FUNNEL CONTROL WITH ONLINEFORESIGHT

FUNNEL CONTROL WITH ONLINEFORESIGHT C.M. Hackl and Dierk Schröder Institute for Electrical Drive Systems Technical University of Munich Arcisstr., D-8333 Munich, Germany email: christoph.hackl@tum.de ABSTRACT This paper presents two approaches to optimize a recent non identifier based time-varying adaptive control strategy Funnel Control. The gain is adapted by weighting the distance between the actual control error and a predefined (decreasing) function the Funnel boundary representing thedesiredlimitoftheerrorevolution. Measurementnoise is admissible. Within this publication the control strategy is improved twice: a) By defining a scaling function only increasing the gain for more accurate and faster control performance b) By introducing an analytic, a numeric and a derivative implementation of the minimal (future) distance an online foresight for the control task, which guarantees accelerated control at efcient use of the control input. KEY WORDS Adaptive Control, Robust Control, Nonlinear Systems Introduction In a wide area of practical control tasks in industry, the control engineer has only rough knowledge of the plant - often only the plant structure is obvious. Also due to parameter uncertainties of the real plant and without time consuming system identification, linear control strategies reach easily their limits. A simple nonidentifier based proportional adaptive control concept can be used to bypass all difficulties of identification/estimation and linear control design. This control strategy - Funnel Control - is applicable to a class of systems S with a known relative degree, a minimum-phase property and a known high-frequency gain [4]. The Funnel controller is designed to cope with all plants of class S. Measurement noise and parameter uncertainties are tolerated. Funnel-Control is adapting its time-varying gain only by the measured control error. Finally, applying Funnel Control the control engineer is not only able to guarantee stability and asymptotic tracking within the boundary, he is also able to predefine the transient behavior of the plant corresponding to the wishes of a customer by a limiting function of time the Funnel boundary, which restricts the control error to evolve inside the Funnel, if the control input is sufficiently dimensioned [4]. If all states are measurable or observable, the restrictive condition for the relative degree may be relaxed by a state feedback like structure reducing the relative degree to one [7]. The robustness of Funnel Control has been shown in [] for industrial standard drive systems with uncertain parameters (nonlinear multi-mass flexible systems). Accurate asymptotic tracking with vanishing error can be assured by implementing an integrating prefilter [3] or auxiliary nonlinear PI-controller like extensions []. As most industrial applications additionally require an optimum in settling time, in this publication Funnel Control itself is optimized by the following concepts: a) Additional condition for the scaling function to ensure only gain increase b) Analytic, numeric and derivative implementation of the online foresight to accelerate the transient response the future (minimal) distance. Funnel Control with Vertical Distance Funnel Control, developed by Ilchmann et al. [4], is a nonidentifier based control strategy which originates from high-gain concepts and employs a time-varying proportional gain α ( ) to control and stabilize all systems of class S. The class S comprises infinite dimensional, nonlinear (and linear) M input u (t ), M output y (t ) systems described by the functional differential equation ẏ (t ) = f ( p (t ), w (t ), u (t )) () inheriting following properties : relative degree of r = as ẏ = f (,, u ) u, minimum phase (stable zerodynamics, w < ) and known high-frequency gain vector ρ = sign(f (,, u )), describing the influence of the control input u on the first derivative ẏ of the output. The System S is driven by the Funnel Controller (see Fig. ), which calculates the control input u (t ) = α ( F ϕ (t ), Ψ(t ), e(t ) ) e(t ) () by evaluating the vertical distance (VD) d V (t ) = F ϕ (t ) e(t ) (3) between the Funnel boundary F ϕ (t ) and the norm 3 of the error e(t ) = y (t) y (t) n(t ) (4) For more explicit information on the system class, the reader is referred to [4], where the subsystems and the Operator are well defined and examples are given for possible systems of class. For exact definitions of these properties refer to [6]. 3 In the SISO case, the absolute value of the error e( t) is sufficient. 55-3 7

y e u α ( F ϕ,ψ, ) Funnel Controller y + n w System of Class S : ẏ = f(p,w,u) : w = Ty Figure. Block Diagram: Funnel Control Loop + F ϕ (t) e() F ϕ (t) d V t t F F ϕ (t F ) d F y + + n Figure. Idea of Funnel-Control with Funnel boundary F ϕ (t), Error Evolution, Vertical d V (t) and Minimal Distance d F (t) (with the reference signal y (t), the output y(t) and bounded measurement noise n(t)) and multiplying with a bounded scaling factor Ψ(t) > for all t. The scaling factor Ψ(t ) allows to e.g. define a minimal control gain. The intra-system variables represent a bounded disturbance p(t) and with w(t) = T(y) a local Lipschitz and causal operator function [4]. The Funnel boundary F ϕ (t) = ϕ(t) is given by the reciprocal of an arbitrarily chosen bounded, continuous and positive function ϕ(t) > for all t with max t ϕ(t) < [4]. Thus the Funnel region itself is defined as the set (5) F ϕ : t { R M ϕ(t) < } (6) which encloses the error for all t. The initial error e(t ) < ϕ(t := ϕ ) must be surrounded by the Funnel (see Fig. ). The overall gain α( ) is adapted in the following manner α ( F ϕ (t),ψ(t), ) = Ψ(t) α D (t) (7) only depending on the scaling function Ψ(t) and the (here: vertical) distance gain α D (t) = dist(, F = ϕ(t)) d V (t) to ensure that the error evolves inside the Funnel F ϕ. Thus the gain α( ) increases more aggressive control, if the error draws close to the boundary F ϕ and decreases more relaxed control, if the error becomes (nearly) zero. In [4] it is assured that both the gain α( ) > and the error stay bounded for all t, if the control input u (t ) is dimensioned sufficiently large but with finite values. Measurement noise n(t) is admissible for any Funnel boundary enclosing its maximal amplitude for (almost) all time. Obviously, the control engineer is able to predefine the Funnel boundary to fit customer wishes with some δ = F ϕ (τ δ ) > for all t τ δ and the desired asymptotic accuracy limit µ = lim t F ϕ (t) > lim t. The exponentially decreasing Funnel boundary F exp (t) is the most appropriate choice for linear systems with exponential transient behavior. The exponential Funnel boundary is given by ( F exp (t) = ϕ exp, exp t T exp ) + ϕ exp, (8) with the initial boundary value F exp (t ) = ϕ exp, + ϕ exp, > e(t ), the predefinable time constant T exp > and the asymptotic tracking accuracy limit lim t F exp (t) = ϕ exp, >. 3 Scaling Function The scaling function Ψ(t) for the distance gain α D (t) is mainly useful to enhance the transient behavior of Funnel Controlled systems with vertical distance d V (t), as e.g. the initial gain α(t ) can be chosen arbitrarily [4]. In [4] the gain was rescaled by the Funnel boundary itself Ψ(t) F ϕ (t) = ϕ(t), which improves the performance drastically, as the gain α(t) t t is bounded away from zero. Besides the positive acceleration effect and the increased accuracy at proportional plants, a disadvantage exists: If the gain is scaled by e.g. the Funnel boundary itself with an asymptotic accuracy limit < µ < (which is desired for almost all industrial applications), the scaling function reduces the overall gain α(t) = Ψ(t) α D (t)! < α D (t) for all t > τ {t F ϕ (t) < }. To remain the advantages of the distance gain and the scaling acceleration, the condition is suggested: Proposition 3.. Scaling Function Ψ(t) For any scaling function Ψ(t) t R + the overall (scaled) gain is α(t) α D (t) t R +. Proof: For any given Ψ(t) and α D (t) >, Prop. 3. follows from Eq. (7). When scaling with the Funnel boundary itself, the factor must be adjusted to Ψ(t) = F ϕ (t) + ( µ), with lim t Ψ(t) = lim t F ϕ (t) + ( µ) =. This additional condition for the choice of the scaling function is permissible, as Ψ(t) still is continuous, bounded, positive for all t and thus conform to the postulations (similar to those for ϕ(t), see Sec. ) in [5]. 7

4 Minimal Future Distance (MD) The vertical distance ensures, that the error evolution stays inside the Funnel region. Although it is obvious that for any (monotone) decreasing limiting function (see Fig. ), there exists a minimal future distance d F (t,t F ) between the control error at the actual time t and some future point t F t on the Funnel boundary F ϕ (t F ), which is smaller than (or equal to) the vertical distance d V (t) [5]. Theorem 4.. Minimal Distance (MD) d F (t,t F ) For any continuous, non-increasing limiting function F ϕ (t) > for all t t, there exists a minimal (future) distance d F (t,t F ) =. min ( F ϕ (t F ) ) + (t F t) t F t (9) with the future time t F t t and the property d F (t,t F ) d H d V (t) t,t F R + () The future time t F t + d H is delimited by the horizon d H = min t H>t F (d V (t),(t H t)) > () with the horizontal time t H min τ>t {τ F ϕ (τ) τ > d V (t) + t}. Proof: The shortest distance between two points and F ϕ (T) is a straight line with length R. For any point F ϕ (T) F ϕ (t) on a continuous function with F ϕ, t,t t, such a line exists with length R(t,T) ( F ϕ (T) ) + (T t). Drawing a circle B R ( ) around with radius R > and increasing R until the circular line B R ( ) firstly contacts { the boundary F ϕ ( ˆt F ), we fix ˆt F = min τ t τ BR ( ) = F ϕ (τ) } and the distance R(t, ˆt F ) R min. Obviously, R min is bounded above with R (t, ˆt F ) d V (t) = F ϕ (t) d H, therefore t ˆt F < t + d H. For all lim T F ϕ (T) = µ >, there exists ˆt H = min τ>t {τ F ϕ (τ) = }, now assume d V + t ˆt F > ˆt H, but R(t, ˆt F ) = ( F ϕ (ˆt F ) ) + (ˆt F t) > R(t, ˆt H ) = ˆt H t d H. Therefore at least ˆt F ˆt H = t + d H and the minimal distance exists with d F (ˆt F ) min(d H,d H ). Corollary 4.. For any scaling function Ψ(t), the maximal Funnel gain is definedby α F (t) = Ψ(t) d F (t,t F ). Proof: With (9) the minimal distance is fixed, as α F (t) d F (t,t F ) the maximum is found. For all F ϕ (t ) < and according to Theorem 4. and Corollary 4. especially the initial control input u F (t ) = Ψ(t ) d F (t,t F ) } {{ } α F (t) e(t ) u V (t ) = Ψ(t ) d V (t ) }{{} α V (t) e(t ) () + F G (t) e() F G (t) d F F G (t F ) t t F τ µ Figure 3. Analytical Method for Calculating the MD forces the controlled system more rapidly towards the reference signal y. Despite a more complex calculation of the control gain α F (t), Funnel-Control becomes much more effective. The transient behavior is generally accelerated and the control input is more efficiently used (see e.g. Sec. 4.). The difficulty hides in determining the optimal time t F t to calculate (or approximate) the minimal distance d F (t,t F ). In the following Subsections an analytical, a numerical and a derivative method to find the minimal (future) distance (MD) are introduced. 4. Analytic Approach A simple limiting function allows to calculate the future distance instantaneously. The chosen exemplary Funnel boundary in Fig. 3 is continuous and nonincreasing, it is defined by { ϕ ϕ F G (t) = τ µ t + ϕ,t < τ µ (3) ϕ,t τ µ! with the initial boundary value ϕ > e(t ) >, the asymptotic accuracy limit µ ϕ > and the cut-off time τ µ > after which the boundary consists only of a (horizontal) tube with radius ϕ. Here the minimal (future) distance is easily identified as the perpendicular to the slope of the Funnel boundary F G (t F ) for all (future) times t F. We derive the condition F G (t F ) t F t d dt F G(t) substituting (3) and solving for t F = t+ ϕ ϕ (ϕ τµ ) t=tf! = (4),t +( ϕ ϕ F t τ µ ) τµ τ µ,t F τ µ t τ µ t,t > τ µ (5) in consideration of the only piecewise differentiable function F G (t), one can calculate the minimal (future) distance analytically with d F (t,t F ) = ( F G (t F ) ) + (t F t) (6) 73

+ F ϕ (t) e() t...ˆt F d F t H + F ϕ (t) e() d V t ˆt F ˆd F ˆF ϕ (ˆt F ) ˆF ϕ (T) F ϕ (t) F ϕ (t) Figure 4. Numerical Method for Approximating the MD Figure 5. Derivative Method for Approximating the MD For all t > τ µ as t F = t, the minimal (future) distance d F (t,t F ) merges with the vertical distance d V (t). The tremendous advantage in using the analytical method is the fast and correct calculation of the minimal distance, as no numerical iterations are needed. Otherwise condition (4) apparently becomes more complex and may even be not analytically solvable for all t F. As Funnel Control also with more general boundaries should benefit from this acceleration, in the next section an iterative numerical approach to approximate the minimal distance is investigated. 4. Numeric Approach For more universal limiting functions e.g. the Exponential (8) Funnel Boundary, condition (4) cannot be solved analytically in general. Therefore the minimal distance d F (t,t F ) must be approximated by approaching t F numerically by t = N t (d H ). In this paper we present an iterative method deploying a simple algorithm to find the optimal foresight, preserving the real-time condition by choosing a maximal N of allowed iteration steps and additionally adjusting N t = dh N online, depending on the minimal horizon d H = H min denedin Eq. (). After initializing the necessary parameters, the actual time t and error, minimal and maximal horizon H m in, H m ax are fixed according to the vertical distance d V (t) and the horizontal time { min τ>t {τ F ϕ (τ) } µ t H = d V + t + ϕ exp, exp( t/t exp ) < µ (7) The time increment N t is properly set and the iterative calculation of the minimal distance is started for i < N with d F,i (t,i N t ) = ( F ϕ (t + i N t ) ) + (i N t ) (8) As the numerical minimum distance d F (t, ˆt F ) exists for t ˆt F = t + i N t d H (see Theorem 4.), as soon as the inequality d F,i (t,i N t ) < d F,i+ (t,(i + ) N t ) (9) is satisfied, the iteration loop is left and the numerical minimal distance d F (t, ˆt F ) = d F,i (t,i N t ) is returned, thus the algorithm assures, that d F (t,t F ) d F (t, ˆt F ) d V (t). The maximal iteration steps N should be chosen as small as possible to preserve the real time applicability. For N = the algorithm coincides with the calculation of the vertical distance d V (t) = d F, (t, N t ). The simulative analysis has shown, that N = shows good precision at acceptable computational time for sampling times h ms. 4.3 Derivative Approach To absolutely avoid an eventual violation of the real time condition, it is desirable to find a method to approximate the minimal future distance without any iterations. In his subsection we present an analytical approximation ˆd F (t, ˆt F ) of the minimal distance. The idea is illustrated in Fig. 5 and is simple: substituting the boundary by its tangent ˆF ϕ (T) for all T t and calculating the perpendicalur distance between and ˆF ϕ (ˆt F ) at some future point ˆt F. The new properties are combined in Theorem 4.3. Approximated MD ˆd F (t, ˆt F ) For any differentiable, monotone decreasing function F ϕ (t) with monotone increasing derivative F ϕ (t) < for all t [, ) and lim t F ϕ (t) =, the approximated minimal distance ˆd F (t, ˆt F ) = ( ˆF ϕ (ˆt F ) ) + (ˆt F t) () exists with the linear approximation ˆF ϕ (T) = F ϕ (t) (T t) + F ϕ (t) T t () and the approximated future time ˆt F = t F ϕ (t) ( d V (t) F ϕ (t) t + F ϕ(t) delimited by t ˆt F t + d H and has the property ) () ˆd F (t, ˆt F ) d F (t,t F ) t, ˆt F R + (3) Proof: As ˆF ϕ (t) is differentiable, its tangent representation ˆF ϕ (t) exists, which can be extended to a straight line 74

ˆF ϕ (T) for T t. The perpendicular onto this approximated boundary calculation with condition (4) at some ˆt F t crossing represents the minimal distance R ˆd F (t, ˆt F ) in the R -plane. For any ˆF ϕ (T) with T t with F ϕ (t ) = ˆFϕ (T) < F ϕ (t ) for all t > t = t, ˆF ϕ (t ) F ϕ (t ) holds true. Finally, the relation d F (t,t ) = ( F ϕ (t ) ) + (t t) ( ˆF ϕ (t ) ) + (t t) R for all t t shows ˆd F (t, ˆt F ) d F (t,t F ) and therefore t ˆt F t + d H equal to the horizon (). Corollary 4.4. For any scaling function Ψ(t ), the approximated maximal Funnel gain is defined by α ˆF(t) = Ψ(t) ˆd F (t,ˆt F ). Proof: With () the approximated minimal distance is fixed, as αˆf(t) the approximated maximum is ˆd F (t,ˆt F ) found. Recapitulatory, this approximated minimal distance ˆd F (t, ˆt F ) exhibits the most aggressive gain α ˆF(t) α F (t) α V (t). 5 Simulation Results All MATLAB/SIMULINK simulations are performed with a sampling time h = ms. To focus only on the results of this publication, the implementation of an integrating prefilter [3] or an auxiliary (nonlinear) PI-controller like extension [] is skipped, although those guarantee an asymptotic vanishing error lim t =. Thus the error here will only remain inside the Funnel with lim t. < µ. The customer wishes are µ =., δ =.8 at τ δ =.3s and ϕ = 5. For intuitive understanding, a simple linear system with its transfer function is chosen: a PT system F PT (s) = (4) + s The Funnel boundaries F A (t) and F exp (t) are designed accordingly to the given values. The maximal iteration number is set to N =. The reference is y = for some initial error e(t ) =. For all Funnel controllers generating its distance gain α D (t) with the vertical distance d V (t), the overall gain α(t) α V (t) is scaled either by Ψ(t) = (Non-Scaled Mode) or by Ψ(t) = F exp (t) + ( µ) or Ψ(t) = F A (t) + ( µ) (Scaled Mode), respectively. Whereas for the minimal (future) distance d F (t,t F ) (MD), the overall gain α(t) α F (t) is not scaled, therefore Ψ(t) = is set. The exhibited positive effect of the minimal distance is more obvious and clear. 5. Analytical Method The simulations are performed for the PT system (4) controlled by three Funnel controllers with the same predefined limiting function F A (t), using the non-scaled and System Output y(t), Reference Signal y * (t) Control Output u(t) Funnel Boundary F(t), Error Figure 6..8.6.4 y(t) with VD (Not Scaled). y(t) with VD (Scaled) y(t) with MD (Analytical Method) y * (t).5.5 5 5 u(t) with VD (Not Scaled) u(t) with VD (Scaled) u(t) with MD (Analytical Method).5.5.5.5.5.5 with VD (Not Scaled) with VD (Scaled) with MD (Analytical Method) ± F A (t).5.5 Comparison between Non-Scaled, Scaled VD and MD Analytical Method. scaled vertical distance (VD) and the analytically determined minimal distance (MD), utilizing Eq. (6). The Funnel boundary F A (t) is given by Eq. (3). In Fig. 6, the system output y(t), the control input u(t) and the error evolution within the Funnel F A (t) are shown. Easily the increased acceleration of the transient behavior is noticed. Without any scaling Ψ(t) = the system response y(t) is slow, using the scaling function Ψ(t) = F A (t) + ( µ) the response is sped up. Finally the MD drives the system output y(t) towards the desired value y = within the shortest period of time. Besides the control inputs u(t) for the VD gains exhibits tremendous peaks in contrast to the more effective use of u(t) for the MD. Here the control input u(t) has its maximum for the initial error e(). 75

System Output y(t), Reference Signal y * (t) Control Output u(t) Funnel Boundary F(t), Error.8.6.4 y(t) with VD (Not Scaled) y(t) with VD (Scaled). y(t) with MD (Iterative Method) y(t) with MD (Derivative Method) y * (t).5.5 9 8 7 6 5 4 3 u(t) with VD (Not Scaled) u(t) with VD (Scaled) u(t) with MD (Iterative Method) u(t) with MD (Derivative Method).5.5.5.5.5.5 with VD (Not Scaled) with VD (Scaled) with MD (Iterative Method) with MD (Derivative Method) ± F exp (t).5.5 Figure 7. Comparison between Non-Scaled, Scaled VD and MD Numerical and Derivative Method. 5. Numerical and Derivative Method The four simulated control loops consist of the PT system (4) and four Funnel controllers with the same predefined Funnel boundary F exp (t), using the non-scaled and scaled vertical distance d V (t) (VD), the numerically determined d F (t, ˆt F ) and approximated ˆd F (t, ˆt F ) minimal distance (MD). The Funnel boundary F exp (t) is given by Eq. (8). The system output y(t), the control input u(t) and the error evolution within the Funnel F exp (t) are depicted in Fig. 7. Again a steady increase of the positive acceleration effect is apparent. The slowest performance shows Funnel Control with non-scaled VD, a slight improved behavior is generated by Funnel Control with scaled VD. The best performances are obtained for the Funnel Controllers with MD. The approximated MD exhibits the fastest transient response. The control input u(t) is most effectively used for the Funnel controllers with MD. At initialization t = the largest control u() most rapidly forces the system output y(t) towards the reference y =. 6 Conclusion In this publication we presented four methods to reduce the settling time of robust adaptive Funnel Controlled systems by using.) a scaling function, which increases the control gain, and three implementations of the online foresight the minimal (future) distance:.) the analytic, 3.) numeric and 4.) derivative approach. Fastest transient behavior exhibit those controllers using the minimal distance, where the approximated minimal distance (derivative approach) shows the best performance. References [] Christoph Hackl and Dierk Schröder. Extension of high-gain controllable systems for improved accuracy. Joint CCA, ISIC and CACSD Proceedings 6, October 4-6, 6, Munich, Germany (CDROM), pages 3 36, 6. [] Christoph Hackl and Dierk Schröder. Funnel-control for nonlinear multi-mass flexible systems. IECON, Paris (to be published in the conference proceedings), 6. [3] C.M. Hackl, H. Schuster, C. Westermaier, and D. Schröder. Funnel-control with integrating prefilter for nonlinear, time-varying two-mass flexible servo systems. The 9th International Workshop on Advanced Motion Control, AMC 6 - Istanbul, pages 456 46, March 7-9 6. [4] A. Ilchmann, E. P. Ryan, and C. J. Sangwin. Tracking with prescribed transient behaviour. ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, SMAI,. [5] A. Ilchmann, E. P. Ryan, and S. Trenn. Tracking control: Performance funnels and prescribed transient behaviour. System and Control Letters (Preprint M9/4), 4. [6] A. Isidori. Nonlinear Control Systems. Springer- Verlag, London, (3rd edition) edition, 995. [7] H. Schuster, C. Westermaier, and D. Schröder. Highgain-control for systems with zero dynamics and arbitrary relative degree. Proceedings of the IEEE Int. Conf. on Mechatronics & Robotics, MechRob, Aachen, Germany, pages 46 43, September 4. 76