Abstract
This paper extends the design and analysis methodology of dynamic surface control (DSC) in Song and Hedrick, 2011, for a more general class of nonlinear systems. When rotational mechanical systems such as lateral vehicle control and robot control are considered for applications, sinusoidal functions are easily included in the equation of motions. If such a sinusoidal function is used as a forcing term for DSC, the stability analysis faces the difficulty due to highly nonlinear functions resulting from the low-pass filter dynamics. With modification of input variables to the filter dynamics, the burden of mathematical analysis can be reduced and stability conditions in linear matrix inequality form to guarantee the quadratic stability via DSC are derived for the given class of nonlinear systems. Finally, the proposed design and analysis approach are applied to lateral vehicle control for forward automated driving and backward parallel parking at a low speed as well as an illustrative example.
1. Introduction
The dynamic surface control (DSC), one of robust nonlinear control techniques, has been developed with a wide spectrum of applications including throttle/brake control on automated vehicles [1], underactuated ship control [2], and robot control [3]. This control technique is a dynamic extension of multiple sliding surface control with a series of first order low-pass filters to avoid an “explosion of terms” [4]. The existence of DSC gains and filter time constants for semiglobal stability was theoretically proved in [4]. Recently, a noble analysis method in the framework of convex optimization has been introduced to allow us to find a quadratic Lyapunov function numerically for a class of nonlinear systems called “strict-feedback” form as follows [3]: Furthermore, if in (1) is replaced by where and are continuous and invertible, the design procedure proposed by Swaroop et al. [4] and Gerdes and Hedrick [5] can be still applied for the given system.
However, this replacement induces another highly nonlinear function resulting from the low-pass filter error dynamics when stability analysis is performed. The following example illustrates the design approach of DSC as well as the difficulty that this paper seeks to solve: where and are continuous on ; for example, ; thus both and are bounded on . The control objective is to stabilize the system; that is, . First, define the first error surface as . After taking its derivative along the trajectory of (2) Then, the synthetic input, which is forced to drive , is derived as where is a controller gain. We now define the second sliding surface , where equals passed through a first order low-pass filter; that is, where is the filter time constant. Finally, the control input is derived as
Next, the stability analysis is investigated based on the closed-loop dynamics as suggested in [3]. If both and are added and subtracted in (2) and in (7) is put in (3), the closed-loop dynamics is written as By use of (5) and definitions of and , (8) is rewritten as Since the first order low-pass filter in (6) is added, the filter dynamics should be included in the closed-loop dynamics for stability analysis. After defining the filter error, , the augmented closed-loop dynamics is summarized as Since the function, , is locally Lipschitz, there exists such that where is a Lipschitz constant on . Using the continuity of and in (2), it is also shown that the last term of the third row in (10) is bounded on . Therefore, the existence of the controller gain and filter time constant for semiglobal stability can be shown as suggested in [5].
However, this fact does not tell us whether the closed-loop system is stable for the given and . To answer the question, we may need to find a Lyapunov function candidate explicitly and one of the possible analysis approaches is based on linear matrix inequality. To apply this approach to (10), it is necessary to write it in matrix form as While the next procedure is to investigate whether (12) is in a class of linear differential inclusions classified in [6], the inclusion of the nonlinear function in (12) results in the mathematical difficulty of stability analysis.
The contribution of this paper is to extend a design and analysis methodology of DSC for a more general class of nonlinear systems as shown in (2) and (3). The consideration of this class of nonlinear systems is motivated when rotational mechanical systems are considered for applications; that is, sinusoidal functions are in general included in the equation of motions. As one of the applications, the proposed control approach is applied to lateral vehicle control for forward automated driving and backward parallel parking at a low speed. Finally, its performance will be validated via simulations.
2. Problem Statement
Consider the class of nonlinear systems where and are continuous on and is in strict-feedback form in the sense that the depends only on . It is implied that is locally Lipschitz and is bounded on [7]. Therefore, there exists a constant such that for all on .
The nonlinear function is also locally Lipschitz; that is, there exists a constant such that In addition, there exist differentiable functions , where , which are inverses of the in the sense that and is bounded on ; that is, there exists a constant such that
3. Analysis and Design of DSC
3.1. Design Procedure
Although the proposed design procedure is quite similar to the standard one described in [4], an outline of the design procedure is as follows. Define the first error surface as , where is the desired value as the control objective. After taking the time derivative of along the trajectory of (13), The surface error will converge to zero if ; however there is no direct control over the surface dynamics. If is considered as the forcing term for the surface dynamics, then the sliding condition outside some boundary layer is satisfied if , where where is the inverse of .
The next step is to force , so define , where and is obtained after passing through a first order low-pass filter; that is, It is noted that this procedure is different from the one explained in the introduction. That is, instead of passes through the filter and the inverse function of the filtered signal is used to define . After taking a derivative of along the trajectory of (13), the resulting synthesis term, , is derived as where and the last equality comes from (20).
Similarly, continuing this process for each consecutive state, define the th error surface as where and is where Then, is obtained by filtering ; that is, After continuing this procedure for , define , where . Finally, the control input is derived as where
3.2. Augmented Error Dynamics
The closed-loop error dynamics will be derived for stability analysis in this section. After subtracting and adding and and using (26) in , the closed-loop dynamics of (13) can be written as By use of (23) and the definition of error surfaces, the above equations can be described in terms of the error surfaces of DSC as follows: where .
In addition, we need to consider the augmented error dynamics due to inclusion of a set of the first order low-pass filters. Let us define the filter error as . Then, the filter dynamics are where the last equality comes from (25). By taking a derivative of (23), we can write as Combining (30) with (31), we have the filter error dynamics,
Therefore, the overall error dynamics, (29) and (32), can be given as Furthermore, (33) can be written in matrix form as follows: where the vectors are defined as and the submatrices are
Since the first block matrix in (34) is invertible such that after multiplying the inverse matrix to both sides in (34), the augmented closed-loop error dynamics are rewritten as where
Since is written in a function of , , , and based on (29) the time derivative of can be decomposed into three parts as follows: Therefore, (38) is rewritten as where the error state , , and .
Finally, we need to determine the upper bound of in (42). Using the assumptions (15) in Section 2, the upper bound of for is Using (14), the upper bound of for is where , , and . Similarly, the upper bound of for is obtained using (17) Combined with (43), (44), and (45), (42) can be written in diagonal norm-bounded LDI form as follows [6]:
3.3. LMI Approach for Stability Analysis
If either stabilization or regulation problem is considered, the first element of in (46) is zero. Furthermore, if for special cases among nonlinear systems in (13), quadratic stability of the resulting closed-loop error dynamics is defined as follows [3].
Definition 1. Let be an exponentially stable equilibrium point of the closed-loop error dynamics in (46) where and is Hurwitz for the given set of controller gains; . Then, a nonlinear system in (13) is quadratically stabilizable via DSC if there exists a positive definite matrix such that
Furthermore, the quadratic stability under the DSC is guaranteed by the following theorem.
Theorem 2. Suppose that the closed-loop error dynamics in (46) are given for the given set of controller gains, for all in a domain . If there exist and such that where and , the origin in (46) is then exponentially stable in . Thus the nonlinear system (13) is quadratically stabilizable via DSC with the given on .
For details of the proof of the theorem, readers are referred to Boyd et al. [6].
If a tracking problem is considered () and it is assumed that is bounded, the ultimate and quadratic boundedness is defined as follows [3].
Definition 3. The error dynamics in (46) is quadratically bounded with Lyapunov matrix if there exists such that for all nonzero .
It is noted that the assumption that is bounded is feasible because the time derivative of the filtered signal is bounded.
Suppose in (46) is norm-bounded such that . After defining and , the error dynamics in (46) is written as Without loss of generality, it can be considered that is a unit-peak input. Then, the following theorem describes the condition for guaranteeing quadratic tracking as well as the computation of the matrix for a given set of controller gains.
Theorem 4. For the given set of controller gains, , suppose that the closed-loop error dynamics in (51) is given on the domain and is a feasible output trajectory. The closed-loop error dynamics is quadratically bounded with Lyapunov matrix if there exist , , and such that where , , and .
Readers are referred to Boyd et al. [6] for the proof and the definition of the feasible output trajectory is explained in [3, 4].
3.4. Illustrative Example
Consider the example in (2) and (3) where and the domain is defined as Then, (5) becomes and a first order low-pass filter is defined as It is noted that instead of is passed through the low-pass filter. After defining the second sliding surface , where , and taking its derivative, the control input is obtained by where and the last equality comes from (55).
If a new filter error, , is defined, the augmented closed-loop dynamics can be written as where and from (55). Therefore, (58) is written in matrix form as The upper bound of can be determined as where the first inequality comes from a Lipschitz condition such that for all and the second inequality comes from a fact that is bounded on .
Finally, the augmented error dynamics can be written in diagonal norm-bounded LDI form as follows: where
When the controller gains are given as and , LMI (49) is solved numerically in the framework of convex optimization using CVX [8]. It is shown that the closed-loop system is quadratically stable by finding the feasible solution of LMI (49). For the given controller gains and initial condition, the time responses of and are shown in Figure 1 and as . Thus, it is validated that the result of quadratic stability analysis based on an LMI approach is equivalent to simulation results.
4. Application to Lateral Vehicle Control
The proposed control approach is applied to design the robust lateral control algorithm for autonomous valet parking (AVP). It is assumed that the position and heading angle information is provided via either infrastructure sensors and vehicle to infrastructure (V2I) communication [9] or an in-vehicle sensor such as DGPS [10]. The objective of the lateral controller is to perform two different maneuvers for AVP, that is, forward driving and backward parallel parking. Therefore, it is necessary for the lateral controller to be robust enough to track desired trajectories for different driving maneuvers.
4.1. Vehicle Model
While the bicycle model has been used widely for design of a lateral controller for high speed driving on highway [11, 12], a vehicle is driving at low speed for AVP and thus slip angle can be neglected in this study. Therefore, the following kinematic model is used for both forward driving and backward parallel parking (refer to Figure 2) [13]: where the subscript represents the driving maneuver; that is, for forward driving and for backward parking, If dynamics of a steering actuator from steering wheel angle command to steering angle of the vehicle is considered, the following equation of motion may be added: where is the maximum steering angle.
4.2. Controller Design
With consideration of operating conditions such as low speed and small slip angle, the nonlinear kinematic model with actuator dynamics in (63) and (65) is used for design of the lateral controller. Then, the proposed control approach based on DSC is applied to the kinematic model as follows. First, the first error surface is defined using the idea of preview control suggested in [14] (see in Figure 2): where the lateral error is defined as the point is the closest point with respect to current position, and the preview distance and desired heading angle are For instance, is negative for the given scenario in Figure 2 because the rotational direction from vector to defined in (67) is clockwise. Thus, the resulting positive lateral error implies a steering wheel angle command in the counterclockwise direction. If it is assumed that the preview distance is a constant, the point in Figure 2 can be calculated with respect to the given desired trajectory and the desired heading angle is then determined.
After taking a derivative of along the trajectory of (63), the derivative of is It is remarked that can be chosen as a variable if necessary. For the simplicity of derivation, it is assumed to be constant. To make go to zero, let , where is a controller gain. Then the desired steering angle is obtained as where It is noted that all of , , , and are known for the given desired trajectory.
Then, the second error surface is defined as , where and is calculated after passing through a first order low-pass filter as follows:
After differentiating and using (65), the resulting equation is where Let , where is a controller gain. Then the desired steering wheel angle is obtained as where and the last equality comes from (72).
4.3. Stability Analysis
If and are added and subtracted in (69) and in (75) is put in (73), the augmented closed-loop error dynamics is where it is assumed that is constant with respect to time. Using (70), (77) is written as where . As done in the example of Section 3.4, the augmented error dynamics is written in matrix form as follows:
Furthermore, the upper bound of is obtained as because it is Lipschitz, and both and in (71) are differentiable on ; is bounded with respect to the desired trajectory. Thus, without loss of generality, it is assumed that is norm-bounded such that . Therefore, the augmented error dynamics can be written in LDI form of (51) as where , , and .
Suppose (m/s), (m) in (79), and the control parameters are assigned as , , and and . When is assumed to be 5, LMI (52) is solved iteratively for a fixed by minimizing the largest semiaxis (i.e., maximizing the smallest eigenvalue of ) [3]. That is, after the 40 logarithmically equally spaced points between and are generated for 's, the minimum of the maximum diameter, which is , is obtained when (in the left plot of Figure 3). Then the 20 linearly equally spaced points between 0.3455 and 0.5541 are generated and the iterative computation of LMI (52) is performed for each . Finally, for , the corresponding maximum diameter of the ellipsoid, , is 0.6925 which is the semiaxis in the axis. It is remarked that the size of the ultimate and quadratic error bound is roughly proportional to the magnitude of and thus more accurate estimation of the error bound relies on better estimate of resulting from the desired trajectory. In consequence, it is expected that is bounded for the given set of control parameters. Furthermore, the relative degree of the error dynamics is one and it is shown that its internal dynamics is input-state stable [14]. Thus, this implies that both and are bounded if is bounded.
4.4. Simulation Results
Suppose 24 waypoints are given a priori as shown in Figure 4. Forward driving maneuver is assigned from the first waypoint to 23rd waypoint and backward parallel parking maneuver is requested from 23rd to 24th waypoint. Moreover, both straight and curved road geometry is considered in this driving scenario; that is, from the first to 7th waypoints are for the straight road, from 7th to 20th waypoint for the curved road with about 12 (m) radius of curvature, and from 20th to 23rd waypoint again for the straight road as shown in Figure 4.
Let the additional system parameters in (63) and (65) be , (degree), and for simulations. When the control parameters used for stability analysis above are applied to the proposed lateral controller, time responses of steering and heading angle are shown in Figure 5 and it is shown that the corresponding lateral error, , is less than 0.3 (m). That is, it is shown that the lateral error is bounded as expected above. It is validated that the proposed controller enables the vehicle to perform two different maneuvers with only different value of .
5. Conclusions
This paper developed the analysis and design method of DSC for a class of nonlinear systems where the nonlinear functions are included as forcing terms of DSC. The proposed control approach was applied to lateral vehicle control for forward driving and parallel parking maneuvers at low speed. With modification of input variables to the filter dynamics, it was shown that most of results in [3] could be used for the new class of nonlinear systems. Thus, the stability conditions in linear matrix inequality form were presented to guarantee the quadratic stability and boundedness via DSC for the given class of nonlinear systems. Furthermore, the quadratic Lyapunov functions were calculated numerically in the framework of convex optimization for a lateral vehicle control problem as well as an illustrative example. It was validated that the analysis results agreed with ones of simulation.
Nomenclature
| Vehicle position in longitudinal direction | |
| : | Vehicle position in lateral direction |
| : | Vehicle velocity |
| : | Length from a center of mass to front wheel axle |
| : | Length from a center of mass to rear wheel axle |
| : | Length of wheelbase, that is, |
| : | Front steering angle |
| : | Steering wheel angle |
| : | Steering wheel angle command |
| : | Steering gear ratio, that is, |
| : | Lateral position error |
| : | Heading angle |
| : | Heading angle error |
| : | Preview distance |
| : | Time constant for steering actuator |
| : | Slip angle. |
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2009-0075110). It was also supported in part by the research project funded by LS Mtron.