Analytical representation of the state-dependent coefficients in the SDRE/SDDRE scheme for multivariable systems

Lin, Li-Gang; Vandewalle, Joos; Liang, Yew-Wen

doi:10.1016/j.automatica.2015.06.015

Cited by 38 publications

(16 citation statements)

References 14 publications

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“…In addition, we denote (•) ⊥ as the orthogonal complement of a vector space, V x = ∇V = (∂ V /∂x) the row vector of the partial derivatives of V : R n → R, C 1 the set of continuously differentiable functions, e 1 (resp., e 2 ) the first (resp., second) standard basis vector in R n , e n+1 the (n + 1)-th standard basis vector in R n+1 , R ≤0 the set of real numbers that are less than or equal to zero, and sgn(x) the sign function (R\{0} → {±1}) that maps to {1}, if x > 0; {−1}, otherwise. Moreover, in accordance with [23], letξ ∈ R n , we defineξ ⊥ ∈ R (n−1)×n as a matrix with orthonormal rows andξ ⊥ξ = 0. Finally, in agreement with [2,4], denote M 0 (resp., M 0), if a matrix M = M T ∈ R n×n is positive definite (resp., semidefinite).…”

Section: Notation and Problem Formulationmentioning

confidence: 99%

“…(10) (resp.,V (x, t f ) in (22)). Recent literature toward this research direction includes [28], with an intention of mutual conversation for the common good [23]. To remain focused in this presentation, we reference the survey [11] for a general picture of the scheme, while the main result of this subsection (Theorem 4.2) gives a preliminary to the optimality recovery, as based on or motivated by more recent findings that, for example, preliminarily and analytically clarify/guarantee the property of global asymptotic stability using SDRE [18,19].…”

Section: Relating To the Optimality Using Sdre/sddrementioning

confidence: 99%

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Convex Quadratic Equation

Liang

Hsieh

2020

J Optim Theory Appl

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Two main results (A) and (B) are presented in algebraic closed forms. (A) Regarding the convex quadratic equation, an analytical equivalent solvability condition and parameterization of all solutions are formulated, for the first time in the literature and in a unified framework. The philosophy is based on the matrix algebra, while facilitated by a novel equivalence/coordinate transformation (with respect to the much more challenging case of rank-deficient Hessian matrix). In addition, the parameter-solution bijection is verified. From the perspective via (A), a major application is reexamined that accounts for the other main result (B), which deals with both the infinite and finite-time horizon nonlinear optimal control. By virtue of (A), the underlying convex quadratic equations associated with the Hamilton-Jacobi equation, Hamilton-Jacobi inequality, and Hamilton-Jacobi-Bellman equation are explicitly solved, respectively. Therefore, the long quest for the constituent of the optimal controller, gradient of the associated value function, can be captured in each solution set. Moving forward, a preliminary to exactly locate the optimality using the state-dependent (resp., differential) Riccati equation scheme is prepared for the remaining symmetry condition.

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Section: Notation and Problem Formulationmentioning

confidence: 99%

Section: Relating To the Optimality Using Sdre/sddrementioning

confidence: 99%

Convex Quadratic Equation

Liang

Hsieh

2020

J Optim Theory Appl

Self Cite

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“…It is worth mentioning that different works have taken advantage of the state-dependent Riccati Eq. (22) in the analysis and synthesis of nonlinear optimal control methodologies and their applications [30][31][32][33][34].…”

Section: Remarkmentioning

confidence: 99%

Reduced‐order Observer for State‐dependent Coefficient Factorized Nonlinear Systems

Ornelas‐Tellez

Alanis

Rios

et al. 2018

Asian Journal of Control

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This paper presents a reduced-order observer for state-dependent coefficient factorized nonlinear systems. By considering that a partial knowledge of the state vector is available from measurements, estimating the full state vector may be unnecessary, which consequently reduces the order of the observer and thus avoids unnecessary implementation issues. In this manuscript, the asymptotic convergence of the proposed reduced-order observer is established when an adequate state-dependent factorization for the nonlinear system is obtained. This paper demonstrates the ease of synthesizing reduced-order observers for state-dependent coefficient factorized nonlinear systems. The effectiveness of the proposed observer is illustrated in real-time for the optimal tracking control of a linear induction motor.

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“…An alternative approach to solve nonlinear optimal control problems is the state-dependent Riccati equation (SDRE) control synthesis which dates back to 1962 [16]. The SDRE method entails factorization of the nonlinear dynamics into the product of a state-dependent matrix and the state vector thereby transforming the original nonlinear system into a pseudo-linear system [17][18][19][20][21]. In [22], a real-time LQR-based suboptimal approach for full six degree-of-freedom spacecraft control was proposed.…”

Section: Introductionmentioning

confidence: 99%

Fixed-time near-optimal control for repointing maneuvers of a spacecraft with nonlinear terminal constraints

Geng

Guo

et al. 2020

ISA Transactions

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The notion of repointing maneuver of a spacecraft operating in staring mode is provided.2.The end conditions of repointing maneuver of a spacecraft both for static target observation and moving target detections are formulated as nonlinear underdetermined equations.3.The finite-time horizon optimal control strategy is proposed to drive the spacecraft to rotate to the desired orientation at a prescribed time.4. The waypoint-based SDRE approach improves the control performance and enhances the computational efficiency.5.The effectiveness and superiorities of the proposed method is verified by numerical simulations.

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Analytical representation of the state-dependent coefficients in the SDRE/SDDRE scheme for multivariable systems

Cited by 38 publications

References 14 publications

Convex Quadratic Equation

Convex Quadratic Equation

Reduced‐order Observer for State‐dependent Coefficient Factorized Nonlinear Systems

Fixed-time near-optimal control for repointing maneuvers of a spacecraft with nonlinear terminal constraints

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