Nonlinear Scheduled Control for Linear Systems ... - Semantic Scholar

Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007

WeB17.1

Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control Sergio Galeani, Simona Onori, Luca Zaccarian∗ Abstract— In this paper we propose a nonlinear state-feedback control law for the global asymptotic stabilization (GAS) of nonexponentially unstable plants. For these types of plants, nonlinear controllers are necessary in general to achieve GAS and several results are available in the literature. The theory is then also used as a solution to the so-called anti-windup problem for these types of plants. The advantages of the approach here proposed, in addition to a formal stability proof, are that it is not computationally intensive, therefore prone for real-time implementation on plants with fast dynamics and that it induces fast transients and converges to the desired equilibrium in set-point regulation. Various examples illustrate the effectiveness of the proposed control laws both in the stabilization and in the antiwindup design task.

I. I NTRODUCTION The problem of global asymptotic stabilization of linear plants with bounded inputs has been widely studied in the literature. Despite the simplicity of its statement, it is quite a complex problem due to its intrinsic nonlinear nature. It is well known in the control community that the global exponential convergence property of a linear plants cannot be modified when using a bounded input, i.e., global exponential stability with bounded inputs can only be achieved on linear plants that are already exponentially stable before the control action. This fact goes together with the issue that even global asymptotic stability is not achievable for plants with exponentially unstable dynamics, so that there’s an extremely peculiar set of linear plants consisting of plants with some poles on the imaginary axis and no poles in the right half plane which are not asymptotically stable and are globally asymptotically stabilizable using bounded inputs but not globally exponentially stabilizable using bounded inputs [18], [17]. These plants are often called ANCBI (asymptotically null-controllable with bounded inputs) and will be addressed in this paper. The fact that global exponential stability cannot be achieved for ANCBI plants makes the global asymptotic stabilization task quite hard. Indeed, long before the works of [18], [17] already in 1969, [5] had shown that a triple integrator couldn’t be globally asymptotically stabilized using a linear control law (this fact was also established, later, in [20]). This important result revealed that in general a nonlinear controller is necessary to induce asymptotic stability on such plants. While it may seem that the class of systems having poles in the closed left half plane but not in the open left half plane is quite small, studying such systems is actually very important to determine high performance controllers for plants having bounded inputs and very slow modes. Indeed, even though these plants are exponentially stable, by continuity of solutions with respect to the plant parameters, they actually generate responses which are arbitrarily close to the non-converging ones generated by their ANCBI sisters. In other words, the investigation of nonlinear laws guaranteeing global asymptotic stability for ANCBI plants is extremely relevant with the goal in mind of finding high performance control law for any plant having slow modes and bounded inputs. The first constructive bounded stabilizers for ANCBI plants were proposed in the 1990s. For example, the nested saturations approach initiated in [21], as well as the results in [19] and [15]. Later on, several approaches have been proposed to guarantee improved performance by way of more or less innovative alternative laws (see, e.g., [16], [12], [1], as well as the monograph [13]). In more recent years, additional approaches have been proposed (see, e.g., [4], [3], [9], [14], [11]) to address nonlinear stabilization of different classes of systems contained in the larger class of ANCBI systems. Most of the approaches are motivated by the double goal of 1) improving the ∗ Dipartimento di Informatica, Sistemi e Produzione, University of Rome, Tor Vergata, 00133 Rome, Italy. [galeani, s.onori, zack]@disp.uniroma2.it Research supported in part by ENEAEuratom and MIUR under PRIN and FIRB projects.

1-4244-1498-9/07/$25.00 ©2007 IEEE.

closed-loop performance and 2) simplifying the control law which often requires the on-line implementation of complex mathematical tasks, such as the solution of algebraic Riccati equations and of linear optimization problems. In this paper we address the global stabilization problem for general ANCBI systems with the goal in mind to provide computationally efficient nonlinear control laws that induce fast transients on the closed-loop system. The approach that we take is that of defining a scheduling parameter which is adjusted on-line and that guarantees an increasingly aggressive control action as the state converges to the origin. The approach consists in advances from our recent results of [7] and [6]. From the point of view of the scheduling parameter, we adopt a similar strategy to that of [16] with the exception that, based on an explicit expression of the parametric Lyapunov function (which is determined here along the lines of [2]), it is possible to come up with a control law that does not require heavy computations in the on-line implementation. The control law is first designed for single-input plants and then extended to the case of MIMO plants in a similar fashion to what was done in [1], except for the fact that we use here a different representation and we exploit a useful structure of the off-diagonal terms to induce an almost decoupled (fully decoupled in the tail of the response where saturation is not anymore active) response among the different single input blocks isolated by a suitable change of coordinates. Unlike several papers in the field, we show the desirable speed of convergence induced by our approach on several examples. Indeed, it is well known that low gain feedbacks often lead to very sluggish and unacceptable transients, despite the formal proof of asymptotic stability. We show here that by suitably adjusting the parameters, the proposed control laws lead to highly desirable transients. The paper is structured as follows: in Section II we illustrate the proposed technique for SISO plants, which requires to determine a suitable pair of functions Kg (·), Pg (·). In Section III we illustrate how to find these functions giving constructive techniques for some special cases and illustrating some heuristics for other classes of systems. In Section IV we extend the results to MIMO plants and in Section V we explain their application for the solution of the L2 anti-windup problem. Finally, in Section VI we develop and comment several simulation examples. All the proofs are omitted due to space constraints. Notation Let R>0 be the set of positive reals, and the scalar saturation function of level a ∈ R>0 be  asign(v), if |v| > a; σa (v) := v, if |v| ≤ a; where sign(·) is the sign function; the (vector, decentralized) saturation function of level w ∈ Rp>0 is defined by saying that its i-th component is [σw (v)]i = σwi (vi ), i = 1, . . . , p where wi and vi are the i-th components of w and v, respectively. The decentralized deadzone function of level w ∈ Rp>0 , is defined as dzw (v) := v − σw (v). If both w and v are real vectors of the same dimension then w > v must be understood componentwise, i.e. w > v means wi > vi for all i = 1, . . . , p. If P = P ′ and Q = Q′ are symmetric matrices of the same dimension, P > 0 means that P is positive definite, and P > Q means that P − Q > 0. A signal q(·) is in LRp , for p = 1, 2, if its Lp norm is bounded, t i.e., kqkpp := limt→∞ 0 |q(τ )|p dτ < ∞. A function is said to 0 belong to C (V) if it is continuous at every point in the set V, and is said to belong to C 1 (V) if its derivative exists and is continuous at every point in the set V.

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Given a matrix P = P ′ > 0, E(P, ρ) := {x : x′ P x ≤ ρ}. Given a square matrix X, HeX := X + X ′ . II. G LOBAL ASYMPTOTIC STABILIZATION OF NON EXPONENTIALLY UNSTABLE SISO PLANTS Consider the system x˙ = Ax + bσm (u) + d n×n

(1)

n×1

with A ∈ R , b ∈ R , d is an external disturbance and, without loss of generality, the pair (A, b) is in canonical controller form, i.e. – » – » −an−1 · · · − a1 −a0 1 b= 0 , (2) A= In−1 0 (where 0 denotes a zero matrix with dimensions clear from the context; in particular, here 0 ∈ Rn−1×1 ). Denote by pA (s) = det(sI − A) = sn + an−1 sn−1 + · · · + a1 s + a0 the characteristic polynomial of A, and by Λ(A) its roots (the eigenvalues of A). Since we will deal with global asymptotic stabilization with bounded inputs, the following assumption is made about the plant. Assumption 1: Plant (1) is asymptotically null controllable with bounded inputs. Remark 1: The requirement in Assumption 1 is equivalent to the ¯ ≤ 0, condition that no eigenvalue of A has positive real part, i.e. ∃λ ¯ ∀λ ∈ Λ(A). such that re(λ) ≤ λ, ◦ A. Achieving GAS by low gain scheduling For the purpose of stabilizing (1), the existence of a suitable Lyapunov function will be assumed; in order to give compact and precise statements in the sequel, the following definition of a parametric Lyapunov function for (1) will be used. In this definition, ε can be considered as a scalar, fixed parameter; later on, ε will be a parameter of x, used to schedule the gain of the state feedback, u = Kg (ε(x))x; as x → ∞, ε → 0, and Kg → 0. Definition 1: Let P (·) : R → Rn×n , P (·) ∈ C 1 ([0, 1]) and A(·) : R → Rn×n , A(·) ∈ C 0 ([0, 1]). The function V˜ (ε, x) = x′ P (ε)x is a parametric Lyapunov function for x˙ = A(ε)x if, for all ε ∈ (0, 1], it holds that P (ε) = P (ε)′ > 0, ∃Q(ε) = Q(ε)′ ≥ 0 : He(P (ε)A(ε)) ≤ −Q(ε),

dP (ε)

g , x′ Pg (ε)x. Moreover, introduce the function ∂Pg (ε) := dε which is well defined for all ε ∈ [0, 1] because Pg (ε) ∈ C 1 ([0, 1]) by Definition 1. Remark 4: The meaning of the properties required in Assumption 2 can be clarified as follows. Item iv) implies that |Kg (ε)x| ≤ m on the set E¯1 , so that saturation never activates in (1); as a consequence, (1) under u = Kg (ε)x behaves as the linear system x˙ = Acl (ε)x, and so item i) guarantees that the set E¯1 is forward invariant, and included in the basin of attraction of the equilibrium xe = 0. By item ii), by choosing a sufficiently small value of ε > 0, it is possible to guarantee that the set E¯1 contains any a priori given, arbitrarily large compact set around the origin (so that i), ii) and iv) guarantee that Kg (ε) yields semiglobal asymptotic stability). Finally, item iii) (implying Pg (ε) increasing in ε ) will be needed to apply an implicit function theorem to show that the equation V˜ (ε, x) − 1 = 0 uniquely defines a Lipschitz continuous function ε(x); this, in turn, will be used to show that the nonlinear state feedback Kg (ε(x))x yields global asymptotic stability (whereas, as said above, the linear state feedback Kg (ε)x only yields semiglobal asymptotic stability, when a suitably small value of ε is adopted). ◦ Similar to what is done, e.g., in [15], [16], [1], the objective of this paper is to propose a scheduled choice of ε, in order to get both global (instead of semiglobal) asymptotic stability, and possibly increasingly better performance as the state converges toward the origin, by choosing the highest value of ε compatible with the current value of the state x; this also yields the additional advantage that the closer the state gets to the set E¯1 , the more aggressive the applied control actions become. Let ( 1, if x ∈ E¯1 , (4) ε(x) := max{ε : Ve (ε, x) ≤ 1} if x ∈ 6 E¯1 .

ε≥0

The following Lemma 1 is a useful stepping stone in order to reach our goal. Lemma 1: Under Assumption 2, The function ε(·) : Rn → R : x 7→ ε(x) is bounded and globally Lipschitz in Rn , whose gradient is given almost everywhere by the bounded function ∇x ε(x) = −

(3a) (3b)

where the pair (A(ε), Q(ε)) is detectable. ◦ Remark 2: As a particular special case (which will be relevant in the following), notice that if ∃δ > 0 : He(P (ε)A(ε)) ≤ −δP (ε),

WeB17.1

(3c)

then (3b) holds with Q(ε) = δP (ε), and the detectability condition required to the pair (A(ε), Q(ε)) in Definition 1 is trivially satisfied (as can be easily seen by using the PBH test and recalling that P (ε) is nonsingular ∀ε ∈ (0, 1]. ◦ Based on the above definition, the following assumption is introduced, by considering a parameterized state feedback u = Kg (ε)x and introducing Acl (ε) := (A + bKg (ε)). Assumption 2: Pg (·) : R → Rn×n and Kg (·) : R → R1×n (where “g” stands for “global”) are such that Kg (·) ∈ C 0 ([0, 1]), and i) V˜ (ε, x) = x′ Pg (ε)x is a parametric Lyapunov function for x˙ = Acl (ε)x; ii) limε→0 Pg (ε) = 0; iii) ∃γ > 0 : ε∇ε Pg (ε) > γPg (ε), ∀ε ∈ [0, 1]; iv) Pg (ε) − m−2 Kg (ε)′ Kg (ε) > 0, ∀ε ∈ [0, 1]. Remark 3: Given any choice of Kg (·) and Pg (·), it is almost immediate to check if Assumption 2 is satisfied by using commercial software. However, in order to ease the design of such functions, constructive recipes will be given in Section III. In particular, such recipes yield polynomial functions Kg (·) and Pg (·), for which it is immediate to compute ∇ε Pg (ε) and to verify if Assumption 2 is satisfied. ◦ For later use, let E1 := {x : x′ Pg (1)x < 1}, ∂E1 := {x : x′ Pg (1)x = 1}, E1c := {x : x′ Pg (1)x > 1}, E¯1 := {x : x′ Pg (1)x ≤ 1}, E¯1c := {x : x′ Pg (1)x ≥ 1} and Ve (ε, x) =

∇x Ve (ε, x) 2x′ Pg (ε) =− ′ . x ∂Pg (ε)x ∇ε Ve (ε, x)

(5)

Moreover, the state feedback Kg (ε(x))x never exceeds the saturation limits given by m, namely Kg (ε(x))x = σm (Kg (ε(x))x), ∀x ∈ Rn . The following theorem is our first important result on global asymptotic stabilization of (1). Theorem 1: Under Assumption 2, the following state-feedback closed-loop system x˙ = Ax + bσm (u) + d, u = Kg (ε(x))x

(6a) (6b)

is globally asymptotically stable when d = 0. Moreover, if d ∈ L1 t+∞ and d(t) −−−→ 0, then all the trajectories of (6) are are bounded t+∞ and x(t) −−−→ 0. Finally, if in addition d ∈ L2 , then also x ∈ L2 . Remark 5: If plant (1) is exponentially stable, i.e. all the eigen¯ < 0 (see Revalues of A have real part not greater than λ mark 1), the above technique can be applied to achieve global exponential stability, by applying the same construction described ¯ Notice that in such above after replacing A by A¯ = A + λI. a case, letting A¯cl (ε) := (A¯ + bKg (ε)), if (3b) is satisfied for A¯cl (ε) (so that He(Pg (ε)A¯cl (ε)) ≤ 0) then the stronger condition ¯ g (ε) is satisfied for (1) under the feedback He(Pg (ε)Acl (ε)) ≤ 2λP u = Kg (ε)x, which implies global exponential stability with a convergence rate which is never worse than the open loop convergence rate; in this sense, the (bounded) control u = Kg (ε(x))x can be seen as always pushing in the most advantageous direction. ◦ B. Achieving high performance by regional scheduled design The scheduled control law proposed above guarantees global asymptotic stability by means of a nonlinear, state dependent rescaling of the state feedback Kg (1)x (which is stabilizing inside

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 the ellipsoid E1 := E(Pg (1), 1) = {x : x′ Pg (1)x ≤ 1}) when the state x is outside the ellipsoid E1 := E(Pg (1), 1) = {x : x′ Pg (1)x ≤ 1}. However, in order to improve performance when the state is inside E1 , it is useful to consider a scheduled control law also inside E1 ; this can be accomplished by using the LMI-based approach proposed in [6], which is now reviewed. The idea is that inside E1 a α parameter, with α ∈ [0, N ], will be used to schedule between the stabilizing gains. When x ∈ 6 E1 we pick ε = N/α with α > N (⇒ ε ∈ [0, 1]). The basic idea is to design (off-line) a family of nested ellipsoids with associated state feedback gains, and then to determine (on-line) the pair of ellipsoids between which the state lies, and to apply a state feedback obtained by interpolating between the pair of gains associated to the pair of ellipsoids. In order to determine the family of nested ellipsoids, the following procedure can be applied: Procedure (Nested ellipsoids design) Step 1. Fix the number N of ellipsoids, and a shrinking factor β ∈ (0, 1). Initialize QN := Pg (1)−1 , X1N = Kg (1)QN , X2N = 0, UN = I, i = N − 1. Step 2. Denoting by σi+1 the smallest eigenvalue of Qi+1 , solve the following generalized eigenvalue problem in the variables {Q, X1 , X2 , U, Y }, where Q = Q′ : max λ subject to Q < Qi+1 Q > βσi+1 I » (A + λI)Q + BX1 He X1 − Y » 2 – m Y ≥ 0. ′ Y Q

B(X2 − U ) X2 − U

– < 0,

(7a) (7b) (7c) (7d) (7e)

Step 3. Set X1,i := X1 , X2,i := X2 and Qi := Q. Step 4. If i = 0, stop; else, set i := i − 1 and go to Step 2. The matrices X1,i , X2,i and Qi , i = 0, . . . , N computed by the above procedure can be used to define a scheduled control law as follows. Let ⌊α⌋ := max{0, floor(α)} (8) ⌈α⌉ := min{N, ceil(α)}, (so that for α ∈ [0, N ], ⌊α⌋ denotes the largest integer smaller than α, whereas ⌈α⌉ denotes the smallest integer greater than α), and P (α) := Q(α)−1 , where Q(α) = (⌈α⌉ − α)Q⌈α⌉ + (α − ⌊α⌋)Q⌊α⌋ U (α) = (⌈α⌉ − α)U⌈α⌉ + (α − ⌊α⌋)U⌊α⌋ X1 (α) = (⌈α⌉ − α)X1⌈α⌉ + (α − ⌊α⌋)X1⌊α⌋ X2 (α) = (⌈α⌉ − α)X2⌈α⌉ + (α − ⌊α⌋)X2⌊α⌋

(9a) (9b) (9c) (9d)

It was shown in [6] that the equation x′ P (α)x = 1 implicitly defines a unique, bounded and Lipschitz continuous function α(x) on E1 , which can be used to define an implicit, nonlinear statefeedback control law u = Kr (α(x))x + Lr (α(x))σm (u)

(10)

where Kr (α) Lr (α)

= =

(I − X2 (α)U (α)−1 )X1 (α)Q(α)−1 −(I − X2 (α)U (α)−1 )X2 (α)U (α)−1 ,

(11)

(where “r” stands for “regional”) which by construction coincides with the globally asymptotically stabilizing law (6b) on ∂E1 . In the following theorem, which is a simple extension of Theorem 1, the global Kg and the regional Kr are blended, so that for x ∈ 6 E1 , ε in Kg (ε) is substituted by N/α, with α > N . Theorem 2: Under Assumption 2, the plant (6a) with the controller u = K(α(x))x + L(α(x))σm (u)  Kr (α) ` N ´ if x ∈ E1 K(α) = if x ∈ / E1 K (12) g α  Lr (α) if x ∈ E1 L(α) = 0 if x ∈ / E1

WeB17.1 satisfies all the properties stated in Theorem 1. C. Computing the scheduling parameter Both the control laws (6b) and (12) of Sections II-A and II-B, respectively, require the on-line computation of the value of the scheduling function. For example, in Section II-B it is necessary to determine the value of α(x) implicitly (and uniquely) defined by the equation V˜ (α, x) = 1, where V˜ (α, x) := x′ P (α)x. While this can be done e.g. by bisection methods, in [6] it was suggested that the value of the scheduling function can be easily determined by integrating the dynamics of α(x) implicitly defined by the condition V˜ (α, x) = 1. The key idea is that, denoting by ∇x V and ∇α V , respectively, the gradient of the function V with respect to x and the derivative of V with respect to α, it holds that V˜˙ (α, x) = 0 implies that ∇x V˜ x˙ + ∇α V˜ α˙ = 0, i.e. α˙ = −

2x′ P (α)(Ax + bσm (K(α)x) + d) ∇x V˜ x˙ =− ˜ x′ ∇α P (α)x ∇α V

(13)

Notice that, in order to implement α, ˙ both x and d need to be measured; we explicitly remark that such an assumption is not restrictive for the applications considered in Sections IV and V. Actually, a more robust implementation of α˙ (involving a correction term injecting the error 1 − V˜ (x, α) possibly due to numerical problems, uncertainties or wrong initialization), also taking into account that α must not fall below zero is given by 8 if α ≤ 0 > <max{0, g(x, α)}, ′ (14) α= ˙ 2x P (α)x˙ > − g(x, α), if α ≥ 0 : ′ x ∇α P (α)x

where g(x, α) := ℓ(V˜ (x, α) − 1), ℓ > 0, and ∇α P (α) can be explicitly computed as 8

N . :− 2 ∂Pg α α III. S OME CONSTRUCTIVE RECIPES

In this section we will provide two procedures for constructing Kg (·) and Pg (·) to achieve global asymptotic stability. A. Hermite matrix based procedure As proposed in [6], a possible choice for Pg (ε) in the parametric Lyapunov function V˜ (ε, x) = x′ Pg (ε)x ensuring GAS of x˙ = Acl (ε)x is given by the Hermite matrix, whose construction will be described below. The procedure consists in pushing to the left in the complex plane (a subset of) the n (not necessarily distinct) eigenvalues of A. For each λi (A) ∈ Λ(A), i = 1, . . . , n, let the corresponding desired eigenvalue be λdi (A, ε) = λi (A) − βi ε, where βi ∈ R≥0 , i = 1, . . . , n, are such that • •

re(λdi (A, ε)) < 0, i = 1, . . . , n; the set {λdi (A, ε)), i = 1, . . . , n} is self conjugated.

With such a choice of λdi (A), define the polynomial pεA (s) :=

n Y

(s − λdi (A, ε))

i=1

= sn + α ¯ n−1 (ε)sn−1 + . . . + α ¯ 0 (ε), which is Hurwitz for all ε ∈ (0, 1]. Considering pεA (s) as the desired closed loop polynomial of Acl (ε), the parameterized state feedback gain

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Kg (ε) = [an−1 − α ¯ n−1 (ε), . . . , a0 − α ¯ 0 (ε)]

46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 is uniquely defined. Associated to pεA (s) is the Hermite matrix Pg (ε) = {pij }i,j=1,...n whose coefficients are pij (ε) =

i X (−1)k+1 α ¯ n−k+1 (ε)α ¯ n−i+j+k (ε), k=1

= pji (ε) =0

j ≥ i, j + i even j ≤ i, j + i even j ≤ i, j + i odd.

(15)

Such a symmetric matrix, (see [2] and references therein) is positive definite for all ε ∈ (0, 1] (because pεA (s) is Hurwitz for all ε ∈ (0, 1]) and satisfies the equality Pg (ε)Acl (ε) + A′cl (ε)Pg (ε) = −R(ε), where R(ε) is a positive semidefinite matrix such that the pair (Acl (ε), R(ε)) is detectable. It is easy to see that with such a Pg (ε) polynomial in ε, items i) and ii) of Assumption 2 are satisfied; iii) can be easily checked (∂Pg (ε) is polynomial in ε as well). Finally, if item iii) is satisfied, item iv) can be easily checked by plotting the scalar function m2 − Kg (ε)Pg−1 (ε)Kg′ (ε) and checking if it is always positive; if not, it is always possible to find a constant c > 1 such that, redefining Pg (ε) as c · Pg (ε), item iv) (as well as all the other items in Assumption 2) is satisfied. B. LMI based procedure This approach is based on defining Pg (ε) and the low gain feedback Kg (ε) by a nonlinear rescaling of the solution of a set of matrix inequalities. Let △ = diag{1, 2, . . . , n}, β > 0, and solve the following generalized eigenvalue problem in the variables {Q, X1 , U }, where Q = Q′ : max λ subject to Q > βIn Q∆ + ∆Q > γQ, γ > 0 » – (A + λI)Q + BX1 −BU He < 0, X1 −U » 2 – m X1 ≥ 0. X1′ Q

(16a) (16b) (16c) (16d)

:= :=

x˙ 1 x˙ 2

x˙ ρ

A1 x1 + σm1 (u1 ) A2 x2 + σm2 (u2 ) + a2,1 x1

= = .. .

(20) Aρ xρ + σmρ (uρ ) +

=

ρ−1 P

aρ,j xj .

j=1

From a structural viewpoint, each state equation in the representation (20) corresponds to the dynamics (6a) where d can be seen as a disturbance acting on each subsystem and depending on the lower substates of the cascade representation. It is then possible to design the feedback stabilizer for the MIMO system as a decentralized state-feedback nonlinear law given by ui = Kgi (εi (xi ))xi ,

(21)

whose effectiveness is stated in the next theorem. Theorem 3: If the plant (19) is such that for each pair (Ai , bi ), i = 1, . . . , ρ, matrix function Pgi (·) and Kgi (·) satisfying Assumption 2 are available, then the decentralized controller (21) globally asymptotically stabilizes the origin. Remark 6: (Improved non-decentralized laws) While Theorem 3 refers to the global asymptotic stabilizer (21) taken from Theorem 1, the same result trivially applies when using the higher performance control law (12) and invoking Theorem 2 to prove GAS of the arising decentralized control system. To further improve performance, it is possible to modify the decentralized controller (21) as ui

= =

udec,i (xi ) − di , i−1 P udec,i (xi ) − ai,j xj

(22)

j=1

εK1 Sε ε2 Sε P1 Sε .

(17) (18)

If all the eigenvalues of A are zero, the above recipe is guaranteed to yield a solution that satisfies Assumption 2. Proposition 1: (Integrator Chain) Given system (2) with A being a matrix with all zeros eigenvalues, if Pg (ε) and Kg (ε) are chosen as in (17) and (18) then Assumption 2 is satisfied. Proposition 1 establishes that a chain of integrators can be made GAS for all ε ∈ (0, 1] by the choice of Pg (·) and Kg (·) in (17) and (18). When the eigenvalues of A are not all zero, the same recipe can be used but it is not a priori guaranteed that Assumption 2 will be satisfied, so that it must be checked numerically. IV. G ENERALIZATION TO MIMO PLANTS In order to apply the results in Section II to MIMO systems, it is useful to consider the following canonical form suggested in [10, page 436], obtainable by a suitable change of coordinates from any linear plant with ρ inputs for which the whole state is reachable and for which the control inputs are all independent (i.e., in any coordinate set, the B matrix is full column rank): x˙ = Ax + BσM (u) 2 2 3 bρ 0 Aρ Aρ,ρ−1 · · · Aρ,1 6 0 A 6 0 b 7 ρ−1 · · · Aρ−1,1 7 ρ−1 6 6 7, B = 6 A=6 6 . 6 7 . . .. . .. .. .. 4 .. 4 .. 5 . . 0 0 0 A1 0 0

where M = [m1 · · · mρ ]T denotes the saturation levels of each entry of the control input u = [u1 · · · uρ ]T , the matrices A1 , . . . , Aρ and b1 , . . . , bρ on theˆmain˜diagonal of A and B have a , i ∈ 2, . . . , ρ, j < i are the form (2) and where Ai,j = i,j 0 all zero except for the terms in the upper row, which is denoted by ai,j . The nice feature of the coordinate representation (19) is that, partitioning the state as x = [xTρ · · · xT1 ]T , accordingly to the partition of the matrices in (19), the plant dynamics satisfy the following cascade structure:

(16e)

Let Sε = diag{1, ε, . . . εn−1 }, P1 = Q−1 , K1 = X1 Q−1 , and define Kg (ε) Pg (ε)

WeB17.1

(19) 3

··· 0 ··· 0 7 7 7 . .. . 7 . . 5 0 b1

so that whenever saturation does not occur, each dynamical system in the cascade structure (20) will act independently of the other ones and separate transients will be assignable via the different decentralized part (at least for small enough signals). In (22), the control law udec,i (·) either correspond to the nonlinear law (6b) of Theorem 1 or to the improved nonlinear law (12) of Theorem 2. ◦ Remark 7: Note that in the presence of exponentially stable plants, the globally exponentially stabilizing technique proposed in Remark 5 can be used to achieve GES of each subsystem, therefore GES of the whole closed-loop. The proposed technique is then comparable to that introduced in our recent paper [6], which also achieves global exponential stability of exponentially stable plants via bounded inputs. The important difference between our new stabilizer and that of [6] is that the latter sees the plant as a whole system (despite the presence of multiple inputs) while here we separate the plant in multiple subsystems and stabilize each of them separately. It is unclear which approach leads to better closed-loop responses. Certainly, the approach of [6] is more desirable in terms of achievable optimality, because the arising controller is fully centralized. However numerical advantages may be experienced with the technique proposed here, in addition to being able to enforce different speeds of response on the different subsystems, which may be desirable, especially in the presence of weakly connected plants. ◦ V. A PPLICATION TO ANTI - WINDUP We consider in this section the application of the proposed SISO and MIMO state feedback laws to the design of an L2 anti-windup compensator. This problem has been also addressed in [6] and the improvements achieved here, as compared to the techniques therein reported, are illustrated in Section VI by way of comparative simulations. It is not the scope of this paper to explain in detail all the aspects of anti-windup compensation, however the key ideas behind the L2

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 anti-windup solution first proposed in [22] and later revisited and improved in several papers (see, e.g., [8], [7] and references therein) are summarized below. w uc

yc

C

w u

+

P

y

+ + −

v1 AW

+ v2 +

Fig. 1.

The anti-windup closed loop system.

With reference to Figure 1, the anti-windup design goal applies to any situation where a controller C has been designed to stabilize a plant P disregarding the fact that the input of the plant is subject to magnitude saturation. Typically, the response obtained by the direct interconnection of C to the saturated plant is not desirable as it often leads to undesired transients and possibly even instability. Indeed, from a formal viewpoint, there’s no guarantee on the closed-loop with saturation except for local asymptotic stability which follows from the fact that C stabilizes P and that the saturation is equal to the identity in a neighborhood of the origin (namely, for small enough signals). As shown in Figure 1, anti-windup compensation aims at recovering stability and, as much as possible, also performance in light of the saturation phenomenon. More specifically, the L2 anti-windup solution consists in the augmentation of the control system by way of an anti-windup compensator AW which, as in Figure 1, senses the occurrence of saturation at its input and produced output signals suitably injected at the controller input and at the controller output. The key goal of L2 anti-windup compensation resemble the intuitive goals arising when wanting to disregard the saturation phenomenon: given any initial conditions and selection of external references r(·) and disturbances d(·), consider the so-called “unconstrained closed-loop response” that one would obtain in the absence of saturation and denote by ycℓ (t) and yℓ (t), t ≥ 0 the corresponding controller output and plant output. Then, 1) for any trajectory that would not exceed the saturation limits at any time, i.e., such that ycℓ (t) = σ(ycℓ (t)), for all t ≥ 0, the anti-windup compensator does nothing and the actual response of the closed-loop with saturation and anti-windup augmentation satisfies y(t) = yℓ (t) for all t ≥ 0; 2) for all other trajectories, kycℓ − σǫ (ycℓ k2 < ∞ implies ky − yℓ k2 < ∞, where k · k2 denotes the L2 norm of the signal at argument, which resembles its energy and where σǫ (·) is a restricted saturation function with saturation limits slightly smaller than that of σ(·). These specifications can be understood as follows: 1) if the unconstrained response never requires more input magnitude than the saturation limits, then it must be perfectly reproduced; 2) in all other cases, as long as the energy spent by the unconstrained controller output ycℓ outside the saturation limits is finite, the actual plant output y must converge (in an L2 sense) to the desirable unconstrained plant output yℓ . 1 In [22] (see also [7], [6]), it has been shown that the antiwindup problem can be reduced to a bounded state feedback stabilization problem for the plant P disturbed by a suitably defined (and measurable) disturbance. In particular, the following dynamics should be inserted in the anti-windup compensator AW of Figure 1: x˙ aw v2 v1

= = =

Axaw + B(σ(yc + v1 ) − yc ) −Cxaw − D(σ(yc + v1 ) − yc ) faw (xaw , σ(yc + v1 ) − yc ),

(23)

Then, given a suitable selection of faw (·, ·), the anti-windup problem is solved by (23) if σǫ (yc ) − yc ∈ L2 implies xaw ∈ L2 . From the viewpoint of this paper, as shown in [7, Lemma 1], this requirement is better appreciated when representing the dynamics 1 Besides trivial cases, it is possible to show that the bounded input is insufficient to asymptotically recover the unconstrained plant output response whenever ycℓ spends infinite energy outside any arbitrary small restriction of the saturation limits.

WeB17.1 of (23) as x˙ aw v1

= =

Axaw + Bσǫ (v1 ) + Bd faw (xaw , σǫ (v1 ) + d),

(24)

where it can be shown by the sector properties of the saturation function that |d| ≤ 2|σǫ (yc ) − yc |, which under minor conditions on the external inputs r and d is an exponentially converging signal therefore belonging to L1 . By the similarity between system (24) and the general representation (6a) for each one of the subsystems of the MIMO plant (19), it is possible employ both the global stabilizers proposed in this paper for anti-windup design Theorem 4: Consider an L2 anti-windup compensation system as in [22], [6] and without loss of generality represent the plant dynamics in the coordinates that yield the form (19). Then the decentralized selection v1i = Kgi (εi (xaw,i ))xaw,i as well as its generalizations proposed in Remark 6 solve the global anti-windup problem. Remark 8: Note that the anti-windup application of Theorem 4 gives an example for which both the assumptions imposed in this paper are quite reasonable: 1) the plant state is available for measurement because it is part of the controller+anti-windup dynamics; 2) the disturbance signal d is also available for measurement because it partly consists in suitable combinations of the anti-windup compensator state (for the part related to the cascade interconnection in (20), partly of the output signal yc produced by the controller C. ◦ VI. S IMULATION EXAMPLES A. Anti Windup case: the triple integrator We consider the case of a triple integrator plant for which an anti-windup filter is designed to prevent disastrous effects due to saturation by the procedure described in Section V. Given " # " # 0 0 0 1 A = 1 0 0 , B = 0 , Cz = Cy = [ 1 0 0 ] . 0 1 0 0 the anti-windup gains are designed according to (12). So, Kr (α) and Lr (α) are chosen following the high performance regional design (as done in [6]), and are given by the scheduled law ) comes from the LMI based procedure of (11), whereas Kg ( N α Section III-B, as well as Pg (1), which is the border ellipsoid between the GAS and high performance designs. With a choice of N = 6, the output and the input responses are plotted in Fig. 2 together with the scheduling parameter α. If α does not exceed the value N = 6 the output and the input signals coincide with those obtained using the design procedure proposed in [6]. (the high performance regional design is the same). As soon as α > N , the low gain controller acts and a better behavior of the new design techinique proposed in this paper can be appreciated. 2 B. MIMO case: stabilization of systems cascade Consider the MIMO system (19) with n = 6 states and ρ = 2 inputs, where (A1 , b1 ) and (A2 , b2 ) are in controller form and have coinciding eigenvalues: Λ(A2 ) = Λ(A1 ) = {0, 0, 0}. Moreover, select a2,1 = [−10 − 10 − 10]. The feedback stabilizer for such a MIMO system has been designed accordingly to the MIMO technique of Section IV using both the decentralized state feedback (21) and also the modification (22). In Fig. 3 the state responses of subsystem 1 and subsystem 2 are both plotted first including the correction term of the law (22) (solid) and then not including it (dashed). It is evident how the presence of such a term produces better results. C. Double Oscillator stabilization The Hermite based procedure described in Section III-A has been applied to the stabilization of a double oscillator system in controller form having open-loop eigenvalues in ±j and having saturation limits ±10. The resulting closed-loop responses from the initial conditions [1.5 3 − 6 0]′ are shown in Fig. 5, where the state 2 The low gain compensation used in [6] was based on the Hermite approach (for a chain of integrators the Hermite approach can be shown to always satisfy Assumption 2).

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007

WeB17.1 10

Position mass [m]

5 0

x

0.005 0

−5

−0.005

−10

0

5

10

15

0

5

10

15

0

5

10

15

−0.01 Unconstrained Scheduled NOLCOS 1 2 Saturated Scheduled

0

Input Force [N]

5

3

4

5

6

7

8

5 u

−0.015

10

0 −5

0 10 8

0

1

2

3

4

5

6

7

α

−5 8

6 4 2

6

α

Time [s]

Fig. 5. State behaviour (above), control input behaviour (center), scheduling parameter behaviour (below).

4 2 0

0

1

2

3

4 Time [s]

5

6

7

8

Fig. 2. Time responses (above) and control inputs (center) of the saturated (dotted), scheduled NOLCOS (dash-dotted), linear L2 anti-windup (dash -dashed ), scheduled nonlinear anti-windup (solid). Below is plotted the behaviour of the scheduled parameter α.

2 1

x1

0 −1 −2 −3

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 Time [s]

6

7

8

9

10

6 4

x2

2 0 −2 −4

Fig. 3. Time behaviour of the states of the first SISO subsystem starting from the initial conditions [1, 1, 1]′ (above) and the second SISO subsystem starting from the initial conditions [0, 0, 0]′ (below): with (solid) and without (dash-dotted) the correction term.

4

2

a)

0

0

−2

u2

u1

b) 2

−2 −4

−6

−6 −8

−4

0

2

4

6

8

Input u2 with correction term

−8

Input u1

−10

10

Input u2 0

2

4

Time [s]

6

8

10

Time [s]

2

2.5

c)

d)

α1

α2 with correction term α2

2

α2

α1

1.5 1.5

1 1

0.5

0

2

4

6 Time [s]

8

10

0.5

0

2

4

6

8

10

Time [s]

Fig. 4. a) Control input of the first SISO subsystem ; b) control input of the second SISO subsystem : with (solid) and without (dash-dotted) the correction term; c) scheduling parameter α1 of the first SISO subsystem; d) scheduling parameter α2 of the second SISO subsystem : with (solid) and without (dash-dotted) the correction term.

response, controller response and scheduling parameter are plotted. The simulation shows that successful stabilization is performed by the proposed stabiliizing law. R EFERENCES [1] Z. Lin adn A. Saberi. Low-and-high gain design technique for linear systems subject to input saturation – a direct method. Internat. J. Robust Nonlinear Control, 7:1071–1101, 1997. [2] B. D. O. Anderson. The reduced Hermite criterion with application to proof of the Li`enard-Chipart criterion. IEEE Trans. Aut. Cont., 17(5):669–672, 1972. [3] A. Casavola and E. Mosca. Global switching regulation of inputsaturated discrete-time linear systems with arbitrary l2 disturbances. IEEE Trans. Aut. Cont., 46(6):915–919, 2001. [4] Z. Ding. Global stabilization of input-saturated systems subject to l2 disturbances. IEE Proceedings: Control Theory and Applications, 147(1):53–58, 2000. [5] A.T. Fuller. In-the-large stability of relay and saturating control systems with linear controllers. Int. J. Contr., 10:457–480, 1969. [6] S. Galeani, S. Onori, A.R. Teel, and L. Zaccarian. Nonlinear l2 antiwindup for enlarged stability regions and regional performance. In Symposium on Nonlinear Control Systems (NOLCOS), Pretoria (South Africa), submitted, August 2007. [7] S. Galeani, S. Onori, A.R. Teel, and L. Zaccarian. Regional, semiglobal, global nonlinear anti-windup via switched design. In European Control Conference, Kos (Greece), July 2007. [8] S. Galeani, A.R. Teel, and L. Zaccarian. Constructive nonlinear antiwindup design for exponentially unstable linear plants. Systems and Control Letters, 2006, to appear. [9] F. Grognard, R. Sepulchre, and G. Bastin. Improving the performance of low-gain designs for bounded control of linear systems. Automatica, 38(10):1777–1782, 2002. [10] T. Kailath. Linear Systems. Prentice-Hall, 1980. [11] G. Kaliora and A. Astolfi. Nonlinear control of feedforward systems with bounded signals. IEEE Trans. Aut. Cont., 49(11):1975–1990, 2004. [12] Z. Lin. Global control of linear systems with saturating actuators. Automatica, 34(7):897–905, 1998. [13] Z. Lin. Low gain feedback. Lecture Notes in Control and Information Sciences. Springer-Verlag, Great Britain, 1998. [14] N. Marchand and H. Ahmad. Global stabilization of multiple integrators with bounded controls. Automatica, 41(12):2147–2152, 2005. [15] A. Megretski. L2 BIBO output feedback stabilization with saturated control. In 13th IFAC World Congress, San Francisco, CA (USA), pages 435–440, 1996. [16] P. Morin, R.M. Murray, and L. Praly. Nonlinear rescaling of control laws with application to stabilization in the presence of magnitude saturation. In IFAC Symposium on Nonlinear Control Systems Design (NOLC OS’98), pages 691–696. Enschede (The Netherlands), July 1998. [17] W.E. Schmitendorf and B.R. Barmish. Null controllability of linear systems with constrained controls. SIAM J. Contr. Opt., 18(4):327– 345, July 1980. [18] E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Int. Journal of Control, 39(1):181–188, 1984. [19] H.J. Sussmann, E.D. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Aut. Cont., 39(12):2411–2424, 1994. [20] H.J. Sussmann and Y. Yang. On the stabilization of multiple integrators by means of bounded feedback controls. In 30th CDC, pages 70–72, Brighton, England, Dec 1991. [21] A.R. Teel. Global stabilization and restricted tracking for multiple integrators with bounded controls. Systems and Control Letters, 18:165–171, 1992. [22] A.R. Teel and N. Kapoor. The L2 anti-windup problem: Its definition and solution. In Proc. 4th ECC, Brussels, Belgium, July 1997.

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