International Journal of Automation and Control Engineering (IJACE) Volume 3 Issue 3, August 2014 doi: 10.14355/ijace.2014.0303.03
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Robust Sliding Mode Controller for Uncertain Nonlinear Systems with Fast Transient Lakshminarayan Chinta V.1, Ma’moun Abu-Ayyad*2, Rickey Dubay3 Center for Addiction and Mental Health, Toronto, ON Canada, M5S 2S1
1
Mechanical Engineering Department, Penn State Harrisburg, Middletown, USA - PA 17057
*2
Mechanical Engineering Deprtment, University of New Brunswick, Fredericton, NB Canada, E3B 5A3
3
lak.chinta@camh.ca; *2maa21@psu.edu; 3dubayr@unb.ca
1
Abstract This paper proposes an improved self-tuning sliding mode control for nonlinear uncertain systems. The proposed approach is based on the principle of approximating the gradient of cost-to-go using the dynamics of the sliding variable and a modified error. Validity of the proposed control design is established through Lyapunov analysis. The effectiveness of the proposed control design with fast transient response was demonstrated on a second order and 2-joint robot manipulator nonlinear systems with uncertainties. The results showed that the robustness of the method is guaranteed and the system arrived at the sliding manifold rapidly. Keywords Nonlinear Control; Sliding Variable; Self-Tuning
Introduction Sliding mode control (SMC) is a control strategy that exhibits robust performance in the presence of large uncertainties in plant dynamics (Lian, 2013), (Xia, 2003), (Park, 2009). Over the years, SMC has been developed and found wide applications in process of control, aerospace and biology (Wang, 2004), (Hess, 2003), (Chang, 2010). The central concept of SMC is the sliding variable , which is a linear combination of the tracking error and its temporal derivatives. SMC has two phases; a reaching phase that drives the system to the sliding manifold (where ), followed by a sliding phase during which the system moves in the sliding manifold, i.e. holding at 0, so that the tracking error decays to zero exponentially. During the sliding phase (i.e., ), the system is insensitive to uncertainties in the plant and disturbance and the closed-loop response is determined by the sliding manifold only (Kuo, 2008), (Huang, 2008). The task in SMC is to design a control law that drives the states to the sliding manifold and stays there. If the
system is driven to in finite time and stays there, the original SMC has the drawback of control chattering (Lee, 2001). The chattering causes the control signal to oscillate around the zero sliding surface resulting in unwanted closed-loop response (Bartolini, 2003). To overcome the chattering effect, an appropriate controller can be designed by using Lyapunov methods. A self-tuning SMC approach using Lyapunov method was developed by (McDonald, 2008) and a PID based sliding mode controller using Lyapunov method was proposed by (Li, 2001). In this work, the PID controller determines a continuous input to replace the discontinuous switching of the SMC in order to eliminate chattering. (Jiang, 2002) developed an adaptive SMC utilizing the system states and perturbation including the influence of system nonlinearities and disturbances. The drawback of this approach is that a steady state error may occur even though the system trajectory remains in the boundary layer. SMC based on a first order plus dead time (FOPDT) model was proposed by (Camacho, 2000). In this approach, the controller has a fixed structure with a set of tuning equations using the model parameters. The disadvantage of this controller is that the FOPDT model is limited to low order nonlinear processes. A fractional interpolation based SMC approach to eliminate the limit cycle of unmodeled systems was proposed by (Xu, 2003). In this method, a conventional SMC with signum function were used to maintain a reasonable tracking boundary in controlling a DC servo motor. The drawback of this approach is that when the unmodeled system has a relative degree more than two, limit cycles are unavoidable even with a perfect switching mechanism. Designing a fuzzy based SMC was studied by (Lin, 2002) where the chattering problem of the SMC was
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International Journal of Automation and Control Engineering (IJACE) Volume 3 Issue 3, August 2014
avoided by fuzzy partition of the manipulated variables. Even though the fuzzy logic schemes are robust controllers for nonlinear processes, it is very difficult to guarantee the stability of the control system especially with the linguistic expression and large number of fuzzy rules (Guo, 2003). Recently, (Eker, 2010) developed a second order SMC approach using a PID sliding surface with constant parameters incorporated with Lyapunov stability method. The main difference between first-order and higher order SMC is that, the control signal acts on higher derivatives of the sliding surface (Mihoub, 2009). Therefore, the idea of this method was to enforce the first derivative of the sliding surface to zero such that . This method was implemented experimentally on controlling the speed of a DC motor connected to a load (Eker, 2010). The complexity of this approach is to generate a secondorder sliding mode on a chosen sliding surface . In this work, an improved reaching control law is proposed for better transient performance during the reaching phase. This approach is based on the principle of approximating the gradient of the optimal cost-to-go, , and self tuning the weighting parameter with a modified error. The details of the proposed control design are given in Section 2 and its effectiveness is demonstrated with examples in Section 3.
(4) is the nominal continuous control and where the reaching control law given by
(5) If the system follows Eq. (5), the system is driven to in infinite time. The control law during the sliding phase (equivalent control) is given by (6) (Kuo, 2008) proposed a control law that is parameter self-tuning making the tracking error go to zero such that (7) This control law was based on self-tuning the parameters and which makes the tracking error go to zero. The characteristics polynomial specifies the output error dynamics in the sliding mode. Furthermore, these self-tuning control laws were designed based on Lyapunov stability method as follow (8) (9) where
and
are positive constants and
Initial Considerations
replaced by
Consider a class of nonlinear uncertain system (Hess, 2003) given by
Controller Design
(1) (2) where , and are the nominal parts of the plant. The uncertain parts are and which are assumed to satisfy the matching conditions to ensure the robustness of the controller (Baik, 2000), (Drazenovic, 1969). If the desired state is , then the tracking error is, . Then, the sliding variable is defined as (3) where , and are chosen such that is the Hurwitz polynomial. An appropriate controller can be designed by Lyapunov methods i.e., if and as makes , . The original SMC control law (Kuo, 2008) is given as
86
is
was
.
The proposed algorithm defines the control signal acting on the real plant as the sum of three quantities as (10) The equivalent control, , as in Eq. (6) is used as a feedback control to restrict the error to the sliding surface and provides the main control action (Mihoub, 2009). The equivalent control is calculated using the process parameters with no disturbance. In SMC scheme, the reaching phase starts when the error moves toward the sliding manifold. The proposed control algorithm introduces a new reaching control law, , that replaces the discontinuous reaching control law in Eq. (5) is given as (11) where R, the control weighting matrix, is a positive definite matrix and is the approximate gradient of
International Journal of Automation and Control Engineering (IJACE) Volume 3 Issue 3, August 2014
the cost-to-go function defined as
(20) (12)
where
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Then by chain rule,
is the loss function given as
(21)
(13) is a positive definite matrix and a where positive semi-definite matrix. Differentiating Eq. (12) along the system trajectories, we get the generalized Hamilton-Jacobi-Bellman (GHJB) equation (AbuKhalaf, 2004) given as (14) Solving Eq. (14) for and then find an improved control based on using Eq. (12). The idea of the new scheme is to approximate the cost-to-go function as (15) The dynamics of the sliding variable is used to approximate the gradient of the cost-to-go which is given as (16) Using Eq. (16) in Eq. (14), we define the error
as
(17) The objective is to make . To solve the GHJB equation, we introduce the model control, , which is given as (18) Next we show how the inverse of the control can be self-tuned. Define weighting matrix then can be self-adapted by
for
(22) (23) From the above analysis, the derivative of the Lyapunov function is negative definite so the stability of the proposed method is guaranteed. Therefore, the system in Eq. (1) with a reaching control law as in Eq. (11) results in an improved transient response during the reaching phase. Results Consider a second-order nonlinear uncertain system (Kuo, 2008) (24) Nominal and uncertain parts of the plant in the simulations of the proposed control design were compared with self-tuning SMC (Kuo, 2008). The same initial conditions were used for both approaches. The parameter values are taken as , , , . The simulation results of the proposed SMC and the self-tuning SMC (Kuo, 2008) are shown in Figures (1-4). Figure 1 (a & b) represents the states and sliding variable of the proposed and original sliding mode control respectively. Figure 2 (a & b) represents the control weighting parameter and the loss function. Figure 3 (a & b) are the equivalent and reaching control. Figure 4 (a & b) are the control gain parameter and control input to the system. The effectiveness of the control design is demonstrated by the superior transient response of the states and the sliding variable. There is no control chattering and the proposed controller shows robust performance.
(19) where the modified error is with for a regulation problem. Equation (19) need not have a discrete representation of
as
is a
function of both and . The validity of the control design Eq. (19) can be analyzed using the Lyapunov stability method. Here, we choose the Lyapunov function,
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International Journal of Automation and Control Engineering (IJACE) Volume 3 Issue 3, August 2014
Control input (proposed)
Control gain parameter
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FIG. 1 (A) STATES (B) SLIDING VARIABLE
2 gamma (proposed)
1.95
1.9
0
1
2
3 Time(sec)
4
5
6
0
1
2
3 Time(sec)
4
5
6
0
-10
-20
-30
FIG. 4 (A) CONTROL GAIN PARAMETER (B) CONTROL INPUT
The second example is a 2-joint rigid robot manipulator where the kinetic energy equation is given as (Murray, 1994) (25) where: FIG. 2 (A) CONTROL WEIGHTING (B) LOSS
,
2 x1 (proposed) x2 (proposed)
States
1
,
0
,
.
The parameters are given as
-1 -2
, 0
1
2
3 Time(sec)
4
5
,
1.5 Sliding variable
,
,
,
6
,
.
Taking the Lagrangian Ï„ of the energy equation T yields:
s (proposed) 1 0.5 0 -0.5
0
1
2
3 Time(sec)
4
5
6
(26)
FIG. 3 (A) EQUIVALENT CONTROL (B) REACHING CONTROL
The control weighting parameter, , is a
identity matrix.
Introducing the uncertainty in the plant as
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,
International Journal of Automation and Control Engineering (IJACE) Volume 3 Issue 3, August 2014
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2
sliding variable
. x1
0.4 0.2
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
x2
4
3 2.5 2 1.5
0.1
0
1
2
3
4
5 6 Time (sec)
7
8
9
10
FIG. 7 (A) SLIDING VARIABLE (B) CONTROL WEIGHTING
0
0
1
2
3
4
5 6 Time(sec)
7
8
9
3
10
FIG. 5 STATES (A) JOINT-1 (B) JOINT-2
Figures (5-9) show simulation results for the 2-joint rigid robot manipulator. Figure 5 (a & b) represents the states of the joint 1 and 2 respectively. Figure 6 (a & b) represents the respective reaching control for joint 1 and 2. Figure 7 (a & b) illustrates the sliding variable and control weight parameter for joints 1 and 2. Figure 8 (a & b) are the equivalent control (joints 1 and 2) and loss function.
Equivalent control
-0.1
joint 1 joint 2
3.5
joint 1 joint 2
2 1 0 -1
0
1
2
3
4
0
1
2
3
4
5
6
7
8
9
10
5 6 Time (sec)
7
8
9
10
15
10 Loss
States- joint 2
x1
0.2
Control weighting parameter
-0.2
0.3
5
0
0
-2
FIG. 8 (A) EQUIVALENT CONTROL (B) LOSS
-4 1
-6 -8
0
1
2
3
4
5
6
7
8
9
10
6
joint 1 joint 2
0.5 0 -0.5 -1
0
1
2
3
4
0
1
2
3
4
6
7
8
9
10
6 5 Time (sec)
7
8
9
10
5
4 6
2 GHJB error, E
Reaching control-joint 1
0.5
0
-0.4
Reaching control-joint 2
1
x2
GHJB model control
States- joint 1
0.6
joint 1 joint 2
1.5
0 -2
0
1
2
3
4
5 6 Time (sec)
7
8
9
FIG. 6 REACHING CONTROL (A) JOINT-1 (B) JOINT-2
4 2 0
10 -2
FIG. 9 (A) MODEL CONTROL (B) GHJB ERROR
Figure 9 (a & b) illustrate the model control (joints 1 & 2) and the GHJB error, E (as in Eq. 17). The effectiveness of the proposed controller is demonstrated by the superior transient response of the states and the sliding variable in the presence of
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International Journal of Automation and Control Engineering (IJACE) Volume 3 Issue 3, August 2014
uncertainties in the plant. There is no chattering and the proposed control scheme shows robust performance. Conclusions A reaching control law based on approximating the gradient of the cost-to-go function was presented. The dynamics of the sliding variable is used in the approximation of the gradient and the weighting parameter is self-tuned with a modified error. The validity of the control design was analyzed using the Lyapunov stability method. The proposed controller was tested in simulation on nonlinear systems with uncertainties. The results of the proposed controller were compared to self-tuning SMC with improved performance. The system arrived at the sliding mode rapidly and improved the character of the reaching phase. The proposed approach has significant potential for other researchers on learning the dynamics of the plant and cost-to-go function.
pneumatic muscle actuators.” Control Engineering Practice, Vol. 18, pp. 13-22, 2010. Drazenovic, B. “The invariance condition in variable structure systems.” Automatica, Vol. 34, pp. 269-277, 1969. Eker, I “Second-order sliding mode control with experimental application.” ISA Transactions, 49, 394-405, 2010. Guo, J., Chiu, F-C., and Huang, C-C. “Design of a sliding mode fuzzy controller for the guidance and control of an autonomous underwater vehicle.” Ocean Engineering, Vol. 30, 2137-2155, 2003. Hess, R.A., and Wells, S.R. “Sliding mode control applied to reconfigurable
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ACKNOWLEDGMENT
Cybernetics-Part B: Cybernetics, Vol. 38, No. 2, pp. 534-
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada and National Scientific Foundation of the United States for the financial assistance to conduct this research investigation.
539, 2008.
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Services at The Centre for Addiction and Mental Health. Ma’moun A. Abu-Ayyad received his B.S. in Mechanical Engineering from Al-Mustansiryia University, Baghdad-Iraq in 1995, his M.S. from Jordan University of Science and Technology, Irbid-Jordan in 1998. Dr. Abu-Ayyad obtained his Ph.D. degree in Mechanical Engineering from the University of New Brunswick, Fredericton-Canada in 2006. His professional experience includes almost 9 years of teaching at academic institutions in the United States and Canada. He worked as a Postdoctoral Fellow in the Advacnced Plastics Laboratory at the University of New Brunswick for two years. His research contributions cover the areas of modeling, simulation and intelliegent control for a wide range of industrial processes. In addition to his publications, he serves the research community in revewing journal papers, conference proceedings, and chairing various sessions at refereed international conferences. Dr. Abu-Ayyad is currently an Associate Professor in Mechanical Engineering Departemt at Pennsylvania State University-Harrisburg. Dr. Abu-Ayyad is a member of ASME since 2006. Rickey Dubay earned his BSc and MSc degrees from West Indies University in Trinidada and Tobago. He received his PhD from DalTech University in Halifax, Nova ScotiaCanada. His research focus is in the areas of designing advanced controllers, process modelling and machine learning methodologies. He has over 25 years of experience in these fields, having worked on power plant systems and the iron and steel industry. Dr. Dubay has a well-established research facility at UNB with open architecture industrialbased equipment. He has provided high quality training to several graduate students on topics including methods of advanced control and system identification, advanced manufacturing, and robotic systems. Dr. Dubay is currently a Professor and Professional Engineer in Mechanical Engineering at the University of New Brunswick (UNB), Fredericton, New Brunswick, Canada. He is an Associate Editor of the ISA Transactions: The Journal of Automation and a member of several academic societies.
Dr. Chinta V. is currently a Senior Manager, Corporate
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