# Time Domain and Linear Quadratic Optimization

Control system synthesis in the time domain, rather than in the frequency domain, is often called modern control theory. Optimal controller design is generally involved.

Although one usually thinks of modern control theory in connection with full automatic control, it is applied as well to the design of stability-augmentation systems.

Linear quadratic (LQ) optimization methods have been used for a number of stability- augmentation system designs. These methods have their origins in the work of R. E. Kalman. Airframe and controller equations are cast in the state matrix form discussed in the previous chapter. The optimal controller is a linear feedback law that minimizes an integral cost function J of the form

J [x T Qx + 8T R8]dt,

where x is the system state vector, 8 is the control vector, and Q and R are weighting matrices that express the designer’s ideas on what constitutes optimal behavior for this case. The optimal control law takes the form of a linear set of feedback gains 8 = Cx, where C is the gain matrix of constants. The gain matrix C is computed by a matrix equation called the Riccati equation.

The linear quadratic approach to controller design is attractive because it is an organized method for finding feedback gains. The method produces an optimal set of feedbacks, but only for the arbitrarily chosen weighting matrix values. One can argue that if the weighting matrix values are poorly chosen, the resulting system can be far from ideal. In fact, it is not uncommon for designers using linear quadratic methods to tinker with weighting matrix values until a reasonable-looking system emerges. This puts the optimal design method on all fours with ordinary cut-and-try methods.

The problem of assigning weighting matrix values aside, there have been numerous variants of the linear quadratic approach to controller design and any number of applications in the literature and in practice. A typical application is to the design of a lateral-directional command augmentation system (Atzhorn and Stengel, 1984). The criterion function includes control system rate as a means of limiting high-frequency or rapid control motions. Displacement and rate saturation are significant nonlinearities that cannot be treated with the linear quadratic approach, except by the use of describing functions (Hanson and Stengel, 1984). Other linear quadratic stability-augmentation designs that may be found in the literature include departure-resistant controls, superaugmented (unstable airplane) pitch controls, and multiloop roll-yaw augmentation.

According to Robert Clarke and his associates at the NASA Dryden Flight Research Center, the Grumman X-29A research airplane’s flight controls were originally designed using an optimal model-following technique. Simplified computer, actuator, and sensor models were used in the original analysis, leading to an unconservative design. A classical approach was chosen in the end, with lags introduced by the actual hardware compensated for by the addition of lead-lag filters (Clarke, Burken, Bosworth, and Bauer, 1994).

Another interesting linear quadratic stability augmentation design adds feed-forward compensation for nonlinear terms that cannot be included in the linearized design. This is the stability-augmentation system for the Rockwell/Deutsche Aerospace X-31 research airplane. Feed-forward compensation is added for nonlinear engine gyroscopic and inertial coupling effects (Beh and Hofinger, 1994).