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# Particle swarm optimization tuned fuzzy terminal sliding mode control for UPS inverters #### Abstract

This paper proposes a particle swarm optimization algorithm tuned fuzzy terminal sliding mode control for the application of UPS inverters. Though classic sliding mode control (SMC) is insensitive to system uncertainties, it possesses an infinite system-state convergence time. For high-accuracy tracking control, a terminal sliding mode control (TSMC) is developed to provide a finite system-state convergence time. However, difficult estimation occurs in TSMC, and incurs high UPS inverter voltage harmonics and slow dynamic response. To obtain high-quality UPS inverter output voltage, a fuzzy logic (FL) with a computationally simple and practically easy estimator is integrated into TSMC to resolve system uncertainties. Simultaneously, the particle swarm optimization (PSO) algorithm is applied to optimally tune the control gains of the TSMC with a fuzzy estimator. Results indicate that the presented combination of PSO, FL and TSMC yields a closed-loop UPS inverter with good performance under various loading conditions. Simulation and experimental results indicate that the proposed control can achieve low total harmonic distortion (THD) under nonlinear loading conditions and fast dynamic response under transient loading conditions.

## 2Mathematical modeling of UPS inverter

As shown in Fig. 1, the output voltage v o of the UPS inverter can be forced to track a sinusoidal reference voltage, v d by applying the proposed control.

As shown in Fig. 1, two equations can be obtained by the use of the KVL and KCL:

##### (1)
iL=ic+io=Cdvodt+voR
##### (2)
LdiLdt+vo=vi

Substituting Equations (1) into (2) yields thefollowing:

##### (3)
d2vodt2+1RCdvodt+1LCvo=1LCvi

Let x 1 = v o , x2=ẋ1=v̇o, ẋ2=ẍ1=v̈o and u = v i . Then, the matrix equation can beexpressed as:

##### (4)
[ẋ1ẋ2]=[01-1LC-1RC][x1x2]+[01LC]u

Suppose the desired output voltage is x d  = v d , and a state variable xe=[xe1xe2]T related to the tracking error is expressed by new state variables asfollows:

##### (5)
xe1=x1-xd=vo-vdxe2=ẋe1=ẋ1-ẋd=vȯ-vḋẍe2=-1LCxe1-1RCxe2+1LCu-1LCvd-1RCvḋ-vd̈

Thus, the error state matrix can be obtained as

##### (6)
[ẋe1ẋe2]=[01-a1-a2][xe1xe2]+[0a1]u+[0-a1vd-a2vḋ-vd̈]w
where a 1 indicates 1/LC, a 2 is 1/RC, and w denotes the interference. Thus, our purpose is to design a control law u. Once the u is accurately designed, the output of the UPS inberter will remain constant at the desired v d .

## 3Control design

For the sake of brevity, system (6) is redefined as follows:

##### (7)
ė1=e2ė2=f(e1,e2)+b(e1,e2)u(t)
where e 1 and e 2 represent the system error states, f (e 1, e 2) and b (e 1, e 2) are nonlinear functions of e 1 and e 2, and u (t) is the control input.

The finite-time convergent sliding function is designed as follows:

##### (8)
s=σ̇0+β1σ0γ1
where σ 0 = e 1 and β 1 is constant.

From Equation (8):

##### (9)
ṡ=σ̈0+β1γ1σ0γ1-1σ̇0=-i=12aiei+bu-w(t)+β1γ1σ0γ1-1σ̇0

Define

##### (10)
ai=ai0+Δai,i=1,2b=b0+Δbw(t)=a1vd+a2v̇d+v̈d
where a i0 denotes the nominal value of a i , b 0 represents the nominal value of b, Δa i indicates the variation of a i , and w (t) is the interference.

For system (6), the control law u can be expressed as follows:

##### (11)
u=ue+uft
where the equivalent control term u e is valid only on the sliding surface, and is defined as follows:
##### (12)
ṡ|u=ue,a1=a10,a2=a20,b=b0,w(t)=0=0
##### (13)
ue=(i=12ai0ei-β1γ1σ0γ1-1σ̇0)/b0

While the system dynamics are in sliding action,s = 0:

##### (14)
σ̇0=-β1σ0γ1

Substituting Equations (14) into (13) yields the following:

##### (15)
ue=(i=12ai0ei+β12γ1σ02γ1-1)/b0

Let

##### (16)
uft=i=12φiei+i=12ψiσ02γ1-1+N
where the u ft guarantees the existence of the finite-time sliding mode, and is used to eliminate systemuncertainties.

The condition for the existence of a sliding motion is, sṡ<0 as follows:

##### (17)
sṡ|u=ue+uft={-i=12Δaiei-Δa2e2+[Δbb0(i=12ai0ei)]+[b(i=12φiei)]+(bψ1-Δbb0β12γ1)σ02γ1-1+(-β12γ1σ02γ1-1+bψ2σ02γ1-1)+[bN-w(t)]}s<0

The conditions φ i , ψ i and N to satisfy Equation (17) can next be obtained. Notice that w (t) in Equation (17) is an interference and in practice should be rewritten as w (k), where k denotes the sample interval. Because w (k) is generally uncertain, the control in this form cannot be implemented and must beestimated.

Therefore, define the uncertainty estimation error w˜e(k) as follows:

##### (18)
w˜e(k)=w(k)-wˆ(k)

Then, an algorithm is expressed as:

##### (19)
wˆ(k)=wˆ(k-1)+β(wˆ(k-1)-wˆ(k-2))+w˜e(k-1)+β(w˜e(k-1)-w˜e(k-2))
where wˆ(k-1) is the estimated value of w (k - 1); the subscript indicates (k - 1) and (k - 2) representing the (k - 1)th and (k - 2)th sampling intervals, respectively; and |β| ≤ 1 is an adjustable parameter.

The FL is used to tune the β in Equation (19), and the fuzzy rule can be expressed as a two-input single-output system as follows:

##### (20)
IFw˜e(k-1)isW˜andΔw˜e(k-1)isΔW˜THENβisbeta
where W˜, ΔW˜, and beta are the fuzzy sets of w˜e(k-1), Δw˜e(k-1), and β, respectively.

The w˜e(k-1) and Δw˜e(k-1) can be easily obtained according to the following processes.

The incremental change of w˜e(k-1) is defined as follows:

##### (21)
Δw˜e(k-1)=w˜e(k-1)-w˜e(k-2)

From Equations (18) through (21), when the parameter β is well tuned, the w˜e(k-1) and Δw˜e(k-1) will be forced to zero and the w˜e(k) will tend toward zero. Thus, the system dynamics of system (6) will be insensitive to system perturbations, and the closed-loop system can stabilize asymptotically. The resulting fuzzy rule base designed with a two-dimensional phase plane for tuning β is displayed in Table 1.

Though the control gains can be tuned by the use of φ i , ψ i and N, thus achieving finite system-state convergence time, Equation (16) implies that the sine function across the surface s, and therefore chatter phenomenon, exists. Thus, to eliminate the chatter, the PSO algorithm represented in Equations (22) and (23) is used to optimally tune the control gains of the fuzzy TSMC. Equations (22) and (23) indicate the evolution models of a particle; then, the speed and position of each particle can be updated when moving toward a destination.

##### (22)
Vi+1=c0Vi+c1λ1(Xipbest-Xi)+c2λ2(Xigbest-Xi)
##### (23)
Xi+1=Xi+Vi+1
where c 0, c 1 and c 2 denote variables, λ 1 and λ 2 are random numbers, V i represents the present velocity, X i is the present position, Xipbest shows the best localized position, and Xigbest is the best global position. The operation of the PSO algorithm is described as follows. First, define the number of particles, and initialize their speeds and positions. Then, calculate the fitness of each particle according to F=k=1M|vo-vd|. For each particle, compare its fitness with its present best fitness. When the former is better than the latter, its present best fitness and best position are updated by its fitness and present position, respectively. Similarly, for each particle, compare its fitness with the global best fitness of the swarm. When the former is better than the latter, the global best fitness and global best position are updated by the former and the best position of the compared particle, respectively. Then, update the position and speed of each particle according to Equations (22) and (23). Finally, repeat this procedure until the terminal condition is completed.

## 5Conclusions

By combining PSO, FL, and TSMC, the presented system has improved the steady-state and dynamic response of the UPS inverter. The TSMC can resolve the classic SMC problem, but the difficult estimation of system uncertainties still exists in TSMC. Such difficulty may cause high voltage harmonics, and slow transient response. The PSO algorithm is used to optimally tune the control gains of the fuzzy TSMC, thus obtaining robust UPS inverter performance. Simulation and experimental results show that THD and dynamic response results from a UPS inverter under the proposed system exceed the results achieved under the classic SMC system with both linear and nonlinear loading.

## Acknowledgments

This work was supported by the Ministry of Science and Technology of Taiwan, R.O.C., under contract number MOST104-2221-E-214-011.

## References

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## Figures and Tables

##### Fig.1

Digitally controlled UPS inverter. ##### Fig.2

Simulated waveforms of output voltage and the load current under (a) TRIAC load with the proposed control; (b) TRIAC load with classic SMC; (c) rectifier load with the proposed control; (d) rectifier load with classic SMC. ##### Fig.3

Experimental waveforms of the output voltage and the load current under (a) TRIAC load with the proposed control. (b) TRIAC load with the classic SMC. (c) Rectifier load with the proposed control. (d) Rectifier load with the classic SMC. ##### Table 1

Fuzzy rule base 