Optimal trajectory tracking controller design: Model based to model ...

2014 International Conference on Mechatronics and Control (ICMC) July 3 - 5, 2014, Jinzhou, China

Optimal Trajectory Tracking Controller Design: Model based to Model Free Methods Haoping WANG and Yang TIAN

Christian VASSEUR

LaFCAS Sino-French Lab, School of Automation Nanjing University of Science and Technology Nanjing, 210094 China Email: {tianyang,hp.wang}@njust.edu.cn

LaFCAS Sino-French Lab, LAGIS Lab Lille University of Science and Technology Villeneuve d’Ascq, France Email: [email protected]

Abstract—This paper presents a new adaptive optimal control which is based on a particular hybrid systems called piecewise linear continuous systems. This referred proposed controller which is based on a L2 norm prescribed to minimize a tracking error and a time-delay estimation technique is developed from model based and finally can be applied to systems without knowing internal dynamics. Thus this proposed Method has adaptive and optimal characters with a very simple ProportionalDerivative (PD) controller structure. It realizes the trajectory tracking of multi-inputs multi-outputs linear continuous systems under systems output feedback. Finally, to validate the proposed method performance, two illustrative examples which contains a knowing dynamic systems and a real-time based unknown motion control example are presented.

I.

I NTRODUCTION

The output trajectory tracking problem is one of the mostly active subjects in control theory design[2, 4, 6, 7]. The difficulties are arising from two sides: - one side is residing how to design a trajectory tracking controller which is able to make the desired trajectory tracking reference to be asymptotically approached by the output of the controlled system, the other side is to ensure the entire system representing good dynamic performance even when existing the perturbations and uncertainties. Recently, the development of optimal control theory helps to offer new type modern approaches regarding the trajectory tracking control of dynamic systems. Generally, under the scheme of ordinary trajectory tracking problem by optimal control theory, the firstly step is to define a trajectory tracking error of the difference between the desired trajectory tracking reference signal and the controlled system output, and the second step is to design a control law which enables to minimize an index of quadratic performance related to the referred tracking error. However, when the uncertainties in the controlled systems exist, the controlled system trajectory may deviate from the desired trajectory tracking reference and the tracking performance would be greatly deteriorated. A very acceptable approach to realize optimal control is called receding-horizon approach in which the optimal sequence driving the system to the origin (or into a neighborhood of the origin) from the current state is recomputed online at each sampling time and only the first input value of the sequence is then applied to the system [1]. However, the implementation of receding-horizon controllers requires the solution online of a convex optimization problem. The difficulties encountered in the existing feedback optimal control

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approaches can be also attributed to the request of realizing optimality for all possible initial conditions of controlled systems [8]. For nonlinear systems whose dynamics have to be known to solve Riccati Equation (RE), Algebraic Riccati Equation (ARE) and HJB equations, are more complex compared to linear systems to solve its corresponding RE, ARE or HJB, while become impractical when the internal dynamics of controlled system are uncertain. Recently, optimal control based online adaptive approximation which refers to an online approximator based Adaptive Critic Designs (ACD) is proposed [9]. While in [16], an optimal adaptive control (OAC) is proposed for MIMO controlled systems in strict feedback form with uncertain internal dynamics. The optimal adaptive feedback control scheme is introduced for the affine-in control systems to estimate online the HJB equation solution which becomes the optimal feedback control input for the closed-loop system. Keeping in mind that developing a new optimal trajectory tracking control which should not be overly dependent on the mathematical tools since they are often unavailable in practice, provides much better performance at low cost and can be implemented intuitively and simply. Therefore in our previous paper, a trajectory sampling tracking controller which is based on a particular hybrid systems theory called PiecewiseContinuous Systems (PCS) is developed [5, 14]. These PCS systems are characterized with autonomous switchings and controlled impulses [5, 10, 14]. Using PCS theory, Piecewise Continuous Controllers (PCC) were first developed, enabling sampled trajectory tracking of linear systems [5, 10, 14]. Then for improving tracking performance, derived piecewisecontinuous controller and recursive model free controller were proposed in [10]. Unfortunately, these referred controllers is defined without any prescribed cost function to minimized. Thus in this present paper based on the referred PCC controller, an optimal output trajectory tracking controller which is based on a L2 norm prescribed to minimiz a trajectory tracking error and a time-delay estimation technique is developed by ensuring PCS systems switching period trends rapidly to zero. This Improved Optimal Controller (IOC) which presents exponentially convergence speed and requires no knowledge of linear systems internal dynamics has a very simple structure and can be easily selected its corresponding parameters. The remainder of this paper is organized as follows: the preliminaries of and PCS hybrid systems are introduced in

Section 2. In Section 3, the design of different trajectory tracking controllers, particularly a IOC controller with output feedback is developed in view of application PCS systems. Then to validate these developed IOC controller in Section 4, two illustrative examples are tested to demonstrate the proposed method performance and robustness. Finally it is followed by some conclusion remarks in Section 5. II.

P RELIMINARIES OF P IECEWISE C ONTINUOUS S YSTEMS

The left limit xs (iTe− ) of xs (t) at t = iTe is deduced from (2) in the interval Φi−1 :

   xs (iTe− ) = eAs Te Bs2 vs (i − 1)Te +

iTe

(i−1)Te

eAs (iTe −τ ) Bs1 us (τ ) d τ .

Thus, in the general case xs (iTe− ) = xs (iTe+ ) . An example state evolution of a first order system with Te = 1s and Bs2 = 1 is shown in Fig. 2. 0.1

For ti = iTe , the Linear Piecewise-Continuous Systems (LPCS) are described as ⎧ x (t + ) = Bs2 vs (ti ), ∀ti ∈ S ⎪ ⎨ s i (1) x˙s (t) = As xs (t) + Bs1 us (t), ∀t ∈ Φt ⎪ ⎩ ys (t) = Cs xs (t), ∀t ∈ ℑ. where xs (t) ∈ ℜn is the system state, ys (t) ∈ ℜm is the system output, vs (t) ∈ ℜ p , us (t) ∈ ℜr are the switching and continuous inputs respectively, and As ∈ ℜn×n , Bs1 ∈ ℜn×r , Bs2 ∈ ℜn×σ , Cs ∈ ℜm×n are constant matrices. At switching instants the system state changes according to the first, algebraic equation of (1) and the continuous-time state evolution is described by the second, linear differential equation of (1). The LPCS structure and symbolic representation are shown in Fig. 1. vs ( t )

us ( t )

vs (t )

Bs1





xs ( t )

³

Cs

ys (t )

0

Ŧ0.05

Ŧ0.1 0

Fig. 2.

1

2

3 Time (s)

4

5

6

Linear piecewise-continuous system state evolution

Based on the PCS theory, piecewise-continuous controllers have been developed in [5, 13]. In this paper, PCS are used in piecewise-continuous state observers for linear systems with sampled and delayed output measurements. III.

T RAJECTORY T RACKING C ONTROL D ESIGNED A PPLICATION OF LPCS

BY

A. Classical Piecewise Continuous Controller With the above introduced LPCS and recalling [5], a particular trajectory tracking controller which is called Classical Piecewise Continuous Controller (CPCC) can be realized by using the only discrete sampling feedback and denoted as   Σ S, Ac , 0, Bc2 ,Cc with uc (t) = 0.

M(Te ) =

 Te 0

(exp A(Te − τ )) BCc exp (Ac τ ) d τ

(3)

and permits to realize the following referred sampling trajectory tracking strategy with one switching period delay Te

(a )

xi+1 = ci

(b)

will

be

By replacing (5) to (4), the CPCC controller state xc (t) at discrete instants S can be calculated as further

denoted

as

Integrating the second equation of (2) in Φi and taking into account that xs (iTe+ ) = Bs2 vs (iTe ), one obtains xs (t) = eAs (t−iTe ) Bs2 vs (iTe ) +

 t

iTe

(4)

where ci is the system desired state reference. Because one has xi+1 = exp (ATe ) xi + M(Te )Bc2 vs (iTe ) (5)

ys (t )

Linear piecewise-continuous system

 The system  (1) Σ S, As , Bs1 , Bs2 ,Cs .

s

As

6[ S , As , Bs1 , Bs 2 , Cs ]

us ( t )

Fig. 1.

xs (t)

x (t) 0.05

Under S = {iTe , i = 0, 1, 2, . . .} and with CPCC controller, one can calculate the following matrix as

Bs 2

S

vs(t)

LPCS state evolution

The Piecewise-Continuous Systems (PCS) which were first proposed in [5] and then developed in [11–13, 15], are hybrid systems with autonomous switchings and controlled impulses. PCS are characterized by two input spaces and two time spaces. The first time space is the discrete time space S = {ti , i = 0, 1, 2, . . .} called switching space, where ti are the switching instants. The second time space is the continuous time space Φt = {ℑ − S}, where ℑ = {t ∈ [0, ∞)}. At each switching instant the system is controlled by a switching input and between two switching instants a continuous input is applied. Two successive switching instants ti and ti+1 delimit an interval denoted Φi = {Φt |∀t ∈ (ti ,ti+1 )}.

eAs (t−τ ) Bs1 us (τ ) d τ .

(2)

618

xc (iTe+ ) = M(Te )−1 (ci − exp (ATe ) xi )

(6)

Then with the selected CPCC state matrix Ac , the CPCC output which is equally the controlled systems control signal can be calculated as u(t) = Cc xc (t) = Cc exp (Act) xc (iTe+ ).

(7)

B. CPCC: Passage to Limit by Supposing Te to 0 . Under the implementation of the proposed above CPCC controller and considering equation (6) by supposing the switching period trends to zero which means (Te → 0), one has (8) M(Te )xc (t + ) = c(t) − x(t) − Te Ax(t) with t = limTe →0 (iTe ). Then according to (3), one has M(Te ) = BCc Te . With application of relation defined in (7), one has Te Bu(t + ) = c(t) − x(t) − Te Ax(t)

(9)

while with the tracking strategy defined in (4) at switching instant(i − 1)Te , one obtains x(t) = c(t − )

(10)

Replacing (10) in (9) and dividing by switching period Te , one obtains c(t) − c(t − ) − Ax(t) Te

(11)

which leads to the following equation by letting Te → 0 ˙ − ) − Ax(t) B.u+ (t) = c(t

(12)

•

(15) 2 In order to minimize the referred trajectory tracking error ey˙ , with respect to u(t+ ), one has the following control Signal  −1 T By (cy˙ (t) −CAx(t)) (16) u(t + ) = BTy By which leads to (17) u(t + ) = Ky (cy˙ (t) −CAx(t))  T −1 T By . And it is important to note that with Ky = By .By Ky By = I. And from the facts that cy˙ (t) and x(t) are continuous, u(t + ) is continuous in consequence, and its corresponding exponent notation + can be removed.

the referred equation (12) supposes that the derivative of the desired state reference should exist. We also assume that this derivative is continuous which will not pose problem in reality implementation. Under (10), equation (12) can be replaced by ˙ − ) − Ac(t − ) which suggests that c(t − ) B.u+ (t) = c(t works as the same equation of the controlled system. This means that it is possible only if the control input u(t + ) is discontinuous at each instant. Unfortunately, this kind control is not physically implementable in reality.

C. Trajectory Tracking of Desired Output Reference Derivative Under this control scheme, by multiplying the controlled system output matrix C to each side of equation (12), one obtains CBu(t + ) = cy˙ (t) −CAx(t) (13) is the derivative reference of controlled whree cy˙ (t) = C d(c(t)) dt system output y(t). ˙ Considering the referred remarks in III-B, the main idea in this sub-section is to propose a feasible control strategy which can minimize a prescribed L2 norm tracking criteria based on a trajectory tracking error between the two items in (12).

By multiplying the matrix Ky to each side of the above equation ˙ = and considering Ky By = I, one obtains the relationship Ky y(t) Ky cy˙ (t), which means that one has ˙ =0 Ky (cy˙ (t) − y(t))

(19)

y(t) ˙ ≡ cy˙ (t)

(20)

Noting that the other possible solutions for (19) which are nonzero vectors nodus of Ky lead to the error norm non zero, thus not corresponding feasible minimal solutions. Reminding that the above calculations have a very general form and is also valuable under state output derivative feedback x(t) ˙ (which means c = I). D. Output Feedback Trajectory Tracking The integral of (20)leads to y(t) − y(t0 ) = cy (t) − cy (t0 ). And in general, the initial conditions of the desired reference and the controlled system are different and has a constant gap for ∀t  t0 . Thus the control proposed in (17) is not sufficient to ensure the trajectory tracking of limt→∞ ey (t) = cy (t) − y(t) → 0 1) Output Trajectory Tracking Controller Design: To overcome the above introduced difficulty, the following Improved Optimal Control (IOC) is proposed as follows u(t) = Ky (cy˙ (t) −CAx(t)) − β Ky (cy (t) − y(t))

(21)

where β is a selected stable square matrix with dimension r ×r, d(c (t)) cy (t) is the desired output reference signal and cy˙ (t) = dty is the derivative of desired output reference. If we replace this new control u(t) to y(t) ˙ = CAx(t) + By u(t), one obtains

This trajectory tracking error is defined as follows ey˙ (t) = (cy˙ (t) −CAx(t)) − By u(t + )

y(t) ˙ = CAx(t) + By u(t) = CAx(t) + By Ky (cy˙ (t) −CAx(t)) (18)

The only possible solution which minimizes the the proposed L2 quadratic norm e2y˙ (t) = eTy˙ (t)ey˙ (t) is

−

) with c(t ˙ − ) = limTe →0 c(t)−c(t . It’s important to note that: Te

•

e2y˙ (t) = eTy˙ (t)ey˙ (t)

If we replace u(t) by its value in the system state equation, one has

with t = limTe →0 (iTe ).

Bu+ (t) =

To facilitate the proposed controller calculation, the referred L2 norm tracking criteria is selected as

(14)

with By = CB.

619

y(t) ˙ = CAx(t) + By Ky (cy˙ −CAx(t)) − By β Ky (cy (t) − y(t)) (22)

By multiplying the above equation by Ky and considering the relationship of Ky By = I, one obtains Ky y˙ (t) = Ky CAx(t) + Ky cy˙ (t) − Ky CAx(t) − β Ky (cy (t) − y(t)), (23) which means that one has ˙ = β Ky (cy (t) − y(t)) . Ky (cy˙ (t) − y(t))

(24) IV.

It is important to note that the above equation has the following form defined as ˙ = β E(t) E(t)

system state matrix A requires the knowledge of only By = CB and has only one parameter of β to select for ensuring its corresponding performance. And compared with classical PID controller whose corresponding parameters are difficult to select, the referred IOC Controller presents more design freedom and simple structure.

(25)

with E = Ky (cy (t) − y(t)). Because the matrix β is a selected matrix with negative eigenvalues, then one obtains lim E → 0

t→∞

which means that system output y(t) convergence exponentially to its corresponding desired output reference cy (t). Reminding that tn practice implementation, the referred matrix β can be selected as −k2 .Ir where Ir is a unity matrix with dimension r × r. And the bigger value of k is, the faster convergence of y → cy is. E. Considerations of Real-time Implementation Problems Recalling de proposed IOC Control in (21), for real-time implementations, the following three cases are considered. 1) Under State Feedback x(t): The proposed IOC Control in (21) can be applied directly without any modification. 2) Under Output Feedback y(t): Under output feedback y(t), because the system state x(t) is not accessible directly, the control (21) should be adapted by combining with some state estimations techniques.

A. Design example 1 To illustrate the proposed controller performance, the following numerical system is introduced as follows

x(t) ˙ = Ax(t) + Bu(t) (29) y = Cx(t)   8 −2 −4 1 3 0 3 7 , B = −0.5 1 ,C = with A = 2 7 1 5 −6

 1 3 2 . 2 −7 −5 Reminding that this referred illustrated multi-input multioutput numerical system has two unstable eigenvalues. With the proposed control defined in (28) with β = −4 ∗ I2×2 where I2 is unit matrix with dimension 2×2. And their corresponding numerical results are illustrated in Fig. (3-7). From these results, it is easily to note that this proposed controller (28) ensures the trajectory tracking not only the system output signal but also the output derivative reference which presents the minimization optimal character of control defined in (16). And the proposed control illustrated in Fig. 7 is moderate and smooth. 150

In the case where x(t) = [CT C]−1Cy(t) with C system output matrix, the referred Control becomes   u(t) = Ky cy˙ (t) −CA[CT C]−1Cy(t) − β Ky (cy (t) − y(t)) . (26)

100

In the case where x(t) = [CT C]−1Cy(t), the following time delay based estimation method is implemented to realize the referred control.

−50

˙ −CBu(t − Te ) CAx(t) = y(t) ˙ −CBu(t) ∼ = y(t) = y(t) ˙ − By u(t − ).

50 0

c

(27)

Reminding that with rapid development of numerical processing and calculation tools, the simulator processing period can be realized easily Te → 0 and t − Te can be denoted as t − for simplification. For continuous control input u(t) = u(t − ) without any problem. Then with the above estimate, the referred IOC controller which becomes   ˙ + By u(t − ) − β Ky (cy (t) − y(t)) u(t) = Ky cy˙ (t) − y(t) ˙ − β Ky (cy (t) − y(t)) (28) = u(t − ) + Ky (cy˙ (t) − y(t)) has a simple structure of PD controller. Reminding that this referred IOC controller (28) which removes the controlled

620

y

−100

1

y1 −150 0

For Te → 0, one has

I LLUSTRATIVE E XAMPLES

Fig. 3.

20

40

Time (s)

60

80

100

Trajectory tracking of y1 to Cy1 by IOC control

B. Design example 2 Reminding that the requirements associated with motion control dictate that the steady-state error be zero, the performance is robust in the face of changes of inertia, friction and torque disturbance. For validation purposes ([3]), the mathematical model of the motion system is derived as follows y(t) ¨ = (−1.41y(t) ˙ + 23.2Td ) + 23.2u(t)

(30)

where y(t) is the system output position, u(t) is the control signal and Td whose variation is illustrated in Fig. 8 is the torque disturbance.

20

100

10

50

0 0

−10 −50

c

−20

y

Td

2

y2 −100 0

Fig. 4.

20

40

Time (s)

60

80

100

−30 0

4

Time (s)

6

8

10

Variable torque disturbance signal Td

Fig. 8.

Trajectory tracking of y2 to Cy2 by IOC control

2

80

To see the performance of control defined in (28) can deal with a totally unknown system, to make the validation more realistic, the variable Ky which is calculated theoretically Ky = 1/b = 0.0431 is selected as Ky = 0.2 (nearly five time) and ε 2 = 0.0001. Their corresponding simulations compared with a classical PID controller are illustrated in Fig. (9-11). It can be seen easily that the proposed IOC controller tracks greatly better than the classical PID controller especially concerning the desired output derivative reference. And the IOC control is less oscillate by comparing the classical PID controller.

60 40 20 0 −20 −40 −60

dy1/dt

−80

dc /dt 1

−100 0

Fig. 5.

20

40

Time (s)

60

80

100 30

Trajectory tracking of y˙1 to C˙y1 by IOC control

c

y

y

25

IOC

yPID

20 15

50

10

0 5 0 0

−50 dy /dt

2

4

2

dc2/dt −100 0

Fig. 6.

20

40

Time (s)

60

6

8

10

Trajectory tracking of y to Cy by recursive controller

Fig. 9.

80

Time (s)

100

Trajectory tracking of y˙2 to C˙y2 by IOC control 10 dcy/dt

400

dyIOC/dt

5

dyPID/dt

200

Control, u(t)

0

0 −5

−200 −10

u1

−400

u −600 0

Fig. 7.

20

40

Time (s)

60

80

−15 0

2

2

4

100

Fig. 10.

Trajectory tracking IOC control inputs

621

Recursive control input signal

Time (s)

6

8

10

30 u

IOC

20

u

PID

10 0 −10 −20 −30 −40 −50 0

Fig. 11.

2

4

Time (s)

6

8

10

Recursive control input signal

V.

C ONCLUSION R EMARKS

A new adaptive and practical optimal control which is based on a particular piecewise linear continuous hybrid systems is developed in this paper. This referred controller which is based on a L2 norm prescribed to minimize a tracking error and a time-delay estimation technique is developed from model based and finally can be applied to systems without knowing internal dynamics. Thus this proposed method has adaptive and optimal characters with a very simple Proportional-Derivative (PD) controller structure, and can be used to realize the trajectory tracking of multi-inputs multi-outputs systems under systems output feedback. Finally, with two illustrative design examples are presented to validate the proposed controller performance and robustness. ACKNOWLEDGMENT This work was partially supported by the National Natural Science Foundation of China (61304077, 61203115), by the Natural Science Foundation of Jiangsu Province (BK20130765), by the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123219120038), by the Chinese Ministry of Education Project of Humanities and Social Sciences (13YJCZH171), by the Fundamental Research Funds for the Central Universities (30920130111014), and by the Zijin Intelligent Program of Nanjing University of Science and Technology (2013 ZJ 0105), the authors would like to express theirs deep appreciations. R EFERENCES [1] F. Blanchini and F.A. Pellegrino. Relatively optimal control and its linear implementation. IEEE Trans. Automat. Contr., 48(12):2151–2162, 2003. [2] R.A. Freeman and P.V. Kokotovic. Optimal nonlinear controllers for feedback linearizable systems. In Proc. American Control Conference, pages 2722–2726. 1995. [3] Z. Gao. From linear to nonlinear control means: a practical progression. 41(1):177–189, 2002. [4] A. Imani, M. Bahrami, and B. Ebrahimi. Optimal sliding mode control for spacecraft formation flying. In 2nd International Conference on Control, Instrumentation and Automation. 2011.

622

[5] V. Koncar and C. Vasseur. Control of linear systems using piecewise continuous systems. 150(6):565–576, Nov. 2003. [6] A. Mannava, S.N. Balakrishnan, T. Lie Tang, and R.G. Landers. Optimal tracking control of motion systems. 20(6):1548–1558, 2012. [7] H.P. Pang and Q. Yang. Optimal sliding mode output tracking control for linear systems with uncertainties. In International Conference on Machine Learning and Cybernetics, pages 942–946. 2010. [8] H. J. Shieh, Y.J. Chiu, and Y.T. Chen. Optimal pid control system of a piezoelectric microospitioner. In IEEE/SICE International Symposium on System Integration, pages 1– 5. 2008. [9] D. Vrabie, K. Vamvoudakis, and F. L. Lewis. Adaptive optimal controllers based on generalized policy iteration in a continous-time framework. In Proc. of the IEEE Mediterranean Conf. on Contr. and Automat. 2009. [10] H.P. Wang, A. Pintea, N. Christov, P. Borne, and D. Popescu. Modelling and recursive power control of horizontal variable speed. Control Engineering And Applied Informatics, 14:4, Dec. 2012. [11] H.P. Wang, Y. Tian, and N. Christov. Piecewisecontinuous observers for linear systems with sampled and delayed output. International Journal of Systems Science - Accepted, 2014. [12] H.P. Wang, C. Vasseur, and V. Koncar. Piecewise continuous systems used in trajectory tracking of a vision based x-y robot. In A. Mahmood T. Sobh, K. Elleithy and M. A. Karim, editors, Novel Algorithms and Techniques In Telecommunications, Automation and Industrial Electronics, pages 255–260. Springer Netherlands, London, 2008. [13] H.P. Wang, C. Vasseur, V. Koncar, A. Chamroo, and N. Christov. Derived piecewise continuous controller: application to nonlinear mechanical x-y robot’s trajectory tracking. Systems Science, 35(4):39–48, 2009. [14] H.P. Wang, C. Vasseur, V. Koncar, A. Chamroo, and N. Christov. Sampled tracking for delayed systems using two time-scales sampled-data controllers. Studies in Informatics and Control, 19(4), 2010. [15] H.P. Wang, C. Vasseur, Y. Tian, V. Koncar, and N. Christov. Recursive model free controller. application to friction compensation and trajectory tracking. International Journal of Control, Automation, and Systems, 9(6):1146– 1153, December 2011. [16] H. Zargarzadeh, T. Dierks, and S. Jagannathan. Optimal adaptive control of nonlinear continuous-time systems in strict feedback form with unknown internal dynamics. In 51st IEEE Conf. on Decision and Control, pages 4127– 4132. December, 2012. Maui, Hawaii, USA.