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Optimal Control
Published in Jitendra R. Raol, Ramakalyan Ayyagari, Control Systems, 2020
Jitendra R. Raol, Ramakalyan Ayyagari
From (6.183), one can easily see that the optimal control law/rule/input signal is essentially the linear state-feedback control law. This necessitates that one must know the entire state vector in order to realize the optimal control. If the state vector is not available, then one must use state estimation or observer methods [5]. Also, even if the dynamic system is time-invariant, the feedback gain in (6.183) is time-varying, since the matrix P(t) is time-varying due to (6.181). One can use the steady-state solution, P of the Riccati equation, if the dynamics (6.173) are time-invariant; however, this will not work well, if the dynamics are time-varying. In that case, one needs to solve the Riccati equation recursively along with the solutions of other equations. The Riccati equation can be solved by using the transition method [5].
Differentials and First-Order Equations
Published in L.M.B.C. Campos, Non-Linear Differential Equations and Dynamical Systems, 2019
The Riccati equation is a non-linear first-order differential equation (section 3.5) that can be transformed via a change of variable (subsection 3.6.1) to a linear second-order differential equation (section 3.6). The general integral of the latter is not known, and indeed a wide variety of special functions (sections 9.4–9.9 and notes 9.1–9.47) are solutions of second-order linear differential equations with simple forms of the variable coefficients (subsection 3.6.2); this leaves scant hope of finding a general integral of the Riccati equation, although several specific cases of solution are known, for example: (i) using known particular integrals (subsections 3.5.2–3.5.5); (ii) using the relation with linear second-order equations, the simplest being constant (homogeneous) coefficients [subsection 3.6.3 (3.6.4)]; and (iii) direct solution for less general sub-cases of the Riccati equation (subsection 3.6.5).
Spacecraft Control Using Magnetic Torques
Published in Yaguang Yang, Spacecraft Modeling, Attitude Determination, and Control Quaternion-based Approach, 2019
This section discusses the attitude control system design using only magnetic torque. We consider linear quadratic regulator (LQR) design method for this problem. Riccati equation plays an important role in the LQR problem [108]. For continuous-time linear systems, the optimal solution of the LQR problem is associated with the differential Riccati equation. For discrete-time linear systems, the optimal solution of the LQR is associated with the algebraic Riccati equation. The numerical algorithms for these Riccati equations have been thoroughly studied since the work of Macfarlane [118], Kleinman [95], and Vaughan [208]. If the linear system is periodic, the optimal solution of the LQR is then associated with the periodic Riccati equation [19]. For continuous-time periodic linear system, algorithms and solutions of the differential periodic Riccati equation have been studied, for example, in [20, 21, 206]. For discrete-time periodic linear system, an efficient algorithm was proposed for the algebraic periodic Riccati equation in [70].
Multihazard life-cycle cost optimization of active mass dampers in tall buildings
Published in Structure and Infrastructure Engineering, 2023
Shalom Kleingesinds, Oren Lavan, Ilaria Venanzi
To ensure the stability of the control system, the gain matrix is computed employing the Riccati Equation. An innovative strategy was developed for this purpose, and this is another contribution of this study. The Riccati Equation considers weighting matrices Q and R, associated with the LQR control algorithm. These matrices are part of the objective function to be minimized in LQR applications, but the true optimization problem of the present research is not well represented by pre-defined Q and R. The LQR-based linear control law must be conciliated with the cost function and constraints defined for the problem. The adopted solution is defining Q and R components as indirect design variables to be optimized. This is done by considering the elements of the triangular matrices resultant from Cholesky decomposition of Q and R as additional design variables. Hence, the Q and R values of the optimized solution are expected to reflect approximately the relative importance of controlling the structure performance and limiting AMDs forces and strokes.
Experimental Validation of LQR Weight Optimization Using Bat Algorithm Applied to Vibration Control of Vehicle Suspension System
Published in IETE Journal of Research, 2022
T. Yuvapriya, P. Lakshmi, Vinodh Kumar Elumalai
The conflicting control requirements of ASS such as ride comfort and road handling motivate control designer to use the optimal state feedback design using linear quadratic theory for its capability to handle the trade-off between the speed of response and control effort [8]. Some of the advantages of LQR design include guaranteed stability, inherent robustness and methodical approach to extend to multiple-input-multiple-output (MIMO) system. Moreover, solving a quadratic cost function, which constitutes the states and control effort along with their respective input and state penalty matrices, LQR offers an optimal performance by solving the celebrated algebraic Riccati equation (ARE). However, one of the major challenges in the LQR design is the choice of state and input penalty matrices. Since the selection of weighting matrices is time-dependent and there is no standard procedure available for selecting these tuning matrices, often the task is transformed to trial and error approach, which is time consuming as well as tedious [9]. Hence, to address the problem of weight selection of LQR, over the last few years, several researchers have explored the use of nature-inspired optimization algorithms and tested the efficacy on several real world applications ranging from fuel cell [10], synchronous motor [11], inverters [12] to unified power-quality conditioner [13].
Nonlinear optimal control for a business cycles macroeconomic model of linked economies
Published in Cyber-Physical Systems, 2018
G. Rigatos, P. Siano, V. Loia, T. Ghosh, A. Krawiec
Remark 4: The linearisation presented in the article takes place at each time-step of the control algorithm, around a time-varying operating point (equilibrium). This operating point is defined by the present value of the financial system’s state vector and by the last value of the control input vector that was applied to it. The linearisation relies on first-order Taylor series expansion and on the computation of the Jacobian matrices of the state-space description of the system. On the other side, in other local model-based approaches for control of financial systems there are multiple linearisation points which are selected in an arbitrary manner. Around these operating points approximate linearisation of the nonlinear state-space description of the system takes place (Ref [14].). As a result multiple local linear models are obtained. To select the stabilising controller’s feedback gains one has (i) either to find the common solution of multiple Riccati equations, each one associated with a local linear model and a local operating point, (ii) or to solve linear matrix inequalities. Comparing the computational burden of the two approaches, one can conclude that the nonlinear optimal (H-infinity) control method which is proposed in the present article is advantageous. This is because the method simply relies the solution on the computation of one single linearisation point, as well as on the solution of one single algebraic Riccati equation[35].