Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering http://pii.sagepub.com/
Second-order sliding mode controllers: an experimental comparative study on a mechatronic actuator Mohamed Harmouche, Salah Laghrouche, Fayez Shakil Ahmed and Mohammed El Bagdouri Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 2012 226: 1231 originally published online 13 August 2012 DOI: 10.1177/0959651812454061 The online version of this article can be found at: http://pii.sagepub.com/content/226/9/1231
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Original Article
Second-order sliding mode controllers: an experimental comparative study on a mechatronic actuator
Proc IMechE Part I: J Systems and Control Engineering 226(9) 1231–1248 Ó IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0959651812454061 pii.sagepub.com
Mohamed Harmouche, Salah Laghrouche, Fayez Shakil Ahmed and Mohammed El Bagdouri
Abstract Numerous advanced control techniques for actuator control of an engine’s air path have been developed in recent years. Among these approaches, the performance of sliding mode controllers stands out due to their robustness against changes in the physical conditions around the engine. This paper presents the design and testing of three high-order sliding mode controllers and a comparison of their performance in controlling a mechatronic engine air path valve. The most effective strategies are further tested for their robustness against variations in temperature and load conditions. The experimental results show the effectiveness and robustness of the controllers. Comparative results are presented that allow the performance of the considered algorithms to be compared both with respect to each other and with respect to sliding surfaces.
Keywords Second-order sliding mode, mechatronic actuator, practical implementation, experimental results, robust control
Date received: 14 January 2012; accepted: 3 May 2012
Introduction The performance of modern diesel engines depends on the regulation of the intake air to control both power output and pollution levels.1,2 Mechatronic actuators or valves play an important role in the regulation of the engine air path.3–5 These actuators have shorter response times and higher tracking accuracy in transient conditions; however, they are difficult to control because of their non-linear behaviour and parametric uncertainties that arise as a result of changes in temperature, external load perturbations and aero-load dynamics.6–11 Friction plays a significant role in the creation of non-linear behaviour, causing stick-slip, dead zones and part wear.9,10 The electrical and mechanical characteristics of actuators also significantly change with changes in temperature resulting from changes in the conditions around the engine. In the long term, temperature and friction effects result in wear and ageing, effects which permanently change the actuator’s characteristics.8,12,13 Hence, modelling and control of mechatronic actuators is a challenging task since an actuator is expected to perform reliably under expected operating conditions over the full lifetime of an engine.8,11,13 Controllers are implemented either in
the engine contol unit (ECU), or integrated with the actuator. Among the control methods for engine air path actuators presented in the literature, conventional control schemes depend on precise identification of actuator parameters and do not take possible parameter uncertainties into account. Examples of these works are time-delay control,7 cascaded velocity and current controllers,8,11 friction compensation-based controllers9 and adaptive control combining a proportional–integral–derivative (PID) controller with a non-linear feed-forward controller.13 Recently, model-based fault detection schemes have been proposed in the literature.14,15 There is an extensive literature on sliding mode control (SMC) including the derivation of robust controllers for air path control. Indeed, SMC and higher-order SMC is considered to be one of the most effective Laboratoire Syste`mes et Transports, Universite´ de Technologie BelfortMontbe´liard, France Corresponding author: Salah Laghrouche, Laboratoire Syste`mes et Transports, Universite´ de Technologie Belfort-Montbe´liard, Belfort, France. Email:
[email protected]
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methods to control uncertain conditions.16–18 Controllers and observers based on SMC have been proposed for mechatronic throttle valves with nonsmooth non-linearities and parameter uncertainties.19 Higher-order SMC (application of a discontinuous control law to a higher time derivative of the sliding variable) significantly reduces chattering while keeping the desirable robustness property of standard SMC.20-23 Detailed work has been performed on second-order SMC (SOSMC) that has resulted in robust control algorithms that have been used to control mechatronic actuators18,20–25 and air path actuators.19,26–28 While SOSMC solutions exist in the literature, their application so far has remained conceptual, limited to prototypes. Practical implementation and performancerelated issues have not been addressed in detail. These aspects are important in order to keep the actuators affordable; in addition the actuator controllers currently used in the automobile industry have limited real-time processing capabilities. In this paper, we address these issues and provided a comparative analysis of experimental results obtained by individually applying three different SOSMC controllers to control an engine air path actuator. This study is partly a synthesis of control techniques and partly presents an applied control viewpoint. The comparison is based on changing the sliding variable as well as varying operating conditions, such as temperature and load changes. Our contribution is therefore of experimental interest, focused on performance validation of the SOSMC controller for our particular application. Three important measures are taken to demonstrate the implementability of the controllers. First, the controllers are tested on a commercial actuator, already in production and use (controller implemented at the ECU level). Second, the controllers are implemented as output feedback devices since commercial actuators are soley equipped with position sensors. Finally, in order to match the mathematical complexity of the controllers to the capability of a real ECU and microcontrollers, no extra observers are added, limiting the control to output feedback. This paper is written from an experimental perspective. Therefore, the issues of design and practical implementation are highlighted. The design and parametrization of SOSMC algorithms is discussed, considering actuator parametric uncertainties. The algorithms are tested on two sliding surfaces. The first surface is of pure output feedback type, and the second surface is described in terms of a Hurwitz polynomial of the system’s error dynamics. The performance of the algorithms is evaluated with respect to response time and chattering.
Figure 1. Air inlet system of a modern diesel engine.
swirled or perturbed by the geometry of the inlet path leading to the second duct, the opening of which is controlled by the valve. The swirl valve is operated by a rotary mechatronic actuator, which has a working range of 0°–110°. As seen in Figure 2, the actuator consists of a DC motor, a return spring and gearing. Figure 3 shows the quasistatic characteristics of the actuator, obtained by powering the motor with slowly evolving ascending and descending voltage ramps (pulse width modulation (PWM)). The system is non-linear, as can be seen in the characteristic curve shown in Figure 3(a). The significant number of mechanical parts result in friction and stick-slip, spring non-linearity (pre-compression) and the actuator model depends upon estimation for parameter identification. Figure 3(b) shows that the actuator parameters also shift with temperature.
Model A detailed and physics-motivated simulation model is presented; in this model motor–spring systems are characterized by non-linearities due to the presence of spring pre-compression and significant friction.8 Keeping all these factors in mind, the actuator can be modelled in the form of the following electrical and mechanical equations8,29 Va Jtot
= ia Ra + La dv dt
dia + Ea dt
= Tm Tspr Tf TL
ð1Þ ð2Þ
where
Actuator model In modern diesel engines, the swirl valve is integrated into the air inlet manifold (Figure 1). The manifold consists of two air ducts for each cylinder. The first duct lets the air to be aspired directly into the cylinder. Air is
Ea Tm
= Kem rv = Ka v = Ka ia , Tspr = Kspr u + Tpc
Note: The definitions of the parameters are given in Table 1. For more details on modelling, please refer to Ahmed et al.29
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Figure 2. Engine air path actuator: (a) internal structure; (b) 3D cutaway diagram.
Figure 3. Characteristic curves of the actuator: (a) nominal characteristics; (b) variations due to temperature.
The spring torque can be expressed as the sum of a linear component and a pre-compression torque constant. Hence, equation (2) becomes Kspr (Tpc + Tf + TL ) dv Ka Va Ka v = u Ra Jtot dt Jtot Jtot ð3Þ
Friction is a natural resistive force that exists between two surfaces in contact, moving relative to each other.30 Its nature is complex and non-linear (see Figure 4) since it depends on a variety of physical conditions. The primary causes are surface irregularities or asperities, that resist the motion between the two bodies.9,10,12 In this paper, friction is modelled using the dynamic LuGre model, represented by the following equations9,31
8 _ v) = so z + s1 z_ + s2 v Tf (z, z, > > < jvj z Tf = z_ = v so p(v) > > : p(v) = Tc + (Ts Tc )e(v=jvs j)
ð4Þ
where, following De Witt et al.31, z is the average deflection of the surface asperities (modelled as elastic bristles. For details on the LuGre model, the reader is referred to Olsson et al.,10 De Witt et al.31 and De Witt and Lischinsky.32 As seen in equation (3), the system dynamics can be represented by a second-order system with a second member. Friction dynamics on the other hand are not accessible through any means. However, from equation (4), it is clear that if u and v are stable, then z is also stable. Therefore, it can be treated as a stable zero dynamic. Using equations (3) and (4), the complete
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Proc IMechE Part I: J Systems and Control Engineering 226(9)
Table 1. Actuator parameters. Quantity
Parameter
Value
Gear ratio Electromagnetic coefficient Spring constant Winding resistance Moment of inertia Spring pre-compression Static friction Coulomb friction Stribeck velocity LuGre stiffness coefficient LuGre damping coefficient LuGre viscous coefficient
r Kem
27.5 13.82 mN.m/A
Kspr Ra Jtot Tpc Ts Tc vs s0 s1 s2
60.88 mN.m/rad 3.35 O 9.65 3 1027 Kg.m2 119.4 mN.m 51.2 mN.m 47.13 mN.m 0.002 rad/s 2800 53 0.012
model, including friction, can be mathematically expressed as follows x1 = u, x2 = x_ 1 = x2
du , x3 = z, u = Va dt
Kspr 3 x1 K2a 3 x2 Tf (x2 , x3 , x_ 3 ) Jtot Jtot Ra Jtot (Tpc + TL ) Ka + u Jtot Ra Jtot
x_ 2 =
x_ 3 = x2 +
so jx2 j x3 p(x2 )
ð5Þ
where x 2 x R3 is the state vector and u 2 U Rðjuj4umax Þ is the control variable (voltage) where umax is the maximum value (15 V in this study). The identified parameters for the actuator under study, based on the actuator identification scheme given in Scattolini et al.8 and Ahmed et al.29 and the friction model identification scheme given in De Witt et al.31 are given in Table 1.
Parametric and load uncertainties As seen in equation (5), the actuator has numerous linear and non-linear physical parameters associated with it. The exact evaluation of these parameters through
experiments is difficult and uncertainty is expected. It can also be seen that the dynamic parameters depend mainly on motor resistance, which can change with a change in temperature. Hence, their value can be uncertain in changing operating conditions. Friction also poses a serious problem in identification and parametrization. In control, these uncertainties would have to be countered with a robust control scheme. Following Laghrouche et al.24 let us formalize these uncertainties as k1
= k01 + dk1
=
k2
= k02 + dk2
=
k3
= k03 (x2 , x3 , x_ 3 ) + dk3
=
k4
= k04 + dk4
=
k5
= k05 + dk5
=
k6
= k06 (x2 ) + dk6
=
Kspr Jtot
K2a Ra Jtot Tf (x2 , x3 , x_ 3 ) Jtot (Tpc + TL ) Jtot Ka Ra Jtot so p(x2 ) ð6Þ
where k0i (1 4i 46) is the nominal value of the parameter ki , based on the physical parameters identified through bench tests in normal conditions. dki is the uncertainty related to ki , representing parameter variations and uncertainties, such that, jdki j4dk0i 4jk0i j, dk0i being a known positive bound. Hence, system (5) can be re-written as 8 < x_ 1 = x2 x_ = k1 x1 + k2 x2 + k3 (x2 , x3 , x_ 3 ) + k4 + k5 u : 2 x_ 3 = x2 + k6 (x2 )jx2 jx3 ð7Þ
Parametric uncertainties are a major obstacle in control design, as conventional controllers are designed for nominal parametric values. Under a changing environment, these controllers fail to maintain their performance levels. An example is presented in Figure 5, where the actuator under consideration is controlled in
Figure 4. Friction: (a) static and coulomb, (b) static, coulomb and viscous and (c) with Stribeck effect.
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a temperature-varying environment, using PID control with friction compensation, as proposed in De Witt and Lischinsky.32 It can be seen that even with compensation, the controller cannot provide the same performance at higher temperatures, as compared to nominal room temperature conditions. In the subsequent sections, we will design secondorder sliding mode controllers that can provide robust control of the actuator position in the presence of these uncertainties, bounded in a known limit. The aim is to design a robust controller which guarantees that the actuator output x1 = u successfully tracks a reference trajectory xref under parameter and load variations.
SOSMC Let us consider an uncertain non-linear system x_ = f(x, t) + g(x, t)u y = s(x, t)
The SOSMC approach allows the finite time stabilization to zero of the sliding variable s by defining a suitable control function u. Let us consider two sliding variables, s1 and s2 . The system (8) has relative degrees r = 2 and r = 1 with s1 and s2 , respectively. We obtain the following two distinct second-order dynamics A. r = 2 ∂s1 ∂s1 + ½s1 ½f(x) + g(x)u, ∂t ∂x ∂ s1 (x, t)g(x)u = 0 with ∂x ∂2 s1 ∂ ∂ s€1 = 2 + s_1 (x, t)f(x) + s_1 (x, t)g(x)u ∂x ∂x ∂t = fA (x, t) + g A (x, t)u
s_1 =
ð9Þ
B. r = 1 ð8Þ
where x 2 Rn and u 2 R is the input control and is a measured smooth output-feedback function (sliding variable). f(x, t) and g(x, t) are uncertain smooth functions. Assuming that H1. The relative degree r of the system with respect to s is constant and known and the associated zero dynamics are stable.
∂s2 ∂s2 + ½s2 ½f(x) + g(x)u, ∂t ∂x ∂ s(x, t)g(x)u 6¼ 0 with ∂x ∂2 ∂ s€2 = 2 s2 (x; t; u) + s_2 (x, t, u)½f(x) + g(x)u ∂x ∂t ∂ _ + s_2 ((x, t, u)u(t) ∂x = fB (x, t, u) + gB (x, t, u)u_
s_2 =
Figure 5. Actuator under PID control experiencing temperature variations.
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ð10Þ
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Proc IMechE Part I: J Systems and Control Engineering 226(9) 8 < v v = am sgn (j1 ) : aM sgn (j1 )
Figure 6. Convergence of the sliding variable using the twisting algorithm.
Let us assume that H2. Functions f(:) and g(:) are bounded uncertain functions and the sign of control gain g is strictly positive. There exist Km 2 R + , KM 2 R + , C 2 Rþ such that ð11Þ
0 \ Km \ g \ KM , jfj4C
The second-order sliding-mode problem may be expressed in terms of the finite time stabilization problem of the following uncertain second-order system j_ 1 = j2 j_ 2 = f(:) + g(:)v
where, for the previously defined two surfaces 8 8 < f(:) = fA (x, t) < f(:) = fB (x, t, u) g(:) = gA (x, t) B : g(:) = gB (x, t, u) A: : : _ v(t) = u(t) v(t) = u(t) where A : j1 = s1 , j2 = s_1 , v(t) = u(t) _ B : j1 = s2 , j2 = s_2 , v(t) = u(t)
Hence, the SOSMC problem is to find a discontinuous control v that is able to return the system trajectories to the sliding surface or variable (s1 and s2), and then exactly keeps them on that trajectory under parametric variations bounded by Km, KM and C. Such a control algorithm would ensure s = s_ = 0. Under the constraints imposed in equation (11), let us now review the control algorithms used in the study.
Twisting algorithm The twisting algorithm is one of the earliest SOSMC algorithms developed to guarantee finite time convergence of the sliding variable dynamics to zero.17 The twisting algorithm is based on adequate commutation of the control between two different gains, that allow the trajectories to converge towards the origin in finite time.17,18,24 This control algorithm is defined as
if jvj . vmax if j1 j2 40 and jvj4vmax if j1 j2 . 0 and jvj4vmax
where vmax is the maximum limit value of the control (in our case, input voltage) that can be physically applied, while aM and am are strictly positive constants. The parameterization is based on the knowledge of the bounds Km , KM and C. The following conditions are sufficient to ensure finite time convergence and keep the sliding variable at a value of zero18 8 C < 0 \ am \ aM , am . Km : Km aM C . KM am + C The stabilization of the sliding variable in the phase plane is shown in Figure 6 The twisting algorithm is one of the initial higherorder sliding mode algorithms, and is notable for its use of switched gains to achieve finite time convergence. However, its disadvantage is that the controller depends on the derivative of the sliding variable. The calculation of signal derivatives in real-time is a well-known problem in controller implementation.33
Super twisting algorithm The super twisting algorithm18,19,26,27,34 is limited to systems with relative degree 1 with the sliding variable. The output is a largely continuous control signal, in which the discontinuity is ‘hidden’ behind an added virtual state. The algorithm is given as follows v if jvj . vmax v_1 = a sgn (j1 ) if jvj4vmax pffiffiffiffiffiffiffi v2 = u jj1 j sgn (j1 ) v = v1 + v2 where a and u are strictly positive constants. The sufficient conditions for convergence of the closed-loop system in the presence of parametric uncertainty in the system have been presented in Levant.18 These conditions are based on knowledge about the bounds on the sliding variable and its derivatives. In order to facilitate control design, a generalized set of conditions has been presented in Fridman and Levant35 and proved in Levant.36 The latter conditions have been used in our study, given as follows a.
C 4CKM (a + C) , u2 5 Km K3m (a C)
The stabilization of the sliding variable is shown in Figure 7 The greatest advantage of the super twisting algorithm is that it does not depend on the derivative of the sliding variable. Also, since the control signal is almost continuous, the level of chattering is considerably lower than for other algorithms. However, the limitation that it can only be applied to variables with relative degree 1
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Harmouche et al.
1237 continuous Lipschitz function L(t)), the first-order differentiator has the following form,39 also illustrated in Figure 8 z_o = l2 L1=2 jzo sj1=2 sgn(zo s) + z1 z_1 = l1 Lsgn(z1 zo )
Figure 7. Convergence of the sliding variable using the super twisting algorithm.
restricts its use, as for systems where the control variable appears in the third or fourth derivative of the output for example, the designer is obliged to calculate multiple derivatives of the output variable, in order to construct the sliding variable itself.
Quasi-continuous algorithm The quasi-continuous second-order sliding mode controller33 is a feedback function of j1 and j2 which is continuous everywhere except on the sliding manifold. It has the following form pffiffiffiffiffiffiffi j2 + jj1 jsgn(j1 ) pffiffiffiffiffiffiffi u= a jj 2 j + jj 1 j where a is a strictly positive constant. The advantage of this controller is that using the homogeneity reasoning for finite time controllers23 results in an improved transient as compared with many other higher-order sliding mode controllers, such as in Levant.37,38 The main disadvantage, however, is that there are no formal conditions to parameterize this controller, and while there is only one parameter to tune, its adjustment is empirical. Also, as for the twisting algorithm, it requires the derivative of the sliding surface. Remark 1. The tuning of parameters with respect to imposed constraints is important because very high controller gains result in unwanted oscillations and chattering.
where zo and z1 are the estimates of s and s_ , respectively. The choice of parameters l1 and l2 is empirical, and in Levant33 the values 1.1 and 1.5 , respectively, were considered to be sufficient. In this case, the differentiator depends only on knowledge of the Lipschitz constant. Since this constant serves as a gain in the differentiator, in cases where it is not exactly known, a large enough approximation is sufficient. However, its order must be comparable to that of the actual value, since taking too large a value would result in larger errors and in extreme cases, oscillations in the derivative output. Due to this dependence on the Lipschitz constant, the differentiator has to be adapted to each application. In this study, the applicable value of L was tuned through simulation by evaluating the differentiator using the following function f(t) = 0:5 sin (0:5t) + 0:5 cos (t)
_ Figure 9 shows the simulation results (f(t)) based on the differentiator L = 260. It can be seen that the derivative calculated by the robust differentiator follows the actual derivative very closely.
Control design The control objective is to design a robust control for the actuator represented by equation (7), which guarantees that the actuator output x1 = u tracks a reference trajectory xref under parameter and load variations. As can be understood from the discusion in the previous section, the SOSMC approach requires two steps. The first step is to select a sliding surface or variable suitable for the control task, the second step is to design a control algorithm that makes the variable and its first derivative converge to the sliding manifold. In order to achieve the control objective, two sliding variables are studied in this paper (defined by s1 and s2 ) along with
Robust differentiator It can be seen in the previous subsections that two of the three control algorithms under consideration depend on the time derivative of the sliding variable. Differentiation of signals in real-time is a well known obstacle in practical implementation of controllers.37 Levant33,37,39 has proposed an arbitrary order exact finite time convergent differentiator to combine with the feedback controller. For a given function s with approximately known Lipschitz constant L (or
Figure 8. Differentiator structure.
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Proc IMechE Part I: J Systems and Control Engineering 226(9) Sliding variable s2. The sliding variable has been chosen as the following Hurwitz polynomial s2 = e_ + le where e = x1 xref
The system has a relative degree r = 1 with respect to the sliding variable. Consider the time derivatives of s2 s_2 = k1 x1 + k2 x2 + k3 (x2 , x3 , x_ 3 ) + k4 + k5 u + l(x2 x_ ref ) x€ref € s2 = A2 + B2 u_
ð13Þ
where A2 = k1 x2 + k2 ½k1 x1 + k2 x2 + k3 (x2 , x3 , x_ 3 ) + k4 + k5 u + k3 (x2 , x3 , x_ 3 ) + l½k1 x1 + k2 x2 ... + k3 (x2 , x3 , x_ 3 ) + k4 + k5 u (l€ xref + xref ) = : Ao2 + dA2 B2 = k5 = : Bo2 + dB2
Figure 9. Differentiator simulation.
where three control algorithms. The algorithms and their parameterization constraints were defined in the previous section. In this section, the sliding variables are defined and the constraints with respect to equation (7) are evaluated in order to parameterize the algorithms.
Ao2 = k01 x2 + k01 k02 x1 + k202 x2 + k02 k03 + k02 k04 + k02 k05 u + k3 + l(k01 + k02 + k03 + k04 + k05 u) dA2 = dk1 x2 + k01 dk2 x1 + k02 dk1 x1 + dk1 dk2 x1 + 2k02 dk2 x2 + dk22 x2 + k02 dk3 + k03 dk2 + dk2 dk3 + k02 dk4 + k04 dk2 + dk2 dk4 + k02 dk5 u + k03 dk5 u + dk2 dk5 u
Sliding variables
+ l(dk1 + dk2 + dk3 + dk4 + dk5 u)
Sliding variable s1
Bo2 = k05 , dB2 = dk5
The first sliding variable has the following form
where, Ao2 , Bo2 , dA2 , dB2 have the same significance as in the case of s1 .
s1 = x1 xref
The system has a relative degree r = 2 with respect to this variable. The time derivatives of this surface are as follows € s1 = k1 x1 + k2 x2 + k3 (x2 , x3 , x_ 3 ) + k4 + k5 u x€ref ð12Þ
€ s1 = A1 + B1 u
where A1 = k1 x1 + k2 x2 + k3 (x2 , x3 , x_ 3 ) + k4 x€ref = : Ao1 + dA1 B1 = k5 = : Bo1 + dB1
where Ao1 = k01 x1 + k02 x2 + k03 (x2 , x3 , x_ 3 ) + + k04 x€ref dA1 = dk1 x1 + dk2 x2 + dk3 (x2 , x3 , x_ 3 ) + dk4 Bo1 = k05 dB1 = dk5
where Ao1 and Bo1 are known nominal expressions (based on k0i ) while dA and dB contain all uncertainties due to parameter and load variations (based on dki ). The disturbance d is uncertain but bounded. The problem of SOSMC now is equivalent to the stabilization of equation (12) to zero in finite time.
Feedback linearization The control problem is equivalent to the finite time stabilization of equations (12) and (13). In order to simplify the considered problem, we have chosen the notations A and B for both surfaces, since the same generalization has been followed. However the actual expressions for A and B are unique to each surface, as derived in the previous sections. In the complete system model, if we consider feedback linearization, applying control to cancel all known non-linearities, we would get For s1 u = B1 o1 ðAo1 + vÞ
For s2 u_ = B1 o2 ðAo2 + vÞ
Applied to equations (12) and (13), we get for both cases s€i = A^i + B^i v
where
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Harmouche et al. dBi ^ dBi , Bi = 1 + A^i = dAi Boi Boi
1239 ð14Þ
where i = 1, 2 refers to the surface. In fact, the control laws u and u_ are functions which would permit feedback linearization if the system had been modelled perfectly, without uncertainties, i.e. (dA = dB = 0). As dA and dB are not equal to zero, the system remains non-linear. Nevertheless, this feedback allows narrow bounds on the parameters A^ and B^ as compared to A and B.
Controller parameters
The values for A^ and B^ and the respective bounding parameters are given in the following sections. For variable s1. The values for A^ and B^ for this variable were calculated to be A^1 = 19:152 3 103 , B^1 = 0:7, 1:3
For these values, the following positive constant parameters were chosen to evaluate the controller parameters Km1 = 0:6, KM1 = 1:4, C1 = 19:560 3 103
Let us recall hypothesis H2, which states that there exist positive constants C, Km , KM , such that A^i 4Ci (c1) 0 \ Kmi \ B^i \ KMi (c2) As seen in the previous section, the parameters of the control algorithm are determined respecting these constraints. The problem of SOSMC now is equivalent to the stabilization of s€1 and €s2 to zero in finite time. The disturbance d is uncertain but bounded. It is hence left to be compensated by the controller parameters. In this section, the parameters of the three control algorithms discussed in the previous section, are determined to satisfy the binding conditions. Figures 3(a) and 3(b) shows the open-loop characteristics of the actuator. The characteristic shift is visible as the temperature increases, i.e. at higher temperatures, more voltage is required to move the motor and the gain (slope of the opening and closing line) also decreases. Friction is seen to increase as well (as suggested by the increase in stick-slip). The actuator’s physical parameters were experimentally determined using the methods given in Scattolini et al.,8 Ahmed et al.29 and De Witt et al.,31 the values are given in Table 1. These values were obtained at room temperature under ideal conditions (proper lubrication, clean environment). Hence, they serve as our set of nominal values to determine k0i . The parametric variations were evaluated by openloop testing of the actuator. Based on the results, a variation of 630% was measured in the resistance and 610% in the static gain. These variations were distributed amongst the parameters of equation (7) as 61% with respect to k01 , 630% with respect to k02 , 610% with respect to k03 , 610% with respect to k04 and 630% with respect to k05 . The changes in parameter k06 were considered to have been incorporated in the friction torque; the nominal value of the latter was taken as the static friction value. All the parameters (nominal and variations) were calculated using the actuator’s physical parameters listed in Table 1. The distribution of these variable was based on the physical characteristics of the actuator, and was validated using simulation and experimental results. The controller parameters were then evaluated, using the condition sets (c1) and (c2). B^ has two values, corresponding to the upper and lower uncertainty bounds.
For variable s2. For the sliding variable parameter l = 100 the following values were found for A^ and B^ A^2 = 2:134 , B^2 = 0:7, 1:3
A significant difference can be seen between the values of A^ in the two variables. B^ is not affected by the choice of surface and remains the same as in the case of s1 . For these values, the following positive constant parameters were chosen to evaluate controller parameters Km2 = 0:6, KM2 = 1:4, C2 = 2:65
The controller parameters were evaluated following these bounding conditions, as expressed in the algorithms. As mentioned previously, if parameters are chosen to be very large, they result in unwanted oscillations and chattering. The parameters of the twisting algorithm were adjusted so that they would satisfy the conditions of both the surfaces. Since no stability conditions are present for the quasi-continuous algorithm, its value was determined empirically, through simulations, which are presented in the next section. The parameters of all the controllers are given in Table 2. The simulation results based on these parameters are presented in the next section.
Simulations In order to choose the control parameters, the three algorithms used in this paper were simulated using the actuator model given in equation (5). The parameters under which the controllers gave the best results are listed in Table 2. As this study is focused on experimental results, the simulations were perfomed solely to determine parameters. Performance and robustness issues were addressed in experimental studies.
Test trajectory The primary control problems for control valves are regulation and tracking. Therefore, a reference trajectory (shown in Figure 10) was designed to tune and test the controller’s performance on step and linear tracking problems. This trajectory is an industrial
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Table 2. Controller parameters. Algorithm
Parameters
Value
Twisting
am aM a a u l1 l2 L
400,000 1000,000 250 5 15 1.1 1.5 260
Quasi-continuous Super twisting Differentiator
Figure 10. Reference trajectory.
benchmark, and is used by a curent manufacturer of EGR valves in the performance evaluation of their systems. The manufacturer uses this trajectory at a sampling rate of 1 kHz. In compliance, the simulations were carried out at1 ms. In the first part of the test, a step input was used to characterize the time response of the actuator, after a low-to-high step and a high-to-low step. This part of the tests allows measurement of the time response (settling time), overshoot and positioning characteristics (steady state error, chattering). In the second part, the ramp was used to evaluate the tracking efficiency of the control law. The different slopes characterize any changes in the system with change in direction. The stationary region between the ramps, provides a zerovelocity region between the two ramps and also helps the comparison between positioning characteristics and tracking characteristics (tracking error, chattering while in motion).
Simulation results The performances of the control algorithms is shown in Figure 11. It can be seen that in the simulations, the super twisting algorithm provides the lowest level of chattering. In terms of time response, the twisting algorithm is the fastest. The quasi-continuous controller cannot achieve a time response better than 180 ms. However, it is important to consider that its parameter’s have to be determined empirically since there are no bounding conditions applicable to this algorithm.
Experimental setup All the controllers except the super twisting controller were tested on both sliding variables. The super twisting algorithm was implemented only on variable s2 since it is the only one with respect to which, the system has relative degree 1. The same reference trajectory was applied on the controllers, as used in simulations.
Test bench The experimental set-up consisted of an actuator test bench which was controlled by LabView, using the
National Instruments CompactRio system. The system (shown in Figure 12 has a high resolution dynamometer and encoder. The dynamometer can measure torque up to 1 Nm with a resolution of 0:001 Nm. The incremental encoder can measure angles with a resolution of 0.088°. In order to test the actuators at different operating temperatures, the test bench has a controlled temperature chamber that can heat the actuators to 150 °C. An external hysteresis brake allows torque variation on the actuator shafts to simulate load variations.
Controller The National Instruments Compact RIO system contains a real-time controller, an FPGA and a configurable chassis on which any combination of input/output (I/O) modules can be installed. In this study, the control algorithms were implemented in the real-time controller to ensure determinism. The FPGA was used to control the I/Os, since it provides more reliable data transfer between the modules and the RT controller, given its speed (up to 40 MHz). The actuator was operated using PWM at a frequency of 500 Hz whereas data acquisition was carried out at 1 kHz, both values chosen to be in compliance with the test conditions proposed by the commercial manufacturer.
The single-edge nibble transmission protocol To keep the operation as close to the real-life application of the actuator, the position sensor integrated in the actuator was used for feedback. The actuator under consideration comes with its own data transmission protocol, called single-edge nibble transmission, or the SENT protocol, intended for use in applications where high-resolution sensor data needs to be communicated from a sensor to an ECU. It is intended as a replacement for the lower resolution methods of 10-bit ADCs and PWM and as a simpler low-cost alternative to CAN or LIN data buses. Designated by the SAE task force as J2716, SENT is a unidirectional communications scheme from sensor/transmitting device to controller/ receiving device. The protocol has been so designed that no coordination signal is required from
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Figure 11. Simulation results: (a) step response; (b) tracking response.
the controller/receiving device. The sensor signal is transmitted as a series of pulses with data measured as falling to falling edge times. The SENT protocol data packet (Figure 13) transmits a sequence of nine pulses repeatedly. Following a 3 ms timing (this period shall henceforth be referred to as ticks, 3 ms = 1 tick), the width of each pulse determines the nibble value coded in it. The pulse width is measured from one falling edge to the next falling edge. Each pulse has a small time of about, but no less than
4 ticks. The rest of the pulse time is large. The transmission sequence is as follows:
calibration pulse (no data); status/serial pulse (4 bit); channel 1: nibble 1 (4 bit); channel 1: nibble 2 (4 bit); channel 1: nibble 3 (4 bit); channel 2: nibble 1 (4 bit); channel 2: nibble 2 (4 bit);
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Figure 12. Actuator test bench.
channel 2: nibble 3 (4 bit); cyclic redundancy check CRC (4 bit).
Experimental results The performance criteria used to judge the controllers were steady state error, settling time and chattering level. Tests were conducted in two stages. In the first stage, all controllers were tested in nominal conditions (room temperature), to determine their performance in terms of the considered criteria. In the next step the controllers whose performance was satisfactory, were tested at a high temperatures to verify their robustness. In this stage, they were expected to maintain their performance over the entire temperature range.
Nominal conditions In the first stage of tests, all controllers were tested at no load and room temperature. These results are shown in Figure 14. There is only one result for the super twisting controller since it can only be applied to s2. It is clear that in terms of steady state error, all sliding mode controllers bring the system error to zero, validating the finite time stabilization claim.
Sliding variable s1. The performance of the controllers on s1 shows that this surface is prone to chattering (Figure 14). The twisting algorithm has a good settling time of 100 ms, and no steady state error is present. However significant chattering oscillations are present, with a magnitude of 0:1 rad. This chattering level is unacceptable since these oscillations are large enough to disrupt air flow through the actuator. The performance of the quasi-continuous algorithm is worse on s1 . As for the twisting algorithm, no static error exists. Its settling time is 300 ms. Its level of chattering is slightly less than for the twisting controller, however, it is still significant enough to rule this controller out for actual application. Sliding variable s2. The performance of all the controllers improves on s2 , with respect to chattering. As can be seen in Figure 14, the chattering level obtained by the twisting algorithm is reduced drastically with the use of this surface, from 0.1 to 0.03 rad. This chattering level, however, is still noticeable and hence higher than acceptable. The settling time of the twisting algorithm deteriorates, increasing from 100 to 210 ms. Improvement is seen in the performance of the quasi-continuous controller, with respect to both chattering level and settling time (Figure 14). On s2 , the
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Figure 13. SENT transmission packet.
quasi-continuous algorithm achieves a settling time of 120 ms and the chattering level is also reduced to 0.01 rad. Hence, the quasi-continuous controller qualifies as a viable option for practical application of our actuator. The super twisting algorithm was tested for the first time on s2 . Its performance is better than both the previous algorithms, achieving a settling response of 110 ms with a chattering level in the order of 0.005 rad (Figure 14). This controller hence qualified for the next stage of experiments.
Temperature tests In the previous section, we saw that two controllers proved themselves as viable options for practical application on the actuator under consideration. The quasicontinuous controller (on s2 ) and the super twisting controller were tested for robustness under parametric drift due to temperature changes. The actuator was heated to temperatures between 20 and 100 °C. Given the imposed conditions of uncertainty bounds, the controllers were expected to maintain their performance level (time response, chattering level) over the temperature range if the controllers compensate for the parametric variations. In Figures 15 and 16, it can be seen that both controllers maintained their performance at high temperatures. The quasi-continuous controller maintained its time response at 120 ms and chattering level to 0.01 rad. The super twisting controller maintained its time response at 110 ms and chattering level to 0.005 rad. Hence, these controllers are robust for application in environments with changing temperatures.
Sinusoid tests In addition to trapezoidal tracking, a sinusoidal trajectory was also used in experimental performance evaluations of the actuators. The sinusoidal trajectory allows us to evaluate the controller performance of mechanical systems during movement or velocity direction inversion, as it demonstrates the effects of friction and backlash on the controller.40 Two frequencies were used in this work: 0.1 and 1 Hz. The results plotted in Figure 17 show how the controllers change their output to compensate for the increased velocity. It can be seen that the controllers are robust against non-linear changes due to a continuously varying trajectory. However, as the frequency increases, the tracking error increases due to the demand for a smaller response time which the controller fails to provide. This is inherently due to the inertial constraints of the actuator itself, which limit the acceleration.
Conclusions In this paper, three sliding mode control algorithms were applied to the control problem of an actuator, in order to evaluate their performance in the presence of uncertainties. Different sliding surfaces were used to control a mechatronic swirl actuator, along with different SOSMC algorithms. The actuator was modelled and friction non-linearities were incorporated into this model. Control strategies were simulated and their parameters determined. Differences were noticeable between simulation and experimental responses in terms of magnitude, but the performances matched qualitatively. While all controllers performed well in tracking, chattering proved to be a problem in their application.
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Figure 14. Comparison: desired and actual trajectories (nominal conditions); (a) step response; (b) tracking response.
The closed-loop system performed better in terms of response time and chattering when a surface of relative degree 1 was considered. Among the control algorithms, the chattering phenomenon was reduced significantly using the quasi-continuous controller and was almost negligible when the super twisting algorithm was applied. Both these controllers also proved their worth in terms of robustness. Hence, sliding mode
control proved to be an effective control strategy in the presence of uncertainties.
Funding This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.
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Figure 15. Quasi-continuous controller experiencing temperature variations: (a) step response; (b) tracking response.
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Figure 16. Super twisting controller experiencing temperature variations: (a) step response; (b) tracking response.
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Figure 17. Tracking a sinusoidal trajectory: (a) super twisting; (b) quasi-continuous.
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