.

Title

SpaceMETA - Global Estabilization of Non-Linear Systems with Output Time Delay

Output-feedback sliding-mode control via cascade observers for global stabilisation of a class of nonlinear systems with output time delay

Camila Lobo CoutinhoTiago Roux Oliveira  ( SpaceMETA www.spacemeta.com.br advisor ) & José Paulo V.S. Cunha

International Journal of Control

http://tandfonline.com/doi/full/10.1080/00207179.2014.913200

Pages 2327-2337 | Received 26 Oct 2012, Accepted 05 Apr 2014, Accepted author version posted online: 14 Apr 2014, Published online: 13 May 2014

In this article

Abstract

This article proposes a sliding-mode control scheme for a class of triangular nonlinear systems with arbitrarily long and known time delay in the output signal. The proposed control strategy guarantees global asymptotic stability of the closed-loop system using only output feedback, without using any kind of approximations. The state of the system is estimated by asymptotic observers connected in cascade. The analysis of such observers in closed-loop feedback is also a contribution of the present manuscript as well as the theoretical demonstration of the chattering elimination even in the presence of delays. Simulation results and a physically motivated example with a full-bridge power converter illustrate the effectiveness of the proposed approach.

Keywords: sliding-mode controloutput-feedbacktime-delay systemnonlinear systemglobal stabilisationhigh-gain observer

1. Introduction

Time delays may be significant in networked controlled systems, communication, teleoperation and cooperative robotics, to name a few examples. The presence of time delay in control systems may cause instability and poor performance. Thus, there is an increasing interest in studying systems with time-delayed signals (Richard, 2003)Richard, J.P. (2003). Time-delay systems: An overview of some recent advances and open problems. Automatica, 39,1667–1694.[CrossRef][Web of Science ®].

Control schemes for stable linear systems with known and constant time delay can be designed by using the Smith predictor (Smith, 1957)Smith, O.J.M. (1957).Closer control of loops with dead time. Chemical Engineering Progress, 53,217–219.. However, for many nonlinear systems with time delay, conventional solutions such as Smith predictor cannot be used or become quite involved.

Sliding-mode control (SMC) is an attractive methodology for nonlinear systems, being robust to parameter uncertainties and disturbances (Edwards & Spurgeon,1998Edwards, C., & Spurgeon,S.K. (1998). Sliding mode control: Theory and applications. London: Taylor & Francis.; Utkin, Guldner, & Shi, 1999Utkin, V.Guldner, J., & Shi,J. (1999). Sliding mode control in electromechanical systems.London: Taylor & Francis.). SMC is also a natural choice for systems with switched actuators, such as power-electronic converters (Solé & Colet,2004Solé, D.B., & Colet, E.F.(2004). SMC applications in power electronics. In A.SabanovicL. Fridman, &S. Spurgeon (Eds.),Variable structure systems: From principles to implementation (Vol. 66, pp. 265–293). London: The IEE.[CrossRef]) and motor drives (Utkin et al., 1999Utkin, V.Guldner, J., & Shi,J. (1999). Sliding mode control in electromechanical systems.London: Taylor & Francis.). On the other hand, the presence of time delay deteriorates the control performance, since it causes chattering and may even destabilise the system. Despite of this, sliding-mode controllers for systems with state delays were proposed by Li and DeCarlo (2003)Li, X., & DeCarlo, R.A. (2003). Robust sliding mode control of uncertain time delay systems.International Journal of Control, 76, 1296–1305.[Taylor & Francis Online],[Web of Science ®], Orlov, Perruquetti, and Richard (2003)Orlov, Y.Perruquetti, W., & Richard, J.P. (2003).Sliding mode control synthesis of uncertain time-delay systems. Asian Journal of Control, 5,568–577.[CrossRef][Web of Science ®], Gouaisbaut, Blanco, and Richard (2004)Gouaisbaut, F.Blanco, Y., & Richard, J.P. (2004).Robust sliding mode control of non-linear systems with delay: A design via polytopic formulation. International Journal of Control, 77,206–215.[Taylor & Francis Online],[Web of Science ®] and Basin, Rodriguez-Gonzalez, and Fridman (2007)Basin, M., Rodriguez-Gonzalez, J., & Fridman, L.(2007). Optimal and robust control for linear state-delay systems.Journal of the Franklin Institute, 344, 830–845.[CrossRef][Web of Science ®] assuming full-state feedback. The use of state observers is an alternative for output-feedback stabilisation of systems with state delay, as developed by Niu, Lam, Wang, and Ho (2004)Niu, Y.Lam, J.Wang, X., &Ho, D.W.C. (2004).Observer-based sliding mode control for nonlinear state-delayed systems. International Journal of Systems Science,35, 139–150.[Taylor & Francis Online],[Web of Science ®] and Yan, Spurgeon, and Edwards (2010)Yan, X.G.Spurgeon, S.K., & Edwards, C. (2010).Sliding mode control for time-varying delayed systems based on a reduced-order observer.Automatica, 46,1354–1362.[CrossRef][Web of Science ®]. However, such observers may not be applied to a wide class of systems. For instance, in (Niu et al., 2004)Niu, Y.Lam, J.Wang, X., &Ho, D.W.C. (2004).Observer-based sliding mode control for nonlinear state-delayed systems. International Journal of Systems Science,35, 139–150.[Taylor & Francis Online],[Web of Science ®], unmatched disturbances are not considered, and in (Yan et al., 2010)Yan, X.G.Spurgeon, S.K., & Edwards, C. (2010).Sliding mode control for time-varying delayed systems based on a reduced-order observer.Automatica, 46,1354–1362.[CrossRef][Web of Science ®], the delay must be known. On the other hand, Han, Fridman, and Spurgeon (2010)Han, X.Fridman, E., &Spurgeon, S.K. (2010).Sliding-mode control of uncertain systems in the presence of unmatched disturbances with applications. International Journal of Control, 83,2413–2426.[Taylor & Francis Online],[Web of Science ®] proposed a static output-feedback SMC for MIMO (multi-input multi-output) linear systems with unmatched disturbances and uncertain state delays. Coutinho, Oliveira, and Cunha (2013)Coutinho, C.L.Oliveira,T.R., & Cunha, J.P.V.S. (2013). Output-feedback sliding-mode control of multivariable systems with uncertain time-varying state delays and unmatched non-linearities. IET Control Theory and Applications, 7,1616–1623.[CrossRef][Web of Science ®] developed an output-feedback SMC for MIMO uncertain nonlinear time-delay systems, using norm observers to estimate the unmeasured state vector since they are more robust to strong uncertainties than state observers (Cunha, Costa, & Hsu, 2008Cunha, J.P.V.S.Costa, R.R., & Hsu, L. (2008). Design of first-order approximation filters for sliding-mode control of uncertain systems. IEEE Transactions on Industrial Electronics, 55,4037–4046.[CrossRef][Web of Science ®]; Oliveira, Peixoto, & Hsu, 2010Oliveira, T.R.Peixoto, A.J., & Hsu, L. (2010). Sliding mode control of uncertain multivariable nonlinear systems with unknown control direction via switching and monitoring function. IEEE Transactions on Automatic Control, 55,1028–1034.[CrossRef][Web of Science ®]). Global tracking was achieved but only state delays were allowed.

The application of observers to SMC was originally proposed by Bondarev, Bondarev, Kostyleva, and Utkin (1985)Bondarev, A.G.Bondarev,S.A.Kostyleva, N.E., &Utkin, V.I. (1985). Sliding modes in systems with asymptotic state observers. Automation and Remote Control, 46,679–684, Pt. 1.[Web of Science ®] for linear systems with known parameters. Bondarev et al. (1985)Bondarev, A.G.Bondarev,S.A.Kostyleva, N.E., &Utkin, V.I. (1985). Sliding modes in systems with asymptotic state observers. Automation and Remote Control, 46,679–684, Pt. 1.[Web of Science ®] have shown the capability of the observer to avoid chattering caused by small time lags due to unmodelled dynamics in the measurement system. Output-feedback stabilisation of uncertain nonlinear systems based on SMC and high-gain observers (HGOs) was proposed by Emelyanov, Korovin, Nersisian, and Nisenzon (1992)Emelyanov, S.V.Korovin,S.K.Nersisian, A.L., &Nisenzon, Y.Y. (1992).Output feedback stabilization of uncertain plants: A variable structure systems approach. International Journal of Control, 55,61–81.[Taylor & Francis Online],[Web of Science ®] and Esfandiari and Khalil(1992)Esfandiari, F., & Khalil, H.K. (1992). Output feedback stabilization of fully linearizable systems.International Journal of Control, 56, 1007–1037.[Taylor & Francis Online],[Web of Science ®] for systems without delays. However, delayed output signals may impair the convergence of the estimated state to the true state and, consequently, the control may become unstable.

Surprisingly, few results are available for SMC of systems with input- or output-delayed signals. Basin, Fridman, Rodríguez-González, and Acosta (2003)Basin, M.Fridman, L.,Rodríguez-González, J., &Acosta, P. (2003). Optimal and robust sliding mode control for linear systems with multiple time delays in control input. Asian Journal of Control, 5,557–567.[CrossRef][Web of Science ®] and Feng, Yu, and Zheng (2006)Feng, Y.Yu, X., & Zheng, X. (2006). Second-order terminal sliding mode control of input-delay systems. Asian Journal of Control, 8, 12–20.[CrossRef][Web of Science ®] proposed state-feedback SMC for systems with delayed input signals as well as input signals free of delay, which facilitate the control. Basin, Rodriguez-Gonzalez, Fridman, and Acosta (2005)Basin, M.Rodriguez-Gonzalez, J.Fridman, L., &Acosta, P. (2005). Integral sliding mode design for robust filtering and control of linear stochastic time-delay systems.International Journal of Robust and Nonlinear Control, 15, 407–421.[CrossRef][Web of Science ®] presented an integral sliding-mode technique for robustifying an optimal controller for linear stochastic systems with input and observation delays, which is based on state feedback. For systems approximated by linear first-order time-delay models, Camacho, Rojas, and García-Gabín (2007)Camacho, O.Rojas, R., &García-Gabín, W. (2007).Some long time delay sliding mode control approaches. ISA Transactions, 46, 95–101.[CrossRef][PubMed],[Web of Science ®] applied SMC to improve the performance and robustness of Smith predictor-based schemes. Liu, Zinober, and Shtessel (2009)Liu, G.Zinober, A., &Shtessel, Y.B. (2009).Second-order SM approach to SISO time-delay system output tracking. IEEE Transactions on Industrial Electronics, 56,3638–3645.[CrossRef][Web of Science ®] applied Padé approximations to transform the SMC of systems with delayed output signal into the problem of controlling nonminimum phase systems. However, it is known that such approximations may be unrealistic for long delays. Furthermore, only local stability could be guaranteed for known parameters and known time delay.

Sun, Xie, and Liu (2013)Sun, Z.Y.Xie, X.J., & Liu,Z.G. (2013). Global stabilisation of high-order nonlinear systems with multiple time delays.International Journal of Control, 86, 768–778.[Taylor & Francis Online],[Web of Science ®] and Sun, Zhang, and Xie (2013)Sun, Z.Y.Zhang, X.H., &Xie, X.J. (2013). Continuous global stabilisation of high-order time-delay nonlinear systems.International Journal of Control, 86, 994–1007.[Taylor & Francis Online],[Web of Science ®] assume full-state measurement to globally stabilise nonlinear time-delay systems, showing how difficult it is to deal with such control problem by using only output feedback. Some recent works on observer design for time-delay systems by Ghanes, De Leon, and Barbot (2013)Ghanes, M.De Leon, J., &Barbot, J. (2013). Observer design for nonlinear systems under unknown time-varying delays. IEEE Transactions on Automatic Control, 58, 1529–1534.[CrossRef][Web of Science ®] and Yan, Spurgeon, and Edwards (2013)Yan, X.G.Spurgeon, S.K., & Edwards, C. (2013). State and parameter estimation for nonlinear delay systems using sliding mode techniques. IEEE Transactions on Automatic Control, 58, 1023–1029.[CrossRef][Web of Science ®] do not consider delays in the output signal. An innovative approach for the state estimation in systems with delayed output signals is the use of cascade observers (Ahmed-Ali, Cherrier, & Lamnabhi-Lagarrigue, 2012Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®]; Germani, Manes, & Pepe,2002Germani, A.Manes, C., &Pepe, P. (2002). A new approach to state observation of nonlinear systems with delayed output. IEEE Transactions on Automatic Control, 47,96–101.[CrossRef][Web of Science ®]), which have not been applied in SMC yet.

This article proposes an output-feedback SMC for triangular nonlinear systems with output delay based on a cascade of HGOs (Ahmed-Ali et al., 2012)Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®] to estimate the system state. The delay can be arbitrary provided that it is known and constant. Moreover, global asymptotic stability of the closed-loop system is guaranteed. It is important to stress that the scheme allows ideal sliding mode, even in the presence of delays. Indeed, the estimated sliding variable provided by the cascade observers becomes null after some finite time, therefore, avoiding chattering phenomena.

The results reported here are relevant since output-feedback controllers for output-delayed systems generally demand delay compensation (Krstic, 2009)Krstic, M. (2009). Delay compensation for nonlinear, adaptive, and PDE systems. Boston, MA: Birkhäuser.[CrossRef]based on prediction, which requires a good knowledge of the system. Signal prediction based on delayed measurements is more difficult to achieve than roughly estimate the magnitudes of signals (current or delayed) needed in SMC of delayed-state systems (Coutinho et al., 2013)Coutinho, C.L.Oliveira,T.R., & Cunha, J.P.V.S. (2013). Output-feedback sliding-mode control of multivariable systems with uncertain time-varying state delays and unmatched non-linearities. IET Control Theory and Applications, 7,1616–1623.[CrossRef][Web of Science ®]. As far as we know, our control scheme with global stability properties is new in the context of SMC for nonlinear systems with output delay, without using any kind of approximation.

1.1 Notation and terminology

The following notation and basic concepts are considered. (1) The maximum and minimum eigenvalues of a symmetric matrix P are denoted by λmax (P) and λmin (P), respectively. (2) The Euclidean norm of a vector x and the corresponding induced norm of a matrix A are denoted by ‖x‖ and ‖A‖, respectively. (3) The definition of Filippov (1964)Filippov, A.F. (1964).Differential equations with discontinuous right-hand side. American Mathematical Society Translations, 42, 199–231. for the solution of discontinuous differential equations is adopted. (4) As is usual in the time-delay literature (Gu, Kharitonov, & Chen,2003Gu, K.Kharitonov, V.L., &Chen, J. (2003). Stability of time-delay systems. Boston, MA: Birkhäuser.[CrossRef], Section 1.2), the initial conditions of a system with state

 are given by z(t) = z0(t), t ∈ [−d, 0], where z0(t) is a vector function continuous in t ∈ [−d, 0] and d is the time delay. (5) A vector signal π(t) is an exponentially decaying term dependent on the initial conditions z0(t), if ∃k, λ > 0 such that ‖π(t)‖ ⩽ ke− λtz*0, ∀t ≥ 0, where z*0 = max {‖z0(t)‖: t ∈ [ − d, 0]}. (6) A vector signal π(t) is an asymptotically decaying term dependent on the initial conditions z0(t), if 

 such that ‖π(t)‖ ⩽ βz(z*0, t), ∀t ≥ 0, where the class of 

 functions is defined by Khalil (2002Khalil, H.K. (2002).Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall., p. 144).

2. Control problem formulation

The control objective is to globally stabilise nonlinear systems with delayed output signal of the following class (Ahmed-Ali et al., 2012)Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®]:

(1)

(2)

where

(3)

(4)

The time delay d > 0 is known and constant. The state vector

 is unmeasurable. The measured output 

 is a linear function of the state x at time t − d

 is the control input and 

 are nonlinear smooth functions with i ∈ {1, … , n}.

According to Ahmed-Ali et al. (2012)Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®], the following assumptions should be satisfied to develop a state observer:

(A1)

The functions φi(xut) are triangular in x:

(5)

where x = [x1x2…xkxn]T.

(A2)

The functions φi(xut) are globally Lipschitz with respect to x, uniformly in u and t, i.e., there exists a constant β > 0 such that

, the inequalities

(6)

hold ∀i ∈ {1, … , n}.

To develop the control law, it is further assumed:

(A3)

The knowledge of a global diffeomorphism (Khalil, 2002Khalil, H.K. (2002).Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall., pp. 515–516)

(7)

which transforms the system (1) and (2) into the normal form:

(8)

(9)

(10)

where 

 and 

 are unmeasured state vectors. The triple {AcBcCc} is a canonical form representation of a chain of l integrators (Marino & Tomei, 1995Marino, R., & Tomei, P.(1995). Nonlinear control design: Geometric, adaptive and robust. London: Prentice Hall International.). The matrices Ac and Cc have the same structure of A and C in Equation (3), and

(11)

(A4)

The subsystem (8) is input-to-state-stable (ISS) (Khalil, 2002Khalil, H.K. (2002).Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall., p. 175) with respect to its input signal ξ.

(A5)

A constant kw ≥ 0 and a piecewise-continuous uniformly bounded scalar function ϕw(t) ≥ 0 are known such that the upper bound,

(12)

is valid 

.

(A6)

The sign of kp(t) is known and constant, and

, ∀t, where 

 is a known constant lower bound for |kp(t)|.

Assumption (A1) is needed to keep constant the relative degree of the system.

The diffeomorphism in assumption (A3) explicitly shows that the system (1) may present matched (wφ) and unmatched (f0) nonlinear terms. In particular, the unmatched nonlinearities in Equation (1) can be represented by a stable zero dynamics (8) according to assumption (A4), i.e., the system is minimum phase. As a consequence of assumption (A2), the matched nonlinearity must satisfy the linear growth condition with respect to ‖x‖ assumed in assumption (A5).

Note that the nonlinear terms in Equations (1), (8) and (9) are not uniformly bounded. In practice, systems with unbounded physical variables can be coped with the proposed control scheme, such as velocity or currents in motor drives. Indeed, the globally Lipschitz assumption (A2) is a boundedness condition for derivatives, such as acceleration and current rate, which are bounded by natural restrictions in industrial applications. Moreover, this class of plants includes many applications, e.g., bioreactors, communication networks, teleoperation and power electronics.

3. Design of high-gain observers connected in cascade

In this section, the cascaded HGOs proposed by Ahmed-Ali et al. (2012)Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®] are briefly reviewed. These observers will be used to estimate the state x of the nonlinear subsystem (1) with delayed output signal (2).

The cascade connection of observers is an approach to deal with larger delays (Ahmed-Ali et al., 2012Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®]; Germani et al., 2002Germani, A.Manes, C., &Pepe, P. (2002). A new approach to state observation of nonlinear systems with delayed output. IEEE Transactions on Automatic Control, 47,96–101.[CrossRef][Web of Science ®]) using the idea of delay distribution (Michiels & Niculescu, 2007Michiels, W., & Niculescu,S.I. (2007). Stability and stabilization of time-delay systems: An Eigenvalue-based approach.Philadelphia, PA: SIAM.[CrossRef], Remark 7.8). The number of observers (m) is proportional to the time delay d and should satisfy (Ahmed-Ali et al.,2012)Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®]

(13)

where d1 > 0 is the maximum delay admitted by a single-stage HGO.

Each observer (j = 1, … , m) estimates a delayed state vector

(14)

where the delay is equally distributed among all observers (d/m). Consider the following notation to represent the delayed control signal:

(15)

Therefore, the observers connected in cascade are given by

(16)

where the vector 

 is chosen such that the matrix A − KC is Hurwitz and

(17)

with θ ≥ 1 being a design constant which sets the feedback gain of the observer output errors.

Remark 1:

The vector

 is an estimate of the delayed state 

. The state vector 

 is an estimate of the current state x(t) of the time-delay system (1)–(4).

After showing the convergence of the state estimated by a single-stage observer for a small delay d1, Ahmed-Ali et al. (2012)Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®] had also proved by using a Lyapunov–Krasovskii functional (Fridman, 2001)Fridman, E. (2001). New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems.Systems & Control Letters,43, 309–319.[CrossRef][Web of Science ®] that a sufficient number of cascade HGOs can estimate the state of the system (1) and (2), for arbitrarily long and constant time delay. This is stated in (Ahmed-Ali et al., 2012Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®], Theorem 1), rewritten below.

Lemma 3.1:

Consider the system described in Equations (1)–(4). Then, for any constant and known delay d, there exist a sufficiently large positive constant θ and an integer msuch that the state estimated by the last observer

 in Equation (16) converges exponentially towards the state x of the system (1)–(4). Moreover, the state estimation errors, 

, ∀j ∈ {1, … , m}, converge exponentially to zero.

4. Parametrisation for controller design

To design the output-feedback SMC, consider the system given in Equations (9)–(10). Then, substituting u by u + Kmξ/kp(t) − Kmξ/kp(t), one has

(18)

(19)

where

(20)

The matrix Am is Hurwitz and Km is chosen as

(21)

where a0, … , al − 1 are coefficients of the characteristic polynomial p(s) of the matrix Am, i.e.,

(22)

The state equation (18) can be rewritten as

(23)

where the equivalent input disturbance is given by

(24)

5. Output-feedback sliding-mode control

If the state vector ξ was available for feedback, one could choose σ = Sξ = 0 as the ideal sliding surface. On the other hand, since only the system output signal y is available for feedback, the sliding surface can be chosen as (Cunha, Costa, Lizarralde, & Hsu, 2009Cunha, J.P.V.S.Costa, R.R.,Lizarralde, F., & Hsu, L. (2009). Peaking free variable structure control of uncertain linear systems based on a high-gain observer. Automatica,45, 1156–1164.[CrossRef][Web of Science ®])

(25)

where b0, … , bl − 1 > 0 are coefficients of the Hurwitz polynomial

(26)

such that the transfer function pS(s)/p(s) = S(sI − Am)−1Bc is strictly positive real (SPR) (Khalil, 2002Khalil, H.K. (2002).Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall., Section 6.3). The signal 

 (see Equation (7)) is an estimate of ξ, with 

 being the estimate of the system state via observer (16).

The proposed control law u is as follows:

(27)

where the modulation function 

 is a scalar function absolutely continuous in 

, piecewise continuous and bounded in t for each fixed 

. It will be shown that, if the following inequality is verified:

(28)

where πϱ(t) is an exponentially decaying term dependent on the initial conditions, then global stabilisation can be guaranteed. The parameter δ ≥ 0 is an arbitrary constant. For instance, a function which satisfies Equation (28) is

(29)

The sliding-mode controller based on HGOs connected in cascade for nonlinear time-delay systems is described by the block diagram in Figure 1.

Figure 1. Block diagram of sliding-mode controller based on high-gain observers connected in cascade for nonlinear time-delay systems. The synthesis of the modulation signal ϱ is not shown to avoid clutter.

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6. Stability analysis

The following lemma shows that the system (23) and (24) with state vector ξ and output signal

 is output-to-state-stable (OSS) (Sontag & Wang, 1997Sontag, E.D., & Wang, Y.(1997). Output-to-state stability and detectability of nonlinear systems.Systems & Control Letters,29, 279–290.[CrossRef][Web of Science ®]), where the partial state estimation error is defined as

(30)

Therefore, according to the lemma, if

, then ‖ξ(t)‖ → 0 and global stability will be proved in the sequence as in (Oliveira, Peixoto, and Hsu, 2013)Oliveira, T.R.Peixoto, A.J., & Hsu, L. (2013). Peaking free output-feedback exact tracking of uncertain nonlinear systems via dwell-time and norm observers. International Journal of Robust and Nonlinear Control, 23,483–513.[CrossRef][Web of Science ®], which only considers systems without delays.

Lemma 6.1 (OSS property from 

 to ξ):

Consider the dynamic system (23) and (24) with state vector ξ, output signal

 and the control law u proposed in Equation (27) with modulation function ϱ given by Equation (29). Then, Equations (23) and (24) are OSS with respect to 

 and ∃ke > 0 such that the following inequality holds:

(31)

where πe(t) ≥ 0 is an asymptotically decaying term dependent on the initial conditions.

Proof:

In the following, ki denote appropriate positive constants and πi denote positive asymptotically decaying terms which depend on the initial conditions of the closed-loop system.

Introducing the transformation,

(32)

and the auxiliary signal σ = Sξ as in Equation (25), such that the system (23) can be represented in the regular form (Khalil, 2002Khalil, H.K. (2002).Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall., p. 564) with state

Since the triple {AmBcS} (see Equations (25) and (26)) is SPR, one can conclude that Equation (23) is OSS from the output σ to the state

. Therefore, 

 satisfies 

. Since from Equation (32), 

, one has

(33)

In what follows, two cases will be considered:

 or 

.
(1)

In the first case,

 implies 

. From the Kalman–Yakubovich Lemma (Khalil, 2002Khalil, H.K. (2002).Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall., Section 6.3), there exist P = PT > 0 and Q = QT > 0 matrices which satisfy ATmP + PAm = −Q and BTcP = S. Using a Lyapunov function candidate V = ξTPξ, one concludes that the time derivative of Valong the solutions in Equation (23) satisfies 

, or equivalently,

(34)

where 

 is the lower bound for |kp(t)| in assumption (A6). Thus, considering that the modulation function ϱ in Equation (29) verifies the inequality (28), one has 

. Then, one can write that ξ → 0 asymptotically and ∃ π3(t) ≥ 0 such that |Sξ(t)| ≤ π3(t) for any initial condition.

(2)

In the second case,

, which in conjunction with the first case leads to the conclusion that 

. Then, applying the last inequality in Equation (33), one concludes that 

. Thus, the dynamics which govern ξ is OSS with respect to 

, according to Equation (31).□

In order to present the main result for global stability, the closed-loop system state (including plant and observer) is defined as

(35)

Theorem 6.2 (Global asymptotic stability):

Consider the nonlinear system with output delay given in Equations (1)–(4), the control law proposed in Equation (27) with modulation function ϱ given by Equation (29), and the observer (16). If assumptions (A1)–(A6) hold, then the equilibrium point z = 0 of the closed-loop system is globally asymptotically stable and all the signals are uniformly bounded.

Proof:

From Lemma 3.1, the estimated states

 converge exponentially to the states xj(t) and, consequently, the estimation errors

, ∀j ∈ {1, … , m}, converge exponentially to zero, i.e.,

(36)

where 

. Since 

 from Equations (7) and (30), then Equation (36) implies the asymptotic convergence of 

. According to inequality (31) in Lemma 6.1, the norm of the state ξ is bounded by the estimation error norm plus an asymptotically decaying term πe(t). Therefore, using Equation (36) in Equation (31) and reminding that 

, one can conclude that ξ(t) and 

 tend asymptotically to zero. From the ISS property (A4) of the η-dynamics (8), it can be concluded that ‖η(t)‖ → 0 asymptotically, since ‖ξ(t)‖ → 0. This implies that the state x and all the estimated states 

 tend asymptotically to zero; consequently z(t) → 0 asymptotically.

Since the estimated state

 tends to zero and the function ϕw(t) is uniformly bounded in view of assumption (A5), the modulation function given by Equation (29) is uniformly bounded. Consequently, the control signal u and all the closed-loop system signals are also bounded.□

Corollary 6.3 (Ideal sliding mode):

In addition to the assumptions in Theorem 6.2, if δ > 0 in Equation (29), then the sliding surface

 is reached in finite time.

Proof:

From the equation of the last observer in Equation (16) and using

, it can be concluded that 

 satisfies

(37)

or equivalently,

(38)

Now, consider the quadratic function

. Then, calculating 

 along the solutions of the dynamics in 

 (see Equation (38)),

(39)

Since SBc = bl − 1 > 0 (see Equations (11) and (25)) and the control signal is given by Equation (27), the function

 can be rewritten as

(40)

Since the modulation function ϱ satisfies Equation (28), the following inequality is valid:

(41)

Note that, according to Theorem 6.2x and

 tend to zero. Then, there exists a finite time T1 > 0 such that

(42)

with some constant 0 < δ1 < δ. Therefore, 

, and the condition 

 for the existence of a sliding mode in some finite time is verified (Utkin et al., 1999Utkin, V.Guldner, J., & Shi,J. (1999). Sliding mode control in electromechanical systems.London: Taylor & Francis., Section 2.5).□

Remark 2:

The ideal sliding mode cannot be reached in the main control loop, due to the time delay. However, the asymptotic observer can eliminate chattering which could be caused by the time delay, since the sliding mode on the surface,

, occurs in an auxiliary observer loop (‘ideal sliding loop’ in Figure 1), where imperfections are absent, as is usual in SMC based on observers for systems without delays (Cunha et al., 2009Cunha, J.P.V.S.Costa, R.R.,Lizarralde, F., & Hsu, L. (2009). Peaking free variable structure control of uncertain linear systems based on a high-gain observer. Automatica,45, 1156–1164.[CrossRef][Web of Science ®]; Utkin et al., 1999Utkin, V.Guldner, J., & Shi,J. (1999). Sliding mode control in electromechanical systems.London: Taylor & Francis., Section 8.3). This loop is ideal for conventional SMC since it is of relative degree one, required for theoretically infinite-frequency switching (Levant, 2003)Levant, A. (2003). Higher-order sliding modes, differentiation and output-feedback control.International Journal of Control, 76, 924–941.[Taylor & Francis Online],[Web of Science ®].

Remark 3:

The dynamic behaviour of the estimated state

 during the sliding-mode phase can be directly obtained from the differential equation 

, which is stable since the polynomial pS(s) in Equation (26) is chosen to be Hurwitz.

7. Numerical example

Consider the following nonlinear system with output delay:

(43)

where the nominal functions are kp(t) ≡ 1 and wφ(xt) = 0.5x1 + 0.5tanh (x2), and the state vector is 

. Here, x1 and x2 are not powers of x, they denote the elements of the state vector x, following the same notation adopted by Ahmed-Ali et al. (2012Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®]). Since this system is in the normal forms (9) and (10) with ξ = x, the diffeomorphism (7) is not needed.

To design the control law (27), the matrix S = [11] is chosen to define the sliding surface as in Equation (25). The modulation function ϱ is given by Equation (29), where Km = [−2 − 3]. The function ϕw(t) = 0.6 and the constant kw = 0.6 were chosen to satisfy assumption (A5), i.e., |wφ(xt)| ≤ 0.6 ‖x‖ + 0.6. The constant

 was chosen to satisfy assumption (A6), since 

. The parameter δ = 0.1 was arbitrarily set as in Corollary 6.3 to guarantee the existence of the sliding mode.

To evaluate the robustness of the proposed scheme to parametric uncertainties,Figures 26 present simulation results obtained for a disturbed system with kp(t) ≡ 0.9, wφ(xt) = 0.6x1 + 0.6 tanh (x2) and d = 0.5 s. In the simulations, the following initial conditions were considered for the system and all observers:

Figure 2. State x1, estimated state

 and output signal y(t) = x1(t − d) for the disturbed system.

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Figure 3. State x2 and estimated state

 for the disturbed system.

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Figure 4. Estimation errors

 and 

 for the disturbed system.

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Figure 5. Switching signal

 for the disturbed system.

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Figure 6. Control signal u for the disturbed system.

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The disturbed parameters are within their bounds assumed in the design of the modulation function. The observer is designed for the nominal system, since the analysis presented by Ahmed-Ali et al. (2012)Ahmed-Ali, T.Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade high gain predictors for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57, 224–229.[Web of Science ®] has not considered uncertain parameters. The parameters of the cascade observers (16) are set to θ = 4 and K = [0.850.24]T. Since the considered nominal output delay is d = 0.4 s, then one observer would not be sufficient to estimate the system state in Equation (43). In this case, the number of observers connected in cascade is m = 2.

The state variables x1 and x2 reach the equilibrium point in the origin as can be seen in Figures 2 and 3, as expected. Moreover, the estimation errors

 and 

 converge to zero, as shown in Figure 4. Note that, after an initial transient, the sliding surface 

 (Figure 5) is reached in finite time, when the high-frequency switching becomes apparent in the control signal (Figure 6). These simulation results indicate that the proposed observer-based control scheme can be robust to parametric uncertainties including the time delay.

8. Application to a full-bridge power converter

The full-bridge power converter presented in Figure 7 illustrates the application of the proposed SMC scheme. It is composed of a low-pass filter (L = 1 mH and C = 100 μF) and power semiconductors represented by the ideal switches in Figure 7, which are controlled according to Table 1 (Solé & Colet, 2004Solé, D.B., & Colet, E.F.(2004). SMC applications in power electronics. In A.SabanovicL. Fridman, &S. Spurgeon (Eds.),Variable structure systems: From principles to implementation (Vol. 66, pp. 265–293). London: The IEE.[CrossRef]). The control objective is to regulate the voltage vl on the load (Rl = 5 Ω) obtained from a DC input voltage ϱ = 10 V.

Figure 7. Full-bridge converter.

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Table 1. Switching logic of the full-bridge converter inFigure 7.

CSVDisplay Table

The dynamics of the full-bridge power converter can be represented by the state equation (43), where the nominal functions are kp(t) ≡ (LC)−1 and

, and the state vector is 

. The output signal y is the load voltage vl delayed by the signal conditioning system (y(t) = vl(t − d)). The time delay d = 0.1 ms has the same order of magnitude of the time constant of the LC network (

 ms).

The control law (27) applies the modified switching signal

(44)

where the reference 

 V is included to set the desired load voltage (Solé & Colet,2004Solé, D.B., & Colet, E.F.(2004). SMC applications in power electronics. In A.SabanovicL. Fridman, &S. Spurgeon (Eds.),Variable structure systems: From principles to implementation (Vol. 66, pp. 265–293). London: The IEE.[CrossRef]). The parameter b1 = 0.1 ms is chosen to specify the desired convergence rate of the load voltage.

If the actual state without delay is available for feedback, then

 can be applied in the switching signal (44) as usually (e.g., Solé & Colet, 2004Solé, D.B., & Colet, E.F.(2004). SMC applications in power electronics. In A.SabanovicL. Fridman, &S. Spurgeon (Eds.),Variable structure systems: From principles to implementation (Vol. 66, pp. 265–293). London: The IEE.[CrossRef]). The simulation results presented in Figure 8 illustrate the exponential convergence of the load voltage to the reference voltage after the sliding mode starts by 0.3 ms. The high-frequency switching of the control signal could be achieved due to the absence of delayed signals.

Figure 8. Load voltage (vl) and control signal (u) for the full-bridge power converter with sliding-mode control based on full-state feedback without delay.

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The feedback of the time-delayed state

 in the switching signal (44) would result in undesirable low-frequency oscillations seen in the simulation results presented in Figure 9. The large amplitude of the oscillations of the load voltage would impair the practical application of this power converter.

Figure 9. Load voltage (vl) and control signal (u) for the full-bridge power converter with control based on the feedback of the time-delayed state.

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If only the delayed load voltage (y) is available for feedback, then the state

 can be estimated by the cascade HGO (16) and applied to the switching signal (44). The parameters of the cascade observer (16) are set to θ = 2 × 104, K = [0.850.24]T and m = 2. The simulation results presented in Figure 10illustrate the convergence of the load voltage to the reference voltage. In the simulations, the following initial conditions were considered for the power converter and all observers:

The differences between the initial conditions of the power converter and the observers cause the oscillations in the initial transient of the load voltage seen inFigure 10, which are not present in the SMC based on full-state feedback (seeFigure 8). The sliding mode starts by 1.13 ms, when the high-frequency switching of the control signal could be achieved due to the ideal sliding loop provided by the HGO.

Figure 10. Load voltage (vl) and control signal (u) for the full-bridge power converter with sliding-mode control based on cascade observers for the time-delayed system.

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9. Conclusions

An output-feedback sliding-mode controller was developed for a class of nonlinear systems with arbitrary, constant and known time delay in the output signal. Since constant time delay in the control signal can be transferred to the system output (Michiels & Niculescu, 2007Michiels, W., & Niculescu,S.I. (2007). Stability and stabilization of time-delay systems: An Eigenvalue-based approach.Philadelphia, PA: SIAM.[CrossRef], Remark 7.8) for analysis purposes, the proposed control scheme can also be applied to some systems with input delays.

The problem of observation and feedback control of uncertain time-delay systems is far from being definitely solved. Although nonlinear systems with known parameters were considered, the results obtained here are relevant due to the difficulty to cope with delayed output signals using only output feedback. Based on cascade observers to estimate the system state, the proposed SMC strategy guarantees global asymptotic stability of the closed-loop system and the ideal sliding mode can be achieved in finite time.

A full-bridge power converter example with delay in the measurement system was presented in order to highlight the application of the proposed controller in real-world problems.

To the best of our knowledge, these results are new in the SMC literature concerning systems with output time delay.

Acknowledgements

This work was supported in part by CAPES, CNPq and FAPERJ, Brazil.

Tiago Roux Oliveira  ( SpaceMETA www.spacemeta.com.br advisor )  ( inserted by Sergio Cabral Cavalcanti , SpaceMETA  Brasil. )

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