Metodi feedforward per la regolazione ad alte prestazioni

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1 Udine, 4//8 Scuola avanzata Giovanni Zappa su tecniche di controllo e identificazione Metodi feedforward per la regolazione ad alte prestazioni Antonio Visioli Dipartimento di Elettronica per l Automazione Universita di Brescia, Italy antonio.visioli@ing.unibs.it

2 Outline Standard design for a two-degree-of-freedom control system. Standard design for the feedforward action for set-point following task. Design of a nonlinear feedforward control law for the minimization of the output transition time with constraints on the manipulated variable (extension to MIMO case). Design of the feedforward action based on input-output inversion. An iterative strategy for the improvement of performance in the presence of uncertainty. Use of Chebyshev polynomials for the design of a feedforward control law for the minimization of the output transition time with constraints on the manipulated and process variable (extension to MIMO case).

3 Two-degree-of-freedom control system y y F r e C d P y The feedback controller is designed to achieve a high performance in the load disturbance rejection task (high bandwidth) and a low pass filter F is employed to smooth the step set-point signal and therefore to reduce the overshoot and the oscillations in the process variable.

4 P (s) = Example ( s + e 4s C(s) = 3 + ) 8s + 2s.s F(s)= F(s)=/(2s+) F(s)=/(s+)

5 PID control with set-point weighting y y F r e C d P y The use of a set-point weight in PID control can be seen into the same framework. ( u(t) = K p br(t) y(t) + t T i ( C(s) = K p + ) T i s + T ds ) de(t) e(v)dv + T d dt F (s) = + bt is + T i T d s 2 + T i s + T i T d s 2

6 P (s) = Example ( s + e 4s C(s) = 3 + ) 8s + 2s.s b=.5 b=..8 reference signal r.6.4 process variable y b=.5 b=

7 Standard design of feedforward action for set-point following y sp M(s) y f G(s) PID u ff u P(s) y M(s) is a reference model that gives the desired response of a set-point change and G(s) is chosen as G(s) = M(s) P (s). where P (s) is the minimum-phase part of the process transfer function P (s). The nominimum-phase part has to be placed in M(s).

8 Nonlinear feedforward action for set-point following y sp y F(s) y f FF PID u ff u P(s) y P (s) = K T s + e Ls. Based on this model, the output u ff of the feedforward block FF is defined as: u ff (t) = { ūff if t < τ y /K if t τ where the value of ū ff is determined, after trivial calculations, in such a way that the process output y (that is necessarily zero until time t = L) is y at time t = τ + L. It results: ū ff = y /K e τ/t.

9 In this way, if the process is described perfectly by the FOPDT model, an output transition in the time interval [L, τ + L] occurs. Then, at time t = τ + L the output settles at value y thanks to the constant value assumed by u ff (t) for t τ. Then, a suitable reference signal y f has to be applied to the closed-loop system. It is desired that y f be equal to the desired process output that would be obtained in case the process is modelled perfectly. Thus, the step reference signal y sp of amplitude y has to be filtered by the system and then saturated at the level y. F (s) = Kū ff/y T s + e Ls

10 Example P (s) = s + e.5s τ =.5 u ff (t) = { 2.54 if t <.5 if t.5 M (s) =.5s + e.5s G(s) = s +.5s + M 2 (s) =.s + e.5s G(s) = s +.s + process output time [s] feedforward signal time [s]

11 P (s) = Example 2 (s + ) 4e.5s P (s) = M(s) = u ff (t) = 2.2s + e 2.38s τ = 2 {.637 if t < 2 if t 2.4s + e 2.38s G(s) = 2.2s +.4s + y(t) and y f (t) time [s] y(t) and y f (t) time [s] 2 6 u(t) and u ff (t) time [s] u(t) and u ff (t) time [s]

12 P (s) = Example 3 (s + ) 4e.5s P (s) = u ff (t) = 2.2s + e 2.38s τ = {. if t < if t M(s) = 2s + e 2.38s G(s) = 2.2s + 2s + y(t) and y f (t) time [s] y(t) and y f (t) time [s].4.4 u(t) and u ff (t).2.8 u(t) and u ff (t) time [s] time [s]

13 Experimental results First Prev Next Last Go Back Full Screen Close Quit

14 M(s) = 4 P (s) =.98 29s + e s τ = 6 { 2.7 if t < 6 u ff (t) = if t 6.2s + e s G(s) = 29s +.2s + process output [V] time [s] control variable [V] time [s]

15 Industrial application First Prev Next Last Go Back Full Screen Close Quit

16 9 TCV55 process variables [C] and manipulated variable [%] TT97 2 TT time [s]

17 MIMO case c P(s) = [P ij (s)] P ij (s) = K ij T ij s + e L ijs i, j =,..., m Denote as H the closed-loop system transfer matrix from r to y, which can be determined as H(s) = P(s)D(s) (I + P(s)D(s)). Ideally, H(s) should have a diagonal form, namely, H (s) H H(s) = 22 (s) H mm (s)

18 Thus, we obtain the decoupler expression where D(s) = P (s) ( H (s) I ) = W(s)diag W(s) = adj(p(s)) det(p(s)) [ Hii (s) ] H ii (s) m m and adj(p(s)) is the adjoint of the process transfer matrix P(s). We can rewrite W(s) as [ ] W(s) = G ij s ij (s)e L i, j =,..., m () where L ij are suitable scalars so that G ij (s) has at least one polynomial term in the denominator that does not include any time delay. Then, a rational expression for G ij (s) is obtained by using the linear fractional Padé expansion.

19 In particular, when the process has no RHP zeros, if the diagonal elements of the decoupled system matrix are chosen as H ii (s) = e θ is T ii s + i =,..., m where θ i = max{l i,..., L im } i =,..., m then, the resulting elements of the decoupler transfer matrix can be determined analytically as D ji (s) = G ij(s)e (θ i L ij)s T ii s + e θ i s T ii s+ j, i =,..., m. Once the decoupler D has been designed, the feedforward action u ff can be determined by applying the method described previously to the m SISO systems H ii (s), i =,..., m.

20 Multiobjective optimisation The transition times can be minimised subject to the saturation constraints of the actuator, that is, subject to limits on the control variable u. However, by taking into account that in the nominal case, the output u c of the feedback controller is zero, the constraints can be equivalently posed to the feedforward vector signal u ff. This actually yields the construction of a Pareto set of output transition times. Formally, the multiobjective optimisation problem can be posed as follows: subject to min τ τ R m + u m,i u ff i (t) u M,i t i =,..., m where u m,i and u M,i, i =,..., m are, respectively, the minimum and maximum given saturation limits of the actuators. An analytical expression of the feedforward vector signal u ff (t) can be derived in order to solve the optimisation problem.

21 P(s) = Example [ s+e 27s 6s+ e 28s s+e 28s 6s+ e 4s ]. u m = u m2 =.6 and u M = u M2 = τ τ

22 y = [ ].5 y,y time u,u time First Prev Next Last Go Back Full Screen Close Quit

23 y = [ ].8 y,y time u,u time First Prev Next Last Go Back Full Screen Close Quit

24 y = [ ].8 y,y time u,u time First Prev Next Last Go Back Full Screen Close Quit

25 Example First Prev Next Last Go Back Full Screen Close Quit P(s) = [.53(s+).s.2429 e e 2s (4s+) 3 (33s+) (43s+)(22s+) e 2.6s (44s+)(2s+) e.7s [.55 ] s+ e.66s 57.6s+ e 2.7s s+e 7.6s 55.5s+ e.2s P(s) = u m = 7, u m2 =, u M = 7, u M2 = ] τ τ

26 y = [ ].4.2 y,y time 5 u,u time First Prev Next Last Go Back Full Screen Close Quit

27 Inversion-based feedforward control design PID y y Command signal generator r e C d P y An input-output inversion procedure can be applied to determine the command input function to be applied to the closed-loop system in order to obtain a predefined output function. If the predefined output function is chosen as a polynomial function, a closed-loop form expression of the command input function is obtained.

28 Modelling of the plant The process to be controlled is modelled as a FOPDT transfer function: P (s; K, T, L) = K T s + e Ls. Then, in order to have a rational transfer function, the dead-time term is approximated by means of a second order Padé approximation P (s; K, T, L) = K Ls/6 + L 2 s 2 /2 T s + + Ls/6 + L 2 s 2 /2

29 Tuning of the PID controller The PID controller can be tuned according to any of the many methods proposed in the literature or even by hand. However, since the purpose of the dynamic inversion procedure is the attainment of high performances in the setpoint following task, disregarding of the controller gains, it is sensible to select the PID parameters aiming only at obtaining good load rejection performances. ( C(s; K p, T i, T d, T f ) = K p + ) T i s + T ds T f s +

30 Output function design -- A polynomial is chosen to define the (desired) transition from a setpoint value y (= ) to another y (to be performed in the time interval [, τ]). The polynomial coefficients can be uniquely found by solving: y() = ; y(τ) = y y () () = ; y () (τ) =. y (k) () = ; y (k) (τ) = The result can be expressed in closed-form as (t [, τ]): y d (t; τ) = y (2k + )! k!τ 2k+ k i= ( ) k i i!(k i)!(2k i + ) τ i t 2k i+.

31 Output function design -2- The order of the polynomial can be selected by imposing the order of continuity of the command input that results from the dynamic inversion procedure Being the relative degree of the closed-loop transfer function equal to one, a third order polynomial (k = ) ensures a continuous command input function: ( y d (t; τ) = y ). τ 3t3 τ 2t2 The value of τ can be selected by solving an optimization problem where the transition time has to be minimized subject to actuator constraints or by the user either directly or through a (possibly) more intuitive reasoning (e.g. by selecting a ratio between the bandwidth of the open-loop system and that of the closed-loop one) In any case, τ allows to handle the trade-off between performance, robustness and control activity.

32 Stable inversion procedure The command signal r(t; K, T, L, K p, T i, T d, T f, τ) that provides the desired output function is found by applying a stable input-output inversion procedure to the closed-loop transfer function H(s) := C(s) P (s) + C(s) P (s) The solution can be written as r(t) = ξ ẏ d (t) + ξ y d (t) + + η NC (t v)y d (v)dv where η NC (t) := (t)η (t) ( t)η+ (t)

33 Preactuation and postactuation time intervals A preactuation time t s and a postactuation time t f can be selected so that the truncated input r a (t; τ) is equal to zero for t < t s and equal to y /H() for t > t f. In particular, t s and t p can be calculated with arbitrary precision by selecting two arbitrary small parameters ε and ε and by subsequently determining Alternatively, t := max{t R : r(t) ε t (, t ]} { t := min t R : r(t) } H() ε t [t, ). t s = min{, t } t f = max{τ, t }. t s = D rhp t f = τ + D lhp where D rhp and D lhp are the minimum distance of the right and left half plane poles respectively from the imaginary axis of the complex plane.

34 Simulation results -- P (s) = s + e 6s τ = ; PID ZN : K p = 2; T i = 2; T d = 3; PI ZN : K p =.5; T i = 8; T d = ; PID IE : K p = 2.4; T i = 7.33; T d = 2.74;.2.8 command signal PID ZN PI ZN PID IE time [s]

35 New method process output.6.4 control signal PID ZN PI ZN PID IE time [s].4.2 Classic method PID ZN PI ZN PID IE time [s] process output control signal PID ZN PI ZN PID IE PID ZN PI ZN PID IE time [s] time [s]

36 P (s) = Simulation results -2- (s + ) 8 τ = 2 (K =, T = 3.3, L = 4.96) PID ZN : K p =.73; T i = 9.93; T d = 2.48; PI ZN : K p =.55; T i = 4.9; T d = ; PID IE : K p =.6; T i = 4.26; T d = 2.4; command signal PID ZN PI ZN PID IE time [s]

37 New method.8.8 process output.6.4 control signal PID ZN PI ZN PID IE.2 PID ZN PI ZN PID IE time [s] Classic method time [s].2 25 PID ZN PI ZN PID IE 2 process output.8.6 control signal PID ZN PI ZN PID IE time [s] time [s]

38 Trade-off between aggressiveness and robustness (and control activity) P (s) = (s + ) 3 Process output Control variable process output y.8.6 τ control variable τ time time

39 Experimental results P (s) =.98 29s + e s K p =.24; T i = 3; T d = ;

40 step dynamic inversion 2 3 command signal process output [V] time [s] time [s]

41 Practical implementation y y F r e C d P y For a practical implementation of the method (namely, with a Distributed Control System) it is convenient to obtain the command input as a step response. The signal to be applied to the closed-loop system is obtained as a step response of a filter F (s). For this purpose, it is necessary first of all to shift the time axis by substituting t = t t s and by taking t as the new time variable. Then, the expression of the filter F (s) can be obtained by applying the Laplace transform operator to r a ( t; τ): R a (s; τ) = L[r a ( t; τ)] and by imposing that R a (s; τ) = F (s; τ). s

42 Practical implementation y y F r e C d P y For a practical implementation of the method (namely, with a Distributed Control System) it is convenient to obtain the command input as a step response. The signal to be applied to the closed-loop system is obtained as a step response of a filter F (s). For this purpose, it is necessary first of all to shift the time axis by substituting t = t t s and by taking t as the new time variable. Then, the expression of the filter F (s) can be obtained by applying the Laplace transform operator to r a ( t; τ): and by imposing that R a (s; τ) = L[r a ( t; τ)] R a (s; τ) = s F (s; τ) F (s; τ) = sr a(s; τ).

43 By performing the required (symbolic) computations and by substituting backwards t = t + t s, the command signal r a (t; τ) is obtained as the step response of the following filters, to be considered in different time intervals: β 2, s 2 +β, s+β, α 2, s 2 +α, s+α, for t < t s β 7,2 s 7 + +β,2 for t s t < t s + τ F (s; τ) = α 7,2 s 7 + +α 3,2 s 3 β 2,3 s 2 +β,3 s+β,3 α 2,3 s 2 +α,3 s+α,3 H () for t s + τ t < t s + t f for t t s + t f where β 2,,..., β,, β 7,2,..., β,2, β 2,3,..., β,3 and α 2,,..., α,, α 7,2,..., α 3,2, α 2,3,..., α,3 are suitable coefficients. In other words, a step signal has to be applied at t = to the four different filters and then the command input to be applied to the closed-loop system is obtained by selecting the step responses of the filters according to the preactuation and postactuation time intervals.

44 (courtesy of Yokogawa Italia s.r.l.) First Prev Next Last Go Back Full Screen Close Quit

45 Possible further simplification A possible simplification for the implementation of the noncausal feedforward control strategy would be the use of a single filter. In general, the determined command input function can have a complex (non monotonic) shape and it can be difficult to represent it as a step response of a single transfer function. However, there are cases for which this can be possible with a good accuracy. In particular, this happens when the command input function is a smooth function and therefore conditions for the occurrence of this situation have to be sought. For this purpose, it is worth stressing that lim τ + H()r( ; τ) y d( ; τ) =

46 From a practical point of view this means that, when the transition time increases, the input function tends to be more similar to the desired output function. From another point of view, increasing the transition time τ yields a more robust system, namely, the obtained system output tends to be more similar to the desired output function. From these considerations, it can be concluded that when the obtained system output has virtually no overshoot, then the corresponding system input is sufficiently smooth to be approximated as a step response of a single filter. By a large number of experimental results, this occurs if τ > 2L. Thus, in this case the command input can be obtained as a step response of a single filter whose transfer function can be obtained by applying a standard least squares procedure by taking a step signal as input and the determined command signal as output (note that there is no noise). A fourth order transfer function is generally sufficient to obtain a satisfactory result. First Prev Next Last Go Back Full Screen Close Quit

47 Illustrative example P (s) = 4s + e 3s K p =.6 T i = 6 T d =.5 T f =.5 τ = L = command input r process variable y time time

48 Illustrative example 2 P (s) = 4s + e 3s K p =.6 T i = 6 T d =.5 T f =.5 τ = 2L = command input r process variable y time time

49 Inversion-based feedforward control design Iterative strategy y y command signal generator r C(s) P^(s;ρ) ˆP (s; ρ) is an estimated scalar LTI rational transfer function of the (unknown) true system P (s) to be controlled, parameterized by a parameter vector ρ R n ρ. The inversion procedure is applied to the estimated closed-loop transfer function ˆT (s; ρ), but the desired output function is not actually obtained. Thus, an iterative strategy is proposed to find the optimal parameter vector defined by ρ := arg min ρ u J(ρ) where J(ρ) is the following integral criterion: J(ρ) = 2T f where T f is the control interval. Tf (y(t; ρ) y d (t)) 2 dt y First Prev Next Last Go Back Full Screen Close Quit

50 Methodology The posed optimization problem can be solved by applying a gradientbased minimization, namely, the following expression can be used after the ith iteration: ρ(i + ) = ρ(i) γ i Ri (i) dj dρ (ρ(i)) where γ i is a positive real scalar that determines the step size and R i is some appropriate positive definite matrix. The cost function gradient can be computed as where dj dρ (ρ(i)) = T f Tf e(t; ρ(i)) e(t; ρ) = y(t; ρ) y d (t). de(t; ρ(i)) dt dρ It can be noted that e(t; ρ(i)) is immediately available after an experiment has been performed.

51 In order to determine an expression of the first derivative of the error signal, it is convenient to consider the Laplace transform of the functions considered: E(s; ρ) = Y (s; ρ) Y d (s) = T (s)r(s; ρ) Y d (s). By considering that R(s; ρ) = ˆT (s; ρ)y d (s) = [ + C(s) ˆP ] (s; ρ) C(s) ˆP Y d (s) (s; ρ) we obtain: de(s; ρ) dρ d ˆP (s;ρ) dρ = T (s)y d (s) C(s) ˆP 2 (s; ρ). By multiplying both the numerator and the denominator by + C(s) ˆP (s; ρ), we have de(s;ρ) dρ = d ˆP (s;ρ) dρ +C(s) ˆP ˆP (s; ρ)t (s) ˆT (s; ρ)y (s;ρ) d (s) = d ˆP (s;ρ) dρ +C(s) ˆP (s;ρ) ˆP (s; ρ)y (s; ρ)

52 Thus, by considering the corresponding time-domain signals, the term de(t; ρ)/dρ can be computed in principle by determining the response of the system d ˆP (s; ρ) dρ + C(s) ˆP ˆP (s; ρ) (s; ρ) to y(t; ρ). However, this is not possible if ˆP (s; ρ) is nonminimum-phase, since ˆP (s; ρ) is unstable. Hence, in general, the gradient of the error signal has to be calculated by first determining the signal û(t; ρ) which, applied to the system ˆP (s; ρ), causes the collected system output y(t; ρ) (i.e., by applying the stable input-output procedure to the system ˆP (s; ρ) with a desired system output y(t; ρ)) and then by determining the response of the system to û(t; ρ). d ˆP (s; ρ) dρ + C(s) ˆP (s; ρ)

53 INFT Algorithm. Determine r(t; ρ) by applying a stable inversion procedure to the closed-loop system ˆT (s; ρ) with output function y d (t). 2. Run a closed-loop system experiment with command input r(t; ρ). 3. Record the output y(t; ρ) and determine e(t; ρ) = y(t; ρ) y d (t). 4. Determine (by applying an inversion procedure) û(t; ρ) that, applied to the system ˆP (s; ρ), causes the output signal y(t; ρ). 5. Determine de(t; ρ)/dρ as the response of the system to the signal û(t; ρ). d ˆP (s; ρ) dρ 6. Calculate the cost function gradient. + C(s) ˆP (s; ρ) 7. If dj(ρ(i))/dρ > ε then update the parameter vector ρ and go to, else terminate.

54 Simulation results -- P (s) = s + (2s + )(3s + ) ˆP (s) = zs + (T s + )(T 2 s + ) ρ() = [z, T, T 2 ] = [.5, 4, ] ( C(s) = K p + ) T i s + T ds T f s + K p =.56; T i = 5.3; T d =.26; T f =. y d (t) := for t < 2 t3 + 3 τ 3 t2 for t τ τ = 2 τ 2 for t > τ = ρ() = [.8, 3.92,.45]

55 .8.8 system output.6 system output time time x command input.8 J(ρ) time iteration

56 Simulation results -2- P (s) = K ˆP (s) = (s + ) 3 T s + e Ls ρ() = [K, T, L] = [.86, 2.2, 2] ( C(s) = K p + ) T i s + T ds T f s + K p = ; T i = 4.37; T d =.9; T f =. y d (t) := for t < 2 t3 + 3 τ 3 t2 for t τ τ = 3 τ 2 for t > τ = ρ() = [., 2.29, 2.2]

57 system output.6 system output time time.6 3 x command input.8.6 J(ρ) time iteration

58 ρ() = [K, T, L] = [.86, 4., 2.2] = ρ(3) = [.97,.9, 2.27].5.8 system output system output time time command input J(ρ) time iteration

59 Experimental results First Prev Next Last Go Back Full Screen Close Quit

60 Command input Error command signal [V].6.4 error time [s] 2.2 First iteration iteration 2.2 Tenth iteration system output [V].6.4 system output [V] time [s] time [s]

61 Chebyshev optimisation for constrained feedforward regulation command signal generator r u C(s) P(s) y y s y P (s) = f K T s + e Ls = C(s) = K p ( + T i s + T ds K L 2 s + T s + L 2 s +, ) T f s + An approach based on Chebyshev polynomials can be employed to determine the signal r(t) to be applied when a transition from an equilibrium point corresponding to a process output value y s (= ) to another equilibrium point corresponding to a process output value y f is required. In particular, a minimum-time (rest-to-rest) transition is required subject to given limits on the control variable and on the process variable.

62 It is convenient to write a minimal state-space realization of the closedloop system: dx(t) = Ax(t) + br(t) dt y(t) = cx(t) where T 2K K A = f LT T b = A 4, A 4,2 A 4,3 A 4,4 The PID controller output is expressed as u(t) = K p T i T f x (t) + 2KK pt d T LT f ( K p T f K p T d T f T dk p T 2 f x 3 (t) + T dk p K T f T c = [ 2K LT ) x 2 (t) x 4(t) + K pt d T r(t) f K T ]

63 The considered time-optimal open-loop constrained control problem can be expressed as follows: min r(t) t f subject to: dx(t) dt = Ax(t) + br(t) t t f x() = x x(t f ) = x f u min u(t) u max t t f y min y(t) y max t t f where u min, u max and y min, y max are evidently the constraints for the control variable and the process variable respectively and x and x f denote the equilibrium state corresponding to the process output y s = and to the process output y f, respectively. This time-optimal control problem has a solution if {, y } f (u min, u max ) and {, y f } (y min, y max ). K

64 Chebyshev polynomials The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted T i (τ). They are normalised such that T i () =, i =,,... and in their trigonometric form they are expressed as: T i (τ) = cos(i arccos(τ)) τ [, ]. They can be also defined by the recurrence relation: T (τ) = T (τ) = τ T i+ (τ) = 2τT i (τ) T i (τ) i >.

65 T i (τ) τ

66 Transformation of the time domain In order to use the Chebyshev polynomials of the first kind for the approximation of the system dynamics, the following time transformation is therefore necessary: t = t f ( + τ). 2 This transformation allows a change from the time domain t [, t f ] to the Chebyshev domain τ [, ]. The new system dynamics expressed in the Chebyshev domain is therefore: dx(τ) dτ = t f 2 with initial and final conditions: (Ax(τ) + Br(τ)) τ x( ) = x =, x() = x f.

67 Approximation through Chebyshev series Once the system dynamics has been rewritten in the Chebyshev domain, the next step is the expansion of both the state vector x and the input vector r through Chebyshev series of order h: x h (τ) = 2 α T (τ) + h α i T i (τ) i= r h (τ) = 2 β T (τ) + h β i T i (τ) where τ [, ] and ᾱ := [α, α,..., α h ] (with α i = [α i, α i2,..., α in ] T, i =,..., h) and β := (β, β,..., β h ) are the unknown coefficients. i= First Prev Next Last Go Back Full Screen Close Quit

68 Equality constraints The approximation of the state and the input variables x h and r h is then substituted into the state-space equation: dx h (τ) dτ = t f 2 (Ax h(τ) + Br h (τ)) τ. as well as into the initial and final condition equations. Note that the derivative of the series x h (τ) with respect to τ is given by where Thus, we obtain dx h (τ) dτ = 2 α + h i= α it i (τ) α r α r+ 2rα r =, r =,..., h. t f 2 (Ax h(τ) + Br h (τ)) = 2 α + h i= α it i (τ)

69 Equality constraints (cont d) By equating the coefficients of same-order Chebyshev polynomials we obtain a system of n h nonlinear equality constraints (note that the final time t f is unknown). The substitution of x h into the initial and final condition expression yields 2 n additional equality constraints, namely, and 2 α + h ( ) i α i x( ) = i= 2 α + h α i x() =. i=

70 Inequality constraints The inequality constraints can be handled by rewriting them as y min cx h (τ) cx h (τ) y max u min u h (τ) u h (τ) u max and by defining four vectors of slack variables w i (τ), i =,..., 4: y min cx h (τ) = w 2 (τ) cx h (τ) y max = w 2 2(τ) u min u h (τ) = w 2 3(τ) u h (τ) u max = w 2 4(τ) Each slack variable can be expanded in a Chebyshev series with unknown coefficients γ and by equating again the coefficients of same-order Chebyshev polynomials, a set of (nonlinear) equality constraint relations in ᾱ, β and γ results..

71 Optimisation The optimal control problem is therefore transformed into a parameter optimisation problem which consists in finding t f, ᾱ and β in order to minimise the transition time t f subject to the posed equality and inequality constraints. To solve this optimisation problem, a sequential quadratic programming (SQP) method, such as the one implemented in the function fmincon of Matlab can be used. In this context the starting values of the parameters t f, ᾱ and β, denoted respectively as t f, ᾱ and β can be selected through the Chebyshev interpolation of the state variables evolution of the system when a step signal (whose amplitude is that required at the final equilibrium point) is applied to each system input.

72 Practical issues The choice of the limits for the control variable can be made by considering the actuator saturation limits. Actually, it might be worth to selecting the value of u min slightly above the minimum saturation limit of the actuator and the value of u max slightly below the maximum saturation limit of the actuator in order for the actuator to be able to compensate for the unavoidable modelling uncertainties. Since the solution of the time-optimal control problem for a linear system with input and output constraints represents a generalized bangbang, there are time intervals in the transient response where the process output saturates at y min and time intervals where the process output saturates at y max. Based on these considerations, it is convenient to select y min = y s.y s and y max = y f +.y f in order to obtain an almost monotonic response (actually, an undershoot and an overshoot of one percent is considered to be negligible).

73 Simulation results P (s) = s + e 3s K p =.2, T i = 8, T d = 2, T f =. u min =, u max =.5, y min =., y max =. t f = 49.2 command input process variable time [s] control variable time [s] time [s]

74 P 2 (s) = (s + ) 8 P 2 (s) = 3s + e 5s K p =.7, T i =, T d =., T f =. u min =, u max = 2, y min =., y max =. t f = process variable command input time [s] control variable time [s] time [s]

75 Experimental results P (s) =.75 3s + e.5s

76 K p =.75, T i = 2, T d =., T f =. u min =, u max = 4.8, y min =.99, y max = 3. t f = 65.3 command input [V] time [s] process variable [V] control variable [V] time [s] time [s]

77 K p = 3, T i =, T d =.5, T f =. u min =, u max = 4.8, y min =.99, y max = 3. t f = 42. command input [V] time [s] process variable [V] control variable time [s] time [s]

78 MIMO case The methodology can be extended easily to the MIMO case. In this case we can pose suitable constraints on the process outputs in order to achieve a decoupling of the system. The optimisation problem is more complex but satisfactory results are achieved in any case.

79 P(s) = Simulation results s + e.2s.64.87s + e.4s s + e.2s 5.8.8s + e.4s u = u 2 =.3 and u+ = u + 2 = r, r time

80 y f = [ ] T.5 y, y time y, y time u, u 2. u, u time time

81 y f = [ ] T.8.6 r, r time y, y 2.5 y, y time time..5 u, u 2.5 u, u time time

82 y f = [ ] T.2.8 r, r time y, y 2.5 y, y time time u, u time u, u time

83 Conclusions There are many different methodologies for the design of a feedforward action for set-point following task. It has to be taken into account that the aim of the feedback is to reduce the effect of external disturbances and of the uncertainties, whereas a high performance can be achieved with feedforward control. Feedforward control can be employed effectively also for load disturbance rejection (provided the disturbance is measured).

84 References A. Visioli, A new design for a PID plus feedforward controller, Journal of Process Control, Vol. 4, No. 4, pp , 24. A. Piazzi, A. Visioli, A noncausal approach for PID Control, Journal of Process Control, Vol. 6, No. 8, pp , 26. A. Visioli, Practical PID Control, Springer, 26. A. Piazzi, A. Visioli, An iterative approach for noncausal feedforward tuning, American Control Conference, New York (NY), pp , July 27. S. Piccagli, A. Visioli, Using a Chebyshev technique for solving the generalized bang-bang control problem, IEEE International Conference on Decision and Control, New Orleans (LA), pp , December 27. S. Piccagli, A. Visioli, Minimum-time feedforward plus PID control using a Chebyshev technique, IEEE International Conference on Decision and Control, New Orleans (LA), pp , December 27. S. Piccagli, A. Visioli, Minimum-time feedforward plus PID control for MIMO systems, 7th IFAC World Congress, Seul (ROK), July 28. A. Piazzi, A. Visioli, Iterative feedforward tuning for residual vibration reduction, 7th IFAC World Congress, Seul (ROK), July 28. S. Piccagli, A. Visioli, Using a Chebyshev approach for the minimum-time open-loop control of constrained MIMO systems, International Conference on Control 28 (UKACC), Manchester (UK), September 28. S. Piccagli, A. Visioli, An optimal feedforward control design for the set-point following of MIMO processes, submitted to Journal of Process Control. First Prev Next Last Go Back Full Screen Close Quit