J = φ(X(T)) + ∫₀ᵀ L(X(t), u(t), t) dt

1.2 Classical Control

Classical or conventional control theory focuses on the analysis and design of single-input single-output (SISO) systems. It is primarily based on the use of Laplace transform methods and block diagram representations to study system behavior in the frequency domain. In a classical feedback control system, the input signal to the plant, denoted by u(t), is generated by processing the error signal e(t) through a compensator or controller. The error signal is the difference between the desired reference input r(t) and the actual output y(t) of the system.

Note: • Only the output of the system is used for feedback; internal state variables are not available. • Classical control is generally applied to linear time-invariant (LTI) systems.

The closed-loop transfer function of a typical classical control system with unity feedback is given by:

Y(s) / R(s) = G(s) / (1 + G(s)H(s))                (1.1)

where:

• Y(s) and R(s) are the Laplace transforms of the output and input, respectively,
• G(s) is the open-loop transfer function,
• H(s) is the transfer function of the feedback path.

If the open-loop system consists of a compensator and a plant, the transfer function G(s) is expressed as:

G(s) = G_c(s) × G_p(s)                               (1.2)

where:

• G_c(s) is the transfer function of the compensator (controller).
• G_p(s) is the transfer function of the plant (the system to be controlled).

Fig. 1. Classical Control System Configuration

1.3 Modern Control Theory

Modern control theory, also referred to as advanced control, extends classical control to handle more complex systems, especially those with multiple inputs and multiple outputs (MIMO). It is based on the state-variable representation, which models systems using first-order differential equations in the time domain. This approach provides a more complete description of system dynamics and is applicable to linear, nonlinear, time-invariant, and time-varying systems.

State-Space Representation of Linear Time-Invariant (LTI) Systems

The state-space model for an LTI system is given by:

Ẋ(t) = A·X(t) + B·U(t)       (1.3)

Y(t) = C·X(t) + D·U(t)       (1.4)

Where:

• X(t) – State vector
• U(t) – Input vector
• Y(t) – Output vector
• A – System matrix
• B – Input matrix
• C – Output matrix
• D – Feedthrough (or direct transmission) matrix

State-Space Representation of Nonlinear Systems

For nonlinear systems, the state-space representation becomes

Ẋ(t) = f(X(t), U(t), t)       (1.5)

Y(t) = g(X(t), U(t), t)       (1.6)

Here, f(·) and g(·) are nonlinear functions that describe the system’s dynamics and output, respectively. This representation enables the use of advanced control techniques such as state feedback, observer design, optimal control, and robust control, making it essential for high-order and complex system analysis and design.

State Feedback in Modern Control Theory

In modern control theory, all state variables are ideally available for feedback. This allows the controller to make accurate and effective decisions. The input u(t) is determined by the controller, which includes an error detector and a compensator, based on the system states X(t) and the reference input r(t), as shown in Fig. 2. This structure supports state feedback control, where each state is multiplied by an appropriate gain to achieve the desired system response — such as improved stability or faster dynamics. Modern control is based on matrix algebra and is well-suited for digital computer implementation and simulations. It provides a flexible and systematic method to model and control multi-input, multi-output (MIMO) systems. It is also important to note that while the state-space model uniquely determines the system’s transfer function, the reverse is not true: a single transfer function can have many valid state-space representations. This allows engineers to choose forms like controllable, observable, or diagonal for convenience in analysis and design. Fig.2. Modern Control System Configuration 1.4. Components of the Modern Control System Fig. 3. Main Components of Modern Control System

Modeling:

The first step in any control theory approach is to model the system dynamics. This involves formulating the system using differential or difference equations. In many cases, the dynamics are derived using the Lagrangian function.

Analysis:

Once the model is established, the system is analyzed to evaluate its performance—especially its stability. A fundamental tool for this analysis is Lyapunov stability theory, which helps determine whether the system will remain stable under various conditions.

Design:

If the system does not meet the desired performance specifications, the next step is control system design. In optimal control theory, this involves designing a controller that minimizes or maximizes a performance index (commonly denoted as J(X) or I(X)) based on system states.

1.5. Optimization Fig.4. Overview of Optimization Optimization can be broadly classified into static optimization and dynamic optimization.

1. Static Optimization

Static optimization involves controlling a plant under steady-state conditions, where system variables do not change with time. The plant is described by algebraic equations. Techniques used: Ordinary calculus, Lagrange multipliers, linear programming, and nonlinear programming.

2. Dynamic Optimization

Dynamic optimization deals with the control of a plant under dynamic conditions, where system variables change over time. The plant is described by differential or difference equations. Techniques used: Search techniques, dynamic programming, calculus of variations, and Pontryagin’s principle.

1.6 Optimal Control

Objective

The objective of optimal control is to determine a control signal u^*(t) that drives the system (plant) in such a way that it:

• Satisfies all physical and operational constraints.
• Minimizes or maximizes a chosen performance index J or I.

Formulation of an Optimal Control Problem

To formulate an optimal control problem, the following components are required:

1. Mathematical Model of the Process A detailed description of the physical system to be controlled, usually expressed in state-space form.
2. Performance Index A scalar function (typically an integral over time) that quantifies the objective to be optimized—such as energy usage, error, time, or cost.
3. Boundary Conditions and Constraints Specifications for initial/final states, along with physical constraints on the system variables and control inputs.

System (Plant) Representation

For optimization purposes, a physical plant is modeled using either linear or nonlinear differential equations. A linear time-invariant (LTI) system is typically described by:

Ẋ(t) = AX(t) + BU(t)     (1.3)
Y(t) = CX(t) + DU(t)     (1.4)

A nonlinear system is generally represented as:

Ẋ(t) = f(X(t), U(t), t)    (1.5)
Y(t) = g(X(t), U(t), t)    (1.6)

Where:

• Ẋ(t) is the state vector
• U(t) is the control input
• Y(t) is the output
• A, B, C, D are system matrices
• f(·), g(·) are nonlinear functions of state, input, and time

Performance Index / Performance Measure (I or J):

1. Classical Control:

Classical control techniques are applied to linear, time-invariant, single-input single-output (SISO) systems. These methods analyze system behavior in both the time and frequency domains.

Time-domain specifications include:

• Rise time
• Settling time
• Peak overshoot
• Steady-state error

Frequency-domain characteristics include:

• Gain margin
• Phase margin
• Bandwidth

Common controllers include PID (Proportional-Integral-Derivative) controllers.

2. Modern Control:

Modern control theory applies to multi-input multi-output (MIMO) systems using state-space representation. It allows feedback using all system state variables. A key objective is optimal control, where the control input drives the system to a desired state or trajectory while optimizing a performance index. Performance index may involve

• Energy consumption
• Time to reach target
• Control effort

Minimizing the performance index I means minimizing the time required to complete the control task.

I = ∫_t₀^t𝑓 1⋅dt = t_f − t₀ = t^*        (1.7)

Modern control methods include state feedback, observer design, and techniques like Linear Quadratic Regulator (LQR). Different Types of Performance Index:

1. Performance Index: Time-Optimal Control System

In a time-optimal control system, the objective is to transfer the system from an initial state x(t₀) to a desired final state x(t_f) in the shortest possible time.

The performance index (PI) for this type of control is defined as the total time taken for the transition:

I = ∫_t₀^t𝑓 1 dt = t_f − t₀ = t^{*                 (1.7)}        

Where:

• t₀: Initial time
• t_f: Final time
• t^*: Minimum (optimal) time

Minimizing the performance index I means minimizing the time required to complete the control task.

I = ∫_t₀^t𝑓 1⋅dt = t_f − t₀ = t^*        (1.7)

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