Analyzing policy iteration in optimal control

In this study, an optimal adaptive control approach is established to solve the robust output tracking problem of a class of continuous time uncertain linear systems based on the policy iteration (PI) in actor-critic algorithm. First, by augmenting the integral variables of the tracking error into state variables, the robust tracking problem is transformed into a robust control problem of an augmented uncertain linear system. It is proven that the robust control law of the augmented system enables the output of the considered system to track a polynomial time signal asymptotically. Second, an optimal control method in the corresponding auxiliary nominal system is established, and based on the Bellman optimality principle, PI algorithms are proposed to solve online tracking controllers for the matched and the mismatched uncertain systems. Finally, for testing the availability of the proposed approach and theoretical results, two numerical experiments are provided.

Q-learning solution for optimal consensus control of discrete-time multiagent systems using reinforcement learning

The paper presents a reinforcement learning approach for optimal tracking control of switched systems with application to a grid-tied hybrid generation system. To enhance interaction with the irregular environment, reference trajectory is learned via controller from states to optimal control. The main issue is to solve the optimal tracking control problem for a hybrid generation system consisting of multiple switched subsystems, and reinforcement learning can seek the globally optimal solution well without knowing accurate system dynamics. The investigated learning algorithm is used to generate an optimum map based on the learned ultimate value without knowledge of system parameters and obtains the optimal control law via deriving of algebraic Riccati equation (ARE) with unnecessary knowing of command generator dynamics. The optimal control solution can converge the online learning algorithm well based on policy iteration as verification in the simulation.

The nonlinear dynamic characteristics and optimal control of a giant magnetostrictive film (GMF)-shaped memory alloy (SMA) composite plate subjected to in-plane stochastic excitation are studied. GMF is prepared based on an SMA plate, and combined into a GMF—SMA composite plate. The Van der Pol item is improved to explain the hysteretic phenomena of GMF and SMA, and the nonlinear dynamics model of a GMF—SMA composite cantilever plate subjected to in-plane stochastic excitation is developed. The stochastic stability of the system is analyzed, and the steady-state probability density function of the dynamic response of the system is obtained. The condition of stochastic Hopf bifurcation is discussed, the reliability function of the system is provided, and then the probability density of the first-passage time is given. Finally, the stochastic optimal control strategy is proposed by the stochastic dynamic programming method. Numerical simulation shows that the stability of the trivial solution varies with bifurcation parameters, and stochastic Hopf bifurcation appears in the process; the system's reliability is improved through stochastic optimal control, and the first-passage time is delayed. A GMF—SMA composite plate combines the advantages of GMF and SMA, and can reduce vibration through passive control and active control effectively. The results are helpful for the engineering applications of GMF—SMA composite plates.

This paper studies data-driven learning-based methods for the finite-horizon optimal control of linear time-varying discrete-time systems. First, a novel finite-horizon Policy Iteration (PI) method for linear time-varying discrete-time systems is presented. Its connections with existing infinite-horizon PI methods are discussed. Then, both data-driven off-policy PI and Value Iteration (VI) algorithms are derived to find approximate optimal controllers when the system dynamics is completely unknown. Under mild conditions, the proposed data-driven off-policy algorithms converge to the optimal solution. Finally, the effectiveness and feasibility of the developed methods are validated by a practical example of spacecraft attitude control.

We present an accelerated algorithm for the solution of static Hamilton--Jacobi--Bellman equations related to optimal control problems. Our scheme is based on a classic policy iteration procedure, which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution. This limitation often degenerates into a behavior similar to a value iteration method, with an increased computation time. The new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling. The method starts with a coarse-mesh value iteration phase and then switches to a fine-mesh policy iteration procedure when a certain error threshold is reached. A delicate point is to determine this threshold in order to avoid cumbersome computations with the value iteration and at the same time to ensure the convergence of the policy iteration method to the optimal solution. We analyze the methods and efficient coupling in a number of examples in different dimensions, illustrating their properties.

In this paper, the Bellman equation is used to solve the optimal adaptive control of quantized linear discrete-time system with unknown dynamics. To mitigate the effect of the quantization errors, the dynamics of the quantization error bound and an update law for tuning the range of the dynamic quantizer are derived. Subsequently, by using adaptive dynamic programming technique, the infinite horizon optimal regulation problem of the uncertain quantized linear discrete-time system is solved in a forward-in-time manner without using value and/or policy iterations. The asymptotic stability of the closed-loop system is verified by standard Lyapunov stability approach in the presence of state and input quantizers. The effectiveness of the proposed method is verified via simulation.

In this paper, we study the optimal control problem for Takagi–Sugeno (T-S) fuzzy systems with disturbances. Due to the presence of disturbances in the T-S fuzzy systems, a fuzzy optimal state feedback control approach is presented by employing a two-player zero-sum game. Since the analytical optimal control solutions and the worst-case disturbance policies of T-S fuzzy systems can be boiled down to solving the algebraic Riccati equations (AREs), which are difficult to be obtained directly, a model-free policy iteration (PI) learning algorithm is proposed to obtain their approximation solutions. It is proved that the developed fuzzy optimal state feedback controller can ensure the fuzzy systems to be asymptotically stable and satisfy the disturbance attenuation condition simultaneously. Also, the designed PI learning algorithm can converge to their optimal solutions. Finally, we apply the developed fuzzy optimal state feedback control method to the truck-trailer system and the simulation results demonstrated the effectiveness of the developed scheme.

Policy iteration and value iteration are at the core of many (approximate) dynamic programming methods. For Markov Decision Processes with finite state and action spaces, we show that they are instances of semismooth Newton-type methods for solving the Bellman equation. In particular, we prove that policy iteration is equivalent to the exact semismooth Newton method and enjoys a local quadratic convergence rate. This finding is corroborated by extensive numerical evidence in the fields of control and operations research, which confirms that policy iteration generally requires relatively few iterations to achieve convergence even in presence of a large number of admissible policies. We then show that value iteration is an instance of the fixed-point iteration method and develop a novel locally accelerated version of value iteration with global convergence guarantees and negligible extra computational costs.

In this paper, an optimal tracking control approach based on adaptive dynamic programming (ADP) algorithm is proposed to solve the linear quadratic regulation (LQR) problems for unknown discrete-time systems in an online fashion. First, we convert the optimal tracking problem into designing infinite-horizon optimal regulator for the tracking error dynamics based on the system transformation. Then we expand the error state equation by the history data of control and state. The iterative ADP algorithm of policy iteration (PI) and value iteration (VI) are introduced to solve the value function of the controlled system. It is shown that the proposed ADP algorithm solves the LQR without requiring any knowledge of the system dynamics. The simulation results show the convergence and effectiveness of the proposed control scheme.

In this chapter, by using neural network (NN)-based online learning optimal control approach, a decentralized control strategy is developed to stabilize a class of continuous-time large-scale interconnected nonlinear systems . It is proven that the decentralized control strategy of the overall system can be established by adding appropriate feedback gains to the optimal control laws of the isolated subsystems . Then, an online policy iteration (PI) algorithm is developed to solve the Hamilton–Jacobi–Bellman equations related to the optimal control problem. By constructing a set of critic NNs, the cost functions can be obtained by NN approximation, followed by the control laws. Furthermore, as a generalization, an NN-based decentralized control law is developed to stabilize the large-scale interconnected nonlinear systems using an online model-free integral PI algorithm. The model-free PI approach can solve the decentralized control problem for large-scale interconnected nonlinear systems with unknown dynamics. Finally, two simulation examples are provided to illustrate the effectiveness of the present decentralized control scheme.

We consider challenging dynamic programming models where the associated Bellman equation, and the value and policy iteration algorithms commonly exhibit complex and even pathological behavior. Our analysis is based on the new notion of regular policies. These are policies that are well-behaved with respect to value and policy iteration, and are patterned after proper policies, which are central in the theory of stochastic shortest path problems. We show that the optimal cost function over regular policies may have favorable value and policy iteration properties, which the optimal cost function over all policies need not have. We accordingly develop a unifying methodology to address long standing analytical and algorithmic issues in broad classes of undiscounted models, including stochastic and minimax shortest path problems, as well as positive cost, negative cost, risk-sensitive, and multiplicative cost problems.

Observer-based dynamic optimal recoil controller design for deepwater drilling riser systems

Online actor–critic algorithm to solve the continuous-time infinite horizon optimal control problem

The innovation of optimal learning control methods is profoundly propelled due to the improvement of the learning ability. In this article, we investigate the synthesis of initialization and acceleration for optimal learning control algorithms. This approach contrasts with traditional methods that concentrate solely on either the improvement of initialization or acceleration. Specifically, we establish a novel relaxed policy iteration (PI) algorithm with self-learning horizon for stochastic optimal control. Notably, by suitably utilizing self-learning horizon, we can directly evaluate inadmissible policies to reduce the initialization burden. Meanwhile, the inadmissible policy can be rapidly optimized with few learning iterations. Then, several critical conclusions of relaxed optimal control are established by discussing algorithm convergence and system stability. Furthermore, to provide the convincing application potentials, a class of unconventional problems is effectively solved by the relaxed PI algorithm, including the dynamics with external noises and nonzero equilibrium. Finally, we present a series of nonlinear benchmarks with practical applications to comprehensively evaluate the performance of relaxed PI. The experimental results obtained from these diverse benchmarks uniformly highlight the effectiveness of self-learning horizon mechanism.