Communication Servo Turntable via a Novel Digital Sliding Mode Controller
3. Design and Simulation of a Velocity Loop Controller Based on Discrete SMC
effect of the static friction force, and the running state is unstable, with obvious amplitude-frequency characteristic curve fluctuation and phase-frequency hysteresis characteristics. The frequency is greater than 2 Hz, and the model frequency characteristic curve can better fit the trend of the actual frequency characteristic curve, however nonlinear disturbances such as friction force and torque imbalance in the system can significantly affect the identification accuracy. The absolute value of the amplitude frequency characteristic curve identification error is less than 8.7 dB. For engineering practice, model identification error and nonlinear disturbance restrict dramatic improvement of the control performance, but a traditional PID controller cannot effectively suppress low and medium frequency nonlinear disturbance, so cannot meet the requirements of high control accuracy of ground-based laser communication. Therefore, it is important to develop a high-precision control algorithm that can better suppress the nonlinear disturbance.
3.1.2. Discrete Sliding Mode Function and the Sliding Mode Surface
In this study, the classical sliding mode function was adopted, which is defined as follows:
s(k) =Cee(k) =Ce(x(k)−Rn(k)), (12) whereCe=[Ce(1), Ce(2),. . ., Ce(n−1),1] is the sliding mode coefficient matrix, ande(k) represents the error matrix between the ideal state and the actual state. Therefore, the sliding mode surface can be defined as:
S=)e(k)Cee(k) =0*
. (13)
Lemma 1.The sliding mode coefficient matrix parameters Ce(1),Ce(2),. . .,Ce(n−1)satisfy the polynomials pn−1+ Ce(n−1)pn−2+. . .+Ce(2)+Ce(1), and must be Hurwitz polynomials; p is the Laplace operator [35].
3.1.3. The Quasi Sliding Mode Domain
Definition 1.For the perturbed system, the system will be in a quasi-sliding-mode (QSM) in theΔvicinity of the sliding surface, not at the sliding surface. This specified domain where the QSM occurs is called the quasi-sliding-mode domain (QSMD) sΔ, the positive constantΔ is the QSMD width, and sΔ is the QSMD [39–41],
sΔ=
s(k)s(k)≤Δ
(14) The condition in which the system state is stable in the QSMD is defined as:
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
−Δ<s(k+1)<s(k),s(k)>Δ s(k)s(k<s(k+1)+1)≤Δ,<Δ,s(k)s(k)<<Δ−Δ .
3.2. A Novel Chatter-Free Approach Law Sliding Mode Control Based on Disturbance Compensator The SMC algorithm is widely applied in control systems because of its simple structure and good dynamic performance. After stabilization of the system states in the QSMD, the SMC is consistent with the nonlinear disturbance, and the smaller the width of the QSMD, the stronger the robustness of the system to the nonlinear factors. However, chattering is the main problem of SMC, which can lead to high frequency noise in the system, with adverse effects on the stability and control accuracy of the system. In this section, a novel chatter-free SMC algorithm is proposed to reduce the effects of chattering on the system and to suppress nonlinear disturbances in the low and medium frequency bands of the system.
The novel chatter-free sliding mode approach law based on exponential approach law is shown
as follows: ⎧
⎪⎪⎪⎨⎪⎪⎪⎩
s(k+1) = (1−λT)s(k)−κT·η(k)·sgn[s(k)] +ω(k) η(k) = |e1(k)|α
δ+
1+|e1(k)|α−1−δe−μ·|s(k)|γ , (15) in this formula,κ>0 is the coefficient of variable structure function;Tis the sampling time of the system;η(k) is the error-related adaptive function, where 0<δ<1,γ>1,μ>0;|e1(k)|represents the first state error of the state error vector, if|e1(k)| >1, 0<α<1,if 0< |e1(k)|≤1,α>1;ω(k) represents the disturbance compensator, which can be expressed as:
ω(k) =Ce(ε(k)−2ε(k−1) +ε(k−2)). (16)
Remark 2.If the system state is far away from the sliding mode surface, according to formula (12), s(k) and
|e1(k)|tend to be great, so the coefficient of the switching function sgn[s(k)] tends to beκT·|e1|α/δ, which is greater thanκT. The approaching movement is the state of the system that gradually approaches the sliding mode surface driven by the reaching law. If the system state is near the sliding surface, s(k)≈0, and the coefficient of the switching function sgn[s(k)] tends toκT·|e1|α/(1+|e1|α), which is far smaller thanκT, so the system state is stable in the QSMD for sliding mode motion. Thus, chattering on the sliding surface is reduced.
Lemma 2.According to [22,42],ε(k)=O(T),ε(k)−ε(k−1)=O(T2), (T is the sampling period), where O(T), and O(T2) represent the disturbance estimation error and are the first-order O(T) and the second-order O(T), respectively; O(T)>O(T2). Therefore, the magnitude of the disturbance estimation error shown in Equation (16) is the third-order O(T),ε(k)−2ε(k−1)+ε(k−2)=O(T3), O(T3)<O(T2).
According to Equation (11), (12), (15), and (16), the sliding mode controller can be deduced as follows:
u(k) = (CeBk)−1[Ce(Rn(k+1)−Akx(k)) + (1−λT)s(k)
−κT·η(k)sgn[s(k)]−Ce(2ε(k−1)−ε(k−2))] (17) Remark 3.The disturbance functionε(k−1) of the sliding mode controller (17) is usually deduced or calculated by the “time delay estimation method” [22,39,42–44]:
ε(k−1) =x(k)−Akx(k−1)−Bku(k−1). (18)
Remark 4.According to Lemma 2, the width of the QSMD of the proposed SMC is related to the disturbance compensator, and its width is O(T3) order, which is smaller than the O(T2) order width described in [23,42,45].
Therefore, the proposed algorithm attains stronger robustness and higher control accuracy.
3.3. Proof of Robustness and Stability of SMC
Theorem 1.The absolute value of the system’s nonlinear function equation as shown in Equation (16) has an upper bound, and this upper bound is assumed asω. The trajectories of the system from any initial state must arrive at the sliding mode surface driven by the proposed algorithm.
Proof.There must be an initial state such that the sign of the sliding mode switching functions s(0), s(1),. . . s(n) does not change, where n is a positive integer. The following proofs are discussed for two cases: s(0)<0 and s(0)≥0.
1. If s(k)≥0 (k=0, 1,. . .n)
Assuming that the system does not cross the sliding surface within thenstep, recursive formulas can be obtained according to formula (15).
s(1) = (1−λT)s(0)−(κT·η(0) +ω(0))
s(2) = (1−λT)2s(0)−(1−λT)(κT·η(0)−ω(0))−κT·η(1)−ω(1) ...
s(n) = (1−λT)ns(0)−n−1
i=0(1−λT)n−1−i(κT·η(i)−ω(i))
(19)
There must be a positive numberδso that the following formula is workable:
n−1 i=0
(1−λT)n−1−i(κT·η(i)−ω(i)) =n−1
i=0
(1−λT)n−1−iδ. (20)
Assuming that the system trajectory reaches the sliding mode surface at timem, then s(m)=0, and according to Equations (19) and (20), the following equation can be obtained,
s(m) = (1−λT)ms(0)−δ·m−1
i=0
(1−λT)m−1−i= (1−λT)ms(0)−δ1−(1−λT)m
λT (21)
Therefore, the arrival timemcan be expressed as:
m=log1−λT 1
λT·s(0)/δ+1. (22)
2. Similarly, ifs(k)<0 (k=0, 1,. . .n), the moment that the system state reaches the sliding mode surface can be expressed as:
m=log1−λT 1
−λT·s(0)/δ+1. (23)
Above all, driven by the sliding mode controller shown in formula (17), the state trajectory of the system from any initial position can reach the sliding mode surface in a limited time, and the arrival time is expressed as follows:
m=log1−λT 1
λT·s(0)/δ+1. (24)
Theorem 1 has been proved.
Theorem 2.Driven by the sliding mode controller, once the system state reaches the sliding mode surface, it will be stable in the QSMD and cannot escape, the control system is bounded. The QSMD can be expressed as:
Φ=
s(k)s(k)≤Δ=κT·η+ω
, (25)
whereω/κT≤η<1,ωis the upper of the disturbance.
Proof.The proof process can be divided into two cases:s(0)<0 ands(k)≥0 (k=0, 1,. . .n).
1. Ifs(k)≥0 (k=0, 1,. . .n), it can be obtained that
s(k+1) = (1−λT)s(k)−κT·η(k)sgn[s(k)] +ω(k)
<s(k)−κT·η+ω <s(k) (26) Therefore, when s(k)≥0, the value of s(k) decreases successively. Assuming that the system state is in the QSMD at timen, the system state at timen+1 must be in the QSMD.
2. Ifs(k)<0 (k=0, 1,. . .n), similar conclusions can be drawn that s(k+1) = (1−λT)s(k)−κT·η(k)sgn[s(k)] +ω(k)
>s(k) +κT·η+ω >s(k) (27) Therefore, whens(k)<0, the value ofs(k) increases successively. When the system state is in the QSMD at timen, the system state at timen+1 must also be in the QSMD.
Above all,|s(k)|deceases with time. Once the trajectory of the system reaches the sliding surface, the system state will stabilize in the QSMD. At this moment, the system is strongly robust to nonlinear disturbances. Therefore, the control algorithm is stable, and Theorem 2 is proved.
Remark 5. In view of [36,44,46–49], the overestimation and underestimation problems may exist when the state is stable in the QSMD, we will study it in the future work.
3.4. Sliding Mode Simulation
In this section, using the pitch axis system model of the ground-based laser communication servo turntable shown in Equation (11) as the simulation model, the stability of the proposed sliding mode algorithm was analyzed as follows. Assuming that the nonlinear disturbance of the system isε(k)= [0;0;2.5sin(2kπ)+0.5], the initial state of the system is [−1;−0.8;0], and the target state is [0;0;0].
According to the discrete state equation of the system, the sliding mode coefficient shown in Equation (12) is a three-order matrix such thatCe=[Ce1,Ce2,1], the parameters should be chosen as [1,2,1] to satisfy lemma 1. The parameters of the sliding mode controller (17) were selected as:λ=5,δ
=0.15,α=2,μ=10, andγ=2. To assess the proposed algorithm, it was compared to the classical exponential reaching law (28) with aO(T2) disturbance compensator and the chatter-reduced reaching law (29) with theO(T3) disturbance compensator described in Ref. [22]. The exponential function coefficients are the same, with values ofλc=5,λ[22]=5, and the variable structure function coefficients κ,κc,κ[22]are determined by the disturbance estimation error in the next section. Other coefficients in the algorithm of Ref. [22] were set toδ[22]=0.15,μ[22]=10, andγ[22]=2.
s(k+1) = (1−λcT)s(k)−κcT·sgn[s(k)] +Ceε(k−1). (28)
⎧⎪⎪⎪⎪
⎨⎪⎪⎪⎪
⎩
s(k+1) =
1−λ[22]T
φ[22](k)·s(k)−φκ[22]T
[22](k)sgn[s(k)] +Ce[ε(k)−2ε(k−1) +ε(k−2)]
φ[22](k) =δ[22]+ 1−δ[22]
e−μ[22]|s(k)|
γ[22] . (29)
The comparison curves of disturbance estimation errors between the three algorithms were determined and are shown in Figure6. The maximumO(T3) amplitude was 0.01 for the algorithm disturbance estimation error curve for both the proposed algorithm and the Ref. [22] algorithm. This value is much smaller than that of the classical exponential reaching law algorithm, which exhibits a maximumO(T2) amplitude of 0.15. Therefore, the variable structure function coefficientsκandκ[22]
of both algorithms are set as 0.012, and the variable structure function coefficientκcof the classical algorithm is set as 0.17. According to Theorem 2, utilization of this parameter, larger than the maximum perturbation estimation, can ensure the stability of the sliding mode trajectory in the QSMD.
Figure 6.Disturbance estimation error contrast curve.
Figure7shows the sliding mode trajectory contrast curves. Three reaching law algorithms can ensure that the system state arrives at the sliding mode surface in a limited time and is stable in the QSMD with a fixed width. Theorems 1 and 2 are proved here. Compared with the traditional exponential reaching law algorithm, the QSMD width of the proposed reaching law algorithm is narrower,Δp≈0.038, compared to the QSMD width of the traditional algorithm ofΔc≈0.31. It should
be noted that a smaller QSMD width correlates to stronger robustness of the system to nonlinear disturbances. Compared with the chatter-reduced reaching law algorithm of Ref. [22], chattering in sliding mode can change to be continuous around the sliding surface, and the widths of both algorithms are almost equal. The proposed algorithm obviously eliminates sliding mode chattering and improves the stability of the system without reducing the robustness of the system.
Figure 7.Sliding mode trajectory contrast curves.