1. Introduction
This paper presents a controller called the proportional-integral sliding mode controller (PISMC). The sliding mode control (SMC) has been widely accepted as an efficient method for the tracking or balance control of an uncertain nonlinear system. The capability of achieving perfect performance in principle is also demonstrated in the presence of parameter uncertainties and bounded external input disturbances. The proposed PISMC combines the proportional sliding mode control (PSMC) and the integral sliding mode control (ISMC). There are two main contributions in this paper. First, the PISMC provides more parameters with which to tune the controller. As a consequence, it reduces adverse effects due to the external disturbances effectively. Second, this paper sets up an experiment as an example to demonstrate the tracking accuracy in a nonlinear system. This example also shows how this algorithm can be implemented to a physical system. Though the name PISMC may sound similar to some algorithms regarding the PID sliding surface, the definition of sliding surface and the mathematical expression of PISMC are different.
The main structure of SMC theory was brought out by Utkin, a former USSR scholar, in 1977 [
1]. After the original theory, various applications and extensions of the theory were proposed and presented. Some of the examples are given and reviewed in the following. In 1994, Sira-Ramirez et al. [
2] proposed a dynamic multivariable discontinuous feedback control strategy of the sliding mode type for the altitude stabilization of a nonlinear helicopter model in vertical flight. In 1996, Utkin and Jingxin [
3] proposed a new sliding mode design concept named integral sliding mode control (ISMC). ISMC is different from the conventional sliding mode design approaches, and the only difference is the integral form of the error term in place of the original error term. In 2000, Baik et al. [
4] designed an ISMC which has one separate boundary surface in order to decrease the chattering and improve the controllers’ responses. In 2003, Lin et. al [
5] proposed a PID sliding surface, where the sliding mode motion occurs with global invariance. In the same year, Hess and Wells [
6] employed the SMC for flight control design. In 2004, Rafimanzelat and Yazdanpanah [
7] investigated the low chattering SMC for balance control design. In 2005, Niu and Ho et al. [
8] applied the robust integral sliding mode control to uncertain stochastic systems. Besides that, Waslander et al. applied the design of ISMC to a multi-agent quadrotor test-bed. Moreover, Bouabdallah, et al. also applied the back-stepping and sliding-mode techniques to an indoor micro-quad-rotor. Koshkouei et al. [
9] in 2005 proposed a dynamic SMC which increases accuracy under high-frequency switching of control. More studies regarding the applications of sliding mode technique and its extensions are discussed in [
10,
11,
12,
13]. In 2009, Soylu et al. [
14] studied the control of an underwater remote vehicle-manipulator system using sliding-mode control and
control. This paper showed that the sliding-mode and
controller combined approach provides superior dynamic coupling reduction performance.
Speaking of PISMC, Nawawi, Sam, and Osman have devoted themselves to the investigation of PISMC [
15,
16,
17,
18] since 2002. In 2011, Lin et al. applied the PISMC to stabilize the attitude and position of a hovering unmanned helicopter [
19]. Sam at al. applied the PISMC to a hydraulically-actuated active suspension system [
17,
18]; Nawawi, et al. focused on the control and manipulation of of robotic systems. [
15,
16]. In [
20], robust integral sliding mode control (ISMC) was proposed for an underactuated rotary hook system in order to deal with parametric uncertainties presenting themselves in the system’s parameters. Pan et al. studied the modification of the integral sliding variable degrades, and suggested that the switching element is smoothed by using a low-pass filter [
21]. More works related to sliding mode control can be found in [
22,
23]. In [
22] a SMC was designed to control a washout filter with constant impedance and nonlinear constant power loads. In [
23] a PDSMC was designed to collaborate with a sliding perturbation observer to control a robot manipulator.
One thing to note is the work, cited in [
15], done by Nawai et al. in 2006. In this reference, Nawawi suggested PISMC for the two-wheel inverted pendulum mobile robot to avoid high chattering drawbacks. Even though [
15] had a similar topic and methodology to this paper, the details were significantly different. In [
15], a linear model was investigated; the proportional and integral sliding modes shared the same coefficient, and the integration applied to the derivative of the states. In this paper, nevertheless, there are three major differences. First, we do not restrict our system to a linear one. Actually, the examples in this paper were presented and controlled using the original nonlinear form. Second, we introduce two parameters, instead of one coefficient as in [
15], in the investigation. This setting provides us more degrees of freedom to tune parameters and improve the system performance. Third, in the integral we include not only the derivative of the states, but also the states themselves through the error term. Consequently, the work described here and our contributions in the PISMC apparently differ from those made by other researchers.
More specifically, we cite the following list the differences from [
15]:
Design of sliding surface for the controlled system: Our paper presents
. In [
15], the sliding surface was designed as
.
The system of control in this paper is nonlinear, whereas the system in [
15] was a linearized system.
Because the system to control in [
15] was linear, their proposed controller was merely a function of the system, given by
where
f is the original form representing a nonlinear function describing the deviation from linearity in term of disturbances and un-modeled dynamics. Our controller, however, has two major differences from theirs: First, we do not employ the remainder of the nonlinearity terms directly. Instead, we employ the bound of the remainder nonlinearity terms. Secondly, the saturation function in our controller is more than a sgn
function; the selection of parameters is different.
The tuning procedures of parameters are discussed in this paper, but not in [
15].
The work presented in this paper extends the work presented in [
19] to a wider range. The following novelties or contributions are presented in this paper. First, differently from the commonly known ISMC, PISMC, and PID sliding surface, which usually each have the sliding surface of
and five parameters or more to tune, our algorithm in [
19] extends the sliding surface to the
nth order derivative of the output error with only two parameters to tune. In [
19], however, a system of unweighted control was considered. Based on the work in [
19], this paper proposes the sufficient condition to design a control for a system of weighted control. Secondly, our algorithm was realized and verified in a two wheel vehicle system. In the literature, researchers either applied their work to nonlinear systems using numerically simulated results [
21,
24], or realized their algorithms in a linear system [
20,
22,
23]. Although nonlinearities were considered in [
22,
23], they were treated as disturbances instead of dynamics themselves. This paper not only verifies our capability of controlling a nonlinear system by numerical simulations, but also realizes the algorithms in a TWVS using its original nonlinear model. Instead of treating the nonlinearities as disturbances like works in the literature, we dealt with the nonlinearity as the dynamics itself. The successful implementation of our algorithm in a practical nonlinear mechanical system indicates that it is possible to extend the application of the PISMC to a wide range of nonlinear systems, such as the control of microfluidics-based devices [
25,
26]. Thirdly, tuning of control parameters is always critical in designing a control. In our proposed algorithms, there are only four parameters to tune for all systems. From analysis and past experience, we roughly understand the effects of these parameters, though they were not rigorously proven. Systematic tuning procedures are also presented in this paper.
In detail, we first briefly review the SMC and PSMC and their drawbacks. The algorithms and control law by PISMC are then introduced. Before applying the PISMC control law to real cases, robustness, systematic tuning procedures, and performance are then discussed. Having shown the robustness and good performance of our algorithm, the control design for a weighted-control system is proposed. This paper then derives the dynamic model of an experimental TWVS with given parameters. The PISMC was designed based on the TWVS model, and all the input information was obtained from the real tests which were conducted by the TWVS. The corresponding control to stabilize the TWVS was derived based on the proposed PISMC algorithm, and tested in experiments. Discussion of experimental results and a conclusion are provided to demonstrate the validity of our proposed algorithm.
2. Control Algorithm
2.1. Conventional Sliding Mode Control
The theory of SMC has been extensively applied in linear and nonlinear systems, and different practical engineering control problems. Studies in the past few decades have demonstrated that the SMC methodology provides a systematic approach to solving control engineering problems. By introducing a sliding mode to the control of a system, one can achieve both stabilization and disturbance rejection.
The conventional PSMC is considered as a proportional type which forces system states to slide into the predetermined sliding surface, and to reach the origin of the phase plane eventually. When the state-trajectory is sliding on the sliding surface, the corresponding dynamic performance of the system is governed by the equations of the sliding surface. Therefore, the PSMC has a strong robustness against external uncertainties and disturbances.
There are two major requirements for the PSMC. The first one is to design a sliding surface. Then the states are enforced to slide into the sliding surface by the switching the control law. Secondly, each of the tracking points on the sliding surface must satisfy the sliding conditions. The switch of the control law is determined by the predefined switching conditions. With the help of the control switch, the trajectory can be guided into the region bounded by boundary layers. Governed by the sliding conditions, the trajectory gradually moves onto the sliding surface and steps forward to the target point.
Consider a desired signal
and the system output
, where both of
and
are in
. The objective of the PSMC is to make the system’s output approach the desired signal when
; i.e.,
Here, we define the tracking error as
where
. Before the PSMC is applied, a sliding surface must be designed in advance. The goal of the PSMC is to guide the system trajectory toward the sliding surface, which naturally leads the trajectory to the target point in the end. Based on the magnitude of errors, we define a time-varying linear differential function in
,
, named the condition space [
27], given by
where
is a positive constant. The equation
represents a sliding surface passing through the origin of the phase plane. According to the above equation,
is an
th-order linear differential equation, and all the eigenvalues of the characteristic equation are located in the left complex plane. Once the sliding surfaces are determined, we may start to design the controller and define the sliding conditions as follows [
27].
where
, and
denotes the time derivative of
s.
2.2. Proportional-Integral Sliding Mode Control
To improve the performance of a SMC, this paper adopts a PISMC as our algorithm. Consider a dynamic system described by an
nth-order nonlinear differential equation
where
is the output of the dynamic system,
f is the system function and must be at least once differentiable,
is the system input, and
t is time. We define the state vector,
, for future derivations.
Assumption 1. Consider (6) and assume that there exists an uncertain parameter in the dynamic function , satisfyingwhere is the estimation and is the variation of function f. Assumegiven the boundary of system uncertainty [27]. The goal of the PISMC proposed in this paper is to design a switching control law to let the output
approach a given
, as shown in (
2). To design the control system, the system states are assumed to be attainable via a state observer, which sometimes requires measurement of an output or output error, as defined in (
3). The sliding surface of the PISMC consists of two sliding surfaces,
and
, the proportional sliding surface and the integral sliding surface, respectively. Here,
as defined in (
4), and
where
,
,
. The operator
is defined as
where
is a parameter to tune, and
denotes the
k-combinations of a set having
n elements. Applying the operator to a function
yields
where the superscript with parentheses denotes the order of derivative, while the superscript without parentheses denotes the power of the parameter
. In our case,
or
. They are two complementary parameters, which provide the capability of tuning the gains of controller. We note that the expansion of (
9) is different from the conventional PSMC or PID sliding surface, since
and
are generally distinct. The PISMC is designed as [
19]:
where, by letting
,
with
and
The controller design is summarized in Theorem 1.
Theorem 1. [19] Consider the system of (6) which satisfies Assumption 1. Let the function be specified in (12) with being equivalent signal and being switching control defined in (13) and (14), respectively. The gain k is designed to satisfy (5) with the definition of given in (15). If there is a k and an such thatthen the tracking error as defined in (3) will tend to a neighborhood of zero within a finite time. Proof. The proof is a brief version of [
19] for further usage. Define the following Lyapunov function
Choose an appropriate
to satisfy
. Then,
The sufficient condition of (
18) to ensure the stability of system and to satisfy the sliding conditions as in (
5) is given by
Let
provided
. We conclude
as stated in (
16). It is also conclusive from (
18) that
converges to the neighborhood around origin of the phase plane for any trajectory. □
In fact, in the control signal there is a the discontinuous contribution , caused by the discontinuous switching function (). This function results in an infinitely large switching frequency. However, this infinitely large switching frequency can not be realized in real physical system. Therefore, a switching component with extremely high-speed is applied. That way, the state trajectory would oscillate around the two sides of which results in the chattering phenomenon.
In order to reduce the chattering problem, the concept of boundary layers for a sliding surface is employed, as depicted in
Figure 1.
The definition of boundary layers is given by
The error phase-plane of boundary layers of a sliding surface,
s, coincides with the boundary range of
. Note that
is the boundary layer. Since the switch action can not be completed in a blink of time and the effect of delay exists, the state trajectory
would oscillate between the boundary layers.
Figure 1 is the 2nd-order error phase plane, and it is the most common way to mitigate the oscillation by replacing the sliding surface with the boundary layers.
is the switching law of the controller, defined as follows
Here sat
is the saturation function defined by
2.3. Alternative from of the Controller
In this paper, we assume that all states are attainable via state observers, but do not touch the design of the observer. The original design of the controller
in (
12) may require additional estimation of
, which may need to employ new gain(s).
In order to avoid this drawback, an alternative design of
is proposed by reducing the order of
. (
3) leads to
To have
, the control
was originally designed as
where
, and
As a result, (
22) can be rewritten as
or,
To eliminate
, we here re-design
as
That is, replace the original
by letting
We claim that this controller works, and this controller does not need to measure .
The proof is given as follows:
2.4. Robustness Discussion
Unlike the design concept of conventional SMC, the insensitivity is included in the dynamics model and also guaranteed in our algorithm. In the conventional SMC design, a precise dynamics model is employed and the perturbation is viewed as an extra input. An example is given in Equation (
22) in [
28]. The change rate of
s is formulated by
and the robustness is discussed based on this formulation. In our approach, however, the perturbations are already included in the parameter
k for
. Recall that
, where
is the bound of the model variation, satisfying
and
Let
, where
is the nominal states. Then,
where
and
includes all the remainder terms. By properly investigating and choosing the bound of
. The influence of perturbations is included and considered in the control. The advantage of our approach is that the robustness of the control is guaranteed with a proper choice of
, and we do not have to worry about specific perturbations. The drawback, of course, it that the control might be inefficient if
is defined too large for the purpose of robustness.
2.5. Parameter Tuning
In the proposed algorithm, there are four parameters to tune in total: and to construct s-surface, to construct , and to determine the saturation function. We have tried to analytically understand the effect of each parameter. However, this is too difficult due to the complexity of nonlinear systems and mathematics. Instead, approximated properties in a 2nd order system are analyzed, and numerical simulations are presented to demonstrate the potential effect of each parameter, so that readers will have some ideas on our tuning procedures. Notably, numerical examples are not solid proofs. An effect may vary in different cases.
The example from [
19] is employed to study the effect of each parameter, and how to tune these parameters. The example, given as follows, was originally from [
27] and employed to demonstrate the validity of the our proposed algorithm:
The function
is unknown but varies within the range
. Let
The control input of SMC is
with
. The control input of PISMC is
with
Notably, we employed this example in our work in [
19], and showed the validity of our work. In this paper, however, we employ this example to show the properties and influences of parameters. As a result, despite the dynamical equation and parameters of the equation being directly employed from [
27], other settings are changed. For example, we used state errors and their derivatives, instead of the output rate in the reference. The parameters of the controller are different from those in the reference. The desired signal
is set as unit step for investigating transient properties, which is different from the original
. Therefore, the comparison occurs between our PISMC and SMC to know the parameters better. It did not occur between our PISMC and the SMC of the reference.
2.5.1. Effect of
is the parameter to determine the switching method and reduce the chattering phenomenon, as shown in (21). However, the larger the value of
, the more sluggish the response. As
, sat(
) =
. As
, sat(
) = sgn(
s), which usually causes chattering phenomenon. Consider the case
, so that sat(
) =
. Since
s is finite in general,
. The smaller the
, the smaller the
. The decreasing rate of the Lyapunov function is influenced by
The original saturation function sat(
) = sgn(
s) brings
. The condition to have a more greatly decreasing rate of the Lyapunov function is given by
or equivalently,
. Therefore, a large
is not helpful in speeding up the converge of the system.
Figure 2 demonstrates the comparison.
in
Figure 2a, whereas
in
Figure 2b. Other parameters are set identical. It is obvious that the responses with smaller
are faster, regardless of the type of controls.
2.5.2. Effect of
It can be expected that the larger the value of
, the more decreasing the rate of the Lyapunov function from
. Suppose
at beginning. The upper bound for
is given in (
33). Since
, we conclude the upper bound of
V satisfies
(
35) implies that
V converges exponentially with time constant
.
Figure 3 demonstrates the comparison.
in
Figure 3a, whereas
in
Figure 3b. Other parameters are set identically. It is obvious that the responses with large
are faster, regardless of the type of controls.
2.5.3. Effect of
The role and influence of on a higher order system is unclear. However, we may approach this problem through a second order system like the example. When , our algorithm suggests and under the SMC assumption, where contains the remainder terms of . From the structure of we conclude that the role of is roughly the gain of a conventional P-controller in a negative feedback. A proper choice of would stabilize the system. Too large a value of may destabilize the system, or have the system converged sluggishly.
On the other hand, the function of SMC is to force first. Then, the property of will force when t approaches infinity. Since , implying that e converges exponentially with time constant , too small a value of will cause the system to converge sluggishly. From our experience, is a better choice.
Figure 4 demonstrates the comparison.
in
Figure 4a,
in
Figure 4b, and
in
Figure 4c. Other parameters are set identically. It is obvious that the responses with too large or too small
are sluggish, if the SMC control is considered.
2.5.4. Effect of
Similarly, the role and influence of on a higher order system are unclear. However, we may approach this problem through a second order system. When , our algorithm suggests . Defining yields . Moreover, when .
A similar statement can be made for
. The function of our algorithm is to force
first. The property of
will then force
when
t approaches infinity. Since
,
z converges exponentially if the discriminant is greater than zero, and converges oscillatorily if the discriminant is less than zero, where the discriminant is defined as
Defining
and
yields
The condition for
z to converge exponentially is
. This approximation is not accurate enough because the actual behavior of
z is also influenced by other parameters. We may conclude
z converges oscillatorily if
.
Given
, the overshoot of
z is influenced by damping ratio of the system. The damping ratio
can be approximated by
when
,
. When
,
.
Since , the oscillation of does not directly equal the oscillation of e. Moreover, the nonlinearity of the system also makes the quantitative prediction inaccurate. However, the trend of the oscillation must be similar. The larger the value is, the smaller the overshoot e has.
Figure 5 demonstrates the comparison.
in
Figure 5a,
in
Figure 5b, and
in
Figure 5c. Other parameters are set identically. It is obvious that a very small
causes no overshooting in the system. This is reasonable. When
, the algorithm degenerates to a conventional SMC. Although large
causes overshooting, the magnitude is small.
The time constant of convergence for an oscillatory
z can be approximated by
when
,
. When
,
.
Although large suppresses overshooting and speeds up the convergence of outputs, grows exponentially compared to the contribution to control signals. Hence, the trade-off of a fast response is a very large control. A very large control is usually energy consuming or impossible in implementation.
2.5.5. Tuning Procedures
From previous analysis and our experience, the tuning procedures applying our algorithm are concluded as follows:
Select . is suggested to speed up the convergence.
Select . Large is helpful to speed up the convergence. However, the value of is proportional to . Too large a value of may cause too large a control signal.
Select . From our experience, is a better choice.
Select and . If an oscillatory response is allowed, can be selected to be larger than 1. If not, a smaller is suggested.
Estimate the transient properties:
- (a)
The overshoot can be estimated from damping ratio for .
- (b)
Estimate the total time constant. Since the convergence procedure is and then , the total time constant of convergence, can be approximated by .
- (c)
The settling time , according to control theory, approximated by .
Build the nonlinear mathematical mode and run numerical simulations with initially selected parameters. Tune the parameters according to the transient performance and the effect of parameters.
Remember to keep as small as possible, because a large usually indicates a very large control input.
2.6. Effectiveness of the Algorithm
As aforementioned, there are only four parameters to tune in total for an arbitrarily high order system: and to construct s-surface, to construct , and to determine the saturation function. Usually, a higher order controller has more parameters to tune. Using only four parameters to control an arbitrarily high order system may sacrifice potential performance more parameters may bring, but it simplifies the tuning procedures.
On the other hand, the proposed algorithm has more degrees of freedom to adjust the system’s performance, compared to the conventional SMC algorithm. Although those characteristics may result in complexity in selecting parameters, with the aforementioned methods and analysis, one is able to tune parameters systematically and efficiently.
Another concern is the cost of the control. Better performance may result from greater effort, such as a larger control signal or greater energy consumption. From our experiences, though not rigorously proven, in some cases the proposed algorithm actually costs less than the conventional SMC, which only tunes to build the sliding surface and to defined the boundary layer.
An example is presented in
Figure 6. The example was simulated using the aforementioned nonlinear system. In order to evaluate the control laws, a cost function
J is defined as follows:
Two control laws were employed to control the system: the SMC and our proposed PISMC. Parameters were tuned so that the two systems had similar settling times. A smaller overshoot occurred in the system using PISMC.
Table 1 lists the parameters:
The simulations in
Figure 6 show that the proposed algorithm costs less while keeping similar transient performance. Notably, under the designated parameters, chattering occurs once using the control from SMC. As a result, it can be concluded that our algorithm performs better than the conventional SMC in this specific case.
2.7. PISMC for a Weighted-Control System
Based on the work in [
19], we extend our work to a nonlinear system of weighted control, which is usually described by the differential equations
where
are the input and output, respectively, and
are smooth functions. The gain
is unknown, yet assumed to be uniformly bounded.
Assumption 2. Assume , where and are given constants, time-varying or state-dependent functions [27]. We assume that the control input may enter multiplicatively in the dynamics, and it is natural to choose our estimate
of gain
as the geometric mean of the above bounds.
Assumption 2 can then be written in the form of
where
Since the PISMC is designed to be robust to the bounded multiplicative uncertainty, as in (
43), we call
the gain margin of our design, by analogy to the terminology used in linear control. Let
The PISMC
is modified from the aforementioned
and it has the form:
where the
and
are defined in (
13) and (14), respectively. We summarize the controller design in Theorem 2.
Theorem 2. Consider the system (41), which satisfies Assumptions 1 and 2. Let the function be specified in (46) with , , and defined in (13), (14), and (15), respectively. Let the gain be designed to satisfy (5). If there exits a such thatthen the tracking error as defined in (3) will tend to the neighborhood of zero within a finite time. Proof. To prove that PISMC
satisfies the sliding conditions, the following Lyapunov function is employed:
If we choose
as in (
46), then follow similar steps as those in (
18):
Finally, let
We obtain
and hence
. This ensures that any trajectory
converges to the neighborhood around of origin in the phase plane. □
2.8. Comparison of Sliding Surfaces
The design of sliding surface is inherited from [
19], whereas the design of control is extended to a weighted-control system. The design of sliding surface in our algorithm extends to the
nth order system, and, thus, is considerably different from conventional SMC and PID sliding surface.
Table 2 presents the differences.