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EE5101 Linear Systems
创建日期：2024-08-15 15:52:07
标签： EE5101
Linear Systems

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EE5101/ME5401 Linear Systems


Late submission is absolutely not allowed as the

grades have to be submitted to the department very soon after the final exam. You may work together

with your classmates. But do write your report independently. And the results are supposed to be

different from each other as the parameters are based upon your matriculation numbers.

1 Background

We all had a tough time when learning to ride a bicycle when we are a teenager. It usually takes

months to master that skill after crashing into walls for hundreds of times. Needless to say even after

that, it is still difficult for us to ride on uneven surface or turn when riding in a high speed. Would

that be excited if such two-wheeled vehicle comes to the market that it can self-balance itself to

improve its stability and driving safety?

Self-sustaining two-wheeled vehicle not only is a proof of how control theory has been

developed during the past decades, but also has a huge market potential. Therefore, a lot of

researchers from universities and companies are working on related topics. Although most of the

study are still in experimental stage, there are research groups and startups that have already published

demonstration video online, such as the C-1 motorcycle from Lit Motors [1].

Figure 1 is a screenshot from a demonstration video on YouTube. As we can see, the vehicle

looks like a motorcycle from outside, but inside the vehicle the driver drives as if it is a car. The

vehicle self-balances itself when running on the road or even when it is still. This two-wheeled self-

balancing vehicle is said to combine the virtues of both the car and the motor: safety and low cost.

Figure 1 Two-wheeled self-balancing electric car/motor [1]

Since there are many more dynamics involved when the vehicle is running, in this mini-project

we only consider the self-balance of the two-wheeled vehicle when it is stationary. We will try to

balance this vehicle using the control methods we have learned in Linear Systems.

2 Modelling

For model-based control, the first step is to build an effective dynamic model for our target plant,

i.e., the two-wheeled vehicle in this project. The detailed procedures to model this vehicle can be

found in [2] and [3]. Here we only give a short introduction and the resulted state space model.

An experimental system for the two-wheeled vehicle prototype is shown in Figure 2. The two-

wheeled vehicle consists of three parts. There is a cart system that corresponds to the rider’s center-

of-gravity movement, a steering system (a front part) for steering, and a body (a rear part). The front

part and the rear part are structures that are movable through a steering axis. A cart system and a

steering system are driven by DC servo motor, and DC motors are controlled by servo amplifier

which contains the velocity control system. Handle angle and cart position are measured by encoders.

Attitude angles of the two-wheeled vehicle (roll angle and yaw angle) are measured by gyroscopes.

Figure 2 Composition of experimental system

Figure 3 Two-wheeled vehicle structure model

The mechanical structure for the two-wheeled vehicle is given in Figure 3. The two-wheeled

vehicle is stabilized by moving the cart position ( )d t and adjusting the handle angle ( )tψ . The

control inputs are the voltages ( )cu t and ( )hu t to two DC servo motors, which drives the cart

system and the steering system correspondingly.


For the dynamic model, the relevant symbols are defined in Table 1. Table 1 Definition of Symbols

, , Mass of each part

, , Vertical length from a floor to a center-of-gravity of each part

, Horizontal length from a front wheel rotation axis to a center-of-gravity of part of front wheel and steering axis.

, Horizontal length from a rear wheel rotation axis to a center-of-gravity of part of rear wheel and steering axis.


Horizontal length from a rear wheel rotation axis to a center-of-gravity of

the cart system

Moment of inertia around center-of-gravity x axially

Moment of inertia for part of front wheel z axially.

Moment of inertia for part of rear wheel that contains cart system z axially.

Viscous coefficient around x axis.

Viscous coefficient for part of front wheel around z axis.


Viscous coefficient for part of rear wheel that contains cart system around

z axis.

A viscosity coefficient of a movement direction of the cart system

Subscript f, r, c Part of front wheel, rear wheel, and cart system respectively

(), (), () Cart position, handle angle and bike angle

In [2], the state space linear model for the two-wheeled vehicle is derived to be


(1)

where the state variable is

and the matrices and the input vector are1


1 Some additional coupling terms are fabricated to facilitate our design.

where g is the gravitational acceleration, 29.8 /g m s≈ .

The physical parameters in (5) can be measured directly or identified by experiments. The value

of all these physical parameters is summarized in Table 2. Table 2 Physical parameters of the two-wheeled vehicle

Parameter Value Parameter Value

[kg] 2.14 + /20 [m] 0.18

[kg] 5.91 ? /10 [m] 0.161

[kg] 1.74 [m] 0.098

[m] 0.05 [m] 0.133

[m] 0.128 [m] 0.308 + ( ? )/100

[m] 0.259

[kgm2] 0.5+( ? )/100 [kgm2/s] 3.33 ? /20 + /60

15.5 ? /3 + /2 27.5 ? /2

11.5 + ( ? )/( + + 3) 60 + ( ? )/10

where in Table 2 a, b, c, d represent the last four digits in your matriculation number. For

example, if your matriculation number is A0162903M, then = 2, = 9, = 0, = 3 and one

of the parameters can be computed as μx = 3.33 ? 9/20 + 2 ? 0/60 = 2.88.


3 Control System Design

After all, we get a linear state space model (1) for the stationary two-wheeled vehicle. In the

following, different control strategies will be explored to stabilize this vehicle to achieve its self-

balance. We will target both the regulation and set point tracking problems. The initial condition for

the two-wheeled vehicle system (1) is assumed to be [ ]0 0.2, 0.1, 0.15, 1, 0.8, 0

Tx = ? ? .

3.1 Design specifications

The transient response performance specifications for all the outputs y in state space model (1)

are as follows:

1) The overshoot is less than 10%.

2) The 2% settling time is less than 5 seconds.

Note: (a) This transient response is checked by giving a step reference signal for each input

channel, i.e., [1, 0] and [0, 1], with zero initial conditions; (b) For all the following task 1) to 5), your

control system should satisfy this performance specification and you are supposed to finish the

required investigation for each task as well.

3.2 Tasks

Your study should include, but not limited to

1) Assume that you can measure all the six state variables, design a state feedback controller using

the pole place method, simulate the designed system and show all the six state responses to non-

zero initial state with zero external inputs. Discuss effects of the positions of the poles on system

performance, and also monitor control signal size. In this step, both the disturbance and set point

can be assumed to be zero. (10 points)

2) Assume that you can measure all the six state variables, design a state feedback controller using

the LQR method, simulate the designed system and show all the state responses to non-zero

initial state with zero external inputs. Discuss effects of weightings Q and R on system

performance, and also monitor control signal size. In this step, both the disturbance and set point

can be assumed to be zero. (10 points)

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