Model Semantics and Simulation for Hybrid Systems Operating in Sliding Regimes Pieter J. Mosterman
Institute of Robotics and System Dynamics German Aerospace Establishment D-82230 Wessling
[email protected]
Feng Zhaoy
Department of Computer and Information Science The Ohio State University Columbus, OH 43210
Department of Computer Science Vanderbilt University Nashville, TN 37235
[email protected]
[email protected]
Abstract We describe a model semantics and a simulation algorithm for characterizing a class of dynamic physical systems operating in the so-called sliding regimes. Many continuous physical systems operate at multiple time scales. To simplify behavior generation, we often abstract away details at faster time scales or near discontinuous boundaries and describe the resulting system as a hybrid system with distinct modes. Mode transitions are induced by internal state changes or external control signals. It is common for such systems to exhibit chattering behaviors at the discontinuous transition boundaries which presents challenges to conventional numerical methods for analyzing system behaviors. We present an ecient, adaptive algorithm for simulating this class of systems, based on a careful analysis of the model semantics at the boundaries of discontinuity. Simulation results show that the algorithm is chatter free and more ecient than conventional integration methods for sliding-mode systems.
Introduction
Nonlinear physical systems often include phenomena that occur at multiple time scales. To simplify behavior analysis, fast time scale phenomena along with associated small, parasitic e ects are commonly abstracted away and system behavior is described as multiple piecewise continuous modes of operation. System models incorporate a meta-level control model operating on top of the data ow model to select active model fragments (Mosterman & Biswas 1997b; Zhao 1994). However, systems with mixed continuous and discrete components, the so-called hybrid systems, Pieter J. Mosterman is supported by grants from PNC, Japan and Hewlett-Packard, Co. y Feng Zhao is supported in part by an NSF Young Investigator Award CCR-9457802, an Alfred P. Sloan Foundation Research Fellowship, and a grant from Xerox Palo Alto Research Center. z Gautam Biswas is supported by grants from PNC, Japan and Hewlett-Packard, Co.
Gautam Biswasz
can exhibit extremely complex behaviors due to nonlinearities and discontinuities (Nerode & Kohn 1993). Examples of hybrid systems include trac control systems, electric power circuits, and economic models. Numerical analysis of these systems is often hampered by the steep gradient near discontinuities. When the system state reaches or exceeds a threshold value, the discrete model is invoked, mode switches occur, and a new model fragment is selected that governs behavior in the active mode of operation. The threshold function speci es switching surfaces in phase space along which discontinuous changes in the system may occur.1 Traditional integration schemes, such as the Runge-Kutta method, are very sensitive to the steep gradients that occur at these discontinuities and may perform poorly when a xed step numerical approximation is applied. To permit ecient computational analysis of hybrid systems, we need a well-de ned semantics for modeling behaviors at discontinuities, and simulation schemes that can seamlessly combine continuous behavior generation with discrete mode switches. This paper presents a model semantics for a class of hybrid systems operating in the so-called sliding regimes, a region where a system chatters between two di erent modes such as in the anti-lock braking system, and describes a method for simulating sliding mode behavior eciently. The switching surfaces in the physical system behavior description arise from modeling artifacts that abstract the hysteresis e ects of small, unmodeled parameters. Our simulation algorithm is based on Filippov's construction of equivalence dynamics in sliding regimes. Simulation results on several examples have shown little error when using a large time step along the switching surface. The sliding mode simulation algorithm relies on the fact that the equivalence dynamics on a sliding sur1 A phase space of a dynamic system is spanned by the independent state variables of the system.
L EV
f in
L
L
Lth
Lth
I, R pipe
C
Rb
f out
f sump
Figure 1: A hydraulics system with two distinct
modes of operation.
p
p
Figure 2: Phase space of evaporator behavior. The right-hand phase space description abstracts away small parameters e ects near the threshold in the left-hand phase space. xball
face is de ned as the limiting behavior when switching tends to be in nitely fast. We have developed an adaptive algorithm elsewhere that can accurately follow a sliding trajectory and generate control signals at discrete times by exploiting the equivalence in control signals (Zhao & Utkin 1996). In contrast, the algorithm described in this paper exploits the equivalence in dynamics for sliding mode systems and presents an alternative method for adaptively following trajectories at discontinuous boundaries.
Examples of Hybrid Systems
As a typical example of hybrid systems, consider the evaporator vessel in Fig. 1 that is part of the secondary cooling system in a fast breeder nuclear reactor (Mosterman & Biswas 1997a). In this system, a sodium pump maintains a sucient ow of coolant in the loop. To keep the level of uid in the evaporator vessel from going over a prespeci ed maximum, an over ow mechanism is activated to drain excessive uid once this level is reached. The ow of liquid through the over ow pipe builds up momentum, and the evaporator behavior can be described by the level of uid in the evaporator, L, and the uid momentum, p. A detailed continuous phase space diagram is shown on the left in Fig. 2 for a given in ow. It exhaustively depicts how relevant system variables relate to each other. The system tends to a steady state, and highly nonlinear but continuous behavior may occur near the over ow level because of adhesive forces in the uid, the intake area of the over ow, and the uid surface area. If the detailed e ects are abstracted away, the over ow mechanism can be modeled to create a model con guration change between two distinct modes where (1) the uid level is below the over ow level, and (2) the uid level is above the over ow level. Phase space behavior is piecewise linear, shown on the right in Fig. 2, and switching occurs at L = Lth which is called the switching surface. We refer to this form of model abstraction as a parameter abstraction (Mosterman & Biswas 1997b).
v
-v
vball
Figure 3: A bouncing ball. Time scale abstraction does not abstract away physical parameters, but collapses time so that the e ects of fast change appear to be instantaneous and discontinuous (Mosterman & Biswas 1997b). An example of this is given by the bouncing ball in Fig. 3. In case of an elastic collision, the ball moves towards the switching surface xball = 0 with negative velocity, vball . When this point is reached, the kinetic energy of the ball is stored in the form of elastic energy in the ball, which is then turned back into kinetic energy as the ball uncompresses and reverses vball in a very short interval of time. Phase space behavior is shown in Fig. 4. In case the velocity reversal occurs on a time scale much smaller than the time scale at which overall behavior occurs, the continuous nonlinear phase space behavior on the left can be modeled as an instantaneous jump in vball upon collision, shown on the right by double arrow heads. In summary, hybrid systems operate in piecewise continuous phase spaces, where governing equations change at switching surfaces, possibly accompanied by jumps in state variables. Behaviors of systems that exhibit multiple modes xball
xball
vball
vball
Figure 4: Phase space of a bouncing ball. The
velocity reversal in a very short period of time in the left-hand phase space is abstracted into instantaneous transitions on the right.
are best represented by an integrated formalism that represents behaviors in the distinct modes of operation using continuous models, speci ed as elds in phase space. Discrete mode switches occur at disctinct points in time to switch the active mode resulting in a new eld description for continuous behavior generation. Fields are often de ned by systems of linear and nonlinear ordinary di erential equations (Guckenheimer & Johnson 1995), and numerical simulation methods may be employed to solve these equations. Similarly, a number of approaches with well de ned execution semantics, such as Petri nets and nite state automata, have been applied to discrete system modeling (Aho, Hopcroft, & Ullman 1974). Many numerical methods rely on the continuity assumptions in the dynamic system models. Near the discontinuous boundary, a wide variety of behaviors may occur. Following the classi cation in (Zhao 1995), Fig. 5 categorizes three types of behaviors in phase space according to the directions of vector- eld normal components in modes and , f n and f n . The simulation may make a large error when a surface is crossed and a mode switch is detected. This is shown in Fig. 6 on the left. To minimize these errors, the simulation time step may have to be reduced to determine precisely where the mode switch occurs. Adopting a variable step approach greatly reduces the simulation error. Using a variable step size works well in many cases, but it may result in problems when chattering along a switching surface occurs (Fig. 7). These chattering motions may be intentionally designed in sliding mode behavior, such as in anti-lock braking systems (Slotine & Li 1991; Utkin 1992). Chattering occurs when the eld vectors are directed towards the switching surface as shown in Fig. 5. During sliding mode operation, the system switches between modes at the switching surface at a very fast rate, producing a fast chattering behavior motion. This causes the aggregate behavior along the switching surface to progress very slowly and simulated time almost comes to a halt. This slow component of the dynamics along the sliding surface has to be approximated either by equivalence in control (Zhao & Utkin 1996) or equivalence in dynamics (Filippov 1960) so that a larger step size could be used by an integration scheme without introducing untolerable errors.
Chattering in Physical Systems
In this section we study multi-mode behavior of two paradigmatic systems. The evaporator over ow mechanism exhibits chattering behavior in its hybrid system model. Similarly, collision phenomena which cause
t
t
fα n fα
n fβ
t
fβ
n fα
t
fα
n fβ
n fα
n fβ
n fα
n fβ
fβ
n fα
n fβ
n
fα
n
fβ
β
α
Figure 5: Types of phase space behaviors near a
switching surface.
ε
ε
δt
δt
Figure 6: Error of a xed step (left) and variable
step (right) integration at a switching surface.
jumps in phase space (e.g., the bouncing ball), are studied using a cam-follower mechanism. The models of these systems are generalized to develop the semantics of physical system behavior at switching surfaces.
The Evaporator
Consider the hydraulics of the evaporator vessel in Fig. 1. There is a constant in ow of liquid into the tank, fin, and an out ow, fout , that depends on the pressure in the tank and the Bernoulli resistance, Rb. The physical system has two distinct modes of operation; mode where the over ow is not active, and mode where it is. The over ow mechanism becomes active when the liquid level in the evaporator, L, exceeds a threshold value, Lth , which causes a ow fsump through a narrow pipe with resistance, Rpipe , and inertia, I. When the over ow mechanism becomes active, the pipe inertia starts to build up ow momentum, p, until its ow maintains a steady level, i.e., fin = fout + fsump . ( Rpipe (1) : Lp__ == ?? I1 Lp + fin Rb C C
ε
δt
Figure 7: A hybrid system may chatter.
Lth
L
β
L
0.06 0.04
p
0.00 0
L th
α
0.02
100
200
t
300
Lth
sα sβ
Figure 8: After an initial transient stage, the
evaporator level reaches steady state. L
L th
L th
p
α
β
0.00 0
100
200
t
300
500
Figure 10: Chattering between modes with an active and inactive over ow, Rb = 1; Rpipe = 0:5; I = 0:5; C = 15; fin = 0:25; T = 0:025. L
p
Figure 9: Phase space of behavior in each mode.
(
p
0.02
p
500
L
L
0.06
β α
2
1
3
p
p_ = ? Rpipe I p+L L_ = ? IC1 p ? Rb1C L + fCin
(2)
Figure 11: Concatenation of pieces of phase spaces from modes and .
Suppose initially the system is in mode , the ow of liquid through the narrow pipe is zero and the tank lls. In steady state, the pressure causes an out ow through Rb that equals the in ow, fin , of uid in the tank. If the level, L, becomes higher than the threshold level, Lth , the system will move into mode as shown in Fig. 8 against time (left) and in phase space (right). Now there is another path of liquid owing out of the tank and it may continue to ll at a slower pace and reach a steady state level, s , which causes a total out ow equal to its in ow. This new steady state liquid level is below what would have been attained had the over ow mechanism not been present. More complicated behavior is exhibited when Lth is higher than the level at steady state in mode , Fig. 9. When the moment the over ow becomes active, the system moves towards a steady state with lower pressure which causes the level to drop and the over ow mechanism turns o . In the separate phase spaces for each mode, and , the grayed out areas represent state vector values that cause a transition to the other mode. The elds in each mode are directed towards the switching border represented by Lth , and, therefore, independent of the initial conditions. In time, the system reaches a point where it moves from one mode to the other and back immediately and chattering occurs (Fig. 10). Analysis of system behavior for this con guration requires a physically consistent treatment of state evolution at Lth . Since the domains of the elds for each mode are mutually exclusive, the phase spaces can be combined into one. Fig. 11 depicts three qualitative scenarios by which Lth may be approached. In the scenario marked 1, the system approaches Lth with a
eld component in the ?p direction in which is the active mode of operation for L = Lth . In scenario 2, the system approaches Lth with a 0 component in the p direction in . In scenario 3, the elds of and have equal angles and opposite direction when approaching Lth from . The objective is to determine which one of these is at equilibrium when Lth is reached. To investigate physical behavior, we rst observe that in reality the border between the modes of operation is not as crisp as modeled. Modeling abstractions disregard small parameters that are present and a ect behavior at the boundary, Lth . For example, though small, forces at the rim of the over ow pipe require the uid level in the tank to be somewhat higher than the rim in order for liquid to start pouring in. During the time interval that this excess level is drained, the over ow mechanism is active, and the level continues to fall. So, after becoming active, in reality a period of time elapses before it turns o again. Other higher order physical phenomena, such as cohesive forces in the liquid, smooth the discrete switching behavior and a small continuous ow of liquid through the over ow is realized. Other physical e ects cause similar hysteresis at the boundary between operational modes, and this can be used to derive correct model semantics for behavior at this boundary. Fig. 12 shows the e ect of a hysteresis band around Lth . Clearly, the system converges to a recurring point on Lth ? and Lth + and starts to oscillate between them. If lim!0 is taken, these recurring points coincide and the eld resultant at the common point on Lth can be determined based on the limit values of the eld in and at this point. If
:
xrod
L
xrod
L th+ ε
vrod
L th L th - ε
vrod
Figure 14: Phase space of the cam-follower. 0.12
4
53
2
1
0.1
p
x
x
x
x
rod base cam
Figure 13: A cam mechanism opens a valve. is taken small, the curvature of the eld in approaches a straight line. If the direction of the eld in is the opposite of the eld in at the boundary Lth , this point is stable. In Fig. 11 this corresponds to the point on Lth reached by trajectory 3.
The Cam-Follower
As illustrated by the bouncing ball, mode switching may involve jumps in the phase space. Consider collision e ects that may occur in a cam-follower system in automobiles (Fig. 13). The cam mechanism is used to translate rotational motion into a linear displacement to open and close valves in the engine cylinders. Typically, a spring mechanism is used to ensure contact between the rod and rotating cam but due to the high velocities of operation (up till several thousands of revolutions per minute) and wear of the spring, the rod may bounce on and o the cam, causing collisions. These collisions can be modeled by Newton's law using a coecient of restitution, , to model loss of energy + ? v+ = ?(v ? v ). Typiduring collision vrod rod cam cam cally, is a function of impact velocities (Brach 1991), and a threshold, th , can be set below which the collision is considered perfectly non-elastic (i.e., = 0). To analyze the phase space of the cam-follower, consider the valve and rod mechanism moving with only the valve spring and rocker arm friction acting. R 1 (3) : vF__rod = g=? 1mvFspring ? m vrod spring C rod The phase space is shown on the left in Fig. 14. The valve spring and combined inertias result in a second
rod
0.04
0.02
0 -1
x
cam
0.06
x(cam)
Figure 12: Iteration across hysteresis e ect.
0.08
-0.5
0
0.5
1
v(rod), v(cam)
Figure 15: The rod may disconnect, R = 10; m = 0:5; C = 0:02; T = 0:01. order system with friction. Therefore, the rod velocity oscillates between positive and negative with a decreasing amplitude. When the cam mechanism is included, the rod velocity follows an ellipsoid path as shown in the right phase space diagram in Fig. 14. (4) : vF_rod = v=cam1 v spring C rod If the rod and cam positions are equal and the rod velocity is more negative than the cam velocity, collision occurs like the bouncing ball in Fig. 4, and in case the collision is perfectly non-elastic, the rod velocity instantaneously equals the cam velocity. This is indicated by the grayed out areas of the phase space. The rod disconnects from the cam if its deceleration is larger, which corresponds to the steeper curve in the left half-plane. This is shown in Fig. 15 for one simulation run. Numerical approximations of the rod parameters may cause the simulation to show the rod disconnects at one time and ends up in the grayed out area of the phase space shown in Fig. 16 where collision occurs and the rod takes on the same velocity as the cam. Now, the rod and cam are connected again, but the next simulation step may result in them disconnecting and chattering occurs along the switching surface Fnormal = 0 and vrod = vcam as shown on the right in Fig. 16. Like the evaporator, chattering is an artifact of the simulation caused by model abstraction. In reality, higher order e ects like the rod's elasticity and adhesive forces between the rod and cam surfaces ensure it is connected for a short while before disconnecting. When the values of these parameters tend to 0, the system behavior starts to slide along the vrod = vcam surface.
0.1
0.09
free
0.06
-0.5
collide
s=0
free
-1
-0.9
-0.8
-0.7
-0.6
-2 -0.02
-0.015
v(rod), v(cam)
-0.01
-0.005
0
v(rod)-v(cam)
Figure 16: Numerical simulation may result in chattering, C = 0:01.
Model Semantics
x = f t t + f t t :
(5)
Then we compute the average velocity of the motion on the surface: v = t x+t t t (6) = ( ff n + ff n )=( f n + f n ) t t = rf + (1 ? r)f where r = (f nf+ f n ) . Thus, the vector v is on the line connecting the end points of f and f , with r and 1 ? r as its barycentric coordinates (Fig. 18). Let c be the di erence vector f ? f , and p be the intersection
n
fβ
fβ
Figure 17: Motion along a sliding surface. fα n fα
s=0
When a system operates in the sliding regime, the dynamics of the system along the sliding surface is not de ned in a continuous sense. We adopt the notion of equivalence dynamics for the sliding regime justi ed by Filippov (Filippov 1960; Utkin 1992). Consider the switching surface as an in nitesimal band rather than a crisp border. The equivalence dynamics on the surface is de ned as the limiting behaviors when the width of the band tends to zero. This construction preserves the physical meaning of the dynamics at the discontinuous boundaries. Furthermore, it serves as a basis for algorithmically determining the direction and magnitude of the sliding motion. Assume is the thickness of the hysteresis band around the sliding surface (see Fig. 17). If is small, the elds f and f on either side of the surface can be represented by their instantaneous vector representations with normal components f n and f n , and tangential components f t and f t . The direction of movement is along the sliding surface, and we need to calculate the average velocity on the surface. We compute the time the system takes to cross the band as t = f n and t = f n , and the tangential distance the system has travelled over two adjacent time intervals (t + t ) as:
n
t
fα
−ε
-1.5
-1
t
fβ
n
fα fα
contact
x(cam)
F(normal)
collide contact
0.05 -1.1
+ε
0
0.08
0.07
δx
0.5
d t
t
fα
e
n
fβ
fβ
fβ
Figure 18: Filippov construction of sliding motion direction and magnitude. of c with the tangent vector. Let p partition c into f n e two segments, d and e. We have d = f n by triangle congruence. Thus, we have shown that the barycentric coordinate for p is the same as that for v (recall 1?r r = f n f n ). This corresponds to Filippov's construction, i.e., the vector v is the same as the tangent vector. In fact, the formula v = rf + (1 ? r)f can be used to compute v, where r is determined from the normal components of the two vector elds if the angle between the sliding surface and vector elds is known.
A Simulation Algorithm
We have implemented the sliding mode algorithm as part of a hybrid system simulation engine using a variable step, forward Euler numerical integration scheme. When a mode switch between elds and is detected (see Fig. 19), a binary search determines the point of switching to within a pre-speci ed accuracy. After the mode switch is executed and the state vector is transferred between modes to x (ts), it is checked whether the new mode, , persists for at least one maximum time step, T. If not, sliding mode simulation is activated. The sliding mode simulation algorithm (Algorithm 1) rst computes x (ts + T) based on f with x (ts ) as initial point. Discontinuous changes may take place, and the resulting x (ts + T) is inferred. Next, x (t+s ) is computed to infer f and x (t+s ), which may di er from x (t+s ) when jumps in the state vector occur between f and f . Next, x (t+s + T) is computed based on x (t+s ) and f , and jumps between f and f are taken into account when x (t+s + T) is inferred.
x β (ts+∆T)
Lth
fα
β α
x β (ts+ ε) x α (ts)
L
0.06
p
0.02 1
0.00 0
3 2
fβ
x α(ts+ ε+∆T)
Figure 19: The sliding mode numerical simula-
tion algorithm.
100
200
Algorithm 1 Sliding Mode Simulation
A mode switch has occurred from f to f Compute x (ts + T ) from x (ts ) and f (ts) Infer f and x (ts +T ) fA discontinuous change between x (ts+ T ) and x (ts + T ) may occur.g Compute x (ts + ) from x (ts) and f (ts ) Infer f and x (ts + ) from x (ts + ) Compute x (ts + + T ) from x (ts + ) and f (ts + ) Infer f and x (ts + + T ) while f 6= f do Compute xd = x (ts + + T ) ? x (ts + T ) g = 12 , gaccuracy = 14 for a given number of iterations do xs = x (ts + T ) + g xd Infer f from xs if f = f then g = g + gaccuracy else g = g ? gaccuracy gaccuracy = 21 gaccuracy end for ts = ts + T x (ts) = xs if f 6= f then f = f Compute x (ts + T ) from x (ts ) and f (ts) Infer f and x (ts + T ) Compute x (ts + ) from x (ts) and f (ts ) Infer f and x (ts + ) from x (ts + ) Compute x (ts + + T ) from x (ts + ) and f (ts + ) Infer f and x (ts + + T )
Require:
end while
The Evaporator
Consider the chattering behavior in Fig. 10 for the evaporator shown in Fig. 1. If the sliding mode algorithm is applied, simulation yields the results in Fig. 20. The temporal, dynamic behavior of the system has not changed, but the error has decreased dramatically. This is even more obvious in the phase space plots in Fig. 21. Note that the mode switch where the over ow mechanism becomes active is detected with a large error from 0:8 because of the xed simulation step. In fact, this error is of the same order of magni-
t
500
Figure 20: Simulation of the evaporater using the
sliding mode algorithm. 0.0805
0.0804 0.0804 0.0803
0.0803
L
L
0.0801
Now, x (ts +T) is taken as the inital point of a vector x (t+s + T) ? x (ts + T). A binary search is performed along this vector given a pre-speci ed number of steps to determine x(ts + T) on the sliding surface. After x(ts + t) and its corresponding eld are determined, it is checked whether chattering persists and the process continues.
300
0.0801
0.08 0.08
0.0799 0
0.005
0.01
0.015
0.02
p
0.03
0.035
0
0.005
0.01
0.015
0.02
p
0.03
0.035
Figure 21: Fixed step Euler (left) vs. variable
step Euler with sliding mode simulation (right).
tude as the chattering behavior. However, a variable step cannot be used if no sliding mode algorithm is available.
The Cam-Follower System
In numerical simulation, when the deceleration of the cam causes the rod to disconnect, it may make contact the following simulation step. This is shown graphically in Fig. 16. In the right-hand gure, the negative plane of the vrod ? vcam axis results in a non-elastic collision and sets vrod = vcam . Mode (where the rod is connected) is operative on the vrod ? vcam axis as long as Fnormal is positive or 0. When Fnormal becomes negative, the system switches to mode (where the rod is moving freely). For certain parameter values this moves the system in the plane where a non-elastic collision occurs, and the system moves back to the vrod ? vcam = 0 axis, with Fnormal > 0. This chattering behavior is due to the numerical time step and for T ! 0 the system would remain at (0; 0). In Fig. 22 this corresponds to movement along the sliding surface, vrod = vcam , which is the continuous curve that represents the movement of the cam in phase space. This illustrates the operation of the sliding mode algorithm on this system, and shows that the non-elastic collision in simulation results in vrod = vcam for mode , where the rod and cam are disconnected. In mode , the rod and cam are connected, and vrod = vcam also. Therefore, the system slides on the switching surface and there is no error due to chattering. This conforms with physical behavior due to unmodeled higher order physical phenomena such as adhesive forces between the rod and cam, which results in the rod and cam moving with the same velocity. Simulation results with a xed step Euler are shown
xrod xcam
Based on a physical model semantics, we have developed a sliding mode simulation algorithm using the Filippov equivalence of dynamics. Our implementation has shown that the simulation introduces a very small error while maintaining consistency of the temporal behavior of the slow component along the sliding surface. The algorithm performs well in several engineering applications where discontinuities in physical behavior cause the system to undergo mode switches.
xβ(ts+ε) xα(ts)
fα
fβ
xβ(ts+∆T) xα(ts+ε+∆T)
xβ(ts+ε+∆T)
vrod, vcam
Figure 22: Sliding mode simulation for the cam-
follower system.
0.12
0.12 0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0 -0.02
0
sliding
chatter error of (v rod - vcam) 0
0.5
1
1.5
time
2
-0.02
0
0.5
1
1.5
time
2
Figure 23: Sliding mode simulation during an interval of time. on the left in Fig. 23. The chatter error can be reduced by using a smaller step size, but a variable step size integration method cannot be applied since it would reduce the step size to its lower bound for a large number of simulation steps. Sliding mode simulation, shown on the right in Fig. 23, has no error because it accounts for the discontinuous jump in rod velocity. The results show that sliding mode simulation correctly turns o when system behavior moves away from the switching surface.
Conclusions
Hybrid systems present distinct challenges for numerical methods that attempt to accurately approximate behaviors of these systems. Near the mode transition boundaries, a simulation often has to reduce time steps in order to produce accurate behaviors. This paper considers an interesting class of behaviors at the boundaries, the sliding motion. Sliding mode systems move along sliding surfaces because of continuous interaction between two alternating operating modes. However, unmodeled small, higher order, dynamic effects or discrete-time numerical simulation could introduce chattering along the surface. To reduce simulation error, the step size in a numerical integration is kept small to capture the fast chattering motion. As a result, simulated time progresses only in small increments, and the slower, sliding movement along the switching surface is not simulated eciently.
References
Aho, A. V.; Hopcroft, J. E.; and Ullman, J. D. 1974. The Design and Analysis of Computer Algorithms. Reading, Massachusetts: Addison-Wesley Publishing Company. ISBN 0-201-00029-6. Brach, R. M. 1991. Mechanical Impact Dynamics. New York: John Wiley and Sons. Filippov, A. F. 1960. Di erential equations with discontinuous right-hand sides. Mathematicheskii Sbornik 51(1). Guckenheimer, J., and Johnson, S. 1995. Planar hybrid systems. In Antsaklis, P.; Kohn, W.; Nerode, A.; and Sastry, S., eds., Hybrid Systems II, volume 999, 202{225. Springer-Verlag. Lecture Notes in Computer Science. Mosterman, P. J., and Biswas, G. 1997a. Monitoring, Prediction, and Fault Isolation in Dynamic Physical Systems. In AAAI-97, 100{105. Rhode Island. Mosterman, P. J., and Biswas, G. 1997b. A theory of discontinuities in dynamic physical systems. Journal of the Franklin Institute 334B(6). Nerode, A., and Kohn, W. 1993. Models for hybrid systems: automata, topologies, controllability, observability. In Hybrid Systems, volume 736. SpringerVerlag. Lecture Notes in Computer Science. Slotine, J. E., and Li, W. 1991. Applied Nonlinear Control. Englewood Cli s, NJ: Prentice Hall. Utkin, V. I. 1992. Sliding Modes in Control and Optimization. Springer-Verlag. Zhao, F., and Utkin, V. I. 1996. Adaptive simulation and control of variable-structure control systems in sliding regimes. Automatica: IFAC Journal 32(7):1037{1042. Zhao, F. 1994. Designing dynamics using geometric constraints. In Proceedings of the 14th IMACS World Congress on Computational & Applied Math. Zhao, F. 1995. Qualitative reasoning about discontinuous control systems. In Proceedings of IJCAI95 Engineering Problems for Qualitative Reasoning Workshop, 83{93.
