Design and Verification of Nonlinear Optimal Controller
Considering Acceleration Constraints based on Invariant
Manifold Calculation
M2015SC016 Shogo SUZUKI
Supervisor : Isao TAKAMI

1Introduction

The ISO 2631-1[1] states that there is a relationshipbetween ride comfort and acceleration of vehicles. The
smaller vehicle acceleration is the better ride comfort becomes. Therefore, in vehicle industry, it is necessary to
impose acceleration constraints to acquire a reasonableride comfort.
Conventionally, a control design that satisﬁes acceleration constraints by tuning weight matrices and gains
has been done. However, this often leads to a degradation on control performance.
In this paper, we consider the optimal control problem of the nonlinear system satisfying acceleration constraints. We propose a nonlinear optimal control design
method and a nonlinear servo control design method via
stable and center-stable manifold methods[2], [3] introducing Lagrange multiplier to handle acceleration constraints. The stable and center-stable manifold methods are recently proposed, which are iterative calculation methods to calculate the approximated solution of a Hamilton-Jacobi equation subject to someconstraints[4]. We verify the eﬀectiveness of the proposed methods on a magnetic levitation system and anumerical example.

2

H1 = pT (f + gu) + xT Qx + uT Ru,
1ū1 = − R−1 g T p,2
λ1 = 0.
• Case 2: h1 is active, which is a = amax .
H2 = H1 + λ2 (l(f + gu) − amax ),1ū2 = ū1 − λ2 R−1 g T lT ,2−2(amax − l(f + g ū1 ))λ2 =.
lgR−1 g T lT

Optimal Regulation Problem with Acceleration Constraints

In this section, we consider the nonlinear optimal regulation problem for nonlinear system subject to acceleration constraints based on the stable manifold theory.
2.1

where amin 6 0 6 amax . l is a vector for extracting
acceleration from the state space representation of thesystem (1). There are 3 possible cases that might happen at a time t, which are• Case 1: all constraints are inactive.
• Case 2: h1 is active, which is a = amax .
• Case 3: h2 is active, which is a = amin .
The optimal control problem considering acceleration
constraints can be handled as the conditional minimization problem by using Lagrange multiplier. Therefore,
the pre Hamiltonian Hi , the optimal input ūi and thevalue of the Lagrange multiplier λi (i = 1, · · · , 3) for eachcase are calculated.
• Case 1: all constraints are inactive.

Problem Definition• Case 3: h2 is active, which is a = amin .
H3 = H1 + λ3 (amin − l(f + gu)),
1ū3 = ū1 + λ3 R−1 g T lT ,22(amin − l(f + g ū1 ))λ3 =.
lgR−1 g T lT

Let us consider a nonlinear system of the physical system of the formẋ = f (x) + g(x)u, x ∈ Rn , u ∈ Rm , x(0) = x0 , (1)where f : Rn → Rn , g : Rn → Rn×m . We will design acontroller that minimize the following cost function∫ ∞J=(xT Qx + uT Ru)dt, Q > 0, R > 0,0

subject to some constraints on acceleration
amin 6 a 6 amax ,Next, we substitute ūi into the pre Hamiltonian Hi toobtain Hamilton-Jacobi equation H̄i = 0(i = 1, · · · , 3).
Then, an associated Hamiltonian system is derived.
And we apply the stable manifold method using Casechoosing algorithm[4] for Hamilton’s canonical equationand obtain an optimal feedback controller u(x).
2.3

Application for Magnetic Levitation System

where a is acceleration of the system (1), amax and amin
are upper and lower limit values of acceleration, respectively.

We verify the eﬀectiveness of the proposed method byapplication for a magnetic levitation system.

2.2

2.3.1

Application of Dynamic Programming

In this section, the acceleration constraints are considered as follows
h1 = a − amax = l(f + gu) − amax 6 0,
h2 = amin − a = amin − l(f + gu) 6 0,Modeling

The schematic diagram of the magnetic levitation system is shown Figure. 1. xb is position of the ball and ic
is input of the system. The equation of motion of thesystem can be derived via Newton’s second law. Considering a new variable x∗b := xb − xb0 , u∗ := u2 − u20

2.5
21.5

input

1
0.5
0-0.5
-1-1.5-2-2.50.05
0x2-0.05

Figure 1 Schematic Diagram of Magnetic Levitation
System-8-6-4-202x1

468×10-3

Figure 2 The nonlinear controller(red surface) and theLQ controller using the weighting matrices of the same
cost function as the nonlinear controller(blue surface).

based on the equilibrium point and choosing state variable x = [x1 x2 ]T = [x∗b ẋ∗b ]T and input u∗ , the equationof motion of the system is shown as followsẋ = f (x) + g(x)u∗ ,where[f (x) =[g(x) =(Cg − mb(C− mbx2
u0
x1 +xb0 +d01
x1 +xb0 +d(2)])2 ,])2 .

Parameters of the magnetic levitation system are listedin Table. 1.
Table 1 Physical parameters of the magnetic levitation
systemGravitational acceleration : g
Mass of ball : mb
Electromagnetic force constant : CActuator parameter : d

9.81[m/s2 ]66[g]1.532×10−4 [Nm2 /A2 ]9.445×10−3 [m]Figure 3 Input modiﬁcation algorithm to satisfy acceleration constraint for LQ controller.

3
2.3.2

Control Design

In control design, the weighting matrices of the costfunction are Q = diag([1, 1]), R = 1 and acceleration
constraints are amax = 0.1[m/s2 ], amin = −0.1[m/s2 ].
Applying the stable manifold method algorithm 3 times,
the optimal feedback controller u(x) is approximated bylinear interpolation (Figure 2).
2.3.3

Optimal Servo Problem with Acceleration Constraints

We consider nonlinear optimal servo problem for nonlinear system with acceleration constraints based on thecenter-stable manifold theory and Lagrange multiplier.
3.1

Problem Definition

We consider the optimal servo problem for the nonlinear system (1) and an error equation is as follows

Simulation Result

The proposed method is compared with a LQ controller using the same weighting matrices Q, R. To satisfy the acceleration constraints, the algorithm in Figure. 3 is applied to modify the LQ controller. The simulation results can be seen in the Fig. 4, 5 and 6. Here, theinitial conditions are x1 (0) = 0.013[m], x2 (0) = 0[m/s]and the equilibrium points x1ep = 0.007[m], x2ep (0) =0[m/s]. It is shown in Fig. 4, 5 and 6 that the responseof proposed method satisﬁes the acceleration constraints
and the nonlinear controller has better convergence tothe origin than the LQ controller with input modiﬁcation.

e = h(x, w),
where the system has relative degree 2. The reference
signal is generated from the exosystem shown as follows
ẇ = s(w), w ∈ Rp , s(0) = 0,(3)

where s : Rp → Rp , h : Rn × Rp → Rr . Since the system
has relative degree of 2, the following cost function ischosen
∫
1 ∞ 2
J=
(|e| + |ė|2 + |ë|2 )dt.
(4)
2 0

0.15

0.02
0.015

0.1

0.01
0.005

0.05

00-0.005
-0.01

-0.05

-0.015-0.02

x1
x2-0.025-0.03

0-0.1NLOR
LinearSaturation

input*10-2

0.5

1-0.15

1.5

00.5

1

1.5

time[s]time[s]Figure 4 Responses of state and input of constrained
nonlinear controller

Figure 6 Response of acceleration of constrained nonlinear and linear controllers• Case 1: all constraints are inactive.

0.02

1
H1 = pTx (f + gu) + pTw s + L(x, w, u),2ū1 = −(Lg Lf h)−1 (L2f h + L2s h + (Lg Lf h)−T g T px ),0.015
0.010.005

λ1 = 0.

0-0.005• Case 2 : h1 is active, which is a = amax .

-0.01

H2 = H1 + λ2 (l(f + gu) − amax ),-0.015-0.02-0.025-0.03

0ū2 = ū1 − λ2 (Lg Lf h)−1 (Lg Lf h)−T g T lT ,−amax + l(f + gū1 ).
λ2 =lg(Lg Lf h)−1 (Lg Lf h)−T g T lT

x1x2

input*10-20.5

1

1.5time[s]• Case 3 : h2 is active, which is a = amin .
H3 = H1 + λ3 (amin − l(f + gu)),

Figure 5 Responses of acceleration of constrained linearcontrollerū3 = ū1 + λ3 (Lg Lf h)−1 (Lg Lf h)−T g T lT ,amin − l(f + g ū1 )λ3 =.
lg(Lg Lf h)−1 (Lg Lf h)−T g T lT

We will design a controller that minimizes the cost function (4). J can be written as∫
1 ∞J=
L(x, w, u)dt,
2 0
L(x, w, u) = |h(x, w)|2 + |Lf h(x, w) + Ls h(x, w)|2+ |L2f h(x, w) + Lg Lf h(x, w)u + L2s h(x, w)|2 ,where Lf h, Ls h, L2f h, Lg Lf h, L2s h are the Lie diﬀerentiations and these are deﬁned as follows∂h∂hf, Ls h =s,∂x∂w∂Lf h∂Lf h∂Ls hL2f h =f, Lg Lf h =g, L2s h =s.
∂x∂x∂w
Lf h =Then, we consider the control design problem of system(1), (3) subject to acceleration constraints (2).
3.2

Application of Dynamic Programming

The pre Hamiltonian Hi , the optimal input ūi andthe value of the Lagrange multiplier λi (i = 1, · · · , 3) for
each case are calculated.

Next, we apply the center-stable manifold method usingcase-choosing algorithm for Hamilton’s canonical equation and obtain a optimal feedback controller u(x, w).
3.3

Numerical Example

We verify the eﬀectiveness of the proposed method on
a nonlinear spring system. The equation of motion of
the system is given bymẍm + kxm + ϵx3m = u(5)where k is the spring constant, ϵ is the nonlinear spring
constant and m is the mass. For simplicity, all of theseparameters have the value of 1. Then, Eq. (5) is rewritten as follows [ ] [] [ ]x20 ẋ1=kϵ 3 + 1 uẋ2(6)−mx1 − mx1me = x1 − w,
where x = [x1 x2 ]T = [xm ẋm ]T . Here, the reference isstep signal, is therefore the exosystem isẇ = 0.

In control design, acceleration constraints are amax =0.1[m/s2 ], amin = −0.1[m/s2 ]. Applying the centerstable manifold method algorithm 10 times, the optimal
feedback controller u(x, w) is approximated by linear interpolation. The proposed method is compared with an
PID controller using the gains KP = 10, KI = 1, KD =15. To satisfy acceleration constraints, the algorithm inFigure. 7 is applied to modify the PID controller.

1.4
1.2
10.8
0.6
0.4
0.2

x1x2

input*10-10reference-0.2

051015time[s]Figure 9 Responses of state and input of constrained
PID controller
0.15
NLOS
PID
Saturation

0.1
0.05
0Figure 7 Input modiﬁcation algorithm to satisfy acceleration constraints for PID controller.

-0.05
-0.1

The simulation results are shown in Figure. 8, 9 and10. Here, the initial conditions are x1 (0) = 0, x2 (0) = 0
and w(0) = 1 which is the reference value of the state x1 .
It is shown in Fig. 8, 9 and 10 that the response of pro-211015time [s]References

0.5x1x2input
reference

051015time[s]Figure 8 Responses of state and input of constrained
nonlinear controller
posed method satisﬁes the acceleration constraints andcorrectly follows the reference and the proposed method
has no overshoot and better response to the reference
than the PID controller.

45controllers were designed via stable manifold and centerstable manifold methods including Lagrange multiplier
to satisfy acceleration constraints. We veriﬁed the eﬀectiveness of the proposed methods on a magnetic levitation system and a numerical example.

1.500Figure 10 Response of acceleration of constrained nonlinear and PID controllers

2.5-0.5-0.15

Conclusion

In this paper, we proposed a nonlinear optimal controller and a nonlinear optimal servo controller designs
for systems with acceleration constraints. The nonlinear[1] ISO 2631-1: Mechanical vibration and shock evaluation of human exposure to whole body vibration
Part 1, General requirements Geneva International
Organization for Standardization (1997)[2] N. Sakamoto and A. J. van der Schaft: Analytical
approximation methods for the stabilizing solutionof the Hamilton-Jacobi equation, IEEE Transactions
on Automatica Control, 53-10, pp. 2335-2350 (2008)[3] N. Sakamoto and B. Rehak: Iterative methods tocompute center and center-stable manifolds with application to the optimal output regulation problem:
Proc. of the 50th Conference on Decision and Control, Orlando, December, pp. 4640-4645 (2011)[4] A. T. Tran and N. Sakamoto: A general framework
for constrained optimal control based on stable manifold method, 55th IEEE Conference on Decisionand Control (2016)