The problems of reachability and construction of estimates of reachable sets are considered for discrete-time systems with initially linear structure and uncertainties in the initial conditions, matrices, and additive input actions. The uncertainties are restricted by given parallelepiped-valued, interval, and integral nonquadratic constraints, respectively. The systems under consideration turn out to be of bilinear type due to the uncertainty in the matrices. The reachable sets are considered not only in the original space \({\mathbb{R}}^{n}\) but also in the “extended” space \({\mathbb{R}}^{n+1}\), where the last coordinate \(\mu\) corresponds to the current reserve of the additive input action. An exact description is given for the reachable sets \({\mathcal{Z}}[k]\) in the “extended” space using multivalued recurrence relations. Here, the representation of sets in the form of the union of their \(\mu\)-sections is used, and the recurrence relations include operations with sets; one of the operations (multiplication by an interval matrix) acts on each cross-section independently, and another combines the Minkowski sum and the union over cross-sections. The reachable sets \({\mathcal{X}}[k]\) in \({\mathbb{R}}^{n}\) are determined by the cross-sections of \({\mathcal{Z}}[k]\) corresponding to \(\mu=0\). However, it is usually difficult to calculate \({\mathcal{Z}}[k]\) exactly from the above relations. Methods are proposed for the construction of parameterized families of external and internal polyhedral estimates of the sets \({\mathcal{Z}}[k]\) in the form of polytopes of a special type. On this basis, external parallelepiped-valued and internal parallelotope-valued estimates of \({\mathcal{X}}[k]\) are constructed. All estimates are found by explicit formulas from systems of recurrence relations.

对于具有初始线性结构和初始条件、矩阵和附加输入动作的不确定性的离散时间系统，考虑了可达集估计的可达性和构造问题。不确定性分别受给定的平行六面体值、区间和积分非二次约束。由于矩阵中的不确定性，所考虑的系统被证明是双线性类型的。可达集不仅在原始空间\({\mathbb{R}}^{n}\)而且在“扩展”空间\({\mathbb{R}}^{n+1}\ )，其中最后一个坐标 \(\mu\)对应于附加输入动作的当前储备。给出了可达集的准确描述\({\mathcal{Z}}[k]\)在“扩展”空间中使用多值递推关系。这里使用了集合的\(\mu\)-节并集的形式表示，递推关系包括对集合的操作；其中一个操作（乘以区间矩阵）独立作用于每个横截面，另一个将 Minkowski 和和横截面上的并集结合起来。可到达集\（{\ mathcal {X}} [K] \）在\（{\ mathbb {R}} ^ {N} \）通过的横截面确定\（{\ mathcal {Z}} [k]\)对应于\(\mu=0\)。然而，通常很难计算\({\mathcal{Z}}[k]\)正是从上述关系。提出了以特殊类型多面体形式构建集合\({\mathcal{Z}}[k]\)的外部和内部多面体估计的参数化族的方法。在此基础上，构建了\({\mathcal{X}}[k]\) 的外部平行六面体值和内部平行面值估计。所有估计都是通过递推关系系统中的显式公式找到的。