# A Novel Robust Control System Design and Its Application to Servo Motor Drive
## 2 Controller Design Using State Feedback and Integral Action  
A conventional state feedback controller with full state feedback is shown in Figure 1. An nth-order single input/single output system is represented as the state variable form, as Equation (1) shows. \[ \begin{array}{l}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{b}u\\ y=\mathbf{c}\mathbf{x}\end{array}  \tag{1}\]  
where \(\mathbf{A}\in R^{n\times n}\), \(\mathbf{b}\in R^{n\times 1}\), \(\mathbf{c}\in R^{n\times 1}\), and \(\mathbf{x}\in R^{n\times 1}\) are the system states. If the system is controllable, state feedback can be used to stabilize the system, and these control law design methodologies, such as pole placement or the linear quadratic (LQ) technique, can be used to decide the feedback control gain. Assume that the control gain \(\mathbf{k}\in R^{n\times 1}\) is decided, the control law can be set as:  
\[u=-\mathbf{k}^{\mathrm{T}}\mathbf{x} \tag{2}\]  
and the system state equation with state feedback can be represented as:  
\[ \begin{array}{ll}\dot{\mathbf{x}}&=\mathbf{A}\mathbf{x}+\mathbf{b}u\\ &=\mathbf{A}\mathbf{x}+\mathbf{b}(-\mathbf{k}^{\mathrm{T}}\mathbf{x})\\ &=\mathbf{A}_{c}\end{array}  \tag{3}\]  
For a velocity-type control system, a system with non-zero reference input, or a system with external perturbation/disturbance, a pure state feedback control law, Equation (2), cannot wholly compensate for those non-ideal cases and will result in a steady-state error. For the control end, integral action is adopted to eliminate those phenomena [33]. Consider the system shown in the block diagram of Figure 2, designed under the integral state feedback compensation control. The system is represented as follows:  
\[ \begin{array}{ll}\dot{\mathbf{x}}&=\mathbf{A}\mathbf{x}+\mathbf{b}u+ \mathbf{w}\\ y&=\mathbf{c}\mathbf{x}\end{array}  \tag{4}\]  
The system is subjected to the reference input \(r(t)\) and input disturbance/perturbation \(w(t)\), where:  
\[\mathbf{w}(t)= \begin{bmatrix}0&\cdots&0&w(t)\end{bmatrix} ^{\mathrm{T}} \tag{5}\]  
Define the new reference states as \(\mathbf{x}_{ref}\), and it has the form:  
\[\mathbf{x}_{ref}= \begin{bmatrix}x_{1 ref}&\cdots&x_{n-1 ref}&x_{nref}\end{bmatrix}  ^{\mathrm{T}} \tag{6}\]  
and  
\[\mathbf{x}_{ref}=\mathbf{n}r \tag{7}\]  
where \(\mathbf{n}\in R^{n\times 1}\) and \(\mathbf{x}_{ref}\in R^{n\times 1}\) are the desired system states in the steady state to the control. The state's error and derivative are defined as:  
\[ \begin{array}{ll}\mathbf{x}_{error}&=\mathbf{x}_{ref}-\mathbf{x}\\ &=\mathbf{n}r-\mathbf{x}\end{array}  \tag{8}\]  
\[ \begin{array}{ll}\dot{\mathbf{x}}_{error}&=\dot{\mathbf{x}}_{ref}-\dot{ \mathbf{x}}\\ &=\dot{\mathbf{n}}\dot{r}-\dot{\mathbf{x}}\\ &=-\dot{\mathbf{x}}\end{array}  \tag{9}\]  
An integrator that accepts the differences between the reference input and system output is added to eliminate the steady-state error, and the integrator can be set as  
\[ \begin{array}{ll}\dot{x}_{\mathrm{t}}&=r-y\\ &=r-\mathbf{c}\mathbf{x}\end{array}  \tag{10}\]  
Figure 1: The control block diagram of state feedback. Figure 2: The control block diagram of the augmented system. The control law design can be accomplished by state feedback to the augmented system, Equations (9) and (10). Assume the new system states are:  
\[\mathbf{z}= \begin{bmatrix}x_{1}\\ \mathbf{x}_{error}\end{bmatrix}  \tag{11}\]  
The augmented system state matrix can be represented as:  
\[ \begin{array}{ll}\dot{\mathbf{z}}&=\begin{bmatrix}\dot{x}_{1}\\ \dot{\mathbf{x}}_{error}\end{bmatrix}=\begin{bmatrix}0&-\mathbf{c}\\ 0&-\mathbf{A}\end{bmatrix}\begin{bmatrix}x_{1}\\ \mathbf{x}\end{bmatrix}+\begin{bmatrix}0\\ -\mathbf{b}\end{bmatrix}u+\begin{bmatrix}r\\ -\mathbf{w}\end{bmatrix}\\ &=\mathbf{A}_{1}\mathbf{z}+\mathbf{b}_{1}u+\mathbf{w}_{1}\end{array}  \tag{12}\]  
where the reference inputs are assumed constant in the control process.