Control System Engineering Basics and Predicting Future States

Control System Engineering Basics

What is control system engineering? Modeling Construct a mathematical model of the application system Analysis Study the behavior (dynamics) of the application system via the model Controller Synthesis Design a control system to direct the future behavior of the system Predict its performance

Consider a control system that fills up a tank Application System Model: h(t) = liquid level in tank at time t Height of tank = H R = rate of filling the tank in inches per minute Dynamic model: h(t) = R t Initial state h(0) = 0

How to predict future states of a system Two ways in generals: 1. Get a model that describes the behavior of the system as a function of time, e.g., a model of the trajectories of the planets in our Solar System. 2. Know how the system changes incrementally in reaction to a change in its environment. Determine the current state of the system. Determine the change in the environment (input) over the next time interval. Compute the change in the system state at the end of the time interval. s(t+ t) = s(t) + s To compute s(?2) from s(?1), divide the interval [?1 ,?2] into a sequence of subintervals and compute the state change for each subinterval, starting from s(?1). h(t+ t) = h(t)+R t Initial state h(0) = 0

A general deterministic system can be described by an operator, H, that maps an input, x(t), as a function of t to an output, y(t), a type of black box description. A system is linear if and only if it satisfies the superposition principle, or equivalently both the homogeneity and additivity properties, for all inputs, all scaling constants and all time.[1][2][3][4] In a system that satisfies the homogeneity property, scaling the input always results in scaling the zero-state response by the same factor.[6] Suppose increasing the input by an amount x will increase the output by an amount y, then Increasing the input from x to a x will increase the output from y to a y. In a system that satisfies the additivity property, adding two inputs always results in adding the corresponding two zero-state responses due to the individual inputs.[6] Mathematically, for a continuous-time system, given two arbitrary inputs x1(t) and x2(t) as well as their zero-state outputs H1(t) and H2(t), then a linear system must satisfy y1(t) + y2(t) = H{ x1(t) + x2(t)} for any scalar values and , for any input signals x1(t) and x2(t), and for all time t. The superposition principle means that a linear combination of inputs to the system produces a linear combination of the individual zero-state outputs (that is, outputs setting the initial conditions to zero) corresponding to the individual inputs.[5][6]

The time it takes to fill up the tank is proportional to the inverse of the input flow rate measured in fraction of the tank filled per second

Use two faucets to fill up the tank at a rate equal to the sum of the two flow rates

Time Invariance We define a delay operatorD so that if x is a function of time, then D (x) is a new function of time where (D (x))(t) = x(t ) Consider the set of all systems that map functions of time into functions of time. Such systems are called time-domain systems. This includes all audio systems, for example. Suppose that H is a time-domain system. H is said to be time invariant if H D = D H. In other words, if for all t Time, y(t) = (H(x))(t), then it must also be true that y(t ) = (H(x))(t ). A time-invariant system is one whose behavior (its response to inputs) does not change with time. Notice that system behavior relates two functions (not two simple variables) and is defined by a mapping of a (input) function to an (output) function.

Examples time-Invariant and time-variant systems Time invariant systems: Push-button door bell On-off buttons on a radio Steering wheel of a car Time-variant systems An athelete s reaction time as (s)he ages Price of a stock in the stock market Reactions of human beings to other human beings Is a bank teller machine time-invariant?

For linear time-invariant systems, the behavior of the system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation. This mathematical property makes the solution of modelling equations simpler than many nonlinear systems. For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function x(t) in terms of unit impulses or frequency components. Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations). Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense. A common use of linear models is to describe a nonlinear system by linearization.

Behavior of a Linear Time Invariant (LTI) system is characterized by its impulse response y(t ) = (H(x))(t ). A time-invariant system is one whose behavior (its response to inputs) does not change with time. Notice that system behavior relates two functions, not two simple variables and is defined by a mapping of a (input) function to an (output) function. For discrete-time systems, the output of a LTI system for any input function can be computed from its response to a unit-time pulse input of height 1. This can be done by treating the input function as the sum of a sequence of delayed unit-time pulse functions of the corresponding scaled up size. y(1) = H(x(1)) y(2) = H(x(2)) y(3) = H(x(3)) x(1), x(2), x(3) y(1), y(2), y(3) Linear Time Invariant (LTI) System H

Decomposition of the impulse response of a continuous LTI system The impulse response of a continuous-time LTI system can be decomposed into its Fourier components. A fundamental theorem in Fourier analysis is that any reasonably well-behaved periodic function f can be written as a discrete sum of trigonometric functions with frequencies that are multiples of the fundamental frequency of the function f. A video that illustrates this theorem can be found at this link: https://www.youtube.com/watch?v=41GuPu3jee4

A Strategy for Driving a System from its Current State to a Target State We want to drive the system from state x1 to state x2. Suppose the system state is denoted by x, initially x = x1. (1) Set controller parameters to stabilize the system at state x1. (2) Move the system from state x1 to state x1+ x by a small state change x (3) Repeat steps (1) and (2) until x = x2. Control theory can help derive the controller parameters as a function of the state x.