On Switched Control Design of Linear Time-Invariant Systems with Polytopic Uncertainties

et al. 2017

Fuzzy Sets and Systems

This paper presents the design and simulation results considering the use of functional electrical stimulation to control the leg position of paraplegic patients. The plant is described by a nonlinear system using Takagi-Sugeno fuzzy models and a closed-loop control is presented. A transfer function represents the mathematical model related to the muscle torque and the electrical stimulation pulse width. Considering that, during the operation, the leg position is between 0 • and 60 • , then two fuzzy regulators were designed, assuming that the state vector is available or using an observer when only the output is available. This design was based on the Lyapunov stability theory and Linear Matrix Inequalities (LMIs). The simulation results show that the proposed procedures are efficient and offer good results for this control problem. Finally, new conditions regarding the design of the output tracking control, using a suitable nonlinear transformation for the description of the plant in an adequate form, is presented.

Section: Discussionmentioning

confidence: 99%

“…The tracking procedure proposed in this manuscript allows the tracking of a broader class of signals than the control strategies presented in [39] and [40], where the operation point is not completely known but is in a given region.…”

Section: Introductionmentioning

confidence: 99%

Electrical stimulation tracking control for paraplegic patients using T–S fuzzy models

Gaino

Covacic

et al. 2017

Fuzzy Sets and Systems

“…Then, the 'smooth switched' control law (37) applied to the system (10) (10), with control laws (11) and (37), respectively. If (16) hold for all i and j 2 I r , and for all D s and D s 2 D, then multiplying (16) by j and taking the sum from j D 1 to r it follows that, from (39) and (48a),…”

Section: Theoremmentioning

confidence: 99%

Smoothing switched control laws for uncertain nonlinear systems subject to actuator saturation

Alves

Adaptive Control & Signal

Oliveira

et al. 2016

Self Cite

This paper considers a class of uncertain nonlinear systems exactly described by Takagi-Sugeno (T-S) fuzzy models, with or without matched uncertainties and/or disturbances, within an operating region in the state space. Considering the plant subject to actuator saturation is proposed, a switched control design method such that the equilibrium point of the controlled systems is locally asymptotically stable with an adequate decay rate, for all initial conditions in a region obtained in the design procedure, that is within a given operating region. Moreover, an exact representation of the minimum function using signal functions is presented. Therefore, it is offered a bridge between the switched control and variable structure control laws, because they are usually based on minimum and signal functions, respectively. Considering the well-known approximation of the signal function by a sigmoid function, this paper introduces the 'smooth minimum' function. The smooth minimum function allowed the design of 'smooth switched' control laws, without chattering, for the aforementioned class of systems. The proposed smooth switched control laws guarantee the uniform ultimate boundedness of the controlled systems. Design examples and simulation results illustrate the control design procedures and the effectiveness of the proposed control laws. Future researches in this subject may explore the presented relationship between the switched control and variable structure control laws, in order to obtain new useful control laws. of the system by choosing a feedback gain, belonging to a given set of gains, which minimizes the time derivative of a Lyapunov function. The design conditions are represented by LMIs, which can be solved in a computationally efficient manner [15,16].Frequent and/or abrupt changes of the feedback gains, however, can generate control signal chattering. Control signal chattering is a generally undesirable phenomenon in real systems. Different control strategies try to avoid the chattering problem, for instance, in sliding mode control [17][18][19][20][21][22][23][24], to name a few, in adaptive control [25][26][27], and in control of switched linear systems [28][29][30].In this context, this paper extends the methodology proposed in [13,14] for uncertain T-S fuzzy models, based on [31,32], taking into account actuators saturation. Additionally, an invariant region E.P; 1/ is estimated, in which for every x.0/ 2 E.P; 1/, the state vector of the controlled system x.t/, t > 0 is in a given operation region where the T-S fuzzy model exactly describes the nonlinear system and also the equilibrium point of the controlled systems is locally asymptotically stable with an adequate decay rate. Moreover, 'smooth switched' control laws are proposed, without using the plant membership functions. A key point of the new control laws is a proposed exact representation of the minimum function using signal functions. Considering that the minimum functions are largely used in switched control laws [13,14,[28][29][30] and the s...

“…Note also that in some practical implementations, this phenomenon of discontinuity may be inconvenient or cannot be allowed. Thus, as in [30], one can modify the function γ ξ to make it continuous and ensure the uniform ultimate boundedness of the system and smoothness of the control input. Thus, for instance, consider a function γ ξ as follows [30]:…”

Section: Remarkmentioning

confidence: 99%

On Switched Regulator Design of Uncertain Nonlinear Systems Using Takagi–Sugeno Fuzzy Models

Souza¹,

IEEE Trans. Fuzzy Syst.

Cardim

et al. 2014

Self Cite

This paper is concerned with the design of state-feedback switched controllers for a class of uncertain nonlinear plants described by Takagi-Sugeno (T-S) fuzzy models. The proposed methodology eliminates the need to find the membership function expressions to implement the control law, which is relevant in cases where the membership function depends on uncertain parameters. The design of the switched controllers is based on a minimum-type Lyapunov function and the minimization of the time derivative of this Lyapunov function. The conditions of the new stability criterion are represented by a kind of bilinear matrix inequalities (BMIs) that has been solved by the path-following method. A numerical example and the nonlinear control design of a magnetic levitator with uncertainties illustrate the procedure. Index Terms-Bilinear matrix inequalities (BMIs), control of uncertain nonlinear systems, linear matrix inequalities (LMIs), switched control, Takagi-Sugeno (T-S) fuzzy model.