On the sensitivity of the linear state regulator†

The sensitivity of the optimal response of a linear system with quadratic performance index to changes in the weighting factors in the performance index is determined. This necessitates finding the sensitivity of the feedback gain matrix K to changes in the weighting factors. It is shown that the sensitivities of K can be obtained as the solution of a set of linear matrix differential equations. Further, for time-invariant systems and an infinite time interval, it is seen that the sensitivities of K are found more simply by solving a set of linear algebraic matrix equations.     

Robust control for linear uncertain systems via linear quadratic state feedback

By the Lyapunov stability criterion and the algebraic Riccati equation, conditions of selecting the weighting matrices in the quadratic cost function are derived so that linear quadratic state feedback can exponentially stabilize a linear uncertain system, provided the uncertainties satisfy the so-called matching conditions and within a given bounding set. Furthermore, two simple but effective algorithms are proposed for systematically selecting the weighting matrices. The main features of this approach are that the uncertain system can be exponentially stabilized with prescribed exponential rate and no precompensator is needed. Two examples are given to illustrate the results.     

Constrained optimal control

The explicit form of the optimal control law of a given linear, discrete-time, time-invariant process subject to a quadratic cost criterion is well known. In some applications it is desirable that the state of a controlled dynamic process be nonnegative, given a certain class of initial disturbances. Using the controllable block companion transformation, sufficient conditions on the weighting matrices of the cost criterion are derived to ensure that the closed-loop response of the original process with the standard, unconstrained optimal feedback law will be nonnegative. It is shown that the nondiagonal elements of the transformed weighting matrices can be chosen to ensure nonnegativity     

Optimal regulators for linear systems with no control cost

This article is concerned with the theory of optimal feedback regulator for the linear system =Ax +Bu with the cost functional given byJ(u) = 1/2(Mx(T),x(T)). Due to absence of the usual positive definite quadratic cost for controls, this is a nonstandard problem.Two sets of results are presented: one for bounded and one for unbounded controls. For bounded controls, the control law is given by solving a system of coupled nonlinear differential equations of the Riccati type; and for unbounded controls, the optimal control law is determined by solving a parameterized family of matrix Riccati differential equations.     

Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems

In this paper we consider the stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The performance criterion is assumed to be formed by a linear combination of a quadratic part and a linear part in the state and control variables. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a necessary and sufficient condition under which the problem is well posed and a state feedback solution can be derived from a set of coupled generalized Riccati difference equations interconnected with a set of coupled linear recursive equations. For the case in which the quadratic-term matrices are non-negative, this necessary and sufficient condition can be written in a more explicit way. The results are applied to a problem of portfolio optimization.     

Linear quadratic networked control of uncertain polytopic systems

This paper investigates the problem of designing robust linear quadratic regulators for uncertain polytopic continuous‐time systems over networks subject to delays. The main contribution is to provide a procedure to determine a discrete‐time representation of the weighting matrices associated to the quadratic criterion and an accurate discretized model, in such a way that a robust state feedback gain computed in the discrete‐time domain assures a guaranteed quadratic cost to the closed‐loop continuous‐time system. The obtained discretized model has matrices with polynomial dependence on the uncertain parameters and an additive norm‐bounded term representing the approximation residual error. A strategy based on linear matrix inequality relaxations is proposed to synthesize, in the discrete‐time domain, a digital robust state feedback control law that stabilizes the original continuous‐time system assuring an upper bound to the quadratic cost of the closed‐loop system. The applicability of the proposed design method is illustrated through a numerical experiment. Copyright © 2015 John Wiley & Sons, Ltd.     

Neighbouring optimal guidance for aeroassisted non-coplanar orbital transfer

The fuel-optimal control problem in aeroassisted non-coplanar orbital transfer is addressed. The equations of motion for the atmospheric manoeuvre are non-linear and the optimal )nominal) trajectory and control are obtained. In order to follow the nominal trajectory under actual conditions, a neighbouring optimum guidance scheme is designed using linear quadratic regulator theory for onboard real-time implementation. One of the state variables is used as the independent variable in preference to the time. The weighting matrices in the performance index are chosen by a combination of a heuristic method and an optimal modal approach. The necessary feedback control law is obtained in order to minimize the deviations from the nominal conditons. The results are presented to a typical aeroassisted non-coplanar orbital transfer problem.     

Analytic solution for a class of linear quadratic open-loop Nash games

A method for solving the asymmetric coupled Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic games is presented. The class of games studied here is one in which the state weighting matrices in player's cost functionals are proportional to each other. By writing in a special order the necessary conditions for open-loop Nash strategy, a matrix with specific properties is derived. These properties are then exploited to solve the two-point boundary-value problem. Some special cases are discussed and a simple example is given to illustrate the solution procedure.     

Exact model-matching control of three-dimensional systems using state and output feedback

For a general state space model of three-dimensional (3-D) systems, the exact model-matching control problem via state and output feedback ia considered. A frequency domain approach is employed in which the 3-D prototype system (model) is given in transfer function matrix of the form G _m(p, w, z). The approach is based on equating the closed-loop transfer matrix function G _c(p, w, z) to G _m(p, w, z) and solving for the required feedback matrix gains through an application of Kronecker matrix product concept. We start with the static feedback case, and then treat the dynamic feedback problem for the important case of proportional plus integral plus derivative (PID) control. The approach leads to a set of linear algebraic equations, which involve the necessary and sufficient conditions for the exact model matching problem to have a solution. Two simple, but non-trivial examples, are computed.     

Risk-sensitive control of Markov jump linear systems: Caveats and difficulties

In this technical note, we revisit the risk-sensitive optimal control problem for Markov jump linear systems (MJLSs). We first demonstrate the inherent difficulty in solving the risk-sensitive optimal control problem even if the system is linear and the cost function is quadratic. This is due to the nonlinear nature of the coupled set of Hamilton-Jacobi-Bellman (HJB) equations, stemming from the presence of the jump process. It thus follows that the standard quadratic form of the value function with a set of coupled Riccati differential equations cannot be a candidate solution to the coupled HJB equations. We subsequently show that there is no equivalence relationship between the problems of risk-sensitive control and H ^∞ control of MJLSs, which are shown to be equivalent in the absence of any jumps. Finally, we show that there does not exist a large deviation limit as well as a risk-neutral limit of the risk-sensitive optimal control problem due to the presence of a nonlinear coupling term in the HJB equations.

     

Right,left characteristic sequences and column,row minimal indices of a singular pencil

For a general singular pencil sF — G?? ^m×n [s] the right, left characteristic sequences _r(F, G), _l(F, G) are defined and they are shown to be of the piecewise arithmetic progression type. The sets of column, row minimal indices ?_c(F, G), ?_r(F, G) are defined by a singular points analysis of _r(F, G), ?_l(F, G) respectively. Thus, ?_c(F, G), ?_r(F, G) emerge as numerical invariants of the ordered pair (F, G) and they may be computed by rank tests on Toeplitz matrices defined on (F, G).     

On attempts to reduce the sensitivity of the optimal linear regulator to a parameter change†

One suggested method is to adjoin a quadratic form in the sensitivity vector to the integrand of the cost functional and to find the feedback, linear in the state and sensitivity vectors, which minimizes the augmented cost functional.

Several authors have incorrectly assumed that the augmented system can be treated as another optimal regulator problem. In this paper it is shown that the problem, when correctly formulated, has no solution in the sense that there can be no finite state-independent feedback matrices which will satisfy the necessary conditions for an optimum.     

Spectral factorization for multiple input delayed discrete-time systems with applications to control

This paper deals with the linear quadratic regulation problems for the linear discrete-time systems with l input delays. The design of the optimal control law is transformed into solving one Diophantine equation and one spectral factorization with delays. A new and simple approach for the spectral factorization is proposed based on reorganized innovation analysis. The calculation of spectral factor comes down to solving l + 1 Riccati equations with the same dimension as the original systems.     

Characterizing all optimal controls for an indefinite stochasticlinear quadratic control problem

This paper is concerned with a stochastic linear quadratic (LQ) control problem in the infinite-time horizon, with indefinite state and control weighting matrices in the cost function. It is shown that the solvability of this problem is equivalent to the existence of a so-called static stabilizing solution to a generalized algebraic Riccati equation. Moreover, another algebraic Riccati equation is introduced and all the possible optimal controls, including the ones in state feedback form, of the underlying LQ problem are explicitly obtained in terms of the two Riccati equations     

Invariants and canonical forms for systems structural and parametric identification

The basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled.With reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set.Then the set of polynomial input-output models of the type P(z)y(t)=Q(z)u(t) and the equivalence relation induced by the premultiplication of P and Q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced.Since the two sets of canonical forms share common sets of complete independent invariants, very simple algebraical links between state space and input-output canonical forms can be deduced.The previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input-output sequences generated by a linear, discrete, time-invariant multivariable system.The results obtained in the identification of a real process are then reported.     

A hierarchy of LMI inner approximations of the set of stable polynomials

Exploiting spectral properties of symmetric banded Toeplitz matrices, we describe simple sufficient conditions for the positivity of a trigonometric polynomial formulated as linear matrix inequalities (LMIs) in the coefficients. As an application of these results, we derive a hierarchy of convex LMI inner approximations (affine sections of the cone of positive definite matrices of size m) of the nonconvex set of Schur stable polynomials of given degree n<m. It is shown that when m tends to infinity the hierarchy converges to a lifted LMI approximation (projection of an LMI set defined in a lifted space of dimension quadratic in n) already studied in the technical literature. An application to robust controller design is described.     

Linear systems with bounded inputs: global stabilization with eigenvalue placement

This work presents a technique for obtaining a bounded continuous feedback control function which stabilizes a linear system in a certain region. If the open-loop system has no eigenvalues with positive real part, the region of attraction of the resulting closed-loop system is all ℝⁿ, i.e., the feedback control is a global stabilizer; otherwise, the region contains an invariant (‘cylindric-like’) set where the controller does not saturate. The proposed control is a linear-like feedback control with state-dependent gains. The gains become implicitly defined in terms of a nonlinear scalar equation. The control function coincides in an ellipsoidal neighbourhood of the origin with a linear feedback law which is a solution of a linear quadratic regulator problem. This design allows eigenvalue placement in a specified region. © 1997 by John Wiley & Sons, Ltd.     

Observer-based controller for robust pole clustering in a vertical strip and disturbance rejection in structured uncertain systems

This paper presents an observer-based multi-objective robust feedback controller to achieve robust pole clustering within a vertical strip and disturbance rejection with an H_∞-norm constraint for the uncertain linear systems. The systems of interest include both matched and mismatched uncertain linear systems with structured uncertainties existing in both the system and input matrices. The controller is obtained by solving two Riccati equations (one for the controller and the other for the observer) and checking three conditions (two for robust pole clustering). A set of tuning parameters is incorporated to enhance flexibility in finding the controller. © 1998 John Wiley & Sons, Ltd.     

Uniting local and global controllers for uncertain nonlinear systems: beyond global inverse optimality

This paper provides a solution to a new problem of global robust control for uncertain nonlinear systems. A new recursive design of stabilizing feedback control is proposed in which inverse optimality is achieved globally through the selection of generalized state-dependent scaling. The inverse optimal control law can always be designed such that its linearization is identical to linear optimal control, i.e. optimal control, for the linearized system with respect to a prescribed quadratic cost functional. Like other backstepping methods, this design is always successful for systems in strict-feedback form. The significance of the result stems from the fact that our controllers achieve desired level of ‘global’ robustness which is prescribed a priori. By uniting locally optimal robust control and global robust control with global inverse optimality, one can obtain global control laws with reasonable robustness without solving Hamilton–Jacobi equations directly.     

Equilibria and steering laws for planar formations

This paper presents a Lie group setting for the problem of control of formations, as a natural outcome of the analysis of a planar two-vehicle formation control law. The vehicle trajectories are described using the planar Frenet–Serret equations of motion, which capture the evolution of both the vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The set of all possible (relative) equilibria for arbitrary G-invariant curvature controls is described (where G=SE(2) is a symmetry group for the control law), and a global convergence result for the two-vehicle control law is proved. An n-vehicle generalization of the two-vehicle control law is also presented, and the corresponding (relative) equilibria for the n-vehicle problem are characterized. Work is on-going to discover stability and convergence results for the n-vehicle problem.