Convex Control Research Articles

Virtual control contraction metrics: Convex nonlinear feedback design via behavioral embedding

AbstractThis article presents a systematic approach to nonlinear state‐feedback control design that has three main advantages: (i) it ensures exponential stability and ‐gain performance with respect to a user‐defined set of reference trajectories, (ii) it provides constructive conditions based on convex optimization and a path‐integral‐based control realization, and (iii) it is less restrictive than previous similar approaches. In the proposed approach, first a virtual representation of the nonlinear dynamics is constructed for which a behavioral (parameter‐varying) embedding is generated. Then, by introducing a virtual control contraction metric, a convex control synthesis formulation is derived. Finally, a control realization with a virtual reference generator is computed, which is guaranteed to achieve exponential stability and ‐gain performance for all trajectories of the targeted reference behavior. We show that the proposed methodology is a unified generalization of the two distinct categories of linear‐parameter‐varying (LPV) state‐feedback control approaches: global and local methods. Moreover, it provides rigorous stability and performance guarantees as a method for nonlinear tracking control, while such properties are not guaranteed for tracking control using standard LPV approaches. Code is available at https://github.com/ruigangwang7/VCCM.

Cooperative convex control of multiagent systems applied to differential drive robots

Cooperative convex control of multiagent systems applied to differential drive robots

Asynchronous rate anti-bump switching convex control for a class of switched positive linear systems

Asynchronous rate anti-bump switching convex control for a class of switched positive linear systems

Stochastic near-optimal control for a system with Markovian switching and Lévy noise

The near-optimal control conditions of a stochastic system are the main subject of this research work, which will extend some previous results. At first, the stochastic system and its adjoint equations are estimated based on certain mild assumptions on a convex control set. Additionally, utilizing various analysis techniques and Ekeland’s variational principle, sufficient and necessary near-optimality conditions are obtained, independent of the second-order adjoint equation. Ultimately, an example of a biological, mathematical model is used to demonstrate the correctness of the theoretical analysis.

Dissipative control of 2‐D discrete‐time switched systems via a point‐dependent convex storage function

AbstractThis article explores the dissipative control of two‐dimensional (2‐D) discrete‐time switched Fornasini–Marchesini local state‐space systems under dwell‐time switching. It introduces a new point‐dependent convex storage function to establish the criteria for the asymptotic stability and strict ‐‐dissipativity of the 2‐D switched Fornasini–Marchesini local state‐space model based on LMIs. From these established findings, special cases of dissipative control such as strict passivity and performance can be derived. The article also presents the design of a state feedback point‐dependent convex controller using the proposed storage function approach, along with a sufficient condition for the dissipativity of the closed‐loop system. Finally, several examples are presented to corroborate the effectiveness and superiority of the proposed method.

Traditional Dual Growing Rods With 2 Different Apical Control Techniques in the Treatment of Early-Onset Scoliosis.

Based on traditional dual growing rods (TDGR), apical control techniques (ACTs) were introduced as adjuvant procedures to improve deformity correction at the apex segment in the treatment of early-onset scoliosis (EOS). We aimed to explore whether TDGR+ ACTs have different indications, attain more deformity correction, have negative effects on spinal growth, and have different complications. Between 2004 and 2019, a retrospective study of EOS patients treated with TDGR with or without ACTs was conducted and divided into 3 groups: TDGR group; hybrid technique (HT) group: Vertebrectomy/hemivertebrectomy with short fusion and TDGR; ACPS group: apical convex control pedicle screws (ACPS) and TDGR. Demographic, radiographic parameters, clinical outcomes, complications, and revisions were analyzed and compared. Seventy-eight EOS patients were enrolled. The preoperative main curve was the largest in the HT group. ACPS group had the smallest residual curve (19° ± 8.9°) and apical vertebral translation (12.0 ± 9.0 mm) at the latest follow-up, followed by the HT group (30° ± 17.4°, 22.1 ± 13.4 mm) and TDGR group (30° ± 13.2°, 32.8 ± 17.1 mm). ACPS group had the largest T1-12 height and T1-S1 height after index surgery. Complications and revisions in the ACTs groups was lower than the TDGR group. Scoliosis Research Society-22 self-image questionnaire was superior in the ACPS group. According to our intermediate results, TDGR+ACTs could improve correction ability of apex deformity. ACTs had little deleterious effects on spinal height during the lengthening procedures, with a lower complication rate than TDGR. TDGR+ACTs might be a supplemental option for suitable EOS patients.

A Geometrical, Reachable Set Approach for Constrained Pursuit–Evasion Games with Multiple Pursuers and Evaders

This paper presents a solution strategy for deterministic time-optimal pursuit–evasion games with linear state constraints, convex control constraints, and linear dynamics that is consistent with linearized relative orbital motion models such as the Clohessy–Wiltshire equations and relative orbital elements. The strategy first generates polytopic inner approximations of the players’ reachable sets by solving a sequence of convex programs. A bisection method then computes the optimal termination time, which is the least time at which a set containment condition is satisfied. The pursuit–evasion games considered are games with (1) a single pursuer and single evader, (2) multiple pursuers and a single evader, and (3) a single pursuer and multiple evaders. Compared to variational methods, this reachable set strategy leads to a tractable formulation even when there are state and control constraints. The efficacy of the strategy is demonstrated in three numerical simulations for a constellation of satellites in close proximity in low earth orbit.

On the Maximum Principle for Optimal Control Problems of Stochastic Volterra Integral Equations with Delay

In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional system. Based on the maximum principle, we prove the existence and uniqueness of the optimal control for this concrete example and obtain a new type of explicit Gaussian state-feedback representation formula for the optimal control.

Dual Growing Rods Combined With the Apical Convex Control Pedicle Screw Technique Versus Traditional Dual Growing Rods for the Surgical Treatment of Early-Onset Scoliosis: A Case-Matched 2-Year Study.

Combination of apical control convex pedicle screws (ACPS) with traditional dual growing rods (TDGRs) had better apex control ability for early-onset scoliosis (EOS); however, there is a paucity of studies on the ACPS technique. To compare 3-dimensional deformity correction parameters and complications between the apical control technique (DGR + ACPS) and TDGR in the treatment of EOS. A retrospective case-match analysis consisting of 12 cases of EOS treated with the DGR + ACPS technique (group A) from 2010 to 2020, and matched with TDGR case (group B) at a ratio of 1:1 by age, sex, curve type, major curve degree, and apical vertebral translation (AVT). Clinical assessment and radiological parameters were measured and compared. Demographic characteristics, preoperative main curve, and AVT were comparable between groups. The correction ability of the main curve, AVT, and apex vertebral rotation were better in group A at index surgery ( P < .05). The increase in T1-S1 and T1-T12 height was large in group A at index surgery ( P = .011, P = .074). The annual increase in spinal height was slower in group A, but without significant difference. The surgical time and estimated blood loss were comparable. Six complications occurred in group A, and 10 occurred in group B. In this preliminary study, ACPS seems to provide better correction of apex deformity, while attaining the comparable spinal height at 2-year follow-up. Larger cases and longer follow-up are needed to achieve reproducible and optimal results.

The application of a universal separating vector lemma to optimal sampled-data control problems with nonsmooth Mayer cost function

&lt;p style='text-indent:20px;'&gt;In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called &lt;i&gt;universal separating vector lemma&lt;/i&gt; is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method.&lt;/p&gt;

Infinite horizon Stackelberg differential games with random coefficients under control input constraint

In this paper, we study Stackelberg stochastic differential game driven by stochastic differential equation (SDE for short) with random coefficients on infinite horizon under convex control constraint. The control enters into the drift and diffusion terms simultaneously. By employing the first-order adjoint equation (backward stochastic differential equation, BSDE for short), we are able to announce Pontryagin's maximum principle for the leader's global Stackelberg solution, within adapted open-loop structure, where the term global means that the leader's domination over the entire game duration. Due to the follower's adjoint equation develops into a BSDE, the leader will be confronted with a control problem where the state equation is a kind of infinite horizon fully coupled forward-backward stochastic differential equation (FBSDE for short). We shed light on an important application in a class of infinite horizon linear-quadratic (LQ for short) Stackelberg games in which the control process of leader's is constrained in a closed convex subset Γ 1 of full space R m 1 . Whenever the control domain is postulated to be full space, we derive a new kind of infinite horizon backward stochastic Riccati equation (BSRE for short) with asymmetric coefficients.

Discontinuous observer-based convex control of the Furuta pendulum

Discontinuous observer-based convex control of an underactuated rotatory system, better known as the Furuta pendulum, is the subject of this report. It is shown that a slight generalization of the traditional discontinuous observer design allows applying it to the referred plant. Since the observer gains are obtained via linear matrix inequalities and convex rewriting of nonlinear terms, convex control comes at hand as a better fit to drive the system to the upright position. Simulation and real-time results are presented which prove the effectiveness of the proposal.

Convex control with H∞ performance of a hexarotor-UAV

This paper is dedicated to the modeling and control for a hexarotor-type unmanned aerial vehicle based in a convex rewrite that it is represents exactly the nonlinear model of the hexarotor. Then, the controller design is considered a parallel-distributed compensation control law to stabilitze of the vehicle and a H∞ performance criterion to aguratee robustness againts distrubances. The stability condition and robust performance are formulated using the function quadratic of Lyapunov where the sufficient conditions are given by a set of linear matrix inequalities. Finally, it is illustrated the numerical simulations of the proposed method through of the hexarotor.

Finite-time H∞ control of discrete-time switched systems based on transferring-dependent Lyapunov function approach

This paper studies the finite-time issues (finite-time stability, finite-time boundedness, finite-time [Formula: see text] control) of discrete-time switched systems. Different from the existing Lyapunov function (LF) approaches only concerning activated subsystem, a novel approach for constructing LF is first proposed, which not only depends on the subsystem the switching reaches but also is closely related to the freshly inactive subsystem. Based on this, a transferring-dependent convex Lyapunov function (CLF) and a transferring-dependent LF are introduced, under which the improved finite-time stability results are deduced with admissible edge-dependent average dwell time (AED-ADT) technique. In light of the merits of established finite-time stability criteria, two transferring-dependent controllers are devised—that is, transferring-dependent convex controller (TD-CC) and transferring-dependent ordinary controller (TD-OC)—to perform finite-time control synthesis. Then, the finite-time [Formula: see text] control is further investigated based on the obtained control synthesis results. The designed transferring–dependent controllers can guarantee that the underlying system is finite-time bounded (FTB) with a prescribed [Formula: see text] performance under tighter AED-ADT switching signals. Besides, it is the first time to use the multiple convex Lyapunov function (MCLF) to investigate the finite-time issues. Finally, two numerical examples and a practical example are used to verify the effectiveness of the developed results.

Reachable set computation of linear systems with nonconvex constraints via convex optimization

This paper addresses the reachable set computation of a linear system with a nonconvex control constraint and other convex control and state constraints. We propose to convexify the nonconvex constraint by a relaxation technique. We prove that the reachable set of the relaxed constrained system at any given final time is equal to that of the original constrained system at the same final time under certain assumptions, and interestingly it holds even though the admissible control set is expanded. The theoretical result can enable us to use convex optimization to efficiently and accurately compute the reachable set of the original constrained system. We will verify and apply the theoretical result to reachable set computation via two planetary precision landing examples.

Output-based optimal [formula omitted]-gain control of weighted edge-dependent switching positive systems by a new copositive Lyapunov function

In this paper, the weighted L1-gain analysis and control synthesis for switched positive linear systems are investigated via a weighted edge-dependent average dwell time approach. First, a novel multiple convex copositive Lyapunov function is devised in a polytopic form with clock-dependent function coefficients, which not merely upgrades the degrees of freedom of the Lyapunov function considerably, but also permits the descent at switching instants. By weighted edge-dependent switching technique, the feasible switching zone is greatly extended, and the dwell time constraint is eliminated for each subsystem. In a particular scenario, the lower bound on the weighted sum of all dwell times can be even erased thoroughly, and the relevant non-weighted L1-gain results can be derived. Furthermore, two L1-gain convex control laws are derived in time-varying forms, comprising state feedback and output feedback, which can be updated continuously during the running time of systems. Finally, a positive circuit system and two numerical examples are presented to conduct comparative studies with other clock-dependent Lyapunov function results, suggesting that the proposed results perform better under low complexity on L1-gain, switching zone, computation time, etc, and improve transient performances such as settling time and overshoot.

A New Convex Controller for Stabilizing of Two Symmetrical Logistic Maps

In this paper, we focused on the coupled logistic map, which is a system of two symmetrically identical single logistic maps. The complicated dynamics of the generated system, which has a wide range of dynamical behavior, grant it much attention for many applications. First, the existence and uniqueness of this system are investigated, where the analytical technique is utilized to find the parametric condition for the local asymptotic stability of its fixed points. It can be obviously seen that it undergoes Niemark-Sacker and Hopf bifurcation in a small neighborhood of the unique positive fixed point and invariant circle. This has been proven based on bifurcation theory and the center manifold theorem. The behavior of this system is stabilized using two conventional controlling methods, the OGY and the pole-placement. As well as, a new control method is proposed to make such stabilization more easier. The numerical simulations are utilized to demonstrate the analytic results.

Maximum principle for discrete-time stochastic control problem of mean-field type

In this work, a discrete-time mean-field type stochastic optimal control problem is studied. The goal is to derive the stochastic maximum principle with convex control domains. L-derivative is applied to handle the mean-field term and a technique of adjoint operator is used to overcome the difficulties of obtaining adjoint equations and duality relation. Then, the stochastic maximum principle for discrete-time mean-field type stochastic optimal control problem is established. Finally, as an illustration of our stochastic maximum principle, a discrete-time mean–variance portfolio selection problem is solved with the decoupling technique which is different from the continuous-time case.

Effective Approximation Methods for Constrained Utility Maximization with Drift Uncertainty

In this paper, we propose a novel and effective approximation method for finding the value function for general utility maximization with closed convex control constraints and partial information. Using the separation principle and the weak duality relation, we transform the stochastic maximum principle of the fully observable dual control problem into an equivalent error minimization stochastic control problem and find the tight lower and upper bounds of the value function and its approximate value. Numerical examples show the goodness and usefulness of the proposed method.

Maximum principle for stochastic optimal control problem of finite state forward‐backward stochastic difference systems

AbstractThis article studies the maximum principle for stochastic optimal control problems of forward‐backward stochastic difference systems (FBSSs) where the uncertainty is modeled by a discrete time, finite state process, rather than white noises. Two distinct forms of FBSSs are investigated. The first one is described by a partially coupled forward‐backward stochastic difference equation (FBSE) and the second one is described by a fully coupled FBSE. We deduce the adjoint difference equation by adopting an appropriate representation of the product rule and a proper formulation of the backward stochastic difference equation (BSE). Finally, the maximum principle for this optimal control problem with the convex control domain is established.