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.

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

2. Performance Index: Fuel-Optimal Control (Minimum Fuel Problem)

Consider a spacecraft control problem where u(t) represents the thrust produced by the rocket engine. The magnitude of the thrust, |u(t)|, is assumed to be proportional to the rate of fuel consumption.

To minimize the total fuel expenditure, the performance index (PI) is defined as:

I = ∫_t₀^t_f |u(t)| dt        (1.8)

This integral represents the total fuel consumed over the time interval from t₀ to t_f.

For a multi-input system with m control inputs u₁(t), u₂(t), …, u_m(t), the performance index becomes:

I = ∫_t₀^t_f ∑_i=1^m R_i |u_i(t)| dt        (1.9)

Where:

• R_i is the weighting factor for the i^th control input
• |u_i(t)| is the magnitude of the control input

These weighting factors allow for prioritization or penalization of fuel usage across different thrusters or control channels.

3. Performance Index: Energy-Optimal Control (Minimum Energy Problem)

Consider u_i(t) as the current in the i^th loop of an electric network, and let r_i be the resistance of that loop. The total power or rate of energy expenditure at time t is given by:

∑_i=1^m u_i²(t)  r_i

To minimize the total energy expended over the time interval [t₀, t_f], the performance index is defined as:

I = ∫_t₀^t_f ∑_i=1^m u_i²(t) r_i dt        (1.10)

This expression represents the total energy consumed by the network during the control interval.

In a more general vector-matrix form, the performance index becomes:

I = ∫_t₀^t_f U^T(t) R U(t) dt        (1.11)

Where:

• U(t) is the control vector containing all inputs u_i(t)
• R is a positive definite matrix representing energy weighting factors (e.g., resistances)

This form is commonly used in optimal control, especially in the Linear Quadratic Regulator (LQR) problem to penalize control energy.

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

Minimization of Tracking Error in Optimal Control

In a tracking system, we aim to minimize the deviation of the actual state from the desired trajectory over time. This objective can be achieved by minimizing the integral of the squared tracking error. I = ∫_t₀^t_f X^T(t) Q X(t) dt            (1.12) Where:

• X_d(t) is the desired state,
• X_a(t) is the actual state,
• X(t) = X_a(t) − X_d(t) is the tracking error,
• Q is a positive semidefinite weighting matrix.

Note: The matrix Q may be time-varying depending on the system requirements. This form is widely used in tracking problems where minimizing the accumulated error over time is essential. 4. Performance Index for Terminal Control System:   In a terminal target problem, we are interested in minimizing the error between the desired target position Xd (tf) and the actual target position Xa (tf) at the end of the maneuver or at the final time tf. The terminal (final) error is X(tf) = Xa(tf) – Xd (tf ). Taking care of positive and negative values of error and weighting factors, we structure the cost function as I= X^T (t_f )PX(t_f ) (1.13) which is also called the terminal cost function. Here, P is a positive semi-definite matrix.

4. Performance Index for Terminal Control System

In a terminal control problem, the objective is to minimize the error between the actual and desired system states at the final time t_f. Specifically, we are concerned with the difference between the actual terminal state X_a(t_f) and the desired terminal state X_d(t_f). The terminal error is defined as: X(t_f) = X_a(t_f) − X_d(t_f) To account for both positive and negative errors and apply appropriate weighting, the performance index (also called the terminal cost function) is defined as: I = X^T(t_f)  P  X(t_f)            (1.13) Where:

• X(t_f) is the terminal state error
• P is a positive semidefinite weighting matrix

This cost function focuses only on the final state, making it suitable for applications where performance at the endpoint is critical.

5. Performance Index for General Optimal Control System

By combining terminal and tracking objectives, the performance index for a general optimal control problem can be expressed as: I = X^T(t_f)  P  X(t_f) + ∫_t₀^t_f [X^T(t) Q X(t) + U^T(t) R U(t)] dt            (1.14) Alternatively, in a more general form, it can be written as: I = S(X(t_f), t_f) + ∫_t₀^t_f F(X(t), U(t), t) dt            (1.15) Where:

• X(t) is the state vector
• U(t) is the control input vector
• Q and P are positive semidefinite weighting matrices
• R is a positive definite weighting matrix
• S(X(t_f), t_f) is the terminal cost function
• F(X(t), U(t), t) is the running (or path) cost function

Note: The matrices Q(t) and R(t) may vary with time depending on the problem requirements. Equation (1.14) is known as the quadratic performance index because it is quadratic in terms of both the states and control inputs. It is widely used in Linear Quadratic Regulator (LQR) and related optimal control problems.

Classification of Optimal Control Problems Based on Performance Index

Optimal control problems are often classified according to the structure of the performance index I. The classification is as follows:

• Mayer Problem:
  If the performance index includes only the terminal cost function,
  I = S(X(t_f), t_f)
  it is called a Mayer-type problem.
• Lagrange Problem:
  If the performance index consists only of the integral (running) cost term,
  I = ∫_t₀^t_f F(X(t), U(t), t) dt
  it is referred to as a Lagrange-type problem.
• Bolza Problem:
  If the performance index includes both the terminal cost and the integral cost term,
  I = S(X(t_f), t_f) + ∫_t₀^t_f F(X(t), U(t), t) dt
  it is known as a Bolza-type problem.

Although various other forms of cost functions can be used depending on performance specifications, the quadratic forms of these indices lead to particularly elegant and tractable solutions in optimal control theory.

Constraints in Optimal Control Problems

In optimal control problems, the control input u(t) and the state vector x(t) may be either unconstrained or constrained, depending on the physical nature of the system.

Unconstrained problems are mathematically simpler and often lead to elegant analytical solutions. However, in practical systems, physical limitations typically impose constraints on both control and state variables.

For example, in systems such as electrical circuits, motors, or rockets, variables like current, voltage, speed, or thrust must lie within specified limits. These constraints are usually expressed as:

U_min ≤ u(t) ≤ U_max,    X_min ≤ x(t) ≤ X_max            (1.16)

Where:

• U_min and U_max: Minimum and maximum allowable control values
• X_min and X_max: Minimum and maximum allowable state values

These constraints must be considered in the formulation and solution of realistic optimal control problems.

Formal Statement of an Optimal Control Problem (Linear Time-Invariant System)

Consider the linear time-invariant (LTI) system:

𝑐̇(t) = A X(t) + B U(t),      t ∈ [t₀, t_f]            (1.17)

with the initial condition:

X(t₀) = X₀

Problem: Find the control input U^*(t) that minimizes the performance index:

I = X^T(t_f)  P  X(t_f) + ∫_t₀^t_f [X^T(t) Q X(t) + U^T(t) R U(t)] dt            (1.18)

The corresponding state trajectory generated by U^*(t) is denoted X^*(t).

Where:

• X(t) ∈ ℝⁿ is the state vector
• U(t) ∈ ℝᵐ is the control vector
• A ∈ ℝⁿˣⁿ and B ∈ ℝⁿˣᵐ are constant system matrices
• P and Q are positive semidefinite weighting matrices (P ≥ 0, Q ≥ 0)
• R is a positive definite weighting matrix (R > 0)

This formulation, which includes both terminal and integral quadratic cost terms, is the standard setup for the Linear Quadratic Regulator (LQR) problem.

Formal Statement of an Optimal Control Problem (Nonlinear System)

Consider a nonlinear dynamic system described by:

𝑐̇(t) = f(X(t), U(t), t),      t ∈ [t₀, t_f]            (1.19)

with the initial condition:

X(t₀) = X₀

Objective:
Find the optimal control U^*(t) that generates the optimal state trajectory X^*(t) such that the following general performance index is minimized (or maximized):

I = S(X(t_f), t_f) + ∫_t₀^t_f F(X(t), U(t), t) dt            (1.20)

Subject to:

• The nonlinear system dynamics (Equation 1.19)
• Constraints on control and/or state variables:
  U_min ≤ U(t) ≤ U_max,    X_min ≤ X(t) ≤ X_max            (1.16)

The final time t_f may be fixed or free, and the final (target) state may be fully specified, partially specified, or free, depending on the nature of the problem.

This formulation represents a general nonlinear optimal control problem and encompasses a wide variety of real-world applications.

The entire problem statement is also shown pictorially in Figure 5. Thus,

                        Fig.5. Formal Representation of Optimal Control

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