SS.01 / Linear Systems Described by State Equations +

The state equations in matrix form explicitly express the derivatives of state variables with respect to themselves and inputs. This formulation represents the state vector as the outcome of a vector integration. The block diagram, as shown in Figure 1.1, displays the matrix operations that connect inputs to outputs using $\mathrm{A}$ , $\mathrm{B}$ , $\mathrm{C}$ , and $\mathrm{D}$ matrices. However, the diagram does not show the path of individual variables.

In this chapter, we will discuss linear systems that are described by state equations. A state-space representation of a system is a set of first-order differential equations that describe the behavior of the system. This representation is particularly useful in control systems, where the inputs and outputs are often described in terms of their rates of change.

State equations can be used to model many different types of systems, including mechanical, electrical, and chemical systems. The basic idea behind a state equation is to describe the system’s behavior in terms of its internal state variables, which are variables that describe the system’s internal state. These variables are typically represented as a vector, called the state vector.

The state equations are represented by a set of differential equations of the form:

$\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)$

where $\dot{\mathbf{x}}(t)$ is the rate of change of the state vector, $\mathbf{x}(t)$ is the state vector, $\mathbf{A}$ is the system matrix, $\mathbf{B}$ is the input matrix, and $\mathbf{u}(t)$ is the input vector.

The state-space representation can be expressed in terms of the output vector $\mathbf{y}(t)$ using the following equation:

$\mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{u}(t)$

where $\mathbf{C}$ is the output matrix and $\mathbf{D}$ is the feedforward matrix.

The state-space representation of a system has several advantages over other methods of representing systems. One advantage is that it is easy to analyze the system’s stability using the eigenvalues of the system matrix $\mathbf{A}$ . The eigenvalues of $\mathbf{A}$ determine whether the system is stable, marginally stable, or unstable.

Another advantage of the state-space representation is that it is easy to analyze the system’s controllability and observability. A system is controllable if it is possible to drive the state vector from any initial state to any desired final state in a finite amount of time. A system is observable if it is possible to estimate the state vector from the input and output signals.

The state-space representation is also useful for designing control systems. The most common approach to control system design is to use a feedback loop, where the output of the system is compared to the desired output, and the error is used to adjust the input to the system. In a state-space representation, this is typically accomplished using a state feedback controller, which computes the input to the system based on the current state of the system.

In state-determined systems, the state variables can be considered as the outputs of integrator blocks. A system of order n has n integrators in its block diagram. The integrator blocks take the derivatives of the state variables as inputs, and each state equation expresses a derivative as a combination of weighted state variables and inputs. A detailed block diagram of a system of order n can be constructed from the state and output equations as follows:

Figure 1.1 [Vector block diagram for a linear system described by state-space system dynamics].

Step 1: Draw n integrator ( $S^{−1}$ ) blocks and assign a state variable to the output of each block.

Step 2: At the input of each block (which represents the derivative of its state variable), draw a summing element.

Step 3: Use the state equations to connect the state variables and inputs to the summing elements through scaling operator blocks.

Step 4: Expand the output equations and combine the state variables and inputs through a set of scaling operators to obtain the components of the output.

In summary, state equations are a powerful tool for modeling and analyzing linear systems. They provide a way to describe the behavior of a system in terms of its internal state variables, and they can be used to analyze the stability, controllability, and observability of a system. Additionally, state-space representations are particularly useful for designing control systems that can drive a system to a desired state and maintain it there.