Robust Stability Analysis

Abstract

Robust control deals explicitly with system uncertainty and how it affects the design of control systems. By dealing with the real parameter uncertainties in control systems we can identify the critical subset of the uncertain parameter set over which the stability will be violated. The analysis depends how system uncertainty is represented. A typical way of representing it is to describe the uncertainty as parameters, which is consistent with the physical parameters of the systems.

Robust control deals explicitly with system uncertainty and how it affects the design of control systems. By dealing with the real parameter uncertainties in control systems we can identify the critical subset of the uncertain parameter set over which the stability will be violated. The analysis depends how system uncertainty is represented. A typical way of representing it is to describe the uncertainty as parameters, which is consistent with the physical parameters of the systems.

In this chapter we consider three applications concerning the robust stability of a linear system, the robust stability of a Daimler-Benz bus, and the analysis of the stability margin a spark ignition engine, the Fiat Dedra.

11.1 Robust Stability Analysis of a Linear System (Robust)

Robust stability analysis of linear systems consists in identifying the largest possible region in the uncertain model parameters space for which the controller stabilizes the system for any disturbances in the system. The stability of a feedback system is determined by the roots of the closed-loop characteristic equation (Floudas et al. 1999).

For the linear dynamic system in Fig. 11.1, in which $H(s,q)$ and $C(s,q)$ are the transfer functions of the plant and the controller, respectively, the characteristic polynomial is

Fig. 11.1

Transfer functions of plant and controller

$$ P(s,q)=\det (I+H(s,q)C(s,q)), $$

where $q\in Q$ is the vector of the uncertain model parameters, which in the most general case can be both in the plant and in the controller.

Expanding the determinant we get

$$ P(s,q)={a_n}(q){s^n}+{a_{n-1 }}(q){s^{n-1 }}+\cdots +{a_1}(q)s+{a_0}(q)=0, $$

where ${a_i}(q)$, $i=0,\ldots,n$, are polynomial functions of $q$. With these, the stability margin ${k_m}$ can be defined as

$$ {k_m}(j\upomega )=\inf \left\{ {k:P(j\upomega, q(k))=0,\;\forall q\in Q} \right\}. $$

The robust stability of the system is then guaranteed if and only if ${k_m}\geq 1$. In real applications the parameter uncertainty is typically expressed as positive and negative deviations of the real parameters from a nominal value. Therefore, checking the stability of a particular system with the characteristic polynomial $P(j\upomega, q)$ involves the solution of the following nonconvex optimization problem:

$$ \mathop{\min}\limits_{{{q_i},k,\upomega \geq 0}}k $$

(11.1)

subject to

$$ \operatorname{Re}\left[ {P(j\upomega, q)} \right]=0, $$

$$ \operatorname{Im}\left[ {P(j\upomega, q)} \right]=0, $$

$$ q_i^N-\Delta q_i^{-}k\leq {q_i}\leq q_i^N+\Delta q_i^{+}k,\;\;i=1,\ldots,{n_q}, $$

where ${q^N}=[q_1^N\cdots q_{{{n_q}}}^N]$ is a stable nominal point for the uncertain parameters of the system in a closed loop, and $\Delta {q^{-}}=[\Delta q_1^{-}\cdots \Delta q_{{{n_q}}}^{-}]$ and $\Delta {q^{+}}=[\Delta q_1^{+}\cdots \Delta q_{{{n_q}}}^{+}]$ are vectors with estimations bounds in which it is supposed that the parameters ${q_i}$, $i=1,\ldots,{n_q}$, belong.

The following example is taken from De Gaston and Safonov (1988) and was analyzed by Psarris and Floudas (1995). Figure 11.2 presents the transfer functions of the system in closed-loop form.

Fig. 11.2

Transfer functions of the closed-loop system

For this system

$$ y=\frac{{{q_1}(s+2)}}{{s(s+{q_2})(s+{q_3})(s+10)+{q_1}(s+2)}}u. $$

Therefore, the characteristic polynomial is:

$$ P(s,q)={s^4}+(10+{q_2}+{q_3}){s^3}+(10{q_2}+10{q_3}+{q_2}{q_3}){s^2}+(10{q_2}{q_3}+{q_1})s+2{q_1}. $$

The stability margin formulation (Eq. 11.1) is

$$ \min k $$

subject to:

$$ {\upomega^4}-(10{q_2}+10{q_3}+{q_2}{q_3}){\upomega^2}+2{q_1}=0, $$

$$ -(10+{q_2}+{q_3}){\upomega^3}+(10{q_2}{q_3}+{q_1})\upomega =0, $$

$$ q_1^N-\Delta q_1^{-}k\leq {q_1}\leq q_1^N+\Delta q_1^{+}k, $$

$$ q_2^N-\Delta q_2^{-}k\leq {q_2}\leq q_2^N+\Delta q_2^{+}k, $$

$$ q_3^N-\Delta q_3^{-}k\leq {q_3}\leq q_3^N+\Delta q_3^{+}k, $$

$$ \upomega \geq 0, $$

where ${q^N}=[800\;\;4\;\;6]$ and $\Delta {q^{-}}=\Delta {q^{+}}=[800\;\;2\;\;3]$. The GAMS expression of the stability margin for the preceding system is presented in Fig. 11.3.

Fig. 11.3

GAMS expression of application 11.1 (Robust)

The solution is:

$$ \begin{array}{lllllllll} {q_1} =1,073.392\;\;\;k=0.342 \\{q_2} =3.317 \upomega =8.228. \\{q_3} =4.975 \end{array} $$

Since $k<1,$ it follows that the uncertainty described by preceding bounds contains unstable points. In other words, the solution of the foregoing problem gives the stability margin $k$, the frequency $\upomega,$ and the most unfavorable combination of parameters for which the system is unstable. The performance of CONOPT, KNITRO, and MINOS are given in Table 11.1.

Table 11.1 $n=6$ (variables), $m=9$ (constraints)

Considering now $\Delta {q^{-}}=\Delta {q^{+}}=[8\;\;1\;\;1],$ we then obtain a margin of stability $k=1.395$ and the frequency $\upomega =7.366$, showing that for this region in the uncertain model parameters the system in closed-loop form is robust stable.

11.2 Robust Stability Analysis of Daimler-Benz 0305 Bus (Benz)

In this application the linearized model of Daimler-Benz 0305 bus is considered with input $\updelta$ as the steering angle rate and output $y$ as the displacement of front antenna with the following transfer function (Ackermann et al. 1991):

$$ H(s,{q_1},{q_2})=\frac{{609.8q_1^2{q_2}{s^2}+388,600{q_1}s+48,280q_1^2}}{{{s^3}(q_1^2q_2^2{s^2}+1,077{q_1}{q_2}s+16.8{q_1}{q_2}+270,000)}}. $$

The parameters of the system are ${q_1}$ as the bus velocity and ${q_2}=m/f$ as the relationship between $m$ the mass of the bus (tons) and $f$ the road friction coefficient (0.5 for a wet road and 1 for a dry road). The transfer function of the controller is:

$$ C(s)=\frac{{2,344{s^2}+10,938s+9,375}}{{{s^3}+50{s^2}+1,250s+15,625}}. $$

Consider ${q^N}=[17.5\;\;20]$ a point from the space of the parameter for which the closed-loop system is stable and $\Delta {q^{-}}=\Delta {q^{+}}=[14.5\;\;15]$ the bounds for parameter domain region for both ${q_1}$ and ${q_2}$. The purpose is to compute the margin of stability for the closed-loop system with uncertain parameters in the previously defined region. Computing the closed-loop characteristic polynomial of the system, problem (Eq. 11.1) for margin of stability is:

$$ \min k $$

subject to

$$ {a_8}(q){\upomega^8}-{a_6}(q){\upomega^6}+{a_4}(q){\upomega^4}-{a_2}(q){\upomega^2}+{a_0}(q)=0, $$

$$ {a_7}(q){\upomega^6}-{a_5}(q){\upomega^4}+{a_3}(q){\upomega^2}-{a_1}(q)=0, $$

$$ 17.5-14.5k\leq {q_1}\leq 17.5+14.5k, $$

$$ 20-15k\leq {q_2}\leq 20+15k, $$

where

$$\begin{array}{lllllllll} {a_0}(q) =453\times {10^6}q_1^2, \\{a_1}(q) =528\times {10^6}q_1^2+3,640\times {10^6}{q_1}, \\{a_2}(q) =5.27\times {10^6}q_1^2{q_2}+113\times {10^6}q_1^2+4,250\times {10^6}{q_1}, \\{a_3}(q) =6.93\times {10^6}q_1^2{q_2}+911\times {10^6}{q_1}+4,220\times {10^6}, \\{a_4}(q) =1.45\times {10^6}q_1^2{q_2}+16.8\times {10^6}{q_1}{q_2}+338\times {10^6}, \\{a_5}(q) =15.6\times {10^3}q_1^2q_2^2+840q_1^2{q_2}+1.35\times {10^6}{q_1}{q_2}+13.5\times {10^6}, \\{a_6}(q) =1.25\times {10^3}q_1^2q_2^2+16.8q_1^2{q_2}+53.9\times {10^3}{q_1}{q_2}+270\times {10^3}, \\{a_7}(q) =50q_1^2q_2^2+1,080{q_1}{q_2}, \\{a_8}(q) =q_1^2q_2^2. \end{array} $$

Table 11.2 presents the performances of considered solvers. The GAMS representation of this application is given in Fig. 11.4.

Fig. 11.4

GAMS expression of application 11.2 (Benz)

Table 11.2 $n=5$ (variables), $m=7$ (constraints)

The solution of this application is

$$ {q_1}=1.6653e-6,\;\;{q_2}=1.897,\;\;\upomega =0.001,\;\;k=1.207. $$

Since $k>1$, it follows that for this region of uncertain parameters and the frequency $\omega =0.001$ the closed-loop system is robust stable. In another set of experiments, let us suppose that the domain of uncertain parameter evolution ${q_1}$ and ${q_2}$ is enlarged as $\Delta {q^{-}}=\Delta {q^{+}}=[18.5\;\;17]$. The corresponding margin of stability is $k=1.059$ at the same frequency.

11.3 Analysis of Stability Margin of Spark Ignition Engine Fiat Dedra (Fiat)

Considering a point in the parameter space for which a closed-loop system is stable, as well as some estimations of the limits in which it is supposed that the parameters ${q_i}$, $i=1,\ldots,7$, are found, the determination of the stability margin is given by the solution of the following problem (Abate et al. 1994; Barmish 1994).

$$ \min k $$

subject to

$$ -{a_6}(q){\upomega^6}+{a_4}(q){\upomega^4}-{a_2}(q){\upomega^2}+{a_0}(q)=0, $$

$${a_7}(q){\upomega^6}-{a_5}(q){\upomega^4}+{a_3}(q){\upomega^2}-{a_1}(q)=0, $$

$$ \quad3.4329-1.02721k\leq {q_1}\leq 3.4329+1.02721k, $$

$$ 0.1627-0.06k\leq {q_2}\leq 0.1627+0.06k, $$

$$ 0.1139-0.0782k\leq {q_3}\leq 0.1139+0.0782k, $$

$$ 1.2539-0.3068k\leq {q_4}\leq 1.2539+0.3068k, $$

$$ 0.0208-0.0108k\leq {q_5}\leq 0.0208+0.0108k, $$

$$ 5.0247-2.4715k\leq {q_6}\leq 5.0247+2.4715k, $$

$$ 1-2k\leq {q_7}\leq 1+2k, $$

where

$$ {a_0}(q)=6.82079\times {10^{-5 }}{q_1}{q_3}q_4^2+6.82079\times {10^{-5 }}{q_1}{q_2}{q_4}{q_5}, $$

$$ \begin{array}{lllllllll} {a_1}(q)=&\ 7.6176\times {10^{-4 }}q_2^2q_5^2+7.6176\times {10^{-4 }}q_3^2q_4^2+4.02141\times {10^{-4 }}{q_1}{q_2}q_5^2 \\& +0.00336706{q_1}{q_3}q_4^2+6.82079\times {10^{-5 }}{q_1}{q_4}{q_5}+5.16120\times {10^{-4 }}q_2^2{q_5}{q_6} \\& +0.00336706{q_1}{q_2}{q_4}{q_5}+6.82079\times {10^{-5 }}{q_1}{q_2}{q_4}{q_7}+6.28987\times {10^{-5 }}{q_1}{q_2}{q_5}{q_6} \\& +4.02141\times {10^{-4 }}{q_1}{q_3}{q_4}{q_5}+6.28987\times {10^{-5 }}{q_1}{q_3}{q_4}{q_6}+0.00152352{q_2}{q_3}{q_4}{q_5} \\& +5.1612\times {10^{-4 }}{q_2}{q_3}{q_4}{q_6},\end{array} $$

$$ \begin{array}{lllllllll} {a_2}(q)=&\ 4.02141\times {10^{-4 }}{q_1}q_5^2+0.00152352{q_2}q_5^2+0.0552q_2^2q_5^2+0.0552q_3^2q_4^2 \\& +0.0189477{q_1}{q_2}q_5^2+0.034862{q_1}{q_3}q_4^2+0.00336706{q_1}{q_4}{q_5} \\& +6.82079\times {10^{-5 }}{q_1}{q_4}{q_7}+6.28987\times {10^{-5 }}{q_1}{q_5}{q_6}+0.00152352{q_3}{q_4}{q_5} \\& +5.1612\times {10^{-4 }}{q_3}{q_4}{q_6}-0.00234048q_3^2{q_4}{q_6}+0.034862{q_1}{q_2}{q_4}{q_5} \\& +0.0237398q_2^2{q_5}{q_6}+0.00152352q_2^2{q_5}{q_7}+5.1612\times {10^{-4 }}q_2^2{q_6}{q_7} \\& +0.00336706{q_1}{q_2}{q_4}{q_7}+0.00287416{q_1}{q_2}{q_5}{q_6}+8.04282\times {10^{-4 }}{q_1}{q_2}{q_5}{q_7} \\& +6.28987\times {10^{-5 }}{q_1}{q_2}{q_6}{q_7}+0.0189477{q_1}{q_3}{q_4}{q_5}+0.00287416{q_1}{q_3}{q_4}{q_6} \\& +4.02141\times {10^{-4 }}{q_1}{q_3}{q_4}{q_7}+0.1104{q_2}{q_3}{q_4}{q_5}+0.0237398{q_2}{q_3}{q_4}{q_6} \\& +0.00152352{q_2}{q_3}{q_4}{q_7}-0.00234048{q_2}{q_3}{q_5}{q_6}+0.00103224{q_2}{q_5}{q_6},\end{array} $$

$$ \begin{array}{lllllllll} {a_3}(q)=&\ 0.0189477{q_1}q_5^2+0.1104{q_2}q_5^2+5.1612\times {10^{-4 }}{q_5}{q_6}+q_2^2q_5^2 \\& +7.6176\times {10^{-4 }}q_2^2q_7^2+q_3^2q_4^2+0.1586{q_1}{q_2}q_5^2+4.02141\times {10^{-4 }}{q_1}{q_2}q_7^2 \\& +0.0872{q_1}{q_3}q_4^2+0.034862{q_1}{q_4}{q_5}+0.00336706{q_1}{q_4}{q_7} \\& +0.00287416{q_1}{q_5}{q_6}+6.28987\times {10^{-5 }}{q_1}{q_6}{q_7}+0.00103224{q_2}{q_6}{q_7} \\& +0.1104{q_3}{q_4}{q_5}+0.0237398{q_3}{q_4}{q_6}+0.00152352{q_3}{q_4}{q_7} \\& -0.00234048{q_3}{q_5}{q_6}+0.1826q_2^2{q_5}{q_6}+0.1104q_2^2{q_5}{q_7} \\& +0.0237398q_2^2{q_6}{q_7}-0.0848q_3^2{q_4}{q_6}+0.0872{q_1}{q_2}{q_4}{q_5} \\& +0.034862{q_1}{q_2}{q_4}{q_7}+0.0215658{q_1}{q_2}{q_5}{q_6}+0.0378954{q_1}{q_2}{q_5}{q_7} \\& +0.00287416{q_1}{q_2}{q_6}{q_7}+0.1586{q_1}{q_3}{q_4}{q_5}+0.0215658{q_1}{q_3}{q_4}{q_6} \\& +0.0189477{q_1}{q_3}{q_4}{q_7}+2{q_2}{q_3}{q_4}{q_5}+0.1826{q_2}{q_3}{q_4}{q_6}+0.1104{q_2}{q_3}{q_4}{q_7} \\& -0.0848{q_2}{q_3}{q_5}{q_6}-0.00234048{q_2}{q_3}{q_6}{q_7}+7.6176\times {10^{-4 }}q_5^2 \\& +0.0474795{q_2}{q_5}{q_6}+8.04282\times {10^{-4 }}{q_1}{q_5}{q_7}+0.00304704{q_2}{q_5}{q_7},\end{array} $$

$$ \begin{array}{lllllllll} {a_4}(q)=&\ 0.1586{q_1}q_5^2+4.02141\times {10^{-4 }}{q_1}q_7^2+2{q_2}q_5^2+0.00152352{q_2}q_7^2 \\& +0.0237398{q_5}{q_6}+0.00152352{q_5}{q_7}+5.1612\times {10^{-4 }}{q_6}{q_7} \\& +0.0552q_2^2q_7^2+0.01898477{q_1}{q_2}q_7^2+0.0872{q_1}{q_4}{q_5} \\& +0.034862{q_1}{q_4}{q_7}+0.0215658{q_1}{q_5}{q_6}+0.00287416{q_1}{q_6}{q_7} \\& +0.0474795{q_2}{q_6}{q_7}+2{q_3}{q_4}{q_5}+0.1826{q_3}{q_4}{q_6}+0.1104{q_3}{q_4}{q_7} \\& -0.0848{q_3}{q_5}{q_6}-0.00234048{q_3}{q_6}{q_7}+2q_2^2{q_5}{q_7}+0.1826q_2^2{q_6}{q_7} \\& +0.0872{q_1}{q_2}{q_4}{q_7}+0.3172{q_1}{q_2}{q_5}{q_7}+0.0215658{q_1}{q_2}{q_6}{q_7} \\& +0.1586{q_1}{q_3}{q_4}{q_7}+2{q_2}{q_3}{q_4}{q_7}-0.0848{q_2}{q_3}{q_6}{q_7}+0.0552q_5^2 \\& +0.3652{q_2}{q_5}{q_6}+0.0378954{q_1}{q_5}{q_7}+0.2208{q_2}{q_5}{q_7},\end{array} $$

$$ \begin{array}{lllllllll} {a_5}(q)=&\ 0.0189477{q_1}q_7^2+0.11104{q_2}q_7^2+0.1826{q_5}{q_6}+0.1104{q_5}{q_7} \\& +0.0237398{q_6}{q_7}+q_2^2q_7^2+0.1586{q_1}{q_2}q_7^2+0.0872{q_1}{q_4}{q_7} \\& +0.0215658{q_1}{q_6}{q_7}+0.3652{q_2}{q_6}{q_7}+2{q_3}{q_4}{q_7}-0.0848{q_3}{q_6}{q_7} \\& +q_5^2+7.6176\times {10^{-4 }}q_7^2+0.3172{q_1}{q_5}{q_7}+4{q_2}{q_5}{q_7},\end{array} $$

$$ {a_6}(q)=0.1586{q_1}q_7^2+2{q_2}q_7^2+2{q_5}{q_7}+0.1826{q_6}{q_7}+0.0552q_7^2, $$

$$ {a_7}(q)=q_7^2. $$

The GAMS expression of this application is given in Fig. 11.5.

Fig. 11.5

GAMS expression of application 11.3 (Fiat)

Table 11.3 presents the performances of the algorithms CONOPT and MINOS. KNITRO, cannot solve the problem.

Table 11.3 $n=18$ (variables), $m=25$ (constraints)

Since $k=1.459>1$, it follows that for this region of the uncertain model parameters the system in closed-loop form is robust stable.