1. Introduction

In control systems , the problem of driving the system output such as it asymptotically tracks a desired reference and rejects undesired disturbances while keeping stability in a closed -loop systems is address frequently due to its wide applicability in mechanical systems, aeronautics and robotics, among others; this task is referred as the output regulation problem (Isidori and Byrnes

, 1990 ). The works of Francis (1977 ) and Francis and Wonham (1976 ) have shown that the solvability of a multivariable linear regulator problem corresponds to a system of two linear matrix equations, called Francis Equations. Later, Isidori and Byrnes (1990)

shown that the result established by Francis is a particular case of the nonlinear problem , providing necessary and sufficient conditions as a set of nonlinear partial differential equations called Francis-Isidori-Byrnes (FIB). Unfortunately , these conditions presents, in many cases,a considerable numerical complexity. Some numerical solutions have been given, most of them in terms of Takagi-Sugeno (TS) fuzzy models (Chiu and Chiang, 2009; Castillo-Toledo et al., 2012; Chen, 2005; Tapia-Herrera et al., 2013; X.-Jun and Z.-Qi, 2000; Lian and Liou, 2006; Ma and qi Sun, 2000;Lee et al., 2003; Hernandez-Cortes et al., 2015; Karamanos et al.,2018) or as dynamic implementations (Armenta et al., 2019 ). The popularity of TS models is due to their capability to exactly represent (via the sector nonlinearity approach) or approximate with an arbitrary exactness nonlinear dynamics (Tanaka and Sugeno

, 1992; Tanaka and Wang, 2001); regardless its origin, a TS model is viewed as a convex combination of linear submodels (vertex models ) together with scalar convex functions ; such structure facilitates the use the direct Lyapunov ’s method and obtaining condintions in terms of linear matrix inequalities (LMIs ). LMI conditions are highly appreciated since its solvability can be determined by convex optimization techniques (Boyd et al., 1994) already implemented in commercial software (Gahinet et al., 1994

; Sturm, 1999). Despite the use of TS models for solving the nonlinear output regulation problem, LMI conditions are only for the stabilizer part of the final control law. Recently in (Bernal et al., 2012 b) an LMI solution for the nonlinear mappings is available . Nevertheless, none of the above works deals with nonlinear descriptor systems; they may appear when the Euler-Lagrange formalism is employed for modelling mechanical , biomechanical , electromechanical systems (Lewis et al., 2003 ; Fantoni et al., 2002 ; Luenberger, 1979).

This work provides a numerical solution for the output regulation of nonlinear descriptor systems via TS models and LMIs. The proposal extends developments from (Lin and Dai, 1996), where linear singular systems are considered; it also takes advantage of descriptor representations since the input matrix remains constant, and as mentioned in (Meda-Campana et al., 2009) it relaxes the fuzzy FIB equations under mild assumptions. The designing conditions are in terms of LMIs. The rest of the work is organized as follows: Section 2 places the problem to be solved and states some standard solutions; Section 3 provides LMI conditions to approximate a solution of the nonlinear output regulation problem for descriptor systems; Section 4 illustrates the performance of the proposal on physical as well as numerical examples. Finally, Section 5 closes the paper and gathers some comments on future work.

2. Problem Statement

Consider the following class of nonlinear descriptor system1:

(1)

where is the state vector, is the input vector, is the output vector , and is the state vector of the exosystem , to be defined later, which generates the reference and/or the perturbation signals; it is assumed that x =0 is an equilibrium point, are sufficiently smooth for all such that are constant matrices . Matrix E( ) is full rank for x Ω x , i.e., from (1) it is always possible to obtain a standard state-space representation:

(2)

with Now let us recall the output regulation problem for system (2) and the following exosystem

(3)

where are sufficiently smooth vector fields holding is called the reference output. The exosystem is poisson stable. Arguments will be omitted when their meaning can be inferred from the context.

In (Isidori and Byrnes, 1990; Isidori, 1995), the nonlinear output regulation problem for systems of the form (2), with a nonlinear output consists in finding a such that

· The equilibrium point x = 0 of the closed-loop system is asymptotically stable.

· The tracking error

(4)

goes asymptotically to zero for any initial condition

Then, the following assumptions are considered: A1) the exosystem (3), with w =0 being an equilibrium point, is Pois- son stable,

that is, the eigenvalues of does not have real part; A2) the linear approximation of (2) at the origin x =0, with w =0, is stabilizable. Thus, the output regulation problem with full information is solvable if and only if there exists mappings (the steady- state zero-error manifold) and (the steady-state input) such that

(5)

(6)

holds. The nonlinear control law that performs output regulation is

(7)

with designed such that the linear approximation of (2), around the origin, is asymptotically stable.

The set of equations (5)-(6) is known as Francis-Isidori- Byrnes (FIB) equations . The next section presents LMI con- ditions to performing output regulation by means of the original descriptor form (1) and convex representations for a nonlinear stabilization gain and using an adaptation of (5)-(6) to find

2.1. Convex representations

There are several methodologies to obtain a Takagi -Sugeno (TS) fuzzy model from (1), among them, the sector nonlinearity approach provides an exact representation (Ohtake et al., 2003). It consists in gathering all the nonlinear terms in A(x), E(x), D(x) in the socalled premise vector such that each entry of z(x) is well-defined and bounded where z0 and z1 are the minimum and maximum of zi(x). Thus, each term can be expressed as convex sums of its

Where

are called weighting functions and fulfill the convex sum property in

Then, the so called membership functions can be defined as

they also hold the convex sum property: 0 ≤ µi(z) ≤ 1 and can be expressed as follows (Taniguchi et al., 1999):

(8)

where are the vertex matrices, r = 2p is the number of vertex (rules, linear submodels). In this work, we assume that the exosystem (3) is linear, this is,

(9)

with S having its eigenvalues on the imaginary axis. The following section presents an adaptation of the traditional FIB equations for nonlinear descriptor systems in an exact TS fuzzy model (8).

1.

In general, descriptor systems are of the form (Duan, 2010). Nevertheless, the class treated in this work (1) naturally appears from state-space representations of systems modeled by Euler-Lagrange formalism where is the vector of generalized coordinates is the inertia matrix is the Coriolis matrix, is the gravity vector, and is the torque vector (Lewis et al., 2003, Section 4.3)

3. Main Results

In (Meda-Campana et al., 2009), it is established that system (2), with a TS representation , performs output regulation if the input distribution matrix and the steady -state mapping are constant , while has a convex form. Based on this and taking advantage of the structure in (8), we Moreover, following the results for singular systems presented by Lin and Dai (1996), we propose the following adapted r-FIB equations for descriptor systems:

(10)

(11)

The r set of linear equations must hold simultaneously for and a single Π. Note that thanks to the structure of the descriptor systems, there are no crossed-products between Γi and Bi (Meda-Campana et al., 2009). Thus, the nonlinear control law that performs output regulation is

(12)

with Kµ designed such that the origin of (1) is asymptotically stable.

Remark 1. The above assumption on matrix B is met when dealing with mechanical systems whose mathematical modeling is obtained from Euler-Lagrange methodology; thus a state- space representation is of the form (1).

Now, in order to design the stabilizer, based on the descriptor convex model, the augmented system redundancy form is used as in (Taniguchi et al., 1999, 2000; Arceo et al., 2016), then

(13)

(14)

Where and . The nonlinear control law under design is a parallel distributed compensation (PDC) one (Wang et al., 1995): where are the controller gains to be designed. To this end consider the following Lyapunov function and being invertible. Indeed, guaranteeing ensures the stability of the closed-loop system; thus we have if

(16)

(16)

Now, considering and multiplying (16) on the left and right by and X, respectively, yields

(17)

with The previous inequality guarantees that V<0, thus asymptotic stability at the origin is achieved. Nevertheless, in practical cases it is important to consider the following : guaranteeing a maximum speed convergence φ> 0 while holding a bound on the control input ||u(t)||≤β, β>0. The former is guaranteed if V˙≤−2φV holds, in terms of LMIs this is implied by

(18)

hold for all . If feasible, the controller gains are computed as As for bounding the control law we have

(19)

where x(0) is a given initial condition.

Remark 2. The Based on the proposed results by Bernal et al. (2012a), conditions (10) and (11) can be written as an LMI problem, thus the design of Π and γ(w(t)) can be seen as anoptimization problem. Let N(x) and R(x) be continuously differentiable linear matrix functions and x as the decision vector. Then, an approximated solution for N(x) − R(x) = 0 can be settled as minimization problem, that is, min stands for element-wise ordinary lower than, in other words:

Thus, by means of Remark 2, conditions (10) and (11) can be expressed as

(20)

with ϵ arbitrarily small.

Remark 3. The numerical complexity of the LMI approach increases as the number of nonlinearities increases, since the number of vertex models is . In Xie et al. (2014) the numerical complexity of the LMI conditions is approximated by where nd is the number of scalar decision variables and nl is the number of LMI rows. In our case, as the complete design involves LMIs (18), (19) and (20) we have nd = 0.5n(n + 1) + 2n2 + mnr + ns + msr + 1 and nl =2nr+n+1+r(n+m)+2(n+s)r+2(o+s), where n is the number of states, m the number of inputs, o the number of outputs, s the size of the exosystem, r= is the number of vertex models.

4. Examples

In this section, the proposed fuzzy regulation approach and method previously derived are applied to two TS descriptor fuzzy models without external disturbances. The first example is numerical while the second on corresponds to the well-known cart-pole system. The LMI conditions have been checked by the LMItoolbox (Gahinet et al., 1994) within Matlab 2018a.

4.1. Numerical example

Consider a nonlinear model of the form (1) with

Applying the sector no linearity approach stated in Section 2.1; two different nonlinear terms can be identified such bounds hold in .

Then, the weighting functions are ; therefore, the four membership functions are and

The vertex matrices are

The considered exosystem is (9) with ˙w1 = w2, ˙w2 = −w1. The task is that x2 tracks w1 + w2, thus Q = [1 1]. Running LMIs (18) with ϕ = 0 (no decay rate condition) together with (20), feasible solutions are found with ϵ = 9.0757 × 10−16, meaning that the FIB equations (10)-(11) are satisfied. The computed gains are

Thus, now we are ready to implement the control law (12) for the nonlinear system. Simulation results for initial conditions x(0) = [1 1.5]Tand w(0) = [0.7 0]T are shown in Figures 1 and 2; it can be seen that the tracking takes place asymptotically

Figure 1:
Output versus reference x2 vs w1 + w2

Figure 2:
control signal u(t) of the system

It is important to notice that the system under study has a linear input distribution matrix and if a standard state-space representation (2) is to be obtained, matrix B would be no longer constant, thus results reported in (Meda-Campana et al., 2009) ˜ cannot be directly applied. Additionally, inverting the matrix E(x) would generate more nonlinear terms and therefore more vertex (rules) models.

4.2. Cart-pole system

Consider the underactuated system in the Figure 3 consisting for a car on the rail and one vertical beam joined to the car (Fantoni et al., 2002). The cart-pole dynamical equations can be represented from the Lagrange equation of motion (Lewis et al., 2003), whose final form is is a vector of generalized coordinates x is the distance of the horizontal rail and θ is the angle from the vertical, i.e.

Figure 3:
Cart-Pole System

Table 1:
Nonlinearities in (21) and their bounds

The motion equation for this system, described by its descriptor model (1) is

(21)

Rewriting the previous nonlinear descriptor as a convex sum in a compact set with their nonlinearities and their bounds settle in the table 1, we have an exact TS representation:

Where and the indexes are the digit binary representation of .

A set of illustration matrices for the cart-pole system are given below:

For this system, the task is that pendulum follows a sinusoidal reference, that is, x3 asymptotically tracks ω1, generated by the exosystem ˙ω1 = ω2, ˙ω2 = −ω1. As mentioned before by using the LMIs (18) with φ = 0 and (20), the controller (12) is computed by the established in Section 3. The LMIs are found feasible, for illustration purposes some of the gains are shown

the minimum ϵ is 0.0135, the common steady state

and some of steady state inputs related to i subsystems Simulation results have been performed for initial conditions , the behavior closed loop trajectories are depicted in Figure 4; it can be observed, the tracking error goes to zero as time increases. The rest of the states and the control input are shown in Figure 5, notice that at the beginning, the control law requires a large amount of energy.

Keep in mind that as we are in the LMI framework, we can directly add performance specifications such that decay rate, input-output constraints, etc. Since the control law would be applied to the real system, lower input control signal is required and the necessity of bounds in order to not affect the actuator. Such improvements can performed by running LMIs (18) for decay rate φ = 0.8 and (19) for the input constrain β = 10 with as initial condition. As expected, the control signal holds the imposed bounds and now it can be implemented directly on the physical system; its performance can be seen in Figures 6 and 7.

Figure 4:
Output x3 versus reference ω1 and tracking error e(t) for the cart-pole

Figure 5:
States and control signal u(t) for the nonlinear cart-pole system

Figure 6:
Output x3 versus reference ω1 and tracking error e(t) for the cart-pole system with decay rate and input constraints

Figure 7:
States and control signal u(t) for the nonlinear cart-pole system with decay rate and input constraints

5. Conclusions

This work has defined the nonlinear output regulation for dynamical systems in descriptor forms; sufficient conditions for the existence of such controller are given in terms of linear matrix inequalities. In particular, the proposed approach are given by two components, the first one is a nonlinear stabilizer which creates a globally attractive steady state and, the second one, the steady-state mappings on the basis on new equations in the sense of Isidori; all of these carried out in a practical way. It has been shown that, the approach can be effective for the implementation in real time. Two systems have been presented in order to show the advantages of the proposed approach.

Acknowledgments

This work has been support via the CONACYT scholarship for CVU 585090, the laboratory LANAVEX of the Universidad Politecnica de Pachuca.

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