Continuous-Time System Representation and Modeling

Sources

1. Control and Optimal Control Theories with Applications
   1. Transfer Functions
   2. State Space
2. Numerical Methods for Linear Control Systems
   1. Linear State-Space Models and Solutions of the State Equations
3. Control System Design
   1. State-Space Representation
4. Linear Feedback Controls
   1. A Tale of Two Poles: The Positioner Example and the Significance of the Poles in the -Plane
5. Spring 2014 Analysis and Design of Feedback Control Systems
   1. Understanding Poles and Zeros

Modeling

Source 3.1

1. State-Space Representation

• The most important task in control systems is the development of a mathematical model of a process of interest. The analysis is founded in this model, and, thus, it is essential to the solution of an analytical design problem.
• Because systems have subsystems with different underlying physical laws, control system engineers face the challenge of harmonizing the operation of those interconnected subsystems.
• Conventionally, the two approaches to the modeling and analysis of linear systems is the transfer function (or frequency domain) approach and the state-space (or time domain) approach.
• What sets the state-space approach apart from the transfer function approach is the use of first-order differential equations when representing the processes of the system being examined.

Characterization of Systems

Transfer Functions

Source 2.1

Transfer Functions

• We can use transfer functions to represent $n^{th}$ order system with a single input and a single output.
• The transfer function of a linear system $G(s)$ is the ratio between a Laplace transform of the output $Y(s)$ and the Laplace transform of the input $U(s)$ with all initial conditions equal to 0.
• They can usually be easily identified in basic block diagrams.

State Space

Source 1.2

State Space

• The issue with transfer functions is that the initial conditions must be 0. Due to the importance of the history of systems in the time domain, the transfer function can sometimes be impractical, and, as such, the state space representation can be used.
• State space systems are applicable to multivariable systems, or systems with multiple inputs and multiple outputs.

Source 2.1

Linear State-Space Models and Solutions of the State Equations

A continuous-time system has an input and output defined in the continuous time over the interval $[0,\infty )$. A linear continuous-time dynamical system can be expressed with the first-order ODE:

$$\begin{array}{rl}\dot{x}(t)& =Ax(t)+Bu(t),\\ y(t)& =Cx(t)+Du(t)\end{array}$$

• $x(t)$ is the system’s state
• $u(t)$ is the system’s input
• $y(t)$ is the system’s output
• $\dot{x}(t)$ is the state equation, while $y(t)$ is the output equation.
• $A,B,C,D$ are time-invariant matrices
• components of $x(t)$ are the state variables
• Even non first-order form physical systems can be reduced to a first-order state-space form.

Poles

Source 4.1

A Tale of Two Poles: The Positioner Example and the Significance of the Poles in the -Plane

Relationship between the poles in the s-plane and the second-order system’s dynamic response

• Poles indicate s-plane locations with a characteristic polynomial that approaches zero.
• A process is stable when it produces a bounded output when given a bounded input.

Source 4.2

Understanding Poles and Zeros

• One of the advantages of transfer functions is that it makes it convenient to discern the system response characteristics even without the solution to the complete differential equation.
• The Laplace output found at the numerator of the transfer function pertains to the system zeros, while the Laplace input found at the denominator pertains to the system poles.
• Because the coefficients of the numerator and denominator is assumed to be real, the poles and zeros must either be real or in complex conjugate pairs.
• With the gain constant $K$, the poles and zeros—transfer function properties—describes the system completely due to them being able to comprehensively characterize and reconstruct the input/output differential equation.
• Pole-zero plots illustrate the dynamics of the system by graphically showing the locations of the poles and zeros on the complex s-plane, wherein the real and imaginary components of the complex variable $s$ is depicted by its axes. In other words, we can gain qualitative insights about the system and its response characteristics by plotting and identifying the locations of the poles and zeros.
• By convention, circle is used to represent zeros, while a cross is used to represent poles.