2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010
WeC07.3
Sampled-data redesign of stabilizing feedback Salvatore Monaco, Doroth´ee Normand-Cyrot and Fernando Tiefensee
Abstract— A sampled-data design procedure ensuring stabilization of passive systems is described in the general framework of Passivity Based Control-PBC. Making reference to a specific output mapping deduced from the initial one and with respect to which passivity under sampling can be preserved, a stabilizing sampled-data controller is designed. An algorithmic procedure for computing the solution is described. The cart pendulum example illustrates the sampled-data controller’s performances.
I. I NTRODUCTION Passivity is a basic argument involved in the design of stabilizing controllers (see [5], [1] and [2]). The first generation of Passivity Based Control-PBC introduced in [11] and inherited from Lyapunov based stabilizing techniques, has been developed in a wide literature. Its impact versus real processes, in particular mechanical or electromechanical ones adopting a Lagrangian formalism, is typically based on its energy balance interpretation. Nowadays, the second generation of PBC emerges in terms of energy shaping or IDA-PBC, Interconnected and Damping Assignment Passivity Based Control (see [12], [14] and the references therein). This refers in a nonlinear context to the continuous-time domain. The situation is somehow different in discrete-time or sampled-data contexts. More in details, the usual inputoutput passivity notion fails in discrete time in the absence of direct input-output link and as a result, usual passivity is lost under sampling. Moreover, the inherent nonlinearity in the control variables of discrete-time dynamics or dynamics issued from sampling, makes the control design a difficult task. The present paper takes part of a more general attempt to explore these difficulties in discrete-time and then sampleddata domains. In this last case, one can exploit the description of sampled-data trajectories or dynamics as asymptotic series rearranged according to the successive powers of δ , the sampling period. The description of these series, solutions or flows associated with nonlinear controlled differential equations, is discussed in [7] in a formal domain. In practice, for computational purposes, these series can be approximated at any fixed degree around what is referred to as the emulated solution or Euler type approximation at the first order. This confers applicability to these methodologies and justifies the used terminology of redesign around the emulated continuous-time solution. S. Monaco is with Dipartimento di Informatica e Sistemistica ’Antonio Ruberti’, Universit`a La Sapienza, via Ariosto 25, 00185, Roma, Italy.
[email protected]
Given a SISO input-affine continuous-time system, assumed passive, and a stabilizing controller uc , we here show the existence of a piecewise constant controller ensuring stabilization at the sampling instants. Based on preliminary result in [8], describing how to modify the output mapping in such a way to preserve passivity under sampling, we show the existence of a sampled-data stabilizing controller. Exploiting the asymptotic series representation of this modified output, we describe an algorithmic procedure for computing the control solution. Finally, the procedure is tested on the cart pendulum example [3] through simulations. Sampled-data stabilizing strategies were developed in [10] and [13] to face the same problem pursuing the idea to match at the sampling instants the Lyapunov behavior of the continuous-time design and even more in [10], to improve its negativity. The here proposed approach is different and its computational simplicity, clearly put in light by the algorithm and the systematic computation of the first terms in the series solution, is a real practical benefit. This is illustrated by the simulated example which demonstrates also the accuracy of computing higher order terms. The paper is organized as follows. Section II sets the problem after recalling the basic tools of the continuoustime design. Section III introduces the output preserving passivity under sampling, describes the digital controller achieving asymptotic stabilization and reports the algorithm for its computation. Section IV deals with the cart pendulum example. II. P ROBLEM SETTLEMENT Let a single input-affine system Σc over Rn x(t) ˙ = y
= h(x)
(1) (2)
where u ∈ U the set of admissible inputs on R, f and g are smooth (i.e. C∞ ) vector fields and h is a smooth mapping on Rn ; xe denotes an equilibrium point f (xe ) = 0, supposed without loss of generality equal to zero. The following usual definitions are recalled. P - Passivity - Σc is passive; i.e. there exists a positive C1 function V on Rn (the storage function) s.t. V (0) = 0 and for all u ∈ U, t ≥ 0 V˙ (x(t)) ≤ yT (t)u(t) or equivalently its integral version for all x0 ∈ Rn
D.Normand-Cyrot and F. Tiefensee are with Laboratoire des Signaux et Syst`emes, CNRS-Supelec, Plateau de Moulon, 91190 Gif-sur-Yvette, France
[email protected],
[email protected]
978-1-4244-7427-1/10/$26.00 ©2010 AACC
f (x) + u(t)g(x)
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V (x(t)) −V (x0 ) ≤
Z t 0
yT (τ )u(τ )d τ .
(3)
ZSD - zero-state detectability property - Σc is locally zero state detectable if no solution of the uncontrolled dynamics x˙ = f (x) can stay in the set Z = {x ∈ Rn s.t h(x) = 0} other than solutions x(t) converging asymptotically to the zero equilibrium.
Proof: It is an immediate consequence of the integrated form of the continuous-time passivity inequality (3) rewritten between kδ and (k + 1)δ and ∀xk ∈ Rn as Z (k+1)δ V (xk+1 ) −V (xk ) ≤ yT (τ )dτ uk . kδ
The following basic stabilizing PB controller is recalled. Negative output feedback gain - Under P, with V > 0 and zero-state detectability, then the storage function V qualifies as a Lyapunov function and the controller uc (t) = −Ky(t) with K > 0 asymptotically stabilizes Σc . Under uc (t), the closed loop dynamics satisfies V˙˜ (t) ≤ −K 2 y(t)
(4)
so making V˙ more negative than under free evolution and thus improving damping performances. The object of this work is to propose a piecewise constant alternate to such a stabilizing strategy.
As expected, the so defined output mapping (5), the average over time intervals of length δ , depends both on the control and the sampling period. On the basis of (6), the goal of the digital controller should be to render its right hand side negative. Mimicing the continuous-time design, one may look for V (xk+1 ) −V (xk ) = −K δ (yδ (xk , uδk ))2 under the action of a suitable uδk of the form
δi uki . i≥1 (i + 1)!
uδk = uk0 + ∑
III. S AMPLED - DATA STABILIZING REDESIGN A. Emulated control A simple calculus shows that, under usual implementation through zero-order holding device, emulated control, namely keeping uc (t) constant over time intervals of amplitude δ > 0, uk := −Kyk with yk = y(t = kδ ) for t ∈ [kδ , (k + 1)δ [, V˙˜ (t) in (4) transforms through integration with x(t = kδ ) = xk into V˜ (xk+1 ) − V˜ (xk ) =
Z (k+1)δ kδ
(L f − Kyk Lg )V˜ (x(t))dt = δ V˙˜k + O(δ 2 )
indicating by L f = ∑ni=1 fi (x) ∂∂xi the operator Lie derivative associated with f . Eventhough V˙˜k = −Ky2k ≤ 0, such a first order difference equation is no more guaranteed to be negative as soon as terms of order ≥ 2 in δ are taken into account. As well known, the emulated control does not maintain the performances of the continuous-time design at the sampling instants except up to approximations of the sampled behaviors at order 1 in δ - Euler approximation that is neglecting terms in O(δ 2 ). This justifies the need for more accurated sampled-data controllers. B. Passivity under sampling Setting in (1), the control variable piecewise constant over time intervals of length δ , u(t) = Cst = uk for t ∈ [kδ , (k + 1)δ [, let us recall a result in [8]. Proposition 3.1: Given the continuous-time dynamics Σc , passive with storage function V satisfying P, then there exists a sufficiently small T ∗ > 0 such that for all δ ∈ ]0, T ∗ ], (xk , uk ) ∈ Rn × R, the sampled equivalent dynamics with output mapping yδ (xk , uk ) :=
1 δ
Z (k+1)δ kδ
y(τ )d τ
(8)
The following result holds true. Theorem 3.1: - (sampled-data negative output gain feedback) - Given Σc satisfying P with V > 0 and ZSD then, for all δ ∈]0, T ∗ ], xk ∈ Rn : • there exists a piecewise constant control of the form (8) which asymptotically stabilizes the equilibrium ; • the solution satisfies at each sampling instant the equality of series below with negative gain K > 0 uδk = −Kyδ (xk , uδk )
(9)
• the proof is constructive, an algorithm for computing the solution up to any order of approximation in δ is given. Proof: The proof works out noting first that (9) ensures at the sampling instants (7). Denoting by Zd := {x ∈ Rn
s.t. ∀δ , yδ (x, uδk ) = 0}
then Zd is included in Z because for δ = 0, yδ = h(x) and asymptotic stabilization under uδk follows from the zero state detectability of y = h(x) with V > 0. In fact, no solution of the sampled closed loop dynamics can stay identically in the set Zd other than solution converging asymptotically to xe = 0. Let us now show the existence of such an uδd . For, rewriting for all δ , xk ∈ Rn , (9) as a formal series equality S(δ , xk , u) = 0 setting S(δ , xk , u) := u + Kyδ (xk , u), one easily verifies that uk0 = −Kh(xk ) satisfies the equality for δ = 0; i.e. S(0, xk , uk0 ) = 0. Then, from the implicit function theorem, one concludes that a solution exists because
∂ S(δ , xk , u) |(δ =0,uk0 ) = 1 6= 0. ∂u The solution is described by the inverse series
(5)
uδk := S−1 (δ , xk )
(6)
which satisfies S(δ , xk , S−1 (δ , xk )) = 0 with S−1 (0, xk ) = uk0 . The solution is thus described by its asymptotic expansion in δ of the form (8).
is passive; i.e. V (xk+1 ) −V (xk ) ≤ (yδ (xk , uk ))T δ uk .
(7)
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An algorithm for computing the solution up to any order of approximation in δ is given. For the first terms one computes the approximate solution below.
In this case studied in [4], the mapping (5) is easily computed according to yδ
Corollary 3.1: The approximate controller uδka
= uk0 +
=
δ δ2 uk1 + uk2 2! 3!
uk2
R
uδk =
= −Ky|t=kδ = −K y(u ˙ k0 )|t=kδ = −K(L f + uko Lg )h|t=kδ . 3K ¨ k0 )|t=kδ = − uk1 Lg h|t=kδ − K y(u 2
Z
R δ R τ sA 0 0 e Bdsdτ . Using the
−KCδ xk . 1 + KDδ
C. The algorithm
satisfies (7) up to the order 3 (error in O(δ 4 )). Corollary 3.1 puts in light that : • the first term uk0 is the emulated continuous-time control and corresponds to the damping injection term proposed in [6]; • the second term uk1 is the time derivative of the continuoustime controller evaluated at the sampling instant; • the strategy does not give back digital implementation of the continuous-time controller through higher order device R (k+1)δ or computing its average value over δ as δ1 kδ uc (τ )d τ , but results in a sampled-data controller properly designed to make negative the first order Lyapunov difference V (xk+1 ) − V (xk ). Remark 3.1: Passivity with respect to yδ , or to satisfy the inequality (7), as well to achieve the control design equality (9), hold asymptotically for all δ ∈]0, T ∗ ], (xk , uk ) ∈ Rn × R. Except for some particular cases for which series expansions simplify in polynomial of finite order, so getting exact but not unique solutions, at present nothing else can be said to relate the sampling period length with the necessary order of approximation. In practice, besides the designer’s ability, one can say that approximations of order one and two with respect to the emulated control already behave quite efficiently. Remark 3.2: By construction yδ being the average of the output one step ahead, confers a predictive effect to the resulting negative output feedback (9). This reflects in the intersample behaviour which improves by increasing the approximation order of the sampled-data controller. Even though intuitively clear, and verified by the simulated example, this aspect deserves a deeper investigation. Remark 3.3: In the linear case, the solution admits a closed form because the right hand side of (5) remains linear in u. Let the linear time invariant system x(t) ˙ = Ax(t) + Bu(t) y
1 (k+1)δ Cx(τ )dτ δ kδ 1 Cδ xk + Dδ uδk δ
with Cδ = C 0δ eτ A d τ and Dδ = C proposed method, one has
with uk0 uk1
=
= Cx(t)
Indicating by ”s” the time-derivative operator and by i si y|t=kδ := d dty(t) i |t=kδ the i-times derivative of the continuoustime output mapping y = h(x), evaluated at time t = kδ , yδ (xk , uδd ) in (5) can be rewritten as eδ s − 1 )y|t=kδ δs δ δi ˙ t=kδ + ... + ∑ y(i) |t=kδ = y|t=kδ + y| 2 i≥2 (i + 1)!
yδ (xk , uδd ) = (
with x(t ˙ = kδ ) = f (xk ) + uδd g(xk ). Substituting uδd with its expansion (8) in both sides of (9), one gets the equality to solve δi δi δi uki = −K y+ ∑ y(i) ( ∑ uki ) |t=kδ . i≥1 (i + 1)! i≥1 (i + 1)! i≥0 (i + 1)!
uk0 + ∑
Then, the computation is performed by equating terms of the same power in δ . Step 0 - It is readily verified that setting δ = 0, one gets uk0 = −Ky|t=kδ O(δ 2 )).
It is readily verified that uk1 should Step 1 - (error in satisfy with an error in O(δ 2 )
δ δ ˙ k0 )|t=kδ )) uk0 + uk1 = −K(y + y(u 2 2 from which one easily deduces because of step 0 that uk1
= −K y(u ˙ k0 )|t=kδ = −K(L f + uko Lg )h|t=kδ .
Step 2 - (error in O(δ 3 )). It is readily verified that uk2 should satisfy with an error in O(δ 3 ) uk0 +
δ δ2 δ δ δ2 uk1 + uk2 = −K y + y(u ˙ k0 + uk1 ) + y(u ¨ k0 ) . 2 3! 2 2 3!
Performing a Taylor’s type expansion of y(u ˙ k0 + δ2 uk1 ) around uko as
δ δ dy˙ y(u ˙ k0 + uk1 ) = y(u ˙ k0 ) + uk1 |u=uk0 2 2 du one deduces because of step 1 the equality to be satisfied δ2 dy˙ δ2 δ2 uk2 = −K uk1 |u=uk0 + y(u ¨ k0 ) 3! 4 du 3! from which one computes
with matrices of appropriate dimensions, assumed passive with quadratic storage V (x) = 12 xT Px, with P > 0, P = PT .
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uk2 = −
3K uk1 Lg h|t=kδ − K y(u ¨ k0 )|t=kδ . 2
Step p - (error in O(δ p+1 )). ukp should satisfy with an error in O(δ p+1 ) the equality p
p p−i δi δi δj uki = −K y + ∑ y(i) ( ∑ uk j ) . i=1 (i + 1)! i=1 (i + 1)! j=0 ( j + 1)!
uk0 + ∑
Performing first order Taylor’s type expansion of each p−1−i δ j δj y(i) (∑ p−i j=0 ( j+1)! uk j ) around ∑ j=0 ( j+1)! uk j ; i.e. p−i
δj uk j ) j=0 ( j + 1)! p−1−i
∑ j=0
δ p−i ukp−i dy(i) δj uk j ) + | p−1−i δ j uk j ( j + 1)! (p − i + 1)! du u=∑ j=0 ( j+1)!
and applying the equality satisfied at step i − 1 with an error in O(δ p ); i.e. p−1
uk0 +
∑ i=1
The storage function V (q, p) is defined by the total system’s energy which is given by the sum of cart and pendulum kinetic energy and pendulum potential energy : 1 T −1 p M (q1 )p + mlg (cos(q1 ) + 1) . 2 Using the Euler-Lagrange equations and neglecting friction, the system’s dynamics takes the form (1); i.e. V (p, q) =
y(i) ( ∑ = y(i) (
the system’s generalized momenta is q˙1 p1 . = M(q) q˙2 p2
δi u (i + 1)! ki
p−1 = −K y + ∑
p−1−i δi δj y(i) ( ∑ uk j ) . (i + 1)! j=0 ( j + 1)!
i=1
(10)
q˙ = M −1 (q1 )p ∂ V (q, p) p˙ = + Gu ∂q with GT = 0 1 . The system has two equilibrium points, a global minimum at (π , 0, 0, 0) and an unstable equilibrium point at (0, 0, 0, 0). Since the time-derivative of the storage function V (q, p) is V˙ (q, p) = uGT M −1 (q1 )p,
one computes for p ≥ 1 p−1
ukp = −K
the system is lossless with respect to the output y = GT M −1 (q1 )p = q˙2 . It is zero state detectable so that the continuous-time controller
dy(i) |u=uk0 − Ky(p) (uk0 ) du −K terms of order p in the r.h.s. of (10). (p + 1)!ukp−i
∑ (i + 1)!(p − i + 1)!
i=1
uc = −KGT M −1 (q1 )p = K q˙2
IV. E XEMPLE - T HE CART PENDULUM SYSTEM
achieves asymptotic stabilization at (π , 0, 0, 0). A. Sampled-data design
m
Specifying on this example the controller given in Corollary 3.1, one computes uδka1 = ud0 + δ2 ud1 and uδka2 = uk0 + δ2 uk1 + δ2 3! uk2 respectively with
q1
uk0 u
Fig. 1.
M
= −KGT M −1 p
= K 2 GT M −1 GGT M −1 p 3K 3 T −1 2 T −1 ... G M G G M p − K q1 . uk2 = − 2 B. Simulation uk1
q2
Cart-Pendulum System
In this section, we show the effectiveness of the proposed sampled-data feedback for stabilizing the inverted pendulum on a cart [3]. Simulations are performed comparing the proposed sampled-data controller with the continuous-time damping controller and the emulated strategy. The system is depicted in Fig.1. q1 and q2 denote respectively the angular deviation from the upright position of the pendulum pivoting around a point fixed on the cart and the position of the cart; M denotes the cart’s mass; m and l denote the mass and length of the pendulum respectively; the gravitational acceleration g is constant. The control u is an horizontal force acting on the cart. Defining the positive and symmetric inertia matrix as ml 2 ml cos(q1 ) M(q1 ) = ml cos(q1 ) M+m
The simulations compare for different values of δ the performances of the continuous-time controller uc , the emulated strategy uk0 , the first and second-order sampled-data stabilizing feedbacks, respectively uδka1 and uδka2 . The system parameters are: m = 0.5kg, M = 0.25kg, l = 0.15m, g = 9.8m/s2 . The initial condition is x0 = (2π /3; 0; 0; 0) and the controller’s gain is equal to K = 5 for both continuous and sampled-data strategies. Since the system is lossless, it is conservative for u = 0 (V˙ = 0), that is the energy V remains constant and pendulum swings depending on the initial conditions (Fig.2(a) and Fig.2(d)). The continuous-time controller uc injects damping and asymptotically stabilizes the system at its equilibrium. Fig.2 depicts the performances of the sampled-data controllers for δ = 90ms. Even if the damping performances of the emulated controller are still acceptable (Fig.2(a) and Fig.2(d)), such a control degrades the output behaviour
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0.4
1
Vd (0) Vd (uc ) Vd (uk0 ) Vd (uδka1 ) Vd (uδka2 )
0.35
0.3
0.25
y(uc )
0.8
0.6
y(uk0 )
0.4
y(uδka1 )
0.2
y(uδka2 )
0.2
0
−0.2 0.15 −0.4 0.1 −0.6 0.05 −0.8
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
−1 0
0.5
1
(a) Storage function V
1.5
2
2.5
3
(b) Output y
5
0.08
uc uk0 uδka1 uδka2
4
3
2
x f ree x(uc ) xk (uk0 ) xk (uδka1 ) xk (uδka2 )
0.06
0.04
0.02 x3
1
0
0
−1
−0.02
−2
−0.04 −3
−0.06 2
2.5
3
−4 0
0.5
1
1.5
2
3.5
4
4.5
x1
2.5
(d) q1 × p1
(c) Control input u Fig. 2.
Performances of Cart Pendulum, δ = 90ms.
0.4
0.8
Vd (0) Vd (uc ) Vd (uk0 ) Vd (uδka1 ) Vd (uδka2 )
0.35
0.3
0.25
y(uc ) y(uδka1 ) y(uδka2 )
0.6
0.4
0.2
0.2 0 0.15 −0.2 0.1
−0.4
0.05
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
−0.6 0
0.5
1
(a) Storage function V
1.5
2
2.5
3
(b) Output y
3
0.08
x f ree x(uc ) xk (uδka1 ) xk (uδka2 )
uc 0.06
2
uδka1 0.04
1
uδka2 0.02 x3
0
0 −1
−0.02 −2
−0.04 −3
−0.06 2
2.5
−4 0
0.5
1
1.5
2
2.5
3.5 x1
(d) q1 × p1
(c) Control input u Fig. 3.
3
Performances of Cart Pendulum, δ = 100ms.
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4
4.5
uuck0)
0.4
y(uc )
Vd (0)
0.35
0.6
y(uδka1 )
Vd (uc ) 0.3
Vd (uδka1 )
y(uδka2 )
0.4
Vd (uδka2 )
0.25
0.2
0.2 0 0.15 −0.2 0.1
−0.4
0.05
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.5
1
1.5
2
2.5
3
(b) Output y
(a) Storage function V 3
0.08
2
1
x f ree
uc
0.06
uδka2
0.04
x(uc ) xk (uδka2 )
0
x3
0.02
−1
0
−2
−0.02 −3
−0.04 −4
−0.06 2
−5 0
0.5
1
1.5
2
2.5
3.5
4
4.5
(d) q1 × p1
(c) Control input u Fig. 4.
3 x1
2.5
Performances of Cart Pendulum, δ = 110ms.
which is time-persistent (Fig.2(b) and Fig.2(c)) - as well known, the validity of emulated strategy uk0 depends on the sampling period length. The first-order sampled-data controller uδka1 efficiently ensures damping performances and output convergency with a smaller control amplitude, uδka2 works well too. For an increasing sampling period, δ = 100ms, the emulated strategy does not work anymore (Fig.reffig3a) while uδka1 and uδka2 work perfectly. The efficiency of the second order term is put in light with a larger δ = 110ms for which uδka1 does not work anymore. Figure 4 depicts the performances still acceptable of the second-order approximate sampleddata controller. V. C ONCLUSION It has been shown how to design a sampled-data passivity based control. An algorithm has been described. The performances of digital controllers with an increasing order of approximation have been tested for various sampling periods. VI. ACKNOWLEDGMENTS Work partially supported by a research mobility grant VINCI- of the UFI/UFI - French/Italian - Italian/French University.
R EFERENCES
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