Design of an Error-Based Adaptive Controller for a Flexible Robot Arm Using Dynamic Pole Motion Approach

Abstract

Design of an adaptive controller for complex dynamic systems is a big challenge faced by the researchers. In this paper, we introduce a novel concept of dynamic pole motion (DPM) for the design of an error-based adaptive controller (E-BAC). The purpose of this novel design approach is to make the system response reasonably fast with no overshoot, where the system may be time varying and nonlinear with only partially known dynamics. The E-BAC is implanted in a system as a nonlinear controller with two dominant dynamic parameters: the dynamic position feedback and the dynamic velocity feedback. For illustrating the strength of this new approach, in this paper we give an example of a flexible robot with nonlinear dynamics. In the design of this feedback adaptive controller, parameters of the controller are designed as a function of the system error. The position feedback K_p(e,t) and the velocity feedback K_v(e,t) are continuously varying and formulated as a function of the system error e(t). This approach for formulating the adaptive controller yields a very fast response with no overshoot.

1. Introduction

Recently, there has been an increasing interest in the design of feedback controllers: from the design of conventional approaches to the design of intelligent-based approaches. One such approach is on the design of adaptive controller for controlling a complex dynamic system containing nonlinearity like flexible joints. During the past, there has been a common practice to approximate a nonlinear system by a linear system in limited operating ranges and then make use of the conventional controller design approaches. However, the nonlinearity of a system is inevitable since many systems in practice involve nonlinear relationships among the variables such as electromechanical systems, hydraulic systems, and pneumatic systems [1]. For decades, various schemes of adaptive control have been proposed, and adaptive control for nonlinear systems with complex dynamics has received great attention. However, not many of these approaches are suitable for complex nonlinear systems [2–4]. Up to the present, inverse optimal controller [5–7] using the Lyapunov function has been considered as one of the most effective way for designing controllers for nonlinear systems.

In this paper, we introduce a new notion of controller called error-based adaptive controller (E-BAC) with a novel conception based upon dynamic pole motion (DPM) approach. In general, for the design of E-BAC, we consider two dominant parameters, the position feedback 𝐾_𝑝(𝑒,𝑡) and velocity feedback 𝐾_𝑣(𝑒,𝑡), and a proper design of these two feedback parameters will yield a faster and stable response of the system with no overshoot. The feedback parameters are adapted by the system error 𝑒(𝑡) and its states 𝐱(𝑡).

The rest of the paper is organized as follows. In Section 2, we introduce some important observations of a step response for a typical linear second-order system. In Section 3, we describe the notion of dynamic pole motion (DPM) and the design of error-based adaptive controller (E-BAC) in detail. A flexible robotic joint control is presented in Section 4 with E-BAC and DPM as a case study. Section 5 concludes this paper with a discussion and future works.

2. Some Important Observations in the Step Response for a Second-Order Linear System

In our study, we consider a typical open-loop second-order plant 𝐺_𝑝(𝑠) defined as𝐺_𝑝(𝑠)=1𝑠²,+𝑎𝑠+𝑏 (1a)  (1b)  𝑥statevector:𝐱=¹𝑥₂=1𝐶𝑦𝑥̇₁,output:𝑦=𝐶𝑥₁=𝑏+𝐾₁𝑥₁. (1c)

Equations (1a), (1b), and (1c) represent a typical second-order system with position (𝐾₁) and velocity (𝐾₂) feedbacks.

As shown in (1b), with position and velocity feedback controller, the transfer function of the closed-loop system is given by𝑌(𝑠)𝑅(𝑠)=𝐶𝑠²+𝑎+𝐾₂𝑠+𝑏+𝐾₁, (2) where 𝐶=𝑏+𝐾₁. This transfer function can be compared with a general linear second-order system model as𝑏+𝐾₁𝑠²+𝑎+𝐾₂𝑠+𝑏+𝐾₁=𝜔²_𝑛𝑠²+2𝜁𝜔_𝑛𝑠+𝜔²_𝑛. (3) Thus, we see that𝜔²_𝑛=𝑏+𝐾₁≜𝐾_𝑝,𝜔_𝑛:systemnaturalfrequency,2𝜁𝜔_𝑛=𝑎+𝐾₂≜𝐾_𝑣,𝜁:systemdampingratio, (4) where the parameters 𝐾_𝑝 and 𝐾_𝑣 are defined as position feedback and velocity feedback, respectively.

Generally the dynamic behavior of a second-order system can be described in terms of two dominant parameters, the natural frequency (𝜔_𝑛) and the damping ratio (𝜁). The transient response of a typical control system often exhibits damped oscillations before reaching the steady state. In specifying the transient response characteristics of a second-order control system to a unit-step input, the following transient parameters in the design of a controller are usually considered [1, 8, 9]:risetime:𝑇_𝑟=𝜋−𝜃𝜔_𝑑=𝜋−𝜃𝜔_𝑛1−𝜁²,𝜃=cos⁻¹(𝜁),settlingtime:𝑇_𝑠=4𝜁𝜔_𝑛(2%criterion),maximumovershoot:𝑀_𝑝=𝑒^{−𝜁𝜋/1−𝜁2}×100(%),bandwidth:𝜔_BW=𝜔_𝑛1−2𝜁²+4𝜁⁴−4𝜁²+2. (5) It is important to note that in the step response of the second-order system, the dominant transient parameters 𝑇_𝑟, 𝑇_𝑠, and 𝑀_𝑝 are dependent upon the natural frequency (𝜔_𝑛) and the damping ratio (𝜁) of the system. Thus, the positions of the poles of the system are determined by the values of 𝜔_𝑛 and 𝜁 as shown in Figure 1.

Figure 1 

Definition of the various parameters in the complex 𝜎-𝑗𝜔 plane: natural frequency: 𝜔𝑛, damping ratio: 𝜁, damped natural frequency: 𝜔𝑑=𝜔𝑛(1−𝜁2), and 𝜃=cos−1𝜁.

As shown in Figure 2, it is also to be noted that, in typical transient responses an underdamped system (𝜁<1) yields a faster rise time (𝑇_𝑟) at the expense of a large overshoot (𝑀_𝑝) and a large settling time (𝑇_𝑠), whereas an overdamped system (𝜁>1) yields no overshoot, that is, 𝑀_𝑝=0, but it yields large 𝑇_𝑟 and 𝑇_𝑠.

(a) The system response curves

(b) The error response curves of the systems

Figure 2 

System responses to a unit-step input with two different locations of poles (i) underdamped case (𝜁<1) and (ii) overdamped case (𝜁>1). The desired system response curve initially follows the underdamped curve for large errors and then settles down to a steady-state value (following the overdamped curve) for decreasing errors.

3. Development of an Error-Based Adaptive Controller (E-BAC): Some Design Criteria

For the design of an appropriate feedback controller, let us consider the system error 𝑒(𝑡) as an important signal in our feedback design. In our design methodology developed in this paper, we will make the parameters of the feedback controller as functions of the error. From the transient responses shown in Figure 2(b), we can emphasize that for large errors a small 𝜁 and a large 𝜔_𝑛, (i.e., an underdamped dynamics with large bandwidth) will yield a very fast response with a very small rise time 𝑇_𝑟. On the other hand, for small errors a large 𝜁 and a small 𝜔_𝑛 (i.e., an overdamped system with a small bandwidth) will inhibit any overshoot. Since 𝜁 and 𝜔_𝑛 are dependent upon the parameters of position feedback (𝐾_𝑝) and velocity feedback (𝐾_𝑣), if we define 𝐾_𝑝(𝑒,𝑡) and 𝐾_𝑣(𝑒,𝑡) as functions of the system error, 𝑒(𝑡)=𝑟(𝑡)−𝑦(𝑡), then we can achieve a very fast dynamic response with no overshoot.

From these qualitative observations on the transient response of the step response, we derive the following design criteria for the E-BAC [10].

Design Criteria for the Error-Based Adaptive Controller (E-BAC)
(i)If the system error is large, then keep the damping ratio 𝜁 very small and natural frequency 𝜔_𝑛 very large. A large  𝜔_𝑛and small 𝜁 will result into a large bandwidth of the system, thereby a shorter rise time and fast response.(ii)If the system error is small, then keep the damping ratio 𝜁 large and natural frequency 𝜔_𝑛 small. This will result into a small bandwidth of the system. For small errors, a large damping ratio in the system will avoid any overshoot in the system response.

Design of Parameters for the E-BAC
(i)Position feedback 𝐾_𝑝 controls the natural frequency 𝜔_𝑛, (𝐾_𝑝=𝜔²_𝑛), and, therefore, the bandwidth of the system.(ii)Velocity feedback 𝐾_𝑣 controls the damping ratio 𝜁, (𝐾_𝑣=2𝜁𝜔_𝑛).

Thus, we design the adaptive controller parameters which, in this case, are the position feedback 𝐾_𝑝(𝑒,𝑡) and velocity feedback 𝐾_𝑣(𝑒,𝑡) as functions of the error, 𝑒(𝑡). This procedure for designing the adaptive controller will introduce a dynamic motion in the poles of the system keeping the system response at an acceptable level. Here thus, we introduce a new notion of the movable poles and give it the name Dynamic Pole Motion (DPM). The proposed novel E-BAC is illustrated in Figure 3.

Figure 3 

The proposed error-based adaptive controller (E-BAC): 𝑥2(𝑡)=𝑥̇1(𝑡). 𝐾𝑝(𝑒,𝑡) and 𝐾𝑣(𝑒,𝑡) are defined in (8) and (12).

This novel design philosophy for adaptive controller is translated into the following linguistic algorithm:

As error decreases from a large value to a small value, 𝐾_𝑝(𝑒,𝑡) (=𝜔²_𝑛(𝑡))is continuously decreased from a very large value to a small value, and simultaneously, 𝐾_𝑣(𝑒,𝑡) (=2𝜁(𝑡)𝜔_𝑛(𝑡))is increased from a small value to a large value.

This linguistic control algorithm causes a larger bandwidth with a smaller damping ratio for large errors and smaller bandwidth with larger damping ratio for small errors. Hence, as discussed above and shown in Figures 2 and 3, during the operation of the system a desired transient response from the systems can be achieved by varying 𝜔_𝑛 and 𝜁 as functions of error. As given in (4), 𝜔_𝑛 and 𝜁 are dependent upon the position feedback 𝐾_𝑝 and velocity feedback 𝐾_𝑣, respectively. Some typical response curves for a second-order closed-loop system with varying 𝐾_𝑝 and 𝐾_𝑣 are shown in Figure 4.

(a) A family of system response curves with various values of Kp and a constant Kv = 3

(b) A family of system response curves with various values of Kv and a constant Kp = 1

Figure 4 

System response curves of a second-order system varying 𝐾𝑝 (position feedback) and 𝐾𝑣 (velocity feedback).

3.1. Design of E-BAC Parameters 𝐾_𝑝(𝑒,𝑡) and 𝐾_𝑣(𝑒,𝑡)

Using the design criteria for the adaptive controller stated above, one can develop many types of functions for 𝐾_𝑝(𝑒,𝑡)and 𝐾_𝑣(𝑒,𝑡), which satisfy the design criteria with respect to the system error and time. Here, we give one such function for 𝐾_𝑝(𝑒,𝑡)and 𝐾_𝑣(𝑒,𝑡) by defining the system error as𝑒(𝑡)=𝑟(𝑡)−𝑦(𝑡), (6) where the system output 𝑦(𝑡) is given by𝑦(𝑡)=𝐾_𝑝(𝑒,𝑡)𝑥₁(𝑡). (7) Thus, we define the position feedback 𝐾_𝑝(𝑒,𝑡) and the velocity feedback 𝐾_𝑣(𝑒,𝑡) gains as functions of 𝑒(𝑡) as𝐾_𝑝(𝑒,𝑡)=𝐾_𝑝𝑓1+𝛼𝑒²(𝑡,𝐾_𝑣)(𝑒,𝑡)=𝐾_𝑣𝑓exp−𝛽𝑒²(𝑡), (8) where 𝛼 and 𝛽 are some gain constants which decide the slope of the functions and affect the system response (see Figure 5), 𝐾_𝑝𝑓 and 𝐾_𝑣𝑓 are the final steady-state values of 𝐾_𝑝(𝑒,𝑡) and 𝐾_𝑣(𝑒,𝑡), and exp(⋅) is the exponential function. The other possible functions for 𝐾_𝑝(𝑒,𝑡) and 𝐾_𝑣(𝑒,𝑡) are given in Table 1.

Table 1 

Various possible functions and their graphic experessions for feedback gains 𝐾_𝑝(𝑒,𝑡) and 𝐾_𝑣(𝑒,𝑡).

(a) Changes in slope for 𝐾𝑝(𝑒,𝑡)=𝐾𝑝𝑓(1+𝛼𝑒2) for various values of 𝛼: the direction of arrows indicates the increasing value of 𝛼 from negative to positive values

(b) Changes in slope for 𝐾𝑣(𝑒,𝑡)=𝐾𝑣𝑓exp[−𝛽𝑒2(𝑡)] for various values of 𝛽: the direction of arrows indicates the increasing value of 𝛽

Figure 5 

The change of the slopes of 𝐾𝑝(𝑒,𝑡) and 𝐾𝑣(𝑒,𝑡) curves for various values of 𝛼 and 𝛽.

3.2. Design of the Error-Based Adaptive Controller (E-BAC)

The error-based adaptive control signal 𝑢(𝑒,𝑡) is derived as a function of the error 𝑒(𝑡) and time 𝑡 using the following two steps:positionfeedbackcontrol:𝑢_𝑝(𝑒,𝑡)=𝐾_𝑝(𝑒,𝑡)𝑥₁(𝑡),velocityfeedbackcontrol:𝑢_𝑣(𝑒,𝑡)=𝐾_𝑣(𝑒,𝑡)𝑥₂(𝑡), (9) where 𝐾_𝑝(𝑒,𝑡) and 𝐾_𝑣(𝑒,𝑡) are defined in (8) and (12), respectively. Thus, the total feedback signal 𝑢(𝑒,𝑡) is given by𝑢(𝑒,𝑡)=𝑢_𝑝(𝑒,𝑡)+𝑢_𝑣(𝑒,𝑡), (10) and the control signal 𝑣(𝑡) (see Figure 3) is defined as𝑣(𝑡)=𝑟(𝑡)−𝑢(𝑒,𝑡). (11)

4. A Case Study: Control of a Flexible Robot Arm Using E-BAC

In this section, we present the design and simulation studies of the proposed error-based adaptive controller (E-BAC) for a flexible joint of a robot arm.

4.1. Modeling of a Single Link Flexible Robot

As shown in Figure 6, a single link manipulator with flexible joint consists of an actuator connected through a gear train (harmonic drive) with the ratio 𝑛 to a rigid link with length 𝑙, mass 𝑚, and moment of inertia 𝑚𝑙²/3.

Figure 6 

A schematic diagram of a single link manipulator with a flexible joint.

Let us symbolize the rotor inertia of the actuator 𝐽_𝑀, the viscous damping of the actuator 𝐵_𝑀, the relative angular displacement of the joint actuator 𝜃_𝑀, a torque to the motor shaft 𝜏_𝑀, and the relative displacement of the end effector (load) 𝜃_𝐿. The joint flexibility is modeled by a linear torsional spring with stiffness 𝑘. The dynamics of the manipulator with a flexible joint can be represented by Euler-Lagrange equation defining 𝜏_𝑀=𝑟 as [11, 12]𝑚𝑙²3𝜃̈_𝐿+𝐵_𝐿𝜃_𝐿̇+𝑚𝑔𝑙2sin𝜃_𝐿+𝑘𝜃_𝐿+𝜃_𝑀𝑛𝐽=0,_𝑀𝜃̈_𝑀+𝐵_𝑀𝜃̇_𝑀+𝑘𝑛𝜃_𝐿+𝜃_𝑀𝑛=𝑟. (12) Equation (12) can be rewritten using the state variables 𝑥_𝑖 (𝑖=1,2,3,4) defining as𝑥₁(𝑡)=𝜃_𝑀,𝑥₂(𝑡)=𝜃̇_𝑀,𝑥₃(𝑡)=𝜃_𝐿,𝑥₄(𝑡)=𝜃̇_𝐿. (13)

Thus, we have 𝑥̇₁(𝑡)=𝑥₂(𝑡),𝑥̇₂(𝑡)=−𝑎₁𝑥₁(𝑡)−𝑎₂𝑥₂(𝑡)−𝑎₃𝑥₃(𝑡)(𝑡),𝑥̇+𝑏𝑟₃(𝑡)=𝑥₄,𝑥̇₄(𝑡)=−𝑎₄𝑥₁(𝑡)−𝑎₅𝑥₃(𝑡)−𝑎₆sin𝑥₃(𝑡)−𝑎₇𝑥₄(𝑡), (14) where1𝑏=𝐽_𝑀,𝑎₁=𝑘𝐽_𝑀𝑛²,𝑎₂=𝐵_𝑀𝐽_𝑀,𝑎₃=𝑘𝐽_𝑀𝑛,𝑎₄=3𝑘𝑚𝑛𝑙²,𝑎₅=3𝑘𝑚𝑙²,𝑎₆=3𝑔2𝑙,𝑎₇=3𝐵_𝐿𝑚𝑙². (15)

The block diagram of the system is shown in Figure 7. This system is a nonlinear and time varying system since the sine function in the feedback loop of the system causes nonlinearity in the system. The output of the system is dependent on the amplitude of the control signal, and if we use the conventional design tools, this nonlinearity causes some problems in designing an effective controller. In this paper, we present a novel approach to the design of a controller for this nonlinear timevarying system by using the error-based adaptive controller (E-BAC) and dynamic pole motion (DPM).

Figure 7 

Block diagram of the single link manipulator with a flexible joint. The system has both linear and nonlinear feedbacks.

4.2. Design of Adaptive Controller for the Systems

In this case study, for simplicity we set the value of the parameters 𝑎_𝑖 (𝑖∈[1,7]) and 𝑏 equal to 1. Thus, (14) can be rewritten as𝑥̇₁(𝑡)=𝑥₂(𝑡),𝑥̇₂(𝑡)=−𝑥₁(𝑡)−𝑥₂(𝑡)−𝑥₃(𝑡)(𝑡),𝑥̇+𝑟₃(𝑡)=𝑥₄(𝑡),𝑥̇₄(𝑡)=−𝑥₁(𝑡)−𝑥1−sin₃(𝑡)𝑥₃(𝑡)𝑥₃(𝑡)−𝑥₄(𝑡). (16) Now we design an error-based adaptive controller for the single link robotic manipulator with a flexible joint. We first formulate the dynamic characteristic equation of the system. In this study for the nonlinear and timevarying robot arm, a new notion of timevarying complex variable 𝑔(𝑡)=𝜎(𝑡)+𝑗𝜔(𝑡) (g-plane) is applied instead of the time invariant complex variable 𝑠=𝜎+𝑗𝜔 (s-plane). The g-plane has the same properties of s-plane with an additional property of timevarying. The dynamic characteristic equation of the single-link manipulator with a flexible joint shown in Figure 7 and described in (16) is given by𝑔⁴(𝑡)+2𝑔³(𝑡)+3+𝜓(𝑡)𝑔²(𝑡)+2+𝜓(𝑡)𝑔(𝑡)+2+𝜓(𝑡)=0, (17) where 𝜓(𝑡)=sin(𝑥₃(𝑡))/𝑥₃(𝑡).

The dynamic roots of this characteristic equation of the transfer function can be calculated as𝑔_1,2(𝑡)=12−1−−2𝜓(𝑡)−3±2𝜓²(𝑡),𝑔−4_3,4(𝑡)=12−1+−2𝜓(𝑡)−3±2𝜓²(𝑡)−4. (18) The nonlinear function, 𝜓(𝑡)=sin(𝑥₃(𝑡))/𝑥₃(𝑡), covers the range −0.22<𝜓(𝑡)<1 for all values of 𝑥₃(𝑡) over[−∞,∞]. Thus, the roots of the dynamic characteristic equation are moving in the g-plane (𝑔(𝑡)=𝜎(𝑡)+𝑗𝜔(𝑡)).

The moving roots of the dynamic characteristic equation are named as dynamic poles. (Note that in linear time-invariant dynamic systems since the parameters of the system remain constant, the poles and zeros of the system are time invariant.) The plot of the four dynamic poles of this flexible robot arm without a controller is shown in Figure 8. From this figure, it is clear that for some values of 𝑥₃ two dynamic poles move towards the right-hand side (RHS) of the 𝑔(𝑡)=𝜎(𝑡)+𝑗𝜔(𝑡) plane causing instability in the system. The design criteria of our proposed error-based adaptive controller (E-BAC) for a system are as below.

Figure 8 

The sketch of dynamic pole motion (DPM) of the single link manipulator with a flexible joint system, (18), without a controller. Four poles are moving in the 𝑔(𝑡)=𝜎(𝑡)+𝑗𝜔(𝑡) plane with varying 𝑥3(𝑡). Note that for certain values of 𝑥3(𝑡) two dynamic poles move towards the right-hand side of the g-plane causing instability in the system.

Design Criteria of E-BAC for the Flexible Joint of a Robot Arm
For designing an E-BAC, we should consider the following important points.(1)For introducing the stability in the robot arm system, we should move the dynamic poles on the left-hand side (LHS) on 𝑔(𝑡)=𝜎(𝑡)+𝑗𝜔(𝑡) plane for all values of 𝑥₃(𝑡).(2) Realization of DPM using E-BAC.(a)For achieving the fast response time, the system must have a large bandwidth for large errors and small bandwidth for small errors. Thus, the position feedback K_p, the bandwidth parameter, must be a function of the system error 𝑒(𝑡).(b)For no overshoot in the system response, damping should be adjusted continuously as a function of the system error. The position feedback 𝐾_𝑝(𝑒,𝑡) and the velocity feedback 𝐾_𝑣(𝑒,𝑡) are designed such that they yield a small damping ratio with large bandwidth for large errors and a large damping ratio with small bandwidth for small errors. For achieving a good controller for this fourth-order flexible robot arm, we must first add a compensator to relocate two dynamic poles far away from the origin in the left-half of the g-plane. In this case study, we relocate 𝑔₁(𝑡) and 𝑔₂(𝑡) far away from the  𝑗𝜔(𝑡)-axis. These two relocated poles far away in the left-side of the g-plane will induce very small time constants, thereby, will have negligible effect in the system dynamic response. The other two poles that are closer to the imaginary axis are dominant poles and will cause an influence in the system dynamic response. Then, an E-BAC is added in the feedback loop with a position feedback 𝐾_𝑝(𝑒,𝑡) and a velocity feedback 𝐾_𝑣(𝑒,𝑡) defined in the previous sections. In this study, the compensator (𝐺_𝑐) used introduces two zeros in the forward loop with the position of zeros being 𝑔(𝑡)=−102±𝑗1.18. This compensator provides a control over the plant poles [𝑔₁(𝑡) and 𝑔₂(𝑡)] keeping them far away from the𝑗𝜔(𝑡)-axis. The feedback controller, E-BAC, provides a control over the two plant poles [𝑔₃(𝑡) and 𝑔₄(𝑡)]. The diagram of the system with a compensator and an E-BAC is illustrated in Figure 9.
The control input signal 𝑣(𝑡) is derived using (11) as 𝑣(𝑡)=𝑟(𝑡)−𝑢(𝑡), (19) where 𝑢(𝑡)=𝑢_𝑝(𝑒,𝑡)+𝑢_𝑣(,𝑢𝑒,𝑡)_𝑝(𝑒,𝑡)=𝐾_𝑝𝑓1+𝛼𝑒²(𝑡)𝑥₃(𝑡),𝑢_𝑣(𝑒,𝑡)=exp−𝛽𝑒²(𝑡)𝑥₄(𝑡). (20)𝑥₃(𝑡)=𝑥 and 𝑥₄(𝑡)=𝑥̇ are the states of the system, 𝐾_𝑝𝑓 and 𝐾_𝑣𝑓 are the steady-state values of feedbacks 𝐾_𝑝(𝑒,𝑡) and 𝐾_𝑣(𝑒,𝑡), respectively, 𝛼 and 𝛽 are some gain constants for 𝐾_𝑝(𝑒,𝑡) and 𝐾_𝑣(𝑒,𝑡), respectively, 𝑟(𝑡) is the reference input of the system, and 𝑒(𝑡)=[𝑦(𝑡)−𝑟(𝑡)]=(𝑟(𝑡)−𝐾_𝑝(𝑒,𝑡)𝑥₃(𝑡))is the system error.
As described in the design criteria, the objective of the embedded E-BAC is to design the control 𝑢(𝑡) to make the system output 𝑦(𝑡) follow the reference input signal 𝑟(𝑡) as closely as possible with fast rise time 𝑇_𝑟 and small settling time 𝑇_𝑠 with no overshoot 𝑀_𝑝. Thus, we continuously change the dynamics of the close-loop system: initially for large errors, we make large bandwidth and very small damping ratio 𝜁(𝑡), and as error decreases, the damping ratio 𝜁(𝑡) is continuously increased and the system bandwidth is decreased.
In the design of the E-BAC, we have arbitrarily chosen the gains 𝛼=2, 𝛽=1, 𝐾_𝑝𝑓=70, and 𝐾_𝑣𝑓=5.5. With these values, the controlled system responded as an underdamped system for large error at 𝑡=0, which continuously moved towards an overdamped system with decreasing error.

Figure 9 

Schematic diagram of the flexible joint of a robot arm with a compensator (𝐺𝑐) and E-BAC.

4.3. Simulation Results

Using the gains 𝛼=2, 𝛽=1, 𝐾_𝑝𝑓=70, and 𝐾_𝑣𝑓=5.5, the initial positions of the dynamic poles of the system are placed at 𝑔_1,2(0)=−100.87±𝑗1.18 (relocated to far from 𝑗𝜔(𝑡)-axis by the compensator), and 𝑔_3,4(0)=−10.24±𝑗45.1. During the operation of the system, as error is decreased to zero, the final positions of the dynamic poles are moved to −100.5639, −99.2695, −37.9557, and −19.2109 on the g-plane. The zeros are located at around −100 near the relocated poles by the compensator, and the zeros attract the relocated poles not to affect the dominant poles. The results of the simulation study of this case study are shown in Figure 10. Further, the maps of the dynamic poles motion (DPM) are illustrated in Figure 11. The output response initially follows the trajectory with a large bandwidth and a small damping ratio, which settles down with a large damping and a smaller bandwidth.

Figure 10 

The system response with the E-BAC to a step reference input. Rise time 𝑇𝑟=0.14 seconds; settling time 𝑇𝑠=0.36 seconds.

(a) The dominant poles are moving from initial position 𝑔3,4(0)=−10.24±𝑗45.14 to final position 𝑔3(0.5)=−37.9557 and 𝑔4(0.5)=−19.2109, respectively. This dynamic pole motion causes the system response to become from underdamped response to overdamped response

(b) 3D sketch of the dynamic poles and zeros motion of the controlled system. The location of the dynamic poles and zeros is a function of the system error

Figure 11 

The sketch of motions of dynamic poles and zeros of the system with a compensator and an E-BAC (×: poles, ◯: zeros).

It is clear from the figure that the dynamic motion of poles of the system is decided by the value of the system error. The initial positions of the dominant dynamic poles are placed to generate a low damping ratio 𝜁(𝑡) and large bandwidth of the system. Thus, initially the system is underdamped. Thereafter, the dynamic poles are optimized and shifted as the system error decreases reducing the bandwidth and increasing 𝜁(𝑡). The final positions of the dominant dynamic poles make the system an overdamped system. Thus, the bandwidth becomes small, but 𝜁(𝑡) becomes high. The variation of the bandwidth at each time interval is shown in Figure 12, and the variation of the damping ratio 𝜁(𝑡) is shown with respect to the system error at each time interval in Figure 13.

(a) Bandwidth, 𝜔BW(𝑡)

(b) 𝜔BW(0)≈70𝐻𝑧

(c) 𝜔BW(0.05)≈32𝐻𝑧

(d) 𝜔BW(0.5)≈18𝐻𝑧

Figure 12 

The variation of the bandwidth of the controlled system. (a) 3D sketch of the variation of the bandwidth with respect to the frequency at each time interval, (b) the bandwidth at 𝑡=0 (sec), (c) the bandwidth at 𝑡=0.05 (sec), and (d) the bandwidth at 𝑡=0.5 (sec).

Figure 13 

The variation of the damping ratio 𝜁(𝑡) with respect to the system error as a function of time. The value of 𝜁(𝑡) changes from a low value, 𝜁(𝑡)=0.221, to a high value, 𝜁(𝑡)=1.

5. Discussion and Conclusions

In this paper, we have proposed the design of an error-based adaptive controller (E-BAC) for controlling the dynamic response of a nonlinear system. The proposed E-BAC is the controller with continuously changing feedback parameters as functions of the system error: initially for large errors an underdamped system with large bandwidth which is forced to approach to become an overdamped system with small bandwidth for small errors. In the beginning, for large errors the system is underdamped, thus, it makes the system faster with a wider bandwidth. As the error decreases, the value of the feedback gains 𝐾_𝑝 decreases and 𝐾_𝑣 increases. The design of this adaptive controller is conceptually error-based and can be used to handle the complexity of systems. In order to support the novel controller, we introduce the notion of dynamic pole motion (DPM).

As a case study, we present a flexible joint of robot arm, which is a nonlinear dynamic system, and this system is controlled by the proposed E-BAC. Without a proper controller, the system is unstable due to the nonlinearity in the feedback loop of the system. However, as shown in Figure 10, with E-BAC the trajectory response of the system is very fast, 𝑇_𝑟 = 0.36 seconds, without any overshoot (𝑀_𝑝 = 0%). Also, as shown in Figures 12 and 13, in this step response, the initial bandwidth of the system is very high (≈70 Hz) which settles down to about 18 Hz in the steady-state situation. The bandwidth of the system changes from a large value to a small value. Similarly but contrarily, the damping ratio 𝜁(𝑡) varies from 0.221 (𝑡=0) to 1 (𝑡=0.5). From the simulation studies, it is shown that the proposed E-BAC is able to control nonlinear time varying systems. Conventionally, a proper design of the controller guarantees that the changing pole position is always positioned in the left-hand side (LHS) on g-plane. In this novel design approach, the dynamic poles are always located in LHS on g-plane, thus the stability of the controlled system is assured. Further work is under way to extend this E-BAC design philosophy for higher-order partially known and unknown complex dynamic systems.