A Smooth Integral Sliding Mode Controller and

International Journal of Control, Automation, and Systems (2015) 13(6):1-11 DOI 10.1007/s12555-013-0544-4

ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555

A Smooth Integral Sliding Mode Controller and Disturbance Estimator Design Muhammad Asad*, Aamer Iqbal Bhatti, Sohail Iqbal, and Yame Asfia Abstract: Integral sliding mode control eliminates the reaching phase of the traditional sliding mode and is therefore robust from the start. However, the phenomenon of chattering inherent to the sliding mode control technique is not eradicated and may result in chattering at the control input. In this work a novel integral sliding mode controller is formulated where the discontinuous control law is based on inverse hyperbolic function and provides variable gain which is a function of the sliding manifold. As the system states converge towards the surface the gain of the discontinuous controller reduces and results in relatively smoother control effort at the steady state. The proposed controller is robust against parameter variations, perturbations and is also used for the disturbance estimation and rejection. Stability of the proposed controller is proved with the help of Lyapunov method. The proposed controller is used to design the controllers for two different problems. The DC motor speed control where the chattering elimination and disturbance cancellation are shown with the help of simulations. In the second problem a digital phase locked loop is designed by using proposed controller where the phenomenon of oscillator pulling is eradicated by the rejection of the injection tone which is treated as a disturbance. Experimental results show eradication of the chattering phenomenon as well as the disturbance. Keywords: Digital phase locked loop, integral sliding mode, inverse hyperbolic function, smooth sliding mode.

1. INTRODUCTION Sliding mode control (SMC) is a variable structure control technique. It is robust against parameter variations and matched uncertainties [1-3]. In SMC, the sliding modes are established in some manifold in the systems state space. These manifolds are created by the intersection of hyperplanes in the state space of the system and are termed as the switching surface. The sliding mode controller switches the control between the hyperplanes along the sliding surface. Theoretically, the switching frequency approaches infinity with near zero amplitude [4]. However, due to the dynamics of the sensors actuators and the plant, the switching frequency is not infinite and has some amplitude. The switching __________ Manuscript received December 11, 2013; revised June 7, 2014; and November 4, 2014, accepted January 27, 2015. Recommended by Associate Editor Guang-Hong Yang under the direction of Editor Ju Hyun Park. The authors are thankful to the Controls and Signal Processing Research Group (CASPR) at Mohammad Ali Jinnah University Islamabad. Muhammad Asad is with the Department of Electrical and Computer Engineering, Center for Advanced Studies in Engineering (CASE), Islamabad, Pakistan (e-mail: [email protected]). Aamer Iqbal Bhatti and Sohail Iqbal are with the Department of Electrical Engineering, Mohammad Ali Jinnah University Islamabad, Pakistan (e-mails: [email protected], siayubi@gmail. com). Yame Asfia is the member of the Control and Signal Processing Research Group Mohammad Ali Jinnah University Islamabad (e-mail: [email protected]). * Corresponding author. © ICROS, KIEE and Springer 2015

phenomenon of system states along the sliding surface is known as chattering [5]. Chattering eradication and its attenuation in the SMC has been a topic of research since the inception of the SMC. The techniques are broadly: first order sliding mode techniques and the higher order sliding mode techniques (HOSM) [6]. The later technique i.e., the HOSM is well known for its simplicity and chattering elimination properties. However the HOSM may exhibit chattering due to the unmodeled fast dynamics [7]. A variant of the HOSM, termed as smooth second order sliding mode (SSOSM) used robust exact differentiator for the estimation and cancellation of the drift term [8,9]. The technique proved to be well suited for the chattering reduction, however its complexity increases due to the observer. Among the first order sliding mode techniques the most famous one was the use of saturation function which helped in reducing the chattering at the cost of robustness. In [10] power rate reaching law was suggested. The power rate reaching law loses robustness as the system states converge towards the surface. In [11-13] robust and chatter free exponential reaching laws with exponential term in the gain as a function of sliding surface were proposed. In [14] the authors proposed a reaching law based on the inverse hyperbolic function (IHF) where it was proved that the proposed law is robust and reduces the chattering as well as the reaching phase. Prior to the establishment of the sliding mode, the system is in the reaching phase during which the traditional sliding mode controller is not robust. Integral sliding mode control (ISMC) [15,16] eliminated the reaching phase and resulted in robust control performance. How-

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Muhammad Asad, Aamer Iqbal Bhatti, Sohail Iqbal, and Yame Asfia

ever the inherent phenomenon of chattering was not solved by the elimination of reaching phase. Integral higher order sliding mode was proposed by [17] where the integral action in the control input reduced the chattering and is also robust from the start. Integral dynamic sliding mode was proposed by [18] where the dynamic sliding mode [19] is used in conjunction with the integral sliding mode (ISM). The technique provided robustness and also reduced the chattering effect. ISMC along with the model predictive control (MPC) was used to reduce the uncertainty of the MPC scheme by [20]. Robustness against the matched and unmatched uncertainties in ISMC has been addressed in [21]. Dynamic ISMC was also applied on the Fuzzy approximated models by [22,23]. ISMC was applied to stabilize the nonlinear stochastic systems by [24]. Uncertain stochastic time delay systems were stabilized using ISMC by [25]. Output feedback ISMC was applied to the time delay systems by [26]. ISMC was applied to control the under actuated mobile robots by [27]. Adaptive higher order sliding mode using ISMC was proposed by [28]. In this paper a smooth ISM controller is proposed and has been successfully applied in the design of phase locked loop (PLL). A PLL is a feedback control system used for phase and frequency tracking in communication systems. It is also widely used in digital clock generation and in sinusoidal frequency synthesis. Traditional sliding mode including the ISM may result in unacceptable performance due to chattering which may introduce the phase noise resulting in spreading of the frequency spectrum [29]. The proposed smooth ISMC is therefore used to design the PLL. The proposed novel controller is an extension of the authors previous work presented in [14] where a novel sliding mode controller was introduced. In that preliminary work the stability analysis was carried out and with the help of simulations the chattering elimination was demonstrated. The work was unable to explain the fundamental reason behind the chattering reduction. In this work the computational complexity of the proposed control law is reduced by using only the IHF instead of the signum function. Mathematical proof of the chattering elimination has been given in this work where it is proved that the proposed reaching law bears low pass filter properties. A novel disturbance estimation and rejection controller has been described. The novel control law has been applied to design the digital PLL where the phenomenon of oscillator pulling has been eradicated [30-35]. To verify the mathematical formulations, a digital PLL using proposed controller is implemented on field programmable gate array (FPGA). Experimental results show smoothing of the chattering effect at the control input, and rejection of the disturbance. The rest of the paper is organized as follows: In Section 2 problem is formulated. In Section 3 the proposed novel ISMC is described. In Section 4 a case study is described where the proposed ISMC is used in the design of digital PLL that rejects the injection signal. Experimental results are also discussed in Section 4. In Section 5 the conclusions are presented.

2. PROBLEM FORMULATION The phenomenon of chattering in SMC is a major issue that prevents the usage of this simple and robust control technique in certain class of control problems such as PLL’s where the key properties of sliding mode based control technique may be used for disturbance rejection and robustness against parametric variations. The second issue that has been addressed in this work is accurate chatter free disturbance estimation and rejection using ISMC. The traditional ISMC when used for disturbance estimation and rejection requires low pass filtration of the discontinuous controller output. The resulting estimates may exhibit following problems: 1) Due to the group delay of the filter the output may be delayed and may not be fully rejected. 2) The bandwidth of the disturbance should be known aprioi otherwise the estimates may be either attenuated or may contain high frequency chattering. The second problem will result if the filter bandwidth is either too low or too high respectively. This work intends to solve the mentioned problems. 3. SMOOTH INTEGRAL SLIDING MODE CONTROLLER 3.1. Integral sliding mode Let a nonlinear system be described by the following state equation [2,15]:

x = f ( x) + Bu + h( x, t ),

(1)

where x ∈ ℝn, u ∈ ℝm are the system states, and the input vector respectively and m 0, a ∈ ℝ, a ≥ 0, b ∈ ℝ, b > 0, so(x) ∈ ℝ, when a single input single output (SISO) system is assumed. For multi input multi output (MIMO) systems, k ∈ ℝm×m, ki,i > 0, a ∈ ℝ m×m, ai,i ≥ 0, i =1..m. so(x) ∈ ℝm×m, and b ∈ ℝ m×m are all block diagonal matrices. Assuming SISO case for analysis the reaching law results in the discontinuous control law of the form: ud = − k sinh −1 ( a + b so ( x ) ) sign ( so ( x ) ) .

(11)

In that preliminary work the authors covered the stability proof and the derivation of the relation for the reaching time of the proposed control law. No analysis of its chattering eradication behavior was discussed. The parameters a, b and their impact was not evaluated. In this work the chattering reduction aspect and the variable boundary layer formation aspect of the control law are discussed and proved rigorously. For a > 0, b = β, β > 0 so→0, (11) can be written as: −1

ud = − k sinh ( a ) sign( so ( x )),

which is the traditional constant gain reaching law. However when a = 0, b = β, β > 0 the control law may be written as: −1

ud = − k sinh ( β so ( x ) ) sign( so ( x )).

The resulting discontinuous controller will form a variable width boundary layer around the sliding surface with the width dependent upon β and k. It also acts as a low pass filter as proved in the next sub sections and hence suppresses the high frequency chattering. The resulting control law is computationally expensive and may be simplified by changing its form by using the following proposition. Proposition 1: If f (x) is a real continuous function then it may be expressed as: f ( x ) = f ( x ) sign ( f ( x ) ) .

(12)

Proof: The proposition can be trivially proved by using mathematical induction. Let a = 0, b = β, β > 0, u1 = ud and s(x) = so(x) then by using proposition 1, (11) can be trivially reduced as: u1 = − k sinh −1 ( β s ( x ) ) .

(13)

Remark 1: The controller given by (13) is computationally more efficient implementation as it eliminates the signum function. Remark 2: The IHF is a monotonic smooth and odd function that can be used as a switching function. As shown in the Fig. 1 the slope of the function varies rapidly when β is increased. This results in the variable width boundary layer around the sliding manifold and acts as a excellent low pass filter resulting in the eradication of chattering effect. 3.3. Stability analysis Using the Lyapunov’s method the candidate Lyapunov function is written as: V=

1 T −1 s P s, 2

(14) −1

where P −1 = ⎡⎣ ∂∂sxo B ⎤⎦ is a positive definite (P.D) matrix. Differentiating (14) with respect to time and by using (3)-(4) and (8), V can be given as: V = sT P −1s, V = sT P −1[ so + z ], ∂s ⎡ ∂s ⎤ V = sT P −1 ⎢ o x − o [ f ( x) + Buo ]⎥ . ∂x ⎣ ∂x ⎦

(15)

Muhammad Asad, Aamer Iqbal Bhatti, Sohail Iqbal, and Yame Asfia

4

Simplifying (15) by using (1)-(4) results as: ∂s ⎡ ∂s ⎤ V = sT P −1 ⎢ o Bu1 + o h ( x, t ) ⎥ . ∂x ⎣ ∂x ⎦

(16)

so = cT x, so = cT x,

Using (13) and h(x,t)=Buh results as:

∂s ⎡ ∂s ⎤ V = sT P −1 ⎢ − o B k sinh −1 ( β s( x) ) + o Buh ⎥ , ∂x ⎣ ∂x ⎦ T − 1 − 1 V = s P ⎡⎣ − Pk sinh ( β s ( x ) ) + Puh ⎤⎦ ,

so = cT [ f ( x) + Bu + h( x, t )].

(17) (18)

∂s

where P = ∂xo B. Let sinh −1 ( β s( x)) = η ( s)sign( s), where η ( s ) = | sinh −1 ( β s( x)) |, η ( s ) ∈ » m×m , V = sT P −1[− Pkη ( s ) sign( s ) + Puh ], V = − sT L( s ) sign( s ) + sT uh ,

(19)

where L(s) = kη(s) is a diagonal P.D matrix. m

V = −∑ L(i ,i ) ( si ) si + si uh,i .

(20)

i =1

Equation (20) will be negative definite i.e., V < 0 if:

L(i ,i ) ( si ) ≥ uh,i + Δ i ,

tem described by (1), assuming a SISO system and considering the traditional sliding mode where the sliding surface is given as:

(21)

where ∆i is strictly positive real constant.

(23)

In the presence of matched uncertainties and the parameter variations etc, the best estimate of the equivalent control can be expressed as: ueq = −(cT B ) −1 cT fˆ ( x),

(24)

where (cTB)−1 ≠ 0 and fˆ ( x) is the best estimate of the uncertain system. The overall controller can be given as: u = ueq − k sinh −1 ( β so ).

(25)

Using this control law in (23) under the assumption of eradication of h(x,t) due to the discontinuous controller, the sliding mode dynamics can be given as: so + cT Bk sinh −1 ( β so ) − cT f ( x) + cT B (cT B ) −1 cT fˆ ( x) = 0.

(26)

Let cTB=ψ ∈ ℝ and f ( x ) − fˆ ( x) = ΔF ( x ), so + kψ sinh −1 ( β so ) = cT ΔF .

(27)

Remark 3: The matrix B is a known constant matrix having rank m. Remark 4: The sliding surface so is a design variable and is designed to ensure the positive definiteness of the P−1 matrix. This can be achieved by using iterative design procedure.

Assuming sinh−1(βso) ≈ βso. This assumption is justified in the vicinity of the sliding surface. Using this assumption (27) can be given as:

3.4. Boundary layer analysis The proposed control law is based on the IHF, which is a continuous nonlinear function as depicted in Fig. 1. It results in the boundary layer around the sliding manifold. The work in this sub section is based on the work of [36,37]. The width of the boundary layer is a function of k and β, and may be formulated by using (21). Let Umax=║uh║∞ be the infinity norm of uh. The corresponding gains are denoted as k(p,p) and β(p,p). Changing the inequality in (21) to equality the boundary layer width can be given as:

where λ = kψβ. (28) represents the low pass filter with bandwidth controlled by λ. Hence the chattering is removed due to the low pass filter properties of the proposed discontinuous controller.

k( p, p ) sinh −1 ( β ( p, p ) s p ( x) = U max , ⎡U ⎤ ( β( p, p ) s p ( x) = sinh ⎢ k max ⎥ , ( p , p ) ⎣ ⎦ 1 U Φ k( p , p ) , β ( p , p ) = sinh ⎡⎢ k max ⎤⎥ . β( p, p) ⎣ ( p, p) ⎦

(

(22)

)

From (22) it is concluded that in order to achieve narrow sliding mode boundary the value of k and β should be high. The proposed discontinuous controller possesses low pass filter properties and hence eradicates the chattering phenomenon. It may be proved by considering the sys-

s o + kψβ so = cT ΔF , s o + λ so = cT ΔF ,

(28)

3.5. Disturbance estimation and rejection 3.5.1 ISM disturbance estimation and rejection ISM has intrinsic ability for the estimation and rejection of the disturbances [2,15,38,39]. As already discussed in previous sections the ISM uses two different controllers, uo is the continuous controller and u1 is the discontinuous controller. After the establishment of sliding mode when s(x)= s (x)=0, the following holds: u1 = −uh ,

(29)

which is the estimate of the disturbance and is also called equivalent control u1eq, as it is the average value of the control that achieves the sliding mode. This can be proved as follows [2,15]. Consider the system defined by (1). The sliding surface is defined in (4) where the auxiliary sliding dynamics are defined as: z = −

∂so [ f ( x) + Bu − Bu1 ] , ∂x

(30)

A Smooth Integral Sliding Mode Controller and Disturbance Estimator Design

5

Table 1. DC motor parameters. Parameter name Inductance L Resistance R Inertia of motor J Torque constant Kt Back emf constant Ke Viscous friction coefficient B

Fig. 2. Disturbance rejection controller. where z(0) = −so(x(0)). (30) is achieved by substituting uo=u−u1. Differentiating (4) w.r.t time and using (30) results as: ∂so ∂s [ f ( x) + Bu + Buh ] − o [ f ( x) + Bu − Bu1 ], ∂x ∂x ∂so ∂so s = Buh + Bu1 , ∂x ∂x

s =

∂s

u1eq = −uh .

(31)

However the chattering present in the u1eq needs to be filtered out using a low pass filter that results in the average value termed as u1av. The low pass filter is based on the first order differential equation with time constant μ > 0, chosen sufficiently large to allow the disturbances and plant dynamics. µ u1av + u1av = u1.

(32)

Remark 5: From (31) it can be inferred that any controller u1 or equivalently u1eq that enforces the ISM i.e., s= s =0 can act as a disturbance estimator. 3.5.2 Proposed disturbance estimation and rejection In the proposed method the equivalent control and thus the average control is obtained by using the proposed control law as: u1av = − k sinh −1 ( β s( x)),

disturbance rejection was made by applying both the techniques on a DC motor problem. A DC motor is a typical example of the application of the phase locked loops where the speed is controlled using the PLL [4042]. The mathematical model of the DC motor is given as [2,8]: di = u − Ri − K eω , dt dω J = Kt i − τ l . dt

L

where ∂xo B ≠ 0. Under the sliding s =0 and following holds: u1 = −uh ,

Value 1 mH 0.5 Ω 0.001 kg·m2 0.008 N·mA-1 0.001 V·sec·rad-1 0.01 N·m·sec·rad-1

(33)

where the value of β is selected very high usually greater than 500. This will keep the shape of the switching function close to the signum function (see Fig. 1), resulting in sufficient gain and bandwidth which may be required to accurately estimate the disturbance and also satisfies (31). If the gain is too high then the value of k may be used to reduce the resulting gain. The need of a separate low pass filter is eliminated due to the low pass characteristics of the proposed controller as proved in the previous sub section. The overall disturbance rejection controller is depicted in the Fig. 2. 3.5.3. Simulation A comparison of the traditional ISM based disturbance rejection framework and the proposed controller based

(34)

where i, ω represent the current and angular speed of the motor respectively, τl=Bω is the load torque and u is the input voltage to the motor. Various parameters of the motor and their values are given in Table 1. Let ωr be the reference speed of motor, the angular speed error can be given as e= ωr−ω. Define x1=e and x2= ė as the state variables. The state space in terms of error dynamics can be given as: x1 = x2 , x2 = −a1 x1 − a2 x2 + f ( x1 , x2 , t ) − bu, kk

(35)

K

t e , a = R and b = t . f ( x , x , t ) is the where a1 = JL 1 2 2 L JL drift term and may be given as:

f ( x1 , x2 , t ) = ωr + a1ωr + a2ω r +

τ R τl + l . JL J

Neglecting the drift term and by using the linear part of (35) the continuous controller is designed using the pole placement technique where the arbitrary poles may be selected in the left half plane. The continuous controller can be given as: uo = − Kx,

where K=[0.2615 −0.05] represents the state feedback gains. The sliding surface so is defined as: so = c1 x1 + x2 ,

where c1=1.2. The overall sliding surface is defined by (4). Using (30) ż may be given as: z = −c1 x2 + a1 x1 + a2 x2 + buo .

z is obtained by the integration of the ż. The controller is the sum of continuous controller and the discontinuous controller. The discontinuous controller for the traditional ISMC is defined as: u1 = − k sign( s ),

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Muhammad Asad, Aamer Iqbal Bhatti, Sohail Iqbal, and Yame Asfia

where the gain term k = 1.2 is selected that ensures correct disturbance estimate. The low pass filter used to filter the discontinuous controller mentioned above is given in (32) where the value of μ = 0.1 is used. The discontinuous control law of the proposed ISMC is given as: u1 = − k sinh −1 ( β s ),

Fig. 3. DC motor speed response.

Fig. 4. Disturbance and its estimates (upper) conventional ISMC, (lower) proposed ISMC.

where k = 0.43 and β = 5. The disturbance term is a square wave signal. The simulation results are shown in the Fig. 3 through Fig. 6. In Fig. 3 the speed tracking is depicted. Both the conventional and proposed controllers are tracking the speed. In Fig. 4 the disturbance and its estimates are depicted which show more accurate estimation of disturbance by using the proposed ISM technique whereas in the case of traditional ISM there is some estimation error and residual chattering which may be removed by further tuning the bandwidth of the low pass filter. It can be inferred that in case of unknown disturbance bandwidth the traditional ISM will not always result in the best estimate of the disturbance as the bandwidth of the low pass filter is fixed. In Fig. 5, the controller effort is depicted. There is chattering at the control input in the case of traditional ISM whereas in the case of proposed ISM there is no chattering at the control input. The surface s for both the cases is depicted in Fig. 6. Chattering around the sliding surface is observable in the case of the traditional ISM (upper) whereas in the case of proposed ISM there is no chattering. 4. APPLICATION TO THE PHASE LOCKED LOOPS: A CASE STUDY A PLL is used for the control of phase and frequency of the oscillators. PLL is used in diverse engineering applications such as motor control, digital communication systems etc. In this case study a digital phase locked loop (DPLL) has been implemented using the novel ISMC technique. The model of the PLL is derived from the work of [43,44]. A detailed derivation of the mathematical model follows.

Fig. 5. Controller effort (upper) conventional ISMC, (lower) proposed ISMC.

4.1. Mathematical formulation In Fig. 7 a proportional plus integral PLL is shown. The voltage controlled oscillator (VCO) is the plant and is modeled as integrator. In discrete time the integrator is replaced by the discrete integrator and the plant is known as digitally controlled oscillator (DCO). The loop filter (LF) which is typically a proportional plus integral controller and the phase detector (PD) which gives the phase difference between reference signal and the VCO signal. For a continuous time PLL the output of the VCO can be expressed as:

( ) s (t ) = sin ( 2π f t + 2π f t + φ (t ) + k ∫ V t

s (t ) = sin 2π ft + φ (t ) + kv ∫ Vctrl (τ )dτ , 0

o

Fig. 6. Sliding surface ‘s’ (upper) conventional ISMC, (lower) proposed ISMC.

ε

s(t ) = sin ( 2π f o t + θ (t ) ) ,

t v 0 ctrl (τ )dτ

) , (36)

A Smooth Integral Sliding Mode Controller and Disturbance Estimator Design

Fig. 7. Continuous time proportional plus integral PLL.

7

Fig. 8. DPLL using ISM.

where f = fo + fε is the frequency of the oscillation, kv is the VCOt gain, fo is the nominal frequency and fε = f − fo. φ (t ) = ∫ z (u ) du is the wiener phase process [45] and z 0 is the white noise. From (36) the phase and its derivative can be given as: t

θ (t ) = 2π fε t + φ (t ) + kv ∫ Vctrl (τ )dτ ,

(37)

θ(t ) = 2π fε + z (t ) + kvVctrl (t ).

(38)

0

Let fd be the desired frequency and may be expressed as: θd (t ) = 2π f d .

(39)

The difference of the desired and actual frequency may be expressed as: θd (t ) − θ(t ) = 2π f d − 2π fε − z (t ) − kvVctrl (t ), Δθ(t ) = 2πΔf − z (t ) − e(t ),

(40)

o

where e(t)=KvVctrl(t) and ∆fo = fd −fε is the initial frequency error between the desired and the actual frequency offset from the center frequency. The error in the frequency will converge to zero with time and is modeled as a first order integrator process. The resulting state equations can be given as: Δθ(t ) = 2πΔf (t ) + w(t ) + u0 (t ), Δf (t ) = u (t ),

(41)

where w(t) = −z(t). The inputs u0(t) and u1(t) are related to e(t) as: t

0

(42)

In state space notation (41) can be written as: ⎡ Δθ(t ) ⎤ ⎡0 2π ⎤ ⎡ Δθ (t ) ⎤ ⎡1 0 ⎤ ⎡u0 ⎤ ⎡1 ⎤ ⎢  ⎥=⎢ ⎥⎢ ⎥+⎢ ⎥ ⎢ ⎥ + ⎢ ⎥ w(t ), ⎣ Δf (t ) ⎦ ⎣0 0 ⎦ ⎣ Δf (t ) ⎦ ⎣0 1 ⎦ ⎣ u1 ⎦ ⎣0 ⎦ ⎡1 0 ⎤ ⎡ Δθ (t ) ⎤ (43) y=⎢ ⎥⎢ ⎥, ⎣0 0 ⎦ ⎣ Δf (t ) ⎦

where ⎡1 0 ⎤ ⎡1 0 ⎤ ⎡0 2π ⎤ and C = ⎢ A=⎢ , B=⎢ ⎥ ⎥. ⎥ ⎣0 1 ⎦ ⎣0 0 ⎦ ⎣0 0 ⎦

The pair (A,B) is controllable and (A,C) is observable. The state equations given by (41) and the control input expressed in (42) are discretized using the Euler’s forward rule. The DPLL can be represented in state space as: ⎡ Δθ k +1 ⎤ ⎡1 2π Ts ⎤ ⎡ Δθ k ⎤ ⎡Ts + ⎢ Δf ⎥ = ⎢ 0 1 ⎥⎦ ⎢⎣ Δf k ⎥⎦ ⎢⎣ 0 ⎣ k +1 ⎦ ⎣ ⎡1 0⎤ ⎡ Δθ k ⎤ yk = ⎢ ⎥, ⎥⎢ ⎣ 0 0 ⎦ ⎣ Δf k ⎦

0 ⎤ ⎡u0,k ⎤ ⎡Ts ⎤ w , ⎢ ⎥+ Ts ⎥⎦ ⎣ u1,k ⎦ ⎢⎣ 0 ⎥⎦ k

(44)

where Ts is the sampling period. The inputs u0,k and u1,k are related as: N −1

ek = −u0, k Ts − 2π Ts 2 ∑ u1, n .

(45)

n=0

1

e(t ) = −u0 (t ) − 2π ∫ u1 (τ )dτ .

Fig. 9. Proposed ISM controller.

The state space given above describes the PD and the DCO which are collectively considered as the plant. 4.2. Control objectives and controller design Using the state space model developed in the previous sub section the controller design objectives are: 1) To design a high speed DPLL using the proposed smooth ISM controller design technique for tracking the input signal. 2) To reject the injection tone thereby eliminating the oscillator pulling phenomenon [30,33]. In Figs. 8 and 9 the DPLL and the proposed ISM controller are depicted. Reference input to the DPLL is a phase domain signal at 1MHz. A disturbance input which is an external sinusoidal signal at 1.5 MHz is injected into the DCO. In the feedback path ∆θk is differentiated to produce ∆fk. The proposed ISM controller is used for the disturbance rejection and control purposes. The conti-

Muhammad Asad, Aamer Iqbal Bhatti, Sohail Iqbal, and Yame Asfia

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Table 2. DCO and controller parameters. Parameter Name DCO gain (Kv) Continuous Controller Q matrix Continuous Controller R matrix β1 , β2 K1 K2 Sampling Period Ts

Value 4 10 radians/volt ⎡0.0015 1⎤ ⎢ 1 1⎥⎦ ⎣ ⎡30 0 ⎤ ⎢ 0 10 ⎥ ⎣ ⎦ 5000 -5 10 -6 10 -8 10 seconds

nuous controller uo is designed using LQR method. The discontinuous controller is based on (13). The model has two inputs and therefore requires two sliding surfaces that are designed using LQR method. The continuous controller and the sliding surfaces may be designed using other methods such as H∞. Various parameters of the plant, continuous controller and sliding surface so are given in Table 2. 4.3. Experimental implementation and results The DPLL was implemented in Verilog hardware design language by using MATLAB®, Simulink®, Xilinx ISE® and ModelSim® softwares. The resulting code was synthesized using Xilinx ISE® tool and verified by running on the ML507 FPGA development board. The experimental setup is shown in Fig. 10. The process of implementation can be summarized in following steps. 1) Design the controller, sliding surface and DCO using MATLAB® and Simulink®. 2) If the desired design specifications are fulfilled then go to step 3 else go to step 1. 3) Convert the Simulink® model into fixed point by using Simulink® fixed point toolbox. 4) Compare the results of fixed point simulation and the floating point simulation and remove the errors or bring the results in acceptable regions. 5) Generate the Verilog code with HDL coder. 6) Perform simulation with ModelSim® and Simulink®. 7) If the cosimulation is correct then generate the FPGA In the Loop (FIL) simulation and synthesize the verilog code.

Fig. 10. Experimental setup.

8) If the Verilog code synthesizes at the correct clock speed then go to step 10. 9) Optimize the verilog code for synthesis at a higher clock frequency and go to step 8. 10) Run the FIL simulation. The results achieved are depicted in Figs. 11 through 17. It is evident from 11 through 13 that the reference signal at 1 MHz is being tracked with near zero steady state error. The settling time is around 100 μsec, which can be further reduced by adjusting the controller parameters. In Fig. 14 the controller effort is plotted. The controller effort may be reduced by selecting a higher value of the DCO gain parameter kv. In Fig. 15 the sliding surfaces for the phase and frequency are depicted. The sliding surface is zoomed to show that there is no chattering effect. Also the convergence of the sliding surface shows good disturbance rejection. In Fig. 16 the disturbance signal at 1.5 MHz is plotted. The amplitude level of the disturbance signal and the reference signal are the same. In Fig. 17 the power spectrum of the output of the DPLL is depicted. It is clear from the Fig. 17 that the disturbance signal at 1.5 MHz is canceled by the proposed controller. However a difference signal at 0.5 MHz which is 70 dBm below the reference signal is visible. It may add some phase noise however for all practical purposes it is well attenuated. It can be further eliminated by using higher sampling rates or by using band pass filter at the output.

Fig. 11. Reference signal and PLL output.

Fig. 12. Frequency error.

A Smooth Integral Sliding Mode Controller and Disturbance Estimator Design

Fig. 13. Time domain representation.

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Fig. 17. Output power spectrum. 5. CONCLUSIONS

Fig. 14. Control effort.

In this work a smooth ISMC technique is proposed which serves two purposes: firstly it significantly reduces the chattering effect and secondly it results in better disturbance estimation and rejection. The proposed controller design scheme uses IHF as a discontinuous controller which eliminates the low pass filter required by the traditional ISM to filter the disturbance from the output of the discontinuous controller. The chattering reduction has been proved with the help of boundary layer analysis. Stability of the controller is proved with the help of Lyapunov method. The effectiveness of the proposed novel control scheme has been proved with the help of two diverse examples. The DC motor speed control where the chattering elimination and disturbance cancellation has been shown with the help of simulations. A case study has been described where the proposed controller is used to design the DPLL and is implemented on FPGA development board. Experimental results show eradication of the chattering phenomenon and disturbance rejection.

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[2] Fig. 15. (a) Sliding surface for phase (top), (b) sliding surface for frequency (lower).

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Fig. 16. Disturbance signal at 1.5 MHz.

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A Smooth Integral Sliding Mode Controller and Disturbance Estimator Design

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Aamer Iqbal Bhatti had a Masters in Control Systems from Imperial College London and a Ph.D. in Control Engineering from Leicester University in 1998. He is currently Head of Controls and Signal Processing Research Group (CASPR) at Mohammad Ali Jinnah University, Islamabad, Pakistan. He has over fifteen years of industrial and academic experience of robust controller design and implementation. His work is mainly focused on the application of Higher Order Sliding Mode on the parameter estimation of industrial systems like automotives and reactors. He is author and coauthor of 123 international journal and conference publications.

Sohail Iqbal received his Masters in Computer Science from International Islamic University Islamabad in 1999, Masters in Engineering (control systems) from Center for Advance Studies in Engineering Islamabad in 2005, and Ph.D. in Control Systems from Mohammad Ali Jinnah University Islamabad in 2011. He has been working with industry in the field of control designing for electromechanical systems for last fifteen years. He was research fellow with Department of Engineering, University of Leicester UK in year 2010. His research interests are control theory & robotics systems emphasizing on Higher Order Sliding Mode theory. He is author and coauthor of five journal papers and twenty five conference papers.

Yame Asfia is a digital systems designer and developer, with an experience of more than seven years in this field. Her major research interests include digital systems design and digital signal processing of communication systems. She received her Masters in Computer Engineering from College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Rawalpindi, Pakistan, where her research focused on the DSP specific optimizations.