Sliding Mode Control and Feedback Linearization for Non-regular Systems

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1 Sliding Mode Control and Feedback Linearization or Non-regular Systems Fu Zhang Benito R. Fernández Pieter J. Mosterman Timothy Josserand The MathWorks, Inc. Natick, MA, University o Texas at Austin, Austin, TX, Abstract: This paper presents an approach or sliding mode control(smc) and eedback linearization(fbl) o systems with relative order singularities. Traditionally, SMC and FBL are designed by taking derivatives o the output until the control signal appears(at the r1 th derivative). When a system does not have a well-deined relative degree (r 1 ), the coeicient that multiplies u vanishes or some region o the state-space S 1. In this instance, conventional SMC and FBL techniques ail. The presented approach dierentiates urther the output until the control input appears again (the secondary relative degree, r 2 ) and a dierential equation in u is acquired. It may be possible to solve or a dynamic compensator, or in the neighborhood o the singularity, N 1, the equations degenerate to a polynomial orm. Preliminary results show that at the singularity region, S 1, the control-derivative term disappears and the dierential equation is degenerated to a center maniold deined by a polynomial (quadratic in general) equation on u. The solution to the quadratic equations is discussed. When this equation has only real roots, the system is well deined at the singularity. A switching controller can be designed to switch rom the r1 th controller when system is ar away rom the singularity to the r2 th controller when the system is in the neighborhood o the singularity. We demonstrate the controller applied to the ball and beam system. 1. INTRODUCTION The ball and beam (B&B) system (Fig. 1) is one o the most popular models or studying control systems because o its simplicity and yet the control techniques that can be studied cover many important control methods (Barbu et al. 1997, Hauser et al. 1992, Lai et al. 1994, Leith and Leithead 2001, Marra et al. 1996, Tomlin and Sastry 1997, Yi et al. 1996). Fig. 1. The ball beam system The (B&B) system is non-regular, i.e., the relative degree o it is not well deined at certain locations in the phase space. This interesting property, common to other diicultto-solve control systems, has motivated much research in the past. Thus, conventional exact eedback linearization(efbl) and sliding mode control(smc), are hard or simply do not apply. The well-known work o Hauser et. al. (Hauser et al. 1992), used approximation eedback linearization(afbl) by disregarding certain terms that lead to the singularity. However, this approach does not work well when the system is away rom the singularity, because o the approximation error that is generated by disregarding the terms. Tomlin et. al. (Tomlin and Sastry 1997) proposed a switching control law: a controller that uses efbl when the system is in the region ar away rom the singularity and a switch to the afbl controller when the system approaches the singularity. Lai et. al. (Lai et al. 1994), proposed a tracking controller based on approximate backstepping (abs) that has better steady state error than other approximation methods. Other approaches or the B&B control problem include a uzzy controller (Yi et al. 1996) and a genetic controller (Marra et al. 1996). This paper tries to generalize the control problem to non-regular systems. The main idea is to deine multiple relative orders r k and propose controllers (FBL or SMC) or each o the relative orders and create a control law that switches when the system approaches the singularity neighborhood S k associated with each relative order. The result is a group o possible switching controllers. The speciic desired controller structure will depend on the particular system characteristics. The technique is applied to the B&B problem, generating a switching control law similar to the switching controller presented in (Tomlin and Sastry 1997). The contribution o this paper is that by taking (r + 1) th (r is the relative degree o the system away rom singularities) derivatives, it shows that the neighborhood o the singularity can be urther divided into two regions. In one o the region, the exact system still have a well-deined relative degree. While approaches in (Hauser et al. 1992) and (Tomlin and Sastry 1997), the exact system is not well-deined in the

2 whole region. Our controller, by dividing the neighborhood into two regions, is able to realize a more precise control. This paper is organized as ollows: Section 2 is a brie description o the supporting methods presented in (Hauser et al. 1992, Tomlin and Sastry 1997). Section 3 presents the high order derivative approach, with the ball and beam system used as an example. Section 4 shows how to extend the high order method to SMC. Section 5 is the discussion o the presented approach and the behavior o a nonregular system, Section 6 gives some simulation results and Section 7 is the generalized ormulation o the presented method to solve eedback linearization with singularities. Section 8 presents the summary and some open questions. 2. APPROXIMATION FEEDBACK LINEARIZATION Figure 1 shows the ball and beam schematic as a speciic sample o the class o systems under investigation(namely, non-regular systems). The controller input τ rotates the beam with the ball on it, around a pivot point. The ball rolls based on the gravitational pull projected by the beam s angle, θ. The objective o the controller is to maintain the ball at a distance r d rom the pivot point. We assume that that the ball is always in contact with the beam. Without loss o generality, we can neglect the contact riction. Using a nonlinear transormation, the equations o the system can be written as (Hauser et al. 1992): 0 ẋ 1 ẋ2 ẋ3 ẋ4 x 2 B(x 1 x 2 4 g sin x 3 0 x u (x) + g(x) u 0 1 y x 1 h(x) (1) where, x 1 (t) r(t) is the ball position measured rom the beam center to the ball center, x 2 (t) ṙ(t) v(t) is the ball relative velocity to the beam, x 3 (t) θ(t) is the beam angle, and x 4 (t) θ(t) ω(t) is the beam s angular velocity. B and g are constants and the input u is a nonlinear transormation o the torque τ. Using the typical steps or eedback linearization (FBL) or sliding mode control (SMC), take the derivatives o y till u appears at the right-hand side, we obtain: y L 0 (h) x 1 h(x) ẏ L 1 (h) x 2 L h(x) ÿ L 2 (h) B(x 1x 2 4 g sin(x 3 ))... y B(x 2 x 2 4 gx 4 cos x 3 ) + 2Bx 1 x 4 u (2) }{{}}{{} L 3 (h) L g (L 2 (h)) where L k (h) is the kth Lie derivative o h along (Fernández-Rodríguez 1998). Here, u appears at the right-hand side o the 3 rd derivative o y, hence the relative degree o the system is 3, except at the singularity S 1 { x R n Lg(L 2 (h(x))) 0}. At this point, u disappears rom the right-hand side o the derivative, and the relative degree is not well deined. I we want to FBL, we can make... y v, then u v B(x 2x 2 4 gx 4 cos(x 3 )) v(t) α(x) (3) 2Bx 1 x 4 The afbl technique presented by Hauser (Hauser et al. 1992) proposes two approximation methods. The irst one is to disregard the x 1 x 2 4 term in Eq. (2) and then dierentiate the output y until u appears (again): ξ 1 y x 1 ξ 1 ξ 2 ẏ x 2 ξ 2 ξ 3 ÿ Bx 1 x 2 4 }{{} Disregards this term ξ 3 ξ 4... y Bgx 4 cos(x 3 ) Bg sin(x 3 ) ξ 4 y 4 Bgx 2 4 sin(x 3 ) Bg cos(x 3 ))u (4) u v Bgx2 4 sin(x 3 ) Bg cos(x 3 ) v(t) α(x) (5) The other approximation is to disregard the 2Bx 1 x 4 u term in Eq. (2) and then take the 4 th derivative o y: y x 1 ẏ x 2 ÿ B(x 1 x 2 4 g sin(x 3 ))... y B(x 2 x 2 4 gx 4 cos(x 3 )) + 2Bx 1 x }{{} 4 u Disregards this term y 4 B 2 x 1 x B(1 B)x 2 4 sin(x 3 ) + ( Bg cos(x 3 ) + 2Bx 2 x 4 )u u v (B2 x 1 x B(1 B)gx 2 4 sin(x 3 )) B(2x 2 x 4 g cos(x 3 )) v(t) α(x) This approximation approach (Hauser et al. 1992) works well when system is ar away rom the S 1. In (Hauser et al. 1992), both approximations disregarded the terms that lead to singularity beore taking the 4 th derivative o y and thus eectively adding modeling errors. These techniques are compared in a later section with our proposed approach and are urther discussed then. 3. APPROXIMATION USING HIGH ORDER DERIVATIVES An alternative is to take the 4 th derivative o y without disregarding the nonlinear terms. This is typically not done because it results in a dierential equation o u that could be diicult to solve and because it creates a dynamic compensator. Following the general eedback linearization procedures and dierentiating the output one more time, we obtain u at the right-hand side: ξ 1 y x 1 ξ 1 ξ 2 ẏ x 2 ξ 2 ξ 3 ÿ B(x 1 x 2 4 g sin(x 3 )) ξ 3 ξ 4... y B(x 2 x 2 4 gx 4 cos(x 3 ) + 2x 1 x 4 u) ξ 4 y 4 Bẋ 2 x x 2 x 4 ẋ 4 g(ẋ 4 cos(x 3 ) x 4 ẋ 3 sin(x 3 )) + 2(ẋ 1 x 4 + ẋ 4 x 1 )u + 2 ux 1 x }{{} 4 (7) u term (6)

3 Substitute the state space equations (1) into (7), yields: y 4 B(Bx 1 x 2 4 Bg sin(x 3 ))x x 2 x 4 u (8) g(u cos(x 3 ) x 2 4 sin(x 3 )) + 2(x 2 x 4 + ux 1 )u + 2 ux 1 x 4 }{{} This term vanishes at singularity S 1 Around the neighborhood o the singular point, N 1, where, N 1 { x R n, z S 1 x z < δ 2} (9) we have x 1 x 4 0 and Eq. (8) becomes: y 4 B Bx 1 x 4 4 Bg sin(x 3 )x 2 4 (10) + 2x 2 x 4 u g(u cos(x 3 ) x 2 4 sin(x 3 )) + 2(x 2 x 4 + ux 1 )u Let v y 4, collecting the terms, a quadratic equation o u at the singular point results: 2Bx 1 u 2 + B(4x 2 x 4 g cos(x 3 )) u (11) + B 2 x 1 x Bg(1 B) sin(x 3 )x 2 4 v 0 I the general FBL procedure is applied and the 4 th order derivative o the output is calculated, this yields Eq. (8), a dierential equation in u. This results in a dynamic controller that may be implemented outside N 1. However, close to the singular point, x N 1, x 1 x 4 0, and the dierential equation degenerates to the quadratic equation Eq. (11) that can be used to solve or u. Further more, Eq. (11) can be used to approximate the system around the neighborhood o the singularity, N 1. By using the switching idea introduced in (Tomlin and Sastry 1997), a switching controller can be designed using Eq. (3) when x N 1 and Eq. (11) when x / N 1. Unlike (Hauser et al. 1992) and (Tomlin and Sastry 1997), Eq. (11) is an exact eedback linearization o the original system at the singularity without disregarding the terms that lead to the singularity. Since Eq. (11) is a quadratic equation, the general solutions are two conjugate complex roots. Deine: a(t) 2Bx 1 b(t) B(4x 2 x 4 g cos(x 3 )) c(t) B 2 x 1 x Bg(1 B)x 2 4 sin(x 3 ) v b 2 4ac u b ± 2a (12) Solutions to Eq. (11) depend on the value o in Eq. (12). In order to implement Eq. (11), 0. It is diicult to ind the conditions that guarantee 0. However, the ollowing section shows that Eq. (11) has only real solutions in some neighborhood o the singularity. In the event o having complex conjugates, one may use the real part o the solution o Eq. (12), i.e., u B(4x 2 x 4 g cos(x 3 )) + v (13) which cancels the open-loop dynamics o the system. 4. SMC THROUGH SINGULARITIES For Sliding Mode Control, we use the same steps or eedback linearization (FBL), taking the derivatives o y until u appears at the right side, we obtain Eq. (2) where u appears at the right-hand side o the 3 rd derivative o y hence the relative degree o the system is 3, except at the singularity S 1. For SMC, we deine a sliding surace as a stable dierential operator o order r 1 1 on y. } S(t) {x R n s(t) s(x) 0 (14) where, s(x) is reerred to as the surace parameter vector. The order o the surace operator is one less than the relative order r k o the system. This is done so the surace attractiveness can be guaranteed. The surace parameter vector s is deined as ollows s(x) D k (y) d i λ i ỹ(t) (15) dti where D k ( ) is a linear dierential operator that deines the surace dynamics with coeicients λ i (coeicients or the characteristic equation), and ỹ(t) y d (t) y(t) is the output tracking error (y d (t) is the desired output trajectory). Existence o the sliding mode requires the attractiveness condition: lim s(t) 0 st (t) ṡ(t) < 0 (16) Dierentiating the surace parameter vector s(t) and orcing its time derivative to a unction in the 2 nd and 4 th quadrant, ( s(t) ) ṡ(t) η σ ṡ d (t) (17) µ the surace satisies the existence condition in Eq. (16). The parameter µ is introduced to adjust the boundary layer o the surace. The squashing unction, σ( ) proposed is the hyperbolic tangent, deined by ṡ d (t) d dt Dk (y) r r σ(ζ) 1 e ζ 1 + e ζ (18) k 2 λ i y i+1 d (t) d i+1 ( s(t) ) λ i dt i+1 ỹ(t) η σ µ λ i y i+1 d (t) λ i y i+1 (t) + λ ry r k (t) λ i L i+1 (h) L g(l r k 1 (h))u Γ(y d ) + α(x) + u (19) next, we solve or u(t) rom Eq. (19) where, u(t) ṡd(t) Γ(y d ) α(x) v(t) α(x) ( s(t) v(t) η σ µ ) (20) λ i y i+1 d (t) (21) Hence, all results rom FBL, including using higher derivatives mentioned in above section, can be extended to SMC by replacing v(t) in the FBL ramework by the v(t) o Eq. (21).

4 5. BEHAVIOR AROUND THE SINGULARITY The singularity and its neighborhood N 1 are shown in Fig. 2. In the x 1 -x 4 phase plane it can be divided into two regions: S a : x 1 0, x 2 R In Fig. 5, it is the x 4 -axis. S b : x 4 0, x 4 R In Fig. 5, it is the x 1 -axis. In this case, the solutions to Eq. (10) is a pair o conjugate complex roots. The condition or the above equation to have only real roots is: (Bg cos(x 3 )) 2 + 8Bvx 1 0 (25) The above condition Eq. (25) is a paraboloid (in the x 1 -x 3 subspace) biurcation surace. Outside the paraboloid sub-region, Eq. (24) will only have real roots and inside only complex-conjugate roots. Further study in this region is needed. Currently, as a heuristic rule, we suggest to use the real part o u. Thus, the neighborhood o the singularity can be approximated by Eq. (11) and Eq. (24). The switching o controllers can be designed so that: When x / N 1, exact linearization Eq. (3) is used. when x S α, Eq. (23), which is also exact linearization, will be used. When x S β, Eq. (24) is used. Previous work (Tomlin and Sastry 1997) provides the applicability o such switching law based on the zero dynamics at the switching boundary. Fig. 2. The singularity region S 1 are both axes. The neighborhood N 1 o the singularity region is enclosed by the hyperbolas x 1 x 4 ±δ 2. In the x 1 -x 4 phase plane, N 1 is the shadowed region surrounded by our hyperbolic curves: x 1 x 4 δ 2. The neighborhood o S 1 can also be divided into two regions: S α {x N 1 x 1 δ } N 1 S β N 1 S α N 1 Condition (1) when system alls in x 1 δ, Eq. (11) can be approximated by 2Bx 1 u }{{} 2 + B(4x 2 x 4 g cos(x 3 )) disregarded, when x 1 0 Thus, +B 2 x 1 x Bg(1 B) sin(x 3 )x 2 4 v 0 (22) u Sa v B2 x 1 x B(1 B)gx 2 4 sin(x 3 ) (23) B(4x 2 x 4 g cos(x 3 )) Comparing with the afbl expression o u (Eq. 6) used in (Hauser et al. 1992), the only dierence o the two equations is the actor o the Bx 2 x 4 term. In Eq. (23), the actor is 4 and in Eq. (6) is 2. This dierence is because o the disregarded term 2Bx 1 x 4 u in afbl. More importantly, it shows that Eq. (6) only captures the system when x 1 0 while it tries to approximate the system when x 1 x 4 0. In other words, approximation by disregarding the nonlinear term beore taking (r+1) th derivatives (Hauser et al. 1992, Tomlin and Sastry 1997) is only a partial approximation o system near the singularity. Condition (2). When system alls into x 4 < δ. Eq. (11) can be approximated by 2Bx 1 u 2 + ub( g cos(x 3 )) v 0 (24) Fig. 3. The switching controller Figure 3 shows the switching controller using the aorementioned scheme. 6. SIMULATION RESULTS Several test cases used in (Hauser et al. 1992, Lai et al. 1994, Tomlin and Sastry 1997, Yi et al. 1996) are also used or comparison purpose. Simulink R (Mat 2004) was used or simulations with the 3 rd order and 4 th order controllers that are designed with all the poles at 2. B and g9.8. For comparison purpose, the ollowing our cases are simulated: ❶ Regulation o the system to the equilibrium point The same examples are used in (Yi et al. 1996). (1a) Initial conditions close to singularity: The switching condition is δ 2 x 1 x The simulation results are shown in Fig. 4. The Results are similar to (Hauser et al. 1992, Yi et al. 1996). (1b) Initial conditions away rom the singularity: The switching condition is δ 2 x 1 x 4 1. The simulations are shown in Fig. 6. As pointed out in (Yi et al. 1996), the approximation method used in (Hauser et al. 1992) ailed on set(1b). Compared to the results in (Yi et al. 1996), (compare Fig. 6 in (Yi et al. 1996) and Fig. 5 in this paper), the presented method gives aster regulation. This test case shows that the presented method can regulate the system, even rom initial conditions away rom the singularity, with aster response. ❷ Tracking periodic unctions, the switching condition is δ 2 x 1 x

5 Fig. 4. Results o set (1a), regulation with x o close to N 1. Fig. 6. Tracking o y d 1.9 sin(1.3t) + 3, x o 3,0,0,0 Fig. 5. Set (1b) regulation to origin results. x o ar rom N 1. Similar simulation with method rom (Hauser et al. 1992) ails or these cases. (2a) y d 1.9 sin(1.3t) + 3, x o 3,0,0,0. This case is used in (Tomlin and Sastry 1997) and the approximation method presented in (Hauser et al. 1992) is unstable. Fig. 6 shows the tracking results, with y d 1.9 sin(1.3t) + 3, x o 3,0,0,0. The transient period is short and then the system quickly tracks the target with a maximum steady state error equals to 5e-5. Although the method in (Tomlin and Sastry 1997) is stable, the steady state error is quite large. (Compare Fig. 3 in (Tomlin and Sastry 1997) and Fig. 6 in this paper). (2b) y d 3 cos( π 5 t), x 3,0,0,0, which is used in (Hauser et al. 1992, Lai et al. 1994), Fig. 7 shows the tracking results or this case. The maximum steady state error is 1.5e-3. The results above demonstrate the beneits o the proposed algorithm as compared to the reviewed literature. 7. GENERAL FORMULATION OF FEEDBACK LINEARIZATION WITH SINGULARITY Here, we try to generalize the ormulation to a class o SISO nonlinear systems o the orm: Fig. 7. Tracking o y d 3 cos( π 5 t), x 3,0,0,0 ẋ(t) (x) + g(x) u y(t) h(x) (26) where, x R n, u R 1, y R 1, are the states, input, and output o the system. I the system has relative order o the output r 1, then, L g (L k (h)) 0 1 < k < r 1 2 L g (L r1 1 (h)) 0 x / S 1 R n For exact I/O linearization, r 1 n. Then, ollowing the general procedure, we obtain: ξ 1 h(x) y L 0 (h) ξ 2 ξ 1 ẏ L 1 (h) ξ 3 ξ 2 ÿ L 2 (h). (27) ξ r1 2 ξ r1 1 y r1 1 L r1 1 (h) ξ r1 1 ξ r1 L r1 (h) + L g(l r1 1 (h))u(t) Let v y r, then v L r1 (h) u L g (L r1 1 (h)) v α 1(x) β 1 (x) x / S 1 (28)

6 where, S 1 is the singularity space or r 1, deined as: { } S 1 x R n Lg (L r1 1 (h(x))) 0 When x x s S 1, then x s is a singularity and the relative degree is not well deined and exact linearization will ail. However, dierentiating the output one more step yields: νy r1+1 d L r1 dt (h(x)) + L g(l r1 1 (h(x)))u(t) L r1+1 (h(x)) + L 2 g(l r1 1 (h(x))) u 2 + L g (L r1 (h(x))) + L (L g(l r1 1 (h(x)))) u + L g (L r1 1 (h(x))) u (29) Where x S 1, then L g (L r1 1 (h(x))) 0, Eq. (29) becomes y r1+1 L r1+1 (h) + L g (L r1 (h)) + L (L g(l r1 1 (h))) u + L 2 g(l r1 1 (h)) u 2 and the system looses relative order. Let y r+1 w, we have: L 2 g(l r1 1 (h)) u L (L g (L r1 1 (h))) + L g (L r1 (h)) u L r1+1 (h) w 0 (30) which is quadratic in u. Generally this equation has two complex conjugate roots. However, i L 2 g(l r1 1 (h)) 0, then Eq. (30) becomes a linear equation in u with a real solution. On the other hand i both L 2 g(l r1 1 (h)) 0 and L g (L r1 (h)) + L (L g(l r1 1 (h))) 0 x Z r1 (31) then, Eq. (29) needs to be dierentiated urther until the next (secondary) relative order r 2, where L g (L k (h)) 0 r 1 < k < r 2 2 L g (L r2 1 (h)) 0 x / S 2 R n where S 2 is the singularity space or r 2, deined as: { } S 2 x S 1 Z 1 Lg (L r2 1 (h)) 0 This means that when x S 1, and r 2 < n then, through u we can control the r2 th derivative o the y. A similar procedure as or r 1 can be used to design the FBL or SMC. A switching controller can be designed that uses an r th controller (28) when the system is away rom singularity S 1, and switches to two (r 2 ) th controller when x S 1 Z 1. On the other hand, i x S 1 S 2, the procedure could be used recursively k-times until r k n ater which the system becomes uncontrollable. 8. CONCLUSIONS This paper presented an approach to eedback linearization(fbl) and sliding mode control(smc) o a class o non-regular systems. As in SMC and FBL, the output is dierentiated r 1 times till the control variable u appears. The coeicient that multiplies u vanishes in a certain subspace S 1 (x). In this region, traditional FBL, and SMC techniques ail. Our claim is that, by urther dierentiating the output, a nonlinear (polynomial) dierential equation o u is obtained. At the singularity region, S 1, the coeicient o the u term disappears and the dierential equation degenerates to a quadratic equation. I the quadratic equation has only real roots, the system has a well deined relative degree at the singularity equal to r Switching controllers can be designed to switch rom an r1 th order controller to two (r 1 + 1) th order controllers when the system is in the neighborhood o the singularity, or even to a r2 th order controller in the singularity space. Further research will ocus on the condition under which the quadratic equation will have only real solutions and the relationship between the type o the roots and the stabilizability o the system. An adaptive switching condition is also under study. REFERENCES C. Barbu, R. Sepulchre, W. Lin, and P.V. Kotovic. Global asymptotic stabilization o the ball-and-beam system. In Proceedings o the 35th IEEE Conerence on Decision and Control, pages , Dec B. Fernández-Rodríguez. Sliding Model Control o Multi- Input Multi-Output Systems. PhD dissertation, Massachusetts Institute o Technology, J. Hauser, S. Sastry, and P. Kokotovi c. Nonlinear control via approximate input-output linearization: the ball and beam example. IEEE Transactions on Automatic Control, 37(3): , March M.C. Lai, C.C. Chien, C.Y. Cheng, Z. Xu,, and Y. Zhang. Nonlinear tracking control via approximate backstepping. In American Control Conerence, volume 2, pages , Jun D.J. Leith and W.E. Leithead. Input-output linearisation o nonlinear systems with ill-deined relative degree: the ball and beam revisited. In American Control Conerence, volume 4, pages , M.A. Marra, B.E. Boling, and B.L. Walcott. Genetic control o a ball-beam system. In Proceedings o the 1996 IEEE International Conerence on Control Applications, pages , Sep Using Simulink R 6. The MathWorks, Inc., Natick MA, C. Tomlin and S. Sastry. Switching through singularities. In Proceedings o the 36 th IEEE Conerence on Decision and Control, volume 1, pages 1 6, G. Yi, D.J. Hill,, and Z.-P. Jiang. Global nonlinear control o the ball and beam system. In Proceedings o the 35th IEEE Conerence on Decision and Control, volume 3, pages , Dec 1996.