Research Express@NCKU - Articles Digest

In this article, we briefly introduce a control design technique which is capable of yielding a wide family of globally asymptotically stabilizing control laws. This method facilitates us to cope with highly nonlinear dynamic systems subjected to unknown uncertainties and to address robustness issues.

Chaotic behavior, as shown in Fig. 1, is a very interesting nonlinear effect and has been intensively studied during the last two decades. There are many particular characteristics generated by chaotic systems, such as excessive sensitivity to initial conditions, unpredictable behavior, and irregular features of the motion in phase plane. Because it is usually difficult to accurately predict the future behavior of a chaotic system, chaos might lead systems to unstable,

Fig 1. Dynamic response of chaotic systems

performance-degraded, and even catastrophic situations such that chaos is sometimes considered an undesirable phenomenon and should be eliminated in many cases. However, in some circumstances, systems performing chaotic phenomena are preferable.

From the view point of mechanism operations, it is desirable to induce regular dynamics or motions in mechanical oscillators as in the case of precise positioning mechanisms where external vibrations might give rise to significant position errors. Regarding measurement at atomic level, atomic force microscope (AFM) is a good alternative. It has been experimentally observed that the vibration of a micro-cantilever can be chaotic under certain conditions. Such the irregular motion definitely causes AFM to give inaccurate measurements. Moreover, chaos also appears in power systems, for example, dc-dc converters.

On the contrary, plenty of applications show that chaos brings about superior contribution. For instance, in the field of chemical engineering, it has been revealed that chaos provides better mixing and yields a more uniform chemical reaction. As for chemical reduction process, the quality of anodes holds a significant impact on the reduction efficiency. To pursue an efficient reduction process, the density of an anode cannot be too high or too low. Under such the scenario, chaotic vibration provides moderate compression to produce well compacted anode. Additionally, with the rapid development of telecommunications, design of chaos-based communication technology has attracted considerable attention during the last decade. Since the broadband nature of chaotic carriers, it is difficult to retrieve hidden messages by simple spectrum analysis. As a result, chaotic signals can be utilized to encode information through masking and it indeed gives a great contribution to secure communication. To put it simply, the chaotic behavior assures security in the communication against exogenous interceptions. Chaos can also be applied to simulate weather variety, population cluster and growth, to name but a few.

Controlling of chaos has attracted a great deal of attention within the engineering society. It can be divided into two chief categories: one is to suppress the chaotic behavior and the other is to generate or enhance chaos in nonlinear systems. For both cases, the use of a robust control algorithm is essential. Backstepping approach, based on the choice of a Lyapunov function, is one of the most popular nonlinear techniques of controller design. This approach is suitable for the design of a class of nonlinear systems in strict feedback form and capable of providing designers to find a set of stabilizing controllers. Backstepping design treats system variables as independent input for subsystems and consequently each step results in an update control law for the next step. Since the control algorithm for each step is adopted with the satisfaction of a prescribed Lyapunov function, the stability for each subsystem can be guaranteed. In the traditional backstepping design, system robustness can be maintained by using high gain control (under the condition that system uncertainties are state dependent). However, systems may subject not only to uncertain parameters, but also to unmodeled system dynamics which are state independent such that the high gain backstepping design cannot achieve asymptotic stability under this condition. To solve this problem, a sliding mode controller design method is proposed based on the backstepping approach. Inheriting advantages of SMC, a developed control law is capable of stabilizing wide class of nonlinear systems subjected to exogenous disturbances and providing asymptotic stability.

Let us consider a second order nonlinear dynamic system

(1.a)

(1.b)
where

and

are continuous function. In this article, we treat the functions in (1.a) are fully known for simplicity and the upper bound of

is available in advance. When dealing with a system, how to derive a suitable control algorithm is definitely taken as a key role in the design process. The control objective is to find a proper control law

such that (1) is stable.
Ignoring the subsystem (1.b) and assuming that the state

is an independent (or virtual) control input

, we can get

(2)
Regarding the system (2), since

is controllable (under the aforementioned assumption), the controller design becomes a regulation problem of the one order subsystem (2). Then the control task now is to come up with a proper

such that the Lyapunov function

for the subsystem satisfies

(3)
Eq. (3) indicates that energy storing in system is going to decrease as time increase. In other words, all the initial states which deviate from origin (i.e., equilibrium point) will finally converge to zero as time approach to infinite. Note that

used in (2)-(3) represents by no means a solution with a specific configuration. It actually stands for a set of controller candidates for stabilizing subsystem (2).
Unfortunately, in normal circumstance, the virtual control input

can not be operated directly and thereby (2)-(3) won’t be able to come into effect. To guarantee the existence of (2)-(3), we define an extra error variable as

(4)
Differentiating (4) and substituting (2) into (1) yields the following transformed system

(5.a)

(5.b)
Comparing (5.a) and (2), one can find that the ideal system dynamics (2)-(3) will occur if

can be suppressed to zero.
From the viewpoint of SMC, one of design approaches is to choose a proper sliding surface. Once system trajectories are trapped on this manifold, an ideal sliding mode occurs. In other words, the extra error

is capable of being the candidate of sliding surface. Therefore, we design

with

(6)
Define a Lyapunov function as

. Applying the control law (6) to the derivative of

, it is easily to obtain

. This is usually referred to as approaching condition, which implies that

in finite time. Therefore, we can deduce that the desired system behavior is achieved. In the following, we take the Genesio system as an example to illustrate the control of chaotic system. The mathematical equations for the Genesio system with control input can be represented by

(7)
where the terms of uncertainty and disturbance are selected as

and

, respectively. Moreover, the Genesio system is chaotic with

and the corresponding dynamic behavior in phase plane

is shown in Fig. 2. A modified design procedure has been presented in our research. Fig. 3 shows that the system states converge to zero after the proposed control algorithm is applied (

). The system states are forced toward zero and kept on it without deviation. In this report, a design of backstepping SMC is simply introduced. The backstepping design provides designers a guide to derive a proper stabilizing controller when dealing with nonlinear systems. Furthermore, control performance can be enhanced by incorporating SMC in the design procedure.

Fig. 2. Genesio Chaotic system subjects to disturbance with initial condition[x₁₀ , x₂₀ , x₃₀] = [3,−4,2]

Fig. 3. The phase plane trajectories of controlled system.