- Email: [email protected]
Adaptive robust PID controller design based on a sliding mode for uncertain chaotic systems
Chaos, Solitons and Fractals 26 (2005) 167–175 www.elsevier.com/locate/chaos
Adaptive robust PID controller design based on a sliding mode for uncertain chaotic systems q Wei-Der Chang a, Jun-Juh Yan b
b,*
a Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan, ROC Department of Electrical Engineering, Far-East College, No. 49, Jung-Hwa Road, Hsin-Shih Town, Tainan 744, Taiwan, ROC
Accepted 7 December 2004
Abstract A robust adaptive PID controller design motivated from the sliding mode control is proposed for a class of uncertain chaotic systems in this paper. Three PID control gains, Kp, Ki, and Kd, are adjustable parameters and will be updated online with an adequate adaptation mechanism to minimize a previously designed sliding condition. By introducing a supervisory controller, the stability of the closed-loop PID control system under with the plant uncertainty and external disturbance can be guaranteed. Finally, a well-known Duffing–Holmes chaotic system is used as an illustrative to show the effectiveness of the proposed robust adaptive PID controller. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The presence of chaos in physical systems has been extensively demonstrated and is very common. Chaotic systems are rather complex dynamical nonlinear system and the classical features appeared in output responses of chaotic systems have, for example, excessive sensitivity to initial conditions, broad spectrums of Fourier transform, and fractal properties of the motion in the phase space [1–3]. Due to its powerful applications in chemical reactions, power converters, and information processing, etc., it is meaningful and practical to design certain control law to cope with these complex chaotic systems for real engineering applications. To do this, different techniques have been developed to achieve the chaotic control. For example, sliding mode control [1–3], bang-bang control [4], optimal control [5,6], intelligent control base on using neural networks [7], feedback linearization [8], differential geometric method [9], adaptive control [10–12], and among many others [13]. On the other hand, the use of PID control has a long history in control engineering and is acceptable for many real applications due to its simplicity in architecture. Hence, in many real industrial applications, the PID controller is still widely used even though lots of new control techniques have been proposed. However, to our knowledge, the control of chaotic system by using simple PID control has not been well discussed.
q *
Supported by the National Science Council of Republic of China under Contract NSC-92-2213-E-269-001. Corresponding author. Fax: +886 6 5977570. E-mail address: [email protected] (J.-J. Yan).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.12.013
168
W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175
The key for designing a PID controller is the determination of three parameters of PID controller, i.e., proportional gain Kp, integral gain Ki, and derivative gain Kd. In the past, the PID self-tuning methods based on the relay feedback technique have often presented for a class of SISO systems [14–17] and MIMO systems [18]. In general, the controlled plants considered in the above-proposed PID control systems mostly belong to linear systems plus time delays. Moreover, in recent years, a new and attractive control strategy that is to combine neural networks and/or fuzzy systems with the traditional adaptive control has been proposed for nonlinear systems. Two types of adaptive controls are generally investigated: • Indirect adaptive control. The unknown model of nonlinear system is first obtained and learned by using the neural/ fuzzy system and then a feedback linearization control law is designed based on this model [19–22]. • Direct adaptive control. The neural/fuzzy system is directly used as a controller in the feedback control system, i.e., the output of neural/fuzzy system is a control input to the nonlinear plant. Usually, an error signal between the desired and actual outputs is employed to online update adjustable parameters in the neural/fuzzy controller [21–25]. In this paper, based on the use of the sliding mode, a robust adaptive PID control tuning is newly proposed to cope with the control problem for a class of uncertain chaotic systems with external disturbance. Three gains of PID controller, i.e., Kp, Ki, and Kd, are regarded as adjustable parameters and will be adjusted during control procedure. We wish to update these parameters with a proper adaptation mechanism such that a previously designed sliding condition is minimized as possibly. The stability of closed-loop PID control system can be guaranteed by using the Lyapunov approach with a supervisory controller [19,21,22,26]. The detailed descriptions for the PID control gains tuning and for the analysis of system stability will be presented in the following.
2. PID controller The continuous form of a PID controller, with input e(Æ) and output upid(Æ), is generally given as Z t 1 d upid ðtÞ ¼ K p eðtÞ þ eðsÞ ds þ T d eðtÞ ; Ti 0 dt
ð1Þ
where Kp is the proportional gain, Ti is the integral time constant, and Td is the derivative time constant. We can also rewrite (1) as Z t d ð2Þ upid ðtÞ ¼ K p eðtÞ þ K i eðsÞ ds þ K d eðtÞ; dt 0 where Ki = Kp/Ti is the integral gain and Kd = KpTd is the derivative gain.
3. System definition for uncertain chaotic systems For simplification, we only consider a second-order uncertain chaotic systems, but the approach can be generalized to high-order systems. With the input uðtÞ 2 R and the output yðtÞ 2 R, the uncertain chaotic system is described as x_ 1 ðtÞ ¼ x2 ðtÞ; x_ 2 ðtÞ ¼ f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ uðtÞ;
ð3Þ
yðtÞ ¼ x1 ðtÞ; where X = [x1, x2]T are measurable states vector of the system, f(Æ) is an unknown nonlinear nominal plant, Df(Æ) is the plant uncertainty applied to the system, and d(Æ) denotes the external noise disturbance. In addition, it is assumed that there exist two positive upper bounds f u(Æ) and Df u(Æ) satisfying |f(Æ)| 6 f u(Æ) and |Df(Æ)| 6 Df u(Æ), and a positive constant a satisfying |d(Æ)| 6 a. Let e = yd y (= yd x1) be an error signal between the desired and actual outputs. Define Y d ¼ ½y d ; y_ d T and assume that both y d and y_ d are bounded, i.e., kYdk1 = suptP0kYd(t)k < 1. Then the error vector of system becomes E ¼ Y d X ¼ ½e; e_ T . Suppose that we can choose a gain vector K = [k0, k1]T such that roots of s2 + k1s + k0 = 0 are in the open left-half complex plane. Now let a feedback linearization controller be given by u ¼ f ðX ; tÞ Df ðX ; tÞ dðtÞ þ €y d þ K T E:
ð4Þ
W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175
169
Substituting (4) into (3), we have €e þ k 1 e_ þ k 0 e ¼ 0:
ð5Þ
Consequently, from (5), we have e(t) ! 0 as t ! 1, i.e., y ! yd asymptotically. However, it is noted that the relative models in (3) are assumed to be unknown in this study. Throughout the paper, the following assumption is made. Assumption 1. Let the constraint set Xx for the state X be defined as Xx ¼ fX 2 R2 : kX k 6 M x g; where Mx is a pre-specified parameter. It is desired that the state trajectory of system X never reach the boundary of Xx during the control procedure. For simplicity of analysis, we may choose Mx P kYdk1. 4. Supervisory controller design and adaptation laws for PID controller Now let the control input in (3) be given by u ¼ upid þ us ;
ð6Þ
where upid is the PID controller as shown in (2) and us is the extra supervisory controller that will be fired only when the states of the system exceeds some bound and guarantees the stability of the system. Based on concepts of sliding mode control, a proper adaptation law that is based on the use of gradient method is presented to minimize a designed sliding condition for updating PID control gains. 4.1. Supervisory controller design To design the supervisory controller us, substituting (6) into (3), we have, in view of (4), x_ 2 ¼ f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ upid þ us ¼ f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ upid þ us þ u u ¼ €y d þ K T E ½u upid us : This implies that €e ¼ K T E þ ½u upid us :
ð7Þ
Let K¼
0 k 0
1 ; k 1
B ¼ ½ 0 1 T
ð8Þ
be a companion form. From (7), we have E_ ¼ KE þ B½u upid us :
ð9Þ
Now consider a Lyapunov function candidate V e ¼ 21 ET PE;
ð10Þ
where P is a positive definite symmetric matrix satisfying the Lyapunov equation KT P þ P K ¼ Q
ð11Þ
and Q is a positive definite symmetric matrix selected by designer. Define V M ¼ 21 kmin ðP ÞðM x kY d k1 Þ2 ; where kmin(P) denotes the minimum eigenvalue of P. Note that if kXkPMx, then, from (10), we have V e P 21 kmin ðP ÞkEk2 P 21 kmin ðP ÞðkX k kY d kÞ2 P 21 kmin ðP ÞðM x kY d k1 Þ2 ¼ V M :
ð12Þ
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Hence if Ve < VM, then kXk < Mx. Moreover, the time derivative of Ve along the trajectories of the closed-loop system (9) satisfies V_ e ¼ 21 ET ðKT P þ P KÞE þ ET PB½u upid us ¼ 21 ET QE þ ET PB½u upid us
ð13Þ
6 21 ET QE þ jET PBjðju j þ jupid jÞ ET PBus : From (4) and hypotheses of |f(Æ)| 6 f u(Æ), |Df(Æ)| 6 Df u, and |d(Æ)| 6 a, we have u 6 f u þ Df u þ a þ j€y d j þ jK T Ej:
ð14Þ
Define the indicator function I* by I* = 1 if Ve P VM and I* = 0 if Ve < VM. Hence, if the supervisory controller is chosen as us ¼ I sgnðET PBÞbf u ðÞ þ Df u ðÞ þ a þ j€y d j þ jK T Ej þ jupid jc;
ð15Þ
then, from (14) and (15), we can guarantee that V_ e < 0 in (13) if Ve P VM [21,22,26]. 4.2. Adaptive laws for PID controller tuning On the other hand, in order to derive a proper adaptation law to update three control gains, let us define a designed signal xr as x_ r ¼ €y d þ k 1 e_ þ k 0 e;
ð16Þ
where k0 and k1 are given as above. Moreover, a sliding surface is defined as S ¼ x2 xr :
ð17Þ
If the sliding mode occurs, i.e., S = 0, then x2 ¼ xr :
ð18Þ
Substituting (18) into (16), we have €e þ k 1 e_ þ k 0 e ¼ 0. This implies that e(t) ! 0 as t ! 1. From viewpoints of the sliding control, the sliding condition that ensures the hitting and existence of a sliding mode is derived according to the Lyapunov stability theory. In general, the Lyapunov function candidate for the sliding mode control is simply given by V ¼ 21 S 2 and therefore the sliding condition is defined as V_ ¼ S S_ < 0:
ð19Þ
From (19), it is guaranteed that S(t) ! 0 as t ! 1. To derive an adequate adaptation mechanism for tuning three PID control gains, a gradient search method is used to minimize the sliding condition S S_ in (19) as possibly in this study. The gradient search algorithm is calculated in the direction opposite to the energy flow, the convergence properties of the PID controller tuning can be also obtained. Moreover, it is quite intuitive to choose S S_ as an error function. From (17) and using (3), we have S_ ¼ x_ 2 x_ r ¼ f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ uðtÞ x_ r :
ð20Þ
Substituting (6) into (20) and multiplying both sides of (20) by S, the following is obtained: S S_ ¼ S½f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ upid þ us x_ r :
ð21Þ
Based on the gradient method and the chain rule, and using (2) and (21), the adaptation laws for three control gains Kp, Ki, and Kd can be easily obtained as follows: oS S_ oS S_ oupid K_ p ¼ c ¼ c ¼ cSe; oK p oupid oK p
ð22Þ
Z t oS S_ oS S_ oupid ¼ c ¼ cS eðsÞ ds; K_ i ¼ c oK i oupid oK i 0
ð23Þ
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171
Fig. 1. Robust adaptive PID control for uncertain chaotic system.
and oS S_ oS S_ oupid de ¼ c ¼ cS ; K_ d ¼ c oK d oupid oK d dt
ð24Þ
where c > 0 is the learning rate. Notice that if the learning rate c or the initial values of PID control gains are not selected adequately, the resulted PID controller will probably make the states of system divergent. Fortunately, the supervisory controller as in (15) will play an important role that provides an extra input to pull the states back to the pre-specified constraint set Xx and also guarantees the stability of the system. The block diagram of the robust adaptive PID control with a supervisory controller for uncertain chaotic system is depicted in Fig. 1. The overall design procedure can be summarized as follows. Data: Uncertain chaotic system in (3) and desired output yd. Goal: Design a controller (6), i.e., u = upid + us, such that the plant output follows the desired output asymptotically. I. Choose a constraint parameter Mx. II. Find positive upper bounds of f u(Æ), Df u (Æ), and a such that |f(Æ)| 6 f u(Æ), |Df(Æ)| 6 Df u(Æ), and |d(Æ)| 6 a to construct a supervisory controller us in (15). III. Choose proper initial values of PID control gains. IV. Choose K and Q and find the solution P of the Lyapunov equation (11). V. Construct the PID controller upid in (2) with adaptation laws in (22)–(24). VI. The desired controller (6) is given by u = upid + us. VII. Stop.
5. An illustrative example The plant of interest considered in this study is a classical Duffing–Holmes chaotic system, and its nominal dynamic equations are described as [2] x_ 1 ðtÞ ¼ x2 ðtÞ; x_ 2 ðtÞ ¼ x1 ðtÞ 0:25x2 ðtÞ x31 ðtÞ þ 0:3 cosðtÞ; yðtÞ ¼ x1 ðtÞ:
ð25Þ
The sampling time is equal to 0.01 and the initial states of the system are assumed to be x1(0) = 0.2 and x2(0) = 0.2, respectively. To demonstrate the dynamic chaotic behavior of (25), the phase portrait of x1(t) and x2(t) is first shown in Fig. 2 from t = 0 to t = 30. Now suppose that let a plant uncertainty Df(X,t), an external disturbance d(t), and a control input u(t) be incorporated into (25). Hence, we have
172
W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175 0.8 0.6 0.4
x2(t)
0.2 0
-0.2 -0.4 -0.6 -0.8 -1.5
-1
-0.5
0
0.5
1
1.5
x1(t)
Fig. 2. Phase portrait of x1(t) and x2(t) for unforced Duffing–Holmes chaotic system with x1(0) = 0.2 and x2(0) = 0.2.
x_ 1 ðtÞ ¼ x2 ðtÞ; x_ 2 ðtÞ ¼ x1 ðtÞ 0:25x2 ðtÞ x31 ðtÞ þ 0:3 cos t þ Df ðX ; tÞ þ dðtÞ þ uðtÞ; yðtÞ ¼ x1 ðtÞ:
ð26Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Without loss of generality, for example, we select Df ðX ; tÞ ¼ 0:1 sinðtÞ x21 þ x22 and dðtÞ ¼ 0:1 sinðtÞ in (26). From (26), the corresponding upper bounds can be obtained as follows: jf ðÞj ¼ jx1 ðtÞ 0:25x2 ðtÞ x31 ðtÞ þ 0:3 cos tj 6 jx1 ðtÞj þ 0:25jx2 ðtÞj þ jx31 ðtÞj þ 0:3 f u ðÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi jDf ðÞj ¼ j0:1 sinðtÞ x21 þ x22 j 6 0:1 x21 þ x22 Df u ðÞ; and jdðÞj ¼ j0:1 sinðtÞj 6 0:1 a: The proposed robust adaptive PID controller is now applied to control the uncertain chaotic system with external disturbance in (26). We wish that the output of uncertain chaotic system y(t) can track the desired output y d ðtÞ ¼ sinð1:1tÞ. Also we choose k0 = 0.1 and k1 = 1 so that the roots of s2 + k1s + k0 = 0 are in the left-half complex plane and let the learning rate c = 5. From (8), we have K¼
0 1 : 0:1 1
From (11) with Q = diag[2,2], we have P¼
11:1 10 10
11
:
Moreover, the adaptation laws in (22)–(24) are used to online update the PID controller with initial gains bKp(0), Ki(0), Kd(0)c = [5, 5, 5]. The value of Mx is deliberately selected to be large enough such that the system state X never reach the boundary of Xx. The results are demonstrated in Figs. 3–6. Fig. 3 shows the time trajectories of the controller u. Figs. 4 and 5, respectively, show the output responses of x1 and x2 of the controlled Duffing–Holmes system. It can be easily seen from these results that the states of such a chaotic system can quickly track the desired states about after 5 s. The phase portrait of x1 versus x2 is also shown in Fig. 6 and seems to be satisfactory.
W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175 20 0
control input
-20
-40
-60
-80
-100
-120
0
5
10
15
20
25
30
time
Fig. 3. Time trajectory of controller u(t).
2
desired x1(t) x1(t)
1.5 1
x1(t)
0.5 0
-0.5 -1 -1.5 -2
0
5
10
15
20
25
30
time
Fig. 4. Output response of x1(t).
2 desired x2(t) x2(t)
1.5
1
x2(t)
0.5
0
-0.5
-1
-1.5 0
5
10
15
20
time
Fig. 5. Output response of x2(t).
25
30
173
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W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175 1.5
1
x2(t)
0.5
0
-0.5
-1
-1.5 -2
-1.5
-1
-0.5
0
0.5
1
1.5
x1(t)
Fig. 6. Phase portrait of x1(t) and x2(t) for controlled uncertain Duffing–Holmes chaotic system with x1(0) = 0.2 and x2(0) = 0.2.
6. Conclusions In this paper, a robust adaptive PID controller for a class of uncertain chaotic systems has been proposed. The proposed adaptation law for PID control gains tuning is to minimize the designed sliding condition that is motivated from the sliding mode control. Moreover, based on the use of by introducing a supervisory controller, it will force the states of the system inside the constraint region that is designed in advance. Therefore, the stability of the closed-loop PID control system is also guaranteed. Finally, the proposed scheme is applied to control a well-known Duffing–Holmes chaotic as an illustrative example. From the simulation results, it can be obviously seen that a satisfactory control performance can be achieved by using the proposed method.
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[20] Park YM, Choi MS, Lee KY. An optimal tracking neuro-controller for nonlinear dynamic systems. IEEE Trans Neural Netw 1996;7:1099–110. [21] Wang LX. Adaptive fuzzy systems and control: design and stability analysis. New Jersey: Prentice-Hall; 1994. [22] Wang LX. A course in fuzzy systems and control. New Jersey: Prentice-Hall; 1997. [23] Chen CT, Peng ST. Intelligent process control using neural fuzzy techniques. J Process Contr 1999;16:493–503. [24] Mayosky MA, Cancelo GIE. Direct adaptive control of wind energy conversion systems using Gaussian networks. IEEE Trans Neural Netw 1999;10:898–905. [25] Chang WD, Hwang RC, Hsieh JG. Stable direct adaptive neural controller of nonlinear systems based on single auto-tuning neuron. Neurocomputing 2002;48:541–54. [26] Lin FJ, Hwang WJ, Wai RJ. A supervisory fuzzy neural network control system for tracking periodic inputs. IEEE Trans Fuzzy Syst 1999;7:41–52.
Adaptive robust PID controller design based on a sliding mode for uncertain chaotic systems q Wei-Der Chang a, Jun-Juh Yan b
b,*
a Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan, ROC Department of Electrical Engineering, Far-East College, No. 49, Jung-Hwa Road, Hsin-Shih Town, Tainan 744, Taiwan, ROC
Accepted 7 December 2004
Abstract A robust adaptive PID controller design motivated from the sliding mode control is proposed for a class of uncertain chaotic systems in this paper. Three PID control gains, Kp, Ki, and Kd, are adjustable parameters and will be updated online with an adequate adaptation mechanism to minimize a previously designed sliding condition. By introducing a supervisory controller, the stability of the closed-loop PID control system under with the plant uncertainty and external disturbance can be guaranteed. Finally, a well-known Duffing–Holmes chaotic system is used as an illustrative to show the effectiveness of the proposed robust adaptive PID controller. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The presence of chaos in physical systems has been extensively demonstrated and is very common. Chaotic systems are rather complex dynamical nonlinear system and the classical features appeared in output responses of chaotic systems have, for example, excessive sensitivity to initial conditions, broad spectrums of Fourier transform, and fractal properties of the motion in the phase space [1–3]. Due to its powerful applications in chemical reactions, power converters, and information processing, etc., it is meaningful and practical to design certain control law to cope with these complex chaotic systems for real engineering applications. To do this, different techniques have been developed to achieve the chaotic control. For example, sliding mode control [1–3], bang-bang control [4], optimal control [5,6], intelligent control base on using neural networks [7], feedback linearization [8], differential geometric method [9], adaptive control [10–12], and among many others [13]. On the other hand, the use of PID control has a long history in control engineering and is acceptable for many real applications due to its simplicity in architecture. Hence, in many real industrial applications, the PID controller is still widely used even though lots of new control techniques have been proposed. However, to our knowledge, the control of chaotic system by using simple PID control has not been well discussed.
q *
Supported by the National Science Council of Republic of China under Contract NSC-92-2213-E-269-001. Corresponding author. Fax: +886 6 5977570. E-mail address: [email protected] (J.-J. Yan).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.12.013
168
W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175
The key for designing a PID controller is the determination of three parameters of PID controller, i.e., proportional gain Kp, integral gain Ki, and derivative gain Kd. In the past, the PID self-tuning methods based on the relay feedback technique have often presented for a class of SISO systems [14–17] and MIMO systems [18]. In general, the controlled plants considered in the above-proposed PID control systems mostly belong to linear systems plus time delays. Moreover, in recent years, a new and attractive control strategy that is to combine neural networks and/or fuzzy systems with the traditional adaptive control has been proposed for nonlinear systems. Two types of adaptive controls are generally investigated: • Indirect adaptive control. The unknown model of nonlinear system is first obtained and learned by using the neural/ fuzzy system and then a feedback linearization control law is designed based on this model [19–22]. • Direct adaptive control. The neural/fuzzy system is directly used as a controller in the feedback control system, i.e., the output of neural/fuzzy system is a control input to the nonlinear plant. Usually, an error signal between the desired and actual outputs is employed to online update adjustable parameters in the neural/fuzzy controller [21–25]. In this paper, based on the use of the sliding mode, a robust adaptive PID control tuning is newly proposed to cope with the control problem for a class of uncertain chaotic systems with external disturbance. Three gains of PID controller, i.e., Kp, Ki, and Kd, are regarded as adjustable parameters and will be adjusted during control procedure. We wish to update these parameters with a proper adaptation mechanism such that a previously designed sliding condition is minimized as possibly. The stability of closed-loop PID control system can be guaranteed by using the Lyapunov approach with a supervisory controller [19,21,22,26]. The detailed descriptions for the PID control gains tuning and for the analysis of system stability will be presented in the following.
2. PID controller The continuous form of a PID controller, with input e(Æ) and output upid(Æ), is generally given as Z t 1 d upid ðtÞ ¼ K p eðtÞ þ eðsÞ ds þ T d eðtÞ ; Ti 0 dt
ð1Þ
where Kp is the proportional gain, Ti is the integral time constant, and Td is the derivative time constant. We can also rewrite (1) as Z t d ð2Þ upid ðtÞ ¼ K p eðtÞ þ K i eðsÞ ds þ K d eðtÞ; dt 0 where Ki = Kp/Ti is the integral gain and Kd = KpTd is the derivative gain.
3. System definition for uncertain chaotic systems For simplification, we only consider a second-order uncertain chaotic systems, but the approach can be generalized to high-order systems. With the input uðtÞ 2 R and the output yðtÞ 2 R, the uncertain chaotic system is described as x_ 1 ðtÞ ¼ x2 ðtÞ; x_ 2 ðtÞ ¼ f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ uðtÞ;
ð3Þ
yðtÞ ¼ x1 ðtÞ; where X = [x1, x2]T are measurable states vector of the system, f(Æ) is an unknown nonlinear nominal plant, Df(Æ) is the plant uncertainty applied to the system, and d(Æ) denotes the external noise disturbance. In addition, it is assumed that there exist two positive upper bounds f u(Æ) and Df u(Æ) satisfying |f(Æ)| 6 f u(Æ) and |Df(Æ)| 6 Df u(Æ), and a positive constant a satisfying |d(Æ)| 6 a. Let e = yd y (= yd x1) be an error signal between the desired and actual outputs. Define Y d ¼ ½y d ; y_ d T and assume that both y d and y_ d are bounded, i.e., kYdk1 = suptP0kYd(t)k < 1. Then the error vector of system becomes E ¼ Y d X ¼ ½e; e_ T . Suppose that we can choose a gain vector K = [k0, k1]T such that roots of s2 + k1s + k0 = 0 are in the open left-half complex plane. Now let a feedback linearization controller be given by u ¼ f ðX ; tÞ Df ðX ; tÞ dðtÞ þ €y d þ K T E:
ð4Þ
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169
Substituting (4) into (3), we have €e þ k 1 e_ þ k 0 e ¼ 0:
ð5Þ
Consequently, from (5), we have e(t) ! 0 as t ! 1, i.e., y ! yd asymptotically. However, it is noted that the relative models in (3) are assumed to be unknown in this study. Throughout the paper, the following assumption is made. Assumption 1. Let the constraint set Xx for the state X be defined as Xx ¼ fX 2 R2 : kX k 6 M x g; where Mx is a pre-specified parameter. It is desired that the state trajectory of system X never reach the boundary of Xx during the control procedure. For simplicity of analysis, we may choose Mx P kYdk1. 4. Supervisory controller design and adaptation laws for PID controller Now let the control input in (3) be given by u ¼ upid þ us ;
ð6Þ
where upid is the PID controller as shown in (2) and us is the extra supervisory controller that will be fired only when the states of the system exceeds some bound and guarantees the stability of the system. Based on concepts of sliding mode control, a proper adaptation law that is based on the use of gradient method is presented to minimize a designed sliding condition for updating PID control gains. 4.1. Supervisory controller design To design the supervisory controller us, substituting (6) into (3), we have, in view of (4), x_ 2 ¼ f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ upid þ us ¼ f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ upid þ us þ u u ¼ €y d þ K T E ½u upid us : This implies that €e ¼ K T E þ ½u upid us :
ð7Þ
Let K¼
0 k 0
1 ; k 1
B ¼ ½ 0 1 T
ð8Þ
be a companion form. From (7), we have E_ ¼ KE þ B½u upid us :
ð9Þ
Now consider a Lyapunov function candidate V e ¼ 21 ET PE;
ð10Þ
where P is a positive definite symmetric matrix satisfying the Lyapunov equation KT P þ P K ¼ Q
ð11Þ
and Q is a positive definite symmetric matrix selected by designer. Define V M ¼ 21 kmin ðP ÞðM x kY d k1 Þ2 ; where kmin(P) denotes the minimum eigenvalue of P. Note that if kXkPMx, then, from (10), we have V e P 21 kmin ðP ÞkEk2 P 21 kmin ðP ÞðkX k kY d kÞ2 P 21 kmin ðP ÞðM x kY d k1 Þ2 ¼ V M :
ð12Þ
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Hence if Ve < VM, then kXk < Mx. Moreover, the time derivative of Ve along the trajectories of the closed-loop system (9) satisfies V_ e ¼ 21 ET ðKT P þ P KÞE þ ET PB½u upid us ¼ 21 ET QE þ ET PB½u upid us
ð13Þ
6 21 ET QE þ jET PBjðju j þ jupid jÞ ET PBus : From (4) and hypotheses of |f(Æ)| 6 f u(Æ), |Df(Æ)| 6 Df u, and |d(Æ)| 6 a, we have u 6 f u þ Df u þ a þ j€y d j þ jK T Ej:
ð14Þ
Define the indicator function I* by I* = 1 if Ve P VM and I* = 0 if Ve < VM. Hence, if the supervisory controller is chosen as us ¼ I sgnðET PBÞbf u ðÞ þ Df u ðÞ þ a þ j€y d j þ jK T Ej þ jupid jc;
ð15Þ
then, from (14) and (15), we can guarantee that V_ e < 0 in (13) if Ve P VM [21,22,26]. 4.2. Adaptive laws for PID controller tuning On the other hand, in order to derive a proper adaptation law to update three control gains, let us define a designed signal xr as x_ r ¼ €y d þ k 1 e_ þ k 0 e;
ð16Þ
where k0 and k1 are given as above. Moreover, a sliding surface is defined as S ¼ x2 xr :
ð17Þ
If the sliding mode occurs, i.e., S = 0, then x2 ¼ xr :
ð18Þ
Substituting (18) into (16), we have €e þ k 1 e_ þ k 0 e ¼ 0. This implies that e(t) ! 0 as t ! 1. From viewpoints of the sliding control, the sliding condition that ensures the hitting and existence of a sliding mode is derived according to the Lyapunov stability theory. In general, the Lyapunov function candidate for the sliding mode control is simply given by V ¼ 21 S 2 and therefore the sliding condition is defined as V_ ¼ S S_ < 0:
ð19Þ
From (19), it is guaranteed that S(t) ! 0 as t ! 1. To derive an adequate adaptation mechanism for tuning three PID control gains, a gradient search method is used to minimize the sliding condition S S_ in (19) as possibly in this study. The gradient search algorithm is calculated in the direction opposite to the energy flow, the convergence properties of the PID controller tuning can be also obtained. Moreover, it is quite intuitive to choose S S_ as an error function. From (17) and using (3), we have S_ ¼ x_ 2 x_ r ¼ f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ uðtÞ x_ r :
ð20Þ
Substituting (6) into (20) and multiplying both sides of (20) by S, the following is obtained: S S_ ¼ S½f ðX ; tÞ þ Df ðX ; tÞ þ dðtÞ þ upid þ us x_ r :
ð21Þ
Based on the gradient method and the chain rule, and using (2) and (21), the adaptation laws for three control gains Kp, Ki, and Kd can be easily obtained as follows: oS S_ oS S_ oupid K_ p ¼ c ¼ c ¼ cSe; oK p oupid oK p
ð22Þ
Z t oS S_ oS S_ oupid ¼ c ¼ cS eðsÞ ds; K_ i ¼ c oK i oupid oK i 0
ð23Þ
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Fig. 1. Robust adaptive PID control for uncertain chaotic system.
and oS S_ oS S_ oupid de ¼ c ¼ cS ; K_ d ¼ c oK d oupid oK d dt
ð24Þ
where c > 0 is the learning rate. Notice that if the learning rate c or the initial values of PID control gains are not selected adequately, the resulted PID controller will probably make the states of system divergent. Fortunately, the supervisory controller as in (15) will play an important role that provides an extra input to pull the states back to the pre-specified constraint set Xx and also guarantees the stability of the system. The block diagram of the robust adaptive PID control with a supervisory controller for uncertain chaotic system is depicted in Fig. 1. The overall design procedure can be summarized as follows. Data: Uncertain chaotic system in (3) and desired output yd. Goal: Design a controller (6), i.e., u = upid + us, such that the plant output follows the desired output asymptotically. I. Choose a constraint parameter Mx. II. Find positive upper bounds of f u(Æ), Df u (Æ), and a such that |f(Æ)| 6 f u(Æ), |Df(Æ)| 6 Df u(Æ), and |d(Æ)| 6 a to construct a supervisory controller us in (15). III. Choose proper initial values of PID control gains. IV. Choose K and Q and find the solution P of the Lyapunov equation (11). V. Construct the PID controller upid in (2) with adaptation laws in (22)–(24). VI. The desired controller (6) is given by u = upid + us. VII. Stop.
5. An illustrative example The plant of interest considered in this study is a classical Duffing–Holmes chaotic system, and its nominal dynamic equations are described as [2] x_ 1 ðtÞ ¼ x2 ðtÞ; x_ 2 ðtÞ ¼ x1 ðtÞ 0:25x2 ðtÞ x31 ðtÞ þ 0:3 cosðtÞ; yðtÞ ¼ x1 ðtÞ:
ð25Þ
The sampling time is equal to 0.01 and the initial states of the system are assumed to be x1(0) = 0.2 and x2(0) = 0.2, respectively. To demonstrate the dynamic chaotic behavior of (25), the phase portrait of x1(t) and x2(t) is first shown in Fig. 2 from t = 0 to t = 30. Now suppose that let a plant uncertainty Df(X,t), an external disturbance d(t), and a control input u(t) be incorporated into (25). Hence, we have
172
W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175 0.8 0.6 0.4
x2(t)
0.2 0
-0.2 -0.4 -0.6 -0.8 -1.5
-1
-0.5
0
0.5
1
1.5
x1(t)
Fig. 2. Phase portrait of x1(t) and x2(t) for unforced Duffing–Holmes chaotic system with x1(0) = 0.2 and x2(0) = 0.2.
x_ 1 ðtÞ ¼ x2 ðtÞ; x_ 2 ðtÞ ¼ x1 ðtÞ 0:25x2 ðtÞ x31 ðtÞ þ 0:3 cos t þ Df ðX ; tÞ þ dðtÞ þ uðtÞ; yðtÞ ¼ x1 ðtÞ:
ð26Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Without loss of generality, for example, we select Df ðX ; tÞ ¼ 0:1 sinðtÞ x21 þ x22 and dðtÞ ¼ 0:1 sinðtÞ in (26). From (26), the corresponding upper bounds can be obtained as follows: jf ðÞj ¼ jx1 ðtÞ 0:25x2 ðtÞ x31 ðtÞ þ 0:3 cos tj 6 jx1 ðtÞj þ 0:25jx2 ðtÞj þ jx31 ðtÞj þ 0:3 f u ðÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi jDf ðÞj ¼ j0:1 sinðtÞ x21 þ x22 j 6 0:1 x21 þ x22 Df u ðÞ; and jdðÞj ¼ j0:1 sinðtÞj 6 0:1 a: The proposed robust adaptive PID controller is now applied to control the uncertain chaotic system with external disturbance in (26). We wish that the output of uncertain chaotic system y(t) can track the desired output y d ðtÞ ¼ sinð1:1tÞ. Also we choose k0 = 0.1 and k1 = 1 so that the roots of s2 + k1s + k0 = 0 are in the left-half complex plane and let the learning rate c = 5. From (8), we have K¼
0 1 : 0:1 1
From (11) with Q = diag[2,2], we have P¼
11:1 10 10
11
:
Moreover, the adaptation laws in (22)–(24) are used to online update the PID controller with initial gains bKp(0), Ki(0), Kd(0)c = [5, 5, 5]. The value of Mx is deliberately selected to be large enough such that the system state X never reach the boundary of Xx. The results are demonstrated in Figs. 3–6. Fig. 3 shows the time trajectories of the controller u. Figs. 4 and 5, respectively, show the output responses of x1 and x2 of the controlled Duffing–Holmes system. It can be easily seen from these results that the states of such a chaotic system can quickly track the desired states about after 5 s. The phase portrait of x1 versus x2 is also shown in Fig. 6 and seems to be satisfactory.
W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175 20 0
control input
-20
-40
-60
-80
-100
-120
0
5
10
15
20
25
30
time
Fig. 3. Time trajectory of controller u(t).
2
desired x1(t) x1(t)
1.5 1
x1(t)
0.5 0
-0.5 -1 -1.5 -2
0
5
10
15
20
25
30
time
Fig. 4. Output response of x1(t).
2 desired x2(t) x2(t)
1.5
1
x2(t)
0.5
0
-0.5
-1
-1.5 0
5
10
15
20
time
Fig. 5. Output response of x2(t).
25
30
173
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W.-D. Chang, J.-J. Yan / Chaos, Solitons and Fractals 26 (2005) 167–175 1.5
1
x2(t)
0.5
0
-0.5
-1
-1.5 -2
-1.5
-1
-0.5
0
0.5
1
1.5
x1(t)
Fig. 6. Phase portrait of x1(t) and x2(t) for controlled uncertain Duffing–Holmes chaotic system with x1(0) = 0.2 and x2(0) = 0.2.
6. Conclusions In this paper, a robust adaptive PID controller for a class of uncertain chaotic systems has been proposed. The proposed adaptation law for PID control gains tuning is to minimize the designed sliding condition that is motivated from the sliding mode control. Moreover, based on the use of by introducing a supervisory controller, it will force the states of the system inside the constraint region that is designed in advance. Therefore, the stability of the closed-loop PID control system is also guaranteed. Finally, the proposed scheme is applied to control a well-known Duffing–Holmes chaotic as an illustrative example. From the simulation results, it can be obviously seen that a satisfactory control performance can be achieved by using the proposed method.
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