Solving a nonlinear output regulation problem: zero miss distanceof pure PNG

Presents a solution to the output regulation problem of a nonlinear system with time-varying disturbance: the system represents the well-known missile-target pursuit situation where the missile is guided by the pure proportional navigation guidance (PPNG) law while the target maneuvers with time-varying normal acceleration, and the problem is to prove the zero miss distance property of PPNG, which has been studied for decades without satisfactory success. To solve this problem, we construct a function by which a time sequence of the missile-to-target range is upper-bounded, and prove that the function is strictly decreasing, which is also proven to guarantee that there is always a sub-sequence that asymptotically converges to zero. The solution is given in the form of a necessary and sufficient condition guaranteeing the zero miss distance of PPNG     

Stabilization of nonlinear systems via designed center manifold

This paper addresses the problem of local state feedback stabilization of a class of nonlinear systems with nonminimum phase zero dynamics. A new technique, namely, the Lyapunov function with homogeneous derivative along solution curves was developed to test the approximate stability of the dynamics on the center manifold. A set of convenient sufficient conditions are provided to test the negativity of the homogeneous derivatives. Using these conditions and assuming the zero dynamics has stable and center linear parts, a method is proposed to design controls such that the dynamics on the designed center manifold of the closed-loop system is approximately stable. It is proved that using this method, the first variables in each of the integral chains of the linearized part of the system do not affect the approximation order of the dynamics on the center manifold. Based on this fact, the concept of injection degree is proposed. According to different kinds of injection degrees certain sufficient conditions are obtained for the stabilizability of the nonminimum phase zero dynamics. Corresponding formulas are presented for the design of controls     

Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems

The so-called problem of almost disturbance decoupling with internal stability (ADDPS) is the following one. Given a system and an (arbitrarily small) number γ > 0, find a feedback law yielding a closed loop system which is stable and in which the gain (in the L₂ sense) between the exogenous input and the regulated output is less than or equal to γ. The complete solution of this problem has been known since a long time in the case of linear systems. In the case of nonlinear systems, the only global results available so far in the literature were about SISO systems having an asymptotically stable zero dynamics. In this paper, a new set of results are presented, dealing with nonlinear SISO systems having a possibly unstable zero dynamics, which include the (general) class of linear SISO systems as a special case.     

Risk-sensitive dual control

In this paper, we develop new results concerning the risk-sensitive dual control problem for output feedback nonlinear systems, with unknown time-varying parameters. These results are not merely immediate specializations of known risk-sensitive control theory for nonlinear systems, but rather, are new formulations which are of interest in their own right. A dynamic programming equation solution is given to an optimal risk-sensitive dual control problem penalizing outputs, rather than the states, for a reasonably general class of nonlinear signal models. This equation, in contrast to earlier formulations in the literature, clearly shows the dual aspects of the risk-sensitive controller regarding control and estimation. The computational task to solve this equation, as has been seen for the risk-neutral dual control problem, suffers from the so-called ‘curse of dimensionality’. This motivates our study of the risk-sensitive version for a suboptimal risk-sensitive dual controller. Explicit controllers are derived for a minimum phase single-input, single-output auto-regressive model with exogenous input and unknown time-varying parameters. Also, simulation studies are carried out for an integrator with a time-varying gain. They show that the risk-sensitive suboptimal dual controller is more robust to uncertain noise environments compared with its risk-neutral counterpart. © 1997 by John Wiley & Sons, Ltd.     

Semiglobal regulation of linear systems in presence of measurement constraint

In this note, we consider the problem of output regulation for linear single-input-single-output systems in presence of time-varying measurement constraint. It is shown how an internal model-based regulator, processing a saturated uncertain function of the regulated error, can be designed in order to secure asymptotic tracking/rejection of exosystem-generated references/disturbances. The design is semiglobal in the initial state of the plant and of the exosystem and requires the zero dynamics of the controlled system to be weakly minimum-phase.     

Path-following for linear systems with unstable zero dynamics

A constructive solution to the path-following problem for MIMO linear systems with unstable zero dynamics is developed. While the original control variable steers the system output along the path, the path parameter θ is used as an additional control to stabilize zero dynamics with a feedback law which is nonlinear due to the path constraint. A sufficient condition for solvability of the path-following problem is given in terms of the geometric properties of the path. When this condition is satisfied, an arbitrary small L₂ norm of path-following error can be achieved, thus avoiding performance limitations of the standard reference tracking problem imposed by unstable zero dynamics.     

Robust backstepping output tracking control for SISO uncertain nonlinear systems with unknown virtual control coefficients

The output tracking controller design problem is dealt with for a class of nonlinear strict-feedback form systems in the presence of nonlinear uncertainties, external disturbance, unmodelled dynamics and unknown time-varying virtual control coefficients. A new method based on signal compensation is proposed to design a linear time-invariant robust controller, which consists of a nominal controller and a robust compensator. It is shown that the closed-loop control system with a controller designed by the proposed method has robust asymptotical practical tracking property for any bounded initial conditions and robust tracking transient property if all initial states are zero.     

STABILIZATION OF A CLASS OF NONLINEAR NON‐MINIMUM PHASE SYSTEMS

This paper tackles the problem of stabilization of a class of non‐minimum phase nonlinear systems which have zero dynamics with an eigenvalue zero of multiplicity 2. By adding some new terms, called cross terms, we are able to generalize the concept of the Lyapunov function with a homogeneous derivative along the trajectory, which was introduced in [4], to produce a suitable Lyapunov function. The Lyapunov function assures that the stability of an approximate system, which consists of some lower order terms of a nonlinear system with an eigenvalue zero of multiplicity 2, implies the stability of the whole system. Applying these to non‐minimum phase zero dynamics of nonlinear systems with such a center, a sufficient condition and a design method of state feedback control are obtained for stabilizing the systems.     

Asymptotic tracking of a nonminimum phase nonlinear system withnonhyperbolic zero dynamics

A two-cart with an inverted-pendulum system is a nonlinear, nonminimum phase system with nonhyperbolic zero dynamics. Devasia introduced this system to study the asymptotic tracking problem for nonlinear systems with nonhyperbolic zero dynamics and pointed out that the nonhyperbolicity may be challenging to the application of the standard inversion-based tracking technique. We first show that nonhyperbolicity is not necessary for the applicability of the output regulation theory. In particular, the problem of asymptotic tracking of the two-cart with an inverted-pendulum system to a class of sinusoidal reference inputs is actually solvable by the standard output regulation theory. Moreover, an approximation method for calculating the center manifold equation associated with the output regulation problem for general nonlinear systems is given. This approach does not rely on the hyperbolicity condition and, hence, applies to a large class of nonlinear systems     

A dynamic feedback control strategy for control loops with time-varying delay

Dynamic systems of nth order with time-varying delay in the control loop are examined in this paper. The infinite-dimensional pure delay problem is approximated using a jth-order Padé approximation. Although the approximation provides a well-matched finite-dimensional configuration, it poses a new challenge in terms of unstable internal dynamics for the resulted non-minimum phase system. Such a non-minimum phase characteristic limits the closed-loop system bandwidth and leads to an imperfect tracking performance. To circumvent this problem, the unstable internal dynamics of the system is captured and a new dynamic compensator is proposed to stabilise it in a systematic framework. A dynamic controller is developed, which provides the overall system stability against unmatched perturbation and meets the desired tracking error dynamics. The proposed approach is then applied to fuelling control in gasoline engines addressing the varying transport delay of the oxygen-sensor measurement in the exhaust. The developed methodology is finally validated on a Ford F-150 SI lean-burn engine model with large time-varying delay in the control loop.     

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

This paper addresses the problem of asymptotic output tracking of a class of semilinear parabolic equations with pointwise in‐domain actuation. First, the assessment of the well‐posedness of the considered systems is performed, and then, the stability of boundary controlled systems is analyzed via Chaffee‐Infante equation and Fisher's equation. The application of the zero dynamics inverse design results in a dynamic control scheme that is implemented by using the technique of trajectory planning for flat systems and the Adomian decomposition method. The convergence of the solution of the original systems to that of the corresponding zero dynamics and the convergence of the solution expressed by an Adomian series are also analyzed. Numerical simulations are carried out to illustrate the effectiveness of the developed approach.     

Time-varying state-feedback stabilisation of stochastic feedforward nonlinear systems with unknown growth rate

We consider the time-varying state-feedback stabilisation problem for a class of stochastic feedforward nonlinear systems with unknown growth rate in this paper. A new LaSalle-type theorem for stochastic time-varying systems is firstly established by using the generalized weakly positive definite function. As an application, to deal with serious uncertainties in the unknown growth rate, a time-varying approach, rather than an adaptive one, is adopted to design the scheme of a state-feedback controller for stochastic feedforward systems. Based on the established LaSalle-type theorem, it is shown that all signals of the resulting closed-loop system converge to zero almost surely. Illustrative examples are given to verify the theoretical findings.     

Consensus of high-order multi-agent systems with large input and communication delays

We study in this paper the consensus problem for multi-agent systems with agents characterized by high-order linear systems with time delays in both the communication network and inputs. Provided that the open-loop dynamics of the agents is not exponentially unstable, but may be polynomially unstable, and the communication topology contains a directed spanning tree, a truncated predictor feedback approach is established to solve the consensus problem. It is shown that, if the delays are constant and exactly known, the consensus problems can be solved by both full state feedback and observer based output feedback protocols for arbitrarily large yet bounded delays. If it is further assumed that the open-loop dynamics of the agents only contains zero eigenvalues, the delays are allowed to be time-varying and unknown. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches.     

Output feedback control of a class of nonminimum‐phase nonlinear systems

The problem of global stabilization by output feedback is investigated in this paper for a class of nonminimum‐phase nonlinear systems. The system under consideration has a cascade configuration that consists of a driven system known as the inverse dynamics and a driving system. It is proved that although the zero dynamics may be unstable, there is an output feedback controller, globally stabilizing the nonminimum‐phase system if both driven and driving systems have a lower‐triangular form and satisfy a Lipschitz‐like condition, and the inverse dynamics satisfy a stronger version of input‐to‐state stabilizability condition. A design procedure is provided for the construction of an n‐dimensional dynamic output feedback compensator. Examples and simulations are also given to validate the effectiveness of the proposed output feedback controller. Copyright © 2016 John Wiley & Sons, Ltd.     

Time-varying line-of-sight rate estimator with a single modified tracking index for RF homing guidance

A practical time-varying line-of-sight (LOS) rate estimator is proposed for surface-to-air missile (SAM) guidance using an RF seeker. The range-dependant LOS rate dynamics and variances of boresight error (BSE) measurements constitute the Kalman filtering problem associated with the timevarying matrix differential Riccati equation (DRE). Since the conventional gain-scheduled steady-state Kalman filter derived by solving an algebraic Riccati equation (ARE) cannot fully consider the time-varying nature of LOS rate dynamics, it might not be appropriate for volatile missile-target engagement scenarios. This motivates us to investigate a pseudo-analytic solution to the time-varying Kalman filtering problem based on the algebraic transform of the given DRE. Apart from the previous works, the explicit form of time-varying Kalman gain is expressed in terms of a single filter design parameter, called modified tracking index. The proposed time-varying filter is not only able to effectively handle the time-varying LOS rate dynamics but is also affordable for real-time implementation. Adjoint analysis results for a homing guidance loop show that the proposed LOS rate estimator could improve guidance performance and be an excellent choice for SAM applications.     

Robust output feedback stabilization of a class of time-varying non-linear systems

This paper deals with the problem of robust output feedback stabilization of a class of time-varying non-linear systems. This class of systems involves two kinds of time-varying uncertainties: those norm-bounded and those bounded by a smooth non-linear function of the output. Under the assumption that the zero dynamics of the system are uniformly asymptotically stable and some additional mild conditions, we show via a Lyapunov function approach that the uncertain system can be robustly stabilized by a time-varying non-linear output feedback controller. The order of this controller turns out to be one less than the relative degree of the uncertain system. A systematic design procedure is given for constructing the controller. Illustrative examples are given. Note that the results generalize several previous results on robust output feedback stabilization.     

时变广义系统线性二次最优控制

研究时变广义系统线性二次最优控制问题.通过引进时变广义系统脉冲能控性及脉冲能 观性等概念,建立了这类问题与标准状态空间系统二次指标问题的等价性.进而证明了解的 存在唯一性,给出了解的表示和最优反馈综合.     

On tracking control for affine nonlinear systems by sliding mode

The stability problem of output tracking of a bounded time-varying reference by decouplable affine nonlinear systems using sliding mode control is investigated. It is shown that when the error dynamics on the sliding surface is chosen to be linear time-invariant, closed loop stability of systems under the presented sliding mode control can be guaranteed only if the systems are minimum phase.     

A nonlinearly-activated neurodynamic model and its finite-time solution to equality-constrained quadratic optimization with nonstationary coefficients

The recently-proposed Zhang dynamics (ZD) has been proven to achieve success for solving the linear-equality constrained time-varying quadratic program ideally when time goes to infinity. The convergence performance is a significant improvement, as compared to the gradient-based dynamics (GD) that cannot make the error converge to zero even after infinitely long time. However, this ZD model with the suggested activation functions cannot reach the theoretical time-varying solution in finite time, which may limit its applications in real-time calculation. Therefore, a nonlinearly-activated neurodynamic model is proposed and studied in this paper for real-time solution of the equality-constrained quadratic optimization with nonstationary coefficients. Compared with existing neurodynamic models (specifically the GD model and the ZD model) for optimization, the proposed neurodynamic model possesses the much superior convergence performance (i.e., finite-time convergence). Furthermore, the upper bound of the finite convergence time is derived analytically according to Lyapunov theory. Both theoretical and simulative results verify the efficacy and superior of the nonlinearly-activated neurodynamic model, as compared to these of the GD and ZD models.     

Optimal control of time-varying singular systems using the RK–Butcher algorithm

The problem of optimal control of time-varying linear singular systems with quadratic performance index has been studied using the Runge–Kutta–Butcher algorithm. The results obtained using the Runge–Kutta (RK) method based on the arithmetic mean (RKAM) and the RK–Butcher algorithms are compared with the exact solutions of the time-varying optimal control of linear singular systems. It is observed that the result obtained using the RK–Butcher algorithm is closer to the true solution of the problem. Stability regions for the RKAM algorithm, the single-term Walsh series method and the RK–Butcher algorithms are presented. Error graphs for the simulated results and exact solutions are presented in graphical form to highlight the efficiency of the RK–Butcher algorithm. This algorithm can easily be implemented using a digital computer. An additional advantage of this method is that the solution can be obtained for any length of time for this type of optimal control of time-varying linear singular systems.