Kirk, D., 1970. Optimal Control Theory, Prentice-Hall Inc., Englewood Cliffs, NJ. 21  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= f (x(t)# u(t)#t)# where x is the state variable vector and u is the control vector, and the objective function t 0 J(x(t)# u(t)#t)dt: We define the Hamiltonian H = J(x(t)# u(t)#t) t) f (x(t)# u(t)#t)# where (t) is the Lagrange multiplier vector. Using the calculus of variations (Kirk [26]# Bryson and Ho [9] we get the necessary condition for optimality u =0: 7) The Lagrange multiplier vector satisfies the following differential equation: x : 8) For the 2 DOF cobot, we define the state variables x = x# y# # v) where (x# y) is the configuration of the cobot, ....

D. E. Kirk. Optimal Control Theory. PrenticeHall Inc., 1970.

....systems that are completely controllable by each component of the input are called normal in the optimal control literature. It is well known that time optimal controllers of normal systems have no singular conditions: conditions where the optimal input is undetermined for a finite time interval [6]. In fact, according to the Pontryagin s Maximum Principle [13] for a normal linear system, the time optimal control exists,isunique, and is piecewise constant that taking values on the vertices of the feasible input set. Moreover, the optimal control has a finite number of switchings if the ....

....to the dynamic game reduces to solving two linear optimal control synthesis problems. Propositions 1 and 2 are fundamental for establishing the well posedness of our controller synthesis methodology. The proofs are due to Pontryagin [13] and can be found in many optimal control texts, such as [6]. Proposition 1 (Nonsingular Optimal Control and Disturbance) If the linear system (10) is normal with respect to both the control and disturbance, then for any x 0 #G, the optimal control u # (x 0 , are unique and piece wise constant taking values on the vertices of U, D. Proposition 2 ....

D.E. Kirk. Optimal Control Theory, An Introduction. Prentice Hall, 1970.

....3 2 1 = P v P v P v P t t P t u t X H (6) where T P P P P P ) 4 3 2 1 = is the costate. Because PMP requires that 5 ) 4 4 t t P t t t P t (7) where is an optimal control, 4 P is the forth component of optimal costate [4], the following bang bang control law corresponding to the above optimal control problem can be obtained readily, 1 ) 0 ( 1 ) 0 , 1 1 1 0 1 ) 4 max 4 max 4 max 4 4 max P for P for P for P for P for t (8) This gives us a guideline to ....

....t t y t x and local trajectory can be calculated as following. Step 1 Calculation of final time f t Case I: max ) f t 7 Because = f t t t f d t v d v t 2 max 2 0 max ) 2 ( tan( tan( 12) so f t can be calculated by numerical approximated method[4]. Case II: max ) f t Because max max tan ) f t and = max 2T t f , so can be calculated easily. Then ) f f t t y t x can be calculated by (1) Step 2 Calculation of final configuration By (12) we can get ) t . Then calling (1) the values of ) ....

D. E. Kirk, Optimal control Theory, Prentice-Hall, INC, Englewood Cliffs, New Jersey, 1970.

....in the measure update. The time propagation is performed on the transformation matrix from aircraft body axes to navigation axes (North, East, Down) denoted . The state derivative then becomes [3] where . The attitude matrix is updated with 30 Hz, using secondorder Runge Kutta integration [4]. The Euler angles are computed as [3] In the measure update, the attitude is represented by three states , describing a small correction in roll, pitch and yaw (body axes) of the attitude matrix from the C B N C B N C B N 0 r q r 0 p q p 0 = p q r = 2 C B N ....

D.E. Kirk, Optimal Control Theory, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1970

....systems that are completely controllable by each component of the input are called normal in the optimal control literature. It is well known that time optimal controllers of normal systems have no singular conditions : conditions where the optimal input is undetermined for a finite time interval [6]. In fact, according to the Pontryagin s Maximum Principle [13] for a normal linear system, the time optimal control exists, is unique, and is piecewise constant that taking values on the vertices of the feasible input set. Moreover, the optimal control has a finite number of switchings if the ....

....to the dynamic game reduces to solving two linear optimal control synthesis problems. Propositions 1 and 2 are fundamental for establishing the well posedness of our controller synthesis methodology. The proofs are due to Pontryagin [13] and can be found in many optimal control texts, such as [6]. Proposition 1 (Nonsingular Optimal Control and Disturbance) If the linear system (10) is normal with respect to both the control and disturbance, then for any x 0 2 G, the optimal control u (x 0 ; Delta) and disturbance d (x 0 ; Delta) are unique and piece wise constant taking values ....

D.E. Kirk. Optimal Control Theory, An Introduction. Prentice Hall, 1970.

....(in direction of motion) optimal applied force line of equivalent forces x y Figure 3: Iso cost force ellipses for w1=w2 = 4. We define the Hamiltonian H = J(x(t) u(t) t) t) T f (x(t) u(t) t) where (t) is the Lagrange multiplier vector. Using the calculus of variations (Kirk [7]; Bryson and Ho [3] we get the necessary condition for optimality H u = 0: 2) The Lagrange multiplier vector satisfies the following differential equation: Gamma H x : 3) For the 2 DOF cobot, we define the state variables x = x; y; v) T , where (x; y) T is the ....

D. E. Kirk. Optimal Control Theory. Prentice-Hall Inc., 1970.

....1 1.5 2 2.5 1.5 1 0.5 0 0.5 1 (b) u t x 2 x 1 Figure 3.1: Performance results for the single agent linear regulator problem: a) control history, b) phase plane. To compare the optimal and satisficing control policies, an unstable second order linear time invariant system example taken from [29] is used. Let R u = 0:05] P = I, A = 0:9974 0:0539 Gamma0:1078 1:1591 # B = 0:0013 0:0539 # Q = 0:25 0 0 0:05 # Since negligible performance is lost by using steady state gains, comparisons were made with results obtained using K1 = Gamma0:5522 Gamma 5:9690] The resulting ....

D. Kirk. Optimal Control Theory. Prentice-Hall, Englewood Cliffs, N.J., 1970.

....this technique in the context of mobile robot layered architectures, and the deliberative reactive architectural spectrum. However, such techniques as model referenced adaptive control and trajectory tracking (Kaufman et al. 1994) modelbased predictive control (Clark, 1994) optimal control (Kirk, 1970), and state space modeling (Ogata, 1967) are well known in the control engineering literature. An excellent reference on all of these topics is (Levine, 1996) None of the control techniques we propose are new. However, as the emphasis in mobile robot research begins to shift from answering ....

Kirk, D.E. 1970. Optimal Control Theory, Prentice Hall: Englewood Cliffs, NJ.

.... (Kawato, 1990; Miyata, 1988; Munro, 1987; Nguyen Widrow, 1989; Robinson Fallside, 1989; Schmidhuber, 1990) based either on the work of Werbos or our own unpublished work (Jordan, 1983; Rumelhart, 1986) There are also close ties between our approach and techniques in optimal control theory (Kirk, 1970) and adaptive control theory (Goodwin Sin, 1984; Narendra Parthasarathy, 1990) We discuss several of these relationships in the remainder of the paper, although we do not attempt to be comprehensive. Distal supervised learning and forward models This section and the following section present ....

....state transition structure (see Figure 9) In the current section we consider a more specialized problem formulation in which the focus is on particular classes of target trajectories. This formulation is based on variational calculus and is closely allied with methods in optimal control theory (Kirk, 1970; LeCun, 1987) The algorithm that results is a form of backpropagation in time (Rumelhart, Hinton, Williams, 1986) in a recurrent network that incorporates a learned forward model. The algorithm differs from the algorithm presented above in that it not only inverts the relationship between ....

Kirk, D. E. (1970). Optimal control theory. Englewood Cliffs, NJ: Prentice-Hall.

....the dots in these graphs which aggravate the situation further. 6 Goal Arbitration This section discusses the motivation behind, and the implementation of our optimal control approach to goal arbitration. The proposed approach is similar in formulation to work in classical optimal control [37] in that it seeks to determine control signals that will both satisfy constraints and optimize a performance criterion. 6.1 Introduction In addition to trajectory search, the local intelligent mobility problem involves an aspect of goal arbitration. For example, given a goal path to follow and ....

D. E. Kirk, Optimal Control Theory, Prentice Hall, 1970.

....initial state of the system at t 0 , x f the final state at time t f and u(t) u 1 (t) u r (t) the control vector. Define a performance function J(x(t) u(t) t) which is continuous over all possible trajectories of system state from x i to x f . The Pontryagin minimum principle [2] provides the optimal control vector u (t) that moves the system state from x i to x f and minimizes the performance function J(x(t) u(t) t) The Pontryagin minimum principle is stated as follows. Let the set of non linear differential equations dx 1 (t) dt = f 1 (x 1 (t) xm ....

Donald E Kirk. Optimal Control Theory. Prentice-Hall, Inc., 1970.

....problem. All these issues will be considered in the current paper. Mainly, there are two classical approaches in optimal control theory. The first approach is the maximum principle, initiated by Pontryagin et al. 17] The original maximum principle has been used and extended by many others [13, 4, 15, 11]. To get a complete treatise of dealing with constrained finite horizon optimal control problems the application of the maximum principle is incorporated. However, for the infinite horizon case a similar result is nontrivial. The convergence results between the finite and infinite horizon problem ....

....the approximations to the exact optimal control is crucial and justifies the proposed algorithm. For more details on implementation aspects, we refer to more specialised books like [12] We discretise our optimization problem in both time and state. Similar techniques can be found, for instance in [11, 12] with some illustrative examples. In later sections, we will see how this method can be used to approximate also the infinite horizon optimal feedback. This method is justified by the convergence results between the finite and infinite horizon problems. The organization of the paper is as ....

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Kirk, D.E., 1970, Optimal Control Theory, An Introduction. Prentice-Hall Inc..

....restrict terminal behaviour of a system by a target set K assuming that x(T ) 2 K and at this stage limiting our consideration to arithmetic Euclidean spaces (2. 4) When the initial conditions (t; x) and the target set are specified we define the performance measure as a time averaging functional [15] J(t; x; u fi 2 fi 1 ) g( x( Z t f 0 (s; x(s) u fi 2 fi 1 (s; Delta) ds (2.16) where x is the solution of (2.5) for the specified initial conditions (t; x) x 2 K; and is the exit time of (s; x(s) from a closure of Omega 0 0 : The function g is defined as g(t; x) ....

....on the type of the problem we consider. In the general case, similar to the control function the adjoint function depends not only on time but on the topology of the state space as well. Having defined the adjoint function, the definition of the Hamiltonian is typically assumed to be of the form [35, 15] H = f 0 f : 3.6) From the mathematical theory of optimal processes developed on the basis of the PMP [30] it follows that, for the process (x(t) u fi 2 fi 1 (t; Delta) to be optimal, it is necessary for an adjoint function (which is not identically zero) to exist such that the ....

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Kirk, D.E. Optimal Control Theory, Englewood Cliffs, N.J.: Prentice-Hall, 1970.

....implementing this result is that equation 4 is extremely difficult, and often impossible, to solve in closed form. There are numerous numerical methods for solving the optimal control problem, however, these methods depend on the initial state of the system and are therefore, open loop (c.f. [2, 3]) The basic idea behind our method is to compute successive approximations to the optimal control given in equation 3, beginning with a known stabilizing control u (0) We accomplish this by successively approximating the HJB equation 4. The remainder of the paper is organized as follows. In ....

....The example in section 4.2 shows that our method is not limited by this type of restriction. 5 0 5 1 0 1 2 3 4 5 (a) W = 1, 1] W V V t , V g 5 0 5 1 0 1 2 3 4 5 (c) W = 3, 3] W V V t V g 5 0 5 1 0 1 2 3 4 5 (d) W = 4.5, 4. 5] W V V t V g 5 0 5 1 0 1 2 3 4 5 (b) W = [ 2, 2]. W V V t V g Figure 9: Comparison between taylor series, y t and Galerkin approximation, y g . The difference between our method and Taylor series based approaches is further illustrated by a simple example. Suppose the actual cost function for a particular example is given by the equation V ....

D. E. Kirk, Optimal Control Theory. Prentice-Hall, 1970.

....with implementing this result is that (4) is extremely difficult, and often impossible, to solve in closed form. There are numerous numerical methods for solving the optimal control problem, however, these methods depend on the initial state of the system and are therefore, open loop (c.f. [8, 9]) From a practical point of view, open loop controls are undesirable. If the solution to equation (4) can be expanded in a Taylor series then [10, 11, 12, 13] provide methods for computing second order approximations of the optimal control. However, higher order approximations become ....

D. E. Kirk, Optimal Control Theory. PrenticeHall, 1970.

....Control, Generalized Hamilton Jacobi Bellman equation, Feedback Synthesis, Galerkin Approximation. Correspondence should be sent to Randy Beard, 444 CB BYU, Provo, Utah 84602, beard Omega ee.byu.edu 1 Introduction The basic mathematical theory for classical optimal control is well established [5, 14, 43, 47, 60]. If we assume full state knowledge and if the dynamics of the system are modeled by linear dynamics and the cost functional to be optimized is quadratic in the state and control, then the optimal control is a linear feedback of the states, where the control gains are obtained by solving a ....

....to solve the optimal control problem. A common approach is to numerically solve for the state and co state equations obtained from a Hamiltonian formulation of the optimal control problem. The problem can be reduced to a two point boundary value problem which can be solved by various methods [43, 60]. In [13] the two point boundary value problem is solved using the Ritz Galerkin approximation theory that is employed (in a different context) in this paper. In [36] the authors reduce the optimal bilinear control problem to successive iterations of a sequence of Riccati equations. In [3] the ....

Donald E. Kirk. Optimal Control Theory. Prentice-Hall, 1970.

....f c , the contact force, f g ; v g , the contact force and cart velocity on the ground (6) v cb , the cart velocity wrt the VMS, x c ; v c the cart position and velocity q; w; s; t are scalar weights. We now follow the derivation of an optimal control based on Pontryagin s Maximum Principle [2]. From the performance index (10) and the underlying cart VMS dynamics (9) the Hamiltonian can be defined as H = 1 2 [q(f c Gamma f g ) 2 w(v cb Gamma v g ) 2 sx 2 c tv 2 c ] p T x where p contains the Lagrange multipliers (costates) With x = H p ; p = Gamma ....

D. E. Kirk. Optimal Control Theory. PrenticeHall, N. J., 1970.

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Kirk, D., 1970. Optimal Control Theory, Prentice-Hall Inc., Englewood Cliffs, NJ. 21

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Kirk, D. E. Optimal Control Theory. Prentice-Hall, Englewood Cliffs, NJ, 1970.

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D.E. Kirk. Optimal Control Theory, An Introduction. Prentice Hall, 1970.

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Kirk, D.E., Optimal Control Theory, Prentice Hall, 1970.

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Kirk, D. E. Optimal Control Theory. Prentice-Hall, Englewood Cliffs, NJ,1970.

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D. E. Kirk, Optimal Control Theory, Prentice Hall, 1970. 26

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D. Kirk, Optimal Control Theory, Prentice-Hall, Englewood Cli#s, 1970.

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D. E. Kirk, Optimal Control Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1970.

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KIRK, D. E. Optimal Control Theory. Prentice-Hall, Englewood Cliffs, New Jersey, 1970.

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