Markov jump linear quadratic dynamic programming is described and analyzed in 2 and the references cited there. This paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty. Kirk 1998, optimal control theory bryson and ho 1975. Linearquadratic approximations to dynamic programs. Lecture notes massachusetts institute of technology. Ee363 winter 200809 lecture 5 linear quadratic stochastic control linearquadratic stochastic control problem solution via dynamic programming. Let u t2rmdenote the action also called the control taken by the system at. Linearquadratic optimal control for unknown meanfield. We study stochastic linear quadratic lq optimal control problems over an infinite time horizon, allowing the cost matrices to be indefinite. Linear quadratic control dynamic programming riccati equation optimal state feedback stability and robustness the sections 9. Similar examples for lateral autopilots could be provided as well. On the relation between the hybrid minimum principle and. The calculations of the optimal control law can be done offline as in the classical linear quadratic gaussian control theory using dynamic programming, which turns out to be a special case of the new theory developed in this technical note.
On the method of dynamic programming for linearquadratic problems of optimal control in hybrid systems article pdf available in automation and remote control 705. This text presents an introduction to linear quadratic lq control theory. The markov decision process mdp is a general stochastic control problem that can be solved in principle using dynamic programming dp 16,17,21. In control theory, the linearquadraticgaussian lqg control problem is one of the most fundamental optimal control problems. Our assumptions are very general, and allow the possibility that the optimal policy may not be stabilizing the system, e. Optimal control and estimation linear quadratic regulation solution to lq optimal control problem the solution u 2 6 6 4 u 0 u 1 u n 1 3 7 7 5 h 1f.
We are concerned with the linear quadratic optimal stochastic control problem where all the coefficients of the control system and the running weighting matrices in the cost functional are allowed to be predictable but essentially bounded processes and the terminal stateweighting matrix in the cost functional is allowed to be random. This feedback is called the linear quadratic regulator lqr, and. One of the most remarkable results in linear control theory and design is that if the cost criterion is quadratic, and the optimization is over an in. Conditional state distribution as a sufficient statistic. The optimal policy is evaluated by solving an optimization problem, one that includes a current stage cost and the expected value of costtogo or value function at the next state. Dynamic programming and linear quadratic lq control discretetime and continuous time cases principle of optimality bellman 1957 linear, non linear, timevarying systems. Lecture 10 linear quadratic stochastic control with. The optimal control law is the one which minimizes the cost criterion. Searchbased motion planning for quadrotors using linear. Dynamic programming for general linear quadratic optimal. In this chapter we derive the lqr using two approaches the minimum principle and dynamic programming. Let x t2rndenote the state 1 of the system at time t. Optimization via calculus of variations differentiation is primary tool for optimization w.
This results in a set of linear constraints, so the underestimators can be found by solving a linear programming problem lp. The results are illustrated through an analytic example with linear dynamics and quadratic costs. One of the most remarkable results in linear control theory and design. However, one problem alluded to at the end of the last lecture was that the method su.
Lq theory represents one of the main approaches to the design of linear multivariable control systems, and is taught in most graduate programs in systems and control. Whilst we have been able to establish some important properties for these algorithms for example, conditions for asymptotic stability, the algorithms remain relatively complex. The optimal control for an lqr problem is easily found l if accurate models of the system and cost. Reinforcement learning for continuous time linear quadratic. Control design objectives are formulated in terms of a cost criterion. Me233 advanced control ii lecture 1 dynamic programming. Drefus, richard bellman on the birth of dynamic programming, operations. However, the main derivation of the lqg controller in appendix 9a is different. Stochastic control systems can be formulated as markov decision problems mdps with continuous state spaces and therefore we can apply the directcomparison based optimization approach to solve.
Lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality. Differential dynamic programming ddp is an iterative method that decomposes a large problem across a control sequence into a recursive series of small problems, each over an individual control at a. We consider the problem of stochastic finite and infinitehorizon linear quadratic control under power constraints. These trajectories are represented via timeparameterized polynomials, which converts the trajectory generation problem into one of nd. Stochastic linearquadratic control via semidefinite. Output measurements are assumed to be corrupted by. Introduction to dynamic programming, examples, problem formulation 2. Under suitable conditions, we prove that the value field. Dynamic programming dp is applied in order to determine the optimal management policy for a water reservoir by modeling the physical problem via a linear quadratic lq structure. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion.
Optimal control and dynamic programming duarte antunes. Introduction there is now an extensive literature on the optimal control of hybrid systems. Lecture slides dynamic programming and stochastic control. Linear quadratic optimal control in this chapter, we study a di. Optimal control and mode are the ones achieving minimum above optimal state feedback switching policy. To begin, it is handy to have the following reminder in mind. The linear quadratic gaussian lqg stochastic control problem traces back to. Markov jump linear quadratic dynamic programming advanced. Great reference optional anderson and moore, linear quadratic methods. Reinforcement learning applied to linear quadratic regulation.
Optimal linearquadratic control martin ellison 1motivation the lectures so far have described a general method value function iterations for solving dynamic programming problems. We develop a systematic approach based on semidefinite programming sdp. Dynamic programming for general linear quadratic optimal stochastic control with random coefficients. We consider a simple information structure, consisting of two interconnected linear systems, and construct the optimal controller subject to a decentralization constraint via a novel dynamic programming method. Dynamic programming was pioneered by bellman in the 1950s 1. Some refer to linear dynamics and involve a quadratic cost. The cost at every time step is a quadratic function of the state and the control signal. Bertsekas these lecture slides are based on the twovolume book.
Adaptive linear quadratic control using policy iteration. Generalized linearquadratic problems of deterministic and. Strong properties of duality are revealed which support the development of iterative approximate techniques of solution in terms of saddlepoints. Stochastic control and linear programming we consider discretetime stochastic control problems involvinga.
In this paper we study the optimization of the discretetime stochastic linear quadratic lq control problem with conic control constraints on an infinite horizon, considering multiplicative noises. The basic idea of dynamic programming is that the optimal control sequence over the entire horizon. Linear quadratic dynamic programming for water reservoir. Multilevel dynamic programming for general multiple linear. As an example, consider the problem of designing an autopilot for the longitudinal movement of an airplane, with the objective of maintaining small vertical acceleration. An upper bound to the optimal cost is obtained by another convex. Kalman filtering and linear quadratic gaussian control. Preface this is the lecture notes for the econ607 course that i am currently teaching at university of hawaii. Consideration was given to the hybrid control systems with autonomous switching, as well as the corresponding problems of the hybrid. Jacobson and mayne, differential dynamic programming, 1970. Linearquadratic programming and optimal control siam. A linear programming oriented procedure for quadratic.
This sp eci cation leads to the widely used optimal linear regulator problem, for whic h the bellman equation can be solv ed quic kly using. The dynamic programming problem with oneperiod concave quadratic returns is analysed. Approximate dynamic programming via iterated bellman. The method has been applied to problems in macroeconomics and monetary economics by 5. Can be accomplished by using a generalization of the differential. This lecture describes markov jump linear quadratic dynamic programming, an extension of the method described in the first lq control lecture. For a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. This also gives an approximation to the optimal control law. A simplified solution to the lq tracking problem is provided under mild assumptions. Moreover, the solution requires inverting a big matrix. Our objective requires differentiation of a real scalar costfunction w. Abstractwe describe a wholebody dynamic walking controller implemented as a convex quadratic program. The controller solves an optimal control problem using an approximate value function derived from a simple walking model while respecting the dynamic, input, and contact constraints of.
Dynamic programming dynamic programming and linear quadratic. This is a special case of the problem of section 4. Dynamic programming dynamic programming and linear. Optimal control for linear dynamical systems and quadratic cost. Introduction to linear quadratic regulation robert platt computer science and engineering suny at buffalo february, 20 1 linear systems a linear system has dynamics that can be represented as a linear equation. It is heavily based on stokey, lucas and prescott 1989. Dynamic economic dispatch using complementary quadratic. We also discuss ways to apply simulationbased pi to the adaptive control of linear systems with unknown model parameters. Discrete time lqr and related problems discrete time linear quadratic. Dynamic programming for general linear quadratic optimal stochastic control with random coefficients authors. However, it can be inconvenient to use the direct least squares method to calculate control because of the need to create those big matrices. This c hapter describ es the class of dynamic programming problems in whic h the return function is quadratic and the transition function is linear. Approximate dynamic programming via iterated bellman inequalities. Optimization of constrained stochastic linearquadratic.
We develop an adaptive dynamic programming adp approach to deal with the linear quadratic lq optimal control problem with unknown discretetime meanfield stochastic system in this paper. Reinforcement learning applied to linear quadratic regulation 297 time t. As wellknown, in this context the fundamental blackwells contraction theorem cannot be applied since the. A constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain. Ee363 winter 200809 lecture 10 linear quadratic stochastic control with partial state observation partially observed linearquadratic stochastic control problem. Machine learning control taming nonlinear dynamics and. The problem we address is how to define an adaptive policy that converges to the optimal control without access to such models. Dynamic programming and optimal control athena scienti. Lecture 5 linear quadratic stochastic control linear quadratic stochastic control problem solution via dynamic programming 51. This sp eci cation leads to the widely used optimal linear regulator problem, for whic h the bellman equation can be solv ed quic kly using linear algebra. On the method of dynamic programming for linearquadratic. Differential dynamic program ming ddp is an iterative method that decomposes a large problem across a control sequence into a recursive series of small problems, each over an individual control at a.
The specific algorithm we analyze is based on qlearning and it is proven to converge to the opti mal controller provided that the underlying system is controllable and a particular signal vector is per sistently excited. Automatic control 2 optimal control and estimation. At first, the meanfield stochastic lq problem is transformed into the deterministic case by system transition. Linear stochastic system linear dynamical system, over. Linearquadratic approximations to dynamic programs 1 a. Weibo gong optimization is ubiquitous in engineering and computer science. Differential dynamic programming with nonlinear constraints.
The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Moreau employ genetic algorithms to optimize linear sensor feedback in a. Ece634 optimal control of dynamic systems new syllabus instructor. Multilevel dynamic programming 61 in section 2, the optimal solution of a general multiple linear quadratic discretetime con trol problem is shown to be attained by a linear control law which is the solution of an auxiliary. The linear quadratic regulator lqr is one of the most basic and powerful methods for designing feedback control systems. Abstract pdf 874 kb 1995 an sqp algorithm for extended linearquadratic problems in stochastic programming.
Discrete time lqr and related problems discrete time linear quadratic gaussian lqg controller. It concerns linear systems driven by additive white gaussian noise. Pdf on the method of dynamic programming for linearquadratic. Lecture 10 linear quadratic stochastic control with partial. Pdf for a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. Multistage problems covering a wide variety of models in dynamic programming and stochastic programming are represented in a new way. Numerous examples highlight this treatment of the use of linear quadratic gaussian methods for control system design. Optimal control, hybrid systems, minimum principle, dynamic programming 1. Examples of stochastic dynamic programming problems. Introduction smooth trajectories obtained by minimizing jerk or snap have been widely used to control differentially at dynamical systems such as quadrotors 1, 2, 3. Dynamic programming and optimal control 4th edition.
Dynamic programming we will use dynamic programming to derive the solution of. Using the dynamic programming principle dpp, we prove that k is a. Stochastic control with a ne dynamics and extended quadratic. In their paper, the authors obtain a value function underestimator by relaxing the bellman equation to an inequality. Cs287 advanced robotics fall 2019 lecture 5 optimal. Dynamic programming and discretetime linear quadratic optimal. Deterministic systems and the shortest path problem 4. We develop optimal controller synthesis algorithms for decentralized control problems, in which individual subsystems are connected over a network. Two fundamental classes of problems in largescale linear and quadratic programming are described. Dynamic programming and linear quadratic lq control discretetime and continuous time cases principle of optimality bellman 1957 linear, nonlinear, timevarying systems. We focus on problems of linearquadratic control, in which the payoff function is quadratic.
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