Linear Quadratic Research Articles

Linear-quadratic optimal control of second-order parabolic systems

This paper considers the linear-quadratic (LQ) optimal control problem for systems governed by a class of second-order parabolic partial differential equations (PDEs). This problem is significant in many areas of mathematics, control, engineering and so on, and thus has received great attention in the past decades. Different from the previous articles where the operator is applied to present the controller, the main contribution of this paper is to propose the discretisation-then-continuousization method, which is explicit and implementable. The solvability condition of the LQ optimal control problems is given based on a set of differential Riccati equations, and an explicit numerical calculation way of these equations and the design of the optimal controller are provided.

Presenting a model based on knowledge management in modeling and optimizing the dynamic viscosity of MWCNT (30%) - ZnO (70%) / SAE 40 oil hybrid nano-lubricant with response surface methodology (RSM) for industrial uses

Thermophysical characteristics of MWCNT-ZnO (30–70%)/SAE40 hybrid nanofluid (HNF) using knowledge management cycle and RSM in different solid volume fractions (SVFs), temperature and shear rates (SRs) have been investigated. Emphasis on knowledge management through the use of fast and high-precision methods has led to the use of the RSM for modeling the viscosity of HNF. Different models such as linear model, 2FI, Quadratic, Cubic, Quartic, Fifth and Sixth models are presented. Fifth order model is better than others. Values of R2, Predicted R² and Adjusted R² for all three parameters are 0.9998, which displays good precision of the modeling. The knowledge management cycle is used to evaluate the model to increase sufficient confidence. Examining the matching graphs, deviation coefficient and standard deviation parameters displays good precision of modeling. MOD value is also within ± 2%. With the help of knowledge management in finding the best mode, the optimal viscosity value is equal to 209.53 mPa.sec in temperature, SVF and SR, respectively 31.156 °C, 0.063 % and 933.923 s−1.

Explainable Artificial Intelligence Using Expressive Boolean Formulas

We propose and implement an interpretable machine learning classification model for Explainable AI (XAI) based on expressive Boolean formulas. Potential applications include credit scoring and diagnosis of medical conditions. The Boolean formula defines a rule with tunable complexity (or interpretability) according to which input data are classified. Such a formula can include any operator that can be applied to one or more Boolean variables, thus providing higher expressivity compared to more rigid rule- and tree-based approaches. The classifier is trained using native local optimization techniques, efficiently searching the space of feasible formulas. Shallow rules can be determined by fast Integer Linear Programming (ILP) or Quadratic Unconstrained Binary Optimization (QUBO) solvers, potentially powered by special-purpose hardware or quantum devices. We combine the expressivity and efficiency of the native local optimizer with the fast operation of these devices by executing non-local moves that optimize over the subtrees of the full Boolean formula. We provide extensive numerical benchmarking results featuring several baselines on well-known public datasets. Based on the results, we find that the native local rule classifier is generally competitive with the other classifiers. The addition of non-local moves achieves similar results with fewer iterations. Therefore, using specialized or quantum hardware could lead to a significant speedup through the rapid proposal of non-local moves.

Linear Quadratic Optimal Robust Multivariable Proportional-Integral Control Design for a Boiler-Turbine Unit

Abstract Boiler-turbine unit is a highly nonlinear, coupled, and ill-conditioned system. Moreover, the presence of physical constraints, such as actuator magnitude and rate limits, makes the system difficult to control. This paper employs a fixed multivariable proportional-integral (PI) controller of the I–P structure for robust linear quadratic (LQ) compensation of a nonlinear boiler-turbine benchmark. In order to ensure that a single PI controller works for the whole operating region of this nonlinear system, the linearized model of the system is represented as a norm-bounded, time-varying uncertain system. The ranges of the uncertain parameters of this linearized model are determined from different operating points of the nonlinear system. To design the PI controller for the uncertain system, first, it is transformed into a state feedback design for an augmented uncertain system and then the state feedback gains satisfying some LQ performance limit are computed by solving a linear matrix inequality (LMI) problem. As the uncertainty in the feedforward matrix of the linearized model cannot be considered in the above design process, an LMI-based method is developed to check if the designed PI controller performance in H∞ sense is close to the one if the neglected uncertainty is included. The performance of the controller is tested on the nonlinear boiler-turbine unit under several operating conditions and physical constraints. Comparisons are also made with some existing PI controllers, to show the superiority of the proposed robust PI controller.

Decentralized control for discrete‐time mean‐field systems with multiple controllers of delayed information

AbstractIn this paper, the finite horizon asymmetric information linear quadratic (LQ) control problem is investigated for a discrete‐time mean field system. Different from previous works, multiple controllers with different information sets are involved in the mean field system dynamics. The coupling of different controllers makes it quite difficult in finding the optimal control strategy. Fortunately, by applying the Pontryagin's maximum principle, the corresponding decentralized control problem of the finite horizon is investigated. The contributions of this paper can be concluded as follows: For the first time, based on the solution of a group of mean‐field forward and backward stochastic difference equations (MF‐FBSDEs), the necessary and sufficient solvability conditions are derived for the asymmetric information LQ control for the mean field system with multiple controllers. Furthermore, by the use of an innovative orthogonal decomposition approach, the optimal decentralized control strategy is derived, which is based on the solution to a non‐symmetric Riccati‐type equation.

Linear Quadratic Optimal Control of Discrete-Time Stochastic Systems Driven by Homogeneous Markov Processes

Random terms in many natural and social science systems have distinct Markovian characteristics, such as Markov jump-taking values in a finite or countable set, and Wiener process-taking values in a continuous set. In general, these systems can be seen as Markov-process-driven systems, which can be used to describe more complex phenomena. In this paper, a discrete-time stochastic linear system driven by a homogeneous Markov process is studied, and the corresponding linear quadratic (LQ) optimal control problem for this system is solved. Firstly, the relations between the well-posedness of LQ problems and some linear matrix inequality (LMI) conditions are established. Then, based on the equivalence between the solvability of the generalized difference Riccati equation (GDRE) and the LMI condition, it is proven that the solvability of the GDRE is sufficient and necessary for the well-posedness of the LQ problem. Moreover, the solvability of the GDRE and the feasibility of the LMI condition are established, and it is proven that the LQ problem is attainable through a certain feedback control when any of the four conditions is satisfied, and the optimal feedback control of the LQ problem is given using the properties of homogeneous Markov processes and the smoothness of the conditional expectation. Finally, a practical example is used to illustrate the validity of the theory.

Radio-Immune Response of Spatially Fractionated Radiotherapy for VMAT Lattice Plans

To evaluate radio-immune response of spatially fractionated radiotherapy (SFRT) for large tumors using VMAT Lattice technique in terms of tumor volume under irradiation and dose fractionation schemes after SFRT. Eleven patients treated with SFRT from a single institution were retrospectively replanned to deliver 15Gy in single fraction using Lattice technique. High dose regions are defined by multiple spheres with the diameter of 1.25 to 1.5cm and their vertex space of 3.0 to 4.0cm inside of GTV. VMAT plans with multiple arcs were developed for SFRT. Four palliative fractionation regimens of 200cGy x 12 (EQD2 = 24Gy with a/b of 10Gy), 400cGy x 5 (23.3Gy), 600cGy x 3 (24Gy) and 800cGy x2 (24Gy) and four definitive regimens of 200cGy x 24 (EQD2 = 48Gy), 400cGy x 10 (46.7Gy), 600cGy x 6 (48Gy) and 800cGy x 4 (48Gy) were considered for radiotherapy to follow SFRT. Linear quadratic (LQ) model is compared with radio-immune (RI) response model in which the activation of cytotoxic T lymphocytes, tumor immune suppression capability and immunotherapy drugs can be considered. Tumor regrowth time (TRT, time to tumor regrowth to the original volume after treatment) from each model was compared as a measure of benefit achieved from the application of SFRT. The average volume of GTVs in this study was 776cc (range 58-2944cc). Three different SFRT plans (2D GRID technique with conventional collimator, 2D GRID with step & shoot IMRT, and 3D Lattice) were developed for each patient but only Lattice plans were considered in this study since they produced comparable dose modulation inside the tumor but only Lattice significantly reduced skin and critical organ dose. Radio-immune response model always expects longer TRT than LQ model. For palliative regimens, TRT of RI model is longer than that of LQ model by 14.5±9.9, 15.1±10.6, 17.2±12.4, 17.5±12.8 days for each fractionation scheme. When Lattice plan of 15Gy is delivered before the palliative treatment, the difference becomes 25.9±15.3, 31.5±23.3, 36.7±27.6, 37.5±28.5 days. The benefit of SFRT from RI response is only about 10-20 days. Interestingly, RI response is inversely proportional to tumor volume. When curative dose is considered, the difference of TRT is drastically changed from 25.9±9.8, 460.7±285.8, 1180.8±985.7, 1512.0±1327.5 days to 20.7±4.4, 449.0±411.7, 1725.4±2171.0, 3517.7±4531.7 days. The benefit of SFRT from RI response appears larger for large tumor with hypo-fractionation in definitive regimens. The benefit of SFRT is significant for large tumors with hypo fractionation in the definitive regimens when radio-immune response model is considered which is not apparent in LQ model. Radio-immune response model may help to guide the development of successful treatment scheme large tumor volumes.

Radiobiological Insights of Staged Brachytherapy Using Eye Plaque for Large Uveal Melanoma

I-125 radioactive seeds embedded in an eye plaque is used to deliver the radiation dose of 85Gy to the apex of the tumor. However, it is challenging to treat a lesion in height (>10 mm) because of a high dose (> 500 Gy) to sclera. A staged radiotherapy (SRT) has been implemented successfully in stereotactic radiosurgery when a single dose is risky for a bigger sized lesion. The hypothesis of this work is that the SRT works for eye plaque brachytherapy of uveal melanoma to break the single fraction into two. To apply the concept of SRT, the following questions need to be answered: What are the two fraction doses? What is the time interval between the two fractions? In this work, the two split fraction doses and time interval are calculated using the "incomplete-repair' model. Equation (1) is the linear-quadratic (LQ) model. f(x) is the cell survival curve (SC). Equation (2) and (3) are radiobiological model developed dealing with the split doses by considering the incomplete repair. Ln(f(x)) = -αx - βx2 (1) Ln(fn(x, Ɵ)) = nLn(f(x))-nβx2hn(Ɵ) (2) hn(Ɵ) = (2/n)[ Ɵ/(1- Ɵ)][n-(1- Ɵn)/(1- Ɵ)] (3) Ɵ = exp(-µΔt) (4) x represents the fraction dose, Ɵx is the initial segment of the survival curve, Ɵ determines the rate at which repair proceeds. n is fraction number and µ represent repair constant which equals to ln2/T1/2. T1/2 is the half-life of I-125. For continuous dose vt delivered at dose rate v for time t, with repair constant µ, the LQ model is modified as Ln (f(vt, µ)) = -α(vt) -β(vt)2g(µt) (5) g(µt) = 2[µt-1+exp(-µt)]/(µt)2 (6) Assuming biological effective dose (BED) is the same for the single fraction and split two fractions, SC when n = 1 should be same as when n = 2. Therefore, the equation (7) is derived. Ɵ = [(α/β)x1 - 2(α/β)(x2)2 + (x1)2]/2(x2)2 - 1 (7) x1 is single fraction dose of 85Gy, x2 is the split dose to two fractions. α/β ratio of 10 Gy is used. By fitting the equation (7), the ranges of doses and time intervals were determined. The fraction doses and time interval to get the same BED as single fraction of 85 Gy are determined from 45 Gy to 50 Gy with time interval of 22 days to 76 days for staged brachytherapy eye plaque (SBEP). Our result is matched the treatment protocol for SBEP reported, e.g., 1st fraction dose of 40Gy and 50 Gy of 2nd fraction with 73 days of time interval are used. The max. dose of scleral dose is 328 Gy compared to the single fraction scleral dose of 543Gy. This work could provide guidelines in SBEP in dosage and time interval. The height of tumor matters the most in the SBEP since the taller the tumor, the higher dose to the sclera is. Reference.

Equivalent Uniform Dose (EUD) and the Evaluation of Cell Survival in Spatially Fractionated Radiotherapy (SFRT)

SFRT has shown promise as a treatment modality to decrease normal tissue sparing without compromising tumor coverage, i.e., an increase in the therapeutic window compared to more conventional uniform radiation therapy (RT). The aim of this work is to examine and test several alternative bio-dosimetric parameters for the prediction of cell survival for normal-tissue and tumor cell lines irradiated in vitro with uniform and microbeam radiotherapy (MRT). A bespoke tungsten collimator with 50 parallel, 50 µm wide slits and 400 µm slit spacing was mounted into an x-ray cabinet. Human fibroblast (MRC5) and two human tumor cell lines (LN18 and A549) were irradiated with a range of doses (< 10 Gy) for uniform and MRT (50um slits, 400um center spacing) using kV X-rays. Average, mean and valley dose as useful predictive metrics of cell survival are compared to the equivalent uniform dose (EUD) with biological parameters estimated from uniform-dose experiments. We find that EUD, with linear-quadratic (LQ) model parameters, is more predictive for survival after SFRT than maximum, minimum or average dose. The maximum and average doses are correlated very poorly with in vitro cell survival. The difference in cell survival between uniform and MRT irradiation as a function of EUD is cell-type and dose dependent. The report results suggest that MRT is more effective at cell killing of tumor-cell lines than uniform irradiation for both tumor cell lines. However, MRT is less effective at killing normal tissue cells than uniform irradiation. EUD is a superior predictor of in vitro cell survival than other metrics sometimes used in the SFRT literature, including mean dose, maximum dose, and valley dose. The reported studies provide some evidence that SFRT may increase the therapeutic ratio by producing spatial dose distributions that effectively reduce normal-tissue damage with little or no change in biological damage to tumor cells. Additional studies are needed to further extend and generalize our results and to test our conclusions against a larger dose range, low and high linear energy transfer (LET) radiations and additional cell lines.

Non-Zero Sum Nash Game for Discrete-Time Infinite Markov Jump Stochastic Systems with Applications

This paper is to study finite horizon linear quadratic (LQ) non-zero sum Nash game for discrete-time infinite Markov jump stochastic systems (IMJSSs). Based on the theory of stochastic analysis, a countably infinite set of coupled generalized algebraic Riccati equations are solved and a necessary and sufficient condition for the existence of Nash equilibrium points is obtained. From a new perspective, the finite horizon mixed robust H2/H∞ control is investigated, and summarize the relationship between Nash game and H2/H∞ control problem. Moreover, the feasibility and validity of the proposed method has been proved by applying it to a numerical example.

A Human-Like Shared Driving Strategy in Lane-Changing Scenario Using Cooperative LPV/MPC

Linear quadratic (LQ) game has been widely used in shared driving to resolve conflicts between drivers and driving automation systems (DAS). However, the system model in LQ game is assumed to be linear time-invariant, which does not consider the effect of changing longitudinal speed on the lateral control. A novel cooperative LPV/MPC approach is proposed, which characterizes the longitudinal and lateral coupled vehicle control in a dynamic game. The stability of the cooperative LPV/MPC system is analyzed in the presence of inconsistent targets of driver and DAS. Driver behavior is studied using 2,701 lane-changing cases extracted from naturalistic driving data (NDD). A human-like risk assessment method is achieved according to the driver behavior characteristics. Finally, a shared driving strategy in lane-changing scenarios is presented based on the cooperative LPV/MPC algorithm and the human-like risk assessment method. Simulation validations show that the shared driving strategy can ensure vehicle stability when longitudinal control, lateral control and dynamic game are performed simultaneously. In addition, the shared driving strategy can realize the driving weight gradual handover, which can help to better assist the driver in risk avoidance.

Robust Optimal Control for Discrete-Time LTI Systems Over Multiple Additive White Gaussian Noise Channels

Unlike the traditional point-to-point control structure, control system design through network media introduces additional constraints due to limited capacities of the used communication channels. In this paper, we address the optimal performance control problem for linear uncertain discrete-time systems over multiple additive white Gaussian noise (AWGN) channels, where mean power constraints are imposed on the actuator side. The desired controller is aimed to robustly stabilize the plant, to minimize the linear quadratic (LQ) cost, and to satisfy diverse pre-specified mean power limits associated with individual elements of the control law, simultaneously. Within the framework of system level synthesis, the performance objective and power constraints, together with relevant upper bounds, over all admissible uncertainties are characterized in terms of system responses of the closed-loop system and statistics of the communication channels. Based on this characterization, numerically tractable algorithms are proposed for controller synthesis. The counterpart for deterministic systems is also discussed as a special case with corresponding algorithms being provided. Moreover, the relative performance loss incurred by the size of model uncertainties and the finite truncation is explicitly established. An example is also included to show the effectiveness of the present results.

Indefinite LQ optimal control for stochastic Takagi–Sugeno fuzzy system under sensor data scheduling: Finite‐horizon case

AbstractThe present paper considers the finite‐horizon indefinite linear quadratic (LQ) control problem for stochastic Takagi–Sugeno (T‐S) fuzzy systems with input delay. In this paper, we consider the presence of sensor data scheduling, which imposes a communication energy constraint and necessitates optimal state estimation for measurements. Then, by utilizing dynamic programming principles, the stochastic LQ problem under consideration can be solved, while the optimal control policy is developed in terms of the unique solutions to a set of coupled difference Riccati equations (CDREs). Specifically, for simple delay‐free case, the linear matrix inequalities based conditions are also proposed, whose feasibility is shown to be equivalent to the well‐posedness of the indefinite LQ control under consideration. As an application, our theoretic analysis is extended to study the intermittent observation model caused by random denial‐of‐service attack.

The mean‐field linear quadratic optimal control problem for stochastic systems controlled by impulses

AbstractIn the present note, we state and solve the linear quadratic (LQ) control problem for a class of McKean–Vlasov stochastic differential equations for which the control actions are of impulsive nature. After reformulating the original optimization problem using an orthogonal decomposition of the state variables, we introduce an adequately defined system of two coupled matrix Lyapunov‐type differential equations with jumps. Such equations are central for the definition of the optimal feedback gains corresponding to the closed‐loop representation of the optimal LQ control.

Decentralized Linear Quadratic Systems With Major and Minor Agents and Non-Gaussian Noise

A decentralized linear quadratic system with a major agent and a collection of minor agents is considered. The major agent affects the minor agents, but not vice versa. The state of the major agent is observed by all agents. In addition, the minor agents have a noisy observation of their local state. The noise processes is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">not</i> assumed to be Gaussian. The structures of the optimal strategy and the best linear strategy are characterized. It is shown that major agent's optimal control action is a linear function of the major agent's MMSE (minimum mean squared error) estimate of the system state while the minor agent's optimal control action is a linear function of the major agent's MMSE estimate of the system state and a “correction term” which depends on the difference of the minor agent's MMSE estimate of its local state and the major agent's MMSE estimate of the minor agent's local state. Since the noise is non-Gaussian, the minor agent's MMSE estimate is a non-linear function of its observation. It is shown that replacing the minor agent's MMSE estimate by its LLMS (linear least mean square) estimate gives the best linear control strategy. The results are proved using a direct method based on conditional independence, common-information-based splitting of state and control actions, and simplifying the per-step cost based on conditional independence, orthogonality principle, and completion of squares.

Radiobiological Meta-Analysis of the Response of Prostate Cancer to Different Fractionations: Evaluation of the Linear-Quadratic Response at Large Doses and the Effect of Risk and ADT.

The purpose of this work was to investigate the response of prostate cancer to different radiotherapy schedules, including hypofractionation, to evaluate potential departures from the linear-quadratic (LQ) response, to obtain the best-fitting parameters for low-(LR), intermediate-(IR), and high-risk (HR) prostate cancer and to investigate the effect of ADT on the radiobiological response. We constructed a dataset of the dose-response containing 87 entries/16,536 patients (35/5181 LR, 32/8146 IR, 20/3209 HR), with doses per fraction ranging from 1.8 to 10 Gy. These data were fit to tumour control probability models based on the LQ model, linear-quadratic-linear (LQL) model, and a modification of the LQ (LQmod) model accounting for increasing radiosensitivity at large doses. Fits were performed with the maximum likelihood expectation methodology, and the Akaike information criterion (AIC) was used to compare the models. The AIC showed that the LQ model was superior to the LQL and LQmod models for all risks, except for IR, where the LQL model outperformed the other models. The analysis showed a low α/β for all risks: 2.0 Gy for LR (95% confidence interval: 1.7-2.3), 3.4 Gy for IR (3.0-4.0), and 2.8 Gy for HR (1.4-4.2). The best fits did not show proliferation for LR and showed moderate proliferation for IR/HR. The addition of ADT was consistent with a suppression of proliferation. In conclusion, the LQ model described the response of prostate cancer better than the alternative models. Only for IR, the LQL model outperformed the LQ model, pointing out a possible saturation of radiation damage with increasing dose. This study confirmed a low α/β for all risks.

Model-free policy iteration approach to NCE-based strategy design for linear quadratic Gaussian games

This paper presents a novel completely model-free method to solve continuous-time linear quadratic (LQ) mean field games (MFGs). We consider the setting where agents can only access their own states and are coupled via their individual infinite-horizon discounted costs. For such games, a set of Nash certainty equivalence-based strategies, determined by two gain matrices and the aggregate population effect, i.e. mean field (MF) state, are developed, providing an ɛ-Nash equilibrium under certain assumptions. Note that the gain matrices are solved from two algebraic Riccati equations (AREs), respectively, and the MF state is given by the linear ordinary differential equation (ODE). All the solutions of these three equations require the exact model of system dynamics. To implement the NCE-based strategy with completely unknown system dynamics, the proposed approach develops a single-agent policy iteration (PI) and a single-agent MF generator to solve the above parameters. Specifically, first, model-based iterative equations with guaranteed convergence are developed to approximate two gain matrices. Then, the requirement of dynamical model is removed from the iteration process via the integral reinforcement learning (IRL) technique. A model-free algorithm is conducted by using measured data from a selected agent as reinforcement signals over a certain time interval. Finally, based on the obtained gain matrices, the MF state is computed offline depending on samples collected from the same agent. A numerical example is given to verify the effectiveness of the proposed method.

Stochastic linear quadratic optimal control problems with expectation-type linear equality constraints on the terminal states

A stochastic linear quadratic (LQ) optimal control problem with an expectation-type linear equality constraint on the terminal state is considered. Under the solvability condition on a stochastic Riccati equation and a surjectivity condition on the linear constraint mapping, the constrained stochastic LQ problem is solved completely by the Lagrangian duality theory. Some equivalent characterizations of the surjectivity condition are discussed by the controllability theory of linear control systems. Especially, the equivalence between the surjectivity condition and a Kalman-type rank condition is proved under proper conditions.

Backward Stackelberg Games with Delay and Related Forward–Backward Stochastic Differential Equations

In this paper, we study a kind of Stackelberg game where the controlled systems are described by backward stochastic differential delayed equations (BSDDEs). By introducing a new kind of adjoint equation, we establish the sufficient verification theorem for the optimal strategies of the leader and the follower in a general case. Then, we focus on the linear–quadratic (LQ) backward Stackelberg game with delay. The backward Stackelberg equilibrium is presented by the generalized fully coupled anticipated forward–backward stochastic differential delayed Equation (AFBSDDE), which is composed of anticipated stochastic differential equations (ASDEs) and BSDDEs. Moreover, we obtain the unique solvability of the AFBSDDE using the continuation method. As an application of the theoretical results, the pension fund problem with delay effect is considered.

UPAFuzzySystems: A Python Library for Control and Simulation with Fuzzy Inference Systems

The main goal of control theory is input tracking or system stabilization. Different feedback-computed controlled systems exist in this area, from deterministic to soft methods. Some examples of deterministic methods are Proportional (P), Proportional Integral (PI), Proportional Derivative (PD), Proportional Integral Derivative (PID), Linear Quadratic (LQ), Linear Quadratic Gaussian (LQG), State Feedback (SF), Adaptative Regulators, and others. Alternatively, Fuzzy Inference Systems (FISs) are soft-computing methods that allow using the human expertise in logic in IF–THEN rules. The fuzzy controllers map the experience of an expert in controlling the plant. Moreover, the literature shows that optimization algorithms allow the adaptation of FISs to control different processes as a black-box problem. Python is the most used programming language, which has seen the most significant growth in recent years. Using open-source libraries in Python offers numerous advantages in software development, including saving time and resources. In this paper, we describe our proposed UPAFuzzySystems library, developed as an FISs library for Python, which allows the design and implementation of fuzzy controllers with transfer-function and state-space simulations. Additionally, we show the use of the library for controlling the position of a DC motor with Mamdani, FLS, Takagi–Sugeno, fuzzy P, fuzzy PD, and fuzzy PD-I controllers.