Open access peer-reviewed chapter

State-Feedback Output Tracking Via a Novel Optimal-Sliding Mode Control

By Ahmadreza Argha and Steven W. Su

Submitted: October 8th 2016Reviewed: January 13th 2017Published: June 28th 2017

DOI: 10.5772/67468

Downloaded: 791

Abstract

This chapter describes a new framework for the design of a novel suboptimal state-feedback-sliding mode control for output tracking while H2/H∞ performances of the closed-loop system are under control. In contrast to most of the current sliding surface design schemes, in this new framework, the level of control effort required to maintain sliding is penalized. The proposed method for the design of optimal-sliding mode control is carried out in two stages. In the first stage, a state-feedback gain is derived using a linear matrix inequality (LMI)-based scheme that can assign a number of the closed-loop eigenvalues to a known value while satisfying performance specifications and ensuring that all the closed-loop poles are located in a preselected subregion. The sliding function matrix related to the particular state feedback derived in the first stage is obtained in the second stage by using one of the two different methods developed for this goal. We present a numerical example to demonstrate the remarkable performance of the proposed scheme.

Keywords

  • optimal H2/H∞-based sliding mode control
  • output tracking
  • partial eigenstructure assignment
  • regional pole placement

1. Introduction

Sliding mode control (SMC) is now a well-developed method of control and its invariance properties against matched uncertainties have inspired researchers to apply this technique to different applications [16]. Traditionally, SMC is designed in two stages. In the first stage, a sliding surface whose sliding motions have suitable dynamics is chosen. Many methods have been proposed in the existing literature for this purpose, for example, eigenstructure assignment, pole placement, optimal quadratic methods, and linear matrix inequality (LMI) methods; see for instance [4, 5, 7, 8]. Then, a controller is designed to induce and maintain the sliding motion.

However, these traditional design methods are unable to limit the available control action required for satisfying the control objective [3], since during the switching-function synthesis, there is no sense of the level of the control action required to persuade and retain sliding [3]. It is worth noting that without having limits on the available control actions, a sliding surface and thereby a control law may always be obtained which is not practically applicable, as it may lead to high level of control efforts.

To tackle this problem, for instance, the authors of [9] propose a scheme to design a sliding surface which minimizes an objective function of the system state and control input, in the meantime. However, since the method in [9] needs to ensure that at least one eigenvalue of the closed-loop system (for single-input systems) is a real value, not necessarily any arbitrary weighting matrices in the objective function may result in a sliding mode control. This reference, therefore, either reselects the weighting matrices or approximates the closed-loop system eigenvalues so that a set of eigenvalues are generated which can be divided into the null-space and range-space dynamics. However, no precise scheme is given on how to reselect the weighting matrices. Further, the approximation of eigenvalues may lead to a loss in optimality and possibly robustness.

For addressing the limitations of [9], Tang and Misawab [10] propose an LQR-like scheme in which a weighting matrix is computed which is closest to the desired one and can result in the desired eigenvalues. Following this, the associated SMC is designed according to the obtained eigenvalues and weighting matrix. Nevertheless, both methods in [9, 10] are suitable to single-input systems. Alternatively, Edwards [3] proposes two new frameworks exploiting two special system coordinate transformations, which are fundamentally different from the aforementioned schemes.

This chapter aims to propose a different way for the sliding surface design while optimizing the control effort associated with the linear part of the control law. This approach is a middle-of-the-road method in that it uses a specific partial eigenstructure assignment method to assign marbitrary stable real eigenvalues while an appropriate sliding motion dynamics will be ensured by enforcing different Lyapunov-type constraints such as the H2/Hand regional pole-placement constraints. The advantages of the proposed approach for the design of sliding surface compared to all the aforementioned references are threefold: (i) it can set the stage for designing SMC while the level of control efforts is taken into account; (ii) it makes it possible to integrate several Lyapunov-type constraints, for example, regional pole-placement constraints, in the SMC design problem; and (iii) the controller can be computed in a numerically very efficient method. The proposed scheme for the design of suboptimal SMC is indeed a two-stage LMI-based approach. In the first stage, while enforcing different Lyapunov-type constraints, for example, the mixed H2/H, a state-feedback gain is derived, using an LMI-based optimization program employing an instrumental matrix variable, which can precisely assign some of the closed-loop eigenvalues to a priori known value. Following this, the sliding surface, associated with the state-feedback gain obtained in the first stage, is determined in the second stage. Two different approaches are presented for deriving the associated switching-function matrix. This chapter indeed examines the problem of designing a state-feedback SMC which utilizes integral action to provide tracking. From the implementation point of view, the simplicity of such a scheme is very advantageous.

The structure of the chapter is as follows. Section 2 is dedicated to the problem statement and preliminaries. Section 3 explains the novel design strategy for the design of H2/H-based SMC. Section 4 discusses two different approaches for deriving the sliding function matrix associated with the linear controller obtained in Section 3. Section 5 summarizes the proposed H2/H-based SMC. In Section 6, we discussed the issue of designing SMC with additional regional pole-placement constraints. Section 7 illustrates this method via an example considering the flight control problem. Section 8 finally concludes the chapter.

2. Problem statement and preliminaries

Consider the following linear-time invariant (LTI) system:

x˜˙(t)=A˜x˜(t)+B˜[u(t)+f(t)],E1

where x˜Rn˜and uRmare the state vector and control input vector, respectively. The matrices in Eq. (1) are constant and of appropriate dimensions. The unknown signal f(t)Rmdenotes matched uncertainty in Eq. (1) whose Euclidean norm is bounded by a known function ρ(t). Without the loss of generality, it is assumed that rank(B˜)=mand the matrix pair (A˜,B˜)are controllable.

In order to provide the problem with a tracking facility, we exploit an integral action as follows. Defining

ξ˙(t)=r(t)y˜(t),E2

where r(t)is the input reference to be tracked by y˜(t)=C˜x˜(t)Rp, and ξrepresents the integral of the tracking error, that is, r(t)y˜(t), and introducing x:=[ξx˜]Rn, an augmented system can be derived as

x˙(t)=Ax(t)+B2u(t)+Brr(t),E3

with

A=[0C˜0A˜],B2=[0B˜],Br=[Ip0].E4

Note that if the matrix pair (A˜,B˜)is controllable and the matrix triplet (A˜,B˜,C˜)has no zeros at the origin, it can be shown that (A,B2)is controllable [11].

Consider a linear switching surface as

S={x:σ(t)Sx(t)=0},E5

where SRm×nis the full-rank-sliding matrix to be designed later so that the associated reduced-order-sliding motions have suitable dynamics.

Let us consider the following controller:

u(t)=(SB2)1(SAΦS)x(t)+ϑ(t),E6

where ΦRm×mis a stable matrix and ϑ(t)Rmdenotes the nonlinear part of the controller with the following form:

ϑ(t)=(SB2)1ρ(t)σ(t)σ(t)ifσ(t)0,E7

in which the scalar function ρ()satisfies ρ(t)SB2f(t); for example, see [2]. It is worth noting that the term (SB2)1ΦSx(t)in the controller Eq. (6) is to govern the convergence rate of the system state to the sliding manifold in association with the nonlinear controller. Further, (SB2)1SAis the so-called equivalent control necessary to maintain sliding in the absence of uncertainty. Here, similar to [3], it is assumed that Φ=λIm, where λ<0is a given constant value. Note that unlike in [3], λcan also belong to the spectrum of A. Because we set Φ=λIm, the control law u(t)in Eq. (6) can be written as

u(t)=(SB2)1SAλx(t)+ϑ(t),E8

where Aλ=λInA. Now assuming that there is no matched uncertainty in Eq. (3) and letting ρ()0, we can consider that the controller in Eq. (8) contains only the linear part. Hence,

x˙(t)=Ax(t)+B2u(t)+Brr(t)+B1w(t)z2(t)=C2x(t)+D2u(t)z(t)=Cx(t)+Du(t),u(t)=(SB2)1SAλx(t),E9

where w(t)is a fictitious exogenous disturbance, z2(t)Rq1and z(t)Rq2are the H2performance output vector and Hperformance output vector of the system, respectively. The matrices in Eq. (9) are constant and of appropriate dimensions. Without the loss of generality, it is also assumed that mqin, i=1,2. Now, the objective can be regarded as finding a sliding matrix Sso that the resulting reduced-order motion, when restricted to S, is stable and meets H2/Hperformance specifications. Indeed, we need to choose S, with a given λ<0, so that the obtained reduced-order-sliding mode

  • guarantees Twz222<δ, where Twz22is the H2norm of the closed-loop transfer function from w(t)to z2(t)and δ>0is a predetermined closed-loop H2performance, and

  • minimizes the Hperformance, subject to the above item.

For this purpose, one may resort to solve a H2/H-state-feedback problem and thereby find the switching matrix associated with the derived optimal state-feedback gain (say F). Broadly speaking, this simple scheme may not necessarily result in any solution, unless the obtained state-feedback gain Fcan ensure that mof the closed-loop poles are exactly located at λ. Hence, in order to design an H2/H-based SMC, we need to address the following two problems:

  1. Blend the mixed H2/Hproblem with the eigenstructure assignment method, that is, design a state-feedback Fenforcing H2/Hconstraints while ensuring that mpoles of the closed-loop system are precisely located at λ.

  2. Obtain the sliding matrix Sassociated with the particular state-feedback F, derived in the previous step.

The abovementioned problems are dealt with in the following two sections.

Remark 1. Note that the linear part of the control law can be considered as

u(t)=[FrF˜][ξ(t)x˜(t)]Fx(t),E10

where F˜∈Rm×n˜ is the state-feedback gain and Fr∈Rm×p is the feed-forward gain due to the reference signal r(t).

The following lemma is recalled from [12], which will be useful in the sequel of this chapter.

Lemma 1 [12]. The following two statements are equivalent:

1.Ψ+S+ST<0.

2. The following LMI is feasible with respect to U.

[Ψ+P(U+UT)ST+UTS+UP]<0,

where P is a positive definite matrix.

It should be noted that Lemma 1 provides a necessary and sufficient condition. However, while imposing some constraints (e.g., structural constraints) on U, the sufficiency of the lemma is not violated; that is, always (2) (1).

3. Partial eigenstructure assignment for the design of H2/H-based SMC

3.1. LMI characterizations

We need to consider the state-feedback synthesis with a combination of H2/Hperformance specifications. In what follows, to avoid the conservatism introduced by the so-called quadratic approach for the design of feedback gains with respect to H2/Hperformance specifications, we need to recall the so-called extended LMI methods developed for the H2and Hcontrol problems from, for example, [12, 13]. This form of LMI characterization will also be shown to be very useful for the novel SMC of this chapter, as it provides us with the possibility to design a certain partial eigenstructure assignment scheme which can ensure precise locations for some of the closed-loop system poles.

3.1.1. H2LMI characterization

The H2control synthesis problem, by assuming the control law as u(t)=Fx(t), can be addressed through the following optimization problem [12]:

Minimizeδsubject to(MH2)

[(G+GT)AG+B2Y+XiXiC2G+D2Y0δIG00Xi]<0,E11
[ZB1Xi]<0,E12
trace(Z)<1,E13

with respect to decision variables Xi, i=1,,N, Z, Y, and G, where Xiand Zare s.p.d matrices. Nhereafter denotes the number of constraints and thus the independent Lyapunov variables. As G+GT>0, Gwill be invertible and the state feedback is obtained as F=YG1.

3.1.2. HLMI characterization

Given scalar 0<ν1, the Hproblem, by assuming the control law as u(t)=Fx(t), can be set as the following minimization problem [13].

Minimizeγsubject to(MHI)

[Xi(G+GT)G+ν(AG+B2Y)XiCG+DY0ν1I0B10γν1I]<0,E14

with respect to decision variables Xi>0, i=1,,N, Y, G, and γ>0. Again, the state feedback is obtained as F=YG1.

Remark 2. It is worth mentioning that the advantage of both LMIs (11) and (14) lies in the fact that the product terms between the matrix A and the Lyapunov matrices Xi disappear which is particularly useful for a wide range of applications such as mixed H2/H∞-feedback gain design and cases where the system matrices belong to a given polytopic region. Besides, as seen from Eqs. (11) and (14), the controller is not dependant on the Lyapunov matrix, but rather the new introduced matrix variable G.

3.1.3. Mixed H2/Hstate feedback using improved LMI characterizations

An interesting application of the mentioned so-called extended LMI characterizations for H2and His the mixed H2/Hstate-feedback problem. The aim is to design-feedback gains such that they

  • ensure the H2performance which means that for a prescribed closed-loop H2performance δ>0, we have Twz222<δ;

  • minimize the Hperformance, subject to the above constraint.

This problem can be formulated through an LMI program in decision variables Xi>0, i=1,,N, Z>0, Y, G, and γ>0:

minimizeγsubject to Eqs(11),(12),(13),and(14),(MHH)

where δ>0and 0<ν1are the given scalars. Notice that another alternative for addressing the mixed H2/Hstate-feedback problem is the so-called quadratic approach (see, e.g., [14]), which is a well-known scheme to address the nonlinearity involved in the matrix inequalities by using a common Lyapunov matrix for all the involved objectives. However, this scheme introduces a significant conservatism to the problem in most of the practical cases. The other alternatives, such as (MHH), which are more computationally expensive, have been basically considered in the literature in order to reduce the conservatism of the quadratic approach.

Remark 3. Another alternative for the mixed control problem is to design a feedback gain that minimizes the H2 norm of one channel while satisfying an H∞-norm constraint on the same or another channel; see, for example, [15]. Hence, in this case, the mixed H2/H∞ problem, given γ>0 and 0<ν≪1, can be set as follows:

minimizeδsubjectto(11),(12),(13)and(14),(MHHN)

where Xi>0, i=1,⋯,N, Z>0, Y, G, and δ>0 are decision variables.

3.2. Partial eigenstructure assignment problem

Locating exactly mpoles at a specific location can fortunately be performed through the LMI characterization presented in the previous section. Our specific partial assignment of the set of eigenvalues

{λ,,λmtimes},E15

by state feedback can be implemented in two steps:

  1. compute the base [MλNλ]of null space of [AλIB2]with conformable partitioning;

  2. with arbitrary ηiRm, i=1,,m, the state feedback will be obtained as F=YG1with

Y=NΣN,G=MΣM,E16

in which

N:=[Nλ,,Nλmtimes,I,,I(nm)times],M:=[Mλ,,Mλmtimes,I,,I(nm)times],ΣN:=diag(η1,,ηm,κ1,,κ(nm)),ΣM:=diag(η1,,ηm,ι1,,ι(nm))E17

with κiRnand ιiRn. Note that only vectors ηkare related to the assignment of the meigenvalues to λ. In other words, other vectors (κkand ιk) are not exploited in the pole-placement purposes and thereby can be employed to meet other Lyapunov-type constraints.

Now, provided by the LMI characterization, for example, (MHH), the first step of our H2/H-based SMC design can be set as an LMI program in the variables Xi>0, i=1,,N, ΣM, ΣN, and γ>0, by recasting (MHH) as:

  minimizeγsubject to (11),(12),(13),(14),and (16).(MHH2)

However, we have not yet shown that the set of closed-loop eigenvalues encompasses Eq. (15). This is the subject of the following lemma.

Lemma 2. Set (15) is a subset of the closed-loop system eigenvalues, acquired by applying the state feedback F=YG−1 with Y and G presented in Eq. (16), to the system in Eq. (3) in the absence of uncertainty, that is, f=0.

Proof. Using Eq. (16), we can write

(A+B2F)Mληi=[A+B2(NΣN)(MΣM)1]Mληi=[A+B2(NΣN)(MΣM)1](MΣM)ei=[A(MΣM)+B2(NΣN)]ei=AMληi+B2Nληi=λMληii=1,,m.

Note that ekhere denotes the canonical basis of Rn.□

4. Deriving the switching-function matrix

This subsection proposes two approaches to obtain the sliding matrix Sassociated with the state feedback F, derived in the previous subsection based on the partial eigenstructure assignment scheme.

4.1. Approach 1

The first approach is represented in the following theorem.

Theorem 1. Let (A,B2) be a controllable matrix pair. Then

  1. λR, there always exists a gain matrix F so that m of the eigenvalues of A+B2F are equivalent to λ, and A+B2F has m-independent eigenvectors associated with λ.

  2. Define S=[v1,⋯,vm]T, where vi is a left eigenvector of A+B2F associated with the eigenvalue λ, then, S(Aλ−B2F)=0 and SB2 is invertible.

Proof. (i) As (A,B2)is controllable, we can claim that (λIA,B2)is also controllable for any λR. Then, it is easy to see that we can always find Fsuch that the null space of AλB2Fhas dimension m, which implies that A+B2Fhas m-independent eigenvectors associated with λ.

(ii) Define S=[v1,,vm]T, it is easy to show that S(AλB2F)=0. Now, assume

SB2:=[v1TvmT]B2=Ω,

where ΩRm×m. If Ωis not full rank, then there exists a nonsingular matrix Λsuch that the first row of ΛΩis zero. This is equivalent to

Λ[v1TvmT]B2:=[v˜1Tv˜mT]B2=ΛΩ,

that is, there exists a vector v˜1such that v˜1TB2=0. On the other hand, we know v˜1T[AλB2F]=0, and so

rank([λI(A+B2F)B2])<n.

This is clearly in contradiction with the controllability of (A,B2). In other words, if we can find a left eigenvector of A+B2Fassociated with λthat is orthogonal to B2, (A,B2)must be uncontrollable, which is obviously a contradiction.

In brief, by virtue of Theorem 1, the switching-function matrix S, associated with the state feedback F, obtained through solving the LMI problem in (MHH2), can be selected as the set of mlinearly independent left eigenvectors of A+B2Fassociated with the (arbitrarily selected) repeated eigenvalue λR.

4.2. Approach 2

An alternative approach to obtain the sliding matrix is to address the equality

(SB2)1SAλ=F,E18

utilizing an LMI optimization method as follows. As the matrix Smust ensure the invertibility of SB2, let us suppose S=B2TP, with Pan s.p.d matrix which will be obtained hereafter. The condition in Eq. (18) can be dealt with a simple relaxation method as

minimizeαsubjecttoB2TP(AλB2F)<α,

where α>0is a scalar variable and Fis a given state-feedback matrix, obtained in the previous subsection, ensuring mof the closed-loop eigenvalues are equal to λ. Simply it can be shown that the above problem is equivalent to the following LMI minimization problem:

minimizeαsubjectto[αIB2TP(AλB2F)αI]<0.E19

Hence, the H2/H-based SMC problem is now to find the global solution of the above minimization problem, and then the sliding matrix is obtained as S=B2TP. In the case of feasibility, this problem will enforce αso that it is an extremely small number associated with the precision of the computational unit.

Notice that this approach is numerically very efficient and attractive compared to the first approach.

5. The summary of H2/H-based SMC design method

Now, we summarize the proposed H2/H-based SMC in the following theorem.

Theorem 2. Assume that the optimization problem in (MHH2) has a solution F for some δ>0 and γ>0. Then, the H2/H∞ performance constraints ‖Twz2‖22<δ and ‖Twz∞‖∞2<γ are ensured, and after the reaching time ts, the resulting reduced n−m-order-sliding mode dynamics, obtained by applying the following control law:

u(t)=Fx(t)+ϑ(t),E20

where ϑ(t) is introduced in Eq. (7), to system (3), is asymptotically stable.

Proof. Consider a change of coordinates xTrx. In this new coordinate system, the new matrix pair (A¯,B¯2)is of the form

A¯=[A11A12A21A22],B¯2=[0Bp]E21

where the square matrix BpRm×mhas full rank and more importantly is nonsingular; see [1]. Suppose also that F¯is the state feedback in the new coordinate that ensures the closed-loop stability, assigns mpoles of the closed-loop system at λ, and satisfies the H2/Hperformance constraints Twz222<δand Twz2<γ. Now, let the switching-function matrix in the original coordinates be parameterized such that [1]

S=S2[MIm]Tr,E22

where S2Rm×m. Notice that theoretically the choice of S2may not influence the sliding motion [1]. According to the discussion given in the previous section, it can be readily shown that there exists a matrix Msuch that F¯=(S¯B¯2)1S¯(A¯λIn), where S¯=S2[MIm]denotes the switching-function matrix in the new coordinate. Let (x¯1,x¯2)be the partition of the system states associated with the certain system coordinates in Eq. (21), then it can be shown that while the system states are confined to the sliding manifold, that is, σ=0, the reduced-order-sliding mode dynamics are governed by the stable reduced-order system matrix A11+A12M.

Moreover, the dynamics of σcan be derived by taking the time derivative of Eq. (5), substituting in the state equation (3), and using controllers (8) and (7), that is,

σ˙(t)=λσ(t)ρ(t)σ(t)σ(t)+SB2f(t).E23

Finally, it follows from SB2f(t)ρ(t)that the reachability condition σTσ˙σ<0holds.□

6. Design of SMC with additional regional pole-placement constraints

Note that the proposed method here offers the advantage of introducing additional convex constraints on the closed-loop dynamics. By locating the closed-loop system poles in a preselected region, an adequate transient response for system trajectories can be guaranteed [14]. Therefore, the objective is to augment the optimization problems previously described by pole-clustering constraints. Note also that as it is already ensured that mof the closed-loop eigenvalues are exactly assigned to a given negative real value (λ), the remaining eigenvalues in fact belong to the spectrum of the reduced nm-order-sliding motion. As a result, a satisfactory transient response for the sliding motion can be achieved by clustering the poles governing the sliding motion.

Let us have a brief introduction to the LMI region. Simply, an LMI region is a subset Dof the complex plane as

D:={zC:fD(z)Ξ+zΠ+z¯ΠT<0}E24

in which Ξ=ΞTRξ×ξand ΠRξ×ξare real matrices. fD(z)is also called the characteristic equation of the region D.

Definition 1 [16]. A real matrix A is said to be D-stable if all its eigenvalues lie within the LMI region D.

Lemma 3 [16]. A real matrix A is said to be D-stable if a symmetric matrix XD>0 exists so that

ΞXD+Π(XDA)+ΠT(ATXD)<0,E25

where ⊗ denotes the Kronecker product.

However, the synthesis problem obtained by imposing the pole-clustering constraints presented in, for example, [14] or [16] to the synthesis problem in (MHH2) would not result in a convex problem. Alternatively, the regional pole-clustering constraints can be reformulated so that the product term between the Lyapunov matrix Xiand the system matrix Ais removed.

An instrumental theorem is represented first and the main theorem will be presented later in Theorem 4.

Theorem 3. Let A be a real matrix. The following statements are equivalent, with s.p.d X, G and given real matrices 0<Ξ∈Rξ×ξ and Π∈Rξ×ξ.

  1. Ais D-stable, where D is given in Eq. (24).

  2. Xsuch that

    [ΞX+Π(XA)+ΠT(XA)T0X]<0.

  3. X>0and G such that

    [(G+GT)(ΠA)G+IξXIξXG0IξXG00Ξ1X]<0.E26

Proof. Refer to the Appendix.□

While Eq. (26) can be seen as a necessary and sufficient condition for D-stability, it is not very useful in terms of control synthesis purposes. Further, since Ξ=0, the result of Theorem 3 cannot cover the standard continuous-time systems stability. However, if we let G=IξGin Eq. (26), a sufficient condition is achieved which is beneficial for the control synthesis purposes.

Theorem 4. Let A, 0≤Ξ∈Rξ×ξ, and Π∈Rξ×ξ be real matrices. A is D-stable if

[Iξ(G+GT)Π(AG)+IξXIξXIξG0IξXΞ12G00IξX]<0,E27

where G is a general matrix and X is an s.p.d matrix.

Proof. The proof can be performed similar to the proof of Theorem 3 by letting G=IξG.□

Clearly, the above theorem does not require Ξ>0, but Ξ0. This is indeed a generalization of the extended Lyapunov theorem presented in Theorem 3.1 of [12], and the usual stability region can be obtained by letting Ξ=0and Π=1in Eq. (27):

[(G+GT)AG+XXG0X]<0.E28

Moreover, the equivalence of Eq. (28) to the standard Lyapunov stability inequality for continuous-time linear systems is presented in [12].

Specifically, let us confine the closed-loop poles to the region S(α,r,θ)(see [14]) which can ensure a minimum decay rate α, a minimum damping ratio ζ=cosθ, and a maximum undamped natural frequency ωd=rsinθ. The LMI region for an αstability, that is, Re(z)<α, can be obtained through Eq. (27), with Ξ=2α, Π=1, AA+B2F, and XXi. Moreover, by letting Ξ=0and Π=[sinθcosθcosθsinθ], the LMI region for a conic sector S(0,0,θ)is achieved. Eventually, a disk centered at the origin with radius rcorresponds to

Ξ=[r00r],Π=[0100].E29

However, for this special pole-clustering constraint, as Ξis not a semi-positive definite matrix, the LMI region cannot be obtained through Eq. (27). We can alternatively state the following theorem.

Theorem 5. Let A be a real matrix. The following conditions are equivalent:

  1. The eigenvalues of A lie in a disk centered at the origin with radius r.

  2. There exists a symmetric matrix X>0 such that

    1rAXATrX<0.E30

  3. There exists a symmetric matrix X>0 such that

    [rXXATrX]<0.E31

  4. There exist a symmetric matrix X>0, and a matrix G such that

    [rXGTAT(G+GT)+1rX]<0.E32

Proof. Refer to the Appendix.□

Notice that the above theorem with r=1reduces to the standard Lyapunov stability inequality for discrete-time linear systems and its extended (robust) version; for example, see [17]. Now, the extended LMI region for a disk centered at the origin with radius ris as follows:

[rXi(AG+B2Y)T(G+GT)+1rXi]<0,E33

which is obtained by replacing A:=A+B2F, X:=Xiin Eq. (32) and introducing Y=FG.

Remark 4. Exploiting a common G may also lead to conservatism compared with the methods, for example, in [18]. However, the methods in the aforementioned references are not beneficial for the control synthesis aims, unless gain-scheduled controllers [19] are considered. Moreover, by employing two instrumental variables, a different sufficient condition for robust D stability has been developed in [20] which is not applicable to the continuous-time control synthesis purposes. Nevertheless, the approach here can achieve less conservative results through employing non-common Lyapunov variables for every involved specification.

7. Numerical examples

This section evaluates the effectiveness of the proposed theory using a numerical example. Consider a two-input, two-output, fourth-order plant describing the motion of a Boeing B-747 aircraft obtained by linearization around an operating condition of 20,000 ft. altitude with a speed of Mach 0.8 [21]. The system matrices are as follows:

A˜=[0.11960.00041.00010.03834.11950.97430.29190.00041.62040.01610.23200.00010.00071.00540.00030.0003],B˜=[0.00040.01260.31030.18320.01240.92190.00010.0002],C˜=[10000001],

and the system state, output, and input vectors are

x˜(t)=[β(t)p(t)r(t)φ(t)]T,y˜(t)=[β(t)φ(t)]T,u(t)=[δa(t)δr(t)]T.

where β(t),p(t),r(t),φ(t),δa(t), and δr(t)denote the sideslip angle, the roll rate, the yaw rate, the roll angle, the aileron deflection, and the rudder deflection, respectively.

We also let

C2=[diag(0.1,0.1,10,10,1,1)02×6],D2=[06×2diag(1,1)],B1=I6.

Note that the last two nonzero terms of C2are associated with the integral action and are less heavily weighted. In addition, the third and fourth terms of C2have strongly been weighted in comparison with the fifth and sixth terms to provide an adequate quick closed-loop response in terms of the angular acceleration in roll and yaw. We also aim to assign the closed-loop poles in the half-plane x<α<0.1.

We solve the minimization problem in (MH2), with λ=3, and the state-feedback gain is obtained as

F=[0.716668.723716.044619.66164.059162.205034.597824.609723.09760.77777.43875.7251].E34

Employing the first proposed approach in Section 4, the associated sliding function matrix for the augmented system is

S=[0.73510.60910.29070.01510.06210.00660.42780.63810.23550.13090.06380.5772].E35

The sliding motion is governed by the set of poles {2.3205±3.0365i,1.9203±1.2377i}, and the associated true value of H2cost from wto zis 28.0959. Assuming the matched uncertainty term in Eq. (1) as f(t)=[0.2sin(t)β(t)0.3sin(t)φ(t)], using the proposed SMC with the obtained linear gain Fin Eq. (34) and the associated switching-function matrix Fin Eq. (35), and letting the switching gain ρ=1, and considering a step of 5° for βduring 30–40 sas well as a step of 2° for ϕduring 5–15 s, Figures 13 show the tracking responses of the system. Note that the discontinuity in the nonlinear control term ϑ(t)in Eq. (7) is smoothed by using a sigmoidal approximation [11] as

ϑε(t)=(SB2)1ρ(t)σ(t)ε+σ(t)E36

Figure 1.

System states.

Figure 2.

Control efforts.

Figure 3.

Switching function.

with the scalar ε=0.01and ρ(t)=1, where this can remove the discontinuity at σ=0and introduce the possibility to accommodate the actuator rate limits.

8. Conclusions

The focus of this chapter was on the development of novel framework for designing a sliding surface for a given system while enforcing a number of Lyapunov-type constraints such as the H2/Hand/or regional pole clustering. We specifically considered the problem of output tracking using a suboptimal state-feedback SMC. In doing so, in the first stage, through a convex optimization approach, a state-feedback gain is designed while assigning a certain number (m) of the closed-loop system eigenvalues to a predetermined value, as well as satisfying H2/H-norm constraints. The advantages of the proposed scheme are threefold: (i) it can set the stage for designing SMC while the level of control efforts is taken into account; (ii) it makes it possible to integrate a number of Lyapunov-type constraints, for example, regional pole-placement constraints, into the SMC design problem; and (iii) the controller can be computed in a numerically very efficient method. The achieved results confirmed the effectiveness of the proposed approach.

A. Proof of Theorem 3

Notice that the equivalence between Eqs. (1) and (2) can be obtained from Lemma 3. We will show the equivalence between Eqs. (2) and (3) here. The use of Lemma 1 with Ψ=ΞX¯, U=G¯and S=Π(X¯A), with X¯=X1, yields

[P(G¯+G¯T)+ΞX¯Π(X¯A)+G¯P]<0,

or equivalently,

[P(G¯+G¯T)+ΞX¯ΠA+(IξX)G¯(IξX)P(IξX)]<0,E37

By performing the congruence transformation [G00I], with G=G¯1, and using the Schur complement, Eq. (37) becomes

[(G+GT)(ΠA)G+IξX(IξX)P(IξX)G0P1G00Ξ1X]<0.

The above inequality finally linearizes to Eq. (26) with the choice P=IξX1.

B. Proof of Theorem 5

The equivalence between Eqs. (1) and (3) is shown in, for example, [14]. Moreover, the equivalence between Eqs. (2) and (3) is simply obtained through applying the Schur complement with respect to the block (2,2) in Eq. (31). The proof can be followed by noticing that if one applies the Schur complement with respect to the block (1,1) in Eq. (32), Eq. (30) is recovered by choosing G=GT=1rX>0, hence Eq. (2) implies Eq. (4). Also, by left and right multiplying Eq. (32) by [IA]and [IA]T, respectively, one can achieve Eq. (30). Hence, Eq. (4) implies Eq. (2), and the proof is completed.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ahmadreza Argha and Steven W. Su (June 28th 2017). State-Feedback Output Tracking Via a Novel Optimal-Sliding Mode Control, Recent Developments in Sliding Mode Control Theory and Applications, Andrzej Bartoszewicz, IntechOpen, DOI: 10.5772/67468. Available from:

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