CONTROL SYSTEM DESIGN CONFORMABLE TO PHYSICAL ACTUALITIES

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1 3rd International Conference on Advanced Micro-Device Engineering, AMDE CONTROL SYSTEM DESIGN CONFORMABLE TO PHYSICAL ACTUALITIES TOSHIYUKI KITAMORI

2 2 Outline Design equation Expression easy to solve design equation Expression easy to measure controlled-object Expression easy to specify desirable control system New control scheme: I-PD control Smooth extension to design of sampled-data control Some examples designed Servo systems, decoupling control Smooth extension to nonlinear control

3 3 Design Equation Controlled-object connection Compensator/Controller = Desirable control system Expression should be easy to indentify the controlled-object to specify the desirable control system to solve the compensator/controller

4 What functions should the expression have for easy solution? 4 We can clarify the functions in the process to solve the design equation below: r e C u P y + - r = W d y e C u =? input output The element is assumed as any element expression which can determine the output for the given input.

5 5 Solution of design equation r e C u P y + - = r W d y r e inv(c) u inv(p) y + + = r inv(w d ) y e inv(c) u inv(p) y = e r + - inv(w d ) y e C u P y = e + + r W d y To be continued

6 = 6 e C u P y = e + + r W d y e u y u C P inv(p) = e r + + W d y inv(p) e u C C=ser(inv(P),inv(par(-I,inv(W d )))) u The function turns out to be computability of series connection of two elements, parallel connection of two elements and inverse of each element. Inverse is an element to give the input from the output.

7 State equation expression is not invertible Forward u B Inverse if u B + + x& D ケ 0 x& + + integrator A D integrator A D - 1 x x C: pエn p C C ( < n) y y x& Ax + Bu y = Cx + Du x ( & = A- BD C)Ax+ BD y u= - D Cx+ D y 7 A physical system does not have the inverse because D= 0 in general.

8 What is the expression easy to compute series and parallel connections and inverse? 8 In a static system the output is the same as the input multiplied by a gain for an arbitrary input. For such static systems three computations are straightforward. A static system is a special case of a dynamical system. So we should be able to extend the expression smoothly from static to dynamical system. In a dynamical system, for an arbitrary input the output is not the same as the input multiplied by any constant. We can find, however, a special class of inputs for witch the output is the same as the input multiplied by a constant. Using the class of inputs as a key, we can get an expression easy to compute series and parallel connections and inverse after some manipulation. It turns out to be the transfer function expression.

9 Expression Easy to Measure Controlled Object 9 The physical system is, in general, a distributed-parameter system. The order of dynamics of a distributed-parameter system is infinitely high. It is by no means linear. It can be time-variant. In addition, the boundary conditions are very much complicated. Thus any model we can get through measurement is an approximation. Yet we know empirically that if time-variation of the system is not so fast, time-invariant approximation is useful, that if the range of operation is narrow enough, linear approximation is useful, and that if dynamical change is not so fast, a lower-order approximation is useful and even static, that is, 0-th order approximation can be useful. Thus we have a lot of approximating expressions for a system, and we naturally think that those approximating expressions are close to each other. To be continued

10 10 In measurement of dynamic characteristics we gather data from phenomena which the system shows us and we estimate parameters in the model expression. So any expressions approximating the same phenomena should be very close to each other. From observations in the following two slides, the transfer function expression is suitable for measurement.

11 step response From the step response data two models are assumed and the parameters of transfer functions are estimated. 11 u R L C y s s 2 Negligible 1.0 u R C y Close to each other s Here are two response curves plotted corresponding to the two models. The two are indistinguishable time R = 10 Ω, C=10 F, L=10 H The transfer function expressions are close to each other and suitable to be measured.

12 From the step response data two models are assumed and the parameters of state equation are estimated. 12 u u R R C L C R = 10 Ω, C=10 F, L=10 H y y 6 鴿 x & 鴿 x1 鴿 0 = + u x 2 8 x & 鴿 x1 y = [ 1 0] + [ 0] u 鸙 x 2 Too big to be neglected [ ] [ ][ ] [ ] x& 1-1 x1 + 1 u y = [ 1][ x1 ] + [ 0] u The state equation expressions are not close enough in spite of step responses are indistinguishably close to each other. Not close to each other

13 step input step response 13 How to specify desirable control system People intuitively evaluate control performance from the shape of step response rather than performance indices. Steady state error is as small as possible, and is zero if possible. Response time is as short as possible. Adequate damping is desirable time time 10.0

14 step response step response How do coefficients operate step response shape? a 2 10% increased unperturbed 1.0 a 3 10% increased a a % increased 10% increased unperturbed (red) and 10% increased a 4 (indistinguishable) time time 1 1 = a + a s+ a s + a s + a s 1+ 4s+ 6s + 4s + 1s Lower-order terms are effective to shape the step response, whereas higher-order terms have little effect on the shape. This enables us partial compensation of lower-order terms only. 20.0

15 15 Design equation for PID control r e cs ( ) u bs ( ) y r bd ( s) = + s as ( ) a ( s) - d y c() s c + c s + c s + L 踐 = = K P 1 TDs s s + + 顏 TIs a() s = a + a s + a s + a s b() s = b + b s + b s + b s d() d0 d1 d2 d3 a s = a + a s + a s + a s d() d0 d1 d2 d3 b s = b + b s + b s + b s + L L L L c( s) b( s) s a( s) bd( s) = c( s) b( s) ad() s 1 + s a() s

16 16 DEF is essential for PID design Inverse of controlled-object Inverse of desirable control system c( s) b( s) s a( s) bd( s) = c( s) b( s) ad() s 1 + s a() s 2 c() s = c + c s + c s + as () = bs () s ad() s b () s d - 1 L Information needed about controlled-object: as () = a () s b() s = a as a s L and about desirable control system: ad() s b () s d = a () s d 2 = a + a s + a s + L d0 d1 d2 Denominator expanded form (DEF) Maclaurin expansion of inverse

17 step response Transfer function and step response shape s+ 0.5s s s s+ 0.75s s s s s+ s + 0.4s s s 1+ 2s+ 1.5s s s s 1+ 3s+ 2.5s s s time 10.0 Many to one correspondence. Transfer function expression is redundant.

18 step response 18 DEF is decisive for step response shape s+ 0.5s s s+ 0.5s s s s s + L s+ 0.5s s s s s + L s+ 0.5s s s s s + L s+ 0.5s s s s s + L time 10.0 Denominator expanded form DEF 1 denominator polynomial numerator polynomial

19 Relations among MacLaurin series, moment series, and numerator expanded form (NEF) dg 1 d G 1 d G G( s) = G(0) + s+ s + s + ds ds ds where s= 0 2! 3! s= 0 s= 0 st 2 3 G( s) - g( t) e dt m0 m1s m2s m3s 0 Gs () = 1 1 = = ! 3! i m = g() t t dt is i-th moment i b0 + b1s + b2s + b3s + L 2 3 a + a s+ a s + a s + L d i 3 of impulse response around t=0 i 1 d G( s) i 1 ( 1) i i ds i s= 0 L L 2 3 = d + d s+ d s + d s = = -!! m i L 19

20 20 Moment series expression of transfer function - st G( s) = g( t) e dt = ( t) g 1- st+ s t - s t + s t - L dt 0 2! 3! 4! = g( t) dt- s g( t) tdt+ s g( t) t dt- s g( t) t dt ! 0 3! = m0 - m1s + m2s - m3s + L 2! 3! L where ( ) i m = g t t dt is i-th moment i 0 of impulse response around t=0

21 21 Relation between DEF and NEF Gs () = = d + d s+ d s + d s + = 2 3 b0 + b1s + b2s + b3s + L 2 3 a + a s+ a s + a s + L c + c s+ c s + c s L L (transfer function) (NEF) (DEF) c0 =, c1 = - c0d1, c2 = - ( c0d2 + c1d1 ) d d d c3 = - ( c0d3 + c1d2 + c2d1 ), L L L d 0

22 22 Equivalence relation among expressions m m m m b b b b d d d d L L c c 0 = =- 1 d 0 1 d c d c c c c Determinable relation independently from successors (Series can be truncated at any term.) Independency from successors (IFS) L 1 c2 = - ( c0d2 + c1d1 ) d 0 L L L

23 Time scale normalization of DEF 23 a a + a s+ a s + a s + a s = a1 a s s s 0 コ, = s a a 0 1 L 1 a a 2 a 3 a s+ 2 s + 3 s + 4 s + a a a a a 踐 a a 踐 a a 踐 a s + s + s + s + a 鉗顏 a a 顏 a a 顏 a = s + a s + a s + a s + 1 L The first moment (average delay time) is set to 1. L L

24 24 The average delay time s of impulse response or the risetime of step response depends on the controlled-object given and the controller/compensator used. Therefore it cannot be specified beforehand. It is to be determined within the process of design or model matching. If we put the rise-time equal to s ( s コ s ), we obtain the DEF of desirable control system as W d ( s) Average delay time set to s = s + a s s + a s s + a s s + L 2 3 4

25 25 Specification of desirable control system W d () 1 1+ s + a s s + a s s + a s s + a s s + L s = Zero offset error in step response: W d(0) = 1 Adequate damping: { a } = { L } i 1, 1, 0.5, 0.15, 0.03, 0.003, Quick response speed: s ョ as small as possible positive value

26 step response 26 PID control for forth-order lag 1.0 PID-r PI-r object I-d PI-d I-r PID-d time s+ 2.4s s s 2 3 4

27 step response 27 PID control for object with pure delay 1.0 PID-r PI-r I-r object PID-d I-d PI-d time s e 1 = s s s s s + L

28 28 Why PID? No better control scheme? A new control scheme Introduction of I-PD control scheme

29 29 What compensation should we use? Given the controlled-object and the desirable system as follows: controlled-object : desirable system: 3 y& + a y& + a y& + a y + a y = b u y& + a y& + a y& + a y + a y = b v Solve the compensator form the equation below: v u y& + a y& + a y& + a y+ a y 3 = b u y compensator = v 3 y& + a y& + a y& + a y + a y = b v y

30 30 Rewriting the desirable system into the form of controlled-object as y& + ( a + a - a ) y& + ( a + a - a ) y& + ( a + a - a ) y y& + a y& + a y& + a y+ a y ( a + a - a ) y = b v 踐 a - a a - a a - a a - a b v 2 2 y 1 1 y 0 0 y = 0 - &- & y - - 顏 b b b b u we get the control input u to the object as follows: 3 a - a a - a a - a a - a u= v- y- y& - y& - y b b b b = v- f y- f y& - f y& - f y 3 3

31 31 Structure of compensation v + - u y& + a y& + a y& + a y+ a y 3 = b u y f y+ f y& + f y& + f y ( ) y& + a y& + a y& + a y+ a y = b v- f y- f y& - f y& - f y ( f) ( f) ( ) ( f ) 3 y& + a + b y& + a + b y& + a + b f y+ a + b y = b v 2 Feedback compensation structure is obtained! One-to-one, additive compensation is very easy to adjust each coefficient to any value.

32 32 r + - e Structure of control : I-PD scheme k e dt Type I controller v + - u a y+ a y& + a y& + L + a y 3 = b u f y+ f y& + f y& + L + f y y v& = ke 3 y& + a 2y& + a 1y& + + a 0y + a 000y = b0v e = r - y 2 y& + a y& + a y& + a y& + 3a y y& + b ky= bkr Zero offset in the step response is assured structurally.

33 step response 33 I-PD control for forth-order lag 1.0 I-PD-r I-P-r object I-d I-r I-P-d I-PD-d time s+ 2.4s s s 2 3 4

34 step response 34 I-PD control for object with pure delay 1.0 I-PD-r I-P-r I-r object I-PD-d I-d I-P-d time s e 1 = s s s s s + L

35 35 Is it digital or analog? Sampled-data control Controlled-object is analog, so sampled-data control system as a whole is analog. If the sampling period approaches to zero, the system should become the continuous-time system. (Continuity of sampleddata control and continuous-time control)

36 36 Design of sampled-data control reference input compensator /controller Controlled -object continuous time base controlled output Rs () discrete time base Ys () sampled-data control system continuous time base Both reference input and controlled output can be described in Laplace transform. Thus, control system should be treated in Laplace transform. Ys () Rs () = Ws ()

37 step response Continuity of sampled data-control and continuous-time control (PID) Sampling period T= object Sampled-data control approaches to continuous-time control as sampling period approaches to zero time 10.0 controlled object: s+ 2.4s s s

38 step response 38 Sampled-data I-PD control sampling perio d object I-PD mode control time s+ 2.4s s s 2 3 4

39 39 Some examples designed Tracking servo for ramp input Tracking servo for parabolic input Tracking servo for sinusoidal input PID and I-PD decoupling control Sampled-data decoupling control with two different sampling periods

40 40 Tracking servo for ramp input

41 41 Tracking servo for parabolic input

42 42 Tracking servo for sinusoidal input

43 Example of 2-input-2-output 43 controlled-object u2 u1 R2 R 1 R3 C y2 y1 R1 : R2 : R3 = 1:2:5 鴿鴿 s 鴿鴿 s + s s s s s s + s

44 44 PID decoupling control [4]

45 45 I-PD decoupling control [3]

46 Sampled-data decoupling control with two different sampling periods 46 Sampling periods are set as 2s for 1 st loop and 20s for 2 nd loop. No deterioration is seen. Mori, et al.[8]

47 47 Nonlinear control If the region of control is narrow enough linear control is sufficient. For wider region nonlinearity becomes unable to be neglected. However, there is no definite boundary of the region. Nonlinear control should be smoothly extended from linear control.

48 48 Continuity of nonlinearity and linearity output input Region where linear approximation seems effective. There, however, is no definite boundary of the region.

49 Smooth extension of linear expression to nonlinear 49 Linear static Nonlinear static Linear dynamic Nonlinear dynamic a y + a y& + a y& + L a yy + a yy& + a yy & + a yy& + L a yyy + a yyy& + a yyy & + a yyy & + a yyy& + L L L L = b u + b u& + b u& + L b uu + b uu& + b uu & + b uu& + L b uuu + b uuu& + b uuu & + b uuu & + b uuu& + L L L L Computation of series and parallel connections and inverse is straightforward.

50 Linear and second degree approximation around the working point of reversed field pinch (RFP) fusion power reactor 50 The state equation of reactor has nonlinearity as high as eighth degree. Shimotohno, et al. [9]

51 Interaction from temperature to power output 51 Shimotohno, et al. [9]

52 Decoupling control with I-P and second-degree compensation 52 With nonlinear compensation Shimotohno, et al. [9]

53 53 Control inputs Fueling rate Impurity injection rate With nonlinear compensation Better control with smaller control input by I-P and second degree compensation Shimotohno, et al. [9]

54 54 References [1] Kitamori, T.(1979a): A method of control system design based upon partial knowledge about controlled processes, Trans. Soc. Instrm. & Control Engrs., Japan, 15, [2] Kitamori, T.(1979b): A design method for sampled-data control systems based upon partial knowledge about controlled processes, Trans. Soc. Instrm. & Control Engrs., Japan, 15, [3] Kitamori, T.(1980a): A design method for I-PD type decoupled control systems based upon partial knowledge about controlled processes, Trans. Soc. Instrm. & Control Engrs., Japan, 16, [4] Kitamori, T.(1980b): A design method for PID type decoupled control systems based upon partial knowledge about controlled processes, Trans. Soc. Instrm. & Control Engrs., Japan, 16, [5] Kitamori, T.(1981): A design method for nonlinear control systems design based upon partial knowledge about controlled objects, Preprints of the Eighth IFAC World Congress, Kyoto, Paper no. 17.5, IV25-IV30.

55 55 [6] Sigemasa, T., Y. Tkagi, Y. Ichikawa and (1983): A practical reference model for control system design, Trans. Soc. Instrm. & Control Engrs., Japan, 19, [7] Kitamori, T.(1983): Unification of continuous-time and sampled-data control theories, Journ. Soc. Instrm. & Control Engrs., Japan, 22, [8] Mori, Y., T. Shigemasa and (1984): A design method for sampled-data decoupled control systems with multirate sampling periods, Trans. Soc. Instrm. & Control Engrs., Japan, 20, [9] Shimotohno, H., and S. Kondo (1989): Power control of RFP fusion power reactor, Bull. of the Society of Plasma Science and Nuclear Fusion Research, Sapporo, 151. [10] Kitamori, T.: Smooth extension of control system design algorithm from linear to nonlinear systems, Preprints of 2nd Japan-China Joint Symposium on Systems Control Theory and Its Applications, Osaka, (1990)

56 56 [11] Kitamori, T.: Partial model matching method conformable to physical and engineering actualities, Proceedings of the IFAC Symposium on System Structure and Control, Prague, Czech Republic, 2001, (2001) Followings are not conformable to physical actualities because of the partial model matching not around s=0 but around s= : [12] Emre, E. and L. M. Silverman (1980): Partial model matching of linear systems, IEEE Trans. Automatic Control, AC-25, [13] Martínez García, J. C., M. Malabre and V. Kučera (1995): The partial model matching problem with stability, Systems Control Letters, 24, [14] Kučera, V., J. C. Martínez García and M. Malabre (1997): Partial model matching: Parametrization of solutions, Automatica, 33, No.5,

57 57 THANK YOU FOR YOUR ATTENTION

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