Block diagram and state variable representations of LTI systems.
Block diagram representations.
The block diagram is a more detailed representation of a system than the impulse response or difference and differential equation descriptions since it describes how the system’s internal computations or operations are ordered. The impulse response and the differential equation or difference descriptions represent only the input-output behavior of a system.
We should show that a system with a given input-output characteristic can be represented by different block diagrams. Each block diagram representation describes a different set of internal computations used to determine the system output.
Block diagram representations has an interconnection of three elementary operations on signals:
- Scalar multiplication: y(t) = cx(t) or y[n] = cx[n], where c is scalar.
- Addition: y(t) = x(t) + w(t) or y[n] = x[n] + w[n].
- Integration for continuous-time LTI systems: y(t)= ∫ x(T) dT; and a time shift for discrete-time LTI systems: y[n] = x[n-1].
In order to express a continuous-time LTI system instead of differentiation, because in block diagrams for continuous-time LTI systems instead of differentiation, because integrators are more easily built from analog components than are differentiators.
Also, integrators smooth out noise in the system, while differentiators accentuate noise.
The integral and difference equation corresponding to the system behavior is obtained by expressing the sequence of operations represented by the block diagram in equation form.
The scalar multiplications and summations imply that:
w[n] = b0x[n] + b1x[n-1] + b2x[n-2].
We can now write the equation in terms of w[n] for y[n]. The block diagram indicates that:
y[n] = w[n] - a1y[n-1] - a2y[n-2].
The output of this system can be expressed as a function of the input x[n] by substituting the respective values:
y[n] = -a1y[n-1] - a2y[n-2] + b0x[n] + b1x[n-1] + b2x[n-2]
Or,
y[n] + a1y[n-1] + a2y[n-2] = b0x[n] + b1x[n-1] + b2x[n-2].
The block diagram represents an LTI system whose input-output characteristic is represented by a second-order difference equation.
Note that the block diagram explicitly represents the operations involved in computing the output from the input and tells us how to simulate the system on a computer. The operations of scalar multiplication and addition are easily evaluated with a computer. The outputs of the time-shift operations correspond to memory locations in a computer.
To compute the current output from the current input, we must have saved the past values of the input and output in memory. To begin a computer simulation at a specified time, we must know the input and past values of the output, The past values of the output are the initial conditions required to solve the difference equation directly.
State-variable description of LTI system
The state-variable description of an LTI system consists of a series of coupled first-order differential or difference equations that describe how the state of the system evolves and an equation that relates the output of the system to the current state variables and input.
The state of a system may be defined as a minimal set of signals that represent the system’s entire memory of the past. That is, given only the value of the state at an initial point in time, n1 (or t1) and the input for times n>= n1 (or t>= t1), we can determine the output for all times n>= n1 (or t>= t1).
The selection of signals indicating the state of a system is not unique and that there are many possible state-variable descriptions corresponding to a system with a given input-output characteristic. The ability to represent a system with different state-variable descriptions is a powerful attribute that finds application in advanced methods for control system analysis and discrete-time system implementations.
The state-variable description
In order to determine the output of the system for n>= n1, we must know the input for n>=n1, and the outputs of the time-shift operations labeled q1[n] and q2[n] at the time n=n1.
This means that we can choose either q1[n] or q2[n] as the state of the system.
We can note that since q1[n] and q2[n] are the outputs of the time-shift operations, the next value of the state, q1[n+1] and q2[n+1]. It must correspond to the variables at the input to the time-shift operations. The block diagram indicates that the next value of the state is obtained from the current state and the input via the two equations:
q1[n+1} = -a1q1[n] - a2q2[n] + x[n]. and, q2[n+1] = q1[n].
Image is taken from slide player
The image represents the direct form II representation of a second-order discrete-time LTI system depicting state variables q1[n] and q2[n].
The block diagram indicates that the system output is expressed in terms of the input and the state of the system as:
y[n] = x[n] - a1q1[n] - a2q2[n] + b1q1[n] + b2q2[n]. Or, y[n] = (b1-a1)q1[n] + (b2-a2)q2[n] + x[n].
Reference
Block diagram and state variable representations of LTI systems