A transformed proportional-integral-derivative controller for a multi-vectored propeller aerostat with independent actuator magnitude and rate saturations

Abstract

The problem of designing a controller for a multi-vectored propeller airship with independent amplitude and rate saturations is addressed. First, a linear Proportional-Integral-Derivative (PID) controller is introduced for position control without considering the input saturations. Then, two design methods are applied to the traditional PID control output to satisfy the independent amplitude and rate constraints: the nested saturated PID controller (N-PID) and the transformed PID controller (T-PID). The bounded magnitudes and rate outputs of the modified controllers are given. Simulation results showed both controllers have good tracking performance while satisfying independent amplitude and rate saturations. However, the transformed PID controller has the advantage of expressing explicitly the relationship of the actuator magnitude and rate saturations with the parameters of the transformed function such that the actuator saturations are suppressed by calculation but not by trial and error.

Keywords

Actuator magnitude saturation actuator rate saturation multi-vectored propeller airship nested saturated PID controller transformed PID controller

Introduction

Given the physical limitations, actuator saturation commonly exists in practical control systems. The stability and stabilization of systems with input saturations are an important field of research in control theory.^1,2 The control of systems with rate and magnitude saturation has been studied in several areas of nonlinear control, including in the dynamic output feedback compensator,^3,4 nested saturation control law,^5,6 gain scheduling technique,^7–9 anti-windup compensator,^10–12 sliding modes,¹³ back-stepping procedure,¹⁴ direct adaptive fuzzy control,¹⁵ p-bounded nested saturations feedback law,¹⁶ optimal covariance control,¹⁷ and model predictive control.¹⁸ Most of the above research work with nonlinear control laws and attempt to show that such laws are globally stabilizing or estimate their domains of attraction. Moreover, the existing solutions are mostly deduced from numerical optimization technologies, and the credible results depend on the accuracy of the mathematical model.

From the practical viewpoint, we hope to identify simpler or more intuitive controllers for a system with rate and magnitude saturation. A Proportional-Integral-Derivative (PID) controller is considered attractive in practice because it is an error-driven rather than a model-based control law, and it has a simple control structure with adjusted controller parameters that have physical meaning.¹⁹ Here, we take a different approach and attempt to find a continuously differentiable function to transform the traditional PID controller to satisfy the actuator rate and magnitude saturation rather than using complex nonlinear controller design skills. The relationship of the actuator magnitude and rate saturations with the transformed function parameters were expressed explicitly, such that actuator saturations are suppressed by calculation but not by trial and error.

This paper is organized as follows. In Section 2, the traditional nested saturated PID controller is introduced, and how this nested saturated PID controller satisfies the magnitude and rate limits is validated. Then, a modified transformed PID controller with a simple controller structure is presented in Section 3. In Section 4, we test the efficiency of the proposed control laws via numerical simulations, and the research conclusions are drawn in Section 5.

Nested saturated PID controller

Nested saturation functions are often used to deal with the problem of amplitude and rate constraints due to their special form suitable for describing multiple saturations. For an n- order integral chain system with a single input, a bounded input nonlinear state feedback control law is given to stabilize the system globally.⁵ The nested saturation controller design of the second-order system is proposed for the nonlinear model of an aerostat, and the third-order nested saturation controller is designed.²⁰

Using a nested saturation controller,²¹ we substitute the state combination feedback with the traditional linear PID controller and develop a nested-saturated PID controller (N-PID). The control structure is shown in Figure 1.

Figure 1.

Nested-saturated PID controller

Here, U is the output of the PID controller and z is the real control input. $σ (\cdot)$ is a linear saturation function defined in.²¹ The nested-saturated PID controller can be drawn as follows:

{\begin{matrix} \overset{\cdot}{z} = R σ (\frac{1}{R} \frac{M σ (\frac{U}{M}) - z_{0}}{Δ t}) \\ z = z_{0} + \overset{\cdot}{z} Δ t \end{matrix}

(1)

where R and M are the rate and magnitude limits of the actuator, respectively. $Δ t$ is the sampling period of the controller. $z_{0}$ is the previous value of the control input.

A. Teel used a nested saturation stabilizer to develop a class of feedforward nonlinear systems that are globally stable.²¹ In this work, we prove how this nested-saturated PID controller satisfies the magnitude and rate limits. The proof is broken down into several cases.

Case1: If $| U | \geq M$ , then

\overset{\cdot}{z} = R σ (\frac{1}{R} \frac{\pm M - z_{0}}{Δ t})

(2)

The left rate saturation function can be handled in two ways.

1) If $| \frac{\pm M - z_{0}}{Δ t} | \geq R$ , then two branches can deduce the limit of real control input z:

① if $\frac{M - z_{0}}{Δ t} \geq R$ then

z = z_{0} + R Δ t \leq z_{0} + \frac{M - z_{0}}{Δ t} Δ t = M

(3)

② if $\frac{- M - z_{0}}{Δ t} \leq - R$ then

z = z_{0} - R Δ t \geq z_{0} + \frac{- M - z_{0}}{Δ t} Δ t = - M

(4)

Thus, we have $| z | \leq M$ .

2) If $| \frac{\pm M - z_{0}}{Δ t} | < R$ , then no rate limit occurred and

z = z_{0} + \frac{\pm M - z_{0}}{Δ t} Δ t = \pm M

(5)

Therefore, only the amplitude limit worked.

Case 2: If $| U | < M$ , then

\overset{\cdot}{z} = R σ (\frac{1}{R} \frac{U - z_{0}}{Δ t})

(6)

The right rate saturation function can also be handled in two ways.

1) If $| \frac{U - z_{0}}{Δ t} | \geq R$ , then two branches can deduce the limit of real control input z:

① if $\frac{U - z_{0}}{Δ t} \geq R$ , then

z = z_{0} + R Δ t \leq z_{0} + \frac{U - z_{0}}{Δ t} Δ t = U

(7)

② if $\frac{U - z_{0}}{Δ t} \leq - R$ , then

z = z_{0} - R Δ t \geq z_{0} + \frac{U - z_{0}}{Δ t} Δ t = U

(8)

Thus, we have $| z | \leq M$ .

2) If $| \frac{z - z_{0}}{Δ t} | < R$ , then, the controller output is within both amplitude and rate limits. We have

z = z_{0} + \frac{U - z_{0}}{Δ t} Δ t = U

(9)

Therefore, the nested saturated PID controller can stabilize a class of feedforward nonlinear systems while satisfying both amplitude and rate limits.

Transformed PID controller

The controller output amplitude $U (t)$ and rate $\overset{\cdot}{U} (t)$ that satisfy the amplitude and rate limits with the nested-saturated function are plotted in Figure 2 with a black solid line. The outputs of the nested-saturated controller $σ (U)$ and $σ (\overset{\cdot}{U})$ have an inflection point when they reach the limit value. The rate limit is a constant when the controller output U is within the limit and switches to zero when U exceeds the maximum. The discontinuity may lead to control oscillation. Therefore, we attempt to identify a continuously differentiable function which can substitute for nested saturated function while producing the controller output and its first derivative continuous.

Figure 2.

Magnitude and rate constraints of the nested saturated function and proposed transformed function.

The proposed continuous differentiable function of U is

z = \frac{kU}{\sqrt{1 + b U^{2}}}, (k > 0)

(10)

The constraint characteristics of the proposed transform function are shown in Figure 2 with a red dashed line. The outputs of the transform function $σ (U)$ and $σ (\overset{\cdot}{U})$ are continuous. The rate attains its maximum value when the amplitude output is near zero and is diminished gradually when U goes to infinity.

The control structure based on the transformed PID controller is shown in Figure 3.

Figure 3.

Transformed PID controller.

Bounded magnitude and rate constraints of the modified controller

Here, we prove how this transformed PID controller satisfies the magnitude and rate limits. The limits of this function as the variable U moves toward positive and negative infinity are

\begin{matrix} lim_{U \to \infty} z = lim_{U \to \infty} \frac{kU}{\sqrt{1 + b U^{2}}} = lim_{U \to \infty} \frac{k}{\sqrt{\frac{1}{U^{2}} + b}} = \frac{k}{\sqrt{b}} \\ lim_{U \to - \infty} z = lim_{U \to - \infty} \frac{kU}{\sqrt{1 + b U^{2}}} = lim_{U \to - \infty} \frac{- k}{\sqrt{\frac{1}{U^{2}} + b}} = - \frac{k}{\sqrt{b}} \end{matrix}

(11)

Thus, the amplitude of the real control input has a maximum value

| z | \leq \frac{k}{\sqrt{b}}

(12)

The differential of the transformed PID controller with time t has an analytic expression

\overset{\cdot}{z} = \frac{k}{{(1 + b U^{2})}^{\frac{3}{2}}} \overset{\cdot}{U} (t)

(13)

The rate limit of the controller can be obtained as the control variable U approaches zero:

lim_{U \to 0} \overset{\cdot}{z} = lim_{U \to 0} \frac{k}{{(1 + b U^{2})}^{\frac{3}{2}}} \overset{\cdot}{U} = k \bar{\overset{\cdot}{U}}

(14)

where $\bar{\overset{\cdot}{U}}$ is the upper bound of the controller output rate and certainly exists for a given control process. Therefore, the transformed controller has both amplitude and rate limits.

Next, we connect this function’s limits with real actuator limits. First, the rate limit is considered: equation (5) indicates that the maximum value is proportional to the real controller output rate $\overset{\cdot}{U}$ , the upper bound of $\overset{\cdot}{U}$ exists but is not fixed. Therefore, we used a pricewise function to determine the variable k:

{\begin{matrix} k = R / | \overset{\cdot}{U} | \begin{matrix} | \overset{\cdot}{U} | > 1 \end{matrix} \\ k = R \begin{matrix} | \overset{\cdot}{U} | \leq 1 \end{matrix} \end{matrix}

(15)

This function guarantees the rate limit of the controller

{| \overset{\cdot}{z} |}_{max} = k | \overset{\cdot}{U} | = R

(16)

From equation (3) we have

{| z |}_{max} = \frac{k}{\sqrt{b}} = M

(17)

Thus, coefficient $b$ can be deduced for a given k:

b = {(\frac{k}{M})}^{2}

(18)

Therefore, the transformed PID controller can satisfy the independent actuator magnitude and rate limits by adjusting two parameters of the transform function.

Stability of a closed system with the transformed PID controller

We prove the stability of a closed system with the transformed PID controller. For two-order feedforward systems

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f (x) + U \end{matrix}

(19)

where $x = [x_{1}, x_{2}]^{T}$ are the states, $f (\cdot)$ is the state-related feedforward term, and $U$ is the input. The unconstrained PID controller is

U = - K_{p} e - K_{d} \overset{\cdot}{e} - K_{I} \int_{0}^{t} edt

(20)

where $e = x_{1} - x_{c}$ and $x_{c}$ is the commanded target. The mathematic proof of global stability and the asymptotic regulation of the PID controller on second-order nonlinear uncertain systems were presented in Zhao and Guo.²²

The modified controller equation (10) can be expressed as

z = k' U

(21)

where $k' = \frac{k}{\sqrt{1 + b U^{2}}} > 0$ for any instant value of U, through it is varied with the tracking process. The coefficient $k'$ only changes the magnitude of the controller and does not change the stability of the closed system.

Application to a multi-vectored propeller airship model

In this study, two controllers, namely, the N-PID and transformed PID controller (T-PID) were tested on a model of a multi-vectored propeller airship to validate the analytic results.

Model introduction

An aerostat with a diameter of 6 m and a volume of 70 m³ is shown in Figure 4. The airship is finless and equipped with four vectored propellers. The equipment tank is suspended under its body to increase pitch and roll stability. Each vectored propeller can change its thrust amplitude and direction independently. Hence, eight control degrees-of-freedom (DOF) can be observed.²³

Figure 4.

Overall structure of the airship.

The aerostat has a mass of 72 kg and volume 70 m³. The maximum thrust of every propeller is about 2 N, and the aerostat with two propellers along one direction reaches a maximum velocity of 2 m/s. The thrust-to-weight ratio is approximately 1/14. Given the lower thrust-to-weight ratio, the aerostats are easily saturated with the actuator output during flight, so it is important to design an anti-windup controller.^24,25 As the aerodynamic characteristics of aerostats are more complicated and susceptible to external disturbances, the robustness of anti-saturation control is crucial.

The dynamics of this vehicle are like those of a conventional airship. External forces and moments are induced by gravity, buoyancy, fluid inertia force, aerodynamics, and thrusters. The following dynamics equation can be constructed in the body-fixed frame through force analysis^26,27:

M {[\overset{\cdot}{u} \overset{\cdot}{v} \overset{\cdot}{w} \overset{\cdot}{p} \overset{\cdot}{q} \overset{\cdot}{r}]}^{T} = F_{T} + F_{GB} + F_{A} + F_{I}

(22)

where $M$ is the mass matrix, $\overset{\cdot}{u}, \overset{\cdot}{v}, \overset{\cdot}{w}$ denote linear accelerations; $\overset{\cdot}{p}, \overset{\cdot}{q}, \overset{\cdot}{r}$ represent the angular accelerations around the body frame, and the right-hand side of the equation corresponds to external forces and moments, including gravity and buoyancy $F_{GB}$ , aerodynamic force $F_{A}$ , coriolis force $F_{I}$ , and vectored thrust $F_{T}$ . The full six-DOF nonlinear dynamics can be found in Chen.²³

Simulation results

If we assume slight excursions in the roll, pitch, yaw, and heave, then the simplified horizontal plane 2-DOF dynamics used for the controller design are described by²⁷

\begin{matrix} (m + m_{11}) \overset{\cdot}{u} = (m + m_{22}) vr + F_{A 1} + F_{T 1} \\ (m + m_{22}) \overset{\cdot}{v} = - (m + m_{11}) ur + {F_{A}}_{2} + F_{T 2} \end{matrix}

(23)

where u and v are the state variables of the forward velocity and lateral velocity, respectively, m is the mass of airship, m₁₁ and m₂₂ are the virtual masses and inertia of the airship along u and v directions, respectively, and the external forces consist of aerodynamic forces, $F_{A i}$ , and vectored thrusts, $F_{T i}$ , for i = 1 and 2.

Two simulations were presented: fixed-point tracking and path tracking. The control period for this aerostat was chosen as $Δ t = 0.5$ . The rate limit of the forward force variation is $R = 2 N / s$ and the amplitude limit of force along on direction $M = 5 N$ .

In a fixed-point tracking (Figures 5 –7), the target is (100 and 120 m) (Figure 5). Gusts of −3 m/s occur from both directions at 200 s (Figure 6). Both controllers have the same control performance (Figure 6); however, because the output amplitude and rate of T-PID are continuous and thus, the control input of the T-PID controller is smoother than that of the N-PID controller (Figure 7).

Figure 5.

Fixed-point tracking with two controllers.

Figure 6.

State variations in fixed-point tracking.

Figure 7.

Actuator inputs in fixed-point tracking.

In the case of path tracking (Figures 8 –10), the target is a square with a side length of 30 m. A steady lateral wind of −1 m/s is present (Figure 9). Therefore, the tracking trajectory deviates from the x-direction (Figure 8). The outputs of both controllers have the same control characteristics but the control input of the T-PID controller is continuous; hence, the state variations of the T-PID controller are slightly smoother.

Figure 8.

Planer path tracking with two controllers.

Figure 9.

State variations in planer path tracking.

Figure 10.

Actuator inputs in planer path tracking

Conclusion

This paper proposes a nested-saturated PID controller and a transformed PID controller for a multi-vectored propeller aerostat with independent amplitude and rate constraints. The stability and boundedness of both controllers are addressed. Simulation results showed both controllers have good tracking performance while satisfying independent amplitude and rate saturations. The output amplitude and rate of the T-PID controller are continuous and thus, the state variations are smoother. The transformed PID controller has the advantage of a simple structure and clear physical meaning.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Science Foundation of China (Grant No. 61733017), by the Foundation of State Key Laboratory of Robotics of China, (Grant No. 2018O13), the Shanghai Pujiang Program through a sponsorship (Grant No. 18PJD018).

ORCID iD

Li Chen

Author biographies

Li Chen, born in 1975, is currently a Professor in the School of Air Transportation, Shanghai University of Engineering Science, Shanghai, China. She received her Ph.D. degree from the State Key Laboratory of Robotics, Shenyang Institute of Automation (SIA), Shenyang, China, in 2004. Her research interests are controller design, dynamic modelling. The e-mail id of is cl200432@tom.com

Yuxiang Deng, born in 1996, is currently a Master Candidate in School of Air Transportation, Shanghai University of Engineering Science. His research interests are robot control. The e-mail id is love296923691@163.com

Qiyuan Gao, born in 1997, is currently a Master Candidate in School of Air Transportation, Shanghai University of Engineering Science. His research interests are robot control. The email id is gaoqiyuan97@foxmail.com

Jinguo Liu, born in 1978, is currently a Professor in Shenyang Institute of Automation. He received his Ph.D. degree in robotics from Shenyang Institute of Automation (SIA), Chinese Academy of Sciences (CAS), Shenyang, China, in 2007. His research interests include bioinspired robot, space robot, human -robot interaction, and their practical applications in extreme environment. The email id is liujinguo@sia.cn

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