Adaptive backstepping control of uncertain systems with unknown input time-delay

In this paper, we establish the robustness of adaptive controllers designed using the standard backstepping technique with respect to unmodeled dynamics involving unknown input time delay. While noting that some results on robust stabilization of non-minimum phase systems using the backstepping technique are available, we realize that the standard adaptive backstepping technique has only been shown applicable to unknown minimum phase systems. Another significance of our result is to enable the class of systems stablizable by adaptive backstepping controllers to cross the boundary of minimum phase systems, since systems with input time delay belong to non-minimum phase systems. Moreover, the L₂ and L_∞ norms of the system output are also established as functions of design parameters. This implies that the transient system performance can be adjusted by choosing suitable design parameters.     

Adaptive tracking of switched nonlinear systems with prescribed performance using a reference-dependent reparametrisation approach

In this paper, adaptive tracking control of switched nonlinear systems in the parametric strict-feedback form is investigated. After defining a reparametrisation lemma in the presence of a non-zero reference signal, we propose a new adaptive backstepping design of the virtual controllers that can handle the extra terms arising from the reparametrisation (and that the state-of-the-art backstepping designs cannot dominate). The proposed adaptive design guarantees, under arbitrarily fast switching, an a priori bound for the steady-state performance of the tracking error and a tunable bound for the transient error. Finally, the proposed method, by overcoming the need for subsystems with common sign of the input vector field, enlarges the class of uncertain switched nonlinear systems for which the adaptive tracking problem can be solved. A numerical example is provided to illustrate the proposed control scheme.     

Decentralized adaptive backstepping stabilization of interconnected systems with dynamic input and output interactions

So far there is still no result available for backstepping based decentralized adaptive stabilization of unknown systems with interactions directly depending on subsystem inputs, even though such interactions commonly exist in practice. In this paper, we provide a solution to this problem by considering both input and output dynamic interactions. To clearly illustrate our approaches, we will start with linear systems and then extend the results to nonlinear systems. Each local controller, designed simply based on the model of each subsystem by using the standard adaptive backstepping technique without any modification, only employs local information to generate control signals. It is shown that the designed decentralized adaptive backstepping controllers can globally stabilize the overall interconnected system asymptotically. The L₂ and L_∞ norms of the system outputs are also established as functions of design parameters. This implies that the transient system performance can be adjusted by choosing suitable design parameters.     

Decentralized adaptive stabilization in the presence of unknown backlash-like hysteresis

Due to the difficulty of handling both hysteresis and interactions between subsystems, there is still no result available on decentralized stabilization of unknown interconnected systems with hysteresis, even though the problem is practical and important. In this paper, we provide solutions to this challenging problem by proposing two new schemes to design decentralized output feedback adaptive controllers using backstepping approach. For each subsystem, a general transfer function with arbitrary relative degree is considered. The interactions between subsystems are allowed to satisfy a nonlinear bound with certain structural conditions. In the first scheme, no knowledge is assumed on the bounds of unknown system parameters. In case that the uncertain parameters are inside known compact sets, we propose an alternative scheme where a projection operation is employed in the adaptive laws. In both schemes, the effects of the hysteresis and the effects due to interactions are taken into consideration in devising local control laws. It is shown that the designed local adaptive controllers can ensure all the signals in the closed-loop system bounded. A root mean square type of bound is obtained for the system states as a function of design parameters. This implies that the transient system performance can be adjusted by choosing suitable design parameters. With Scheme II, the proposed control laws allow arbitrarily strong interactions provided their upper bounds are available. In the absence of hysteresis, perfect stabilization is ensured and the L₂ norm of the system states is also shown to be bounded by a function of design parameters when the second scheme is applied.     

Adaptive robust stabilisation of uncertain nonlinear dynamical systems: an improved backstepping approach

The main purpose of this paper is to propose a design approach by which some simple adaptive robust controllers can be synthesised for a class of uncertain nonlinear dynamical systems which can be transformed into uncertain strict-feedback nonlinear systems. In this paper, an improved backstepping design approach is presented to synthesising a class of continuous adaptive robust state-feedback controllers with a rather simple structure. The improved backstepping design approach can avoid the repeated differentiation problem which appears in using the conventional backstepping algorithm. In particular, it is not required to know the nonlinear upper bound functions of uncertainties. In the light of the presented approach, the state-feedback controllers can be constructed to be linear in the state, with the time-varying control gains which can be self-tuned by the adaptive laws. Similar to the conventional backstepping algorithm, the improved backstepping approach can be extended to a rather large class of uncertain nonlinear systems, and by combining the improved backstepping approach with other control methods, it may be expected to obtain a number of interesting results.     

Mixed H∞ and passive control for linear switched systems via hybrid control approach

This paper investigates the mixed H_∞ and passive control problem for linear switched systems based on a hybrid control strategy. To solve this problem, first, a new performance index is proposed. This performance index can be viewed as the mixed weighted H_∞ and passivity performance. Then, the hybrid controllers are used to stabilise the switched systems. The hybrid controllers consist of dynamic output-feedback controllers for every subsystem and state updating controllers at the switching instant. The design of state updating controllers not only depends on the pre-switching subsystem and the post-switching subsystem, but also depends on the measurable output signal. The hybrid controllers proposed in this paper can include some existing ones as special cases. Combine the multiple Lyapunov functions approach with the average dwell time technique, new sufficient conditions are obtained. Under the new conditions, the closed-loop linear switched systems are globally uniformly asymptotically stable with a mixed H_∞ and passivity performance index. Moreover, the desired hybrid controllers can be constructed by solving a set of linear matrix inequalities. Finally, a numerical example and a practical example are given.     

Robust H∞ control for uncertain discrete singular systems with pole placement in a disk

This paper addresses the design of robust H_∞ controllers for uncertain discrete singular systems with time-invariant uncertainty in both the state and measurement matrices. The singular system to be controlled is not assumed to be regular. A regular dynamic output feedback controller is designed such that a prescribed H_∞ performance condition is satisfied and the closed-loop poles are placed in a specified disk while the regularity, causality and stability of the closed-loop system can be guaranteed for all admissible uncertainties. The desired controller can be obtained by solving a set of matrix inequalities. A numerical example is given to demonstrate the application of the proposed method.     

Robust ℋ2/ℋ∞/reference model dynamic output-feedback control synthesis

This article presents a new strategy to design robust model matching dynamic output-feedback controllers that guarantee tracking response specifications, disturbance rejection and noise attenuation. The proposed synthesis methodology, based on a multi-objective optimisation problem, can be applied to uncertain continuous or discrete-time linear time-invariant systems with polytopic uncertainty, leading to both full-order and reduced-order robust-performance dynamic controllers. The objective functions represent the ?_∞-norm of the difference between the closed-loop transfer function matrix, from the reference signals and the plant outputs and the reference model matrix, the ?_∞-norm of the closed-loop transfer function matrix from the disturbances and the plant outputs and the ?₂-norm of the closed-loop transfer function matrix from the measurement noises and the control inputs. An integral control action is also introduced in order to achieve zero steady-state error. In the case of MIMO systems, the proposed strategy can be applied to decouple the closed-loop control system choosing an appropriated reference model matrix. Two examples are presented to illustrate both SISO and MIMO systems control synthesis.     

Two-degree-of-freedom ?2-optimal tracking with preview

A simple SISO two-degree-of-freedom pole-placement design method is presented that provides ?₂-optimal tracking of a given reference signal. The closed-loop pole locations are first chosen by the system designer. The closed-loop zeros are then placed in an optimal fashion by a computationally inexpensive algorithm to achieve asymptotic tracking with an optimal transient response. The preview approach, which has become a common method for dealing with systems which have non-minimum phase behavior, can then optionally be used to further improve the transient behavior for both minimum phase and non-minimum phase systems. Unlike previous results based on the preview approach, the solution presented here takes into consideration the closed-loop pole dynamics, and is ?₂ optimal with respect to all other two-degree-of-freedom preview controllers with the same closed-loop poles. A simple solution to the H₂ model matching problem, where the design parameter Q is not rational, but polynomial, is the heart of the solution method.     

Adaptive dynamic surface control for a class of output-feedback nonlinear systems with guaranteed ℒ∞ tracking performance

In this article, an output-feedback adaptive dynamic surface control (DSC) is proposed for a class of nonlinear systems. It is proved that, by using the new scheme, the explosion of the complexity problem in a traditional backstepping design can be eliminated, the semi-global stability of a closed-loop system can be guaranteed and, in particular, by choosing the design parameters and initialising the filters and the update law properly, we show that the ?_∞ performance of the system-tracking error can be achieved without over-parametrisation. Another advantage of the proposed scheme compared with those traditional backstepping control and current adaptive DSC schemes, whose adaptive control law is obtained through a series of steps recursively, is that the adaptive law is needed only at the first design step, and therefore significantly reduces the design procedure.     

Design of finite-time H∞ controller for uncertain nonlinear systems and its application

This paper studies the problem of finite-time H _∞ control for strict feedback nonlinear systems with external disturbance. The finite-time stability theory, H _∞ control method, backstepping technique, together with adding a power integrator tool are combined to design a finite-time H _∞ state feedback controller. The obtained controller can make the closed-loop system finite-time convergent, and the influence of the external disturbance is attenuated to a given degree. Two numerical examples are presented to show the effectiveness and feasibility of the proposed method. Meanwhile, the proposed method is also applied to robot manipulators.     

Output feedback decentralized control of large-scale systems using weighted sensitivity functions minimization

This paper considers the problem of achieving stability and desired dynamical transient behavior for linear large-scale systems, by decentralized control. It can be done by making the effects of the interconnections between the subsystems arbitrarily small. Sufficient conditions for stability and diagonal dominance of the closed-loop system are introduced. These conditions are in terms of decentralized subsystems and directly make a constructive H_∞ control design possible. A mixed H_∞ pole region placement is suggested, such that by assigning the closed-loop eigenvalues of the isolated subsystems appropriately, the eigenvalues of the overall closed-loop system are assigned in desirable range. The designs are illustrated by an example.     

H ∞ output-feedback control for switched linear discrete-time systems with time-varying delays

In this paper, the problem of H _∞ output feedback control for switched linear discrete-time systems with time delays is investigated. The time delay is assumed to be time-varying and bounded. By constructing a switched quadratic Lyapunov function for the underlying system, both static and dynamic H _∞ output feedback controllers are designed respectively such that the corresponding closed-loop system under arbitrary switching signals is asymptotically stable and a prescribed H _∞ noise-attenuation level bound is guaranteed. A cone complementary linearization algorithm is exploited to design the controllers. A numerical example is presented to show the effectiveness of the developed theoretical results.     

Event-triggered Networked H∞ Output Tracking Control Based on Dynamic Compensation Controller

In this paper, the event-triggered H_∞ output tracking problem is investigated for networked control systems. In order to reduce the output tracking error as well as to improve network resource utilization, we propose an idea of dynamic compensation controller with the discrete-time event-triggered mechanism, that is, the integral term of tracking error and the state of the reference system are introduced to form states of the augmented system. We first examine the dynamic compensation idea by the H_∞ output tracking control problem for linear time-invariant (LTI) systems. Then, we model the closed-loop event-triggered networked control system as a time-delay augmented linear system. By constructing a Lyapunov-Krasovskii functional with the delay fractioning technique, the stability conditions with lower conservatism are derived in the form of the linear matrix inequalities (LMIs). Furthermore, a method is proposed to design the H_∞ dynamic compensation controllers and the discrete-time event-triggered mechanisms. Finally, the satellite tracking control problem is used as an example to show that the dynamical compensation idea is effective in reducing the tracking error and that the proposed method in this paper can achieve better performance than that in the existing literature.

     

Integrated design of a linear/neuro-adaptive controller in the presence of norm-bounded uncertainties

Adaptive neural controllers are often criticised for the lack of clear and easy design methodologies that relate adaptive neural network (NN) design parameters to performance requirements. This study proposes a methodology for the design of an integrated linear-adaptive model reference controller that guarantees component-wise boundedness of the tracking error within an a priori specified compact domain. The approach is based on the design of a robust invariant ellipsoidal set where both the NN reconstruction error and the neuro-adaptive control are considered as bounded persistent uncertainties. We show that all the performance and control requirements for the closed-loop system can be expressed as linear matrix inequality constraints. This brings the advantage that feasibility and optimal design parameters can be effectively computed while solving a linear optimisation problem. An advantage of the method is that it allows a systematic and quantitative evaluation of the interplay between the design parameters and their impact on the requirements. This produces an integrated linear/neuro-adaptive performance-oriented design methodology. A numerical example is used to illustrate the approach.     

H2 performance of continuous time periodically time varying controllers

This paper uses a frequency domain approach to the analysis of H₂ performance of continuous time periodically time varying controllers. For control of linear time invariant plants, it is shown that the time varying dynamics of the periodic controller deteriorates the closed-loop system H₂ performance and a linear time invariant controller can be found to provide strictly better H₂ control of the system.     

Decentralized adaptive stabilization for interconnected systems with dynamic input-output and nonlinear interactions

This paper considers the problem of robust decentralized adaptive output feedback stabilization for a class of interconnected systems with dynamic input and output interactions and nonlinear interactions by using MT-filters and the backstepping design method. It is shown that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the L₂ and L_∞ norms of the system outputs are also be bounded by functions of design parameters. The scheme is demonstrated by a simulation example.     

A robust output-feedback adaptive dynamic surface control for linear systems with input disturbance

In this paper, a robust output-feedback adaptive control is proposed for linear time-invariant (LTI) singleinput single-output (SISO) plants with unmeasurable input disturbance. Using dynamic surface control (DSC) technique, it is shown that the explosion of complexity problem in backstepping control can be eliminated. Furthermore, the proposed adaptive DSC scheme has the following merits: 1) by introducing an initialization technique, the L∞ performance of system tracking error can be guaranteed even if the plant high-frequency gain is unknown and the input disturbance exists, and 2) the adaptive law is necessary only at the first design step, which significantly reduces the design procedure. It is proved that with the proposed scheme, all the closed-loop signals are semiglobally uniformly ultimately bounded. Simulation results are presented to demonstrate the effectiveness of the proposed scheme.     

Single-approximation-based adaptive control of a class of nonlinear time-delay systems

This paper presents a simple adaptive control approach for uncertain strict-feedback nonlinear systems with unknown time-varying delays. All nonlinear functions and time delays in the systems are assumed to be unknown. Compared with the existing works, the contribution of this study is the design of a simple adaptive control law using single function approximator, without the implementation of virtual controllers derived from the backstepping design procedure. Unlike the existing backstepping methods, virtual controllers are only used as intermediate signals for designing the actual control. Therefore, the proposed control scheme is simpler than the existing methods for strict-feedback time-delay systems because the problems of using multiple approximators and calculating virtual controllers are eliminated. In addition, it is shown that all signals in the closed-loop system are uniformly ultimately bounded.     

Output tracking control of a parallel-flow heat exchange process

In this paper, we consider an output tracking problem of a parallel-flow heat exchange process with distributed and boundary inputs. As the distributed inputs to the system, the output feedback control is first applied. Under zero boundary inputs, it is shown that the C₀-semigroup describing the closed-loop system satisfies the spectrum determined growth condition. Next, we apply a backstepping method to the design of the boundary inputs for output tracking. Our main result shows that the output of the system reaches a reference signal in finite time under both the output feedback control and the boundary control law derived by the backstepping method.