Robust Control of Chaotic Systems with Magnitude and Rate Actuator Constraints Kouamana Bousson University of Beira Interior, Portugal Carlos Velosa University of Beira Interior, Portugal ABSTRACT The present chapter proposes a robust control approach for the class of chaotic systems subject to magnitude and rate actuator constraints. The approach consists in decomposing the chaotic system into a linear part plus a nonlinear part and form an augmented system comprising the system itself and the integral of the output error. The resulting system is posteriorly seen as a linear system plus a bounded disturbance and two robust controllers are applied: firstly, a controller based on a generalization of the Lyapunov function, and secondly, the LQR with a prescribed degree of stability. Numerical simulations are performed to validate the approach applying it to the Lorenz system and to an aeroelastic system, and parameter uncertainties are also considered to prove its robustness. The results confirm the effectiveness of the approach and the constraints are guaranteed as opposed to other control techniques which do not consider any kind of constraints. 1. INTRODUCTION The problem of control in the presence of magnitude and rate actuator constraints has received a large amount of attention over the past years but still very few control techniques are opened to community and the few that exist are kept secret by top companies such NASA, ESA, Airbus, Boeing, among others. On the other hand, one has noticed that the class of chaotic systems in particular has been controlled successfully by several control techniques but actuator constraints are almost never or very rarely taken into account. It is well-known, and it can also be proved whether through analytical, numerical or experimental techniques, that chaotic oscillations may arise in a wide range of engineering problems such as in aerospace systems, aeronautical systems, electronic circuits, optical systems, among others, if the system concerned is subject to specific disturbances. There is therefore a need to formulate control strategies to control chaotic systems considering also actuator constraints, and, moreover, those strategies must be robust against parameter uncertainties and disturbances given the particular characteristics that chaotic systems have. The objective of the chapter is to propose a robust, easy to implement, and low computational methodology for the output control of chaotic systems assuming bounded controls. Specifically, it is proposed a control subject to magnitude and rate actuators constraints - the type of constraints that most of control engineering applications require, as opposed to other control techniques for this class of systems which do not consider any kind of constraints. Given the importance of the topic for today’s most advanced engineering problems, the chapter includes practical examples of the application of the method to real-world physical systems. This way, one believes that the approach of control proposed is an interesting tool for the target audience: practicing control engineers, graduate, postgraduate students and researchers. The chapter is organized as follows: the problem to be solved is stated in section 3; in section 4, a solution to the problem, which is a robust control with actuator constraints based on a generalization of the Lyapunov function and on the LQR with a prescribed degree of stability, is proposed; in section 5, numerical simulations are presented to validate the approach - it is used to control the classic Lorenz system and an aeroelastic system, and in both applications parameter uncertainties are also considered to prove the robustness of the control; section 6 addresses possible future research directions; and, lastly, the chapter ends with the conclusion of the approach proposed in section 4. 2. BACKGROUND Chaotic systems are deterministic nonlinear systems mostly with a relatively simple structure, which despite being governed by well-defined dynamical laws, exhibit an unpredictable motion and are characterized as being highly sensitive to the initial conditions and to parameter changes. The same chaotic system started at nearly initial states or with slightly different parameters produces trajectories in the phase-space which quickly become uncorrelated. Due to this, it is practically impossible to construct two identical chaotic systems in laboratory without an active control because the trajectories will never be equal or even similar. Systems with a chaotic behaviour can arise in a large variety of fields such as in engineering problems, economics, biology, chemistry, medicine, among others (Boccaletti, Grebogi, Lai, Mancini, & Maza, 2000; F. Chen, Zhou, & Chen, 2011; Ferreira, de Paula, & Savi, 2011; Moon, 1992; Schuster, 1999), and it can also be proven, through analytical methods, numerical techniques or experimental tests, that any nonlinear system may exhibit chaotic motions even if the parameters have been previously chosen to trigger a regular behaviour. It sounds unthinkable at the first glance but that can really be possible if the system is exposed to external disturbances with particular characteristics. A small perturbation with a specific power and frequency(ies) is sufficient to trigger a chaotic dynamics. As an example, the attitude of a satellite in a circular orbit near the equatorial plane of the Earth, despite being planned to exhibit a regular motion as it happens with the communications satellites, it may become chaotic if the spacecraft has magnetic elements. The terrestrial magnetic field acts as perturbing torques that are sufficient to trigger a chaotic attitude (Cheng & Liu, 1999). The fact that an undesirable behaviour may appear in dynamical systems and that can lead to a catastrophe in some circumstances, led the chaos theory and its control to become an important research field in the last two decades and an increasingly important subject in the scientific community. On the other hand, any control system has to deal inevitably with control constraints regardless if the controlled system exhibits a regular or a chaotic behaviour. The voltages or currents that can be applied to an electrical circuit are bounded, the range of the control surfaces of an aircraft is bounded, the thrust of a rocket is bounded, and so on. These are few examples of actuator constraints that should be considered when designing a controller. Moreover, there are applications where not only the magnitude of the actuators plays an important role but also their rates: the slew-rate of an operational amplifier is bounded, the responsiveness of a pneumatic is bounded, valves and pumps have maximum throughputs, etc., which should also be considered. Otherwise, actuator saturations can lead to a degradation of the performance of the system or even to a catastrophe scenario in the extreme case. It was what happened in 1992 regarding to flight control systems. An F-22 Raptor prototype crash occurred due pilot-induced oscillations partly caused by rate saturation of the control surfaces which in turn induced time-delay effects in the control loop. In 1993, a similar aircraft crashed occurred with the Gripen JAS 39 where saturations played again a critical role (Murray, 1999). It was reported that actuator saturations were also blamed as one of several unfortunate mishaps which led to the 1986 Chernobyl nuclear power plant disaster, where rate limitations aggravated an already hazardous situation (Stein, 2003). One of the most common techniques used to control nonlinear systems is the feedback linearization wherein the objective consists, in a first stage, in formulating a control law such that the closed loop system becomes linear and then resorting to powerful and well-known techniques of the linear control to finish the job. Feedback linearization has been also used to control chaotic systems (Fradkov & Evans, 2005; Liqun & Yanzhu, 1998; Shi & Zhu, 2007). However, this technique requires the exact mathematical model of the system and consequently does not assure robustness of the control against model uncertainties neither to external disturbances. Since in most real-world physical systems the model contains parameters that are known only with some degree of accuracy (ex.: geometric parameters, inertia parameters, friction parameters, among others), and particularly for the class of chaotic systems that are extremely sensitive to parameter changes, feedback linearization often turns out to be not successful. Another remarkable technique to control nonlinear systems is the backstepping control. Based on a Lyapunov function, it resorts to the computation of fictitious controls in intermediate steps which allows in a last step the computation of the real control law. Oppositely to the feedback linearization where nonlinearities are completely eliminated, backstepping control is more flexible in the sense that global stability can be achieved avoiding that “useful nonlinearities” are eliminated. Backstepping control was also used to control chaotic systems (Idowu, Vincent, & Njah, 2008; Kareem, Vincent, Laoye, & Akinola, 2008; Yassen, 2006), but it has been reported that besides being cumbersome to find the Lyapunov function, it is also sensitive to parameter changes. A successful control can be achieved but the model has to be free of noise. When a controller is designed based purely on the mathematical model of the system such is the case of the feedback linearization or the backstepping control, the control law is formulated based on a nominal model of the physical system. As a result, if the controlled system contains uncertainties on its dynamics or if it is subject to external disturbances, the controller stops being effective precisely because it is designed assuming that the model is accurate and does not suffer any change. Taking into account that uncertainties are almost always present in most dynamical systems (ex.: uncertainties in parameters, noises in sensors, unmodeled dynamics, undesired fluctuations in actuators), the robust control, whose control law is designed considering not only the nominal model but also some uncertainties, is a very effective tool. SMC (Sliding Mode Control), LQG/LTR (Linear-Quadratic-Gaussian / Loop Transfer Recovery), QFT (Quantitative Feedback Theory), -Synthesis Control and are techniques which can be used to ensure a robust control. Robust control has been also used to control chaotic systems (Huang & Feng, 2008; Xu & Yang, 2009; Yan, Yang, Chiang, & Chen, 2007). Nevertheless, robust control has limitations when used to control systems with actuator constraints. Adaptive control is an excellent technique to deal with systems subject to uncertainties or whose parameters are time-varying. Oppositely to robust control that guarantees that the control law needs not to be modified if the uncertainties are within pre-specified limits, adaptive control does not require any a priori information about parameter or uncertainty limits, and the control law is constantly modified over the time. MRAC (Model Reference Adaptive Control), STR (Self-Tuning Regulator), Gain Scheduling and Dual Control are examples of adaptive techniques that can be used to control uncertain models. However, dealing with uncertainties is not an easy task and in that sense adaptive control has a drawback: the structure of the model parameters must be known. Such requirement may limit its application to the control of chaotic systems. For example, it was not yet shown that two oscillators with very different structures can be synchronized through adaptive control (Femat & Solis-Perales, 2008). Even so, chaotic systems have been also controlled through this technique (Alstrom, Marzocca, Bollt, & Ahmadi, 2010; Ayati & K-Sedigh, 2008; Koofigar, Sheikholeslam, & Hosseinnia, 2011) but, once again, not considering actuator constraints. MPC (Model Predictive Control), also known as RHC (Receding Horizon Control), is one of the most advanced and successfully techniques for the control of dynamical systems. Its principle consists in the successive resolution in real-time of a finite horizon optimal control problem based on measurements or estimates of the current system states. The major advantage of predictive control lies in the possibility to include several restrictions on the optimization criterion such as output constraints, state constraints, and mostly important for the present work, control magnitude constraints and constraints on the variation of those signals. At first glance, MPC seems to be the best technique to achieve the purpose proposed in this chapter. Nonetheless, although it has been used also to control chaotic systems (Ahn, Lee, & Song, 2012; Mohammadbagheri & Yaghoobi, 2011; Polyak, 2005) the models concerned are free of noise and the control may fail. On the other hand predictive control may require, particularly for systems with a fast dynamics, a large power of computational calculus because the resolution time of the optimization problem between each sample becomes a requisite that can be on the order of milliseconds or even microseconds. Techniques specifically designed to control chaotic systems such as the renowned OGY method (Ott, Grebogi, & Yorke, 1990), the Continuous Delayed Feedback Control (Pyragas, 1992) and the Otani-Jones Control (Otani & Jones, 1997) have been used successfully in practise although having some unfavourable points. One of the drawbacks that stands out most is that actuator constraints are not taken into account. Control based on LMIs (Linear Matrix Inequalities) is an interesting control approach and it has been also used to control chaotic systems (Deng & Xu, 2010; Tanaka, Ikeda, & Wang, 1998; Yu, Zhong, Li, Yu, & Liao, 2007). The technique allows calculating the gain of the system in such a way that the controlled system satisfies certain performance requirements. That includes the possibility to consider stability conditions, conditions for a specific decay rate, conditions in order to reduce the effect of disturbances on the outputs (disturbance rejection), output constraints, and also input constraints. It is an effective approach but at same time it requires a special care and also some experience by the designer because some parameters should be relaxed in order to obtain an admissible control. When using toolboxes to solve LMIs, the decision variables are optimized by the resolution algorithms, and sometimes it is necessary to relax some parameters so that the system responds as intended. If used to control a chaotic system subject to control constraints it may happen that saturation occurs so easily, when actually it is needless, and can lead to instability. A common approach to deal with actuator constraints consists in designing a controller considering no input constraints and in adding a saturation block after the controller output followed by anti-windup techniques (Hippe, 2006). However, this is not the best approach because it introduces nonlinearities on the system, degrades its performance and in some cases can lead to instability. To sum up, several control techniques have been used to control chaotic systems. However, this class of dynamical systems in particular requires a special attention due to its characteristics. Essentially, due to its high sensitivity to initial conditions and to its high sensitivity to parameter changes. Its control should be therefore robust against parameter uncertainties in order to be effective. On the other hand, one has noticed that there is no much work opened to community regarding to control of chaotic systems with actuator constraints. Thus, this chapter contributes in that sense. Not only with robust control with magnitude constraints but also with rate constraints. 3. PROBLEM STATEMENT Consider an output tracking problem of a time-invariant nonlinear system in a continuous-time approach. Let that system be described by differential equations of the form (1) and the reference signals to be tracked generated by the reference system (2): ( ̇ Controlled system: ̇ Reference system: ) ( (1) ) (2) where , represent state vectors, , vectors of parameters, the outputs of each system (vectors with same dimension) where are the reference signals to be tracked, the control input vector, , the output matrices, the input matrix, two smooth nonlinear functions, and ̇ . For the class of chaotic systems, a nonlinear system of the form (1) can exhibit a regular motion (ex.: equilibrium points, periodic orbits, multi-periodic orbits) or a chaotic motion depending on the values of the parameters. Lorenz, Chua, Rössler, Chen, Genesio-Tesi and Coullet systems are well-known examples of that. On the other hand, even if the parameters have been set to trigger a regular behaviour, it can be proven through a Melnikov analysis that the trajectory of any nonlinear system may become chaotic if it is subject to external disturbances with particular characteristics. Chaos may be desirable or not depending on the application, but an unexpected behavioural change is no longer desirable. Most of control approaches to control systems of the form (1) such that their outputs follow given reference signals consider that the parameters are kept constant. However, in practical applications, the parameters can suffer unavoidably slight variations due to external causes which in turn can force the system to exhibit a different behaviour from the desired one. As an example, consider the well-known Chua circuit whose parameters are set by electronic components (resistors, inductors and capacitors). If the parameters are chosen so that the system exhibits a chaotic oscillation, for example to be applied in a secure communication, it is possible that the oscillation can be no longer chaotic because chaotic systems are extremely sensitive to parameter changes, and factors such as temperature, humidity and age cause variations in the component values. On the other hand, real-world physical systems have to deal necessarily with actuator constraints. The classic approach to limit the magnitude of the control consists in adding a saturation block at the controller output and resort to anti-windup techniques to prevent both controller and plant windups. Nevertheless, despite being a workable solution, the approach is not the best because it introduces nonlinearities in the system and the resulting performance is not the best. Moreover, due to electrical/mechanical restrictions, some applications require not only magnitude constraints but also rate constraints. The problem to be solved consists in forcing the outputs of the system (1) to follow bounded reference signals even if the parameters are not known accurately and considering both magnitude and rate actuator constraints. Mathematically, the purpose is to find ( ) such that conditions (3) and (4) remain: Problem to solve: Assume ‖ ( )‖ and find ( ) such that: ‖ ( )‖ for any wherein , and ‖ ( ) ( )‖ (3) ‖ ( )‖ , ‖ ̇ ( )‖ from the time-instant at which the control is turned on (4) . In inequality (3), ‖ ‖ represents an appropriate norm and is desirable to be as small as possible for a good tracking. Parameters and denote respectively the magnitude and rate saturation bounds of the control vector. 4. THE APPROACH PROPOSED Consider a dynamic system of the form: ̇ ( ) ( ) , ( ) (5) where ( ) for all , is continuously differentiable in and in the components of for near the origin, and is a bounded disturbance continuously differentiable with respect to the components of and Lebesgue measurable with respect to . Definition 1: Stability in the Sense of Lyapunov The unperturbed system (5), that is, with therein, is said to be stable in the sense of Lyapunov with respect to the equilibrium , if for any and for any initial time , there exists a positive ( ) constant , such that: ‖ ( )‖ ‖ ( )‖ , for all (6) Definition 2: Asymptotic Stability The unperturbed system (5), that is, with therein, is said to be asymptotically stable about its equilibrium , if it is stable in the sense of Lyapunov and, furthermore, there exists a positive ( ) constant , such that: ‖ ( )‖ ( ) (7) Definition 3: Uniform Stability The unperturbed system (5), that is, with therein, is said to be uniformly stable about its equilibrium , if it is stable (asymptotically or in the Lyapunov sense) with the constraint being ( ). independent of , Definition 4: Stability under Persistent Disturbances The perturbed system (5) is said to be stable under persistent disturbances (or also totally stable) about its equilibrium , if given a tolerance there are two positive constants, and , such that: ‖ ̃( )‖ ‖ ( ̃ )‖ } ‖ ̃( )‖ where ̃( ) denotes the solution of the perturbed system. Theorem 1 (Malkin Theorem): , for all (8) If the unperturbed system (5) is uniformly and asymptotically stable about its equilibrium , then it is stable under persistent disturbances. Namely, the persistently perturbed system remains to be stable in the sense of Lyapunov, (G. Chen, 2004; Hoppensteadt, 2000). Chaotic systems have by nature a simple structure. That is, the differential equations are constituted mainly by linear terms and by some nonlinear terms. In that sense, the controlled system (1) can be decomposed into a linear part plus a nonlinear part as follows: ( ) ̇ (9) where, in turn, can be seen as a state matrix and ( ) as disturbances. The parameter vector was dropped to simplify the terminology but the parameters still implicitly on matrix and/or on . Remark: For nonlinear systems wherein the linear part does not exist explicitly, ( ) in (1) can be written into the same form of (9) linearizing it about a specified state and writing the remaining terms as: ( ) ( ) . Typically, the origin can be chosen as the linearization point. However, some precautions must be taken into account. There are some particular nonlinear systems which are controllable at the origin but that when linearized about the same point are no longer controllable. It is the case of the underactuated systems. In these specific systems, the point should be chosen in such a way that the linearized system still to be controllable. Alternatively, an optimal linearization could be performed instead of the classic linearization (Jacobian matrix). With the optimal linearization the controllability characteristic is not lost during the linearization process, (Bousson & Quintiãs, 2008). Define a vector as the integral of the output error: ∫ ( ) (10) Let denotes an operator which will be used to enforce the effective control input ( ) to satisfy amplitude and rate constraints of the form (4). Let that operator be given by equation (11), (Saberi, Stoorvogel, & Sannuti, 2011): ( ) (11) where: ̇ ( ) and ( ) ( ( )) (12) ( ( )) is a saturation function of limit , defined as: ( ) {| | } ( ) (13) Consider now an augmented system composed by (9) and by the time-derivative of equation (10). Together with the operator (11-13), that system yields: ̇ [ ] ̇ where can be written in the form: [ ][ ] [ ] ( ) [ ( ) ] ( ) (14) ̇ with ( ) denoting the augmented state vector, the augmented input matrix and ( ) ( ) ( ) ( (15) ) the augmented state matrix, the vector of external disturbances. At this point, the output tracking problem with constraints (3-4) was transformed into a regulation problem of the form (14), where the goal is to find the control ( ) such that ( ̇ ̇ ) ( ). Note that this implies, in turn, that ( ) ( ). One of the characteristics of chaotic systems is that their trajectories in the phase space are bounded. If one considers that the reference signals to be tracked are also bounded, all components of the disturbance vector ( ) remain consequently bounded. Assuming that is a continuous-time vector function, and actually it is, one has, by definition (continuous and bounded function), that is Lebesgue measurable with respect to . Therefore, a robust controller can be design to stabilize asymptotically the unperturbed system (15), that is, with therein, because the persistently perturbed system will remains stable in the sense of Lyapunov, (theorem 1). Note that if a time-invariant system, which is the case of the unperturbed system (15), is designed to be asymptotically stable, it will be asymptotically- and also uniformly stable because a time-invariant system has the constraint independent of . The robust control approach proposed in this chapter to stabilize the system (15) is presented in the following subsections and is divided in two parts: a control based on the Korobov technique (Korobov & Skoryk, 2002) and a control through an extension of the LQR - the LQR with a prescribed degree of stability (Naidu, 2002). In both cases, it is assumed that the augmented system (15) is controllable, or in other words, that the pair ( ) is controllable which implies that the rank of the controllability matrix is equal to the order of the system (15): [ ] (16) 4.1. Korobov control (Korobov & Skoryk, 2002) developed a control law based on a controllability function (a generalization of the Lyapunov function) that steers, under certain conditions, the trajectory of linear systems of the form ̇ to the origin. The technique takes into account control constraints which can be chosen until a given order, that is, constraints in ̇ ̈ , but the trajectory must to start inside a small neighbourhood of the origin so that those constraints are satisfied. Nevertheless, the assumptions made by Korobov allow realizing that the controlled trajectory is steered to the origin even if its initial point has not started on that attraction domain and even if the model is perturbed. Bounded and Lebesgue measurable disturbances are acceptable. The drawback that arises as a consequence is that the saturation limits of the controls are no longer satisfied. Once the trajectory can be far away from the reference when the control is turned on, alone, this technique ( ), has not power to ensure a constrained control. Therefore, one attached the operator (11-13), which is already included in the augmented system (15), to guarantee an effective constrained control. Consider the augmented system (15). Let the fictitious control (Korobov & Skoryk, 2002): be computed by the control law (17), ( where the way to find matrices , , ( ) and ( ) ) (17) is resumed below: is an upper triangular matrix of dimension and the remaining defined as the vector is the th column of the matrix condition (18): in which the elements of the main diagonal are , . For the calculation of each element , , the values of are chosen so as to satisfy the ∑ ( ) (18) and each vector , , is chosen so as to be orthogonal to all linearly independent vectors (19) ⟩ with the exception of the vector , and such that the scalar product ⟨ is satisfied. (19) is a matrix defined by (20): ( is a matrix of dimension each is defined for as ) ( with elements ( ∑ . ) let be a matrix of dimension given by is a matrix in which the elements of the first off-diagonal equal ) (20) and the others equal to zero, wherein ( ) where each and all the other elements equal . ( ): is given by the inverse matrix of (21) where the parameter can assume any value . Here, denotes the order of the derivative of the control and one considers for constraints on the magnitude and on the rates. ( ) Controllability function ( the equation (22): ∫ ( ) ): the controllability function ⟨ ( ) where ⟨ ⟩ denotes the dot product, and its unique solution following conditions: ( ) ( . ) for all (21) . ( ⟩ ) is defined, for any , by (22) is a scalar function that satisfies the ( ∑ ( ) ( ( ) is continuous at all In equation (22), ( ( )) )) , and continuously differentiable at any is a scalar number chosen within the interval: ̅ where matrix and : except at the origin. ‖ is computed by the integral (24), (23) ‖ defined by the expressions (25-27), ∫ ‖ , (24) ‖ (∑ ‖ ‖ ‖) ‖ (25) (26) ( ‖ ) ‖ ∏ In (25), { } and if matrix of the dimension of and in (27) ( (27) ) and if ( ) calculated with . In (26), . denotes the identity In practise, the solution that satisfies relation (22) is obtained numerically and the calculation process throughout the entire trajectory may become time consuming. This can be seen as a drawback because the frequency at which the control can be applied to the system is limited by the time necessary to compute . However, it can be proven (Korobov & Skoryk, 2002) that the controllability function ( ) is equal to the solution of the differential equation ̇ if its initial condition is equal ⟨ ( ) ⟩ to . Thus, expression (22) can be solved only once, and once ̇ obtained the solution ( ( )), can be solved simultaneously with system (15) and ( ) substituted by ( ( )) when calculating the control. The control law (17) has the power to steer the trajectory of the system (15) to the equilibrium state but has not the power to hold it after that. That is, if the equilibrium is the origin, the trajectory is steered just until reach the point immediately before. This is why there is a need to switch to another controller when the equilibrium is reached. As a criterion to do that, let us define a tolerance and carry out the switch when the value of the controllability function is less than or equal to , ( ) . Note that as approaches to zero means that the controlled trajectory approaches to the reference. 4.2. LQR with a specified degree of stability For a linear time-invariant system of the form ̇ with initial condition ( ) and no final time constraint, it is well-known from the optimal control theory that the control law which steers the trajectory to the origin minimizing the performance index: ( ∫ ) , (28) is given by (29), (Naidu, 2002): (29) where (ARE): is a symmetric and positive definite matrix, solution of the Algebraic Riccati Equation ( ) ( ) wherein and are two weighting matrices, being definite ( ) and symmetric and positive definite ( number denoting the specified degree of stability. (30) symmetric and positive semi), and a scalar Regarding to the system to be controlled, system (15), let the fictitious control be computed by the control law (29) from the time-instant at which the trajectory ( ) is close to the equilibrium state. Under this condition, both trajectories ( ) and ( ) are close to each other, and as a result, the effort of the control required to hold the output error small is not too large. The largest effort occurs when the error is large but for that situation the Korobov control undertakes to decrease it. The robustness of the LQR is assured by the parameter . Once all components of the disturbance vector ( ) are bounded and Lebesgue measurable, can be chosen such that the controller supports such disturbance because causes a -shift in the eigenvalues of the closed-loop matrix ̅ to the left in the direction of the real axis. ( ) applied to Note that from the time-instant at which the LQR is turned on the effective control system (15) does not saturate, and in that sense, one can say that the control is optimal in the sense it minimizes the performance index (28). 5. SIMULATION RESULTS In this section one considers two engineering applications and presents numerical simulations to validate the approach proposed. The first application deals with the output tracking of a classic chaotic system and the second one with the flutter or chaotic motion suppression of an aeroelastic system. In both applications the parameters of the controlled system are also changed over the time in order to validate the robustness of the control. 5.1. Application 1: Lorenz system It was considered two well-known chaotic systems, Lorenz and Chua, and the goal is to force the outputs of the Lorenz system to follow reference signals generated by the outputs of the Chua system. In a first stage, the parameters of both systems are set to nominal values which lead them both to exhibit chaotic motions and the methodology is applied to validate its effectiveness when tracking chaotic signals. In a second stage, the parameters of the Lorenz system are changed over the time with the purpose to verify the robustness of the control against parameter uncertainties. Here, the parameters of the Chua system are kept unchanged, once what the controller needs are the reference signals and not the reference model. The Chua system is a nonlinear chaotic oscillator described, in its normalized form, by the following set of differential equations (Chua, Wu, Huang, & Zhong, 1993): ̇ ( ( )) ̇ (31) ̇ where: ( ) ( )| | | | The parameters of system (31) are and for the values presented below the system exhibits the chaotic behaviour shown in figure (1), a double-scroll attractor: , , , 50 4 45 3 40 2 35 30 x 3 1 3,r (32) Lorenz attractor Chua attractor x , 25 0 20 -1 15 -2 0.5 -3 5 0 -4 -3 10 -2 -1 0 x 1,r 1 2 3 x 2,r -15 -0.5 Figure 1. Phase-space of the Chua system. Considers that the first state variable unique reference signal to be tracked: 40 20 0 -20 0 -10 -5 0 x1 5 -20 10 15 20 x2 -40 Figure 2. Phase-space of the Lorenz system. ( ) is the unique output ( ) of system (31) and therefore the (33) The Lorenz system is a simplified mathematical model of atmospheric convection governed by the following set of differential equations (Lorenz, 1963): ( ̇ ) ̇ (34) ̇ which for the values of parameters (2): given below gives rise to the chaotic motion shown in figure , , Decomposing the Lorenz system into a linear part plus a nonlinear part, yields: (35) ̇ [ ̇ ] ̇ [ ][ ] [ ] (36) where, if one considers two control inputs, on the first state equation and trajectory of the Lorenz system becomes controllable and the system written as: ̇ on the third one, the ( ) (37) with: [ ] [ , ] (38) ( ) is the unique output ( ) of system (36): Considers that the first state variable (39) Writing the Lorenz system together with the integral of the output error and applying the operator to ensure bounded controls, one has the augmented system: ̇ [ ] ̇ [ ][ ] [ ] ( ) [ ( ) ] ( ) ( ) (40) which takes the form (15) and is ready to be controlled with control laws (17) and (29). Note that both the Lorenz as the Chua systems are both chaotic and therefore the term ( ) is a bounded disturbance and Lebesgue measurable with respect to as assumed in (5). On the other hand, system (40) can be controlled because the pair ( ) is controllable, that is, the rank of the controllability matrix is equal to its order, ( ) . For simulation purposes, the ordinary differential equations of systems (12), (31) and (40) were solve together using the RK-Butcher method between and seconds, with a step of s, and with initial conditions (41). In (12), the value of the required parameter is . The parameters of the Korobov controller and of the LQR are the ones presented in (42) and switching from one to another occurs when the controllability function is ( ) . The constraints of the actuators are presented in (43). To ensure that both systems, Lorenz and Chua, are already in a chaotic regime the control was turned on at s, a time-instant wherein the transient period has already gone. Initial conditions: , , , (41) Controller parameters: Korobov LQR , (42) Actuator constraints: ‖ ‖ , ‖ ̇‖ (43) Simulation 1: In the following simulation the parameters of both systems are kept unchanged. Figure (3) shows the time evolution of the state variables and of the outputs of each system. The first vertical dashed line indicates the time-instant at which the Korobov control is turned on and the second one the time-instant at which the control is switched to the LQR. The output error is shown in figure (4). The magnitude and rates of the control variables are shown in figures (5) and (6), respectively. The rates / derivatives of the control variables were computed numerically by the method of the centered finite differences. Figure (7) shows the integral of the output error , and figure (8) the controllability function of the Korobov control. x 1 = y Lorenz x 1 = y Chua ey 40 30 20 0 20 -20 -40 0 10 x 2,Lorenz 20 x 2,Chua 30 t [s] 40 50 60 10 40 20 0 0 -20 -40 0 10 x 3,Lorenz 20 x 3,Chua 30 t [s] 40 50 60 -10 60 40 -20 20 0 -20 0 10 20 30 t [s] 40 50 60 -30 0 10 Figure 3. State variables: Lorenz and Chua. 20 30 t [s] 40 50 Figure 4. Output error: 300 60 . 3000 u1 u1,dot 200 2000 100 1000 0 0 -100 -1000 -200 -300 0 10 20 30 t [s] 40 50 60 140 -2000 0 10 20 30 t [s] 40 50 60 20 30 t [s] 40 50 60 2000 u2,dot u2 120 1500 100 1000 80 60 500 40 0 20 -500 0 -20 0 10 20 30 t [s] 40 Figure 5. Control variables: ‖ ‖ 50 60 . -1000 0 10 Figure 6. Control rates: ‖ ̇ ‖ . 45 100 (t) q 40 50 35 30 0 25 20 -50 15 -100 10 5 -150 0 -200 0 10 20 30 t [s] 40 50 60 Figure 7. Integral of the output error. -5 0 10 20 30 t [s] 40 50 60 Figure 8. Korobov controllability function. Simulation 2: In this simulation the parameters of the Lorenz system are slightly changed over the time to demonstrate the robustness of the control against parameter uncertainties. One considers that the parameters are known within specified intervals, which is what happens in most real applications. Considering that each parameter may suffer a change around its nominal value in about , one has , and . Let the parameter vector be perturbed according with a multivariate normal distribution with their nominal values as means and covariance matrix computed such that each parameter belongs to its respective interval with sure. ( , where denotes the mean of each parameter and the respective standard deviation). The correlation coefficients assumed between each two parameters are the ones presented in (45): Mean vector of parameters and vector of standard deviations: [ ] [ ] [ ] , [ ] (44) Pearson correlation coefficients: , , (45) Covariance matrix: [ ] [ ] (46) Figure (9) shows the time evolution of the perturbed parameters. Figure (10) shows the state variables of the Lorenz and of the Chua systems, figure (11) the magnitude of the control variables and figure (12) their respective rates. Lorenz x 1 = y Lorenz 12 40 11 20 10 0 9 -20 8 0 10 20 Lorenz 30 t [s] 40 50 60 -40 40 30 20 28 0 26 -20 0 10 20 Lorenz 30 t [s] 40 50 60 -40 60 2.8 40 2.6 20 2.4 0 0 10 0 10 x 3,Lorenz 3 2.2 10 x 2,Lorenz 32 24 0 20 30 t [s] 40 50 60 -20 0 Figure 9. Perturbed parameters (Lorenz). 10 x 1 = y Chua 20 x 2,Chua 20 x 3,Chua 20 30 t [s] 40 50 60 30 t [s] 40 50 60 30 t [s] 40 50 60 Figure 10. State variables: Lorenz and Chua. 300 3000 u1 u1,dot 200 2000 100 1000 0 0 -100 -1000 -200 -300 0 10 20 30 t [s] 40 50 60 140 -2000 0 10 20 30 t [s] 40 50 60 20 30 t [s] 40 50 60 2000 u2 120 u2,dot 1500 100 1000 80 60 500 40 0 20 -500 0 -20 0 10 20 30 t [s] 40 Figure 11. Control variables: ‖ ‖ 50 60 . -1000 0 10 Figure 12. Control rates: ‖ ̇ ‖ . 5.2. Application 2: Aeroelastic system In the aeronautical field an aeroelastic system refers to an aircraft system including a wing or a tail in which the interaction of the elastic structure, the inertia and the aerodynamic forces can cause many complicated phenomena presumably due to nonlinear effects of the structure. It is well-known that from a specific/critical flight speed, a limit cycle oscillation, which typically is referred as flutter, occurs. The phenomenon arises either in subsonic flow as in supersonic flow, and under some particular conditions can degenerate to chaos, (Alstrom et al., 2010; Bousson, 2010; F. Chen et al., 2011; Ramesh & Narayanan, 2001; Wang, Chen, & Yau, 2013; Zhao & Yang, 1990; Zheng & Yang, 2008). Furthermore, if the controlled aeroelastic system is subject to actuator constraints, structural instabilities may also arise (Demenkov, 2008). Because limit cycle oscillations and possible chaotic motions are dangerous phenomena which can lead eventually to a structural failure due to material fatigue and consequently to an aircraft disaster in the worst scenario, a robust controller is needed to supress any undesired motion during the flight. Moreover, the controller must deal with actuator constraints because it is well-known that the amplitudes and the velocities at which the control surfaces can operate are bounded due to mechanical, electrical or aerodynamic limitations. In this sub-section, one applies the control approach proposed in section (4) to suppress effectively the undesired motions of an aeroelastic system. Following references (Platanitis & Strganac, 2004; Wang et al., 2013), the model of a two-degree of freedom wing structure includes a linear spring oriented along the plunge displacement, a rotational spring along the pitch angle, corresponding dampers, and at a given flight speed the wing oscillates along the plunge displacement direction and rotates at the pitch angle about the elastic axis. The control of both motions is done through two control surfaces: one at the leading edge and one at the trailing edge. Figure (13) shows the respective two-dimensional aeroelastic system. Figure 13. Aeroelastic system with two degrees of freedom. The mathematical model for the aeroelastic system depicted in figure (13) is (Wang et al., 2013): ̈ ][ ] ̈ [ ̇ ][ ] ̇ [ [ ][ ] ( ) [ ] (47) where the plunge displacement and pitch angle are the variables to be controlled and and the angles of the leading- and trailing-edge of the control surfaces deflection, respectively. The aerodynamic terms and (lift force and moment respectively) can be modelled suitably for a lowfrequency and subsonic flight scenario including a leading-edge and a trailing-edge control surface as (48), the well-known Theodorsen functions: [ ̇ ( [ ( ( ̇ ) ̇ )] (48) ( ) ̇ )] where is the air density, the flight speed, the wing section half-chord, the wing section span, the dimensionless distance from the mid-chord to the elastic axis position, and the lift and moment coefficients per unit angle of attack, respectively, and the lift and moment coefficients per unit angle of wing trailing-edge control surface deflection, respectively, and the lift and moment coefficients per unit angle of wing leading-edge control surface deflection, respectively, and , , are the effective control moment derivatives due to angle of attack and trailing- and leading-edge control surfaces deflection respectively about the elastic axis, defined as: ( ) ( , ) ( , ) (49) In system (47), is the total mass of the pitch-plunge system (wing plus suporting body), the mass of the wing section, the dimensionless distance between the center of gravity and the neutral axis, the moment of inertia (where denotes the pitch cam moment of inertia, the wing section moment of inertia about the center of gravity and ), and and the plunge displacement and the pitch angle damping coefficients, respectively, and and ( ) the plunge displacement and the picth spring stiffness coefficients, respectively. ̇, Defining the state variables: , of ordinary differential equations as: ̇ , system (47) can be converted into the form , ̇ ̇ ̇ ̇ (50) ̇ ̇ ̇ ̇ which in turn can be written into the form of a linear part plus a nonlinear part ̇ ( ): ̇ ̇ [ ] ̇ ̇ with the coefficients () ̇ ̇ ̇ ̇ [ ][ ] [ ] (51) ] ( , ( ) ) )( {[( ̇ [ defined as: , ̇ [ ] ( ) ] } ) ( ) ( ) (52) ( ) )( {[( ̇ ̇ ( ) ] } ) ( ) ( ) Since the purpose of the present application is to supress undesired motions whenever they occur (limit cycle or chaotic oscillations), the goal in terms of control can be simplified to find the control that forces the state variables and to be zero instead of forcing the trajectory of system (51) to the origin. Note that ̇ and ̇ and given the nature of these relations the entire trajectory is steered to the origin because ( ) . Being and the outputs of system (51), the output matrix [ ] yields: [ ] (53) and the reference signals to be tracked is this particular case are constant, augmented system as required in (15), one has: ̇ [ ] ̇ [ ][ ] [ ] ( ) [ ( ) ( ) ] ( ) . Forming the (54) ( ) where it is now in the form of a perturbed system ̇ ( ) and can be perfectly controlled through the control strategy proposed in section (4), i.e., with control laws (17) and then with (29), because the disturbance term ( ) is bounded and the pair ( ) is controllable, ( ) . Hereafter one presents two simulation cases wherein the aeroelastic system is already in a limit cycle oscillation (the wing section moving vertically and rotating, continuously) and the control is turned on to supress the undesired motion. For that, the augmented system (54) is solved together with equation (12) using the RK-Butcher method between and s, with a step of s, and departing from initial conditions (55). The parameters of the aeroelastic system are the ones stated in table (1) and the required parameter in (12) is . The control is turned on at s, a time-instant at which the system exhibits already a limit cycle, and switching from the Korobov controller to the LQR controller occurs when the controllability function is ( ) . The parameters of both controllers are presented in (56). Regarding to the actuator constraints, one considers that each control surface, leading-edge and trailing-edge, can has a maximum deflection of , , and that ̇ the speed at which each surface can act is , ̇ . Parameter Value Parameter Value ( ) Table 1. Parameters of the aeroelastic system. Initial conditions: , , (55) Controller parameters: Korobov LQR , (56) Actuator constraints: ‖ ‖ ‖ ̇‖ , (57) Simulation 1: In this simulation the free stream velocity is assumed constant and selected such that triggers the flutter phenomenon/limit cycle oscillation. It was set to as considered by (Wang et al., 2013). In figure (14) one shows the time evolution of the state variables of the aeroelastic system. The plots on the left represent also the system outputs, and . The first vertical dashed line indicates the time-instant when the Korobov control is turned on and the second one the time-instant when the control is switched to the LQR. Figures (15) and (16) show respectively the amplitudes and rates of the control surfaces (leading- and trailing-edge). Figure (17) shows the phase space of the system subdivided in two phase planes: the plane ( ̇ ) on the left and the plane ( ̇ ) on the right. 300 20 200 dot [º/s] [º] 10 0 100 0 -100 -10 -200 -20 0 5 10 15 20 -300 25 0 5 10 t [s] 15 20 25 15 20 25 t [s] 200 15 10 100 [cm/s] 0 dot 0 -5 h h [cm] 5 -100 -10 -15 0 5 10 15 20 25 -200 0 5 10 t [s] Figure 14. State variables of the aeroelastic system. t [s] 10 100 = [º/s] dot 200 0 1,dot u = [º] 1 20 -20 u -10 0 5 10 15 20 0 -100 -200 25 0 5 10 10 = [º/s] dot 0 2,dot -5 u u = [º] 2 20 25 15 20 25 200 5 -10 -15 15 t [s] t [s] 0 5 10 15 20 100 0 -100 -200 25 0 5 10 t [s] t [s] Figure 15. Control variables: ‖ ‖ Figure 16. Control rates: ‖ ̇ ‖ . . 200 250 200 150 150 100 100 50 [cm/s] 0 dot 0 h dot [º/s] 50 -50 -50 -100 -100 -150 -150 -200 -250 -20 -15 -10 -5 0 5 10 15 -200 -15 20 -10 -5 0 h [cm] [º] Figure 17. Phase space subdivided in two phase planes: the plane ( on the right. 5 10 ̇ ) on the left and the plane ( 15 ̇) Simulation 2: To validate the approach against parameter uncertainties one considers in the following simulation a change over the time in the free stream velocity according with function (58), figure (18). The results are presented in figures (19) to (21). ( ) (58) 26 24 V [m/s] 22 20 18 16 14 12 0 5 10 15 20 t [s] Figure 18. Perturbed free stream velocity. 25 20 300 200 dot [º/s] [º] 10 0 100 0 -100 -10 -200 -20 0 5 10 15 20 -300 25 0 5 10 t [s] 15 20 25 15 20 25 15 20 25 15 20 25 t [s] 15 200 10 100 [cm/s] 0 dot 0 -5 h h [cm] 5 -100 -10 -15 0 5 10 15 20 -200 25 0 5 10 t [s] t [s] 200 10 100 = [º/s] dot 20 0 1,dot u = [º] 1 Figure 19. State variables of the aeroelastic system (with parameter uncertainty). -20 u -10 0 5 10 15 20 0 -100 -200 25 0 5 10 t [s] t [s] 200 25 = [º/s] dot 15 2,dot 10 u u = [º] 2 20 5 0 0 5 10 15 20 25 0 -100 -200 0 5 10 t [s] t [s] Figure 20. Control variables: ‖ ‖ 100 . Figure 21. Control rates: ‖ ̇ ‖ . 6. FUTURE RESEARCH DIRECTIONS The problem addressed in this chapter concerns the robust control of chaotic systems with bounded controls. Specifically, the approach proposed deals with the control of this class of systems assuming that the system to be controlled is subject both to magnitude and rate actuator constraints. Nonetheless, although the objective has been successfully achieved, there are interesting ideas which are not yet solved, at least that is authors’ knowledge to date, and that should be improved. Notice that in the approach proposed the actuators may saturate! However, it is well-known that chaotic systems are highly sensitive to initial conditions, and in that sense, one believes that is possible to steer the trajectory to a given point in the phase space through small amounts of control without reach the saturation limits. Thus, here stays an interesting problem for researchers and engineers: How to control (or synchronize) chaotic systems without even saturate the actuators? 7. CONCLUSION In the present chapter a robust control approach is proposed to control the class of chaotic systems assuming magnitude and rate actuators constraints. The purpose of the control precisely is to force the outputs of a chaotic system to track specified reference signals. The approach consists in decomposing the nonlinear system into a linear part plus a nonlinear part and form an augmented system comprising the system itself and the integral of the output error. Thus, the tracking problem is transformed into a regulation problem wherein the system to be controlled is seen as a linear system plus a bounded disturbance. This way, the mathematical model of the reference system (who generates the reference signals) is not required as opposed to other control techniques. Two robust controllers are posteriorly designed to stabilize the system about its equilibrium state and an operator is employed to ensure a constrained control. Two controllers are used because the trajectory of the system can be anywhere in the phase space when the control is turned on and it is also intended a control with global stability. This is overcome starting the control with a robust control of Korobov (a control based on a generalization of the Lyapunov function) and then substituting it by the LQR with a prescribed degree of stability. Numerical simulations are presented to validate the approach proposed. The control is applied to the classic Lorenz system and to an aeroelastic system. In both applications some parameters are also slightly changed over the time to validate the robustness of the control against parameters uncertainties. 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Dearborn, MI. 10. KEY TERMS AND DEFINITIONS Chaotic: Represents the type of motion that is highly sensitive to changes in the initial conditions, that is extremely sensitive to parameter changes, and that its behaviour is unpredictable for medium/longterm. Two trajectories initiated with slightly different initial conditions diverge exponentially over the time. Limit cycle: An isolated closed trajectory in the phase space. The term isolated means that the neighbouring trajectories are not closed. Lyapunov exponents: Numbers that measure the rate of exponential attraction or separation over the time of two adjacent trajectories started with different initial conditions in the phase space. For a dynamical system with bounded trajectories, a positive Lyapunov exponent indicates the existence of a chaotic motion and if two or more exponents are positive the system is said to be hyperchaotic. Melnikov function: A function that measures the distance between a stable manifold and an unstable manifold associated with a saddle point of a given Poincaré section resulting from a continuous flow. According to the Melnikov theory when the two manifolds intersect the function has a simple zero indicating the existence of chaos, Phase space: For dynamical systems governed by a set of first order differential equations is a space whose coordinates are the state variables or some components of the state vector. It is used to represent the system behaviour and a point in the phase space defines a potential state of the system. Poincaré section: A sequence of points in the phase space generated by the intersection of a continuous flow with a given surface/section. Strange/Chaotic attractor: Refers to an attractor set in the phase space with a fractal dimension in which the motions of the trajectories are chaotic. Trajectory/Orbit: A path (sequence of points) generated by a continuous or discrete system in its phase space. Underactuated systems: Systems in which the number of control variables is less than the number of degrees of freedom.