Robust stabilization via Hit-and-Run techniques

Robust stabilization via Hit-and-Run techniques B.T. Polyak and E.N.Gryazina

Abstract— In previous works the authors proposed to use Hit-and-Run versions of Markov-chain Monte-Carlo algorithms for various problems of control and optimization. In this paper we focus on robust stabilization applications of the method. The crucial notion for this algorithm is a Boundary Oracle (BO), and we start with constructing BO for robustness problems, including robust stability of polynomials and robust LMIs. Numerical results for various control applications are presented. In particular, we consider a problem arising in control of helicopters. Simulations confirm that the randomized approach can be an effective tool for solving robust stability and robust stabilization problems.

I. I NTRODUCTION Recent years exhibited the growing interest to randomized algorithms in control and optimization; see, e.g., [23]. There are numerous reasons for this interest, from philosophical to computational ones. The present paper continues this line of research for robust stability analysis and design. Our approach is based on Hit-and-Run (H&R) versions of the Monte-Carlo method. The H&R algorithm has been proposed by Turchin [24] and independently later by Smith [22]. It is aimed at approximately uniform generation of points in a body via random walks. Its properties have been studied in numerous works by Lov´asz and co-authors (see, e.g., the survey [5]); roughly speaking, it mixes fast [14]. H&R has been applied to various control and optimization problems in several publications, see [19], [18], [9], [12], [16]. In the present paper we focus on robust stabilization applications of the method. The typical problem we deal with is as follows. Given a linear system with uncertainty and some specifications and/or some performance index. We describe the set of robustly stabilizing controllers of fixed structure (say, PID controllers). By use of H&R technique we generate random points in this set, and for each point check whether all specifications are met. Among admissible points we choose one with the best value of the performance index. On this way we can solve approximately various hard design problems with non-analytical specifications (such as overshoot, settling time, decay rate etc.) The set of robustly stabilizing controllers can be nonconvex and not simplyconnected, nevertheless H&R algorithm allows to deal with some of these sets. In the simpler case of quadratical stabilization the set of controllers is described by LMIs and happens to be convex; the description of H&R technique for this case is one of the goals of the paper. Thus, if compared B.T. Polyak ([email protected]) and E.N. Gryazina ([email protected]) are with Institute for Control Sciences RAS, Moscow, Russia

with our previous publications in H&R to control [16][19] we extend the applications to robust versions of design problems. The paper is organized as follows. In Section 2 we remind the general scheme of H&R method and its implementation to control problems. Section 3 contains main results on robustness problems. Robust boundary oracle (RBO) is presented for various classes of uncertainty for polynomials and matrices. Results of numerical experiments with the proposed algorithm are described in Section 4. II. H IT AND RUN ALGORITHM We start with presenting the idea and results relating to H&R method in general setting. Suppose there is a bounded closed set K ⊂ R` and a point k 0 ∈ K. In every step we choose a random vector d uniformly distributed on the unit sphere in Rn . We call boundary oracle (BO) an algorithm which provides the intersection of the straight line k 0 + td, −∞ < t < +∞ with K, i.e., the set L = {t ∈ R : k 0 + td ∈ K}. In the simplest case, when K is convex, this set is the interval [t, t], where t = maxt>0 {t : k 0 + td ∈ K}, t = mint0

t(ω) = max{t : t g(jω) ∈ −S(ω)}. t 0 (or ∀t < 0). Finally, RBO is Lrob = {t ≤ t ≤ t}, where t = maxω t(ω) and t = minω t(ω). The technique for calculation of value sets is well developed; see, e.g., [2]. In particular, for Q = {||q||2 ≤ 1} the set S(ω) is an ellipse and Lrob can be obtained analytically. Note that this boundary oracle describes just the component of robust stability domain containing nominal polynomial p0 (s). Complete robust boundary oracle is hard to calculate. B. LMIs with norm uncertainty There are various formulations of LMI problems and assumptions on uncertainties available in the literature; e.g., [8], [4]. Here we concentrate mostly on the situation where uncertainty is bounded in the spectral norm. Consider A(k, ∆) = A0 (∆0 ) +

` X ki Ai (∆i ),

(3)

i=1

where Ai (∆) = Ai + ∆i , Ai , ∆i are real symmetric n × n matrices, . ∆ ∈ D = (∆0 , . . . , ∆` ) : ∆i ∈ Sn×n , k∆i k 6 εi ,

k · k is the spectral norm and εi > 0 are given numbers. We consider only symmetric perturbations ∆ in order to retain the LMI structure of the problem. Robustly feasible domain is defined by:  Krob = k ∈ R` : A(k, ∆) ≤ 0, ∀ ∆ ∈ D . (4) Consider the construction of the boundary oracle for robust stability domain Krob (4) for the family (3) with ` ≥ 2. We seek for the intersection points of a ray and the boundary of robust stability domain. Let k 0 ∈ Krob be robustly feasible point and d ∈ R` be a certain direction. Consider the straight line k 0 +td and compute t rob , trob , the minimal and maximal

values of t, which guarantee the negative definiteness of the matrix A(k 0 + td, ∆) for all ∆ ∈ D. We have

`

(ki0 + tdi )∆i .

The boundaries of the regions (intervals) of the robust stability are defined from the condition that the matrix F (t) + ∆(t) be singular for some ∆ ∈ D , i.e., the problem is to determine the radius of nonsingularity of the matrix F (t) + ∆(t). The theorem below is the main tool: Theorem 1 [17]: For a nonsingular matrix M ∈ Sn×n , its symmetric radius of nonsingularity . ρ(M ) = inf{k∆k : ∆ ∈ Sn×n , M + ∆ is singular}, is equal to ρ(M ) = 1/kM −1 k = min |λi (M )|. i

The critical value of ∆ is given by ∆ = −λeeT , where λ is the minimal (in absolute value) eigenvalue of M , and e is the associated eigenvector. By the theorem the matrix F (t) + ∆(t) remains nonsingular (hence,

negative definite) for all ∆ ∈ D satisfying

−1

(F (t)) < 1/k∆(t)k. Since the perturbations ∆i are independent, the estimate k∆(t)k 6 k∆0 k+

|ki0 +tdi | k∆i k = ε0 +

i=1

|ki0 +tdi |εi

i=1

is sharp. Hence, by considering the two scalar functions

!−1

` X

. 0

, ϕ(t) = A + (k + td )A (5) 0 i i i



i=1 1

. ε(t) =

ε0 +

` P

i=1

|ki0

,

|ki0 |εi < 1/k(A(k 0 , 0))−1 k holds.

C. LMIs with parametric uncertainty

i=1

` X

ki0 Ai and check if the inequality

The boundary oracle for LMI without uncertainty is described in [18],[17]. For given symmetric matrices F < 0 and G interval of negative definiteness of F +tG is (−t, t), where t = min λi , t = min µi , λi are positive real eigenvalues of matrix pencil F, −G, µi are positive real eigenvalues of matrix pencil F, G.

X . F (t) = A0 + (ki0 + tdi )Ai ,

` X

` P

i=1

where it is denoted

. ∆(t) = ∆0 +

` P

i=1

ε0 +

A(k 0 + td, ∆) = F (t) + ∆(t),

i=1 ` X

k 0 : A(k 0 , 0) = A0 +

(6)

+ tdi | εi

the interval (trob , trob ) of robust definiteness of the family A(k 0 +td, ∆) can be found numerically as {t : ϕ(t) 6 ε(t)}. Theorem 2 : Let A(k 0 , 0) < 0. For any d ∈ R` , the maximal and minimum values of t retaining the negative definiteness of the matrix A(k 0 + td, ∆) for all admissible perturbations ∆ are given by the two solutions of the equation ϕ(t) = ε(t) (5)-(6) over the segment [t, t], where [t, t] are the bounds of negative definiteness of the matrix A(k 0 + td, 0). Similarly, to solve a simpler problem of checking if k 0 ∈ rob K for some k 0 ∈ R` , it suffices to check ϕ(0) < ε(0). That is, we are to consider the unperturbed matrix at the point

Consider another class of uncertain LMIs ` N X X qi Bi , A(k, q) = A0 + ki Ai +

(7)

q ∈ Q,

i=1

i=1

where Ai , Bi are real symmetric n × n matrices, uncertainty 0 set Q = {q : |qi | ≤ αP i }. Consider Pthe straight P line k + td 0 then A(k, q) = A0 + ki Ai + t di Ai + qi Bi and it is sufficient to check feasibility of a certain number of LMIs corresponding to uncertain bounds. For every vertex q v of Q find BO Lv = {t : F + tG < 0}, (8) P 0 P v P where F = A0 + ki Ai + qi Bi , G = di Ai , i.e., the bounds of negative definiteness of one parameter matrix family. Robust boundary oracle is an intersection \ Lrob = Lv . v

D. LMIs with matrix variables

In the above considerations the variables in LMIs were vector ones: k ∈ R` . Below we analyse LMIs with matrix variables, which we denote P ∈ Sn×n . For simplicity we deal with Lyapunov-like inequalities only. Two classes of uncertainty are investigated: either uncertainty appears in system matrix A Krob = {P > 0 : X H + (A0 + qi Ai )P

(9) + P (A0 +

X

T

qi Ai ) ≤ 0,

q ∈ Q},

where Q = {q : ||q||1 ≤ γ} or Q = {q : |qi | ≤ αi }, or we have norm-bounded uncertainty: (10)

Krob = {P > 0 : T

H + M ∆N + N ∆M

T

T

+ AP + P A ≤ 0, ||∆|| ≤ 1},

where A is a nominal n × n system matrix without uncertainty, H < 0 is real symmetric n × n matrix, ∆ is r × m matrix of uncertainty bounded in Frobenius norm, M and N are matrices of the size n × r, m × n correspondingly. For the family (9) the approach is similar to the case of uncertain LMI (7). Indeed, take P0 ∈ Krob , generate matrix D = DT , ||D||F = 1, that is uniformly distributed on the

unit sphere in Frobenius norm. Then for P = P0 + tD we have X X F = H + (A0 + qiv Ai )P0 + P0 (A0 + qiv Ai )T , X X G = (A0 + qiv Ai )D + D(A0 + qiv Ai )T ,

where S(s) is sensitivity of the closed-loop plant we obtain

38.3286s + 41.6555 , J ∗ = 0.8781. s + 15.0023 Example 2. Consider the classical example for a helicopter control thatis described in many papers; see, e.g., [6],  [21], −0.0366 0.271 0.0188 −0.4555 v where q are vertices of Q, and robust boundary oracle is an   intersection of the sets (8) corresponding to vertex uncertain [10]. A =  0.0482 −1.01 0.0024 −4.0208 ,  0.1002 q1 −0.707 q2  samples. 0 0 1 0  Now we consider another class of uncertainties (10). The 0.4422 0.1761    analysis is based on the following result of Peterson. q3 −7.5922  , C = 0 1 0 0 , B= T   Theorem 3 [15]: Let G = G , M 6= 0, N 6= 0 be −5.52 4.49 matrices of the size n × n, n × r, m × n correspondingly. 0 0 Then inequality q ∈ Q = {q : |qi − qi0 | ≤ γi }, q 0 = [0.3681, 1.42, 3.5446]; γ = [0.05, 0.01, 0.04]. G + M ∆N + N T ∆T M T < 0 Output feedback stabilizing controllers for the nominal system are provided in [18]. The robustness was investigated hold for all matrices ∆ of size r × m, k∆k < 1 iff there in a probabilistic manner checking whether the selected exist ε > 0 such that controller stabilizes 1000 random points uniformly generated 1 in the box Q. G + εM M T + N T N < 0. ε Now we modify the example; our aim is to generate Various generalizations and applications of this result are state feedback robustly stabilizing controllers. Note that these addressed in [20]. For our case, for the family (10) taking controllers can be exploited as a starting point in order P = P0 + tD we have to initialize the algorithm for H2 (H2 /H∞ ) static output A(t, ∆) = H+M ∆N +N T ∆M +AP0 +P0 AT +t(AD+DAT ). feedback synthesis proposed in [1]. In this example to provide robust stability it is sufficient Denote Aˆ = H + AP0 + P0 AT , B = AD + DAT , then to check quadratic stabilizability of 8 vertex samples, i.e., A(t, ∆) < 0 for all k∆k 6 1 if Aˆ + tB < −εM M T − find such K that for Av = Av + B v K there exist P > 0: c 1 T N N for some ε > 0. Thus we can find the intervals in t (Avc )T P + P Avc < 0, index v = 1, . . . , 8 corresponds to ε the vertex of Q. Multiplying by Q = P −1 and denoting that preserve robust sign-definiteness of A(t, ∆). Y = KQ we have LMI in Q and Y . Thus: IV. N UMERICAL E XPERIMENTS 8 \ rob K = {Q > 0, Q(Av )T +Av Q+B v Y +Y T (B v )T < 0}. Example 1. This example is originally proposed in [7] (p. v 557). The problem is to find a controller robustly stabilizing the interval plant For generating quadratic robust stabilizing controllers the boundary oracle for the set (9) is exploited taking Q = Q0 + s + 1 + 6q1 G(s, q) = 3 , tD, Y = Y0 + tR and F = Q0 ATi + Ai Q0 + Bi Y0 + Y0T BiT , s + 8s2 + 22s + 20 + 6(q2 s2 + q3 s + q4 ) G = DATi + Ai D + Bi R + RT BiT , where matrices D and |qi | ≤ 1, i = 1, 2, 3, 4. R specify random direction in a corresponding space. We generate 1000 state feedback stabilizing controllers In [7] the third order controller was found via complicated K : 2 × 4 and check the one that maximazes the decay for technique based on Nevanlinna-Pick theorem. Later in [13] it the nominal system: was noted that the controller C(s) = 1 is robustly stabilizing   and the first order controller was chosen in order to guarantee −34.0614 −11.1304 3.6047 21.8239 ∗ K = robust stability for the expanded uncertainty intervals. Here −1.5488 0.0514 0.6915 1.3711 we strive to generate robustly stabilizing controllers for and the decay is α = 1.94. the original uncertainty intervals and to describe the robust stability domain in 3D parameter space of the first order V. C ONCLUSIONS 1 s+k2 controller C(s, k) = ks+k . 3 In this paper we describe the opportunity to apply Since the numerator h(s, q) contains only one uncertainly H&R method for robust stabilization problems. Namely, we q1 there are 8 Kharitonov plants to consider. discuss the construction of the robust boundary oracle for We generate 1000 robustly stabilizing controllers via robust stability of polynomials and matrices with structured H&R and choose the one that optimized H∞ criteria for the and parametric uncertainty. Numerical results confirm that nominal plant G(s, 0). Namely, for the problem the algorithms can be easily implemented and give an s + 1 effective tool for solving problems of robustness analysis J ∗ = min ||W S||∞ , W (s) = , and synthesis. C(s,k) 10s + 1 C ∗ (s) =

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