Analysis and synthesis of nonlinear systems with uncertain initial ...
1
Analysis and synthesis of nonlinear systems with uncertain initial conditions G. Chesi (Senior Member) and Y.S. Hung (Senior Member)
Abstract—This paper considers, for polynomial and nonpolynomial systems, the problems of computing extremal values of the trajectories over a given set of initial conditions as well as finding output controllers minimizing these extremal values under time-domain constraints. It is shown that upper bounds of the sought extremal values as well as candidates of the sought controllers can be computed by solving a one-parameter sequence of BMI optimizations obtained through the Square Matricial Representation (SMR) of polynomials. Moreover, a necessary and sufficient condition is proposed to establish the tightness of the found upper bound in spite of the conservatism introduced by the non-convexity of BMI optimizations and the chosen degree of the Lyapunov function and relaxing polynomials. It is worthwhile to note that other approaches such as state augmentation and approximation techniques used to recast non-polynomial into polynomial systems can easily yield conservative results.
systems whose state updating law depends polynomially on the state and affinely on the input, and then it is extended to deal with non-polynomial dependence on the state by using truncated Taylor expansions and taking into account the worstcase remainders. It is shown that upper bounds of the sought extremal values as well as candidates of the sought controllers can be computed by solving a one-parameter sequence of bilinear matrix inequality (BMI) optimizations by using Lyapunov functions (LFs) and polynomial relaxations based on the Square Matricial Representation (SMR). In order to deal with the conservatism introduced by the non-convexity of the problem as well as the chosen degree of the LF, a necessary and sufficient condition is proposed for the analysis part to establish the tightness of the found upper bound.
Index Terms—Nonlinear system, Uncertain initial condition, SMR, BMI, tightness.
I. I NTRODUCTION Stability is not the only important issue in the analysis and synthesis of closed-loop systems, in fact it is often mandatory also to deal with time-domain constraints on the system signals. For linear systems, the analysis problem can be solved by exploiting closed-form solutions for the trajectory, while the synthesis problem with time-domain constraints can be addressed via frequency-domain constraints on weighted transfer functions or via model predictive control (MPC). For nonlinear systems the problem is more involved. First, there do not exist closed-form solutions for the trajectory except few special cases. Second, the synthesis must deal with the nonlinearities of the state updating law and measurable output, which affect convergence and efficiency of strategies such as nonlinear MPC [1], [2]. And, clearly, the problem is even more difficult if the initial condition is unknown and if the nonlinearities are non-polynomial. In this respect, it should be noted that approaches such as state augmentation and approximation techniques for recasting the non-polynomial system into a polynomial system can easily yield conservative results. In this paper a new approach to deal with time-domain constraints in the analysis and synthesis of nonlinear systems is proposed. In particular, the proposed framework considers the computation of extremal values of the trajectories over a given set of initial conditions and the computation of output controllers minimizing these extremal values under time-domain constraints. The approach is first described for G. Chesi and Y.S. Hung are with the Department of Electrical and Electronic Engineering, University of Hong Kong. Email: {chesi,yshung}@eee.hku.hk
II. P RELIMINARIES The notation is as follows: N, R: natural and real numbers sets; C n (X ): set of functions whose first n derivatives are continuous over the set X ; ∂X : boundary of set X ; 0n : origin of Rn ; I: identity matrix (of size specified by the context); Φ′ : transpose of matrix Φ; Φ > 0 (Φ ≥ 0): symmetric positive definite (semidefinite) matrix Φ; φ > 0 (φ ≥ 0): positive (non-negative) entries vector φ; φ|max : max{φ1 , . . . , φn } for φ ∈ Rn . Consider the class of continuous-time systems ˙ = f (x(t)) + ϕ(x(t)) + g(x(t))u(t) x(t) y(t) = h(x(t)) (1) x(0) = xinit
where x(t) ∈ Rn is the state, u(t) ∈ Rnu is the input, y(t) ∈ Rny is the output, xinit is the initial condition, f (x(t)), g(x(t)) and h(x(t)) are polynomial functions of x(t), and ϕ(x(t)) is a non-polynomial function of x(t). The set of initial conditions of interest is defined as A = {x ∈ Rn : ai (x) ≥ 0 ∀i = 1, . . . , na }
(2)
where a1 (x), . . . , ana (x) are polynomials. We assume that A is compact and define the set {k(y) : Rny → Rnu such that, for u(t) = k(y(t)), limt→+∞ x(t) = 0n , and w(t) ≤ w0 for all t ∈ [0, +∞) and for all xinit ∈ A} (3) where k is a polynomial function, w(t) ∈ Rnw is a selectable design signal to constrain the controller, and w0 ∈ Rnw is a given vector. Let us define the cost achieved by u = k(y) as K
=
σ(k(y)) =
sup xinit ∈A,t∈[0,+∞)
z(t)|max
(4)
2
where z(t) ∈ Rnz is another selectable signal for optimization. It is assumed that: 1) the signals z(t) and w(t) are expressed by [qz (x(t))′ , lz (u(t))′ ]′ [qw (x(t))′ , lw (u(t))′ ]′
z(t) = w(t) =
(5)
where qz , qw are polynomial functions and lz , lw are linear functions; 2) the origin is the equilibrium point of interest and the output vanishes in the origin. The problems we address are: 1) Problem P1 (analysis): to compute the maximum of z(t)|max for the autonomous system: σ = σ(0nu );
(6)
III. A NALYSIS
We consider first the case ϕ(x) = 0n . In order to simplify the description we assume that: - (A1) the linearized autonomous system is asymptotically stable, that is A is Hurwitz where d (f (x) + ϕ(x)) | dg(x) | dh(x) A|B|C= (10) dx x=0n
The basic idea is to look for LFs whose unitary sublevel sets are invariant sets and contain A. Then, we also require that these sublevel sets are contained in the region where the supremum of qz (x)|max is bounded by a certain quantity ζ ∈ R. If we can find this LF, it is guaranteed that ζ ≥ σ. More formally, let us denote with v : Rn → R the LF candidate, and define with V(c) its sublevel set
2) Problem P2 (synthesis): to compute the controller that minimizes the maximum of z(t)|max : k ∗ (y) = σ∗
=
arg min σ(k(y))
(7)
σ(k ∗ (y)).
(8)
k(y)∈K
In the sequel the dependence on the time t will be generally omitted for ease of notation. Let us observe that, depending on the choice of z, one can select several costs. such as: z = [h(x)′ B1′ B1 h(x), u′ B2′ , −u′ B2′ , B3 x]′ ⇒ z|max = max{kB1 yk22 , kB2 uk∞ , b3,1 x, . . . , b3,m x} where B1 , B2 , B3 are weighting matrices of suitable dimensions, and b3,1 , . . . , b3,m are the rows of B5 . Moreover, one can analogously define several constraints by similarly defining w and w0 . Before proceeding let us introduce the complete square matricial representation (SMR) [3], [4]. Let x{m} ∈ Rτ1 (m) contain all monomials of degree not greater than m in x, and let p1 (x) be a polynomial of degree 2m. The complete SMR of p1 (x) with respect to x{m} is ′
p1 (x) = x{m} P1 (α1 )x{m}
P1 (α1 ) = P1 + N1 (α1 ) (9) where P1 is any symmetric matrix such that p1 (x) = ′ x{m} P1 x{m} , α1 ∈ Rν1 (m) is a vector of free parameters and N1 (α1 ) is a linear parametrization of the set N1 (m) = ′ {N = N ′ : x{m} N x{m} = 0 ∀x}. As explained in [3], [4] 1 τ1 (m) = (n+m)! n!m! and ν1 (m) = 2 τ1 (m)(τ1 (m) + 1) − τ1 (2m). In the case of polynomials having special structures, more compact representations can be derived. Indeed, polynomials without constant and linear terms can be represented with respect to the vector x[m] ∈ Rτ2 (m) containing all monomials of degree less than or equal to m in x but the constant term. For these polynomials the SMR is analogously defined by substituting N1 (m) with the set N2 (m) = {N = ′ N ′ : x[m] N x[m] = 0 ∀x}. We have τ2 (m) = τ1 (m) − 1 and the dimension of N2 (m) is ν2 (m) = ν1 (m) − τ1 (m) + n + 1. In the sequel we will refer to P1 and P1 (α) as SMR matrices of p1 (x). Unless explicitly specified otherwise, it will be assumed that these matrices are defined with respect to x{m} . with
PROBLEM
V(c) = {x ∈ Rn : v(x) ≤ c} Q(ζ)
n
= {x ∈ R : qz (x)|max ≤ ζ} .
(11) (12)
Suppose there exists v(x) and ζ ∈ R such that: - (C1) v(x) is radially unbounded, v(0) = 0 and v(x) > 0 ∀x ∈ Rn \ {0n }; - (C2) ∇v(x)f (x) < 0 ∀x ∈ V(1) \ {0n }; - (C3) A ⊆ V(1); - (C4) Q(ζ) ⊇ V(1). Then, ζ ≥ σ. Now, let us select v(x) as a polynomial of degree 2δv and introduce t1 (x) = ∇v(x)f (x) P + s1 (x)(1 − v(x)) na t2 (x) = v(x) − 1 + i=1 s2,i (x)ai (x) t3,i (x) = s3,i (x)(qz,i (x) − ζ) + 1 − v(x), i = 1, . . . , nqz (13) where s∗ (x) are auxiliary polynomials known as Hilbert’s polynomials (see for example [5]). Theorem 1: Let ζ > 0 be a given real scalar, and let 2δv be the chosen degree of v(x). Let V, S∗ , T1 (V, S1 , α1 ), T2 (V, S2,∗ , α2 ), T3,∗ (V, S3,∗ , α3,∗ ) be SMR matrices of the polynomials v(x), s∗ (x), t1 (x), t2 (x), t3,∗ (x) respectively, with the matrices V, S1 , T1 (V, S1 , α1 ) defined with respect to extended vectors without constant term. Define λ∗ = sup λ s.t. (15) (14) λ∈R,V >0,S∗ >0,α∗
T1 (V, S1 , α1 ) + λI T2 (V, S2,∗ , α2 ) + λI T3,i (V, S3,i , α3,i ) + λI
< 0 ≤ 0 ≤ 0 ∀i = 1, . . . , nqz
(15)
If λ∗ ≥ 0, then ζ ≥ σ. Proof Suppose that (15) is satisfied with λ ≥ 0. Let x[δt,1 ] be the extended vector in the representation of t1 (x). From the first inequality we have ′
0 > x[δt,1 ] (T1 (V, S1 , α1 ) + λI) x[δt,1 ] = t1 (x) + λkx[δt,1 ] k22 that is t1 (x) < 0 for all x ∈ Rn \ {0n }. From the other inequalities we similarly obtain that v(x) > 0 and s1 (x) > 0 for all x ∈ Rn \ {0n }, and that t2 (x) ≤ 0, t3,∗ (x) ≤ 0, s2,∗ (x) > 0 and s3,∗ (x) > 0 for all x. Now, from t1 (x) < 0 and s1 (x) > 0 it follows that ∇v(x)f (x) < 0 for all x ∈ V(1) \ {0n }. Analogously we prove that x ∈ V(1) for all
3
x ∈ A, and x ∈ Q(ζ) for all x ∈ V(1). Moreover, v(x) is radially unbounded because V > 0, hence C1–C4 hold. Theorem 1 provides a sufficient condition to establish if ζ is an upper bound of σ by solving the optimization (14) which is a nonconvex optimization because the first inequality in (15) is a BMI since T1 (V, S1 , α1 ) is a bilinear function of V and S1 . BMI optimizations can be locally solved through dedicated software, alternatively they can be approached via a sequence of convex LMI optimizations by alternatively fixing one variable and optimizing with respect to the other (we refer to this solution as V -S iterations) as done in Section V. In order to find σ, we can simply adopt a bisection algorithm minimizing ζ subject to the condition of Theorem 1. Now, we describe how Theorem 1 can be extended to deal with the case ϕ(x) 6= 0n . First of all, it is worthwhile to observe that one may attempt to deal with this case by performing state augmentations to system into polynomial, or by substituting the non-polynomial terms with their truncated Taylor expansions. However, it is known that these attempts can easily lead to conservative and disastrous results. Our idea is to introduce truncated Taylor expansions taking into account the worst-case remainder. We suppose that assumptions A1 previously introduced and A2 below hold: - (A2) the function ϕ(x) has the form ϕ(x) =
r X
pi (x)ψi (xki )
(16)
i=1
where pi (x) are polynomials, k1 , . . . , kr are integers in [1, n], and ψi : C di ([ci,min , ci,max ]) → R are nonpolynomial functions for integers di ≥ 1 and scalars ci,min < 0, ci,max > 0. Let us write ψi (xki ) via the Taylor expansion centered in xki = 0 up to the βi -th power for a given integer βi in [1, di − 1] and express the remainder in the Lagrange form: ψi (xki ) = µi (xki ) + ̺i (xki )θi (bi )
(17)
where µi (xki ) is a polynomial of degree βi , ̺i (xki ) is a monomial of degree βi + 1, θi (bi ) is the βi + 1-th derivative of ψi (xki ) evaluated for xki = bi , and bi ∈ [0, xki ] if xki ≥ 0 or bi ∈ [xki , 0] otherwise. Let us introduce the polynomials Pr t1,m (x) = ∇v(x) (f (x) + i=1 piP (x)µi (xki )) r +s1,m (x)(1 − v(x)) + i=1 (θi,mi mi +(−1) s1,m,i (x)) ∇v(x)pi (x)̺i (xki ), (18) m ∈ {0, 1}r t4,i (x) = s4,i (x)(xki − ci,max )(xki − ci,min ) +1 − v(x), i = 1, . . . , r where s1,m (x), s1,m,i (x), s4,i (x) are polynomials, and supbi ∈[ci,min ,ci,max ] θi (bi ) if mi = 0 θi,mi = inf bi ∈[ci,min ,ci,max ] θi (bi ) if mi = 1
(19)
Theorem 2: Let ζ > 0 be a given real scalar, and let 2δv be the chosen degree of v(x). Let S∗ , T1,m (V, S1,∗ , α1,m ), T4,i (V, S4,i , α4,i ) be SMR matrices of the polynomials s∗ (x), t1,m (x), t4,i (x) respectively. Define λ∗ =
sup λ∈R,V >0,S∗ >0,α∗
λ s.t. (21)
(20)
ineq. (15) without the first inequality T1,m (V, S1,∗ , α1,m ) + λI < 0 ∀m ∈ {0, 1}r T4,i (V, S4,i , α4,i ) + λI ≤ 0 ∀i = 1, . . . , r
(21) If λ∗ ≥ 0, then ζ ≥ σ. ¯ be any Proof Suppose that (21) is satisfied with λ ≥ 0. Let x point in V(1) and define 0 if ∇v(¯ x)pi (¯ x)̺i (¯ xki ) ≥ 0 m ¯ = [m ¯ 1, . . . , m ¯ r ]′ , m ¯i = 1 otherwise From t4,i (¯ x) ≤ 0 it follows that x ¯ki ∈ [ci,max , ci,max ]. Moreover, since s1,m (¯ x ) > 0 and s1,m,i x) > 0, from ¯ ¯ (¯ t1,m (¯ x ) < 0 we have ¯ Pr 0 > ∇v(¯ x) + i=1 pi (¯ x)µi (¯ xki )) + s1,m x)(1 − v(¯ x)) ¯ (¯ Pxr) (f (¯ m ¯i + i=1 (θi,m (−1) s (x)) ∇v(¯ x )p (¯ x )̺ (¯ x ¯i + 1, m,i ¯ i i k i) Pr > ∇v(¯ x ) (f (¯ x ) + p (¯ x )µ (¯ x )) i i k i i=1 Pr + i=1 θi,m x)pi (¯ x)̺i (¯ xki ) ¯ i ∇v(¯ = ∇v(¯ x) (f (¯ x) + ϕ(¯ x)) that is v(¯ ˙ x) < 0. Following the remaining proof of Theorem 1, we conclude that ζ ≥ σ. Theorem 2 provides a sufficient condition for computing an upper bound of σ through BMI optimizations in spite of the presence of non-polynomial terms in (1). This is made possible by taking into account the worst-case remainder of the truncated Taylor expansion in the time-derivative of v(x). In particular, the unitary sublevel set V(1) is constrained within the set C = {x ∈ Rn : xki ∈ [ci,min , ci,max ], i = 1, . . . , r}
(22)
where ci,min , ci,max delimit the region of validity of the worst-case remainder computed through the bounds θi,mi . If no information are available a priori about the extension of V(1) and/or the computation of θi,mi is difficult, conservative estimates of ci,min , ci,max , θi,mi can be used. In fact, let us observe that the effect of these conservative quantities, represented by term ωi,mi (x) = θi,mi ∇v(x)pi (x)̺i (xki )
(23)
in the polynomials t1,m (x), can be compensated by increasing the degree βi of the truncated Taylor expansion which makes this term convergent to zero. IV. S YNTHESIS
AND TIGHTNESS PROBLEMS
Let us consider first problem P2. In order to simplify the description we assume that: - (A3) the linearized system can be asymptotically stabilized through a static output feedback, i.e. there exists K ∈ Rni ×nu such that A + BKC is Hurwitz where A, B, C are as in (10). Let us express the controller k(y) in the class K in (3) as k(y) = ξ ′ y [δk ]
(24)
where δk is the degree of the controller, and ξ ∈ Rτ2 (δk ) is the coefficient vector to be determined. Since the state updating law depends linearly on the input, one can exploit the condition derived in Theorem 2 by letting the vector ξ vary together with
4
the other variables of the optimization (20). In particular this can be done by adding t1,u (x) to t1,m (x) where t1,u (x) = ∇v(x)g(x)ξ ′ y [δk ] .
(25)
Then, in order to take into account the presence of the input in the cost signal z, we introduce for i = 1, . . . , nlz t5,i (x) = lz,i (ξ ′ (h(x))[δk ] ) − ζ + s4,i (x)(1 − v(x)).
(26)
In order to take into account the time-domain constraint on the signal w we also define the polynomials tj,∗ and sj,∗ for j = 6, 7 analogous to those for j = 3, 5 for the cost signal z by replacing qz,i , lz,i , ζ with qw,i , lw,i , w0,i respectively. Theorem 3: Let ζ > 0 be a given real scalar, and let 2δv be the chosen degree of v(x). Let T1,u (V, ξ), S∗ , T5,∗ (V, ξ, S5,∗ , α5,∗ ), T6,∗ (V, S6,∗ , α6,∗ ), T7,∗ (V, ξ, S7,∗ , α7,∗ ) be SMR matrices of the polynomials t1,u (x), s∗ , t5,∗ (x), t6,∗ (x), t7,∗ (x) respectively, with T1,u (V, ξ) defined with respect to the extended vector used for defining T1 (V, S1 , α1 ). Define λ s.t. (28) (27) λ∗ = sup λ∈R,V >0,ξ,S∗ >0,α∗
ineq. (21) appending T1,u (V, ξ) T5,i (V, ξ, S5,i , α5,i ) + λI ≤ T6,i (V, S6,i , α6,i ) + λI ≤ T7,i (V, ξ, S7,i , α7,i ) + λI ≤
to 0 0 0
T1,m (V, S1,∗ , α1,m ) ∀i = 1, . . . , nlz ∀i = 1, . . . , nqw ∀i = 1, . . . , nlw (28)
If λ∗ ≥ 0, then ζ ≥ σ ∗ . Proof Analogous to the proof of Theorem 2 by observing that (28) implies t1,m (x) + t1,u (x) < 0. Hence, we have ∇v(x)(f (x) + g(x)ξ ′ (h(x))[δk ] < 0 for all x ∈ V(1) \ {0n }. Moreover, from (28) we have lz (ξ ′ (h(x))[δk ] ) max ≤ ζ and [qw (x)′ , lw (ξ ′ (h(x))[δk ] )′ ]′ ≤ w0 for all x ∈ V(1). Therefore, V(1) is an invariant set for the controlled system where z|max ≤ ζ and w ≤ w0 . Theorem 3 allows one to compute an upper bound of σ ∗ and a controller guaranteeing this upper bound by solving the optimization (27). In the sequel we will indicate with ζ2δv ,δk the best upper bound of σ ∗ found with a LF of degree 2δv and a controller of degree δk . Now we consider the problem of establishing if the best upper bound found for σ by using Theorem 2 with a LF of degree 2δv is tight or not. Let us indicate with ζ2δv this upper bound, and let us observe that ζ2δv can be conservative for several reasons: - the degree of the LF or auxiliary polynomials s∗ (x) is too low; - the degree of the Taylor expansion for ϕ(x) is too low; - BMI optimizations are nonconvex. The following result provides a necessary and sufficient condition to establish if ζ2δv is tight. Theorem 4: Let V2δv be the optimal value of V corresponding to the found ζ2δv , and let v2δv (x) and V2δv (1) be the LF and its unitary sublevel set corresponding to V2δv . Then, ζ2δv = σ ⇐⇒ ∃xR,init ∈ T , t¯ ∈ [0, ∞) : xR (t¯) ∈ A (29) where T = ∂V2δv (1) ∩ ∂Q(ζ2δv )
(30)
and xR (t) is the solution of the system x˙ R (t) = −f (xR (t)) − ϕ(xR (t)) xR (0) = xR,init
(31)
Proof “⇐” Suppose that ∃xR,init ∈ T and t¯ ∈ [0, ∞) such that xR (t¯) ∈ A. Then, since the system (31) evolves reversely with respect to the system (1), it follows that by initializing (1) with xinit = xR (t¯) we have that x(t¯) = xR,init . Since xinit ∈ A, it follows from the definition of σ in (6) that z(t¯)|max ≤ σ. On the other hand, z(t¯)|max = qz (x(t¯))|max = qz (xR,init )|max = ζ2δv because xR,init ∈ ∂Q(ζ2δv ), and hence ζ2δv ≤ σ. But ζ2δv ≥ σ. Therefore, ζ2δv = σ. “⇒” Suppose that ζ2δv = σ. From the definition of σ in (6) and since A is compact, it follows that ∃xinit ∈ A and t¯ ∈ [0, ∞) such that z(t¯)|max = σ. Observe that x(t¯) ∈ ∂Q(ζ2δv ) because ζ2δv = σ. Suppose now for contradiction that x(t¯) does not belong to ∂V2δv (1). Then, this implies either that v2δv (x(t¯)) > 1 or that v2δv (x(t¯)) < 1. But the former implies that x(t¯) lies outside an invariant set containing the initial condition xinit ∈ A, and the second implies that V2δv (1) is not included in Q(ζ2δv ) as ensured by the third inequality in (15). Hence, both hypotheses are impossible and hence x(t¯) ∈ ∂V2δv (1). Therefore, x(t¯) ∈ T . Then, by initializing the reverse system (31) with xR,init = x(t¯) we obtain that xR (t¯) = xinit ∈ A which concludes the proof. The condition of Theorem 4 can be checked in two steps. First, computing T which is the intersection of the boundaries of V2δv (1) and Q(ζ2δv ). Second, computing the trajectories of the reverse system (31) initialized with the points in T and checking if at least one of these trajectories intersects A. Observe that T is composed by a finite number of points, typically one, being the intersection of two tangent surfaces. The set T can be found by solving v2δv (x) − 1 = 0 (32) qz (x)|max − ζ2δv = 0 However, solving the system (32) can be a difficult task because it is nonlinear system. Theorem 5: Define Mi = T3,i (V2δv , S3,i,2δv , α3,i,2δv ), i = 1, . . . , nqz
(33)
where S3,i,2δv and α3,i,2δv are the found optimal values of S3,i and α3,i . Then, ( ) nqz [ T = x∈ Mi : x satisfies (32) (34) Mi
=
i=1 n o x ∈ Rn : x{δt,3 } ∈ ker(Mi ) .
(35)
Proof Consider x ∈ T . Since x satisfies (32) there exists j such that t3,j (x) = 0. Now, from (21) it follows that }′ Mj ≤ 0. Hence: 0 = t3,j (x) = x{δt,3 Mj x{δt,3 } =
2 ′ ˜ ′M ˜ j x{δt,3 } = − ˜ j x{δt,3 } ˜ j is x{δt,3 } −M
M
where M j 2
2
˜ {δt,3 } any Cholesky factor of −Mj . Clearly, M
= 0 if jx 2
5
The trajectory xR (t) of (31) with initial condition xR,init = x¯ intersects A in xR (t¯) as shown in Figure 1b. Therefore, the upper bound is tight. 2δv \ δk 2 4 6
1 2 +∞ +∞ +∞ 2.727 +∞ 2.463 TABLE I
3 +∞ 2.687 2.191
E XAMPLE 1: UPPER BOUNDS ζ2δv ,δk
COMPUTED THROUGH SIMPLE V ITERATIONS .
4
-S
3
3 2 2 1
x2
1
x2
˜ j ). Since ker(M ˜ j ) = ker(Mj ), we and only if x{δt,3 } ∈ ker(M conclude the proof. Theorem 5 provides an alternative way of computing the set T which consists of finding the vectors x ∈ Rn satisfying x{δt,3 } ∈ ker(Mi ) for i = 1, . . . , nqz . This can be trivially done if the dimension of ker(Mi ) is 1 corresponding to a unique element in the set Mi as shown in Example 1 in Section V. In other cases one can use, for example, the approaches described in [6]–[9]. Once the sets Mi have been found, one can simply the set T by verifying if the Snobtain qz vectors contained in i=1 Mi satisfy the equations in (32) via trivial substitution. Before introducing some examples in the next section, let us observe that the BMI optimization can be initialized with any matrix K satisfying assumption A3 and any LF proving the asymptotical stability of the linearized system controlled by such a matrix.
0
0
−1 −1
xR (t¯)
−2
V. E XAMPLES
−2 −3
A. Example 1
−4 −6
Let us consider the non-polynomial system x˙ 1 = x2 + 0.2x21 x2 + 0.2 (1 − ex1 ) x˙ 2 = −1.5x1 − 2x2 − 1.1x22 + u y = x1 xinit ∈ A = {x : kxk22 ≤ 4} min
sup
stabilizing u(t)=k(y(t)), |u(t)| −5.977 and ξ1 < −0.072 in the optimization, where ξ1 is the coefficient of the linear term of the controller. Table IIb shows the obtained upper bounds, and Figure 2d the corresponding performance. Observe that the convergence is much faster, clearly at the expense of a larger amplitude of the control signal. 2δv \ δk 2 4
1 2 0.0783 0.0783 0.0772 0.0772 (a)
3 0.0503 0.0494
2δv \ δk 2 4
1 2 3 0.1136 0.1136 0.0582 0.1131 0.1131 0.0581 (b) TABLE II E XAMPLE 2: UPPER BOUNDS ζ2δv ,δk OBTAINED WITH UNCONSTRAINED ( A ) AND CONSTRAINED ( B ) CONTROLLER LINEAR TERM .
0.6 0.4
θ
x3 [A]
0.2 0
−0.2
+
Ra
La
−0.4 −0.6
u
−0.8 −0.2 −0.1 0 0.1 0.2
2
−1
x2 [rad·s
1.5
0
−0.5
−1
−1.5
−2
(b) 0.06
0.04
0.04
u [v·rad−1 ]
0.06
u [v·rad−1 ]
0.5
] x1 [rad]
(a)
0.02
0.02
0
0
−0.02
−0.02
−0.04
−0.04
−0.06
1
0
50
t [s] (c)
100
150
−0.06
0
5
10
15
t [s]
20
25
30
(d)
Fig. 2. Example 2. (a) Inverted pendulum controlled with a DC motor. (b) Trajectories starting in A with u = k4,3 (y). (c) Control input with u = k4,3 (y). (d) Control input achieved by constraining the linear term of the controller.
VI. C ONCLUSION The problem of computing extremal values of the trajectories over a given set of initial conditions and the problem of computing output controllers minimizing these extremal values under time-domain constraints have been addressed for polynomial and non-polynomial systems. It has been shown that an upper bound of the sought extremal values as well as candidates of the sought controllers can be found by solving a one-parameter sequence of BMI optimizations, which can be approached through either recently developed software or simple iterative convex LMI optimizations. Since the found upper bound may be conservative due to the nonconvexity of BMI optimizations and the chosen degree of the Lyapunov function, a necessary and sufficient condition has been proposed for the analysis part to establish the tightness of this upper bound in spite of all these sources of conservatism. R EFERENCES [1] F. Allgower and A. Zheng, Nonlinear Model Predictive Control: Assessment and Future Directions for Research, ser. Progress in Systems and Control Series. Basel: Birkhauser Verlag, 2000. [2] B. Kouvaritakis and M. Cannon, Non-Linear Predictive Control: Theory and Practice, ser. IEE Control Series. London: IEE, 2001. [3] G. Chesi, A. Tesi, A. Vicino, and R. Genesio, “On convexification of some minimum distance problems,” in 5th European Control Conf., Karlsruhe, Germany, 1999. [4] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Solving quadratic distance problems: an LMI-based approach,” IEEE Trans. on Automatic Control, vol. 48, no. 2, pp. 200–212, 2003. [5] J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry. Springer, 1998. [6] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “An LMI-based approach for characterizing the solution set of polynomial systems,” in 39th IEEE Conf. on Decision and Control, Sydney, Australia, 2000, pp. 1501–1506. [7] ——, “Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach,” Int. Journal of Robust and Nonlinear Control, vol. 13, no. 13, pp. 1239–1257, 2003. [8] H. J. Stetter, Numerical Polynomial Algebra. Philadelphia: SIAM, 2004. [9] D. Henrion and J. Lasserre, “Detecting global optimality and extracting solutions in gloptipoly,” in Positive Polynomials in Control, ser. Lecture Notes in Control and Information Sciences, D. Henrion and A. Garulli, Eds. London: Springer-Verlag, 2005, no. 312.
Analysis and synthesis of nonlinear systems with uncertain initial conditions G. Chesi (Senior Member) and Y.S. Hung (Senior Member)
Abstract—This paper considers, for polynomial and nonpolynomial systems, the problems of computing extremal values of the trajectories over a given set of initial conditions as well as finding output controllers minimizing these extremal values under time-domain constraints. It is shown that upper bounds of the sought extremal values as well as candidates of the sought controllers can be computed by solving a one-parameter sequence of BMI optimizations obtained through the Square Matricial Representation (SMR) of polynomials. Moreover, a necessary and sufficient condition is proposed to establish the tightness of the found upper bound in spite of the conservatism introduced by the non-convexity of BMI optimizations and the chosen degree of the Lyapunov function and relaxing polynomials. It is worthwhile to note that other approaches such as state augmentation and approximation techniques used to recast non-polynomial into polynomial systems can easily yield conservative results.
systems whose state updating law depends polynomially on the state and affinely on the input, and then it is extended to deal with non-polynomial dependence on the state by using truncated Taylor expansions and taking into account the worstcase remainders. It is shown that upper bounds of the sought extremal values as well as candidates of the sought controllers can be computed by solving a one-parameter sequence of bilinear matrix inequality (BMI) optimizations by using Lyapunov functions (LFs) and polynomial relaxations based on the Square Matricial Representation (SMR). In order to deal with the conservatism introduced by the non-convexity of the problem as well as the chosen degree of the LF, a necessary and sufficient condition is proposed for the analysis part to establish the tightness of the found upper bound.
Index Terms—Nonlinear system, Uncertain initial condition, SMR, BMI, tightness.
I. I NTRODUCTION Stability is not the only important issue in the analysis and synthesis of closed-loop systems, in fact it is often mandatory also to deal with time-domain constraints on the system signals. For linear systems, the analysis problem can be solved by exploiting closed-form solutions for the trajectory, while the synthesis problem with time-domain constraints can be addressed via frequency-domain constraints on weighted transfer functions or via model predictive control (MPC). For nonlinear systems the problem is more involved. First, there do not exist closed-form solutions for the trajectory except few special cases. Second, the synthesis must deal with the nonlinearities of the state updating law and measurable output, which affect convergence and efficiency of strategies such as nonlinear MPC [1], [2]. And, clearly, the problem is even more difficult if the initial condition is unknown and if the nonlinearities are non-polynomial. In this respect, it should be noted that approaches such as state augmentation and approximation techniques for recasting the non-polynomial system into a polynomial system can easily yield conservative results. In this paper a new approach to deal with time-domain constraints in the analysis and synthesis of nonlinear systems is proposed. In particular, the proposed framework considers the computation of extremal values of the trajectories over a given set of initial conditions and the computation of output controllers minimizing these extremal values under time-domain constraints. The approach is first described for G. Chesi and Y.S. Hung are with the Department of Electrical and Electronic Engineering, University of Hong Kong. Email: {chesi,yshung}@eee.hku.hk
II. P RELIMINARIES The notation is as follows: N, R: natural and real numbers sets; C n (X ): set of functions whose first n derivatives are continuous over the set X ; ∂X : boundary of set X ; 0n : origin of Rn ; I: identity matrix (of size specified by the context); Φ′ : transpose of matrix Φ; Φ > 0 (Φ ≥ 0): symmetric positive definite (semidefinite) matrix Φ; φ > 0 (φ ≥ 0): positive (non-negative) entries vector φ; φ|max : max{φ1 , . . . , φn } for φ ∈ Rn . Consider the class of continuous-time systems ˙ = f (x(t)) + ϕ(x(t)) + g(x(t))u(t) x(t) y(t) = h(x(t)) (1) x(0) = xinit
where x(t) ∈ Rn is the state, u(t) ∈ Rnu is the input, y(t) ∈ Rny is the output, xinit is the initial condition, f (x(t)), g(x(t)) and h(x(t)) are polynomial functions of x(t), and ϕ(x(t)) is a non-polynomial function of x(t). The set of initial conditions of interest is defined as A = {x ∈ Rn : ai (x) ≥ 0 ∀i = 1, . . . , na }
(2)
where a1 (x), . . . , ana (x) are polynomials. We assume that A is compact and define the set {k(y) : Rny → Rnu such that, for u(t) = k(y(t)), limt→+∞ x(t) = 0n , and w(t) ≤ w0 for all t ∈ [0, +∞) and for all xinit ∈ A} (3) where k is a polynomial function, w(t) ∈ Rnw is a selectable design signal to constrain the controller, and w0 ∈ Rnw is a given vector. Let us define the cost achieved by u = k(y) as K
=
σ(k(y)) =
sup xinit ∈A,t∈[0,+∞)
z(t)|max
(4)
2
where z(t) ∈ Rnz is another selectable signal for optimization. It is assumed that: 1) the signals z(t) and w(t) are expressed by [qz (x(t))′ , lz (u(t))′ ]′ [qw (x(t))′ , lw (u(t))′ ]′
z(t) = w(t) =
(5)
where qz , qw are polynomial functions and lz , lw are linear functions; 2) the origin is the equilibrium point of interest and the output vanishes in the origin. The problems we address are: 1) Problem P1 (analysis): to compute the maximum of z(t)|max for the autonomous system: σ = σ(0nu );
(6)
III. A NALYSIS
We consider first the case ϕ(x) = 0n . In order to simplify the description we assume that: - (A1) the linearized autonomous system is asymptotically stable, that is A is Hurwitz where d (f (x) + ϕ(x)) | dg(x) | dh(x) A|B|C= (10) dx x=0n
The basic idea is to look for LFs whose unitary sublevel sets are invariant sets and contain A. Then, we also require that these sublevel sets are contained in the region where the supremum of qz (x)|max is bounded by a certain quantity ζ ∈ R. If we can find this LF, it is guaranteed that ζ ≥ σ. More formally, let us denote with v : Rn → R the LF candidate, and define with V(c) its sublevel set
2) Problem P2 (synthesis): to compute the controller that minimizes the maximum of z(t)|max : k ∗ (y) = σ∗
=
arg min σ(k(y))
(7)
σ(k ∗ (y)).
(8)
k(y)∈K
In the sequel the dependence on the time t will be generally omitted for ease of notation. Let us observe that, depending on the choice of z, one can select several costs. such as: z = [h(x)′ B1′ B1 h(x), u′ B2′ , −u′ B2′ , B3 x]′ ⇒ z|max = max{kB1 yk22 , kB2 uk∞ , b3,1 x, . . . , b3,m x} where B1 , B2 , B3 are weighting matrices of suitable dimensions, and b3,1 , . . . , b3,m are the rows of B5 . Moreover, one can analogously define several constraints by similarly defining w and w0 . Before proceeding let us introduce the complete square matricial representation (SMR) [3], [4]. Let x{m} ∈ Rτ1 (m) contain all monomials of degree not greater than m in x, and let p1 (x) be a polynomial of degree 2m. The complete SMR of p1 (x) with respect to x{m} is ′
p1 (x) = x{m} P1 (α1 )x{m}
P1 (α1 ) = P1 + N1 (α1 ) (9) where P1 is any symmetric matrix such that p1 (x) = ′ x{m} P1 x{m} , α1 ∈ Rν1 (m) is a vector of free parameters and N1 (α1 ) is a linear parametrization of the set N1 (m) = ′ {N = N ′ : x{m} N x{m} = 0 ∀x}. As explained in [3], [4] 1 τ1 (m) = (n+m)! n!m! and ν1 (m) = 2 τ1 (m)(τ1 (m) + 1) − τ1 (2m). In the case of polynomials having special structures, more compact representations can be derived. Indeed, polynomials without constant and linear terms can be represented with respect to the vector x[m] ∈ Rτ2 (m) containing all monomials of degree less than or equal to m in x but the constant term. For these polynomials the SMR is analogously defined by substituting N1 (m) with the set N2 (m) = {N = ′ N ′ : x[m] N x[m] = 0 ∀x}. We have τ2 (m) = τ1 (m) − 1 and the dimension of N2 (m) is ν2 (m) = ν1 (m) − τ1 (m) + n + 1. In the sequel we will refer to P1 and P1 (α) as SMR matrices of p1 (x). Unless explicitly specified otherwise, it will be assumed that these matrices are defined with respect to x{m} . with
PROBLEM
V(c) = {x ∈ Rn : v(x) ≤ c} Q(ζ)
n
= {x ∈ R : qz (x)|max ≤ ζ} .
(11) (12)
Suppose there exists v(x) and ζ ∈ R such that: - (C1) v(x) is radially unbounded, v(0) = 0 and v(x) > 0 ∀x ∈ Rn \ {0n }; - (C2) ∇v(x)f (x) < 0 ∀x ∈ V(1) \ {0n }; - (C3) A ⊆ V(1); - (C4) Q(ζ) ⊇ V(1). Then, ζ ≥ σ. Now, let us select v(x) as a polynomial of degree 2δv and introduce t1 (x) = ∇v(x)f (x) P + s1 (x)(1 − v(x)) na t2 (x) = v(x) − 1 + i=1 s2,i (x)ai (x) t3,i (x) = s3,i (x)(qz,i (x) − ζ) + 1 − v(x), i = 1, . . . , nqz (13) where s∗ (x) are auxiliary polynomials known as Hilbert’s polynomials (see for example [5]). Theorem 1: Let ζ > 0 be a given real scalar, and let 2δv be the chosen degree of v(x). Let V, S∗ , T1 (V, S1 , α1 ), T2 (V, S2,∗ , α2 ), T3,∗ (V, S3,∗ , α3,∗ ) be SMR matrices of the polynomials v(x), s∗ (x), t1 (x), t2 (x), t3,∗ (x) respectively, with the matrices V, S1 , T1 (V, S1 , α1 ) defined with respect to extended vectors without constant term. Define λ∗ = sup λ s.t. (15) (14) λ∈R,V >0,S∗ >0,α∗
T1 (V, S1 , α1 ) + λI T2 (V, S2,∗ , α2 ) + λI T3,i (V, S3,i , α3,i ) + λI
< 0 ≤ 0 ≤ 0 ∀i = 1, . . . , nqz
(15)
If λ∗ ≥ 0, then ζ ≥ σ. Proof Suppose that (15) is satisfied with λ ≥ 0. Let x[δt,1 ] be the extended vector in the representation of t1 (x). From the first inequality we have ′
0 > x[δt,1 ] (T1 (V, S1 , α1 ) + λI) x[δt,1 ] = t1 (x) + λkx[δt,1 ] k22 that is t1 (x) < 0 for all x ∈ Rn \ {0n }. From the other inequalities we similarly obtain that v(x) > 0 and s1 (x) > 0 for all x ∈ Rn \ {0n }, and that t2 (x) ≤ 0, t3,∗ (x) ≤ 0, s2,∗ (x) > 0 and s3,∗ (x) > 0 for all x. Now, from t1 (x) < 0 and s1 (x) > 0 it follows that ∇v(x)f (x) < 0 for all x ∈ V(1) \ {0n }. Analogously we prove that x ∈ V(1) for all
3
x ∈ A, and x ∈ Q(ζ) for all x ∈ V(1). Moreover, v(x) is radially unbounded because V > 0, hence C1–C4 hold. Theorem 1 provides a sufficient condition to establish if ζ is an upper bound of σ by solving the optimization (14) which is a nonconvex optimization because the first inequality in (15) is a BMI since T1 (V, S1 , α1 ) is a bilinear function of V and S1 . BMI optimizations can be locally solved through dedicated software, alternatively they can be approached via a sequence of convex LMI optimizations by alternatively fixing one variable and optimizing with respect to the other (we refer to this solution as V -S iterations) as done in Section V. In order to find σ, we can simply adopt a bisection algorithm minimizing ζ subject to the condition of Theorem 1. Now, we describe how Theorem 1 can be extended to deal with the case ϕ(x) 6= 0n . First of all, it is worthwhile to observe that one may attempt to deal with this case by performing state augmentations to system into polynomial, or by substituting the non-polynomial terms with their truncated Taylor expansions. However, it is known that these attempts can easily lead to conservative and disastrous results. Our idea is to introduce truncated Taylor expansions taking into account the worst-case remainder. We suppose that assumptions A1 previously introduced and A2 below hold: - (A2) the function ϕ(x) has the form ϕ(x) =
r X
pi (x)ψi (xki )
(16)
i=1
where pi (x) are polynomials, k1 , . . . , kr are integers in [1, n], and ψi : C di ([ci,min , ci,max ]) → R are nonpolynomial functions for integers di ≥ 1 and scalars ci,min < 0, ci,max > 0. Let us write ψi (xki ) via the Taylor expansion centered in xki = 0 up to the βi -th power for a given integer βi in [1, di − 1] and express the remainder in the Lagrange form: ψi (xki ) = µi (xki ) + ̺i (xki )θi (bi )
(17)
where µi (xki ) is a polynomial of degree βi , ̺i (xki ) is a monomial of degree βi + 1, θi (bi ) is the βi + 1-th derivative of ψi (xki ) evaluated for xki = bi , and bi ∈ [0, xki ] if xki ≥ 0 or bi ∈ [xki , 0] otherwise. Let us introduce the polynomials Pr t1,m (x) = ∇v(x) (f (x) + i=1 piP (x)µi (xki )) r +s1,m (x)(1 − v(x)) + i=1 (θi,mi mi +(−1) s1,m,i (x)) ∇v(x)pi (x)̺i (xki ), (18) m ∈ {0, 1}r t4,i (x) = s4,i (x)(xki − ci,max )(xki − ci,min ) +1 − v(x), i = 1, . . . , r where s1,m (x), s1,m,i (x), s4,i (x) are polynomials, and supbi ∈[ci,min ,ci,max ] θi (bi ) if mi = 0 θi,mi = inf bi ∈[ci,min ,ci,max ] θi (bi ) if mi = 1
(19)
Theorem 2: Let ζ > 0 be a given real scalar, and let 2δv be the chosen degree of v(x). Let S∗ , T1,m (V, S1,∗ , α1,m ), T4,i (V, S4,i , α4,i ) be SMR matrices of the polynomials s∗ (x), t1,m (x), t4,i (x) respectively. Define λ∗ =
sup λ∈R,V >0,S∗ >0,α∗
λ s.t. (21)
(20)
ineq. (15) without the first inequality T1,m (V, S1,∗ , α1,m ) + λI < 0 ∀m ∈ {0, 1}r T4,i (V, S4,i , α4,i ) + λI ≤ 0 ∀i = 1, . . . , r
(21) If λ∗ ≥ 0, then ζ ≥ σ. ¯ be any Proof Suppose that (21) is satisfied with λ ≥ 0. Let x point in V(1) and define 0 if ∇v(¯ x)pi (¯ x)̺i (¯ xki ) ≥ 0 m ¯ = [m ¯ 1, . . . , m ¯ r ]′ , m ¯i = 1 otherwise From t4,i (¯ x) ≤ 0 it follows that x ¯ki ∈ [ci,max , ci,max ]. Moreover, since s1,m (¯ x ) > 0 and s1,m,i x) > 0, from ¯ ¯ (¯ t1,m (¯ x ) < 0 we have ¯ Pr 0 > ∇v(¯ x) + i=1 pi (¯ x)µi (¯ xki )) + s1,m x)(1 − v(¯ x)) ¯ (¯ Pxr) (f (¯ m ¯i + i=1 (θi,m (−1) s (x)) ∇v(¯ x )p (¯ x )̺ (¯ x ¯i + 1, m,i ¯ i i k i) Pr > ∇v(¯ x ) (f (¯ x ) + p (¯ x )µ (¯ x )) i i k i i=1 Pr + i=1 θi,m x)pi (¯ x)̺i (¯ xki ) ¯ i ∇v(¯ = ∇v(¯ x) (f (¯ x) + ϕ(¯ x)) that is v(¯ ˙ x) < 0. Following the remaining proof of Theorem 1, we conclude that ζ ≥ σ. Theorem 2 provides a sufficient condition for computing an upper bound of σ through BMI optimizations in spite of the presence of non-polynomial terms in (1). This is made possible by taking into account the worst-case remainder of the truncated Taylor expansion in the time-derivative of v(x). In particular, the unitary sublevel set V(1) is constrained within the set C = {x ∈ Rn : xki ∈ [ci,min , ci,max ], i = 1, . . . , r}
(22)
where ci,min , ci,max delimit the region of validity of the worst-case remainder computed through the bounds θi,mi . If no information are available a priori about the extension of V(1) and/or the computation of θi,mi is difficult, conservative estimates of ci,min , ci,max , θi,mi can be used. In fact, let us observe that the effect of these conservative quantities, represented by term ωi,mi (x) = θi,mi ∇v(x)pi (x)̺i (xki )
(23)
in the polynomials t1,m (x), can be compensated by increasing the degree βi of the truncated Taylor expansion which makes this term convergent to zero. IV. S YNTHESIS
AND TIGHTNESS PROBLEMS
Let us consider first problem P2. In order to simplify the description we assume that: - (A3) the linearized system can be asymptotically stabilized through a static output feedback, i.e. there exists K ∈ Rni ×nu such that A + BKC is Hurwitz where A, B, C are as in (10). Let us express the controller k(y) in the class K in (3) as k(y) = ξ ′ y [δk ]
(24)
where δk is the degree of the controller, and ξ ∈ Rτ2 (δk ) is the coefficient vector to be determined. Since the state updating law depends linearly on the input, one can exploit the condition derived in Theorem 2 by letting the vector ξ vary together with
4
the other variables of the optimization (20). In particular this can be done by adding t1,u (x) to t1,m (x) where t1,u (x) = ∇v(x)g(x)ξ ′ y [δk ] .
(25)
Then, in order to take into account the presence of the input in the cost signal z, we introduce for i = 1, . . . , nlz t5,i (x) = lz,i (ξ ′ (h(x))[δk ] ) − ζ + s4,i (x)(1 − v(x)).
(26)
In order to take into account the time-domain constraint on the signal w we also define the polynomials tj,∗ and sj,∗ for j = 6, 7 analogous to those for j = 3, 5 for the cost signal z by replacing qz,i , lz,i , ζ with qw,i , lw,i , w0,i respectively. Theorem 3: Let ζ > 0 be a given real scalar, and let 2δv be the chosen degree of v(x). Let T1,u (V, ξ), S∗ , T5,∗ (V, ξ, S5,∗ , α5,∗ ), T6,∗ (V, S6,∗ , α6,∗ ), T7,∗ (V, ξ, S7,∗ , α7,∗ ) be SMR matrices of the polynomials t1,u (x), s∗ , t5,∗ (x), t6,∗ (x), t7,∗ (x) respectively, with T1,u (V, ξ) defined with respect to the extended vector used for defining T1 (V, S1 , α1 ). Define λ s.t. (28) (27) λ∗ = sup λ∈R,V >0,ξ,S∗ >0,α∗
ineq. (21) appending T1,u (V, ξ) T5,i (V, ξ, S5,i , α5,i ) + λI ≤ T6,i (V, S6,i , α6,i ) + λI ≤ T7,i (V, ξ, S7,i , α7,i ) + λI ≤
to 0 0 0
T1,m (V, S1,∗ , α1,m ) ∀i = 1, . . . , nlz ∀i = 1, . . . , nqw ∀i = 1, . . . , nlw (28)
If λ∗ ≥ 0, then ζ ≥ σ ∗ . Proof Analogous to the proof of Theorem 2 by observing that (28) implies t1,m (x) + t1,u (x) < 0. Hence, we have ∇v(x)(f (x) + g(x)ξ ′ (h(x))[δk ] < 0 for all x ∈ V(1) \ {0n }. Moreover, from (28) we have lz (ξ ′ (h(x))[δk ] ) max ≤ ζ and [qw (x)′ , lw (ξ ′ (h(x))[δk ] )′ ]′ ≤ w0 for all x ∈ V(1). Therefore, V(1) is an invariant set for the controlled system where z|max ≤ ζ and w ≤ w0 . Theorem 3 allows one to compute an upper bound of σ ∗ and a controller guaranteeing this upper bound by solving the optimization (27). In the sequel we will indicate with ζ2δv ,δk the best upper bound of σ ∗ found with a LF of degree 2δv and a controller of degree δk . Now we consider the problem of establishing if the best upper bound found for σ by using Theorem 2 with a LF of degree 2δv is tight or not. Let us indicate with ζ2δv this upper bound, and let us observe that ζ2δv can be conservative for several reasons: - the degree of the LF or auxiliary polynomials s∗ (x) is too low; - the degree of the Taylor expansion for ϕ(x) is too low; - BMI optimizations are nonconvex. The following result provides a necessary and sufficient condition to establish if ζ2δv is tight. Theorem 4: Let V2δv be the optimal value of V corresponding to the found ζ2δv , and let v2δv (x) and V2δv (1) be the LF and its unitary sublevel set corresponding to V2δv . Then, ζ2δv = σ ⇐⇒ ∃xR,init ∈ T , t¯ ∈ [0, ∞) : xR (t¯) ∈ A (29) where T = ∂V2δv (1) ∩ ∂Q(ζ2δv )
(30)
and xR (t) is the solution of the system x˙ R (t) = −f (xR (t)) − ϕ(xR (t)) xR (0) = xR,init
(31)
Proof “⇐” Suppose that ∃xR,init ∈ T and t¯ ∈ [0, ∞) such that xR (t¯) ∈ A. Then, since the system (31) evolves reversely with respect to the system (1), it follows that by initializing (1) with xinit = xR (t¯) we have that x(t¯) = xR,init . Since xinit ∈ A, it follows from the definition of σ in (6) that z(t¯)|max ≤ σ. On the other hand, z(t¯)|max = qz (x(t¯))|max = qz (xR,init )|max = ζ2δv because xR,init ∈ ∂Q(ζ2δv ), and hence ζ2δv ≤ σ. But ζ2δv ≥ σ. Therefore, ζ2δv = σ. “⇒” Suppose that ζ2δv = σ. From the definition of σ in (6) and since A is compact, it follows that ∃xinit ∈ A and t¯ ∈ [0, ∞) such that z(t¯)|max = σ. Observe that x(t¯) ∈ ∂Q(ζ2δv ) because ζ2δv = σ. Suppose now for contradiction that x(t¯) does not belong to ∂V2δv (1). Then, this implies either that v2δv (x(t¯)) > 1 or that v2δv (x(t¯)) < 1. But the former implies that x(t¯) lies outside an invariant set containing the initial condition xinit ∈ A, and the second implies that V2δv (1) is not included in Q(ζ2δv ) as ensured by the third inequality in (15). Hence, both hypotheses are impossible and hence x(t¯) ∈ ∂V2δv (1). Therefore, x(t¯) ∈ T . Then, by initializing the reverse system (31) with xR,init = x(t¯) we obtain that xR (t¯) = xinit ∈ A which concludes the proof. The condition of Theorem 4 can be checked in two steps. First, computing T which is the intersection of the boundaries of V2δv (1) and Q(ζ2δv ). Second, computing the trajectories of the reverse system (31) initialized with the points in T and checking if at least one of these trajectories intersects A. Observe that T is composed by a finite number of points, typically one, being the intersection of two tangent surfaces. The set T can be found by solving v2δv (x) − 1 = 0 (32) qz (x)|max − ζ2δv = 0 However, solving the system (32) can be a difficult task because it is nonlinear system. Theorem 5: Define Mi = T3,i (V2δv , S3,i,2δv , α3,i,2δv ), i = 1, . . . , nqz
(33)
where S3,i,2δv and α3,i,2δv are the found optimal values of S3,i and α3,i . Then, ( ) nqz [ T = x∈ Mi : x satisfies (32) (34) Mi
=
i=1 n o x ∈ Rn : x{δt,3 } ∈ ker(Mi ) .
(35)
Proof Consider x ∈ T . Since x satisfies (32) there exists j such that t3,j (x) = 0. Now, from (21) it follows that }′ Mj ≤ 0. Hence: 0 = t3,j (x) = x{δt,3 Mj x{δt,3 } =
2 ′ ˜ ′M ˜ j x{δt,3 } = − ˜ j x{δt,3 } ˜ j is x{δt,3 } −M
M
where M j 2
2
˜ {δt,3 } any Cholesky factor of −Mj . Clearly, M
= 0 if jx 2
5
The trajectory xR (t) of (31) with initial condition xR,init = x¯ intersects A in xR (t¯) as shown in Figure 1b. Therefore, the upper bound is tight. 2δv \ δk 2 4 6
1 2 +∞ +∞ +∞ 2.727 +∞ 2.463 TABLE I
3 +∞ 2.687 2.191
E XAMPLE 1: UPPER BOUNDS ζ2δv ,δk
COMPUTED THROUGH SIMPLE V ITERATIONS .
4
-S
3
3 2 2 1
x2
1
x2
˜ j ). Since ker(M ˜ j ) = ker(Mj ), we and only if x{δt,3 } ∈ ker(M conclude the proof. Theorem 5 provides an alternative way of computing the set T which consists of finding the vectors x ∈ Rn satisfying x{δt,3 } ∈ ker(Mi ) for i = 1, . . . , nqz . This can be trivially done if the dimension of ker(Mi ) is 1 corresponding to a unique element in the set Mi as shown in Example 1 in Section V. In other cases one can use, for example, the approaches described in [6]–[9]. Once the sets Mi have been found, one can simply the set T by verifying if the Snobtain qz vectors contained in i=1 Mi satisfy the equations in (32) via trivial substitution. Before introducing some examples in the next section, let us observe that the BMI optimization can be initialized with any matrix K satisfying assumption A3 and any LF proving the asymptotical stability of the linearized system controlled by such a matrix.
0
0
−1 −1
xR (t¯)
−2
V. E XAMPLES
−2 −3
A. Example 1
−4 −6
Let us consider the non-polynomial system x˙ 1 = x2 + 0.2x21 x2 + 0.2 (1 − ex1 ) x˙ 2 = −1.5x1 − 2x2 − 1.1x22 + u y = x1 xinit ∈ A = {x : kxk22 ≤ 4} min
sup
stabilizing u(t)=k(y(t)), |u(t)| −5.977 and ξ1 < −0.072 in the optimization, where ξ1 is the coefficient of the linear term of the controller. Table IIb shows the obtained upper bounds, and Figure 2d the corresponding performance. Observe that the convergence is much faster, clearly at the expense of a larger amplitude of the control signal. 2δv \ δk 2 4
1 2 0.0783 0.0783 0.0772 0.0772 (a)
3 0.0503 0.0494
2δv \ δk 2 4
1 2 3 0.1136 0.1136 0.0582 0.1131 0.1131 0.0581 (b) TABLE II E XAMPLE 2: UPPER BOUNDS ζ2δv ,δk OBTAINED WITH UNCONSTRAINED ( A ) AND CONSTRAINED ( B ) CONTROLLER LINEAR TERM .
0.6 0.4
θ
x3 [A]
0.2 0
−0.2
+
Ra
La
−0.4 −0.6
u
−0.8 −0.2 −0.1 0 0.1 0.2
2
−1
x2 [rad·s
1.5
0
−0.5
−1
−1.5
−2
(b) 0.06
0.04
0.04
u [v·rad−1 ]
0.06
u [v·rad−1 ]
0.5
] x1 [rad]
(a)
0.02
0.02
0
0
−0.02
−0.02
−0.04
−0.04
−0.06
1
0
50
t [s] (c)
100
150
−0.06
0
5
10
15
t [s]
20
25
30
(d)
Fig. 2. Example 2. (a) Inverted pendulum controlled with a DC motor. (b) Trajectories starting in A with u = k4,3 (y). (c) Control input with u = k4,3 (y). (d) Control input achieved by constraining the linear term of the controller.
VI. C ONCLUSION The problem of computing extremal values of the trajectories over a given set of initial conditions and the problem of computing output controllers minimizing these extremal values under time-domain constraints have been addressed for polynomial and non-polynomial systems. It has been shown that an upper bound of the sought extremal values as well as candidates of the sought controllers can be found by solving a one-parameter sequence of BMI optimizations, which can be approached through either recently developed software or simple iterative convex LMI optimizations. Since the found upper bound may be conservative due to the nonconvexity of BMI optimizations and the chosen degree of the Lyapunov function, a necessary and sufficient condition has been proposed for the analysis part to establish the tightness of this upper bound in spite of all these sources of conservatism. R EFERENCES [1] F. Allgower and A. Zheng, Nonlinear Model Predictive Control: Assessment and Future Directions for Research, ser. Progress in Systems and Control Series. Basel: Birkhauser Verlag, 2000. [2] B. Kouvaritakis and M. Cannon, Non-Linear Predictive Control: Theory and Practice, ser. IEE Control Series. London: IEE, 2001. [3] G. Chesi, A. Tesi, A. Vicino, and R. Genesio, “On convexification of some minimum distance problems,” in 5th European Control Conf., Karlsruhe, Germany, 1999. [4] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Solving quadratic distance problems: an LMI-based approach,” IEEE Trans. on Automatic Control, vol. 48, no. 2, pp. 200–212, 2003. [5] J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry. Springer, 1998. [6] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “An LMI-based approach for characterizing the solution set of polynomial systems,” in 39th IEEE Conf. on Decision and Control, Sydney, Australia, 2000, pp. 1501–1506. [7] ——, “Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach,” Int. Journal of Robust and Nonlinear Control, vol. 13, no. 13, pp. 1239–1257, 2003. [8] H. J. Stetter, Numerical Polynomial Algebra. Philadelphia: SIAM, 2004. [9] D. Henrion and J. Lasserre, “Detecting global optimality and extracting solutions in gloptipoly,” in Positive Polynomials in Control, ser. Lecture Notes in Control and Information Sciences, D. Henrion and A. Garulli, Eds. London: Springer-Verlag, 2005, no. 312.
