Optimal adaptive higher order controllers subject to sliding modes for a carrier system

Abstract

Due to costly space projects, affordable flight models and test prototypes are of incomparable importance in academic and research applications, for example, data acquisition and subsystems testing. In this regard, CanSat could be used as a low-cost, high-tech, and lightweight model. CanSat carrier launch system is a simple second-order aerospace system. Aerospace systems require the highest level of effective controller performance. Adding second-order integral and second-order derivative terms to proportional–integral–derivative controller leads to the elimination of steady-state errors and yields to a faster systems convergence. Moreover, sliding mode control is considered as a robust controller that has appropriate features to track. Thus, this article tends to present an adaptive hybrid of higher order proportional–integral–derivative and sliding mode control optimized by multi-objective genetic algorithm to control a CanSat carrier launch system.

Keywords

CanSat carrier launch system PID controller SMC adaptive hybrid of higher order PID and sliding mode control

Introduction

CanSat, a similar technology used in miniaturized satellites, is a type of autonomous space technology that is employed for research and educational purposes. In this regard, CanSat can be considered as an intelligent space robot. Carrier systems are used to carry CanSats to a certain altitude where they are released afterward. The carrier system, which has a vertical takeoff, releases the CanSat in a predefined altitude with a certain path angle relative to the local horizon. Then, the parachute is opened for a safe landing. In this regard, in order to launch the CanSat payload to a certain flight altitude, designing the CanSat controller is one of the key issues.

Proportional–integral–derivative (PID) controller is the most common controller in feedback loops and has been used in most recent control processes.¹ Determining three control parameters, proportional (K_P ), integral (K_I ), and derivative (K_D ) gains, is the main concern in the design of PID controllers. In order to specify these control gains, Kosari et al. used the Gaussian and triangular fuzzy membership functions to design fuzzy PID controllers for a CanSat carrier launch system.² They have implemented the genetic algorithm (GA) to optimize fuzzy parameters. Although the fuzzy controller can cover a number of uncertainties, the design of an adaptive controller, which can provide an appropriate response to unknown uncertainties and disturbances, is essential. In order to increase the performance of the control system, hybrid controllers such as PID control and fuzzy control,^3

–7 sliding mode control (SMC) and fuzzy control,^8

–11 SMC and H2/H∞ control,^12

–16 and SMC and PID control^17

–21 have been used abundantly by researchers. Although the PID controller has higher reliability and stability compared to SMC, SMC can achieve better tracking performance and robustness to the disturbances.²²,²³ In order to benefit from the advantages of both PID controller and SMC, a combination of these controllers is used. Zhang et al. proposed a hybrid of PID control and SMC and demonstrated that the suggested controller has better stable precision than each controller is implemented separately.²⁴ Li et al. combined backstepping kinematic control and PID sliding mode dynamic control to control nonholonomic mobile robots using an adaptive neural network gain of SMC.²⁵ In order to guarantee the static and dynamic system performance, Zhang et al. proposed a control method based on neuron PID and sliding mode.²⁶ Besides, in order to increase the efficiency of the PID controller, fractional calculus has been discussed.^27
–29 Taking into account unknown models, the fractional order PID controller with the non-integer gains’ order³⁰ has a more robust performance compared to PID controller.³¹ On the other side, adding second-order derivative and integral terms to the PID controller leads to enhancement of the systems’ performance and robustness with considering existing uncertainties.³² Therefore, in order to control the chaotic behavior of chaotic systems such as pendulum-driven spherical robot,³³ using the second-order integral and derivative terms can improve the performance of the control system.

The parameters of the controllers have to be determined using an appropriate optimization approach. To choose these parameters effectively, the evolutionary algorithms,³⁴ such as the GA,³⁵ are employed.

The main contribution of this article is to propose novel adaptive proportional plus first- and second-order derivative and integral (PI²D, PID², PIDD², PI²ID, and PI²IDD²) controllers for the CanSat carrier launch system as a simple model of a second-order system. The gains of designed higher order controllers are tuned by a sliding mode–based adaptation mechanism in which a gradient search algorithm with opposite direction of energy flow is implemented. The learning rates of adaption laws are considered as design variables and optimized with the GA. Finally, the performance of the designed controllers is compared with each other.

The rest of this article is organized as follows. A complete description of the mathematical model of the CanSat carrier launch system is presented in the second section. In the third section, the higher order control algorithm is discussed. Adaptation laws and optimization process are demonstrated in this section. In the fourth section, simulation results are presented and a detailed and comprehensive discussion is made on them, followed by conclusion in the last section.

Dynamic modeling

The governing dynamic equations of a CanSat carrier launch system are derived from the Newtonian method along with the rule that in the separation stage, the projection of satellite velocity vector must be tangent to the horizontal plane. Figure 1 shows a simplified model of a launch vehicle in which θ is the angle of the longitudinal vector of the vehicle in the perpendicular direction (toward the ground) and Φ is the angle of the thrust vector with body centerline. The dynamics of the system can be summarized as follows

\sum M_{CM} = I α

in which $M_{CM}$ is the moment around the center of mass, I is the moment of inertia, and α is the angular acceleration about an axis perpendicular to the plane. Equation (1) can be expanded as follows

\frac{l}{2} \times F_{n} = I \ddot{θ}

in which l represents the length of the vehicle, F shows the propulsion force, and F_n is the projection perpendicular to the longitudinal direction of the launch vehicle. It is known that the vehicle moves along the vertical axis with acceleration of a. Therefore, Newton equation for the vertical axis is derived as follows

\sum F_{z} = m a

where $F_{z}$ and m are the force along the vertical axis and the mass of the launch vehicle, respectively. Equation (3) can be rewritten as follows

- m g + F_{t} cos θ = m a

Meanwhile, geometric relations dictate the following equations in the vertical plane

F_{n} = F sin (Φ)

F_{t} = F cos (Φ)

Figure 1.

Carrier system scheme.

By substituting equation (6) in equation (4), we have the following equation

F = \frac{- m (a + g)}{cos (Φ) cos (θ)}

In the same way, by substituting equation (5) in equation (2), we have the following equation

\overset{˙˙}{θ} = \frac{1}{2 I} l F sin (Φ)

Considering $θ = 0$ , substitution of equation (7) in equation (8) results in

\overset{˙˙}{θ} = \frac{- 1}{2 I} m l (a + g) tan (Φ)

By substituting $\frac{- 1}{2 I} m l (a + g)$ by b and $tan (Φ)$ by $u_{t}$ , the dynamic equation of the system leads to

\overset{˙˙}{θ} = b u_{t}

in which $u_{t}$ is the control parameter. Therefore, state equations of the system take the following form

\begin{array}{l} {\dot{x}}_{1} (t) = x_{2} (t) \\ y (t) = x_{1} (t) \end{array}

where $x_{1} (t)$ and $x_{2} (t)$ indicate θ and $\dot{θ}$ , respectively. In addition, $X = {[x_{1}, x_{2}]}^{T}$ is the measurable state vector. Signal error between the desired output and actual output is equal to $e = y_{d} - y = y_{d} - x_{1}$ . The desired output of the system is defined as $Y_{d} = {[y_{d}, {\dot{y}}_{d}]}^{T}$ and let us suppose that $y_{d}$ and ${\dot{y}}_{d}$ are bound to satisfy $∥ Y_{d} ∥_{\infty} = {sup}_{\geq 0} ∥ Y_{d} (t) ∥ < \infty$ . Therefore, the system error vector is defined as $E = Y_{d} - X = {[e, \dot{e}]}^{T}$ . The gain vector of $K = {[k_{0}, k_{1}]}^{T}$ is selected so that the roots of the equation $s^{2} + k_{1} s + k_{0} = 0$ are placed on the left side of the complex plate. Therefore, the feedback linearization controller is defined as follows

u^{*} = - f (X, t) + y_{d} + K^{T} E

By replacing equation (12) in equation (11), it would be as follows

\overset{˙˙}{e} + k_{1} \dot{e} + k_{0} e = 0

It can be concluded from equation (13) that $e (t)$ tends to zero when t tends to infinity which means y asymptotically tends to $y_{d}$ .

Higher order controllers design

The PI²IDD², PI²ID, PIDD², PID², and PI2D controllers with input $e (t)$ and outputs of $u_{{PI}^{2} {IDD}^{2}} (t)$ , $u_{{PI}^{2} ID} (t)$ , $u_{{PIDD}^{2}} (t)$ , $u_{{PID}^{2}} (t)$ , and $u_{{PI}^{2} D} (t)$ are defined using the following equations

u_{{PI}^{2} {IDD}^{2}} (t) = K_{P} e (t) + \frac{1}{T_{I}} \int_{0}^{t} e (τ) d τ + \frac{1}{T_{I^{2}}} \iint_{0}^{t} e (τ) d τ + T_{D} \frac{d}{d t} e (t) + T_{D^{2}} \frac{d^{2}}{d t^{2}} e (t)

u_{{PI}^{2} ID} (t) = K_{P} e (t) + \frac{1}{T_{I}} \int_{0}^{t} e (τ) d τ + \frac{1}{T_{I^{2}}} \iint_{0}^{t} e (τ) d τ + T_{D} \frac{d}{d t} e (t)

u_{{PIDD}^{2}} (t) = K_{P} [e (t) + \frac{1}{T_{I}} \int_{0}^{t} e (τ) d τ + T_{D} \frac{d}{d t} e (t) + T_{D^{2}} \frac{d^{2}}{d t^{2}} e (t)]

u_{{PID}^{2}} (t) = K_{P} [e (t) + \frac{1}{T_{I}} \int_{0}^{t} e (τ) d τ + T_{D^{2}} \frac{d^{2}}{d t^{2}} e (t)]

u_{{PI}^{2} D} (t) = K_{P} [e (t) + \frac{1}{T_{I^{2}}} \iint_{0}^{t} e (τ) d τ + T_{D} \frac{d}{d t} e (t)]

where K_P is proportional gain, T_I is the integral time constant, $T_{I^{2}}$ is the second-order integral time constant, T_D is the derivative time constant, and $T_{D^{2}}$ is the second-order derivative time constant. In addition, the abovementioned equations can be rewritten as follows

u_{{PI}^{2} {IDD}^{2}} (t) = K_{P} e (t) + K_{I} \int_{0}^{t} e (τ) d τ + K_{I^{2}} \iint_{0}^{t} e (τ) d τ + K_{D} \frac{d}{d t} e (t) + K_{D^{2}} \frac{d^{2}}{d t^{2}} e (t)

u_{{PI}^{2} ID} (t) = K_{P} e (t) + K_{I} \int_{0}^{t} e (τ) d τ + K_{I^{2}} \iint_{0}^{t} e (τ) d τ + K_{D} \frac{d}{d t} e (t)

u_{{PIDD}^{2}} (t) = K_{P} e (t) + K_{I} \int_{0}^{t} e (τ) d τ + K_{D} \frac{d}{d t} e (t) + K_{D^{2}} \frac{d^{2}}{d t^{2}} e (t)

u_{{PID}^{2}} (t) = K_{P} e (t) + K_{I} \int_{0}^{t} e (τ) d τ + K_{D^{2}} \frac{d^{2}}{d t^{2}} e (t)

u_{{PI}^{2} D} (t) = K_{P} e (t) + K_{I^{2}} \iint_{0}^{t} e (τ) d τ + K_{D} \frac{d}{d t} e (t)

where $K_{I} = \frac{K_{P}}{T_{I}}$ is the integral gain, $K_{I^{2}} = \frac{K_{P}}{T_{I^{2}}}$ is the second-order integral gain, $K_{D} = \frac{K_{P}}{T_{D}}$ is the derivative gain, and $K_{D^{2}} = \frac{K_{P}}{T_{D^{2}}}$ is the second-order derivative gain.

Adaptive rules

In order to update the gains of the designed higher order controllers, an adaption law is implemented. The adaptation law takes advantage of sliding mode concepts and is designed based on the gradient search method. Here, for example, adaptive PI²IDD² is presented and other controllers are obtained in the same way. To provide proper tuning laws for updating control gains, the designing signal xr is defined in the following form

{\dot{x}}_{r} = {\ddot{y}}_{d} + k_{1} \dot{e} + k_{0} e

where $k_{1}$ and $k_{0}$ were previously defined. The sliding surface is defined as follows

S = x_{2} - x_{r}

If the sliding mode occurs, that is, $S = 0$ , then $x_{2} - x_{r}$ . By substituting in equation (24), we have the following equation

{\dot{x}}_{2} = {\ddot{y}}_{d} + k_{1} \dot{e} + k_{0} e

By investigating equation (26), it can be concluded that if t tends to infinity, then $e (t)$ tends to zero. From SMC viewpoint, the conditions that ensure sliding mode are derived from Lyapunov stability theory. In general, the Lyapunov function of SMC is defined as follows

V = \frac{1}{2} S^{2}

By this, the condition of sliding mode includes

\dot{V} = S \dot{S} < 0

Equation (28) ensures that $S (t)$ tends to zero when t tends to infinity. In order to tune the control gains of PI²IDD², PI²ID, PIDD², PID², and PI²D, the gradient search method is used to derive an appropriate tuning mechanism. Using equations (11) and (25), it would be as follows

\dot{S} = {\dot{x}}_{2} - {\dot{x}}_{r} = f (X, t) + Δ f (X, t) + δ (t) + u_{{PI}^{2} {IDD}^{2}} (t) - {\dot{x}}_{r}

From the multiplication of both sides of the abovementioned equation in S, the following equation is obtained

S S = S [f (X, t) + Δ f (X, t) + δ t + u_{{PI}^{2} {IDD}^{2}} - \dot{x} r]

Using gradient method and the chain rule, equation (19), as well as equation (30), the adaptive laws are calculated for control gains of $K_{P}$ , $K_{I}$ , $K_{I^{2}}$ , $K_{D}$ , and $K_{D^{2}}$ as follows

{\dot{K}}_{P} = - γ_{(c, 1)} \frac{\partial S \dot{S}}{\partial K_{P}} = - γ_{(c, 1)} \frac{\partial S \dot{S}}{\partial u_{{PI}^{2} {IDD}^{2}}} \times \frac{\partial u_{{PI}^{2} {IDD}^{2}}}{\partial K_{P}} = - γ_{(c, 1)} S e

{\dot{K}}_{I} = - γ_{(c, 2)} \frac{\partial S \dot{S}}{\partial K_{I}} = - γ_{(c, 2)} \frac{\partial S \dot{S}}{\partial u_{{PI}^{2} {IDD}^{2}}} \times \frac{\partial u_{{PI}^{2} {IDD}^{2}}}{\partial K_{I}} = - γ_{(c, 2)} S \int_{0}^{t} e (τ) d τ

{\dot{K}}_{I^{2}} = - γ_{(c, 3)} \frac{\partial S \dot{S}}{\partial K_{I^{2}}} = - γ_{(c, 3)} \frac{\partial S \dot{S}}{\partial u_{{PI}^{2} {IDD}^{2}}} \times \frac{\partial u_{{PI}^{2} {IDD}^{2}}}{\partial K_{I^{2}}} = - γ_{(c, 3)} S \iint_{0}^{t} e (τ) d τ

{\dot{K}}_{D} = - γ_{(c, 4)} \frac{\partial S \dot{S}}{\partial K_{D}} = - γ_{(c, 4)} \frac{\partial S \dot{S}}{\partial u_{{PI}^{2} {IDD}^{2}}} \times \frac{\partial u_{{PI}^{2} {IDD}^{2}}}{\partial K_{D}} = - γ_{(c, 4)} S \frac{d e}{d t}

{\dot{K}}_{D^{2}} = - γ_{(c, 5)} \frac{\partial S \dot{S}}{\partial K_{D^{2}}} = - γ_{(c, 5)} \frac{\partial S \dot{S}}{\partial u_{{PI}^{2} {IDD}^{2}}} \times \frac{\partial u_{{PI}^{2} {IDD}^{2}}}{\partial K_{D^{2}}} = - γ_{(c, 5)} S \frac{d^{2} e}{d t^{2}}

in which $γ_{c}$ is introduced as the learning rate. If the learning rate and the gains’ initial values of PI²IDD², PI²ID, PIDD², PID², and PI²D controllers are not selected properly, the controller would not correctly follow up the system’s state.

Figure 2 represents the block diagram of the adaptive higher order controllers for the CanSat carrier launch system.

Figure 2.

Block diagram of adaptive higher order controllers for the CanSat carrier launch system. MOGA: multi-objective genetic algorithm.

Optimization

In this article, multi-objective GA (MOGA) optimization is utilized as a stochastic search algorithm that imitates the process of evolution primarily based on Darwin’s survival-of-the-fittest approach to obtain the optimal values of the controllers’ parameters. In GA optimization, a population of candidate solutions called individual are evolved into superior solutions. Indeed, the population is evaluated and the best solutions are selected to reproduce and mate to create the next generation. After a number of generations, elite individuals lead the population and increase the quality of solutions. A straightforward GA involves new individual selection from population based on the fitness, crossover, and mutation regarding a number of probabilities to produce new individuals.

In this article, the value set of GA operators is set in Table 1. Learning rates are considered as design variables in the optimization process of the proposed controllers. Furthermore, the system’s diversion from equilibrium state and the system’s thrust vector deviation are represented by OF1 and OF2, respectively. These objective functions can be written as follows

OF 1 = \int | θ | d t

OF 2 = \int | Φ | d t

Table 1.

Genetic algorithm configuration parameters.

Parameter	Value
Crossover fraction	0.8
Population size	200
Selection function	Tournament
Mutation function	Constraint-dependent
Crossover function	Intermediate
Migration direction	Forward
Migration fraction	0.2
Migration interval	20
Stopping criteria	800

Results and discussion

The results obtained from applying the proposed controllers on the CanSat carrier launch system indicate the effective performance of the higher order controllers. The parameters used in the dynamic model of equation (9) are as follows: $m = 100 kg$ , $g = 10 \frac{m}{s^{2}}$ , $l = 1 m$ , $a = 100 \frac{m}{s^{2}}$ , and $I = 1000 kg {.m}^{2}$ . The initial values of the state variables of equation (11) are measured in radians, $X = {[\int θ d t, θ, \dot{θ}]}^{T} = {[0, 0.2, 0]}^{T}$ , and the initial values of adaptive control gains of equations (31) to (35) are $K = {[K_{P}, K_{I}, K_{I^{2}}, K_{D}, K_{D^{2}}]}^{T} = {[2, 4, 4, 1, 1]}^{T}$ .

For each controller, the best point among all the output points of GA from the viewpoints of two objective functions is selected as optimum design point. Table 2 shows the design variables related to the optimum design point. Additionally, the values of corresponding objective functions have been represented in Table 3.

Table 2.

Design variables of the optimum design point for higher order controllers.

Controller	Design variables
Controller	$γ_{(C, 1)}$	$γ_{(C, 2)}$	$γ_{(C, 3)}$	$γ_{(C, 4)}$	$γ_{(C, 5)}$
PI²IDD²	1.3756	1.1275	1.2530	1.3495	1.4455
PI²ID	1.1908	1.3395	1.0635	1.4786	—
PIDD²	1.4529	1.4853	—	1.0179	1.0159
PID²	1.3530	1.2194	—	—	1.1380
PI²D	1.4074	—	1.0788	1.3279	—

Table 3.

Objective functions of the optimum design point for higher order controllers.

Controller	Objective function
Controller	OF1	OF2
PI²IDD²	0.2409	4.1518
PI²ID	0.2362	4.6171
PIDD²	0.2397	4.3832
PID²	0.2391	4.6981
PI²D	0.2347	4.7351

Figures 3 and 4 represent the angular position and angular velocity of the CanSat carrier launch system under performance of the PI²IDD² controller, respectively. Additionally, the angular position of the CanSat carrier launch system under performance of the PI²ID, PIDD², PID², and PI²D controllers is shown in Figures 5 to 8, respectively. Also, the angular velocity of the CanSat carrier launch system under performance of the PI²ID, PIDD², PID², and PI²D controllers is illustrated in Figures 9 to 12, respectively. As it can be seen from the figures, the designed controllers reach the desired state. Figure 13 shows the time response of the angular position of CanSat carrier launch system thrust vector under performance of the PI²IDD² controller. Besides, Figures 14 to 17 represent the time response of CanSat carrier launch system thrust vector under performance of the PI²ID, PIDD², PID², and PI²D controllers, respectively. Regarding the theory of sliding modes, if the sliding condition is satisfied, according to equations (31) to (35), the time derivative of PI²IDD² gains will be zero and the changes in $K_{P}$ , $K_{I}$ , $K_{I^{2}}$ , $K_{D}$ , and $K_{D^{2}}$ will be limited. This also applies to the other designed higher order controllers. Moreover, the diagrams of control gains with respect to time in Figures 18 to 22 confirm this concept.

Figure 3.

Time response of the angular position of the CanSat carrier launch system under performance of the adaptive PI²IDD² controller.

Figure 4.

Time response of the angular velocity of the CanSat carrier launch system under performance of the adaptive PI²IDD² controller.

Figure 5.

Time response of the angular position of the CanSat carrier launch system under performance of the adaptive PI²ID controller.

Figure 6.

Time response of the angular position of the CanSat carrier launch system under performance of the adaptive PIDD² controller.

Figure 7.

Time response of the angular position of the CanSat carrier launch system under performance of the adaptive PID² controller.

Figure 8.

Time response of the angular position of the CanSat carrier launch system under performance of the adaptive PI²D controller.

Figure 9.

Time response of the angular velocity of the CanSat carrier launch system under performance of the adaptive PI²ID controller.

Figure 10.

Time response of the angular velocity of the CanSat carrier launch system under performance of the adaptive PIDD² controller.

Figure 11.

Time response of the angular velocity of the CanSat carrier launch system under performance of the adaptive PID² controller.

Figure 12.

Time response of the angular velocity of the CanSat carrier launch system under performance of the adaptive PI²D controller.

Figure 13.

Time response of the angular position of CanSat carrier launch system thrust vector under performance of the adaptive PI²IDD² controller.

Figure 14.

Time response of the angular position of CanSat carrier launch system thrust vector under performance of the adaptive PI²ID controller.

Figure 15.

Time response of the angular position of CanSat carrier launch system thrust vector under performance of the adaptive PIDD² controller.

Figure 16.

Time response of the angular position of CanSat carrier launch system thrust vector under performance of the adaptive PID² controller.

Figure 17.

Time response of the angular position of CanSat carrier launch system thrust vector under performance of the adaptive PI²D controller.

Figure 18.

Time response of the control gains of the adaptive PI²IDD² controller.

Figure 19.

Time response of the control gains of the adaptive PI²ID controller.

Figure 20.

Time response of the control gains of the adaptive PIDD² controller.

Figure 21.

Time response of the control gains of the adaptive PID² controller.

Figure 22.

Time response of the control gains of the adaptive PI²D controller.

Figures 3 and 5 to 8, which represent the time response of the angular position of the CanSat carrier launch system under performance of higher order controllers, indicate the high efficiency of these controllers. All of the controllers properly control the system in about 5 s. In addition, comparison of Figures 4 and 9 to 12, which represent the angular velocity of the CanSat carrier launch system under performance of PI²IDD², PI²ID, PIDD², PID², and PI²D, shows that the system has higher stability under performance of PI²ID and PI²D controllers. So, the time response of the system under performance of these two controllers has the least instability and chattering.

As it can be seen in Figures 13 to 17, it can be found that PI²IDD² and PIDD² are not recommended due to their high control efforts, while PI²ID, PID², and PI²D controllers seem to be more cost-effective. All in all, the PID² controller can be recommended as a proper controller due to its optimum control effort and higher stability.

Conclusion

In this article, adaptive higher order controllers have been implemented on the CanSat carrier launch system optimized with GA. Choosing proper Lyapunov function and initial values of control gains was the first steps of controllers’ design. Furthermore, the control gains were adapted with the aid of an adequate adaptation mechanism. By choosing the system’s diversion from the equilibrium state and the system’s thrust vector deviation as the objective functions, the GA was implemented to achieve the optimal control conditions. The results confirmed that the optimal adaptive PID² controller has the best control performance compared to other controllers for a second-order dynamical system. As a future suggestion, in order to improve the output response of the system, it is essential to update control gains by fuzzy logic.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Viet-Thanh Pham

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