Genetic Algorithm Optimized Predictive PID Controller Mohammad H. Moradi, Hemen Showkati Electrical Engineering Department, Bu Ali Sina University Hamedan, Iran

[email protected], [email protected] Abstract- In this paper a new predictive PID controller has been introduced with similar performance and features as Generalized Predictive Controller (GPC). A predictive type PID structure is defined and a Genetic algorithm (GA) approach is used to obtain optimal gains of predictive PID controller. The performance of this method is then compared with GPC, gradient optimized predictive PID and conventional PID controller. The results show that the proposed method gives similar performance as GPC. The GA optimized predictive PID can be implemented on existing process control hardware. So this method integrates the simplicity of implementation of conventional PID controllers and complex process handling of advanced methods like GPC.

I.

INTRODUCTION

Proportional-Integral-Derivative (PID) controllers are control algorithms commonly found in industrial applications. Because of their robust performance, simplicity of structure and easy comprehension in principle, PID controllers are extensively used in process industries. [16], [6].The three parameters of PID controllers must be tuned to the process to obtain a satisfactory closed loop performance. But having few parameters to be tuned despite the simplicity of tuning deprives the control action from sufficient degree of flexibility needed to control complex processes such as systems with long time delay and non-minimum phase systems. Over the years numerous techniques have been suggested for tuning of PID parameters. Some well known techniques can be found in [18], [1]. On the other hand, the Generalized Predictive Controller (GPC) was introduced by [4]. And this has become one of the most popular Model Predictive Control (MPC) algorithms both in industry and academia. It has been successfully implemented in many industrial applications [4] showing good performance and certain degree of robustness. It can handle many different control problems for a wide range of plants with a reasonable number of design variables, which have to be specified by the user depending upon prior knowledge of plant and control objectives. GPC is an optimal method which can deal with unstable and non-minimum phase plants. Because of hardware and software restrictions and practical implementation issues for using GPC and other advanced control techniques, many researchers have attempted to restrict these advanced methods to retrieve a PID controller with similar features. In [14] an Internal Model Control (IMC) based controller design for first order process model is proposed and [3] extended IMC-PID controller to cover the

second order process model. The limitation of these methods is that tuning rules are derived for the delay free system. In [9] a mathematically equivalent PID control law to GPC was proposed. This was done by equating the discrete PID control law with linear form of GPC. Although there is no restriction on choice of GPC tuning parameters, the process model order is restricted to maximum of two. In [17] a least square algorithm was used to compute the closest equivalent PID controller to an IMC design, but still ineffective for timedelay and unstable systems. In [8] a model-based PID controller was used by utilizing multiple Distributed Control System (DCS) PID blocks to implement a model based predictive control strategy. In [7] a predictive PID algorithm which has similar features to model-based predictive control (MPC) is presented. Corresponding to a prediction horizon of size M, a bank of M parallel conventional PID controllers is defined. Three term PID gains are calculated using optimal signal matching with GPC. In [13] this Predictive PID scheme has been extended to MIMO predictive PID control in which can cover most of practical MIMO control applications in industries. In [10] a two layer auto-tuning method was proposed in which the lower layer is a conventional PID controller and upper level is identification and tuning modules. In [11] by use of Radial Basis Function Neural Network (RBFNN) an adaptive predictive PID controller was introduced which has high degree of robustness. Based on Lagrangian Particle Swarm Optimization [15] introduced a robust PID tuning strategy. Although it is difficult to guarantee it’s effectiveness in theoretical way because of stochastic approach of particle swarm optimization. This paper presents a new PID type controller with similar performance in comparison with GPC. This method is applicable to time delay, unstable and non-minimum phase systems. We use a genetic algorithm optimization procedure to tune the predictive PID controller to obtain a similar performance compared with GPC. This controller has high degree of adaptability for being applied in various systems. This method integrates simplicity of implementation of conventional PID controller and complex process handling capability of GPC. In configuration of this paper, the system transient behavior is directly managed by the predicted trajectories. The actual closed-loop is a combination of the PID response and the GPC output predicted system trajectories. The prediction can be computed off-line or can be updated at a

slower rate than in GPC approach, reducing the computational burden implied by receding horizon strategy (without instability problems). The problem of unfeasibility can be overcome in this way. Note that this approach considers also similar assumptions to those of GPC approach. Another important feature of this scheme is related to hardware implementation and current operator’s practice. If there exists a PID controller regulating the system, changing to a MPC strategy implies the removal of all the components currently in use. Such changes involve the approval of the operator and training in the use of new tuning rules of new control algorithm. The approach of this paper is based on existing PID controller configuration and requires only little extra additional hardware. Because for most of typical industrial applications the calculations can be done off-line and be tested on a computer away from the actual system and then downloaded to the appropriate storage device. And more ever there are no changes from the point of view of operator’s training; the operator can continue to manage the plant in the same manner they used before. The main idea is to use Genetic Algorithms (GAs) to obtain optimal predictive PID gains to give a similar closed-loop performance compared with GPC. In this way an optimal controller is developed which has a simple and desirable structure but yields the level of performance expected from GPC. The organization of this paper is as follows: section 2 describes the structure of predictive PID controller and calculates the control law. In section 3 optimal values of predictive PID parameters will be calculated. Section 4 discusses the stability issues of this method. In section 5 performance of the controller will be discussed and the performance of it will be compared with 3 other controllers. Finally conclusions close the paper. II. STRUCTURE OF PREDICTIVE PID CONTROLLER

D

[e(k ) − e(k − 1)]

+ k D [e(k + i ) − e(k + i − 1)]

)

(4) The controller consists of M parallel PID controllers as shown in figure (1). For M=0, the controller is identical to the conventional PID in equation (1). For M>0 the proposed controller has prediction capability similar to MBPC where M is the prediction horizon of PID controller. The horizon M will be selected to find the best approximation to GPC solution. Using (1) equation (4) can be decomposed into M control signals as follows: u (k ) = u~(k ) + u~(k + 1) + ... + u~ (k + M ) (5) Where k

u~ (k + i ) = k P e(k + i ) + k I ∑ e( j + i ) + j =1

k D [e(k + i ) − e(k + i − 1)] (i=0,…,M) It is assured that e = yset − y = − y ( yset = 0) and if incremental form of control signal is considered, ∆u (k ) = u (k ) − u (k − 1) , and after some straight forward algebra, the control signal can be written as: ∆u (k ) = ∆u~ (k ) + ∆u~ (k + 1) + ... + ∆u~ (k + M ) (6) Where k

u~ (k + i ) = k P e(k + i ) + k I ∑ e( j + i ) j =1

(i=0,…,M)

+ k D [e(k + i ) + e(k + i − 1)] ~ (k + i ) can be written as: In compact form ∆u ∆u~ (k + i ) = − KY (k + i ) (7) Where K = [k D − k P − 2k D k P + k I + k D ] and

y (k + i − 1)

y (k + i )]

T

Using equation (7) in equation (5):

k

∑ e( j ) + k

j =1

Y (k + i ) = [ y (k + i − 2)

A. General form of PID A discrete PID controller has the following form:

u~ (k ) = k P e(k ) + k I

k

u~ (k ) = ∑ ( k P e(k + i ) + k I ∑ e( j + i )

(1)

j =1

Where kP , k I , and kD are the proportional, integral, and derivates gains, respectively. Taking difference on both sides of equation (1) at step (k) and (k-1) leads to:

∆u~ (k ) = u~ (k ) − u~ (k − 1) = k p [e(k ) − e(k − 1)]

+ k I e(k ) + k D [e(k ) − 2e(k − 1) + e(k − 2)]

∆u (k ) = ∆u~ (k ) + ∆u~ (k + 1) + ... + ∆u~ (k + M )

= − K {Y (k ) + Y (k + 1) + ... + Y (k + M )}

This implies that the current control signal value is a linear combination of the future predicted outputs. The i-th step ahead prediction of output can be obtained from the following equation [2]:

(2)

Transforming equation (2) into z domain gives:

[ q + q z −1 + q z −2 ] ~ U ( z ) = 0 1 −1 2 E( z) (3) 1− z Where q0 = ( k p + k I + k D ), q1 = −( k p + 2k D ), q2 = k d B. Predictive form of PID A type of predictive PID controller is defined as follows:

(8)

y (k + i ) = [g i −1

gi −2

⎡ ∆u (t ) ⎤ ⎢ ∆u (t + 1) ⎥ ⎥ ... g 0 ]⎢ ⎥ ⎢ M ⎥ ⎢ ⎣∆u (t + i − 1)⎦

[

+ f i1

[

+ g i′1

fi2

g i′2

⎡ g (′i − 2)1 ⎢ Gi′ = ⎢ g (′i −1)1 ⎢ gi′1 ⎣

⎡ y (t ) ⎤ ⎢ y (t − 1) ⎥ ⎥ (9) ... f i ( na +1) ⎢ ⎢ ⎥ M ⎢ ⎥ ⎣ y (t − n a )⎦ ⎡ ∆u (t − 1) ⎤ ⎢ ∆u (t − 2) ⎥ ⎥ ... g in′ b ⎢ ⎢ ⎥ M ⎢ ⎥ ⎣∆u (t − nb )⎦

]

∆uˆ (k ) = [∆u (k ) ∆u (k + 1) ... ∆u (k + N u − 1)]

T

∆u0 (k ) = [∆u (k − 1) ∆u (k − 2) ... ∆u (k − nb )]

T

y0 = [ y ( k )

]

⎡ g i −3 ⎢g ⎢ i−2 ⎢⎣ g i −1

g0

0

g i −3 gi−2

... g i −3

g0 ...

⎡ f ( i − 2 )1 ⎢ + ⎢ f (i −1)1 ⎢ f (i )1 ⎣

f (i − 2 ) 2 ... f ( i −1) 2 ...

⎡ g (′i − 2 )1 ⎢ + ⎢ g (′i −1)1 ⎢ g i′1 ⎣

g (′i − 2 ) 2 ... g (′i −1) 2 ... g i′2 ...

f (i ) 2

...

M

i =0

In equation (10), future control inputs are needed to calculate {∆u (k + i ) i = 1 : ( N u − 1)} . Rewriting the output prediction in compact form gives:

Where:

g0 0 0⎤ ⎡ g i − 3 ... ⎢ Gi = ⎢ g i − 2 g i − 3 ... g 0 0 ⎥⎥ ⎢⎣ g i −1 g i − 2 g i − 3 ... g 0 ⎥⎦ ⎡ f ( i − 2 )1 f ( i − 2 ) 2 ... f ( i − 2 )( na +1) ⎤ ⎢ ⎥ Fi = ⎢ f (i −1)1 f ( i −1) 2 ... f ( i −1)( na +1) ⎥ ⎢ f ( i )1 f ( i ) 2 ... f i ( na +1) ⎥⎦ ⎣

y (k + i )]

T

y (k + i − 1)

∆u (k ) = − K ∑ Y (k + 1) =

∆u ( k ) ⎤ ⎡ 0 ⎤⎢ ∆u (k + 1) ⎥⎥ 0 ⎥⎥ ⎢ ⎥ ⎢ M g 0 ⎥⎦ ⎢ ⎥ ⎣∆u (k + N u − 1)⎦ ⎡ y (k ) ⎤ f (i − 2 )( na +1) ⎤ ⎢ ⎥ ⎥ ⎢ y (k − 1) ⎥ f ( i −1)( na +1) ⎥ ⎥ ⎢ M f i ( na +1) ⎥⎦ ⎢ ⎥ ⎣ y ( k − na ) ⎦ ⎡ ∆u (k − 1) ⎤ g (′i − 2 ) nb ⎤ ⎢ ⎥ ⎥ ∆u (k − 2) ⎥ g (′i −1) nb ⎥ ⎢ ⎥ ⎢ M g in′ b ⎥⎦ ⎢ ⎥ ⎣∆u (k − nb )⎦

Y (k + i ) = Gi ∆uˆ (k ) + Fi y0 (k ) + G′∆u0 (k )

T

Substituting equation (11) in equation (8):

(10)

...

y (k − 1) ... y (k − na )]

Y (k + i ) = [ y (k + i − 2)

Therefore, the output prediction for i-th PID will be:

⎡ y ( k + i − 2) ⎤ Y (k + i ) = ⎢⎢ y (k + i − 1) ⎥⎥ = ⎢⎣ y (k + i ) ⎥⎦

g (′i − 2) 2 ... g (′i − 2) nb ⎤ ⎥ g (′i −1) 2 ... g (′i −1) nb ⎥ gi′2 ... gin′ b ⎥⎦

(11)

M

M

i =0

i =0

− K {∑ Gi ∆uˆ (k ) + ∑ Fi y0 (k ) + ∑ Gi′∆u0 (k )} (12) Rewriting the control signal in compact form for PID type predictive control gives: ∆u (k ) = − K α∆uˆ (k ) + F f y0 (k ) + G g ∆u0 (k ) (13)

{

}

Where

α=

M

∑

M

Ff =

Gi

i =0

∑

M

Gg =

Fi

i =0

∑ G′ i

i =0

For systems with time delay, d, the output of the process will not be affected by ∆u (k ) until the time instant (k+d+1), the previous outputs will be a part of free response. We have shown that control signal increment for systems with timedelay can be written as follows [12]: ∆u (k − d ) = − K α∆uˆ (k − d ) + F f y0 (k ) + Gg ∆u0 (k ) (14)

{

}

Shifting the control signal for d step ahead gives: ∆u (k ) = − K α d ∆uˆ (k ) + F fd y0 (k ) + Ggd ∆u0 (k )

{

}

(15)

Where coefficients matrices are:

αd =

d +M

∑ i =d

d +M

Gi

F fd =

∑ i=d

d +M

Fi

Ggd =

∑ G′ i

i =d

III. OPTIMAL VALUES OF PREDICTIVE PID GAINS A. Genetic algorithms Genetic Algorithms have been introduced in early 70s by Holland [5]. In last decades GAs have attracted lot of attention. These algorithms simulate the biological theories of “survival of the fittest” and “natural selection”. Genetic algorithms provide a powerful search tool for a wide range of science and engineering optimizations. Instead of starting from just one point to find optimal values, GAs have a set of initial values called “initial population”. Because of this, GAs have a high degree of robustness. GAs work with the coded version of problem instead of optimization problem itself, i.e. the optimization parameters are encoded in a string of binary or real digits called “chromosome”. This property makes GAs

appropriate for controller parameter tuning. A. Optimal values of predictive PID gains In section II structure of predictive PID controller has been discussed. The controller gets M future errors as inputs and takes a control action according to those errors signals. The control action took here is a conventional PID action. The control signal can be written as a linear combination of each conventional PID’s control signal. Sets of three controller parameters are the same for each PID controller. These three parameters of predictive PID should be tuned to the process to achieve a desirable closed-loop performance. In our previous works we have found these parameters by optimal signal matching to GPC. These optimal values were determined by use of gradient-based methods [7]. Using gradient-based methods several approximations must be used to achieve a straightforward answer to optimization problem. In this section parameters of predictive PID controller are found using Genetic Algorithms. Here no approximations will be used and we use the output of the plant directly in optimization formulation instead of control signal increments. The objective is to minimize the difference between output of the plant with GPC control and plant predictive PID control. So fitness function will be introduced as follows: n

i i F = ∑ yGPC − yPPID

(16)

i =1

i

In which yGPC is the output of the plant under GPC control, i

and y PPID is the output of the plant under predictive PID control. Genetic optimization was applied to a second order system. Moradi et al [13] have shown that for a general second order system two layers of PID controllers are sufficient to approximate GPC performance. So the prediction horizon of predictive PID (M) is 1. So in our optimization for a second order process model we selected the prediction horizon M=1. Each time the genetic algorithm produces a population, the individuals inside that population are used to simulate the performance of the control system. At the end of the simulation the value of the fitness function is calculated. The values of future error signals are found using [2]. These error signals are fed to predictive PID and appropriate control signal is produced. In this paper instead of control signal increment matching between predictive PID and GPC, the output of the plant is used to calculate the fitness function of genetic algorithm. One advantage of this method is that no approximations will be used to simplify the equations in order to obtain a satisfactory and straightforward answer. Several set of GA parameters were used to run the optimization (table 1). Most of them show good convergence. It should be considered that the optimal predictive PID gains obtained in each run of GA are not the same and vary a lot. Which means that for a given performance criterion (I.e. cost function); several different set of gains could be obtained. The objective function can be defined as complex as

needed to solve the optimization problem. Optimization of the predictive PID gains is done offline. That’s because the genetic algorithms are usually computationally expensive. But once the PID gains are obtained they could be used in process control without any extra computational burden. The three Predictive PID gains are tuned to the process away from the real process itself, and then will be simulated to examine its efficiency. Then Predictive PID gains are saved in an appropriate storage device. These gains are used in Predictive PID structure presented in section II, to generate appropriate control signal. The goal of GA optimization stage is to minimize the difference between process output under Predictive PID controller and under GPC control. So in each run of genetic algorithm we have to simulate the process for example for a step test signal. In our optimization we have minimized the difference between two outputs for a double step means a step in t=1 and another in t=250. In this way the result could be better generalized for other responses. For example when we minimize the cost function for a single step response, and then obtain optimal gains from the optimization, the obtained gains work well in a single step response. But maybe for another test signals the response will not be satisfactory (multiple steps or ramp signal). So in our optimization we have used two steps to optimize the response. The results show that the optimal gains obtained here could be used in a general form for any other test signals like multiple steps or ramp signal. In first run of GA, the first population is produced in which the individuals inside the population are coded versions of Predictive PID gains. These individuals are used to simulate the process for a double step test signal. The value of the fitness function is then calculated for each individual. According to the value of the fitness function, the next generation of the Predictive PID gains is produced. In first generations of Predictive PID gains, the mutation operator plays an important rule. Because the value of the fitness function in first generations is very high for example in 200 order and some times approaches infinity. This is so because the first population individuals are generated randomly. By mutating the individuals better population could be achieved. After several generations the population starts to converge. The order of the fitness function for the best individual of the population becomes as less as order of two or three. So adaptive mutation will be best suited mutation function for propose of our optimization. Because mutations in middle stage of the optimization process make the fitness function diverge. So an adaptive mutation function in which has a high mutation rate when fitness function order is high and low mutation rate when the population starts to converge. IV. STABILITY STUDY FOR PROPOSED METHOD To study the stability for proposed method, the closed loop transfer function of system is calculated. It has been shown in [12] that the closed loop transfer function of the system can

be written as follows: B( z −1 ) z − d −1 y= w* −1 −1 − d −1 −1 ~ −1 Ac ( z ) B ( z ) z + Bc ( z ) A( z )

(17) The closed loop poles are the roots of characteristic equation: ~ Ac ( z −1 ) B ( z −1 ) z − d −1 + Bc ( z −1 ) A( z −1 ) = 0 (18)

The poles of closed loop system depend on PID gains. On the other hand, the PID gains are affected by selection of prediction horizon of proposed method (M). It has been shown in [12], [13] that for unstable second order systems bigger M causes bigger stability regions. For non-minimum phase system bigger M causes the smaller stability region (see 13 for further discussions). The GA optimized Predictive PID controller can be implemented using the following procedure: Step 1: initialization 1. Find the system model and calculate the discrete polynomials. 2. Choose the value of prediction horizon, M, and formulate the future set point vectors. Step2: Genetic Algorithm optimization 1. Calculate the MPC gain. 2. Calculate optimal value of PID gains using procedure of section (III). 3. Iterate over M to find the best approximation to GPC. 4. Check the closed loop stability using equation (17). Change M or use another set of optimal predictive PID gains to achieve stability if necessary. V. PERFORMANCE STUDY The performance of the GA optimized Predictive PID controller has been studied through several benchmark system (Astrom and Hugglund). simulations. Performance of GA optimized predictive PID controller is compared with three other methods which are GPC, Gradient Optimized Predictive PID and conventional PID controller Fig (2). This method shows good tracking performance in comparison with gradient based method. This method shows better performance than GPC in settling time point of view. The GPC response oscillated in its steady state but Predictive PID response oscillations are better damped. The results are for a second order system. VI. CONCLUSION A new predictive PID controller was proposed which has important similar features to the model based predictive control. The optimal values of predictive PID gains were obtained. In this method instead of just one set of optimal gains, several set of optimal predictive PID gains have been calculated which have similar performance, i.e. for a desired measure of performance several set of optimal values can be obtained. The performance of this method was compared with GPC, gradient based optimized predictive PID and

conventional PID controller showing good performance. This method integrates the simplicity of implementation of conventional PID controllers and complex process handling of GPC. Which means that similar performance as GPC can be achieved by use of just a little bit of extra hardware than conventional PID. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

Astrom, K.J. and T. Hugglund, (1995): PID controllers: Theory, Design and Tuning, Instrument Society of America, Research Triangle Park, NC, USA. Camacho, E.F. and C. Bordons. Model Predictive Control. SpringerVerlag London, 1999. Chien, I.L. (1988). IMC-PID Controller Design-An Extension, IFAC Proceeding Series, 6, pp. 147-152. Clarke, D.W., C. Mohtadi and P.S. Tuffs (1987). Generalised Predictive Control' I & II, Automatica, 23(2): 137-160. Holland J.H Adaptation in natural and artificial systems. Ann Arbor: university of Michigan press, 1975 Johnson, M.A. and Moradi M.H, PID Control, Springer-Verlag London, 2005. Katebi, M.R. and Moradi, M.H. Predictive PID Controllers, IEE Proceeding of Control Theory and Application, Vol. 148, No 6, November 2001, pp. 478-487. Kwok, K.E. Ping, M.C. Li, P. A Model-based Augmented PID Algorithm. Journal of Process Control Miller , R.M. Shah, S.L. Wood, R.K. Kwok, E.K. Predictive PID, ISA Transaction, Volume 38, Issue 1, January 1999, PP 11-23. Min Xu. Shaoyuan, Li. Chenkun Qi. Wenjian Cai. Auto-tuning of PID controller parameters with supervised receding horizon optimization. ISA Transactions, Vol. 48, Issue 4, October 2005, pp491-500 Minghe, Li. Wang, Meng. Shi, Yanyan. Adaptive Prediction PID Control Based on RBFNN. Electronic Measurement and Instruments, 2007, ICEMI. Moradi M.H. and Katebi M.R. Predictive PID: A new Algorithm, 2001, The 27th annual conference of the IEEE Industrial Electronic Society, IECON'01, 29th November 2nd Dec 2001, pp 764-769. Moradi M.H., M.R. Katebi and M.A. Johnson,2002a, ‘The MIMO Predictive PID Controller Design’, “Advances in PID Control” Special Issue of Asian Journal of Control, Vol.4, No. 4, 2002 Rivera, D.E., S. Skogestad and M. Morari (1986). Internal Model Control 4. PID controller Design, Ind. Eng Chem. Proc. Des & Dev, 25, pp. 252-265 Tae-Hyoung Kim. Ichiro Maruta. Toshiharu Sugie. Robust PID Controller Tuning Based on the Constrained Particle Swarm Optimization, Automatica, Vol. 44, Issue 4 April 2008, pp. 1104-1110. Tan K.K. Q-G, Wang, C.C. Hang and T.J Hugglund, Advances in PID Control, Springer-Verlag London 1999. Wang, Q.G., C.C. Hang and X.P. Yang (2000), Single -Loop Controller Design Via IMC Principles, In Proceeding Asian Control Conference. Shanghai, P.R.China. Ziegler, J.G. and N.B. Nichols (1942). Optimum setting for automatic controllers, Trans. ASME 64, 759-768.

Prediction d step

Matrix GG

Prediction d+1 step r(k)

Set-point Prediction

∆ u(k)

w(k) +

e(k)

+

PID

SYSTEM

y(k)

r(k)

-

Set-point Prediction

Prediction d+M step

z −1

z −1 z −1

nb

2

1

w(k) +

e(k)

+

PID(k)

y(k)

1β

SYSTEM

∆ u(k)

-

z −1 z −1 1

2

z −1

Matrix FF

na

Fig (1): The block diagram of proposed method, a) M horizon prediction, b) structure of the controller

Fig (6): Performance study for proposed method, two changes in set point in t=0 and t=250.

Table (1): Results of GA optimization, 12 different optimal predictive PID gains

NO.

Number of generations

1 2 3 4

100 722 664 600

Best fitness Value 24.316 15.378 13.48 10.480

5 6 7 8 9 10 11 12

50 30 100 100 100 100 100 100

15.335 21.173 18.929 19.089 21.938 15.995 17.917 24.431

Alteration

Predictive PID Gains

Non Non Rolette selection Rolette selection & adaptive mutation Non Non Non Population of size 30 Adaptive mutation Non Fitness top 0.4 Elite count 5

[0.1063 0.00605 -0.50074] [0.08762 0.0062 o.89802] [0.05061 0.00458 0.63824] [0.0557 0.00465 1.0798] [0.00577 0.00221 -0.25191] [0.1017 0.0079 0.57925] [0.13526 0.00905 -0.01513] [0.05666 0.00374 0.02411] [0.13164 0.00662 0.11251] [0.06622 0.004797 0.12987] [0.02002 0.00238 0.20715] [0.24766 0.01389 -1.42644]