New Integrated Control Design Method based on … · New Integrated Control Design Method based on...

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New Integrated Control Design Method based on Receding Horizon Control with Adaptive DA Converter Takuto MOTOYAMA , Tohru KAWABE , Kazuo TORAICHI , Kazuki KATAGISHI Graduate School of Systems and Information Engineering University of Tsukuba Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8573 JAPAN http://www.wslab.risk.tsukuba.ac.jp/index-e.html Abstract: - In this paper, a new integrated control design method based on RHC (receding horizon control) and Fluency analysis is proposed for systems with relatively fast-moving dynamics. In the Fluency analysis which has been proposed by a co-author of this paper, signal spaces are well-classified by its smoothness, and Fluency sampling functions with good properties have been proposed in each space. However, it has been difficult to apply the Fluency analysis approach for a DA (digital-to-analog) converter in sampled-data control systems, since the interpolation using sampling function needs the information of future samples. We, therefore, propose a new interpolation method using predicted information calculated on RHC controller in each step. Moreover, the way to switch the Fluency sampling functions according to the system status is proposed. This approach is realized as an adaptive DA converter. A numerical example is also included to show the proposed method enables us to improve the control performance of systems including a controller and the adaptive DA converter. Key-Words: - RHC (Receding horizon control), Sampling function, DA (digital-to-analog) converter 1 Introduction The recent dramatic advance of computer has made it possible to control the continuous-time objects by a discrete-time controller designed on computer. At the same time, digital control has become increas- ingly important. In such systems, that is a kind of sampled-data control systems, analog-to-digital(AD) and digital-to-analog(DA) conversion of signals are indispensable operations. Generally, the zero-order hold is used as a DA conversion, while the way to sample the analog signal at a constant frequency is used as an AD conversion [1]. However, to improve the performance of sampled-data control systems, it’s very important to take account of the behavior of sys- tems between sampling intervals. On this issue, some notable methods to design the discrete-time controller for continuous-time objects with AD/DA conversion have been proposed [2][3][4]. We also have studied this type of sampled-data con- trol systems from the viewpoint of DA conversion based on Fluency analysis [5]. In the Fluency analysis which has been proposed by a co-author of this paper, signal spaces are well-classified by its smoothness, and sampling functions with sampling bases in each space are given in complying with requested specifi- cations [6]. However, it has been difficult to apply these sam- pling functions to DA conversion directly in sampled- data control system, since the interpolation based on sampling function needs the information of future sampling points. Generally, in the case of DA con- version in sampled-data control, it is impossible to obtain the information of future sampling points, and DA converter has to interpolate discrete control input generated from discrete-time controller based on past information. Therefore, we have been forced to tol- erate the long time-delay during the DA conversion using sampling function in our previous works. How- ever, in the case of control of the systems with rel- atively fast-moving dynamics, such as robots or ve- hicles, the method with long time-delay is unable to be applied. Hence, we need to develop a new control method without long time-delay. In this paper, therefore, we propose a new inte- grated control design method based on RHC (reced- ing horizon control) with the Fluency analysis for such systems with relatively fast-moving dynamics. RHC is one of digital control methods which have gotten a lot of attention recently because of its high practicality [7]. In RHC algorithm, a prediction of the future system status is executed, and future con- trol inputs based on the prediction are also calculated in each step. Hence, we propose a new idea to use these future control inputs based on the prediction for interpolation by sampling function. This idea en- ables us to apply sampling functions for DA conver- Proceedings of the 8th WSEAS Int. Conference on Automatic Control, Modeling and Simulation, Prague, Czech Republic, March 12-14, 2006 (pp6-11)

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New Integrated Control Design Method based on Receding HorizonControl with Adaptive DA Converter

Takuto MOTOYAMA†, Tohru KAWABE†, Kazuo TORAICHI†, Kazuki KATAGISHI††Graduate School of Systems and Information Engineering

University of TsukubaTennodai 1-1-1, Tsukuba, Ibaraki, 305-8573

JAPANhttp://www.wslab.risk.tsukuba.ac.jp/index-e.html

Abstract: - In this paper, a new integrated control design method based on RHC (receding horizon control) andFluency analysis is proposed for systems with relatively fast-moving dynamics. In the Fluency analysis whichhas been proposed by a co-author of this paper, signal spaces are well-classified by its smoothness, and Fluencysampling functions with good properties have been proposed in each space. However, it has been difficult toapply the Fluency analysis approach for a DA (digital-to-analog) converter in sampled-data control systems, sincethe interpolation using sampling function needs the information of future samples. We, therefore, propose a newinterpolation method using predicted information calculated on RHC controller in each step. Moreover, the wayto switch the Fluency sampling functions according to the system status is proposed. This approach is realizedas an adaptive DA converter. A numerical example is also included to show the proposed method enables us toimprove the control performance of systems including a controller and the adaptive DA converter.

Key-Words: -RHC (Receding horizon control), Sampling function, DA (digital-to-analog) converter

1 IntroductionThe recent dramatic advance of computer has madeit possible to control the continuous-time objects bya discrete-time controller designed on computer. Atthe same time, digital control has become increas-ingly important. In such systems, that is a kind ofsampled-data control systems, analog-to-digital(AD)and digital-to-analog(DA) conversion of signals areindispensable operations. Generally, the zero-orderhold is used as a DA conversion, while the way tosample the analog signal at a constant frequency isused as an AD conversion [1]. However, to improvethe performance of sampled-data control systems, it’svery important to take account of the behavior of sys-tems between sampling intervals. On this issue, somenotable methods to design the discrete-time controllerfor continuous-time objects with AD/DA conversionhave been proposed [2][3][4].

We also have studied this type of sampled-data con-trol systems from the viewpoint of DA conversionbased on Fluency analysis [5]. In the Fluency analysiswhich has been proposed by a co-author of this paper,signal spaces are well-classified by its smoothness,and sampling functions with sampling bases in eachspace are given in complying with requested specifi-cations [6].

However, it has been difficult to apply these sam-pling functions to DA conversion directly in sampled-

data control system, since the interpolation based onsampling function needs the information of futuresampling points. Generally, in the case of DA con-version in sampled-data control, it is impossible toobtain the information of future sampling points, andDA converter has to interpolate discrete control inputgenerated from discrete-time controller based on pastinformation. Therefore, we have been forced to tol-erate the long time-delay during the DA conversionusing sampling function in our previous works. How-ever, in the case of control of the systems with rel-atively fast-moving dynamics, such as robots or ve-hicles, the method with long time-delay is unable tobe applied. Hence, we need to develop a new controlmethod without long time-delay.

In this paper, therefore, we propose a new inte-grated control design method based on RHC (reced-ing horizon control) with the Fluency analysis forsuch systems with relatively fast-moving dynamics.RHC is one of digital control methods which havegotten a lot of attention recently because of its highpracticality [7]. In RHC algorithm, a prediction ofthe future system status is executed, and future con-trol inputs based on the prediction are also calculatedin each step. Hence, we propose a new idea to usethese future control inputs based on the predictionfor interpolation by sampling function. This idea en-ables us to apply sampling functions for DA conver-

Proceedings of the 8th WSEAS Int. Conference on Automatic Control, Modeling and Simulation, Prague, Czech Republic, March 12-14, 2006 (pp6-11)

sion in sampled-data control system without so longtime-delay. Moreover, we propose a new method toswitch the Fluency sampling function optimally ac-cording to the system status, since Fluency analysishas provided sampling functions with good proper-ties. This approach is realized as the adaptive DAconverter. A numerical example is also included toshow the proposed method gives the improvement ofcontrol performance of the entire system.

2 PreliminariesLet’s consider a discrete-time linear time-invariantsystem as follows,

x(k + 1) = Ax(k) + Bu(k) (1)

y(k) = Cx(k) (2)

whereu(k) ∈ R1, x(k) ∈ Rn andy(k) ∈ R1 meancontrol input, state values and observed output at stepk respectively, andA ∈ Rn×n, B ∈ Rn×1 andC ∈R1×n are coefficient matrices.

The outlines of the Fluency analysis and RHC aregiven briefly in next sections.

2.1 Fluency analysisIn Fluency analysis, the signal spaces are defined asthe function spaces composed of piecewise polynomi-als of degree(m−1) with (m−2) times differentiabil-ity [6]. Moreover, the existence of the sampling basisin each space has been proven. For example, whenm = 1, the sampling basis is equivalent to the sam-pling basis in the signal space composed by staircasefunction, and whenm goes to infinity, the samplingbasis is equivalent to the Shannon’s sampling basiswhich characterizes the band limited signal space.By selecting appropriate signal spaces characterizedby parameterm according to the object, the Fluencyanalysis enables us to deal with signals flexibly andprecisely. In the field of audiology and image pro-cessing, our new approaches based on the Fluencyanalysis have already been highly-acclaimed.

In this paper, we assume the signal spaces withm = 1, 2, 3, since the effectiveness of these threespaces is known in a series of our studies. Further-more, in the case thatm is 4 or more, it’s diffi-cult to apply to fast-moving dynamic systems due tothe bigger calculation amount of interpolation. Fig.1shows the Fluency sampling functions(m = 1, 2, 3)we use in this paper and their interpolations. Espe-cially, the sampling function withm = 3 in fig.1,which is called as the compactly supported Fluency

m Sampling function Interpolated signal

1

2

3

time(t)2/τ0

1)(1 tψ

2/τ−

time(t)0

1)(2 tψ

ττ−

time(t)

)(3][ tc ψ1

0 τ τ2τ−τ2−

time(t)

time(t)

time(t)

Fig. 1. Fluency sampling functions and their interpo-lations (m = 1, 2, 3), (τ : sampling interval).

sampling function with 2nd degree, has already beenput to practical use in a DA converter of DVD-Audio[8]. The interpolated signal in the closed-open inter-val [(i − 1)τ, iτ) using the compactly supported Flu-ency sampling function with 2nd degree is calculatedas follows,

u(t) =i+1∑

l=i−2

{u(l) · 3

[c]ψ(t − lτ)}

(3)

whereu(t) andu(l) are analog signal and digital sig-nal respectively, andτ is sampling interval.

2.2 RHC (receding horizon control)RHC is an online powerful control method whichsolves a finite horizon open-loop optimal problemwith respect to each sampling frequency [7]. Al-though RHC has been used in systems with relativelyslow-moving dynamics such as petrochemical plantsdue to its big calculation amount, the recent advanceof computer performance has made it possible to ap-ply for the systems with relatively fast-moving dy-namics.

Now, let’s consider the finite-time constrained op-timal control problem with the state space model asfollows,

min{u(k|k),···,u(k+N−1|k)}

J(k) = ||x(k + N |k)||2P

+N−1∑i=0

{||x(k + i|k)||2Q + ||u(k + i|k)||2R

}(4)

Proceedings of the 8th WSEAS Int. Conference on Automatic Control, Modeling and Simulation, Prague, Czech Republic, March 12-14, 2006 (pp6-11)

subject to:

u(k) ∈ U , x(k) ∈ X (5)

whereP , Q andR are positive definite matrices, andN is the length of prediction horizon. Eq.(5) meansconstraint conditions for the control input and thestate values. In practice, since this problem is equiva-lent to the quadratic programming problem, the opti-mal solution{u(k|k), · · · , u(k + N − 1|k)} is easilysolved. Finally, only the first solutionu(k|k) is usedas a control input for control object at stepk, and then,the current step goes on to next step.

Several kinds of RHC method have been also pro-posed until now [9][10].

3 Integrated control design methodIn this section, the detail of the proposed integratedcontrol design method based on RHC and Fluencyanalysis is given.

3.1 Interpolation based on sampling func-tion using predicted control inputs

As we mentioned above, the interpolation based onsampling function needs the information about futuresampling points. However, it’s generally impossibleto know the future without any foresight information.We therefore propose a new idea to use the predictedcontrol inputs obtained by RHC for interpolation.

In RHC, the optimal control inputs{u(k|k), u(k +1|k), · · · , u(k+N−1|k)} are calculated in each step,and only the first control inputu(k|k) is used as a realcontrol input. Therefore, we consider to use the otheroptimal control inputs{u(k + 1|k), u(k + 2|k), · · ·}as virtual future sampling points. Actually, it is onlynecessary to use the optimal control inputs which areneeded for interpolation according to the samplingfunction. Fig.2 shows this way using the compactlysupported Fluency sampling function with 2nd de-gree. Onlyu(k + 1|k) is used as a virtual future sam-pling point in this case. By using the predicted futurecontrol inputs for interpolation, it becomes possibleto reduce the time-delay in DA conversion, and thetotal time-delay to be needed is just only computationtime of optimization in current step.

Of course, it needs to take account that there is adifference between virtual future sampling points andreal sampling points likeu(k+1|k) = u(k+1) in fu-ture step. However, we consider that this point is not acritical problem because the influence on interpolatedwaveform due to prediction error is not so big com-pared to the scale of prediction error. Although the

time(t)

Now Future Past

u(k−2)

u(k−1)

u(k|k) (=u(k))u(k+1|k)^

(k−2)τ (k−1)τ (k)τ (k+1)τ

^

Fig. 2. Interpolation based on sampling function us-ing predicted future control inputs.

differentiability which each sampling function has islost at sampling points, this also does not become acritical problem compared to the zero-order hold, andit is possible to keep a certain level of smoothness.

3.2 Switching Fluency sampling functionsFluency analysis has provided various sampling func-tions with good properties. Therefore, we con-sider switching these Fluency sampling functions op-timally according to the system status in the adaptiveDA converter. Fig.3 shows the proposed integratedcontrol design. As this figure shows, the adaptiveDA converter has internal model with sampling inter-val τ/d. Please take notice that this internal model2 is different from internal model 1 with samplinginterval τ of controller. The interval to be interpo-lated is also divided tod sections, and the dividingpointsum(j; k), (j = 1, 2, · · · , d− 1) on interpolatedwaveforms are used for the selection of parameterm,that indicates the degree of Fluency sampling func-tions. Fig.4 shows the difference of the interpolationand dividing points according to the sampling func-tion whenm is limited from1 to 3 andd = 5. Thedividing pointsum(j; k) are calculated as follows,

um(j; k) =k+α−1∑l=k−α

{u(l) · mψ

((k − 1)τ +

τ

d· j − lτ

)}(j = 1, 2, · · · , d − 1) (6)

whereα is the number of samples which the samplingfunction needs for interpolation, and it is adjusted ac-cording to the sampling function.

Proceedings of the 8th WSEAS Int. Conference on Automatic Control, Modeling and Simulation, Prague, Czech Republic, March 12-14, 2006 (pp6-11)

Optimizedcalculation

Internal model 1Sampling interval:τ

Internal model 2Sampling interval: d/τ

Select m Controlobject

Output

State value

Controller(MPC) Adaptive DA converter

Sampling )(τ

Discrete time

Discretecontrolinputs

Continuouscontrolinput

Fig. 3. Proposed integrated control design.

t

Interpolated interval

)(tu );1( kum

);2( kum

);3( kum

);4( kum

)1( −ku

)(ku

τ)1( −k τ)(k

1)1( =m

2)2( =m 3)3( =m

Fig. 4. Interpolation ways (d = 5).

Then, we summarize the algorithm to switch theFluency sampling functions for the adaptive DA con-verter as follows,

(step1) Step= k.(step2) The dividing pointsum(j; k) are calculated

as eq.(6).(step3) The predicted state valuesxm(j+1; k) in this

interval are calculated using internal model ofDA converter and the dividing pointsum(j; k).

(step4) If the interpolation wave exceeds the con-strained conditions of control input due to theovershoot or undershoot, thism is excluded.

(step5) The evaluation values using evaluation func-tion Jm(k) are calculated in eachm.

(step6) The parameterm whose evaluation value isthe smallest is selected as an interpolation wayin this interval, and thenk = k + 1 and go backto (step1).

In this paper, the evaluation function in (step 5) is

used as follows,

Jm(k) =d−1∑j=1

{||xm(j + 1; k)||2Q1

+ ||um(j; k)||2R1

}(7)

whereQ1 andR1 are positive definite matrices.From several test simulation results, it seems to be

appropriate that the divided number of interval,d, set5 due to the trade-off of computation time and preci-sion. If d = 5, the calculation amount in the adaptiveDA converter is also vanishingly small compared tothe calculation in RHC controller keeping a certainlevel of accuracy.

It is considered that the proposed method enablesus to give the improvement of control performance ofthe entire system. Longer the sampling interval, thisimprovement of control performance is more conspic-uous.

4 Numerical exampleIn this section, a numerical example is givento demonstrate the effectiveness of the proposedmethod. As the numerical example, let’s consider acontrol problem of the inverted pendulum. Moreover,the following three methods are compared through asimulation.

1. LQ optimal control with zero-order hold.2. RHC with zero-order hold.3. Proposed method.

LQ optimal control is one of efficient control methodsfor systems with relatively fast-moving dynamics [1].

4.1 Inverted pendulumAs fig.5 shows, inverted pendulum control problemis to control the wheel truck position without taking

Proceedings of the 8th WSEAS Int. Conference on Automatic Control, Modeling and Simulation, Prague, Czech Republic, March 12-14, 2006 (pp6-11)

l

gmg

gm ),(

GGyx

θ

u

X

Y

x

O

M

Fig. 5. An example of inverted pendulum.

down the bar with weight.M , mg, and l mean thewheel truck mass, the weight mass, and the length ofthe bar respectively, and when theO−XY coordinatesystem is defined like fig.5, the center of gravity pointand the wheel truck position is defined as(xG, yG)andx. The mass of the bar and the frictions are verytiny vanishingly.

Then, when the state values are taken as,

x1 = θ, x2 = θ, x3 = x, x4 = x (8)

the state space model is expressed as follows,

d

dt

x1

x2

x3

x4

=

0 1 0 0

M+mg

Ml g 0 0 00 0 0 1

−mg

M g 0 0 0

x1

x2

x3

x4

+

0

− 1Ml01M

u (9)

y =[

0 0 1 0]

x1

x2

x3

x4

(10)

4.2 SimulationEach parameter in the simulation is set as follows,

wheel truck mass : M = 2weight mass : mg = 0.1length of the bar : l = 0.5gravity acceleration : g = 9.81division number : d = 5sampling interval of controller : τ = 0.05sprediction horizon : N = 55Q,R, Q1, R1 are identity matrices.

P is obtained by solving the Riccati difference equa-tion [7]. Moreover, discrete-time models for RHC

Table 1. Experimental environment.

CPU Intel(R)Pentium(R)4 2.80GHzMemory 760MB RAM

OS Windows XP ProfessionalSoftware MATLAB 6.5.1Toolbox Control System Toolbox 5.2.1

controller and the adaptive DA converter are derivedby a general discretization of the continuous-timemodel with the sampling intervalτ = 0.05s andτ/d = 0.01s respectively. In the simulation, weuse a discrete-time model with the sampling interval0.0005s as a control object and the DA conversionmeans 100 times up-sampling. Control inputs are alsoconstrained as−0.3 ≤ u(t) ≤ 0.3, and 0.02s time-delay is added to the proposed adaptive DA converteras waiting time for optimized calculation. Moreover,since it’s usually difficult to obtain the state valuex1

due to air resistance, measurement error and so on,x1 is assumed to be perturbed by a disturbance. Thisdisturbance acts as a force to take down the bar.

Figs.6 through 8 show the appearances of the simu-lation. Fig.6 shows added disturbance and the controlinput signals and observed output signals are shownin fig.7 and fig.8 respectively. From this simulationresults, we can easily see that both input and outputby the proposed method converged to the equilibriumposition 0 faster than the other ones. The LQ con-troller can not give its ability into full play due toconstraint conditions. From comparison between theRHC with zero-order hold and the proposed method,there is no doubt about the effectiveness of the pro-posed adaptive DA converter. In the case of the sys-tems with relatively fast-moving dynamics like the in-verted pendulum, the tiny difference of DA conver-sion causes a big influence for result. If the sam-pling interval of controller is set longer, this tendencybecomes clearer. Therefore, we think the proposedmethod is efficient for any other general systems withrelatively fast-moving dynamics.

5 ConclusionIn this paper, a new integrated control design methodbased on RHC and Fluency analysis for the systemswith relatively fast-moving dynamics has been pro-posed. The proposed method gives us improvementsof control performance of systems including a con-troller and the adaptive DA converter.

Proceedings of the 8th WSEAS Int. Conference on Automatic Control, Modeling and Simulation, Prague, Czech Republic, March 12-14, 2006 (pp6-11)

0 5 10 15−5

0

5

10

15

20x 10

−4

time(t)

dist

urba

nce(

t)

disturbance

Fig. 6. Disturbance used in simulation.

0 5 10 15−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time(t)

u(t)

LQ with 0−order holdMPC with 0−order holdProposed

Fig. 7. Control input signals.

time(t)time(t)time(t)time(t)time(t)time(t)time(t)time(t)time(t)time(t)time(t)time(t)0 5 10 15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

y(t)

LQ with 0−order holdMPC with 0−order holdProposed

time(t)

Fig. 8. Observed output signals.

As future works, we need to develop the selectionmethod of the best sampling function according tocontrol object. Moreover, we need to make an exper-iment using a real control object and make sure theeffectiveness of the proposed method.

References:[1] J. Ackermann,Sampled-Data Control Systems –

Analysis and Synthesis, Robust System Design,Springer-Verlag, 1985.

[2] P. T. Kabamba, Control of Linear Systems Us-ing Generalized Sampled-Data Hold Functions,IEEE Trans. on Automatic Control, Vol.32,No.9, 1987, pp.772-783.

[3] B. D. O. Anderson, Controller Design: Movingfrom Theory to Practice,IEEE Control SystemsMagazine, Vol.13, No.4, 1993, pp.16-25.

[4] Y. Yamamoto, A Function Space Approach toSampled-Data Control Systems And TrackingProblems,IEEE Trans. on Automatic Control,Vol.39, No.4, 1994, pp.703-713.

[5] K. Toraichi, B. Sridadi, and H. Inaba, A series ofdiscrete-time models of a continuous-time sys-tem based on fluency signal approximation,In-ternational Journal of Systems Science, Vol.26,No.4, 1995, pp.871-881.

[6] M. Kamada, K. Toraichi, and R. Mori, PeriodicSpline Orthonormal Bases,Journal of Approxi-mation Theory, Vol.55, No.1, 1988, pp.27-34.

[7] D. Q. Mayne, J. B. Rawlings, C. V. Rao, andP. O. M. Scokaert, Constrained model predic-tive control: Stability and optimality,Automat-ica, Vol.36, No.6, 2000, pp.789-814.

[8] K. Nakamura, K.Toraichi, K. Katagishi, and S.L. Lee, Compactly Supported Sampling Func-tions of Degree 2 for Applying to ReproducingDVD-Audio, Proc. of IEEE Pacific Rim Con-ference on Communication, Computer and Sig-nal Processing, Victoria, Canada, 2001, pp.670-673.

[9] A. Bemporad, M. Morari, V. Dua, E. N. Pis-tikopoulos, The explicit linear quadratic regula-tor for constrained systems,Automatica, Vol.38,No.1, 2002, pp.3-20.

[10] T. Ohtsuka, A continuation/GMRES method forfast computation of nonlinear receding hori-zon control,Automatica, Vol.40, No.4, 2004,pp.563-574.

Proceedings of the 8th WSEAS Int. Conference on Automatic Control, Modeling and Simulation, Prague, Czech Republic, March 12-14, 2006 (pp6-11)