A Youla–Kučera parameterization approach to output feedback relatively optimal control

This paper presents a continuous time solution to the problem of designing a relatively optimal control, precisely, a dynamic control which is optimal with respect to a given initial condition and is stabilizing for any other initial state. This technique provides a drastic reduction of the complexity of the controller and successfully applies to systems in which (constrained) optimality is necessary for some “nominal operation” only. The technique is combined with a pole assignment procedure. It is shown that once the closed-loop poles have been fixed and an optimal trajectory originating from the nominal initial state compatible with these poles is computed, a stabilizing compensator which drives the system along this trajectory can be derived in closed form. There is no restriction on the optimality criterion and the constraints. The optimization is carried out over a finite-dimensional parameterization of the trajectories. The technique has been presented for state feedback. We propose here a technique based on the Youla–Kučera parameterization which works for output feedback. The main result is that we provide conditions for solvability in terms of a set of linear algebraic equations.     

Relatively optimal control with characteristic polynomial assignment and output feedback

A relatively optimal control is a stabilizing controller such that, if initialized at its zero state, produces the optimal (constrained) behavior for the nominal initial condition of the plant (without feedforwarding and tracking the optimal trajectory). In this paper, we prove that a relatively optimal control can be obtained under quite general constraints and objective function, in particular without imposing 0-terminal constraints as previously done. The main result is that stability of the closed-loop system can be achieved by assigning an arbitrary closed-loop characteristic stable polynomial to the plant. An explicit solution is provided. We also show how to choose the characteristic polynomial in such a way that the constraints (which are enforced on a finite horizon) can be globally or ultimately satisfied (i.e., satisfied from a certain time on). We provide conditions to achieve strong stabilization (stabilization by means of a stable compensator) precisely, we show how to assign both compensator and closed-loop poles. We consider the output feedback problem, and we show that it can be successfully solved by means of a proper observer initialization (based on output measurements only). We discuss several applications of the technique and provide experimental results on a cart-pendulum system.     

On the design of approximate non‐linear parametric controllers

This paper focuses on the design of non‐linear parametric controllers, around a nominal input/output trajectory of a discrete‐time non‐linear system. The main result provided herein is a relationship between the tracking performance of the closed‐loop control system in the neighbourhood of a nominal trajectory, and some local features (the first‐order linear approximations about the nominal trajectory) of the non‐linear mappings which characterize the plant and the feedback controller. Such a result can be used to predict the dynamic behaviour of the control system, and to reduce the computational complexity of the optimization task associated with the tuning of the parametric feedback controller. Copyright © 2000 John Wiley & Sons, Ltd.     

A robust model predictive control algorithm for incrementally conic uncertain/nonlinear systems

This paper presents a robustly stabilizing model predictive control algorithm for systems with incrementally conic uncertain/nonlinear terms and bounded disturbances. The resulting control input consists of feedforward and feedback components. The feedforward control generates a nominal trajectory from online solution of a finite‐horizon constrained optimal control problem for a nominal system model. The feedback control policy is designed off‐line by utilizing a model of the uncertainty/nonlinearity and establishes invariant ‘state tubes’ around the nominal system trajectories. The entire controller is shown to be robustly stabilizing with a region of attraction composed of the initial states for which the finite‐horizon constrained optimal control problem is feasible for the nominal system. Synthesis of the feedback control policy involves solution of linear matrix inequalities. An illustrative numerical example is provided to demonstrate the control design and the resulting closed‐loop system performance. Copyright © 2010 John Wiley & Sons, Ltd.     

六自由度机械臂约束预测控制系统的设计

本文提出了一种基于约束预测控制的机械臂实时运动控制方法.该控制方法分为两层,分别设计了约束预测控制器和跟踪控制器.其中,约束预测控制器在考虑系统物理约束的条件下,在线为跟踪控制器生成参考轨迹;跟踪控制器采用最优反馈控制律,使机械臂沿参考轨迹运动.为了简化控制器的设计和在线求解,本文采用输入输出线性化的方式简化机械臂动力学模型.同时,为了克服扰动,在约束预测控制器中引入前馈策略,提出了带前馈一反馈控制结构的预测控制设计.因此,本文设计的控制器可以使机械臂在满足物理约束的条件下快速稳定地跟踪到目标位置.通过在PUMA560机理模型上进行仿真实验,验证了预测控制算法的可行性和有效性.     

Suboptimal local feedback control for a class of constrained discrete time nonlinear control problems

In this paper, we consider a class of constrained discrete time optimal control problems involving general nonlinear dynamics with fixed terminal time. A method to solve the feedback control problem for a class of unconstrained continuous time nonlinear systems has been proposed previously. In that work, the solution is based on synthesizing an approximate suboptimal feedback controller locally in the neighbourhood of a certain nominal optimal trajectory. This paper expands on the same theme by considering problems involving discrete time systems. Taking advantage of the nature of discrete time systems, a further reduction on the computational effort of synthesising the feedback controller is made possible. Also, this paper extends the applicability of the method to constrained systems. For illustration, a numerical example is solved using the proposed method.     

Autonomous mobile robots navigation using RBF neural compensator

This paper presents an approach to adaptive trajectory tracking of mobile robots which combines a feedback linearization based on a nominal model and a RBF-NN adaptive dynamic compensation. For a robot with uncertain dynamic parameters, two controllers are implemented separately: a kinematics controller and an inverse dynamics controller. The uncertainty in the nominal dynamics model is compensated by a neural adaptive feedback controller. The resulting adaptive controller is efficient and robust in the sense that it succeeds to achieve a good tracking performance with a small computational effort. The analysis of the RBF-NN approximation error on the control errors is included. Finally, the performance of the control system is verified through experiments.     

A probabilistic likelihood approach to trajectory sensitivity

Comparative trajectory sensitivity is investigated with respect to small probabilistic perturbations in initial state and plant parameters when a neighboring feedback control rather than the nominally equivalent open-loop control is used.The uncertainty in initial state and plant parameters is modeled by jointly normally distributed random variables with known mean and known covariance matrix. The nominal control is assumed to yield a satisfactory nominal trajectory and it is desired to preserve this shape in non-nominal situations. The likelihood of the state provides a measure of insensitivity at time t.Using the system equations linearized around the nominal trajectory, the joint distributions of the incremental state are computed in both open-loop and closed-loop cases. It is established that the augmented nominal state when a neighboring optimal feedback control is used is at least as likely as that when the nominally equivalent open-loop control is used. The domain where the augmented incremental state for the optimal closed-loop control is more densely distributed than that corresponding to open-loop control is shown to be a hyper-hyperboloid. This domain is unbounded in certain directions, thus pointing out a potential disadvantage in using optimal feedback control.     

Dynamic Modeling and Inverse Optimal PID with Feed-forward Control in <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Framework for a Novel 3D Pantograph Manipulator

This paper affords dynamic modeling and control for a new 3D pantograph manipulator. The new manipulator possesses pure decoupled translational motions and it is characterized by large workspace to size ratio, high speed, rigidity, and accuracy. Euler-Lagrange first type method is used to get the dynamic model. However, the resulted dynamic model is too complex to be used in model-based control techniques. Therefore, a simplified nominal plant is proposed. It allows the inverse dynamic solution efficiently. However, an explicit form of the nominal Coriolis and centrifugal matrix cannot be obtained due to the complicated kinematic terms. Considering these dynamic characteristics as well as the required robust trajectory tracking performance of the manipulator, a new controller is proposed. The new controller is called inverse optimal PID with Feed-Forward Control which is designed in H_∞ framework. The new controller has the following merits; robustness, optimality, simple implementation, and efficient execution without the need of explicit forms of dynamic matrices. The extended disturbance in the proposed controller is smaller than that in the inverse optimal PID control (IPID) and contains one type of error contrary to the nonlinear robust motion controller (NRIC). The performance of the proposed controller is compared with those of IPID and NRIC controllers for different trajectories and payloads. The dynamic simulation results via co-simulation of MSC-ADAMS® and MATLAB®/Simulink software prove the robustness of the proposed controller against speed/payload variations. The proposed controller is found to have higher performance compared with IPID and NRIC controllers. These results assure the feasibility of the 3D pantograph manipulator with the proposed controller for pure translational tracking applications.     

Aggressive Longitudinal Aircraft Path Tracking Using Nonlinear Control

We study the problem of converting a trajectory tracking controller to a path tracking controller for a nonlinear non-minimum phase longitudinal aircraft model. The solution of the trajectory tracking problem is based on the requirement that the aircraft follows a given time parameterized trajectory in inertial frame. In this paper we introduce an alternative nonlinear control design approach called path tracking control. The path tracking approach is based on designing a nonlinear state feedback controller that maintains a desired speed along a desired path with closed loop stability. This design approach is different from the trajectory tracking approach where aircraft speed and position are regulated along the desired path. The path tracking controller regulates the position errors transverse to the desired path but it does not regulate the position error along the desired path. First, a trajectory tracking controller, consisting of feedforward and static state feedback, is designed to guarantee uniform asymptotic trajectory tracking. The feedforward is determined by solving a stable noncausal inversion problem. Constant feedback gains are determined based on LQR with singular perturbation approach. A path tracking controller is then obtained from the trajectory tracking controller by introducing a suitable state projection.     

An embedding approach for the design of state‐feedback tracking controllers for references with jumps

We study the problem of designing state‐feedback controllers to track time‐varying state trajectories that may exhibit jumps. Both plants and controllers considered are modeled as hybrid dynamical systems, which are systems with both continuous and discrete dynamics, given in terms of a flow set, a flow map, a jump set, and a jump map. Using recently developed tools for the study of stability in hybrid systems, we recast the tracking problem as the task of asymptotically stabilizing a set, the tracking set, and derive conditions for the design of state‐feedback tracking controllers with the property that the jump times of the plant coincide with those of the given reference trajectories. The resulting tracking controllers guarantee that solutions of the plant starting close to the reference trajectory stay close to it and that the difference between each solution of the controlled plant and the reference trajectory converges to zero asymptotically. Constructive conditions for tracking control design in terms of LMIs are proposed for a class of hybrid systems with linear maps and input‐triggered jumps. The results are illustrated by various examples. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparison and application of DDC algorithms for a heat exchanger

In recent years, several algorithms for Direct Digital Control (DDC) have been proposed in literature. Although some of these, such as PID or cascade controllers, are very commonly used in industrial applications, the more recent ones like optimal state feedback controllers using an observer or parameter adaptive controllers have rarely been applied to a real plant. The primary difficulty behind this application has been perhaps the lack of testing such algorithms on a pilot plant. Moreover, there has been no serious attempt to make a comparative study of the merits of such algorithms for an existing plant under actual operating conditions. In this paper, seven DDC algorithms are applied to the temperature control of a heat exchanger. These algorithms are: PID, cascade, compensation (pole assignment), deadbeat, half-proportional, adaptive and optimal state feedback controller using an observer. The system performance and sensitivity with respect to changes of the plant parameters, disturbances and set point variations are investigated for the heat exchanger using these algorithms. The results indicate that the more sophisticated algorithms, e.g. optimal state feedback, compensation and adaptive controllers, requiring more computer time and memory, yield relatively less improvement when applied to a low-order plant than do the simple algorithms such as PID or cascade. It was deduced that the PID controller with anti-windup is the most suitable one.     

Wiener-Hopf design of optimal decoupling one-degree-of-freedom controllers

This reasonably self-contained paper is directed to the design of a multivariable optimal one-degree-of-freedom feedback loop which incorporates a decoupling controller in the forward path. The criterion for optimality is a quadratic-cost functional that penalizes both tracking error and saturation. The controllers on which optimization is based are general enough to allow for non-unity feedback and rectangular plant transfer matrices possessing normal row rank. Nevertheless, for the sake of brevity and clarity, attention is focused mainly on the square case. Earlier treatments of the problem have employed multi-degree-of-freedom controllers. The solution we present for the one-degree-of-freedom case is considerably more difficult to obtain, especially when saturation is taken into account. Explicit formulas are derived for the set of all decoupling controllers yielding finite cost, as well as those that are optimal. It is shown that these controllers are strictly-proper under conditions usually prevailing in practice. Four fully worked examples serve to illustrate many important numerical aspects of the theory and all major proofs are transferred to the Appendix.     

Inverse dynamics of articulated flexible structures: Simultaneous trajectory tracking and vibration reduction

This paper addresses the problem of inverse dynamics for articulated flexible structures with both lumped and distributed actuators. This problem arises, for example, in the combined vibration minimization and trajectory control of space robots and structures. For such flexible structures, closed loop passive joint based controllers have been shown to be effective in trajectory control by Paden et al. Crucial to the development of such closed loop controllers, which are robust to external perturbations, is the problem of dynamic inversion which yields the nominal state trajectories and the feedforward inputs. In this paper we propose a new inverse dynamics scheme for computing the nominal lumped and distributed feedforward inputs for tracking a prescribed trajectory.     

Output deadbeat controllers with stability for discrete-time multivariable linear systems

This short paper Treats the problem of designing output deadbeat controllers having the property that the control input to the system converges to zero as time goes to infinity, for discrete-time multivariable linear systems. Two configurations of controllers are considered: one is of state feedback; the other is a dynamic controller using an observer. The existence of such controllers is examined, and the methods are presented for designing such controllers when they exist. The controller using a state feedback obtained in this paper is optimal in the sense that the controller settles the output in zero for any initial state in the minimum number of steps. On the other hand, the dynamic controller is not optimal in that sense, but it minimizest, wheretis defined as an integer such that the controller drives the output to zero in no more thantsteps for any set of initial conditions of the system and the observer.     

Robust output feedback model predictive control of constrained linear systems

This paper provides a solution to the problem of robust output feedback model predictive control of constrained, linear, discrete-time systems in the presence of bounded state and output disturbances. The proposed output feedback controller consists of a simple, stable Luenberger state estimator and a recently developed, robustly stabilizing, tube-based, model predictive controller. The state estimation error is bounded by an invariant set. The tube-based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the ‘center’ of which is the solution of a nominal (disturbance-free) system and the ‘cross-section’ of which is also invariant. Satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system. The complexity of the resultant controller is similar to that required for nominal model predictive control.     

Robust tracking and disturbance rejection of bounded rate signals for uncertain/non-linear systems

We consider the problem of designing controllers for non-linear/uncertain systems to achieve tracking of a reference output signal in the presence of a disturbance input signal. When the exogenous signal (combined reference and disturbance) is constant, we require that the tracking error eventually go to zero. When the exogenous signal has a bounded rate, we require that the tracking error be eventually bounded with a bound which only depends on the bound on the magnitude of the rate of the exogenous signal. We also require that the state and control input are bounded when the exogenous signal is bounded. We propose controllers which have a classical PI (proportional integral) structure using state feedback. For linear systems and specific classes of non-linear/uncertain systems we present conditions whose satisfaction guarantees the existence of controllers which achieve the desired behaviour. Satisfaction of these conditions also yields the controller gain matrices.     

指数稳定的机器人鲁棒跟踪控制

本文提出了一种用于控制具有参数不确性机器人轨迹踊跃的鲁棒控制策略，该控制器的设计根据Ｌｙａｐｕｎｏｖ理论，由一个基于Ｓｌｏｔｉｎｅ方法的标称控制器和一个非线性连续反馈补偿器组成。     

Global approximate output tracking for nonlinear systems

Addresses the global output tracking problem for nonlinear systems with singular points. For nonlinear systems which satisfy a suitable observability condition, the authors identify a class of smooth output trajectories which the system can track using continuous open-loop controls. This class includes all output trajectories generated by smooth state feedback. They then study the problem of approximate output tracking using discontinuous time-varying feedback controllers. Given a smooth output trajectory for which exact tracking is possible, the authors construct a discontinuous feedback controller which achieves robust tracking of the desired output trajectory in the face of perturbations. Finally, it is shown that their results can be applied to the control of a chain system, and some numerical results are presented to illustrate the performance of their controller     

Controller switching based on output prediction errors

We consider a switched nonlinear feedback control strategy for controlling a plant with unknown parameters so that the output asymptotically tracks a reference signal. The controller is selected online from a given set of controllers according to a switching rule based on output prediction errors. For control problems requiring asymptotic tracking of a reference input we provide sufficient conditions under which the switched closed-loop control system is exponentially stable and asymptotically achieves good control even if the switching does not stop, Our results are illustrated with three examples