Proportional-Integral Sliding Mode Control with an Application in the Balance Control of a Two-Wheel Vehicle System
Round 1
Reviewer 1 Report
The paper is fairly well written although containing numerous mistakes or unclarities. The contribution seems limited as it evolves around nonlinear control of an inverted pendulum cart with adding a second sliding surface in the state transformation (s_2) also including integral action of the output error. The topic of integral sliding mode control is far from new and the innovation in this paper seems cumbersome. All the terms in s_1 and s_2 are common except from the integral part, such that you end up, sort of, increasing the gain on each derivative of \tilde(x) as a sum of \lambda and \alpha parts plus \alpha^2\int e d\tau. Wouldn’t it be easier to realize it as s=s_1+\alpha \int e d\tau which is normally done when designing integral sliding mode controllers? That makes tuning seemingly much easier.
I have the following comments/suggestions (numbers at start of each point is referring to line in original manuscript):
- Normally introducing extra parameters makes tuning more complex and is thus not necessarily a pro but rather a con, if it isn’t strictly necessary.
- The dimensions of all defined states are missing. Is \tilde x \in R or R^n? What about \tilde x_d? if \tilde x \in R and as (d/dt+\lambda)^(n-1) \in R then according to Eq. (3) s(\tilde x,t) \in R but in 93 it is stated that s(\tilde x,t) \in R^n.
- It is stated that \tilde x is output, which normally means the measured state. On the other hand in Eq. (12) the knowledge of all output derivatives up to n is needed (middle term on RHS) which is really full state plus derivative of last state. This is also evident from the example given below 157, and is thus far from output only?
- The notation in (12) seems cumbersome. Why split the \alpha related terms? Couldn’t it be expressed as \sum_m=0^(m=n-1) (n-1&m) \alpha^(n-m-1)e^(m)(\alpha+e) or similar (if my derivation was correct)?
- (11) it seems here that k is constant but in 135.5 and example 157.5 it is state dependent (varying with the whole state vector \tilde X plus extra derivative).
- 5 sgn function is not defined at s=0.
- 170 There is a significant difference in control action also explaining why PISMC is faster. I assume that is due to the increase in gains due to definition of s=s_1+s_2 as discussed earlier.
- (19)-(20) Some definitions are missing (T, F, H, f_d, L/R). Also, from Figure 6 there seems to be something odd with signs. If F_l,r is negative it would mean they cause deceleration in \ddot x_L,R but in Fig. 6 x_L,R and F_L,R are pointing in same direction.
- It is not explicitly mentioned, but is it assumed that the ground is flat, i.e. no roll motion?
- F_p is missing from Fig. 6. What is difference between F_p and f_p? Also, it seems that positive \Theta is rotating in opposite direction as is normal for right handed system. This will impact sign of eq. (21) and possibly others. Also, the force F_\Theta which seems to point along negative z direction seems odd, is there any transversal force present?
- 221 It is stated that Eq. (33) is derived from Eqs. (19), (20) and (22) although none of them contains any trigonometric functions. Thus the derivation seems not straight forward.
- 8. Normally a fair comparison is to tune the controller such that they have similar performance and then look at the energy spent or tune them such that they are spending the same energy and then compare transient and steady state performance. As now as both are different, it is difficult to evaluate which is actually performing better.
- 12 an oscillation starts at about 12 seconds on the SMC test while the PISMC plot is cut just at that point. Is this a coincidence or is there something going on also in the PISMC? Can you comment on the cause of this oscillation?
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Reviewer 2 Report
- The novelty is not well stressed within the current research context.
- In the conventional SMC control systems, insensitivity to (matched) perturbations is guaranteed once the system enters the sliding phase. ISMC has been proposed to extend this property over the reaching phase, at the expense of enlarged space (order reduction does not occur). Consequences of this fact for the investigated two-wheel vehicle system are not discussed.
- Does the matching condition need not be imposed with respect to disturbance (6) so that the properties stated in the theorems hold?
- More flexibility in the controller tuning in the proposed method means more effort to adjust the system performance. No guidelines have been provided. Be specific.
- k in (13) requires measurement of perturbation, which is difficult to obtain in the physical systems. Feasibility issues are not discussed.
- In order to give a proper perspective on the benefits of the proposed method a comparison with [15] should be provided in the numerical/experimental section. In the analytical section, the conceptual difference between PISMC and PID sliding surface design could be illustrated by a figure?
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Reviewer 3 Report
The paper "PI Sliding Mode Control with Application to Balance Control of a Two-Wheel Vehicle System" presents a control method for an inverted pendulum type system, which the authors call a Two-Wheel Vehicle System (TWVS).
Inverted pendulum systems are often used as testbeds for various control algorithms. In this case, the authors propose a PI Sliding Mode Control (PISMC) algorithm and a self built physical setup to test the algorithm. Both of these components are discussed in the paper.
In the Introduction the authors present the state of the art with respect to sliding mode control (SMC) and PISMC. Except [19] (2011), all the cited references are more than 10 years old. This indicates the low interest these algorithms have received from the scientific community over the past years.
With respect to the novelty of the approach, this paper builds upon a paper from 2006 ([15]). The authors claim there are major differences between this paper and the one previously mentioned. The new contributions compared to [15] should be clearly stated and better emphasized throughout the paper. They are only briefly mentioned in the Introduction, then, as the model of the controller and the plant develop, no other references are made to paper [15].
In Section 2, the authors introduce the theoretical aspects related to the proposed control methods. They start with conventional SMC, then move on to PISMC. This section is heavily theoretical and ends with a simple example intended to show the performance of the PISMC.
The paper continues with the presentation of the physical setup. Some details are given regarding the hardware. However, these are not well documented and the authors simply rely on some photos. Instead of images of the boards, a hardware architecture of the system would have been more useful.
By reading the content of Section 3.1 it is unclear how many PIC18F258 microcontrollers (2 or 3) the board uses and what functions do they implement in the system (apart from reading the sensors). This section, relating to the hardware, is confusing and incomplete. It remains unclear where the PIMSC controller is implemented and how it was implemented. Some software implementation details would complete this section. As a side argument, according to Microchip, PIC18F258 controller "is not recommended for new designs".
An extensive modeling of the plant is shown which is later used for simulation purposes. The simulation details need improvement. While the modeling part is quite thorough, there is a gap between the resulted models, the available simulation details and the experimental setup.
For instance, how are the dynamics equations of the plant and the equations of the controller comprised in the block schematic of Fig. 7. Also, different notations are used across the paper which are not consistent with the simulations model. It was not specified which simulation environment was used and how the developed models were implemented.
The tunning process of the controller's parameters for the given experimental plant/simulation model is missing. The tuning parameters of the PISMC are advertised as the paper's main contribution. They need more attention.
At the end of the paper simulation and experimental results are shown and discussed.
In the case of the experimental tests in the presence of disturbances it is not specified how were the disturbances introduced to the system, or what were these disturbances like.
Author Response
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Round 2
Reviewer 1 Report
- The authors state that the suggested structure is fundamentally different than standard ISMC as the proposed structure has two tuning parameters instead of n tuning parameters. This reduction in tuning parameters is due to the relationships between the gains in a standard n-dimensional surface reducing the number of degrees of freedom for the tuning parameters to two, which is more a matter of preference than a fundamental difference. What is different is that the proposed solution also requires d^n e/dt^n in the control law, e.g. \ddot e in \tilde u below line 222 compared to \dot e in the standard ISMC, which seems like a drawback as it is difficult to measure and thus requires an estimator with new gain(s) to tune.
- Figure 3 (b) clearly shows the lack of proportional feedback in the controller when the disturbance rejection margin (\eta) is small. Thus inspecting \tilde u below line 221 shows that the k_p e-term is missing from the equation, which is included in the example collected from the book of Slotine & Li. This affect the whole analysis of comparison. Also, in the author feedback an example of PISMC and SMC is included, and it is seen that the SMC is tuned more aggressively and thus converges faster at a cost of higher control action. Especially, the (bad) tuning causes (excessive) control effort at the start and when reaching reference which is heavily penalized due the quadratic nature of the cost function.
- The model in Section 3.1 can be found in many standard textbooks and thus considerably space can be saved by restating the model with a reference.
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Reviewer 2 Report
The concerns raised in the original review have been dispersed. I have no further remarks. The paper may be published.
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Reviewer 3 Report
The structure of the paper has been extensively changed in order to provide a more logical flow. The new form of the paper is more clear and it manages to answer my initial comments to a great extend.
Moreover, some aspects raised by all three reviewers, such as the novelty of the approach, its relation with [ref 15], the tunning process have been better addressed in this version of the paper.
Author Response
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Round 3
Reviewer 1 Report
Dear authors, my concerns have been properly addressed and the structure of the paper is now much better. Regarding comment #3 in previous feedback the controller from the book of Slotine and Li you refer to as Eq. (7.13) is designed based on the sliding surface s = \dot\tilde x +\lambda\tilde x in Eq. (7.11) while for the integral case s = \dot\tilde x + 2\lambda\tilde x + \lambda^2\int\tilde x dr (bottom of p. 286) leads to \hat u = -\hat f + \ddot x_d – 2\lambda \dot\tilde x - \lambda^2\tilde x (top of p. 287) i.e. when differentiating the sliding surface the integral term shows up as a proportional term in the dynamics of the sliding surface which then becomes part of the controller.
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