Sliding Mode Control With Closed-Loop Filtering ... - IEEE Xplore

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 4, APRIL 2004

Sliding Mode Control With Closed-Loop Filtering Architecture for a Class of Nonlinear Systems Jian-Xin Xu, Senior Member, IEEE, Ya-Jun Pan, Member, IEEE, and Tong-Heng Lee, Member, IEEE

Abstract—It is well known by equivalent control theory, that in the sliding mode a low-pass filter, working as an average operator, can capture the desired control fairly well from the switching control signals. However, directly adding the filtered signals in parallel with the switching control does not warrant any improvement in the control system performance. In this work, we make clear the underlying reason why the sliding mode control (SMC) system does not function properly with such a simple closed-loop filtering structure. The correct way of incorporating the closed-loop filtering into SMC requires a second low-pass filter, which works concurrently with the first low pass filter to scale down the gain of the switching control. The SMC system with the new closed-loop filtering is able to realize the acquisition of equivalent control or estimate the disturbance, effectively reduce the switching gain to the minimum level, and as a result to eliminate chattering. Complementary to existing SMC, the proposed control system can easily incorporate feed-forward control into the closed-loop for bounded system perturbations. In addition, the frequency-domain knowledge can be easily used to construct the two filters. Index Terms—Closed-loop filtering, nonlinear systems, sliding mode control, Van der Pol circuits.

I. INTRODUCTION

O

VER the past two decades, increasing attention has been drawn to sliding mode control schemes which warrant the system robustness in the presence of norm-bounded system uncertainties or disturbances [1]–[3]. The sliding mode control theory has been widely and successfully applied in many real systems such as robotics, servo mechanisms, power systems, circuits systems, etc. In order to perform perfect tracking, from the viewpoint of internal model principle, the acquisition of the equivalent control signals is imperative in the sliding mode control. However, the existence of system perturbations prevents the possibility of direct acquisition of the equivalent control. It has been shown in [1] that equivalent control can be obtained using a first-order filter provided the following two conditions are satisfied: 1) system in sliding mode; and 2) infinite bandwidth of the first-order filter. Note that the second condition is needed only when equivalent control signals possess infinite bandwidth. In most practical circumstances, however, it is not necessary to consider such extreme cases as requiring infinite bandwidth. Inherently associating with the equivalent control, another important issue constantly being addressed in the sliding mode control is how to avert the use of high gain

Manuscript received August 16, 2002; revised December 15, 2002. This paper was recommended by Associate Editor X. Yu. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2004.824066

such as signum function. Passive smoothing schemes such as the introduction of boundary layer make compromise between the tracking accuracy and the alleviation of chattering, since the reduction of switching gain leads to the reduction of feedback effect. By virtue of internal model principle, in order to retain good tracking performance with much lower control gain, one has to incorporate the “internal model” in the closed-loop, i.e., provide equivalent control signals. Adaptive control and learning control combined with SMC are two main strategies to produce “internal model.” However, learning control can only handle periodic control environment and adaptive control can only handle constant parametric uncertainties. In this paper, a new SMC scheme with closed-loop filtering architecture using signal processing techniques, is proposed for tracking control tasks. Based upon the reasonable assumption that actual frequency band in equivalent control is far lower than the sampling frequency, a low-pass filter (LPF1) is employed to acquire equivalent control signals from the switching control signals by “averaging” operation. Nevertheless this low-pass filter alone cannot fully extract the equivalent control signals if its output is to be fed back to the process as the feedforward compensation. This is because the existence of the switching control part will interfere with the low-pass filter. Moreover, adding the feedforward compensation does not immediately warrant an auto-shaping of the switching control. Hence the chattering phenomenon incurred by high gain still remains. To facilitate the acquisition of equivalent control and reduce the switching gain, a second filter (LPF2) is incorporated to shape the switching gain directly. Note that the acquisition of the equivalent control is in a closed-loop architecture which is different from the traditional open-loop approaches used in state estimation and parameter identification. The new SMC scheme is complementary to existing passive smoothing schemes in that a feedforward loop is incorporated, in the sequel the feedback gain can be lowered. The new SMC scheme is also complementary to existing active SMC schemes such as adaptive VSC and learning VSC in that it directly estimates the lumped system perturbations that may neither be constant parameters nor be periodic signals in nature. II. EQUIVALENT CONTROL AND SMC WITH CLOSED-LOOP FILTERING A. Problem Formulation Consider the th order nonlinear system

1057-7130/04$20.00 © 2004 IEEE

(1)

XU et al.: SMC WITH CLOSED-LOOP FILTERING ARCHITECTURE FOR A CLASS OF NONLINEAR SYSTEMS

, is the measurable state, , are known functions with respect to the is the system control input, is an disarguments, turbance. In addition, , and are assumed , to be continuously differentiable, . The system is required to track the trajectory as , , , where , with respect to all arguments and is a reference input. The tracking er, , . rors are defined as and its derivative are upperbounded Assumption 1: and , where and are by known constants. is positive definite which is bounded Assumption 2: , and by known positive constants and , i.e., , where is a known constant.

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where

B. SMC With Closed-Loop Filtering Architecture

Fig. 1.

The schematic diagram of new sliding mode controlled system.

scaled down in LPF2 when is sufficiently small in order to facilitate the acquisition of the equivalent control by LPF1. Theorem 1: If the parameters of the two low-pass filters, , , and , are chosen to satisfy the following conditions

The new SMC with closed-loop filtering is (2)

(4)

and , where and is a constant. denotes the feedforward compensation term which can be achieved according the conception of the equivalent is a switching quantity as . The control [1]. is generated by a first-order low-pass filter continuous term , where and are (denoted as LPF1) positive constants. LPF1 is activated at the time with . The switching gain is zero initial condition chosen as below

. then the sliding motion is maintained for all , both filters LPF1 and LPF2 start Proof: When working concurrently, and in particular the switching gain will be shaped by LPF2. The key issue, in such circumstance, is to prove that the sliding mode retains after . The proof consists of three parts. First we show the ideal conditions under which LFP1 is able to approximate the disturbance, in the sense of equivalent control. Since these conditions with equivalent control are practically not implementable, we look for alternative conditions, which are related to the design parameters of LPF1 and LPF2, such that the sliding mode holds at least for a suf. Then we show that the ficiently small interval sliding mode can be maintained over the subsequent intervals , where is not infinitesimal. . Hence the sliding mode exists over Part A: We first look into the approximation property of LPF1 to the disturbance . Substituting the control law , leads to the closed-loop dynamics (2), with . Substituting the closed-loop dynamics into LPF1

where

with

(3) , is the reaching time and is the output where , of another first-order filter LPF2 , where , , and are all constants. The initial condition of LPF2 is where is a constant. The schematic diagram is shown in Fig. 1. III. CONVERGENCE ANALYSIS A. Convergence Analysis in Reaching Phase , , , (2) . The Lyapunov candidate is chosen . The derivative of becomes . Thus the system is convergent during the reaching phase. The reaching . In this work, time can be calculated as . In this phase, becomes as

B. Existence Conditions of the Sliding Motion In this subsection we will derive the sufficient conditions for maintaining the sliding mode when using the proposed controller. When introducing the two filters, the switching gain is

where is

. Define

, the solution of

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where

and

. Calculating and using integral by part, and noticing the fact , we have

(5)

(6) Using (5) and (6), and substituting (from LPF2), we have

Furthermore, let

, then

,

and . Note that the conditions (8) are similar to the conditions for deriving equivalent control [4]. Clearly, shaping the gain is necessary in order to capture the disturbance. However, when . Thus the estimate of disturbance by LPF1 will converge asymptotically. Part B: The preceding part did not provide the assurance for the existence of the sliding mode. In this part, we investigate the system behavior for a sufficiently small interval right after , and demonstrate that the system state cannot move away from the switching surface in the interval. Suppose that is in anyway moving away from , . In such circumstance, there will be no switching and the . Therefore, for a given system trajectory is continuous, so is , such sufficiently small , there exists a sufficiently small , monotonically increases and reaches the that maximum . Likewise, with the given , , there exists from the continuity of , such that , where a sufficiently small and are the maximum and minimum values of in the time interval . Choose which meets both requirements. Taking the absolute values on both sides of (7), taking the upper bounds on the right-hand side and using , where

, we

have

(7) Now let us show that are satisfied ,

if the following conditions (9) (8)

Let

and

, (7) renders to

Now we are in a position to derive the sliding mode in . Note that tends to 0 when , i.e., is finite . From the closed-loop dynamics when and the switching control law , the derivative of is

(10)


Hence, if can be ensured sliding mode is retained. Now let us prove that holds . Note that

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, the

(11) From (9) and (11), we can derive

Fig. 2. The schematic diagram of Van der Pol circuit.

where, analogous to (13), that

can be made sufficiently small so

(15) Since from (14) we can further derive

and

,

(12) For all terms related to (12), choose a small constant . There exists a sufficiently small , in the sequel is sufficiently small such that the following condition holds

where . Hence, from (10) we , where . Thus the sliding mode have for . does exist, that is, Part C: Now let us investigate the system behavior in the . Suppose that the system diverges in interval . Because of the existence of the sliding mode as proven in Part B, at time , we in have

(16)

(13) Using (13) and choosing

, (12) becomes

when beHere we consider that function at the point cause there is no definition for the . Analogous to preceding discussions, let be sufficiently small, such that increases monotonically and reach . For simplicity, assume the maximum at and . In the following, we will . At time prove that sliding mode exists , LPF1 and LPF2 can be

(17)

(14)

(18)

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Fig. 3.


(a) Solid line: u = u + k

+ u ; dash-dotted line

Using (16), (17) and (18), the derivative of

0u

0

. (b)Solid line: u ; dash-dotted line: (x; t).

becomes

, the first term in (20) is negative. We can define the second term in (20) as . Note that and if is selected, we have ,. Hence and

(19) Because (19) becomes

, where

,

. Thus sliding mode exists . Similarly, for , we can derive the same results. By repeating the same derivations shown in this part during with , we the interval , can once again derive . This implies that where . By repeating the same derivation infinite times, , we can reach the conclusion that the sliding motion is maintained . for all IV. ILLUSTRATIVE EXAMPLE Consider a Van der Pol circuits system with an additive disas shown in Fig. 2. turbance By states transformation, the system dynamics becomes

(21)

, . and . The system (21) is to track a model of oscillator—another Van der Pol equation , where

(20)

,


Fig. 4.

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(a) Switching surface . (b) Gain shaping of switching gain k (t).

, where , . The switching surface is chosen to be . The control law is

,

as limited sampling rate, the switching gain will not completely , disappear. However, comparing with the initial value the effect of gain shaping is obvious. V. CONCLUSIONS

,

, where and , , are generated by the following filters , , where and are selected according to Theorem I. , with and . The sampling period is 1 ms. Fig. 3(a) demonstrates the appreciated results: chattering has been reduced to the minimum and the control profile dominated by a 5-Hz component is consistent with that of exogenous disturbance. From Fig. 3(b), we can see that the LPF1 estimates the disturbance almost perfectly except for the reaching phase. is approxiThis is possible because the bandwidth of mately 5 Hz and that of LPF1 is 10 Hz. Consequently is able to compensate the most part of . Fig. 4(a) further confirms that the produced smooth control waveform is the desired equivalent control which achieves almost perfect tracking. Fig. 4(b) shows the fast attenuation of the switching gain using LPF2. Due to imperfect factors in the sliding mode, such

The filtering techniques with two low-pass filters incorporated in the closed loop are adopted into the SMC to acquire equivalent control and shape the switching control gain in the presence of bounded system uncertainties. It achieves that: (1) system disturbances can be compensated quite accurately; (2) the switching gain can be shaped to the very low level in the sliding mode; (3) both time and frequency domain knowledge can be used in the filter design; and (4) the new scheme is easy to be implemented as only two extra simple first-order filters are employed. REFERENCES [1] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Germany: Springer-Verlag, 1992, vol. 34. [2] K. D. Young and U. Ozguner, Lecture Notes In Control And Information Sciences 247, Variable Structure Systems, Sliding Mode And Nonlinear Control. New York: Springer-Verlag, 1999, vol. 51. [3] X. H. Yu and J. X. Xu, Advances in Variable Structure Systems—Analysis, Integration and Applications, Singapore: World Scientific, 2000. [4] V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems. Moscow, Russia: MIR, 1978.