Zero state response

In electrical circuit theory, the zero state response (ZSR), is the behaviour or response of a circuit with initial state of zero. The ZSR results only from the external inputs or driving functions of the circuit and not from the initial state.
The total response of the circuit is the superposition of the ZSR and the ZIR, or Zero Input Response. The ZIR results only from the initial state of the circuit and not from any external drive. The ZIR is also called the natural response, and the resonant frequencies of the ZIR are called the "natural frequencies". Given a description of a system in the s-domain, the zero-state response can be described as Y(s)=Init(s)/a(s) where a(s) and Init(s) are system-specific.
Zero state response and zero input response in integrator and differentiator circuits.
One example of zero state response being used is in integrator and differentiator circuits. By examining a simple integrator circuit it can be demonstrated that when a function is put into a linear time-invariant (LTI) system, an output can be characterized by a superposition or sum of the Zero Input Response and the zero state response. 
A system can be represented as
formula_1 formula_2
with the input formula_3 on the left and the output formula_4 on the right.
The output formula_4 can be separated into a zero input and a zero state solution with
formula_6
The contributions of formula_7 and formula_1 to output formula_9 are additive and each contribution formula_7 and formula_11 vanishes with vanishing formula_7 and formula_13
This behavior constitutes a linear system. A linear system has an output that is a sum of distinct zero-input and zero-state components, each varying linearly, with the initial state of the system and the input of the system respectively.
The zero input response and zero state response are independent of each other and therefore each component can be computed independently of the other.
Zero state response in integrator and differentiator circuits.
The Zero State Response formula_11 represents the system output formula_9 when formula_16
When there is no influence from internal voltages or currents due to previously charged components
formula_17
Zero state response varies with the system input and under zero-state conditions we could say that two independent inputs results in two independent outputs:
formula_18formula_19
and
formula_20formula_21
Because of linearity we can then apply the principles of superposition to achieve
formula_22formula_23
Verifications of zero state response in integrator and differentiator circuits.
To arrive at general equation.
The circuit to the right acts as a simple integrator circuit and will be used to verify the equation formula_24 as the zero state response of an integrator circuit.
Capacitors have the current-voltage relation formula_25
where "C" is the capacitance, measured in farads, of the capacitor.
By manipulating the above equation the capacitor can be shown to effectively integrate the current through it. The resulting equation also demonstrates the zero state and zero input responses to the integrator circuit.
First, by integrating both sides of the above equation
formula_26
Second, by integrating the right side
formula_27
Third, distribute and subtract formula_28 to get
formula_29
Fourth, divide by formula_30 to achieve
formula_31
By substituting formula_32 for formula_33 and formula_34 for formula_35 and by using the dummy variable formula_36 as the variable of integration the general equation
formula_37
is found.
To arrive at circuit specific example.
The general equation can then be used to further demonstrate this verification by using the conditions of the simple integrator circuit above.
By using the capacitance of 1 farad as shown in the integrator circuit above
formula_38
which is the equation containing the zero input and zero state response seen at the top of the page.
To verify zero state linearity.
To verify its zero state linearity set the voltage around the capacitor at time 0 equal to 0, or formula_39, meaning that there is no initial voltage. This eliminates the first term forming the equation
formula_40.
In accordance with the methods of linear time-invariant systems, by putting two different inputs into the integrator circuit, formula_41 and formula_42, the two different outputs
formula_43
and
formula_44
are found respectively.
By using the superposition principle the inputs formula_41 and formula_42 can be combined to get a new input
formula_47
and a new output
formula_48
By integrating the right side of
formula_49
formula_50
is found, which implies the system is linear at zero state, formula_39.
This entire verification example could also have been done with a voltage source in place of the current source and an inductor in place of the capacitor. We would have then been solving for a current instead of a voltage.
Zero state response industry uses.
The circuit analysis method of breaking a system output down into a zero state and zero input response is used industry wide including circuits, control systems, signal processing, and electromagnetics. Also most circuit simulation software, such as SPICE, support the method in one form or another.