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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY
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LECTURE NOTES
ON
ADVANCED CONTROL SYSTEMS

UNIT - I
INTRODUCTION
Introduction The classical control theory and methods (such as root locus) that we have been using in class to
date are based on a simple input-output description of the plant, usually expressed as a transfer function.
These methods do not use any knowledge of the interior structure of the plant, and limit us to single-input
single-output (SISO) systems, and as we have seen allows only limited control of the closed-loop behavior
when feedback control is used. Modern control theory solves many of the limitations by using a much
“richer” description of the plant dynamics. The so-called state-space description provide the dynamics as a set
of coupled first-order differential equations in a set of internal variables known as state variables, together
with a set of algebraic equations that combine the state variables into physical output variables.
In a state space system, the internal state of the system is explicitly accounted for by an equation known as
the state equation. The system output is given in terms of a combination of the current system state, and the
current system input, through the output equation. These two equations form a system of equations known
collectively as state-space equations. The state-space is the vector space that consists of all the possible
internal states of the system. For a system to be modeled using the state-space method, the system must meet
this requirement: The system must be "lumped""Lumped" in this context, means that we can find a finitedimensional state-space vector which fully characterizes all such internal states of the system. this text
mostly considers linear state space systems, where the state and output equations satisfy the superposition
principle and the state space is linear. However, the state-space approach is equally valid for nonlinear
systems although some specific methods are not applicable to nonlinear systems. Central to the state-space
notation are the idea of a state. A state of a system is the current value of internal elements of the system,
that change separately (but not completely unrelated) to the output of the system. In essence, the state of a
system is an explicit account of the values of the internal system components. Here are some examples:
Consider an electric circuit with both an input and an output terminal. This circuit may contain any number of
inductors and capacitors. The state variables may represent the magnetic and electric fields of the inductors
and capacitors, respectively.
Consider a spring-mass-dashpot system. The state variables may represent the compression of the spring, or
the acceleration at the dashpot.
Consider a chemical reaction where certain reagents are poured into a mixing container, and the output is the
amount of the chemical product produced over time. The state variables may represent the amounts of unreacted chemicals in the container, or other properties such as the quantity of thermal energy in the container
(that can serve to facilitate the reaction).
When modeling a system using a state-space equation, we first need to define three vectors:
Input variables
A SISO (Single Input Single Output) system will only have a single input value, but a MIMO system may
have multiple inputs. We need to define all the inputs to the system, and we need to arrange them into a
vector.
Output variables
This is the system output value, and in the case of MIMO systems, we may have several. Output variables

should be independent of one another, and only dependent on a linear combination of the input vector and the
state vector.
State Variables
The state variables represent values from inside the system, that can change over time. In an electric circuit,
for instance, the node voltages or the mesh currents can be state variables. In a mechanical system, the forces
applied by springs, gravity, and dashpots can be state variables.
We denote the input variables with u, the output variables with y, and the state variables with x. In essence,
we have the following relationship:
Where f(x, u) is our system. Also, the state variables can change with respect to the current state and the
system input:
Where x' is the rate of change of the state variables. We will define f(u, x) and g(u, x).
The state equations and the output equations of systems can be expressed in terms of matrices A, B, C, and D.
Because the form of these equations is always the same, we can use an ordered quadruplet to denote a system.
We can use the shorthand (A, B, C, D) to denote a complete state-space representation. Also, because the state
equation is very important for our later analyis, we can write an ordered pair (A, B) to refer to the state
equation:
Obtaining the State-Space Equations
The beauty of state equations, is that they can be used to transparently describe systems that are both
continuous and discrete in nature. Some texts will differentiate notation between discrete and continuous
cases, but this text will not make such a distinction. Instead we will opt to use the generic coefficient
matrices A, B, C and D for both continuous and discrete systems. Occasionally this book may employ the
subscript C to denote a continuous-time version of the matrix, and the subscript D to denote the discrete-time
version of the same matrix. Other texts may use the letters F, H, and G for continuous systems and Γ,
and Θ for use in discrete systems. However, if we keep track of our time-domain system, we don't need to
worry about such notations.
IMPORTANCE

The state space model of a continuous-time dynamic system can be derived either from the system model
given in the time domain by a differential equation or from its transfer function representation. Both cases
will be considered in this section. Four state space forms—the phase variable form (controller form), the
observer form, the modal form, and the Jordan form—which are often used in modern control theory and
practice, are presented.
APPLICATIONS
the analysis and design of the following systems can be carried using state space method
1.linear systems
2.non-linear systems
3.time varying systems
4.multiple i/p and multiple output systems
State-space methods of feedback control system design and design optimization for invariant and timevarying deterministic, continuous systems; pole positioning, observability, controllability, modal control,
observer design, the theory of optimal processes and Pontryagin's Maximum principle, the linear quadratic
optimal regulator problem, Lyapunov’s functions and stability theorems, linear optimal open loop control;
introduction to the calculus of variations. Intended for engineers with a variety of backgrounds. Examples will
be drawn from mechanical, electrical and chemical engineering applications. MATLAB is used extensively
during the course for the analysis, design and simulation. Transfer functions ↔ state-space representations Solution of linear differential equations, linearization - Canonical systems, modes, modal signal-flow
diagrams - Observability & Controllability - Observability & Controllability, Rank tests - Stability - State
feedback control; Accommodating reference inputs - Linear observer design - Separation principle

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