On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints

We deal with a problem of target control synthesis for dynamical bilinear discrete-time systems under uncertainties (which describe disturbances, perturbations or unmodelled dynamics) and state constraints. Namely we consider systems with controls that appear not only additively in the right hand sides of the system equations but also in the coefficients of the system. We assume that there are uncertainties of a set-membership kind when we know only the bounding sets of the unknown terms. We presume that we have uncertain terms of two kinds, namely, a parallelotope-bounded additive uncertain term and interval-bounded uncertainties in the coefficients. Moreover the systems are considered under constraints on the state ("under viability constraints"). We continue to develop the method of control synthesis using polyhedral (parallelotope-valued) solvability tubes. The technique for calculation of the mentioned polyhedral tubes by the recurrent relations is presented. Control strategies, which can be constructed on the base of the polyhedral solvability tubes, are proposed. Illustrative examples are considered.

1. Introduction. We deal with a problem of target control synthesis for dynamical discrete-time systems with a bilinear structure under uncertainties and state constraints (SC). We use a deterministic model of uncertainty with set-membership description of the uncertain items (which describe disturbances, perturbations or unmodelled dynamics) when they are taken to be unknown but bounded with given bounds and we have no statistical information whatever [23,24,25]. The setmembership model of uncertainty is appropriate for many applied problems. The problems in dynamics and control with such model of uncertainty are described through set-valued functions known as trajectory tubes and are to be treated through set-valued analysis. The problem of construction of the mentioned trajectory tubes that describe the dynamics of reachable sets, solvability sets, informational domains [24,25] may be called one of the fundamental problems of the mathematical control theory. Note that there are many different problem statements in the theory of dynamical systems concerning investigations of an influence of perturbations, and there were obtained deep and surprising results, in particular, in KAM theory (see, for example, [28,12]).

ELENA K. KOSTOUSOVA
There are known approaches for solving the problems of terminal target feedback control for differential systems based on construction of so called solvability tubes (in other terms, maximal stable bridges, Krasovskii's bridges, backward reachable tubes) and there are known descriptions of trajectory tubes for differential systems through multivalued integrals and through evolution equations of several types (in particular, through funnel equations, through the evolution of support functions or through level sets of appropriate Hamilton-Jacobi-Bellman equations) [24,25] (here and below, we note, as examples, only some references from numerous publications; see also references therein). SC further complicate the problem [13,25] and lead to so called viability tubes. A close target control problem, which is connected with constructing an appropriate set of initial states (so called capture basin), was considered in viability theory [3,6]. Since practical construction of the trajectory tubes (in particular, the solvability tubes, reachable tubes, viable trajectory tubes) as well as of capture basins and other so called "kernels" from viability theory may be cumbersome, various numerical methods have been developed for approximation of the set-valued solutions and for numerical solution of the above mentioned evolution equations and multivalued integrals including methods based on approximations of sets either by arbitrary polytopes with a large number of vertices or by unions of a large number of points [4,5,6,7,29,34,36]. Such methods are devised to obtain approximations as accurate as possible. But they can require much calculations, especially for large dimensional systems. Other techniques are based on estimates of sets by domains of some fixed shape such as ellipsoids and parallelepipeds, including boxes aligned with coordinate axes as in interval analysis [7,8,10,14,15,17,18,19,20,21,24,25,26,35]. The main advantage of the last techniques is that they enable to obtain approximate/particular solutions using relatively simple tools. More accurate approximations may be obtained by using the whole families (varieties) of such simple estimates (as suggested by A.B. Kurzhanski) [8,15,17,18,19,20,21,24,25,26,35]. The methods of interval analysis which use subpavings (unions of non-overlapping boxes) [14] serve the same purpose, but may require much computations and memory for large dimensional systems.
As for solving the feedback target control problems for differential and discretetime systems, constructive computation schemes using ellipsoidal techniques were proposed [8,24,25,26,35] and then expanded to the polyhedral techniques that use polyhedral (parallelotope-valued) solvability tubes [15,17,19,20,21,22] (this had required the development of a quite different techniques). In [30,31], the polyhedral technique from [19,Sec. 3] was applied for constructing a real-time control of aircraft take-off in windshear using a sequential linearization of highly nonlinear point mass aircraft model equations around appropriate ascending paths. Also in [30], the example with the linear differential game known as "Boy and Crocodile" is considered, where the mentioned parallelotope technique is compared with a grid method which computes the maximal solvability tube almost exactly if the grid is sufficiently fine.
An important for investigations class of dynamical systems is produced by systems with a bilinear structure, where matrices of coefficients may contain not only uncertainties but also controls. Such models may be useful in many applied areas from physics and engineering to biology, ecology, socioeconomics [32,33], [22,Ref. 23]. Different techniques to solving some control and stabilization problems for bilinear systems, including ellipsoidal and interval ones, can be found, for example in [1,2,5,11,25,32,33] and in references therein.
Here we develop the polyhedral method of control synthesis for bilinear discretetime systems on a given fixed time interval. We consider systems with controls that appear not only additively in the right hand sides of the system equations but also in the coefficients of the system. We assume that there are uncertainties of two kinds, namely, the parallelotope-bounded additive uncertain term and interval-bounded uncertainties in the coefficients. Note that the systems with controls (or/and uncertainties) in the system matrix are of bilinear type and have properties of nonlinear systems (in particular reachable sets and solvability sets of such systems can be nonconvex). Moreover we consider the systems under constraints on the state ("under viability constraints"). To solve the problem of target control synthesis under these conditions we expand the techniques from [17,19,20,21,22] (some additional comments about these works will be given below). The technique for calculation of the polyhedral tubes by the recurrent relations is presented. Corresponding control strategies are described. Results of computer simulations in the illustrative examples are presented confirming the operability of the proposed method. We use the following notation: R n is the n-dimensional vector space; (x, y) = x y is the scalar product for x, y ∈ R n ; is the transposition symbol; x 2 = (x x) 1/2 , x ∞ = max 1≤i≤n |x i | are the vector norms for x = (x 1 , . . . , x n ) ∈ R n ; e i = (0, . . . , 0, 1, 0, . . . , 0) is the unit vector oriented along the axis x i (the unit stands at position i); e = (1, 1, . . . , 1) ; R n×m is the space of real n×m-matrices A = {a j i } = {a j } with elements a j i and columns a j (the upper index numbers the columns and the lower index numbers the components of vectors); I is the identity matrix; 0 is the zero matrix (vector); E = {e j i } is the matrix with all elements equal to the unity: e j i = 1; Abs A = {|a j i |} for A = {a j i } ∈ R n×m ; diag π, diag {π i } are the diagonal matrix A with a i i = π i , where π i are the components of the vector π; det A is the determinant of A ∈ R n×n ; A * B = {a j i b j i } ∈ R n×n is the Hadamard product of n × n-matrices A = {a j i } and B = {b j i } (elementwise product); int X is the set of interior points of the set X ⊂ R n ; Pr [a,a] (z) is a projection of the real z on the segment [a, a] ⊂ R 1 , namely it is equal to a, z, a for z < a, a ≤ z ≤ a, z > a respectively; the notation k = 1, . . . , N is used instead of k = 1, 2, . . . , N for brevity.
2. Problem formulation. Consider the controlled system (x ∈ R n is the state): with a given terminal target set M.
∈ R n×nv are given matrices; U [k] ∈ R n×n and u[k] ∈ R nu serve as controls that satisfy the following constraints with given bounding sets: (3) v[k] ∈ R nv (unknown but bounded disturbances) and V [k] ∈ R n×n (matrix uncertainties) are subjected to given set-valued constraints: Below we consider the following cases: (A) without uncertainty, when v and V are given functions, i.e.,Q ≡ 0,V ≡ 0; (B) under uncertainty including two subcases: The system is considered under the state constraints (SC) We presume the sets R[k], Q[k], M, Y[k] to be given and accept the following.
By a parallelepiped P(p, P , π) ⊂ R n we mean a set such that P = P(p, P , π) = {x ∈ R n | x = p + P diag π ξ, ξ ∞ ≤ 1}, where p ∈ R n ; P = {p i } ∈ R n×n is a nonsingular matrix (det P = 0) such that p i 2 = 1; π ∈ R n , π ≥ 0; the condition p i 2 = 1 may be omitted to simplify formulas. It may be said that p determines the center of the parallelepiped, P is the orientation matrix, p i are the "directions", and π i are the values of its "semi-axes". We call a parallelepiped nondegenerate if all π i > 0. By We call a parallelotope P nondegenerate if m = n and detP = 0. By a zone (or m-zone) S = S(c, S, σ, m) ⊂ R n we mean an intersection of m ≤ n strips Each parallelepiped P(p, P , π) is a parallelotope P[p,P ] withP = P diag π. Each nondegenerate parallelotope is a parallelepiped with P =P diag { p i −1 2 }, π i = p i 2 or, in a different way, with P =P , π = e, where e = (1, . . . , 1) . Each parallelepiped is a zone, and vice versa if m = n.
The following Problem 1, which is similar to ones from [24,25,26,35], was investigated earlier [17,26,35]. Similarly to [24,25], we say that the multivalued function The solution to Problem 1 for cases (A), (B,i) (i.e., without matrix uncertainty) is known (see [17,20], and see also [35] for the case (A)). It contains recurrent relations for W[·], which involve operations with sets such as Minkowski's sum ( affine transformation, and intersection of sets. Thus exact construction of W[·] by the mentioned relations can be very cumbersome. Even more difficulties arise for the cases with uncertainties/controls in matrices. Therefore, the ellipsoidal methods for solving Problem 1 were elaborated (see [35] for case (A) under SC and [26] for cases (A), (B,i) without SC). Polyhedral techniques were also proposed [17], which use parallelotope-valued estimates for the solvability tubes.
We call P − (P + ) an internal (external) estimate for Q⊂R n if P − ⊆ Q (P + ⊇ Q).
In [17], for cases (A) and (B,i), the families of external P + [·] and internal P − [·] parallelepiped-valued and parallelotope-valued (shorter, polyhedral ) estimates for Recall that in [19,21] for cases (A), (B,i), (B,ii) without SC the polyhedral techniques were proposed for synthesis of controls which appear either additively or in the system matrix; such controls can be constructed by explicit formulas. For systems with U ≡ 0, this technique was expanded in [20] to systems with SC and also the technique from [17] was expanded for the case (B,ii) (the case under matrix uncertainties). It appears that both mentioned techniques provide one the same families of tubes P − [·], while control strategies, generally speaking, are different.
To solve Problem 2 for the general case we expand the techniques from [19,20,21,22]. As a result we can construct controls u and U by explicit formulas and obtain more rich family of the tubes P − [·] than it could be obtained using constructions from [18] as it was done in [19,22]. Note that expanding the technique from [17] for the general case with both controls u and U is an open question yet.
Here v and V serve as parameters, which specify a parametric family of estimates. It was assumed above that the point v ∈ int Q is known. It is not difficult to find such point by some explicit formulas for some types of sets. In particular, let Q = P ∩ S, i.e., Q is the intersection of the parallelepiped P with the zone S = m i=1 Σ i . If S consists of the unique strip S = Σ 1 , then such point can be found by some explicit formulas [20,Ref. 10], otherwise it can be found successively via m steps starting from P, where at each step an internal estimate is constructed for the intersection of a current parallelepiped with the strip Σ i . It should be mentioned that, generally speaking, such successive procedure can produce rather rough estimates for m > 1.
An alternative attractive ways for calculating v (when V is fixed) is to find v ∈ Argmax {vol P − v,V (Q) | v ∈ Q} (for example, using the Nelder-Mead simplex method [27]). Now let us consider Problem 2. Let us introduce the parametric family of tubes P − [·] that satisfy the following system of recurrent relations: where ] satisfies the following relations (k = N, . . . , 1):   (14) where λ[k, x] are either scalar multipliers of the form or diagonal matrices of the form and Here, the index k in control strategies u[k, x] and U [k, x] indicates that they are used in system (7) at the kth step. The dependence on k − 1 (through parallelotope P 0− [k−1]) is not indicated to simplify the notation. The similar remark is true for the notation for η and β in (12).
Indeed, it follows from the inclusions LetŪ [k, x] be constructed by (17), where the projection operations are omitted. Then it can be verified using estimates similar to ones from [21, Proof of Theo- In order to prove (19) it is sufficient to verify that if x j = 0, then ϕ lj = 0, because then we will have, using (17) and . The situation with x j = 0 and ϕ lj = 0 is impossible because if ϕ lj = 0, then (18) gives η j = 0, and according to (20) and (12), we can obtain the contradiction: be the solution of (7) that corresponds to (17) and u = u[k, x] from (14), (15) or (14), (16), and to arbitrary admissible v . It follows from (7), (9), (20), and (4) that Note that due to Γ[k] ≤ 1 we have for x ∈ P − [k−1] ⊆ P 0− [k−1] that both pairs of formulas (14), (15) and (14), (16) give one and the same equality ). Taking into account (19), (20), (10), (11), we can conclude that In [19], the technique was proposed for solving Problem 2 with u ≡ 0 without SC using a parametric family of polyhedral tubes with one parameter J[·]. The following Corollary 1 from Theorem 3.1 (where L * serves as an analogue of J) provides the extension of the mentioned technique for the case of systems under SC and also shows that Theorem 3.1 generally gives more rich family of tubes P − [·].  [k]) . In this case matrices Φ from (12) become diagonal of the following form Remark 2. Let the discrete-time system be obtained by the Euler approximations 1 of some differential equations: and Ω[k] may be constructed by some special formulas using arguments of "local" volume optimization similarly to [15,21]. (iv) put Ω = Ω(L) = Π(L) and find L (and, consequently, Ω) by solving a problem of the type f (L, Ω(L)) → max L similarly to the above case (iii).

Remark 4.
With an unsuccessful choice of admissible parameters in formulas (8)-(13), it is not excluded a case when we can obtain the empty set P − [k] for some k and therefore we can not construct the solution of Problem 2 using such parameters.
We would like to emphasize once more a relatively simplicity of the proposed method. It may be efficient (in the sense of a number of operations and of memory consumption) for systems with not only small dimensions. Namely, the controls can be calculated by explicit formulas on the base of a polyhedral solvability tube. Cross-sections of such tubes are parallelotopes in R n whose centers and shape are determined by only n + n 2 numbers in contrast to maximal solvability tubes whose cross-sections are polytopes of general form, where the number of verticies and faces may increase greatly from each time step k to the next one, in particular due to the operation of the Minkowsky sum 2 . A whole number of examples shows that the method of simple iteration can be successfully applied for solving nonlinear matrix equations  It worth to be mentioned that the method of simple iteration was used for solving nonlinear equations (10), and it was sufficient a small number of iterations (approximately from 5 to 10 for case (B,ii;SC)).

5.
Conclusion. The problem of feedback terminal target control for bilinear discrete-time uncertain systems under state constraints is considered. Polyhedral control synthesis using polyhedral (parallelotope-valued) solvability tubes is elaborated. This technique provides guaranteed results under set-bounded uncertainties which appear in additive and matrix terms in right hand sides of the system equations. The method allows to calculate guaranteed polyhedral solvability tubes in advance (similarly to constructing maximal solvability tubes) and then to construct control strategies by explicit formulas using these tubes. Although the proposed polyhedral solvability tubes may turn out to be smaller than maximal solvability tubes, they are useful, especially on not too long time intervals, because we can rather easily calculate them, while it is hard to calculate maximal solvability tubes.