Control Of Continuous-time Nonlinear Systems Research Articles

T–S Fuzzy-Affine-Model-Based Reliable Output Feedback Control of Nonlinear Systems with Actuator Faults

This article presents a singular system approach to the reliable $$\mathscr {H}_{\infty }$$ static output feedback (SOF) control for continuous-time nonlinear systems with Markovian jumping actuator faults. The approximation of nonlinear plants is carried out in a Takagi–Sugeno fuzzy-affine (FA) environment, and the Markov chain is adopted to characterize the actuator failure phenomenon. Specifically, by utilizing a singular model transformation strategy, the traditional closed-loop system is converted into a singular FA system. With the construction of a mode-dependent Lyapunov function, and invoking S-procedure and some convexifying techniques, the reliable piecewise SOF controller design is then carried out for the underlying systems via a convex program. An illustrative example is finally given to show the efficacy of the developed method.

Robust Switched Control of Nonlinear Continuous-Time Systems

Robust Switched Control of Nonlinear Continuous-Time Systems

H2−H∞ control of continuous-time nonlinear systems using the state-dependent Riccati equation approach

ABSTRACTThis paper presents a novel state-dependent Riccati equation (SDRE) control approach with the purpose of providing a more effective control design framework for continuous-time nonlinear systems to achieve a mixed nonlinear quadratic regulator and H∞ control performance criteria. By solving the generalized SDRE, the optimal control solution is found to achieve mixed performance objectives guaranteeing nonlinear quadratic optimality with inherent stability property in combination with H∞ type of disturbance reduction. An efficient computational algorithm is given to find the solution to the SDRE. The efficacy of the proposed technique is used to design the control system for inverted pendulum, an under-actuated nonlinear mechanical system.

Approximate Optimal Control of Affine Nonlinear Continuous-Time Systems Using Event-Sampled Neurodynamic Programming.

This paper presents an approximate optimal control of nonlinear continuous-time systems in affine form by using the adaptive dynamic programming (ADP) with event-sampled state and input vectors. The knowledge of the system dynamics is relaxed by using a neural network (NN) identifier with event-sampled inputs. The value function, which becomes an approximate solution to the Hamilton-Jacobi-Bellman equation, is generated by using event-sampled NN approximator. Subsequently, the NN identifier and the approximated value function are utilized to obtain the optimal control policy. Both the identifier and value function approximator weights are tuned only at the event-sampled instants leading to an aperiodic update scheme. A novel adaptive event sampling condition is designed to determine the sampling instants, such that the approximation accuracy and the stability are maintained. A positive lower bound on the minimum inter-sample time is guaranteed to avoid accumulation point, and the dependence of inter-sample time upon the NN weight estimates is analyzed. A local ultimate boundedness of the resulting nonlinear impulsive dynamical closed-loop system is shown. Finally, a numerical example is utilized to evaluate the performance of the near-optimal design. The net result is the design of an event-sampled ADP-based controller for nonlinear continuous-time systems.

An extremum seeking approach to sampled-data iterative learning control of continuous-time nonlinear systems

Abstract: Iterative learning control (ILC) of continuous-time nonlinear plants with periodic sampled-data inputs is considered via an extremum seeking approach. ILC is performed without exploiting knowledge about any plant model, whereby the input signal is constructed recursively so that the corresponding plant output tracks a prescribed reference trajectory as closely as possible on a finite horizon. The ILC is formulated in terms of a non-model-based extremum seeking control problem, to which local optimisation methods such as gradient descent and Newton are applicable. Sufficient conditions on convergence to a neighbourhood of the reference trajectory are given.

Online adaptive optimal control for continuous-time nonlinear systems with completely unknown dynamics

An online adaptive optimal control is proposed for continuous-time nonlinear systems with completely unknown dynamics, which is achieved by developing a novel identifier-critic-based approximate dynamic programming algorithm with a dual neural network (NN) approximation structure. First, an adaptive NN identifier is designed to obviate the requirement of complete knowledge of system dynamics, and a critic NN is employed to approximate the optimal value function. Then, the optimal control law is computed based on the information from the identifier NN and the critic NN, so that the actor NN is not needed. In particular, a novel adaptive law design method with the parameter estimation error is proposed to online update the weights of both identifier NN and critic NN simultaneously, which converge to small neighbourhoods around their ideal values. The closed-loop system stability and the convergence to small vicinity around the optimal solution are all proved by means of the Lyapunov theory. The proposed adaptation algorithm is also improved to achieve finite-time convergence of the NN weights. Finally, simulation results are provided to exemplify the efficacy of the proposed methods.

A novel auto-tuning PID control mechanism for nonlinear systems

In this paper, a novel Runge–Kutta (RK) discretization-based model-predictive auto-tuning proportional-integral-derivative controller (RK-PID) is introduced for the control of continuous-time nonlinear systems. The parameters of the PID controller are tuned using RK model of the system through prediction error-square minimization where the predicted information of tracking error provides an enhanced tuning of the parameters. Based on the model-predictive control (MPC) approach, the proposed mechanism provides necessary PID parameter adaptations while generating additive correction terms to assist the initially inadequate PID controller. Efficiency of the proposed mechanism has been tested on two experimental real-time systems: an unstable single-input single-output (SISO) nonlinear magnetic-levitation system and a nonlinear multi-input multi-output (MIMO) liquid-level system. RK-PID has been compared to standard PID, standard nonlinear MPC (NMPC), RK-MPC and conventional sliding-mode control (SMC) methods in terms of control performance, robustness, computational complexity and design issue. The proposed mechanism exhibits acceptable tuning and control performance with very small steady-state tracking errors, and provides very short settling time for parameter convergence.

Optimal control of nonlinear continuous-time systems in strict-feedback form.

This paper proposes a novel optimal tracking control scheme for nonlinear continuous-time systems in strict-feedback form with uncertain dynamics. The optimal tracking problem is transformed into an equivalent optimal regulation problem through a feedforward adaptive control input that is generated by modifying the standard backstepping technique. Subsequently, a neural network-based optimal control scheme is introduced to estimate the cost, or value function, over an infinite horizon for the resulting nonlinear continuous-time systems in affine form when the internal dynamics are unknown. The estimated cost function is then used to obtain the optimal feedback control input; therefore, the overall optimal control input for the nonlinear continuous-time system in strict-feedback form includes the feedforward plus the optimal feedback terms. It is shown that the estimated cost function minimizes the Hamilton-Jacobi-Bellman estimation error in a forward-in-time manner without using any value or policy iterations. Finally, optimal output feedback control is introduced through the design of a suitable observer. Lyapunov theory is utilized to show the overall stability of the proposed schemes without requiring an initial admissible controller. Simulation examples are provided to validate the theoretical results.

Neural Network-Based Event-Triggered State Feedback Control of Nonlinear Continuous-Time Systems

This paper presents a novel approximation-based event-triggered control of multi-input multi-output uncertain nonlinear continuous-time systems in affine form. The controller is approximated using a linearly parameterized neural network (NN) in the context of event-based sampling. After revisiting the NN approximation property in the context of event-based sampling, an event-triggered condition is proposed using the Lyapunov technique to reduce the network resource utilization and to generate the required number of events for the NN approximation. In addition, a novel weight update law for aperiodic tuning of the NN weights at triggered instants is proposed to relax the knowledge of complete system dynamics and to reduce the computation when compared with the traditional NN-based control. Nonetheless, a nonzero positive lower bound for the inter-event times is guaranteed to avoid the accumulation of events or Zeno behavior. For analyzing the stability, the event-triggered system is modeled as a nonlinear impulsive dynamical system and the Lyapunov technique is used to show local ultimate boundedness of all signals. Furthermore, in order to overcome the unnecessary triggered events when the system states are inside the ultimate bound, a dead-zone operator is used to reset the event-trigger errors to zero. Finally, the analytical design is substantiated with numerical results.

Model-free optimal controller design for continuous-time nonlinear systems by adaptive dynamic programming based on a precompensator

In this paper, we consider the problem of developing a controller for continuous-time nonlinear systems where the equations governing the system are unknown. Using the measurements, two new online schemes are presented for synthesizing a controller without building or assuming a model for the system, by two new implementation schemes based on adaptive dynamic programming (ADP). To circumvent the requirement of the prior knowledge for systems, a precompensator is introduced to construct an augmented system. The corresponding Hamilton–Jacobi–Bellman (HJB) equation is solved by adaptive dynamic programming, which consists of the least-squared technique, neural network approximator and policy iteration (PI) algorithm. The main idea of our method is to sample the information of state, state derivative and input to update the weighs of neural network by least-squared technique. The update process is implemented in the framework of PI. In this paper, two new implementation schemes are presented. Finally, several examples are given to illustrate the effectiveness of our schemes.

Infinite-Time Optimal Control of Nonlinear Continuous-Time Systems with Nonlinear Feedback Gains

The optimal control problem for continuous-time nonlinear systems with nonlinearity in state is formulated and solved. A solution to the problem using state-dependent Riccati equation method (SDRE) is established, procedure for solving the problem is proposed and illustrated with a numerical example.

Global Exact Quadratization of Continuous-Time Nonlinear Control Systems

We consider a class, denoted by ${\cal Q}$, of the nonlinear control systems which can be densely represented as a subsystem of a certain kind of quadratic system, namely a quadratic target. We say that a system in $\cal Q$ undergoes a globally exact quadratization. Here “globally” adds up to a slight extension of the notion of $C^\infty$ immersion (of systems), namely a dense immersion, which amounts to saying that it is defined on the whole manifold of the system states, except possibly a zero-measure set. It is proven that the class ${\cal Q}$ includes all systems characterized by vector fields whose components are analytic integral closed-form functions (ICFFs). The result is first proven for algebraic system functions, by means of a constructive proof, and next extended up to analytic ICFFs. For nonanalytic ICFFs a weaker result is proven as well. Also the case of a partially observed system is considered, as well as the internal structure of every quadratic representation, which is proven to be alwa...

Event-triggered robust model predictive control of continuous-time nonlinear systems

The event-triggered control is of compelling features in efficiently exploiting system resources, and thus has found many applications in sensor networks, networked control systems, multi-agent systems and so on. In this paper, we study the event-triggered model predictive control (MPC) problem for continuous-time nonlinear systems subject to bounded disturbances. An event-triggered mechanism is first designed by measuring the error between the system state and its optimal prediction; the event-triggered MPC algorithm that is built upon the triggering mechanism and the dual-mode approach is then designed. The rigorous analysis of the feasibility and stability is conducted, and the sufficient conditions for ensuring the feasibility and stability are developed. We show that the feasibility of the event-triggered MPC algorithm can be guaranteed if, the prediction horizon is designed properly and the disturbances are small enough. Furthermore, it is shown that the stability is related to the prediction horizon, the disturbance bound and the triggering level, and that the state trajectory converges to a robust invariant set under the proposed conditions. Finally, a case study is provided to verify the theoretical results.

Robust Tracking Control for Takagi–Sugeno Fuzzy Systems With Unmeasurable Premise Variables: Application to Tank System

In this paper, a robust fuzzy observer-based tracking controller for continuous-time nonlinear systems presented by Takagi–Sugeno (TS) models with unmeasurable premise variables, is synthesized. Using the H∞ norm and Lyapunov approach, the control design for TS fuzzy systems with both unmeasurable premises and system states is developed to guarantee tracking performance of closed loop systems. Sufficient relaxed conditions for synthesis of the fuzzy observer and the fuzzy control are driven in terms of linear matrix inequalities (LMIs) constraints. The proposed method allows simplifying the design procedure and gives the observer and controller gains in only one step. Numerical simulation on a two tank system is provided to illustrate the tracking control design procedure and to confirm the efficiency of the proposed method.

Robust and resilient state-dependent control of continuous-time nonlinear systems with general performance criteria

A novel state-dependent control approach for continuous-time nonlinear systems with general performance criteria is presented in this paper. This controller is optimally robust for model uncertainties and resilient against control feedback gain perturbations in achieving general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. For the system model, unstructured uncertainty description is assumed, which incorporates commonly used types of uncertainties, such as norm-bounded and positive real uncertainties as special cases. By solving a state-dependent linear matrix inequality at each time, sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this paper unify existing results on nonlinear quadratic regulator, H∞ and positive real control. The efficacy of the proposed technique is demonstrated by numerical simulations of the nonlinear control of the inverted pendulum on a cart system.

Non-fragile finite-time guaranteed cost fuzzy control for continuous-time nonlinear systems

AbstractThis paper considers the problem of non-fragile finite-time guaranteed cost fuzzy control for continuous-time nonlinear systems. Firstly, the definition on finite-time stability (FTS) for continuous-time nonlinear systems is provided and we give a novel and explicit interpretation for non-fragile finite-time guaranteed cost control. Secondly, sufficient conditions for the existence of state feedback controller are derived in terms of linear matrix inequities (LMIs), which guarantee the requirements of the provided performance criterion. The related optimization problem is also offered to minimize the guaranteed cost performance bound. Finally, an illustrative example is presented to show the validity of the proposed scheme.

Robust digital design of continuous-time nonlinear control systems using adaptive prediction and random-local-optimal NARMAX model

In this paper the time-delay and uncertainty of continuous-time (CT) systems are considered, and it is suggested that input and output of a discrete-time (DT) Neural Plant Model (NPM) and recursive neural controller have scaling factors which limit the value zone of measured data from a system. Adapted scaling factors cause the tuned parameters to converge to obtain a robust control performance. However, the proposed Random-Local-Optimization (RLO) design for a model/controller uses off-line initialization to obtain a near global optimal model/controller. Other important issues are the considerations of cost, greater flexibility, and highly reliable digital products for these control problems. This issue of DT control design for CT plant is more difficult than that of CT control design for CT plant, because of the need to process the modeling error between the CT plant and DT model. The input-delay, uncertainty, and sampling distortion of a CT nonlinear power system need to be solved by developing a digital model-based controller. Here, this is called the DT tracking control design of CT systems (DT–CT).Therefore, the DT structure of the adaptive controller for the CT nonlinear power system should be designed as a kind of feed-forward-Recursive-Predictive controller (FRP). First, due to the problem of delays, a digital neural controller with feed-forward of the reference signal and a Nonlinear Auto-Regressive Moving Average eXogenous (NARMAX) neural model design is adopted to reduce this difficulty. The most important contribution is that the more reasonable and systematic two-stage control design, the CT nonlinear delayed system to be controlled is modeled using a NARMAX technique with the first-stage (off-line) method by the proposed global optimal network algorithm and second-stage (on-line) adaptive steps. Second, the dynamic response of the system is controlled by an adaptive NARMAX neural controller via a sensitivity function. A theorizing method is then proposed to replace the sensitivity calculation, which reduces the calculation of Jacobin matrices of the BP method. Finally, the feed-forward input of reference signals helps the digital neural controller to improve the control performance, and the technique works to control the CT systems precisely.

Generalization of an input-to-state stability preserving Runge–Kutta method for nonlinear control systems

Given a nonlinear control system that is input-to-state stable (ISS) it is assured that the states of the system remain bounded if the input is bounded and the system is global asymptotic stable (GAS) in the sense of Lyapunov in the absence of inputs. Recently, Hu and Liu (2007) [7] studied under which condition ISS of a continuous time nonlinear control system implies ISS of the discrete time system obtained by an implicit Runge–Kutta (RK) method. In this contribution, we extend those results to explicit RK methods. This represents an important extension with respect to applications like system identification where explicit RK methods are presupposed or real-time applications where the computational burden of implicit RK methods is prohibitive.

Reliable H ∞ non-uniform sampling tracking control for continuous-time non-linear systems with stochastic actuator faults

This study is concerned with the fuzzy H ∞ non-uniform sampling tracking control problem of non-linear systems described by Takagi-Sugeno fuzzy systems. Considering the stochastic actuator failures, together with the input delay approach, a unified tracking control model of the non-linear systems with stochastic parameter matrix is first established. The unified model has three features: (I) the different premise variables and timescales are employed in the T-S fuzzy models and the controller rules; (II) non-uniform sampling controller is designed and (III) the fault statistics of each actuator is individually quantified. Based on Lyapunov functional approach and stochastic analysis technique, a control design scheme that guarantees the prescribed H ∞ tracking performance of the non-linear systems is proposed in terms of linear-matrix-inequalities. Finally, two application examples are given to show the effectiveness of the proposed method.

Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems

This paper considers the problem of finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems. Firstly, the definition on finite-time stability (FTS) for continuous-time nonlinear systems is provided and we give a novel and explicit interpretation for finite-time quantized guaranteed cost control. Secondly, sufficient conditions for the existence of state feedback controller are derived in terms of linear matrix inequities (LMIs), which guarantee the requirements of the provided performance criterion. The related optimization problem is also offered to minimize the guaranteed cost performance bound. Finally, an illustrative example is presented to show the validity of the proposed scheme.