Null Controllable Region Research Articles

A hybrid control design for stabilisation of nonlinear systems from the null controllable region*

The present paper addresses the problem of designing a controller to enable stabilisation from the entire null controllable region (NCR) – the set of states from where stabilisation is possible). A recently developed approach for the numerical construction of the NCR is utilised, and first, the validity of the use of the NCR as a constrained control Lyapunov function (CCLF) is established. The analysis reveals the ability of a CCLF-based control design to drive the system to an equilibrium manifold. The CCLF-based control design is then utilised within a hybrid control framework that guarantees the ability to stabilise to the origin. Simulation results are used to illustrate the implementation of the control design.

Time-Fuel Efficient Intermittent Feedback Control of LTI Systems

To derive analytical feedback solution of time- $L_{1}$ optimal control problem is a challenging task. In this regard, this letter proposes an alternative strategy that involves intermittent application of open loop time- $L_{1}$ optimal solution. When disturbances affect the system, such policy is shown to retain the system states to within a small bounded safe region about the origin. The key idea is to switch the processor in between active and idle mode. During the active mode, the states are measured, open-loop optimal control policy is computed and transmitted along the communication link to the actuator. While during the idle mode, the precomputed control profile is applied and rest of the system resources remain inactive. The open loop solution is obtained by solving a set of optimization problems with an additional bound constraint on the update time instances. The bound constraints are derived using an algorithm to ensure that the safe region (defined in terms of attainable set) is always inside the null controllable region or the reachable set. Since attainable and reachable sets are approximated as convex polytopes, the bound constraints are obtained through linear inequalities. Simulation results show that under the proposed control strategy, system achieves practical stability with reduced usage of system resources.

Controllability minimum principle based construction of the null controllable region for nonlinear systems

AbstractThe present manuscript addresses the problem of designing a procedure for constructing the null controllable Region (the set of states from where stabilization is achievable subject to constraints). To this end, a controllability minimum principle based computationally tractable approach for constructing the null controllable region is presented. Simulation results are used to illustrate the computation of the NCR for several examples.

Definition and Utilization of the Null Controllable Region for Model Predictive Control of Multi-Input Linear Systems

The problem of guaranteeing stability from the entire null controllable region (NCR) for multi-input linear dynamical systems is addressed in the present manuscript. The proposed controller design is inspired by results for single input systems and generalized to multiple input systems. The approach relies on utilizing the level sets of the NCR as level sets of a Lyapunov function. A contractive constraint is incorporated into a model predictive control design, guaranteeing feasibility for any horizon length, and resulting in the NCR as the closed-loop stability region. The proposed method is illustrated using a simulation example.

Output regulation for decentralized networked control systems under jamming attacks and actuator saturation

Output regulation problem for a decentralized networked control system under jamming attacks and actuator saturation is solved by state feedback in this paper. A jammer’s power allocation strategy aiming at degrading signal-interference-plus-noise ratios and leading to packets dropout in different wireless transmission channels of the decentralized networked control system is derived. In the presence of attack-induced packets dropout, a relation between null controllable region and regulatable region is described. For a plant and an exosystem, an asymptotically regulatable region is constructed based on a null controllable region. At last, an illustrative example certificates effectiveness of the presented techniques.

A trajectory‐based method for constructing null controllable regions

SummaryThe problem of determining a system's set of stabilizable states, the null controllable region (NCR), is intricately related to the problem of determining control Lyapunov functions. In this paper, we address the problem of construction of the NCR for control‐affine nonlinear systems with input constraints. To this end, we explain how the boundary of the NCR is covered by time‐optimal trajectories. To construct the NCR, we employ an algorithm based on Pontryagin's minimum principle, which integrates optimal trajectories of a special smooth system in reverse time from the boundary of an initial NCR estimate. We illustrate this algorithm with linear and nonlinear system examples.

Local null controllability of the control-affine nonlinear systems with time-varying disturbances

The problem of local null controllability for the control-affine nonlinear systems x˙(t)=f(x(t))+Bu(t)+w(t),t ∈ [0, T] is considered in this paper. The principal requirements on the system are that the LTI pair ((∂f/∂x)(0), B) is controllable and the disturbance is limited by the constraint |f(0)+w(t)|≤Md(1−tT)η,Md ≥ 0 and η > 0. These properties together with one technical assumption yield an answer to the problem of deciding when the null controllable region has a nonempty interior. The obtained criterion is built on the purely algebraic and/or differential manipulations with vector field f, input matrix B and a bound on the disturbance w(t). To prove the main result we have derived a new Gronwall-type inequality allowing the fine estimates of the closed-loop solutions. The theory is illustrated and the efficacy of proposed controller is demonstrated by the example where the null controllable region is explicitly calculated. Finally, we established the sufficient conditions to be the system under consideration with w(t) ≡ 0 globally null controllable.

Constrained control Lyapunov function-based control of nonlinear systems

In this paper, we present an algorithm to address the problem of guaranteed stabilization of nonlinear systems from the largest set of initial conditions from where stabilization is possible, the so-called null controllable region (NCR), while also handling state constraints and partial unboundedness of the NCR. The controller uses the geometry of the NCR to inform a constrained control Lyapunov function which in turn provides feasibility and stability guarantees. The NCR is produced from a novel iterative growth algorithm that exploits a certain invariance property. The new controller implementation is illustrated using a chemical reactor system and compared to a conventional nonlinear model predictive control (MPC) implementation.

Utilizing null controllable regions to stabilize input-constrained nonlinear systems

In this paper, we present a method for control of input-constrained nonlinear systems that offers guaranteed stabilization from the entire null controllable region (NCR). The controller achieves stabilization by using a constrained control Lyapunov function (CCLF) based on this NCR. Prior to online implementation, the level sets the CCLF are constructed using an iterative algorithm. The algorithm works by using an invariance principle to expand an initial quadratic Lyapunov function-based region of attraction. The level sets of this CCLF are then utilized in the control calculations, and in particular, an MPC is formulated that requires the system to go to lower level sets of the CCLF. The proposed MPC thus achieves stabilization from the entire NCR. The proposed approach is first corroborated against existing results for linear systems using two- and three-dimensional linear systems examples. Subsequently, the implementation is shown for two and three dimensional nonlinear systems.

Output regulation for linear delta operator systems subject to actuator saturation

Summary In this paper, output regulation for linear delta operator systems subject to actuator saturation is investigated by state feedback. The relation between the regulatable regions and the null controllable region is described in this paper. A set of all initial conditions of a plant and a exosystem is called asymptotically regulatable region for which output regulation is possible. An asymptotically regulatable region is characterized according to a null controllable region of an anti-stable subsystem of the plant. Feedback laws are constructed to solve the problems on output regulation. A numerical example is given to illustrate the effectiveness and potential for the developed techniques. Copyright © 2016 John Wiley & Sons, Ltd.

Semi‐global stabilisation with guaranteed regional performance for delta operator systems subject to actuator saturation

In this study, the domain of attraction is further enlarged for delta operator systems subject to actuator saturation by considering semi-global stabilisation with guaranteed regional performance. An original feedback law is designed to guarantee certain close-loop performance in a fixed region which is a compact subset of the null controllable region. Some fixed regions that cover compact subsets of the null controllable region are obtained by a finite sequence of state feedback laws. The union of all obtained fixed regions is the final domain of attraction for the delta operator systems with actuator saturation. A numerical example is provided to illustrate the effectiveness of the proposed techniques.

Stabilization on null controllable region of delta operator systems subject to actuator saturation

Summary This paper focuses on stabilization on null controllable region of delta operator systems subject to actuator saturation via constructing suitable controllers. Under a saturated linear feedback law, a set of stable equilibrium points is obtained for the delta operator system with actuator saturation. A comprehensive controller is given to show global stabilization of the delta operator system. Thereby, semi-global stabilization of the delta operator system is also achieved. For higher-order systems with no more than two unstable exponentially poles, a new feedback law is constructed to achieve semi-global stabilization on the null controllable region. Finally, three examples are given to illustrate the effectiveness of the proposed techniques. Copyright © 2016 John Wiley & Sons, Ltd.

Null controllable region of delta operator systems subject to actuator saturation

ABSTRACTIn this paper, we give exact description of null controllable regions for delta operator systems subject to actuator saturation. The null controllable region is in terms of a set of extremal trajectories of anti-stable subsystems. For the delta operator system with real eigenvalues or complex eigenvalues, the description is simplified to an explicit formula which is used to characterise the boundary of a null controllable region. The relations of null controllable regions are shown separately for continuous-time systems, discrete-time systems and delta operator systems. Two numerical examples are given to illustrate the effectiveness of the proposed techniques on null controllable regions.

Robust stabilization of T–S fuzzy discrete systems with actuator saturation via PDC and non-PDC law

In this paper, the problem of robust stabilization of T–S fuzzy discrete systems with actuator saturation is studied. A set invariance condition is established as a null controllable region. The robust stabilization results of the closed-loop T–S fuzzy system have a domain of attraction which is arbitrarily close to the null controllable region. The problem of estimating the domain of attraction of the T–S fuzzy closed-loop systems is formulated and solved as an optimization problem. By using parameter-dependent Lyapunov function, both parallel-distributed compensation (PDC) control law and non-parallel-distributed compensation control law are designed. And there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T–S fuzzy system with actuator saturation. A numerical example is provided to illustrate the effectiveness of the proposed design techniques.

Computation of Time Optimal Feedback Control Using Groebner Basis

The synthesis of time optimal feedback control of a single input continuous time linear time invariant (LTI) system is considered. If the control input u(t) is constrained to obey |u(t)| ≤ 1, then it is known that the optimal input switches between the extreme values ±1, according to some “switching surfaces” in the state space. It is shown that for systems with non-zero, distinct and rational eigenvalues, these switching surfaces are semi-algebraic sets and a method to compute them using Groebner basis, is proposed. In the process the null-controllable region for such systems is characterized and computed. Numerical simulations illustrate the proposed computational methods.

Robust fuzzy-scheduling control for nonlinear systems subject to actuator saturation via delta operator approach

This paper focuses on robust fuzzy-scheduling control for nonlinear systems with actuation saturation via delta operator approach. A class of T–S fuzzy delta operator systems is extended to describe nonlinear systems subject to actuator saturation and linear fractional uncertainty. Moreover, a set invariance condition is established for T–S fuzzy delta operator systems. Based on the set invariance condition, an optimization approach is proposed to estimate the domain of attraction for the T–S fuzzy delta operator systems. The domain of attraction is arbitrarily close to a null controllable region by the designed PDC control law or non-PDC control law. Furthermore, the non-PDC control law is proposed by using fuzzy weighting-dependent Lyapunov function. The effectiveness of the developed techniques is shown through a numerical example.

Computation of null controllable regions for antistable linear systems subject to input saturation: an iterative approach

In this study an alternative characterisation of the null controllable region for linear systems subject to input saturation with strictly positive real eigenvalues is proposed. First an outer estimate of the null controllable region is obtained and then an iterated convexification technique is developed.

Output Regulation with Actuator Saturation for the Benchmark Active Control Technology Model

Output Regulation with Actuator Saturation for the Benchmark Active Control Technology Model

Enhanced stability regions for model predictive control of nonlinear process systems

AbstractThe problem of predictive control of nonlinear process systems subject to input constraints is considered. The key idea in the proposed approach is to use control‐law independent characterization of the process dynamics subject to constraints via model predicative controllers to expand on the set of initial conditions for which closed–loop stability can be achieved. An application of this idea is presented to the case of linear process systems for which characterizations of the null controllable region (the set of initial conditions from where closed–loop stability can be achieved subject to input constraints) are available, but not practically implementable control laws that achieve stability from the entire null controllable region. A predictive controller is designed that achieves closed–loop stability for every initial condition in the null controllable region. For nonlinear process systems, while the characterization of the null controllable region remains an open problem, the set of initial conditions for which a (given) Lyapunov function can be made to decay is analytically computed. Constraints are formulated requiring the process to evolve within the region from where continued decay of the Lyapunov function value is achievable and incorporated in the predictive control design, thereby expanding on the set of initial conditions from where closed–loop stability can be achieved. The proposed method is illustrated using a chemical reactor example, and the robustness with respect to parametric uncertainty and disturbances demonstrated via application to a styrene polymerization process. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Control of an Aeroelastic System with Actuator Saturation

The objective of this study is the semiglobal stabilization of the benchmark active control technology wind-tunnel model developed by NASA Langley Research Center. This study applies established theoretical results to the design of a saturation linear state feedback control law. The framework of this investigation is the construction of the null-controllable region and domains of attraction of the planar antistable subsystem of the higher order model in its canonical modal form. Numerical simulations at different densities of the unstable wind-tunnel system are presented that demonstrate construction and performance of the control law applied to an a priori defined subset of the null-controllable region.