This work was done at the college of engineering, trivandrum
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1. INTRODUCTION

Reusable launch vehicles are designed to be used repeatedly. Since they are reused, RLV’s are expected to dramatically reduce the cost of accessing the space. Current design objectives are to make access to space a routine event, to achieve airplane like operation. The RLV flight trajectory consists of different phases such as ascent phase, coast phase and descent phase. At these different phases, the control of the vehicle is provided by the aerodynamic control surfaces (elevon, rudder, body flap). Since the trajectories are predefined the vehicle is vulnerable to changes in the flight conditions. The navigation must be much precise and the flight control system shall steer the vehicle over the constrained trajectory, not only in the nominal operation but also with the occurrence of a fault in the vehicle. It is the task of a fault-tolerant controller to minimize the impact of the fault on the mission objectives. It is desirable to have a vehicle with automatic and self-adjusting capabilities, detecting the fault and reorganizing the demands in a manner that the predefined trajectory can be own. There is considerable motivation for devising control systems which are able to survive major equipment failures or damage. Conventionally, flight control systems are designed through linearization of the system at a series of operating points and designing separate controllers at each of those operating points. Finally, the overall flight control system is realized by bringing in the philosophy of ‘Gain Scheduling’, where the individual gains are interpolated online. Such a gain scheduling approach is found to be ineffective under extreme dispersions in aero parameter and fault occurrence. Model predictive control (MPC) offers a promising basis for fault-tolerant control. The basis for this proposal is very straightforward: since MPC relies on an explicit internal model, one can imagine dealing with failures by updating the internal model, and letting the on-line optimizer work out how to control the system in its new condition or by modifying the constraints in the MPC problem definition.
The focus of this thesis is the design of Model Predictive Controller for RLV in the pitch plane in order to track a predefined trajectory, even in the presence of actuator faults. The essence of MPC is to optimize, over the manipulated inputs, forecasts of system behavior. MPC has a distinct advantage over other types of control methodologies, such as PID, LQ and . A PID controller is also easily tuned but can only be applied to lower order SISO systems. LQ and H. can easily be applied to MIMO systems, but cannot handle system constraints in an adequate way. MPC has the ability to take into account hard constraints on the system such as actuator position limits, actuator rate limits, and state or output limits. Classical control strategies (such as PID) normally do not explicitly consider the future implications of the current control action. MPC on the other hand explicitly computes the predicted behavior over some time horizon. The MPC makes use of a process model to obtain the optimal control value by minimizing a cost function.MPC algorithms differ among themselves mainly due to the model used to represent the process being controlled, the representation of measurement noise and exogenous disturbances and the objective function to be minimized.

1.1 Literature Survey
With over 2000 industrial installations, model predictive control (MPC) is currently the most widely implemented advanced process control technology for process plants. In [8] a survey of the progress in theory and application of MPC has been done. Some of the earliest applications of MPC have been in chemical engineering dating as far back as 1970’s. In the late seventies, various articles appeared showing an interest in MPC by industry. In 1994 Berlin and Frank used MPC to control a 3 tank system applying multi input multi output systems with two inputs and two outputs [7].
Applications of MPC have been made recently to aerospace industry. Because of increased computing rates of today’s computer’s, MPC research is gaining popularity in aerospace industry.
The work [13] presents a novel method of fault-tolerant model predictive control where the response of a reference closed-loop model is followed even in the presence of an actuator
fault. This architecture is capable of redistributing, in a stable manner, the control efforts among healthy actuators, respecting their limitations. Also, a constrained guidance system is proposed that considers the calculated limitations of the autopilot as input constraints, in order to smooth the transition between two consecutive navigation legs defined by waypoints. The trajectory tracking system composed of the constrained guidance and the fault-tolerant model predictive controller is demonstrated through numerical simulations and experimental results on the VFW-614 ATTAS (Advanced Technologies Testing Aircraft System), showing adequate performance.


In [6] R.K Mehra presented a fuzzy supervised model predictive controller for tilt rotor aircraft. Current design practice for obtaining closed loop control law is based on single input/single output transfer functions with the loops being closed one a time. The resulting control law is then gain scheduled as a function of different parameters. It is widely recognized that this approach does not widely address the critical issues. The feasibility of designing a robust multivariable control system for a rigid body modes using Model Predictive control approach is demonstrated. Numerical results show that MPC can achieve excellent performance over a wide range of flight conditions.

The application of MPC for control of an unmanned helicopter is carried out in [12]. The design of controller includes an attitude controller using model predictive control and a position controller .The advantage of MPC is that it can deal with the limits of the actuators and the existing time delay of the plant. It also increases the performance of the closed loop system due to the minimization of the control error using the prediction of the future system behavior. The real flight experiments verified the performance of the controller.

A flight control system for reusable launch vehicle based on MPC has been found in [1] which shows the potential of MPC for reconfigurable control. The MPC architecture is presented first, followed by a mathematical background. Next, two controllers are presented. The first is a simplified model utilizing only the longitudinal vehicle dynamics where the focus is on the selection of the MPC design parameters: prediction horizon, simulation rates, and weighting matrices for controlling an RLV during final approach and landing. The second design controls both longitudinal and lateral dynamics where basis functions are employed to reduce the control complexity and computational time. Finally, constraints are applied to the MPC control architecture and an online quadratic programming solver is used to find the optimal control. By constraining the control input to the plant, a failed control surface in a possible abort scenario is simulated, thereby demonstrating MPC’s potential for reconfigurable control.


1.2 Objective
The main objective of this thesis is to develop a fault tolerant Model Predictive Controller in the pitch plane for RLV in the descent phase in the presence of an actuator fault.
-To obtain favorable tracking performance.
-To demonstrate constraint handling capability of MPC.
-To show the aptitude of the controller for reconfigurability

1.3 Thesis Organization
Chapter 2 deals with basic theory of Model Predictive controller, Chapter 3 deals with the mathematical model of RLV. Chapter 4 deals with development of MPC controller for RLV. Chapter 5 contains the simulation results, Chapter 6 contains the conclusion and Future works and finally chapter 7 contains the references.












2. MODEL PREDICTIVE CONTROL
Model predictive control theory is a model base optimal control technique .Every MPC system has the same general architecture composed of an internal plant model and an optimizer .Fig is a block diagram showing this structure. The MPC controller has an internal model of plant dynamics, which it uses to predict future output of the actual plant. The MPC’s optimizer differences the predicted outputs with a reference trajectory. It takes the error signal, the past input to the actual plant and any constraint imposed on the system and calculates a set of optimal future inputs for the actual plant according to a defined cost function. The set of future control inputs are the inputs that will drive the output to the desired reference set points.

Figure 2.1 General MPC Architecture
Fig 2.2 and fig 2.3 illustrate MPC theory in greater detail through a plot of the predicted output and the future inputs. The vertical axis represents the current time with the shaded and unshaded sections signifying the past and future time, respectively. The horizontal dashed line in fig .2 shows the desired reference output projected P steps into the future from time t to time t+p. The open circles represent the p predicted outputs ∆t apart. The horizontal dashed lines in Fig .3 represent hard constraints that the input values may not exceed. The output constraints in fig 2 and the input reference in fig .3 are omitted from the figures.


Figure 2.2 MPC output problem definition

Figure 2.3 MPC input problem definition
The following sections describe the MPC structure more explicitly. The discussion starts with an explanation of the prediction concept followed by a description of how the internal plant model and the optimizer make predictions and find optimal control inputs. Finally, potential benefits and problems with using MPC are assessed.

2.1 MPC Design Components
2.1.1 Prediction & Control Horizon Description
The prediction Horizon (p) is the number of predictions in the future the MPC uses to gain understanding of the effects of the control input on the system. Each prediction is equally spaced in time as determined by the designer. The control Horizon (M) is the number of free movements for the control inputs calculated in the future. A free movement is the ability of the control to assume a new value. The respective control inputs are held constant for a length of time of ∆t until the next sampling prediction. When the control horizon is an integer it must be less than or equal to the prediction horizon. If both horizons are equal, this indicates that the control is allowed to take on a different value at every prediction calculation. If on the other hand, M<P then the first M controls input are allowed to vary, and subsequent control values are held equal to the last input for the remaining prediction horizon. Fig 4 shows an example of r the control horizon with M<P. Note how the control input is allowed to vary for first M moves and is held constant thereafter. A plant with a time delay gives reason to selecting a control horizon less than the prediction horizon as it requires additional prediction samples to see the effects of the M control values.
For controllers that run at a faster rate than the prediction rate, the MPC controller calculates a set of M control inputs for its prediction, but only applies the first control value at the current time. It then discards the remaining calculated control values and calculates an entirely new set in the next iteration. In the next iteration, MPC has the benefits of seeing the effects of applying the first control value and the benefit of looking one time sample farther into the future and than it did in the previous iteration. On the other hand, for controllers that run at a slower rate than prediction rate multiple control inputs may be used.
Although MPC discards most calculated control inputs, it must optimize over the entire prediction horizon because the current control value applied will likely depend on set point changes in the future. If a constraint is imposed affecting the future, the MPC must adjust the applied control at the current time to meet that constraint. In addition ,for non-minimum phase plants (known for their characteristic initial inverse response), the MPC need to know the short and long term effects of a given control input, so optimizing over the prediction horizon is critical . Similarly plants with time delays require additional predictions to fully understand the impact of the applied control.
2.1.2 Internal Model
The internal plant model is essential to model protective control because it is the mechanism used to estimate the future output. The MPC uses the internal model to simulate how the actual plant will react to the MPC generated input signal. The input signal is calculated based on the internal plant dynamics the vehicle’s current state. Previous control value, and the future state and control reference values. By knowing how the actual plants reacts, the MPC can selected the optimal control value that will minimize errors throughout the entire prediction horizon. Compare to other optimal controllers such as LQR. MPC accepts a grater error in tracking at the current time. If by the end of prediction horizon the overall error is reduced as a result of accepting the early error.
The internal model may be linear or non linear, time varying or time invariant. The more the internal model accurately represents the actual model the more accurate the predictions and hence the control will be. Full state or partial state feedback may be employed. If partial state feedback is used, a state estimator is required. If the state estimator or the plant model is very inaccurate, the MPC will issue poor control commands resulting in poor performance. Robustness to plant uncertainty may be a design and implementation issue when perfect plant knowledge is not assumed.
2.1.3 Optimizer
The optimizer calculates the optimal control inputs for the actual plant by minimizing the cost function. When the system is constrained a closed form solution is derived. When constraints are present, the optimizer numerically finds the input minimizing the constrained cost function.
2.1.4 MPC Optimizer formulation & cost function
Consider the discrete linear time invariant plant with state and output equations:
(2.1)
(2.2)
Where is the vector of states , is the vector of inputs, is the vector of outputs.
The model predictive control is based on the solution of following optimization problem
(2.3)
The first term of equation (2.3) penalizes deviation of the tracking error. The first term penalizes the change in the control input.
The equation (2.3) can be solved with or without constraints.
2.1.5 Unconstrained closed form solution
The control input that minimizes the cost function described in (2.3) to be found.
The derivative of cost function with respect to the change in control input is set to zero and solution is found.
At the sampling instant k, k>0, the state variable vector is available through measurement which provides the current plant information.
(2.4)
From the predicted state variables, the predicted output variables are,
(2.5)
Define vectors
Y=

The equations () and () can be together represented as
(2.6)
Where
; H=
The derivative of cost function is found :
The necessary condition of the minimum J is obtained as


From which the optimal solution for the control signal as

(2.8)
where,
2.1.6 Constrained solution
The constrained problem solves the same problem as the unconstrained subjected to different constraints. Hard constraints are imposed on the control input and change in control input.Hard constraints means the MPC will not violate the constraints under any condition. Soft constraints are placed on state output and may be violated, but at a high penalty cost. When constraints are imposed on the system no closed form solution exists so an optimizer should be used
2.2 Potential Benefits and Problems of MPC
2.2.1 Potential Benefits
Model predictive control like any other controller has potential benefits and problems associated with it. The benefits of using an MPC controller are discussed first. A useful advantage of MPC is its prediction feature. MPC optimizes over the prediction horizon instead of an optimization of a current time. For this reason the MPC allows errors at the current time if allowing the errors enables the MPC to reduce future errors significantly. The MPC then uses anticipative action to its advantage. For example, if an MPC control input to the plant to accommodate for that step before it actually receives the command. This anticipation is useful as it can avoid large errors in overshoot by starting early and moving less aggressively. If constraints are imposed on the control input, the MPC will see the constraint in the prediction horizon and move preemptively to meet those constraints on the output to tailor the output to specific values. Hard constraints may also be used to avoid saturating the control system.
Often constraints on the input or output of a system will make it impossible to achieve the absolute minimum of the cost function. The controller must then find the minimum of the cost subject to the constraints. In such a situation it is advantageous to operate as close to the constraint as possible. Using anticipation, the MPC is capable of operating nominally very close to constraints because it can predict if it I going to violate a constraint. If it predicts it will violate s constraint, it takes corrective action to insure the constraint is met.
The MPC uses an internal model control framework for defining sufficient condition guaranteeing stability for stable and unstable plants under all constraints consideration. In [8] a survey of a variety of stability formulations proposed in recent MPC literature is conduced. According to the survey, the specific method used to show stability must be chosen carefully to insure that the control system being analyzed meets all of the method’s assumptions. As a consequence, ad hoc tuning of the MPC from a comprehensive set of simulations over a variety of operating conditions is found to be technique that may be applied to all MPC control laws.
MPC can control a variety of systems ranging from ones with simple dynamics to others with complex nonlinear dynamics, including unstable plants and systems with long delay times or of nonlinear dynamics, including unstable plants and system with long delay times or of non-minimum phase. The SISO case may be easily extended to MIMO. It is an established and proven control algorithm in industry. Its concepts are well known and have been used by process plant engineers and by chemical engineers for years. However, MPC is relatively new to the aerospace industry.
MPC may compensate for measurable disturbances through feed forward control. MPC controllers have multiple uses for a given problem. It can be used to augment an existing inner loop to create an MPC outer loop for the states with slow dynamics, or it may be used as the sole controller without an inner loop. These two architectures are demonstrated in the following chapter. MPC’s flexibility expands its utility.
2.2.2 Potential Problems
With any controller there exists some disadvantages. The look ahead feature of MPC comes at the cost of greater complexity when compared to other modern controllers such as LQR. MPC requires the addition of the prediction routine to the control law. Organizing the large amount of data MP requires for predictions for the unconstrained case are organized using matrices within matrices may be confusing at times. The matrices easily become very large and the associated computation time increases significantly.
Another disadvantage is that the reference output in the future must be known. This information is not always readily available, so MPC may only be applied to certain problems where that information is known.
The MPC uses the internal plant model to make predictions about the output of the real plant. With that information it selects the control inputs that minimize the deviation from reference according to the cost function. This dependence on the internal plant model requires the model to closely match the actual plant. Robustness to model uncertainty is an issue.
Finally, one of the problems with MPC lies in the selection of the various parameters. This is where MPC application can become more of an art form than a science.
Selecting the prediction horizon, control horizon, simulation rates, applying the proper constraints and populating the weighting matrices becomes increasingly more difficult. The parameters cannot be calculated directly in a closed form solution, however there do exist procedures a designer may perform to select adequate horizon lengths, weighting values, and other selection parameters.










3. MATHEMATICAL MODEL OF RLV
The mathematical modeling of rigid body dynamics of RLV during descent phase is given in this chapter. The vehicle resembles an aircraft in structure. Unlike launch vehicles there is a strong coupling between roll and yaw channels. Because of this rotational dynamics is divided in two groups. The first group comprises of pitch plane dynamics which is referred to as longitudinal plane dynamics. During longitudinal plane dynamics vehicle remain in the plane of symmetry. Yaw and roll plane dynamics are kept in second group which are modeled together and referred as lateral plane dynamics. RCS thrusters and aero control surfaces are used to control the vehicle during the descent phase. The RLV has three types of aero surfaces: an elevon, rudder and body flap. The elevons function as both elevators and ailerons. When the elevons are in unison they act as elevators and the differential deflection achieves the aileron control. Aerospace vehicles experiences aerodynamic force and moments. These forces and moments act as a disturbance and change the dynamic of the vehicle and deviates the vehicle from its desired path. The dynamics has to be modeled properly to study the dynamic behavior of the vehicle during their motion. Although the dynamics has lateral dynamics along with longitudinal dynamics, in this work the model predictive controller is designed only to control the longitudinal dynamics.
Different axis systems exist. In this work the body axis system is used. The origin is located at the center of gravity of the vehicle with the x-axis parallel to RLV base towards vehicle nose tip, the y-axis perpendicular to plane of symmetry towards the right wing and z-axis completes right-handed coordinate system.
A common design procedure for determining a control system for launch vehicle is to linearize about a nominal operating point and decouple the equations on motion for the sake of mathematical tractability. The classical control method typically consist of studying a short period model, evolved using a time-slice approach, which is assumed time invariant for a small duration, so that linear time-invariant control gains so that the desired performance is achieved. The gains are chosen for good tracking, rapid response and good damping ensuring the system stability.
In this section the mathematical model of the vehicle in the pitch plane is formulated in two levels. In the first level the plant dynamics with deal actuator is modeled and in the second level a second order actuator dynamics is added to the plant dynamics.
Figure 3.1 Geometry of vehicle in pitch plane

3.1 Plant dynamics with Ideal Actuator
A general launch vehicle can be described by two vehicle equations of motion, the translational motion which describes the total force acting on the launch vehicle and the rotational motion which describes the total moment acting on the launch vehicle.
In this level of modeling a simplified model of the vehicle dynamics can be formulated with the following assumptions:
• Rigid body alone with actuator dynamics not taken into account,
• Attitude and rate feedbacks are considered, with unity gain in rate feedback path.
The geometry of the vehicle in pitch plane is shown in the Fig.1. To compute the total pitching moment, the rate measurement is available from the sensors reading; the derivative of this rate gives the angular acceleration .The variation of moment of inertia in pitch plane, Iyy with respect to time is obtained from the moment of inertia table of flight. The total pitching moment can be computed as Iyy
From fig.1 the total pitching moment is given by,
(3.1)
The dynamics equation describing the perturbed motion of the vehicle along the nominal path is obtained as:
(3.2)
where, is the control command to elevon, µc is the control moment coefficient and µc, aerodynamic moment coefficient
Choosing the state variables as angle of attack , pitch rate and attitude angle , the space model of the launch vehicle can be obtained as:
(3.3)
And the output is given by
(3.4)




3.2 Plant Dynamics with Second Order Actuator
In this level of modeling a second order actuator is added to the plant dynamics given in (3.2) and all the assumption in the first level holds good.
To maintain the attitude of the vehicle control power plants are required. The control power plants are deflected using actuators and the actuator dynamics can be modeled by considering simple second dynamics,
(3.5)
The state space model of the launch vehicle in the pitch plane ,with actuator dynamics is;

(3.6)
And the output is given by,
(3.9)
where ,is the input to the actuator, is the actuator frequency, is the actuator damping ratio.


4. MODEL PREDICTIVE CONTROL OF RLV

In this section a Model Predictive Controller is designed for RLV in the longitudinal plane without taking into consideration any constraints. The MPC controller for the unconstrained system has a closed form solution. MPC calculates the control directly from the prediction matrices, reference signals, current state vector, and previous control vector. , angle of attack is the only state controlled by MPC leaving the remaining states uncontrolled and subsequently elevon is the only control input. Model Predictive Control has many design variables that must be chosen. These parameters include: prediction horizon, appropriate simulation rates, relative state and control weightings. The first parameter selected is the prediction horizon. It is an important parameter to select correctly because it represents the length of time the MPC will predict into the future. If the prediction horizon is too small, MPC will not have adequate knowledge of the plant and the response will not track the command well or may go unstable. If the prediction horizon is too long, the computational time becomes too great. The designed controller is analyzed using short period analysis. The short period analysis is carried out for 1) Only the plant and 2) Plant with actuator.

In short period analysis the controller is designed for the linear time invariant model of the vehicle. The system is subjected to a command signal for angle of attack. The plant with and without actuator has been simulated to track the commanded angle of attack for various time instants.
The model predictive controller is designed for the RLV considering only the longitudinal dynamics. A reference angle of attack command is provided which is system should track. Two cases are considered, one with plant dynamics only and the other including actuator dynamics along with plant. The system is simulated at different flight points. Fig. 5.1 & Fig. 5.2 show the step response of the two systems at different flight points for the angle of attack command.


Figure 5.1 Step response of the system considering only plant at time t=200s


Figure 5.2 Step response of the system considering actuator dynamics along with plant at time t=200s

Figure 5.3 Step response of the system considering only plant

Figure 5.4 Step response of the system considering actuator dynamics along with plant at time t=100s
5. CONCLUSION AND FUTURE WORK
Conclusion
Model Predictive controller was developed for RLV in the pitch plane without taking into consideration the constraints and any faults in the system. The short period analysis of system has been carried out and satisfactory performance was obtained while simulating the same in Matlab-Simulink.

Future Work
The long period analysis of the system has to be carried out. Then the control law has to be derived taking into consideration the constraints for the system input. The constraints can be placed in the control input and change in control input. Following the fault tolerant capability of model predictive controller has to be demonstrated.











6. REFERENCE
1] Dr.Piero Mitotto, Robert .C. Lepone, ”Design of a model predictive control system for a reusable launch vehicle”, AIAA Guidance, Navigation & Control Conference, August 2003,Austin,Texas
2] Liuping Wang, “Model Predictive Control System Design and Implementation Using MATLAB” , Springer-Verlag London Limited,2009.
3]D.W Clark, C .Mohtadi, ” Generalized predictive control part 1 & 2” Automaica, vol.23, 1987, pp.137-160
4] D.W Clarke, C.Mohtadi, ”Proprieties of generalized predictive control”, Automatica, vol.23, 1987,pp.137-160
5]Camacho, Eduardo F, Carlos Bordons, ”Model Predictive Control”, Springer-verlag, London, Engand 1999.
6] R. K. Mehra, R. K. Prasanth, S. Gopalaswamy ,”XV-15 Tilt rotor Flight Control System Design Using Model Predictive Control” 1998 IEEE

7]Berlin F, P.M Frank,”Design and Realization of a MIMO predictive Controller for a 3-tank System” Advances in Model Predictive Control, Oxford University Press Inc. June 1994.
8] C. E. Garcia, D. M. Prett and M. Morari, Model Predictive Control: Theory and Practice- A survey, Automatica, VoI. 25, pp335-348, 1989

9] LePome, Robert C., “Model Predictive Control for Terminal Area Energy Management and
Approach and Landing for a Reusable Launch Vehicle”. Masters Thesis, Department of Aeronautics and Astronautics, MIT, June 2002.

10] James B. Rawlings ,”Tutorial Model Predictive Control Technology” Proceedings of the
American Control Conference, San Diego, California, June 1999.

11] Qin, Joe S. and Badgwell, Thomas A. "A survey of industrial model predictive control
technology". [ed.] Elsevier Science Ltd. Control Engineering practice 11. s.l. : Control
Engineering practice 11, 2003, pp. 733‐764.

12]Jianfu Du, Yaou Zhang, ”Unmanned Helicopter Flight Controller Design by use of Model Predictive Control” WSEAS Transactions on Systems, vol. 7,February 2008.

13] Fabio A. de Almeida, Dirk Leibling,” Fault-Tolerant Model Predictive Control with Flight
Test Results on ATTAS”, AIAA Guidance, Navigation, and Control Conference10 - 13 August 2009, Chicago, Illinois