EST7

• The cruise control system of a car is a common feedback system encountered in everyday life.
• Insulin in the bloodstream is considered an input.
• The model uses two compartments, one representing the concentration of glucose in the bloodstream and the other representing the concentration of insulin in the interstitial fluid.
• The cruise control system attempts to maintain a constant velocity in the presence of disturbances primarily caused by changes in the slope of a road.
• The controller compensates for these unknowns by measuring the speed of the car and adjusting the throttle appropriately.
• The throttle in turn controls the torque T delivered by the engine, which is transmitted through the gears and the wheels, generating a force F that moves the car.
• There are disturbance forces F d due to variations in the slope of the road, the rolling resistance and aerodynamic forces.
• The cruise controller also has a human-machine interface that allows the driver to set and modify the desired speed.
• There are also functions that disconnect the cruise control when the brake is touched.
• The system has many individual components — actuator, engine, transmission, wheels and car body — and a detailed model can be very complicated.
• In spite of this, the model required to design the cruise controller can be quite simple.
• To develop a mathematical model, a force balance for the car body is started with the forces F and F d.
• The throttle-controlled engine generates a torque T that is transmitted to the ground through the gearbox and wheels.
• Combined with the external forces from the environment, such as aerodynamic drag and gravitational forces on hills, the net force causes the car to move.
• The velocity of the car v is measured by a control system that adjusts the throttle through an actuation mechanism.
• A driver interface allows the system to be turned on and off and the reference speed v r to be established.
• The force F is generated by the engine, whose torque is proportional to the rate of fuel injection, which is itself proportional to a control signal 0 ≤ u ≤ 1 that controls the throttle position.
• The torque also depends on engine speed ω.
• A simple representation of the torque at full throttle is given by the torque curve where the maximum torque T m is obtained at engine speed ω m.
• The bicycle rolls around the point O with the angular velocity v0δ/b.
• The tilt of the bicycle is modeled by considering the rigid body obtained when the wheels, the rider and the front fork assembly are fixed to the bicycle frame.
• The bicycle is an interesting dynamical system with the feature that one of its key properties is due to a feedback mechanism created by the design of the front fork.
• Applications of control in automotive systems include emissions control, traction control, power control (especially in hybrid vehicles) and adaptive cruise control.
• The cruise control system operates in four modes: on, off, set, resume or cancel.
• The steering angle is influenced by the torque the rider applies to the handlebar.
• The operation of the system is governed by a finite state machine that controls the modes of the PI controller and the reference generator.
• The system is nonlinear due to the torque curve, the gravity term and the nonlinear character of rolling friction and aerodynamic drag.
• The steering properties of a bicycle depend critically on the trail.
• Typical parameters are T m = 190 Nm, ω m = 420 rad/s (about 4000 RPM) and β = 0.4.
• The equations of motion for the bicycle are derived by assuming that the bicycle rolls on the horizontal xy plane.
• An observer fixed to the bicycle experiences forces due to the motion of the coordinate system.
• The torques acting on the system are due to gravity and centripetal action.
• A feedback controller is added to the model to regulate the speed of the car in the presence of disturbances.
• The controller is a proportional-integral controller.
• The controller can be realized as an input/output dynamical system by defining a controller state z and implementing the differential equation where vr is the desired (reference) speed.
• The parameters of the system can vary, for example, the mass of the car depends on the number of passengers and the load being carried in the car.
• The integrator (represented by the state z) ensures that in steady state the error will be driven to zero, even when there are disturbances or modelling errors.
• The input to the throttle position control system is the signal u that controls the throttle position, and the disturbance is the force Fd, which depends on the slope of the road.
• Letting the slope of the road be θ, gravity gives the force F g = mg sin θ, as illustrated in Figure 3.3a, where g = 9.8 m/s 2 is the gravitational constant.
• A simple model of rolling friction is where C r is the coefficient of rolling friction and sgn( v) is the sign of v (±1) or zero if v = 0.