Control Systems: Design Steps and System Equations

• Control System Design Steps involve a series of steps to achieve the desired system performance.
• The first step is to establish control goals.
• Next, identify the variables that need to be controlled.
• Write specifications for these variables.
• Then, establish the system configuration and identify the actuators.
• Obtain a model of the system.
• Describe a controller and select its key parameters.
• Optimize and analyze the system's performance.
• Finalize the design.

Physical Systems

• Equations describing the dynamic performance of a physical system are obtained through physical laws of the process.
• This approach applies to mechanical and electrical systems.
• It also extends to electronic, fluid, and thermodynamic systems.

System Equations

• A fundamental property of basic physics laws is that certain fundamental quantities can be defined by numerical values.
• Physical laws define relationships between these fundamental quantities.
• An example is Newton's 2nd Law: F = ma, where F is force, m is mass, and a is acceleration.
• Angular rate difference is expressed as ω(t) = ωs(t) - ωa(t).

Torsional Spring-Mass System

• The system consists of a mass (M) connected to a spring, subject to an applied torque (Ta).

Spring Element

• In the spring element, the applied torque Ta(t) is transmitted through the torsional spring.
• Because of this, the torque is called a "through-variable".
• Ta(t) - Ts(t) = 0 where Ta(t) is the applied torque and Ts(t) is the spring torque.
• The angular rate difference is measured across the torsional spring element.
• The angular rate difference is referred to as an "across-variable".

Summary Table

• Electrical systems have current (i) as the through-variable and charge (q) as the integrated through-variable.
• Their across-variable is voltage difference (V21) and integrated across-variable is flux linkage (λ21).
• Mechanical translational systems have force (F) as the through-variable and translational momentum (P) as the integrated through-variable.
• Their across-variable is velocity difference (V21) and integrated across-variable is differential difference (Y21).
• Mechanical rotational systems have torque (T) as the through-variable and angular momentum (h) as the integrated through-variable.
• Their across-variable is angular velocity(ω21) and integrated across-variable is angular displacement difference (θ21).
• Fluid systems have fluid volumetric rate of flow (Q) as the through-variable and volume (V) as the integrated through-variable.
• Their across-variable is pressure difference (P21) and integrated across-variable is pressure momentum (γ21).
• Thermal systems have heat flow rate (q) as the through-variable and heat/energy (H) as the integrated through-variable.
• Their across-variable is temperature difference (T21) and integrated across-variable is not mentioned.

Block Diagram

• A block diagram is a pictorial representation of the cause-and-effect relationship between the input and output of a physical system.
• It visually shows the relationship between the input and output.

Differential Equations of Physical Systems

• First order differential equations have the form 𝑑𝑦/𝑑𝑥 + P(x)y = g(x).
• The integrating factor is 𝐼(𝑥) = e ∫ 𝑃(𝑥)𝑑𝑥.
• Engineers use quantitative models to design and analyze control systems as the dynamic behavior is generally described by equations.
• For an inductor where V = L(di/dt).
• For a capacitor where i = C(dv/dt).
• For a resistor i = V/R.