Over the last decades, autonomous robots have been considered in a variety of applications such  as persistent monitoring, package delivery and cooperative transportation. These applications often require the satisfaction of a set of complex tasks that need to be possibly performed in a timely manner. For example, in search and rescue missions, UAVs are expected to cover a set of regions within predetermined time intervals in order to increase the probability of identifying the victims of an accident. Spatio-temporal tasks of this form can be easily expressed in Signal Temporal Logic (STL), a predicate language that allow us to formally introduce time-constrained tasks such as visit area A between 0 and 5 min or robot 1 should move in a formation with robot 2 until robot 1 reaches region B between 5 and 20 sec. Existing approaches in control under spatio-temporal tasks encode the STL constraints using mixed-integer expressions. In the majority of these works, receding horizon schemes are designed and long planning horizons are considered that depend on the temporal constraints of the STL tasks. As a result, the complexity of these problems may increase with the number of the tasks or the length of the time interval within which a STL task needs to be satisfied. Other approaches, consider a limited STL fragment and propose computationally efficient feedback controllers that ensure the satisfaction of the STL task with a minimum, desired robustness. Nevertheless, these approaches do not consider actuation limitations that are always present in real-world systems and thus, yield controllers of arbitrarily large magnitude.  In this thesis, we consider the control problem under spatio-temporal constraints for systems that are subject to actuation limitations. In the first part, receding horizon control schemes (RHS) are proposed that ensure the satisfaction or minimal violation of a given set of STL tasks. Contrary to existing approaches, the planning horizon of the RHS scheme can be chosen independent of the STL task and hence, arbitrarily small, given the initial feasibility of the problem. Combining the advantages of the RHS and feedback strategies, we encode the STL tasks using control barrier functions that are designed either online or offline and design controllers that aim at maximizing the robustness of the STL task. The recursive feasibility property of the framework is established and a lower bound on the violation of the STL formula is derived. In the next part, we consider a multi-agent system that is subject to a STL task whose satisfaction may involve a large number of agents in the team. Then, the goal is to decompose the global task into local ones the satisfaction of each one of which  depends only on a given sub-team of agents. The proposed decomposition method enables the design of decentralized controllers under local STL tasks avoiding unnecessary communication among agents.  In the last part of the thesis, the coordination problem of multiple platoons is considered and related tasks such as splitting, merging and distance maintenance are expressed as Signal Temporal Logic tasks. Then, feedback control techniques are employed ensuring the satisfaction the STL formula, or alternatively minimal violation in presence of actuation limitations. / De senaste ̊artiondena har autonoma robotar sett en rad nya användningsområden, såsom ̈overvakning, paketleverans och kooperativ transport. Dessa innebär ofta att en samling komplexa uppgifter måste lösas på kort tid. Inom Search and Rescue (SAR), till exempel, krävs att drönare hinner genomsöka vissa geografiska regioner inom givna tidsintervall. Detta för att ̈oka chansen att identifierade drabbade vid en olycka. Den här typen av uppgift i tid och rum (spatio-temporal) kan enkelt uttryckas med hjälp av Signal Temporal Logic (STL). STL ̈är ett språk som tillåter oss att på ett formellt sätt formulera tidsbegränsade uppgifter, såsom besök område A mellan o och 5 minuter, eller robot 1 ska röra sig i formationtillsammans med robot 2 till dess att robot 1 når område B mellan 5 och 20 sekunder. Nuvarande lösningar till styrproblem av spatio-temporal-typen kodar STL-begränsningar med hjälp av mixed-integer-uttryck. Majoriteten av lösningarna involverar receding-horizon-metoder med långa tidshorisonter som beror av tidsbegränsningarna i STL-uppgifterna. Detta leder till att problemens komplexitet ̈ökar med antalet deluppgifter inom och tiden för STL-uppgifterna. Andra lösningar bygger på restriktiva STL-fragment och beräkningsmässigt effektiva ̊aterkopplingsregulatorer som garanterar STL-begränsningarna med minimal önskad robusthet. Dessvärre tar dessa sällan hänsyn till fysiska begräsningar hos regulatorn och ger ofta godtyckligt stora styrsignaler. I den här licentiatuppsatsen behandlar vi styrproblem med begräsningar i rum och tid, samt den ovan nämnda typen av fysiska regulatorbegränsningar. I den första delen presenterar vi receding-horizon-metoder (RHS) som uppfyller kraven i STL-uppgifter, eller minimalt bryter mot dessa. Till skillnad från tidigare lösningar så kan tidshorisonten i våra RHS-metoder väljas oberoende av STL-uppgifterna och därmed göras godtyckligt kort, så länge ursprungsproblemet ̈ar lösbart. Genom att formulera STL-uppgifterna som control barrier funktioner kan vi kombinera fördelarna hos RHS och ̊återkoppling. Vi härleder en rekursiv lösbarhetsegenskap och en undre gräns på ̈overträdelsen av STL-kraven. I den andra delen behandlar vi multi-agent-system med uppgifter i tid och rum som berör många agenter. Målet är att bryta ner den globala uppgiften i fler men enklare lokala uppgifter som var och en bara involverar en given delmängd av agenterna. Vår nedbrytning till ̊åter oss att konstruera decentraliserade regulatorer som löser lokala STL-uppgifter, och kan i och med det markant minska kommunikationskostnaderna i j̈ämförelse med centraliserad styrning. I den sista delen av uppsatsen behandlar vi samordning av flera grupper. Vi uttrycker uppgifter såsom delning, sammanslagning och avståndshållning med hjälp av STL, och utnyttjar sedan ̊aterkoppling för att uppfylla eller minimalt bryta mot kraven. / <p>QC 20220311</p>

There is a demand for renewable energy in today’s society. Wave energy is a nearly untapped source of renewable energy. Ocean Harvesting Technologies AB (OHT) is currently developing a device that can be used to convert wave energy into electricity. The device is a Wave Energy Converter of the type point absorber. Their concept is a floating buoy that is connected to the seafloor via a Power Take-Off (PTO) unit. The PTO unit is equipped with generators, which are used to convert kinetic energy of the buoy into electricity. The objective of this thesis is to control the generators to optimize the performance of the system. OHT was interested in knowing how their system performs under the influence of a controller based on MPC. Hence an MPC-controller is constructed in this thesis. The developed controller functions by predicting the states (position and velocity) of the buoy over a finite time (e.g. $5s$). Then the controller uses the predictions to find a control force that makes the system behave optimally for the next $5$ seconds. A requirement from the company is that the controller should find the control force based on how the buoy is predicted to move in 3 Degrees Of Freedom (DOF). Further, the controller should be able to operate in real-time. To meet the company’s requirements, the following is done. A linear 3ODF model of the system is derived. This is used to predict the states of the buoy in the controller. An MPC algorithm is constructed. In this, the linear model and constraints of the system are included. Then, a simulation environment is built. This is including a non-linear model of OHT’s system. The performance of the controller is tested in the simulation environment. Real-time implementation is an important aspect of the controller. The computational time required by the controller is measured in the simulations. The results imply that the controller stands a chance of being real-time implementable. However, make sure that it can be run in real-time it should be tested on the control unit that OHT plans to use in their system. A linear model of the system is used in the controller to predict the future states o the buoy. It is important that the predictions are accurate for the controller to control the system in an optimal way. Hence, the validity of the linear model is investigated. The controller is managing to predict some states better than others. However, the controller is doing a fine job with controlling the system in terms of generated power. Thus the linear model is considered to be valid for the application. An advantage with controllers based on MPC is the simplicity of tuning the controller. Changes of settings in the controller have a predictable effect on the results. For the settings found in this thesis, the system is performing fine in terms of power generation. However, more work is needed to find more optimal settings.

With the ongoing trend in moving the upper levels of the automation hierarchy to the cloud, there has been investigation into supplying industrial automation as a cloud based service. There are many practical considerations which pose limitations on the feasibility of the idea. This research investigates some of the requirements which would be needed to implement a platform which would facilitate competition between different controllers which would compete to control a process in real-time. This work considers only the issues relating to implementation of the philosophy from a control theoretic perspective, issues relating to hardware/communications infrastructure and cyber security are beyond the scope of this work. A platform is formulated and all the relevant control requirements of the system are discussed. It is found that in order for such a platform to determine the behaviour of a controller, it would need to simulate the controller on a model of the process over an extended period of time. This would require a measure of the disturbance to be available, or at least an estimate thereof. This therefore increases the complexity of the platform. The practicality of implementing such a platform is discussed in terms of system identification and model/controller maintenance. A model of the surge tank from SibanyeStillwater’s Platinum bulk tailings treatment (BTT) plant, the aim of which is to keep the density of the tank outflow constant while maintaining a steady tank level, was derived, linearised and an input-output controllability analysis performed on the model. Six controllers were developed for the process, including four conventional feedback controllers (decentralised PI, inverse, modified inverse and H¥) and two Model Predictive Controllers (MPC) (one linear and another nonlinear). It was shown that both the inverse based and H¥ controllers fail to control the tank level to set-point in the event of an unmeasured disturbance. The competing concept was successfully illustrated on this process with the linear MPC controller being the most often selected controller, and the overall performance of the plant substantially improved by having access to more advanced control techniques, which is facilitated by the proposed platform. A first appendix presents an investigation into a previously proposed switching philosophy [15] in terms of its ability to determine the best controller, as well as the stability of the switching scheme. It is found that this philosophy cannot provide an accurate measure of controller performance owing to the use of one step ahead predictions to analyse controller behaviour. Owing to this, the philosophy can select an unstable controller when there is a stable, well tuned controller competing to control the process. A second appendix shows that there are cases where overall system performance can be improved through the use of the proposed platform. In the presence of constraints on the rate of change of the inputs, a more aggressive controller is shown to be selected so long as the disturbance or reference changes do not cause the controller to violate these input constraints. This means that switching back to a less aggressive controller is necessary in the event that the controller attempts to violate these constraints. This is demonstrated on a simple first order plant as well as the surge tank process. Overall it is concluded that, while there are practical issues surrounding plant and system identification and model/controller maintenance, it would be possible to implement such a platform which would allow a given plant access to advanced process control solutions without the need for procuring the services of a large vendor. / Dissertation (MEng)--University of Pretoria, 2020. / Electrical, Electronic and Computer Engineering / MEng / Unrestricted

<p>Automated Vehicle Path Following Control (PFC) is an advanced control system that can regulate the vehicle into a collision-free region in the presence of other objects on the road. Common collision avoidance functions, such as forward collision warning and automatic emergency braking, have recently been developed and equipped on production vehicles. However, it is impossible to develop a perfectly precise vehicle model when the vehicle is driving. The most PFC did not consider uncertainties in the vehicle model, external disturbances, and parameter variations at the same time. To address the issues associated with this important feature and function in autonomous driving, a new vehicle PFC is proposed using a robust model predictive control (MPC) design technique based on matrix inequality and the theoretical approach of the hybrid $\&$ switched system. The proposed methodology requires a combination of continuous and discrete states, e.g. regulating the continuous states of the AV (e.g., velocity and yaw angle) and discrete switching of the control strategy that affects the dynamic behaviors of the AV under different driving speeds. Firstly, considering bounded model uncertainties, norm-bounded external disturbances, the system states and control matrices are modified. In addition, the vehicle time-varying longitudinal speed is considered, and a vehicle lateral dynamic model based on Linear Parameter Varying (LPV) is established by utilizing a polytope with finite vertices. Then the Min-Max robust MPC state feedback control law is obtained at every timestamp by solving a set of matrix inequalities which are derived from Lyapunov stability and the minimization of the worst-case in infinite-horizon quadratic objective function. Compared to adaptive MPC, nonlinear MPC, and cascade LPV control, the proposed robust LPV MPC shows improved tracing accuracy along vehicle lateral dynamics. Finally, the state feedback switched LPV control theory with separate Lyapunov functions under both arbitrary switching and average-dwell-time (ADT) switching conditions are studied and applied to cover the path following control in full speed range. Numerical examples, tracking effectiveness, and convergence analysis are provided to demonstrate and ensure the control effectiveness and strong robustness of the proposed algorithms.</p>

Cette thèse est dédiée à l’analyse du retard (de calcul ou induit par la communication), qui représente un des paramètres sensibles, et qui doit être pris en compte, pour la mise en œuvre de la Commande Prédictive en temps réel d’un processus dynamique. Dans la première partie, nous avons abordé le problème d’existence des ensembles D-invariants et avons fourni par la suite des conditions nécessaires et/ou suffisantes pour l’existence de ces ensembles. En outre, nous avons détaillé quelques nouvelles idées sur la construction des ensembles D-invariants en utilisant des algorithmes itératifs et d’autres algorithmes basés sur des techniques d’optimisation à deux niveaux. La seconde partie a été consacrée à l’étude du problème de robustesse des systèmes linéaires discrets affectés par un retard variable en boucle fermée avec un contrôleur affine par morceaux défini sur une partition polyédrale de l’espace d’état. L’étude a porté sur l’analyse de la fragilité d’une telle loi commande en présence du retard dans la boucle. Nous avons décrit les marges d’invariance robustes définies comme étant le plus grand sous-ensemble de l’incertitude paramétrique pour lequel l’invariance positive est garantie par rapport à la dynamique en boucle fermée en présence du retard. La dernière partie de cette thèse s’est articulée autour de la conception des lois de commande prédictives avec un attention particulière aux modèles linéaires discrets décrivant des dynamiques affectées par des contraintes en présence du retard. Nous avons proposé plusieurs méthodes offrant différentes solutions au problème de stabilisation locale sans contrainte. Afin d’assurer la stabilité et de garantir la satisfaction des contraintes, nous avons exploité le concept d’invariance et à l’aide du formalisme "ensemble terminal-coût terminal", un problème d’optimisation a été formulé où les états sont forcés d’atteindre l’ensemble maximal admissible d’états retardés/D-invariant à la fin de l’horizon de prédiction. Enfin, nous avons étudié le problème de stabilisation des systèmes continus commandés en réseau soumis à des retards incertains et éventuellement variant dans le temps. Nous avons montré que les ensembles λ-D-contractifs peuvent être utilisés comme ensembles cibles où la stratégie de commande consiste en un simple problème de programmation linéaire ’LP’ qui peut être résolu en ligne. / The research conducted in this thesis has been focusing on Model Predictive Control (MPC) and the implication of network induced time-varying delays. We have addressed, in the first part of this manuscript, the existence problem and the algorithmic computation of positive invariant sets in the state space of the original discrete delay difference equation. The second part of these thesis has been devoted to the study of the robustness problem for a specific class of dynamical systems, namely the piecewise affine systems, defined over a polyhedral partition of the state space in the presence of variable input delay. The starting point was the construction of a predictive control law which guarantees the existence of a non-empty robust positive invariant set with respect to the closed-loop dynamic. The variable delay inducing in fact a model uncertainty, the objective was to describe the robust invariance margins defined as the largest subset of the parametric uncertainty for which the positive invariance is guaranteed with respect to the closed-loop dynamics in the presence of small and large delays. The last part has been dedicated to Model Predictive Control design with a specific attention to linear discrete time-delay models affected by input/state constraints. The starting point in the analysis was the design of a local stabilizing control law using different feedback structures. We proposed several design methods offering different solutions to the local unconstrained stabilization problem. In order to ensure stability and guarantee input and state constraints satisfaction of the moving horizon controller, the concept of positive invariance related to time-delay systems was exploited. Using the "terminal setterminal cost" design, the states were forced to attain the maximal delayed-state admissible set at the end of the prediction horizon. Finally, we have investigated the stabilization problem of Networked Control Systems ’NCSs’ subject to uncertain, possibly time-varying, network-induced delays. We showed that λ-D-contractive sets can be used as a target sets in a set induced Lyapunov function control fashion where a simple Linear Programming ’LP’ problem is required to be solved at each sampling instance.

Les dernières années ont permis des développements très rapides, tant au niveau de l’élaboration que de l’application, d’algorithmes de commande prédictive non linéaire (CPNL), avec une gamme relativement large de réalisations industrielles. Un des obstacles les plus significatifs rencontré lors du développement de cette commande est lié aux incertitudes sur le modèle du système. Dans ce contexte, l’objectif principal de cette thèse est la conception de lois de commande prédictives non linéaires robustes vis-à-vis des incertitudes sur le modèle. Classiquement, cette synthèse peut s’obtenir via la résolution d’un problème d’optimisation min-max. L’idée est alors de minimiser l’erreur de suivi de la trajectoire optimale pour la pire réalisation d'incertitudes possible. Cependant, cette formulation de la commande prédictive robuste induit une complexité qui peut être élevée ainsi qu’une charge de calcul importante, notamment dans le cas de systèmes multivariables, avec un nombre de paramètres incertains élevé. Pour y remédier, une approche proposée dans ces travaux consiste à simplifier le problème d’optimisation min-max, via l’analyse de sensibilité du modèle vis-à-vis de ses paramètres afin d’en réduire le temps de calcul. Dans un premier temps, le critère est linéarisé autour des valeurs nominales des paramètres du modèle. Les variables d’optimisation sont soit les commandes du système soit l’incrément de commande sur l’horizon temporel. Le problème d’optimisation initial est alors transformé soit en un problème convexe, soit en un problème de minimisation unidimensionnel, en fonction des contraintes imposées sur les états et les commandes. Une analyse de la stabilité du système en boucle fermée est également proposée. En dernier lieu, une structure de commande hiérarchisée combinant la commande prédictive robuste linéarisée et une commande par mode glissant intégral est développée afin d’éliminer toute erreur statique en suivi de trajectoire de référence. L'ensemble des stratégies proposées est appliqué à deux cas d'études de commande de bioréacteurs de culture de microorganismes. / The last few years have led to very rapid developments, both in the formulation and the application of Nonlinear Model Predictive Control (NMPC) algorithms, with a relatively wide range of industrial achievements. One of the most significant challenges encountered during the development of this control law is due to uncertainties in the model of the system. In this context, the thesis addresses the design of NMPC control laws robust towards model uncertainties. Usually, the above design can be achieved through solving a min-max optimization problem. In this case, the idea is to minimize the tracking error for the worst possible uncertainty realization. However, this robust approach tends to become too complex to be solved numerically online, especially in the case of multivariable systems with a large number of uncertain parameters. To address this shortfall, the proposed approach consists in simplifying the min-max optimization problem through a sensitivity analysis of the model with respect to its parameters, in order to reduce the calculation time. First, the criterion is linearized around the model parameters nominal values. The optimization variables are either the system control inputs or the control increments over the prediction horizon. The initial optimization problem is then converted either into a convex optimization problem, or a one-dimensional minimization problem, depending on the nature of the constraints on the states and commands. The stability analysis of the closed-loop system is also addressed. Finally, a hierarchical control strategy is developed, that combines a robust model predictive control law with an integral sliding mode controller, in order to cancel any tracking error. The proposed approaches are applied through two case studies to the control of microorganisms culture in bioreactors.

Recent decades have witnessed the phenomenal success of model predictive control (MPC) in a wide spectrum of domains, such as process industries, intelligent transportation, automotive applications, power systems, cyber security, and robotics. For constrained dynamic systems subject to uncertainties, robust MPC is attractive due to its capability of effectively dealing with various types of uncertainties while ensuring optimal performance concerning prescribed performance indices. But most robust MPC schemes require prior knowledge on the uncertainty, which may not be satisfied in practical applications. Therefore, it is desired to design robust MPC algorithms that proactively update the uncertainty description based on the history of inputs and measurements, motivating the development of adaptive MPC. This dissertation investigates four problems in robust and adaptive MPC from theoretical and application points of view. New algorithms are developed to address these issues efficiently with theoretical guarantees of closed-loop performance. Chapter 1 provides an overview of robust MPC, adaptive MPC, and self-triggered MPC, where the recent advances in these fields are reviewed. Chapter 2 presents notations and preliminary results that are used in this dissertation. Chapter 3 investigates adaptive MPC for a class of constrained linear systems with unknown model parameters. Based on the recursive least-squares (RLS) technique, we design an online set-membership system identification scheme to estimate unknown parameters. Then a novel integration of the proposed estimator and homothetic tube MPC is developed to improve closed-loop performance and reduce conservatism. In Chapter 4, a self-triggered adaptive MPC method is proposed for constrained discrete-time nonlinear systems subject to parametric uncertainties and additive disturbances. Based on the zonotope-based reachable set computation, a set-membership parameter estimator is developed to refine a set-valued description of the time-varying parametric uncertainty under the self-triggered scheduling. We leverage this estimation scheme to design a novel self-triggered adaptive MPC approach for uncertain nonlinear systems. The resultant adaptive MPC method can reduce the average sampling frequency further while preserving comparable closed-loop performance compared with the periodic adaptive MPC method. Chapter 5 proposes a robust nonlinear MPC scheme for the visual servoing of quadrotors subject to external disturbances. By using the virtual camera approach, an image-based visual servoing (IBVS) system model is established with decoupled image kinematics and quadrotor dynamics. A robust MPC scheme is developed to maintain the visual target stay within the field of view of the camera, where the tightened state constraints are constructed based on the Lipschitz condition to tackle external disturbances. In Chapter 6, an adaptive MPC scheme is proposed for the trajectory tracking of perturbed autonomous ground vehicles (AGVs) subject to input constraints. We develop an RLS-based set-membership based parameter to improve the prediction accuracy. In the proposed adaptive MPC scheme, a robustness constraint is designed to handle parametric and additive uncertainties. The proposed constraint has the offline computed shape and online updated shrinkage rate, leading to further reduced conservatism and slightly increased computational complexity compared with the robust MPC methods. Chapter 7 shows some conclusion remarks and future research directions. / Graduate

No description available.

Medium-Voltage Direct-Current (MVDC) power system has been considered as the trending technology for future All-Electric Ships (AES) to produce, convert and distribute electrical power. With the wide employment of highrequency power electronics converters and motor drives in DC system, accurate and fast assessment of system dynamic behaviors , as well as the optimization of system transient performance have become serious concerns for system-level studies, high-level control designs and power management algorithm development. The proposed technique presents a coordinated and automated approach to determine the system adjustment strategy for naval power systems to improve the transient performance and prevent potential instability following a system contingency. In contrast with the conventional design schemes that heavily rely on the human operators and pre-specified rules/set points, we focus on the development of the capability to automatically and efficiently detect and react to system state changes following disturbances and or damages by incooperating different system components to formulate an overall system-level solution. To achieve this objective, we propose a generic model-based predictive management framework that can be applied to a variety of Shipboard Power System (SPS) applications to meet the stringent performance requirements under different operating conditions. The proposed technique is proven to effectively prevent the system from instability caused by known and unknown disturbances with little or none human intervention under a variety of operation conditions. The management framework proposed in this dissertation is designed based on the concept of Model Predictive Control (MPC) techniques. A numerical approximation of the actual system is used to predict future system behaviors based on the current states and the candidate control input sequences. Based on the predictions the optimal control solution is chosen and applied as the current control input. The effectiveness and efficiency of the proposed framework can be evaluated conveniently based on a series of performance criteria such as fitness, robustness and computational overhead. An automatic system modeling, analysis and synthesis software environment is also introduced in this dissertation to facilitate the rapid implementation of the proposed performance management framework according to various testing scenarios.

Shipboard Power System (SPS) is known as an independent controlled small electric network powered by the distributed onboard generation system. Since many electric components are tightly coupled in a small space and the system is not supported with a relatively stronger grid, SPS is more susceptible to unexpected disturbances and physical damages compared to conventional terrestrial power systems. Among different distribution configurations, power-electronic based DC distribution is considered the trending technology for the next-generation U.S. Navy fleet design to replace the conventional AC-based distribution. This research presents appropriate control management frameworks to improve the Medium-Voltage DC (MVDC) shipboard power system performance. Model Predictive Control (MPC) is an advanced model-based approach which uses the system model to predict the future output states and generates an optimal control sequence over the prediction horizon. In this research, at first, a centralized MPC is developed for a nonlinear MVDC SPS when a high-power pulsed load exists in the system. The closed-loop stability analysis is considered in the MPC optimization problem. A comparison is presented for different cases of load prediction for MPC, namely, no prediction, perfect prediction, and Autoregressive Integrated Moving Average (ARIMA) prediction. Another centralized MPC controller is also designed to address the reconfiguration problem of the MVDC system in abnormal conditions. The reconfiguration goal is to maximize the power delivered to the loads with respect to power balance, generation limits and load priorities. Moreover, a distributed control structure is proposed for a nonlinear MVDC SPS to develop a scalable power management architecture. In this framework, each subsystem is controlled by a local MPC using its state variables, parameters and interaction variables from other subsystems communicated through a coordinator. The Goal Coordination principle is used to manage interactions between subsystems. The developed distributed control structure brings out several significant advantages including less computational overhead, higher flexibility and a good error tolerance behavior as well as a good overall system performance. To demonstrate the efficiency of the proposed approach, a performance analysis is accomplished by comparing centralized and distributed control of global and partitioned MVDC models for two cases of continuous and discretized control inputs.