# Control Systems Homework Answers

The modern control systems will feature in most robotics classes. It is not one of the easiest topics to understand. It becomes more challenging when you have to handle homework alone and sometimes away from the class. Where do you find solutions that are reliable and easy to understand?

From Course Books

At the beginning of the unit, the teacher issues a list of learning materials and books to be used. These books are designed for your level and have been approved for reference purposes. Refer to these books when looking for a solution. Furthermore, the examples and methodologies used in these books are consistent with what your teacher uses. This eliminates the possibility of confusion. These course books can be obtained from the library since you are not required to buy all of them. Consult your teacher in case you do not understand the directions provided by these books.

Class Notes

Teachers provide notes or handouts when teaching. The notes contain examples and explanations on different aspects of Modern Control Systems. Since the notes cover the chapters of topics listed in the course outline, they will provide direction when completing your assignment. The fact that these notes come from your teacher is an assurance that they are accurate and relevant to your unit.

Local Assistants

There are local assistants who provide mobile homework services on demand. They advertise their services on websites and local platforms. Since they are within your neighborhood, it is convenient to summon them for a one-on-one session. Ensure that the person is well trained and understands the topic. You will have to pay for their services. You should meet at populated or open places to guarantee your safety.

Online

There are numerous resources online that make modern control systems homework easy to complete. These resources include tutorials, text materials, video demonstrations and recorded lectures. There are writing agencies that offer their services online where you upload your assignment and they complete it. If you have to use such agencies, ensure that they are genuine and will offer quality services. They might waste your time only to end up with substandard work.

Classmates

The unit on modern control systems is not challenging to everyone in your class. Join with your classmates to discuss the homework. Use free hours in the morning, during breaks or before leaving school to tackle the work. Since you learnt in the same session, you are likely to use similar methodology which helps to avoid confusion.

## System Identiﬁcation

### Systems

Systems, in one sense, are devices that take input and produce an output. A system can be thought to operate on the input to produce the output. The output is related to the input by a certain relationship known as the system response. The system response usually can be modeled with a mathematical relationship between the system input and the system output.

### System Properties

Physical systems can be divided up into a number of diﬀerent categories, depending on particular properties that the system exhibits. Some of these system classiﬁcations are very easy to work with and have a large theory base for analysis. Some system classiﬁcations are very complex and have still not been investigated with any degree of success. By properly identifying the properties of a system, certain analysis and design tools can be selected for use with the system.

### Initial Time

The initial time of a system is the time before which there is no input. Typically, the initial time of a system is deﬁned to be zero, which will simplify the analysis signiﬁcantly. Some techniques, such as the Laplace Transform require that the initial time of the system be zero. The initial time of a system is typically denoted by t0. The value of any variable at the initial time t0 will be denoted with a 0 subscript. For instance, the value of variable x at time t0 is given by:

x(t0)=x0 Likewise, any time t with a positive subscript are points in time after t0, in ascending order: t0 ≤ t1 ≤t2 ≤···≤ tn So t1 occurs after t0, and t2 occurs after both points. In a similar fashion above, a variable with a positive subscript (unless specifying an index into a vector) also occurs at that point in time: x(t1)=x1 x(t2)=x2 This is valid for all points in time t.### Additivity

A system satisﬁes the property of additivity, if a sum of inputs results in a sum of outputs. By deﬁnition: an input of x3(t)=x1(t)+x2(t) results in an output of y3(t)=y1(t)+y2(t). To determine whether a system is additive, use the following test: Given a system f that takes an input x and outputs a value y, assume two inputs (x1 and x2) produce two outputs:

y1 =f(x1) y2 =f(x2) Now, create a composite input that is the sum of the previous inputs: x3 =x1+x2 Then the system is additive if the following equation is true: y3 =f(x3)=f(x1+x2)=f(x1)+f(x2)=y1+y2### Homogeneity

A system satisﬁes the condition of homogeneity if an input scaled by a certain factor produces an output scaled by that same factor. By deﬁnition: an input of ax1 results in an output of ay1. In other words, to see if function f() is homogeneous, perform the following test: Stimulate the system f with an arbitrary input x to produce an output y:

y =f(x) Now, create input x1, scale it by a multiplicative factor C(C is an arbitrary constant value), and produce a corresponding output y1: y1 =f(Cx1) Now, assign x to be equal to x1: x1 =x Then, for the system to be homogeneous, the following equation must be true: y1 =f(Cx)=Cf(x)=Cy### Linearity

A system is considered linear if it satisﬁes the conditions of Additivity and Homogeneity. In short, a system is linear if the following is true: Take two arbitrary inputs, and produce two arbitrary outputs:

y1 =f(x1) y2 =f(x2) Now, a linear combination of the inputs should produce a linear combination of the outputs: f(Ax+By) = f(Ax) + f(By) = Af(x) + Bf(y) This condition of additivity and homogeneity is called superposition. A system is linear if it satisﬁes the condition of superposition.### Memory

A system is said to have memory if the output from the system is dependent on past inputs (or future inputs!) to the system. A system is called memoryless if the output is only dependent on the current input. Memoryless systems are easier to work with, but systems with memory are more common in digital signal processing applications. Systems that have memory are called dynamic systems, and systems that do not have memory are static systems.

### Causality

Causality is a property that is very similar to memory. A system is called causal if it is only dependent on past and/or current inputs. A system is called anti-causal if the output of the system is dependent only on future inputs. A system is called non-causal if the output depends on past and/or current and future inputs. A system design that is not causal cannot be physically implemented. If the system can't be built, the design is generally worthless.

### Time-Invariance

A system is called time-invariant if the system relationship between the input and output signals is not dependent on the passage of time. If the input signal x(t) produces an output y(t) then any time shifted input, x(t+δ), results in a time-shifted output y(t+δ) This property can be satisﬁed if the transfer function of the system is not a function of time except expressed by the input and output. If a system is time-invariant then the system block is commutative with an arbitrary delay.

## System Modeling

### The Control Process

It is the job of a control engineer to analyze existing systems, and to design new systems to meet speciﬁc needs. Sometimes new systems need to be designed, but more frequently a controller unit needs to be designed to improve the performance of existing systems. When designing a system, or implementing a controller to augment an existing system, we need to follow some basic steps:

- Model the system mathematically
- Analyze the mathematical model
- Design system/controller
- Implement system/controller and test

### External Description

An external description of a system relates the system input to the system output without explicitly taking into account the internal workings of the system. The external description of a system is sometimes also referred to as the Input-Output Description of the system, because it only deals with the inputs and the outputs to the system

If the system can be represented by a mathematical function h(t, r), where t is the time that the output is observed, and r is the time that the input is applied. We can relate the system function h(t, r) to the input x and the output y through the use of an integral:

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