Nonlinear Design of Active Suspensions. Jung-Shan Lin and Ioannis Kanellakopoulos y

Transcription

1 IEEE Control Systems Magazine, vol. 17, pp Nonlinear Design of Active Suspensions Jung-Shan Lin and Ioannis Kanellakopoulos y This paper develops a new nonlinear backstepping design for the control of active suspension systems, which improves the inherent tradeoff between ride quality and suspension travel. The novelty is in the use of a nonlinear filter whose effective bandwidth depends on the magnitude of the suspension travel. This intentional introduction of nonlinearity, which is readily accommodated by backstepping, results in a design that is fundamentally different from previous ones: as the suspension travel changes, the controller smoothly shifts its focus between the conflicting objectives of ride comfort and rattlespace utilization, softening the suspension when suspension travel is small and stiffening it as it approaches the travel limits. Thus, our nonlinear design allows the closed-loop system to behave differently in different operating regions, thereby eliminating the dilemma of whether to use a soft or stiff suspension setting. The improvement achieved with our design is illustrated through comparative simulations. INTRODUCTION When designing vehicle suspensions, the dual objective is to minimize the vertical forces transmitted to the passengers (i.e., to minimize vertical car body acceleration) for passenger comfort, and to maximize the tire-to-road contact (i.e., to minimize wheelhop) for handling and safety. While traditional passive suspensions can negotiate this tradeoff effectively, active suspension systems have the potential to improve both ride quality and handling performance, with the important secondary benefits of better braking and cornering because of reduced weight transfer. This improvement, of course, is conditional upon the use of feedback to control the hydraulic actuators; several control methods have been applied to this problem, most notably skyhook designs [2, 3, 14, 15] and optimal control [5 7, 13, 17, 18]. In the process of enhancing passenger comfort and road handling, active suspensions introduce the additional considerations of suspension travel and power consumption, which must be factored into the overall design goals. While the ride/handling tradeoff is prevalent in most approaches [5 7], the ride/rattlespace tradeoff is not explicitly addressed, even though it may be a more fundamental one: to decrease the vertical acceleration of the car body, it is necessary to use more suspension travel. This increases the likelihood of hitting the suspension travel limits when driving over a speed bump or into a pothole, which causes not only considerable passenger discomfort but also increased wear and tear of vehicle components. Hence, active suspensions should have the ability to behave differently on smooth and rough roads; the desired response should be soft in order to enhance ride comfort, but when the road surface is too rough the suspension should stiffen up to avoid hitting its limits. A version of this article was originally presented at the 34th IEEE Conference on Decision and Control, New Orleans, LA, December 11 13, This work was supported in part by NSF through Grants ECS and ECS and in part by UCLA through the SEAS Dean s Fund. y The authors are with the Department of Electrical Engineering, UCLA, Los Angeles, CA jlin@ee.ucla.edu, ioannis@ee.ucla.edu.

2 In this paper we propose a new nonlinear design aimed at accommodating and improving the tradeoff between ride quality and suspension travel. The starting point is the backstepping design methodology [8, 1], whose inherent flexibility can be exploited to satisfy these conflicting control objectives. The novel feature of our backstepping design is the intentional introduction of nonlinearity into the definition of the regulated variable, which is defined as the difference between the car body displacement and the output of a nonlinear filter. The input to this filter is the wheel displacement, and its effective bandwidth is a nonlinear function of suspension travel. The resulting response is soft for enhancing passenger comfort when the suspension travel is small, but stiffens up very quickly as the suspension approaches its travel limits, to avoid hitting them. Hence, the intentional introduction of nonlinearity endows the closed-loop system with the ability to emphasize different objectives in different operating regions relative to response amplitude. This amplitude dependence is a property which linear control can never attain, regardless of the design method used: the response of a linear controller is always proportional to the amplitude of the error. Thus, if one chooses the gains so that the suspension travel limits are not reached when going over a large bump, the design will be too conservative and the ride quality over smooth roads will suffer. The remainder of the paper is organized as follows. First of all, we introduce the basic passive quarter-car suspension system and discuss the design of parallel active suspensions in which the hydraulic actuator force is viewed as the control input. This simplification allows us to illustrate the limitations and basic properties of active suspensions via frequency-domain analysis. Then, we consider the full-blown quarter-car active suspension model, including the dynamics of the hydraulic actuator, and use it to design a nonlinear backstepping controller which meets the desired control objectives. This controller is evaluated using the simulation results which compare the performance of our nonlinear active design to that of a passive suspension system. Finally, some concluding remarks are given for further research. Passive Suspension SIMPLIFIED MODELS Quarter-car models are very often used for suspension analysis and design, because they are simple yet capture many important characteristics of the full model. Fig. 1 shows a quarter-car model of a passive suspension system, in which the single wheel and axle are connected to the quarter portion of the car body through a passive spring-damper combination, while the tire is modeled as a simple spring without damping. The motion equations of this system are x s + K a (x s? x w ) + C a ( _x s? _x w ) = x w + K a (x w? x s ) + C a ( _x w? _x s ) + K t (x w? r) = ; (1) where and are the masses of car body and wheel, x s and x w are the displacements of car body and wheel, K a and K t are the spring coefficients, C a is the damper coefficient and r is the road disturbance. Using the state variables x 1 = x s, x 2 = _x s, x 3 = x w and x 4 = _x w, we can rewrite (1) as _x 1 = x 2 _x 2 =? 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )] _x 3 = x 4 _x 4 = 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )? K t (x 3? r)] : (2) 2

3 car body 6x s e C a K a 6x w wheel K t 6r?????????????? Figure 1: Quarter-car model with passive spring/damper only. Since (2) is a linear system, we can use frequency domain analysis. If we consider the displacements of car body and wheel as the outputs and the road disturbance r as the input, and denote the constants a 1 = Ca, a 2 = M Ka b, a 3 = M Ca us, a 4 = M Ka us and! 2 = M Kt us (! is the natural frequency of the unsprung subsystem), then from (2) we have the following transfer functions: H 1p (s) = X 1(s) R(s) H 3p (s) = X 3(s) R(s) =!2 (a 1s + a 2 ) p (s) =!2 (s2 + a 1 s + a 2 ) p (s) ; (3) where X 1 (s), X 3 (s) and R(s) are the Laplace transform functions of x 1 (t), x 3 (t) and r(t), and p (s) = s 4 + (a 1 + a 3 )s 3 + (a 2 + a 4 +! 2 )s2 + a 1! 2 s + a 2! 2 : (4) These transfer functions will be used for comparison purposes later on. Active Suspension Active suspension systems add hydraulic actuators to the passive components of Fig. 1. Fig. 2 shows a parallel configuration, whose equations of motion are x s + K a (x s? x w ) + C a ( _x s? _x w )? u a = x w + K a (x w? x s ) + C a ( _x w? _x s ) + K t (x w? r) + u a = ; (5) where u a is the control force from the hydraulic actuator and the other terms are the same as those defined in (1). Note that if the control force u a =, then (5) is the same as (1), that is, the active suspension becomes passive. Many existing results on active suspension design ignore the hydraulic dynamics of the active element, and use u a as the control input [4 7, 9, 13, 15, 17, 18, 2]. We will also consider this simplified case, in order to 3

4 car body 6x s e Q - hydraulic actuator C a K a 6x w wheel K t 6r?????????????? Figure 2: Quarter-car model for active suspension design with parallel connection of hydraulic actuator to passive spring/damper. illustrate some of the limitations and basic properties of active suspensions and to motivate our further control development. Viewing u a as the control input, the state-space representation of (5) is: _x 1 = x 2 _x 2 =? 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )? u a ] _x 3 = x 4 _x 4 = 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )? K t (x 3? r)? u a ] ; (6) where x 1 = x s, x 2 = _x s, x 3 = x w and x 4 = _x w. Backstepping design. The first step in the design of a backstepping controller is the choice of a quantity to be regulated. The choice of this variable is crucial to the performance of the closed-loop system, and one of the main focal points of this paper will be to show how to exploit the design flexibility built into this choice in order to achieve the desired closed-loop behavior. First, let us show that this design freedom can lead to undesirable results if used carelessly: If we consider the minimization of the forces transmitted to the passengers as our control objective, then the corresponding choice of regulated variable is the car body acceleration _x 2. Hence, the desired value for the actuator force is u a = K a (x 1? x 3 ) + C a (x 2? x 4 ) to yield _x 2 =. Substituting this expression into (6) yields the closed-loop system _x 1 = x 2 _x 2 = _x 3 = x 4 (7) _x 4 =? K t (x 3? r) : 4

5 This represents the zero dynamics of the closed-loop system, which consist of an unstable subsystem (double integrator for body position and velocity) and an oscillatory subsystem (wheel position and velocity). Clearly, this system is not stable, and road disturbance inputs will result in sustained wheel oscillations and even diverging car body displacement. If we instead choose the body position x 1 as the regulated variable, then the (x 1 ; x 2 )-subsystem is stabilized, while the (x 3 ; x 4 )-subsystem represents the zero dynamics. A choice of u a which yields will also result in _x 3 = x 4 _x 1 = x 2 _x 2 =?c 1 x 1? c 2 x 2 ; _x 4 = (c 1 x 1 + c 2 x K t 2 )? (x 3? r) : Substituting x 1 = x 2 = into (9), we obtain the oscillatory zero dynamics (8) (9) _x 3 = x 4 _x 4 =? K t (x 3? r) : (1) The unstable subsystem of (7) has been eliminated with this second choice of regulated variable, but the resulting closed-loop behavior is still not acceptable, since it contains an undamped oscillatory subsystem. If we shift our attention to the minimization of the suspension travel, then the regulated variable becomes x 1? x 3 : the suspension travel is the difference between the body position x 1 and the wheel position x 3. However, the zero dynamics are again oscillatory as in (1), and hence this design is still not acceptable. We must therefore choose the regulated variable so as to avoid the oscillatory zero dynamics. One such choice is the variable z 1 = x 1? x 3 ; (11) where x 1 is the car body displacement and x 3 is a filtered version of the wheel displacement x 3 : x 3 = s + x 3 : (12) This choice represents the first step towards the design of a controller which will accommodate the inherent tradeoff between ride quality and rattlespace usage. Let us see how the choice of the positive constant affects the properties of our active suspension: For small values of, (12) is just a low-pass filter. Hence, the regulated variable z 1 is essentially equal to the car body displacement x 1 as long as the road input contains only high-frequency components which are rejected; however, at very low frequencies (constant or slowly changing road elevations) and in steady state, z 1 becomes almost identical to the suspension travel x 1? x 3. Thus, as we will see later on, the sustained oscillations are eliminated, and the active suspension rejects only high-frequency road disturbances, namely the ones which generate large vertical accelerations and cause passenger discomfort. 5

6 As the value of becomes larger, more high-frequency components of the road input are allowed to pass through the filter (12). Hence, the regulated variable z 1 approximates the suspension travel x 1? x 3 : the high filter bandwidth renders x 3 x 3. As a result, the active suspension becomes stiffer and reduces its rattlespace use, at the price of significantly reduced passenger comfort. With this choice of regulated variable z 1 defined in (11), the backstepping design procedure consists of two steps: Step 1: We compute the derivative of z 1 as _z 1 = _x 1? _x 3 = x 2 + (x 3? x 3 ) = x 2 + (x 1? z 1? x 3 ) = x 2 + (x 1? x 3 )? z 1 ; (13) and use x 2 as the first virtual control variable, for which we choose the stabilizing function 1 =?c 1 z 1? (x 1? x 3 ) ; (14) with c 1 a positive design constant. The corresponding error variable is z 2 = x 2? 1, and the resulting error equation is _z 1 =?(c 1 + )z 1 + z 2 : (15) Step 2: The derivative of z 2 is computed as _z 2 = _x 2? _ 1 =? 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )? u a ]? [?c 1 (?c 1 z 1? z 1 + z 2 )? (x 2? x 4 )] : (16) Since the actual control u a appears in (16), we choose our control law as u a = [?(c 2 + c 1 )z 2 + (c 2 1? 1 + c 1)z 1? (x 2? x 4 )] + K a (x 1? x 3 ) + C a (x 2? x 4 ) ; (17) where c 2 is a positive design constant, to render the derivative of the Lyapunov function V a = 1 2 z z2 2 (18) negative definite: _V a =?(c 1 + )z 2 1? c 2z 2 2 : (19) This implies that the error system has a globally exponentially stable equilibrium at (z 1 ; z 2 ) = (; ). _z 1 =?(c 1 + )z 1 + z 2 _z 2 =?c 2 z 2? z 1 (2) 6

7 Zero dynamics. We started out with the fifth-order system consisting of the active suspension (6) and the linear filter (12), and ended up with the second-order error system (2). The remaining three states are thus the zero dynamics subsystem of the closed-loop system. To find the zero dynamics, we set the output identically equal to zero, i.e., y = z 1 = x 1? x 3. Hence, we have _y = x 2 + (x 3? x 3 ) = y =? 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )? u a ] + [?(x 3? x 3 )? x 4 ] = : (21) Using the last equation of (21) we substitute K a (x 1? x 3 ) + C a (x 2? x 4 )? u a = [?(x 3? x 3 )? x 4 ] (22) into the _x 4 -equation of (6) to obtain the zero dynamics: _x 3 _x 4 _x 3 =?(x 3? x 3 ) which is then rewritten in the following matrix form: _x = 4 _x 3 = x 4 (23) _x 4 = [?(x 3? x 3 )? x 4 ] K t? (x 3? r) ;? 1? 2 2? Kt? x 3 x 3 x K t r : (24) Using the Ruth-Hurwitz criterion, it is easy to show that the 3 3 matrix in (24) is Hurwitz if and only if >. Therefore, the zero dynamics are exponentially stable for all >. Frequency domain analysis. If we rewrite the control law (17) as u a =? (c 2 + c 1 )fx 2? [?c 1 (x 1? x 3 )? (x 1? x 3 )]g + (c 2 1? 1 + c 1)(x 1? x 3 )? (x 2? x 4 ) + K a (x 1? x 3 ) + C a (x 2? x 4 ) =? (c 2 + c 1 )x 2 + [K a? (c 2 + c 1 )](x 1? x 3 ) + (C a? )(x 2? x 4 ) + [c 1 (? c 2 )? 1](x 1? x 3 ) ; (25) then the resulting closed-loop system becomes _x 1 _x 2 _x 3 _x 3 _x = ?q 1?(q 2 + )?q 3 q 2? 1 m r q 1 m r (q 2 + ) m r q 3?(m r q 2 +! 2 )?m r x 1 x 2 x 3 x 3 x ! 2 3 r ; (26) 7 5 where m r =, q 1 = c 2 (c 1 + ) + 1, q 2 = c 2 + c 1 and q 3 = c 1 (? c 2 )? 1. Now we want to compute the transfer functions relating the road input r to the car body displacement x 1 and the wheel travel x 3. It is a tedious but straightforward task to obtain these transfer functions directly from the state-space representation (26). An equivalent and much simpler way is to utilize the results of our control design, which guarantees that the error variable z 1 converges to zero exponentially, which means that x 1 approaches x 3 exponentially 7

8 + - Kt - 1 s 2 +! 2 1-1?M r - M 6?- m r x 1 = x 3 s + x 3 Figure 3: The block diagram of the active suspension with backstepping design. fast. Since this result is independent of frequency content, it implies that the transfer function from x 3 to x 1 is equal to one. Using this information and the variables 1 = x 1 + x 3 and 2 = x 2 + x 4, whose derivatives are computed from (6) as _ 1 = 2 _ 2 =?! K t (m r x 1 + r) ; (27) we can represent the closed-loop system in the equivalent simplified block diagram given in Fig. 3, from which we obtain that is, x 3 = = = 1 1? m r x K s 2 +! 2 t (m r x 3 + r)? m r x 3! 2 s 2 +! 2 r? m rs 2 s 2 +! 2 As a result, the transfer functions of interest are computed as s + x 3 ; (28)! m r s x (s 2 +! 2 3 =!2 r : (29) )(s + ) s 2 +! 2 H 1a (s) = X 1(s) R(s) H 3a (s) = X 3(s) R(s) =!2 a (s) =!2 (s + ) ; a (s) (3) where X 1 (s), X 3 (s) and R(s) are the Laplace transforms of x 1 (t), x 3 (t) and r(t), and a (s) = (s 2 +! 2 )(s + ) + m rs 2 = s 3 + (m r + 1)s 2 +! 2 s +!2 : (31) Note that the design constants c 1 and c 2 from our backstepping design do not appear in the transfer functions (3) although they are used in the state equations (26). These design constants influence only the rate of convergence of exponentially decaying initial condition terms, and do not affect the closed-loop transfer functions. 8

9 Let us now compare the transfer functions of the passive suspension and of our active suspension with those of the ideal suspension: H 1i (s) = X 1(s) R(s) H 3i (s) = X 3(s) R(s) p 2 2 = = 1 s opt s (n 2 s opt ns + 1)(s opt s + 1) ; (32) s where n = 1, opt = and s =!. The reasons for viewing (32) as the ideal response are lucidly explained in [9], where it is also shown that this response can never be achieved by any suspension, be it passive or active, which exerts forces only between the wheel and the car body, i.e., by any suspension which can be implemented on a road vehicle. Using the normalized Laplace variable s, we can rewrite (3) as where a 1 = a 1!, a 2 = a 2! 2 H 1p (s) = a 1s + a 2 p (s), a 3 = a 3!, a 4 = a 4! 2 and ; H 3p (s) = s2 + a 1 s + a 2 p (s) ; (33) p (s) = s 4 + (a 1 + a 3 )s 3 + (a 2 + a 4 + 1)s 2 + a 1 s + a 2 ; (34) and (3) as where =! and H 1a (s) = a (s) ; H 3a(s) = s + a (s) ; (35) a (s) = s 3 + (m r + 1)s 2 + s + : (36) These transfer functions are plotted in Figs. 4 7 with the following standard values taken from [2, 3]: = 29 kg K a = N=m K t = 19 N=m : = 59 kg C a = 1 N=(m=sec) (37) Each of these four figures contains frequency response plots for four different configurations: the ideal suspension ( dashed line), the passive suspension ( dashdot line), and our active suspension with = 1:5 ( thin solid line) and = 1 ( thick solid line). Figs. 4 and 5 show the frequency response plots of X 1(j!) R(j!) (car body displacement) and X 3(j!) (wheel displacement). The frequency response plots of?!2 X 1 (j!) (car R(j!) R(j!) body acceleration) and X 1(j!)?X 3 (j!) (suspension travel) are shown in Figs. 6 and 7 respectively. R(j!) As shown in [4, 16, 2], the frequency response plots of any real suspension must pass through certain invariant points: In both Fig. 4 and Fig. 6, there is an invariant point at the natural frequency of the unsprung mass! = q Kt (i.e., at! = 1, where the normalized frequency is defined as! =!! ). In particular, at that point we have jh 1p (j! )j = jh 1a (j! )j = 1 m r ( :2) (38) j?! 2 H 1p(j! )j = j?! 2 H 1a(j! )j =!2 m r ( 655) : (39) 9

10 ! = 1 * X Figure 4: Frequency response plots of 1 (j!) R(j!) (car body displacement). 1 1!! ! ! X Figure 5: Frequency response plots of 3 (j!) R(j!) (wheel displacement). 1

11 ! = 1 *?! Figure 6: Frequency response plots of 2 X 1 (j!) R(j!) (car body acceleration). 1 1!! ! 2 :41 X Figure 7: Frequency response plots of 1 (j!)? X 3 (j!) R(j!) (suspension travel). 6!! 11

12 In q Fig. 7, there is another q invariant point at the natural frequency of the whole quarter-car system! 2 = Kt 1 + (i.e.,! 2 = 1+m r :41). The corresponding value of the transfer function magnitude is jh 1p (j! 2 )? H 3p (j! 2 )j = jh 1a (j! 2 )? H 3a (j! 2 )j = m r ( 1:2) : (4) As explained in [9], these invariant properties are a result of the fact that the suspension forces are applied only between a wheel and the car body, and they place insurmountable limitations to what can be achieved by active suspension designs. In particular, as demonstrated by Figs. 4 7, our active suspension design cannot completely coincide with the ideal suspension in (32). Nevertheless, it is also clear that with the appropriate choice of the filter bandwidth our active suspension design is superior not only to the passive suspension but also to the ideal one in some frequency ranges. With small ( = 1:5), the active suspension design reduces both car body displacement and acceleration compared to the passive one (cf. Figs. 4 and 6), but increases the suspension travel as seen in Fig. 7. On the other hand, if is increased to 1, then the suspension travel can be significantly reduced, as seen in Fig. 7, but then the car body displacement and acceleration are increased (cf. Figs. 4 and 6). FULL-ORDER NONLINEAR DESIGN It is clear from the above analysis that any fixed value of leads to a compromise between ride quality and rattlespace usage. It is also clear from Figs. 4, 6 and 7 that we should use small values of to improve ride quality and large values to reduce suspension travel. But how can the two be combined? One possibility is to use the magnitude of suspension travel as the criterion for emphasizing one control objective more than the other: As long as the suspension travel is small, the controller should be allowed to focus on passenger comfort, which means that the bandwidth of the filter should be low. When the suspension travel becomes large, the control objective should shift to preventing the suspension from hitting its travel limits, by using a high bandwidth filter. Nonlinear Filter To realize this nonlinear control objective, we replace the linear filter (12) by a nonlinear one, whose effective bandwidth changes with the magnitude of the suspension travel: _x 3 =?( + 1 '())(x 3? x 3 ) : (41) In (41), is a positive constant, 1 is a nonnegative constant, = x 1? x 3 is the suspension travel, and the nonlinear function '(), shown in Fig. 8, is defined as 8 4? m1 ; > m 1 '() = >< >: m 2 ; jj m m1 ; <?m 1 ; m 2 (42) 12

13 6 '() 1 m2 - m1 - m1 - m2 - - Figure 8: The nonlinear function '() defined in (42). where m 1 and m 2 >. Note that if 1 =, this nonlinear filter (41) is identical to the linear filter in (12) with =. This nonlinearity contains a deadzone defined by?m 1 m 1 ; hence it remains dormant as long as the suspension travel is smaller than m 1 in magnitude. In that region, the bandwidth of the nonlinear filter (41) is constant and equal to, which can be chosen small to satisfy the passenger comfort requirement; the nonlinear filter then becomes identical to the linear filter shown to possess good ride quality properties in the previous section. As soon as the suspension travel leaves this deadzone, the nonlinearity '() is activated, and it rapidly increases the effective bandwidth of the filter, thereby shifting the control objective to minimization of suspension travel. Thus, the intentional introduction of nonlinearity into the control objective allows the controller to react differently in different operating regimes. Hydraulic Dynamics In the previous section we neglected the dynamics of the hydraulic actuator, because we wanted to illustrate the basic ideas which led to the development of the nonlinear filter (41) (42). Now that we are ready to embark on the actual nonlinear control design, we include the hydraulic dynamics and design a controller for the full-order quarter-car model. The model we use for the hydraulic actuator and its spool valve is the same as in [2, 3]; the following discussion of the basic concepts is adapted from [1, 12, 16, 19]. The hydraulic actuator we use here is a four-way valve-piston system. We know the force u a from the actuator is u a = AP L ; (43) where A is the piston area and P L is the pressure drop across the piston. Following Merritt [12], the derivative of P L is given by: V t P_ L = Q? C tp P L? A( _x s? _x w ) ; (44) 4 e where V t is the total actuator volume, e is the effective bulk modulus, Q is the hydraulic load flow, and C tp is the total leakage coefficient of the piston. In addition, the servovalve load flow equation is given by Q = C d wx v s 1 [P s? sgn(x v )P L ] ; (45) 13

14 where C d is the discharge coefficient, w is the spool valve area gradient, x v is the valve displacement from its closed position, is the hydraulic fluid density, and P s is the supply pressure. However, since here we want to include the possibility of the term P s? sgn(x v )P L becoming negative, we replace (45) with the corrected flow equation: s 1 Q = sgn[p s? sgn(x v )P L ]C d wx v jp s? sgn(x v )P L j : (46) Finally, the spool valve displacement is controlled by the input to the servovalve u, which could be a current or a voltage. The valve dynamics are approximated by a linear filter with time constant. This is a good approximation if the frequency is not too high, and it is regularly used by active suspension designers in industry. A more elaborate model would include stiction and deadzone nonlinearities commonly found in inexpensive valves. Without loss of generality, the steady-state gain is taken to be one: _x v = 1 (?x v + u) : (47) Choosing the state variables x 1 = x s, x 2 = _x s, x 3 = x w, x 4 = _x w, x 5 = P L and x 6 = x v, we rewrite (5), (43) (44) and (46) (47) as follows: _x 1 = x 2 _x 2 =? 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )? Ax 5 ] _x 3 = x 4 _x 4 = 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )? K t (x 3? r)? Ax 5 ] (48) _x 5 =?x 5? A(x 2? x 4 ) + x 6 w 3 _x 6 = 1 (?x 6 + u) ; q where = 4e 1 V t, = C tp, = C d w and q w 3 = sgn[p s? sgn(x 6 )x 5 ] jp s? sgn(x 6 )x 5 j : (49) Backstepping Design The choice of regulated variable we use here is the same as (11), that is, z 1 = x 1? x 3 ; (5) where x 1 is the car body displacement, but here x 3 is the output of the nonlinear filter (41). The presence of the nonlinearity '() can easily be handled by the backstepping design procedure, which was developed for nonlinear systems; the design remains qualitatively the same as it would be with the linear filter (12), and the only effect is the appearance of some new terms in the expressions. Our backstepping design procedure which contains four steps is outlined next, and its details are presented in the Appendix. Step 1: Starting with the regulated variable z 1 from (5), we use x 2 as the virtual control in the _z 1 -equation, and introduce the error variable z 2 = x 2? 1, where 1 is the first stabilizing function. Step 2: Let x 5 = x 5, with a positive constant which rescales x 5 (this rescaling is very useful for reducing the numerical integration errors in our simulations). We use x 5 as the virtual control in the _z 2 -equation, define z 3 = x 5? 2, and choose the second stabilizing function 2. 14

15 Step 3: Our choice of virtual control in the _z 3 -equation is x 6 w 3, with w 3 defined in (49). We then introduce the error variable z 4 = x 6 w 3? 3 and choose the third stabilizing function 3. Step 4: Since the actual control u appears in the _z 4 -equation, we can finally determine the control law u in this step. The resulting control law is chosen as u = w 3 4 ; (51) where 4 is the last stabilizing function. With this choice, we have the closed-loop error system (in the z- coordinates) as follows: _z 1 =?c 1 z 1? ( + 1 '())z 1 + z 2 _z 2 =?c 2 z 2? z 1 + A z 3 _z 3 =?c 3 z 3? A z 2 + z 4 + d 3 r + n 3 h 3 r? b 3 h 2 3z 3 (52) _z 4 =?c 4 z 4? z 3 + d 4 r + n 4 h 4 r? b 4 h 2 4 z 4 ; where c 1, c 2, c 3, c 4, b 3 and b 4 are positive design constants and n 3, d 3, h 3, n 4, d 4 and h 4 are defined in the Appendix. Now let us consider the partial Lyapunov function From (52), the derivative of (53) is computed as V = 1 2 (z2 1 + z2 2 + z2 3 + z2 4 ) : (53) _V = z 1 _z 1 + z 2 _z 2 + z 3 _z 3 + z 4 _z 4 =?(c '())z 2 1? c 2z 2 2? c 3z 2 3? c 4z d 3 z 3 r + n 3 h 3 z 3 r? b 3 h 2 3z d 4z 4 r + n 4 h 4 z 4 r? b 4 h 2 4z 2 4 : (54) Since the road disturbance r is unknown, we can not cancel the cross terms in (54) by using the control u. However, the boundedness of all the error signals is guaranteed for any positive values of the design constants c 1, c 2, c 3, c 4, b 3 and b 4. Indeed, by completing squares, we rewrite (54) as _V =?(c '())z 2 1? c 2z 2 2? 1 c 3z 2 3 2? 1 c 4z c 3? z d 2 3 3? r + d2 3 r c 2 4? z d 4 4? 2 c 3 2c 3 2? b 3 h 3 z 3? n 3 2b 3 r 2 + n2 3 r 2? b 4 4b 3 2 r + d2 4 c 4 h 4 z 4? n 4 2b 4 r?(c '())z 2 1? c 2z 2 2? 1 c 3z 2 3 2? 1 c 4z 2 4! 2 + d2 3 2c 3 + d2 4 2c 4 + n2 3 4b 3 + n2 4 4b 4 r 2 2c n2 4 r 2 4b 4 r 2 : (55) It should be clear from (55) that any bounded road disturbance r will generate bounded error signals, since _ V will become negative for large enough values of the error states z 1, z 2, z 3 and z 4. This is true with any positive choices for c 1, c 2, c 3, c 4, b 3 and b 4, although very small values of these constants may lead to unacceptably large errors. 15

16 Zero Dynamics The backstepping design procedure involved four steps, and resulted in a fourth-order error system with states z 1, z 2, z 3 and z 4. However, the original suspension system contains a total of seven states (including the state x 3, the output of the nonlinear filter in (41)), so the zero dynamics of the resulting closed-loop system still consist of three states. To find the zero dynamics, we set the output y = z 1 = x 1? x 3. Hence, we obtain _y = x 2 + ( + 1 '())(x 3? x 3 ) = y =? 1 w + 1 d (x 2? x 4 )(x 3? x 3 ) (56) + ( + 1 '())[?( + 1 '())(x 3? x 3 )? x 4 ] = : Substitution of (56) into (41) and (48) yields the zero dynamics: _x 3 =?( + 1 '( )) _x 3 = x 4 _x 4 = [?( + 1 '( )) M? x 4 ]( + 1 '( ) + 1 us d ) K t? (x 3? r) ; (57) where = x 3? x 3. If 1 =, then (57) becomes _x 3? _x 3 5 = 4 1 _x 4? 2 2 M M? Kt us? b x 3 x 3 x K t r : (58) This is the same as the zero dynamics in (24) with =, so we conclude that the zero dynamics (58) are stable for all >. If 1 >, then we consider the following system (obtained from (57) with r = ): _ =?? x 4 _x 3 = x 4 (59) _x 4 = (? M? x 4 ) ' K t? x 3 ; us where = + 1 '( ) (> ) and ' = + 1 (> ). To show that (59) is asymptotically stable, we consider d the Lyapunov function V = K t x M + 1 x 2 b 2 M ; (6) 4 b whose derivative is computed as _V = 1 d (?? x 4 ) (?? x 4 ) + K t x 3 x 4 + x 4 [(?? x 4 ) '? K t x 3 ] =? 2 1 d 3? 1 d x 4 2? 3 2? 2 x 4? 'x 4? 'x 2 4 =? 2 ( + 1 d ) 2? ( ' d ) x 4? 'x 2 4 =? 2 ' 2? 2 ' x 4? 'x 2 4 =? '( + x 4 ) 2 : (61) 16

17 Using LaSalle s invariance theorem, we conclude that the equilibrium point ( ; x 3 ; x 4 ) = (; ; ) of the system (59) is asymptotically stable, since it is n the largest invariant set of o(59) contained in the set where the Lyapunov derivative is zero, i.e., in the set E = ( ; x 3 ; x 4 ) 2 IR 3 j V _ =. SIMULATION RESULTS To verify that our new nonlinear control design achieves the desired objective, we simulated the resulting closed-loop system and compare it to both a passive suspension system and to an active design with the linear filter; we used the parameter values from (37) and the following standard values from [2, 3]: = 4: N=m 5 = 1 sec?1 = 1: N=(m 5=2 kg 1=2 ) = 1=3 sec P s = Pa (15 psi) A = 3:35 1?4 m 2 : (62) Since the value of the supply pressure P s is so large, we used = 1?7 to rescale the state x 5 for improved numerical accuracy. We also assumed the following limits: Suspension travel limits: 8 cm. Spool valve displacement limits: 1 cm. Furthermore, since w 3 in (49) appears in the denominator of (51), we employed the following modification to avoid division by zero: Set w 3 = 1 if w 3 1 and set w 3 =?1 if?1 w 3 < in the denominator (only) of the control law (51). We chose the road disturbance r as a single bump represented in the following form: r = ( a(1? cos 8t) ; :5 t :75 ; otherwise ; (63) and ran three simulations with a set to :25, :38 and :55 m, that is, with the height of the bump equal to 5, 7:6 and 11 cm. In addition, we chose the design constants as follows: c 1 = c 2 = c 3 = c 4 = 2 ; b 3 = b 4 = :1 = 1:5 ; m 1 = :55 ; m 2 = :5 : (64) The choice of m 1 is particularly important, since it defines the width of the deadzone in which the nonlinearity of the filter remains inactive. Since our suspension travel limits here are 8 cm, we chose the deadzone to be 5:5 cm. Our simulations compare a standard passive suspension (dotted line) with two active suspension designs: one that does not attempt to prevent the suspension from hitting its travel limits, i.e, uses 1 = (dashed line), and one that uses 1 = :125 (solid line). We show plots of body acceleration, body travel, wheel travel and suspension travel for three different magnitudes of the road disturbance: 1. Fig. 9 uses a = :25 m (road bump height 5 cm), 2. Fig. 1 is with a = :38 m (road bump height 7.6 cm), and 17

18 3. Fig. 11 is with a = :55 m (road bump height 11 cm). In Fig. 9, where none of the three configurations hits the suspension travel limits, we can see the improvement of ride quality achieved by the active suspension: the body acceleration is reduced by almost 7% and the body travel by almost 8% compared to the passive suspension. However, the suspension travel is increased slightly for the active designs. Note that since jj m 1 throughout this simulation, the two active designs behave in an identical manner and therefore their plots overlap. As the height of the road bump is increased in Fig. 1, the active suspension with 1 = becomes the first one to hit its travel limits. As a result, a very large acceleration is transmitted to the car body. The active suspension with 1 = :125 does not hit its travel limits, but it still generates a slightly larger acceleration than the passive suspension around t = :6 sec. This is due to the fact that in the first part of the simulation this configuration isolated its passengers much better than the passive one, yet still managed to avoid hitting its travel limits; this can only be achieved by generating a slightly larger acceleration as soon as the suspension travel exceeds 5:5 cm, causing the control objective to shift from the regulation of the car body displacement x 1 to the regulation of the suspension travel. A further increase in the height of the road bump causes both the passive and the active suspension with 1 = to hit their travel limits in Fig. 11, generating very large car body accelerations. The active design with 1 = :125 is the only one which does not hit its travel limits and results in accelerations that are significantly smaller than for the other two configurations. 18

19 5 body acceleration.7 body travel 4.6 m sec 2 2 m time (sec).85.5 suspension travel time (sec) wheel travel m m time (sec) time (sec) Figure 9: The height of the bump is 5 cm. None of the three configurations hits the suspension travel limits (note that the dashed and solid lines overlap). 19

20 15 body acceleration.9 body travel 1.6 m sec 2 5 m time (sec).85 suspension travel time (sec).9 wheel travel.5.6 m m time (sec) time (sec) Figure 1: The height of the bump is 7.6 cm. Only the active suspension with 1 = hits its travel limits (note the different time scale of the body acceleration plot). 2

21 75 body acceleration.13 body travel 5 25 m sec 2 25 m time (sec).85 suspension travel time (sec).13 wheel travel.5.8 m m time (sec) time (sec) Figure 11: The height of the bump is 11 cm. Only the active suspension with 1 = :125 avoids hitting its travel limits (the time scale of the body acceleration plot is again focused on the interval [:5 :8]). 21

22 CONCLUDING REMARKS Backstepping designs feature significant flexibility, which can be used to successfully resolve many of the tradeoffs inherent in real-world control applications. In this paper we demonstrated this capability for active suspension systems: the introduction of a nonlinear control objective allowed us to successfully negotiate and improve the fundamental tradeoff between ride quality and suspension travel. There are several issues related to this design that need further investigation. A critical one is the sensitivity of the response to measurement errors and uncertainty in the plant parameters. Here we designed a full-state feedback controller, thus assuming the availability of position and velocity measurements for both the car body and the wheel (x 1 ; : : : ; x 4 ), as well as actuator pressure (x 5 ) and valve position (x 6 ). In real active suspension designs, the last two signals are usually measured; the first four, however, are estimated from direct measurements of body acceleration, suspension travel, and possibly wheel acceleration. Clearly, the estimation process can introduce bias and errors in addition to the noise included in the original measurements. The closed-loop stability properties achieved with our controller guarantee its robustness to small measurement and estimation errors, but further research is required to determine how small the errors have to be. As for uncertainty in the plant parameters, our first results reported in [11] indicate that our controller is quite robust to variations in many plant parameters; adaptation of a single gain is enough to robustify the controller against uncertainty in the remaining parameters. Another important issue is the choice of the critical nonlinear filter parameters, m 1 and 1, which specify how soft the suspension is when its travel is small, how soon the nonlinearity starts stiffening the suspension, and how fast it stiffens it. The main advantage of employing an active suspension is the associated adaptation potential: the suspension characteristics can be adjusted while driving, to match the profile of the road being traversed. The effective and efficient adaptation of the nonlinear filter parameters can realize this potential. APPENDIX The backstepping design procedure for our nonlinear active suspension controller begins with the choice of regulated variable defined by (5), (41) and (42), and is now presented in detail: Step 1: The derivative of z 1 is _z 1 = _x 1? _x 3 The choice of the first stabilizing function = x 2 + ( + 1 '())(x 3? x 3 ) = x 2 + ( + 1 '())(x 1? z 1? x 3 ) = z ( + 1 '())? ( + 1 '())z 1 : (65) 1 =?c 1 z 1? ( + 1 '()) ; (66) yields _z 1 =?c 1 z 1? ( + 1 '())z 1 + z 2 : (67) Step 2: The derivative of z 2 is computed as _z 2 = _x 2? _ 1 =? 1 [K a (x 1? x 3 ) + C a (x 2? x 4 )? A ( z {z } x 5 )] + g 2 ; (68) 22

23 where g 2 =? _ 1 We choose the second stabilizing function 2 as 2 = A = c 1 [?c 1 z 1? ( + 1 '())z 1 + z 2 ] + ( + 1 '())(x 2? x 4 ) + 1 d (x 2? x 4 ) :?c 2 z 2? z Mb [K a (x 1? x 3 ) + C a (x 2? x 4 )]? g 2 ; (69) so (68) becomes _z 2 =?c 2 z 2? z 1 + A z 3 : (7) Step 3: The derivative of z 3 is _z 3 = _x 5? _ 2 =? x 5? A(x 2? x 4 ) + ( z {z } x 6 w 3 ) + g 3 + (d 3 + n 3 h 3 )r ; (71) where w 3 is defined by (49), n 3 = K t A and d 3 = n 3 ( C a? ) h 3 =? 1 '()? 1 d g 3 =? A?(c 2 + c 1 )(?c 2 z 2? z 1 + A z 3 ) + c 1 1 d (x 2? x 4 )z 1 + [c 2 1? 1 + c 1( + 1 '())][?c 1 z 1? ( + 1 '())z 1 + z 2 ] + 1 [K a (x 2? x 4 ) + C a w 1 ]? ( + 1 '())w 1 #? 2 1 d (x d 2 ' 2? x 4 ) 2? 1 d (x 2 2? x 4 ) 2? 1 d w 1 w 1 =?m t [K a (x 1? x 3 ) + C a (x 2? x 4 )? Ax 5 ] + K t x 3 m t = : With the choice of the third stabilizing function = _x 2? _x 4 + K t r 3 = 1?c 3 z A 3? z 2? b 3 h 2 M 3z 3 + x 5 + A(x 2? x 4 )? g 3 ; (72) b (71) becomes _z 3 =?c 3 z 3? A z 2 + z 4 + d 3 r + n 3 h 3 r? b 3 h 2 3z 3 : (73) 23

24 Step 4: The derivative of z 4 is computed as where n 4 = n 3 = K t A _z 4 = d dt (x 6w 3 )? _ 3 and = 1 (?x 6 + u)w 3? 1 2jw 3 j jx 6jw 2 + g 4 + (d 4 + n 4 h 4 )r ; (74) w 2 =?x 5? A(x 2? x 4 ) + x 6 w 3 = 1 _ x 5 = _x 5 d 4 = (c 3 + c 2 + c 1 ) d 3 + K t (A 2 + K a? m t C 2 a AM + m t C a ) us h 4 = 1 (c 3 + c 2 + c 1 + b 3 h 2 )h b 3h C 2 3 ( a? ) M " b? 1 d 2 ' 2 1 d (x 2 2? x 4 )? m t C a 1 d? c 1 1 d z 1? m t C a 1 '() d (x 2? x 4 ) g 4 =? 1?(c 3 + c 2 + c 1 + b 3 h 2 3 )(?c 3z 3? g 4 = A " A z 2 + z 4? b 3 h 2 M 3z 3 ) b? A z 2 + w 2 + Aw 1? 2b 3 z 3 h 3 h 3 + g 4?(c 2 + c 1 )(?c 2 z 2? z 1 ) + c 1 1 d 2 ' d 2 (x 2? x 4 ) 2 z 1 + c 1 1 d w 1z 1 + 2c 1 1 d (x 2? x 4 )z 1 + [c 2 1? 1 + c 1( + 1 '())]z (K a w 1 + C a w 1 )? 6 1 d (x 2? x 4 )w 1? ( + 1 '()) w 1? 3 1 d 2 ' d 2 (x 2? x 4 ) 3? 1 d 3 ' d 3 (x 2? x 4 ) 3? 3 1 d 2 ' d 2 (x 2? x 4 )w 1? 1 d w 1 h 3 = _ h 3 =?2 1 d (x 2? x 4 )? 1 d 2 ' d 2 (x 2? x 4 ) z 2 = _z 2 =?c 2 z 2? z A 1 + z 3 z 1 = _z 1 =?c 1 z 1? ( + 1 '())z 1 + z 2 z 1 = _z 1 =?(c '())z 1? 1 d (x 2? x 4 )z 1 + z 2 w 1 =?m t [K a (x 2? x 4 ) + C a w 1? Aw 2 ] + K t x 4 The resulting control law u is chosen as # = _w 1? m t C a K t r : u = w 3 4 ; (75) 24

25 with the last stabilizing function 4 =?c 4 z 4? z 3? b 4 h 2 4z x 6w jw 3 j jx 6jw 2? g 4 : (76) Hence, (74) becomes _z 4 =?c 4 z 4? z 3 + d 4 r + n 4 h 4 r? b 4 h 2 4z 4 : (77) REFERENCES [1] W. R. Anderson, Controlling Electrohydraulic Systems, New York, NY: Marcel Dekker, [2] A. Alleyne and J. K. Hedrick, Nonlinear control of a quarter car active suspension, Proceedings of the 1992 American Control Conference, Chicago IL, pp [3] A. Alleyne, P. D. Neuhaus, and J. K. Hedrick, Application of nonlinear control theory to electronically controlled suspensions, Vehicle System Dynamics, vol. 22, pp , [4] J. K. Hedrick and T. Butsuen, Invariant properties of automotive suspensions, Proceedings of the Institution of Mechanical Engineers, vol. 24, pp , 199. [5] D. Hrovat, Influence of unsprung weight on vehicle ride quality, Journal of Sound and Vibration, vol. 124, pp , [6] D. Hrovat, Optimal active suspension structures for quarter-car vehicle models, Automatica, vol. 26, pp , 199. [7] D. Hrovat, Applications of optimal control to advanced automotive suspension design, Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, vol. 115, pp , [8] I. Kanellakopoulos, P. V. Kokotović, and A. S. Morse, A toolkit for nonlinear feedback design, Systems & Control Letters, vol. 18, pp , [9] D. Karnopp, Theoretical limitations in active vehicle suspensions, Vehicle System Dynamics, vol. 15, pp , [1] M. Krstić, I. Kanellakopoulos, and P. V. Kokotović, Nonlinear and Adaptive Control Design, New York, NY: John Wiley & Sons, [11] J.-S. Lin and I. Kanellakopoulos, Adaptive nonlinear control in active suspensions, Proceedings of the 13th World Congress of International Federation of Automatic Control, San Francisco, CA, vol. F, pp , [12] H. E. Merritt, Hydraulic Control Systems, New York, NY: John Wiley & Sons, [13] L. R. Ray, Robust linear-optimal control laws for active suspension systems, Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, vol. 114, pp , [14] M. Satoh, N. Fukushima, Y. Akatsu, I. Fujimura, and K. Fukuyama, An active suspension employing an electrohydraulic pressure control system, Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, HI, pp ,

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