9 - Optimal and robust control system design

This paper investigates the robust optimal control problem of a class of continuous-time, partially linear, interconnected systems. In addition to the dynamic uncertainties resulted from the interconnected dynamic system, unknown bounded disturbances are taken into account throughout the learning process, wherein the system’s dynamics and the disturbances are assumed unknown. These challenges lead the collected online data to be imperfect. In this scenario, traditional data-driven control techniques, such as adaptive dynamic programming (ADP) and robust ADP, encounter a challenge in approximating the optimal control policy precisely due to imperfect data and computational errors. In this paper, a novel data-driven robust policy iteration method is proposed to simultaneously solve the robust optimal control problems. Without relying on the knowledge of the system’s dynamics, the external disturbances or the complete state, the implementation of the proposed method only needs to access the input and partial state information. Based on the small-gain theorem, the notions of strong unboundedness observability and input-to-output stability, it is guaranteed that the learned robust optimal control gain is stabilizing and that the solution of the closed-loop system is uniformly ultimately bounded despite the existence of dynamic uncertainties and unknown external disturbances. The simulation results reveal the efficiency and practicality of the proposed data-driven control method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work is motivated by the use of reinforcement learning to improve the quality of designing adaptive optimal controllers for engineering applications. Adaptive dynamic programming methods, in particular policy iteration (PI), are widely used in solving optimal control problems. However, due to the iterative nature of PI, the approximated optimal control policy may be inaccurate and imprecise. Especially, when using imperfect system’s measurements instead of the modelling information. This can result in causing the learned control policy to deviate from the actual optimal policy. This becomes more challenging in the existence of dynamic uncertainties and unknown external disturbances which corrupt the measurements and result in imperfect data. This work investigates the conditions on the uncertainties such that the proposed novel data-driven PI algorithm is robust to system’s uncertainties, unknown external disturbances and imperfect measurements. The approximated robust optimal control policy performs robustly in the existences of imperfect data and uncertainties, and at the same time is close enough to the optimal control policy.

In this paper, the problem of the robust optimal control (ROC) for a class of nonlinear systems with uncertainties and external disturbances is studied. For this purpose, the State Dependent Riccati Equation (SDRE) is considered as a solution of the ROC. The proposed method is to transform a robust control problem into an optimal control problem, where the uncertainties and disturbances are reflected in the performance index. Hence, the performance index is selected as an energy function and using the proposed constraints in this paper the Lyapunov stability for this class of nonlinear systems is proved and the robust optimal controller is designed. In order to illustrate the high capability of the ROC method in optimal control of complex systems such as Hyper-chaotic system, the time response of state variables in the proposed method and the Global Robust Optimal Sliding Mode Control (GROSMC) method are compared together. The simulation results show that the time response of state variables in the ROC method is more sufficient than the GROSMC method.

Optimal control of aluminum electrolysis production process (AEPP) has long been a challenging industrial issue due to its inherent difficulty in establishing an accurate dynamic model. In this paper, a novel robust optimal control algorithm based on adaptive dynamic programming (ADP) is proposed for the AEPP, where the system subjects to input constraints. First, to establish an accurate dynamic model for the AEPP system, recursive neural network (RNN) is employed to reconstruct the system dynamic using the input-output production data. To ensure input constraints are not to exceed the bound of the actuator, the optimal control problem of the AEPP is formulated under a new nonquadratic form performance index function. Then, considering the perturbation of the AEPP, the robust control problem is effectively converted to the constrained optimal control problem via system transformation. Furthermore, a single critic network framework is developed to obtain the approximate solution of the Hamilton-Jacobi-Bellman (HJB) equation. Finally, the proposed ADP controller is applied to the AEPP system to validate the effectiveness and performance.

In this paper, the engine idle speed control (ISC) problem is revisited with the l/sub 1/ optimization design methodology. A controller configuration consisting of a feedback linear quadratic Gaussian (LQG) optimal controller and a feedforward l/sub 1/ optimal controller is chosen to satisfy the design objectives. The main goal of this work is to demonstrate the application of the l/sub 1/ optimal control methodology on a standard automotive control problem and evaluate the advantages and disadvantages of the l/sub 1/ design method. Using the idle bypass valve as the only actuator, the l/sub 1/ optimal feedforward control is shown to be effective in reducing the idle speed errors at the expense of extra control energy being spent during the transient. The controller design process is described along with the simulation and experimental results on a 4.6L 2 valve V8 production engine. A method to address tradeoff issues in evaluating the performance of a feedforward control system is also discussed.

In this paper, an infinite horizon optimal robust guaranteed cost control scheme of a class of continuous-time uncertain nonlinear systems is established based on adaptive dynamic programming. The main idea lies in that the optimal robust guaranteed cost control problem can be transformed into an optimal control problem. Actually, the optimal cost function of the nominal system is nothing but the optimal guaranteed cost of the original uncertain system. A critic neural network is constructed to help solving the modified Hamilton-Jacobi-Bellman equation corresponding to the nominal system. Then, an additional stabilizing term is introduced to reinforce the updating process of the weight vector and reduce the requirement of an initial stabilizing control. An example is provided to illustrate the effectiveness of the present control approach.

The design method of a quantitative robust optimal control system is developed. Wiener-Hopf linear quadratic optimal control and quantitative feedback theory are applied to achieve the optimal system control. In this design, two frequency-dependent weighting functions are shaped and two-degree-of-freedom compensators are designed to achieve the robust optimal control.

Time-optimal maneuvers of a spacecraft between two arbitrary states in proximity of a circular reference orbit

We considered a non-stationary optimal control system with delay. Non-stationary optimal control systems are described by systems of differential or differential-algebraic equations. Variable coefficients do not allow, in general, to construct a solution to such systems in quadratures. Numerical or asymptotic methods are used to solve such systems. In this paper, asymptotic methods are used, in particular, the Feshchenko-Shkil' method for integrating of singularly perturbed systems and the Wasow’s method for systems with an unstable spectrum. In this paper, we construct a transformation that reduces the optimal delay control system to a system that does not contain terms without delay. This transformation makes it possible to integrate the system by the method of steps without solving systems of differential equations at each step. The system of equations obtained as a result of the transformation of the original system is somewhat easier to study in terms of building a solution. However, the problem of optimizing the control of both systems requires both a separate mathematical study and clarification of the practical reality of spectrum instability in such systems. If the instability of the spectrum is caused by the degeneracy of the main matrix, this leads to the unboundedness of the system solution as the small parameter approaches zero. The aforementioned growth of the solution can create emergency situations in real systems. Future research concerns solving the problem of optimal control of systems with an unstable spectrum and studying the question of the reality and physical meaning of turning points in specific systems. In future studies, it is also planned to construct an asymptotic representation of the solution of a control system with delays and degeneracies. Keywords: optimal control system, systems of differential equations with delay, asymptotic solution, turning point.

An interceptor missile with tail fins and pulse thrusters has two different control inputs: one is continuous and the other one is discontinuous. This brings some challenges to the autopilot design and stability analysis. Indirect robust control strategy is applied to handle this problem. Based on the nonlinear dynamics of the interceptor, a nonlinear system for tracking acceleration commands is presented. Treating the error between the actual inputs and the computed inputs as input uncertainty, a robust control problem is formed. It is equivalent to an optimal control problem of the nominal system with a modified cost functional. The state-dependent Riccati equation approach is applied to solve the nonlinear optimal control to obtain a solution to the robust control problem. Numerical results show the effectiveness of the proposed strategy.

The paper considers a train operation which minimizes energy consumption for leading the train from one station to the next station at the specified instant and stopping times. The minimization of the energy consumption is defined as an optimal trajectory used as a reference signal for the robust optimal control to regulate an AGT train in Automatic People Movers (APM) system. In this case, we consider the optimal control system where the performance index is of general form containing a final (terminal) cost function in addition to the integral cost function. Since the final time and the final state are fixed or specified, there are no extra boundary conditions to be used other than those given in the problem formulation. Full state feedback control law is selected to design Linear Quadratic Regulator (LQR) and PID controller. Here we also consider the parameter selection problems of the PID controller within the framework of robust control theory which provides an alternative method to optimize the setting of the PID controller. To handle uncertainty of the train parameter such as mass, robust design is applied to find a controller, for a given system, such that the closed loop system is robust. By considering the uncertain passenger arrival flow, the robust optimal control law is designed to suppress the effect of this uncertainty and the uncertain disturbances to the nominal mass of the train system. Simulations show the robustness of the proposed train control algorithms against disturbances occurring on a train parameter.

This paper deals with various duality results for a new class of constrained robust nonlinear control problems. Concretely, we introduce a minimisation problem associated with a vector of functionals of multiple integral type. In addition, the considered multi-objective controlled optimisation problem is governed by several mixed constraints generated by partial derivatives of second order and uncertain data. For this new class of control problems, we establish and characterise Wolfe, Mond-Weir and mixed type robust dual models. To this aim, by using the notion of convex multiple integral functional, generated by a controlled second-order Lagrangian with uncertain data, and the concept of robust weak efficient solution to the problem under study, we build a new mathematical context to state and prove the duality theorems.

In this paper, the nonlinear optimal control problem is investigated for spacecraft formation flying with collision avoidance. Based on a nonlinear model of formation flying, two optimal tracking control laws are proposed to ensure the formation pattern converges to the predetermined configuration. The first optimal control law which combines Lyapunov optimizing control with a trajectory-following optimization technique is developed to solve the finite-time nonlinear optimal control problem. For the second controller, an extended [Formula: see text]-[Formula: see text] method is applied to design a closed-form feedback control scheme for nonlinear control problem of spacecraft formation flying with non-standard cost functions. By taking into account the flying safety, a repulsive control scheme is incorporated in the optimal controller to ensure collision avoidance and minimize the performance index. Finally, a numerical example is performed to demonstrate the effectiveness of the proposed approaches.

In this paper, the infinite horizon optimal robust guaranteed cost control of continuous-time uncertain nonlinear systems is investigated using neural-network-based online solution of Hamilton-Jacobi-Bellman (HJB) equation. By establishing an appropriate bounded function and defining a modified cost function, the optimal robust guaranteed cost control problem is transformed into an optimal control problem. It can be observed that the optimal cost function of the nominal system is nothing but the optimal guaranteed cost of the original uncertain system. A critic neural network is constructed to facilitate the solution of the modified HJB equation corresponding to the nominal system. More importantly, an additional stabilizing term is introduced for helping to verify the stability, which reinforces the updating process of the weight vector and reduces the requirement of an initial stabilizing control. The uniform ultimate boundedness of the closed-loop system is analyzed by using the Lyapunov approach as well. Two simulation examples are provided to verify the effectiveness of the present control approach.

The authors present a state-space solution to the discrete H/sub 2/ control problem and give direct formulas for an H/sub 2/-optimal sampled-data control problem with state feedback and disturbance feedforward using the powerful lifting technique. The discrete-time H/sub 2/-optimal control is first treated via state feedback and disturbance feedforward and then via dynamic output feedback. New direct formulas for a sampled-data H/sub 2/ problem using state measurement are presented. The optimal sampled-data control is applied to a two-motor system and a comparison is made with the optimal analog control. >

We consider the robust (H???) control problem for linear time-invariant systems over unreliable communication channels that are subject to packet losses from sensors to the controller and from the controller to the actuators. Moreover, the acknowledgments of control packet transmissions are also sent over reverse unreliable communication channels, which is known as the quasi-TCP model. The robust control problem is formulated within a stochastic zero-sum dynamic game framework, from which we obtain an H??? controller that minimizes the worst-case cost function in the presence of an adversary in the dynamical system. We characterize a set of existence conditions of the H??? controller in terms of the disturbance attenuation parameter ?? and packet loss rates. For the least disturbance attenuation scenario (as ?? ??? 8), we show that the H??? control system converges to the corresponding LQG system. We also show that the H??? controller for the TCP- and UDP-cases can be arrived at as appropriate limits of the solution for the quasi-TCP problem. Numerical examples are presented to illustrate the theoretical results.