in^{[15]}; The decentralized adaptive integral SMC scheme was developed to stabilize large-scale interconnected systems in^{[16]}, see more in^{[17-21]} For the issue (b), a lot of advanced methods have been incorporated into the sliding mode controller design, for instance the fuzzy logic and the adaptive algorithm approaches. The proportional-derivative-based fuzzy SMC was developed to deal with un-modeled dynamics and external disturbances in the human-exoskeleton system in^{[22]}; An adaptive SMC was proposed in^{[23]} in order to adapt switching gain such that to cope with possibly unknown system uncertainty. Regarding the application of SMC in the field of flexible joint manipulators is also appealing due to its flexibility in the controller design. Recently, some nice works have appeared, for example, an adaptive SMC method was proposed for a single-link joint robot taking consideration of mismatched uncertainties in^{[24]}; In^{[25]}, a hierarchical SMC was presented for a rotary flexible joint manipulator through Lyapunov function theory; In^{[26]}, the SMC method introduced to deal with the fault-tolerant tracking control for a single-link joint manipulator, etc. However, these results did not take the advantages of T-S fuzzy approaches.

On one hand, the system state components are not always available because of various limitations in practice, which means the analysis and feedback control of such systems based on observers is imperative. On the other hand, it happens that the system premise variables are the same with the state components, thus to achieve an effective sliding mode observer design by T-S fuzzy method, the estimated fuzzy dynamics combined with state-dependent premise variables must be considered in many practical problems. So far, a few pioneer works have taken efforts on this problem, for instance, the fuzzy state observer design in the sense uncertain input was proposed in^{[27]}. To the authors' knowledge, the issue of adaptive integral SMC for nonlinear T-S system with unmeasurable variables is interesting and an open issue to be studied.

In view of above discussion, this paper intents to investigate the issue of observer-based fuzzy integral SMC for nonlinear T-S systems. Based on the single-link flexible joint robotics models, a fuzzy model approach is introduced to obtain universal mathematical model. By designing an adaptive compensator in the state observer, an integral-type hyperplane function is proposed on the estimated space. The reachability of switching surface in finite-time is ensured by an adaptive sliding mode controller. A strict LMI condition is developed ensure stability and an _{∞} performance for the whole closed-loop dynamics.

Notations: In this paper, _{min}^{T} + P.

Let's consider the following single-link flexible joint robotic dynamics^{[28]}:

in which _{m}(_{l}(_{m}_{l}_{m}_{l}

NOTATIONS

Symbols | Meaning |
---|---|

m (kg) | Pointer mass |

h (m) | Link length |

k (N.m.rad^{–1}) |
Torsional spring constant |

k_{l} (N.m.V^{–1}) |
Viscous friction coefficient |

B_{m} (N.m.V^{–1}) |
Amplifier gain |

Now, denote _{1}(_{m}(_{2}(_{m}_{3}_{l}(_{4}(_{l}^{[29]} that under certain angle position, the function sin(_{3}(

where _{1}(_{3}(_{2}(_{3}(_{1}(_{3}(_{2}(_{3}(_{1}(_{3}(_{2}(_{3}(

It is easily seen in above functions that _{1}(_{3}(_{2}(_{3}(_{3}(_{1}(_{3}(_{2}(_{3}(_{3}(

Plant Rule 1: IF _{3}(

THEN

Plant Rule 2: IF _{3}(

THEN

where

More generally, by taking consideration of unknown local perturbations, the following universal model is considered:

Plant Rule _{1}(_{i}_{1} and _{2}(_{i}_{2} and ··· and _{n}_{in}

where _{1} (_{n}_{ij}_{i}, B

with

By fuzzy blending, the overall system is depicted as:

in which _{i}_{ij}_{j}_{j}_{ij}.

In this paper, the purposes are to construct a fuzzy observer-based SMC strategy for the fuzzy system (_{∞} performance and good stability property can be obtained.

^{[30]} and DC-DC converter^{[31]}, etc.

In this section, due to the premise variables are also unmeasurable. Then, the following fuzzy state observer is constructed:

Plant Rule _{i}_{1} and _{i}_{2} and ··· and _{in}

THEN

where _{s}_{i}

Similarly, the fuzzy observer (

Let

where _{2}

In this part, relying on the estimated component (

with

Taking the systems (

When the switching surface

Then, combining (

Up to now, we can conclude the aim of this work is to proposed a fuzzy sliding mode controller based on the fuzzy observer (

The systems (

An _{∞}

is satisfied with zero-initial condition, where

Now, as we can see, in order to attenuate the unknown local perturbation, _{s}

So far, we can propose the compensator _{s}

in which it is required that ^{T}P = NC, ε

with _{1} and _{2} are positive scalars to be selected in advance.

^{T}P = NC

Therefore, in the process of implementing the compensator, unavailable errors will be replaced by the fuzzy observer components and the original output variables.

In this part, we are to deal with the reachability of the predefined switching surface

with

Then,

By substituting (

Therefore, we can conclude from (

In this part, we are going to give a sufficient condition to check if the systems (_{∞} disturbance attenuation level

_{∞}_{i}

and the controller gain is computed by _{1} is given such that _{i}_{i}C

in which

Taking the compensator _{s}

Overall, it obtains

where ^{T}^{T}^{T},with

in which ^{-1}, then pre-multiplying Γ_{i} with _{i} with _{i} <

Therefore, for any

So it yields

which means the system (

Next, lets consider the _{∞} performance for the systems (

in which

Therefore, according to the Schur complement and the condition (is), it obtains that _{∞}

^{T}P_{2} = NC

Therefore, there is a scalar

In view of the Schur complement, we have

Therefore, the _{∞} performance issue we proposed before is now turned into by finding a global optimal solution in the following way:

with the condition in (is) satisfied simultaneously.

Therefore, the design of sliding mode observer in this paper is given as follows: For the plant (_{i}_{i} – L_{i}C_{i}_{s}

Consider the single-link flexible joint robotic presented in model (

SYSTEM PARAMETERS

Parameter(Units) | Value |
---|---|

m(kg) | 2.1×10^{-3} |

h(m) | 1.5×10^{-1} |

k(N·m·rad ^{-1}) |
1.8 ×10^{-2} |

k_{j}(N·m·V-^{1}) |
8.0 ×10^{-2} |

B_{m} (N·m·V-^{1}) |
4.6 × 10^{-3} |

J_{m} |
3.7 × 10^{-3} |

J_{l} |
9.23×10^{-3} |

By setting the attenuation level _{1} _{2}

^{-4}.

Therefore, the controller gain matrices can be computed as

With the help of above parameters, the simulation is performed in the following. Firstly, the initial conditions of original system and observer system are provided as ^{T} and _{1}(_{1} _{2} _{1} (

The state trajectories for the original system.

The state trajectories for the observer system.

The trajectory response for sliding surface function.

The control input.

Estimated value for

Estimated value for

The state trajectories of the observer system under state feedback control.

In this paper, the topic of T-S fuzzy model-based state observer design for SMC of nonlinear systems with application to robotics model has been investigated. Firstly a state observer relying on estimated premise variables was constructed, based on which an integral switching surface has been developed. Secondly, sliding mode dynamics was derived through equivalent theory. Then, the arrival of switching surface was ensured by designing an adaptive sliding mode controller. Furthermore, stability analysis with an _{∞} performance were undertaken for the resulting systems. Finally, simulation study based on the robotics model has been conducted to confirm the validity of the fuzzy controller. In the near future, our attention will be focused on proposing more advanced control strategy to n-link robotic manipulator^{[32]} in terms of robustness, computation, operation cost and system limitation etc.

Made substantial contributions to supervision, writing, review, editing and methodology: Jiang B Performed writing-original draft, software, validation and visualization: Liu Q, Cai Z, Chen J

Not applicable.

This work is supported by The National Natural Science Foundation of China under grant 62003231; partially supported by The Natural Science Foundation of Jiangsu Province under grant no. BK20200989, in part by the NCF for colleges and universities in Jiangsu Province under grant 20KJB120005, in part by Anhui Province Key Laboratory of Intelligent Building and Building Energy Saving under grant IBES2020KF01, and in part by The China Postdoctoral Science Foundation (2021M692369).

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

© The Author(s) 2021.

_{∞}controller designs based on fuzzy observers for TS fuzzy systems via LMI.

_{∞}control for T-S fuzzy systems.

_{∞}-Sliding mode control of one-sided Lipschitz nonlinear systems subject to input nonlinearities and polytopic uncertainties.