Experimental Robustness Stuy of a Secon-Orer Sliing Moe Controller Anré Blom, Bram e Jager Einhoven University of Technology Department of Mechanical Engineering P.O. Box 513, 5600 MB Einhoven, The Netherlans Email: A.G.e.Jager@wfw.wtb.tue.nl Fax: +31 40 2461418 The esign of a Secon-Orer Sliing Moe Controller is iscusse an guielines are given for tuning. The robustness to unmoele ynamics an parameter-errors is investigate an teste in an experimental case stuy. The experimental results, as far as robustness to unmoele ynamics is concerne, are not better than for a traitional PD-controller. When robustness to parameter-errors is concerne the Secon-Orer Sliing Moe Controller performs slightly better. 1 Introuction Mechanical manipulators are controlle to make their en-effector track a esire trajectory. The control is often base on a mathematical moel that represents the ynamic behavior of the system. In practice, this moel is never an exact representation of reality. There are always phenomena like unmoele ynamics, inaccurate parameters an measurement noise that cause moel imprecisions. These imprecisions may come from actual uncertainty about the system, or from the eliberate choice for a simplifie representation of the system ynamics. The presence of these imprecisions often requires a robust control algorithm. One class of robust controllers is VSS (Variable Structure System), see Utkin (1977). This class of controllers can be use when the moel structure itself is inaccurate, but the inaccuracies are boune with known bouns. VSS controllers are often use as Sliing Moe Controllers. Characteristic components of Sliing Moe Controllers are sliing hyperplanes s(x, t) = 0 in the state space. Recently the traitional sliing control (applying a first-orer sliing conition) has been further enhance, resulting in a Secon-Orer Sliing Moe Control (SOSMC). The SOSMC was presente by Chang (1990) for MIMO systems in Controllability Canonical Form. Elmali an Olgac (1992) extene the SOSMC-technique to general nonlinear MIMO systems, applying I/O-linearization first. In this paper the theory of SOSMC in combination with I/O-linearization is teste in an experimental environment to investigate the robustness to unmoele ynamics (i.e., system of higher orer than the moel) an parametererrors. To tune the SOSMC, a set of guielines is given to achieve best performance. An expression is given for the tracking accuracy an from this expression we recognize a trae-off between tracking accuracy an amping. 2 Preliminaries Consier a nonlinear system that can be escribe (exactly) by a MIMO moel in Controllability Canonical Form, linear in the control u (affine): x (n) = f (x) + B(x)u (1) Publishe in J. Dynamic Systems, Measurement, an Control, Vol. 118, No. 1, March 1996, pp. 184 187. Author to whom all corresponence shoul be sen. 1
with x (n) = [y (n 1) 1, y (n 2) 2,...,y (n k) k ] T, n 1 + n 2 + +n k = n with: x R n (state vector) u R m (input vector) y = [y 1,...,y k ] T R k (output vector) Superscripts in parenthesis inicate the orer of time erivatives. Uner certain conitions, a nonlinear system can be transforme into the Controllability Canonical Form by a technique calle I/O-linearization. This technique an the conitions are escribe by Elmali an Olgac (1992), base on Isiori (1989). Assume that to control the system a mathematical moel of the system is available: x (n) = ˆ f (x) + ˆB(x)u (2) For simplicity we assume that the states can be ientifie with those of (1). This is not necessary. Because sliing control requires the uncertainties to be boune with known bouns, a general assumption is: f (x) = f ˆ(x) + f (x) with f (x) α x B(x) = ˆB(x) + B(x) with B(x) β x (3) In properly controlle systems the state vector x will behave boune, so the uncertainty bouns can be etermine. 3 MIMO Sliing Control With Secon-Orer Sliing Conition A secon-orer Sliing Moe Control strategy by Chang (1990) efines a zero z 0, in the error ynamics : ( ) t + z 0 i s i = n i j=1 ( ) t t + λ ji e i τ (4) 0 e i = y i y i for i = 1,...,k This equation is a set of ban-pass filters where the break-frequencies are etermine by the selection of the poles (λ ji ) an zero (z 0i ). An integral term in the equation assures zero steay-state errors. Writing (4) in the unfactore (polynomial) form we get (see Chang (1990)): t ṡ + Z 0 s = e (n 1) + C n 1 e (n 2) + +C 1 e + C 0 e τ (5) where ṡ = 0 represents the sliing hyperplanes. Taking time erivatives of (5) yiels: s + Z 0 ṡ = e (n) + e p (6) with e p = C n 1 e (n 1) + +C 1 ė + C 0 e anthisisusetorelates with the control input u. The error-vector e p can be compute if all states are measure. Substituting (1) in (6) yiels: s + Z 0 ṡ = f + Bu x (n) + e p (7) Stability is guarantee if the control is esigne as: u = ˆB 1 (û k sign(ṡ)) (8) û = ˆ f + x (n) e p + Z 0 ṡ s with = iag(ω 2 n ) 0 2
an the gain k is quantifie as (see Chang (1990) an Elmali an Olgac (1992)): α + β ˆB 1 û k > if β ˆB 1 sign(ṡ) < 1 (9) 1 β ˆB 1 sign(ṡ) We get the s-ynamics by substituting the control (8) in (6): s + k sign(ṡ) + s + B ˆB 1 (k sign(ṡ) Z 0 ṡ + s) = f + B ˆB 1 (x (n) ˆ f e p ) (10) As we can see from (8) the control law is iscontinuous across ṡ = 0, which leas to chattering. In general, chattering must be eliminate for the controller to perform properly. This can be achieve by smoothing out the control iscontinuity in a bounary layer neighboring the switching surface (ṡ = 0). Therefore we on t use the signum -function (sign), but we apply the saturation -function (sat) instea for each element ṡ i of ṡ: 1 ṡ i >φ ṡ i sat(ṡ i,φ) = ṡ i φ (11) φ 1 ṡ i < φ Then, the s-ynamics within the bounary layer become: s + k 1 ṡ + s + B ˆB 1 (k 1 ṡ Z 0 ṡ + s) = f + B ˆB 1 (x (n) This equation represents a set of secon-orer low-pass filters. ˆ f e p ) (12) 4 Tuning the SOSMC The lack of tools in nonlinear systems theory now creates a problem; a systematic way of selecting the parameters Z 0, ω n,, C n 1,...,C 0 an k oes not exist. Yet, the influence of the tuning parameters can be evaluate. First, consier the gain k. This controlparameteris completelyetermineby the confinementson the uncertainties an can be calculate with (9). For control esign purposes, the minimum value of k is selecte, since the least control effort is esire. Next, we investigate the influence of the zero z 0 in the error-ynamics. Chang (1990) suggeste that higher values of the zeros provie more amping in the s-ynamics. As we can see from (10/12), z 0 inee contributes in the amping of the s-ynamics, but the exact influence is not clear, since the total amping can increase an ecrease, epening on the sign of the matrix B ˆB 1 Z 0. Simulations, however, inee show more amping when z 0 is increase. This is very likely ue to the error-ynamics (4), in whose response z 0 has a amping contribution, recognize as a reuction of the rise-time. Thir, we investigate the influence of ω n. As we can see from the s-ynamics insie the bounary layer (12), ω n sets the break-frequency. Preferably we choose this break-frequency (etermining the s-ynamics banwith) smaller than the lowest unmoele structural resonant moe. Fourth, to simplify the choice of the poles in the error ynamics (by setting C n 1,..., C 0 ) we can choose the banwith of the error ynamics the same as the banwith (ω n )ofthes-ynamics. This is not necessary. As we will show, a larger error-ynamics-banwith has a positive effect on the tracking error, but may involve a higher control effort. Fifth, we will motivate the choice of, by eriving an expression for the tracking error. Within a finite time ṡ <. Noting that ṡ contains no frequencies higher than ω n (as an approximation) we fin: s <ω n,sothat the maximal tracking error will be (once ṡ < an for the SISO case): ε < φ(ω n + z 0 ) (13) λ 1 λ 2 λ n with a guarantee precision ε (Blom, 1992). has a minimum value, since the implementation has a limite sample frequency. We see from (13) that the zero affects the tracking accuracy in a negative sense also. However, this zero provies amping, which is especially important uring transient. We see that there exists a trae-off between tracking accuracy an amping. 3
5 Experiments To investigate the SOSMC in an experimental environment, the control law was implemente in the control software of an XY-table. In Fig. 1 a schematic top view representation of the XY-table is given. The en effector is a slie with mass m e, which can move in the XY-plane by three slieways. Two of them slie in x-irection an one in y-irection. The belt wheels of both slieways are riven by servomotors, exerting torques T 1 an T 3. Coulomb friction appears in all slies an is represente by friction-torques W 1, W 2,anW 3 [Nm]. Viscous amping is represente by D 1, D 2,anD 3 [Nms]. Figure 1: Schematic representation XY-table Unfortunately it is not possible to measure the position of the en effector irectly. Only three encoer signals are available: x 1, x 2,any. We therefore restrict ourselves to the control of motor positions: x 1 an y. The belt wheels of the slieways in x-irections can be connecte in two ways: 1. With a rigi bar (k 1 = ), resulting in a (stiff) moel with two egrees of freeom: x 1, y, since the translations x 1 an x 2 are equal. 2. With a torsion spring with stiffness k 1, resulting in a (flexible) moel with three egrees of freeom: x 1, x 2,any. The equations of motion in case of the two egrees of freeom (stiff) moel are represente by: a 1 ẍ 1 + a 3 sign(ẋ 1 ) + a 5 ẋ 1 = T 1 /r x a 2 ÿ + a 4 sign(ẏ) + a 6 ẏ = T 3 /r y (14) with ientifie parameters: a 1 = J 1 /rx 2 + 2m s + m e + m y = 34 [kg] a 2 = J 3 /ry 2 + m e = 2.7 [kg] a 3 = (W 1 + W 2 )/r x = 36 [N] a 4 = W 3 /r y = 9 [N] a 5 = (D 1 + D 2 )/rx 2 = 50 [Ns/m] a 6 = D 3 /ry 2 = 8 [Ns/m] r x = = 0.01 [m] r y = = 0.01 [m] The equations of motion in case of the three egrees of freeom (flexible) moel are much more complex an will therefore not be presente. 4
The esire trajectory to be tracke by the en-effector uring all experiments is chosen to be a circle: x = 0.5 r cos(ωt) y = 0.5 + r sin(ωt) with r = 0.25 [m] an ω = π [ra/s]. The setting of the SMC is tune up, to get the best results, i.e., the controlle system banwith is chosen maximal, with a relative amping in the error ynamics of β = 0.71. The gain k has been etermine by assuming that the available three egrees of freeom moel is an exact representation of reality, so that the uncertainty bouns on the two egrees of freeom moel can be calculate (3). The control parameters are liste below. c 1 = 63.6 [ra/s] c 0 = 2025 [ra/s 2 ] ω n = 45 [ra/s] z 0 = 90 [ra/s] k = 70 [m/s 2 ] = 1.11 [m/s] To assess robustness of the SOSMC to unmoele ynamics, experiments are one with several torsion springs k 1. The results are compare with a traitional PD-controller, whose setting is also tune up, resulting in a controlle system break frequency of 45 [ra/s]. To eliminate the trae-off between amping an tracking accuracy, experiments are one while moifying the zero on-line, from the initial (i.e., high) value uring transient (amping) to half the initial value (i.e., low) afterwars (high tracking accuracy). If switching is only one once, stability is guarantee. The results for the x-irection only are shown in Fig. 2. In this figure the tracking error RMS is plotte against the stiffness of torsion spring k 1 (8 springs were available with stiffnesses from 0.19 to 213 [Nm/ra]). RMS of tracking error [mm] 10 0 10-1 10-1 10 0 10 1 10 2 10 3 Spring stiffness [Nm/ra] Figure 2: Robustness to unmoele ynamics; soli: SOSMC, ashe: SOSMC with moification z 0, otte: PD control We see that for all stiffnesses k 1 the SOSMC with moification of z 0 realizes a smaller tracking error than an orinary SOSMC, as expecte (13). For relatively stiff torsion springs (right sie) both SOSMC controllers (with an without moification of z 0 ) realize a smaller tracking error RMS than a PD controller. However, the tracking error RMS for the PD-controller is more constant for a wie range of stiffnesses, in contrast with the SOSMC. For one torsion spring (k 1 = 0.5 [Nm/ra]) the tracking error realize with a PD-controller is even smaller than realize with a SOSMC. We therefore conclue that a PD controller is more robust to unmoele ynamics (ue to stiffness k 1 ) than SOSMC. The level of robustness for both SOSMC is approximately the same. A strange phenomenon is that for a ecreasing stiffness the tracking error RMS increases, but the weakest spring k 1 again yiels a small tracking error RMS. This phenomenon can be seen in all experiments an is because we control the motor positions. 5
To assess robustness to parameter variations, experiments are one with aitional mass attache to the eneffector (m e ). The results are shown in Fig. 3. In this figure the tracking error RMS is plotte against the aitional mass. RMS of tracking error [mm] 10 0 10-1 0 1 2 3 Aitional mass [kg] Figure 3: Robustness to parameter variations; soli: SOSMC, ashe: SOSMC with moification z 0, otte: PD control For all mass-variations the SOSMC with moification of z 0 realizes the smallest, an the PD-controller the highest tracking error. The tracking error, realize with both SOSMC is more constant for variations in the aitional mass, in contrast with the PD-controller. We conclue that both SOSMC are more robust to mass-variations than a PD-controller. Again, the robustness of both SOSMC is approximately the same. 6 Conclusions The main conclusion of this investigation into robustness of the SOSMC is that as far as robustness to unmoele ynamics is concerne, there is no avantage in using a SOSMC technique in favor of a traitional PD-controller. As far as robustness to parameter errors is concerne, the SOSMC performs slightly better than a PD controller. However, for a large range of operating conitions the SOSMC has far better tracking properties, ue to its moel base structure, an shoul be given preference. References Blom, A., 1993, Robustness of a Secon Orer Sliing Moe Controller, Masters thesis, Einhoven University of Technology, Department of Mechanical Engineering, WFW Report 93.007. Chang, L.-W., 1990, A MIMO Sliing Control with a Secon-Orer Sliing Conition, ASME WAM, paper no. 90 WA/DSC-5, Dallas, Texas. Elmali, H an Olgac, N., 1992, Robust Output Tracking Control of Nonlinear MIMO Systems via Sliing Moe Technique, Automatica, Vol 28, No. 1, pp. 145 151. Isiori, A., 1989, Nonlinear control systems, an introuction, Berlin: Springer-Verlag. Utkin, V. I., 1997, Variable Structure Systems with Sliing Moes, IEEE Transactions on Automatic Control, Vol. AC-22, Apr., pp. 212 222. 6