Nonlinear Model Predictive Path-Following Control 1

Nonlinear Model Predictive Path-Following Control Timm Faulwasser and Rolf Findeisen Institute for Automation Engineering, Otto-von-Guericke University Magdeburg, Universit¨ atsplatz 2, 39104 Magdeburg, Germany {timm.faulwasser, rolf.findeisen}@ovgu.de Keywords : reference tracking, path-following, nonlinear systems, model predictive control, parametrized reference, stability Abstract : In the frame of this work, the problem of following parametrized reference paths via nonlinear model predictive control is considered. It is shown how the use of parametrized paths introduces new degrees of freedom into the controller design. Sufficient stability conditions for the proposed model predictive path-following control are presented. The method proposed is evaluated via simulations of an autonomous mobil robot.

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Introduction

The design of feedback controllers for dynamical systems is usually subject to one of the following purposes: either suppress disturbances to stabilize a system around a fixed reference state via an appropriate input, or influence its dynamic behavior such that the system states or outputs converge to a timevarying reference signal. The presence of input and state constraints makes both problems considerably tougher. Even in the absence of input and state constraints the controller design for tracking and tracking-related problems is a non-trivial task, especially for nonlinear systems. For example the design of output tracking controllers for non-minimum phase systems is subject to fundamental limits of achieveable tracking performance. These limits can arise from unstable zero-dynamics of non-minimum phase systems – see inter alia [12] for details on the linear case and [14] for the nonlinear case –. Recently, path-following approaches have shown their ability to circumvent these fundamental performance limits [1, 15]. Most of the existing methods are based on the idea of using parametrized reference signals instead of time-dependent reference trajectories. However, the aforementioned results on path-following are limited, since input and state constraints are not considered. One control strategy that allows to take constraints on states and inputs into account is nonlinear model predictive control (NMPC). In [11] one of the first results on the application of NMPC to tracking problems is outlined. Robust output feedback tracking for time discrete systems is discussed in [8]. [9] presents results on the tracking of asymptotically constant references for the continuous case. Furthermore, NMPC can be applied to the tracking of non-holonomous Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008

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wheeled robots as well, e.g. [6]. All these NMPC approaches towards the tracking problem rely on the definition of the tracking error as the difference between the output or state and a time-depending reference. Parametrized references in the context of (N)MPC have previously been discussed inter alia in [7]. There, tracking of piecewise constant reference signals is considered. As it will be shown, the use of parametrized reference signals leads to a control structure which affects both the system inputs and the evolution of reference signals. Feedback control of reference evolution can be also considered by applying reference governors. Reference governors, as presented inter alia in [2, 13], are usually hierarchically structured. An inner control loop stabilizes the system, while an outer loop controls the reference evolution such that input and state constraints are fulfilled. Unlike these approaches the results presented here only rely on one control loop. The contribution of this paper is a scheme for model predictive path-following control (MPFC) which implements parametrized references into a NMPC setup while guaranteeing stability. Combining NMPC and the core idea of pathfollowing leads to additional degrees of freedom in the controller design. These can be utilized to guarantee stability and to achieve better performance. In contrast to other works on path-following [1, 15], which apply back-stepping techniques to construct output-feedback controllers, here the results are based on state-feedback. The remain of this work is structured as follows: Section 2 introduces the path-following problem and shows how parametrized references can be utilized in NMPC. Furthermore, results on the stability of the proposed model predictive path-following control are given. In Section 3 MPFC is applied to a model of a simple wheeled robot. Section 4 gives final conclusions and remarks.

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Model Predictive Path-Following Control

Consider a continuous time nonlinear system, subject to input and state constraints: x˙ = f (x, u), x(0) = x0 , (1) where x ∈ X ⊆ Rn , u ∈ U ⊂ Rm . Tracking of system (1) refers to the design of a controller such that the difference between the system state x(t) and a timevarying reference signal r(t) vanishes. Furthermore, it has to be guaranteed that the state x(t) is in the state constraint set X ⊆ Rn and that the inputs u(t) are taken out of the set of admissible inputs U ⊂ Rm . The tracking problem is often defined in terms of the time-dependent tracking error: eT (t) = x(t) − r(t).

(2)

Usually, the time-dependent reference signal r(t) is assumed to be generated by an exo-system. Hence, the tracking problem can be reformulated as a stabilization problem. Typical applications of tracking are synchronization tasks, movement of robots or tracking of optimal state trajectories, which have been calculated previously. Path-following refers to a different problem. Instead of a reference trajectory, a parametrized reference r(Θ) is considered. This reference is called path and

Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008

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it is often given by a regular curve in the state space Rn : r(Θ) :

ˆ 0] ⊂ R 7→ r(Θ) ∈ Rn , [Θ,

r(0) = 0.

(3)

It should be noted that in this paper the path as stated by (3) is negatively parametrized and ends in the origin. It is assumed that r(Θ) is sufficiently often continuously differentiable with respect to the parameter Θ. This path formulation does not distinguish between finitely and infinitely long paths, since ˆ 0] can be mapped onto both. The considered system (1) is the real interval [Θ, subject to constraints, hence the path has to fulfill the state constraints r(Θ) ∈ ˆ 0]. A path is denoted as regular if it is a non-singular curve X for all Θ ∈ [Θ, ˜ such that the and for each state x ∈ X there exists a unique path parameter Θ, ˜ distance between the path point r(Θ) and x is minimal. Combining (2) and (3) yields to the path-following error: eP (t) = x(t) − r(Θ).

(4)

Comparing the definition of eP (t) to (2) reveals important differences between tracking and path-following. Trajectory tracking implies that the reference signal inheres an explicit requirement when to be where in the state space. This arises from the fact that r(t) is a reference trajectory. In path-following these requirements are relaxed. In general, the path parameter Θ = Θ(t) is time ˙ is not given a priori, it has to be dependent, but since its time evolution Θ ˙ serves as an additional obtained in the controller. Therefore, the timing law Θ degree of freedom in the design of path-following controllers. Typical practical path-following problems are the car-parking problem, autonomous ship control, the control of CNC-machines or the control of batch crystallisation processes. In these and many other applications it is desireable to stay very close to a given path even if this implies slower movement. Considered Path-Following Problem In the frame of this work, the subsequently stated path-following problem is considered. Find a controller, such that the following is satisfied: P1 Path convergence: The path-following error vanishes, lim eP (t) = 0. t→∞

˙ P2 Forward motion: The system moves forward in path direction. Θ(t) >0 ˆ holds for all t > 0 and all Θ ∈ [Θ, 0). P3 Constraint satisfaction: The state and input constraints x ∈ X ⊆ Rn , u ∈ U ⊂ Rm are fulfilled. Proposed Control Strategy Since a predictive control strategy is considered, predicted states and inputs are referred as x ¯ and u ¯. At the sampling instants tk = kδ, where δ is the constant sampling time, the cost functional to be minimized over the prediction horizon TP is given by: J (¯ x, u ¯, Θ, v) =

Z

tk +TP

F (¯ x, u ¯, Θ, v) dτ + E (¯ x(tk + TP ), Θ(tk + TP )) .

(5)

tk

In contrast to standard NMPC approaches, this cost functional does not only depend on the predicted states and inputs (¯ x and u ¯) but also on the path Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008

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parameter Θ and a to be defined path parameter input v. Θ is regarded as an internal or virtual state, hence the end penalty E is a function of x ¯(tk + TP ) and Θ(tk + TP ). Similarly, the stage cost F depends on the variables (¯ x, u ¯, Θ, v). Nonlinear model predictive path-following control can then be stated as the repeated solution of the following open-loop optimal control problem: minimize J (¯ x, u ¯, Θ, v) u ¯(·), v(·)

(6)

subject to the usual constraints x ¯˙ = f (¯ x(τ ), u ¯(τ )),

x ¯(tk ) = x(tk ),

∀τ ∈ [tk , tk + TP ] : x ¯(τ ) ∈ X , u ¯(τ ) ∈ U, n x ¯(tk + TP ) ∈ E ⊆ X ⊆ R .

(7a) (7b) (7c)

The constraint (7c) indicates that at the end of each prediction the predicted state x ¯(tk + TP ) has to be in the terminal region E and E ⊆ X is a closed subset of Rn . Additonal path-following constraints must also be respected: ˙ = g(Θ(τ ), v(τ )), Θ

Θ(tk ) = argminkx(tk ) − r(Θ)k,

(8a)

Θ

ˆ 0] ⊂ R, v(τ ) ∈ V ⊆ R. ∀τ ∈ [tk , tk + TP ] : Θ(τ ) ∈ [Θ,

(8b)

These additional constraints state that the path parameter evolution is given ˙ = g(Θ, v), where g is called timing law and v ∈ V is a virtual input which by Θ influences the path parameter evolution. To solve the open-loop control problem at any sampling instance tk , it is necessary to calculate the path point closest ˙ = g(Θ, v), see to the system state x(tk ), since it serves as initial condition for Θ (8a). This initial condition is calculated via an extra minimisation. To ensure that the system moves forward along the path, this minimisation is subject to ˆ 0). The timing law g(Θ, v) the constraint Θ(tk ) > Θ(tk−1 ) for all Θ(tk ) ∈ [Θ, in (8a) provides an additional degree of freedom in the controller design. This ˙ = g(Θ, v) > 0 holds for all Θ ∈ [Θ, ˆ 0) function has to be chosen such that Θ and all v ∈ V (compare with requirement P2). The solution to the optimal control problem defined by (5) – (8) leads to the optimal input trajectory u ¯⋆ (t, x(tk )) which defines the input to be applied: u(t) = u ¯⋆ (t, x(tk )), t ∈ [tk , tk + δ].

(9)

It should be noted that the additional virtual input v and the path parameter Θ are internal variables of the controller. The MPFC algorithm, as defined by (5) – (9), is a modified NMPC scheme, which chooses the velocity on the path such that the system stays close to it. Metaphorically speaking, if the system is far away from the path, first approach the path and then try to follow it along. Considering this MPFC scheme path convergence is more important than speed. In general, the considered approach does not lead to the fastest feasible path evolution. In contrast to the tracking of previously calculated optimal reference trajectories (which might be time-optimal), the MPFC scheme will iteratively adjust the reference evolution such that good path convergence is achieved. This online adjustment can be used to compensate disturbances or model-plant mismatch.


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Stability Sufficient conditions to prove the stability of NMPC schemes have been discussed widely throughout literature, e. g. in [4, 5, 10]. Since the proposed scheme is a modified NMPC scheme with expanded states and inputs, the following usual assumptions are made [3]: A1 X ⊆ Rn contains the origin in its interior. X is closed and connected. A2 U ⊂ Rm is compact and the origin is contained in the interior of U. A3 f : Rn × Rm 7→ Rn is a continuous and locally Lipschitz vector field. Furthermore, f (0, 0) = 0. A4 For all initial conditions in the region of interest and any piecewise continuous input function u(·) : [0, TP ] 7→ U, (1) has a unique continuous solution. ˆ 0] × V 7→ R is continuous and positive A5 The cost function F : X × U × [Θ, ˆ 0] × V. definite in the domain X × U × [Θ, Subsequently it is shown that the following additional assumptions are required to ensure stability of the proposed MPFC scheme: A6 The path r(Θ) is regular and negatively parametrized, such that ∀Θ ∈ ˆ 0] : r(Θ) ∈ X ⊆ Rn and r(0) = 0 hold. [Θ, A7 The timing law g(Θ, v) has equivalent properties as required for f (x, u) ˆ 0) : in assumptions A3 and A4. Furthermore, ∀ v ∈ V and ∀ Θ ∈ [Θ, g(Θ, v) > 0, where V ⊆ R is compact and 0 ∈ V. If these conditions are fulfilled, then the following theorem holds. Theorem 1 (Stability of Model Predictive Path-Following Control). Consider the path-following problem for (1) as given by P1–P3 and assume that assumptions A1–A7 are fulfilled. Suppose that: ˆ 0] is closed and ∀Θ ∈ [Θ, ˆ 0] : r(Θ) ∈ E. (i) The terminal region E × [Θ, The terminal penalty E(x, Θ) is continuously differentiable and positive semidefinite. Furthermore E(0, 0) = 0 holds. ˆ 0] there exists a pair of admissible inputs (uE , vE ) ∈ U ×V (ii) ∀(x, Θ) ∈ E ×[Θ, such that f (x, uE ) + F x, Θ, uE , vE ≤ 0 (10) ∇E(x, Θ) · g(Θ, vE ) ˙ = g(Θ, vE ) starting at (x, Θ) ∈ and the solutions of x˙ = f (x, uE ) and Θ ˆ ˆ E × [Θ, 0] stay in E × [Θ, 0] for all times. (iii) The NMPC open-loop optimal control problem is feasible for t0 . Then, for the closed-loop system defined by (1), (5)–(9), the path-following error eP (t) = x(t) − r(Θ) converges to zero for t → ∞. Furthermore the region of attraction is given by the set of states for which the open-loop optimal control problem (5)–(8) has a feasible solution. The stability conditions contained in Theorem 1 are very similar to wellknown conditions for standard sampled data NMPC and the approach to establish stablity is very similar to [5, 10]. Hence, only a concise draft of the proof is provided. Mainly it is shown how the MPFC problem, as given by (5)–(9), can be reformulated such that it is equal to a standard NMPC problem. Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008

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Proof. Consider the following coordinate changes: y= w=

(x − r(Θ), Θ)T , (u, v)T ,

y ∈ Rn+1 w ∈ Rm+1 .

(11)

The problem specified by (5) – (8) can be reformulated for the expanded state and input y and w. In the new coordinates the state to be stabilized is the origin 0 ∈ Rn+1 . Hence, the problem is equivalent to a stabilization NMPC problem. Start by applying the sufficient conditions for nominal stability of NMPC as given inter alia by [3, 5, 10] to the NMPC problem in y, w-coordinates. This leads to straightforward conditions on the feasibility of the open-loop optimal control problem at t0 (compare with (iii)) and the invariance of x and Θ in the ˆ 0] in (ii). With respect to the coordinate transformation terminal region E × [Θ, (11), the sufficient stability condition in y and w can be reformulated in the orginal coordinates x, u, Θ and v. This approach directly yields the condition as given in (ii) of Theorem 1. Theorem 1 implies that the definition of suitable terminal penalties is relaxed in the MPFC setup compared to a straightforward tracking via NMPC. Starting with E(x, Θ) = E1 (x) + E2 (Θ), the degrees of freedom to choose the timing law g(Θ, v) and E2 (Θ) can be utilized to assure that (10) holds.

3

Simulation Results

To illustrate the performance properties of the method proposed, an autonomous mobil robot in a fixed coordinate frame is considered: x˙ 1 u1 ·cos(x3 ) x˙ 2 = u1 ·sin(x3 ) . (12) x˙ 3

u2

x1 and x2 refer to the vehicle position in the x1 -x2 plane and x3 denotes the yaw angle. u1 is the velocity of the vehicle and u2 is the time derivative of the vehicles steering angle. The inputs u1 , u2 are subject to the contraints (u1 , u2 )T ∈ U = ([0, 12.5], [− π4 , π4 ])T . For the path-following case the timing ˙ = −λΘ + 0.6 − v, λ = 10−3 , v ∈ V = [0, 0.6]. The law is chosen to be: Θ cost function is given by F (·) = y T Qy + wT Rw, where y and w are taken from (11) and Q = diag(105 , 105 , 105 , 10), R = diag(10−1 , 10−2 , 103 ). In [6] a NMPC controller, which solves the tracking task for this system, was presented. Plot a) of Figure 1 shows the movement of the vehicle (12) for tracking and path-following. While the projection of the reference trajectory into the x1 -x2 plane is depicted by the thick grey line, the black line marks the vehicle movement when a NMPC tracking controller is applied. The dashed line shows the movement if the proposed scheme for predicitve path-following is considered. The triangles indicate the yaw angle at selected locations. In plot b) of Figure 1 the corresponding input signals for tracking (solid black lines), path-following (dashed lines) and the considered input constraints (solid grey lines) are shown. The reference trajectory in the tracking case of Figure 1 is such that the approach used by [6] fails, since inputs constraints have to be violated to stay on the reference trajetory. Metaphorically speaking, the last turn of the reference, as depicted in plot a) of Figure 1, is too sharp to be realized by an admissible input in the tracking case. The time plots for all states of (12) are depicted Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008

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b) input signals

a) movement in x1 -x2 plane

u110

1

u1 , track u1 , path

5 0 0.5

u2 1

x2 0

reference in x1 -x2 plane tracking result

−0.5

−2

x1

−1 0.6 0.3

−1

v, path

0 0

0

2

c) tracking

4

t

6

8

10

d) path-following

−1

−1

x1

r 1 (t) x1 (t)

−3

x1

r 2 (Θ(t))

1

x2 (t)

x2

r 1 (Θ(t)) x1 (t)

−3

r 2 (t)

1

x2 (t)

x2

0

0

1

r 3 (Θ(t))

1

x3

x3

r 3 (t) x3 (t)

−1 0

u2 , path

v

path-following result −3

u2 , track

2

4

t

6

8

x3 (t)

−1 10

0

2

4

t

6

8

10

Figure 1: Plot a) shows the motion of the vehicle in the x1 –x2 plane for tracking and path-following. Plot b) depicts the corresponding input signals. Plots c) and d) show state and reference signals for tracking and path-following.

in plots c) and d) of Figure 1. c) refers to the tracking case, d) shows the path-following results. The MPFC results, as depicted by the dashed lines in plots a)-d) of Figure 1, show that the system under MPFC feedback is able to stay on the path. It needs to be pointed out that the MPFC controller requires less time to accomplished the whole path, therefore all states for path-following are stabilized at the final path point in plot d) of Figure 1. Nevertheless path-following slows down the path evolution in the turns to make the vehicle stay on the path. The acceleration of the reference evolution in easy sections of the path – as it can be observed in plot d) of Figure 1 – is a direct consequence of the chosen reference value for the virtual input v. Not in every practical application this speed-up property might be desired. To avoide this, the reference evolution can be bounded by choosing the constant reference value for the path parameter ˙ = g(Θ, v) matches a desired input v such that the corresponding evolution Θ reference evolution.


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4

Conclusions

It has been shown how the main idea of path-following can be implemented into a NMPC framework. Combining these approaches leads to a control scheme which computes the evolution of a reference signal and the input signals to follow this reference at the same time. Sufficient stability conditions for the proposed MPFC scheme have been presented. To investigate the performance of the method a simple vehicle has been considered as an example. Future work will investigate the robustness of the proposed scheme as well as the robust design. In particular, the differences between the proposed approach and the tracking of offline calculated optimal trajectories will be discussed.

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