This article proposes for the first time a novel adaptive nonlinear gain based composite feedback controller. In the proposed method an adaptive nonlinear gain is introduced in the composite feedback controller to improve the system dynamic response. Using the Lyapunov analysis and the adaptive law for the nonlinear gain, it is shown that all the closed loop signals of the system are bounded. Finally, we use a double integrator single-input single-output system and twin rotor multi-input multi-output (MIMO) system to demonstrate the application of the proposed scheme. In both the applications, the improvement in settling time and root mean square error is scalable by the adaptive nonlinear gain based composite feedback control. Further the controller is successfully tested for a set of desired set points for MIMO system example which will create large initial errors.

## 1. Introduction

Composite nonlinear feedback (CNF) control law is a blend of linear feedback law and nonlinear feedback law. This control law quickly manoeuvres the output of the system to track the reference target with fast settling time and less overshoot subjected to constrained control input. The nonlinear gain in CNF control theory has its roots in Lin *et al.*, (1998) where nonlinear tracking control laws are constructed for second order linear systems with saturation input. The CNF control technique in (Turner & Postlethwaite, 2001) was extended to multivariable systems where regions of guaranteed stability for arbitrary linear quadratic control laws are developed. In Chen *et al.*, (2003), CNF control technique is applied to hard disk drive servo system. Here it is shown that the performance of CNF controller is better than the time optimal controller. A CNF controller for robot manipulators with bounded torques is presented comprising of the computed torque control (CTC) and the nonlinear feedback law in Peng *et al.*, (2009). The CTC based CNF controller is compared with simple CTC controller and sectorial fuzzy controller (SFC). It is shown that with simple CTC and SFC controller stability is not guaranteed when actuator saturation occurs but with CTC based CNF controller stability is guaranteed. In Fulwani *et al.*, (2012), a nonlinear sliding surface is designed by using the principle of variable damping concept to achieve better transient response for a class of single-input single-output (SISO) nonlinear uncertain system. The work in Chen *et al.*, (2003) has been extended in Cheng & Peng, (2007) by incorporating disturbance estimation and compensation into the CNF control framework. Further the technique has been applied to solve the position control problem for a dc servomotor. In Mobayen & Majid, (2012), the algebraic Riccati equation and CNF technique have been used for robust tracking and model following of a class of linear systems with time varying uncertainties and disturbances. The work in Lan *et al.*, (2010) investigates design of CNF control law for a hard-disk drive servo system. Here an auto tuning method is implemented to tune the parameters of the nonlinear function in the CNF control law.

In the design of the CNF control law, we need to construct a linear control law and a nonlinear control law. Once the linear law is fixed, the performance of the CNF control relies on the selection of the nonlinear function in the CNF control law. Different nonlinear functions will result in different transient performances of the closed loop system. There are several versions of the nonlinear function in the literature. The adaptation of the nonlinear gain in the CNF control law has not yet been explored. However, the adaptation of the control law for nonlinear systems has been widely studied in the literature (Krstic, 2008; Fischer *et al.*, 2014; Liu *et al.*, 2014; Zhou *et al.*, 2014; Ma *et al.*, 2015) and the references therein. The main contribution of this article is to develop a novel adaptive nonlinear gain based CNF controller. Using the Lyapunov analysis and the adaptive law for the nonlinear gain, it is shown that all the closed loop signals of the system and the nonlinear gains are finite and the overall closed loop system is asymptotically stable. This control law is applied to a double integrator SISO system and Twin Rotor multi-input multi-output (MIMO) system to illustrate the efficacy in terms of settling time and minimum overshoot. As the complexity of the system increases, the adaptive nonlinear gain CNF (ANG-CNF) controller exhibits improvement as compared to classical CNF in the performance of the overall closed loop system in terms of settling time and peak overshoot. The control efforts are almost same in case of classical CNF and ANG-CNF.

The article is organized as follows: Section 2 presents the steps involved in designing the adaptive nonlinear gain based CNF controller. Sections 3 and 4 present the numerical simulation results of double integrator SISO system and Twin Rotor MIMO system respectively. The ANG-CNF is compared to the classical CNF and the detailed controller analysis of both the controllers is reported in these sections. Section 5 concludes the article.

## 2. Adaptive nonlinear gain based composite nonlinear feedback control (ANG-CNF) using modified Lyapunov function

A linear system with input saturation is considered for which ANG-CNF control is designed. Consider a linear system as given below:

(1) $$ \left({{{A,B}}} \right) $$ is controllable.

(2) $$ \left({{{A,B,C}}} \right) $$ is right invertible and has no invariant zeros at $$ s=0. $$

### 2.1. Objective and control development

The objective is to design a control law such that the system output tracks a desired step signal smoothly and quickly subjected to constraints on controller output amplitude. This is accomplished through a combination of a linear feedback law and an adaptive nonlinear feedback law. Using Lyapunov stability analysis it is shown that the closed loop system tracks the desired step change quickly.

The design of the controller is presented in three steps.

**Step1**: Linear control law (Cheng & Peng,, 2007)

Further we define one more matrix

For a given positive definite matrix $$ {{W}}\in {{ R}}^{nxn} $$ one solves the Lyapunov equation:

Note that such $$ P $$ exists as $$ \left({A+BF} \right) $$ is Hurwitz.

**Step 2**: Nonlinear control law

The nonlinear control law then takes the form:

(On selection of $$ W $$ matrix:) With $$ \tilde{{x}}=x-x_{e} $$, poles of the closed loop system comprising of plant in (2.1) with CNF control law in Chen *et al.*, (2003) reproduced in (2.10) approaches locations of invariant zeros of auxiliary system reproduced in (2.11) as the magnitude of $$ \rho $$ increases.

Such poles are selected by judicious choice of $$ W $$ matrix. It is observed in case of MIMO system, the $$ W $$ matrix elements affect the settling time as well as peak overshoot. Note that in Section 4, the $$W$$ matrix elements are unique and need to be chosen more carefully.

**Step 3**: The combination of the linear and nonlinear control law generates ANG-CNF controller

Now the asymptotic stability of closed loop system involving equation (2.1) with the control law in equation (2.12) is proven in Theorem 1.

Consider a given linear system in equation (2.1) and ANG-CNF control law in equation (2.12) satisfying assumptions (2.1) and (2.2). Let the nonlinear gain $$ \rho $$ be adaptively updated by (2.9). Then by application of the proposed control law (2.12), output $$ y $$ tracks the desired step signal $$ {{x}}_{{{d}}} $$ in minimum time and with acceptable overshoot such that the initial condition $$ x_{0} $$ satisfy the following condition Chen *et al.*, (2003)

The tracking error is defined as $$ {{\tilde{{x}}}}={{ x}}-{{x}}_{{{e}}} $$. Using (2.5) and (2.6) the linear control law (2.3) is given by:

Equation (2.1) can be written as

From Chen *et al.*, (2003) it is clear that $$ \left({{{Ax}}_{{{e}}} {{+BHx}}_{{{d}}} } \right) = 0 $$. Then adding and subtracting the term $$ {{BF}}{{\tilde{{x}}}}+{{BHx}}_{{{d}}} $$, equation (2.15) becomes:

Thus the closed loop system can be written as

Now a Lyapunov function is proposed for the fast convergence of errors. To proceed with the Lyapunov stability analysis, let us choose the Lyapunov function as follows:

Using (2.1) and (2.7) the time derivative of equation (2.19) is given by:

After substituting (2.9) in above equation, one gets

Assuming $$ k_{3} \gg k_{2} $$ and $$ \frac{k_{1} }{k_{2} +k_{3} \rho }\approx k_{4} \rho $$

Now three cases of saturation function are considered for Lyapunov stability analysis. □

**Case (1)**$$ \left| {F\tilde{{x}}+Hx_{d} +u_{N} } \right|<u_{\max } $$ (*Linear region*). In this case from equation (2.18)

Now equation (2.22) can be written as

**Case (2)**$$ F\tilde{{x}}+Hx_{d} +u_{N} \leqslant -u_{\max } $$ (*negative saturation region*). In this case from equation (2.18)

Now the cases are considered for $$ \rho =0;\rho <0\,\,\rho >0 $$.

(a) $$ \left| {u_{s} } \right|\,<u_{\max } \,,\,\rho =0. $$ Note that $$ \dot{{V}} $$ from equation (2.22) becomes

$$ \Rightarrow -\,W_{\min } \left\| {\tilde{{x}}} \right\|^{2}+2PBu_{s} \left\| {\tilde{{x}}} \right\|<0 $$ where $$ W_{\min } $$ is minimum singular value of $$ W $$.

(b) $$ u_{s} <0\,,\,\rho >0 $$

So first and third terms in equation (2.22) remain negative, further omitting third term, the remaining terms are considered as $$ \dot{{V}}_{b} $$.

$$ \Rightarrow\! \left({-W_{\min } +2\rho_{b1} \,\theta_{\min } } \right)\left\| {\tilde{{x}}} \right\|^{2}+N\left\| {\tilde{{x}}} \right\|\leqslant 0\Rightarrow \left({-W_{\min } +2\rho_{b1} \,\theta _{\min } } \right)\left\| {\tilde{{x}}} \right\|<-N. $$ As $$ 2\rho_{b1} \,\theta_{\min } \ll W_{\min }$$ so,

So $$ \dot{{V}}<0\, $$ subjected to fulfilment of condition (2.29).

(c) $$ u_{s} <0\,,\,\rho <0 $$

Note that third and fourth term becomes negative in equation (2.22). The remaining terms are then indicated as $$\dot{{V}}_{c} $$:

So $$ \dot{{V}}<0\, $$ subjected to fulfilment of condition (2.31).

**Case (3)**$$ F\tilde{{x}}+Hx_{d} +u_{N} \geqslant u_{\max } $$ (*positive saturation region*). In this case, from equation (2.18)

(d) $$ u_{s} >0;\,\rho =0 $$

Then last two terms in (2.22) becomes zero so resulting equation is:

Hence $$ \dot{{V}}<0 $$ subjected to fulfilment of condition (2.33).

(e) $$ u_{s} >0;\,\rho >0. $$ After omitting third negative term resulting from (2.22), a condition similar to Case 2 (b) is formed and $$ \dot{{V}}<0 $$.

(f) $$ u_{s} >0;\,\rho <0. $$ After omitting third and fourth negative terms resulting from (2.22), a condition similar to Case 2 (c) is formed and $$ \dot{{V}}<0 $$.

Hence the closed loop system (2.17) will be asymptotically stable.

The design of adaptive nonlinear gain based CNF controller is summarized in following steps:

(i) Select state feedback gain matrix $$ F $$ such that ($$ A+ $$

*BF*) is asymptotically stable.(ii) For a desired signal value $$ {{x}}_{{{d}}} $$, calculate $$ G $$ and $$ {{x}}_{{{e}}} $$ from (2.4) and (2.6) respectively.

(iii) Select W and find P from (2.7) for $$ P>0 $$ and compute $$ {{u}}_{{{L}}} $$ from (2.3).

(iv) Compute the adaptive nonlinear gain ($$ \rho ) $$ from (2.9) with initial condition $$ \rho (0){{=}}\rho_{0} $$.

(v) Compute $$ {{u}}_{{{N}}} $$ from (2.8) and finally the ANG-CNF control law is given by (2.12).

## 3. Example1: Numerical simulation—with double integrator

It is an attempt to develop a control law that will provide faster response with acceptable overshoot. In another words the settling time is to be improved, which is very important in many applications. The ANG-CNF achieves the objectives which are presented and discussed in the numerical simulations. Two examples are presented to illustrate the ability of the controller to perform well. These are SISO system and MIMO system applications. In this section, for the double integrator SISO system the ANG-CNF control is applied and the results are compared with classical CNF control (Chen *et al.*, 2003):

The settling time in this context is taken as the time it takes for the controlled output to enter the $$ \pm 1\% $$ region of the set point. With the classical CNF (the control law stated in Chen *et al.*, (2003)) the output reaches the set point in 1.845 s. Referring to Fig. 1 one can verify; however, it takes 1.752 s for ANG-CNF control to achieve the same. The associated control signals are shown in Fig. 2. Note that these control efforts are similar in magnitude. Figure 3 shows error convergence because of classical CNF and ANG-CNF. The peak overshoot with ANG-CNF is within acceptable limit of 5% and with classical CNF controller as given in Chen *et al.*, (2003) is 2%. Hence it is noteworthy to mention that there subsists a control technique that accomplishes better settling time than the classical CNF in asymptotic tracking.

## 4. Example 2: Numerical simulation–with TRMS

Now the proposed control law is applied to Twin Rotor MIMO System (TRMS) with input saturation. Original system is a highly nonlinear system with dynamic coupling between input and output. The TRMS is a laboratory helicopter practical model that acts as excellent source for research study [14]. The linearized MIMO system and the associated matrices are given below [15]:

*et al.*, (2003), i.e. the total time it takes for the output to enter the $$ \pm 1\% $$ region of the set-point.

For this system, ANG-CNF control law with equations (2.9) and (2.12) is developed as below: the selected state feedback gain matrix:

The set of constants for two nonlinear gain update laws (i.e. for pitch and yaw) are as given:

Matrix $$ {{P}}_{\mathrm{ }} $$ is solution of Lyapunov equation (2.7) is given by

The ANG-CNF control provides a slight improvement in the pitch response. However, the yaw angle response is much improved as compared to classical CNF controller. Error convergence of the pitch and yaw angle are also plotted in Figs 6 and 7. Yaw error convergence is faster than pitch error.

In case of pitch angle, the settling time is 7.673 s and 8.170 s using classical CNF and ANG-CNF respectively. As per the classical CNF control, yaw angle has the settling time of 9.626 s where as it is 4.604 s using ANG-CNF control. Settling time is slightly more in case of pitch response however, for yaw response ANG-CNF has settling time that is less than half of that taken by classical CNF. The overshoots are listed in Table 3 and are well within acceptable level of 1%. Hence the proposed control successfully achieves the set point in less time as compared to the classical CNF control in case of MIMO system as well. The control effort magnitudes are within saturation level and is maximum about 1.5 V in case of pitch response while it is less than about $$ - $$1 V for yaw response.

Figure 8 shows that the control efforts are comparable and within the bounds (i.e. 2.5 V). The results are summarised in Tables 1, 2 and 3. From results, it is clear that in case of double integrator and MIMO system application, the settling time gets improved reasonably with acceptable overshoot by application of ANG-CNF control. Root mean square (RMS) error for pitch response is same and for yaw response is reduced with ANG-CNF.

Control technique | RMS error Double integrator | RMS error Pitch angle (TRMS) | RMS error Yaw angle (TRMS) |
---|---|---|---|

Classical CNF | 0.510 | 0.062 | 0.156 |

ANG-CNF | 0.510 | 0.062 | 0.133 |

Control technique | RMS error Double integrator | RMS error Pitch angle (TRMS) | RMS error Yaw angle (TRMS) |
---|---|---|---|

Classical CNF | 0.510 | 0.062 | 0.156 |

ANG-CNF | 0.510 | 0.062 | 0.133 |

Control technique | Double integrator | Pitch angle (TRMS) | Yaw angle (TRMS) |
---|---|---|---|

Classical CNF | 1.845$$ ^{\mathrm{\dagger }} $$ | 7.673 | 9.626 |

ANG-CNF | 1.752 | 8.170 | 4.604 |

Control technique | Double integrator | Pitch angle (TRMS) | Yaw angle (TRMS) |
---|---|---|---|

Classical CNF | 1.845$$ ^{\mathrm{\dagger }} $$ | 7.673 | 9.626 |

ANG-CNF | 1.752 | 8.170 | 4.604 |

$$ ^{\dagger} $$ (Chen *et al.*, 2003)

Control technique | Double integrator | Pitch angle (TRMS) | Yaw angle (TRMS) |
---|---|---|---|

Classical CNF | 2 | Nil | Nil |

ANG-CNF | 5 | 1 | 1 |

Control technique | Double integrator | Pitch angle (TRMS) | Yaw angle (TRMS) |
---|---|---|---|

Classical CNF | 2 | Nil | Nil |

ANG-CNF | 5 | 1 | 1 |

(Checking the performance of controller with large tracking errors:) The controller works satisfactorily for large tracking errors. For the sake of example, we consider MIMO TRMS. The desired signal values are increased to: [0.75–1.25], which creates large initial tracking errors. The plot of pitch and yaw error convergence given in Fig. 9 establishes satisfactory performance of the proposed controller.

## 5. Conclusion

A novel adaptive nonlinear gain based CNF (ANG-CNF) control design is presented for linear systems with actuator saturation. The controller design is based on the Lyapunov stability analysis for all the possibilities associated with the nonlinear gain values (.i.e. zero, positive and negative). Simulation results are proven in terms of better settling times and reduction in RMS errors as compared to those achieved with classical CNF control technique with acceptable overshoot. The double integrator SISO system and the Twin Rotor MIMO system are used to demonstrate the application of the proposed control law. Further the controller is also tested for tracking with new set of desired values which will create large initial errors. It is observed that as the complexity of the system increases, the adaptive nonlinear gain CNF controller exhibits improvement in the performance of the overall closed loop system.