Automatic Control

- State Vector, - Input vector, - State Equation, - Output Vector
Where is the state matrix, is the input matrix, is the output matrix and is the input-output matrix
MATLAB: sys(A,B,C,D) - Creates continuous-time system from matrices A,B,C,D
LTI Systems may be solved using the laplace transform, with the responses of the system, how it reacts to an input, being defined in two contributions, first given for the state response
and for the output response,
However, these are much more conveniently defined in the Laplace Domain, the transformed responses.

For the transformed state response in the Laplace Domain we have

and for the transformed output response we have

where is the transfer function

The inverse Laplace Transform when the PFE as a whole is of form is given by the function , where we have and

An LTI system is SISO(Single Input Single Output) if it has one input and one output (duh).
The transfer function mentioned earlier, is the ratio between the output and input Laplace transforms, with

The transfer function then can be expressed as a ratio of two polynomials of the complex variable , with form
The roots of the numerator polynomial, when that value is approached, send the transfer function towards zero, meaning it attenuates the output, and are promptly called the Zeroes of the transfer function. Similarly, the roots of the denominator polynomial send the transfer function towards infinity, meaning it amplifies the output, and these are called the Poles of the system(transfer function).

• If there are as many zeroes as there are poles the system is called proper
• If there are less zeroes than poles, the system is strictly proper

So, for a given state space representation of , we derive the transfer function (tf) representation as
MATLAB: tf(sys) - Obtains tf from ss

We can use matlab to convert with the following sequence of commands from tf to ss form:

s = tf('s'); % Declare laplace variable
H = (s^2+3*s+1)/(s^3+s^2+s+1); % Define transfer fxn
sys = ss(H) % Get state-space from transfer fxn
Copy

Stability is the most important property of a dynamical system. It prevents system damage.
The zero-input state response is a linear combination of some functions of time, called the natural modes of the system. The natural modes depend on the eigenvalues of the matrix , each eigenvalue is associated to some natural mode.

A system is stable if its output remains bounded (In control) eventually. We have two types of stability, internal stablity and BIBO stability, and they depend on the convergence of the natural modes.

Internal stability depends on the system's eigenvalues, while BIBO stability depends on the system poles. The system eigenvalues are the roots of the characteristic polynomial of the matrix , while the system poles are the roots of the denominator polynomial of the transfer functions after simplifying all cancellations between numerator and denominator, called zero-pole cancellations. We denote to be the minimal polynomial of the matrix , also defined as the least common multiple of the denominators of . The roots of the minimal polynomial are the same as the roots of the characteristic polynomial, the eigenvalues of . The multiplicity of an eigenvalue as a root of the minimal polynomial is called the minimal polynomial multiplicity
A system is minimal if no zero-pole cancellations occur during calculating the transfer function. Unless specified in this course we will only deal with minimal transfer functions.

For some convergent natural mode associated with an eigenvalue with negative real part, the time constant is defined as , and is a measure of the "convergence speed" of the mode. The smaller the time constant, the faster the natural mode converges.
We are now ready to define stability of LTI systems

An LTI system is

• Internally stable if and only if all eigenvalues of the matrix have non-positive real part, and all eigenvalues of real part zero have multiplicity , that is, they don't repeat.
• Asymptotically stable if and only if all the eigenvalues of the matrix have strictly negative real part
• Unstable if some eigenvalue has positive real part or zero real part of multiplicity higher than 1

BIBO Stability (Bounded Input-Bounded Output) means that the output remains bounded for any values of the input whatsoever, and an LTI system is BIBO stable if and only if

• All the poles of the transfer function have strictly negative real parts

The output response can be broken down into two parts:

• The transient response which dies out eventually as
• The steady-state response which is the system's response as the transient has died out for
• The value is the steady state value and is the asymptotic value of the output response as
  The steady-state response can be calculated using the Final Value Theorem if and only if the system is minimal and BIBO stable

For a stable first order system described by the transfer function where is the gain and is a negative real pole. Let and , then the transfer function can be rewritten as where is the generalized steady-state gain or the DC Gain, the gain for a constant input signal.

• is called the 10%-90% rise time and is, obviously, the time needed for the response to go from 10% to 90T of the steady-state value
• is called the % settling time and is the amount of time required for the response to reach and stay within of the steady-state value

For a stable second order system described by the transfer function , where is the gain, is the natural frequency, is the damping ratio, with and the time constant, with poles and the output response in the laplace domain is given as .
The natural frequency and damping coefficient of a complex conjugate pair are defined as

• The steady-state value is the asymptotic value of the output response as
• The peak value is the highest value the output response takes anywhere
• The peak time is the time at which the output response is at its highest
• The rise time is the time required for the response to reach the steady-state value FOR THE FIRST TIME
• The maximum overshoot is defined as how much the output response overshoots the steady-state value during the initial transient phase,
• The 10%-90% rise time is again the time required for the response to go from 10% to 90% of the steady-state output response value
• The % settling time, again is the amount of time required for the response to reach and stay within % of the steady-state value
  The parameters we defined for the response can be expresed as functions of and , where
  In the special case that we end up with two coinciding real poles at , and the transfer function is simplified to
  

Time delay is when there's a delay between the commanded input and the start of the output response. An LTI system with dynamic behavior described by the transfer function in the presence of a time delay of is defined
MATLAB: Time delay to a tf can be introduced using .inputdelay, e.g. H.inputdelay=2;
But, such function is not real and rational. To obtain a real rational representation for such function the Pade approximation is defined to approximate the time delay term , with MATLAB: PADE(sys,N) returns the delay-free approximation of a transfer function by replacing delays with their -th order Pade approximation.

Bode magnitude plots represent the gain in magnitude v.s. the frequency of the complex variable . Alternatively, bode phase plots represent the phase angle v.s. the frequency.
Magnitude plot:

The polar diagram is a representation of the imaginary part of and its real part, both parametrized by , in the sense that if we plot and as the frequency changes. Quite literally, just plots the complex variable.

A singularity is a pole and zero at the same point.

• are the zeroes of
• are the number of poles of at the origin
• is the generalized dc-gain, calculated as

A feedback control system is a control system that acts on feedback, a guiding signal that adapts the control depending on desired parameters.

The controller, based on reference signal provides control input to the actuator that realizes the control action and modifies the input signal onto the plant which produces the output . The measured output gathered by the sensor is suppied to a filter that performs amplification to make the signal distinguishable from the noise. This signal is used as feedback for the controller.
Systems are not invulnerable. We define three disturbances:

• The actuator disturbance that affects actuator
• The plant disturbance which hinders the input-output relationship of the plant
• The sensor disturbance which adds noise or otherwise hinders the output of the sensor.
  We start by naming and defining the main blocks(elements) of the system, the main signals and its disturbances, where we have: