Discrete-Time Inverse Optimal Control for Nonlinear Systems ...

49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA

Discrete-time Inverse Optimal Control for Nonlinear Systems Trajectory Tracking Fernando Ornelas, Edgar N. Sanchez and Alexander G. Loukianov Abstract— This paper presents an inverse optimal control approach for output tracking of discrete-time nonlinear systems, avoiding to solve the associated Hamilton-Jacobi-Bellman (HJB) equation, and minimizing a meaningful cost function. This stabilizing optimal controller is based on discrete-time passivity theory. The applicability of the proposed approach is illustrated via simulations by trajectory tracking control of a planar robot.

I. INTRODUCTION In optimal nonlinear control, we deal with the problem of finding a stabilizing control law for a given system such that a criterion, which is a function of the state variables and the control inputs is minimized; the major drawback is the requirement to solve the associated HJB equation [1], [2]. The aim of the inverse optimal control is to avoid the solution of this HJB equation [3]. In the inverse approach, a stabilizing feedback control law, based on a priori knowledge of a control Lyapunov function (CLF), is designed first, and then it is established that this control law optimize a meaningful cost functional. The main characteristic of the inverse problem is that the meaningful cost function is a posteriori determined for the stabilizing feedback control law. For continuous-time inverse optimal control applicability, we refer to the results presented in [2], [3], [4], [5], [6], [7], [8]. For the discretetime framework see [9]. The existence of CLF implies stabilizability [2] and every CLF is a meaningful cost function [8], [10]. Systematic techniques for finding CLFs do not exist for general nonlinear systems; however, this approach has been applied successfully to classes of systems for which CLFs can be found such as: feedback linearizable, strict feedback and feed-forward systems, etc. [11], [12]. In this paper, using passivity, these system structure constraints are relaxed. Passivity-based control (PBC) was introduced in [13] to define a controller synthesis methodology, which achieves stabilization by passivation. The advantage of passivity is that it can be used to design stable and robust feedback controllers. Despite the fact that nonlinear PBC for continuous-time has attracted considerable attention, and many developments in this direction have been obtained [14], [15], [16], discretetime nonlinear passivity theory has only a few results [17], [18], [19], [20]; however, the passivation is achieved by a state feedback control in these works, while we relaxed this This work is supported by CONACYT under projects 57801 and 46069. All authors are with CINVESTAV, Unidad Guadalajara, Jalisco 45015, México. [email protected]

978-1-4244-7746-3/10/$26.00 ©2010 IEEE

condition by using a discrete-time CLF (DTCLF) for system passivation, by using a synthesized output. In this paper, we propose a novel discrete-time inverse optimal controller based on a quadratic storage function, which can be selected as a DTCLF, and on the synthesis of an output such that the system is rendered passive. Thus, asymptotic stability of the passive system is achieved with output feedback under detectability conditions. Finally, this DTCLF is modified in order to achieve asymptotic tracking of given output reference trajectories. II. M ATHEMATICAL P RELIMINARIES Let consider a nonlinear affine system and an output (to achieve passivity) given as xk+1 = f (xk ) + g(xk ) uk

(1)

yk = h(xk ) + J(xk ) uk

(2)

where xk ∈ Rn is the state of the system at time k ∈ N , u, y ∈ Rm f : Rn → Rn , g : Rn → Rn×m , h : Rn → Rm , and J : Rn → Rm×m are smooth and bounded mappings. We assume f (0) = 0 and h(0) = 0. N denotes the set of nonnegative integers. It is worth to note that, the output that renders the system passive is not in general the variable we wish to control. The first problem considered in this paper is to find a feedback control law which stabilizes system (1) at its equilibrium point and to establish that this controller is inverse optimal with respect to a meaningful cost functional given as ∞ X L(xk , uk ) (3) J = k=0

where L(xk , uk ) is a non-negative function. Similar to the continuous-time case, the discrete-time Hamiltonian becomes [21] H(xk , uk ) = L(xk , uk ) + V (xk+1 ) − V (xk )

(4)

where H(xk , uk ) = 0 for x ∈ Rn and the optimal control law u ∈ Rm ; V : Rn → R is a nonnegative definite function such that V (0) = 0 and V (xk ) > 0 (positive definite function), ∀xk 6= 0. The main characteristic of the inverse problem is that the meaningful cost functional (3) is a posteriori determined and minimized by the stabilizing feedback. Before to design a stabilizing and inverse optimal control law, we present definitions, sufficient conditions and key results, which help us to solve the stabilization and the inverse optimal control problems.

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Due to the fact that the inverse optimal control is based on a Lyapunov function, we establish the following definition:

if

Definition 1. (DTCLF [22]) Let V1 be a radially unbounded, definite positive function, with V1 (xk ) > 0, ∀xk 6= 0 and V1 (0) = 0. If for any xk ∈ Rn , there exist real values uk such that ∆V1 (xk , uk ) < 0

where Im is the m×m identity matrix, (·)−1 denotes inverse, and P is a definite positive matrix. Moreover, with V as a DTCLF, this control law is inverse optimal in the sense that minimizes the meaningful functional given as ∞ X J = L(xk , uk ). (7)

where ∆V1 (xk , uk ) is defined as V1 (f (xk ) + g(xk ) uk ) − V1 (xk ). Then V1 (·) is said to be a “discrete-time control Lyapunov function” (DTCLF) for system (1). Definition 2. (Passivity [23]) The system (1)-(2) is said to be passive if there exists a non-negative function V , called storage function, such that for all uk V (xk+1 ) − V (xk ) ≤ ykT uk

(5)

where (·)T denotes transpose. This storage function can be selected as a DTCLF if it is a definite positive function. Definition 3. [17] A system (1)-(2) is locally zero-state observable (respectively locally zero-state detectable) if there exists a neighborhood Z of xk = 0 such that x0 = xk ∈ Z. yk |uk =0 = h(Φ(k, xk , 0)) = 0

∀k ⇒ xk = 0

f T (xk ) P f (xk ) − xTk P xk ≤ 0,

k=0

Proof: Let

1 T x P xk (8) 2 k be a candidate DTCLF. System (1)-(2) must be rendered passive, such that the inequality V (xk+1 ) − V (xk ) ≤ ykT uk is fulfilled with yk = h(xk ) + J(xk ) uk . Thus, from (5) we have V (xk ) =

f T (xk )P f (xk ) − xTk P xk 2 2f T (xk )P g(xk )uk + uTk g T (xk )P g(xk )uk + 2 ≤ hT (xk )uk + uTk J T (xk )uk .

To achieve passivity, we rewrite inequality (9) as follows: 1) From the first term of (9), we have f T (xk ) P f (xk ) − xTk P xk ≤ 0,

(resp. lim Φ(k, xk , 0) = 0) k→∞

To this end, we proceed to develop the stabilizing inverse optimal control law for system (1), which can be globally asymptotically stabilized by the output feedback uk = −yk . It is worth to mention that, the output with respect to which the system is rendered passive will not be the variable which we wish to control. The passive output will only be a preliminary step for control synthesis; additionally, we have to define the signals which ensure the output variables, which we want to control, behaves as desired. For the sake of completeness, the following result, established in [24], is included. Theorem 1. Assume an affine discrete-time nonlinear system (1)-(2), which is zero-state detectable, with V (xk ) = 1 T 2 xk P xk as a DTCLF, and satisfies the passivity condition (5). Then, system (1)-(2) is globally asymptotically stabilized by the output feedback uk = −yk and thus h(xk )

h(xk ) = g T (xk ) P f (xk ),

(11)

3) uT g T (xk ) P g(xk ) uk = 2 uTk J T (xk ) uk , thus 1 T g (xk ) P g(xk ). (12) 2 If system (1)-(2) fulfill the zero-state detectability property, from 1), 2), and 3), we deduce that, if there exist a P such that f T (xk ) P f (xk ) − xTk P xk ≤ 0, then the system (1)(2) is passive. To guarantee asymptotic stability, we select uk = −yk and then ∆V (xk , uk ) ≤ −ykT yk ≤ 0, which satisfies the Lyapunov forward difference of V . In order to establish the inverse optimality, we minimize (4) w.r.t. uk , with J(xk ) =

III. I NVERSE O PTIMAL C ONTROL A NALYSIS

uk = − (Im + J(xk ))

(10)

2) 2f T (xk ) P g(xk ) uk = 2 hT (xk ) uk , thus

where φ(k, xk , 0) = f k (xk ) is the trajectory of the unforced dynamics xk+1 = f (xk ) from x0 = x; Z is in general a neighborhood of the origin in Rn . If Z = Rn , the system is zero-state observable (respectively zero-state detectable).

−1

(9)

L(xk , uk )

= =

l(xk ) + uTk uk l(xk ) −

ykT

(13)

uk

(14) where we define l(xk ) = − f T (xk ) P f (xk ) − xTk P xk ≥ 0; thus, we have1 0 = =

(6)

=

with

=

1 h(xk ) = g T (xk ) P f (xk ) and J(xk ) = g T (xk ) P g(xk ) 2

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1 The

min {L(xk , uk ) + V (xk+1 ) − V (xk )} uk min l(xk ) − ykT uk + V (xk+1 ) − V (xk ) uk

−hT − uTk (J + J T ) + (f T − ykT g T ) P g T

−h −

uTk (J

T

T

+J )+f P g−

ykT

T

g Pg

arguments in the functions are omitted to easy the writing.

(15)

Considering hT = f T P g and J = J T , we obtain 0 = T T uk (J + J J) = (J + J 2 )uk

=

and

−uTk J − hT J − uTk J T J −hT J (16)

=

f T P f + xTδ,k+1 P xδ,k+1 − f T P xδ,k+1 +

− (Im + J(xk ))

−1

h(xk ).

Now, solving (4) for L(xk , uk ) and summing over [0,N], where N ∈ N , yields N X

L(xk , uk ) = −V (xN ) + V (x0 ) +

N X

xTδ,k+1 P f − (xk − xδ,k )T P (xk − xδ,k ) ≤ 0,

H(xk , uk ).

Corollary 1. If (1)-(2) is a single input single output, the control law is given by

(xk+1 − xδ,k+1 )T K T P K (xk+1 − xδ,k+1 ) 2 (xk − xδ,k )T K T P K (xk − xδ,k ) − 2 ≤ hT (xk , xδ,k )uk + uTk J T (xk )uk .

(17)

In this section, we modify the DTCLF (8) such that the new energy function (storage function) has a global minimum on the desired trajectory xδ,k . To achieve tracking, first we redefine the DTCLF (8) as

(f + g uk − xδ,k+1 )T P (f + g uk − xδ,k+1 ) 2 (xk − xδ,k )T P (xk − xδ,k ) − 2 ≤ hT uk + uTk J T uk

(19)

xTδ,k+1 P f − (xk − xδ,k )T P (xk − xδ,k ) + T

(2f P g −

xTδ,k+1 P f − (xk − xδ,k )T P (xk − xδ,k ) ≤ 0, (27) 2) (2f T P g − 2xTδ,k+1 P g)uk = 2 hT uk , thus

(20)

with h(xk , xδ,k+1 ) = g T (xk ) P (f (xk ) − xδ,k+1 )

(21)

P g)uk + uTk g T P g uk ≤ 2 hT uk + 2 uTk J T uk .


h(xk , xδ,k+1 ) = g T (xk ) P (f (xk ) − xδ,k+1 ),

Then, system (1) with output (19), is globally asymptotically stabilized by the output feedback uk = −yk and thus h(xk , xδ,k+1 )

(26)

2xTδ,k+1

From (26), passivity is achieved if: 1) from the first term of (26), we can find P > 0 such that

which is zero-state detectable with a candidate DTCLF defined by (18), and satisfies the modified passivity condition

uk = − (Im + J(xk ))

(25)


1 (xk − xδ,k )T K T P K (xk − xδ,k ) (18) 2 where xδ,k is the desired trajectory and K is an additional gain matrix to modify the convergence rate of the tracking error. Theorem 2. Assume an affine discrete-time nonlinear system (1), and define an output as

(24)

thus, (25) becomes

V (xk , xδ,k ) =

−1

(23)

Defining P = K T P K and substituting (1) in (24), we have

IV. T RAJECTORY T RACKING

V (xk+1 , xδ,k+1 ) − V (xk , xδ,k ) ≤ ykT uk .

L(xk , xδ,k , uk ).

Proof: Let (18) be a candidate DTCLF. System (1) with output (19), must be rendered passive, such that the inequality (20) is fulfilled. Thus, from (20), and considering one step ahead for xδ,k , we have

where h(xk ) and J(xk ) are defined as in Theorem 1.

yk = h(xk , xδ,k+1 ) + J(xk ) uk

∞ X

k=0

Letting N → ∞ and noting that V (xN ) → 0 for all x0 , and H(xk , uk ) = 0 for inverse optimal control uk , then J (x0 , uk ) = V (x0 ), which is called the optimal value function. Finally, if V (xk ) is a radially unbounded function, i.e., V (xk ) → ∞ as kxk k → ∞, then the solution xk = 0, k ∈ N , of the closed-loop system (1) is globally asymptotically stable.

uk = −(1 + J(xk ))−1 h(xk )

(22)

where P = K T P K is a definite positive matrix. Moreover, with (18) as a DTCLF, this control law is inverse optimal in the sense that minimizes the meaningful functional given as J =

k=0

k=0

1 T g (xk ) P g(xk ) 2

if

−J h

and solving for uk , the proposed inverse optimal control law is given as uk

J(xk ) =

(28)

3) uT g T P g uk = 2 uTk J T uk , thus 1 T (29) g (xk ) P g(xk ). 2 If system (1) with output (19) fulfill the zero-state detectability property, from 1), 2), and 3) we deduce that, if

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J(xk ) =

there exist a P , such that satisfies (27), then the system (1) with output (2) is passive. To guarantee asymptotic trajectory tracking, we choose uk = −yk and then V (xk+1 , xδ,k+1 ) − V (xk , xδ,k ) ≤ −ykT yk ≤ 0, which satisfies the Lyapunov forward difference of V . The meaningful cost functional is synthesized analogous as in Theorem 1 and hence this proof is omitted.

where xk = [xT12,k xT34,k ]T , being x12,k = [x1,k x2,k ]T the position variables, and x34,k = [x3,k x4,k ]T the velocity variables, for link 1 and link 2 respectively; x1,k + x3,k T f1 (xk ) = , x2,k + x4,k T

Comment 1. Theorem 2 guarantees that system (1) indeed tracks the desired trajectory xδ,k .

f2 (xk ) =

Comment 2. Theorem 2 also constitutes our main contribution.

x3,k + c(−D22 (V1 + F1 ) + D12 (V2 + F2 )) x4,k + c(D12 (V1 + F1 ) − D11 (V2 + F2 )) D22 −D12 g(xk ) = −D12 D11

In this section, we apply Section IV results, to synthesize position tracking control for a two DOF planar rigid robot. The approach presented in this paper comes from energy balancing and passivity viewpoint, where trajectory tracking of the planar robot is reduced to find P and the respective synthesized output yk to achieve passivity, such that the shape of the total energy (kinetic plus potential energy) V (xk , xδ,k ), has a minimum value at the desired equilibrium, and thus, the plant output variables behave are as desire. A. Robot Model After discretizing by means of the Euler approximation the robot dynamics, the discrete-time planar robot model can be written as:

B. Control Synthesis For trajectory tracking, we propose the desired storage function as 1 V (xk , xδ,k ) = (xk − xδ,k )T K T P K (xk − xδ,k ) 2 where xδ,k are the reference trajectories; K is an additional gain to modify the convergence rate and P is synthesized to achieve passivity, according to Section IV, which can be written, respectively, with a block structure as: K1 0 K= 0 K2 and P =

= =

x3,k+1

=

x4,k+1

=

x1,k + x3,k T x2,k + x4,k T −D22 (V1 + F1 ) + D12 (V2 + F2 ) x3,k + 2 D11 D22 − D12 D22 u1,k − D12 u2,k T (30) + 2 D11 D22 − D12 D12 (V1 + F1 ) − D11 (V2 + F2 ) x4,k + 2 D11 D22 − D12 −D12 u1,k + D11 u2,k T + 2 D11 D22 − D12

where T is the sampling time, u1,k and u2,k are the applied torques; x1 = θ1 , x2 = θ2 are the positions; x3 = θ˙1 , x4 = θ˙2 are the velocities; i = 1, 2; s2 = sin(x2 ), c2 = cos(x2 ) and, with entries in (30) as 2 2 D11 (Θ) = m1 lc1 + m2 (l12 + lc2 + 2l1 lc2 c2 ) + Izz1 + Izz2 2 D12 (Θ) = m2 lc2 + m2 l1 lc2 c2 + Izz2 2 D22 (Θ) = m2 lc2 + Izz2 ˙ = −m2 l1 lc2 s2 (θ˙1 + θ˙2 )θ˙2 − m2 l1 lc2 θ˙1 θ˙2 s2 V1 (Θ, Θ) ˙ = m2 l1 lc2 s2 (θ˙1 )2 V2 (Θ, Θ) ˙ = µ1 θ˙1 F1 (Θ, Θ) ˙ = µ2 θ˙2 . F2 (Θ, Θ) 1) Robot as a Affine System: In order to easy the controller synthesis, we rewrite (30) in a block structure form as x12,k+1 = f1 (xk ) x34,k+1 = f2 (xk ) + g(xk ) u(xk ),

xk=0 = x0

,

2 with c = T /(D11 D22 − D12 ).

V. P LANAR ROBOT E XAMPLE

x1,k+1 x2,k+1

P11 P21

P12 P22

where K2 is chosen as the 2 × 2 identity matrix. Thus, developing passivity condition (20) for system (31), and according to (28)-(29), the output is established as y(xk , xδ,k+1 ) = h(xk , xδ,k+1 ) + J(xk )uk where h(xk , xδ,k+1 )

=

and

g T (xk ) (P22 f2 (xk )− K1T P12 x12δ,k+1 + K1T P12 f1 (xk )

1 T g (xk ) P22 g(xk ). 2 Global asymptotic convergence to state reference trajectory is guaranteed with (21), (28)-(29), if we can find a positive definite matrix P satisfying (22). J(xk ) =

C. Simulation Results The parameters of the plant model used for simulation (MATLAB2 ) are given in the Table I. The reference signals are x1δ,k = 2.0 sin(1.0 k T ) rad

(31)

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x2δ,k = 1.5 sin(1.2 k T ) rad. References x3δ,k and x4δ,k , are defined accordingly. 2 It

is a trademark of the MathWorks Inc.

VALUE 0.3 m 0.2 m 0.25 m 0.1 m 1 Kg 0.3 Kg 0.05 Kg − m2 0.004 Kg − m2 0.005 Kg/s 0.0047 Kg/s

DESCRIPTION Length of the link 1 Mean length of the link 1 Length of the link 2 Mean length of the link 2 Mass of the link 1 Mass of the link 2 Moment of inertia 1 Moment of inertia 2 Friction coefficient 1 Friction coefficient 2

Amplitude(N−m)

PARAMETER l1 lc1 l2 lc2 m1 m2 Izz1 Izz2 µ1 µ2

Amplitude(N−m)

TABLE I M ODEL PARAMETERS

Amplitude(rad)

Tracking performance (Link 1) 2

0 −20

0

1

2

3 Time(s) Control signal 2

4

5

6

0

1

2

3 Time(s)

4

5

6

10 0 −10

0 Fig. 2.

Performance of the control signals u1 and u2 .

−2 0

1

2

3 Time(s)

4

5

6

to apply the proposed control law in real-time. Work is progressing to complete the design for non-affine systems.

Tracking performance (Link 2) Amplitude(rad)

Control signal 1

20

2

R EFERENCES 0 −2

0

1

2

3 Time(s)

4

5

6

Fig. 1. Tracking performance of Link 1 and Link 2, respectively. x12δ (solid line) are the reference signal and x12 (dashed line) are the Link positions

These signals are selected to illustrate the ability of the proposed algorithm to track nonlinear trajectories. Constraint (22) is satisfied with 2 1 4 3 ; P12 = 100 ∗ P11 = 100 ∗ 3 2 3 4 170 0 T T P21 = P12 ; P22 = P11 ; K1 = . 0 110 The tracking performance for both, link 1 and link 2 position, are shown in the Fig. 1, with initial conditions x1,k = 0.4 rad; x2,k = −0.5 rad; x3,k = 0 rad/s; x4,k = 0; and T = 0.001. The control signals u1 and u2 responses are displayed in Fig. 2. VI. CONCLUSIONS This paper has presented a novel discrete-time inverse optimal control, which achieve trajectory tracking for nonlinear system and is inverse optimal in the sense that, a posteriori, minimizes a meaningful cost functional. The controller synthesis is based on the selection of a DTCLF and an output to render passive the system. An example is considered to illustrate the results: a planar robot. In the example, the control goal is achieved, i.e., the proposed controller ensures trajectory tracking of a planar robot. Research will continue

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