Preprint - Control of renewable energy systems - TUM
Contributions to Non-identifier Based Adaptive Control in Mechatronics Christoph M. Hackl∗,a , Christian Endischa , Dierk Schrödera a Institute
for Electrical Drive Systems and Power Electronics Technische Universität München Arcisstr. 21, 80333 München, Germany
Abstract Funnel-Control (FC) — an adaptive (time-varying) MIMO/SISO control strategy — is re-introduced and its applicability introductory for position control of nonlinear, coupled (rigid) robotic systems and for speed control of nonlinear two-mass flexible servo systems is shown. Additionally Error Reference Control (ERC) — a direct derivative of FC — is established. ERC is specially designed with asymmetric boundaries and auxiliary reference, ensuring that the control error evolves within a prespecified tube (a shrinked funnel region, achieving even better tracking performance). FC and hence ERC are based on the high-gain stabilizability of minimum-phase systems with relative degree one and known sign of the high-frequency gain. Although the plant only needs to be known in structure, both concepts assure prescribed transient behavior without identification and/or parameter estimation. As most industrial applications, also the considered robotic and two-mass flexible servo systems, exhibit relative degrees greater than one, FC and ERC cannot directly be applied. Therefore a special state-feedback is introduced, reducing the relative degree and retaining the minimum-phase property. The additional implementation of a nominal PI-like extension guarantees good disturbance rejection and asymptotic tracking of (constant) velocity and position reference trajectories. Simulation results for a 6-DOF Manutec r3 robot underpin and compare the achievable position control performance of one overall MIMO Funnel Controller for all joints and one SISO Funnel Controller for each joint (6 controllers). For nonlinear two-mass flexible servo systems, measurement results demonstrate the achievable load speed control performance of FC and ERC in comparison to optimal LQ state feedback. Key words: Non-identifier Adaptive Control, Funnel Control, Error Reference Control, Relative Degree, Minimum-phase, Mechatronics
1. Introduction In industry the control engineer often has only (very) rough knowledge of the plant under consideration — often only the system structure is obvious. In addition most applications in mechatronics inhere nonlinear friction, backlash of gears and damping ratios, which are not exactly known or identifiable. Due to parameter deviations (or uncertainties) of the real plant and without time-consuming (cost-intensive) precise system identification, linear control strategies reach very easily ∗ Corresponding
author Email address: [email protected] (Christoph M. Hackl) URL: http://www.eat.ei.tum.de/people/ch.html (Christoph M. Hackl) Preprint submitted to Robotics and Autonomous Systems
their limits: Besides unsatisfying control performance, global stability cannot be shown in general. In this paper a robust, adaptive (time-varying) control concept is re-examined to bypass these difficulties in linear control design. This control concept — Funnel-Control (FC) — can be applied to a wide class S of minimum-phase systems with relative degree one and known (e.g. positive) sign of the high-frequency gain [1, 2]. The controller is able to cope with all plants of class S, without parameter estimation/identification. Measurement noise and parameter uncertainties are tolerated. This non-identifier based approach allows not only to guarantee global stability and good tracking performance of the closed-loop system, but also the predefinition of the desired transient behaviour of the closed-loop system by a limiting function of time — the funnel boundary meetDecember 29, 2008
ing for e.g. customer specifications (if the control input is sufficiently dimensioned [1, 3, 4, 5, 6]). FC adjusts its time-varying proportional gain only by the measured control error and its distance to the prescribed funnel boundary — and therefore absolutely guarantees an error evolution within this funnel region, however turbulent error evolutions (oscillations & overshoots) due to the necessary initial funnel width may occur. In this paper we establish and introduce Error Reference Control (ERC) — a direct derivative and special design of FC, which additionally allows to guide the error evolution along a desired, prescribed error reference trajectory surrounded by a virtual tube. The ERC controller design combines the overall reference with the desired error reference to an auxiliary reference signal and uses asymmetric boundaries — establishing a tube, not a funnel. The (actual) distance evaluation between the measured error and either the upper or lower boundary of the tube correctly adapts the time-varying proportional gain (similar to that of Funnel-Control). If saturated control inputs cause severe performance issues (neglected in this paper), Saturated Control Input Compensation can be applied to bypass these problems [4, 6]. Since the system class S restricts direct use of both controller designs to most industrial plants (e.g. robotic and nonlinear two-mass flexible servo systems), a state-feedback like extension leads to an auxiliary system being element of class S (following the idea in [7, 8, 9]). The auxiliary system exhibits a reduced relative degree of one and retains the minimum-phase property. The sign of the high-frequency gain stays untouched. The inherent proportional characteristic of both controllers does not allow asymptotic tracking and/or good disturbance rejection in general, therefore a PI-like structure is supplementary implemented attaining both goals [10, 11]. As first main result, this paper shows introductory the applicability of FC to robotic systems: An example 6–DOF robotic system (Manutec r3 robot modelled in Modelica) is comparatively controlled by i) one MIMO Funnel-Controller for all 6 joints and ii) one SISO Funnel-Controller for each joint (6 controllers). The simulation results underpin the achievable performance of the MIMO and SISO approach in tracking a given (smooth) position reference trajectory without any knowledge or the need for identification of the robot parameters. The second main results focuses on load speed control of a nonlinear two-mass flexible servo system. FC and ERC are compared and implemented at a real plant in the laboratory of the Institute for Electrical Drive Systems and Power Electronics. Both approaches show
better performance in velocity tracking than linear LQ state feedback control. Besides, ERC beats FC due to better damped system response. In [8], it was already shown, that the non-identifier based approach exhibits great robustness especially for nonlinear multimass flexible servo systems in contrast to state feedback design. 2. Mathematical Preliminaries 2.1. Affine, Nonlinear MIMO Systems This paper focuses on multi-mass-systems, therefore affine, nonlinear M-input u ∈ R M , M-output y ∈ R M MIMO systems are introduced. Such systems of order N without direct feedthrough and system state vector x ∈ RN are given by the set of ODEs: ) x˙ (t) = f(x(t)) + G(x(t))u(t); x(0) = x0 (1) y(t) = h(x(t)) with sufficiently often differentiable vector fields f(x), G(x) = [g1 (x), . . . , g M (x)] and h(x) = [hT1 (x)), . . . , hTM (x)] for all x ∈ RN (all element of C k (RN ; R M ) with appropriate k). The following holds true G(x)) , 0 for all x ∈ D. The system has its equilibrium at x0 = 0 (w.l.o.g.), such that h(x0 ) = 0 and f(x0 ) = 0. 2.2. Relative Degree An affine, nonlinear MIMO system of form (1) is said to have (vector) relative degree [r1 , r2 , . . . , r M ]T for all x(t) in a region D0 ⊆ D ⊆ RN , if both conditions hold: (1) Lg j Lfk hi (x) = 0 for all 1 ≤ i, j ≤ M and 0 ≤ k < ¯ ¯ ri − 1 and (2) det(G(x)) , 0 ⇐⇒ G(x) non-singular M×M ¯ where G(x) = {¯gi j (x)}1≤i, j≤M ∈ R with g¯ i j (x) := Lg j Lfri hi (x)1 for all x ∈ D0 . The scalar relative degree is PM rk . If the region D0 may be expanded the sum r = k=1 to the whole domain D ⊆ RN , the relative degree is said to be strong, otherwise weak. Is additionally D0 = RN , the relative degree is said to be global and if r1 = · · · = r M , then the relative degree is strict [12]. 2.3. Byrnes-Isidori Normal Form Any MIMO system of the form (1) having (vector) relative degree [r1 , . . . , r M ]T for all x in a region D0 ⊆ D ⊆ RN with r = r1 + · · · + r M ≤ N may be locally 1 where Li h(x) represents the i-th Lie-Derivative or directed grak dient of h(x) along the vector field k(x) (here f or g). The recursion formula yields Lki h(x) = Lk (Lki−1 h(x)) for i = 1, 2 . . . the desired i-th derivative
2
rM uM M P g¯ M j u j f¯M + j=1
.. .
..
u1
.. .
M P f¯1 + g¯ 1 j u j j=1
(M) ξr M
(M) ξ˙r M .
r1 (1) ξr 1
(1) ξ˙r 1
(M) ξ˙ 1
...
...
(1) ξ˙ 1
ξ
..
.
ξ
(1) = y1 1
(M) = yM 1
y∗
η˙ i = ni (ξ, η) 1≤i≤N−r
y+n
y
: w = Ty
+ +n
Figure 2: Funnel-Control Block Diagram
.. .
.. .
.. .
matrix Φ ∈ RN×N ), hence the (vector) relative degrees is globally defined and the representation in BINF is global even under output feedback.
Figure 1: Byrnes-Isidori Normalform of an affine, nonlinear MIMO system: Seperation in M–external states ξ(i) (i = 1, . . . , M) and N − r– internal states η
2.4. Zero-Dynamics and Minimum-Phase Systems The term η˙ = n(ξ, η) represents the internal dynamics of the transformed system and may be analysed due to stability aspects by setting the output and its derivatives to zero, that is yi = ξ˙1(i) = · · · = ξ˙r(i)i = 0 for all i = 1, . . . , M. Now, if the Zero-Dynamics η˙ = n(0, η) show asymptotic stability for all initial values η(0), the MIMO system is called minimum-phase [12].
transformed into Byrnes-Isidori Normal Form (BINF). Therefore, fix for i = 1, . . . , M, φ(i) (= yi ) 1 (x) = hi (x) φ(i) (x) = L h (x) (= y ˙ ) f i i 2 (2) .. . (ri −1) φ(i) (x) = Lri −1 h (x) (= y ) ri
P
2
ξ
η
System of Class S u e P 1 : y˙ = f(p, w, u) α ∂Fϕ , Ψ, kek − Funnel Controller w
f
i
i
3. Funnel-Control (FC)
If r < N, it is always possible to find N−r more mapping functions φr+1 (x), . . . , φNh (x), such that the overall map-i ping [ξ, η]T = Φ(x) = φ(1) (x), . . . , φ(M) (x), . . . , φN (x)
Funnel-Control, developed by Ilchmann et al. [1, 2], is a recent control strategy which is based on highgain concepts and employs an adjustable proportional (time-varying) gain α(·) to control nonlinear systems of class S2 with strict relative degree r = 1, stable zerodynamics (minimum-phase property) [12] and known (positive) sign of the high-frequency gain. The system S is governed by the Funnel Controller (see Fig. 2), which calculates the control input u(t) = α ∂Fϕ (t), Ψ(t), ke(t)k · e(t) (5)
where φ(k) (x) = [φ1(k) (x), . . . , φr(k) (x)] for all k = k 1, . . . , M has a jacobian matrix which is nonsingular for all x ∈ D0 and therefore qualifies as a local coordinate transformation in the region D0 . Moreover, if the distribution G = span{g1 , . . . , g M } is involutive for all x ∈ D0 , it is always possible to choose φr+1 (x), . . . , φN (x), such that Lg j φi (x) = 0 (3)
for all i = r + 1, . . . , N, j = 1, . . . , M and all x ∈ D0 [12]. Finally, the system can be given in new coordinates ξ and η of the BINF M P (i) ξ˙ ri = f¯i (ξ, η) + g¯ i j (ξ, η)u j j=1 (4) η˙ = n(ξ, η) i h (1) T y = ξ1 , . . . , ξ1(M)
simply by weighting — the difference between reference signal y∗ (·) ∈ W 1,∞ (R≥0 ; R M ) and actual system output y(t) deteriorated by (bounded) measurement noise n(t) — the control error e(t) = y∗ (t) − y(t) − n(t)
(6)
with the time-varying control gain Ψ(t) Ψ(t) α ∂Fϕ (t), Ψ(t), ke(t)k = = (7) dV (t) ∂Fϕ (t) − ke(t)k
for all i = 1, . . . , M where f¯i (ξ, η) := Lfri hi (Φ−1 (ξ, η) g¯ i j (ξ, η) := Lg j Lfri hi (Φ−1 (ξ, η)). The M branches (or interconnected subsystems) may be severly coupled due to the influence of the internal states η and the input vector u(t) on each ξ˙r(i)i for all i = 1, . . . , M (see Fig. 1). In this paper, we only apply linear coordinate transformations (e.g. Φ(x) = Φx, with constant and non-singular
2 For more explicit information on the system class S, the reader is P P referred to [1, 2, 13], where the subsystems 1 and 2 with the Operator T are well defined and examples are given for possible systems of class S.
3
+∂Fϕ (·) δ ke(0)k
and assures asymptotic steady-state accuracy and good disturbance rejection [10, 11]. A proper choice of the funnel boundary is given by an exponentially decaying function of time ∂Fϕ (t) =: ∂FE (t), where ! ! 1 1 t 1 ∂FE (t) = − exp − + (8) ϕE,0 ϕE,∞ TE ϕE,∞
∂Fϕ (τδ ) dF
dV ke(·)k
µ
τδ t t F
Time [s]
with the initial value ∂FE (0) = 1/ϕE,0 > ke(0)k enclosing the initial error, the arbitrarily predefined time constant T E > 0 and limt→∞ ∂FE (t) = 1/ϕE,∞ > 0 the steady-state accuracy. It seems obvious that the initial error has to be known when initializing/predefining the funnel boundary. For industrial control tasks this holds true for any given reference signal y∗ (0) and measurable output y(0). Mathematically this condition may be relaxed by choosing the initial value of the boundary to ∂Fϕ (0) = ∞ [1, 2]. Obviously, costumer specifications (δ, τδ , µ)4 — depicted in Fig. 3 — can be easily met by an adequate definition of the funnel boundary with ∂Fϕ (τδ ) ≡ δ and limt→∞ ∂Fϕ (t) ≡ µ.
−∂Fϕ (·) Figure 3: Basic Idea of Funnel-Control
The gain is reciprocally proportional to the (vertical) distance dV (t) = ∂Fϕ (t) − ke(t)k at the actual time t (see Fig. 3) between the funnel boundary ∂Fϕ (t) and the Euclidian norm 3 k · k of the error (the minimal future distance dF at some future time tF ≥ t might also be applied [2, 3, 5], but not used in this paper for simplicity). It can additionally be scaled by a (bounded) scaling factor Ψ(·) ∈ W 1,∞ R≥0 ; R>0 to e.g. define a minimal control gain. The intra-system variables in Fig. 2 represent a bounded disturbance p(·) and with w(·) = (Ty)(·) a (locally) Lipschitz and causal operator function [1, 2]. The funnel boundary ∂Fϕ (t) = 1/ϕ(t) is given by the reciprocal of an arbitrarily chosen bounded, continuous and positive function ϕ(t) > 0 for all t ≥ 0 with supt≥0 ϕ(t) < ∞ [2]. Thus the Funnel itself is defined as the set Fϕ : t → {e ∈ R M | ϕ(t) · kek < 1} which encloses the error e(t) for all t ≥ 0, if the initial error e(0) is surrounded by the Funnel. The gain adaption (7) ensures that the error e(·) evolves inside the Funnel Fϕ . Therefore, the gain α(·) increases, if the error e(·) draws close to the boundary ∂Fϕ (·) (more aggressive control) and decreases, if the error e(·) becomes small (more relaxed control). In [1] it is proven that both the gain α(·) > 0 and the error e(·) stay bounded for all t ≥ 0, if the control input u(·) can adopt sufficiently large but finite values. Due to the condition of 1 the funnel boundary ∂Fϕ (t) = ϕ(t) > 0 for all t ≥ 0, the steady-state control error might be arbitrarily small, but eventually limt→∞ e(t) , 0. This drawback is typically for proportional control loops without integrating components in plant or controller and is therefore not a typical disadvantage of FC. This drawback may be overcome by introducing a (nonlinear) PI-like extension, which does not deteriorate the affiliation to class S
4. Robotics 4.1. Dynamics of Rigid Robotic Systems In robotics, a common approach to derive the system dynamics is to utilize generalized coordinates q = [q1 , . . . , q M ]T ∈ R M and q˙ = [q˙ 1 , . . . , q˙ M ]T ∈ R M (here: angular position and the angular velocity, respectively) and Lagrangian Dynamics. The Lagrangian ˙ = K(q, q) ˙ − function of the system is given by L(q, q) ˙ = 12 q˙ T M(q)q˙ P(q) where the kinetic energy K(q, q) depends on the real, symmetric, positive definite nonsingular and uniformaly bounded [14] inertia matrix M(q) = M(q)T ∈ R M×M with det(M(q)) , 0 and λm (M(q)) · 1 ≤ M(q) ≤ λ M (M(q)) · 1 (with the minimal and maximal eigenvalues of the inertia matrix, 0 < λm < λ M < ∞) for all q ∈ R M [15, 16, 17] and the potential energy P(q). Following partial differential equation can be formulated ! d ∂L ∂L ˙ − = τ − f(q) (9) dt ∂q˙ ∂q where the vector τ = [τ1 , . . . , τ M ]T ∈ R M describes external applied inputs (e.g. joint motor torques), the ˙ ∈ R M contains viscous and/or dyvector function f(q) namic frictions terms, unstructured friction effects (e.g.
3 In the SISO case, the absolute value | · | of the error e(·) is sufficient.
4δ
racy
4
is the accuracy at time τδ and µ is the desired steady-state accu-
u sat,M
u′M
e′M .. .
q ROBOT .. . u sat,1
′ u e′ PI u1 + FC
q˙ −
y′
−
KD
u1
+
q∗
PI M
.. .
FC M
e′1
[see Eq. (11)]
KP
uM
PI1
u sat,M
u′M
q .. ROBOT . u sat,1
u′1
FC1
−
KP
y′
q˙ ∗
q˙
[see Eq. (11)]
q∗ −
KD
q˙ ∗
(b) One SISO Controller for Each Joint
(a) One MIMO Controller for All Joints
Figure 4: MIMO/SISO Funnel Controller Structures for Nonlinear M–DOF Robotic Systems of Form (11)
50
Axis Axis Axis Axis Axis Axis
0 −50 0
Position Errors eq [◦ ] - Comparison 0.5
0.5
1
1.5
1 2 3 4 5 6
FC (MIMO)
FC (MIMO)
Position q [◦ ] & Reference q∗ [◦ ] - Comparison 100
−0.25 0.5
1
1.5
Axis Axis Axis Axis Axis Axis
1 2 3 4 5 6
Axis Axis Axis Axis Axis Axis
1 2 3 4 5 6
2
0.5
50
Axis Axis Axis Axis Axis Axis
0 −50 0.5
1 Time [s]
1.5
1 2 3 4 5 6
FC (SISO)
FC (SISO)
0
−0.5 0
2
100
0
0.25
0.25 0 −0.25 −0.5 0
2
(a) Position (colored) & Reference (dotted)
0.5
1 Time [s]
1.5
2
(b) Position Errors (colored)
Figure 5: Comparison between MIMO and SISO: Angular Position q with Corresponding Reference and Position Error eq for each Axis, i = 1, . . . , 6 Funnel Gain α [1] - Comparison FC (MIMO)
FC (MIMO)
Aux. Error e′ & Boundary ±∂F - Comparison 1 0.5 ′
ke (·)k ±∂F(·)
0 −0.5 −1 0
0.5
1
1.5
40 α(·) 20
0 0
2
0.5
1
1.5
2
e′1 (·) e′2 (·) e′3 (·) e′4 (·) e′5 (·) e′6 (·) ±∂F(·)
0 −0.5 −1 0
0.5
1 Time [s]
1.5
FC (SISO)
FC (SISO)
1 0.5
20
0 0
2
(a) Auxiliary Error (colored) & Funnel Boundary (red)
α1 (·) α2 (·) α3 (·) α4 (·) α5 (·) α6 (·)
40
0.5
1 Time [s]
1.5
2
(b) Funnel Gain (colored)
Figure 6: Comparison between MIMO and SISO (i = 1, . . . , 6): Auxiliary Error e′ (MIMO) & e′i (SISO) within prescribed funnel boundary ∂F (MIMO) & ∂Fi (SISO) and corresponding Funnel Gain α (MIMO) & αi (SISO)
5
the auxiliary system is set up, where KP ∈ R M×M and KD ∈ R M×M are constant proportional and derivative gain matrices amplifying the angular positions and angular velocities, respectively. Analyzing the (vector) relative degree of the auxiliary system yields y˙ ′ = KP x˙ 1 + KD x˙ 2 = KP x2 + KD −M(x1 )−1 n(x1 , x2 ) + M(x1 )−1 · τ which proofs that the auxiliary output y′i is a function of its corresponding joint/motor torque τi , that is each branch ′ of the MIMO-system has global relative degree ri = 1. The relative degree is successfully reduced for all fullranked KD , 0 and the overall system has one integrator in each of its M forward branches (see Fig. 1). To assure the minimum-phase condition, the internal dynamics must be analysed by transforming the auxiliary ′ system into BINF with the external states ξ1i := yi and the internal states ηk := qk for all k = 1, . . . , M. One gets [ξ, η]T = Φx with the transformation matrix " # # " Φξ KP KD (14) Φ= = Φη 1 0
stiction) and external disturbances or unmodeled dynamics. The dynamics of a (rigid) robotic system can be deduced by differentiating Eq. (9) and are given by the following second-order ODE with M-inputs and Moutputs ˙ τ = M(q)q¨ + n(q, q) (10) ˙ = C(q, q) ˙ q˙ + g(q) + f(q) ˙ is the sum of all where n(q, q) nonlinearities where the uniformaly bounded [18] ma˙ ∈ R M×M includes all Centripetal and Coritrix C(q, q) olis terms, the vector g(q) ∈ R M represents the gravity force. 4.2. State-Space Realization For the later analysis, define x(t) = [x1 (t), x2 (t)]T = ˙ T ∈ RN as state vector, then the convenient [q(t), q(t)] state space representation of the robotic MIMO system # " # " 0 x2 + ·τ (11) x˙ = M(x1 )−1 −M(x1 )−1 n(x1 , x2 ) | {z } | {z } f(x)∈RN
G(x)∈RN×M
−1 As K−1 P and KD may be arbitrarily chosen to be nonsingular, the inverse transformation matrix Φ−1 exists. This transformation fulfills condition (3), since " # dΦη x 0 [1 · G = 0] =0 LG Φη x = M(x1 )−1 dxT (15)
of order N = 2M is deduced from (10). The main interest is robot positioning, thus the output is fixed to y = h(x) = x1 = q
(12)
The state-space representation (11) is reasonable, since the matrix M(x1 ) is non-singular and its inverse M(x1 )−1 exists and is uniformaly bounded away from zero for all x1 ∈ R M . In real world the working space of the robot manipulator is constrained to some D ⊆ R M and cannot be left due to physical and geometrical means.
for all x ∈ RN . The seperation of coordinates in external ξ := y′ (see Eq. (13)) and internal η := x1 = q states, allows a compact representation of the robotic system in BINF −1 −1 −1 ξ˙ = KP K−1 D ξ − KD KP η − KD M(η) n(Φ (ξ, η))
4.3. Vector Relative Degree The application of FC requires the affiliation to class S with strict vector relative degree [r1 , . . . , r M ] = [1, . . . , 1], stable Zero-Dynamics (minimum-phase property) and known (positive) sign of the highfrequency gain. By differentiating the output with respect to time i) y˙ = x˙ 1 = x2 and ii) y¨ = x˙ 2 = −M(x1 )−1 n(x1 , x2 ) + M(x1 )−1 · τ until it becomes a function of the control input u = τ, the (vector) relative degree is easily derived to [r1 , . . . , r M ] = [2, . . . , 2] with PM the sum r = i=1 ri = 2M. Hence, no direct application of FC is possible.
+ KD M(η)−1 τ
−1 η˙ = K−1 D ξ − KD KP η iT h ′ y = ξ1(1) , . . . , ξ1(M)
conform to the general form (4). The Zero-Dynamics η˙ = −K−1 D KP η := Qη
(17)
(setting ξ = 0) are asymptotically stable, if the matrix Q = −K−1 D KP exhibits eigenvalues with negative real part, that is a matrix spectrum σ(Q) ⊂ C− . As both matrices KP , 0 and KD , 0 are free design parameters, they can be set accordingly and non-singular by the control engineer without any knowledge of system parameters. Additionally, for any scaled positive definite derivative gain identity matrix e.g. KD = β·1 with β > 0,
4.4. Auxiliary System and Affiliation to Class S By introducing a state-feedback like structure defining an auxiliary output (following the idea in [7, 8, 9]) y′ = h′ (x) = KP x1 + KD x2
(16)
(13) 6
5. Error Reference Control (ERC)
the positive high-frequency gain is not touched. Since KD ·M−1 (η) = M−1 (η)·KD = β·M−1 (η) > 0 is also positive definite as the matrices commute. This reveals with (16) for every non-empty compact set C ⊂ R M ×R M and sequence (un ) = (τn ) ⊂ R M \ {0} with kun k → ∞ as n → ˙ M ∞ that min[ξ,η]∈C hukunn,ξi k → ∞ holds true for all η ∈ R (generalization of the high-frequency gain concept, see [1]). In conclusion, the introduced auxiliary system is element of class S and may be controlled by FC.
ERC is directly derived from FC (so far for the SISO case only). Its special asymmetric design of the funnel boundaries (“virtual tube”, see Fig. 7) and adequate choice of its auxiliary reference enable ERC not only to predefine transient behavior within the given limits, but also to guide the error along a desired error reference trajectory (see Fig. 7). The control-loop conforms to that shown in Fig. 2 where the Funnel Controller is replaced by the Error Reference Controller and all signals by scalar signals. The same class S of high-gain controllable systems can be stabilized and asymptotic tracking within the virtual tube TERC = {e ∈ R| e∗ (t) < e < e∗ (t) ∀t ≥ 0} (see Fig. 7 for a possible and illustrative example tube/boundary design and error evolution) guaranteed, if the initial error lies inside TERC . The er-
4.5. MIMO/SISO Control-Loop
Error
Fig. 4 illustrates possible control-loops: R a MIMO and a SISO approach with u′ (t) = u(t) + T1I u(τ)dτ where u is defined as in Eq. (5). In Fig. 4(a) the MIMO variant with one controller (FC+PI) for all M joints is depicted. Fig. 4(b) shows M–SISO controllers (FCi + PIi , i = 1, . . . , M) for the M joints, each axis is driven by one SISO control-loop.
e(0)
4.6. Simulation Results
∗
e (·)
The introduced MIMO/SISO FC+PI controllers with exponential funnel boundary are now tested for a 6– DOF Manutec r3 Robot implemented in Modelica [19] with adequate simulation interface to Matlab/Simulink. The PI-like structure is implemented as nominal linear PI block with F PI (s) = 1 + sT1 I . Due to a diagonal choice of KP = 15 and KD = 0.01 · 1, the Zero-Dynamics are asymptotically stable with σ(Q) = {−100, . . . , −100} ⊂ C− . The realistic model also exhibits elasticity due to gears, nevertheless the application of Funnel-Control is not deteriotated as the minimum-phase condition is not touched (passive influence of the gear elasticity! [20]). Although none of the system parameters are known or estimated, the controlled robot follows the reference trajectories q∗i (t)6 quite nicely, both for the MIMO and the SISO controller7 (see Fig. 5(a)). The maximal control error does not exceed the pre-set accuracy limit µ = 0.5◦ (see Fig. 5(b)). Both strategies assure an auxiliary error evolution within the prescribed funnel boundary (see Fig. 6(a)) by their instantenous gain adaption (see Fig. 6(b)) indirect proportional to the measured distance to the funnel boundary.
e∗ (·)
e∗ (·) dV (t) ∆e(t)
e(·) Time [s]
t dV (t)
Figure 7: Principle Idea of Error Reference Control (ERC) - SISO Case for initial errors e(0) ≥ 0 (for initial errors e(0) < 0 analogous)
ror evolution now is even more strictly bounded by the virtual tube (see Fig. 7) around a predefined desired error evolution — the Error Reference trajectory e∗ (·) ∈ C 1 (R≥0 ; R). The tube must be designed in such way, that the (known!) initial error e(0) = y∗ (0) − y(0) − n(0) is enclosed. In this paper, the error reference trajectory is exemplary set to ! t e∗ (t) = y∗ (0) − y(0) − n(0) · exp − (18) | {z } T exp e(0)
where its time constant T exp > 0 may be chosen arbitrarily. The desired error reference e∗ (·) should clearly tend to zero. Error Reference Control ensures guidance of the error along e∗ (·) within the virtual tube TERC by applying the proportional control law (utilizing (6) in the SISO case)
51
= diag(1, . . . , 1) ∈ R M represents the diagonal identity matrix references for each joint angles [◦ ]: q∗1 : −30 → 50◦ , q∗2 : 80 → −15◦ ,q∗3 : 65 → −55◦ ,q∗4 : −20 → −75◦ ,q∗5 : −90 → 100◦ and q∗6 : 10 → −70◦ 7 Controller Design: i) Costumer Specifications (µ = 0.5◦ , τ = δ 0.3s, δ = 0.75◦ /s); ii) Initial Boundary ∂Fexp (0) = 1; iii) PI Integrator Time Constant T I = 0.1s and iv) Scaling Factor Ψ(t) = 10 6 Smooth
uERC (t) = αERC (t) · e(t) − e∗ (t) ∗
∗
= αERC (t) · [y (t) − e (t) −y(t) − n(t)] | {z } =:y∗ERC (t)
7
(19) (20)
1/gr
with the specially chosen auxiliary reference y∗ERC = y∗ − e∗ ∈ W 1,∞ (R≥0 ; R) (corresponding to the SISO case of FC [1, 2]) where the time-varying gain αERC (t) =
Ψ(t) min dV (t), dV (t)
ML kD
MM −
(21)
∆ϕ˙ c
(22)
1 e∗ (t) = e∗ (t) − µ 1 + κ(t) =: ∂F ϕ (t) = ϕ(t)
(23)
−
ΩA
BL ∆ϕBL
∆ϕ
1/sΘ M
1 ϕ(t)
e∗ (t) = e∗ (t) + µ (1 + κ(t)) =: ∂F ϕ (t) =
1/gr ΩM
−
— similar to FC, see Eq. (7) — is a function of the scaling factor Ψ(t) and indirect proportional to the evaluation of the minimal (vertical) distance dV (t) = e∗ (t)−e(t) or dV (t) = e(t) − e∗ (t) between the upper e∗ (t) or lower e∗ (t) limit of the virtual tube and the error e(t), respectively. The tube limits are generated according to the desired error reference trajectory in the following manner: the upper limit
NF
d
MC + MD
1/sΘA
Figure 8: Block Diagram of Nonlinear T MS
that the error will now even closer coincide with the desired error reference e∗ (·), as any deviation will additionally increase the control gain αERC (·) and accelerate the closed-loop system towards the predefined track. Besides that, possible oscillations around e∗ (·) are seemingly damped as well. The scaling function (24) can obviously also be applied to standard FC but is much less effective — particularly at startup where the Funnel is still wide — as the control input (5) will solely change its sign with the control error.
and the lower limit
6. Nonlinear Two-Mass-System (T MS)
where κ, κ ∈ C(R≥0 ; R≥0 ) are free design functions with limt→∞ κ(t), κ(t) = 0. Upper and lower limit tend to the asymptotic accuracy limt→∞ e∗ (t) = µ and limt→∞ e∗ (t) = −µ, respectively. The tube TERC is a specially designed asymmetric funnel boundary with upper ∂F ϕ (·) and lower ∂F ϕ (·) limiting function. Due to the choice e∗ (t0 ) < e(t0 ) = e∗ (t0 ) < e∗ (t0 ), the initial error starts within the tube. Whether the error is above or below the desired error trajectory e∗ (·), the control input changes its sign and therefore allows appropriate acceleration or deceleration of the system. This property ensures in contrast to standard FC design not only the evolution within the virtual tube but also the possible guidance along the desired error reference e∗ (·). The following proposal for a damping algorithm has indicated in simulations and experiments to have beneficial effects — overshoots and oscillations are drastically reduced — on the system response, but yet the mathematical proof for applicability and general improvement is not finished. The suggested damping algorithm is realized by the implementation of a saturated scaling function Ψmax Ψ (t) = sat ΨD (e(t) − e∗ (t))(˙e(t) − e˙ ∗ (t)) 0 D +Ψ0 (24)
The considered plant (shown in Fig. 8) is a nonlinear two-mass flexible servo system (T MS ), which is a common example for an electrical drive connected to a load machine via gear and flexible shaft. The system has the state vector xT = [Ω M ∆ϕ ΩA ] with the revolution speed of the motor Ω M and of R tthe1 load machine ΩA and the angle of twist ∆ϕ(t) = 0 ( gr Ω M (τ) − ΩA (τ))dτ = 1 gr ϕ M − ϕA . The system’s input is the motor torque M M (t) = u(t) + uD (t), depending on the chosen control law (5) or (20) and an active damping term uD (to be specified later). The load velocity is the controlled variable y(t) = ΩA (t). The nonlinear state space ODE is given by x˙ =
1 ΘM
1 ΘA
M M − gdr
∂BL(∆ϕ) − gcr ∂t
1 gr
BL(∆ϕ)
Ω M −ΩA −ML −NF (ΩA )+d ∂BL(∆ϕ) +cBL(∆ϕ) ∂t
(25)
with the stiffness c > 0, the damping d > 0, the (constant) gear ratio gr > 0, the inertia of the motor Θ M and of the load ΘA [21]. The nonlinearities BL(·) and NF (·) represent the backlash of gear with the backlash angle ∆ϕBL (t) := BL(∆ϕ(t)) where
with the minimal offset Ψ0 ≥ 1, the damper gain ΨD ≥ 0 und the maximal scaling value Ψmax D , which limits the damper influence on e.g. measurement noise amplification. All made simulations and experiments have shown
8
∆ϕ + aBL 0 BL(∆ϕ(t)) = ∆ϕ − a BL
, ∆ϕ ≤ −aBL , |∆ϕ| < aBL , ∆ϕ ≥ aBL
(26)
and the friction torque MNF (t) := NF (ΩA (t)) where 0 NF (ΩA ) = mNF ΩA + arctan(103 · ΩA )MNF
which corresponds to the inherent frequency of a T MS [22] and is seemingly not changed. But on the other hand, the system damping may be altered arbitrarily, which becomes obvious by evaluation of the damping coefficient ω0 ΘA d + kD D= (31) 2c gr Θ M + Θg2A
(27)
with the viscous friction coefficient mNF > 0 and the 0 Coulomb friction offset MNF > 0. Both nonlinearities are modelled continuously. Therefore those do not affect affiliation of the T MS to system class S and do not deteriorate the latter control performance of FC and ERC [1, 8]. The current control loop is assumed to be fast enough and is thus neglected (adequate for modern power converters and drives [8]). So the torque M M can be an arbitrary function of time, whereas the conmax max >0 is constrained by M M trol input |M M (t)| ≤ M M to protect the motor and the power electronics unit (converter). Within this paper the control inputs will not exceed the allowed range.
r
Therefore, for any kD > 0 the damping coefficient of the T MS system is increased (for simulative illustration, see Fig. 9). Constraints of this damping approach are: i) the available control input (how much of the overall motor torque should be reserved for damping?), ii) the achievable performance (dynamic) of the power converter and drive (how fast can the damping torque be generated) and iii) the higher the chosen gain kD , the slower the system response (since we penalize any change ∆ϕ˙ of the torsional angle). All three constraints are not to severe for real world application, if well specified actuators and power electronics are used, being able to allow damping. Important to note that, we must not choose negative gains kD < 0, as negative values will decrease damping or evendestabilize the system for all
6.1. Active Damping by Static Feedback If the nonlinear two-mass flexible servo system exhibits serious oscillations between motor and load speed (or position), explicit necessity for active damping is indicated. Therefore we present a simple approach, which allows to tune a static damping feedback # " 1 (28) Ω M − ΩA uD = −kD gr
r feedback gains k∗D ≤ − dg ΘA Θ M +
to increase and adjust the system‘s damping to one‘s need, if the underlying control loop achieves global stability (like FC and ERC). Since we induce the change (time derivative) of the angle of twist ∆ϕ, ˙ any (foremost fast) changes in the shaft‘s torsion are penalized by the weighting factor kD ≥ 0 (free design parameter, e.g. can be determined emperically during system run). The influence of this static damping feedback is easily analyzed by investigation of the transfer function of the linear system [22], see (25) neglecting backlash and friction: FTΩMS (s) = 1 + dc s Θ M ΘA 1 + s ΘA c d +
ΘA g2r
2 s Θ
y′ (t) = kT x(t)
M+ 2 gr
c
gr
(32)
resulting in a linear combination of the constant feedback vector kT = [k1 k2 k3 ] with the state vector xT = [x1 x2 x3 ] = [Ω M ∆ϕ ΩA ]. By choosing the feedback structure to fulfill the relations: k1 > 0, k2 > −d/ΘA (k1 + k3 ) and k3 > −k1 , it can be guaranteed that the relative degree of the T MS (for any parameter sets of ΘA , Θ M , c, d, gr — all greater than zero by physical means) is reduced to rT′ MS = 1 and the required minimum phase property is retained [9]. Besides, the above introduced static damping feedback with kD ≥ 0 does not touch the minimum-phase condition. The numerator of (29) does not depend on the parameter kD and hence remains Hurwitz for all c, d > 0 (this result can be easily derived for the nonlinear case as well).
ΩA (s) = M M (s)
ΘA kD Θ gr (Θ M + 2A )
for Electrical Drive Systems and Power Electronics Technische Universität München Arcisstr. 21, 80333 München, Germany
Abstract Funnel-Control (FC) — an adaptive (time-varying) MIMO/SISO control strategy — is re-introduced and its applicability introductory for position control of nonlinear, coupled (rigid) robotic systems and for speed control of nonlinear two-mass flexible servo systems is shown. Additionally Error Reference Control (ERC) — a direct derivative of FC — is established. ERC is specially designed with asymmetric boundaries and auxiliary reference, ensuring that the control error evolves within a prespecified tube (a shrinked funnel region, achieving even better tracking performance). FC and hence ERC are based on the high-gain stabilizability of minimum-phase systems with relative degree one and known sign of the high-frequency gain. Although the plant only needs to be known in structure, both concepts assure prescribed transient behavior without identification and/or parameter estimation. As most industrial applications, also the considered robotic and two-mass flexible servo systems, exhibit relative degrees greater than one, FC and ERC cannot directly be applied. Therefore a special state-feedback is introduced, reducing the relative degree and retaining the minimum-phase property. The additional implementation of a nominal PI-like extension guarantees good disturbance rejection and asymptotic tracking of (constant) velocity and position reference trajectories. Simulation results for a 6-DOF Manutec r3 robot underpin and compare the achievable position control performance of one overall MIMO Funnel Controller for all joints and one SISO Funnel Controller for each joint (6 controllers). For nonlinear two-mass flexible servo systems, measurement results demonstrate the achievable load speed control performance of FC and ERC in comparison to optimal LQ state feedback. Key words: Non-identifier Adaptive Control, Funnel Control, Error Reference Control, Relative Degree, Minimum-phase, Mechatronics
1. Introduction In industry the control engineer often has only (very) rough knowledge of the plant under consideration — often only the system structure is obvious. In addition most applications in mechatronics inhere nonlinear friction, backlash of gears and damping ratios, which are not exactly known or identifiable. Due to parameter deviations (or uncertainties) of the real plant and without time-consuming (cost-intensive) precise system identification, linear control strategies reach very easily ∗ Corresponding
author Email address: [email protected] (Christoph M. Hackl) URL: http://www.eat.ei.tum.de/people/ch.html (Christoph M. Hackl) Preprint submitted to Robotics and Autonomous Systems
their limits: Besides unsatisfying control performance, global stability cannot be shown in general. In this paper a robust, adaptive (time-varying) control concept is re-examined to bypass these difficulties in linear control design. This control concept — Funnel-Control (FC) — can be applied to a wide class S of minimum-phase systems with relative degree one and known (e.g. positive) sign of the high-frequency gain [1, 2]. The controller is able to cope with all plants of class S, without parameter estimation/identification. Measurement noise and parameter uncertainties are tolerated. This non-identifier based approach allows not only to guarantee global stability and good tracking performance of the closed-loop system, but also the predefinition of the desired transient behaviour of the closed-loop system by a limiting function of time — the funnel boundary meetDecember 29, 2008
ing for e.g. customer specifications (if the control input is sufficiently dimensioned [1, 3, 4, 5, 6]). FC adjusts its time-varying proportional gain only by the measured control error and its distance to the prescribed funnel boundary — and therefore absolutely guarantees an error evolution within this funnel region, however turbulent error evolutions (oscillations & overshoots) due to the necessary initial funnel width may occur. In this paper we establish and introduce Error Reference Control (ERC) — a direct derivative and special design of FC, which additionally allows to guide the error evolution along a desired, prescribed error reference trajectory surrounded by a virtual tube. The ERC controller design combines the overall reference with the desired error reference to an auxiliary reference signal and uses asymmetric boundaries — establishing a tube, not a funnel. The (actual) distance evaluation between the measured error and either the upper or lower boundary of the tube correctly adapts the time-varying proportional gain (similar to that of Funnel-Control). If saturated control inputs cause severe performance issues (neglected in this paper), Saturated Control Input Compensation can be applied to bypass these problems [4, 6]. Since the system class S restricts direct use of both controller designs to most industrial plants (e.g. robotic and nonlinear two-mass flexible servo systems), a state-feedback like extension leads to an auxiliary system being element of class S (following the idea in [7, 8, 9]). The auxiliary system exhibits a reduced relative degree of one and retains the minimum-phase property. The sign of the high-frequency gain stays untouched. The inherent proportional characteristic of both controllers does not allow asymptotic tracking and/or good disturbance rejection in general, therefore a PI-like structure is supplementary implemented attaining both goals [10, 11]. As first main result, this paper shows introductory the applicability of FC to robotic systems: An example 6–DOF robotic system (Manutec r3 robot modelled in Modelica) is comparatively controlled by i) one MIMO Funnel-Controller for all 6 joints and ii) one SISO Funnel-Controller for each joint (6 controllers). The simulation results underpin the achievable performance of the MIMO and SISO approach in tracking a given (smooth) position reference trajectory without any knowledge or the need for identification of the robot parameters. The second main results focuses on load speed control of a nonlinear two-mass flexible servo system. FC and ERC are compared and implemented at a real plant in the laboratory of the Institute for Electrical Drive Systems and Power Electronics. Both approaches show
better performance in velocity tracking than linear LQ state feedback control. Besides, ERC beats FC due to better damped system response. In [8], it was already shown, that the non-identifier based approach exhibits great robustness especially for nonlinear multimass flexible servo systems in contrast to state feedback design. 2. Mathematical Preliminaries 2.1. Affine, Nonlinear MIMO Systems This paper focuses on multi-mass-systems, therefore affine, nonlinear M-input u ∈ R M , M-output y ∈ R M MIMO systems are introduced. Such systems of order N without direct feedthrough and system state vector x ∈ RN are given by the set of ODEs: ) x˙ (t) = f(x(t)) + G(x(t))u(t); x(0) = x0 (1) y(t) = h(x(t)) with sufficiently often differentiable vector fields f(x), G(x) = [g1 (x), . . . , g M (x)] and h(x) = [hT1 (x)), . . . , hTM (x)] for all x ∈ RN (all element of C k (RN ; R M ) with appropriate k). The following holds true G(x)) , 0 for all x ∈ D. The system has its equilibrium at x0 = 0 (w.l.o.g.), such that h(x0 ) = 0 and f(x0 ) = 0. 2.2. Relative Degree An affine, nonlinear MIMO system of form (1) is said to have (vector) relative degree [r1 , r2 , . . . , r M ]T for all x(t) in a region D0 ⊆ D ⊆ RN , if both conditions hold: (1) Lg j Lfk hi (x) = 0 for all 1 ≤ i, j ≤ M and 0 ≤ k < ¯ ¯ ri − 1 and (2) det(G(x)) , 0 ⇐⇒ G(x) non-singular M×M ¯ where G(x) = {¯gi j (x)}1≤i, j≤M ∈ R with g¯ i j (x) := Lg j Lfri hi (x)1 for all x ∈ D0 . The scalar relative degree is PM rk . If the region D0 may be expanded the sum r = k=1 to the whole domain D ⊆ RN , the relative degree is said to be strong, otherwise weak. Is additionally D0 = RN , the relative degree is said to be global and if r1 = · · · = r M , then the relative degree is strict [12]. 2.3. Byrnes-Isidori Normal Form Any MIMO system of the form (1) having (vector) relative degree [r1 , . . . , r M ]T for all x in a region D0 ⊆ D ⊆ RN with r = r1 + · · · + r M ≤ N may be locally 1 where Li h(x) represents the i-th Lie-Derivative or directed grak dient of h(x) along the vector field k(x) (here f or g). The recursion formula yields Lki h(x) = Lk (Lki−1 h(x)) for i = 1, 2 . . . the desired i-th derivative
2
rM uM M P g¯ M j u j f¯M + j=1
.. .
..
u1
.. .
M P f¯1 + g¯ 1 j u j j=1
(M) ξr M
(M) ξ˙r M .
r1 (1) ξr 1
(1) ξ˙r 1
(M) ξ˙ 1
...
...
(1) ξ˙ 1
ξ
..
.
ξ
(1) = y1 1
(M) = yM 1
y∗
η˙ i = ni (ξ, η) 1≤i≤N−r
y+n
y
: w = Ty
+ +n
Figure 2: Funnel-Control Block Diagram
.. .
.. .
.. .
matrix Φ ∈ RN×N ), hence the (vector) relative degrees is globally defined and the representation in BINF is global even under output feedback.
Figure 1: Byrnes-Isidori Normalform of an affine, nonlinear MIMO system: Seperation in M–external states ξ(i) (i = 1, . . . , M) and N − r– internal states η
2.4. Zero-Dynamics and Minimum-Phase Systems The term η˙ = n(ξ, η) represents the internal dynamics of the transformed system and may be analysed due to stability aspects by setting the output and its derivatives to zero, that is yi = ξ˙1(i) = · · · = ξ˙r(i)i = 0 for all i = 1, . . . , M. Now, if the Zero-Dynamics η˙ = n(0, η) show asymptotic stability for all initial values η(0), the MIMO system is called minimum-phase [12].
transformed into Byrnes-Isidori Normal Form (BINF). Therefore, fix for i = 1, . . . , M, φ(i) (= yi ) 1 (x) = hi (x) φ(i) (x) = L h (x) (= y ˙ ) f i i 2 (2) .. . (ri −1) φ(i) (x) = Lri −1 h (x) (= y ) ri
P
2
ξ
η
System of Class S u e P 1 : y˙ = f(p, w, u) α ∂Fϕ , Ψ, kek − Funnel Controller w
f
i
i
3. Funnel-Control (FC)
If r < N, it is always possible to find N−r more mapping functions φr+1 (x), . . . , φNh (x), such that the overall map-i ping [ξ, η]T = Φ(x) = φ(1) (x), . . . , φ(M) (x), . . . , φN (x)
Funnel-Control, developed by Ilchmann et al. [1, 2], is a recent control strategy which is based on highgain concepts and employs an adjustable proportional (time-varying) gain α(·) to control nonlinear systems of class S2 with strict relative degree r = 1, stable zerodynamics (minimum-phase property) [12] and known (positive) sign of the high-frequency gain. The system S is governed by the Funnel Controller (see Fig. 2), which calculates the control input u(t) = α ∂Fϕ (t), Ψ(t), ke(t)k · e(t) (5)
where φ(k) (x) = [φ1(k) (x), . . . , φr(k) (x)] for all k = k 1, . . . , M has a jacobian matrix which is nonsingular for all x ∈ D0 and therefore qualifies as a local coordinate transformation in the region D0 . Moreover, if the distribution G = span{g1 , . . . , g M } is involutive for all x ∈ D0 , it is always possible to choose φr+1 (x), . . . , φN (x), such that Lg j φi (x) = 0 (3)
for all i = r + 1, . . . , N, j = 1, . . . , M and all x ∈ D0 [12]. Finally, the system can be given in new coordinates ξ and η of the BINF M P (i) ξ˙ ri = f¯i (ξ, η) + g¯ i j (ξ, η)u j j=1 (4) η˙ = n(ξ, η) i h (1) T y = ξ1 , . . . , ξ1(M)
simply by weighting — the difference between reference signal y∗ (·) ∈ W 1,∞ (R≥0 ; R M ) and actual system output y(t) deteriorated by (bounded) measurement noise n(t) — the control error e(t) = y∗ (t) − y(t) − n(t)
(6)
with the time-varying control gain Ψ(t) Ψ(t) α ∂Fϕ (t), Ψ(t), ke(t)k = = (7) dV (t) ∂Fϕ (t) − ke(t)k
for all i = 1, . . . , M where f¯i (ξ, η) := Lfri hi (Φ−1 (ξ, η) g¯ i j (ξ, η) := Lg j Lfri hi (Φ−1 (ξ, η)). The M branches (or interconnected subsystems) may be severly coupled due to the influence of the internal states η and the input vector u(t) on each ξ˙r(i)i for all i = 1, . . . , M (see Fig. 1). In this paper, we only apply linear coordinate transformations (e.g. Φ(x) = Φx, with constant and non-singular
2 For more explicit information on the system class S, the reader is P P referred to [1, 2, 13], where the subsystems 1 and 2 with the Operator T are well defined and examples are given for possible systems of class S.
3
+∂Fϕ (·) δ ke(0)k
and assures asymptotic steady-state accuracy and good disturbance rejection [10, 11]. A proper choice of the funnel boundary is given by an exponentially decaying function of time ∂Fϕ (t) =: ∂FE (t), where ! ! 1 1 t 1 ∂FE (t) = − exp − + (8) ϕE,0 ϕE,∞ TE ϕE,∞
∂Fϕ (τδ ) dF
dV ke(·)k
µ
τδ t t F
Time [s]
with the initial value ∂FE (0) = 1/ϕE,0 > ke(0)k enclosing the initial error, the arbitrarily predefined time constant T E > 0 and limt→∞ ∂FE (t) = 1/ϕE,∞ > 0 the steady-state accuracy. It seems obvious that the initial error has to be known when initializing/predefining the funnel boundary. For industrial control tasks this holds true for any given reference signal y∗ (0) and measurable output y(0). Mathematically this condition may be relaxed by choosing the initial value of the boundary to ∂Fϕ (0) = ∞ [1, 2]. Obviously, costumer specifications (δ, τδ , µ)4 — depicted in Fig. 3 — can be easily met by an adequate definition of the funnel boundary with ∂Fϕ (τδ ) ≡ δ and limt→∞ ∂Fϕ (t) ≡ µ.
−∂Fϕ (·) Figure 3: Basic Idea of Funnel-Control
The gain is reciprocally proportional to the (vertical) distance dV (t) = ∂Fϕ (t) − ke(t)k at the actual time t (see Fig. 3) between the funnel boundary ∂Fϕ (t) and the Euclidian norm 3 k · k of the error (the minimal future distance dF at some future time tF ≥ t might also be applied [2, 3, 5], but not used in this paper for simplicity). It can additionally be scaled by a (bounded) scaling factor Ψ(·) ∈ W 1,∞ R≥0 ; R>0 to e.g. define a minimal control gain. The intra-system variables in Fig. 2 represent a bounded disturbance p(·) and with w(·) = (Ty)(·) a (locally) Lipschitz and causal operator function [1, 2]. The funnel boundary ∂Fϕ (t) = 1/ϕ(t) is given by the reciprocal of an arbitrarily chosen bounded, continuous and positive function ϕ(t) > 0 for all t ≥ 0 with supt≥0 ϕ(t) < ∞ [2]. Thus the Funnel itself is defined as the set Fϕ : t → {e ∈ R M | ϕ(t) · kek < 1} which encloses the error e(t) for all t ≥ 0, if the initial error e(0) is surrounded by the Funnel. The gain adaption (7) ensures that the error e(·) evolves inside the Funnel Fϕ . Therefore, the gain α(·) increases, if the error e(·) draws close to the boundary ∂Fϕ (·) (more aggressive control) and decreases, if the error e(·) becomes small (more relaxed control). In [1] it is proven that both the gain α(·) > 0 and the error e(·) stay bounded for all t ≥ 0, if the control input u(·) can adopt sufficiently large but finite values. Due to the condition of 1 the funnel boundary ∂Fϕ (t) = ϕ(t) > 0 for all t ≥ 0, the steady-state control error might be arbitrarily small, but eventually limt→∞ e(t) , 0. This drawback is typically for proportional control loops without integrating components in plant or controller and is therefore not a typical disadvantage of FC. This drawback may be overcome by introducing a (nonlinear) PI-like extension, which does not deteriorate the affiliation to class S
4. Robotics 4.1. Dynamics of Rigid Robotic Systems In robotics, a common approach to derive the system dynamics is to utilize generalized coordinates q = [q1 , . . . , q M ]T ∈ R M and q˙ = [q˙ 1 , . . . , q˙ M ]T ∈ R M (here: angular position and the angular velocity, respectively) and Lagrangian Dynamics. The Lagrangian ˙ = K(q, q) ˙ − function of the system is given by L(q, q) ˙ = 12 q˙ T M(q)q˙ P(q) where the kinetic energy K(q, q) depends on the real, symmetric, positive definite nonsingular and uniformaly bounded [14] inertia matrix M(q) = M(q)T ∈ R M×M with det(M(q)) , 0 and λm (M(q)) · 1 ≤ M(q) ≤ λ M (M(q)) · 1 (with the minimal and maximal eigenvalues of the inertia matrix, 0 < λm < λ M < ∞) for all q ∈ R M [15, 16, 17] and the potential energy P(q). Following partial differential equation can be formulated ! d ∂L ∂L ˙ − = τ − f(q) (9) dt ∂q˙ ∂q where the vector τ = [τ1 , . . . , τ M ]T ∈ R M describes external applied inputs (e.g. joint motor torques), the ˙ ∈ R M contains viscous and/or dyvector function f(q) namic frictions terms, unstructured friction effects (e.g.
3 In the SISO case, the absolute value | · | of the error e(·) is sufficient.
4δ
racy
4
is the accuracy at time τδ and µ is the desired steady-state accu-
u sat,M
u′M
e′M .. .
q ROBOT .. . u sat,1
′ u e′ PI u1 + FC
q˙ −
y′
−
KD
u1
+
q∗
PI M
.. .
FC M
e′1
[see Eq. (11)]
KP
uM
PI1
u sat,M
u′M
q .. ROBOT . u sat,1
u′1
FC1
−
KP
y′
q˙ ∗
q˙
[see Eq. (11)]
q∗ −
KD
q˙ ∗
(b) One SISO Controller for Each Joint
(a) One MIMO Controller for All Joints
Figure 4: MIMO/SISO Funnel Controller Structures for Nonlinear M–DOF Robotic Systems of Form (11)
50
Axis Axis Axis Axis Axis Axis
0 −50 0
Position Errors eq [◦ ] - Comparison 0.5
0.5
1
1.5
1 2 3 4 5 6
FC (MIMO)
FC (MIMO)
Position q [◦ ] & Reference q∗ [◦ ] - Comparison 100
−0.25 0.5
1
1.5
Axis Axis Axis Axis Axis Axis
1 2 3 4 5 6
Axis Axis Axis Axis Axis Axis
1 2 3 4 5 6
2
0.5
50
Axis Axis Axis Axis Axis Axis
0 −50 0.5
1 Time [s]
1.5
1 2 3 4 5 6
FC (SISO)
FC (SISO)
0
−0.5 0
2
100
0
0.25
0.25 0 −0.25 −0.5 0
2
(a) Position (colored) & Reference (dotted)
0.5
1 Time [s]
1.5
2
(b) Position Errors (colored)
Figure 5: Comparison between MIMO and SISO: Angular Position q with Corresponding Reference and Position Error eq for each Axis, i = 1, . . . , 6 Funnel Gain α [1] - Comparison FC (MIMO)
FC (MIMO)
Aux. Error e′ & Boundary ±∂F - Comparison 1 0.5 ′
ke (·)k ±∂F(·)
0 −0.5 −1 0
0.5
1
1.5
40 α(·) 20
0 0
2
0.5
1
1.5
2
e′1 (·) e′2 (·) e′3 (·) e′4 (·) e′5 (·) e′6 (·) ±∂F(·)
0 −0.5 −1 0
0.5
1 Time [s]
1.5
FC (SISO)
FC (SISO)
1 0.5
20
0 0
2
(a) Auxiliary Error (colored) & Funnel Boundary (red)
α1 (·) α2 (·) α3 (·) α4 (·) α5 (·) α6 (·)
40
0.5
1 Time [s]
1.5
2
(b) Funnel Gain (colored)
Figure 6: Comparison between MIMO and SISO (i = 1, . . . , 6): Auxiliary Error e′ (MIMO) & e′i (SISO) within prescribed funnel boundary ∂F (MIMO) & ∂Fi (SISO) and corresponding Funnel Gain α (MIMO) & αi (SISO)
5
the auxiliary system is set up, where KP ∈ R M×M and KD ∈ R M×M are constant proportional and derivative gain matrices amplifying the angular positions and angular velocities, respectively. Analyzing the (vector) relative degree of the auxiliary system yields y˙ ′ = KP x˙ 1 + KD x˙ 2 = KP x2 + KD −M(x1 )−1 n(x1 , x2 ) + M(x1 )−1 · τ which proofs that the auxiliary output y′i is a function of its corresponding joint/motor torque τi , that is each branch ′ of the MIMO-system has global relative degree ri = 1. The relative degree is successfully reduced for all fullranked KD , 0 and the overall system has one integrator in each of its M forward branches (see Fig. 1). To assure the minimum-phase condition, the internal dynamics must be analysed by transforming the auxiliary ′ system into BINF with the external states ξ1i := yi and the internal states ηk := qk for all k = 1, . . . , M. One gets [ξ, η]T = Φx with the transformation matrix " # # " Φξ KP KD (14) Φ= = Φη 1 0
stiction) and external disturbances or unmodeled dynamics. The dynamics of a (rigid) robotic system can be deduced by differentiating Eq. (9) and are given by the following second-order ODE with M-inputs and Moutputs ˙ τ = M(q)q¨ + n(q, q) (10) ˙ = C(q, q) ˙ q˙ + g(q) + f(q) ˙ is the sum of all where n(q, q) nonlinearities where the uniformaly bounded [18] ma˙ ∈ R M×M includes all Centripetal and Coritrix C(q, q) olis terms, the vector g(q) ∈ R M represents the gravity force. 4.2. State-Space Realization For the later analysis, define x(t) = [x1 (t), x2 (t)]T = ˙ T ∈ RN as state vector, then the convenient [q(t), q(t)] state space representation of the robotic MIMO system # " # " 0 x2 + ·τ (11) x˙ = M(x1 )−1 −M(x1 )−1 n(x1 , x2 ) | {z } | {z } f(x)∈RN
G(x)∈RN×M
−1 As K−1 P and KD may be arbitrarily chosen to be nonsingular, the inverse transformation matrix Φ−1 exists. This transformation fulfills condition (3), since " # dΦη x 0 [1 · G = 0] =0 LG Φη x = M(x1 )−1 dxT (15)
of order N = 2M is deduced from (10). The main interest is robot positioning, thus the output is fixed to y = h(x) = x1 = q
(12)
The state-space representation (11) is reasonable, since the matrix M(x1 ) is non-singular and its inverse M(x1 )−1 exists and is uniformaly bounded away from zero for all x1 ∈ R M . In real world the working space of the robot manipulator is constrained to some D ⊆ R M and cannot be left due to physical and geometrical means.
for all x ∈ RN . The seperation of coordinates in external ξ := y′ (see Eq. (13)) and internal η := x1 = q states, allows a compact representation of the robotic system in BINF −1 −1 −1 ξ˙ = KP K−1 D ξ − KD KP η − KD M(η) n(Φ (ξ, η))
4.3. Vector Relative Degree The application of FC requires the affiliation to class S with strict vector relative degree [r1 , . . . , r M ] = [1, . . . , 1], stable Zero-Dynamics (minimum-phase property) and known (positive) sign of the highfrequency gain. By differentiating the output with respect to time i) y˙ = x˙ 1 = x2 and ii) y¨ = x˙ 2 = −M(x1 )−1 n(x1 , x2 ) + M(x1 )−1 · τ until it becomes a function of the control input u = τ, the (vector) relative degree is easily derived to [r1 , . . . , r M ] = [2, . . . , 2] with PM the sum r = i=1 ri = 2M. Hence, no direct application of FC is possible.
+ KD M(η)−1 τ
−1 η˙ = K−1 D ξ − KD KP η iT h ′ y = ξ1(1) , . . . , ξ1(M)
conform to the general form (4). The Zero-Dynamics η˙ = −K−1 D KP η := Qη
(17)
(setting ξ = 0) are asymptotically stable, if the matrix Q = −K−1 D KP exhibits eigenvalues with negative real part, that is a matrix spectrum σ(Q) ⊂ C− . As both matrices KP , 0 and KD , 0 are free design parameters, they can be set accordingly and non-singular by the control engineer without any knowledge of system parameters. Additionally, for any scaled positive definite derivative gain identity matrix e.g. KD = β·1 with β > 0,
4.4. Auxiliary System and Affiliation to Class S By introducing a state-feedback like structure defining an auxiliary output (following the idea in [7, 8, 9]) y′ = h′ (x) = KP x1 + KD x2
(16)
(13) 6
5. Error Reference Control (ERC)
the positive high-frequency gain is not touched. Since KD ·M−1 (η) = M−1 (η)·KD = β·M−1 (η) > 0 is also positive definite as the matrices commute. This reveals with (16) for every non-empty compact set C ⊂ R M ×R M and sequence (un ) = (τn ) ⊂ R M \ {0} with kun k → ∞ as n → ˙ M ∞ that min[ξ,η]∈C hukunn,ξi k → ∞ holds true for all η ∈ R (generalization of the high-frequency gain concept, see [1]). In conclusion, the introduced auxiliary system is element of class S and may be controlled by FC.
ERC is directly derived from FC (so far for the SISO case only). Its special asymmetric design of the funnel boundaries (“virtual tube”, see Fig. 7) and adequate choice of its auxiliary reference enable ERC not only to predefine transient behavior within the given limits, but also to guide the error along a desired error reference trajectory (see Fig. 7). The control-loop conforms to that shown in Fig. 2 where the Funnel Controller is replaced by the Error Reference Controller and all signals by scalar signals. The same class S of high-gain controllable systems can be stabilized and asymptotic tracking within the virtual tube TERC = {e ∈ R| e∗ (t) < e < e∗ (t) ∀t ≥ 0} (see Fig. 7 for a possible and illustrative example tube/boundary design and error evolution) guaranteed, if the initial error lies inside TERC . The er-
4.5. MIMO/SISO Control-Loop
Error
Fig. 4 illustrates possible control-loops: R a MIMO and a SISO approach with u′ (t) = u(t) + T1I u(τ)dτ where u is defined as in Eq. (5). In Fig. 4(a) the MIMO variant with one controller (FC+PI) for all M joints is depicted. Fig. 4(b) shows M–SISO controllers (FCi + PIi , i = 1, . . . , M) for the M joints, each axis is driven by one SISO control-loop.
e(0)
4.6. Simulation Results
∗
e (·)
The introduced MIMO/SISO FC+PI controllers with exponential funnel boundary are now tested for a 6– DOF Manutec r3 Robot implemented in Modelica [19] with adequate simulation interface to Matlab/Simulink. The PI-like structure is implemented as nominal linear PI block with F PI (s) = 1 + sT1 I . Due to a diagonal choice of KP = 15 and KD = 0.01 · 1, the Zero-Dynamics are asymptotically stable with σ(Q) = {−100, . . . , −100} ⊂ C− . The realistic model also exhibits elasticity due to gears, nevertheless the application of Funnel-Control is not deteriotated as the minimum-phase condition is not touched (passive influence of the gear elasticity! [20]). Although none of the system parameters are known or estimated, the controlled robot follows the reference trajectories q∗i (t)6 quite nicely, both for the MIMO and the SISO controller7 (see Fig. 5(a)). The maximal control error does not exceed the pre-set accuracy limit µ = 0.5◦ (see Fig. 5(b)). Both strategies assure an auxiliary error evolution within the prescribed funnel boundary (see Fig. 6(a)) by their instantenous gain adaption (see Fig. 6(b)) indirect proportional to the measured distance to the funnel boundary.
e∗ (·)
e∗ (·) dV (t) ∆e(t)
e(·) Time [s]
t dV (t)
Figure 7: Principle Idea of Error Reference Control (ERC) - SISO Case for initial errors e(0) ≥ 0 (for initial errors e(0) < 0 analogous)
ror evolution now is even more strictly bounded by the virtual tube (see Fig. 7) around a predefined desired error evolution — the Error Reference trajectory e∗ (·) ∈ C 1 (R≥0 ; R). The tube must be designed in such way, that the (known!) initial error e(0) = y∗ (0) − y(0) − n(0) is enclosed. In this paper, the error reference trajectory is exemplary set to ! t e∗ (t) = y∗ (0) − y(0) − n(0) · exp − (18) | {z } T exp e(0)
where its time constant T exp > 0 may be chosen arbitrarily. The desired error reference e∗ (·) should clearly tend to zero. Error Reference Control ensures guidance of the error along e∗ (·) within the virtual tube TERC by applying the proportional control law (utilizing (6) in the SISO case)
51
= diag(1, . . . , 1) ∈ R M represents the diagonal identity matrix references for each joint angles [◦ ]: q∗1 : −30 → 50◦ , q∗2 : 80 → −15◦ ,q∗3 : 65 → −55◦ ,q∗4 : −20 → −75◦ ,q∗5 : −90 → 100◦ and q∗6 : 10 → −70◦ 7 Controller Design: i) Costumer Specifications (µ = 0.5◦ , τ = δ 0.3s, δ = 0.75◦ /s); ii) Initial Boundary ∂Fexp (0) = 1; iii) PI Integrator Time Constant T I = 0.1s and iv) Scaling Factor Ψ(t) = 10 6 Smooth
uERC (t) = αERC (t) · e(t) − e∗ (t) ∗
∗
= αERC (t) · [y (t) − e (t) −y(t) − n(t)] | {z } =:y∗ERC (t)
7
(19) (20)
1/gr
with the specially chosen auxiliary reference y∗ERC = y∗ − e∗ ∈ W 1,∞ (R≥0 ; R) (corresponding to the SISO case of FC [1, 2]) where the time-varying gain αERC (t) =
Ψ(t) min dV (t), dV (t)
ML kD
MM −
(21)
∆ϕ˙ c
(22)
1 e∗ (t) = e∗ (t) − µ 1 + κ(t) =: ∂F ϕ (t) = ϕ(t)
(23)
−
ΩA
BL ∆ϕBL
∆ϕ
1/sΘ M
1 ϕ(t)
e∗ (t) = e∗ (t) + µ (1 + κ(t)) =: ∂F ϕ (t) =
1/gr ΩM
−
— similar to FC, see Eq. (7) — is a function of the scaling factor Ψ(t) and indirect proportional to the evaluation of the minimal (vertical) distance dV (t) = e∗ (t)−e(t) or dV (t) = e(t) − e∗ (t) between the upper e∗ (t) or lower e∗ (t) limit of the virtual tube and the error e(t), respectively. The tube limits are generated according to the desired error reference trajectory in the following manner: the upper limit
NF
d
MC + MD
1/sΘA
Figure 8: Block Diagram of Nonlinear T MS
that the error will now even closer coincide with the desired error reference e∗ (·), as any deviation will additionally increase the control gain αERC (·) and accelerate the closed-loop system towards the predefined track. Besides that, possible oscillations around e∗ (·) are seemingly damped as well. The scaling function (24) can obviously also be applied to standard FC but is much less effective — particularly at startup where the Funnel is still wide — as the control input (5) will solely change its sign with the control error.
and the lower limit
6. Nonlinear Two-Mass-System (T MS)
where κ, κ ∈ C(R≥0 ; R≥0 ) are free design functions with limt→∞ κ(t), κ(t) = 0. Upper and lower limit tend to the asymptotic accuracy limt→∞ e∗ (t) = µ and limt→∞ e∗ (t) = −µ, respectively. The tube TERC is a specially designed asymmetric funnel boundary with upper ∂F ϕ (·) and lower ∂F ϕ (·) limiting function. Due to the choice e∗ (t0 ) < e(t0 ) = e∗ (t0 ) < e∗ (t0 ), the initial error starts within the tube. Whether the error is above or below the desired error trajectory e∗ (·), the control input changes its sign and therefore allows appropriate acceleration or deceleration of the system. This property ensures in contrast to standard FC design not only the evolution within the virtual tube but also the possible guidance along the desired error reference e∗ (·). The following proposal for a damping algorithm has indicated in simulations and experiments to have beneficial effects — overshoots and oscillations are drastically reduced — on the system response, but yet the mathematical proof for applicability and general improvement is not finished. The suggested damping algorithm is realized by the implementation of a saturated scaling function Ψmax Ψ (t) = sat ΨD (e(t) − e∗ (t))(˙e(t) − e˙ ∗ (t)) 0 D +Ψ0 (24)
The considered plant (shown in Fig. 8) is a nonlinear two-mass flexible servo system (T MS ), which is a common example for an electrical drive connected to a load machine via gear and flexible shaft. The system has the state vector xT = [Ω M ∆ϕ ΩA ] with the revolution speed of the motor Ω M and of R tthe1 load machine ΩA and the angle of twist ∆ϕ(t) = 0 ( gr Ω M (τ) − ΩA (τ))dτ = 1 gr ϕ M − ϕA . The system’s input is the motor torque M M (t) = u(t) + uD (t), depending on the chosen control law (5) or (20) and an active damping term uD (to be specified later). The load velocity is the controlled variable y(t) = ΩA (t). The nonlinear state space ODE is given by x˙ =
1 ΘM
1 ΘA
M M − gdr
∂BL(∆ϕ) − gcr ∂t
1 gr
BL(∆ϕ)
Ω M −ΩA −ML −NF (ΩA )+d ∂BL(∆ϕ) +cBL(∆ϕ) ∂t
(25)
with the stiffness c > 0, the damping d > 0, the (constant) gear ratio gr > 0, the inertia of the motor Θ M and of the load ΘA [21]. The nonlinearities BL(·) and NF (·) represent the backlash of gear with the backlash angle ∆ϕBL (t) := BL(∆ϕ(t)) where
with the minimal offset Ψ0 ≥ 1, the damper gain ΨD ≥ 0 und the maximal scaling value Ψmax D , which limits the damper influence on e.g. measurement noise amplification. All made simulations and experiments have shown
8
∆ϕ + aBL 0 BL(∆ϕ(t)) = ∆ϕ − a BL
, ∆ϕ ≤ −aBL , |∆ϕ| < aBL , ∆ϕ ≥ aBL
(26)
and the friction torque MNF (t) := NF (ΩA (t)) where 0 NF (ΩA ) = mNF ΩA + arctan(103 · ΩA )MNF
which corresponds to the inherent frequency of a T MS [22] and is seemingly not changed. But on the other hand, the system damping may be altered arbitrarily, which becomes obvious by evaluation of the damping coefficient ω0 ΘA d + kD D= (31) 2c gr Θ M + Θg2A
(27)
with the viscous friction coefficient mNF > 0 and the 0 Coulomb friction offset MNF > 0. Both nonlinearities are modelled continuously. Therefore those do not affect affiliation of the T MS to system class S and do not deteriorate the latter control performance of FC and ERC [1, 8]. The current control loop is assumed to be fast enough and is thus neglected (adequate for modern power converters and drives [8]). So the torque M M can be an arbitrary function of time, whereas the conmax max >0 is constrained by M M trol input |M M (t)| ≤ M M to protect the motor and the power electronics unit (converter). Within this paper the control inputs will not exceed the allowed range.
r
Therefore, for any kD > 0 the damping coefficient of the T MS system is increased (for simulative illustration, see Fig. 9). Constraints of this damping approach are: i) the available control input (how much of the overall motor torque should be reserved for damping?), ii) the achievable performance (dynamic) of the power converter and drive (how fast can the damping torque be generated) and iii) the higher the chosen gain kD , the slower the system response (since we penalize any change ∆ϕ˙ of the torsional angle). All three constraints are not to severe for real world application, if well specified actuators and power electronics are used, being able to allow damping. Important to note that, we must not choose negative gains kD < 0, as negative values will decrease damping or evendestabilize the system for all
6.1. Active Damping by Static Feedback If the nonlinear two-mass flexible servo system exhibits serious oscillations between motor and load speed (or position), explicit necessity for active damping is indicated. Therefore we present a simple approach, which allows to tune a static damping feedback # " 1 (28) Ω M − ΩA uD = −kD gr
r feedback gains k∗D ≤ − dg ΘA Θ M +
to increase and adjust the system‘s damping to one‘s need, if the underlying control loop achieves global stability (like FC and ERC). Since we induce the change (time derivative) of the angle of twist ∆ϕ, ˙ any (foremost fast) changes in the shaft‘s torsion are penalized by the weighting factor kD ≥ 0 (free design parameter, e.g. can be determined emperically during system run). The influence of this static damping feedback is easily analyzed by investigation of the transfer function of the linear system [22], see (25) neglecting backlash and friction: FTΩMS (s) = 1 + dc s Θ M ΘA 1 + s ΘA c d +
ΘA g2r
2 s Θ
y′ (t) = kT x(t)
M+ 2 gr
c
gr
(32)
resulting in a linear combination of the constant feedback vector kT = [k1 k2 k3 ] with the state vector xT = [x1 x2 x3 ] = [Ω M ∆ϕ ΩA ]. By choosing the feedback structure to fulfill the relations: k1 > 0, k2 > −d/ΘA (k1 + k3 ) and k3 > −k1 , it can be guaranteed that the relative degree of the T MS (for any parameter sets of ΘA , Θ M , c, d, gr — all greater than zero by physical means) is reduced to rT′ MS = 1 and the required minimum phase property is retained [9]. Besides, the above introduced static damping feedback with kD ≥ 0 does not touch the minimum-phase condition. The numerator of (29) does not depend on the parameter kD and hence remains Hurwitz for all c, d > 0 (this result can be easily derived for the nonlinear case as well).
ΩA (s) = M M (s)
ΘA kD Θ gr (Θ M + 2A )
