State space averaging [18] is used to model switching converters by calculating a weighted average of the two sets of state equations using the nominal values of the time spent in each state as the weights. A number of key modelling assumptions underlie the validity of the use of this modelling method. One of these is the small ripple approximation, which assumes that variations in inductor currents and capacitor voltages (caused by switching) are small relative to their respective average values. This allows the use of time-averaged values of currents and voltages to represent the dynamics, rather than dealing with high-frequency switching components. A second important assumption is the use of a high switching frequency is assumed to be much higher than the natural frequencies of the converter. This allows the high-frequency switching behavior to be averaged out over a switching period, resulting in a low-frequency equivalent model.

We will define the state vector as follows:

$x={\left[\begin{array}{llll}{v}_2\hfill & v_1\hfill & i_2\hfill & i_1\hfill \end{array}\right]}^{\prime }$

which are the capacitor voltages and inductor currents as represented in Figure 1 and Figure 2 .

$A_1=\left[\begin{array}{cccc}-\frac{1}{R{C}_2}& 0& -\frac{1}{C_2}& \frac{1}{C_2}\\ 0& 0& \frac{1}{C_1}& 0\\ \frac{1}{L_2}& -\frac{1}{L_2}& 0& 0\\ -\frac{1}{L_1}& 0& 0& 0\end{array}\right],B_1=\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{L_1}\end{array}\right],C_1=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]$(1)

Conversely, when switch SW₂ is ON, with switch SW₁ OFF, the circuit state matrices are given by:

$A_2=\left[\begin{array}{cccc}-\frac{1}{R{C}_2}& 0& -\frac{1}{C_2}& \frac{1}{C_2}\\ 0& 0& 0& \frac{1}{C_1}\\ \frac{1}{L_2}& 0& 0& 0\\ -\frac{1}{L_1}& -\frac{1}{L_1}& 0& 0\end{array}\right],B_2=\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{L_1}\end{array}\right],C_2=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]$(2)

By representing the steady-state value of the duty cycle of switch SW₁ as $D$, the following results may be obtained for a state space averaged model of the C1 converter:

$D^{\prime }\triangleq 1-D$

$\stackrel{\dot{}}{X}=AX+B{V}_g$

$Y=CX$

where the averaged matrices are given by:

$A=D{A}_1+D^{\prime }{A}_2$

$B=D{B}_1+D^{\prime }{B}_2$

$C=D{C}_1+D^{\prime }{C}_2$

With the understanding that in steady state $\stackrel{\dot{}}{X}=0$, we can solve for the steady state vector:

$X=-A^{-1}B{V}_g$

For the C1 converter this vector is determined to be:

$X=\left[\begin{array}{c}{V}_2\\ V_1\\ I_2\\ I_1\end{array}\right]=\left[\begin{array}{c}D{V}_g\\ V_g\\ \frac{D{V}_g}{R}\\ \frac{D^2{V}_g}{R}\end{array}\right]$

Of particular importance here is that the output voltage, which we will designate as $V_o$ is the output capacitor voltage, i.e. $V_o=V_2$, indicating that the voltage conversion ratio is given by:

$\frac{V_o}{V_g}=D$(3)

Subsequently the small-signal model can be derived. The caret symbol () indicates a small deviation from the steady state value, i.e.

$x=X+\hat{x}$

$v_g=V_g+{\hat{v}}_g$

$d=D+\hat{d}$

$v_o=V_o+{\hat{v}}_o$

With the condition that the output matrices $C_1=C_2$, which appears for the C1 converter, the small signal model is given by:

$\stackrel{\dot{}}{\hat{x}}=A\hat{x}+B{\hat{v}}_g+B_d\hat{d}$(4)

${\hat{v}}_o=C\hat{x}$(5)

where

$B_d=\left(A_1-A_2\right)X+\left(B_1-B_2\right)V_g$(6)

Equations (4) and (5) lead to the general small-signal model block diagram shown in Figure 3 .

We will now examine the open loop small-signal performance of the converter. To this end a unit step signal will be applied to the ${\hat{v}}_g$ input and the output response of ${\hat{v}}_o$ will be examined. As we are examining the open loop response, no control signal will be applied to the $\hat{d}$ input, so that we have $\hat{d}=0$. The converter parameters used are the same as in the design presented in [4] . Specifically, these are summarized in Table 1 .

Note that the model used is a small signal model so any signals that appear will deviate around a zero value. To put this in the context of the actual system a DC voltage of 5 V is added to the output signal. Also, note that a unit step of the input voltage, ${\hat{v}}_g=1$, represents a 10% change of the nominal input voltage of 10 V. The resulting response is shown in Figure 4 .

In order to clearly see the initial step response, which would be obscured by the vertical axis when the input is applied at $t=0$, the step is applied slightly delayed at $t=1$ ms. Prior to the step’s application, the output voltage is at a 5 V level. Subsequently, a peak oscillation value of 5.7 V is reached before settling down to a 5.5 V level, as dictated by the voltage conversion ratio: $V_o=D\cdot \left(V_g+{\hat{v}}_g\right)$.

For a perfect voltage regulator system, when given a step input voltage disturbance we would like the output to remain unchanged. However, this is not achievable practically and a non-zero step response will be seen in the output. With the incorporation of integral control, which guarantees zero steady state error, the output will revert to its set value in the steady state. The rate, and the manner, that this reversion is achieved will be dictated, but the overall feedback loop design and the speed of response will be primarily dictated by the loop bandwidth. Consequently, measures such as settling time, percentage overshoot, and unity gain bandwidth will be of interest.

In the next section, the basic converter model is augmented to achieve full state feedback with integral control.

The state of a system is defined as a minimal set of variables which together with the input, permits any output to be obtained in a finite time span (if the system is fully controllable). An integrator will also be added to achieve zero steady state error. This will increase the state dimension by one. Our basic converter model will now be modified by adding full state feedback along with integral control. This is shown in Figure 5 . The cross (X) at input $\hat{d}$ indicates where the feedback loop may be broken to determine the loop gain of the system.

We will now derive the state equations for this system. As before, for the basic converter we have:

$\stackrel{\dot{}}{\hat{x}}=A\hat{x}+B{\hat{v}}_g+B_d\hat{d}$(7)

The output of the added integrator is the added state, denoted as ${\hat{x}}_i$, so that at its input we have

${\stackrel{\dot{}}{\hat{x}}}_i=-C\hat{x}$(8)

These two expressions can be combined into one state equation

$\left[\begin{array}{l}\stackrel{\dot{}}{\hat{x}}\hfill \\ {\stackrel{\dot{}}{\hat{x}}}_i\hfill \end{array}\right]=\left[\begin{array}{ll}A\hfill & 0\hfill \\ -C\hfill & 0\hfill \end{array}\right]\left[\begin{array}{l}\hat{x}\hfill \\ {\hat{x}}_i\hfill \end{array}\right]+\left[\begin{array}{l}B\hfill \\ 0\hfill \end{array}\right]{\hat{v}}_g+\left[\begin{array}{l}{B}_d\hfill \\ 0\hfill \end{array}\right]\hat{d}$(9)

Converter and integral state feedback is achieved with

$\hat{d}=-\left[\begin{array}{ll}k\hfill & k_i\hfill \end{array}\right]\left[\begin{array}{l}\hat{x}\hfill \\ {\hat{x}}_i\hfill \end{array}\right]$(10)

In summary, the above equations can be represented as follows:

$\stackrel{\dot{}}{\bar{x}}=\bar{A}\bar{x}+\bar{B}{\hat{v}}_g+{\bar{B}}_d\hat{d}$(11)

${\hat{v}}_o=\bar{C}\bar{x}$(12)

where the augmented state, $\bar{x}$, is given by

$\bar{x}=\left[\begin{array}{l}\hat{x}\hfill \\ {\hat{x}}_i\hfill \end{array}\right]$(13)

and

$\bar{A}=\left[\begin{array}{ll}A\hfill & 0\hfill \\ -C\hfill & 0\hfill \end{array}\right],\bar{B}=\left[\begin{array}{l}B\hfill \\ 0\hfill \end{array}\right],{\bar{B}}_d=\left[\begin{array}{l}{B}_d\hfill \\ 0\hfill \end{array}\right],\bar{C}=\left[\begin{array}{ll}C\hfill & 0\hfill \end{array}\right]$(14)

Also, equation (10) becomes

$\hat{d}=-\bar{k}\bar{x}$(15)

where

$\bar{k}=\left[\begin{array}{ll}k\hfill & k_i\hfill \end{array}\right]$(16)

Substituting (15) into (11) results in the closed loop system state equations:

$\stackrel{\dot{}}{\bar{x}}=\left(\bar{A}-{\bar{B}}_d\bar{k}\right)\bar{x}+\bar{B}{\hat{v}}_g$(17)

${\hat{v}}_o=\bar{C}\bar{x}$(18)

The eigenvalues of matrix $\bar{A}-{\bar{B}}_d\bar{k}$ represent the poles of the closed loop system. These poles can be placed appropriately by varying the feedback gain vector $\bar{k}$. What follows are the various approaches by which this could be done optimally. For the C1 converter there are a total of five feedback gains to be optimized.

As mentioned previously, the cross (X) between the signals $\hat{d}$ and ${\hat{d}}^{\prime }$ in Figure 5 represents the point in the loop which is broken in order to measure the loop gain. With the loop broken at that point, the input signal $\hat{d}$ is fed into the system, (with ${\hat{v}}_g=0$), and the output is ${\hat{d}}^{\prime }$. The corresponding state equations are

$\stackrel{\dot{}}{\bar{x}}=\bar{A}\bar{x}+{\bar{B}}_d\hat{d}$(19)

${\hat{d}}^{\prime }=-\bar{k}\bar{x}$(20)

where the $\bar{A}$, ${\bar{B}}_d$ and $\bar{k}$ matrices are given in Equations (14) and (16). This results in a loop gain expression given by

$G_{loop\text{\_}gain}=-\bar{k}{\left(sI-\bar{A}\right)}^{-1}{\bar{B}}_d$(21)

In this work we will primarily be looking at the output signal ${\hat{v}}_o$ in response to a disturbance step input ${\hat{v}}_g$. State Equations (17) and (18) will be utilized to achieve this. We will also examine the control effort in closed loop where the input signal $\hat{d}$ is monitored. The state equations used for this purpose are therefore (17) and (15).

As previously stated, the vector of gains $\bar{k}$ determine the poles of the closed loop system. In the work presented here the poles are initially assigned and the gains $\bar{k}$ are subsequently determined. The Matlab functions acker or place may be used for this purpose. However, for pole placement to be achievable the system must first be determined to be fully controllable. Controllability is defined as the ability to move the state of a system from an initial value $x_0$ to any arbitrary state $x_f$ within a finite time period $t$ using the input signal $u$. (Note that controllability says nothing about the magnitude of the input signal $u$, i.e., the control effort, nor the time $t$ required to accomplish this transition). Essentially, it is a test to determine if the closed-loop system poles may be arbitrarily placed in the complex plane.

A test for controllability can be conducted by first constructing the controllability matrix $M_c$ using the system and input matrices, $\left(A,B\right)$ as follows:

$M_c=\left[\begin{array}{lllll}B\hfill & AB\hfill & A^{2}B\hfill & \cdots \hfill & A^{n-1}B\hfill \end{array}\right]$(22)

where $n$ is the order of the system. If $M_c$ is a full rank matrix, the system is fully controllable. Matlab provides the function ctrb to obtain the controllability matrix. The rank deficiency of $M_c$ indicates how many modes are uncontrollable. Using the couple $\left(\bar{A},{\bar{B}}_d\right)$, as defined in (14), our system was determined to be fully controllable.

In this paper, several pole placement methods will be explored and compared. To form a basis for comparison, we first examine the traditional and widely accepted ITAE prototype pole placement method.

Graham and Lathrop [15] discuss assigning the system poles to prototype locations that minimize a performance metric (or cost function), $J$, known as the integral of the time-weighted absolute error (ITAE) to an input signal:

$J={\displaystyle {\int }_0^{\infty }t\cdot \left|e\left(t\right)\right|\text{d}t}$(23)

For a system of order $n$ the characteristic polynomial has form $s^n+a_{n-1}{a}^{n-1}+\cdots +a_2{s}^2+a_{1}s+1$. Note here that the coefficients of the highest and lowest order terms, i.e., the terms involving $s^n$ and $s^0$, are 1. To minimize the ITAE performance metric, all other coefficients are optimized. Table 2 shows the traditional values used for these coefficients and corresponding pole positions, $p_0$, corresponding to a cutoff frequency ${\omega }_0=1$ [15] and [16] . New pole positions, $p_N$, are then obtained by simple scaling with a different value of ${\omega }_0$: $p_N={\omega }_0\cdot p_0$.

The ITAE pole placement method sets the system pole locations such that this integral is minimized for a given system cutoff frequency ${\omega }_0$. The traditional normalized pole locations, which are easily scaled for different ${\omega }_0$, are given in Table 2 for up to 7^th order systems.

More recent work [19] has highlighted the fact that the original ITAE polynomials and poles were inaccurately determined using an analog computer and an effort to more precisely determine these was made. To illustrate how the updated polynomials are determined, we’ll consider the 5th order case, which we’ll use later on. The general form of a 5^th order polynomial is given as $s^5+a_4{a}^4+a_3{a}^3+a_2{s}^2+a_{1}s+1$. Thus a search is required to determine the four coefficients $a_4$, $a_3$, $a_2$ and $a_1$ which minimize the ITAE metric as defined in (23). Use is made of the trapezoidal method of integral approximation: given a time series $t_1,t_2,\cdots ,t_n$ and corresponding errors $e_1,e_2,\cdots ,e_n$, the ITAE can be approximated as:

$\text{ITAE}\approx \underset{i=1}{\overset{n-1}{{\displaystyle \sum }}}\frac{t_{i+1}-t_i}{2}\left(t_i\left|e_i\right|+t_{i+1}\left|e_{i+1}\right|\right)$(24)

Use of the Matlab trapz function is made to perform the integration. If we consider the system response to a unit step, the error signal $e$, is defined as $e\left(t\right)=1-y\left(t\right)$ where $y\left(t\right)$ is the system output. Appendix A shows a Matlab script, incorporating the ideas discussed here, which can be used to determine the updated coefficients for polynomials of order 2 to 7. The updated set of ITAE polynomials and pole locations are summarized in Table 3 . The values displayed in Table 3 are slightly different from those appearing in [19] , due to an unfortunate error in coding used to produce the values given in [19] .

As mentioned above, the system discussed in this paper is 5^th order, so traditionally to make use of the ITAE pole placement method, an appropriate ${\omega }_0$ value needs first to be identified to scale the five poles given in the tables. For our purposes, this was done by simply sweeping ${\omega }_0$ through a range of values, evaluating the step response of the system, and identifying the ${\omega }_0$ that resulted in the smallest output variation. The sweep interval used was [800, 100,000] rad/s. The left plot of Figure 6 shows the value of the ITAE metric over this interval. The lowest value of ITAE of the order of 2e⁻⁸ appears at the upper ${\omega }_0$ boundary of 100,000 rad/s. A simulation using this value of ${\omega }_0$ to determine the output response to a unit step, is shown in the right plot of Figure 6 . Unfortunately, a peak to peak variation of 30 volts is seen in the output. This is untenable for a practical design. It appears that the ITAE metric has been minimized by increasing the speed of response without consideration of output voltage deviations.

In consideration of the above results, we next decided to monitor the MaxMin metric defined by

$\text{MaxMin}=e_{\max }\left(t\right)-e_{\min }\left(t\right)$(25)

Accordingly, over the ${\omega }_0$ sweep range where the ITAE system polynomial is used for design, the MaxMin metric is evaluated, as seen in the left-side plot of Figure 7 . Here we find that in order to achieve minimum output variation, as defined by Equation (25), a value of ${\omega }_0=11640$ rad/s is needed. The resulting output plot for this value of ${\omega }_0$, for a unit step input, is shown in the right-side plot of Figure 7 . The peak to peak output amplitude variation has been reduced to around 1.2 volts. Whilst this is a big improvement from the former case, it is also not acceptable.