LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C₀-semigroup generation property of such an operator is proven and it is shown that the generated C₀-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.     

Insensitivity of optimal linear discrete-time regulators †

Insensitivity of the optimal linear discrete-time regulator x_k+1 = A_xk+ B_uk with quadratic performance index is investigated. Expressions are derived for finite changes which can occur in A or B without affecting the solution P of the algebraic matrix Riccati equation, and for simultaneous changes in A and B which leave both P and the optimal feedback law uk=— Kxt unaltered. If there is a predetermined variation in P, changes in A and B which leave K fixed are also given. The work complements an earlier treatment of the continuous-time case, and comparisons are noted.     

Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S^−1/2B^*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H_∞ for positive real transfer functions of the form D+S^−1/2B^*(author−A)⁻¹,B.     

Structure-preserving numerical algorithm for solving discrete-time LMI and DARS

We present a numerical algorithm to solve a discrete-time linear matrix inequality (LMI) and discrete-time algebraic Riccati system (DARS). With a given system (A,B,C,D) we associate a para-hermitian matrix pencil. Then we transform it by an orthogonal transformation matrix into a block-triangular para-hermitian form. Under either of the two assumptions (1) matrix pair (A,B) is controllable or (2) matrix pair (A,B) is reachable and (A,B,C,D) is a left invertible system, we extract the solution of LMI and DARS by the entries of the orthogonal transformation matrix.     

Optimization and pole placement for a single input controllable system

Given the system [xdot]=Ax+bu and the cost function J=dt, relations are to be determined among the open-loop characteristic polynomial, the closed-loop characteristic polynomial and the matrices A and Q. Those relations take a simple form if the system is in the standard controllable form. In this case the optimal control law can be found easily without solving the matrix Riccati equation while the minimum value of the cost function, if it is required, can be determined by solving a matrix equation of the form C ^T. X+XC= ?D     

Positivity properties in pencils of matrices†

If A and B are positive definite self-adjoint matrices, then all the matrices P_i, occurring in the expansion of an exterior power Λ^k(A+ λB) = P ₀ + P ₁λ + … + P _kλ^k are positive definite.     

Local Approximability of Max-Min and Min-Max Linear Programs

In a max-min LP, the objective is to maximise ω subject to A x≤1, C x≥ω 1, and x≥0. In a min-max LP, the objective is to minimise ρ subject to A x≤ρ 1, C x≥1, and x≥0. The matrices A and C are nonnegative and sparse: each row a _i of A has at most Δ _I positive elements, and each row c _k of C has at most Δ _K positive elements.     

On additive decompositions of solutions of the matrix equation <Emphasis Type="Italic">AXB</Emphasis>=<Emphasis Type="Italic">C</Emphasis>

This paper considers decompositions of solutions of the linear matrix equation AXB=C into the sums of solutions of other two linear matrix equations A ₁ X ₁ B ₁=C ₁ and A ₂ X ₂ B ₂=C ₂. Some applications are also given on additive decompositions of generalized inverses, as well as decompositions of solutions of matrix equations into the sums of solutions of their small equations.     

Linear systems with input nonlinearities: Global stabilization by scheduling a family of H∞-type controllers

A new global stabilization algorithm is presented for linear systems that have (A, B) stabilizable, the eigenvalues of A in the closed left-half plane, but where the controls pass through nonlinearities of a general type. This algorithm is based on a semi-global stabilization solution for the same class of systems. The global solution is constructed by dynamically scheduling the adjustable parameter of the semi-global solution according to the size of the state. The semi-global solution is a family of linear control laws generated by a family of H_∞-type algebraic Riccati equations. We show that the closed-loop state feedback system has the property that additive disturbances which converge exponentially to zero cannot produce unbounded states. This fact is used to show that, under a linear detectability assumption, the state feedback results are recovered when the solution is implemented via a standard linear-based observer.     

Model‐Free H∞ Control Design for Unknown Continuous‐Time Linear System Using Adaptive Dynamic Programming

In this paper, a new online model‐free adaptive dynamic programming algorithm is developed to solve the H_∞ control problem of the continuous‐time linear system with completely unknown system dynamics. Solving the game algebraic Riccati equation, commonly used in H_∞ state feedback control design, is often referred to as a two‐player differential game where one player tries to minimize the predefined performance index while the other tries to maximize it. Using data generated in real time along the system trajectories, this new method can solve online the game algebraic Riccati equation without requiring the full knowledge of system dynamics. A rigorous proof of convergence of the proposed algorithm is given. Finally, simulation studies on two examples demonstrate the effectiveness of the proposed method.     

Further oscillation criteria for delay partial difference equations

This paper is concerned with the partial difference equation A_m+1,n+A_m,n+1−A_m,n+p_m,nA_m−k,n−l=0,m,n=0,1,2,…,, where k and l are two positive integers, {p_m,n} is a real double sequence. Some new oscillation criteria for this equation are obtained.     

LQ tracking and disturbance rejecting with invariant zeros on the unit circle

A solution of the problem of optimal linear-quadratic (LQ) tracking and disturbance rejecting with invariant zeros on the unit circle of the plant is given, under a quite general assumption. For that purpose, we transform this problem to a problem of LQ control of an unstabilisable plant by augmentation, and then deal with weakly stabilising controls, defined as the controls such that the unstable modes of the closed-loop system are at most the unstabilisable modes of the augmented pair (A, B).

Then we solve the transformed problem by the newly introduced minimal rank weakly stabilising solution of the most general discrete-time algebraic Riccati system (DARS), associated with the system given by matrix quadruple (A, B, C, D), with unstabilisable matrix pair (A, B).

We show and illustrate by examples that there is a class of LQ tracking problems in the presence of disturbances, which cannot be solved by the existing methods, but can be solved by the introduced minimal rank weakly stabilising solution of the DARS.     

H∞ estimation for uncertain systems

This paper deals with the problem of H_∞ estimation for linear systems with a certain type of time-varying norm-bounded parameter uncertainty in both the state and output matrices. We address the problem of designing an asymptotically stable estimator that guarantees a prescribed level of H_∞ noise attenuation for all admissible parameter uncertainties. Both an interpolation theory approach and a Riccati equation approach are proposed to solve the estimation problem, with each method having its own advantages. The first approach seems more numerically attractive whilst the second one provides a simple structure for the estimator with its solution given in terms of two algebraic Riccati equations and a parameterization of a class of suitable H_∞ estimators. The Riccati equation approach also pinpoints the ‘worst-case’ uncertainty.     

Robust filtering for a class of discrete-time uncertain nonlinear systems: An H∞ approach

This paper deals with the H_∞ filtering problem for a class of discrete-time nonlinear systems with or without real time-varying parameter uncertainty and unknown initial state. For the case when there is no parametric uncertainty in the system, we are concerned with designing a nonlinear H_∞ filter such that the induced l₂ norm of the mapping from the noise signal to the estimation error is within a specified bound. It is shown that this problem can be solved via one Riccati equation. We also consider the design of nonlinear filters which guarantee a prescribed H_∞ performance in the presence of parametric uncertainties. In this situation, a solution is obtained in terms of two Riccati equations.     

Dependence-space-based attribute reduction in consistent decision tables

This paper proposes a novel approach to attribute reduction in consistent decision tables within the framework of dependence spaces. For a consistent decision table (U,Aè{d}),(U,A\cup \{d\}), an equivalence relation r on the conditional attribute set A and a congruence relation R on the power set of A are constructed, respectively. Two closure operators, T _r and T _R , and two families of closed sets, C_r{\mathcal C}_r and C_R,{\mathcal C}_R, are then constructed with respect to the two equivalence relations. After discussing the properties of C_r{\mathcal C}_r and C_R,{\mathcal C}_R, the necessary and sufficient condition for C_r=C_R{\mathcal C}_r={\mathcal C}_R is obtained and employed to formulate an approach to attribute reduction in consistent decision tables. It is also proved, under the condition C_r=C_R,{\mathcal C}_r={\mathcal C}_R, that a relative reduct is equivalent to a RR-reduction defined by Novotny and Pawlak (Fundam Inform 16:275–287, 1992).     

Feedback representations of critical controls for well-posed linear systems

This is the first part in a three part study of the suboptimal full information H^∞ problem for a well-posed linear system with input space U, state space H, and output space Y. We define a cost function Q(x₀,u)=∫〈y(s),Jy(s)〉_Yds, where y∈L²_loc( R ⁺; Y) is the output of the system with initial state x₀∈H and control u∈L²_loc( R ⁺; U), and J is a self-adjoint operator on Y. The cost function Qis quadratic in x₀ and u, and we suppose (in the stable case) that the second derivative of Q(x₀, u) with respect to u is non-singular. This implies that, for each x₀∈H, there is a unique critical control u^crit such that the derivative of Q(x₀, u) with respect to u vanishes at u=u^crit. We show that u^crit can be written in feedback form whenever the input/output map of the system has a coprime factorization with a (J, S)-inner numerator; here S is a particular self-adjoint operator on U. A number of properties of this feedback representation are established, such as the equivalence of the (J, S)-losslessness of the factorization and the positivity of the Riccati operator on the reachable subspace. © 1998 John Wiley & Sons, Ltd.     

Linear Operator Inequalities for Strongly Stable Weakly Regular Linear Systems

We consider the question of the existence of solutions to certain linear operator inequalities (Lur'e equations) for strongly stable, weakly regular linear systems with generating operators A, B, C, 0. These operator inequalities are related to the spectral factorization of an associated Popov function and to singular optimal control problems with a nonnegative definite quadratic cost functional. We split our problem into two subproblems: the existence of spectral factors of the nonnegative Popov function and the existence of a certain extended output map. Sufficient conditions for the solvability of the first problem are known and for the case that A has compact resolvent and its eigenvectors form a Riesz basis for the state space, we give an explicit solution to the second problem in terms of A, B, C and the spectral factor. The applicability of these results is demonstrated by various heat equation examples satisfying a positive-real condition. If (A, B) is approximately controllable, we obtain an alternative criterion for the existence of an extended output operator which is applicable to retarded systems. The above results have been used to design adaptive observers for positive-real infinite-dimensional systems. Date received: July 25, 1997. Date revised: February 10, 2001.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.