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Modeling in the Time Domain
Two approaches are available for the analysis and design of feedback control
systems.
- The first is known as the classical or frequency-domain technique which is
based on converting a system’s differential equation to a transfer function,
thus generating a mathematical model of the system that algebraically
relates a representation of the output to a representation of the input.
- The second is known as the state-space approach (also referred to as the
modern, or time-domain, approach) which is a unified method for
modeling, analyzing, and designing a wide range of systems.
- The state-space approach can be used to represent nonlinear systems that
have backlash, saturation, and dead zone.
- Also, it can handle systems with nonzero initial conditions. Time-varying
systems, (for example, missiles with varying fuel levels or lift in an aircraft
flying through a wide range of altitudes) can be represented in state
space.
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Modeling in the Time Domain
- Multiple-input, multiple-output systems (such as a vehicle with input
direction and input velocity yielding an output direction and an output
velocity) can be compactly represented in state space with a model similar
in form and complexity to that used for single-input, single-output
systems.
- The time-domain approach can be used to represent systems with a digital
computer in the loop or to model systems for digital simulation. With a
simulated system, system response can be obtained for changes in system
parameters—an important design tool.
- The time-domain approach can also be used for the same class of systems
modeled by the classical approach. This alternate model gives the control
systems designer another perspective from which to create a design.
- The designer has to engage in several calculations before the physical
interpretation of the model is apparent, whereas in classical control a few
quick calculations or a graphic presentation of data rapidly yields the
physical interpretation.
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Modeling in the Time Domain
The function i(t) is a subset of all possible network variables that we are
able to find from Eq. (3.4) if we know its initial condition, i(0), and the
input, v(t). Thus, i(t) is a state variable, and the differential equation (3.1)
is a state equation.
4. We can now solve for all of the other network variables algebraically in
terms of i(t) and the applied voltage, v(t). For example, the voltage across
the resistor is
The voltage across the inductor is
The derivative of the current is
Thus, knowing the state variable, i(t), and the input, v(t), we can find the
value, or state, of any network variable at any time,. Hence, the
algebraic equations, Eqs. (3.5) through (3.7), are output equations.
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Modeling in the Time Domain
5. Since the variables of interest are completely described by Eq. (3.1) and
Eqs. (3.5) through (3.7), we say that the combined state equation (3.1)
and the output equations (3.5 through 3.7) form a viable representation
of the network, which we call a state-space representation.
6. Equation (3.1), which describes the dynamics of the network, is not
unique. This equation could be written in terms of any other network
variable. For example, substituting into Eq. (3.1) yields
which can be solved knowing that the initial condition and
knowing v(t). In this case, the state variable is. Similarly, all other
network variables can now be written in terms of the state variable, ,
and the input, v(t).
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The General State-Space Representation
where the state variables are assumed to be a resistor voltage, V (^) R, and a capacitor voltage, VC. These variables form the axes of the state space. A trajectory can be thought of as being mapped out by the state vector, x(t), for a range of t. Also shown is the state vector at the particular time t =4. State equations: A set of n simultaneous, first-order differential equations with n variables, where the n variables to be solved are the state variables. Output equation: The algebraic equation that expresses the output variables of a system as linear combinations of the state variables and the inputs. We define the state-space representation of a system. A system is represented in state space by the following equations:
x = state vector x = derivative of the state vector with respect to time y = output vector u = input or control vector A = system matrix B = input matrix C = output matrix D = feedforward matrix
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Applying the State-Space Representation
- In order to apply the state-space formulation to the representation of more complicated physical systems. The first step in representing a system is to select the state vector, which must be chosen according to the following considerations: - A minimum number of state variables must be selected as components of the state vector. This minimum number of state variables is sufficient to describe completely the state of the system. - The components of the state vector (that is, this minimum number of state variables) must be linearly independent.
Linearly Independent State Variables:
The components of the state vector must be linearly independent.
- For example, if x 1 , x 2 , and x 3 are chosen as state variables, but , then x 3 is not linearly independent of x 1 and x 2 , since knowledge of the values of x 1 and x 2 will yield the value of x 3.
- Variables and their successive derivatives are linearly independent. For example, the voltage across an inductor, v (^) L , is linearly independent of the current through the inductor, iL , since. Thus, v (^) L cannot be evaluated as a linear combination of the current, iL.
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Applying the State-Space Representation
Our approach for selecting state variables and representing a system in state space:
- First, we write the simple derivative equation for each energy-storage element and
solve for each derivative term as a linear combination of any of the system variables and the input that are present in the equation.
- Next we select each differentiated variable as a state variable.
- Then we express all other system variables in the equations in terms of the state
variables and the input.
- Finally, we write the output variables as linear combinations of the state variables
and the input.
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Example 3.1 - Representing an Electrical Network
Given the electrical network in
find a state-space representation if the output is the current through the resistor. Solution:
The following steps will yield a viable representation of the network in state space.
- Label all of the branch currents in the network. These include iL , iR , and iC, as shown in Figure 3.5.
- Select the state variables by writing the derivative equation for all energy storage elements, that is, the inductor and the capacitor. Thus,
- From Eqs. (3.22) and (3.23), choose the state variables as the quantities that are differentiated, namely vC and iL. Using Eq. (3.20) as a guide, we see that the state- space representation is complete if the right-hand sides of Eqs. (3.22) and (3.23) can be written as linear combinations of the state variables and the input. Since i (^) C and v (^) L are not state variables, our next step is to express iC and v (^) L as linear combinations of the state variables, v (^) C and iL , and the input, v(t).
Example 3.3 - Representing a Translational
Mechanical System
- Find the state equations for the translational mechanical system shown in
Solution:
- First write the differential equations for the network to find the Laplace- transformed equations of motion.
- Next take the inverse Laplace transform of these equations, assuming zero initial conditions, and obtain
- Now let and , and then select x 1 , v 1 , x 2 , and v 2 as state variables.
- Next form two of the state equations by solving Eq. (3.44) for and Eq. (3.45) for.
Example 3.3 - Representing a Translational
Mechanical System
- Finally, add and to complete the set of state equations. Hence,
- In vector-matrix form,
where the dot indicates differentiation with respect to time.
Example 3.5 - Converting a Transfer Function with
Polynomial in Numerator
- Find the state-space representation of the transfer function shown in
Example 3.5 - Converting a Transfer Function with
Polynomial in Numerator
Solution: Separate the system into two cascaded blocks, as shown in Figure 3.12(b).The first block contains the denominator and the second block contains the numerator.
Find the state equations for the block containing the denominator. We notice that the first block’s numerator is 1/24 that of Example 3.4. Thus, the state equations are the same except that this system’s input matrix is 1/24 that of Example 3.4. Hence, the state equation is
Introduce the effect of the block with the numerator. The second block of Figure 3.12(b), where b 2 =1; b 1 =7, and b 0 =2, states that
Taking the inverse Laplace transform with zero initial conditions, we get
Example 3.6 - State-Space Representation to
Transfer Function
Given the system defined by Eq. (3.74), find the transfer function, , where U(s) is the input and Y(s) is the output.
Solution: The solution revolves around finding the term. All other terms are already defined. Hence, first find :
Now form :
we obtain the final result for the transfer function:
