Optimal output feedback without trace

For a linear system (C,A,B) with integral quadratic cost, an optimal control problem is presented which has as its solution an output feedback control. The output feedback chosen ensures that the closed-loop cost is not worse than the open-loop cost for any initial condition, which is not guaranteed by the standard optimization method for finding output feedback (optimization with respect to the feedback matrix of an average over initial conditions of the closed-loop cost). The most severe restriction involved is thatker[C]? R[B]. Finite- and infinite-time cases are discussed.     

A vector measure approach to state feedback control in Banach resolution spaces I

An abstract version of the linear regulator-quadratic cost problem is considered for a dynamical system S, where input and output are elements of various Banach resolution spaces. Our main result is the representation of the optimal control in memoryless state feedback form. This representation is obtained as an integral with respect to a vector measure defined on the state space of S.     

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.     

Optimal output tracking control for nonlinear systems via successive approximation approach

This paper deals with the optimal output tracking control (OOTC) problem of a class of nonlinear systems whose reference input to be tracked is produced by a generalized linear exosystem. To solve the nonlinear OOTC problem, the nonlinear two-point boundary value (TPBV) problem derived from the necessary conditions of the OOTC problem is transformed into two decoupled iterative sequences of linear differential equations. The OOTC law obtained consists of accurate feedforward and feedback terms and a nonlinear compensation term. The former can be found by solving a Sylvester equation and a Riccati equation, and the latter can be approximated using a successive approximation approach (SAA). A reduced-order reference input observer is constructed to make the feedforward control law physically realizable. A simulation example is employed to illustrate the validity of the results.     

On the equivalence of maximum principle open-loop controllers and Carathéodory feedback controllers for time-delay systems

The optimal feedback gain matrix derived from the maximum principle for linear time-delay systems with quadratic cost satisfies an integral equation. On the other hand, if the extended Carathéodory lemma is used to solve the same problem, the optimal feedback gains satisfy a set of partial differential equations. It is shown that the resulting feedback gains are equivalent.     

Dynamic feedback over principal ideal domains and quotient rings

Let R be a principal ideal domain. In this paper we prove that, for a large class of linear systems, dynamic feedback over R is equivalent to static feedback over a quotient ring of R. In particular, when R is the ring of integers Z one has that the static feedback classification problem over finite rings is equivalent to the dynamic feedback classification problem over Z restricted to a special type of system.     

Optimal control of time delay parabolic systems

Various optimal control problems for linear parabolic systems with multiple constant time delays are considered. Necessary and sufficient conditions of optimality are derived for the Neumann problem. The optimal control is obtained in the feedback formMaking use of the results of Schwartz's, the representation of the optimal feedback control is given. A simple example of application is also provided     

Optimal syntheses for infinite-dimensional linear delayed state-output systems: A semicausality approach

Quadratic optimal control synthesis for infinite-dimensional delayed dynamical systems with output time delay involved in the cost functional is described. By introducing a truncation operator and associated semicausal trajectory, a new dynamical optimality principle was established. The closed-loop optimal control is given in three parts as a linear feedback: real-time state feedback, retarded state integral feedback, and initial data feedback which effects only a small time interval. The main feedback operator can be determined by solving a linear Fredholm integral equation.This research was supported by the National Science Foundation under Grant No. 8607687 and the Airforce Office of Scientific Research under Grant No. 860088.     

Geometric existence theory for the control-affine nonlinear optimal regulator

For infinite horizon nonlinear optimal control problems in which the control term enters linearly in the dynamics and quadratically in the cost, well-known conditions on the linearised problem guarantee existence of a smooth globally optimal feedback solution on a certain region of state space containing the equilibrium point. The method of proof is to demonstrate existence of a stable Lagrangian manifold M and then construct the solution from M in the region where M has a well-defined projection onto state space. We show that the same conditions also guarantee existence of a nonsmooth viscosity solution and globally optimal set-valued feedback on a much larger region. The method of proof is to extend the construction of a solution from M into the region where M no-longer has a well-defined projection onto state space.     

The sufficient conditions for continuous ɛ-optimal feedbacks in control problems with a terminal cost

The existence of continuous positional strategies of ?-optimal feedback is proved for linear optimal control problems with a convex terminal cost. These continuous feedbacks are determined from Bellman's equation in ?-perturbed control problems with an integral-terminal cost and a smooth value function. An example is given in which an ?-optimal continuous feedback does not exist. It is shown that the point limit of the ?-optimal feedbacks when ?→0 determines the optimal feedback, that is, a positional strategy and, possibly, a discontinuous strategy.     

A novel feedforward–feedback suboptimal control of linear time-delay systems

This paper presents a new approach for solving the optimal control problem of linear time-delay systems with a quadratic cost functional. In this approach, a method of successive substitution is employed to convert the original time-delay optimal control problem into a sequence of linear time-invariant ordinary differential equations (ODEs) without delay and advance terms. The obtained optimal control consists of a linear state feedback term and a forward term. The feedback term is determined by solving a matrix Riccati differential equation. The forward term is an infinite sum of adjoint vectors, which can be obtained by solving recursively the above-mentioned sequence of linear non-delay ODEs. A fast-converging iterative algorithm for this purpose is presented which provides a promising possible reduction of computational efforts. Numerical examples demonstrating the efficiency, simplicity and high accuracy of the suggested technique have been included. Simulation results reveal that just a few iterations of the proposed algorithm are required to find an accurate enough feedforward–feedback suboptimal control.     

Optimal persistent disturbance attenuation control for linear hybrid systems

In this paper, the disturbance attenuation properties for a class of linear hybrid systems are investigated, and a hybrid optimal persistent disturbance attenuation control problem is studied. First, a procedure is developed to determine the minimal l^∞$l^{\infty }$ induced gain of linear hybrid systems. However, for general hybrid systems, the termination of the procedure is not guaranteed. Then, the decidability issues are discussed. The termination of the procedure in a finite number of steps is shown for a subclass of hybrid systems with simplified discrete event dynamics, called switched linear systems. Finally, the optimal persistent disturbance attenuation controller synthesis problem is studied. It is shown that the optimal performance level can be achieved by a piecewise linear state feedback control law, and a systematic approach is proposed to design such feedback control.     

INFINITE DIMENSIONAL TRACKING OPTIMAL CONTROL VIA DYNAMIC OUTPUT FEEDBACK~*

This paper explores implemantation problems of infinite dimensional linear-quadratictracking optimal control.Based on the closed-loop result,a new formula of optimal controlexpressed by past-time state feedback is proved.From this,on the conditions of observa-bility,expressions of optimal control via dynamic output feedback are derived.The mainfeedback operator functions are given by solution of linear integral equations.     

To the Synthesis of Optimal Control Systems

The linear-quadratic optimal control problem subject to linear terminal constraints is considered. An optimal feedback control that is linear in the state variables is constructed.     

The explicit synthesis of stabilizing (time optimal) feedback controls for the attitude control of a rotating satelite

The attitude stabilization problem for a spinning satellite controlled by two small jets may be modelled as a four-dimensional, nonlinear control system, linear in the controls. The recent feedback linearization theorem of Hunt and Su may be applied to transform this system, via state feedback and a local coordinate change, to a pair of uncoupled, two-dimensional, linear systems. Feedback controls for the problem of time optimal transfer to the origin for these linear systems are explicitly calculated and then transformed to give explicit feedback controls for time optimal stabilization in the original nonlinear problem. The theory is illustrated by sample calculations.     

Determination of the optimal constant output feedback gains for linear discrete systems

The optimal constant gain output feedback control of linear time-invariant discrete systems with respect to an averaged performance criterion is discussed. The algebraic conditions necessary for minimizing the averaged performance cost are derived and a method for computing the constant optimal feedback gain is presented.     

Suboptimal linear feedback for a class of stochastic discrete-time systems

The well-known optimal linear feedback of the linear quadratic problem with jump Markov and independent disturbances is also a suboptimal feedback of the problem perturbed by a nonlinear element with a small parameter if the element is continuous. The assertion is shown and an asymptotic expansion is given.     

Stability analysis of linear parabolic systems and removement of singularities in substructure: Static feedback

We analyze stability property of a class of linear parabolic systems via static feedback. Stabilization via static feedback scheme is most difficult and challenging when both actuators and observation weights admit spillovers. This arises typically in the boundary observation-boundary feedback scheme. We propose a simple static feedback law containing a parameter γ, and enhance the stability property or achieve (slightly) stabilization. In some situations, the evolution of the substructure of finite dimension contains singularities regarding γ. We show that these singularities are removed as long as the dimension is not large.     

Sensitivity optimization problem for uncertain discrete SISO system

We consider an optimal stabilization problem for an uncertain system described as a linear time-invariant plant closed by an uncertain feedback which is a conic nonlinearity. We apply the so-called S-procedure losslessness theorem by Yakubovich to reduce the original problem to a parametrized H-infinity optimization problem. We prove that the minimal cost in this parameterized optimization is a concave function of the parameter. This fact makes applicable the standard methods of the H-infinity control theory and the convex analysis in developing optimal linear feedback design algorithms for the original problem.This work was supported by the Swedish Academy of Science and by the Göran Gustafsson Foundation. The author is grateful to the anonymous referees for many useful remarks.